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

1. Problem Statement

Antagonistic artificial muscles (e.g., Pneumatic Artificial Muscles, HASELs, DEAs) mimic biological muscles by using co-contraction to tune stiffness and bias to generate torque. While this offers the potential for Variable Impedance Control (VIC), achieving independent, real-time regulation of torque and stiffness is difficult due to:

• Nonlinear Dynamics: Actuator force depends nonlinearly on strain and input.
• Contact Transients: During unexpected contact (impact), viscoelastic effects and rate limits degrade the independence of torque and stiffness control within the first tens of milliseconds.
• Model Complexity: High-fidelity simulators for specific actuators are often too computationally expensive for real-time control, while simple quasi-static models fail during dynamic impact events.

The core challenge is to create a unified framework that allows musculoskeletal robots to track independent torque and stiffness commands robustly, even during rapid contact transitions, without relying on explicit surface classification.

2. Methodology

The authors propose a Unified Real-Time Framework comprising three main components:

A. Unified Low-Order Dynamic Model

Instead of creating unique models for each actuator type, the authors use a separable Padé [2/1] force model augmented with a minimal two-state dynamic wrapper.

• Force Law: $F(\alpha, x) = \alpha F_{base}(x)$, where $\alpha$ is normalized activation and $x$ is internal deformation. The base force $F_{base}(x)$ is a rational function (Padé form) capable of capturing nonlinear force-length behaviors (including saturation-like roll-off in HASELs via the denominator term).
• Dynamic Wrapper:
  • Drive State ($a$): Filters the physical command (voltage/pressure) to account for input bandwidth.
  • Viscoelastic State ($x$): A Kelvin-Voigt branch in series with the active element models internal viscoelasticity and finite bandwidth.
• Applicability: The same mathematical structure applies to PAMs, HASELs, and DEAs by adjusting the activation mapping ($\alpha$) and coefficients.

B. Cascaded Inverse-Dynamics Controller

The control architecture operates in co-contraction/bias coordinates ($c_\alpha, b_\alpha$) to decouple the physical inputs:

• Outer Loop (Optional): Standard impedance control generating desired torque ($T_{des}$) based on position errors.
• Inner Loop (Command Solver): A cascaded inverse-dynamics controller that solves for normalized commands ($u$) to track specific targets: Desired Torque ($T_{des}$) and Desired Mechanical Stiffness ($K_{mech,des}$).
• Decoupling Mechanism:
  • Bias ($b_\alpha$): Primarily controls net torque.
  • Co-contraction ($c_\alpha$): Primarily controls mechanical stiffness.
• Performance: The solver uses a damped Gauss-Newton (Levenberg-Marquardt) iteration with warm-starting, achieving convergence in 1–2 iterations with a computation time of < 1 ms per tick.

C. Depth-Adaptive Interaction Policy

To handle unknown environments, the authors propose a bio-inspired policy that schedules stiffness based on penetration depth ($\delta$) rather than explicit surface labels:

• Policy: $K_{mech,des}(\delta) = K_{low} + (K_{high} - K_{low}) \cdot \min(1, \delta/\delta_{char})$.
• Logic: Low stiffness for small penetrations (soft contacts/impact mitigation) and high stiffness for large penetrations (rigid contacts/stability).

3. Key Contributions

1. Unified Modeling Framework: A single, low-order Padé-based dynamic model with a two-state wrapper that accurately represents diverse artificial muscle families (validated on HASELs) while remaining computationally efficient for real-time control.
2. Independent Torque-Stiffness Control: A cascaded controller that successfully decouples torque and stiffness tracking in real-time (<1 ms), demonstrated by a fixed-torque stiffness-step test where torque regulation is preserved even during rapid stiffness transitions.
3. Depth-Adaptive Policy: A control strategy that automatically balances the shock/load vs. stability tradeoff across soft-to-rigid environments without requiring prior knowledge of surface stiffness.
4. Attribution Ablation Study: A rigorous experimental design (2x2 audit) that isolates the effects of "virtual" impedance scheduling (outer loop) versus "mechanical" plant stiffness modulation (inner loop), proving that plant-level stiffness modulation is critical for transient robustness.

4. Results

The framework was evaluated via simulation on a single-DOF antagonistic joint (using parameters identified from HASEL hardware) under various contact protocols:

• Model Accuracy: The identified model achieved sub-Newton accuracy on hardware data (RMSE = 0.78 N, $R^2$ = 0.83), capturing both quasi-static trends and transient responses.
• Decoupling Performance:
  • In the fixed-torque stiffness-step test, the decoupled controller reduced torque disturbance by 29× compared to an open-loop baseline during stiffness transitions.
  • Closed-loop control maintained independent tracking of torque and stiffness trajectories despite external disturbances.
• Contact Interaction (Depth-Adaptive vs. Fixed):
  • Rigid Surfaces: Fixed-low stiffness reduced shock peaks but caused instability; Fixed-high ensured stability but generated high impact loads. The Depth-Adaptive policy achieved a middle ground, reducing shock compared to Fixed-High while maintaining better stability than Fixed-Low.
  • Soft Surfaces: Fixed-high stiffness caused excessive sinking (instability), while Fixed-low was stable but less responsive. The Adaptive policy interpolated effectively.
  • Ablation: The study confirmed that mechanical stiffness modulation (changing $K_{mech,des}$) is the primary driver for reducing rigid impact loads, while virtual scheduling plays a secondary role.

5. Significance and Future Work

• Significance: This work moves beyond nominal decoupling to address the early-contact regime, where dynamics are most volatile. It provides a practical, control-oriented foundation for musculoskeletal robots to execute adaptive impedance behaviors in unstructured environments. By unifying the modeling of different muscle types, it lowers the barrier to deploying complex antagonistic robots.
• Limitations & Future Work:
  • Current contact results are simulation-based using simplified linear spring-damper environments.
  • Model identification was performed only on HASELs (though the framework is general).
  • Future work includes hardware validation on PAMs and DEAs, extension to multi-DOF coupled systems, and comparison with learning-based policies.

In summary, the paper establishes that independent torque and stiffness control is achievable in real-time for antagonistic artificial muscles using a unified low-order model and a cascaded controller, enabling robots to adaptively manage the trade-off between impact mitigation and stability during dynamic interactions.