cs.ROFeb 27, 2026
SavePhysics-Embedded Neural ODEs for Learning Antagonistic Pneumatic Artificial Muscle Dynamics
Organizations: Dept. of Mechanical Engineering, University of California, Riverside, CA, USA
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. \rev{The forward model was trained on 29 selected co-contraction conditions and predicted joint motion and chamber pressures over 196 held-out conditions with a mean R of 0.88.} 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 & tables
Fig. 1: System schematic and modeling ingredients. a Antagonistic PAM joint with flexor (f) and extensor (e) chamber pressures, volumes, and air masses, pulley inertia , joint angle , and external torque . b Equivalent translational model: reflected mass , tendon coordinate , external force , and nominal muscle geometry . c Embedded physics: ideal-gas law and Newtonian mechanics. d Neural ODE concept: a neural network (NN) maps the state and an exogenous input to the state derivative, which is integrated to predict the state trajectory. In our hybrid model, the network represents only the antagonistic force term within the physics-based dynamics of a – c .
Fig. 2: Offline-to-online validation workflow. Desired motion and stiffness are converted by inverse synthesis into desired air masses and , then into pressure commands and using the ideal gas law. The commands are stored in a lookup table and tracked online by the pressure controller using discrete valve pulsing.
| Stage | (kg/s) | Training datasets (kPa) |
|---|---|---|
| 1–7 | 170-170 | |
| 8 | 0 | 170-170, 412-412, 653-653 |
| 9 | 0 | previous + 377-446, 446-377 |
| 10 | 0 | previous + 308-515, 515-308 |
| 11 | 0 | previous + 239-584, 584-239 |
| 12 | 0 | previous + 170-653, 653-170 |
TABLE I: Staged training curriculum. Each stage starts from the best checkpoint of the previous stage; is the prescribed auxiliary damping. Dataset conditions are reported as rounded absolute pressures in kPa and were originally defined using gauge-pressure labels in 5-psi increments
Fig. 3: Experimental antagonistic PAM platform. a Physical setup with antagonistic PAM pair, pulley joint, and DC motor for external torque excitation. b Pneumatic and sensing schematic: flexor and extensor PAM chambers are independently regulated by three-way solenoid valves, supplied from a common air tank, and measured by inline pressure sensors.
| Symbol | Description | Value | Unit |
|---|---|---|---|
| Nominal PAM radius | m | ||
| Nominal PAM length | m | ||
| Pulley radius | m | ||
| Motor rotor inertia | kg m 2 | ||
| Gearbox inertia | kg m 2 | ||
| Gear ratio | - |
TABLE II: System parameters used in the model. Measured quantities come from hardware specifications or direct measurement; derived quantities are computed from measured values; marks initialization values for training.
Fig. 4: of the hybrid Neural ODE over 225 co-contraction conditions. Circles mark the 29 training datasets; the remaining datasets are held-out validation cases. The star marks the condition used for the time-series comparison in Fig. 5 . a Desired flexor/extensor pressure space . b Corresponding flexor/extensor air-mass space .
Fig. 5: Example comparison between the hybrid Neural ODE prediction and experimental measurements for one training dataset at kPa absolute (30, 60 psi gauge). The model predicts displacement , velocity , and flexor/extensor pressures and . Right panels show measured and predicted hysteresis loops for - , - , and - at 0.5, 1, and 2 Hz.
Fig. 6: Learned scalar force model versus joint displacement for discrete flexor/extensor air masses and velocities mm/s. a Symmetric mass pairs show stiffness changes with total air mass. b Asymmetric mass pairs show equilibrium shifts from flexor-extensor imbalance. Dotted lines mark zero-force equilibrium positions.
Fig. 7: Displacement-only comparison between the hybrid Neural ODE and baseline models. a Training datasets. b Held-out validation datasets. Bars show mean and error bars show one standard deviation.
| Model type | Prediction | Force | Stiffness | Interpretability |
|---|---|---|---|---|
| Analytical | Weak | Strong | Strong | High |
| Koopman/EDMDc | Strong | Indirect | Indirect | Moderate |
| GRU | Strong | Weak | Indirect | Low |
| Neural ODE | Strong | Strong | Strong | High |
TABLE III: Qualitative comparison of baseline model classes. “Force” and “Stiffness” indicate direct force representation and stiffness extraction capability.
Fig. 8: Feedforward tracking validation for sinusoidal and triangular trajectories. a – c Sinusoidal tracking under amplitude variation ( mm), frequency variation ( Hz), and stiffness variation ( N/mm). d – f Triangular tracking under the same amplitude, frequency, and stiffness conditions.
Fig. 9: Stiffness synthesis validation. a Time-varying stiffness tracking during sinusoidal motion with N/mm; stiffness is identified from closed-valve positive and negative perturbation responses at mm. b Stiffness range verification at equilibrium for N/mm, showing identified stiffness, flexor/extensor pressures and , and representative perturbation responses.
Fig. 10: Stiffness comparison between Neural ODE (NN) and equilibrium-point (EP) models. Circles and triangles denote perturbations at zero and maximum velocity, respectively; large markers show mean standard deviation, and faded markers show individual trials. Asterisks indicate significant velocity-dependent stiffness changes for the EP model ( ).
| Model | Part | Nominal (N/mm) | -value | |||
|---|---|---|---|---|---|---|
| NN | K low | 140 | 139.54 | 140.07 | 1.06 | 0.6036 |
| K mid | 150 | 146.96 | 145.83 | 2.26 | 0.1075 | |
| K high | 160 | 152.39 | 153.36 | 1.94 | 0.2585 | |
| EP | K low | 140 | 140.66 | 136.64 | 8.04 | |
| K mid | 150 | 146.98 | 143.09 | 7.78 | ||
| K high | 160 | 151.39 | 147.46 | 7.86 |
TABLE IV: Identified stiffness for the Neural ODE (NN) and equilibrium-point (EP) models at zero and maximum velocity.
Fig. S1: Representative displacement response comparison for the 308-515 kPa absolute-pressure dataset (30-60 psi gauge). The ground-truth trajectory is the same dataset used for the main-paper time-series example. Reported values are displacement-only scores over the full trajectory. The Kang-type baseline uses measured chamber pressures, while the Koopman/EDMDc, GRU, and hybrid Neural ODE models use recursive rollout.