On certain hyperelliptic signals that are natural controls for nonholonomic motion planning

In this paper we address the general problem of approximating, in a certain optimal way, non admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider systems modeled by a subriemannian Goursat structure, a particular case being the well known system of a car with trailers, along with the associated parallel parking problem. Several authors approximate the successive Lie brackets by using trigonometric functions. By contrast, we show that the more natural optimal motions are related with closed hyperelliptic plane curves with a certain number of loops.