Group 7 - Revision history

Cjcrowley at 14:01, 12 January 2012

2012-01-12T14:01:11Z

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

			[[Media: The_Inverted_Pendulum.pdf \| Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf \| Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf \| Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf \| Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf \| Balachandra]]

	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.		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.

Cjcrowley at 16:18, 11 January 2012

2012-01-11T16:18:42Z

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			'''Inverted pendulum'''

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

Cjcrowley: Protected "Group 7" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2012-01-09T16:52:35Z

Protected "Group 7" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

← Older revision	Revision as of 12:52, 9 January 2012
(No difference)

Group7: /* Discussion */

2011-12-23T05:17:40Z

Discussion

← Older revision		Revision as of 01:17, 23 December 2011
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	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.		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 ===		=== Flipping State of the Double Pendulum ===

Group7: /* Simulation */

2011-12-16T16:55:30Z

Simulation

← Older revision		Revision as of 12:55, 16 December 2011
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	[[File:HorizontalMovement.jpg \| center \| 500px]]		[[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.		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 ==		== Results ==

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

2011-12-16T16:50:48Z

Flipping State of the Double Pendulum

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

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

2011-12-16T16:45:51Z

Flipping State of the Double Pendulum

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

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

2011-12-16T16:42:10Z

Flipping State of the Double Pendulum

← Older revision		Revision as of 12:42, 16 December 2011
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	[[Image: Stabledoubleexpflipexp.png \| figure \| down \| 500px]]		[[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.		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.

Group7 at 17:17, 15 December 2011

2011-12-15T17:17:18Z

← Older revision		Revision as of 13:17, 15 December 2011
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	===== The Original =====		===== The Original =====

	*Effective Length: 6.88 cm		*Effective Length: 6.9 cm
	*Material: Aluminum		*Material: Aluminum

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

	*Effective Length: 3.13 cm		*Effective Length: 3.1 cm
	*Material: Aluminum		*Material: Aluminum

Group7: /* Determination of damping parameter */

2011-12-15T12:43:49Z

Determination of damping parameter

← Older revision		Revision as of 08:43, 15 December 2011
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	[[Image:Decay profile.png \| figure \| right \| 500px]]		[[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.		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]]		[[Image: prediction.png \| figure \| right \| 500px]]

← Older revision		Revision as of 08:43, 15 December 2011
Line 271:		Line 271:
	[[Image:Decay profile.png \| figure \| right \| 500px]]		[[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.		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]]		[[Image: prediction.png \| figure \| right \| 500px]]