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Data-driven Model Reduction to Control Soft Swimming Robots

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<strong>Figure 1.</strong> Anguilliform soft robot design and simulation. <strong>1a.</strong> The anguilliform soft robot is composed of soft, underactuated head and tail segments; three actuated body segments that are driven by antagonistic fluid elastomer actuators; and rigid couples that securely connect the soft segments. Because the two fluidic chambers in the actuated segments are pressurized with inputs \(\mathbf{u}_i,\) we can induce bending moments along the robot’s body to affect its shape. <strong>1b.</strong> The robot accepts six pressure inputs that are physically coupled, such that \(\mathbf{u}_{2i} = -\mathbf{u}_{2i+1}\) for \(i \in \{0, 1, 2\}.\) We constructed the output of the robot as the \(x\)-\(z\) position of 20 equally spaced points along the centerline of the robot’s body (red dots). <strong>1c.</strong> To simulate the system’s dynamics, we constructed a custom finite element model in an open-source simulation framework called SOFA, where the high-dimensional state \(\mathbf{x}(t) \in {\mathbb{R}}^n\) represents the spatial coordinates of each node in a high-dimensional mesh of the robot’s geometry. Figure courtesy of [1].
Figure 1. Anguilliform soft robot design and simulation. 1a. The anguilliform soft robot is composed of soft, underactuated head and tail segments; three actuated body segments that are driven by antagonistic fluid elastomer actuators; and rigid couples that securely connect the soft segments. Because the two fluidic chambers in the actuated segments are pressurized with inputs \(\mathbf{u}_i,\) we can induce bending moments along the robot’s body to affect its shape. 1b. The robot accepts six pressure inputs that are physically coupled, such that \(\mathbf{u}_{2i} = -\mathbf{u}_{2i+1}\) for \(i \in \{0, 1, 2\}.\) We constructed the output of the robot as the \(x\)-\(z\) position of 20 equally spaced points along the centerline of the robot’s body (red dots). 1c. To simulate the system’s dynamics, we constructed a custom finite element model in an open-source simulation framework called SOFA, where the high-dimensional state \(\mathbf{x}(t) \in {\mathbb{R}}^n\) represents the spatial coordinates of each node in a high-dimensional mesh of the robot’s geometry. Figure courtesy of [1].

In the field of robotics, soft robots are created with stretchable materials including silicone and gels, which gives them a continuously deformable structure that enables muscle-like actuation. Their pliant build enables them to mimic biological systems [6] and exhibit unprecedented adaptation, sensitivity, and agility by leveraging dynamic control of their entire bodies. Their unique capabilities and properties offer several potential advantages over their rigid counterparts, including heightened safety during interactions with humans and delicate objects; a strong potential for medical applications; and the ability to perform well in complex, uncertain environments.

Although their build equips them with a unique set of capabilities, soft robots come with their own set of mathematical and computational challenges [4]. First, unlike hard-bodied robots, models of soft robots usually involve spatially one-dimensional (1D) to three-dimensional (3D) partial differential equations (PDEs) that allow for continuous deformation. This stipulation results in a large number of degrees of freedom after spatial discretization, thus necessitating the need for control methods that are capable of acting on high-dimensional models. Second, soft robot dynamics include many material nonlinearities (e.g., hysteresis and viscoelastic effects) as well as geometric nonlinearities under loading, all of which pose additional challenges for efficient real-time control. And third, the creation of muscle-like actuators and entire robots with nature-like material properties—as well as the study of biological locomotion and navigation—remains an active area of research, which complicates the functionality of engineered robots that need this information to mimic the low-energy, elegant behavior of biological systems.

Our recent work explores data-driven model reduction techniques for the model-predictive dynamic shape control of soft robots [1]. Here, we explain the selection and creation of our experimental platform, explore the necessity of simulations when constructing reduced-order models (ROMs), and share results that use ROM-based controllers to track biologically inspired and experimentally obtained trajectories.

Soft Robot Design

Anguilliform swimmers—such as eels, lampreys, and sea snakes—are among the most efficient natural swimmers, undulating their long, narrow bodies to propel themselves efficiently through water. Their motion is highly dynamic and requires careful coordination of the entire body to effectively produce thrust [9]. This movement makes the dynamic shape control of a simulated soft, anguilliform robot an interesting case study, as such a robot must maintain control of its entire body (rather than just a caudal fin, for example). We consider a physical, eel-inspired robot that comprises five soft segments—passive soft head, passive tail, and three actuated body segments—all of which are coupled via custom rigid clamps [2, 5] (see Figure 1a). Each actuated body segment is driven by two antagonistic fluid elastomer actuators (FEAs) that are embedded inside the segment and separated by a thin, strain-limiting layer that is aligned with the robot’s sagittal plane [2]. Pressurizing the FEAs within a given body segment thus induces a bending movement along the segment, and connecting multiple body segments enables complex bending across the robot’s whole body.

Soft Robot Simulation

<strong>Figure 2.</strong> Process flow that synthesizes 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 define to be broadly applicable to a variety of soft robots. We focus specifically on an anguilliform-inspired soft robot that we designed through a bioinspiration process. Using data 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 (ROMs) that are amenable to control. We then used these ROMs to construct a closed-loop state observer and ROM predictive controller. Figure courtesy of [1].
Figure 2. Process flow that synthesizes 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 define to be broadly applicable to a variety of soft robots. We focus specifically on an anguilliform-inspired soft robot that we designed through a bioinspiration process. Using data 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 (ROMs) that are amenable to control. We then used these ROMs to construct a closed-loop state observer and ROM predictive controller. Figure courtesy of [1].

We utilized an open-source finite element simulation tool called SOFA [3] to develop a simulation model, meshing the robot as one continuous body and defining regions of varying elastic moduli across the body. We used linear elasticity models across the entire system’s structure to ensure that any nonlinear, dynamic behavior would stem from geometric effects. The resulting finite-dimensional dynamical system is then

\[\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t),\mathbf{u}(t)), \quad \quad {\mathbf{y}}(t) = {\mathbf{Cx}}(t).\]

At time \(t,\) the full-order state \(\mathbf{x}(t) \in {\mathbb{R}}^n\) represents the spatial coordinates of every node in the robot’s meshed geometry, the control input \(\mathbf{u}(t) \in {\mathbb{R}}^m\) represents the pressure inputs to each fluidic chamber in the robot’s actuators, and the output \(\mathbf{y}(t) \in {\mathbb{R}}^p\) prepresents the \(x\)-\(z\) coordinates of 20 equally spaced points along the dorsal centerline of the robot (see Figure 1b). The ensuing full-order model of the robot’s structural dynamics has state of dimension \(n = 243{,}789\), output dimension \(p = 40\), and input dimension \(m = 6\). The aforementioned antagonistic coupling constraints only require the selection of three decoupled control inputs at each timestep.

Data-driven Reduced-order Models

Because control techniques for linear time-invariant systems are mature and fast, we opted to learn a reduced-order, linear time-invariant system of soft robot dynamics for model-predictive control (MPC). These ROMs take the form

\[\begin{eqnarray}\tilde{\mathbf{x}}_{k+1}&=&\tilde{\mathbf{A}}\tilde{\mathbf{x}}_k+\tilde{\mathbf{B}}\mathbf{u}_k, \\ \tilde{\mathbf{y}}_{k+1}&=&\tilde{\mathbf{C}}\tilde{\mathbf{x}}_{k+1} + \tilde{\mathbf{D}}\mathbf{u}_{k+1}. \end{eqnarray}\]

We then compared three popular algorithms to learn such models: the eigensystem realization algorithm (ERA), dynamic mode decomposition with control (DMDc), and Lagrangian operator inference (LOpInf) [7]. We found that the LOpInf structure-preserving ROM method performed best in prediction, estimation, and control settings. This outcome is not surprising, as the robot’s dynamics are of a second-order nature, and building ROMs at the second-order level—including matrix symmetries that arise from energy terms—should intuitively yield better models. For LOpInf, we first used SOFA to generate high-dimensional simulation data from the nonlinear finite element model. Doing so produced \(\mathbf{x}(t_0), ..., \mathbf{x}(t_K)\), from which we computed a linear subspace that best fit the data. We next projected the high-dimensional snapshot data onto said subspace, after which the method fits the linear reduced operators that preserve an assumed Lagrangian structure of the system’s dynamics. This action results in a learned ROM with the aforementioned linear time-invariant structure.

Reduced-order Model-predictive Control

To control the soft robot, we follow the workflow in Figure 2. Figure 3 further illustrates the closed-loop operation, demonstrating the use of a Luenberger observer for reduced-order state estimation from outputs \(\mathbf{y}(t)\), as well as the reduced-order MPC policy as an input to the SOFA simulation. Given the reference trajectory \(\mathbf{y}^*_{k+1:k+T}\) over control horizon \(T\), MPC determines an optimal control sequence \(\mathbf{u}^*_{k+1:k+T}\) by solving the following quadratic program at every timestep \(k\):

\[\begin{eqnarray} \mathbf{u}^*_{k+1:k+T} &=& \underset{{\mathbf{u}_{k+1:k+T}}}{\textrm{arg min}}\ {\sum^{k+T}_{i=k+1}}\left(\|\tilde{\mathbf{y}}_i-{\mathbf{y}^*_i}\|^2_{\mathbf{W}_{\mathbf{y}}}+\|\mathbf{u}_{i}\|^2_{\mathbf{W}_{\mathbf{u}}}+\|{\mathbf{u}_i}-{\mathbf{u}_{i-1}}\|^2_{\mathbf{W}_{\delta\mathbf{u}}}\right){\textrm{s.t.}} \\ \tilde{\mathbf{x}}_{-1}&=&\tilde{\mathbf{x}}_{k} \tag1\\ \mathbf{u}_{-1}&=&\mathbf{u}_{k} \tag2 \\ \tilde{\mathbf{x}}_{i}&=&\tilde{\mathbf{A}}\tilde{\mathbf{x}}_{i-1} + \tilde{\mathbf{B}}\mathbf{u}_{i-1}, \tag3 \\ \tilde{\mathbf{y}}_{i}&=&\tilde{\mathbf{C}}\tilde{\mathbf{x}}_{i} + \tilde{\mathbf{D}}\mathbf{u}_{i}, \tag4 \\  \|\mathbf{u}_i\|_{\infty}&<&{u_{\textrm{max}}}, \tag5 \\ \mathbf{u}_{i, 2j}&=&- {\mathbf{u}}_{i, 2j+1} \qquad \forall{j}\in\textrm{{0,1,2}}. \tag6  \end{eqnarray}\]

<strong>Figure 3.</strong> Block diagram of the proposed reduced-order model (ROM) predictive control loop. For each tested ROM, we synthesized a closed-loop state observer that takes measurements \(\mathbf{y}(t)\) of the simulated robot’s centerline to produce estimates of the reduced-order state. We then use these estimates and a desired reference trajectory to seed an optimization-based control policy that produces a trajectory of input pressures to the fluid elastomer actuators. At each timestep, we utilize the pressures at the first timestep of this trajectory as input for the robot. Figure courtesy of [1].
Figure 3. Block diagram of the proposed reduced-order model (ROM) predictive control loop. For each tested ROM, we synthesized a closed-loop state observer that takes measurements \(\mathbf{y}(t)\) of the simulated robot’s centerline to produce estimates of the reduced-order state. We then use these estimates and a desired reference trajectory to seed an optimization-based control policy that produces a trajectory of input pressures to the fluid elastomer actuators. At each timestep, we utilize the pressures at the first timestep of this trajectory as input for the robot. Figure courtesy of [1].

Here, \(\|\cdot\|_{\mathbf{W}_*}\) indicates the weighted 2-norm of a vector with diagonal weighting matrix \(\mathbf{W}_*.\) The objective of this optimization is to determine the sequence of inputs that simultaneously minimizes the weighted magnitudes of the tracking error \(\|\tilde{\mathbf{y}}_i-{\mathbf{y}^*_i}\|^2_{\mathbf{W}_{\mathbf{y}}}\), weighted control energy \(\|\mathbf{u}_i\|^2_{\mathbf{W}_{\mathbf{u}}}\), and discrete rate of change of the control input \(\|{\mathbf{u}_i}-{\mathbf{u}_{i-1}}\|^2_{\mathbf{W}_{\delta\mathbf{u}}}\). We impose the constraints to use the initial state from the Luenberger observer, set a maximum possible pressure value for control and the constraint, and account for the antagonistic coupling of pressures in the robot’s closed fluidic drive system.

After computing the optimizing input sequence \(\mathbf{u}^*_{k+1:k+T}\) from this constrained optimization, we apply the first input in the sequence \((\mathbf{u}^*_{k+1})\) to the simulated robot, compute the system’s response to the optimized control input, and generate the simulated system state at the next timestep. We continue to iterate this process until the end of the simulation horizon.

Closing the Loop for Reference Tracking

With the ROM-MPC architecture, we performed three increasingly realistic tests for the simulation platform to track reference trajectories that are (i) simulated, (e.g., guaranteed to be dynamically feasible), (ii) generated from a biological model of eel kinematics, and (iii) generated by a reduced-scale physical robot. Figure 4 illustrates results for case (i) by comparing ERA, DMDc and LOpInf ROMs. More detailed results are available in the literature [1].

<strong>Figure 4.</strong>Tracking the performance of the reduced-order model-predictive control scheme for three reduced-order model (ROM) types, dimensions, quantities of training data, and objective function tuning schemes. The average tracking error for eigensystem realization algorithm (ERA)-based controllers is shown in red, the error for dynamic mode decomposition with control (DMDc)-based controllers is in blue, and the error for Lagrangian operator inference (LOpInf)-based controllers is in green. We generated these controllers through an extensive tuning process that considered the effects of <strong>(4a)</strong> reduced-order dimension \(r\) for ROMs that were trained on one trial of data and <strong>(4b)</strong> the quantity of training data for generated ROMs of dimension 18 on tracking performance. <strong>4c.</strong> Our tuning process also explored the effects of varying coefficients in the model-predictive control objective function when applying equal weighting to tracking penalties across the robot body, as well as weights that are defined by a posterior-focused scheme. <strong>4d.</strong> Relative tracking error of the best observed controllers based on each ROM generation method and computed over the robot’s body length. We computed tracking error over known feasible trajectories from the trials that we used to train each ROM (top panel) and a set of eight test trials from our dataset (middle panel). Dotted lines indicate the average tracking error for every best-case controller over each set of trials, while the surrounding shaded regions represent one standard deviation of tracking error at a given point. Output trajectories from each controller are shown at control points at the robot’s head (lower panel, left), midpoint (lower panel, middle), and tail (lower panel, right) — all for an example reference trajectory (in black) from trial 33 of our dataset. The full reference centerline (in black) of the simulated robot is shown for the same trial (bottom) and overlaid with centerlines that were produced by each best-case controller in their respective colors (deflections in the \(z\)-direction are amplified by a factor of five for sake of visualization). Figure courtesy of [1].
Figure 4.Tracking the performance of the reduced-order model-predictive control scheme for three reduced-order model (ROM) types, dimensions, quantities of training data, and objective function tuning schemes. The average tracking error for eigensystem realization algorithm (ERA)-based controllers is shown in red, the error for dynamic mode decomposition with control (DMDc)-based controllers is in blue, and the error for Lagrangian operator inference (LOpInf)-based controllers is in green. We generated these controllers through an extensive tuning process that considered the effects of (4a) reduced-order dimension \(r\) for ROMs that were trained on one trial of data and (4b) the quantity of training data for generated ROMs of dimension 18 on tracking performance. 4c. Our tuning process also explored the effects of varying coefficients in the model-predictive control objective function when applying equal weighting to tracking penalties across the robot body, as well as weights that are defined by a posterior-focused scheme. 4d. Relative tracking error of the best observed controllers based on each ROM generation method and computed over the robot’s body length. We computed tracking error over known feasible trajectories from the trials that we used to train each ROM (top panel) and a set of eight test trials from our dataset (middle panel). Dotted lines indicate the average tracking error for every best-case controller over each set of trials, while the surrounding shaded regions represent one standard deviation of tracking error at a given point. Output trajectories from each controller are shown at control points at the robot’s head (lower panel, left), midpoint (lower panel, middle), and tail (lower panel, right) — all for an example reference trajectory (in black) from trial 33 of our dataset. The full reference centerline (in black) of the simulated robot is shown for the same trial (bottom) and overlaid with centerlines that were produced by each best-case controller in their respective colors (deflections in the \(z\)-direction are amplified by a factor of five for sake of visualization). Figure courtesy of [1].

An Exciting Field for High-dimensional Control

Our work highlights the strong potential of ROMs for the control of soft robots. The use of these models is appealing, as they can leverage much of the experimental robot’s complex 3D geometry and dynamics. Of course, other strategies do exist, such as starting with 1D PDEs for the centerline (e.g., Cosserat and slender beam theories) or empirically deriving ordinary differential equations for certain mode shapes to avoid high-dimensional discretization altogether.

This is by no means the end of the story. In the future, we plan to further reduce the discrepancies between the nonlinear 3D SOFA model and the experimental robot. We see the highest benefit in the development of accurate, full-order 3D finite element models that incorporate fluid-structure interaction and account for various nonlinear effects, such as geometric nonlinearity and nonlinear material behaviors.

Additionally, while we investigated bioinspired and experimentally generated trajectories, it is not clear whether these trajectories are dynamically feasible for the SOFA simulation. As such (assuming better full-order models), future work could leverage various extensions of the LOpInf method that account for nonlinear dynamics in the full-order model while preserving the physics-based structure of the system [8]. The use of these methods in an online dynamic shape control of feasible—or approximately feasible—reference trajectories would then require nonlinear, model-based control techniques that appropriately incorporate the resulting ROMs. To stimulate further research in this area, we made the SOFA simulation data that we used for training available online.  

Boris Kramer delivered a minisymposium presentation on this research at the 2025 SIAM Conference on Control and Its Applications, which was co-located with The Third Joint SIAM/CAIMS Annual Meetings and took place last year in Montréal, Québec, Canada.

References    
[1] Adibnazari, I., Sharma, H., Park, M., Cervera-Torralba, J., Kramer, B., & Tolley, M.T. (2025). Dynamic shape control of soft robots enabled by data-driven model reduction. Preprint, arxiv:2511.03931.
[2] Cervera-Torralba, J., Kang, Y., Khan, E.M., Adibnazari, I., & Tolley, M.T. (2024). Lost-core injection molding of fluidic elastomer actuators for the fabrication of a modular eel-Inspired soft robot. In 2024 IEEE 7th international conference on soft robotics (RoboSoft) (pp. 971-976). San Diego, CA: Institute of Electrical and Electronics Engineers.
[3] Coevoet, E., Morales-Bieze, T., Largilliere, F., Zhang, Z., Thieffry, M., Sanz-Lopez, M., … Duriez, C. (2017). Software toolkit for modeling, simulation, and control of soft robots. Adv. Robot., 31(22), 1208-1224.
[4] Della Santina, C., Duriez, C., & Rus, D. (2023). Model-based control of soft robots: A survey of the state of the art and open challenges. IEEE Contr. Syst., 43(3), 30-65.
[5] Park, M., Cervera-Torralba, J., Adibnazari, I., Pawlak, G., & Tolley, M.T. (2025). Analysis on kinematics and propulsion of a self-sensing multi-DoF undulating robotic fish. In 2025 IEEE international conference on robotics and automation (ICRA) (pp. 14692). Atlanta, GA: Institute of Electrical and Electronics Engineers.
[6] Rus, D., & Tolley, M.T. (2015). Design, fabrication and control of soft robots. Nature, 521(7553), 467-475. 
[7] Sharma, H., & Kramer, B. (2024). Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems. Phys. D: Nonlinear Phenom., 462, 134128. 
[8] Sharma, H., Najera-Flores, D.A., Todd, M.D., & Kramer, B. (2024). Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems. Comput. Method. Appl. Mech. Eng., 423, 116865.
[9] Tytell, E.D., & Lauder, G.V. (2004). The hydrodynamics of eel swimming: I. Wake structure. J. Exp. Biol., 207(Pt. 11),1825-1841.

About the Authors

Boris Kramer

Associate professor, University of California San Diego

Boris Kramer is an associate professor of mechanical and aerospace engineering at the University of California San Diego. He develops reduced-order models to enable the control, design, optimization, and uncertainty quantification of complex and high-dimensional systems.

Michael T. Tolley

Professor, University of California San Diego

Michael T. Tolley is a professor of mechanical and aerospace engineering and director of the Bioinspired Robotics and Design Lab at the University of California San Diego. He seeks inspiration from nature to design robotic systems with the versatility, resilience, and efficiency of biological organisms.