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Control-Oriented Learning for Dynamic Tracking and Stability Analysis of Soft Pneumatic Actuators
Nithin S. Kumar, Eric J. Barth
TL;DR
Soft pneumatic actuators are difficult to control because their nonlinear, distributed, hysteretic, and variable behavior challenges conventional modeling approaches. The paper combines a learned nonlinear static equilibrium model with EDMDc-based linear residual dynamics for feedforward and feedback control, then validates trajectory tracking, interactive motion, and local stability analysis. Experiments report accurate tracking, highly dynamic user-command tracking with obstacle avoidance, and prediction of stable, marginal, and unstable regimes.
Problem
Soft actuators remain difficult to model and control because nonlinear, distributed, hysteretic, and parameter-varying behavior limits dynamic task-space control.
Method
The framework combines a learned nonlinear static equilibrium model with EDMDc-identified linear residual dynamics for feedback control and local stability analysis.
Results
Experiments validate accurate trajectory tracking, stable highly dynamic user-guided motion with obstacle avoidance, and reliable prediction of stable, marginal, and unstable regimes.
Takeaways & Limitations
Control-oriented learning provides a practical middle ground combining closed-loop performance with computational efficiency, interpretability, and analytical tractability.
Takeaways & Limitations
The models are learned from freespace motion and may perform less reliably outside the sampled workspace, pressure limits, and motion bandwidth or during contact-rich manipulation.
Abstract
from arXiv · showhide
Soft pneumatic actuators offer inherent compliance and safe interaction but remain difficult to model and control because of their highly nonlinear, distributed dynamics. We present a control-oriented data-driven modeling and control framework that decomposes actuator behavior into a nonlinear static equilibrium model and a linear residual dynamics model identified using Extended Dynamic Mode Decomposition with control (EDMDc). This representation enables feedforward compensation, task-space feedback control, and local closed-loop stability analysis through an augmented linear model. Experiments achieve approximately 1 mm root mean square error (RMSE) during low-speed (approximately 10 mm/s) trajectory tracking and below 10 mm RMSE at higher speeds (approximately 100 mm/s). The framework further achieves stable tracking of highly dynamic user-generated references with peak accelerations exceeding 25 m/s^2 while simultaneously performing real-time obstacle avoidance. Finally, the proposed stability analysis is experimentally validated by accurately predicting stable, marginal, and unstable operating regimes. These results demonstrate that structured, control-oriented learning provides an accurate and practical framework for soft actuator control.
I. INTRODUCTION
Soft pneumatic actuators are safe and versatile but difficult to model and control because of nonlinear, distributed, hysteretic, and variable behavior. The paper proposes structured control-oriented learning that combines nonlinear equilibrium modeling with linear residual dynamics for feedback and stability analysis.
- Motivation: Soft robots enable safe interaction and operation in challenging environments through intrinsic compliance and large continuous deformations.Applications include minimally invasive surgery, rehabilitation, wearable robotics, industrial manipulation, and confined or unstructured environments.
- Motivation: Soft actuators exhibit effectively infinitedimensional kinematics, geometric nonlinearities, hysteresis, and parameter variations, making dynamic task-space control difficult beyond quasi-static operation.These challenges arise despite the capabilities enabled by actuator compliance.
- Prior approaches: Physics-based models require analytical effort, actuator-specific parameter identification, and simplifying assumptions, while increasing complexity makes real-time computational efficiency harder to maintain.Reduced-order continuum, finite element, Cosserat rod, and discrete elastic rod models provide physical interpretability but impose practical burdens.
- Prior approaches: Data-driven methods reduce reliance on first-principles modeling but can behave as black-box models, require large datasets, and offer limited interpretability or feedback-control guarantees.The paper frames these trade-offs as motivation for combining structure with learning.
- Contributions: The proposed framework combines a learned nonlinear static equilibrium model with EDMDc-identified linear residual dynamics for task-space feedback and local stability analysis.It is experimentally validated through trajectory tracking, interactive motion, and stability experiments.
- Contributions: The experiments validate accurate trajectory tracking, stable tracking of highly dynamic user-generated references, real-time obstacle avoidance, and prediction of stable, marginal, and unstable regimes.These capabilities are presented as the framework’s principal experimental contributions.
A. Pneumatic soft actuator
The experimental platform is a three-bellows spatial pneumatic actuator whose pressure-dependent deformation provides a challenging benchmark for the learned control framework. The setup combines optical tip-position measurement, pressure regulation, workspace projection, nonlinear feedforward, and residual-model feedback.
- Actuator: The actuator uses three parallel linear bellows between rigid base and tip plates; independent pressurization elongates it, while differential pressurization produces three-dimensional bending.This configuration supplies coupled axial and bending motion for experimental evaluation.
- Actuator: The actuator is a representative benchmark rather than an application-specific device, combining large deformation, nonlinear pressure-to-position behavior, and coupled pneumatic dynamics.These properties make it experimentally challenging while remaining tractable for data-driven modeling and control.
- Workspace: The reachable workspace is approximated by a fitted half-ellipsoid used to project references outside the workspace during interactive motion generation.The workspace projections include three-dimensional, x-z, and x-y views.
- Instrumentation: Six optical markers define the actuator-fixed frame and compute tip position, while motion capture, feedforward, and feedback run at 50 Hz and pressure control runs at 1 kHz.The marker system provides Cartesian measurements for task-space control.
- Control framework: The framework decomposes behavior into a nonlinear static equilibrium mapping and a linear residual dynamics model, using feedforward compensation and task-space feedback to generate pressure commands.The residual model also supports local closed-loop stability analysis.
A. Static Input-Output Mapping
The static model maps chamber pressures to equilibrium tip position using a second-order polynomial, then numerically inverts that map to generate feedforward pressure commands. Degree two balances fidelity, interpretability, and real-time implementation while residual feedback compensates remaining errors.
- The equilibrium behavior is represented by a nonlinear static mapping from chamber pressure to actuator tip position.
- A second-order polynomial basis captures dominant nonlinear pressure-to-position behavior while maintaining computational efficiency.The basis contains q = 10 functions.
- 500 s of staircase pressure excitation produced 161 equilibrium pressure-position pairs spanning the actuator workspace.Pressure commands were held for 3 s, and equilibrium positions were averaged over the final 0.5 s.
- Degree two provides an effective balance between model fidelity, interpretability, and real-time implementation because residual feedback compensates remaining modeling error.Higher-order models reduce fitting error but substantially increase basis terms.
- Feedforward pressure is obtained by numerically inverting the learned static map for each desired reference position.Precomputed references use fmincon; user-guided experiments use a warm-started Jacobian-based Gauss–Newton solver with backtracking line search.
B. Residual Dynamic Model
Residual dynamics model transient deviations from the learned static equilibrium prediction using a compact, normalized lifted state and an EDMDc-identified linear system. The model supports state propagation and local stability analysis.
- Residual dynamics represent transient deviations from the static equilibrium prediction after feedforward tip position is computed.The static map uses 161 equilibrium pairs, whereas residual dynamics use the complete 500 s staircase dataset.
- Time-delay coordinates augment the residual state, with residual history length tr = 2 selected to capture dominant actuator and pneumatic dynamics compactly.
- Inputs and lifted states are normalized using training-set means and standard deviations before EDMDc identification.
- The residual dynamics are modeled as a discrete-time linear system with matrices A and B identified using EDMDc.
- The residual model supports state propagation and local stability analysis, while feedback is computed directly from measured tip position.
C. Controller
The controller converts task-space tracking errors into residual corrections and pressure feedback through the learned pressure-to-residual mapping. The final pressure command combines feedforward and feedback terms and is saturated at allowable limits.
- A learned mapping from pressure changes to residual changes converts desired residual corrections into pressure feedback using a regularized pseudo-inverse.
- A task-space PI controller generates the desired residual correction from position error and integral error.
- The final pressure command combines feedforward and feedback pressure components before actuation.
- Cartesian position histories evaluate tracking in all three coordinates while reporting RMSE for each component and smooth transient behavior.
- Pressure commands are saturated to allowable limits of 0–35 psi/241 kPa before being sent to the pressure valve.
IV. RESULTS
Experiments evaluate baseline trajectory tracking, highly dynamic interactive behavior, and agreement between predicted and observed closed-loop stability. The same controller gains are used across precomputed reference experiments.
- The experiments assess baseline closed-loop tracking, real-time highly dynamic interaction, and experimentally observed versus predicted stability behavior.
- The precomputed reference experiments use controller gains KP = diag(2,2,2) and KI = diag(5,5,5).
A. Reference trajectory tracking
The framework evaluates task-space tracking across independent and coupled spatial trajectories, then extends the same controller to dynamic user references and obstacle avoidance through reference modification.
- Precomputed reference tracking: Three task-space trajectories assess independent Cartesian motion, coupled circular motion with axial translation, and simultaneous excitation of all three coordinates.Together, they test both independent-axis and coupled spatial motion with continuously varying curvature.
- Precomputed reference tracking: The actuator accurately reproduces prescribed three-dimensional trajectories with little steady-state error and no observable oscillatory behavior.
- Interactive user tracking: 9.18 m/s2 and 25.92 m/s2 peak reference accelerations produced approximately 1 cm and below 2 cm RMSE during smooth and step-like user-guided tracking, respectively.The smooth and step-like references produced overall RMSE values of 10.1 mm and 16.8 mm, respectively.
- Dynamic obstacle avoidance: Obstacle avoidance modifies the reference when the obstacle enters the 100 mm outer safety distance and uses a 25 mm minimum separation.The modified reference is projected onto the reachable workspace when necessary before being supplied to the controller.
- Dynamic obstacle avoidance: During 0.1 Hz circular tracking, the actuator maintained the prescribed minimum separation while smoothly switching between nominal and modified references without instability or discontinuities.The nominal trajectory was temporarily displaced four times and restored once the obstacle exceeded the outer safety distance.
C. Stability Analysis
The framework predicts local closed-loop stability from an augmented linear model, and hardware experiments distinguish stable, marginal, and unstable operating regimes as predicted by its spectral radius.
- Stability criterion: The augmented state-transition matrix Aaug predicts local closed-loop stability when its spectral radius satisfies ρ(Aaug) < 1.The criterion applies to the learned linearized closed-loop model about an operating point, not to global nonlinear stability.
- Experimental validation: Randomly perturbed feedback gains were evaluated by constructing the augmented system, computing ρ(Aaug), and executing the resulting controller on hardware.
- Experimental validation: Stable, marginal, and unstable regimes corresponded respectively to ρ(Aaug) < 1, ρ(Aaug) ≈ 1, and ρ(Aaug) > 1 during 0.1 Hz circular tracking.Tracking-error histories showed the transition from stable convergence to divergent behavior.
- Experimental validation: The learned augmented model reliably predicts local closed-loop behavior across stable, marginal, and unstable operating regimes.
- Implication: The framework aims to achieve accurate dynamic control without sacrificing analytical tractability.
A. Stability analysis as a design tool
The learned residual representation supports pre-deployment stability screening and separates reference generation from low-level control, while its conclusions remain bounded by training conditions and local modeling assumptions.
- Stability analysis as a design tool: The residual representation permits spectral-radius evaluation of an augmented linear closed-loop system before hardware deployment, reducing reliance on empirical gain tuning.
- Control architecture: The same learned models and PI controller support reference tracking, interactive user following, and obstacle avoidance through reference modification.
- Limitations: The framework is identified from freespace motion, so it does not explicitly account for external loading or contact interactions.Performance is expected to be strongest within the sampled workspace, pressure limits, and motion bandwidth; contact-rich use may require representative interaction data or explicit contact modeling.
- Limitations: The stability analysis is local and may not accurately predict behavior under large operating-point changes or long-term material drift.
- Limitations: The framework was evaluated on a pneumatic bellows actuator, while applicability to tendon-driven and concentric-tube systems remains to be investigated.
- Conclusion: The framework combines accurate dynamic control with local stability analysis without requiring high-fidelity physics-based models.
APPENDIX
The appendix derives augmented closed-loop dynamics by combining residual dynamics, feedback input, tracking error, and accumulated error. It shows that local stability depends on the augmented state transition matrix.
- Closed-loop derivation: The derivation substitutes the normalized feedback input into the residual dynamics to obtain the closed-loop system.The feedback gains are expressed in normalized coordinates before substitution.
- Closed-loop derivation: The tracking error combines the reference, feedforward position, and residual state, while the augmented state includes accumulated tracking error.The accumulated error evolves as E_k+1 = E_k + e_k.
- Augmented stability model: The resulting augmented closed-loop dynamics isolate exogenous feedforward and reference terms from the state-transition structure.The appendix notes that the exogenous terms do not affect the eigenvalue-based stability calculation.
- Augmented stability model: Local closed-loop stability is governed solely by the augmented state transition matrix Aaug.The spectral radius ρ(·) is the stability-analysis quantity used for the matrix criterion.