Trajectory Tracking on Lie groups and Riemannian manifolds

Francesco Bullo, Control and Dynamical Systems, Caltech

We describe the generalization of the classic notion of proportional derivative (PD) feedback to the case of mechanical systems defined on Lie groups and on Riemannian manifolds. This work applies to the case of fully actuated systems, that is, we assume that one actuator is available for each degree of freedom: robotic manipulators, satellites and 6DOF underwater vehicles belong to this class.

In order to avoid local parametrizations of the configuration space, our control laws rely on geometric properties of the configuration manifold and possess intrinsic meaning. We therefore achieve both an general formulation as well as improved global and local properties. For Lie groups, we provide a natural solution in the SO(3) case and we generalize it to SE(3) case, where some geometric properties make the control design more difficult and challenging. Regarding the general case of Riemannian manifold, we provide a complete solution to the trajectory tracking problem for systems on spheres and then we introduce a methodology for dealing with the general case.