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Symmetries in Motion: Geometric Foundations of Motion Control

Marsden, Jerrold E. and Ostrowski, Jim (1998) Symmetries in Motion: Geometric Foundations of Motion Control. In: Motion, Control, and Geometry: Proceedings of a Symposium. National Academies Press , pp. 3-19. ISBN 9780309057851 http://resolver.caltech.edu/CaltechAUTHORS:20100902-113526349

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Abstract

Some interesting aspects of motion and control, such as those found in biological and robotic locomotion and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the Whole object can result. This property can be used for control purposes; the position and attitude Of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is used extensively in general relativity and other parts of theoretical physics. This tool, part of the general subject Of geometric mechanics, has been helpful in the study of both the stability and instability of a system and system bifurcations, that is, changes in the nature of the system dynamics, as some parameter changes. Geometric mechanics, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in attitude dynamics. Theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This paper explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation.


Item Type:Book Section
Additional Information:© 1998. Thanks are extended to John Tucker, Tony Bloch, Roger Brockett, Joel Burdick, Jaydev Desai, Bill Goodwine, Scott Kelly, P.S. Krishnaprasad, Vijay Kumar, Naomi Leonard, Andrew Lewis, Richard Montgomery, Richard Murray, Tudor Ratiu, Shankar Sastry, Greg Walsh, and Je Wentlandt for their kind advice and help. Thanks to Joel Burdick and Bill Goodwine for help in the preparation of Figure 3.2. Also, thanks to Andy Ruina and an anonymous reviewer for providing suggestions that helped improve this paper. The list of references provides a large sample of related work in this area. For additional material related to this paper, including MPEG movies of the snakeboard, a postscript copy of the paper by Bloch, Krishnaprasad, Marsden and Murray [1996], and current work in optimal control, the reader is referred to http://www.cis.upenn.edu/~jpo/papers.html and http://cds.caltech.edu/~marsden/
Record Number:CaltechAUTHORS:20100902-113526349
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100902-113526349
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ID Code:19770
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Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 20:23
Last Modified:26 Dec 2012 12:23

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