Direct Informed Sampling on Riemannian Manifolds via Loewner Order Lower Bounds

2026-06-01

AI Summary

Existing heuristic sampling methods based on Euclidean space fail under configuration-dependent Riemannian metrics, while conventional scalar eigenvalue bounds—though admissible—neglect the directional structure of the metric, yielding overly conservative sampling regions. This work proposes a matrix-valued admissible heuristic that, for the first time, leverages the Loewner order on symmetric positive-definite matrices to compute an optimal constant lower bound for the metric tensor. By preserving full directional information and employing Cholesky decomposition to map the resulting Riemannian information set into a standard ellipsoid in isotropic Euclidean space, the method enables efficient rejection-free sampling. Experiments on UR5, Franka, and PR2 robots across three Riemannian metrics demonstrate that the generated information sets are significantly tighter than those from Euclidean or scalar bounds, substantially accelerating convergence in multiple asymptotically optimal motion planners.

Abstract

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.

Problem

• informed sampling
• Riemannian manifolds
• Loewner order
• metric tensor
• motion planning

Innovation

• Riemannian manifold
• informed sampling
• Loewner order
• matrix-valued heuristic
• Cholesky factorization