A Coordinate-Free Approach to Tracking for Simple Mechanical Systems On Lie Groups

Abstract

A coordinate-free approach to the tracking problem for simple mechanical systems on Lie groups highlights the exploitation of geometrical and topological structure in controller design. First, the algebraic structure of the Lie group is used to construct an intrinsic and coordinate-free notion of configuration error. This construction leads through geometric ideas to a state feedback “inner” controller that gives the error dynamics the structure of a simple mechanical system on the original Lie group. This structure in turn allows the stability properties of the identity element of the error system to be characterized in terms of a potential energy-like error function, through implicit definition of a locally exponentially stabilizing state feedback “outer” controller. Stabilization of the identity element of the error system is equivalent to the original system tracking a smooth and bounded, but otherwise arbitrary, reference trajectory. Finally, suitable choices for the error function may be identified based on purely topological considerations. This means that a class of error functions may be defined for any simple mechanical system on the configuration space, greatly simplifying the design task. As an example of the power of this result, it is known that error functions corresponding to almost-global tracking may be found on any compact connected Lie group, and such functions have been explicitly constructed for many important special cases. The intrinsic, coordinate-free, structure-based approach is used to show a separation principle for this tracking problem, namely that the system velocities in the tracking controller may be replaced by their estimates, obtained from any locally exponentially convergent observer, without affecting the convergence properties of the closed-loop system.