Controller design for a class of underactuated nonlinear systems

We design a controller for a class of underactuated nonlinear systems. First, we try to find an appropriate global change of coordinates to transform the dynamics of the system into a desired form which consists of a lower order nonlinear subsystem plus a chain of integrators. Then, we find a control Lyapunov function (CLF) and its associated control law for the lower order subsystem. Finally, using a backstepping procedure we derive the control Lyapunov function and the controller for the whole system. The obtained controller renders the origin semiglobally asymptotically stable. As an special case, we demonstrate this procedure for the Acrobot example which is a two-link planar robot with a single actuator at the elbow