Revisiting the Asymptotic Optimality of RRT$^*$

TL;DR

This paper revisits the asymptotic optimality of RRT* algorithm, providing a rigorous proof and suggesting an adjustment to the connection radius to account for the temporal ordering of samples.

Contribution

It identifies a logical gap in the original proof of RRT*'s optimality and offers a new, mathematically rigorous proof with an improved connection radius formula.

Findings

The original proof of RRT*'s optimality has a logical gap.

A revised connection radius formula is proposed for better asymptotic optimality.

The new proof accounts for the additional dimension of time in sample ordering.

Abstract

RRT* is one of the most widely used sampling-based algorithms for asymptotically-optimal motion planning. This algorithm laid the foundations for optimality in motion planning as a whole, and inspired the development of numerous new algorithms in the field, many of which build upon RRT* itself. In this paper, we first identify a logical gap in the optimality proof of RRT*, which was developed in Karaman and Frazzoli (2011). Then, we present an alternative and mathematically-rigorous proof for asymptotic optimality. Our proof suggests that the connection radius used by RRT* should be increased from $\gamma \left(\frac{\log n}{n}\right)^{1/d}$ to $\gamma' \left(\frac{\log n}{n}\right)^{1/(d+1)}$ in order to account for the additional dimension of time that dictates the samples' ordering. Here $\gamma$, $\gamma'$, are constants, and $n$, $d$, are the number of samples and the dimension of the problem, respectively.