High-order averaging on Lie groups and control of an autonomous underwater vehicle

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.