## Collective Motion of Ring-Coupled Planar Particles

### James Jeanne, Naomi Ehrich Leonard and Derek Paley

*Proc. 44th IEEE Conf. Decision and Control and European Control Conference*,
2005.
We study stabilization of collective motion of N constant-speed, planar particles
with less than all-to-all coupling. Our interest is in circular motions of the particles around the
fixed center of mass of the group, as has been studied previously
with all-to-all coupling. We focus on coupling
defined by a ring, i.e., each particle communicates with exactly two other particles.
The Kuramoto model of coupled oscillators, restricted to "ring" coupling, serves as our
model for controlling the relative headings of the particles. Each phase oscillator represents the
heading of a particle. We prove convergence to a set
of solutions that correspond to symmetric patterns of the phases about the unit circle.
The exponentially stable patterns are generalized regular polygons, determined by the
sign of the coupling strength parameter K.