Task-space model-based control of pneumatic soft actuators

The gist 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<ref:2608.27186#pg2> 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<ref:2608.27186#pg4> The resulting structure preserves distributed mechanics while maintaining computational efficiency through sparse system matrices, enabling real-time control with up to 10 discretized rods<ref:2608.27186#pg5>

Model Formulation and Efficiency

The backbone of the soft continuum actuator is discretized as a chain of N rigid rods with rotational elasticity and damping as shown in Fig. 2<ref:2608.27186#pg6> A non-minimal (maximal) set of generalized coordinates is used to describe the configuration: q = [x1, y1, θ1, xN, yN, θN]<ref:2608.27186#pg6> Adjacent rods are connected through holonomic constraints enforcing geometric continuity, and the base of rod 1 is pinned to the origin<ref:2608.27186#pg6> Using a constrained Lagrangian formulation, the equations of motion are d/dt ∂L/∂q˙ − ∂L/∂q = QNC + A T(q)λH<ref:2608.27186#pg6> The resulting index-1 differentialalgebraic system is M AT A 0 q¨ −λH = Q −A˙q˙ − 2ζBωBAq˙ − ω2BϕH<ref:2608.27186#pg6> Internal viscoelastic torques (τk, τb in Fig. 2) arise from relative angular displacements and velocities between adjacent rods and are modeled as linear rotational springs and dampers, yielding block-tridiagonal stiffness and damping contributions in Q<ref:2608.27186#pg6> Pneumatic actuation is modeled as equivalent external generalized torques (τext in Fig. 2) applied at discrete rod indices<ref:2608.27186#pg6> The resulting constrained system preserves a sparse, block-structured form that enables efficient linear solves at each integration step<ref:2608.27186#pg6>

Parameter Estimation

The discrete rod model is parameterized by rod mass (mi), length (Li), and the rotational stiffness (ki) and damping (bi)<ref:2608.27186#pg6> The rotational stiffness parameter (ki) was identified by pressurizing individual chambers and performing a least-squares fit to the measured static deflections, following a procedure similar to [20]<ref:2608.27186#pg6> By construction, the inertia matrix is diagonal, M = diag(mi, mi, miL2i12) ∈ R 3N×3N, where each rod is modeled as a uniform slender body<ref:2608.27186#pg6> The damping constant (bi) was estimated from the measured free oscillation of the actuator tip using logarithmic decrement to obtain the dominant modal damping ratio ζ<ref:2608.27186#pg6> A Rayleigh damping model was assumed, B = αM + βK, with (α, β) selected to bound the damping ratios within a narrow range around the measured ζ<ref:2608.27186#pg6>

Control Architecture

The overall control structure is shown in Fig. 3<ref:2608.27186#pg6> A quasistatic model-based feedforward term is combined with a taskspace PI controller and a dynamic observer to generate the input vector u = [u1, u2]T ∈ R 2×1<ref:2608.27186#pg6> 1) Quasi-Static Feedforward Control: For a desired tip reference pref ∈ R 2×1, the feedforward input uF F is obtained by solving the static inverse problem pmodel(u) = pref<ref:2608.27186#pg6> This is formulated as a nonlinear least-squares problem minimizing the task-space residual, r(u) = pmodel(u) − pref<ref:2608.27186#pg6> A Gauss–Newton iteration is used, based on the local input–output sensitivity, S(θ), given by S(θ) = ∂pmodel/∂u = J(θ)K−1θT ∈ R 2×2<ref:2608.27186#pg6> The update term at iteration k is ∆uF F,k = -S−1r(uk), if S is well-conditioned, otherwise, -(ST S + ϵI)−1STr(uk)<ref:2608.27186#pg6> 2) Dynamic observer: Tip and base positions are measured via optical tracking and transformed into the task-space frame<ref:2608.27186#pg6> State correction is achieved by injecting measurement residuals as generalized forces through the vector Q<ref:2608.27186#pg6> Specifically, this virtual tip force is applied at the distal rod to steer the states of the observer toward the configuration of the physical actuator<ref:2608.27186#pg6> Qobs = Ko(pdata − pmodel) + Bo(˙pdata − p˙model)<ref:2608.27186#pg6> 3) Task-space PI control: Feedback is implemented in task space by mapping a desired task-space effort (fdes) to actuator inputs via the sensitivity matrix S(θ), which is computed online from the dynamic observer states<ref:2608.27186#pg6> The total control input is u = uF F + uF B<ref:2608.27186#pg6>

Experimental Results and Performance

Across five representative tasks we validate the approach: 1) Precision trajectory tracking (3-18 mm/s), 2) High-speed periodic motion (37 cm/s), 3) Generalization to actuators with differing geometries (P2 and P3 in Fig. 1a), 4) Trajectory tracking with intermediate-point sensing only, and 5) Real-time user-defined reference tracking<ref:2608.27186#pg6> Task 1: Precise Tracking of Slow Trajectories achieved an RMSE of 1.9 mm for the letter “VU” task<ref:2608.27186#pg6> Task 3: Generalization to Varying Actuator Geometries showed that the controller achieves an RMSE of 3.7 mm for the “VU” task and 9.1 mm for the 60 mm, 1 Hz circular trajectory<ref:2608.27186#pg6> Task 4: Reduced Sensing showed that incorporating intermediate backbone sensing significantly improves performance relative to open-loop control<ref:2608.27186#pg6> Task 5: Real-Time User-Defined Tracking demonstrated stable tracking under non-periodic and user-defined motion<ref:2608.27186#pg6>

Conclusion

This work demonstrates that non-minimal coordinate discrete elastic rod models can be used for real-time planar free-space task-space control of soft pneumatic continuum actuators<ref:2608.27186#pg6> Across five experimental tasks—including precision tracking, high-speed motion, actuator generalization, reduced sensing, and real-time user-driven references—the proposed controller architecture achieved stable and accurate performance without relying on reduced-order curvature assumptions or learning-based inverse mappings<ref:2608.27186#pg6> More broadly, the results show that when sparsity is exploited, constrained non-minimal formulations can retain distributed mechanical fidelity while remaining computationally viable for planar free-space control<ref:2608.27186#pg6> Future work will extend this framework to spatial rod models and evaluate contact-rich manipulation, further establishing physics-grounded structured dynamics as a scalable foundation for high-performance soft robotic control<ref:2608.27186#pg6>

Comparison with Prior Work

In contrast, this work demonstrates mm-scale Cartesian accuracy (1.5–2.3 mm RMSE) for slow trajectories and subcm accuracy (5.5–12.4 mm RMSE) for periodic tracking at 1–2 Hz<ref:2608.27186#pg6> This work represents one of the highest reported closed-loop task-space tracking bandwidths for pneumatically actuated soft continuum robots while maintaining mm-level precision<ref:2608.27186#pg6> The results suggest a shift in the prevailing modeling–control tradeoff: real-time task-space control with dynamic model-based state estimation does not require either minimal-coordinate reduction or heavy datadriven augmentation when the mechanics are structured appropriately<ref:2608.27186#pg6> This capability is significant for practical deployment<ref:2608.

1. A real-time dynamic model-based task-space feedback and estimation framework can be implemented for pneumatic soft actuators by combining a quasi-static model-based feedforward term with a taskspace PI controller and a dynamic observer. This allows the system to achieve real-time, high-precision, moderate-bandwidth task-space control of planar soft pneumatic actuators in free space.

2. A sparse, block tridiagonal structure in the equations of motion is exploited by using a non-minimal coordinate discrete elastic rod formulation, which enables real-time control with up to 10 discretized rods without forming dense inertia matrices. This addresses the computational intractability associated with higher spatial resolutions in traditional models.

3. A force-based dynamic observer can be used to reconstruct full-state estimates from sparse sensing (e.g., tip-only measurements), which is achieved by injecting measurement residuals as generalized forces through the vector Q. This enables the system to maintain state estimation even under reduced sensing conditions.

4. The controller architecture can generate a total input as u = uF F + uF B, where uF F is a quasistatic feedforward term and uF B is a feedback correction. This structure ensures that the system compensates for model uncertainties while maintaining the benefits of the feedforward term, which does not explicitly compensate inertial terms such as Mq¨.

5. The controller can handle arbitrary user-defined tracking by replacing the precomputed feedforward input with an online inverse update computed at a lower rate (100 Hz), allowing for stable real-time tracking under non-periodic and user-defined motion.