Optimal, stable switching between arcs during low-speed Ackerman path tracking

In numerous applications an autonomous Ackerman (conventional front-wheel steer) vehicle must be made to follow a desired path. Early motion planning methods for cars used only straight lines and circular arcs, although more modern methods use clothoids and more general curves. We have chosen to plan with the simpler primitives due to the computational speed benefit and ease of implementation. The problem then is how to best transition from one arc to the next in the presence of a finite-speed steering servo. In this paper we describe a simple approach involving the adjustment of a single parameter, which is the time to stop tracking the current arc segment and to begin tracking the segment which follows. We present a return map associated with the periodic tracking of a slalom path. It has a fixed point corresponding to the periodic solution, and we numerically evaluate the stability of this fixed point to determine the stability of the associated periodic, continuous-time trajectory. We show that the map's stability is well-predicted by the continuous (unswitched) system. We have also derived an approximate expression for the optimal switching time using singular perturbation theory, and it agrees well with the numerical results. Experimental results are also included, which show good tracking performance when using the approach described in the paper.