An intrinsic observer for a class of simple mechanical systems on a Lie group

This work presents an intrinsic formulation of an observer for an important class of simple mechanical systems on a Lie group. Recently, Aghannan and Rouchon have formulated an observer for a simple mechanical system on a general Riemannian manifold. The current paper specializes their result to the case where the manifold is a Lie group, the kinetic energy is left invariant, and the velocity variables are to be estimated based on measurement of the configuration variables. These restrictions allow a greatly simplified result, of interest in its own right. Most significantly, no coordinates need be introduced on the Lie group, hence a single formulation is valid for all coordinate patches. To illustrate the method, observers are computed for two simple mechanical systems, on the rotation group SO(3) and on the Euclidian motion group SE(3). Simulations of an example on SO(3) show excellent performance.