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Topological constraints in search-based robot path planning

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Abstract

There are many applications in motion planning where it is important to consider and distinguish between different topological classes of trajectories. The two important, but related, topological concepts for classifying manifolds that are of importance to us are those of homotopy and homology. In this paper we consider the problem of robot exploration and planning in Euclidean configuration spaces with obstaclees to (a) identify and represent different homology classes of trajectories; (b) plan trajectories constrained to certain homology classes or avoiding specified homology classes; and (c) explore different homotopy classes of trajectories in an environment and determine the least cost trajectories in each class. We exploit theorems from complex analysis and the theory of electromagnetism to solve the problem 2-dimensional and 3-dimensional configuration spaces respectively. Finally, we describe the extension of these ideas to arbitrary D-dimensional configuration spaces. We incorporate these basic concepts to develop a practical graph-search based planning tool with theoretical guarantees by combining integration theory with search techniques, and illustrate it with several examples.

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Notes

  1. Parts of this paper have appeared elsewhere—the 2-dimensional case was introduced in Bhattacharya et al. (2010) and the 3-dimensional case was analyzed in Bhattacharya et al. (2011c).

  2. The genus of an obstacle refers to the number of holes or handles (Munkres 1999).

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Acknowledgements

We gratefully acknowledge support from the ONR Antidote MURI project, grant no. N00014-09-1-1031; ONR Grants N00014-08-1-0696 and N00014-09-1-1052; and NSF Grant IIP-0742304.

We would like to thank Prof. Robert Ghrist and Dr. David Lipsky, University of Pennsylvania, for providing valuable insights on homology theory.

We would like to thank Mr. Dimitar Simeonov and Mr. Michael Fleder, Computer Science and Artificial Intelligence Laboratory, MIT, for providing us with the example described in Fig. 2(a) that illustrates the distinction between the equivalence of homotopy and that of homology classes of trajectories in 2-D.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Applied Mechanics, School of Engineering and Applied Science, University of Pennsylvania, Towne 229, 220 S. 33rd Street, Philadelphia, PA, 19104, USA

    S. Bhattacharya & V. Kumar

  2. Robotics Institute, Carnegie Mellon University, NSH 3211, 5000 Forbes Ave, Pittsburgh, PA, 15213, USA

    M. Likhachev

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  1. S. Bhattacharya
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  2. M. Likhachev
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  3. V. Kumar
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Correspondence to S. Bhattacharya.

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Bhattacharya, S., Likhachev, M. & Kumar, V. Topological constraints in search-based robot path planning. Auton Robot 33, 273–290 (2012). https://doi.org/10.1007/s10514-012-9304-1

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