Shapes of cyclic pursuit and their evolution

We study the geometry of a 2-dimensional cyclic pursuit problem where n identical mobile agents modeled as unicycles are driven by a distributed control law. The agent i pursuits the agent i + 1 modulo n with the same constant forward speed. We propose, for the first time, a stable relative (1 : n)-periodic trajectory (RPT) for the polygonal chain formed by the system with a sufficiently large n. Unlike regular polygon, the polygonal chain evolves into the shape of figure-eight while the system follows this RPT. The shape of the figure-eight can be approximated by closed Euler elastica. We show several geometrical and dynamical properties of this polygonal chain such as curve length and configuration precession. In addition, it reveals that the rotation number of an unbroken polygonal chain is a geometric invariant when it converges to a stable formation.