Nonuniform Discretization for Kinodynamic Motion Planning and its Applications

The first main result of this paper is a novel nonuniform discretization approximation method for the kinodynamic motion-planning problem. The kinodynamic motion-planning problem is to compute a collision-free, time-optimal trajectory for a robot whose accelerations and velocities are bounded. Previous approximation methods are all based on a uniform discretization in the time space. On the contrary, our method employs a nonuniform discretization in the configuration space (thus also a nonuniform one in the time space). Compared to the previously best algorithm of Donald and Xavier, the running time of our algorithm reduces in terms of $1/\varepsilon$, roughly from $O((1/\varepsilon)^{6d-1})$ to $O((1/\varepsilon)^{4d-2})$, in computing a trajectory in a $d$-dimensional configuration space, such that the time length of the trajectory is within a factor of $(1+\varepsilon)$ of the optimal. More importantly, our algorithm is able to take advantage of the obstacle distribution and is expected to perform much better than the analytical result. This is because our nonuniform discretization has the property that it is coarser in regions that are farther from all obstacles. So for situations where the obstacles are sparse, or the obstacles are unevenly distributed, the size of the discretization is significantly smaller. Our second main result is the first known polynomial-time approximation algorithm for the curvature-constrained shortest-path problem in three and higher dimensions. We achieved this by showing that the approximation techniques for the kinodynamic motion-planning problem are applicable to this problem.