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Parametrized Motion Planning and Topological Complexity

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Algorithmic Foundations of Robotics XV (WAFR 2022)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 25))

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Abstract

In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing the recent paper [2], we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles in \({\mathbb R}^d\). This algorithm is optimal, it has minimal possible topological complexity for any \(d\ge 3 \) odd. Besides, we describe a modification of this algorithm which is optimal for \(d\ge 2\) even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes.

Both authors were partially supported by an EPSRC—NSF grant.

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Notes

  1. 1.

    Note that in [11] we used a non-reduced notion of topological complexity which is greater by 1 than the reduced version.

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Author information

Authors and Affiliations

  1. Queen Mary University of London, London, UK

    Michael Farber

  2. The University of Chicago, Chicago, USA

    Shmuel Weinberger

Authors
  1. Michael Farber
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  2. Shmuel Weinberger
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Corresponding author

Correspondence to Michael Farber .

Editor information

Editors and Affiliations

  1. Center for Ubiquitous Computing, University of Oulu, Oulu, Finland

    Steven M. LaValle

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA

    Jason M. O’Kane

  3. Department of Aerospace Engineering, University of Maryland, College Park, MD, USA

    Michael Otte

  4. Gates Computer Science, Stanford University, Stanford, CA, USA

    Dorsa Sadigh

  5. Department of Computer Science, University of Maryland, College Park, MD, USA

    Pratap Tokekar

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Farber, M., Weinberger, S. (2023). Parametrized Motion Planning and Topological Complexity. In: LaValle, S.M., O’Kane, J.M., Otte, M., Sadigh, D., Tokekar, P. (eds) Algorithmic Foundations of Robotics XV. WAFR 2022. Springer Proceedings in Advanced Robotics, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-031-21090-7_1

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