Symmetry and reduction for coordinated rigid bodies

Motivated by interest in the collective behavior of autonomous agents, we lay foundations for a study of networks of rigid bodies and, specifically, the problem of aligning orientation and controlling relative position across the group. Our main result is the reduction of the (networked) system for the case of two individuals coupled by control inputs that depend only on relative configuration. We use reduction theory based on semi-direct products, yielding Poisson spaces that enable efficient formulation of control laws. We apply these reduction results to satellite and underwater vehicle dynamics, proving stability of coordinated behaviors such as two underwater vehicles moving at constant speed with their orientations stably aligned.