Learning Analytical Models for Soft Robot Kinematic Control

Learning Analytical Models for Soft Robot Kinematic Control Daniele Somma1,∗, Enrico Donato1,∗, Francesco Iori1, Carlo Alessi2, Alessandro Lucantonio3, Egidio Falotico1 1BioRobotics Institute and Department of Excellence in Robotics and AI, Scuola Superiore Sant’Anna, Pisa, Italy 2Istituto Italiano di Tecnologia, Genova, Italy 3Department of Mechanical and Production Engineering, Aarhus University, Aarhus, Denmark ∗These authors contributed equally to this work Abstract—The challenge of controlling soft robots, despite their potential, has limited their widespread use. Traditional methods are either too complex for real-time applications or lack the necessary interpretability and reliability. We introduce a novel approach using symbolic regression to learn simple, yet accurate, forward kinematic models from data. Our models, validated on a soft robotic arm, can be integrated into a controller to achieve precise trajectory tracking. This research marks a step toward turning data-driven modelling into an efficient and interpretable tool for controlling nonlinear systems. Keywords—Soft Robots, Machine Learning, Robot Control, Modelling, Symbolic Regression I. INTRODUCTION Soft manipulators, renowned for their compliance and adaptability, hold great promise in their ability to continuously deform and safely interact with daily-world environments [1]. Nonetheless, these advantages come as challenges to the design of soft robot controllers, owing to non-linear behaviours which make their modelling hard to derive or learn [2]. Over the past decade, both data-driven and analytical modelling approaches for soft robots have been extensively investigated. Machine Learning (ML) methods - such as recurrent neural networks [3] and generative modelling [4], have achieved notable accuracy and generalization capabilities. Nonetheless, they offer limited theoretical guarantees. Conversely, analytical approaches provide stability guarantees and strong interpretability, but they are typically system-specific, and computational expensive - making them impractical for real-time control [5]. Most existing results focus on longitudinal actuators , whereas patterns that induce complex deformations [6] remain an open challenge. More recently, a third, hybrid paradigm, Symbolic Regression (SR), has emerged, aiming to uncover analytical equations directly from data [7]. These models offer a balance between accuracy and interpretability. To date, symbolic computation has been applied to soft manipulators with bending and elongation behaviours [8], but its potential extends towards enabling more dexterous and versatile soft robotic systems. Therefore, the need for an accurate and interpretable robot modelling is fundamental for soft robot control advancement. This work was supported by the projects PNRR-PRIN 2022 DISCOVER CUP: J53D23002310006 and SHARE-CP P.N. 300/22 FONDAZIONE PISA - RICERCA SCIENTIFICA E TECNOLOGICA – CUP J83C24000530007. Corresponding author: [email protected] In this work, we introduce a SR-based approach to derive a forward kinematic model for soft robots. The method is first validated on a soft arm with longitudinal actuation. To further highlight its generality, we extend the evaluation to a helicoidal pattern, showcasing the model’s applicability to non-trivial actuation routing. II. SYMBOLIC SOFT ROBOT MODELLING AND CONTROL SR provides explicit and interpretable equations that can be directly exploited for physical insight and analytical computations, including the derivation of Jacobian matrices for control. 1) Forward kinematics via SR: Considering a soft arm, the forward model x=f(τ;S)defines a mapping from actuation τ∈R3to task-space position x∈R3via a set of mathematical operators S. For each component of x: (x, y, z), we normalize the data and train a different regressor, using the gplearn Python library. The regressors worked over a set of four binary operators S:= {add,sub,mul,div}, ensuring that the resulting expressions remain interpretable and computationally tractable. The symbolic regressors were instantiated with a population size of 2500 and evolved over 20 generations. The evolutionary process was guided by genetic operators such as crossover (p=0.7), subtree mutation (p=0.1), hoist mutation (p=0.05), and point mutation (p=0.1). To avoid overfitting, Fig. 1. Soft robotic platform. The soft arm is mounted on a fixed base, housing a servomotor per each tendon, whose routing from base to tip is pointed out. 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629874 235 Fig. 2. Results of trajectory following with the soft arm. (A) Actual and desired trajectories over circular and infinite-like shapes. A single demonstration is reported since performance is similar among trials. (B) Actuation values (motor unit m.u) for circular trajectory following for both triads. a maximum sample ratio of 0.9was used together with a parsimony coefficient of 0.01, which penalizes overly complex expressions. The optimization was performed using meansquared error (MSE) as the evaluation metric. 2) Robot controller: Resulting equations from SR were used to compute the end-effector Jacobian matrix, defined as J. At each timestep, the actuation update is computed as ∆τ=J#(τ)·ut·dt, (1) where J#denotes the pseudo-inverse of J. The control input utis provided by a PI controller. III. SOFT ROBOTIC ARM We evaluate the proposed methodology on a tendon-driven soft continuum robot composed of 3D-printed modular soft elements - see Fig.1. The fabricated arm is 10 cm long and 3 cm in diameter. Actuation is provided by tendons routed from base to tip, controlled by Dynamixel XM430-W210-R servomotors. Three tendon triads are defined: longitudinal tendons for pure bending and compression, and clockwise/counterclockwise helical tendons inducing combined bending and torsion. SR for forward modelling will be separetely evaluated on longitudinal and counterclockwise helical tendon routings. Data collection is performed via motor babbling, independently actuating each triad with random tendon displacements (0–25% of the resting length). Robot shapes are recorded at 10 Hz using a VICON motion-capture system. The dataset has been collected over 1000 timesteps for each triad, providing a comprehensive basis for model evaluation. IV. RESULTS The symbolic regression converges to simple, yet effective analytical equations with high R2 on the normalized dataset [bending →x: 0.93, y: 0.87, z: 0.75; CCW helicoidal →x: 0.88, y: 0.80, z: 0.82]. To validate the use of such models for control, they have been employed for trajectory tracking with PI control. Figure 2(A) illustrates the desired trajectories (dashed line) compared with the actual end-effector positions. The manipulator successfully followed the reference trajectories with good accuracy over the entire duration of the test. The average root MSE for the circle is 5.62 mm, and 7.78 mm for the infinite. Examining the robot actuation for both models is also important. Figure 2(B) shows the actuation signals for the circular trajectory. As expected, the bending triad exhibits three sinusoidal waves, each shifted by 120◦and with similar amplitudes. In contrast, the CCW helicoidal triad displays a less regular pattern, particularly in the phase shifts, but still converges to a stable configuration. Differences in peak amplitudes arise from variations in tendon pulling offsets. This confirms the model’s ability to capture and reproduce appropriate actuation strategies. V. FUTURE WORKS This work presented a method for analytical, data-driven modelling of soft arms via symbolic regression. Preliminary results promise good performance and applicability for kinematic tasks - despite simplifying assumptions. Future work will extend this framework to dynamic modeling, incorporating the system’s forces and inertial effects. This advancement will be essential to enable the robot to perform safe and precise physical interactions. REFERENCES [1] Laschi, C., Rossiter, J., Iida, F., Cianchetti, M., & Margheri, L. (2017). Soft robotics: trends, applications and challenges. Proc. Soft Robot. [2] Falotico, E., Donato, E., Alessi, ... & Killpack, M. (2025). Learning controllers for continuum soft manipulators: Impact of modeling and looming challenges. Wiley Advanced Intelligent Systems [3] Centurelli, A., Arleo, L., Rizzo, A., Tolu, S., Laschi, C., & Falotico, E. (2022). Closed-loop dynamic control of a soft manipulator using deep reinforcement learning. 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