Control of Underactuated Mechanical Systems with two Degrees of Freedom and Symmetry

In this paper, we consider a special class of underactuated mechanical systems with two degrees of freedom and symmetry. By symmetry, we mean the inertia matrix of the system is independent of the unactuated degree of freedom. We show that there exists a natural global change of coordinates obtained from the Lagrangian of the system that transforms the system into a partially linear cascade nonlinear system that is strict feedback. The nonlinear part of this system is non-affine in control and this highly complicates control design for the system. We provide conditions under which this nonlinear subsystem can be globally stabilized and give globally stabilizing control laws for it. The strict feedback structure of the system in new coordinates allows us to obtain a globally stabilizing control law for the composite system using standard backstepping. We apply our result to global asymptotic stabilization of the Acrobot.