cs.ROAug 25, 2026

Control-Oriented Learning for Dynamic Tracking and Stability Analysis of Soft Pneumatic Actuators

Authors: Nithin S. Kumar, Eric J. Barth

Organizations: Department of Mechanical Engineering, Vanderbilt University, Nashville TN 37215

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Abstract

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.

Figures & tables

Fig. 1: Spatial soft pneumatic actuator comprised of three linear bellows (LB) and experimental platform. (a) Actuator-fixed coordinate frame {XA,YA,ZA}\{X_{A},Y_{A},Z_{A}\} defined by the three base optical markers. (b) Example deformation produced by independent actuation of a single bellow. (c) Linear extension produced by simultaneous actuation of all three bellows together with the six optical tracking markers used for tip position estimation. (d) Bellows geometry and overall actuator dimensions. (e) Cross-sectional view of a single compliant bellows segment.
Fig. 2: Reachable workspace identified from the training dataset together with the best-fit half-ellipsoid used for real-time workspace projection. (a) Three-dimensional workspace. (b) x-z projection showing the workspace height. (c) x-y projection showing the workspace footprint. The fitted ellipsoid is used to project user references outside the reachable workspace during interactive motion generation.
Fig. 3: Overview of the proposed control-oriented learning framework. Different reference-generation modules (reference trajectories, user-guided motion, and obstacle avoidance) provide the desired tip position, prefp_{\mathrm{ref}} . The learned control core generates the equilibrium pressure PFFP_{\mathrm{FF}} from the static model and the feedback pressure PFBP_{\mathrm{FB}} from the residual dynamic model, yielding the desired pressure command Pref=PFF+PFBP_{\mathrm{ref}}=P_{\mathrm{FF}}+P_{\mathrm{FB}} . A low-level pressure controller regulates the chamber pressures PP , producing the actuator tip position pp , which is fed back to the controller. The learned residual model also enables local closed-loop stability analysis.
Fig. 4: Representative 50 s segment of the staircase training dataset used to identify the static input-output mapping. (Left) Independent chamber pressure excitations. (Right) Corresponding actuator tip position. Equilibrium samples (circles) are obtained by averaging the final 0.5 s of each 3 s hold period before fitting the static polynomial model.
Degree ddRMSE [mm]
 xtipx_{tip}ytipy_{tip}ztipz_{tip}Total
19.129.129.309.3017.3717.3721.7121.71
27.947.948.348.345.795.7912.8912.89
36.166.166.606.605.615.6110.6310.63
45.005.005.725.725.625.629.459.45
TABLE I: Static Map RMSE for Polynomial Fit Degrees dd
Fig. 5: Reference trajectory tracking task. Desired (black) and measured (red) tip trajectories for the staircase (6.5 mm/s), helix (10 mm/s), and lemniscate (10 mm/s) references. Trajectories begin at ∘\circ and end at □\square .
Fig. 6: Cartesian position histories corresponding to the trajectories in Fig. 5 . The proposed controller accurately tracks the desired motion in all three coordinates while maintaining smooth transient behavior. Reported RMSE values correspond to each Cartesian component.
TrajectoryRef. Speed [mm/s]Avg. RMSE [mm]
Helix101.091.09
Helix1009.479.47
Lemniscate101.361.36
Lemniscate1006.606.60
Staircase6.51.791.79
TABLE II: Reference Tracking Performance
Fig. 7: Real-time user-guided trajectory tracking. Smooth (left) and step-like (right) references generated from a handheld optical marker together with the measured actuator response. The smooth reference resulted in a peak reference acceleration and overall RMSE of 9.18 m/s 2 and 10.1 mm, respectively. The corresponding values for the step reference were 25.92 m/s 2 and 16.8 mm.
Fig. 8: Representative snapshots of the interactive user-tracking experiment. The desired reference is generated in real-time from the handheld optical marker while the actuator tracks the projected reference. Please refer to the supplemental materials for a video.
Fig. 9: Dynamic obstacle avoidance during tracking of a 0.1 Hz circular reference. (a) Cartesian position histories. Orange shaded regions indicate intervals when the obstacle is within 100 mm of the actuator tip (reference modification is active), whereas green regions correspond to nominal tracking. (b) Obstacle-tip Euclidean distance ( LL ). The dashed orange and red lines denote the outer (100 mm) and inner safety distances (25 mm), respectively. The proposed reference modification maintains the prescribed minimum separation while returning smoothly to the nominal reference once the obstacle-tip distance exceeds 100 mm.
Fig. 10: Experimental validation of the proposed stability analysis. Representative closed-loop responses for (a) stable ( ρ(Aaug)<1\rho(A_{aug})<1 ), (b) marginal ( ρ(Aaug)≈1\rho(A_{aug})\approx 1 ), and (c) unstable ( ρ(Aaug)>1\rho(A_{aug})>1 ) operating regimes during 0.1 Hz circular tracking with randomly sampled feedback gains. (d) The corresponding tracking-error (Euclidean distance) histories illustrate the transition from stable convergence to divergent behavior.
TrajectoryRef. SpeedRMSE [mm]
 [mm/s]FF onlyFB onlyFF+FB
Helix2518.4318.437.147.142.142.14
Lemniscate2519.8119.816.576.572.372.37
Staircase6.520.6720.675.145.141.941.94
TABLE III: Ablation study comparing static feedforward (FF), residual feedback (FB), and the complete controller (FF+FB)
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