Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds

2026-02-01

AI Summary

This work addresses the suboptimality of motion plans in robotics that arises from neglecting the non-Euclidean geometric structure of configuration spaces. To this end, we propose a sampling-based planning framework that operates directly on Riemannian manifolds. Our method efficiently approximates Riemannian geodesic distances using a third-order accurate midpoint scheme and, for the first time, integrates Riemannian natural gradients with first-order retraction operations into local path generation. This approach preserves geometric fidelity while ensuring scalability to high-dimensional systems. Experimental results demonstrate that our planner consistently produces trajectories with significantly lower cost—measured under the kinetic energy metric—than both Euclidean planners and conventional numerical geodesic solvers across diverse platforms, including a planar two-link arm, a 7-DoF Franka manipulator, and an SE(2) nonholonomic system.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

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.

Problem

• Riemannian manifolds
• motion planning
• geodesic distance
• non-Euclidean geometry
• configuration space

Innovation

• Riemannian manifold
• sampling-based motion planning
• geodesic approximation
• natural gradient
• non-Euclidean geometry