Double-graph formation control for co-leader vehicle networks

The paper studies the directed formation control problem for co-leader vehicle networks. Double graphs are considered: one for sensing graph and the other for communication graph. Complex-valued Laplacians for graphs whose edges are attributed with complex weights are introduced for the purpose of formation control while real-valued Laplacians are involved for synchronizing the velocities. For both single-integrator kinematics and double-integrator dynamics, necessary and sufficient conditions are obtained for complex-valued Laplacian such that the resulting moving formation will not be deformed. Moreover, since the control law with certain choice of complex weights could not always ensure asymptotic stability, we show the existence of a stabilizing matrix that updates the complex weights, but preserves the equilibrium formation and makes the system asymptotically stable. Simulations are presented to illustrate our results.