Group 3 2014 - Revision history

Astrojax2014 at 01:32, 18 February 2015

2015-02-18T01:32:54Z

← Older revision		Revision as of 21:32, 17 February 2015
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	==Experiment Videos and Photos==		==Experiment Videos and Photos==
	{\|style="margin: 0 auto;"		{\|style="margin: 0 auto;"
	\| [[File:FalconGlove.jpg\|thumb\| alt+text\| Figure 8: ~~The~~]]		\| [[File:FalconGlove.jpg\|thumb\| alt+text\| Figure 8: An image showing the method for capturing Astrojax motion via human forcing. An infrared reflective marker was attached to the top of the hand to allow OptiTrack to track the forcing.]]
	\|}		\|}
	{\|style="margin: 0 auto;"		{\|style="margin: 0 auto;"
			\| <videoflash>qvpj1DxlpX0</videoflash>
	\| <videoflash>KlnBpO_LThE</videoflash>		\| <videoflash>KlnBpO_LThE</videoflash>
	~~\| <videoflash>qvpj1DxlpX0</videoflash>~~
	\| <videoflash>lGg5QRCDRCs</videoflash>		\| <videoflash>lGg5QRCDRCs</videoflash>
	\| <videoflash>IeAsMpIIsPw</videoflash>		\| <videoflash>IeAsMpIIsPw</videoflash>
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	{\|style="margin: 0 auto;"		{\|style="margin: 0 auto;"
	\| <videoflash>wU8yi3MdGhc</videoflash>		\| <videoflash>wU8yi3MdGhc</videoflash>
			\| <videoflash>rrZDmMzerEo</videoflash>
			\| <videoflash>_tt-gAX3mq4</videoflash>
	\|}		\|}
	==References==		==References==
	{{reflist}}		{{reflist}}

Astrojax2014 at 01:26, 18 February 2015

2015-02-18T01:26:51Z

← Older revision		Revision as of 21:26, 17 February 2015
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	\| [[File:FalconGlove.jpg\|thumb\| alt+text\| Figure 8: The]]		\| [[File:FalconGlove.jpg\|thumb\| alt+text\| Figure 8: The]]
	\|}		\|}
	<videoflash>KlnBpO_LThE</videoflash>		{\|style="margin: 0 auto;"
			\| <videoflash>KlnBpO_LThE</videoflash>
			\| <videoflash>qvpj1DxlpX0</videoflash>
			\| <videoflash>lGg5QRCDRCs</videoflash>
			\| <videoflash>IeAsMpIIsPw</videoflash>
			\|}
			{\|style="margin: 0 auto;"
			\| <videoflash>wU8yi3MdGhc</videoflash>
			\|}
	==References==		==References==
	{{reflist}}		{{reflist}}

Astrojax2014 at 01:11, 18 February 2015

2015-02-18T01:11:28Z

← Older revision		Revision as of 21:11, 17 February 2015
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	\| [[File: Take1Horiz3D.png\|thumb\| alt=text\| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]		\| [[File: Take1Horiz3D.png\|thumb\| alt=text\| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
	\|}		\|}

			==Experiment Videos and Photos==
			{\|style="margin: 0 auto;"
			\| [[File:FalconGlove.jpg\|thumb\| alt+text\| Figure 8: The]]
			\|}
			<videoflash>KlnBpO_LThE</videoflash>
	==References==		==References==
	{{reflist}}		{{reflist}}

PatrickChang at 15:51, 12 December 2014

2014-12-12T15:51:32Z

← Older revision		Revision as of 11:51, 12 December 2014
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	''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''		''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''

	[[Media:NLD.pdf\|Report]]		[[Media:NLD.pdf\|Final Report]]

	[[File:astrojax_balls.png\|thumb\| alt+text\| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]		[[File:astrojax_balls.png\|thumb\| alt+text\| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]

PatrickChang at 15:50, 12 December 2014

2014-12-12T15:50:20Z

← Older revision		Revision as of 11:50, 12 December 2014
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	''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''		''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''

	[[Media:NLD.pdf\|Report]]		[[Media:NLD.pdf\|Report]]

PatrickChang at 15:49, 12 December 2014

2014-12-12T15:49:42Z

← Older revision		Revision as of 11:49, 12 December 2014
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	''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''		''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''
			[[Media:NLD.pdf\|Report]]

	[[File:astrojax_balls.png\|thumb\| alt+text\| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]		[[File:astrojax_balls.png\|thumb\| alt+text\| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]

Astrojax2014 at 04:37, 11 December 2014

2014-12-11T04:37:02Z

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 </math>
 ==Results==
 '''All Figures are located in the gallery below'''

Astrojax2014 at 04:29, 11 December 2014

2014-12-11T04:29:38Z

← Older revision		Revision as of 00:29, 11 December 2014
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	=====Physical Interpretation of the System=====		=====Physical Interpretation of the System=====
	Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf ~~Philip du Toit~~</ref> and the Newtonian equations of motion have been derived in previous works. We provide a summary of these equations below.		Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf</ref> and the Newtonian<ref>http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf</ref> equations of motion have been derived in previous works. We provide a summary of these equations below.
	The generalized cooridnates are given as:		The generalized cooridnates are given as:
	<math>		<math>

Astrojax2014 at 04:28, 11 December 2014

2014-12-11T04:28:24Z

Astrojax2014 at 04:23, 11 December 2014

2014-12-11T04:23:49Z

← Older revision		Revision as of 00:23, 11 December 2014
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	==Figure Gallery==		==Figure Gallery==
	{\|style="margin: 0 auto;"		{\|style="margin: 0 auto;"
	\| [[File: ActiveTimevsSplit90.png\|thumb\|]]		\| [[File: ActiveTimevsSplit90.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]
			\| [[File: ActiveTimevsSplit150.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]
			\| [[File: ActiveTimevsSplit180.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]
			\| [[File: ActiveTimevsForcingAmp.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. forcing amplitude. ]]
			\| [[File: ActiveTimevsFreq.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. forcing frequency. ]]
			\| [[File: FreqPlot.png\|thumb\| alt=text\| A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]
			\| [[File: Take1Horiz3D.png\|thumb\| alt=text\| A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
	\|}		\|}
	==References==		==References==
	{{reflist}}		{{reflist}}

← Older revision		Revision as of 00:28, 11 December 2014
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	''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''		''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''

	[[File:astrojax_balls.png\|thumb\| alt+text\| ~~Fig 1:~~ A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]		[[File:astrojax_balls.png\|thumb\| alt+text\| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]
	<blockquote>We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.</blockquote>		<blockquote>We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.</blockquote>

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	==Results==		==Results==
	'''All Figures are located in the gallery below'''		'''All Figures are located in the gallery below'''

	We recorded an average of three takes each for each amplitude of robotic forcing, and five takes each for each type of human forcing (Horizontal, Vertical, Butterfly). In general, the naive method of having the robot arm force the Astrojax with a periodic motion with no feedback did not result in any stable orbits occurring, only quasi-stable and/or chaotic orbits that rapidly change or decay. In the robotic forcing, we found that a sufficient degree of acceleration was required in order to separate the two bobs from one another and cause the system to evolve away from the trivial state (i.e. the two bobs remain in contact with one another at the string's nadir).		We recorded an average of three takes each for each amplitude of robotic forcing, and five takes each for each type of human forcing (Horizontal, Vertical, Butterfly). In general, the naive method of having the robot arm force the Astrojax with a periodic motion with no feedback did not result in any stable orbits occurring, only quasi-stable and/or chaotic orbits that rapidly change or decay. In the robotic forcing, we found that a sufficient degree of acceleration was required in order to separate the two bobs from one another and cause the system to evolve away from the trivial state (i.e. the two bobs remain in contact with one another at the string's nadir).

	Once separated in the robotic forcing, we found the Astrojax did not follow any predictable behaviors; their motion was mostly chaotic under this simple periodic forcing. The duration of time where the bobs were in this chaotic behavior vs. the time they spent in the trivial state increased overall as the forcing progressed, as shown in the plots below.		Once separated in the robotic forcing, we found the Astrojax did not follow any predictable behaviors; their motion was mostly chaotic under this simple periodic forcing. The duration of time where the bobs were in this chaotic behavior vs. the time they spent in the trivial state increased overall as the forcing progressed, as shown in the plots below.

	We also plot the mean active time of each take vs. each forcing's respective amplitude and frequency. As seen in Fig. 6, there is high variance in the mean active time solely for a forcing amplitude of 150mm, while the remaining amplitudes cluster around having 30 seconds. Figure 7 shows the same data with respect with the forcing frequency. Note that a small difference in forcing frequency near 2.5 Hertz seemed to generate a wider spread of mean active times.		We also plot the mean active time of each take vs. each forcing's respective amplitude and frequency. As seen in Fig. 4, there is high variance in the mean active time solely for a forcing amplitude of 150mm, while the remaining amplitudes cluster around having 30 seconds. Figure 5 shows the same data with respect with the forcing frequency. Note that a small difference in forcing frequency near 2.5 Hertz seemed to generate a wider spread of mean active times.

	Using MATLAB's Fast Fourier Transform (~~fft~~), we found the most dominant frequency in the recorded oscillations in all three Cartesian dimensions. The resultant frequencies were often symmetrical about the Cartesian $x$ \& $y$ coordinates, while ~~ffts~~ of the $z$ coordinate often failed or were too noisy for some unknown reason. The plot below shows the resultant frequencies of each bob as a function of each take's forcing frequency.		Using MATLAB's Fast Fourier Transform (FFT), we found the most dominant frequency in the recorded oscillations in all three Cartesian dimensions. The resultant frequencies were often symmetrical about the Cartesian $\scriptsize{x}$ & $\scriptsize{y}$ coordinates, while FFTs of the $\scriptsize{z}$ coordinate often failed or were too noisy for some unknown reason. The plot below shows the resultant frequencies of each bob as a function of each take's forcing frequency.

	As seen in Fig. 8, as the forcing frequency increases, the resultant dominant oscillation frequency for each bob tends to increase in a roughly linear manner, with the end bob increasing slightly faster than the middle bob.		As seen in Fig. 6, as the forcing frequency increases, the resultant dominant oscillation frequency for each bob tends to increase in a roughly linear manner, with the end bob increasing slightly faster than the middle bob.

	With M. Kingsbury's tracking program, we created full 3-D plots and movies of the tracked trajectories. We provide one of the more interesting plots below: a trajectory plot of human forcing creating a stable horizontal orbit (Fig. 9).		With M. Kingsbury's tracking program, we created full 3-D plots and movies of the tracked trajectories. We provide one of the more interesting plots below: a trajectory plot of human forcing creating a stable horizontal orbit (Fig. 7).

	The interesting part of the forcing is the green trajectory created by the human forcing marker. As seen in the plot, a complicated, helical path in order to maintain a steady horizontal orbit. The dynamic path that the human forcing creates with changing frequencies and amplitudes implies that some human 'feedback' mechanism is necessary to create such stable orbits.		The interesting part of the forcing is the green trajectory created by the human forcing marker. As seen in the plot, a complicated, helical path in order to maintain a steady horizontal orbit. The dynamic path that the human forcing creates with changing frequencies and amplitudes implies that some human 'feedback' mechanism is necessary to create such stable orbits.
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	From our results, we can qualitatively confirm certain aspects of the Astrojax system. As seen in Figure 8, the resultant frequencies of the middle & end bobs increase linearly as the forcing frequency increases. We suspect that at higher forcing frequencies this linear relation will no longer be true as the forcing frequency becomes too fast and the end bobs no longer have time to fall sufficient distances to split.		From our results, we can qualitatively confirm certain aspects of the Astrojax system. As seen in Figure 8, the resultant frequencies of the middle & end bobs increase linearly as the forcing frequency increases. We suspect that at higher forcing frequencies this linear relation will no longer be true as the forcing frequency becomes too fast and the end bobs no longer have time to fall sufficient distances to split.

	Figures 3 ~~thru 5~~ demonstrate that the active time of the end bobs increased as the Astrojax split from each other. These results match our qualitative observations with the robotic forcing. Often what would happen is that the forcing would generate small, short-lived splits of the jax that would either rapidly decay or create small, unstable orbits. It was only after some time of forcing and a few of these small splits that the jax would separate enough to create the full, rich, chaotic orbits with active times in the dozens of seconds.		Figures 1 thru 3 demonstrate that the active time of the end bobs increased as the Astrojax split from each other. These results match our qualitative observations with the robotic forcing. Often what would happen is that the forcing would generate small, short-lived splits of the jax that would either rapidly decay or create small, unstable orbits. It was only after some time of forcing and a few of these small splits that the jax would separate enough to create the full, rich, chaotic orbits with active times in the dozens of seconds.

	Another observation we made in our experiments was that under certain smooth motions of the robotic arm's end affector, the Astrojax would not deign to split under any attempted forcing frequency or amplitude. This occurred because the type of motion interpolation used was too smooth, and the robot arm was slowing down before reaching its assigned endpoint. Using a different kind of interpolation generated a much jerkier motion and successful splitting of the Astrojax. Thus, we strongly suspect that a sufficient degree of acceleration is required to separate the bobs, an acceleration which most likely must exceed $g$. This acceleration is generated by a sufficiently jerky motion, since acceleration is the antiderivative of jerk.		Another observation we made in our experiments was that under certain smooth motions of the robotic arm's end affector, the Astrojax would not deign to split under any attempted forcing frequency or amplitude. This occurred because the type of motion interpolation used was too smooth, and the robot arm was slowing down before reaching its assigned endpoint. Using a different kind of interpolation generated a much jerkier motion and successful splitting of the Astrojax. Thus, we strongly suspect that a sufficient degree of acceleration is required to separate the bobs, an acceleration which most likely must exceed $\scriptsize{g}$. This acceleration is generated by a sufficiently jerky motion, since acceleration is the antiderivative of jerk.

	==Conclusion==		==Conclusion==
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	==Figure Gallery==		==Figure Gallery==
	{\|style="margin: 0 auto;"		{\|style="margin: 0 auto;"
	\| [[File: ActiveTimevsSplit90.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]		\| [[File: ActiveTimevsSplit90.png\|thumb\| alt=text\| FIG. 1: A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]
	\| [[File: ActiveTimevsSplit150.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]		\| [[File: ActiveTimevsSplit150.png\|thumb\| alt=text\| FIG 2: A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]
	\| [[File: ActiveTimevsSplit180.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]		\| [[File: ActiveTimevsSplit180.png\|thumb\| alt=text\| FIG 3: A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]
	\| [[File: ActiveTimevsForcingAmp.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. forcing amplitude. ]]		\| [[File: ActiveTimevsForcingAmp.png\|thumb\| alt=text\| FIG 4: A plot of the Astrojax's active time vs. forcing amplitude. ]]
	\| [[File: ActiveTimevsFreq.png\|thumb\| alt=text\| A plot of the Astrojax's active time vs. forcing frequency. ]]		\| [[File: ActiveTimevsFreq.png\|thumb\| alt=text\| FIG 5: A plot of the Astrojax's active time vs. forcing frequency. ]]
	\| [[File: FreqPlot.png\|thumb\| alt=text\| A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]		\| [[File: FreqPlot.png\|thumb\| alt=text\| FIG 6: A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]
	\| [[File: Take1Horiz3D.png\|thumb\| alt=text\| A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]		\| [[File: Take1Horiz3D.png\|thumb\| alt=text\| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
	\|}		\|}
	==References==		==References==
	{{reflist}}		{{reflist}}