Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds

• URL: http://arxiv.org/abs/2602.00992v1
• Date: Sun, 01 Feb 2026 03:14:46 GMT
• Title: Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds
• Abstract summary: We address the problem of planning collision-free motions that minimize length under configuration-dependent metrics.<n>Our approach consistently produces lower-cost trajectories than Euclidean-based planners and classical numerical geodesic-solver baselines.
• Score: 9.081925829245455
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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