Controllability and motion algorithms for underactuated Lagrangian systemson Lie groups

In this paper, we provide controllability tests and motion control algorith ms for underactuated mechanical control systems on Lie groups with Lagrangi an equal to kinetic energy. Examples include satellite and underwater vehic le control systems with the number of control inputs less than the dimensio n of the configuration space. Local controllability properties of these sys tems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturba tion theory is applied to compute approximate solutions for the system unde r small-amplitude forcing; in-phase signals play a crucial role in achievin g motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static i nterpolation and exponential stabilization. We illustrate the theoretical r esults and the algorithms with applications to models of planar rigid bodie s, satellites and underwater vehicles.