Hybrid Geometric Controllers for Fully-Actuated Left-invariant Systems on Matrix Lie Groups

This paper proposes a hybrid geometric control scheme for a system defined on a matrix Lie group in the form of a left-invariant vector field. Our solution to the point stabilization problem is coordinate free (or geometric). Specifically, we propose a hybrid geometric controller that uses a controller from a local class of geometric controllers and an open-loop geometric controller. Our method guarantees that the given point in the manifold is robustly globally asymptotically stable for the closed-loop system when each controller from the local geometric class is combined with the geometric open-loop controller using a hybrid systems framework.