Equivariant Morse theory and formation control

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    18 Citations (Scopus)

    Abstract

    In this paper we study the critical points of potential functions for distance-based formation shape of a finite number of point agents in Euclidean space ℝd with d ≤ 3. The analysis of critical formations proceeds using equivariant Morse theory for equivariant Morse functions on manifolds of configuration spaces. We establish lower bounds for the number of critical formations. For d = 2 these bounds agree with the bounds announced in [3], while for d = 3 we obtain new bounds. We also propose a control law of the form of a decentralized gradient flow that evolves on a configuration manifold for agents in ℝd such that collisions among the agents do not occur. By computing the equivariant cohomology of the configurations spaces we establish new lower bounds for the number of critical collision-free formations in the configuration space. Our work parallels earlier research in geometric mechanics by Pacella [19] and McCord [18] on enumerating central configurations for the N-body problem.

    Original languageEnglish
    Title of host publication2013 51st Annual Allerton Conference on Communication, Control, and Computing, Allerton 2013
    PublisherIEEE Computer Society
    Pages1576-1583
    Number of pages8
    ISBN (Print)9781479934096
    DOIs
    Publication statusPublished - 2013
    Event51st Annual Allerton Conference on Communication, Control, and Computing, Allerton 2013 - Monticello, IL, United States
    Duration: 2 Oct 20134 Oct 2013

    Publication series

    Name2013 51st Annual Allerton Conference on Communication, Control, and Computing, Allerton 2013

    Conference

    Conference51st Annual Allerton Conference on Communication, Control, and Computing, Allerton 2013
    Country/TerritoryUnited States
    CityMonticello, IL
    Period2/10/134/10/13

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