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Motion Planning via Manifold Samples

Abstract

We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably simpler sampling-based approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire low-dimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of low-dimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generic programming. This in turn enabled us to implement our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. We show that this instance of the framework is probabilistically complete. Moreover, we demonstrate that the integration of several carefully engineered components leads to significant speedup over the popular PRM sampling-based algorithm, which represents the more simplistic approach that is prevalent in practice.

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Notes

  1. 1.

    A subdivision of the plane into zero-dimensional, one-dimensional and two-dimensional cells, called vertices, edges and faces, respectively induced by the curves.

  2. 2.

    Considering vertical lines and not vertical slabs as discussed in Sect. 3 is done for simplifying the proof. This is obviously a special case of the same family of manifolds thus the presented proof applies for the case of vertical slabs.

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Correspondence to Oren Salzman.

Additional information

This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.

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Salzman, O., Hemmer, M., Raveh, B. et al. Motion Planning via Manifold Samples. Algorithmica 67, 547–565 (2013). https://doi.org/10.1007/s00453-012-9736-1

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Keywords

  • Motion planning
  • Computational geometry
  • Manifolds