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

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 low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The scheme is particularly suitable for applications in manufacturing, such as assembly planning, where typically motion planning needs to be carried out in very tight quarters. 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 Probabilistic Roadmaps (PRM), with over 40-fold speedup in a six-dimensional coordination-tight setting. (ii) A probabilistic completeness proof for the case of 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. Note to Practitioners-Highly constrained motion-planning scenarios, even of low degree of freedom, arise in various applications such as assembly planning and manufacturing applications. Our approach, which emphasizes high precision over any known sampling-based technique that we are aware of, allows to cope with exactly such cases. For instance, we show that our framework can be applied to tight scenarios that arise in three-handed assembly planning. The ability to cope with tight scenarios is possible, in part, due to recent improvements in exact geometric software such as the publicly available Computational Geometry Algorithms Library [43] (CGAL).