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The problem of orbital stabilization of underactuated mechanical systems withone passive degree-of-freedom (DOF) is revisited. Virtual holonomic constraintsare enforced using a continuous controller; this results in a dense set ofclosed orbits on a constraint manifold. A desired orbit is selected on themanifold and a Poincare section is constructed at a fixed point on the orbit.The corresponding Poincare map is linearized about the fixed point; thisresults in a discrete linear time-invariant system. To stabilize the desiredorbit, impulsive inputs are applied when the system trajectory crosses thePoincare section; these inputs can be designed using standard techniques suchas LQR. The Impulse Controlled Poincare Map (ICPM) based control design haslower complexity and computational cost than control designs proposed earlier.The generality of the ICPM approach is demonstrated using the 2-DOFcart-pendulum and the 3-DOF tiptoebot.

Orbital Stabilization of Underactuated Systems using Virtual Holonomic Constraints and Impulse Controlled Poincare Maps