Biological Modeling of Bionic Fish
1. Multi-Scale Anatomical Modeling of Biological Fish Body
1.1 Macroscopic Topological Anatomical Modeling of Teleost Fish
A typical bony fish consists of three functional units: axial locomotion unit, appendage regulation unit and physiological regulation unit.
Rigid cranial module The skull encapsulates the brain, vestibular balance system, visual organs, olfactory sacs and the front-end neural structure of the lateral line system. Physiologically constrained by the vestibulo-ocular reflex, the lateral swing amplitude of the fish head is less than 3% of body length during swimming to guarantee spatial positioning and visual stability, which serves as the boundary constraint for attitude control of bionic fish.
Axial myomere-driven trunk The body is lined with W-shaped myomeres on both sides. Red muscles sustain low-speed cruising with aerobic metabolism, while white muscles support burst high-speed locomotion via anaerobic metabolism. Alternating rhythmic contraction of bilateral myomeres generates a backward-propagating bending traveling wave along the body, featuring three core nonlinear biological characteristics: phase delay, dynamically adjustable muscle stiffness and neural activation time lag. These properties cannot be precisely reproduced by conventional geometric sinusoidal traveling wave models.
Caudal peduncle and caudal fin propulsion unit The caudal fin is a flexible thin membrane supported by primary and secondary fin rays. Under hydrodynamic loading, it undergoes coupled pitch and deflection deformation to form a variable angle of attack, producing three typical vortex structures: leading-edge vortex, reverse Kármán vortex street and tip vortex, which account for the high propulsive efficiency of fish.
Paired and stabilizing fins
Pectoral fins: 6-DOF flexible appendages for propulsion, depth adjustment, lateral translation, braking and flow control;
Dorsal and anal fins: passive roll-damping structures to suppress rolling instability during locomotion.
Buoyancy regulation unit (swim bladder) Fish are classified into physostomous and physoclistous types according to swim bladder structures. The former adjusts buoyancy by gulping or expelling gas; the latter relies on gas glands and oval windows for precise slow buoyancy tuning. Modeling couples hydrostatic pressure, ideal gas thermodynamic equations and dynamic force balance.
1.2 Mesoscopic Mechanical Modeling of Biological Soft Tissues
1.2.1 Hill Three-Element Skeletal Muscle Model
Each muscle unit includes a contractile element, a parallel elastic element and a series elastic element. The muscle output tension is expressed as: Fmuscle=a⋅Fmax⋅fl(l)⋅fv(v)+Fpass(l) Where:
a∈[0,1]: neural activation signal from central pattern generator;
fl(l): length-dependent tension function;
fv(v): contraction velocity-dependent force attenuation term;
Fpass(l): passive elastic tension of fascia and connective tissues, described by exponential constitutive relation.
Symmetric bilateral muscle pairs drive joint rotation, and adjacent myomere activation signals carry fixed phase lag to reproduce the biological propagation of body traveling waves.
1.2.2 Hyperelastic Constitutive Model for Trunk Soft Tissue
Fish skin and connective tissues adopt Mooney-Rivlin or Ogden hyperelastic models to characterize large-deformation nonlinear stress-strain behavior, which is essential for fluid-structure interaction (FSI) simulation of soft bionic fish to quantify local trunk deformation induced by muscle contraction.
1.3 Microscopic Modeling of Lateral Line Sensing System
The fish lateral line contains superficial neuromasts and canal neuromasts:
Superficial neuromasts exposed on the epidermis detect low-frequency flow velocity and wall shear disturbance;
Canal neuromasts embedded in subcutaneous pipelines sense pressure gradient, vortex perturbation and flow distortion induced by obstacles.
In numerical modeling, distributed pressure-shear sensor arrays are deployed on the fish body surface. A transfer function is established to convert flow field pressure into neural electrical signals, incorporating sensory noise, temporal filtering and neural threshold characteristics to mimic biological turbulence filtering and hydrodynamic imaging capability.
2. Refined Kinematic Modeling of Axial Propulsion
2.1 Large-Amplitude Elongated Body Theory (LAEBT)
For high-speed swimming with large trunk deflection, the curvature distribution along the fish body is defined as: κ(s,t)=A(s)sin(ωt−λ2πs+φ) Where:
s∈[0,L]: arc-length coordinate along fish body;
A(s): position-dependent amplitude distribution (near-zero at head, maximum at caudal peduncle);
ω: angular frequency of myomere contraction;
λ: body wave wavelength (0.8–1.2 body lengths for most teleost cruising).
The continuous curvature field is discretized into multi-rigid articulated joint angles. Inertial lag phase compensation is introduced to eliminate the phase lead error between traditional sinusoidal kinematic models and real biological locomotion.
2.2 Classification and Biological Constraints of BCF & MPF Propulsion Modes
2.2.1 Body and Caudal Fin (BCF) Modes
Anguilliform: full-body traveling wave oscillation, suitable for low-speed high-maneuverability locomotion;
Carangiform: only the posterior half of the body bends, optimal for high-efficiency steady cruising with minimum frictional drag;
Thunniform: oscillation limited to caudal peduncle and caudal fin, driven by fast-twitch white muscles for burst high-speed swimming.
Each locomotion mode corresponds to unique amplitude distribution, frequency range and phase gradient constraints.
2.2.2 Median and Paired Fin (MPF) Modes
Pectoral fins are actuated by multiple independent flexible fin rays with chordwise traveling wave motion. Kinematics couples pitch, deflection and twist, combined with unsteady thin-airfoil theory to analyze leading-edge vortex adhesion, supporting hovering, lateral displacement and precise low-speed manipulation.
2.3 Biological Constraint Modeling of Maneuver Motions
Steady turning: asymmetric body wave modulation with attenuated amplitude on the inner side and amplified amplitude on the outer side, coordinated with differential pectoral fin angle of attack. The minimum turning radius is biologically constrained to approximately 0.3 body lengths;
Emergency braking: reverse-propagating body wave combined with large-angle symmetric pectoral fin deflection to generate reverse drag;
Pitch control: coupled differential deflection of pectoral fins and dynamic buoyancy adjustment of the swim bladder.
3. Multi-Field Coupled Hydrodynamic Modeling Based on Vortex Dynamics
3.1 Full Decomposition of Hydrodynamic Forces
The 6-DOF dynamic forces acting on bionic fish include:
Potential-flow added mass inertial force: induced by accelerated entrainment of surrounding fluid, expressed as a posture-dependent time-varying added mass matrix;
Viscous drag: skin frictional drag and pressure drag caused by flow separation;
Unsteady lift and thrust of fins: determined by instantaneous angle of attack under the Kutta condition with periodic vortex shedding;
Vortex-induced force: disturbance from self-shed vortices, ambient shear flow and obstacle-induced flow field distortion.
3.2 Vortex Dynamics Mechanism of Efficient Propulsion
Steady cruising: the caudal fin periodically sheds reverse Kármán vortex streets, forming downstream jet flow to generate forward thrust via momentum conservation;
Low-speed braking: forward Kármán vortex street is produced to achieve deceleration;
Passive flexible deformation of the caudal fin regulates vortex shedding frequency and circulation, enabling high propulsive efficiency over a wide range of swimming frequencies.
Large Eddy Simulation (LES) is adopted to resolve boundary layer evolution, vortex generation, fusion and dissipation, quantifying thrust coefficient, drag coefficient, slip ratio and propulsive efficiency under different oscillation parameters.
3.3 Bidirectional Fluid-Structure Interaction (FSI) Modeling
One-way FSI: pre-defined fish body kinematics to solve pressure load distribution for structural optimization and fin profile design;
Two-way FSI closed-loop coupling: muscle contraction drives body deformation → flow field pressure redistribution → hydrodynamic load inversely alters trunk curvature, which accurately reproduces the dynamic interaction between flexible biological body and ambient fluid.
3.4 Ambient Flow and Depth Coupled Modeling
The model incorporates shear incoming flow, turbulent boundary effects and depth-dependent hydrostatic pressure as well as swim bladder thermodynamic equations to establish a closed-loop system that couples depth, buoyancy, attitude and swimming kinematics.
4. Hierarchical Neurobiological Control Modeling
4.1 Refined Central Pattern Generator (CPG) Model
4.1.1 Matsuoka Mutually Inhibitory Neuron CPG Model
{τx˙i=−xi−βyi+∑wijyj+uτ′y˙i=−yi+max(xi,0)
Where xi denotes neuron membrane potential, yi represents normalized neural activation output directly transmitted to the Hill muscle model, wij is synaptic coupling weight, τ,τ′ are time constants, and u is the central drive signal. Reciprocal inhibitory synaptic connections between bilateral neurons realize alternating contraction of left-right trunk muscles, while segmental excitatory coupling introduces fixed phase lag for backward body wave generation.
High-fidelity modeling can further adopt Hodgkin–Huxley neuron models with sodium, potassium and leakage ion channels to characterize action potential firing, refractory period and neural transmission delay.
4.2 Three-Tier Biological Motion Regulation Architecture
High-level decision layer: implements behavioral rules including rheotaxis, obstacle avoidance and schooling control, modulating CPG frequency, amplitude and phase offset based on multi-sensor fusion data;
Mid-level vestibular balance layer: receives inertial attitude feedback to suppress roll and pitch instability via closed-loop correction, mimicking vertebrate vestibular reflex;
Low-level spinal CPG rhythm layer: generates robust periodic locomotion signals with smooth mode transition between cruising, acceleration, turning and braking.
4.3 Neural-Muscular Nonlinear Time-Delay and Fatigue Modeling
Neural conduction and muscle activation introduce tens of milliseconds of physiological time delay, which may cause body wave instability and efficiency degradation under high-frequency oscillation. First-order time-delay links, activation thresholds and fatigue attenuation terms are incorporated to describe maximum muscle tension decay under prolonged high-intensity contraction.
5. Energy Metabolism and Physiological Constraint Modeling
Aerobic metabolism model for red muscle low-speed cruising: energy consumption is positively correlated with the square of oscillation frequency and amplitude. A global optimal cruising frequency exists to achieve minimum energy cost;
Anaerobic metabolism model for white muscle burst locomotion: It features a sharp surge in energy consumption and has a limited sustainable duration;
A four-dimensional constraint framework covering swimming speed, power consumption, endurance and muscle fatigue is established for energy-optimal trajectory planning and locomotion parameter optimization.
6. Biological Schooling Modeling Based on Individual Hydrodynamic Interaction
Three classical collective behavioral rules are applied to each bionic fish agent:
Separation: short-range repulsion to avoid intra-school collision;
Alignment: synchronization of velocity and heading with neighboring individuals;
Cohesion: long-range attraction to maintain group topological structure.
Vortex wake energy utilization is integrated into the model: follower fish exploit the Kármán vortex street shed by leading individuals to reduce muscular energy expenditure, combined with long-distance lateral-line hydrodynamic perception to realize vision-free coordinated schooling in complex flow environments.
7. Biological Calibration and Model Validation Framework
Kinematic calibration: high-speed camera and Particle Image Velocimetry (PIV) extract body curvature, joint timing and fin deformation of real fish for parameter identification;
Hydrodynamic calibration: steady and transient force measurement in circulating water tanks to correct added mass, lift and drag empirical coefficients;
Neuro-muscular calibration: electromyography (EMG) signal acquisition to align CPG output timing with in-vivo muscle activation sequence;
Energy calibration: prototype power consumption tests to fit biological metabolic energy consumption parameters.
8. Frontier Research Directions
Integrated modeling of flexible sensing and actuation: closed-loop control combining epidermal pressure sensor arrays and lateral-line hydrodynamic perception;
Active flow control via small deflection of dorsal and anal fins to suppress flow separation and reduce cruising drag;
Biomimetic fault-tolerant adaptive modeling: neural network reconstruction of rhythmic locomotion to realize movement compensation under partial actuator failure.