In this paper, we propose a new swing-up strategy for cart inverted pendulums with restricted rail length. The proposed swing-up strategy is derived from a new Lyapunov function. The Lyapunov function is defined as the sum of the square of the pendulum energy and the weighted square of the cart's velocity. The resulting swing-up strategy is represented in a compact form and has two design parameters. By adjusting these design parameters, we can affect the swing-up strategy such that the restriction on the rail length is satisfied. We also provide a state-dependent transformation to obtain voltage input to a DC motor required to generate the cart's acceleration obtained from the proposed swing-up strategy. Finally, we illustrate the performance of the proposed swing-up law through simulation and experiments. It is shown that there is quite good correspondence between theory and experiments.

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