Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping

We extend the method of controlled Lagrangians (CL) to include potential sh aping, which achieves complete state-space asymptotic stabilization of mech anical systems. The CL method deals with mechanical systems with symmetry a nd provides symmetry-preserving kinetic shaping and feedback-controlled dis sipation for state-space stabilization in all but the symmetry variables. P otential shaping complements the kinetic shaping by breaking symmetry and s tabilizing the remaining state variables. The approach also extends the met hod of controlled Lagrangians to include a class of mechanical systems with out symmetry such as the inverted pendulum on a cart that travels along an incline.