Incremental Sampling-based Algorithms for Optimal Motion Planning

TL;DR

This paper analyzes the theoretical properties of incremental sampling-based motion planning algorithms, proving that RRT does not guarantee optimality while introducing RRG and RRT* algorithms that do, with comparable computational complexity.

Contribution

It establishes the non-optimality of RRT, introduces RRG and RRT* algorithms with proven asymptotic optimality, and links these algorithms to random geometric graph theory.

Findings

RRT's best path cost converges to a non-optimal value.

RRG's best path cost converges to the optimal solution.

RRT* maintains asymptotic optimality with similar complexity to RRT.

Abstract

During the last decade, incremental sampling-based motion planning algorithms, such as the Rapidly-exploring Random Trees (RRTs) have been shown to work well in practice and to possess theoretical guarantees such as probabilistic completeness. However, no theoretical bounds on the quality of the solution obtained by these algorithms have been established so far. The first contribution of this paper is a negative result: it is proven that, under mild technical conditions, the cost of the best path in the RRT converges almost surely to a non-optimal value. Second, a new algorithm is considered, called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost of the best path in the RRG converges to the optimum almost surely. Third, a tree version of RRG is introduced, called the RRT$^*$ algorithm, which preserves the asymptotic optimality of RRG while maintaining a tree structure like RRT. The analysis of the new algorithms hinges on novel connections between sampling-based motion planning algorithms and the theory of random geometric graphs. In terms of computational complexity, it is shown that the number of simple operations required by both the RRG and RRT$^*$ algorithms is asymptotically within a constant factor of that required by RRT.