Walk on Spheres for PDE-based Path Planning

TL;DR

This paper explores the use of the Walk on Spheres algorithm, a Monte Carlo method, for robot motion planning in configuration spaces, demonstrating its efficiency, parallelizability, and convergence properties.

Contribution

It is the first to analyze WoS for robot motion planning in configuration spaces with potential fields solving screened Poisson equations.

Findings

Method is trivially parallelizable

Converges independently of dimension at rate O(1/N)

Validated on the RR platform

Abstract

In this paper, we investigate the Walk on Spheres algorithm (WoS) for motion planning in robotics. WoS is a Monte Carlo method to solve the Dirichlet problem developed in the 50s by Muller and has recently been repopularized by Sawhney and Crane, who showed its applicability for geometry processing in volumetric domains. This paper provides a first study into the applicability of WoS for robot motion planning in configuration spaces, with potential fields defined as the solution of screened Poisson equations. The experiments in this paper empirically indicate the method's trivial parallelization, its dimension-independent convergence characteristic of $O(1/N)$ in the number of walks, and a validation experiment on the RR platform.