Morse theory and formation control

Formation shape control for a collection of point agents is concerned with devising decentralized control laws which will ensure that the formation will move so that certain inter-agent distances assume prescribed values. A number of algorithms based on steepest descent of an error function have been suggested for various problems, and all display the existence of incorrect equilibria, though often the equilibria are saddle points or unstable. This paper introduces Morse theory as a tool for analyzing the number of such equilibria. A key conclusion is that for two-dimensional rigid formations of point agents, there will always be incorrect equilibria associated with any steepest descent law.