Critical Points of Potential Functions for Circular Formation Control

Circular formation shape control is concerned with the design of decentralized control laws that achieve desired formations of one-dimensional agents on the circle. Natural potential functions for circular formation control are smooth functions on an N-torus that achieve their global minima at the desired formations. Critical points of the potential function correspond to critical formations, i.e. to equilibrium points of the associated gradient flow. This work extends the analysis by Anderson and Helmke (2013) for critical formations on a line to the circular formation case. Using tools from global analysis such as Morse theory and Betti numbers we establish lower bounds on the number of S1-orbits of critical formations on an N-torus.