Controller Design for Systems on Manifolds in Euclidean Space

Given a control system on a manifold that is embedded in Euclidean space, it is sometimes convenient to use a single global coordinate system in the ambient Euclidean space for controller design rather than to use multiple local charts on the manifold or coordinate-free tools from differential geometry. In this paper, we develop a theory about this and apply it to the fully actuated rigid body system for stabilization and tracking. A noteworthy point in this theory is that we legitimately modify the system dynamics outside its state-space manifold before controller design so as to add attractiveness to the manifold in the resulting dynamics.