リーマン多様体上の幾何学を考慮したサンプリングベース運動計画
Geometry-Aware Sampling-Based Motion Planning on Riemannian Manifolds
リーマン多様体上で直接動作するサンプリングベース運動計画を提案し、中点近似による測地線距離と自然勾配に基づく局所計画器を開発した。
詳しい要約
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著者: Phone Thiha Kyaw, Jonathan Kelly
分類: cs.RO
原文アブストラクト
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.
