Shortest paths for the Reeds-Shepp car: A worked out example of the use of geometric techniques in nonlinear optimal control

We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion could always be achieved by means of trajectories of a special kind, namely, concatenations of at most ve pieces, each of which is either a straight line or a circle, and (b) that these concatenations can be classied into 48 three-parameter families. We show how these results t in a much more general framework, and can be discovered and proved by applying in a systematic way the techniques of Optimal Control Theory. It turns out that the \classical " optimal control tools developed in the 1960's, such as the Pontryagin Maximum Principle and theorems on the existence of optimal trajectories, are helpful to go part of the way and get some information on the shortest paths, but do not suce to get the full result. On the other hand, when these classical techniques are combined with the use of a more recently developed body of theory, namely, geometric methods based on the Lie algebraic analysis of trajectories, then one can recover the full power of the Reeds-Shepp results, and in fact slightly improve upon them by lowering their 48 to a 46