Collective motion from consensus with Cartesian coordinate coupling - Part II: Double-integrator dynamics

This is the second part of a two-part paper on collective motion from consensus with Cartesian coordinate coupling. In this part, 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. In particular, we show a necessary and sufficient condition on the damping gain for rendezvous when there is no Cartesian coordinate coupling. We also explicitly show the critical value for the Euler angle when there is Cartesian coordinate coupling and quantitatively characterize the resulting collective motions, namely, rendezvous, circular patterns, and logarithmic spiral patterns. Simulation results are presented to demonstrate the theoretical results.