💻 computer science

AquaROM: shape optimization pipeline for soft swimmers using parametric reduced order models

This paper presents AquaROM, a novel optimization pipeline that utilizes a tensorial parametric reduced order model with analytical gradients to efficiently optimize the shapes of soft robotic swimmers subject to complex nonlinear hydrodynamic forces, significantly reducing computational costs compared to traditional finite element methods.

Original authors: Mathieu Dubied, Paolo Tiso, Robert K. Katzschmann

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mathieu Dubied, Paolo Tiso, Robert K. Katzschmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to design the perfect swimming fish out of soft, squishy clay. You want it to be fast, efficient, and able to wiggle through the water like a real fish.

In the past, designing something like this was like trying to find a needle in a haystack while wearing heavy winter gloves. Here is why:

The Clay is Complicated: Soft robots bend and twist in crazy ways. To simulate this on a computer, engineers have to break the robot down into thousands of tiny Lego blocks (called Finite Elements).
The Water is Tricky: The water pushes back, creating drag and helping the fish move (thrust). Calculating how water interacts with a squishy, twisting shape is incredibly hard math.
The Computer Gets Tired: If you try to run a simulation to test one shape, it takes a long time. If you want to find the best shape, you have to test thousands of variations. Doing this with the "heavy Lego" method would take your computer weeks or months to finish. It's like trying to paint a masterpiece by moving every single pixel one by one.

The Solution: AquaROM (The "Smart Sketch" Method)

The authors of this paper, Dubied, Tiso, and Katzschmann, created a new tool called AquaROM. Think of it as a "Smart Sketch" that lets you design the perfect fish in minutes instead of months.

Here is how they did it, using some simple analogies:

1. The "Reduced Order Model" (The Shadow Puppet)

Instead of simulating every single tiny Lego block of the fish, the team created a "shadow puppet" version.

The Old Way: Simulating the whole fish is like trying to calculate the movement of every single muscle fiber in a real human body.
The New Way (ROM): They figured out that the fish mostly moves in a few specific, predictable ways (like bending its tail up and down). They built a simplified model that only tracks these main movements. It's like watching a shadow puppet show; you don't need to see the puppeteer's fingers to understand the story. The shadow moves fast, but it still looks like the fish.

2. The "Parametric" Twist (The Shape-Shifter)

Usually, if you wanted to change the shape of your shadow puppet (make the tail longer or the head wider), you'd have to rebuild the whole thing from scratch.

The Innovation: The authors made their "shadow puppet" parametric. This means they built a "master puppet" that can stretch, shrink, and morph instantly based on a few control knobs (parameters).
The Analogy: Imagine a clay model on a potter's wheel. Instead of smashing the clay and starting over to make a new shape, you just turn a few dials, and the clay instantly morphs into a new shape. The math behind this is complex, but the result is that the computer can "morph" the fish shape in a split second.

3. The "Tensorial" Secret Sauce (The Recipe Book)

The hardest part of soft robotics is that the forces (pushing against water, the rubber stretching) are non-linear. They don't behave in a straight line; they get messy and complicated.

The Problem: Usually, when things get messy, you can't easily calculate how to fix them. It's like trying to bake a cake without knowing how much sugar changes the taste.
The Solution: The team used a "Tensorial" approach. Think of this as writing a perfect recipe book. Instead of just guessing, they wrote down the exact mathematical "recipe" for how the water pushes and the rubber stretches. Because they wrote it down in this specific format, the computer can instantly calculate the "gradient" (the direction to tweak the shape to make it better).
The Result: It's like having a GPS that doesn't just tell you where you are, but instantly tells you exactly which way to turn to get to the destination faster.

The Experiment: Designing the Ultimate Fish

To test their method, they asked the computer: "What is the best shape for a soft robotic fish to swim the furthest in 2 seconds?"

Starting Point: They began with a boring, rectangular block of soft material (like a brick of jelly).
The Process: The computer used their "Smart Sketch" to try thousands of shapes. It morphed the jelly brick, testing how fast it swam, and used the "gradient GPS" to nudge the shape slightly better each time.
The Outcome:
- In just 5 to 12 minutes, the computer designed a fish that was 8 times faster than the original brick.
- The new fish looked like a real fish: it had a sleek body, a distinct head, and a long, flexible tail.
- They even tested it with different numbers of "control knobs" (3, 5, or 8 parameters) and found that even with more complex shapes, the computer solved it incredibly fast.

Why This Matters

This paper is a big deal because it bridges the gap between theory and real-world design.

Before: Engineers had to build a physical robot, test it, break it, build a new one, and repeat. It was slow and expensive.
Now: With AquaROM, engineers can simulate, optimize, and find the perfect design on a laptop in minutes. They can then build only the best version.

In a nutshell: The authors built a super-fast, mathematically smart "virtual clay" that can instantly morph into the perfect swimming shape, saving engineers from weeks of trial and error. It turns the slow, painful process of designing soft robots into a quick, efficient dance.

1. Problem Statement

The design and optimization of soft robotic systems, particularly those operating in complex fluid environments (e.g., soft robotic fish), face significant computational hurdles.

High Computational Cost: Accurate simulation of soft robots requires Finite Element Method (FEM) to capture large deformations and nonlinear material behaviors. However, FEM models (Full Order Models or FOMs) involve a massive number of degrees of freedom (DoFs), making them computationally expensive.
Nonlinear Forces: Soft swimmers are subject to complex nonlinear forces, including internal elastic forces (due to large deformations) and external hydrodynamic forces (thrust and drag). These nonlinearities further increase simulation costs and complicate gradient-based optimization.
Optimization Bottleneck: Traditional optimization approaches often rely on full simulations or data-driven models that require extensive offline training. This prevents rapid prototyping and efficient design space exploration, especially when analytical gradients are needed for gradient-based optimization.

2. Methodology

The authors propose AquaROM, a novel optimization pipeline that integrates a Tensorial Parametric Reduced Order Model (PROM) with a gradient-based optimization algorithm.

A. Mathematical Formulation

Full Order Model (FOM): The system is modeled using nonlinear FEM. The equations of motion (EoMs) include mass, damping, internal nonlinear forces, and external forces (thrust, drag, actuation).
Parametric Shape Variations: The geometry of the robot is defined by a nominal mesh and a set of shape variations. These variations are linearly combined using a parameter vector $\xi$ to define the search space for optimization.
Tensorial PROM Construction:
- Basis Construction: The Reduced Order Basis (ROB) is constructed using a data-free approach, combining Vibration Modes (VMs), Modal Derivatives (MDs), and Parameter Sensitivities (PSs). This avoids the need for pre-simulated datasets.
- Polynomial Approximation: Internal elastic forces are expressed as high-order tensors using a Neumann expansion of the deformation gradient.
- Hydrodynamic Reduction: The authors extend this tensorial approach to external forces. They model thrust using Lighthill's large-amplitude elongated-body theory (LAEBT) and drag using a shape-dependent coefficient. Crucially, the drag force (originally non-polynomial) is approximated via a Taylor series expansion to fit the polynomial structure required for the PROM.
- Galerkin Projection: All forces (internal, thrust, drag, actuation) are projected onto the ROB to create a low-dimensional system of equations ( $m \ll n$ ) that retains the analytical structure of the original forces.

B. Optimization Pipeline

Objective: Maximize the swimming distance of the soft fish over a fixed time horizon.
Gradient-Based Search: Because the PROM expresses forces as structured polynomials, analytical gradients of the cost function with respect to the shape parameters ( $\xi$ ) can be computed efficiently.
Adaptive Strategy: The pipeline uses a sensitivity-based approximation to update the solution between optimization steps. If the shape parameters deviate too far from the current nominal shape, the PROM is rebuilt with the new shape as the reference to maintain accuracy.
Multi-Objective Extension: The framework supports co-optimization of shape and actuation signals (e.g., minimizing energy while maximizing distance) by treating actuation amplitude as an additional parameter.

3. Key Contributions

Data-Free Tensorial PROM for Nonlinear Systems: The paper develops a method to embed both internal nonlinearities and complex external hydrodynamic forces (specifically Lighthill's theory) into a parametric reduced order model without requiring offline training data.
Analytical Gradient Computation: By maintaining a tensorial structure, the method enables the calculation of exact analytical gradients for the optimization problem, significantly accelerating convergence compared to finite-difference or black-box methods.
Generalized Hydrodynamic Reduction: The authors provide a blueprint for reducing diverse nonlinear forces (thrust and drag) into a PROM, extending previous work that focused primarily on internal structural forces.
Efficient Optimization Pipeline: A complete workflow is demonstrated that allows for rapid shape optimization of soft swimmers, handling constraints and multi-objective trade-offs efficiently.

4. Results

The method was validated through numerical experiments on a soft robotic fish with a rigid head and a deformable tail equipped with muscle actuators.

Accuracy vs. Efficiency:
- Accuracy: The PROM results showed high fidelity compared to the FOM, with relative errors in swimming distance typically between -10% and +15%. The error was minimal for the specific operating regime tested.
- Speed: The PROM reduced simulation time by approximately one order of magnitude compared to the FOM. While building the PROM takes time, the subsequent simulation and gradient evaluation are extremely fast.
Shape Optimization Performance:
- The pipeline successfully optimized fish shapes with 3, 5, and 8 design parameters.
- Performance Gain: The optimized shapes achieved swimming distances 3.8x to 8x greater than the nominal (unoptimized) rectangular block shape.
- Convergence Time: Optimizations converged in minutes (e.g., ~5.7 minutes for 3 parameters, ~12.6 minutes for 8 parameters), whereas a grid search on a standard ROM would take hours or days due to the curse of dimensionality.
Multi-Objective Co-Optimization: The system successfully generated a Pareto front balancing swimming distance against energy consumption, demonstrating the ability to handle competing objectives.

5. Significance and Future Work

Impact: This work bridges the gap between high-fidelity physics-based simulation and practical design optimization. It enables engineers to rapidly iterate on soft robot morphologies without the prohibitive cost of full FEM simulations or the data requirements of deep learning surrogates.
Limitations:
- The hydrodynamic model relies on Lighthill's theory (inviscid flow), which is a simplification of real fluid-structure interaction.
- The constitutive model used (Saint Venant-Kirchhoff) is valid for small-to-moderate strains; very large strains might require more advanced models (e.g., Neo-Hookean).
- The energy cost function for actuation was simplified and did not fully account for shape-dependent energy changes during optimization.
Future Directions:
- Validating the optimal shapes against physical robotic prototypes to assess sim-to-real gaps.
- Extending the reduction method to more complex constitutive models and advanced hydrodynamic solvers (e.g., CFD).
- Applying the pipeline to more complex control and design scenarios beyond simple forward swimming.

In conclusion, AquaROM demonstrates that tensorial parametric reduced order models are a powerful tool for the efficient, gradient-based design of complex, nonlinear soft robotic systems, unlocking new possibilities for rapid, physics-informed robotic design.