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

N.E. Leonard and P.S. Krishnaprasad

Abstract

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.