Energy-based controller for parallel-type double inverted pendulums with restricted cart rail length

In this paper, the Newton-Euler method is used to establish the mathematical model of a parallel-type double inverted pendulums system(PDIPS). A new energy-based swinging up strategy is proposed for the PDIPS with restricted cart rail length. The controller is developed by a Lyapunov function in the form of two pendulums energy and the carts velocity. The way to swing up two pendulums is to drive the total energy of the two pendulums to its potential energy at the vertical. Since the pendulum is cart-type, this paper gives an analysis of traveling distance of the cart. The proposed controller can also make the speed of the cart converge to zero such that the restriction on the rail length is satisfied. The corresponding stability analysis is guaranteed by Lyapunov techniques and LaSalles invariance theorem. The designed controller has a simple structure, good convergence performance and easy tuning parameters. We also illustrate the good performance of the proposed swing-up law through simulation and experiment.