Averaging for attitude control and motion planning

Shows how to use periodic forcing to solve the constructive controllability problem for drift-free, left-invariant systems on matrix Lie groups with fewer controls than states. In particular, the authors prove a second-order averaging theorem applicable to systems evolving on general matrix Lie groups. Using this theorem, the authors show how to construct open loop controls for complete controllability of systems that require up to depth-one Lie brackets to satisfy the Lie algebra controllability rank condition. The authors apply these results to the attitude control problem with only two controls available and to the unicycle motion planning problem