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Summary  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    R  n   , extend the system from  M  to the ambient space    R  n   , and modify it outside  M  to add transversal stability to  M  in the final dynamics in    R  n   . Controllers are designed for the final system in the ambient space    R  n   . 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    R  n    is used for controller synthesis, and any controller design method in    R  n   , 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.

On controller design for systems on manifolds in Euclidean space