Analysis of the Energy Based Swing-up Control for a Double Pendulum on a Cart

Designing and analyzing controllers for underactuated systems with underactuation degree greater than one is still an open and challenging problem. In this paper, we study an unsolved problem of analyzing energy based swing-up control for a double pendulum on a cart, which has three degrees of freedom and only one control input. We present an original analysis of the convergence of the energy of the cart-double pendulum system. We show that for all initial states of the cart-double pendulum system, if the convergent value of the energy is not equal to the energy at the upright (up-up) equilibrium point, then the cart-double pendulum remains at its up-down, down-up, and down-down equilibrium points. Moreover, we show that these three equilibrium points are unstable. This shows that for almost all initial states of the cart-double pendulum system, as time approaches infinity, the energy of the cart-double pendulum system can be controlled to its energy at the upright equilibrium point. This paper provides insight into the energy based control approach to underactuated systems with underactuation degree greater than one.