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.

Adding elastic elements to the mechanical structure should enable robots to perform efficient oscillatory tasks. Still, even characterizing natural oscillations in nonlinear systems is a challenge in itself, which nonlinear modal theory promises to solve. Therein eigenmanifolds generalize eigenspaces to mechanical systems with non-Euclidean metrics and thus characterize families of oscillations that are autonomous evolutions of the robot. Eigenmanifolds likewise provide a framework for deriving feedback controllers to excite and sustain these oscillations. Nevertheless, these results have been so far essentially theoretical. They have been applied on relatively low dimensional systems and almost exclusively in simulation. We aim to bridge the theory to the real-world gap with the present work and show that we can excite nonlinear modes in complex systems. To this end, we propose control strategies that can simultaneously stabilize numerically evaluated eigenmanifolds and sustain oscillations in the presence of dissipation. We then focus on the KUKA iiwa with simulated parallel springs as an example of the highly nonlinear and articulated system. We calculate all the nonlinear modes of the system, and we use the proposed strategies to excite the associated natural oscillations.