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Task-space model-based control of pneumatic soft actuators

Nithin S. Kumar, Joshua Gaston, D. Caleb Rucker, Eric J. Barth

arXiv:2608.27186v1cs.RO

TL;DR

Closed-loop task-space control of soft pneumatic actuators is difficult because their nonlinear, distributed, and uncertain dynamics must be handled under underactuation and pneumatic bandwidth limits. The paper uses a sparse non-minimal discrete elastic rod model with quasi-static feedforward, task-space PI feedback, and a dynamic observer for sparse-sensing estimation. Across five planar free-space tasks, it reports 1.5–2.3 mm RMSE for precision motions and 5.5–12.4 mm RMSE at 1–2 Hz.

  • Problem

    Soft pneumatic actuators have underactuated, highly deformable, effectively infinite-dimensional dynamics, making real-time closed-loop task-space control challenging.

  • Method

    The framework combines a non-minimal constrained discrete elastic rod model with quasi-static model-based feedforward, task-space PI feedback, and a dynamic observer.

  • Results

    1.5–2.3 mm RMSE was achieved for slow precision motions, while periodic tracking at 1–2 Hz achieved 5.5–12.4 mm RMSE.

  • Takeaways & Limitations

    Structured non-minimal dynamics supported real-time planar free-space task-space control across precision, speed, generalization, reduced-sensing, and user-defined-reference tasks.

  • Takeaways & Limitations

    Experimental validation is restricted to planar free-space tracking and does not include contact interaction, force regulation, manipulation, full 3D motion, torsional dynamics, or long-term material degradation.

Abstract

from arXiv · show

Soft actuators enable dexterous and compliant interaction, but closed-loop task-space control remains challenging due to strong nonlinearities, distributed deformation, and uncertainty in their dynamics. This paper presents a real-time dynamic-model-based task-space feedback and estimation framework based on a non-minimal coordinate discrete elastic rod model formulated in absolute coordinates with holonomic constraints. The resulting structure preserves distributed mechanics while maintaining computational efficiency through sparse system matrices, enabling real-time control with up to 10 discretized rods. A quasi-static feedforward inverse model is combined with a task-space PI controller and a dynamic observer that fuses measurement residuals as virtual forces, enabling full-state estimation from sparse sensing. The approach is experimentally validated on three planar pneumatic soft actuators with varying geometries. Across five tasks, including drawing the digits 0-9 across the workspace (3-18 mm/s tip speed), tracking periodic motion (up to 37 cm/s), cross-platform generalization, reduced sensing conditions, and real-time user-defined references, our method achieves 1.5-2.3 mm root mean square error (RMSE) for precision motions and 5.5-12.4 mm RMSE at 1-2 Hz. Results demonstrate that structured, non-minimal dynamic models can enable real-time, high-precision, moderate-bandwidth task-space control of planar soft pneumatic actuators in free space.

I. INTRODUCTION

Closed-loop task-space control of soft pneumatic actuators is difficult because their underactuated, highly deformable dynamics combine distributed elasticity, model uncertainty, and pneumatic bandwidth limits. This work addresses the challenge with a computationally efficient non-minimal dynamic model, model-based feedback, and sparse-sensing state estimation evaluated across five practical tasks.

  • Motivation: Soft pneumatic actuators provide dexterity, passive compliance, and conformability, but their underactuated and effectively infinite-dimensional dynamics complicate real-time task-space control.Controllers must address nonlinear distributed elasticity, model uncertainty, and pneumatic bandwidth limitations.
  • Limitations of Prior Models: Reduced-order piecewise-constant-curvature models can miss distributed dynamic effects at coarse resolutions, while dense inertia matrices make finer real-time discretization computationally impractical.The cited prior approaches are typically limited to only a few segments because computation scales cubically.
  • Proposed Direction: Non-minimal discrete elastic rods preserve distributed mechanics without predefined curvature assumptions and remain tractable through sparse, block-tridiagonal system structure.The formulation supports real-time control with up to 10 discretized rod elements without forming dense inertia matrices.
  • Control and Estimation: The control architecture combines quasi-static model-based feedforward, task-space PI feedback, and a dynamic observer that reconstructs full-state estimates from sparse sensing.The framework is designed to support measurements such as tip-only sensing.
  • Evaluation: Five experiments evaluate precision tracking, high-speed motion, actuator-geometry generalization, reduced sensing, and real-time user-defined reference tracking.The reported task conditions include 3–18 mm/s precision motion and periodic motion at 37 cm/s.

A. Constrained Dynamic Model

The actuator is modeled as a chain of rigid rods described by absolute positions and orientations, with holonomic constraints enforcing geometric continuity. Elasticity, damping, pneumatic torques, and stabilization are incorporated in a sparse constrained-dynamics formulation suitable for efficient integration.

  • Coordinate Representation: The backbone is discretized into N rigid rods with rotational elasticity and damping, using a non-minimal coordinate vector of absolute rod positions and orientations.Each rod contributes x_i, y_i, and θ_i coordinates.
  • Holonomic Constraints: Holonomic constraints connect adjacent rods to enforce geometric continuity, while the first rod is pinned to the origin.The constraints are represented as ϕ_H(q) = 0 with Jacobian A.
  • Constrained Dynamics: The constrained Lagrangian equations use Baumgarte stabilization to control numerical drift, producing an index-1 differential-algebraic system.The stabilization operates at the acceleration level through second-order constraint dynamics.
  • Forces and Actuation: Internal spring and damper torques produce block-tridiagonal stiffness and damping contributions, while pneumatic actuation enters as generalized torques at selected rod indices.These contributions preserve the sparse, block-structured form used for efficient linear solves.
  • Parameterization: The model uses measured rod mass and length, stiffness fitted from static chamber-induced deflections, and damping estimated from free oscillation through logarithmic decrement.Rayleigh damping combines mass and stiffness matrices while avoiding excessive damping of higher-frequency modes.

C. Controller Architecture

The controller combines quasi-static model-based feedforward with task-space PI feedback, using a local sensitivity map to convert task-space corrections into actuator inputs. The feedforward inverse is solved pointwise and does not compensate inertial dynamics.

  • The control architecture combines a quasi-static feedforward input with task-space PI feedback to generate the actuator command.
  • 1) Quasi-Static Feedforward Control:: The feedforward input solves pmodel(u) = pref by minimizing the task-space residual through nonlinear least squares.A Gauss–Newton iteration uses the local input–output sensitivity S(θ).
  • 1) Quasi-Static Feedforward Control:: The sensitivity matrix maps actuator-input perturbations to model-tip-position perturbations through the Jacobian, stiffness, and actuation-torque mappings.
  • 1) Quasi-Static Feedforward Control:: Regularized least-squares updates maintain robustness when the local sensitivity becomes ill-conditioned near kinematic or actuation singularities.The regularization parameter is 1e−8, and ill-conditioning is detected when the reciprocal condition number is below 1e−10.
  • 1) Quasi-Static Feedforward Control:: Backtracking selects the update step size to ensure monotonic reduction of the task-space residual along the reference trajectory.
  • 1) Quasi-Static Feedforward Control:: The feedforward term is quasi-static rather than inverse-dynamic and therefore does not explicitly compensate inertial terms such as M ¨q.Dynamic tracking errors are handled by the observer and task-space PI feedback.

2) Dynamic observer:

The dynamic observer integrates the constrained rod model and corrects its state using measured tip residuals injected as virtual generalized forces. Online observer states also provide the sensitivity matrix used for task-space feedback.

  • 2) Dynamic observer:: The observer integrates the dynamic model from the unactuated equilibrium using measured tip and base positions, with N = 10 rods balancing fidelity and computation.
  • 2) Dynamic observer:: Measurement residuals are injected through Qobs as a virtual force at the distal rod, steering observer states toward the physical actuator configuration.
  • 2) Dynamic observer:: The observer uses proportional correction with Ko = diag([100, 100]) and Bo = 0; derivative correction was omitted because proportional correction produced low observer error.
  • 2) Dynamic observer:: Pneumatic dynamics are approximated by a 50 ms first-order time constant and approximately 50 ms pure delay, with mismatch treated as model error corrected by feedback.
  • 2) Dynamic observer:: The online observer states determine the sensitivity matrix that maps desired task-space effort into actuator inputs.The desired effort combines proportional–integral feedback with a velocity feedforward term useful for dynamic trajectory following.
  • 2) Dynamic observer:: The total input is saturated and mapped to pressure setpoints regulated by low-level valve PI loops, while only feedback uses the dynamic observer.The feedforward component remains a pointwise quasi-static inverse without inverse-dynamics compensation.

D. Stability and Passivity Analysis

The nominal constrained rod model has a passive mechanical structure: damping dissipates energy, while actuation and observer-force injection supply mechanical work. This establishes nominal-model passivity and dissipativity, not stability of the complete implemented closed loop.

  • The constrained rod dynamics can be written with mass, damping, stiffness, actuation, observer forces, and holonomic constraint forces.
  • Holonomic constraint forces do no work because admissible velocities satisfy A(q) ˙q = 0.
  • The model is passive from actuator inputs u to T^T ˙q and observer forces Qobs to distal-tip velocity ˙pN.
  • In the unforced case, energy satisfies ˙H = −˙q^T B ˙q ≤ 0, making the nominal model Lyapunov stable and dissipative.
  • Bounded mechanical work from pneumatic actuation and observer-force injection yields bounded stored mechanical energy.
  • The analysis does not prove stability for the implemented closed loop, which includes the observer, PI controller, delays, saturation, noise, and numerical implementation.

E. Experimental Platform

Experiments use three planar Pneunet actuators with differing geometries and independently regulated pneumatic chambers. The platform supports 1 kHz control and evaluates step, drawing, and response-error characterization tasks.

  • E. Experimental Platform: The platform includes three planar pneumatic actuators, with P1 used primarily and P2–P3 used to evaluate generalizability.
  • 1) Soft Actuator:: Step-response experiments compare FF, FF+P, FF+I, and FF+PI controllers on a four-step diamond trajectory.
  • E. Experimental Platform: P2 varies wall thickness from 3 mm at the base to 1.1 mm at the tip, while P3 uses 10 triangular segments per chamber and is 193 mm long.
  • E. Experimental Platform: Each actuator has four independently actuated chambers controlled through proportional spool valves with inline pressure sensing and low-level PI regulation.
  • E. Experimental Platform: The controller, observer integration, pressure sensing, and valve commands run at 1 kHz, while optical tracking streams Cartesian position at 100 Hz.
  • 2) Hardware setup:: The digit-drawing task commands the actuator tip to draw all 10 digits in 10 seconds, with corresponding RMSE values reported for each digit.
  • E. Experimental Platform: The mean computation time per step is 0.97 ms.

III. EXPERIMENTAL RESULTS

The experiments compare feedforward and feedback variants before evaluating the selected FF+PI controller across five tracking and generalization tasks.

  • Integral feedback particularly reduced steady-state error, while FF+PI provided the best overall transient and steady-state performance.FF+PI was used in all subsequent experiments.
  • Task 3 tested cross-platform generalization on actuators P2 and P3.
  • Task 4 evaluated tip tracking using only intermediate backbone sensing.
  • Task 5 evaluated real-time tracking of user-defined trajectories.

Task 1: Precise Tracking of Slow Trajectories

The controller achieved millimeter-scale precision on slow drawing trajectories and maintained stable tracking for faster periodic motions, with larger deviations for vertical-line motion.

  • Task 1: Precise Tracking of Slow Trajectories: Tracking errors remained below 5 mm along each axis while drawing the digits 0–9 at reference speeds of 3–18 mm/s.Each digit was drawn over 10 s using trajectories including an ellipse, vertical line, and square-bounded paths.
  • Task 1: Precise Tracking of Slow Trajectories: 1.9 mm RMSE was achieved while drawing “VU” over 25 s, with tracking error below 4 mm throughout the motion.The trajectory spanned 125 mm at a constant reference speed of 5 mm/s.
  • Task 2: High-speed tracking: Stable tracking was maintained for 60 mm, 1 Hz circular and vertical-line trajectories after transition from the unactuated configuration.
  • Task 2: High-speed tracking: Larger deviations occurred for the 1 Hz vertical line, because coordinated curvature variation is more challenging for the bending-dominated actuator.
  • Task 2: High-speed tracking: Slow trajectories achieved millimeter-scale accuracy near 10 mm/s, while dynamic trajectories maintained tracking near 19 cm/s with 7–12 mm RMSE.The summary reports moderate-bandwidth control without instability or excessive phase lag.

Task 3: Generalization to Varying Actuator Geometries

The same controller generalized to actuators with different geometries using updated model parameters, while intermediate sensing improved performance over open-loop control.

  • Task 3: Generalization to Varying Actuator Geometries: P2 achieved 3.7 mm RMSE on “VU” and 9.1 mm RMSE on the 60 mm, 1 Hz circle despite nonuniform curvature distribution.No appreciable phase lag was observed.
  • Task 3: Generalization to Varying Actuator Geometries: P3 achieved 2.6 mm RMSE on “VU” and 5.5 mm RMSE on the 1 Hz circle after vertical reference shifts accommodated its reduced workspace.
  • Task 3: Generalization to Varying Actuator Geometries: At 2 Hz, P3 tracked a 60 mm circle at 37 cm/s peak speed and 4.7 m/s^2 peak acceleration, but higher frequency produced 11.4 mm RMSE and 27° x-direction phase lag.
  • Task 5: Real-Time User-Defined Tracking: The experiments also included arbitrary online reference tracking generated by an optical marker and a 30 s trial.
  • Task 4: Reduced Sensing: Using rod 6 position sensing reduced RMSE relative to feedforward-only control by 27%, 74%, and 34% for sweep, circle, and line trajectories, respectively.Combining rods 6 and 10 yielded no statistically significant improvement over tip-only sensing.

Task 5: Real-Time User-Defined Tracking

The discussion presents structured non-minimal rod modeling as a practical alternative for real-time soft-actuator control, retaining distributed mechanics through sparse constrained dynamics.

  • Non-minimal coordinate discrete elastic rod models are presented as a practical alternative to reduced-order PCC and learning-dominant approaches.
  • Prior work is characterized as focusing largely on curvature tracking, quasi-static task-space control, or lower-bandwidth dynamic tracking.
  • The reported results demonstrate 1.5–2.3 mm RMSE for slow trajectories and 5.5–12.4 mm RMSE for periodic tracking at 1–2 Hz.
  • The formulation preserves absolute coordinates and holonomic constraints while maintaining block-sparse structure, enabling real-time integration with 10 segments.This captures distributed curvature variation without predefined shape constraints.

2) Comparison to Learning-Based Control Approaches:

The framework favors structured mechanics-based control over learning-based inverse mappings, using sparse sensing and observer residuals for free-space tracking. Its demonstrated scope remains planar, free-space operation, with contact interaction, full 3D motion, and long-term adaptation left for future validation.

  • Learning-based controllers can depend on training coverage, generalize poorly across geometries or regimes, and typically lack formal stability guarantees.
  • Parameter updates enabled generalization across P1, P2, and P3 without retraining or structural controller modification.
  • A virtual force-injected observer reconstructed full-state behavior from sparse Cartesian measurements rather than dense strain sensing or full-field vision.
  • 45% average RMSE reduction relative to feedforward-only control was achieved with intermediate-point sensing and no direct tip sensing.
  • The observer residual is an aggregate correction, not a contact-force estimate, because position-only sensing cannot distinguish contact forces from model or actuation errors.
  • Real-time arbitrary-reference tracking remained stable under aggressive non-periodic excitation, though pneumatic delay and omitted velocity feedforward produced measurable phase lag.
  • The experiments validate planar free-space tracking, excluding contact interaction, force regulation, manipulation, full 3D motion, torsional dynamics, and long-term material adaptation.

V. CONCLUSION

The paper demonstrates real-time planar free-space task-space control using constrained non-minimal discrete elastic rod models. Across five tasks, the framework combines distributed mechanical fidelity with computational viability without reduced-order curvature assumptions or learned inverse mappings.

  • The controller achieved stable and accurate performance across precision tracking, high-speed motion, actuator generalization, reduced sensing, and real-time user-driven references.
  • Sparse system structure allowed constrained non-minimal dynamics to retain distributed mechanics while remaining computationally viable for planar free-space control.
  • The study reports funding from a Vanderbilt University Seeding Success grant and no conflicts of interest.
  • The authors’ research interests include design, modeling, and control of soft or continuum robots and fluid-power systems.
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