Swing-Up of a Weakly Actuated Double Pendulum via Nonlinear Normal Modes

TL;DR

The paper addresses the problem of swing-up for a gravity-driven double pendulum under very weak actuation by exploiting nonlinear normal modes (NNMs) and their eigenmanifolds to guide energy injection with minimal torque. It identifies two NNMs that connect the stable downward equilibrium to a neighborhood of the upright equilibrium via energy-increasing brake orbits and parametrize their eigenmanifolds by energy $E$ and phase $phi$, enabling a control law that stabilizes onto the manifold while incrementally increasing energy. The proposed controller combines an eigenmanifold stabilizer with an energy-injection term, computed under torque bounds through a simple optimization, and is implemented in a four-state machine (Bootstrap, Start, SwingUp, Hold). Simulation results in MuJoCo demonstrate swing-up at actuator limits as low as a few percent of the maximum gravitational torque, given sufficient time, validating the nonlinear-modal approach for energy-efficient maneuvers and outlining future hardware validation and generalization to other link configurations. The work highlights the practical impact of NNMs and homoclinic connections for designing efficient, robust control strategies in underactuated or weakly actuated robotic systems.

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.

Figures (8)

• Figure 1: Two pairs of nonlinear normal mode generators of the double pendulum shown on the torus. Both generators (blue, red) start at the downward stable equilibrium (white dot) and meet at the upright equilibrium (purple dot) approaching homoclinic orbits. A version of this illustration ironed onto the $q_1 q_2$-plane is shown in Fig. \ref{['fig:gens']}.
• Figure 2: Double pendulum under gravity. The links are assumed thin rods.
• Figure 3: Sketch of one periodic orbit of an NNM for the energy level $E$ in configuration space. The system periodically oscillates between two turning points $\boldsymbol{q}_{\curvearrowleft{}i}$ where it synchronously comes to rest before reversing direction.
• Figure 4: NNMs of the double pendulum. (a) Generators. The bold and bold-dashed lines show the two pairs of generators and the thin gray lines show some trajectories of the NNMs projected into configuration space. The white/purple dots match the ones in Fig. \ref{['fig:torus_and_pendula']}; (b) period times for different energy levels; (c) exemplary angle evolution and angular velocity evolution for both NNMs at $E = 11.0J$. The symbols $\blacksquare$ and $\bigstar$ mark the points on the generators. (d) modal oscillations for a selection of energy levels on both modes shown in Cartesian representation.
• Figure 5: Projection of modal trajectories onto sections of the state space. The right lower triangular matrix of plots shows mode 1 and the upper left mode 2. The color indicates the corresponding energy level. A combination of energy $E$ and the angle $\varphi = \mathop{\mathrm{atan2}}\nolimits(\dot{q}_1, q_1)$ in the $q_1\dot{q}_1$-plane (greenisch boxes) can be used to parametrize the eigenmanifolds.