Minimum-Length Trajectories for a Car: An Example of the Use of Boltianskii´s Sufficient Conditions for Optimality

We consider the kinematic model of a car which describes the rolling-without-slipping constraint of the wheels on an horizontal floor and the bound on the angle of steering of front wheels. The problem of determining shortest paths for such a vehicle is known as the Reeds and Shepp´s problem. Ten years ago, a complete solution to this problem was determined on the basis of a complex reasoning grounded on the necessary conditions of Pontryagin´s Maximum Principle and completed with a set of geometric arguments. In this note, we provide a simple new proof of the optimality of this construction by using a verification theorem based on Boltianskii´s sufficient regularity conditions. To our knowledge, it is the first example of a regular synthesis for a nonholonomic system in a three-dimensional space