Discrete Geometric Optimal Control on Lie Groups

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-pre...

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Bibliographic Details
Published in:IEEE transactions on robotics 2011-08, Vol.27 (4), p.641-655
Main Authors: Kobilarov, M. B., Marsden, J. E.
Format: Article
Language:English
Subjects:
Aerospace electronics
Algorithms
Approximation methods
Control algorithms
Control systems
Discrete mechanics
Dynamical systems
Dynamics
Equations
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
geometric optimization
Invariants
Lie groups
Mechanical systems
Numerical analysis
Optimal control
Optimization
Physics
Preserves
Robotics
Satellites
Solid dynamics (ballistics, collision, multibody system, stabilization...)
Solid mechanics
Trajectory
underactuated systems
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Summary: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.
ISSN:1552-3098
1941-0468
DOI:10.1109/TRO.2011.2139130