Nearly Time-optimal Paths for a Ground Vehicle

It is well known that the sufficient family of time-optimal paths for both Dubins’ as well as Reeds-Shepp’s car models consist of the concatenation of circular arcs with maximum curvature and straight line segments, all tangentially connected. These time-optimal solutions suffer from some drawbacks. Their discontinuous curvature profile, together with the wear and impairment on the control equipment that the bang-bang solutions induce, calls for “smoother” and more supple reference paths to follow. Avoiding the bang-bang solutions also enhances the robustness with respect to any possible uncertainties. In this paper, our main tool for generating these nearly time-optimal, but nevertheless continuous-curvature paths, is to use the Pontryagin Maximum Principle (PMP) and make an appropriate choice of the Lagrangian function. Despite some rewarding simulation results, this concept turns out to be numerically divergent at some instances. Upon a more careful investigation, it can be concluded that the problem at hand is nearly singular. This is seen by applying the PMP to Dubins’ car and studying the corresponding two point boundary value problem, which turn out to be singular. This is thus a counterexample to the widespread belief that all the information about the motion of a mobile platform lies in the initial values of the auxiliary variables associated with the PMP.