Intrinsic Observer-Based Stabilization for Simple Mechanical Systems on Lie Groups

This paper presents a dynamic observer for a class of simple mechanical systems on Lie groups. This observer provides velocity estimates based on configuration measurements. The observer is \textit{intrinsic}, so its performance does not depend on the choice of coordinates, and it is \textit{coordinate free}, in the sense that the equations may be written explicitly without specifying coordinates for the configuration space. Our main result is obtained by specializing a previous result of Aghannan and Rouchon concerning velocity estimation of simple mechanical systems on Riemannian manifolds to such systems on Lie groups. This specialization is nonobvious and extremely powerful. Further we extend the original result to include velocity-dependent external forces. This estimator, combined with a coordinate-free formulation of passivity-based state-feedback control, allows the construction of a coordinate-free, intrinsic dynamic output feedback compensator. This is, to our knowledge, the first time such a result has been reported. Explicit expressions are computed for the Lie groups $SO(3)$ and $SE(3)$, allowing easy specialization to practical problems of rigid body motion. The theory is illustrated via application to the axisymmetric top and to a six-degrees-of-freedom microelectromechanical system.