Physics-Embedded Neural ODEs for Learning Antagonistic Pneumatic Artificial Muscle Dynamics

TL;DR

A hybrid neural ordinary differential equation (Neural ODE) framework that embeds physical structure into a learned model of antagonistic PAM dynamics, and combines parametric joint mechanics and pneumatic state dynamics with a neural network force component that captures antagonistic coupling and rate-dependent hysteresis.

Abstract

Pneumatic artificial muscles (PAMs) enable compliant actuation for soft wearable, assistive, and interactive robots. When arranged antagonistically, PAMs can provide variable impedance through co-contraction but exhibit coupled, nonlinear, and hysteretic dynamics that challenge modeling and control. This paper presents a hybrid neural ordinary differential equation (Neural ODE) framework that embeds physical structure into a learned model of antagonistic PAM dynamics. The formulation combines parametric joint mechanics and pneumatic state dynamics with a neural network force component that captures antagonistic coupling and rate-dependent hysteresis. The forward model predicts joint motion and chamber pressures with a mean R$^2$ of 0.88 across 225 co-contraction conditions. An inverse formulation, derived from the learned dynamics, computes pressure commands offline for desired motion and stiffness profiles, tracked in closed loop during execution. Experimental validation demonstrates reliable stiffness control across 126-176 N/mm and consistent impedance behavior across operating velocities, in contrast to a static model, which shows degraded stiffness consistency at higher velocities.

Figures (9)

• Figure 1: System schematic and hybrid Neural ODE modeling components. a Antagonistic PAM joint with flexor (f) and extensor (e) muscles coupled through a pulley. b Equivalent translational model with chamber state variables and geometric parameters. c Embedded physics submodels: isothermal ideal gas law for pressure dynamics and Newton's second law for joint mechanics. d Neural ODE architecture with a learned vector field $f_\theta(\mathbf{x},\mathbf{u})$, integrated forward in time to predict state trajectories.
• Figure 2: Offline-to-online validation workflow. Desired motion $x_d(t)$ and stiffness $K_d(t)$ profiles are processed through the inverse formulation to produce air mass trajectories, converted to pressure commands via the ideal gas law, and stored in a lookup table. During online execution, the pressure controller tracks the precomputed commands using discrete valve pulsing to drive the antagonistic PAM joint.
• Figure 3: Experimental antagonistic PAM platform. a Physical setup showing the antagonistic PAM pair, pulley joint, and DC motor. b System schematic: each PAM chamber is independently regulated through a pair of three-way solenoid valves supplied from a common air tank, with chamber pressures measured by inline pressure sensors. An embedded controller coordinates valve actuation, motor commands, and sensor acquisition.
• Figure 4: Coefficient of determination ($R^2$) of the hybrid Neural ODE model over 225 operating conditions. Each grid intersection represents one dataset. The model achieves consistently high accuracy across the full co-contraction range, with an overall mean $R^2 = 0.88$. a$R^2$ in the desired pressure space $(P_{f,d}, P_{e,d})$. b$R^2$ in the corresponding air-mass space $(m_f, m_e)$. Circles denote the operating condition used for training; other conditions were used to evaluate the forward model's accuracy. The star marks the condition shown in Fig. \ref{['fig:response_example']}.
• Figure 5: Example comparison between the hybrid Neural ODE prediction and experimental measurements for one training dataset at $(P_{f,d}, P_{e,d}) = (206.8, 413.7)~\mathrm{kPa}$ (30, 60 $\mathrm{psi}$). The model accurately reproduces the measured displacement $x$, velocity $\dot{x}$, and PAM pressures $P_f$ and $P_e$. Right panels show measured (black dashed) and model-predicted (colored solid) hysteresis loops for force versus displacement ($x$-$F$), flexor pressure ($P_f$-$F$), and extensor pressure ($P_e$-$F$) at 0.5, 1, and 2 Hz.