Stabilization of Homoclinic Orbits of Two Degree-of-Freedom Underactuated Systems

A hybrid controller for stabilization of homoclinic orbits of two degree-of-freedom (DOF) underactuated systems is proposed. The controller is comprised of continuous-time inputs, impulsive brakings, and virtual impulsive inputs for resetting of the passive coordinate. Impulsive brakings of the active coordinate result in instantaneous negative changes in the mechanical energy of the system. An impulsive dynamical system framework is adopted for modeling the hybrid dynamics and a Lyapunov function is defined for stabilization of the orbit. Sufficient conditions for stabilization are presented such that the Lyapunov function decreases monotonically under the action of the continuous inputs and undergoes negative jumps due to impulsive brakings. The control design is implemented on an inverted pendulum on a cart example. Simulation results indicate fast convergence of system trajectories to the homoclinic orbit corresponding to the upright equilibrium configuration.