Learning Low-Dimensional Strain Models of Soft Robots by Looking at the Evolution of Their Shape with Application to Model-Based Control

TL;DR

This work tackles the difficulty of obtaining accurate, interpretable dynamic models for continuum soft robots by learning low-dimensional, physics-based models directly from backbone shape data. It introduces a two-part framework: Kinematic Fusion to automatically minimize PCS segments while preserving shape fidelity, and Dynamic Regression with Strain Sparsification to identify a Lagrangian-consistent dynamic model and prune insignificant strains. Across diverse planar robot configurations, the method yields models that are more accurate out-of-distribution than state-of-the-art ML baselines and can be integrated directly into model-based controllers. The approach improves data efficiency, preserves physical interpretability, and demonstrates practical control capabilities, paving the way for scalable, physics-informed soft-robot modeling. Extensions to 3D and more complex actuation schemes are outlined for future work.

Abstract

Obtaining dynamic models of continuum soft robots is central to the analysis and control of soft robots, and researchers have devoted much attention to the challenge of proposing both data-driven and first-principle solutions. Both avenues have, however, shown their limitations; the former lacks structure and performs poorly outside training data, while the latter requires significant simplifications and extensive expert knowledge to be used in practice. This paper introduces a streamlined method for learning low-dimensional, physics-based models that are both accurate and easy to interpret. We start with an algorithm that uses image data (i.e., shape evolutions) to determine the minimal necessary segments for describing a soft robot's movement. Following this, we apply a dynamic regression and strain sparsification algorithm to identify relevant strains and define the model's dynamics. We validate our approach through simulations with various planar soft manipulators, comparing its performance against other learning strategies, showing that our models are both computationally efficient and 25x more accurate on out-of-training distribution inputs. Finally, we demonstrate that thanks to the capability of the method of generating physically compatible models, the learned models can be straightforwardly combined with model-based control policies.

Figures (8)

• Figure 1: Overview of the proposed methodology with the key contributions (Kinematic Fusion and Dynamic Regression and Strain Sparsification) highlighted in orange. Inputs: We consider $N$ Cartesian pose measurements $\chi$ distributed along the soft robot backbone, for example, obtained using Computer Vision (CV) techniques from video recordings as inputs. Kinematic Fusion: We apply an iterative procedure that involves (i) computing the robot configuration $q$ using PCS inverse kinematics, and (ii), to avoid overly complex and high-dimensional models, we merge adjacent segments with similar strains across the dataset into one segment of constant strain. Dynamic Identification: We identify the PCS dynamic model by iteratively regressing coefficients using linear least squares and further reduce the model complexity by neglecting insignificant strains. Output: The identified dynamic model has a Lagrangian structure suitable, for example, for model-based control applications.
• Figure 2: Panel (a): Schematic of the kinematic fusion algorithm. As inputs serve a sequence of $N$ discrete poses along the backbone of the soft robot. Next, we execute inverse kinematics in closed form with a $n_\mathrm{s} = N-1$ segment PCS model to identify the (unmerged) configuration of the robot. Initially, each constant strain segment connects two neighboring backbone poses. Subsequently, we compute a strain similarity measure $\bar{d}_{i,i+1}$ between each pair of adjacent segments. If segments exhibit a similar strain (i.e., the metric falls below a threshold $h$), we merge them into one constant strain segment. This process is repeated until no more merging is possible, resulting in a kinematic model with (hopefully) fewer segments: $1 \leq n_\mathrm{s} \leq N-1$. Panel (b): Schematic of the dynamic model identification process that simultaneously regresses the dynamic parameters and neglects unimportant strains. Based on a $n_\mathrm{s}$-segment PCS model, a library of basis functions is constructed to parameterize the system’s Lagrangian and EOM. A regression framework is established on a dataset of configuration-space positions $\dot{q}(k)$, velocities $\dot{q}(k)$, accelerations $\ddot{q}(k)$, and actuation torques $\tau(k)$ that estimates the dynamic parameters $\hat{\pi}^+$ with closed-form, linear least squares. Strains that exhibit a stiffness higher than a predefined threshold are neglected, prompting adjustments to the basis functions. Subsequently, this procedure is repeated until all strain stiffnesses lie below the threshold.
• Figure 3: Kinematic Fusion Results: Panels (a) & (b): Average strain distances between pairs of adjacent segments for Cases 2 & 3. The poses of $20$ markers along the manipulators are tracked, resulting in $19$ pairs of segments to be evaluated for strain similarity. The threshold is represented by a dashed line, and the background shading marks the resulting segments (separate segments are shaded in different colors). Panel (c): Average strain distances between segments after the first iteration of the kinematic fusion algorithm for Case 6. Panel (b): Pareto front that describes the trade-off between the DOF of the kinematic model (i.e., the number of segments) and the shape reconstruction error for Case 6. The blue and orange lines represent the average kinematic body position error $e_\mathrm{p}^\mathrm{body}$ and the body orientation error $e_\theta^\mathrm{body}$, respectively. The strain distance threshold $h$ that is used for separating segments is plotted on the upper x-axis.
• Figure 4: Verification of the dynamical model with noise for a two-segment PCS soft robot (Case 2). The dotted lines denote the ground-truth (GT) trajectory. The blue lines refer to the first segment, while the orange lines are associated with the second segment.
• Figure 5: Sequence of stills for the test rollout of the regressed dynamic model trained on the two-segment PCS dataset (Case 2). The blue and red dots represent the ground truth and estimated shape of the soft robot, respectively. The orange line represents the shape estimated by a model trained on a training set with added measurement noise.