On Controller Design for Systems on Manifolds in Euclidean Space

A new method is developed to design controllers in Euclidean space for systems defined on manifolds. The idea is to embed the state-space manifold M of a given control system into some Euclidean space mathbb Rn, extend the system from M to the ambient space mathbb Rn, and modify it outside M to add transversal stability to M in the final dynamics in mathbb Rn. Controllers are designed for the final system in the ambient space mathbb Rn. Then, their restriction to M produces controllers for the original system on M. This method has the merit that only one single global Cartesian coordinate system in the ambient space mathbb Rn is used for controller synthesis, and any controller design method in mathbb Rn, such as the linearization method, can be globally applied for the controller synthesis. The proposed method is successfully applied to the tracking problem for the following two benchmark systems: the fully actuated rigid body system and the quadcopter drone system.