Initial Configuration Analysis of Multi-Agent Systems that Converge to Evenly Spacing Points on Unit Circle

Abstract

The purpose of this paper is to clarify initial configuration conditions of multi-agent systems that converge to evenly spacing points on the unit circle embedded in the control space. The agents can converge to points on the submanifold by the gradient method to maximize the distance between the agents that are mutually connected. This paper considers the cycle graph as the communication topology and the unit circle as the submanifold on the plane. Then the paper introduces the first integral of the arguments of the agents and gives a necessary and sufficient condition that the agents converge to evenly spaced points on the unit circle with the order of the cycle graph provided that the agents converge to the unit circle by the gradient method. Finally, numerical examples confirm the final configuration of the agents that satisfy the proposed initial configuration condition.