Asymptotically Optimal Feedback Planning: FMM Meets Adaptive Mesh Refinement

摘要

The first asymptotically optimal feedback motion planning algorithm is presented. Our algorithm is based on two well-established numerical practices: (1) the Fast Marching Method (FMM), which is a numerical Hamilton-Jacobi-Bellman solver, and (2) the adaptive mesh refinement algorithm designed to improve the resolution of a simplicial mesh and, consequently, reduce the numerical error. Using the uniform mesh refinement, we show that the resulting sequence of numerical solutions converges to the optimal one. According to the dynamic programming principle, it is sufficient to refine the discretization within a small region around an optimal trajectory in order to reduce the computational cost. Numerical experiments confirm that our algorithm outperforms previous asymptotically optimal planning algorithms, such as PRM~* and RRT~*, in that it uses fewer discretization points to achieve similar quality approximate optimal paths.