Swing-Up of a Weakly Actuated Double Pendulum via Nonlinear Normal Modes
Arne Sachtler (Deutsches Zentrum für Luft- und Raumfahrt (DLR), Technische Universität München)
Davide Calzolari (Technische Universität München, Deutsches Zentrum für Luft- und Raumfahrt (DLR))
Maximilian Raff (University of Stuttgart)
Annika Schmidt (Deutsches Zentrum für Luft- und Raumfahrt (DLR), Technische Universität München)
Yannik P. Wotte (University of Twente)
Cosimo Della Santina (Deutsches Zentrum für Luft- und Raumfahrt (DLR), TU Delft - Mechanical Engineering)
C. David Remy (University of Stuttgart)
Alin Albu-Schäffer (Technische Universität München, Deutsches Zentrum für Luft- und Raumfahrt (DLR))
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Abstract
We identify the nonlinear normal modes spawning from the stable equilibrium of a double pendulum under gravity, and we establish their connection to homoclinic orbits through the unstable upright position as energy increases. This result is exploited to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators. Our approach involves stabilizing the system onto periodic orbits associated with the nonlinear modes while gradually injecting energy. Since these modes are autonomous system evolutions, the required control effort for stabilization is minimal. Even with actuator limitations of less than 1% of the maximum gravitational torque, the proposed method accomplishes the swing-up of the double pendulum by allowing sufficient time.