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

Zhangjingyi Jiang; Mark Campbell

cs.ROAug 14, 2026

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

Authors: Zhangjingyi Jiang, Mark Campbell

Organizations: Department of Mechanical and Aerospace Engineering, Cornell University

Abstract

Anguilliform locomotion is a highly efficient swimming mode; the advent of new materials for soft robots enables the development of an eel-inspired soft robot. This paper presents a simulation model of an eel-inspired soft robot designed for anguilliform swimming. This model can aid in design optimization and the development of model-based estimation, reasoning, and control systems. A Finite Element Method (FEM) model of an elastic rod is used to capture the soft materials of the robotic fish, which makes it particularly amenable to variation over time as the material properties change. The material model is coupled with a hydrodynamic force model to simulate the behavior of a soft, elongated robot in water. The model is used to demonstrate the effectiveness of the proposed control approaches in achieving desired swimming behaviors. It also provides insights into design decisions, including the robustness of different system configurations and the impact of material degradation and failure. The results show that slightly asymmetric designs are advantageous, offering comparable swimming velocities but greater maneuverability. This model can be used to guide future robotic design decisions aimed at optimizing performance for specific tasks.

Figures & tables

Fig. 1: Physical fish robot from UCSD Bioinspired Robotics and Design Lab [ 10 ]

Fig. 2: FEM model of fish robot with $n_{\tau}$ torque inputs.

Variables	Values
Number of Elements $n_{e}$	$20$
Number of Torque Inputs (Actuators) $n_{\tau}$	$5$
Material Density $\rho_{r}\ (kg\cdot m^{-3})$	$1200$
Young’s Modulus of Material $E\ (N\cdot m^{-2})$	$1\ x\ 10^{6}$
Rod Length $L\ (m)$	$1$
Rod Cross-Sectional Area $A_{C}\ (m^{2})$	$7.85\ x\ 10^{-3}$

TABLE I: FEM Model Parameters

Variables	Values
Drag Coefficient $C_{D}$	$0.167$ [ 6 ]
Fluid Density $\rho_{f}\ (kg/m^{3})$	$10^{3}$

TABLE II: Hydrodynamic Force Parameters

*all actuators are oscillating at the same frequency
Variables	Values
Amplitude Ratio $A_{1:n\tau}$	$[20\ 20\ 16\ 4\ 4]$
Oscillation Frequency $\omega$ * $\ (Hz)$	$1.5$
Phase Shift $\phi\ (rads)$	$2\pi/n_{\tau}\approx 1.2566$
Joint Index $j_{1:n\tau}$	$[1\ 2\ 3\ 4\ 5]$
Torque Offset Ratio $\tau^{O}_{1:n\tau}$	$[3.5\ 5.5\ 6.5\ 6.5\ 3.5]$

TABLE III: Control Parameters

Swimming	$C_{A}$	$C_{\omega}$	$C^{O}_{\tau}$	Time $\Delta t$ (s)
Swimming Straight	$1$	$1$	$0$	$7$
Slight Left Turn	$1$	$1$	$-0.2$	$7$
Hard Right Turn	$0.75$	$1$	$1$	$7$

TABLE IV: Control Parameters - Demo 1

Fig. 3: Demonstration of the robot swimming straight (blue), slight turning left (green), and hard turning right (red) for $t=15s$ each

*When breaking, $\phi$ becomes negative and $A_{1:n\tau}$ is flipped
Swimming	$C_{A}$	$C_{\omega}$	$C^{O}_{\tau}$	Time $\Delta t$ (s)
Accelerate	$1$	$1$	$0$	$12$
Cruise	$0$	$0$	$0$	$4$
Break*	$1$	$1$	$0$	$3.55$
Stopped	$0$	$0$	$0$	$10.45$

TABLE V: Control Parameters Demo 2

Fig. 4: Demonstration of the robot accelerating (green), cruising (blue), breaking (red), and stopped (cyan) for $12s,4s,4s,10s$ respectively

Fig. 5: Steady swimming linear velocities and turning radii when torque inputs are operating below maximum

$A_{H}$ ( $m^{2}$ )	$A_{T}$ ( $m^{2}$ )	$A_{H}/A_{T}$	$SA$ ( $m^{2}$ )	V ( $10^{3}m^{3}$ )
0.30 $\pi$	0.10 $\pi$	1.5/0.5	0.31	8.5
0.29 $\pi$	0.11 $\pi$	1.45/0.45	0.31	8.4
0.28 $\pi$	0.12 $\pi$	1.4/0.6	0.31	8.3
0.27 $\pi$	0.13 $\pi$	1.35/0.65	0.31	8.2
0.26 $\pi$	0.14 $\pi$	1.3/0.7	0.31	8.1
0.25 $\pi$	0.15 $\pi$	1.25/0.75	0.31	8.0

TABLE VI: Design Parameters

Fig. 6: A comparison of steady swimming linear velocities for symmetric and asymmetric robot designs at $C_{A}=1$ , $C_{A}=0.75$ , and $C_{A}=0.5$ (left column). A comparison of turning radii for symmetric and asymmetric robot designs at $C^{O}_{\tau}=1$ , $C^{O}_{\tau}=0.75$ , and $C^{O}_{\tau}=0.5$ (right column).

Fig. 7: Linear velocities and turning radii for design 1 (rod with length l = 1m rod with $A_{H}=0.30\pi m^{2}$ and $A_{T}=0.10\pi m^{2}$ - blue), design 2 (l = 1m, $A_{H}=0.25\pi m^{2}$ and $A_{T}=0.15\pi m^{2}$ - green), and design 3 (l = 1m, $A_{H}=0.20\pi m^{2}$ and $A_{T}=0.20\pi m^{2}$ - red) for forward (solid line) and reverse (dotted line) steady swimming.

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In this paper, we introduce the Soft Lamprey-Inspired Dual Environment Robot (SLIDER) and a proper modeling and optimization procedure employed to design the robot. We represent the primary fluid environment actions - inertial effects, vortex forces, and viscous dissipation - using Lighthill's theory for large-amplitude elongated bodies. For structural design parameters such as internal pressure, tail size, and body stiffness, a fast, geometrically and materially nonlinear model is developed and validated. The fluid-structure interaction equations are solved implicitly with an efficient second-order box method. A pneumatic manifold robotic system is employed to actuate SLIDER in a quiescent water tank environment, allowing cross-comparison of computational and experimental results. We find that low-frequency swimming is dominated by resistant environmental forces, whereas higher-frequency swimming is primarily affected by inertial fluid forces. Using our efficient model alongside a genetic algorithm, we co-optimize a swimming control pattern and caudal fin design (subject to SLIDER's climbing morphology) to achieve a tethered swimming speed of 21.7 +/- 0.4 cm/s (0.59 Bl/s). Furthermore, we investigate the optimization procedure for a multimodal robot performing both swimming and climbing tasks.

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