Energy-Based Orbital Stabilization of Underactuated Systems Using Impulse Controlled Poincaré Maps

Abstract

The problem of energy-based orbital stabilization of underactuated mechanical systems with one passive degree-of-freedom is addressed. The orbit is a manifold where the active generalized coordinates are fixed and the total mechanical energy is equal to some desired value. A hybrid control strategy comprised of continuous and intermittent impulsive inputs is presented. The continuous controller is designed using partial feedback inearization to converge the active generalized coordinates to their desired values. The choice of desired energy characterizes a unique orbit which is stable but not asymptotically stable. To stabilize the desired orbit, a Poincaré section is constructed at a fixed point and the Poincaré map is linearized about the fixed point. This results in a discrete LTI system. To stabilize the desired orbit, impulsive inputs are applied when the system trajectory crosses the Poincaré section. The applicability of the control design is demonstrated by stabilization of the homoclinic orbit of the cart-pendulum system.