Asymptotic Optimality of Rapidly Exploring Random Tree

TL;DR

This paper analyzes the asymptotic optimality of the RRT motion planner, proving it is not asymptotically optimal and revealing its degree distribution follows a power law, supported by simulations.

Contribution

It provides a theoretical proof that RRT is not asymptotically optimal and characterizes its degree distribution, offering insights into sampling-based motion planning.

Findings

RRT is not asymptotically optimal.

The degree distribution of RRT follows a power law asymptotically.

A necessary condition for asymptotic optimality in sampling-based planners.

Abstract

In this paper we investigate the asymptotic optimality property of a randomized sampling based motion planner, namely RRT. We prove that a RRT planner is not an asymptotically optimal motion planner. Our result, while being consistent with similar results which exist in the literature, however, brings out an important characteristics of a RRT planner. We show that the degree distribution of the tree vertices follows a power law in an asymptotic sense. A simulation result is presented to support the theoretical claim. Based on these results we also try to establish a simple necessary condition for sampling based motion planners to be asymptotically optimal.