Collective motion from consensus with Cartesian coordinate coupling - Part I: Single-integrator kinematics

This is the first part of a two-part paper on 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. In this first part, we study the collective motions of a team of vehicles in 3D by introducing a rotation matrix to an existing consensus algorithm for single-integrator kinematics. It is shown that both the network topology and the value of the Euler angle affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties 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. In particular, when the vehicles eventually move on circular orbits, the relative radius of the orbits (respectively, the relative phase of the vehicles on their orbits) is equal to the relative magnitude (respectively, the relative phase) 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.