Swing-Up of a Double, Triple and Quadruple Pendulum via Nonlinear Normal Modes

Abstract

Identifying the nonlinear normal modes (NNMs) of a double, triple and quadruple pendulum, we found that all NNMs approach homoclinic orbits through the unstable upright equilibrium. The NNMs are computed by numerical continuation of conser- vative orbits initialized at small-amplitude linear modes around the downward stable equilibrium. Based on this insight we develop a swing-up control scheme bringing the pendula from the downward equilibrium to the upright position. As NNMs collect natural trajectories, sustaining a modal oscillation does not need control. The available control force is used to walk up the NNM, gradually injecting energy, until the homoclinic orbit is approached, and therefore the upright position is reached.