Energy Level Stabilization of Pendulum on a Cart with Restricted Cart Track Based on Elliptic Functions and Integrals

This study presents an energy level stabilization algorithm for the pendulum on a cart system with restricted cart track length. The objective is to bring the pendulum to its unstable equilibrium. To do so the energy level of the pendulum is increased or decreased by accelerating the cart in the appropriate direction while keeping within imposed limit positions. To achieve, the equation of motion of the pendulum is solved by means of elliptic integrals considering the cart is moved with constant acceleration, and all the calculations are performed numerically to obtain the bounds for elapsed time and change of the energy level of the pendulum. The algorithm is tested on the system by means of numerical simulations.