提出新方法在黎曼流形上实现高效采样，显著提升运动规划速度。
Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds
- 用矩阵形式的洛弗纳序下界保留度量方向信息，构建可接受启发式。
- 在3种机器人和3种度量下，启发式集比欧氏与标量方法更紧致。
- 支持无拒绝采样，适合高自由度机械臂的实时运动规划场景。
知情采样技术通过聚焦状态空间中的有希望区域来加速基于采样的运动规划器，但现有方法多依赖于欧几里得启发式，在配置相关的黎曼度量下会失去可接受性。虽然标量特征值下界能通过均匀缩放欧几里得距离恢复可接受性，却忽略了度量的方向结构，导致知情集过于保守。本文提出一种矩阵值可接受启发式，利用对称正定矩阵上的洛弗纳序计算度量张量的最紧常数下界，同时保持其完整方向结构。该下界的乔列斯基分解定义了一个线性映射，将黎曼空间转换为各向同性的欧几里得空间，在此空间中黎曼知情集退化为标准的拉长超椭球体，从而可直接使用现有算法进行无拒绝采样。在6-DoF UR5、7-DoF Franka和14-DoF PR2三种机器人上，针对三种不同黎曼度量的抓取任务实验表明，该启发式生成的知情集始终比欧氏及标量特征值下界更紧，显著加速了多个最先进的渐近最优规划器的收敛速度。
原文摘要 · Abstract (English)
Informed sampling techniques accelerate sampling-based motion planners by focusing the search on promising regions of the state space, yet most existing methods rely on Euclidean heuristics that become inadmissible under configuration-dependent Riemannian metrics. While scalar eigenvalue bounds restore admissibility by uniformly scaling the Euclidean distance, they discard the directional structure of the metric, producing overly conservative informed sets. We propose a matrix-valued admissible heuristic that exploits the Loewner order on symmetric positive definite matrices to compute the tightest constant lower bound on the metric tensor while preserving its full directional structure. The Cholesky factorization of this bound defines a linear map to an isotropic Euclidean space in which the Riemannian informed set reduces to a standard prolate hyperspheroid, enabling direct, rejection-free sampling using existing algorithms. Experiments on manipulation tasks with a 6-DoF UR5, 7-DoF Franka, and 14-DoF PR2 under three distinct Riemannian metrics show that our heuristic produces consistently tighter informed sets than both the Euclidean and scalar eigenvalue bounds, accelerating convergence across multiple state-of-the-art asymptotically optimal planners.
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