A rigid-flexible coupling bionic chest fin wave beating cooperative motion parameter optimization method

Abstract

The application discloses a rigid-flexible coupling bionic pectoral fin wave-slap cooperative motion parameter optimization method, generates a pose and a speed by coupling a three-degree-of-freedom rigid wave-slap and a span / chordwise flexible traveling wave under a unified coordinate and a parameterized framework, and constructs a generator layer combining a reduced-order dynamics and physical supervision; a hydrodynamic mapping model is formed by using a numerical simulation and a pool experiment in parallel, a multi-fidelity decomposition and a Gaussian process prior based on correlation and difference, outputting a hydrodynamic coefficient posterior mean and uncertainty, and mapping into a mechanical system resultant force and resultant moment; finally, three types of multi-objective robust closed-loop optimization of propulsion priority, disturbance priority and balanced propulsion are constructed, a multi-objective evolutionary algorithm and realizability projection and online updating are combined, and robust optimal parameters are obtained; the method can realize high-precision modeling, real-time optimization and intelligent control of the pectoral fin wave-slap cooperative motion, and significantly improves the propulsion efficiency, attitude stability and environmental adaptability of the bionic robotic fish.

Description

Technical Field

[0001] This invention relates to biomimetic robot control technology, and in particular to a method for optimizing the motion parameters of a rigid-flexible coupled biomimetic pectoral fin flapping wave coordinated motion, applicable to hydrodynamic analysis and motion parameter optimization in the biomimetic pectoral fin flapping wave coordinated propulsion of robotic fish. Background Technology

[0002] Multi-finned flapping fish rely heavily on the coordinated flapping of the pectoral fin bones and the wavy movement of the fin surface to achieve low-speed maneuverability and posture stability. The rigid flapping of the pectoral fin bones provides the main propulsion and torque, while the spanwise and chordal flexible wavy movement of the fin surface adjusts the near-wall vortex volume and vortex structure, thereby coupling and generating fine-tuning of thrust, lateral force and lift under unsteady conditions. The flapping / wave coordination manifests as a dual role of rigid induction and flexible buffering. The former determines the fish's posture, while the latter regulates the energy path of the wake vortex street, achieving a dynamic balance between propulsion, disturbance resistance and energy consumption.

[0003] Inspired by fish pectoral fins, biomimetic pectoral fin structures mainly include rigid flapping and flexible undulation. The former achieves three-degree-of-freedom rigid body motion through servo motors or gear linkages, which is simple in structure and precise in control, but has limited ability to modulate the flow field on the fin surface. The latter often uses flexible materials, shape memory alloys, or smart actuators to simulate traveling wave propagation, which can achieve a fluid coupling effect with good compliance, but is limited in terms of motion controllability and stability. At present, pectoral fin structure research still focuses on a single flapping or undulation mode, and lacks systematic research on the comprehensive mechanism and parameter optimization of flapping-wave coordination. At the same time, existing mechanisms often cannot describe rigid body flapping and flexible undulation simultaneously under a unified coordinate and parameterization framework, resulting in a disconnect in information transmission between fluid, structure, and control, and failing to fully explore the synergistic potential of pectoral fins in thrust, lift, and attitude control.

[0004] To address the aforementioned shortcomings, it is necessary to conduct systematic research on biomimetic pectoral fin flapping coordinated motion. On the one hand, pectoral fin flapping coordinated motion can simultaneously consider thrust generation and fluid disturbance suppression within the same cycle, achieving a dynamic balance between propulsion efficiency and attitude stability. On the other hand, by establishing a rigid-flexible coupled unified motion generation model and hydrodynamic mapping model, closed-loop optimization can be achieved between the fish body, pectoral fins, and flow field, improving the motion performance of biomimetic fish in multi-task scenarios. In summary, research on rigid-flexible coupled biomimetic pectoral fin flapping coordinated motion methods can not only deepen the understanding of the unsteady propulsion mechanism of fish, but also provide key theoretical and technical support for the design of high-performance, multi-functional biomimetic underwater robots. Summary of the Invention

[0005] To address the problems in the technical background, the present invention aims to provide a rigid-flexible coupled biomimetic pectoral fin flapping wave collaborative motion parameter optimization method. Under a unified coordinate and parameterized framework, it can collaboratively model the three-degree-of-freedom rigid body flapping motion of the fin bone and the spanwise / chordal flexible traveling wave of the fin membrane, forming a rigid-flexible coupled hydrodynamic model of the body, pectoral fin, and flow field. Based on the parallel acquisition and mutual calibration of unsteady computational fluid dynamics and pool experiments, online parameter identification and calibration are implemented, thereby improving the accuracy and robustness of thrust and torque prediction.

[0006] The purpose of this invention is to provide a method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin flapping motion. This method includes the following steps: S1. Inspired by the structure and movement of fish pectoral fins, a biomimetic pectoral fin structure is designed, consisting of rigid fin bones and flexible fin surfaces. The rigid fin bones realize three-degree-of-freedom rigid flapping motion, and the flexible fin surfaces realize spanwise and chordwise flexible wave motion. The rigid flapping motion consists of forward and backward flapping motion, up and down flapping motion, and rocking motion. The flexible wave motion is a superposition and propagation of spanwise and chordwise fin surface motions. A unified motion generation framework is established for the pectoral fin-body-flow field, defining the pectoral fin system as... The system is Adopting intrinsic Euler angle sequences are used to establish coordinate transformation relationships, and the rigid flapping motion is mapped to the mechanical system through the active direction matrix of the pectoral fin. An inverse mapping form to avoid gimbal lock is also given. The pectoral fin motion model consists of a rigid flapping equation of the fin bone and a flexible wave equation of the fin surface. The rigid flapping equation of the fin bone is determined by the flapping amplitude, flapping envelope function, flapping frequency, and flapping phase. The flexible wave equation of the fin surface is described by the fin surface spatial amplitude envelope function, the fin surface motion time envelope function, and the cosine relationship of the total phase of the fin surface. Through the above unified description, seamless coupling between the rigid flapping motion of the pectoral fin and the flexible traveling wave of the fin surface is achieved. S2. Under the mechanical system, the rigid flapping equation of the fin bone and the flexible wave equation of the fin surface are coupled. A unified pectoral fin flapping wave coupled displacement equation is generated, which includes the fin root reference point position, the initial position of the pectoral fin, the flexible wave equation of the fin surface, and the passive rotation direction matrix of the pectoral fin. The pectoral fin flapping wave velocity equation is obtained by combining the pectoral fin flapping angular velocity and the time derivative of the normal basis vector within the system. This equation is then decomposed into normal and tangential components within the system to obtain the coupled normal and tangential velocities of the pectoral fin flapping wave. To provide a unified interface, the pectoral fin flapping wave velocity equation is written as a linear combination of the rigid body geometric Jacobian matrix multiplied by the pectoral fin flapping angular velocity and the flexible body geometric Jacobian matrix multiplied by the minimum parameter velocity of the flexible wave. This allows the controller and the computational fluid dynamics mesh to directly access the same set of state variables. To ensure engineering feasibility, the trajectory satisfies feasible region constraints within the generation layer. Soft saturation is used to limit the commanded angular displacement and angular velocity, and power gating is employed for same-scale contraction to ensure that instantaneous power does not exceed the power upper limit, while simultaneously ensuring that the pectoral fin flapping angular velocity does not exceed the limit. S3. A generator layer scheme combining a reduced-order dynamic model and physical supervision is proposed. In the offline stage, eigenorthogonal decomposition is used to extract the shape basis from high-fidelity snapshots, training to obtain a low-order shape basis and a reduced-order dynamic model constructed using dynamic mode decomposition. In the online stage, a low-frequency physical snapshot generation operator outputs a gold-standard deformation snapshot. At each sparse correction time, the... The deformation snapshot of the gold standard is mapped to modal coordinates through weighted least squares projection. A supervised correction method with residual minimization and prior covariance regularization is used to fuse the modal coordinates obtained by projection with the predicted modal coordinates to update the corrected modal coordinates, forming an intermittent closed loop of online prediction-correction. S4. Based on the kinematics of S2, a parallel acquisition and cross-calibration process for numerical simulation and water tank experiment is established, and the consistency of the calculation is ensured through unified data reduction and deviation calibration. The hydrodynamic coefficient vector is defined as consisting of thrust coefficient, lateral force coefficient, lift coefficient and power coefficient, and the hydrodynamic coefficient is converted into the instantaneous fluid force and instantaneous power of the machine system using fluid density, reference velocity and effective area.In the numerical simulation section, the unsteady Reynolds-averaged Navier-Stokes method was used for solution. Geometric motion was realized using a moving mesh, and no-slip boundary conditions were applied to the fluid-solid interface. The inlet flow velocity was set, the outlet pressure was constant, and the solid wall was no-slip. Based on this, a numerical simulation dataset was constructed, which consisted of pectoral fin motion control parameters and their corresponding simulated hydrodynamic coefficient vectors. To control numerical accuracy, mesh independence and time independence indices were verified. In the experimental measurement section, a six-dimensional force sensor was used to synchronously acquire data with the pose, forming an experimental calibration dataset, which is a pairwise set of pectoral fin motion control parameters and their corresponding experimental hydrodynamic coefficients. To compare the thrust coefficients on the same frequency band and time reference, the simulated thrust coefficients and experimental thrust coefficients were first compared. Bandpass filtering operators are applied to obtain filtered sequences. Then, cross-correlation is used to find the optimal time delay that maximizes the cross-correlation value among candidate time delays, completing phase alignment. S5: Utilizing a multi-fidelity decomposition based on "correlation-difference," a system residual function based on the pectoral fin control parameters is introduced to correct systematic biases on the global structure of the low-fidelity model, thus forming a generalizable prior-posterior mapping during the training phase. In the prediction phase, the hydrodynamic mapping model uses both numerical simulation and experimental calibration datasets as joint support, inputting pectoral fin control parameters and outputting the posterior mean of hydrodynamic coefficients and the uncertainty covariance matrix. To characterize the systematic differences between the numerical simulation and experimental calibration datasets, autoregressive coefficients are used to analyze the low-fidelity scalar response. The system is scaled and superimposed with system residuals dependent on pectoral fin control parameters to obtain a high-fidelity scalar response. Independent Gaussian process priors are applied to the low-fidelity scalar response and system residuals to construct a learnable prior-posterior mapping model. To improve stability and reliability under complex conditions, residual regularization terms constructed from momentum and energy conservation control equations are superimposed on the loss function, and the data fitting error and physical residuals are jointly constrained by a tradeoff coefficient within a given time window. Subsequently, the posterior mean of the hydrodynamic coefficients is converted into the resultant force of the pectoral fins of the system based on reference velocity, fluid density, and effective area. The moment is calculated by integrating the lever arm, and then superimposed with the additional mass model and viscous drag torque model to obtain the resultant torque of the pectoral fins of the system. The resulting parameter transfer chain is formed by… The pectoral fin control parameters are obtained by the hydrodynamic mapping model to obtain the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. Then, the resultant force and resultant torque of the pectoral fins of the machine system are obtained by mechanical mapping, which serve as the unified interface quantity for subsequent multi-objective optimization. S6. Based on the hydrodynamic mapping model, a multi-objective robust closed-loop optimization model for different task scenarios is constructed. In order to make the target definition correspond to the task scenario, the average value and root mean square of the hydrodynamic coefficient interval are calculated within a unified evaluation window. Then, three target layers are switched according to the task scenario: one is the propulsion priority model, which maximizes the average thrust under the premise of ensuring power control, and uses the relative thrust ratio and jitter threshold to constrain the lateral force and lift at the same time.The second is the disturbance suppression priority model, which primarily minimizes the average and root mean square of lateral force and lift, while simultaneously reducing net propulsion to near zero and limiting the power ceiling. The third is the balanced propulsion model, which uses a weighted summation of the same family of indices to achieve a trade-off between propulsion, energy consumption, and disturbance suppression. The multi-objective robust closed-loop optimization model consists of the propulsion priority model, the disturbance suppression priority model, and the balanced propulsion model. A multi-objective evolutionary algorithm is used to solve the multi-objective robust closed-loop optimization model, and all candidate solutions always satisfy the engineering feasible region. To ensure engineering feasibility, a projection metric weighting matrix is ​​used to project the candidate solutions into the engineering feasible region, ensuring they strictly fall within the feasible region. Simultaneously, the hydrodynamic mapping model employs an online data update method, updating the surrogate parameters in small steps at an adaptive learning rate along the gradient of the instantaneous loss in each evaluation round to resist distribution drift and improve adaptability to other operating conditions.

[0007] As a further technical solution, the front and rear flapping wing movements around the pectoral fin system The axis rotates, and the upper and lower flapping wings move around the pectoral fin system. The axis rotates, and the rocker motion revolves around... The axis rotates; the spanwise motion of the fin surface is transmitted from the fin root to the fin tip, and the chordal motion of the fin surface is transmitted from the leading edge to the trailing edge; the transformation matrix from the pectoral fin system to the muscular system is defined as the active rotation direction matrix of the pectoral fin, and the transformation matrix from the muscular system to the pectoral fin system is defined as the passive rotation direction matrix of the pectoral fin; the expression for the passive rotation direction matrix of the pectoral fin is:

[0008] ;

[0009] In the formula, For time, This is the matrix representing the passive rotation direction of the pectoral fins. This refers to the angular displacement of the front and rear flaps. This refers to the angular displacement of the upper and lower flaps. This is the angular displacement of the wing. For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of the axis;

[0010] The expression for the cosine matrix of the active rotation direction of the pectoral fin is:

[0011] ;

[0012] In the formula, The matrix represents the active rotation direction of the pectoral fin, with symbols... This indicates transpose; the active rotation direction matrix of the pectoral fin is used for the rotation vector itself, and the passive rotation direction matrix of the pectoral fin is used for coordinate transformation; the two are transposes of each other.

[0013] The forward and aft flap angular displacements, the vertical flap angular displacements, and the rocking wing angular displacements are mapped to the pectoral fin flapping angular velocity of the aircraft system, and the expression is as follows:

[0014] ;

[0015] In the formula, This refers to the displacement of the pectoral fin flapping angle; The angular velocity of the pectoral fin flapping motion. For the front and rear flapping angular velocities, The angular velocity of the upper and lower flapping wings, This refers to the angular velocity of the wing. The angular velocity of the pectoral fin flapping of the machine system, For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction; The pectoral fin angular velocity mapping matrix represents the result of... Sequential Euler angular velocity Mapped to the pectoral fin flapping angular velocity of the machine system ;

[0016] The equation for the rigid flapping motion of the fin bones in the pectoral fin motion model is:

[0017] ;

[0018] In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first The flapping angular displacement of the degree of freedom For the first Bounce amplitude of degrees of freedom For the first The beat envelope function with degrees of freedom is used for smooth modulation of amplitude over time and asymmetric shaping. For the first Beating frequency of degrees of freedom For the first The beat phase of the degree of freedom; the expression for the beat envelope function is:

[0019] ;

[0020] In the formula, Pi As a constant bias term, it determines The baseline; For the first The cosine coefficient of each harmonic. ; For the first The sinusoidal coefficient of each harmonic. ; , Indicates harmonic index, The number of harmonic terms controls the spectral resolution; spline basis functions For node vectors, For the first The weights of the individual spline basis are used for amplitude modulation in local time intervals; , Represents a spline base index. Number of spline bases;

[0021] Taking the time derivative of the rigid flapping equation of the fin bone, we obtain the angular velocity equation for the flapping of the pectoral fin, which is expressed as:

[0022] ;

[0023] In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first Angular velocity of flapping motion of degrees of freedom The time first derivative of the flapping envelope function;

[0024] The equation for the flexible wave pattern of the fin surface in the pectoral fin motion model is:

[0025] ;

[0026] In the formula, For time, The displacement is the flexible wave motion of the fin, indicating the location of... The displacement of the material points on the fin surface along the local normal; The spatial amplitude envelope function of the fin describes the distribution of deformation intensity along the spanwise and chordwise directions. Let be the time envelope function of the fin motion, describing the gradual in and out modulation over time; The overall phase of the fin determines the propagation direction and velocity of the fin crests and troughs; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , Let be the chord length of the pectoral fin; the expression for the spatial amplitude envelope function of the fin surface is:

[0027] ;

[0028] In the formula, To display the envelope coefficient of the fin surface, The tangential envelope coefficient of the fin surface. The nominal amplitude; the expression for the fin motion time envelope function is:

[0029] ;

[0030] In the formula, For time, The angular frequency of the traveling wave. , The frequency of the traveling wave; , Indicates the harmonic envelope index. The harmonic order; For the first Time envelope harmonic amplitude coefficient, adjustment The intensity of fluctuations; For the first The first-order time envelope harmonic phase is used to control the position of the envelope peak and valley on the time axis; the total phase of the fin is:

[0031] ;

[0032] In the formula, To determine the spanwise wavenumber, along Directional phase propulsion rate; This is the initial phase. For spanwise phase gradient, The chordal phase gradient is: The spanwise phase gradient is:

[0033] ;

[0034] In the formula, Let be the spanwise phase density function, and let be the chordwise phase gradient.

[0035] ;

[0036] In the formula, The phase density function is a chordal direction.

[0037] To calculate the normal velocity, the time derivative of the fin flexible wave equation is taken to obtain the fin flexible wave velocity equation; the fin flexible wave velocity equation is:

[0038] ;

[0039] In the formula, The fin surface flexible wave velocity, Let be the time first derivative of the fin motion time envelope function. Let be the first time derivative of the total phase of the fin, and denoted as the local phase angular velocity; the first time derivative of the total phase of the fin is:

[0040] ;

[0041] In the formula, The time first derivative of the spanwise phase gradient, It is the time first derivative of the chordal phase gradient.

[0042] As a further technical solution, the pectoral fin beat wave coupled displacement equation is:

[0043] ;

[0044] In the formula, For time, These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; Control points Spatial position vector in the machine system The location of the fin root reference point in the machine system. This is the matrix representing the passive rotation direction of the pectoral fins. The initial position of the fin, i.e. Time control point Coordinates within the pectoral fin system; Control points Unit normal in the pectoral fin system , The normal basis vector of the pectoral fin of the machine system; This refers to the flexible wave displacement of the fin surface;

[0045] Taking the time derivative of the pectoral fin beat wave coupled displacement equation, we obtain the pectoral fin beat wave coupled velocity equation, which is expressed as follows:

[0046] ;

[0047] In the formula, The coupling velocity of the pectoral fin beat wave. The fin surface flexible wave velocity, The normal basis vector of the pectoral fin system; The first term represents the angular velocity of the pectoral fin flapping motion of the entire system; the second term represents rigid body induction, indicating the point velocity caused by the overall flapping motion of the pectoral fin. The first term is the rotational speed around the instantaneous hinge; the second term is the flexibility change, representing the change in normal deflection caused by the flexible traveling wave over time and the contribution of the normal direction itself as it evolves with deformation.

[0048] The normal velocity of the pectoral fin beat wave coupling is:

[0049] ;

[0050] In the formula, The normal velocity of the pectoral fin beat wave coupling;

[0051] The tangential velocity of the pectoral fin beat wave coupling is:

[0052] ;

[0053] In the formula, The pectoral fin beat wave coupled tangential velocity; the pectoral fin beat wave coupled normal velocity is used for immersion boundary conditions, and the pectoral fin beat wave coupled tangential velocity is used for no-slip condition construction and dynamic mesh update;

[0054] To ensure consistency with the hardware controller and dynamic mesh interface, the pectoral fin beat wave velocity equation is rewritten as follows:

[0055] ;

[0056] In the formula, The angular velocity of the pectoral fin flapping motion. For the minimum parameter of flexible fluctuation, velocity, For a rigid body, the geometric Jacobian matrix is... The Jacobian matrix of the flexible body geometry;

[0057] To ensure project feasibility, the trajectory inherently satisfies feasibility constraints at the generation layer, and is subject to real-time pruning using soft saturation and power gating when necessary; soft saturation is applied to the pectoral fin flapping angular displacement and pectoral fin flapping angular velocity.

[0058] ;

[0059] In the formula, the subscript "cmd" represents the corresponding instruction quantity, and the subscript "max" represents the corresponding upper limit. This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. This is the commanded angular displacement after soft saturation. The command angular velocity after soft saturation. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in; through Saturation softly limits angular displacement and angular velocity to a range achievable by hardware.

[0060] After soft saturation, a contraction of the same scale is performed to ensure that the estimated instantaneous power does not exceed the upper limit and the pectoral fin flapping angular velocity does not exceed the limit:

[0061] ;

[0062] In the formula, The equivalent actuation torque of the rigid body joint; This is the equivalent generalized force of fin surface flexible wave; For instantaneous power estimation; This is the upper limit of power.

[0063] As a further technical solution, intrinsic orthogonal decomposition is used to extract the intrinsic orthogonal decomposition shape basis from the high-fidelity snapshot; the intrinsic orthogonal decomposition shape basis is reconstructed as follows:

[0064] ;

[0065] In the formula, For time, , Indicates modal index, The modal number; For the first A spatial mode, For the first Modal coordinates; This refers to the flexible wave displacement of the fin surface; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; It is an intrinsic orthogonal decomposition shape basis. To be Column vectors stacked on a discrete grid; For modal coordinates, ;

[0066] The reduced-order dynamic model is as follows:

[0067] ;

[0068] In the formula, For modal coordinates, the time derivative is... For the intermodal coupling system matrix, For external driving input matrix; The driving vector is derived from the pectoral fin control parameters; This refers to continuous-time process noise.

[0069] The gold label deformation snapshot is as follows:

[0070] ;

[0071] In the formula, Indicates the index value. No. Secondary sparse correction time. At any moment A snapshot of the gold standard deformation. For pectoral fin control parameters, For generating operators for physical snapshots, given pectoral fin control parameters With time Below, a numerical snapshot of the fin surface flexible wave displacement is output by a high-fidelity simulator;

[0072] The weighted least squares estimate of the modal coordinates is:

[0073] ;

[0074] In the formula, exist Modal coordinates, It is an intrinsic orthogonal decomposition shape basis. A positive definite weighting matrix used to define the projection metric; As a weighted pseudo-inverse operator, the gold standard deformation snapshot is projected onto modal coordinates;

[0075] The expression for the supervised correction method is:

[0076] ;

[0077] In the formula, To predict modal coordinates, To correct the modal coordinates, To predict covariance, reflecting Uncertainty; For regularization weights, ;

[0078] The expression for the online cyclic prediction-correction is:

[0079] ;

[0080] Thus, while maintaining real-time performance, extrapolation drift is suppressed and physical consistency is preserved.

[0081] As a further technical solution, the hydrodynamic coefficient vector is:

[0082] ;

[0083] In the formula, For time, This is the hydrodynamic coefficient vector; The thrust coefficient represents the thrust along the fuselage. Instantaneous value of axial hydrodynamic coefficient; The lateral force coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; The lift coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; For power coefficient, Indicates vector transpose;

[0084] The instantaneous fluid force is:

[0085] ;

[0086] In the formula, For thrust, it indicates the force along the body. Instantaneous fluid force in the axial direction; The force is lateral, representing the force along the body. Instantaneous fluid force in the axial direction; For lift, it means along the body Instantaneous fluid force in the axial direction; For fluid density, For reference speed, Effective area;

[0087] The instantaneous power is:

[0088] ;

[0089] In the formula, Instantaneous power;

[0090] In the numerical simulation section, pectoral fin control parameters are generated based on computational fluid dynamics. The numerical simulation dataset corresponding to the hydrodynamic coefficient vector is expressed as follows:

[0091] ;

[0092] In the formula, For time, This is a numerical simulation dataset; , Indicates the working condition number index. This represents the total number of simulation conditions. For the first Pectoral fin control parameters for the group's operating conditions; To and The corresponding simulated hydrodynamic coefficient vector;

[0093] In terms of numerical accuracy control, constructing mesh sequences With time step sequence Convergence is evaluated using grid independence and time step independence indices; the grid independence index is:

[0094] ;

[0095] In the formula, Indicates the values ​​taken in each direction. As a grid independence index, Coarse grid Medium grid For fine mesh; when Less than the preset threshold and , , When the mesh exhibits a monotonically converging trend, it is determined that the mesh division satisfies the independence requirement. For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients; the time independence index is:

[0096] ;

[0097] In the formula, As a time independence indicator, For long-term steps, For medium time steps, For short time steps; when Less than the preset threshold and , , When there is a consistent convergence trend, the time step is considered to meet the independence requirement. In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients;

[0098] In the experimental measurement section, a six-dimensional force sensor was used to synchronously acquire hydrodynamic coefficients under different experimental conditions, forming an experimental calibration dataset; the experimental calibration dataset is as follows:

[0099] ;

[0100] In the formula, To calibrate the dataset for the experiment; , Indicates the working condition number index. This represents the total number of experimental conditions. and The corresponding experimental hydrodynamic coefficient vector;

[0101] To compare simulated and experimental thrust coefficients on the same frequency band and time base, a bandpass filter is first applied to both the simulated and experimental thrust coefficients, expressed as follows:

[0102] ;

[0103] In the formula, To simulate the thrust coefficient, The experimental thrust coefficient, For bandpass filtering operators, The simulated thrust coefficients after filtering. The filtered experimental thrust coefficients are used; the optimal time delay is obtained through cross-correlation to achieve phase alignment, and the optimal time delay is:

[0104] ;

[0105] In the formula, As candidate time delay variables, Indicates all Select the one with the highest cross-correlation , For optimal latency; if This indicates that the experimental sequence lags behind the simulation sequence. ;like This indicates that the experimental sequence is ahead of the simulation. .

[0106] As a further technical solution, in numerical simulation datasets With experimental calibration dataset With the joint support of [unclear], a hydrodynamic mapping model was constructed, with the pectoral fin control parameters as input. The output is the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix; the hydrodynamic mapping model is:

[0107] ;

[0108] In the formula, For time, The overall output of the model consists of two parts: the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. The posterior mean of the hydrodynamic coefficients. , The posterior mean of the thrust coefficient. The posterior mean of the lateral force coefficients. The posterior mean of the lift coefficient. The power coefficient is the posterior mean. The uncertainty covariance matrix, For hydrodynamic mapping model, For pectoral fin control parameters, For numerical datasets, This is the experimental dataset;

[0109] To characterize the systematic differences between the numerical simulation dataset and the experimental calibration dataset, the high-fidelity scalar response is expressed as the sum of the scaled low-fidelity scalar response and the system residual term using autoregressive coefficients; the high-fidelity scalar response is:

[0110] ;

[0111] In the formula, For high-fidelity scalar response, represents the experimentally measured hydrodynamic coefficients. The low-fidelity scalar response represents the hydrodynamic coefficients in the numerical simulation. These are scalar autoregressive coefficients. For system residuals;

[0112] Subsequently, independent Gaussian process priors are applied to the low-fidelity scalar response and the system residual term to obtain a learnable prior-posterior mapping model; the prior-posterior mapping model is as follows:

[0113] ;

[0114] In the formula, This is a prior-posterior mapping model. For low-fidelity process mean function, The mean function of the difference process, For low-fidelity process covariance kernel function, The covariance kernel function for the difference process; This represents another set of pectoral fin control parameters;

[0115] The residual regularization term is:

[0116] ;

[0117] In the formula, At the start time, For time window, For the experimental hydrodynamic coefficient vector, The posterior mean of the hydrodynamic coefficients. For weighted matrices, The physical residuals are constructed from the governing equations; For the weighting factor;

[0118] The equation for the resultant force of the pectoral fins of the aforementioned system is:

[0119] ;

[0120] In the formula, For fluid density, For reference speed, Effective area; The resultant force of the pectoral fins of the machine system is calculated by integrating the force arms and compensating with the additional mass model and the viscous drag torque model. The resultant torque of the pectoral fins of the machine system is then obtained.

[0121] ;

[0122] In the formula, The resultant torque of the pectoral fins of the machine system, The position vector of the point of application relative to the reference point. For the addition of quality model, For viscous drag torque model, For the speed of the machine system, Accelerate the machine system.

[0123] As a further technical solution, the average value of the hydrodynamic coefficient range is

[0124] ;

[0125] In the formula, To evaluate the start time of the window, To evaluate the window duration, For the first One hydrodynamic coefficient, For interval The average value of the hydrodynamic coefficient over the range, , Indicates different hydrodynamic coefficient indices. Represents the thrust coefficient. Lateral force coefficient, Represents the lift coefficient. Represents the power factor;

[0126] ;

[0127] In the formula, For interval The root mean square of the hydrodynamic coefficients;

[0128] First, when there is a clear need to advance, the advancement priority model is adopted, and the expression is:

[0129] ;

[0130] The constraints are:

[0131] ;

[0132] In the formula, min represents finding the minimum value. To advance the weighting function of the priority model, For feasible regions, For pectoral fin control parameters, To and The corresponding power-hydrodynamic coefficient, To and The corresponding average thrust coefficient To and The corresponding average lateral force coefficient, To and The corresponding average lift coefficient, To and The corresponding root mean square of the lateral force coefficient, To and The corresponding root mean square lift coefficient, To avoid small positive numbers with a denominator of zero, The relative threshold of the lateral force coefficient. The relative threshold of the lift coefficient. The threshold for lateral force coefficient jitter. The lift coefficient jitter threshold;

[0133] Secondly, when the objective is stable movement, the aforementioned disturbance priority model is adopted, and its expression is:

[0134] ;

[0135] Constraints:

[0136] ;

[0137] In the formula, The weighting function for the disturbance suppression priority model, The relative threshold of the thrust coefficient, This represents the upper limit of the power factor.

[0138] Finally, when a trade-off is needed between propulsion and stability, the aforementioned balanced propulsion model is adopted, with the expression:

[0139] ;

[0140] In the formula, To balance the weighting function of the propulsion model, As the weight value, when taking The time is the aforementioned priority model; when and When the primary factor is the disturbance priority model, the model will be biased towards the disturbance priority model.

[0141] The multi-objective robust closed-loop optimization model, during optimization iteration, obtains the coefficient time series through the hydrodynamic mapping model based on the design variable input, and then calculates the interval average and root mean square, substituting them into the corresponding objective and constraint evaluation sequence; the engineering projection is:

[0142] ;

[0143] In the formula, For engineering projection design variables, As candidate design variables, As decision variables, For the engineering feasible region, This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in the data; the online data update method is as follows:

[0144] ;

[0145] In the formula, In the first The data-driven model parameter vector for each evaluation round represents the current parameters of the hydrodynamic mapping model; In the first The updated parameter vector obtained after one round of online calibration; For the first The learning rate for each round; For the first The instantaneous loss function for each round. instantaneous loss function about The gradient of the parameters.

[0146] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.

[0147] The beneficial effects of this invention are as follows:

[0148] (1) Under the unified coordinate and parameterization framework, the three-degree-of-freedom rigid beat motion is coupled with the span / chordal flexible traveling wave. The “geometric Jacobian unified interface” is used to directly connect to the controller computational fluid dynamic mesh, and soft saturation and power gating are used to ensure that the trajectory always meets the feasible domain constraints.

[0149] (2) The generator layer based on the combination of reduced-order dynamics model and physical supervision takes into account both long-term stability and real-time performance, and effectively suppresses extrapolation drift of online prediction.

[0150] (3) By parallel acquisition and cross-calibration of numerical simulation and water tank experiment, as well as bandpass filtering and cross-correlation phase alignment, and supplemented by unified data reduction and deviation calibration, the consistency and accuracy of hydrodynamic coefficient calculation are significantly improved.

[0151] (4) Using the "correlation-difference" multifidelity decomposition and Gaussian process prior, the posterior mean and uncertainty of the hydrodynamic coefficients are obtained and mapped to the resultant force / resultant torque of the machine system. Under the three types of multi-objective robust closed-loop optimization of propulsion priority, disturbance priority and balanced propulsion, combined with engineering feasibility projection and online update, the optimal parameters that satisfy power constraints, high thrust efficiency, small side / lift force and robustness are obtained. Attached Figure Description

[0152] Figure 1 This is a flowchart of a method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, according to an embodiment of the present invention.

[0153] Figure 2 This is a schematic diagram of a biomimetic pectoral fin driven by wave-beating in an embodiment of the present invention.

[0154] Figure 3 This is a schematic diagram of the pectoral fin beat wave coupled displacement equation structure in an embodiment of the present invention.

[0155] Figure 4 This is a schematic diagram of the intermittent closed-loop prediction-correction of the pectoral fin in an embodiment of the present invention.

[0156] Figure 5 This is a flowchart of the parallel acquisition and cross-calibration process for numerical simulation and water tank experiment in this embodiment of the invention.

[0157] Figure 6 This is a schematic diagram of the prior-posterior mapping model in an embodiment of the present invention.

[0158] Figure 7This is a schematic diagram of a multi-objective robust closed-loop optimization model in an embodiment of the present invention. Detailed Implementation

[0159] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this invention, not all embodiments. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0160] Please refer to the attached document. Figures 1 to 7 This invention provides a method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin flapping wave. The method includes the following steps: S1. Inspired by the structure and movement of fish pectoral fins, a biomimetic pectoral fin structure is designed, consisting of rigid fin bones and flexible fin surfaces. The rigid fin bones realize three-degree-of-freedom rigid flapping motion, and the flexible fin surfaces realize spanwise and chordwise flexible wave motion. The rigid flapping motion consists of forward and backward flapping wing motion, up and down flapping wing motion, and rocking wing motion. The flexible wave motion is a superposition and propagation of spanwise and chordwise fin surface motions. A unified motion generation framework is established for the pectoral fin-body-flow field, defining the pectoral fin system as... The system is Adopting intrinsic Euler angle sequences are used to establish coordinate transformation relationships, and the rigid flapping motion is mapped to the mechanical system through the active direction matrix of the pectoral fin. An inverse mapping form to avoid gimbal lock is also given. The pectoral fin motion model consists of the rigid flapping equation of the fin bone and the flexible wave equation of the fin surface. The rigid flapping equation of the fin bone is determined by the flapping amplitude, flapping envelope function, flapping frequency, and flapping phase. The flexible wave equation of the fin surface is described by the spatial amplitude envelope function of the fin surface, the temporal envelope function of the fin surface motion, and the cosine relationship of the total phase of the fin surface. Through the above unified description, seamless coupling between the rigid flapping motion of the pectoral fin and the flexible traveling wave of the fin surface is achieved. S2. Under the mechanical system, the rigid flapping equation of the fin bone and the flexible wave equation of the fin surface are coupled to generate a unified pectoral fin flapping wave coupled displacement equation. The process includes the fin root reference point position, pectoral fin initial position, fin surface flexible wave equation, and pectoral fin passive rotation direction matrix. The pectoral fin beat wave velocity equation is obtained by combining the pectoral fin beat wave coupled displacement equation with the time derivative of the pectoral fin beat wave angular velocity and normal basis vector within the system. This equation is then decomposed into normal and tangential components within the system to obtain the coupled normal and tangential velocities of the pectoral fin beat wave. To provide a unified interface, the pectoral fin beat wave velocity equation is written as a linear combination of the rigid body geometric Jacobian matrix multiplied by the pectoral fin beat wave angular velocity and the flexible body geometric Jacobian matrix multiplied by the minimum parameter velocity of the flexible wave, allowing the controller and the computational fluid dynamics mesh to directly access the same set of state variables. To ensure engineering feasibility, the trajectory satisfies the following conditions within the generation layer: Feasibility region constraints are implemented by limiting the commanded angular displacement and angular velocity through soft saturation, and power gating is used for same-scale contraction to ensure that the instantaneous power does not exceed the power upper limit, while ensuring that the pectoral fin flapping angular velocity does not exceed the limit. S3. A generator layer scheme combining a reduced-order dynamic model and physical supervision is proposed. In the offline stage, eigenorthogonal decomposition is used to extract the shape basis from high-fidelity snapshots, training a low-order shape basis constructed from dynamic mode decomposition and a reduced-order dynamic model. In the online stage, a low-frequency physical snapshot generation operator outputs a gold standard deformation snapshot. At each sparse correction time, the gold standard deformation snapshot is mapped to modal coordinates through weighted least squares projection, and a supervised correction method using residual minimization and prior covariance regularization is used to map the projected modal coordinates. The predicted modal coordinates are fused with the calibrated modal coordinates to update the corrected modal coordinates, forming an intermittent closed loop of online prediction-correction. S4. Based on the kinematics of S2, a parallel acquisition and cross-calibration process for numerical simulation and pool experiments is established, and measurement consistency is ensured through unified data reduction and deviation calibration. The hydrodynamic coefficient vector is defined as consisting of thrust coefficient, lateral force coefficient, lift coefficient, and power coefficient, and the hydrodynamic coefficients are converted into instantaneous fluid force and instantaneous power of the system using fluid density, reference velocity, and effective area. In the numerical simulation, the unsteady Reynolds-averaged Navier-Stokes method is used for solution. Geometric motion is achieved using a moving mesh, and no-slip boundary conditions are applied to the fluid-solid interface. The inlet velocity is set, the outlet pressure is constant, and the solid wall has no slip.Based on this, a numerical simulation dataset was constructed, consisting of pectoral fin motion control parameters and their corresponding simulated hydrodynamic coefficient vectors. To control numerical accuracy, grid independence and time independence indices were verified. In the experimental measurement section, a six-dimensional force sensor was used to synchronously acquire data with pose, forming an experimental calibration dataset, which is a pairwise set of pectoral fin motion control parameters and their corresponding experimental hydrodynamic coefficients. To compare thrust coefficients on the same frequency band and time reference, bandpass filtering operators were first applied to the simulated and experimental thrust coefficients to obtain filtered sequences. Then, cross-correlation was used to find the optimal time delay that maximizes the cross-correlation value among candidate time delays, completing phase alignment. S5. Using "correlation-difference" multi-fidelity decomposition, in low-fidelity... Based on the global structure of the model, a system residual function input according to the pectoral fin control parameters is introduced to correct systematic biases, thereby forming a generalizable prior-posterior mapping during the training phase. During the prediction phase, the hydrodynamic mapping model is jointly supported by numerical simulation and experimental calibration datasets, inputting pectoral fin control parameters and outputting the posterior mean of hydrodynamic coefficients and the uncertainty covariance matrix. To characterize the systematic differences between the numerical simulation and experimental calibration datasets, the low-fidelity scalar response is scaled using autoregressive coefficients and superimposed with system residual terms dependent on the pectoral fin control parameters to obtain the high-fidelity scalar response. Independent Gaussian process priors are applied to both the low-fidelity scalar response and the system residual terms to construct a learnable prior-posterior mapping. To improve stability and reliability under complex operating conditions, a residual regularization term constructed from the momentum and energy conservation control equations is superimposed on the loss function. Within a given time window, the data fitting error and physical residual are jointly constrained by a trade-off coefficient. Subsequently, the posterior mean of the hydrodynamic coefficients is converted into the resultant force of the pectoral fins of the machine system based on the reference velocity, fluid density, and effective area. The moment is obtained by integrating the lever arm and superimposing the additional mass model and viscous drag moment model. The resulting parameter transfer chain is as follows: the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix are obtained from the pectoral fin control parameters through the hydrodynamic mapping model, and then the resultant force and moment of the pectoral fins of the machine system are obtained through mechanical mapping, serving as a unified interface for subsequent multi-objective optimization. Quantity; S6. Based on the hydrodynamic mapping model, construct a multi-objective robust closed-loop optimization model for different mission scenarios; to ensure that the objective definition corresponds to the mission scenario, calculate the average value and root mean square of the hydrodynamic coefficient interval within a unified evaluation window; then switch between three objective layers according to the mission scenario: the first is the propulsion priority model, which maximizes the average thrust while ensuring power control, and uses the relative thrust ratio and jitter threshold to constrain lateral force and lift simultaneously; the second is the disturbance suppression priority model, which focuses on minimizing the average value and root mean square of lateral force and lift, while reducing net propulsion to near zero and limiting the upper limit of power; the third is the balanced propulsion model, which performs weighted summation on the same family of indicators to achieve a trade-off between propulsion, energy consumption and disturbance suppression.The multi-objective robust closed-loop optimization model consists of a propulsion-first model, a disturbance-first model, and a balanced propulsion model. A multi-objective evolutionary algorithm is used to solve the multi-objective robust closed-loop optimization model, and all candidate solutions always satisfy the engineering feasible region. To ensure engineering feasibility, a projection metric weighting matrix is ​​used to project the candidate solutions into the engineering feasible region, ensuring they strictly fall within the feasible region. Simultaneously, the hydrodynamic mapping model employs an online data update method, updating the surrogate parameters in small steps at each evaluation round with an adaptive learning rate along the gradient of the instantaneous loss to resist distribution drift and improve adaptability to other operating conditions.

[0161] The following describes specific embodiments of the present invention.

[0162] See Figure 1 Starting with the three-degree-of-freedom flapping motion of rigid fins and the spanwise / chord-oriented traveling wave of flexible fins, an intrinsic Z–X–Y Euler unified kinematics of the pectoral fin system and the mechanical system is established, and the Jacobian interface and feasible region constraints for the flapping / wave coupled pose and velocity are realized. On this basis, a generator layer combining reduced-order dynamics and physical supervision is constructed. Unsteady Reynolds-averaged Navier-Stokes numerical simulation and pool experiments are combined to form consistent data reduction and cross-calibration. Based on this, a hydrodynamic mapping model is established using the "correlation-difference" multifidelity decomposition and Gaussian process prior. Finally, multi-objective robust closed-loop optimization is carried out under three objective layers: propulsion priority, disturbance priority, and balanced propulsion. Engineering projection and online data updates are implemented through a projection metric weighted matrix to ensure the feasibility and cross-condition adaptability of the solution.

[0163] S1, see reference Figure 2 This step is based on a biomimetic pectoral fin with rigid fin bones and flexible fin surfaces: the rigid fin bones realize three-degree-of-freedom rigid flapping, and the flexible fin surfaces realize flexible waves propagating in the spanwise and chordwise directions; under the unified motion generation framework of pectoral fin-body-flow field, the pectoral fin system and the body system are defined, the coordinate transformation is established using the intrinsic Euler angle sequence, and the rigid flapping is mapped to the body system through the active direction matrix of the pectoral fin and an inverse mapping to avoid gimbal lock is given; the pectoral fin motion model consists of the rigid flapping equation of the fin bones and the flexible wave equation of the fin surface, thereby realizing the seamless coupling of the rigid body flapping of the pectoral fin and the flexible traveling wave of the fin surface.

[0164] Specifically, the fore and aft flapping wing movements circulate around the pectoral fin system The axis rotates, and the upper and lower flapping wings move around the pectoral fin system. The axis rotates, and the rocker arm moves around. Axial rotation; spanwise motion of the fin surface is transmitted from the fin root to the fin tip, and chordal motion of the fin surface is transmitted from the leading edge to the trailing edge; the conversion matrix from the pectoral fin system to the body system is defined as the active rotation direction matrix of the pectoral fin, and the conversion matrix from the body system to the pectoral fin system is defined as the passive rotation direction matrix of the pectoral fin;

[0165] Specifically, the expression for the passive rotation direction matrix of the pectoral fin is:

[0166] ;

[0167] In the formula, For time, This is the matrix representing the passive rotation direction of the pectoral fins. This refers to the angular displacement of the front and rear flaps. This refers to the angular displacement of the upper and lower flaps. This is the angular displacement of the wing. For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of the axis;

[0168] Specifically, the expression for the cosine matrix of the active rotation direction of the pectoral fin is:

[0169] ;

[0170] In the formula, The matrix represents the active rotation direction of the pectoral fin, with symbols... This indicates transpose; the active rotation direction matrix of the pectoral fin is used for the rotation vector itself, and the passive rotation direction matrix of the pectoral fin is used for coordinate transformation. The two are transposes of each other.

[0171] Specifically, around the pectoral fin system The expression for the intrinsic rotation matrix of the axis is:

[0172] ;

[0173] Specifically, the pectoral fin system The expression for the intrinsic rotation matrix of the axis is:

[0174] ;

[0175] Specifically, around the pectoral fin system The expression for the intrinsic rotation matrix of the axis is:

[0176] ;

[0177] Furthermore, mapping the front and rear flap angular displacements, upper and lower flap angular displacements, and wing angular displacements to the pectoral fin flapping angular velocity of the aircraft system, its expression is:

[0178] ;

[0179] In the formula, This refers to the displacement of the pectoral fin flapping angle; The angular velocity of the pectoral fin flapping motion. For the front and rear flapping angular velocities, The angular velocity of the upper and lower flapping wings, This refers to the angular velocity of the wing. The angular velocity of the pectoral fin flapping of the machine system, For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction; The pectoral fin angular velocity mapping matrix represents the result of... Sequential Euler angular velocity Mapped to the pectoral fin flapping angular velocity of the machine system ;

[0180] Specifically, the expression for the pectoral fin angular velocity mapping matrix is:

[0181] ;

[0182] In the formula, ;when At that time, the inverse mapping matrix of the pectoral fin angular velocity avoids gimbal lock;

[0183] Specifically, the inverse mapping matrix of the pectoral fin angular velocity is:

[0184] ;

[0185] Furthermore, the equation for the rigid flapping motion of the fin bones in the pectoral fin motion model is:

[0186] ;

[0187] In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first The flapping angular displacement of the degree of freedom For the first Bounce amplitude of degrees of freedom For the first The beat envelope function with degrees of freedom is used for smooth modulation of amplitude over time and asymmetric shaping. For the first Beating frequency of degrees of freedom For the first Bounce phase of degrees of freedom;

[0188] Specifically, the expression for the beat envelope function is:

[0189] ;

[0190] In the formula, Pi As a constant bias term, it determines The baseline; For the first The cosine coefficient of each harmonic. ; For the first The sinusoidal coefficient of each harmonic. ; , Indicates harmonic index, The number of harmonic terms controls the spectral resolution; spline basis functions For node vectors, For the first The weights of the individual spline basis are used for amplitude modulation in local time intervals; , Represents a spline base index. Number of spline bases;

[0191] Specifically, by differentiating the rigid flapping equation of the fin bones over time, we obtain the angular velocity equation for the pectoral fin flapping motion, which is expressed as follows:

[0192] ;

[0193] In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first Angular velocity of flapping motion of degrees of freedom The time first derivative of the beating envelope function;

[0194] Furthermore, the equation for the flexible wave motion of the fin surface in the pectoral fin motion model is:

[0195] ;

[0196] In the formula, For time, The displacement is the flexible wave motion of the fin, indicating the location of... The displacement of the material points on the fin surface along the local normal; The spatial amplitude envelope function of the fin describes the distribution of deformation intensity along the spanwise and chordwise directions. Let be the time envelope function of the fin motion, describing the gradual in and out modulation over time; The overall phase of the fin determines the propagation direction and velocity of the fin crests and troughs; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord;

[0197] Specifically, the expression for the fin spatial amplitude envelope function is:

[0198] ;

[0199] In the formula, To display the envelope coefficient of the fin surface, The tangential envelope coefficient of the fin surface. Nominal amplitude;

[0200] Specifically, the expression for the fin motion time envelope function is:

[0201] ;

[0202] In the formula, For time, The angular frequency of the traveling wave. , The frequency of the traveling wave; , Indicates the harmonic envelope index. The harmonic order; For the first Time envelope harmonic amplitude coefficient, adjustment The intensity of fluctuations; For the first The first time envelope harmonic phase is used to control the position of the envelope peak and the envelope trough on the time axis;

[0203] Specifically, the total phase of the fin is:

[0204] ;

[0205] In the formula, To determine the spanwise wavenumber, along Directional phase propulsion rate; This is the initial phase. For spanwise phase gradient, The phase gradient is in the chord direction;

[0206] Specifically, the spanwise phase gradient is:

[0207] ;

[0208] In the formula, The spanwise phase density function;

[0209] Specifically, the chordal phase gradient is:

[0210] ;

[0211] In the formula, The phase density function is a chordal direction.

[0212] Furthermore, in order to calculate the normal velocity, the time derivative of the fin flexible wave equation is taken to obtain the fin flexible wave velocity equation;

[0213] Specifically, the equation for the fin flexible wave velocity is:

[0214] ;

[0215] In the formula, The fin surface flexible wave velocity, Let be the time first derivative of the fin motion time envelope function. The first time derivative of the total phase of the fin is expressed as the local phase angular velocity;

[0216] Specifically, the first time derivative of the total phase of the fin is:

[0217] ;

[0218] In the formula, The time first derivative of the spanwise phase gradient, It is the time first derivative of the chordal phase gradient.

[0219] S2, see reference Figure 3 Under the system, the rigid flapping motion of the fin bone is coupled with the flexible wave motion of the fin surface to generate the coupled displacement equation of the pectoral fin flapping wave. From this, the velocity equation of the pectoral fin flapping wave is derived and decomposed into normal / tangential velocities. Externally, it is written as a unified interface that provides state variables by multiplying the rigid body geometric Jacobian matrix and the pectoral fin flapping angular velocity, and the flexible body geometric Jacobian matrix and the minimum parameter velocity of the flexible wave. At the same time, the feasible region constraint is satisfied within the generation layer, and soft saturation and power gating are used to ensure that the power and angular velocity do not exceed the limits.

[0220] Furthermore, the equation for the coupled displacement of the pectoral fin beat wave is:

[0221] ;

[0222] In the formula, For time, These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; Control points Spatial position vector in the machine system The location of the fin root reference point in the machine system. This is the matrix representing the passive rotation direction of the pectoral fins. The initial position of the fin, i.e. Time control point Coordinates within the pectoral fin system; Control points Unit normal in the pectoral fin system , The normal basis vector of the pectoral fin of the machine system; This refers to the flexible wave displacement of the fin surface;

[0223] Furthermore, by differentiating the time-dependent displacement equation of the pectoral fin beat wave, we obtain the coupled velocity equation of the pectoral fin beat wave, which is expressed as follows:

[0224] ;

[0225] In the formula, The coupling velocity of the pectoral fin beat wave. The fin surface flexible wave velocity, The normal basis vector of the pectoral fin system; The first term represents the angular velocity of the pectoral fin flapping motion of the entire system; the second term represents rigid body induction, indicating the point velocity caused by the overall flapping motion of the pectoral fin. The first term is the rotational speed around the instantaneous hinge; the second term is the flexibility change, representing the change in normal deflection caused by the flexible traveling wave over time and the contribution of the normal direction itself as it evolves with deformation.

[0226] Specifically, the coupling normal velocity of the pectoral fin beat wave is:

[0227] ;

[0228] In the formula, The normal velocity of the pectoral fin beat wave coupling;

[0229] Specifically, the coupled tangential velocity of the pectoral fin beat wave is:

[0230] ;

[0231] In the formula, The pectoral fin beat wave coupled tangential velocity; the pectoral fin beat wave coupled normal velocity is used for immersion boundary conditions, and the pectoral fin beat wave coupled tangential velocity is used for no-slip condition construction and dynamic mesh update;

[0232] Furthermore, to ensure consistency with the hardware controller and dynamic mesh interface, the pectoral fin beat wave velocity equation is rewritten as follows:

[0233] ;

[0234] In the formula, The angular velocity of the pectoral fin flapping motion. For the minimum parameter of flexible fluctuation, velocity, For a rigid body, the geometric Jacobian matrix is... The Jacobian matrix of the flexible body geometry;

[0235] Specifically, the expression for the Jacobian matrix of a rigid body geometry is:

[0236] ;

[0237] In the formula, Represents the cross product of vectors. The unit vector of the local coordinate axes of the pectoral fin, with the symbol... Indicates matrix transpose;

[0238] Specifically, the expression for the flexible body geometric Jacobian matrix is:

[0239] ;

[0240] In the formula, , Indicates an index. For its dimensions, For the first The minimum parameter for flexible fluctuation is velocity; Let be the column vector of geometric sensitivity with respect to each flexibility parameter, representing the partial derivative with respect to deformation;

[0241] Furthermore, to ensure the feasibility of the project, the trajectory inherently meets the feasibility constraints at the generation layer, and is subject to real-time pruning through soft saturation and power gating when necessary; soft saturation is applied to the pectoral fin flapping angular displacement and pectoral fin flapping angular velocity.

[0242] ;

[0243] In the formula, the subscript "cmd" represents the corresponding instruction quantity, and the subscript "max" represents the corresponding upper limit. This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. This is the commanded angular displacement after soft saturation. The command angular velocity after soft saturation. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in; through Saturation softly limits angular displacement and angular velocity to a range achievable by hardware.

[0244] Furthermore, after soft saturation, a contraction of the same scale is performed to ensure that the estimated instantaneous power does not exceed the upper limit and the pectoral fin flapping angular velocity does not exceed the limit:

[0245] ;

[0246] In the formula, The equivalent actuation torque of the rigid body joint; This is the equivalent generalized force of fin surface flexible wave; For instantaneous power estimation; This is the upper limit of power.

[0247] S3, see reference Figure 4 A generator layer combining a reduced-order dynamic model and physical supervision is proposed. Offline, the shape basis is extracted from high-fidelity snapshots using intrinsic orthogonal decomposition, and trained to obtain a low-order shape basis and a reduced-order dynamic model based on dynamic mode decomposition. Online, the low-frequency output of the gold standard deformation snapshot is generated by the physical snapshot generator, and at the sparse correction time, it is projected onto the modal coordinates using weighted least squares. The correction is supervised by residual minimization and prior covariance regularization. The projected mode and the predicted mode are fused and updated to form an online "prediction-correction" closed loop.

[0248] Specifically, intrinsic orthogonal decomposition is used to extract the intrinsic orthogonal decomposition shape basis from the high-fidelity snapshot; the intrinsic orthogonal decomposition shape basis is reconstructed as:

[0249] ;

[0250] In the formula, For time, , Indicates modal index, The modal number; For the first A spatial mode, For the first Modal coordinates; This refers to the flexible wave displacement of the fin surface; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; It is an intrinsic orthogonal decomposition shape basis. To be Column vectors stacked on a discrete grid; For modal coordinates, ;

[0251] Furthermore, the reduced-order dynamics model is as follows:

[0252] ;

[0253] In the formula, For modal coordinates, the time derivative is... For the intermodal coupling system matrix, For external driving input matrix; The driving vector is derived from the pectoral fin control parameters; This refers to continuous-time process noise.

[0254] Furthermore, the gold standard deformation snapshot is as follows:

[0255] ;

[0256] In the formula, Indicates the index value. No. Secondary sparse correction time. At any moment A snapshot of the gold standard deformation. For pectoral fin control parameters, For generating operators for physical snapshots, given pectoral fin control parameters With time Below is a numerical snapshot of the fin surface flexible wave displacement output by a high-fidelity simulator;

[0257] Furthermore, the weighted least squares estimation of modal coordinates is as follows:

[0258] ;

[0259] In the formula, exist Weighted least squares estimation of modal coordinates, It is an intrinsic orthogonal decomposition shape basis. A positive definite weighting matrix used to define the projection metric; As a weighted pseudo-inverse operator, the gold standard deformation snapshot is projected onto modal coordinates;

[0260] Furthermore, the expression for the supervised correction method is:

[0261] ;

[0262] In the formula, To predict modal coordinates, To correct the modal coordinates, To predict covariance, reflecting Uncertainty; For regularization weights, ;

[0263] Furthermore, the expression for online cyclic prediction-correction is:

[0264] ;

[0265] Thus, while maintaining real-time performance, extrapolation drift is suppressed and physical consistency is preserved.

[0266] S4, see reference Figure 5 A parallel acquisition and cross-calibration process for numerical simulation and pool experiments was established. Consistency of calculations was ensured through unified data reduction and deviation calibration. Hydrodynamic coefficient vectors were defined, and instantaneous fluid force and instantaneous power were calculated based on fluid density, reference velocity, and effective area. The numerical simulation employed an unsteady Reynolds-averaged Navier-Stokes method, a moving mesh, and a fluid-structure interface with no-slip boundaries to form a numerical simulation dataset of pectoral fin motion control parameters and simulated hydrodynamic coefficient vectors. Accuracy was verified through mesh independence and time independence indices. Experimental measurements were performed using a six-dimensional force sensor and synchronous pose acquisition to form an experimental calibration dataset. Phase alignment was achieved by applying bandpass filters to the simulated and experimental thrust coefficients and using cross-correlation to obtain the optimal time delay.

[0267] Specifically, the hydrodynamic coefficient vector is:

[0268] ;

[0269] In the formula, For time, This is the hydrodynamic coefficient vector; The thrust coefficient represents the thrust along the fuselage. Instantaneous value of axial hydrodynamic coefficient; The lateral force coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; The lift coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; For power coefficient, Indicates vector transpose;

[0270] Specifically, the instantaneous fluid force is:

[0271] ;

[0272] In the formula, For thrust, it indicates the force along the body. Instantaneous fluid force in the axial direction; The force is lateral, representing the force along the body. Instantaneous fluid force in the axial direction; For lift, it means along the body Instantaneous fluid force in the axial direction; For fluid density, For reference speed, Effective area;

[0273] Specifically, the instantaneous power is:

[0274] ;

[0275] In the formula, Instantaneous power;

[0276] Furthermore, in the numerical simulation section, the control parameters for the pectoral fins are generated based on computational fluid dynamics. The numerical simulation dataset corresponding to the hydrodynamic coefficient vector is expressed as follows:

[0277] ;

[0278] In the formula, For time, This is a numerical simulation dataset; , Indicates the working condition number index. This represents the total number of simulation conditions. For the first Pectoral fin control parameters for the group's operating conditions; To and The corresponding simulated hydrodynamic coefficient vector;

[0279] Furthermore, in terms of numerical precision control, constructing mesh sequences... With time step sequence Convergence was evaluated using grid independence and time step independence indices.

[0280] Specifically, the grid independence index is:

[0281] ;

[0282] In the formula, Indicates the values ​​taken in each direction. As a grid independence index, Coarse grid Medium grid For fine mesh; when Less than the preset threshold and , , When the mesh exhibits a monotonically converging trend, it is determined that the mesh division satisfies the independence requirement. For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients;

[0283] Specifically, the time independence indicator is:

[0284] ;

[0285] In the formula, As a time independence indicator, For long-term steps, For medium time steps, For short time steps; when Less than the preset threshold and , , When there is a consistent convergence trend, the time step is determined. satisfy Independence requirements ; In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients;

[0286] Furthermore, in the experimental measurement section, a six-dimensional force sensor was used to synchronously acquire hydrodynamic coefficients under different experimental conditions, forming an experimental calibration dataset; the experimental calibration dataset is as follows:

[0287] ;

[0288] In the formula, To calibrate the dataset for the experiment; , Indicates the working condition number index. This represents the total number of experimental conditions. and The corresponding experimental hydrodynamic coefficient vector;

[0289] Furthermore, to compare the simulated thrust coefficient and the experimental thrust coefficient on the same frequency band and time reference, a bandpass filter is first applied to both the simulated and experimental thrust coefficients, expressed as follows:

[0290] ;

[0291] In the formula, To simulate the thrust coefficient, The experimental thrust coefficient, For bandpass filtering operators, The simulated thrust coefficients after filtering. The filtered experimental thrust coefficients are used; the optimal time delay is obtained through cross-correlation to achieve phase alignment.

[0292] Specifically, the optimal latency is:

[0293] ;

[0294] In the formula, As candidate time delay variables, Indicates all Select the one with the highest cross-correlation , For optimal latency; if This indicates that the experimental sequence lags behind the simulation sequence. ;like This indicates that the experimental sequence is ahead of the simulation. .

[0295] S5, see reference Figure 6 The model employs a "correlation-difference" multi-fidelity decomposition. On the global structure of the low-fidelity model, a system residual function input according to the pectoral fin control parameters is introduced, and independent Gaussian process priors are applied. Simultaneously, a residual regularization term constructed from the momentum and energy conservation control equations is added to the loss, resulting in a generalizable prior-posterior mapping. During prediction, the hydrodynamic mapping model is jointly supported by numerical simulation and experimental calibration datasets. The input pectoral fin control parameters output the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. Subsequently, this posterior mean is calculated as the resultant force of the pectoral fin of the system based on the reference velocity, fluid density, and effective area. This resultant force is then obtained by integrating the lever arm and superimposing the added mass model and the viscous drag torque model, forming a unified parameter transfer chain for subsequent multi-objective optimization.

[0296] Specifically, in numerical simulation datasets With experimental calibration dataset With the joint support of [unclear], a hydrodynamic mapping model was constructed, with the pectoral fin control parameters as input. The output is the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix;

[0297] Specifically, the hydrodynamic mapping model is as follows:

[0298] ;

[0299] In the formula, For time, The overall output of the model consists of two parts: the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. The posterior mean of the hydrodynamic coefficients. , The posterior mean of the thrust coefficient. The posterior mean of the lateral force coefficients. The posterior mean of the lift coefficient. The power coefficient is the posterior mean. The uncertainty covariance matrix, For hydrodynamic mapping model, For pectoral fin control parameters, For numerical datasets, This is the experimental dataset;

[0300] Furthermore, to characterize the systematic differences between the numerical simulation dataset and the experimental calibration dataset, the high-fidelity scalar response is expressed by the autoregressive coefficient as the sum of the scaled low-fidelity scalar response and the system residual term;

[0301] Specifically, the high-fidelity scalar response is:

[0302] ;

[0303] In the formula, For high-fidelity scalar response, represents the experimentally measured hydrodynamic coefficients. The low-fidelity scalar response represents the hydrodynamic coefficients in the numerical simulation. These are scalar autoregressive coefficients. For system residuals;

[0304] Furthermore, independent Gaussian process priors are applied to the low-fidelity scalar response and the system residual term to obtain a learnable prior-posterior mapping model.

[0305] Specifically, the prior-posterior mapping model is as follows:

[0306] ;

[0307] In the formula, This is a prior-posterior mapping model. For low-fidelity process mean function, The mean function of the difference process, For low-fidelity process covariance kernel function, The covariance kernel function for the difference process; This represents another set of pectoral fin control parameters;

[0308] Furthermore, the residual regularization term is:

[0309] ;

[0310] In the formula, At the start time, For time window, For the experimental hydrodynamic coefficient vector, The posterior mean of the hydrodynamic coefficients. For weighted matrices, The physical residuals are constructed from the governing equations; For the weighting factor;

[0311] Furthermore, the equation for the resultant force of the pectoral fins of the machine system is:

[0312] ;

[0313] In the formula, For fluid density, For reference speed, Effective area; The resultant force of the pectoral fins of the machine system is obtained by integrating the moment through the lever arm and compensating with the additional mass model and the viscous drag moment model.

[0314] Furthermore, the resultant torque of the pectoral fins of the machine system is:

[0315] ;

[0316] In the formula, The resultant torque of the pectoral fins of the machine system, The position vector of the point of application relative to the reference point. For the addition of quality model, For viscous drag torque model, For the speed of the machine system, Accelerate the machine system.

[0317] S6, see reference Figure 7 Based on the hydrodynamic mapping model, a multi-objective robust closed-loop optimization is constructed for different task scenarios. First, the interval average and interval root mean square of the hydrodynamic coefficients are calculated within a unified evaluation window. Then, the model switches between three objective layers: "propulsion priority model", "disturbance priority model" and "balanced propulsion model" according to the scenario. A multi-objective evolutionary algorithm is used to solve the problem and force all candidate solutions to satisfy the engineering feasible region. At the same time, the solution is projected into the engineering system using a projection metric weighted matrix. The hydrodynamic mapping model updates the surrogate parameters in small steps along the gradient of the instantaneous loss using an adaptive learning rate according to the online data update method to resist distribution drift and improve adaptability to other working conditions.

[0318] Specifically, the average value of the hydrodynamic coefficient range is:

[0319] ;

[0320] In the formula, To evaluate the start time of the window, To evaluate the window duration, For the first One hydrodynamic coefficient, For interval The average value of the hydrodynamic coefficient over the range, , Indicates different hydrodynamic coefficient indices. Represents the thrust coefficient. Lateral force coefficient, Represents the lift coefficient. Represents the power factor;

[0321] ;

[0322] In the formula, For interval The root mean square of the hydrodynamic coefficients;

[0323] Furthermore, when there is a clear need to advance, the advancement-first model is adopted, expressed as:

[0324] ;

[0325] Specifically, the constraints are:

[0326] ;

[0327] In the formula, min represents finding the minimum value. To advance the weighting function of the priority model, For feasible regions, For pectoral fin control parameters, To and The corresponding power-hydrodynamic coefficient, To and The corresponding average thrust coefficient To and The corresponding average lateral force coefficient, To and The corresponding average lift coefficient, To and The corresponding root mean square of the lateral force coefficient, To and The corresponding root mean square lift coefficient, To avoid small positive numbers with a denominator of zero, The relative threshold of the lateral force coefficient. The relative threshold of the lift coefficient. The threshold for lateral force coefficient jitter. The lift coefficient jitter threshold;

[0328] Furthermore, when the objective is stable movement, a disturbance-priority model is adopted, expressed as:

[0329] ;

[0330] Specifically, the constraints are:

[0331] ;

[0332] In the formula, The weighting function for the disturbance suppression priority model, The relative threshold of the thrust coefficient, This represents the upper limit of the power factor.

[0333] Furthermore, when a trade-off needs to be struck between propulsion and stability, a balanced propulsion model is adopted, expressed as:

[0334] ;

[0335] In the formula, To balance the weighting function of the propulsion model, As the weight value, when taking The time is for advancing the priority model; when and When the primary factor is the disturbance, the model tends to prioritize the disturbance.

[0336] Furthermore, when the multi-objective robust closed-loop optimization model performs optimization iterations, each optimization iteration obtains the coefficient time series by inputting the design variables through the hydrodynamic mapping model, and then calculates the interval average and root mean square and substitutes them into the corresponding objectives and constraints for evaluation in sequence.

[0337] Specifically, the engineering projection is as follows:

[0338] ;

[0339] In the formula, For engineering projection design variables, As candidate design variables, As decision variables, For the engineering feasible region, This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in;

[0340] To elaborate further, the method for updating data online is as follows:

[0341] ;

[0342] In the formula, In the first The data-driven model parameter vector for each evaluation round represents the current parameters of the hydrodynamic mapping model; In the first The updated parameter vector obtained after one round of online calibration; For the first The learning rate for each round; For the first The instantaneous loss function for each round. instantaneous loss function about The gradient of the parameters.

[0343] The background section of this invention may include background information about the problems or environment in which the invention is being developed, and is not necessarily a description of prior art. Therefore, the content included in the background section does not constitute an admission of prior art by the applicant.

[0344] The above description, in conjunction with specific / preferred embodiments, provides a further detailed explanation of the present invention, but it should not be construed that the specific implementation of the present invention is limited to these descriptions. Obviously, those skilled in the art can make various modifications and variations to the present invention without departing from its spirit and scope. Therefore, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention also intends to include these modifications and variations.

Claims

1. A method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, characterized in that, Includes the following steps: S1. Rigid flapping motion consists of forward and backward flapping fin motion, up and down flapping fin motion, and rocking fin motion; flexible wave motion is a superposition and propagation of spanwise and chordwise motion of the fin surface; a unified motion generation framework is established for the pectoral fin-body-flow field, defining the pectoral fin system as... The system is Adopting intrinsic The Euler angle sequence establishes the coordinate transformation relationship, and the pectoral fin system is mapped to the machine system through the active rotation direction matrix of the pectoral fin, and the inverse mapping form to avoid gimbal lock is given; The pectoral fin motion model consists of a rigid flapping equation of the fin bones and a flexible wave equation of the fin surface. The rigid flapping equation of the fin bones is determined by the flapping amplitude, flapping envelope function, flapping frequency, and flapping phase. The flexible wave equation of the fin surface is described by the spatial amplitude envelope function of the fin surface, the time envelope function of the fin surface motion, and the cosine relationship of the total phase of the fin surface. S2. Within the system, the rigid flapping equation of the fin bone and the flexible wave equation of the fin surface are coupled to generate a unified pectoral fin flapping wave coupled displacement equation. The pectoral fin flapping wave coupled displacement equation includes the fin root reference point position, the initial position of the pectoral fin, the flexible wave equation of the fin surface, and the passive rotation direction matrix of the pectoral fin. By combining the pectoral fin flapping wave coupled displacement equation with the pectoral fin flapping angular velocity and the time derivative of the normal basis vector within the system, the pectoral fin flapping wave velocity equation is obtained. The pectoral fin flapping wave velocity equation is then decomposed into normal and tangential components within the system to obtain the pectoral fin flapping wave coupled normal velocity and the pectoral fin flapping wave coupled... Tangential velocity; To provide a unified interface, the pectoral fin flapping velocity equation is written as a linear combination of the rigid body geometric Jacobian matrix multiplied by the pectoral fin flapping angular velocity and the flexible body geometric Jacobian matrix multiplied by the minimum parameter velocity of the flexible wave, so that the controller and the computational fluid dynamics mesh can directly call the same set of state variables; To ensure engineering feasibility, the trajectory satisfies the feasible region constraint within the generator layer, and the commanded angular displacement and commanded angular velocity are limited by soft saturation, and power gating is used for same-scale shrinkage to ensure that the instantaneous power does not exceed the power upper limit, while ensuring that the pectoral fin flapping angular velocity does not exceed the limit; S3. A generator layer scheme combining a reduced-order dynamic model and physical supervision is proposed. In the offline stage, the shape basis is extracted from the high-fidelity snapshot using intrinsic orthogonal decomposition, and the low-order shape basis and the reduced-order dynamic model constructed by dynamic mode decomposition are trained. In the online stage, the gold standard deformation snapshot is output by the low-frequency physical snapshot generator operator. At each sparse correction time, the gold standard deformation snapshot is mapped to the modal coordinates through weighted least squares projection. The modal coordinates obtained by projection are fused with the predicted modal coordinates by the supervision correction method of residual minimization and prior covariance regularization, and the corrected modal coordinates are updated to form an intermittent closed loop of online prediction-correction. S4. Based on the kinematics of S2, establish a parallel acquisition and cross-calibration process for numerical simulation and water tank experiment, and ensure the consistency of the calculation through unified data reduction and deviation calibration. The hydrodynamic coefficient vector is defined as consisting of thrust coefficient, lateral force coefficient, lift coefficient, and power coefficient. These hydrodynamic coefficients are converted into instantaneous fluid forces and instantaneous power of the system using fluid density, reference velocity, and effective area. In the numerical simulation, the unsteady Reynolds-averaged Navier-Stokes method is used. Geometric motion is implemented using a moving mesh, and no-slip boundary conditions are applied to the fluid-solid interface. The inlet flow velocity is set, the outlet pressure is constant, and there is no slip on the solid wall. Based on this, a numerical simulation dataset is constructed, consisting of pectoral fin motion control parameters and their corresponding simulated water... The dynamic coefficient vector is composed of parameters. To control numerical accuracy, grid independence and time independence indices are verified. In the experimental measurement section, a six-dimensional force sensor is used to synchronously acquire data with the pose, forming an experimental calibration dataset. The dataset consists of pairs of pectoral fin motion control parameters and corresponding experimental hydrodynamic coefficients. To compare thrust coefficients on the same frequency band and time reference, bandpass filtering operators are first applied to the simulated thrust coefficients and experimental thrust coefficients to obtain filtered sequences. Then, cross-correlation is used to find the optimal time delay that maximizes the cross-correlation value on the candidate time delays to complete phase alignment. S5. Using the "association-difference" multifidelity decomposition, based on the global structure of the low-fidelity model, the systematic bias is corrected by introducing the system residual function input according to the pectoral fin control parameters, thereby forming a generalizable prior-posterior mapping during the training phase. In the prediction phase, the hydrodynamic mapping model is jointly supported by the numerical simulation dataset and the experimental calibration dataset. It takes the pectoral fin control parameters as input and outputs the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. In order to characterize the system differences between the numerical simulation dataset and the experimental calibration dataset, the low-fidelity scalar response is scaled by the autoregressive coefficients and the system residual terms dependent on the pectoral fin control parameters are superimposed to obtain the high-fidelity scalar response. Independent Gaussian process priors are applied to the low-fidelity scalar response and the system residual term, respectively, to construct a learnable prior-posterior mapping model. To improve the stability and reliability under complex conditions, residual regularization terms constructed by the momentum and energy conservation control equations are superimposed on the loss function, and the data fitting error and physical residual are jointly constrained by a trade-off coefficient within a given time window. Subsequently, the posterior mean of the hydrodynamic coefficients is converted into the resultant force of the pectoral fins of the machine system according to the reference velocity, fluid density, and effective area. The moment is obtained by integrating the lever arm and superimposing the additional mass model and viscous drag moment model. The parameter transfer chain formed here is that the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix are obtained from the pectoral fin control parameters through the hydrodynamic mapping model, and then the resultant force and moment of the pectoral fins of the machine system are obtained through mechanical mapping, which serve as the unified interface quantity for subsequent multi-objective optimization. S6. Based on the hydrodynamic mapping model, construct a multi-objective robust closed-loop optimization model for different mission scenarios. To ensure that the objective definition corresponds to the mission scenario, calculate the average value and root mean square of the hydrodynamic coefficient interval within a unified evaluation window. Then, switch between three objective layers according to the mission scenario: the first is the thrust-priority model, which maximizes the average thrust while ensuring power control, and uses the relative thrust ratio and jitter threshold to constrain lateral force and lift simultaneously; the second is the disturbance suppression-priority model, which focuses on minimizing the average value and root mean square of lateral force and lift, while reducing net thrust to near zero and limiting the upper limit of power. Thirdly, the balanced propulsion model uses a weighted summation of the same family of indicators to achieve a trade-off between propulsion, energy consumption, and disturbance resistance. The multi-objective robust closed-loop optimization model consists of the propulsion-first model, the disturbance-first model, and the balanced propulsion model. A multi-objective evolutionary algorithm is used to solve the multi-objective robust closed-loop optimization model, and all candidate solutions always satisfy the engineering feasible region. To ensure engineering feasibility, the candidate solutions are projected onto the engineering projected matrix through a projection metric to ensure that they strictly fall within the feasible region. At the same time, the hydrodynamic mapping model uses an online data update method, updating the surrogate parameters in small steps along the gradient of the instantaneous loss with an adaptive learning rate in each evaluation round to resist distribution drift and improve adaptability to other operating conditions.

2. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... The fore and aft flapping wing movements around the pectoral fin system The axis rotates, and the upper and lower flapping wings move around the pectoral fin system. The axis rotates, and the rocker motion revolves around... The axis rotates; the spanwise motion of the fin surface is transmitted from the fin root to the fin tip, and the chordal motion of the fin surface is transmitted from the leading edge to the trailing edge; the transformation matrix from the pectoral fin system to the muscular system is defined as the active rotation direction matrix of the pectoral fin, and the transformation matrix from the muscular system to the pectoral fin system is defined as the passive rotation direction matrix of the pectoral fin; the expression for the passive rotation direction matrix of the pectoral fin is: ； In the formula, For time, This is the matrix representing the passive rotation direction of the pectoral fins. This refers to the angular displacement of the front and rear flaps. This refers to the angular displacement of the upper and lower flaps. This is the angular displacement of the wing. For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of axis For the pectoral fin system Intrinsic rotation matrix of the axis; The expression for the cosine matrix of the active rotation direction of the pectoral fin is: ； In the formula, The matrix represents the active rotation direction of the pectoral fin, with symbols... This indicates transpose; the active rotation direction matrix of the pectoral fin is used for the rotation vector itself, and the passive rotation direction matrix of the pectoral fin is used for coordinate transformation; the two are transposes of each other. The forward and aft flap angular displacements, the vertical flap angular displacements, and the rocking wing angular displacements are mapped to the pectoral fin flapping angular velocity of the aircraft system, and the expression is as follows: ； In the formula, This refers to the displacement of the pectoral fin flapping angle; The angular velocity of the pectoral fin flapping motion. For the front and rear flapping angular velocities, The angular velocity of the upper and lower flapping wings, This refers to the angular velocity of the wing. The angular velocity of the pectoral fin flapping of the machine system, For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction For the system along the machine Angular velocity component of pectoral fin flapping direction; The pectoral fin angular velocity mapping matrix represents the result of... Sequential Euler angular velocity Mapped to the pectoral fin flapping angular velocity of the machine system ; The equation for the rigid flapping motion of the fin bones in the pectoral fin motion model is: ； In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first The flapping angular displacement of the degree of freedom For the first Bounce amplitude of degrees of freedom For the first The beat envelope function with degrees of freedom is used for smooth modulation of amplitude over time and asymmetric shaping. For the first Beating frequency of degrees of freedom For the first Bounce phase of degrees of freedom; The expression for the beat envelope function is: ； In the formula, Pi As a constant bias term, it determines The baseline; For the first The cosine coefficient of each harmonic. ; For the first The sinusoidal coefficient of each harmonic. ; , Indicates harmonic index, The number of harmonic terms controls the spectral resolution; spline basis functions For node vectors, For the first The weights of the individual spline basis are used for amplitude modulation in local time intervals; , Represents a spline base index. Number of spline bases; Taking the time derivative of the rigid flapping equation of the fin bone, we obtain the angular velocity equation for the flapping of the pectoral fin, which is expressed as: ； In the formula, Indicates the index of the degrees of freedom of the flapping motion. For the first Angular velocity of flapping motion of degrees of freedom The time first derivative of the flapping envelope function; The equation for the flexible wave pattern of the fin surface in the pectoral fin motion model is: ； In the formula, For time, The displacement is the flexible wave motion of the fin, indicating the location of... The displacement of the material points on the fin surface along the local normal; The spatial amplitude envelope function of the fin describes the distribution of deformation intensity along the spanwise and chordwise directions. Let be the time envelope function of the fin motion, describing the gradual in and out modulation over time; The overall phase of the fin determines the propagation direction and velocity of the fin crests and troughs; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , Let be the chord length of the pectoral fin; the expression for the spatial amplitude envelope function of the fin surface is: ； In the formula, To display the envelope coefficient of the fin surface, The tangential envelope coefficient of the fin surface. The nominal amplitude; the expression for the fin motion time envelope function is: ； In the formula, For time, The angular frequency of the traveling wave. , The frequency of the traveling wave; , Indicates the harmonic envelope index. The harmonic order; For the first Time envelope harmonic amplitude coefficient, adjustment The intensity of fluctuations; For the first The first-order time envelope harmonic phase is used to control the position of the envelope peak and valley on the time axis; the total phase of the fin is... ； In the formula, To determine the spanwise wavenumber, along Directional phase propulsion rate; This is the initial phase. For spanwise phase gradient, The chordal phase gradient is: The spanwise phase gradient is: ； In the formula, Let be the spanwise phase density function, and let be the chordwise phase gradient. ； In the formula, The phase density function is a chordal direction. To calculate the normal velocity, the time derivative of the fin flexible wave equation is taken to obtain the fin flexible wave velocity equation, which is: ； In the formula, The fin surface flexible wave velocity, Let be the time first derivative of the fin motion time envelope function. Let be the first time derivative of the total phase of the fin, and denoted as the local phase angular velocity; the first time derivative of the total phase of the fin is: ； In the formula, The time first derivative of the spanwise phase gradient, It is the time first derivative of the chordal phase gradient.

3. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... The equation for the coupled displacement of the pectoral fin beat wave is: ， In the formula, For time, These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; Control points Spatial position vector in the machine system The location of the fin root reference point in the machine system. This is the matrix representing the passive rotation direction of the pectoral fins. The initial position of the fin, i.e. Time control point Coordinates within the pectoral fin system; Control points Unit normal in the pectoral fin system , The normal basis vector of the pectoral fin of the machine system; This refers to the flexible wave displacement of the fin surface; Taking the time derivative of the pectoral fin beat wave coupled displacement equation, we obtain the pectoral fin beat wave coupled velocity equation, which is expressed as follows: ； In the formula, The coupling velocity of the pectoral fin beat wave. The fin surface flexible wave velocity, The normal basis vector of the pectoral fin system; The first term represents the angular velocity of the pectoral fin flapping motion of the entire system; the second term represents rigid body induction, indicating the point velocity caused by the overall flapping motion of the pectoral fin. The first term is the rotational speed around the instantaneous hinge; the second term is the flexibility change, representing the change in normal deflection caused by the flexible traveling wave over time and the contribution of the normal direction itself as it evolves with deformation. The normal velocity of the pectoral fin beat wave coupling is: ； In the formula, The normal velocity of the pectoral fin beat wave coupling; The tangential velocity of the pectoral fin beat wave coupling is: ； In the formula, The pectoral fin beat wave coupled tangential velocity; the pectoral fin beat wave coupled normal velocity is used for immersion boundary conditions, and the pectoral fin beat wave coupled tangential velocity is used for no-slip condition construction and dynamic mesh update; To ensure consistency with the hardware controller and dynamic mesh interface, the pectoral fin beat wave velocity equation is rewritten as follows: ； In the formula, The angular velocity of the pectoral fin flapping motion. For the minimum parameter of flexible fluctuation, velocity, For a rigid body, the geometric Jacobian matrix is... The Jacobian matrix of the flexible body geometry; To ensure project feasibility, the trajectory inherently satisfies feasibility constraints at the generation layer, and is subject to real-time pruning using soft saturation and power gating when necessary; soft saturation is applied to the pectoral fin flapping angular displacement and pectoral fin flapping angular velocity. ； In the formula, the subscript "cmd" represents the corresponding instruction quantity, and the subscript "max" represents the corresponding upper limit. This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. This is the commanded angular displacement after soft saturation. The command angular velocity after soft saturation. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in; through Saturation softly limits angular displacement and angular velocity to a range achievable by hardware. After soft saturation, a contraction of the same scale is performed to ensure that the estimated instantaneous power does not exceed the upper limit and the pectoral fin flapping angular velocity does not exceed the limit: ； In the formula, The equivalent actuation torque of the rigid body joint; This is the equivalent generalized force of fin surface flexible wave; For instantaneous power estimation; This is the upper limit of power.

4. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... The intrinsic orthogonal decomposition shape basis is extracted from the high-fidelity snapshot using intrinsic orthogonal decomposition; the intrinsic orthogonal decomposition shape basis is reconstructed as follows: ； In the formula, For time, , Indicates modal index, The modal number; For the first A spatial mode, For the first Modal coordinates; This refers to the flexible wave displacement of the fin surface; These are the coordinates of the fin surface parameters; This is the span coordinate of the pectoral fin, along the fin base to the fin tip. , For the elongation of the pectoral fins; The coordinates for the pectoral fin are chordal, from the anterior edge to the posterior edge. , The length of the pectoral fin chord; It is an intrinsic orthogonal decomposition shape basis. To be Column vectors stacked on a discrete grid; For modal coordinates, ; The reduced-order dynamic model is as follows: ； In the formula, For modal coordinates, the time derivative is... For the intermodal coupling system matrix, For external driving input matrix; The driving vector is derived from the pectoral fin control parameters; This refers to continuous-time process noise. The gold label deformation snapshot is as follows: ； In the formula, Indicates the index value. No. Secondary sparse correction time. At any moment A snapshot of the gold standard deformation. For pectoral fin control parameters, For generating operators for physical snapshots, given pectoral fin control parameters With time Below, a numerical snapshot of the fin's flexible wave displacement is output by a high-fidelity simulator; the weighted least squares estimated modal coordinates are: ； In the formula, exist Weighted least squares estimation of modal coordinates, It is an intrinsic orthogonal decomposition shape basis. A positive definite weighting matrix used to define the projection metric; As a weighted pseudo-inverse operator, the gold standard deformation snapshot is projected onto modal coordinates; The expression for the supervised correction method is: ； In the formula, To predict modal coordinates, To correct the modal coordinates, To predict covariance, reflecting Uncertainty; For regularization weights, ; The expression for the online cyclic prediction-correction is: ； Thus, while maintaining real-time performance, extrapolation drift is suppressed and physical consistency is preserved.

5. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... The hydrodynamic coefficient vector is: ； In the formula, For time, This is the hydrodynamic coefficient vector; The thrust coefficient represents the thrust along the fuselage. Instantaneous value of axial hydrodynamic coefficient; The lateral force coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; The lift coefficient represents the force along the body. Instantaneous value of axial hydrodynamic coefficient; For power coefficient, Indicates vector transpose; the instantaneous fluid force is: ； In the formula, For thrust, it indicates the force along the body. Instantaneous fluid force in the axial direction; The force is lateral, representing the force along the body. Instantaneous fluid force in the axial direction; For lift, it means along the body Instantaneous fluid force in the axial direction; For fluid density, For reference speed, The effective area is; the instantaneous power is: ； In the formula, Instantaneous power; In the numerical simulation section, pectoral fin control parameters are generated based on computational fluid dynamics. The numerical simulation dataset corresponding to the hydrodynamic coefficient vector is expressed as follows: ； In the formula, For time, This is a numerical simulation dataset; , Indicates the working condition number index. This represents the total number of simulation conditions. For the first Pectoral fin control parameters for the group's operating conditions; To and The corresponding simulated hydrodynamic coefficient vector; In terms of numerical accuracy control, constructing mesh sequences With time step sequence Convergence is evaluated using grid independence and time step independence indices; the grid independence index is: ； In the formula, Indicates the values ​​taken in each direction. As a grid independence index, Coarse grid Medium grid For fine mesh; when Less than the preset threshold and , , When the mesh exhibits a monotonically converging trend, it is determined that the mesh division satisfies the independence requirement. For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients, For in the grid Upper The periodic average of the coefficients; the time independence index is: ； In the formula, As a time independence indicator, For long-term steps, For medium time steps, For short time steps; when Less than the preset threshold and , , When there is a consistent convergence trend, the time step is considered to meet the independence requirement. In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients, In time step Next The periodic average of the coefficients; In the experimental measurement section, a six-dimensional force sensor was used to synchronously acquire hydrodynamic coefficients under different experimental conditions, forming an experimental calibration dataset; the experimental calibration dataset is as follows: ； In the formula, To calibrate the dataset for the experiment; , Indicates the working condition number index. This represents the total number of experimental conditions. and The corresponding experimental hydrodynamic coefficient vector; To compare the simulated thrust coefficients and experimental thrust coefficients on the same frequency band and time base, a bandpass filter is first applied to both the simulated and experimental thrust coefficients, expressed as follows: ； In the formula, To simulate the thrust coefficient, The experimental thrust coefficient, For bandpass filtering operators, The simulated thrust coefficients after filtering. The filtered experimental thrust coefficients are used; the optimal time delay is obtained through cross-correlation to achieve phase alignment, and the optimal time delay is: ； In the formula, As candidate time delay variables, Indicates all Select the one with the highest cross-correlation , For optimal latency; if This indicates that the experimental sequence lags behind the simulation sequence. ;like This indicates that the experimental sequence is ahead of the simulation. .

6. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... In numerical simulation datasets With experimental calibration dataset With the joint support of [unclear], a hydrodynamic mapping model was constructed, with the pectoral fin control parameters as input. The output is the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix; the hydrodynamic mapping model is: ； In the formula, For time, The overall output of the model consists of two parts: the posterior mean of the hydrodynamic coefficients and the uncertainty covariance matrix. The posterior mean of the hydrodynamic coefficients. , The posterior mean of the thrust coefficient. The posterior mean of the lateral force coefficients. The posterior mean of the lift coefficient. The power coefficient is the posterior mean. The uncertainty covariance matrix, For hydrodynamic mapping model, For pectoral fin control parameters, For numerical datasets, This is the experimental dataset; To characterize the systematic differences between the numerical simulation dataset and the experimental calibration dataset, the high-fidelity scalar response is expressed as the sum of the scaled low-fidelity scalar response and the system residual term using autoregressive coefficients; the high-fidelity scalar response is: ； In the formula, For high-fidelity scalar response, represents the experimentally measured hydrodynamic coefficients. The low-fidelity scalar response represents the hydrodynamic coefficients in the numerical simulation. These are scalar autoregressive coefficients. For system residuals; Subsequently, independent Gaussian process priors are applied to the low-fidelity scalar response and the system residual term to obtain a learnable prior-posterior mapping model; the prior-posterior mapping model is as follows: ； In the formula, This is a prior-posterior mapping model. For low-fidelity process mean function, The mean function of the difference process, For low-fidelity process covariance kernel function, The covariance kernel function for the difference process; This represents another set of pectoral fin control parameters; The residual regularization term is: ； In the formula, At the start time, For time window, For the experimental hydrodynamic coefficient vector, The posterior mean of the hydrodynamic coefficients. For weighted matrices, The physical residuals are constructed from the governing equations; For the weighting factor; The equation for the resultant force of the pectoral fins of the aforementioned system is: ； In the formula, For fluid density, For reference speed, Effective area; The resultant force of the pectoral fins of the machine system is calculated by integrating the force arms and compensating with the additional mass model and the viscous drag torque model. The resultant torque of the pectoral fins of the machine system is then obtained. ； In the formula, The resultant torque of the pectoral fins of the machine system, The position vector of the point of application relative to the reference point. For the addition of quality model, For viscous drag torque model, For the speed of the machine system, Accelerate the machine system.

7. The method for optimizing the coordinated motion parameters of a rigid-flexible coupled biomimetic pectoral fin beat wave, as described in claim 1, is characterized in that... The average value of the hydrodynamic coefficient range is: ； In the formula, To evaluate the start time of the window, To evaluate the window duration, For the first One hydrodynamic coefficient, For interval The average value of the hydrodynamic coefficient over the range, , Indicates different hydrodynamic coefficient indices. Represents the thrust coefficient. Lateral force coefficient, Represents the lift coefficient. Represents the power factor; ； In the formula, For interval The root mean square of the hydrodynamic coefficients; First, when there is a clear need to advance, the advancement priority model is adopted, and the expression is: ； The constraints are: ； In the formula, min represents finding the minimum value. To advance the weighting function of the priority model, For feasible regions, For pectoral fin control parameters, To and The corresponding power-hydrodynamic coefficient, To and The corresponding average thrust coefficient To and The corresponding average lateral force coefficient, To and The corresponding average lift coefficient, To and The corresponding root mean square of the lateral force coefficient, To and The corresponding root mean square lift coefficient, To avoid small positive numbers with a denominator of zero, The relative threshold of the lateral force coefficient. The relative threshold of the lift coefficient. The threshold for lateral force coefficient jitter. The lift coefficient jitter threshold; Secondly, when the objective is stable movement, the aforementioned disturbance priority model is adopted, and its expression is: ； Constraints: ； In the formula, The weighting function for the disturbance suppression priority model, The relative threshold of the thrust coefficient, This represents the upper limit of the power factor. Finally, when a trade-off is needed between propulsion and stability, the aforementioned balanced propulsion model is adopted, with the expression: ； In the formula, To balance the weighting function of the propulsion model, As the weight value, when taking The time is the aforementioned priority model; when and When the primary factor is the disturbance priority model, the model will be biased towards the disturbance priority model. The multi-objective robust closed-loop optimization model, during optimization iteration, obtains the coefficient time series through the hydrodynamic mapping model based on the design variable input, and then calculates the interval average and root mean square, substituting them into the corresponding objective and constraint evaluation sequence; the engineering projection is: ； In the formula, For engineering projection design variables, As candidate design variables, As decision variables, For the engineering feasible region, This refers to the displacement of the pectoral fin flapping angle. The angular velocity of the pectoral fin flapping motion. For feasible regions The upper limit of angular displacement in For feasible regions The upper limit of angular velocity in the data; the online data update method is as follows: ； In the formula, In the first The data-driven model parameter vector for each evaluation round represents the current parameters of the hydrodynamic mapping model; In the first The updated parameter vector obtained after one round of online calibration; For the first The learning rate for each round; For the first The instantaneous loss function for each round. instantaneous loss function about The gradient of the parameters.

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