Optimal Sampling-Based Motion Planning under Differential Constraints: the Driftless Case

TL;DR

This paper develops a theoretical framework for assessing and guaranteeing the asymptotic optimality of sampling-based motion planning algorithms for driftless control-affine robotic systems, providing convergence rate bounds.

Contribution

It introduces two novel algorithms, Differential Probabilistic RoadMap and Differential Fast Marching Tree, with proven convergence guarantees and rate bounds for optimal solutions.

Findings

Algorithms converge to optimal solutions as samples increase

Provided convergence rate bounds for the algorithms

Numerical experiments validate theoretical guarantees

Abstract

Motion planning under differential constraints is a classic problem in robotics. To date, the state of the art is represented by sampling-based techniques, with the Rapidly-exploring Random Tree algorithm as a leading example. Yet, the problem is still open in many aspects, including guarantees on the quality of the obtained solution. In this paper we provide a thorough theoretical framework to assess optimality guarantees of sampling-based algorithms for planning under differential constraints. We exploit this framework to design and analyze two novel sampling-based algorithms that are guaranteed to converge, as the number of samples increases, to an optimal solution (namely, the Differential Probabilistic RoadMap algorithm and the Differential Fast Marching Tree algorithm). Our focus is on driftless control-affine dynamical models, which accurately model a large class of robotic systems. In this paper we use the notion of convergence in probability (as opposed to convergence almost surely): the extra mathematical flexibility of this approach yields convergence rate bounds - a first in the field of optimal sampling-based motion planning under differential constraints. Numerical experiments corroborating our theoretical results are presented and discussed.