Passivity-based 3D attitude coordination: Convergence and connectivity

In this paper we consider the problem of attitude coordination on balanced graphs, where the kinematics of each rigid-body is modeled on SE(3). We show that the kinematics on SE(3) satisfies a passivity property with a positive definite storage function and we propose an angular velocity control law for each rigid-body that decreases the sum of the storage functions of the individual rigid-bodies. Attitude coordination results if all rigid-bodies´ rotation matrices are positive definite in the initial conditions and the communication graph is connected. We show that the speed of convergence in the attitude coordination problem is determined by the second smallest eigenvalue of graph Laplacian. Our results are also extended to the case when there are delays in communication among rigid-bodies, a leader and communication failures. Using the concept of brief instability we show that, for a given failure rate and bound, attitude coordination is still achieved. Finally, the results are validated in simulations and experiments.