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