Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds

TL;DR

This paper introduces a scalable sampling-based motion planning method on Riemannian manifolds that computes geodesic paths efficiently, improving trajectory quality in complex robotic systems with non-Euclidean configuration spaces.

Contribution

It proposes a midpoint-based approximation for Riemannian geodesic distances and a local planner using Riemannian natural gradients, enabling efficient planning on manifolds.

Findings

The method produces lower-cost trajectories than Euclidean planners.

It scales well to high-dimensional systems like robotic arms.

Experiments demonstrate improved geometric fidelity in motion planning.

Abstract

In many robot motion planning problems, task objectives and physical constraints induce non-Euclidean geometry on the configuration space, yet many planners operate using Euclidean distances that ignore this structure. We address the problem of planning collision-free motions that minimize length under configuration-dependent Riemannian metrics, corresponding to geodesics on the configuration manifold. Conventional numerical methods for computing such paths do not scale well to high-dimensional systems, while sampling-based planners trade scalability for geometric fidelity. To bridge this gap, we propose a sampling-based motion planning framework that operates directly on Riemannian manifolds. We introduce a computationally efficient midpoint-based approximation of the Riemannian geodesic distance and prove that it matches the true Riemannian distance with third-order accuracy. Building on this approximation, we design a local planner that traces the manifold using first-order retractions guided by Riemannian natural gradients. Experiments on a two-link planar arm and a 7-DoF Franka manipulator under a kinetic-energy metric, as well as on rigid-body planning in $\mathrm{SE}(2)$ with non-holonomic motion constraints, demonstrate that our approach consistently produces lower-cost trajectories than Euclidean-based planners and classical numerical geodesic-solver baselines.