Kinematic nonholonomic optimal control: the skate example

The motion of a skate on a plane is one of the simplest kinematic nonholonomic systems and yet the problem of determining optimal motions exhibits many important features common to a broad class of kinematic optimal control problems. Although these problems for the skate have been studied in the literature, the present paper takes the analysis to a new level of detail. This article describes the analytic structure and geometry of the optimal controls for point to point kinematic control of the skate. It is shown that there are four types of optimal trajectories for the skate. These are characterized in terms of a parameterization obtained by careful use of invariants of the optimal motion. Numerical computations establish key qualitative features of optimal motions such as the geometry of geodesic neighborhoods and conjugate points. We briefly indicate how these results may be extended and applied to other problems in the optimal control of kinematic nonholonomic mechanical systems