On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

TL;DR

This paper advances motion planning by using manifold samples to better explore configuration spaces, especially narrow passages, demonstrating significant speedups and providing a probabilistic completeness proof.

Contribution

It introduces a recursive MMS approach in 6D spaces, proves probabilistic completeness for affine subspace samples, and characterizes narrow passages by their dimensionality.

Findings

MMS outperforms PRM with over 20-fold speedup in complex scenarios.

Probabilistic completeness is established for MMS with affine subspace samples.

MMS has a significant advantage in high-dimensional narrow passage scenarios.

Abstract

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.