On Asymptotic Optimality Property of a Sampling based Motion Planner

In this paper, we study the asymptotic optimality property of a randomized incremental sampling based motion planner, namely RRT. The study proves that an RRT planner is not an asymptotically optimal motion planner. The result, while being consistent with similar results that exist in the literature, however, brings out an important characteristic of an RRT planner, that is, the degree distribution of the tree vertices follows a power law in an asymptotic sense. Simulation results are shown to support the theoretical claim. Based on these results, this work establishes a simple necessary and sufficient condition for sampling based motion planners to be asymptotically optimal.