Stable synchronization of mechanical system networks

In this paper we address stabilization of a network of underactuated mechanical systems with unstable dynamics. The coordinating control law stabilizes the unstable dynamics with a term derived from the method of controlled Lagrangians and synchronizes the dynamics across the network with potential shaping designed to couple the mechanical systems. The coupled system is Lagrangian with symmetry, and energy methods are used to prove stability and coordinated behavior. Two cases of asymptotic stabilization are discussed; one yields convergence to synchronized motion staying on a constant momentum surface, and the other yields convergence to a relative equilibrium. We illustrate the results in the case of synchronization of n carts, each balancing an inverted pendulum.