Capped Repulsion Limits the Size of Safely Separated Swarms in Potential-Game Flocking

This article introduces Flockwork, a distributed flocking law for drone swarms, and analyzes its energy and safety properties. Each drone follows the saturated negative gradient of a local cost combining a goal term and a coupling term to nearby neighbors. The pair repulsion is capped at a finite value, and the attraction between them switches off once they exceed a fixed sensing range. We first show that the individual costs form an exact potential game, so a single function decreases along every trajectory. We then prove that the sensing-range cutoff is harmless: for almost every starting condition the solution is unique and spends zero time on the switching surface. Next, we give a condition on the step size under which the discrete-time implementation still decreases the potential, and show that once the step size exceeds this bound, equilibria go unstable through period doubling. Finally, capping the repulsion at a finite value limits the size of a safely separated swarm: once the swarm exceeds a threshold that grows as the square of the cap-to-navigation-speed ratio, every equilibrium has a pair pushed into the saturated zone, where repulsion can no longer maintain the safety gap. This order is sharp: we build safe equilibria of the same order from concentric rings of drones. Numerical experiments on a literature parameter set and a deployed controller confirm the results.