Symmetric Motion Planning

Michael Farber*, Mark Grant

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions of topological complexity of such algorithms and compare them with each other and with the usual (non-symmetric) concept of topological complexity. Using equivariant cohomology and the theory of Schwarz genus we obtain cohomological lower bounds for symmetric topological complexity. One of our main results states that, in the case of aspherical manifolds, the complexity of symmetric motion planning algorithms with fixed midpoint map exceeds twice the cup-length.

We introduce a new concept, the sectional category weight of a cohomology class, which generalises the notion of category weight developed earlier by E. Fadell and S. Husseini. We apply this notion to study the symmetric topological complexity of aspherical manifolds.

Original languageEnglish
Title of host publicationTopology and Robotics
EditorsM Farber, R Ghrist, M Burger, D Koditschek
Place of PublicationProvidence
PublisherAmerican Mathematical Society
Pages85-104
Number of pages20
ISBN (Print)978-0-8218-4246-1
Publication statusPublished - 2007
EventWorkshop on Topology and Robotics - Zurich, Switzerland
Duration: 10 Jul 200614 Jul 2006

Publication series

NameContemporary Mathematics Series
PublisherAmerican Mathematical Society
Volume438
ISSN (Print)0271-4132

Conference

ConferenceWorkshop on Topology and Robotics
CountrySwitzerland
Period10/07/0614/07/06

Keywords

  • Motion planning
  • equivariant cohomology
  • Schwarz genus

Cite this

Farber, M., & Grant, M. (2007). Symmetric Motion Planning. In M. Farber, R. Ghrist, M. Burger, & D. Koditschek (Eds.), Topology and Robotics (pp. 85-104). (Contemporary Mathematics Series; Vol. 438). Providence: American Mathematical Society.

Symmetric Motion Planning. / Farber, Michael; Grant, Mark.

Topology and Robotics. ed. / M Farber; R Ghrist; M Burger; D Koditschek. Providence : American Mathematical Society, 2007. p. 85-104 (Contemporary Mathematics Series; Vol. 438).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Farber, M & Grant, M 2007, Symmetric Motion Planning. in M Farber, R Ghrist, M Burger & D Koditschek (eds), Topology and Robotics. Contemporary Mathematics Series, vol. 438, American Mathematical Society, Providence, pp. 85-104, Workshop on Topology and Robotics, Switzerland, 10/07/06.
Farber M, Grant M. Symmetric Motion Planning. In Farber M, Ghrist R, Burger M, Koditschek D, editors, Topology and Robotics. Providence: American Mathematical Society. 2007. p. 85-104. (Contemporary Mathematics Series).
Farber, Michael ; Grant, Mark. / Symmetric Motion Planning. Topology and Robotics. editor / M Farber ; R Ghrist ; M Burger ; D Koditschek. Providence : American Mathematical Society, 2007. pp. 85-104 (Contemporary Mathematics Series).
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