Non-Uniform Discretization Approximations for Kinodynamic Motion Planning and its Applications

The rst main result of this paper is a novel non-uniform approximation method for the kinody-namic motion-planning problem. The kinodynamic motion-planning problem is to compute a collision-free, minimum-time trajectory for a robot whose accelerations and velocities are constrained. Previous approximation methods are all based on a uniform dis-cretization in time space. On the contrary, our method employs a non-uniform discretization in connguration space (thus also a non-uniform one in time space). Compared to the previously best algorithm of Don-ald and Xavier, the running time of our algorithm reduces in terms of 1=", roughly from O((1=") 6d?1) to O((1=") 4d?2), in computing a trajectory in a d-dimensional connguration space, such that the time length of the trajectory is within a factor of (1 + ") of the optimal. More importantly, our algorithm is able to take advantage of the obstacle arrangement, and is expected to perform much better than the analytical result in many cases when the obstacles are sparse or when the obstacles are unevenly distributed. This is because our non-uniform discretization has the property that it is coarser in regions that are farther away from all obstacles. So for the above mentioned situations, the total number of discretized grids will be signiicantly smaller. Our second main result is the rst known polynomial-time approximation algorithm for the curvature-constrained shortest-path problem in 3 and higher dimensions. We achieved this by showing that the approximation techniques for the kinodynamic motion-planning problem are applicable to this problem.