Stabilization of relative equilibria

This paper discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We show how to stabilize an internally unstable relative equilibrium using internal actuators. The methodology is that of potential shaping, but the system is allowed to be underactuated, i.e., have fewer actuators than the dimension of the shape space. The theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum.