Equivariant Morse theory and formation control
Uwe Helmke, Brian D. O. Anderson · 2013
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 ℝdwith 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 ℝdsuch 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.