Geometric Integration - Revision history

Mpw6 at 16:12, 24 November 2015

2015-11-24T16:12:19Z

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			__NOTOC__

	==What is Geometric Integration?==		==What is Geometric Integration?==

Msandrle at 17:09, 20 November 2012

2012-11-20T17:09:04Z

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	Significant improvement is observed using a fourth-order CG algorithm with larger time-steps (~> 1s). The quaternion norm constraint is maintained essentially to within machine precision and attitude errors are reduced by approximately two orders of magnitude.		Significant improvement is observed using a fourth-order CG algorithm with larger time-steps (~> 1s). The quaternion norm constraint is maintained essentially to within machine precision and attitude errors are reduced by approximately two orders of magnitude.


	==Applications==		==Applications==

Msandrle at 17:07, 20 November 2012

2012-11-20T17:07:53Z

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	==Motivation==		==Motivation==

	[[File:sat_cloud.png\|thumbnail\|right\|150px\|Alt=SSA\|Space Object Cloud]]One prevailing field in which this research is very pertinent is [[Space_Situational_Awareness\|Space Situational Awareness]] (SSA). The U.S. Air Force is collecting data in order to catalogue Earth orbiting satellites, objects and debris. Accurate tracking of all Earth-orbiting objects is important since their presence can be extremely hazardous to current space missions as well as disruptive for any future mission. This danger can either be passive in the case of debris or active intentional threats. However, the data necessary to estimate the attitude of such objects is sparse as a result of the number of sensors available and the large number of objects in orbit. In particular, for objects in low-Earth orbit (LEO), observation windows may last only a few minutes and opportunities to obtain data are infrequent. As a result, an object's attitude may often need to be propagated very accurately over long time intervals before an update is possible.[[File:step_size.png\|thumbnail\|left\|~~400px~~\|Alt=Error Tradeoff\|Truncation and Roundoff Error Tradeoff]]		[[File:sat_cloud.png\|thumbnail\|right\|150px\|Alt=SSA\|Space Object Cloud]]One prevailing field in which this research is very pertinent is [[Space_Situational_Awareness\|Space Situational Awareness]] (SSA). The U.S. Air Force is collecting data in order to catalogue Earth orbiting satellites, objects and debris. Accurate tracking of all Earth-orbiting objects is important since their presence can be extremely hazardous to current space missions as well as disruptive for any future mission. This danger can either be passive in the case of debris or active intentional threats. However, the data necessary to estimate the attitude of such objects is sparse as a result of the number of sensors available and the large number of objects in orbit. In particular, for objects in low-Earth orbit (LEO), observation windows may last only a few minutes and opportunities to obtain data are infrequent. As a result, an object's attitude may often need to be propagated very accurately over long time intervals before an update is possible.[[File:step_size.png\|thumbnail\|left\|380px\|Alt=Error Tradeoff\|Truncation and Roundoff Error Tradeoff]]
	In summary, the problem is that		In summary, the problem is that
	*there are an increasing number of space objects to track		*there are an increasing number of space objects to track

Mpw6 at 20:46, 14 November 2012

2012-11-14T20:46:45Z

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	The results are particularly important for high fidelity attitude dynamics. Most perturbing forces, such as drag, solar radiation pressure, magnetic forces and gravity gradients, are a direct function of attitude. Attitude dynamics are much faster than orbital dynamics and therefore dictate the integration time-step in any simulator. Improvements made using a geometric integration method significantly increased accuracy at larger time-steps. The great computational cost of high fidelity models and long-term propagation intervals (days, weeks or months) is therefore reduced.		The results are particularly important for high fidelity attitude dynamics. Most perturbing forces, such as drag, solar radiation pressure, magnetic forces and gravity gradients, are a direct function of attitude. Attitude dynamics are much faster than orbital dynamics and therefore dictate the integration time-step in any simulator. Improvements made using a geometric integration method significantly increased accuracy at larger time-steps. The great computational cost of high fidelity models and long-term propagation intervals (days, weeks or months) is therefore reduced.

	*Attribute Estimation		*[[Attribute_Estimation\|Attribute Estimation]]
	[[File:shape.png\|thumbnail\|right\|300px\|Alt=Satellite Shape\|Perturbing Forces Depend on Attitude]]		[[File:shape.png\|thumbnail\|right\|300px\|Alt=Satellite Shape\|Perturbing Forces Depend on Attitude]]
	In applications involving satellite attribute estimation, the size, shape, surface materials and attitude may be unknown. Accurate attitude estimate propagation between measurement updates is essential. Falling well within the domain of SSA, satellite attribute estimation suffers from the same challenges: limited sensor time due to over-tasking, infrequent pass opportunities and longer propagation intervals. While sparse data must be used most effectively, attitude propagation between measurement updates must be as accurate as possible.		In applications involving satellite attribute estimation, the size, shape, surface materials and attitude may be unknown. Accurate attitude estimate propagation between measurement updates is essential. Falling well within the domain of SSA, satellite attribute estimation suffers from the same challenges: limited sensor time due to over-tasking, infrequent pass opportunities and longer propagation intervals. While sparse data must be used most effectively, attitude propagation between measurement updates must be as accurate as possible.

Msandrle: /* What is Geometric Integration? */

2012-11-13T20:54:44Z

What is Geometric Integration?

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	==What is Geometric Integration?==		==What is Geometric Integration?==

			Geometric integration employs a numerical method that preserves geometric or group properties of the exact differential equation solution trajectory.

	==Motivation==		==Motivation==

Msandrle: /* Findings */

2012-11-13T20:30:05Z

Findings

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	For small time-steps (< 1s) the CG geometric approach did not perform as well as classical Runge-Kutta (RK) methods. Geometric approaches require more calculations and in this step-size regime, roundoff error and machine precision dominate.		For small time-steps (< 1s) the CG geometric approach did not perform as well as classical Runge-Kutta (RK) methods. Geometric approaches require more calculations and in this step-size regime, roundoff error and machine precision dominate.

			Significant improvement is observed using a fourth-order CG algorithm with larger time-steps (~> 1s). The quaternion norm constraint is maintained essentially to within machine precision and attitude errors are reduced by approximately two orders of magnitude.

	==Applications==		==Applications==

Msandrle: /* Findings */

2012-11-13T20:20:47Z

Findings

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	==Findings==		==Findings==

	~~The quick brown fox jumps over~~ the ~~lazy dog~~.~~<ref name="LazyDog" />~~		For small time-steps (< 1s) the CG geometric approach did not perform as well as classical Runge-Kutta (RK) methods. Geometric approaches require more calculations and in this step-size regime, roundoff error and machine precision dominate.

	==Applications==		==Applications==

Msandrle: /* References */

2012-11-13T20:08:56Z

References

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	==References==		==References==

	~~<references>~~
	~~<ref name="LazyDog">This is the lazy dog reference.</ref>~~
	~~</references>~~

	Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.		Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.

Msandrle at 20:06, 13 November 2012

2012-11-13T20:06:07Z

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	==Findings==		==Findings==

	<ref name="~~andrle~~"~~>"John Chrysostom", ''Encyclopaedia Judaica''<~~/~~ref~~>		The quick brown fox jumps over the lazy dog.<ref name="LazyDog" />

	==Applications==		==Applications==
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	==References==		==References==

	<references />		<references>
			<ref name="LazyDog">This is the lazy dog reference.</ref>
	~~{{Reflist}}~~		</references>

	Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.		Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.

Msandrle: /* References */

2012-11-13T20:04:05Z

References

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	==References==		==References==

			<references />

	{{Reflist}}		{{Reflist}}

	Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.		Andrle, M.S., and Crassidis, J.L., “[http://ancs.eng.buffalo.edu/pdf/ancs_papers/2012/geom_int.pdf Geometric Integration of Quaternions],” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, Aug. 2012, AIAA Paper #2012-4421.