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Published August 2011 | public
Journal Article

Discrete Geometric Optimal Control on Lie Groups

Abstract

We consider the optimal control of mechanical systems on Lie groups and develop numerical methods that exploit the structure of the state space and preserve the system motion invariants. Our approach is based on a coordinate-free variational discretization of the dynamics that leads to structure-preserving discrete equations of motion. We construct necessary conditions for optimal trajectories that correspond to discrete geodesics of a higher order system and develop numerical methods for their computation. The resulting algorithms are simple to implement and converge to a solution in very few iterations. A general software implementation is provided and applied to two example systems: an underactuated boat and a satellite with thrusters.

Additional Information

© 2011 IEEE. Manuscript received June 11, 2010; revised December 4, 2010; accepted March 29, 2011. Date of publication June 9, 2011; date of current version August 10, 2011. This paper was recommended for publication by Associate Editor T. Murphey and Editor K. Lynch upon evaluation of the reviewers' comments. The authors thank L. Noakes and D. M. de Diego for interesting discussions on closely related topics, and M. Desbrun, G. Johnson, and the paper reviewers for their useful feedback regarding this paper.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023