Convergence of rotating consensus algorithm for second-order dynamics in three dimensions

Abstract

This paper investigates the collective rotating motions of a group of agents with second-order dynamics in three dimensions. A rotating consensus algorithm for such agents was proposed by introducing a Cartesian Coordinate Coupling and a velocity consensus item to an existing consensus protocol. This note shows that the resulting rotating motions can be affected by network topology, the damping gain and the Euler angle. It is also shown that convergence of the rotating consensus can be achieved when the damping gain is above a certain bound, the Euler angle is below a critical value and the communication network topology has a directed spanning tree. Furthermore, we show that the agents will eventually keep moving on a straight-line path, cylindrical spirals, and logarithmic columnar curves when the damping gain is above a certain bound, the Euler angle is below, equal and above a critical value, respectively. In particular, when the agents eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the agents on their orbits) are equal to the relative magnitudes (respectively, the relative phases) of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric Laplacian matrix. Simulation results are presented to demonstrate the proposed algorithm.