nldlab - User contributions [en]

Group 7

2011-12-23T05:17:40Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

A simulink model was also made for the double pendulum based on its equations of motion. Below is a sample simulation at an unstable state.

<videoflash>LigbdKmFp_E</videoflash>

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.
=== Stability of the Double Pendulum ===

<videoflash>JW7DvCpUbTY</videoflash>

Not only did we stabilize the inverted state of the single pendulum, but the inverted double pendulum was also stabilized.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

<videoflash>H3BEuaeerRo</videoflash>

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-16T16:55:30Z

Group7: /* Simulation */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

A simulink model was also made for the double pendulum based on its equations of motion. Below is a sample simulation at an unstable state.

<videoflash>LigbdKmFp_E</videoflash>

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

<videoflash>H3BEuaeerRo</videoflash>

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-16T16:50:48Z

Group7: /* Flipping State of the Double Pendulum */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

<videoflash>H3BEuaeerRo</videoflash>

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-16T16:45:51Z

Group7: /* Flipping State of the Double Pendulum */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-16T16:42:10Z

Group7: /* Flipping State of the Double Pendulum */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

{{#ev:youtube|H3BEuaeerRo}}

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-15T17:17:18Z

Group7:

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.9 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.1 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-15T12:43:49Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional <ref name=Marchewka>{{cite journal |
author = A. Marchewka, D. Abbott, & R. Beichner |
year = 2005 |
title = Oscillator damped by a constant-magnitude friction force |
journal = American Journal of Physics|
volume = 72 |
issue = 4 |
pages = 477-483}}
</ref>.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-15T12:39:59Z

Group7: /* Basins Of Attraction */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution)<ref name=Tongue>{{cite journal |
author = B.H. Tongue and K. Gu |
year = 1988 |
title = Interpolated Cell Mapping of Dynamical Systems |
journal = Journal of Applied Mechanics|
volume = 55 |
issue = 2 |
pages = 461-466}}
</ref>. I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-14T19:54:06Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a = 0.008294 \pm 0.000687, 95\%$, $b = -0.1979 \pm 0.0054, 95\%$ and $c = 1.28 \pm 0.009, 95\%$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-14T19:18:44Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-\omega^{-2}$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

Group 7

2011-12-14T17:53:37Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

=== Dependence of the Frequency of Small Oscillations about the Stable Inverted State on Driving Parameters ===

[[Image: Frequencies of Oscillation.png | figure | right | 600px]]

Depicted at right, we observed a positive correlation in the frequencies of small oscillation (vertical axis) with the amplitude of driving acceleration (horizontal axis). Further, we suspect a (slightly less obvious) correlation of the frequency of small oscillation with the driving frequency (depicted by color). However, due to the implicit requirement that we be able to find and maintain the inverted stable state over long time scales, the density of points on this graph is subject to the same material limitations we ran into for fleshing out the stability boundary.

== References ==
<references/>

File:Frequencies of Oscillation.png

2011-12-14T17:45:40Z

Group7:

Group 7

2011-12-14T17:43:59Z

Group7: /* Flipping State of the Double Pendulum */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===

[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T17:43:04Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===
[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.
[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T17:37:32Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

Visually, our experience in the laboratory suggests the existence of separable dynamics in the anti-symmetric states $(0, \pi)$, $(\pi, 0)$. Since in either of these anti-symmetric states the small-angle approximation is valid, it should be possible to simplify the dynamics of the system treating slight perturbations of the inner pendulum as inertial restoring forces on the outer pendulum (which explains why the symmetric states $(0, 0)$, $(\pi, \pi)$ do not exhibit this phenomenon). Intuitively, the gauge freedom of the potential energy implies that for these symmetry preserving states the position of the inner pendulum simply defines a higher or lower potential level of the outer pendulum for the $\theta_1 = 0$ and $\theta_1 = \pi$ states, respectively. As long as the symmetry is not broken, or weakly broken, or is strongly contracting, the outer pendulum is expected to behave as an independent single pendulum. However, further theoretical work is needed to prove this conjecture.

=== Flipping State of the Double Pendulum ===
[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T17:21:00Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally, by a minimization of the RMSE of the two series with equal weighting. Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

=== Flipping State of the Double Pendulum ===
[[Image: 24HzFlippingPhase.png | figure | up | 500px]]

Next, we present a novel set of anti-symmetric states for the double pendulum: the "Flipping Modes". Above is a phase portrait for the double pendulum being driven at 24 Hz, which shows the inner pendulum in the up state, and outer pendulum in the down state. This anti-symmetric state is then perturbed, and inverts; the inner pendulum moves to the down state, and the outer pendulum to the up state. The time series $\{\theta_{1,n}, \theta_{2,n}\}$ for this transition is presented below.

[[Image: Stabledoubleexpflipexp.png | figure | down | 500px]]

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:Stabledoubleexpflipexp.png

2011-12-14T17:18:35Z

Group7:

File:24HzFlippingPhase.png

2011-12-14T17:14:04Z

Group7:

Group 7

2011-12-14T17:11:02Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 500px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 500px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally (i.e. a minimization of the RMSE of the two series). Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the right is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.

** phase portraits

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T17:10:09Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 600px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | left | 600px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally (i.e. a minimization of the RMSE of the two series). Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$. On the left is a comparison of the two time-series. Note the divergence of the two series near the tail; this is to be expected if the damping is only mostly frictional, consistent with the parameter magnitudes determined above.
* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T17:08:42Z

Group7: /* Determination of damping parameter */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 600px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

[[Image: prediction.png | figure | right | 600px]]

From the previously determined functional form of the damping, the damping parameter is determined by fitting the first oscillatory period of ring-down tracking data to a sine wave. This is used to first determine a natural frequency of the pendulum, which analytically describes an initial condition for a simulation of the system. In the figure to the right, an explicit Euler numeric method was used to integrate the dynamics and compare the time-series output to the experimentally determined data. The damping parameter is then determined variationally (i.e. a minimization of the RMSE of the two series). Using this method, the damping coefficient, from Eq. (1), is determined to be $\gamma \approx 2.5$.
* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:Prediction.png

2011-12-14T16:57:22Z

Group7:

Group 7

2011-12-14T16:55:48Z

Group7: /* Single Inverted Pendulum, $N = 1$ */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{equation} \label{single pend dynamics}
0 = \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}
With scaled, dimensionless parameters $\alpha = g/\ell \omega ^{2}$, $\beta = b/\ell$, $\tau = \omega t$.

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 600px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

* compare sims to experiment
** phase portraits
** stability boundaries
=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T16:54:03Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned} \label{single pend dynamics}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

=== Determination of damping parameter ===
[[Image:Decay profile.png | figure | right | 600px]]

Ring-down tracking of the single pendulum was performed, and the trajectory of $\theta_{n}$ as a time-series is presented to the right. The profile of the series determines, implicitly, the form of the damping in the dynamics, Eq. (1). A second order polynomial in $t$, $f(t; a,b,c) = at^2 + bt + c$ was used for approximating the profile. The fit converged with an RMSE value of 0.006664, with parameter values determined to be $a= 0.008294 \pm 0.000687$, $b = -0.1979 \pm 0.0054$ and $c = 1.28 \pm 0.009$. Since $|a/b| \approx \mathcal{O}(|b|/10)$, we can be confident that the decay profile is linear, and the damping is frictional.

* compare sims to experiment
** phase portraits
** stability boundaries
=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T16:44:02Z

Group7:

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned} \label{single pend dynamics}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T16:43:29Z

Group7: /* Single Inverted Pendulum, $N = 1$ */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned} \label{single pend dynamics}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned*}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-14T16:43:12Z

Group7: /* Single Inverted Pendulum, $N = 1$ */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned} \label{single pend dynamics}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:Decay profile.png

2011-12-14T16:39:11Z

Group7:

Group 7

2011-12-14T04:58:26Z

Group7: /* Stability Boundary */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.79 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-13T01:04:38Z

Group7: /* Experiment */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.9 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T22:02:21Z

Group7: /* Stability Boundary */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.9 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:58:50Z

Group7: /* Stability Boundary */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

[[File:ExpStab7.jpg | center | 600px]]

When comparing this figure alone in its present state, it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.9 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose them on the parameter space with the theoretical boundaries:

[[File:TheoryStab7.jpg | center | 600px]]

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials used prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:TheoryStab7.jpg

2011-12-12T21:56:50Z

Group7:

File:ExpStab7.jpg

2011-12-12T21:56:37Z

Group7:

Group 7

2011-12-12T21:55:58Z

Group7: /* Results */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==
=== Stability Boundary ===
Below is a figure of the results for the experiment of mapping the stability boundary of the modified single pendulum.

insert figure here

When comparing this figure alone in its present state it is difficult to discern agreement with theory. This data looks nearly linear when compared with the predicted lower boundary of stability from theory. However, upon determining $f_{0}$ from experimental ring down data (which was found to be apprixmately 1.9 Hz), we can then transform frequency and amplitude into the reduced units of $\Omega$ and $\epsilon$, respectively, and superimpose then on the parameter space with the theoretical boundaries:

insert other figure here

It is clearly seen that, for our range of frequencies, there is quantitative agreement between theory and experiment. Unfortunately, even with the improved pendulum, our constraints with the shaker and materials used prevented us from being able to fully explore the parameter space.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:44:03Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Results ==

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:42:14Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature<ref name=Blackburn>{{cite journal |
author = J. A. Blackburn, H. J. T. Smith and N. Gronbech-Jensen |
year = 1992 |
title = Stability and Hopf Bifurcations in an Inverted Pendulum |
journal =American Journal of Physics |
volume = 60 |
issue = 10|
pages = 903–908}}
</ref>. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:34:44Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

[[Image:GronFig.jpg | thumb | right | 500px]]
A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:GronFig.jpg

2011-12-12T21:33:22Z

Group7:

Group 7

2011-12-12T21:32:35Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure on the right by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

where $f$ is the frequency in Hz, $f_{0}$ is the natural frequency of the pendulum, $A$ is the amplitude in meters, $\omega_{0}$ is the natural frequency in radians/sec, and $g$ is gravity in m/s^2.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:27:15Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

indicated in the figure by the solid lines. The forcing frequency, $\Omega$, and the the forcing amplitude, $\epsilon$, are both reduced units defined by the following equations:

\begin{equation}
\Omega = \frac{f}{f_{0}}
\end{equation}

\begin{equation}
\epsilon = \frac{A\omega_{0}^{2}}{g}
\end{equation}

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T21:18:14Z

Group7: /* Linear Stability Analysis */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

A study on the stability of the inverted state within the frequency/amplitude parameter space has been done in previous literature. It was calculated that the inverted state was stable as long as the frequency and amplitude were within the following two curves:

\begin{equation}
\epsilon = \sqrt{2}/\Omega
\end{equation}

\begin{equation}
\epsilon = 0.450 + 1.799/\Omega^{2}
\end{equation}

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T17:03:33Z

Group7: /* Basins Of Attraction */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color representing the final angular position to a pixel at the appropriate initial position/velocity coordinates in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T05:04:05Z

Group7: /* Basins Of Attraction */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these basins with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case, black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for a pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolve when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T04:58:13Z

Group7: /* Double Pendulum */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axles of rotation were also both made from lego plastic, and with significant forcing, these axles would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these attractions with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolves when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T04:55:40Z

Group7: /* The Original */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about an axle the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axels of rotation were also both made from lego plastic, and with significant forcing, these axels would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these attractions with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolves when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T03:19:06Z

Group7: /* Basins Of Attraction */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about a axel the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axels of rotation were also both made from lego plastic, and with significant forcing, these axels would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these attractions with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolves when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure2.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

File:BasinsFigure2.png

2011-12-12T03:18:53Z

Group7:

Group 7

2011-12-12T03:15:12Z

Group7: /* Basins Of Attraction */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about a axel the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axels of rotation were also both made from lego plastic, and with significant forcing, these axels would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these attractions with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolves when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure.png | figure | center | 900px]]

Again, white indicates settling at the bottom, and black indicates settling at the inverted state. The large expanses of gray are areas in which the trajectory of those initial conditions eventually mapped to points outside of the visible phase space, and thus had no mapping and could not be mapped further, even if the trajectory eventually returned. Stripes and noise indicate spinning modes that have yet to settle.

== References ==
<references/>

Group 7

2011-12-12T03:08:49Z

Group7: /* Discussion */

''Group Members: J. J. Aguilar, G. Lee, C. Marcotte, B. Suri''

The inverted pendulum is a simple mechanical system which models a canonical problem in both Control Theory and Nonlinear Dynamics. It is possible to control this system through either horizontal or vertical control of the pivot point. This article explores control of the planar pendulum through oscillation of the vertical position of the pivot point.

== Background ==
Originally investigated by [http://en.wikipedia.org/wiki/Pyotr_Kapitsa P. L. Kapitza], the inverted pendulum and the larger phenomenon of dynamical stability was all but unknown to the majority of physicists at the time. On the subject, Kapitza had this to say:
<blockquote>the striking and instructive phenomenon of dynamical stability of the turned pendulum not only entered no contemporary handbook on mechanics but is also nearly unknown to the wide circle of specialists... ...not less striking than the spinning top and as instructive. <ref name=Kapitza>{{cite journal |
author = P. L. Kapitza, edited by T. Der Haar |
year = 1965 |
title = Collected papers of P. L. Kapitza |
journal = |
volume = 2 |
issue = |
pages = 714, 726
doi = }}
</ref></blockquote>
For the record, the problem was later included in Landau & Lifschitz' introductory text ''Mechanics''.

==Theory==
=== Single Inverted Pendulum, $N = 1$ ===
The inverted pendulum with vertical driving has as it's Lagrangian
\begin{equation}
\mathcal{L} = \frac{m}{2}(\ell^{2} \dot{\theta^{2}} + \dot{y}^2 + 2\ell\dot{y}\dot{\theta}\sin\theta)- m g [y(t) + \ell\cos(\theta)]
\end{equation}

Where $\ell$ represents the length of the pendulum, $m$, it's mass, $\theta$ is the angle the pivot arm makes with the upward vertical, $g$ is the acceleration due to gravity, $y(t)$ is the vertical displacement of the pivot point, and $\dot{\theta}$ represents the derivative of the angle $\theta$ with respect to the time, $t$.

Solving the first-order Lagrange-Euler equation in $\theta, \dot{\theta}$ and rescaling, we find The Vertically Driven Pendulum equation <ref name=Bartuccelli>{{cite journal |
author = M. V. Bartuccelli, G. Gentile and K. V. Georgiou |
year = 2001 |
title = On the dynamics of a vertically driven damped planar pendulum |
journal =Proc. R. Soc. Lond. A |
volume = 458 |
issue = |
pages = 3007–3022 |
doi = 10.1098/rspa.2001.0841}}
</ref>.

\begin{aligned}
0 &= \ddot{ \theta} + (\beta f(\tau) - \alpha)\sin\theta \\
\alpha &= \frac{g}{\ell \omega ^{2}} \\
\beta &= \frac{b}{\ell} \\
\tau &= \omega t \\
\end{aligned}

Where $f(\tau)$ corresponds to the normalized driving function, such that $ \partial^{2}_{\tau}{y}(t) = b f(\tau)$, and $\ddot{\theta} $ is understood to be differentiated with respect to $\tau$, now.
This simple second-order system can be generalized to include damping terms thus:

\begin{equation}
0 = \ddot{ \theta} + \gamma \zeta(\dot{\theta}) + (\beta f(\tau) - \alpha)\sin\theta
\end{equation}

Where $\gamma$ represents a constant, scaled, friction coefficient and $\zeta$ is some function on the angular velocity $\dot{\theta}$, for example, $\dot{\theta}/|\dot{\theta}|$.

==== Linear Stability Analysis ====
Clearly, the fixed points are given by $(\theta^{*}, \dot{\theta^{*}})_{-} = (\pi,0)$ and $(\theta^{*}, \dot{\theta^{*}})_{+} = (0,0)$. Making the local transformation $\eta_{\pm} = \theta^{*}_{\pm} + \delta\theta_{\pm}$ we derive the Mathieu equations.<ref name = Bartuccelli></ref>

\begin{equation}
\delta\ddot{\theta}_{\pm} \mp (\beta f(\tau) - \alpha)\delta\theta_{\pm} = 0
\end{equation}

Note that the stability of the fixed point is determined by the sign of the linear prefactor, and that, should $y(t)$ be an eigensolution of the operator $\partial^{2}_{\tau}$ with eigenvalue $-1$, that the stability of the fixed points are not determined by the sign of $\beta$ as $-\beta$ corresponds to a phase shift of $\pi$ relative to $\beta$.

=== Double Inverted Pendulum, $N = 2$ ===
The Double Inverted Pendulum is governed following dynamical equations in the variables $\theta_{1}$, $\theta_{2}$.
\begin{align}
\ddot{\theta}_{1} &= -\frac{m_{2}\, (L_{1}\,\dot{\theta_{1}}^2\sin\!\left(2\,\theta_{1} - 2\, \theta_{2}\right)\ + 2\,L_{2}\,\dot{\theta_{2}}^2\sin\!\left(\theta_{1} - \theta_{2}\right)\,) + g\, \left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right) - \ddot{y}\left(\left(2\, m_{1} + m_{2}\right)\sin\!\left(\theta_{1}\right)\ + m_{2}\, \sin\!\left(\theta_{1} - 2\, \theta_{2}\right)\right)}{2\, L_{1}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}
\\
\ddot{\theta}_{2} &=\frac{L_{2}\, m_{2}\, {\dot{\theta_{2}}}^2 \sin\!\left(2\, \theta_{1} - 2\, \theta_{2}\right)\ + \left(m_{1} + m_{2}\right)\, \left(2\, L_{1}{\dot{\theta_{1}}}^2 \sin\!\left(\theta_{1} - \theta_{2}\right)\ + g\, \left(\sin\!\left(2\, \theta_{1} - \theta_{2}\right) - \sin\!\left(\theta_{2}\right)\right)\right) + \ddot{y}\left(m_{1} + m_{2}\right)\, \left(\sin\!\left(\theta_{2}\right) - \sin\!\left(2\, \theta_{1} - \theta_{2}\right)\right)}{2\, L_{2}\, \left(m_{2}\, {\sin\!\left(\theta_{1} - \theta_{2}\right)}^2 + m_{1}\right)}

\end{align}

These governing equations were used in the simulation of the Inverted Double Pendulum, in both Simulink and using the internal ode45 Matlab routine.

== Experiment ==
[[Image:GeneralSetup7.jpg | thumb | right | 500px]]
Through the use of a motor we attempt to force the stability of the free pendulum, such that the downward vertical $(\theta^{*}_{-} = \pi)$ becomes unstable and the upward vertical $(\theta^{*}_{+} = 0)$ becomes stable.
We intend to investigate extending this to the case of coupled (series) pendula. We assume the limit $\gamma \rightarrow 0$.

=== Materials ===

# The Oscillating Base
# Rod or Pendulum Simulacrum
# Reflective Tracking Material
# Camera
# Matlab, no doubt

The general setup includes the above mentioned parts and are laid out in the figure to the right. They will be addressed in turn in the following sections.

==== The Oscillating Base ====

[[Image:shaker7.png | figure | 500px]][[Image:osc7.png | figure | 500px]]

The oscillating base is the device used to vertically shake the pivot of the pendulum. It consists of a function generator, amplifier, motor, air bearing, accelerometer, and oscilloscope. The function generator creates the sine wave at a set desired frequency. However, the amplitude of this signal is not powerful enough to meaningfully drive the motor in the base. So the signal is passed through an amplifier that adds current and increases the amplitude of the signal. This signal is then sent to the motor, which converts that current into force. The motor is allowed to move vertically with near frictionless motion with the air bearing. The motor also has an accelerometer attached to it which sends a signal to the oscilloscope in order provide feedback about the actual amplitude of the forcing. The relationship between the output peak to peak voltage from the oscilloscope and the actual displacement amplitude is as follows:

\begin{equation}
A_{disp} = \frac{9.8*V_{pk}/2*10}{(2\pi*Frequency)^{2}}
\end{equation}

==== Trackers ====

The physical trackers that were used varied for camera tracking included 10mm white plastic balls as well as white-out painted on black electrical tape. Additionally, it was especially important to include a black backdrop for added contrast in the image when we used the 1000 fps camera setup.

==== Camera Setup ====

[[Image:pg7.png | thumb | left | 200px| First Camera]][[Image:mx7.png | thumb | right | 400px| Second Camera]]
The first camera that we used was a PointGrey high speed camera. It ran at a maximum of 200 fps, and communicated via firewire to the computer. With this setup, tracking was done in real-time with a LabView program. The advantage to this method is that the LabView software was readily available and reduced the amount of time required to get position data from the video capture. However, actual frame rate times varied from frame to frame due to realtime calculations, which had to be taken into account during later data analysis. Also, this method really only worked for single pendulum experiments when the movement of the pendulum was fairly slow and the tracking point was never obscured.

The second camera we used alleviated the above problems encountered with the first camera. The MotionXtra was used at 1000 fps, and the video was saved on the camera itself and then transfered to the computer as a single video file. This allowed for capture of faster motions and required offline point tracking in Matlab. This post-processing method of tracking also allowed for more intelligent tracking schemes and manual point tracking as needed.

==== Matlab Tracking ====

The following steps are the basic elements of the algorithm used to track points in matlab. This initial algorithm is similar to what is used in the LabView setup:

# Click on markers in the first frame: To track a single a point, you must first select the coordinates of the search point from where the program will search for the centroid of the tracker in the image.
# Threshold data: For a grayscale image in which you have white trackers, you must determine a good 0-255 grayscale value to distinguish tracker pixels from non-tracker pixels. Each pixel is now a binary true or false value.
# For a square of points around the selected search point (the dimension of the square is a number of pixels of your discretionary choosing), you average the locations of all the "true" points to get the actual centroid position.
# Make the centroid the updated search point for the next frame.
# Iteratively repeat steps 2-4 for all frames.

[[Image:track1.png | figure |left| 400px]][[Image:track2.png | figure | 400px]]

With this simple algorithm, however, tracking points would still get lost when they were obscured from the view of the camera. For example, in a double pendulum setup with equal rod lengths, there were instances when the second rod could obscure the pivot tracking point. Under the above algorithm, the pivot would usually effectively be lost, tracking the pivot point as though it was the second pendulum. This specific situation was fixed by adding a constraint to the position of the pivot, assuming that it would only be moving in the y direction. And during moments of obscurity, the program would use this constraint as well as known values such as the first pendulum postion and the length of the first pendulum to estimate the pivot's location. This did not correct all situations of obscurity, nor did it correct for when the pendulum would be whirling too fast even for 1000 fps. So for these general situations, the option of manual correction and tracking on a frame by frame basis was included. This was very painstaking at times, but necessary.

[[Image:overl1.png | figure | 400px]][[Image:overl2.png | figure | 380px]][[Image:overl3.png | figure | 500px]]

==== The Pendula ====

Although there is a main pendulum from which the large majority of our experimental results are derived, we used 3 different pendula in our experiments.

===== The Original =====

*Effective Length: 6.88 cm
*Material: Aluminum

This pendulum's pivot was a high end ball bearing for skate boards which rotated about a axel the had supports on both ends. There were two problems with this pendulum. The first was that the effective length was too large, which decreased the range of frequencies that the shaker could handle to perform the stability map experiment of the inverted fixed point. The second was a strange frictional issue with the bearing itself that caused the pendulum to randomly settle slightly to the left or to the right of the actual fixed point in a notched sort of manner for either the top or the bottom equilibrium point. The phenomenon is illustrated in the following figure.

[[Image:original7.png | figure | left | 400px]]
[[Image:frict7.png | thumb | center | 500px]]

===== Modified Version ('''MAIN''') =====

*Effective Length: 3.13 cm
*Material: Aluminum

This single pendulum was our solution to the problems of the original pendulum. Its shorter rod length slightly increased the range of possible forcing frequencies for the stability map. And the skate bearing, which actually was comprised of two separate ball bearings, was modified such that one of the bearings was removed. This modification did not completely resolve the strange bearing problems; it did however significantly improve it. The pendulum would more consistently stabilize at the actual fixed point.
[[Image:modified7.png | figure | left | 500px]]
[[Image:damp7.png | thumb | center | 500px]]

In characterizing the decay that is observed in this pendulum, it was noticed that, for a significant range of angles about the bottom fixed point, the zero forcing ring down trajectory exhibited geometric decay instead of exponential decay, which would suggest frictional damping instead of viscous damping. And thus a frictional damping model was used in our simulations.

===== Double Pendulum =====

[[Image:Double7.png | figure | right | 400px]]

*Effective Length: 3.2 cm
*Material: Plastic

This pendulum was made out of legos. Since it had no ball bearings, it exhibited no strange bearing issues near the fixed points, and thus, it had regular, even friction. From a design perspective, its axels of rotation were also both made from lego plastic, and with significant forcing, these axels would bend the entire double pendulum in and out of the plane of rotation, causing an added degree of freedom as well the occasional self-destruction of the setup. However, given more thought and time, we discovered that one could easily design an excellent pendulum out of mere legos!

=== Methods ===
Given a pendulum of predetermined moment, affixed at it's pivot to an oscillating table, several methods were utilized to collect data on the Stability Boundary of the single pendulum, the dynamics in general, and damping of the system.

==== Stability Boundary ====
The algorithm for determining the Stability Boundary of the single pendulum is as follows:

#Choose a driving frequency for the sinusoidal movement of the shaking table.
#Move and support the pendulum at some small angular displacement from top fixed point.
#Slowly increase the driving amplitude until: the pendulum is stable to small perturbations from the inverted state (+).
#Determine forcing amplitude from oscilloscope read out.

Reduce the driving amplitude to zero, and vary the driving frequency. Repeat the previous steps until the parameter space $(\omega, A)$ is explored.
[[Image:StabilityMappingProcedure.png | left | 900px]]

==== General Dynamics Tracking ====
By affixing high-contrast white dots to the pendulum at the pivot and the pendulum end, visual tracking was performed using LabView and Matlab for real-time and offline data generation, respectively.

== Simulation ==
[[File:Diagram.png | thumb | 400px]]
A simulation was construction based on the single inverted pendulum model with a vertically oscillating pivot. The simulation was done through Matlab with Simulink. The equations of motion for such a model reduce to the following equation:

\begin{equation}
\ddot{ \theta} = \sin(\theta)\cdot(g/l - \ddot{y}/l)
\end{equation}

where $\theta$ is the angular position of the rod, $g$ is gravity, and $l$ is the rod length. And for a vertical forcing of the following form: $y(t) = A\sin(\omega t)$, the resulting equation used in the model is the following:

\begin{equation}
\ddot{ \theta} = \sin(\theta)(g/l-(A/l)\omega^2\cdot\sin(\omega t))
\end{equation}

where $A$ and $\omega$ are the amplitude and angular frequency, respectively.

Based on these equations, the following Simulink model was created.
[[File:Model.png | center | 500px]]
This model is a block diagram representation of the equation of motion formulated above. At the first time step, $t=0$, it uses an initial angular position of the rod to calculate the angular acceleration and then integrates twice into the next time step as the shaker also moves to its next position in time. The output is an array of angular positions in time, which can be transformed to their appropriate x and y positions.

With this model, one can easily vary the shaker frequency and amplitude and observe the pendulum’s behavior for different values. For example, a useful metric easily attainable with this model is the range of the horizontal bounds of the pendulum’s trajectory, or xmax - xmin. The following figure is a sample result of the horizontal range for 10 cm long rod starting at $\theta_o = \pi/30$ for a sweep of frequencies and amplitudes.
[[File:HorizontalMovement.jpg | center | 500px]]
Notice that there is a point when the range jumps to 0.2 m (twice the rod length). This indicates that the pendulum was not able to exhibit a stable oscillation about the vertical fixed point and instead fell. Keeping the amplitude constant and sweeping only through frequencies, one can observe how the critical stabilization frequency would vary as the initial angular position changes.

== Discussion ==

* Quadratic fit of ring-down profile
** a/b << 1 => Linear profile => Friction, not viscous
** grants damping parameter for simulation

* compare sims to experiment
** phase portraits
** stability boundaries

=== Basins Of Attraction ===

When analyzing the stability of the fixed points with different amounts of vertical forcing, it is instructive to see how the basins of attraction change when changing parameters such as amplitude or frequency. It is possible to visualize these attractions with our Simulink model. To do this, one must take an array of different initial conditions (angular position and angular velocity) in phase space and record the final position of the trajectory after a certain amount of time for each of these initial conditions. This is easily visualized by assigning a color to a pixel positioned at the appropriate initial position/velocity coordinate in phase space. In our case black represents 0 degrees, or the top position, and white represents 180 degrees, or the bottom fixed point.

The problem arises in getting a decent resolution of initial states, and integrating the equation out (simulating) for a large enough end time. The computational time to achieve this makes this simulation cost prohibitive. For example, if I want to get an image for the basins of attraction for pendulum running a 50 Hz forcing frequency, a reasonable time would be 50 cycles, or 1 second. And the resolution could be a set of 500x500 different initial conditions in the phase space. This would take at least a day to simulate. However, if I simulate only the first forcing cycle, it takes much less time. And with this first cycle, I now have a mapping of each individual initial condition as it evolves 1 cycle at a time. With this mapping, one can use a method called Interpolated Cell Mapping (ICM) to iterate through successive cycles of forcing and get to 1 second of "simulation" much faster and still quite accurately (especially with a high resolution). I was able to double the resolution and output basin figures many times faster. Below is a series of plots for a pendulum with an effective rod length of 2.72 cm, gravity at 9.8 m/s^2, damping factor of 10, and forcing frequency of 50 Hz, the figure shows how the basins evolves when increasing the forcing amplitude from 0 to 1 cm.

[[Image:BasinsFigure.png | figure | center]]

== References ==
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