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.

翻译：我们识别了重力作用下双摆从稳定平衡衍生的非线性正常模态，并建立了它们随能量增加而通过不稳定直立位置与同宿轨道的关联。利用这一结果，我们设计了一种针对具有弱饱和执行器双摆的高效起摆策略。该方法在逐步注入能量的同时，将系统稳定在与非线性模态相关的周期轨道上。由于这些模态是自治系统的演化，稳定所需的控制力极小。即使执行器限制小于最大重力扭矩的1%，所提方法通过给予足够时间也能实现双摆的起摆。