Shortest paths to obstacles for a polygonal car-like robot

The paper shows how to compute the nonholonomic distance between a car-like robot of polygonal shape and polygonal obstacles. Adopting an optimal control point of view, we use transversality conditions to get information about the structure of paths that are admissible solutions. With this information, the problem of minimizing the length of a path that is, in general, a function of three parameters, is reduced to that of minimizing a function of one variable, namely, the robot final orientation. To solve the problem, we decompose it into three subproblems and find sufficient families of shortest paths, solving each of the subproblems.