Geometric Mechanics and Control

Abstract

We present a study of the optimal time of flight of a vertical rolling disk on a surface of revolution. This is a twist on the classic problem of a disk rolling without slipping on the plane, which is a canonical example of a nonholonomic dynamical system. Such systems are characterized by the presence of non-integrable constraints on the velocities, which results in path-dependent behavior. Geometrically, these constraints appear as a distribution on the tangent bundle to the configuration manifold of the system. Optimal control problems on such systems are particularly interesting, since the distribution is non-involutive. Briefly, this means that the distribution defined by the nonholonomic constraints is not closed under the Lie bracket operation on a set of spanning vector-fields. This lack of closure results in a larger attainable set of configurations than would be possible if one only naively considered integral curves along the generalized coordinate curves.