Exact minimum control switch motion planning for the snakeboard

We study the problem of computing an exact motion plan for the snakeboard by exploiting its kinematic controllability properties and its decoupling vector fields. Decoupling vector fields allow us to treat the underactuated dynamic system as a kinematic one, and rest-to-rest paths are the concatenation of integral curves of the decoupling vector fields. These paths can then be time-scaled according to actuator limits to yield fast trajectories. Switches between decoupling vector fields must occur at zero velocity, so to find fast trajectories, we wish to find paths minimizing the number of switches. In this paper we solve the minimum switch path planning problem for the snakeboard. We consider two problems: (1) finding motion plans achieving a desired position and orientation of the body of the snakeboard, and (2) the full problem of motion planning for all five configuration variables of the snakeboard. The first problem is solved in closed form by geometric considerations, while the second problem is solved by a numerical approach with guaranteed convergence. We present a complete characterization of the snakeboard's optimal paths in terms of the number of switches.