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

Core Concepts

Exploiting the nonlinear normal modes of a double pendulum to devise an efficient swing-up strategy for a system with weak, saturating actuators.

Abstract

The authors identify the nonlinear normal modes (NNMs) of a double pendulum under gravity and establish their connection to homoclinic orbits through the unstable upright position as energy increases. This insight is then leveraged to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators.
The key steps are:

Compute the NNMs of the double pendulum system, which form continuous families of periodic brake-orbits connecting the stable downward equilibrium to configurations arbitrarily close to the unstable upright position.
Parameterize the eigenmanifolds of the NNMs using the total energy and a modal phase angle.
Design a controller that stabilizes the system onto the NNM eigenmanifolds while gradually injecting energy. Since the NNMs are autonomous system evolutions, the required control effort for stabilization is minimal.
Validate the approach in simulation, showing that the double pendulum can be swung up even with actuator limitations of less than 1% of the maximum gravitational torque, by allowing sufficient time.

The authors highlight that the NNMs approach homoclinic orbits asymptotically, allowing the system to be driven arbitrarily close to the upright unstable equilibrium in a controlled and energy-efficient manner.

Stats

The maximum gravitational torque occurs when the pendulum is stretched out horizontally at ±6.49 N·m.
The motor torque limits are set to 0.5 N·m (7.7% of the maximum gravitational torque), 0.3 N·m (4.6%), 0.2 N·m (3.0%), 0.1 N·m (1.5%), 0.05 N·m (0.8%), and 0.02 N·m (0.3%).
The critical angles at which the regulation controller can take over are 175.58°, 177.35°, 178.23°, 179.11°, 179.55°, and 179.82°, respectively.

Quotes

"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."
"Both NNMs approach homoclinic orbits passing through the unstable upright position: the two NNMs connect the downward equilibrium to the unstable one upright."

Key Insights Distilled From

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

by Arne... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08478.pdf

Deeper Inquiries

How would the swing-up strategy need to be adapted if the double pendulum had asymmetric mass distribution or link lengths

If the double pendulum had an asymmetric mass distribution or varying link lengths, the swing-up strategy would need to be adapted to account for these changes in the system dynamics. The nonlinear normal modes (NNMs) identified in the study are based on the specific parameters of the double pendulum model provided in the context.
For asymmetric mass distribution, the NNMs may not approach homoclinic orbits through the upright position as smoothly as in the symmetric case. The generators of the NNMs could be affected, leading to different trajectories and stability properties. The swing-up controller would need to be recalibrated to accommodate these variations in the system's behavior.
Similarly, if the link lengths were different, the modal analysis and energy injection strategy would have to be adjusted to reflect the new system dynamics. The eigenmanifold parametrization and the energy controller would need to be redefined based on the updated system parameters to ensure effective swing-up control.

What are the potential limitations of the NNM-based approach in the presence of dissipative effects, such as friction, that are not captured in the conservative system model

The NNM-based approach presented in the study may face limitations in the presence of dissipative effects, such as friction, that are not considered in the conservative system model. These limitations could impact the effectiveness and robustness of the swing-up control strategy in real-world applications.

Sensitivity to External Disturbances: Dissipative effects like friction can introduce external disturbances that are not accounted for in the idealized model. These disturbances can perturb the system dynamics and affect the accuracy of the swing-up control.

Energy Dissipation: Friction and other dissipative effects can lead to energy dissipation in the system, causing deviations from the ideal energy-injection profile designed by the controller. This could result in slower swing-up times or incomplete swing-up maneuvers.

Stability Concerns: The presence of dissipative effects can alter the stability properties of the system, potentially leading to instability or undesired behavior during the swing-up process. The characteristic multipliers of the orbits may change, affecting the overall control performance.

Model-Reality Discrepancy: The discrepancy between the idealized model and the real-world system with dissipative effects can lead to inaccuracies in the control strategy. Calibration and adaptation of the controller to account for these effects would be necessary for practical implementation.

Could the insights from this work on exploiting nonlinear modal dynamics be extended to other underactuated mechanical systems beyond the double pendulum

The insights from exploiting nonlinear modal dynamics, particularly the concept of nonlinear normal modes (NNMs), can be extended to other underactuated mechanical systems beyond the double pendulum. By leveraging the inherent dynamics and energy-efficient properties of NNMs, similar control strategies can be developed for a variety of systems with limited actuation.

Inverted Pendulum Systems: Systems like inverted pendulums, cart-pole systems, or acrobot mechanisms can benefit from NNM-based control strategies for swing-up and stabilization. The identification of NNMs and energy-efficient control methods can enhance the performance of these underactuated systems.

Walking Robots: Underactuated walking robots, such as bipedal or quadrupedal robots, could utilize NNMs for gait generation, energy-efficient locomotion, and stability control. By exploiting the natural dynamics encoded in NNMs, these robots can achieve robust and agile locomotion.

Aerial Vehicles: Unmanned aerial vehicles (UAVs) or drones with underactuated dynamics could also benefit from NNM-based control approaches for trajectory tracking, energy optimization, and maneuvering in complex environments. The inherent stability properties of NNMs can enhance the flight performance of such systems.

Marine Systems: Underwater vehicles or surface vessels with underactuated configurations could leverage NNMs for energy-efficient propulsion, path following, and station-keeping tasks. The application of NNM-based control strategies can improve the maneuverability and efficiency of marine systems.