Walk on Spheres for PDE-based Path Planning

TL;DR

The paper addresses high-dimensional robotic path planning by replacing classic potential-field approaches with PDE-based navigation fields derived from the screened Poisson equation. It introduces Walk on Spheres (WoS), a grid-free Monte Carlo solver that provides online estimates of the solution $u$ and its gradient $\\nabla u$ via recursive sampling on largest empty spheres, enabling gradient-ascent navigation without full domain discretization. The authors establish convergence properties ($O(1/n)$ variance, $O(1/\\sqrt{n})$ std) and demonstrate that computation scales nearly linearly with dimension and is trivially parallelizable, then validate the method on a planar environment and a RR platform with both IK-based and Lipschitz-based distance bounds. The results suggest WoS as a scalable, online, single-query approach for PDE-based path planning in arbitrarily shaped configuration spaces, motivating further exploration of biases, hardware acceleration (e.g., GPUs), and extensions to kinodynamic settings.

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.

Figures (13)

• Figure 1: The table shows a non-exhaustive categorization of path planning strategies applicable in high-dimensional domains. The main contribution of this paper is the introduction of the method in the bottom left corner to robotics, i.e., to compute the solution of PDEs for path planning without prior offline computations leveraging the Walk on Spheres algorithm. The bottom right plot shows the example resulting path in task space, computed using the WoS method.
• Figure 2: Walk on Spheres, algorithm visualization. The random walker starting at $\mathbf{x}_i$ is simulated by recursively jumping to a random uniformly sampled point $\mathbf{x}_1$ on the largest empty sphere (radius $R_i$) centered at $\mathbf{x}_i$. The boundary reaching point is approximated by $\bar{\mathbf{x}_k}$, which is the closest point on the boundary $\partial \Omega$ to the first sampled point ($\mathbf{x}_k$) at a distance lower then $\epsilon$ from the domain boundary. The unit vector $\mathbf{v}$ is used to estimate the gradient, and the source point $\mathbf{z}$ appears in the consideration of source function contributions, both for the value and gradient expressions.
• Figure 3: The task space distance $d_\tau(\mathbf{x})$ is the length of the red line between the RR planar manipulator in configuration $\mathbf{x}$ (black) and a point obstacle (red), using four obstacle locations with increasing distance. In each quadrant, the boundary circle of the approximated safe disk of radius ($K^{-1}d_\tau(\mathbf{x})$) is visualized through the task space representation of $500$ equidistant points (light green) on the boundary circle.
• Figure 4: Notation used to describe the WoS evaluation environment.
• Figure 5: Demonstrating the landscape through multiple start points. Screening coefficient fixed to $c_{\mathrm{screen}} = 1$ for comparability. Radius scaling parameter $k_r=0.2$.