Nilpotentization and Motion Control for Under-actuated Systems on Matrix Lie Groups

In this paper we consider under-actuated, drift-free, invariant systems on matrix Lie groups and show how motion control results for nilpotent systems can be extended to invariant systems on non-nilpotent Lie groups by applying the method of nilpotentization. An algorithm for computing nilpotent approximations for invariant systems on Lie groups is presented. These approximations are used to construct a nilpotent model system for nilpotentization of a generic system in the class deened above. Using this method the corresponding feedback and state transformations can be computed in cases to which none of the previously known suucient conditions apply. New applications of this feedback equivalence are introduced by showing how approximate tracking control laws and exponentially stabilizing feedback control laws can be improved by making use of nilpotentization. The proposed methods are illustrated with a kinematic model of an under-actuated rigid body.