💻 computer science

Modeling and Control of an Eel-Inspired Soft Robot for Design Optimization

This paper presents a Finite Element Method-based simulation model coupled with hydrodynamic forces to optimize the design and control of an eel-inspired soft robot, revealing that slightly asymmetric configurations enhance maneuverability while maintaining swimming velocity.

Original authors: Zhangjingyi Jiang, Mark Campbell

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhangjingyi Jiang, Mark Campbell

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

Deep in the ocean, where light fades and pressure mounts, nature has perfected a way of moving that machines struggle to copy. For centuries, engineers have looked to the eel, an elongated fish that swims by rippling its entire body in a wave-like motion. This style of movement, known as undulatory swimming, is remarkably efficient. While a trout might burn through energy quickly to maintain a steady pace, an eel can travel the same distance using a fraction of the fuel. This efficiency makes the eel an ideal blueprint for building autonomous underwater vehicles designed to explore the deep, inspect underwater infrastructure, or gather data beneath the ice. However, building a robot that mimics this soft, flexible motion is difficult. Real eels are made of muscle and bone that bend and twist in complex ways, whereas traditional robots are often built from rigid metal and plastic. To create a machine that truly behaves like a living creature, researchers must first understand how soft materials interact with water, how to control them without getting tangled, and how to ensure they keep working even if parts of them break or wear out.

At the Cornell University Department of Mechanical and Aerospace Engineering, researchers Zhangjingyi Jiang and Mark Campbell have developed a powerful new tool to solve these problems. Instead of building a physical robot immediately, they created a detailed computer simulation of an eel-inspired soft robot. This digital model acts as a virtual testing ground where they can tweak the shape, material, and control systems of a robot fish before a single piece of hardware is manufactured. The simulation treats the robot's body as a long, flexible rod made of many small segments, similar to how a chain is made of links. By using a method that calculates how these segments bend and stretch under pressure, the model captures the behavior of soft materials that change over time. This approach allows the team to see how the robot would move through water, how it would react to currents, and how it would perform if a motor failed or the material began to degrade.

The researchers used this simulation to test how to make the robot swim straight, turn, and stop. They found that the most effective way to control the robot was to send a traveling wave of twisting force down its body, much like the ripple that moves through a snake. By adjusting the strength and timing of these twists, they could make the robot accelerate from a standstill, cruise at a steady speed, or brake to a halt. The simulation showed that the robot could also turn by adding a slight offset to the twisting forces, allowing it to make wide, gentle curves or sharp, tight turns. Crucially, the model allowed the team to test how the robot would behave if it were damaged. In the real world, an underwater robot might get snagged by a rock, have a motor fail, or suffer from material wear over a long mission. The simulation revealed that if the motors near the front of the robot lost power, the robot's ability to swim forward would suffer significantly. However, if the motors near the tail failed, the robot would struggle more with turning. This insight helps engineers design systems that can keep working even when parts of them are compromised, ensuring the robot can still return to base for repairs.

One of the most surprising discoveries from the study concerned the shape of the robot. For a long time, engineers might have assumed that a perfectly symmetrical robot, with a head and tail of equal size, would be the most balanced and efficient. The simulation, however, told a different story. The researchers tested designs where the head was slightly larger than the tail and found that these slightly asymmetrical shapes performed just as well as the symmetrical ones when swimming forward. In fact, the asymmetrical designs were much better at turning. They could make tighter turns with greater ease, a vital skill for navigating around obstacles in a complex underwater environment. The study suggests that for a robot intended to explore the ocean, a slightly uneven shape is not a flaw but a feature, offering a better balance between speed and maneuverability.

The value of this work lies in its ability to guide the future of robotic design. By running thousands of simulations in a fraction of the time it would take to build and test physical prototypes, the researchers can optimize every aspect of the robot's design. They can determine the best materials to use, the ideal number of motors, and the most robust shape for specific tasks. This digital framework does more than just predict how a robot will move; it provides a way to anticipate failure and design for resilience. As the technology moves from the computer screen to the ocean floor, these insights will be essential for creating machines that can survive the harsh conditions of the deep sea, continuing their missions even when faced with damage or the slow wear of time. The result is a clearer path toward building autonomous explorers that are not only efficient but also tough enough to handle the unknown challenges of the underwater world.

Technical Summary: Modeling and Control of an Eel-Inspired Soft Robot for Design Optimization

Problem Statement

Anguilliform (eel-like) locomotion is recognized as a highly efficient swimming mechanism, requiring significantly less energy than other aquatic propulsion methods. However, existing eel-inspired robotic prototypes often prioritize basic motion demonstration over design optimization, efficiency enhancement, or advanced goal-seeking capabilities. Furthermore, autonomous underwater vehicles (AUVs) operating in harsh, unexplored environments face unique challenges, including communication difficulties and the risk of physical damage or material degradation. Retrieving damaged robots from such environments is often impossible or prohibitively expensive. Consequently, there is a need for a simulation framework that can optimize design parameters for robustness and efficiency before physical prototyping, and support the development of model-based estimation and control systems capable of handling actuator failure or material deterioration.

Methodology

The authors present a simulation model designed to capture the physics of an eel-inspired soft robot and its interaction with fluid dynamics. The methodology integrates two primary components:

Finite Element Method (FEM) Model:
- The soft robot is modeled as a free-floating elastic rod divided into $n_e$ elements.
- The system utilizes a 2D FEM approach where each node possesses three degrees of freedom ( $x, y, \theta$ ).
- The global dynamics are governed by the equation $M \ddot{q}(t) + Kq(t) = Q(t)$ , where $M$ and $K$ are the global mass and stiffness matrices derived from individual beam element properties (density, Young's modulus, cross-sectional area).
- A key feature of this model is its parameterization; each element's properties can be independently defined. This allows the simulation to model spatial variations in material properties along the robot's body and temporal changes due to degradation or failure.
- The robot is actuated by $n_\tau$ torque inputs, where each actuator controls a collection of nodes.
Hydrodynamic Force Model:
- A simplified hydrodynamic model simulates external fluid forces.
- Drag forces are assumed to act perpendicular to the robot's surface, neglecting shear forces at high Reynolds numbers.
- The model calculates drag on the side of each finite element and a separate drag force on the robot's head based on effective surface area and velocity vectors.
- The total drag force is the sum of head and side components.
Control Strategy:
- The control approach utilizes an open-loop traveling sinusoidal wave with bias as the torque input, mimicking natural undulatory motion.
- Linear Velocity Control: Achieved by modulating the amplitude ( $A_j$ ) and frequency ( $\omega$ ) of the sinusoidal wave across joints.
- Angular Velocity Control: Achieved by adding a torque offset ( $\tau_O$ ) to the linear control signal.
- Robustness Testing: The model simulates actuator degradation by reducing the functionality percentage of specific torque inputs (0% to 100%) to observe impacts on velocity and turning radius.

Key Contributions

A Parametric Simulation Framework: The paper introduces a coupled FEM and hydrodynamic model specifically designed for soft, elongated robots. Unlike rigid-body models, this framework captures the continuous deformation of soft materials and allows for the independent parameterization of each element.
Design Optimization Tool: The model serves as a platform to optimize design parameters (e.g., head-to-tail cross-sectional area ratios) and control inputs (amplitude, frequency, phase, offset) to meet specific system goals such as cruising speed and maneuverability.
Robustness Analysis: The framework enables the evaluation of system resilience against actuator degradation and failure, as well as the impact of material property changes over time.
Asymmetric Design Insights: The study investigates the performance trade-offs between symmetric and slightly asymmetric robot geometries.

Results

The simulation results, executed on a standard processor with a time-step rate exceeding 3300 Hz, demonstrate the following:

Control Effectiveness: The proposed control laws successfully enable the robot to accelerate from rest, cruise at steady speeds, perform soft and hard turns, and decelerate to a stop. Adjusting control constants ( $C_A, C_\omega, C_{O\tau}$ ) allows for intuitive modulation of velocity and turning radius.
Actuator Degradation Impact:
- Degradation of actuators in the front of the fish (specifically $\tau_2$ ) has the most significant negative impact on linear swimming velocity.
- Degradation of actuators toward the rear (specifically $\tau_3$ and $\tau_4$ ) has a greater impact on the turning radius.
Symmetric vs. Asymmetric Designs:
- Slightly asymmetric designs (e.g., a head-to-tail cross-sectional area ratio of 1.25/0.75) demonstrated comparable forward swimming velocities to symmetric designs.
- However, asymmetric designs offered significantly lower forward turning radii, indicating superior maneuverability.
- Asymmetric designs showed higher reverse velocities at lower control amplitudes ( $C_A = 0.5$ ) compared to symmetric designs.
- When the head-to-tail ratio dropped below 1.3/0.7, reverse turning radii increased significantly, suggesting an optimal range for asymmetry.

Significance and Claims

The paper claims that the proposed model provides a comprehensive framework for the design and development of eel-inspired soft robots. Its primary significance lies in its ability to:

Guide Design Decisions: By simulating various configurations, the model helps identify robust system configurations that can withstand material degradation or actuator failure, which is critical for long-duration missions in inaccessible environments.
Optimize Performance: The model allows for the fine-tuning of geometric parameters (such as slight asymmetry) to achieve a balance between cruising efficiency and high maneuverability without the need for immediate physical prototyping.
Support Advanced Control: The simulation environment facilitates the development of model-based estimation, reasoning, and control systems, including fault detection and recovery strategies.

The authors emphasize that the model's modular nature and high simulation speed make it suitable for rapid iteration, enabling researchers to assess resilience under diverse conditions and optimize performance for specific tasks such as undersea infrastructure maintenance, deep-sea exploration, and under-ice data collection.