Paper Info
Title
Equivariant Morse theory and formation control.
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 R-d 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 R-d 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.
Year
DOI
Venue
2013
10.1109/Allerton.2013.6736716
2013 51ST ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING (ALLERTON)
Field
DocType
ISSN
Geometric mechanics,Mathematical optimization,Combinatorics,Equivariant map,Circle-valued Morse theory,Pure mathematics,Euclidean space,Equivariant cohomology,Mathematics,Manifold,Morse theory,Configuration space
Conference
2474-0195
Citations 
PageRank 
References 
9
0.68
4
Authors
2
Name
Order
Citations
PageRank
Uwe Helmke133742.53
Brian D. O. Anderson23727471.00