Predicting the Size of Spring Network Swarm in Quadratic Potential Fields

Formation control of multiple autonomous agents or a swarm of robots have become popular in robotics. Formation control is to maintain specific connections among multiple autonomous robots while performing tasks such as traversing trajectories, exploring environments, or covering spaces. We assume that robots are virtually connected by Delaunay triangulation, which is a general form of acute angle connection. This virtual connection forms a spring network, and we find tight upper bounds of the swarm size when we apply quadratic potential fields to the group of robots for the position and formation control. We present a method of Delaunay triangulation using circle bounds. By assuming hexagonal shape of each cell of swarm connection, we present a method of predicting a tight size of an ellipse that encircles the swarm.