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Path homotopy invariants and their application to optimal trajectory planning

Abstract

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.

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Acknowledgements

The authors acknowledge the support of federal contracts FA9550-12-1-0416 and FA9550-09-1-0643. The first author acknowledges the support of ONR grant number N00014-14-1-0510 and University of Pennsylvania subaward number 564436.

Author information

Correspondence to Subhrajit Bhattacharya.

Additional information

Parts of this paper has appeared in the proceedings of the IMA Conference on Mathematics of Robotics (IMAMR), 2015.

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Bhattacharya, S., Ghrist, R. Path homotopy invariants and their application to optimal trajectory planning. Ann Math Artif Intell 84, 139–160 (2018) doi:10.1007/s10472-018-9596-8

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Keywords

  • Mathematical robotics
  • Path homotopy invariants
  • Knot and link complements
  • Coordination spaces
  • Graph search
  • Topological path planning

Mathematics Subject Classification (2010)

  • 55-04
  • 55Q99
  • 90C35