On the trajectories and coordination of steered particles with time-periodic speed profiles

Motivated by observations of fish and possibilities for mobile robotics, we study collective motion of networks of agents that move with periodically time-varying speed. Each agent is modeled as a particle with constant turning rate and time-periodic speed profile at steady state. Expressions are derived for the trajectories of such particles, emphasizing the variation from the constant-speed circular orbit. We show that trajectories remain bounded if the speed profile contains no frequency content at the turning rate. Steering and speed control laws are derived that stabilize a rich family of collective motion patterns of a many-particle system about a common center point, where headings and speed phases are coordinated.