Swing-up and positioning control of an inverted wheeled cart pendulum system with chaotic balancing motions

This paper explores the problem of swinging-up an inverted pendulum formed by a rod attached to a wheeled cart with a hanging bob at its opposite end. The system is driven by the wheeled cart platform system, which is formed by a cart, wheels with counterbalance and connecting-rods. The model of the system is initially obtained under the assumption of rolling without slipping of the wheels, which is then verified by computing the reaction forces. The motion of the wheeled cart is initially oscillating, whereas the rod can move freely giving rise to an under-actuated mechanical system. From the harmonic prescribed motion for the wheeled cart, necessary conditions for chaotic rod motion are deduced by means of the Melnikov function. Once the chaotic oscillation has been reached and the rod is close to the upright position, the force over the wheeled cart is commutated to a control law based on the pole-placement plus integrator technique. This procedure allows driving the rod and the wheeled cart system to the upright position and to a prescribed set point respectively. The onset of strange attractors is crucial in the design of the control law, whose performance to obtain rolling without slipping is researched by means of sensitive dependence, power spectral density, Lyapunov exponents and reaction forces. The results of the analytical calculations are verified by full numerical simulations.