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

Nithin S. Kumar; Eric J. Barth

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

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 $\{X_{A},Y_{A},Z_{A}\}$ 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, $p_{\mathrm{ref}}$ . The learned control core generates the equilibrium pressure $P_{\mathrm{FF}}$ from the static model and the feedback pressure $P_{\mathrm{FB}}$ from the residual dynamic model, yielding the desired pressure command $P_{\mathrm{ref}}=P_{\mathrm{FF}}+P_{\mathrm{FB}}$ . A low-level pressure controller regulates the chamber pressures $P$ , producing the actuator tip position $p$ , 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 $d$	RMSE [mm]
	$x_{tip}$	$y_{tip}$	$z_{tip}$	Total
1	$9.12$	$9.30$	$17.37$	$21.71$
2	$7.94$	$8.34$	$5.79$	$12.89$
3	$6.16$	$6.60$	$5.61$	$10.63$
4	$5.00$	$5.72$	$5.62$	$9.45$

TABLE I: Static Map RMSE for Polynomial Fit Degrees

d

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.

Trajectory	Ref. Speed [mm/s]	Avg. RMSE [mm]
Helix	10	$1.09$
Helix	100	$9.47$
Lemniscate	10	$1.36$
Lemniscate	100	$6.60$
Staircase	6.5	$1.79$

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 (

L

). 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 ( $\rho(A_{aug})<1$ ), (b) marginal ( $\rho(A_{aug})\approx 1$ ), and (c) unstable ( $\rho(A_{aug})>1$ ) 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.

Trajectory	Ref. Speed	RMSE [mm]
	[mm/s]	FF only	FB only	FF+FB
Helix	25	$18.43$	$7.14$	$2.14$
Lemniscate	25	$19.81$	$6.57$	$2.37$
Staircase	6.5	$20.67$	$5.14$	$1.94$

TABLE III: Ablation study comparing static feedforward (FF), residual feedback (FB), and the complete controller (FF+FB)

Extended Dynamic Mode Decomposition

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Aug 27, 2026J/K move · Enter open · S save

Task-space model-based control of pneumatic soft actuators

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

Department of Mechanical Engineering, Vanderbilt University, Nashville, Tennessee 37235 · Department of Mechanical, Aerospace, and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996

Robotic Control Soft Robotics Soft Pneumatic Actuators Continuum Robot Control Pneumatic Actuation

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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.

Abstract

Extended Dynamic Mode Decomposition

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