Revisiting a Double Inverted Pendulum

Abstract

Abstract Previously a double inverted pendulum was built with the cg of the lower pendulum located at the pinned joint connecting the lower and upper pendulum. A full-order estimator and state feedback controller were designed for this system and, with some difficulty, the pendulum was stabilized but operated in a limit cycle. The controllability of this system is analyzed by transforming a linearized state-space model of the pendulum to the modal domain and evaluating the modal control vector. It is shown that changing the lower pendulum cg location from the pinned joint to the middle of the lower pendulum significantly increases controllability and makes it easier to control the upper pendulum. These results invite the question: Would it have been easier to balance the pendulum if the cg was located in the middle of the lower pendulum? To answer this question, the pendulum system has been analyzed to determine the effects of this change in cg location on system controllability, observability, and design of the state estimator and state feedback controller. A state estimator and state feedback controller are designed for the new cg location and compared with those of the original system and it is found that to achieve the same closed-loop bandwidth, the state feedback gains for the model with the new cg location are reduced by the same factor that the controllability is increased, as expected. The steady-state operation of the two models is compared in the presence of angle measurement biases and Coulomb friction that caused a limit cycle in the original system. The results indicate that the sensitivities to measurement biases are very similar for both models but the amplitude of the steady-state limit cycle due to Coulomb friction is twice as large for the model with the new cg location. It is concluded that changing the cg of the lower pendulum from the pinned joint to the middle of the pendulum to increase controllability would not have made it easier to stabilize the pendulum. In analyzing these models it became apparent that biases in the angle measurements result in significantly larger the steady-state displacements of the base when the estimator is used than when direct state feedback is used. This sensitivity contributed to the difficulty of stabilizing the original pendulum.