Averaging and Motion Control On Lie Groups

Abstract

The deeper investigation of problems of feedback stabilization and constructive controllability has drawn increased attention to the question of structuring control systems. Thus, for instance, it is interesting to know how to combine periodic open loop controls with intermittent feedback corrections to achieve prescribed behavior in robotic motion planning systems. As a first step towards understanding this type of question, it would be useful to obtain some insight into the average behavior of a periodically forced system. In the present paper we are primarily interested in periodic forcing of left-invariant systems on Lie groups such as would arise in spacecraft attitude control. We prove averaging theorems applicable to systems evolving on general matrix Lie groups with particular focus on the attitude control problem. The results of this paper also yield useful formulae for motion planning of a variety of other systems such as an underwater vehicle which can be modeled as a control system evolving on the Lie group SE (3).