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.

翻译：在许多机器人运动规划问题中，任务目标与物理约束会在构型空间上诱导出非欧几里得几何结构，然而多数规划器仍使用忽略该结构的欧氏距离进行运算。本文研究在构型相关的黎曼度量下规划无碰撞运动轨迹以最小化路径长度的问题，该问题对应于构型流形上的测地线计算。传统数值方法在计算此类路径时难以扩展到高维系统，而基于采样的规划器虽具有可扩展性却牺牲了几何保真度。为弥合这一差距，我们提出一种直接在黎曼流形上操作的基于采样的运动规划框架。我们引入一种计算高效的基于中点的黎曼测地距离近似方法，并证明该近似能以三阶精度逼近真实黎曼距离。基于此近似，我们设计了一种局部规划器，该规划器利用黎曼自然梯度引导的一阶收缩运算沿流形进行轨迹追踪。在动能度量下的二连杆平面机械臂与7自由度Franka机械臂实验，以及具有非完整运动约束的$\mathrm{SE}(2)$刚体规划实验中，本方法均能持续生成比基于欧氏距离的规划器及经典数值测地线求解基线更低成本的轨迹。