Decoupling Torque and Stiffness: A Unified Modeling and Control Framework for Antagonistic Artificial Muscles

TL;DR

This work tackles the challenge of decoupling torque and stiffness in antagonistic soft actuators to enable safe, adaptive interaction with unstructured environments. It introduces a unified framework built on a separable Padé force law, a minimal two-state dynamic wrapper, and a cascaded control architecture with analytical inverse dynamics, enabling independent torque and stiffness commands in real time. Key contributions include actuator-agnostic modeling for PAMs, HASELs, and DEAs, a co-contraction/bias-based decoupling strategy, and simulation-backed validation showing depth-based impedance improves contact performance on both soft and rigid surfaces, with substantial gains in settling time, stability, and interaction force reduction. The framework lays a computational foundation for soft robotic systems to emulate biological impedance strategies, enhancing safety and robustness in human–robot collaboration and unstructured manipulation, while highlighting the need for hardware validation across multiple actuators and degrees of freedom.

Abstract

Antagonistic soft actuators built from artificial muscles (PAMs, HASELs, DEAs) promise plant-level torque-stiffness decoupling, yet existing controllers for soft muscles struggle to maintain independent control through dynamic contact transients. We present a unified framework enabling independent torque and stiffness commands in real-time for diverse soft actuator types. Our unified force law captures diverse soft muscle physics in a single model with sub-ms computation, while our cascaded controller with analytical inverse dynamics maintains decoupling despite model errors and disturbances. Using co-contraction/bias coordinates, the controller independently modulates torque via bias and stiffness via co-contraction-replicating biological impedance strategies. Simulation-based validation through contact experiments demonstrates maintained independence: 200x faster settling on soft surfaces, 81% force reduction on rigid surfaces, and stable interaction vs 22-54% stability for fixed policies. This framework provides a foundation for enabling musculoskeletal antagonistic systems to execute adaptive impedance control for safe human-robot interaction.

Figures (6)

• Figure 1: (A) Biological Inspiration: Antagonistic muscle pairs (e.g., quads-hamstring) achieve independent torque-stiffness control through co-contraction and bias. We replicate this principle in musculoskeletal robotic systems using artificial muscles. (B) Decoupling Problem: DC motor joints suffer from entangled torque-stiffness control (stiffness is merely a software gain, left), while antagonistic artificial muscles offer plant-level decoupling through dual independent activation channels ($\alpha_1$, $\alpha_2$, right). Using co-contraction/bias coordinates ($c = (\alpha_1+\alpha_2)/2$, $b = (\alpha_1-\alpha_2)/2$), we achieve independent control: co-contraction $c$ modulates stiffness $K$, while bias $b$ generates torque $T$. (C) Unified Framework: Our separable Padé [2/1] force law, experimentally identified from HASEL data, captures PAMs, HASELs, and DEAs in a single model that enables real-time control ($<$1ms computation), maintains decoupling through transients, and provides independent torque and stiffness commands. (D) Simulation Performance: Adaptive impedance control modulates stiffness according to penetration depth and achieves substantial improvements over fixed-stiffness policies in simulation: 200× faster settling on soft surfaces ($<$10ms vs 2.02s) and 81% force reduction on rigid surfaces, demonstrating potential for biological-level robustness across unknown environments.
• Figure 2: (A) Single Muscle Model: Command $u$ flows through drive dynamics $\dot{a}=(\phi(u)-a)/\tau_a$ with feedback (dashed), then activation mapping $\alpha=g(a)$ to produce active force $\alpha F_{\text{base}}(x)$. The viscoelastic series branch $[k_s, \eta]$ couples external strain $\varepsilon$ to internal deformation $x$ via $\eta\dot{x}=\eta\dot{\varepsilon}+k_s(\varepsilon-x)-\alpha F_{\text{base}}(x)$, with internal feedback (dashed). Output force is $F=k_s(\varepsilon-x)+\eta(\dot{\varepsilon}-\dot{x})$. Two-state dynamics $(a,x)$ capture bandwidth and viscoelasticity; separable Padé law $F_{\text{base}}(x)$ is universal across PAMs, HASELs, DEAs. (B) Cascaded Control Architecture: Optional outer impedance loop generates $T_{\text{des}}=K_{\text{imp}}(e_\theta)+D_{\text{imp}}(e_{\dot\theta})+\hat{\tau}_g$ from position errors for trajectory tracking tasks. Inner activation solver combines feedforward (FF) Newton inverse $\mathbf{y}(\boldsymbol{\alpha})=\mathbf{y}_{\text{des}}$ with optional feedback (FB) PI controllers $\Delta b=\text{PI}(e_T)$, $\Delta c=\text{PI}(e_K)$ operating in co-contraction/bias $(c,b)$ coordinates. Coordinate transform $\boldsymbol{\alpha}=c\pm b$ combines FF and FB. Plant executes muscle model ($\times 2$) and joint dynamics $J\ddot\theta=r(F_1-F_2)+\tau_{\text{ext}}$. Dashed lines show feedback paths. Outer loop can be bypassed to directly specify $(T_{\text{des}},K_{\text{des}})$, demonstrating plant-level decoupling independent of any policy-level coupling from impedance control.
• Figure 3: Experimental validation on held-out test trajectory. Model prediction (red, dashed) tracks measured force (blue) with sub-Newton error (RMSE = 0.78N, $R^2{=}0.83$). Residuals remain bounded, validating the separable Padé structure (\ref{['eq:pade']}) and viscoelastic wrapper (\ref{['eq:series_branch', 'eq:x_dynamics']}) capture both quasi-static and transient dynamics.
• Figure 4: Torque-stiffness decoupling at $\theta = 0^\circ$. Blue: vary co-contraction $c$ at fixed bias $b$ (stiffness control). Orange: vary bias $b$ at fixed $c$ (torque control). Gray boundary shows feasibility limits. Vertical blue sweeps and horizontal orange sweeps confirm independent control of both quantities.
• Figure 5: Controller comparison under $\pm$1.0Nm disturbances. Open-loop (orange) loses decoupling during disturbances; closed-loop (blue) maintains independent torque-stiffness tracking with 29$\times$ lower torque error (Table \ref{['tab:controller']}). The activation signals show smooth, chatter-free control despite aggressive disturbances. Shading indicates disturbance periods.