[Paper Review] Modeling, Reduction, and Control of a Helically Actuated Inertial Soft Robotic Arm via the Koopman Operator

David A. Haggerty, Michael J Banks+3 more|arXiv (Cornell University)|Nov 16, 2020

Soft Robotics and Applications27 references17 citations

TL;DR

This paper proposes a data-driven modeling and control framework for a highly inertial, helically actuated soft robotic arm using the Koopman operator via Hankel Dynamic Mode Decomposition (HDMD). By identifying dominant Koopman modes from spatial tracking data with time delays, the method reduces the system to a low-dimensional linear model that enables effective open-loop control via Linear Quadratic Regulator (LQR), achieving 25% RMS error in pose tracking with only 35 modes out of 150.

ABSTRACT

Soft robots promise improved safety and capability over rigid robots when deployed in complex, delicate, and dynamic environments. However, the infinite degrees of freedom and highly nonlinear dynamics of these systems severely complicate their modeling and control. As a step toward addressing this open challenge, we apply the data-driven, Hankel Dynamic Mode Decomposition (HDMD) with time delay observables to the model identification of a highly inertial, helical soft robotic arm with a high number of underactuated degrees of freedom. The resulting model is linear and hence amenable to control via a Linear Quadratic Regulator (LQR). Using our test bed device, a dynamic, lightweight pneumatic fabric arm with an inertial mass at the tip, we show that the combination of HDMD and LQR allows us to command our robot to achieve arbitrary poses using only open loop control. We further show that Koopman spectral analysis gives us a dimensionally reduced basis of modes which decreases computational complexity without sacrificing predictive power.

Motivation & Objective

• To address the challenge of modeling and controlling soft robots with infinite degrees of freedom, high nonlinearity, and inertial dynamics.
• To develop a data-driven, linear model of a soft robotic arm using Koopman operator theory without relying on closed-form dynamic equations.
• To reduce model dimensionality while preserving predictive power through Koopman mode analysis.
• To enable effective control using standard linear control techniques like LQR on a nonlinear soft robot.
• To demonstrate that a small number of Koopman modes can capture the dominant dynamics of a complex soft robotic system.

Proposed method

• Apply Hankel Dynamic Mode Decomposition (HDMD) to time-series data from 15 motion tracking points and their time delays to identify Koopman operator approximations.
• Use Koopman spectral analysis to compute eigenvalues and mode powers, identifying the most dynamically relevant modes.
• Select the 35 modes with highest average mode power to form a reduced basis for model projection.
• Project the full-state system matrices onto the reduced Koopman mode basis to obtain a low-order linear model.
• Design an LQR controller using the reduced-order model for open-loop trajectory tracking.
• Deploy the controller in simulation and on the physical robot to evaluate pose tracking performance.

Experimental results

Research questions

• RQ1 Can Koopman operator-based modeling via HDMD produce a low-dimensional, accurate, and linear approximation of a highly nonlinear, inertial soft robotic arm?
• RQ2 To what extent can Koopman mode analysis reduce model complexity without sacrificing predictive accuracy or controllability?
• RQ3 Can a reduced-order Koopman model enable effective open-loop control of a soft robot using standard linear control techniques like LQR?
• RQ4 How does the number of retained Koopman modes affect control performance and model robustness?
• RQ5 What is the relationship between mode power and trajectory-specific performance in model-based control?

Key findings

• The Koopman-based model reduced from 150 modes to 35 while maintaining high predictive fidelity, significantly lowering computational cost.
• Open-loop control using the LQR controller with the 35-mode reduced model achieved an RMS error of 25% in pose tracking, outperforming the full-state model.
• The full-state model performed worse than trimmed models, likely due to noise amplification in high-dimensional representations.
• Performance varied across trajectories, indicating that mode importance is trajectory-dependent, enabling informed mode selection for specific tasks.
• The 35-mode model successfully guided the robot through dynamic transient poses to final target positions, closely matching desired trajectories.
• The method required only ~10^4 training samples and no pre-optimization of observables, demonstrating data efficiency and practical feasibility.