Formation control on lines, circles and ellipses: Genericity results and Morse theoretic ideas

In this work we consider natural potential functions for 1-dimensional formation control problems on the real line, the circle and ellipses. It is shown that generically such functions on the line and the circle are Morse functions, i.e., their critical points are nondegenerate. This property is important in order to establish sharp upper and lower bounds for the number of critical points. For formations of higher dimensional agents it is an open problem to decide whether the Morse property is satisfied for generic choices of desired distances. For the circular case we apply methods from algebraic geometry, such as Bézout's theorem and the Bernstein-Kushnirenko-Khovanski theorem, to provide novel upper bounds on the number of critical points. These upper bounds grow exponentially in the number N of point agents, which indicates the underlying complexity of the problem of characterizing critical formations. Studying formations on an arbitrary curve is much more complicated and may lead to the generic appearance of degenerate critical points. We provide an example of a family of potential functions on an ellipse that is never a Morse function.