  A method of designing observers and observer-based tracking controllers is
proposed for nonlinear systems on manifolds via embedding into Euclidean space
and transversal stabilization. Given a system on a manifold, we first embed the
manifold and the system into Euclidean space and extend the system dynamics to
the ambient Euclidean space in such a way that the manifold becomes an
invariant attractor of the extended system, thus securing the transversal
stability of the manifold in the extended dynamics. After the embedding, we
design state observers and observer-based controllers for the extended system
in one single global coordinate system in the ambient Euclidean space, and then
restrict them to the original state-space manifold to produce observers and
observer-based controllers for the original system on the manifold. This
procedure has the merit that any existing control method that has been
developed in Euclidean space can be applied globally to systems defined on
nonlinear manifolds, thus making nonlinear controller design on manifolds
easier. The detail of the method is demonstrated on the fully actuated rigid
body system.
