Nonlinear Spectral Modeling and Control of Soft-Robotic Muscles from Data

TL;DR

This work addresses the control of soft robotic muscles that exhibit nonlinear memory and hysteresis by developing a data driven spectral-submanifold reduction framework. It introduces and validates both adiabatic (aSSM) and slow-manifold (SM) reductions to learn a low dimensional input–output map from forced responses, enabling a fast inverse map for feedforward control that is augmented with PI feedback for robustness. The approach is demonstrated on HASEL actuators in an antagonistic joint, achieving substantial tracking-error reductions compared to baselines and showing that a one dimensional slow manifold suffices under strong time scale separation. The findings highlight a practical pathway to rapid characterization and high performance control of soft muscles without detailed physics based models, with potential applicability to other soft actuators that exhibit clear time scale separation.

Abstract

Artificial muscles are essential for compliant musculoskeletal robotics but complicate control due to nonlinear multiphysics dynamics. Hydraulically amplified electrostatic (HASEL) actuators, a class of soft artificial muscles, offer high performance but exhibit memory effects and hysteresis. Here we present a data-driven reduction and control strategy grounded in spectral submanifold (SSM) theory. In the adiabatic regime, where inputs vary slowly relative to intrinsic transients, trajectories rapidly converge to a low-dimensional slow manifold. We learn an explicit input-to-output map on this manifold from forced-response trajectories alone, avoiding decay experiments that can trigger hysteresis. We deploy the SSM-based model for real-time control of an antagonistic HASEL-clutch joint. This approach yields a substantial reduction in tracking error compared to feedback-only and feedforward-only baselines under identical settings. This record-and-control workflow enables rapid characterization and high-performance control of soft muscles and muscle-driven joints without detailed physics-based modeling.

Figures (8)

• Figure 1: Overview of data-driven slow manifold modeling and control for antagonistic artificial muscles. (a) HASEL actuator working principle: a flexible polymer shell filled with dielectric oil and covered by electrodes; applying voltage (up to $U=8$ kV) generates electrostatic pressure that redistributes the fluid, producing axial contraction $\Delta x$ against a tendon load. (b) Experimental platform: antagonistic musculoskeletal joint actuated by HASEL artificial muscles paired with electrostatic clutches, enabling bidirectional motion and variable stiffness; $\theta$ denotes the joint angle. The inset shows the contraction $\Delta x$ at 0 kV and 8 kV. (c) Data-driven modeling and control: pronounced time-scale separation allows trajectories to rapidly converge onto a low-dimensional slow manifold $\mathcal{L}_\epsilon$, learned directly from forced-response trajectories. The learned polynomial map $x = g(u)$ relates control input $u(t)$ to system state $x(t)$. The inverse slow-manifold model provides feedforward control, combined with proportional-integral (PI) feedback for disturbance rejection.
• Figure 2: (a) Geometry of the attracting invariant manifold $\mathcal{M}_0$ formed by a family of SSMs, $\mathcal{W}(E(u))$, parametrized by the external forcing parameter $u$, for $\epsilon = 0$. For simplicity, $u$ is assumed to be a scalar in this plot. The critical manifold $\mathcal{L}_0$ consists of the collection of the fixed points corresponding to each value of $u$. (b) For $\epsilon \ne 0$, the manifold $\mathcal{M}_0$ perturbs into an aSSM, $\mathcal{M}_\epsilon$, and $\mathcal{L}_0$ perturbs into a slow manifold, $\mathcal{L}_\epsilon$. The dynamics of the external forcing is associated with the hollow arrowhead. (c) Trajectories rapidly converge to $\mathcal{L}_\epsilon$ when the timescales $\tau_1$ and $\tau_2$ of the autonomous dynamics are much faster than the timescale $\tau_3$ of the external excitation.
• Figure 3: (a) Autonomous system ($\text{I}$) and externally forced system ($\text{II}$). (b) Comparison of trajectories decaying to zero from the same initial point $\bar{h}$ through autonomous transient decay and under slower ($\rho < 1$) and faster ($\rho \approx 1$) control in a regulation task.
• Figure 4: (a) Comparison between the aSSM-based and the SM-based reduced-order model of the SDOF of Eq. \ref{['SDOF_oscillator']} under random external excitations with increasing speed ($\rho = 0.1$ (top), $\rho = 0.5$ (middle) and $\rho = 1.2$ (bottom)). As predicted by theory, the slow manifold approximation is accurate when $\rho \ll 1$, while in the last case even the leading-order aSSM-based approximation breaks down. (b) Legend and table reporting the NMTE values for the different cases.
• Figure 5: (a) Geometry of a single HASEL pouch actuator in the inactive state (left) and actuated state (right). Adapted from Kellaris_2019. (b) Autonomous response of the analytic HASEL actuator model \ref{['governing_eq_analytic_HASEL']} under different constant voltage levels. (c) Predictions of the data-driven aSSM-based and SM-based models for the actuator stroke in system \ref{['governing_eq_analytic_HASEL']} under random excitation.