Dynamic Shape Control of Soft Robots Enabled by Data-Driven Model Reduction

TL;DR

This work addresses dynamic shape control for soft robots by learning fast, linear reduced-order models from data to enable model predictive control. It conducts a rigorous, side-by-side evaluation of three non-intrusive ROM classes—LOpInf, DMDc, and ERA/OKID—on a high-fidelity anguilliform soft-robot testbed implemented in SOFA, showcasing their estimation and control performance. Across feasible references, bioinspired gaits, and data-derived trajectories from a physical analog, LOpInf-based ROMPC consistently achieves the lowest tracking errors, highlighting the value of structure-preserving, second-order dynamics for soft-robot control under limited data. The study provides a publicly available dataset to spur further data-driven modeling; it also outlines limitations related to boundary conditions, fluid-structure interactions, and nonlinear effects, pointing to future work in nonlinear ROMs and online adaptation for simulation-to-real-world transfer.

Abstract

Soft robots have shown immense promise in settings where they can leverage dynamic control of their entire bodies. However, effective dynamic shape control requires a controller that accounts for the robot's high-dimensional dynamics--a challenge exacerbated by a lack of general-purpose tools for modeling soft robots amenably for control. In this work, we conduct a comparative study of data-driven model reduction techniques for generating linear models amendable to dynamic shape control. We focus on three methods--the eigensystem realization algorithm, dynamic mode decomposition with control, and the Lagrangian operator inference (LOpInf) method. Using each class of model, we explored their efficacy in model predictive control policies for the dynamic shape control of a simulated eel-inspired soft robot in three experiments: 1) tracking simulated reference trajectories guaranteed to be feasible, 2) tracking reference trajectories generated from a biological model of eel kinematics, and 3) tracking reference trajectories generated by a reduced-scale physical analog. In all experiments, the LOpInf-based policies generated lower tracking errors than policies based on other models.

Figures (8)

• Figure 1: Process flow for synthesizing a reduced-order model predictive controller for dynamic shape control of a soft robot. Our proposed process spans three stages---design (top), modeling (middle), and control (bottom)---that we defined to be broadly applicable to a variety of soft robots. In this work, we focus on an anguilliform-inspired soft robot, designed through a process of bioinspiration (top). Using data generated from a high-fidelity finite element simulation of the robot, we explored various techniques for data-driven model reduction to generate linear reduced-order models that are amenable to control. We then used these ROMs to construct a closed-loop state observer and reduced-order model predictive controller.
• Figure 2: Anguilliform soft robot design and simulation. (a) We demonstrate our proposed methods on a simulated version of an eel-inspired soft robot developed in cervera-torralba_lost-core_2024park_analysis_2025 (top right). This robot is comprised of soft, unactuated head and tail segments, three actuated segments driven by antagonistic fluid elastomer actuators, and rigid couples, used to securely connect the soft segments. As the two fluidic chambers in the actuated segments are pressurized with inputs, $\mathbf{u}_i$, bending moments can be induced along the robot's body to affect its shape. b) The robot accepts six pressure inputs that are physically coupled such that $\mathbf{u}_{2i} = -\mathbf{u}_{2i+1}$ for $i\in \{0,1,2\}$. In our setting, we constructed the output of the robot as the $x$-$z$ position of 20 equally spaced points placed along the centerline of the robot's body, shown as red dots. c) We simulated the system's dynamics through a custom finite element model constructed in an open source simulation framework (SOFA), with high-dimensional state $\mathbf{x}(t)\in\mathbb{R}^n$ representing the spatial coordinates of each node in a high-dimensional mesh of the robot's geometry. We applied fixed constraints to two points at the base of the robot's head, allowing it to pivot about the line passing through these points (dorsal constrained point and line passing through both constrained points shown in red on the mesh of the robot). We selected this constraint based on past work on kinematics of anguilliform swimming indicating that this region of eels exhibits little lateral motion during forward swimming.
• Figure 3: Block diagram of proposed reduced-order model predictive control loop. For each tested ROM, we synthesized a closed-loop state observer that takes measurements of the simulated robot's centerline to produce estimates of the reduced-order state. These estimates and a desired reference trajectory are then used to seed an optimization-based control policy, which produces a trajectory of input pressures. At each timestep, the pressures at the first timestep of this trajectory are used as input to the robot.
• Figure 4: Prediction error of open-loop and closed-loop estimation laws based on ROMs generated via the ERA (red), DMDc (blue), and LOpInf (green) methods. a, i-iii) Comparison across ROM estimation methods of relative estimation error, $e_y$, as a function of ROM dimension, $r$, grouped by the amount of training data used to train ROMs of each method: i) one trial, ii) two trials, iii) three trials. For nearly all synthesized ROMs, except higher dimensional ROMs generated via LOpInf, open-loop relative estimation error over a trial remained approximately at or above $e_y=1$ (black dotted line). Values for DMDc-based ROMs trained on three trials of data and with state dimensions of $r=2$ (${e_y=2946.6\pm3601.7}$) and $r=4$ (${e_y=2150.3\pm2665.9}$) were off the scale and are omitted for visual clarity. In contrast to the open-loop estimation schemes, the closed-loop state observers significantly improve estimation accuracy for nearly all models. b, i-iii) Estimation accuracy of each ROM as a function of the state dimensions tested in our proposed control loops, grouped by ROM synthesis method: i) ERA-based ROMs, ii) DMDc-based ROMs, iii) LOpInf-based ROMs.
• Figure 5: Tracking performance of ROMPC scheme over ROM types, dimensions, quantities of training data, and objective function tuning schemes. The average tracking error for ERA-based (red), DMDc-based (blue), and LOpInf-based (green) controllers generated through this process followed from an extensive tuning process that considered the effects of (a) reduced-order dimension, $r$, for ROMs trained on one trial of data and (b) quantity of training data used for generated ROMs of dimension 18 on tracking performance. (c) Our tuning process also explored the effects of varying coefficients in the MPC objective function to apply equal weighting to tracking penalties across the body as well as weights defined by a posterior-focused scheme. (d) Relative tracking error of the best observed controllers based on each ROM generation method computed over the robot's body length. Tracking error was computed over known feasible trajectories from (i) the trials used to train each ROM and (ii) a set of eight test trials from our dataset. The average tracking error for each best-case controller over each set of trials is shown as dotted lines, with surrounding shaded regions representing one standard deviation of tracking error at a given point. (iii) Output trajectories generated by each controller are shown at control points located at the head (top left), midpoint (top middle), and tail (top right) of the robot for an example reference trajectory (black) extracted from Trial 33 of our dataset. The full reference centerline (black) of the simulated robot are shown for the same trial (bottom) overlaid with centerlines produced by each best-case controller shown in their respective colors (deflections in the $z$-direction amplified by a factor of five for sake of visualization).