High-Order Averaging on Lie Groups
and Control of an Autonomous Underwater Vehicle

N.E. Leonard and P.S. Krishnaprasad

In Proceedings of the 1994 American Control Conference, Baltimore, MD, June, 1994, p.157-162.


In this paper, extending our previous work on averaging on Lie groups, we present a third-order averaging theorem for periodically forced, drift-free, left-invariant systems on Lie groups and use it to demonstrate constructive controllability for a class of problems. Specifically, this class includes the case for which depth-two Lie brackets are needed for complete controllability. We illustrate this via an example on the group $SE(3)$, appropriate as a model of kinematic control of an underwater vehicle.

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