Stabilization of trajectories for systems on Lie groups. Application to the rolling sphere.

This paper addresses the stabilization of admissible reference trajectories generated with constant inputs for driftless systems on Lie groups. The general expression of the linear approximation of the tracking error system is derived from the system's constants of structure and a necessary condition for the controllability of this approximation is specified in terms of the growth of the filtration of the Lie Algebra generated by the system's vector fields. This condition is illustrated with examples of mechanical systems whose control inputs correspond to velocity variables. By contrast with nonholonomic mobile robots whose kinematic equations can be transformed into the chained form, the linearized system associated with the rolling sphere is never controllable. Consequences of this lack of controllability as for stabilization problems are discussed from a general viewpoint and addressed more specifically for the rolling sphere. Finally, a practical stabilizer for this system based on the transverse function approach is proposed.