Collective Motion From Consensus With Cartesian Coordinate Coupling

Collective motions including rendezvous, circular patterns, and logarithmic spiral patterns can be achieved by introducing Cartesian coordinate coupling to existing consensus algorithms. We study the collective motions of a team of vehicles in 3D by introducing a rotation matrix to an existing consensus algorithm for double-integrator dynamics. It is shown that the network topology, the damping gain, and the value of the Euler angle all affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound, and the Euler angle is below, equal, or above a critical value, the vehicles will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane perpendicular to the Euler axis. In particular, when the vehicles eventually move on circular orbits, the relative radii of the orbits (respectively, the relative phases of the vehicles 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 theoretical results.