Modification via averaging of partial-energy-shaping control for orbital stabilization: cart-pendulum example

We consider the problem of creating oscillations with respect to a part of state variables in underactuated mechanical systems. The main contribution is a modification of a recently proposed control strategy, exploiting passivity to shape the energy of the system with respect to a subset of the state variables and neglecting the dynamics of the other ones. We propose a way to design an additional control action, which guarantees boundedness of the motion of these other degrees of freedom, which otherwise almost always evolve without any bound. The idea is presented on the well-known two-degrees-of-freedom benchmark example of inverted pendulum on a cart