Optimal Trajectories of Curvature Constrained Motion in the Hamilton-Jacobi Formulation

We propose a Hamilton-Jacobi equation approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton-Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins’ and Reeds-Shepp’s cars, and gives further generalizations. We show that the value function of the control problem solves the Hamilton-Jacobi equation in the viscosity sense. Numerical schemes and results that illustrate the versatility of the Hamilton-Jacobi approach are presented for the two dimensional case.