Calibration method for eddy current sensor of small diameter metal ball

Abstract

The application relates to the technical field of eddy current sensors and voltage calibration. The eddy current sensor calibration method for small-diameter metal balls comprises four parts of coordinate system establishment, calibration data acquisition, calibration sample set construction and calibration equation coefficient optimization. Through innovative coordinate system definition and position parameterization, combined with an intelligent optimization algorithm, the application successfully establishes an accurate voltage-position relationship model capable of accurately representing the small-diameter metal ball center deviating from the actual effective measurement axis of the sensor, and effectively compensates for the nonlinear measurement error caused by the ball center eccentricity, surface high curvature and sensor axis deviation.

Description

Technical Field

[0001] This application relates to the fields of eddy current sensors and voltage calibration technology, and more specifically, to a calibration method for an eddy current sensor for a small-diameter metal sphere. Background Technology

[0002] The content in this section provides only background information related to this application and may not constitute prior art.

[0003] Eddy current sensors operate based on the eddy current effect. A high-frequency alternating current is passed through the excitation coil inside the probe, generating an alternating magnetic field. When this magnetic field acts on the surface of a metal sphere, it induces a closed loop current, i.e., an eddy current, on the surface of the sphere. This eddy current itself generates a reverse magnetic field, which in turn alters the equivalent impedance of the probe coil (mainly manifested as changes in inductance and resistance). By detecting the change in coil impedance through a measuring circuit (usually converted into a voltage signal output), the distance information between the probe and the surface of the metal sphere can be reflected. When measuring a flat surface or a metal sphere with a sufficiently large diameter (typically 5 to 10 times the probe diameter), the magnetic field lines are relatively uniformly distributed, the eddy current coupling is stable, and a relatively linear correspondence can be established between the sensor output voltage and the distance, thus achieving high-precision distance measurement.

[0004] When measuring small-diameter metal spheres (diameter less than 5 times the probe diameter), the large surface curvature of the sphere leads to severely uneven distribution of the magnetic field generated by the probe on the sphere's surface, resulting in significant changes in the eddy current coupling region and intensity. Particularly when the center of the sphere is not precisely aligned with the sensor's theoretical axis, the local curvature at the measurement point further increases, causing a complex nonlinear relationship between the sensor's output voltage and the actual distance, resulting in significant measurement errors. Furthermore, the eddy current sensor itself may experience zero drift, differences in coil material, uneven winding processes, or imperfect internal magnetic field distribution, causing its theoretical central axis to not perfectly coincide with the actual effective measurement axis. This axial deviation directly affects the judgment of the measurement point's position relative to the sphere's center, further exacerbating the deviation between the output voltage and the ideal voltage.

[0005] Currently, a precise model has not been effectively established for the relationship between the spatial position (including distance and eccentricity) of a small-diameter metal sphere (especially when the center of the sphere is off-axis) and the output voltage of an eddy current sensor. Furthermore, mature compensation methods are lacking to effectively reduce the nonlinear effects of highly curved surfaces and sensor axial deviations on the measured voltage. Summary of the Invention

[0006] The summary section of this application is intended to provide a brief overview of the concepts, which will be described in detail in the detailed description section below. This summary section is not intended to identify key or essential features of the claimed technical solutions, nor is it intended to limit the scope of the claimed technical solutions.

[0007] Some embodiments of this application propose a calibration method for eddy current sensors for small-diameter metal spheres to address the technical problems mentioned in the background section above.

[0008] As a first aspect of this application, some embodiments of this application provide a method for calibrating an eddy current sensor for a small-diameter metal sphere, comprising the following steps:

[0009] Step 1: Take the lowest point of the range on the central axis of the eddy current sensor as the origin O. C Construct a calibration coordinate system O C -X C Y C Z C X C Axis, Y C Axis, Z C The axis is parallel to the direction of motion of the degrees of freedom of the calibration instrument;

[0010] The origin O is the point whose center coordinates within the range of the small-diameter metal sphere are closest to the probe plane of the eddy current sensor. S Construct a measurement coordinate system O S -X S Y S Z S And take the direction of the largest voltage gradient as Z. S Axis, O S There are cases where the axis is not collinear with the central axis;

[0011] The relative positional relationship between the center of the small-diameter metal sphere and the eddy current sensor is described by l and r;

[0012] Where l represents the distance between the center of the small-diameter metal sphere and the X coordinate system. S O S Y S The distance between the planes, r, represents the distance between the center of the small-diameter metal sphere and the Z-axis in the measurement coordinate system. S Distance between axes;

[0013] Based on the positional relationship between the center of the small-diameter metal sphere and the center of the eddy current sensor, a voltage-position calibration equation is constructed.

[0014] Step 2: Move the small-diameter metal ball along the preset calibration trajectory in the calibration coordinate system to obtain several calibration points and the voltage readings corresponding to the calibration points;

[0015] Step 3: Take the calibration point and the voltage reading at the calibration point as a calibration sample, and obtain all the calibration samples to get the calibration sample set;

[0016] Step 4: Use the particle swarm optimization algorithm to dynamically update the fitting coefficients in the voltage calibration equation to obtain an accurate voltage-position calibration equation.

[0017] This calibration method effectively overcomes the strong nonlinear effects caused by the high curvature of the small-diameter metal sphere surface, the deviation of the sphere center from the sensor axis, and the deviation of the sensor's own axis by constructing a measurement coordinate system and voltage-position calibration equations that include the eccentricity r, and uses the particle swarm optimization algorithm to dynamically optimize the fitting coefficients. It significantly improves the measurement accuracy of the distance l from the probe to the sphere center and the eccentricity r, and provides reliable technical support for high-precision eddy current ranging under small-size and non-ideal alignment conditions.

[0018] Furthermore, the voltage-position calibration equation is as follows:

[0019] ;

[0020] k1~k 10 These are 10 fitting coefficients to be calibrated.

[0021] The voltage-position calibration equation provided in this application can accurately describe the relationship between the position of the ball and the voltage, thereby establishing an accurate and precise relationship model by calibrating the corresponding coefficients.

[0022] Furthermore, step 2 includes the following steps:

[0023] Step 21: In the X coordinate system of the calibration system C Axis, Y C Axis, Z C The effective range of the eddy current sensor is divided into calibration points based on fixed intervals Δx, Δy, and Δz, forming equally spaced calibration points.

[0024] Step 22: Position the small-diameter metal ball at a Z-axis angle. C The axis starts with a column and passes through equally spaced calibration points in sequence.

[0025] Step 23: Obtain the eddy current sensor readings of the small-diameter metal ball at each equally spaced calibration point.

[0026] This equally spaced gridded calibration path design, through three-axis fixed step size division and systematic traversal prioritizing the central axis, achieves full coverage of the sensor's measurement range space and high-density uniform sampling, providing a structured data foundation for particle swarm optimization and significantly improving the calibration accuracy and efficiency in complex nonlinear regions.

[0027] Furthermore, in step 23, a voltage threshold is preset, and the readings of the small-diameter metal ball near the equally spaced calibration points are monitored in real time. When the reading of the eddy current sensor is less than the voltage threshold, the reading of the eddy current sensor at this time is taken as the effective calibration voltage of the calibration point, and the average value of the effective calibration voltage is taken as the calibration voltage of the calibration point.

[0028] By setting a voltage threshold and monitoring the reading changes in real time, this method can intelligently determine the moment when a small-diameter metal ball stably arrives at the calibration point during its movement (when the reading change is less than the threshold), thereby quickly and accurately capturing the stable calibration voltage value corresponding to that position. This effectively avoids reading deviations caused by mechanical vibration or moving inertia, and significantly improves the efficiency and reliability of calibration data acquisition.

[0029] Furthermore, step 4 includes the following steps:

[0030] Step 41: Apply the voltage-position calibration equation and define the fitting coefficient k. j Domain U j , j represents the index of the fitting coefficient, j∈{1、2...10}, U j This represents the domain of the j-th fitting coefficient;

[0031] Step 42: Randomly select a value for each fitting coefficient within its domain to generate one particle x. i Randomly generate N particles to obtain the initial population;

[0032] , This represents the value of the j-th fitting parameter in the i-th particle;

[0033] ;

[0034] Step 43: Define a fitness function F(x) such that after substituting the particle swarm parameters into the calibration equation, the error between the calculated voltage and the measured voltage of each calibration sample is minimized;

[0035] Step 44: Calculate the fitness function value of each particle, and iterate based on the fitness function value of the population until the preset termination condition is reached.

[0036] This application achieves global collaborative optimization of 10 fitting coefficients by defining a coefficient constraint domain, randomly initializing the population, and using a fitness function that minimizes voltage error. This effectively overcomes the strong nonlinearity problem caused by high curvature and axis deviation, ensuring that the voltage-position calibration equation accurately approximates the measured data and significantly improves the spatial positioning accuracy of small-diameter metal spheres.

[0037] Step 44 includes the following steps:

[0038] Step 441: Calculate the fitness function value F(x) of all particles in the initial population N0. i This yields the probability that each particle is selected as part of the crossover population. , Represents particles The probability of being selected as a crossover population;

[0039] ;

[0040] ;

[0041] This represents the sum of the fitness function values ​​of all particles;

[0042] Step 442: Based on the probability of each particle Each particle is mapped to a closed circle, and the area of ​​the sector mapped to each particle is... ; ; Represents the total area of ​​the circle;

[0043] Step 443: Randomly generate discrete points within the circle that have the same number of particles as in the population at the t-th iteration, extract the particles corresponding to the region of each discrete point, and obtain the crossover population.

[0044] Step 444: Randomly pair up the particles in the crossover population. For each pair of particles x... a x b Randomly generate the intersection point G;

[0045] ; Represents particle x a The j-th fitted parameter;

[0046] ; Represents particle x b The j-th fitted parameter;

[0047] Particle x a x b At the intersection point G, a new particle c is formed. a c b ;

[0048] ; c a The G-th fitted parameter;

[0049] ; c b The G-th fitted parameter;

[0050] Step 445: Fitting parameters for the new particles Based on the update probability up j Choose whether to perform an update; if not, fit the parameters. This is the original value. If an update is needed, the update formula is as follows:

[0051] ;

[0052] Represents the fitting parameters The updated value, L j U represents the lower bound of the j-th fitted parameter. j This represents the upper limit of the j-th fitted parameter. This represents a random number, ranging from 0 to 1.

[0053] Step 446: Repeat steps 442 to 445 until the termination condition is met.

[0054] The adaptive crossover and mutation mechanism in this particle swarm optimization effectively maintains population diversity by selecting dominant particles through a circular mapping based on fitness probability, reorganizing the parameter space through random crossover points, and performing random updates to parameters according to probability. This avoids the problem of traditional algorithms easily getting trapped in local optima in strongly nonlinear optimization and significantly improves the global fitting accuracy and convergence stability of the calibration equation for complex working conditions with high curvature and axis deviation.

[0055] Furthermore, in step 444, the random intersection point G is generated as follows:

[0056] Step 4441: Assign all fitting parameters k1~k 10 Generate intersection points and generate initial selection factors p1~p 10 The initial selection factor gradually increases;

[0057] Step 4442: In each iteration, update the selection factor p for the previously selected intersection point G. G A new selection factor p is obtained. G `;

[0058] ;

[0059] in, This represents the number of particles in the population at the t-th iteration. The number of particles in the population at the next iteration. This represents the updated selection factor. Let G represent the selection factor before the update, and let G = {1, 2, ..., 10}.

[0060] Step 4443: Add the new selection factor p G Replace the selection factor of the corresponding fitting parameter, and randomly select a new fitting parameter as the crossover point based on the roulette algorithm;

[0061] The probability that the fitting parameter j is chosen as the intersection point is Pt. j ;

[0062] ; This represents the selection factor for the G-th fitting parameter in the t-th iteration.

[0063] This dynamic crossover point generation mechanism iteratively updates the parameter selection probability and adaptively adjusts the crossover position using a roulette wheel method, enabling the optimization process to prioritize key fitting parameters (such as high sensitivity coefficients) while ensuring a balanced exploration of all parameter spaces. This effectively solves the problem of insufficient optimization of some parameters caused by traditional fixed crossover points, and significantly improves the efficiency of parameter co-optimization and global convergence accuracy of the particle swarm optimization algorithm in strongly nonlinear calibration equations.

[0064] Furthermore, in step 445, the update probability is up. j Related to the number of iterations and the number of terms;

[0065] ;

[0066] Where t represents the number of iterations, m represents the sum of the number of terms of the fitted parameters, e represents the natural constant, and f(t) represents the iteration function;

[0067] When t < 80 ;

[0068] When 120≥t≥80, ;

[0069] When T≥t>120 b0 is a constant factor.

[0070] This dynamic update probability mechanism intelligently couples the iterative process with the nonlinearity of the parameters. In the early stages of optimization, it maintains a high mutation intensity to fully explore the solution space. In the middle stages, it smoothly reduces the perturbation amplitude as iteration progresses to balance global search and local exploitation. In the later stages, it significantly suppresses random updates to achieve stable convergence. That is, in the early stages of iteration, most of the fitted parameters undergo random changes, while in the later stages, the focus shifts to the interaction between fitted parameters, with only a few parameters themselves being updated. Furthermore, this scheme automatically strengthens the perturbation probability of higher-order terms based on the degree of parameter complexity. This effectively avoids the premature convergence risk of traditional fixed-probability strategies in strongly nonlinear optimization, while ensuring fine-tuning of key parameters under high curvature and eccentric conditions. Ultimately, it achieves a synergistic improvement in the convergence speed and fitting accuracy of the calibration equations, especially enhancing the robustness of complex spatial pose measurements.

[0071] Furthermore, in step 441, the fitness function value F(x) i ) represents the mean squared error of all samples.

[0072] ;

[0073] Where q represents the sample index and R represents the total number of samples. This represents the voltage corresponding to the calibration point of the q-th sample. The voltage calculated for the voltage-position calibration equation.

[0074] This fitness function calculates the mean square error between the measured and predicted voltages of all calibration samples, forcing the optimization process to improve the overall fitting accuracy in a balanced manner. Its squared-term characteristic significantly amplifies the prediction deviations in key working conditions such as high curvature regions and axis eccentricity points, driving the particle swarm to prioritize the correction of weak links in the model. At the same time, the error integration mechanism of the entire sample effectively suppresses local overfitting, and combined with the continuous differentiable mathematical properties, it efficiently supports gradient optimization. Ultimately, it ensures that the calibration equation can output highly consistent voltage prediction values ​​under any spatial pose of the metal sphere, providing a core guarantee for sub-millimeter positioning accuracy.

[0075] Furthermore, the termination condition is that the number of iterations reaches a preset threshold or the gradient of the optimal direction of change of the fitness function value is less than a preset value.

[0076] The technical solution of this application embodiment has at least the following advantages and beneficial effects:

[0077] This method expands the application scenarios of eddy current sensors. Under certain conditions, when using eddy current sensors to measure small-diameter spherical surfaces, it can reduce the influence of the curvature of the measured surface on the measurement results of the eddy current sensors, thus achieving effective measurement. Attached Figure Description

[0078] Figure 1This is a flowchart of a calibration method for an eddy current sensor for a small-diameter metal sphere.

[0079] Figure 2 Schematic diagram of coordinate system and parameter definition

[0080] Figure 3 Schematic diagram of calibration points

[0081] Figure 4 Schematic diagram of calibration trajectory

[0082] Figure 5 Schematic diagram of calibration voltage preprocessing results Detailed Implementation

[0083] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions of this application will be clearly and completely described below in conjunction with specific embodiments. The same reference numerals in the accompanying drawings represent the same components. It should be noted that the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the described embodiments of this application without creative effort are within the scope of protection of this application.

[0084] Compared to the embodiments shown in the accompanying drawings, feasible embodiments within the scope of this application may have fewer components, other components not shown in the drawings, different components, differently arranged components, or components with different connections, etc. Furthermore, two or more components in the drawings may be implemented in a single component, or a single component shown in the drawings may be implemented as multiple separate components.

[0085] When measuring the spatial pose of a metal sphere, the output voltage of an eddy current sensor exhibits a strong nonlinearity in its mapping relationship with the spatial position (especially in high curvature regions and under axial eccentricity conditions). Furthermore, individual sensor variations, temperature drift, and assembly errors make it difficult for traditional linear calibration models to meet sub-millimeter positioning accuracy requirements. Moreover, current solutions often treat the spatial position of the eddy current sensor and the metal sphere simply as the vertical distance from the center of the metal sphere to the plane containing the sensor end. In other words, the center of the metal sphere is approximated as being on the sensor's axis. This approach significantly reduces the measurement accuracy of the eddy current sensor. Therefore, this application provides Implementation Method 1:

[0086] refer to Figure 1 Example 1: A calibration method for an eddy current sensor for a small-diameter metal sphere, comprising the following steps:

[0087] like Figure 2 As shown: Step 1: Take the lowest point of the range on the central axis of the eddy current sensor as the origin O. C Construct a calibration coordinate system O C-X C Y C Z C X C Axis, Y C Axis, Z C The axis is parallel to the direction of motion of the degrees of freedom of the calibration instrument.

[0088] Calibration coordinate system O C -X C Y C Z C It is established with the lowest point of the range on the geometric axis of the eddy current sensor as the origin, and is used to describe the position of the center of the small diameter metal ball and the spatial pose of the eddy current sensor. Its axis directions are parallel to the motion direction of the degrees of freedom of the calibration instrument.

[0089] The calibration instrument is a device with X C Axis, Y C Axis, Z C A robotic arm with axial degrees of freedom can move a small-diameter metal ball to the required precise position. The eddy current sensor is calibrated by using a calibration instrument to move the small-diameter metal ball. For this purpose, a calibration coordinate system O is set. C -X C Y C Z C At that time, let X C Axis, Y C Axis, Z C The axis is parallel to the direction of motion of the degrees of freedom of the calibration instrument.

[0090] The origin O is the point whose center coordinates within the range of the small-diameter metal sphere are closest to the probe plane of the eddy current sensor. S Construct a measurement coordinate system O S -X S Y S Z S And take the direction of the largest voltage gradient as Z. S Axis, O S It is not collinear with the central axis.

[0091] Measurement coordinate system O S -X S Y S Z S It is a coordinate system that describes the voltage measured by an eddy current sensor, considering that the eddy current sensing axis and the geometric axis of the eddy current sensor do not coincide.

[0092] After establishing two coordinate systems, the center coordinates of the small-diameter metal sphere can be described with reference to both systems. In this scheme, the relative positional relationship between the center of the small-diameter metal sphere and the eddy current sensor is described by l and r; where l represents the position of the center of the small-diameter metal sphere relative to the X coordinate system. S O S Y S The distance between the planes, r, represents the distance between the center of the small-diameter metal sphere and the Z-axis in the measurement coordinate system. S The distance between the axes; based on the positional relationship between the center of the small-diameter metal sphere and the center of the eddy current sensor, a voltage-position calibration equation is constructed.

[0093] The voltage-position calibration equation is:

[0094] ;

[0095] k1~k 10 These are 10 fitting coefficients to be calibrated.

[0096] The calculation process for the voltage-position calibration equation is as follows:

[0097] refer to Figure 2 The calibration coordinate system in the Z-axis of the measurement coordinate system S Axis and X S O S Y S The expression for a plane is:

[0098] (1);

[0099] Then the calibration point P i To Z S Axis and X S O S Y S Planar distance l i r i The mathematical relationship is as follows:

[0100] (2);

[0101] Among them, (x si , y si , z si (x) represents the coordinates of the center point of the small-diameter metal sphere at the i-th calibration point. s0 , y s0 , z s0 (a) represents the coordinates of the measurement coordinate system in the calibration coordinate system. s0 , b s0 , c s0 ) represents the measurement coordinate system Z S The axis is a vector in the calibrated coordinate system.

[0102] The calibration voltage V i Distance l i With r i Establish functional relationship V i (l i ,r i Specifically:

[0103] (3);

[0104] The calibrated voltage V at each calibration point i Distance l i With r i Substitute them into formula (3) in sequence, and adjust the coefficients k1 to k 10 The solution involves establishing a functional relationship V(l,r) between the calibrated voltage V (the voltage measured by the eddy current sensor) and the distance l and r, specifically:

[0105] ;

[0106] Step 2: Move the small-diameter metal ball along the preset calibration trajectory in the calibration coordinate system to obtain several calibration points and the voltage readings corresponding to the calibration points.

[0107] Step 2 includes the following steps:

[0108] Step 21: In the X coordinate system of the calibration system C Axis, Y C Axis, Z C The effective range of the eddy current sensor is divided into calibration points based on fixed intervals Δx, Δy, and Δz, forming equally spaced calibration points; for example... Figure 3 As shown.

[0109] Step 22: Position the small-diameter metal sphere at an X-axis angle. C Y C =0 is the starting column, and the columns pass through equally spaced calibration points in sequence; for example... Figure 4 As shown.

[0110] Step 23: Obtain the eddy current sensor readings of the small-diameter metal ball at each equally spaced calibration point.

[0111] In step 23, a voltage threshold is preset, and the readings of the small-diameter metal ball near the equally spaced calibration points are monitored in real time. When the reading of the eddy current sensor is less than the voltage threshold, the reading of the eddy current sensor at this time is taken as the effective calibration voltage of the calibration point, and the average value of the effective calibration voltage is taken as the calibration voltage of the calibration point.

[0112] Step 3: Take the calibration point and the voltage reading at the calibration point as a calibration sample, and obtain all the calibration samples to get the calibration sample set;

[0113] Step 4: Use the particle swarm optimization algorithm to dynamically update the fitting coefficients in the voltage calibration equation to obtain an accurate voltage-position calibration equation.

[0114] The mapping relationship between the output voltage of the eddy current sensor and the spatial pose of the metal sphere exhibits strong nonlinear characteristics (caused by the superposition of magnetic field distortion, material eddy current effects, and sensor nonlinear response). This complex mapping must be accurately described by a calibration equation (such as a 10-parameter model) that includes higher-order terms, coupling terms, and compensation coefficients. The calibration principle involves collecting measured position-voltage data from sample points across the entire spatial domain. The goal is to minimize the global error between the measured voltage and the voltage predicted by the equation. Experimental data is used to solve for the undetermined coefficients in the calibration equation, thereby establishing a deterministic transformation relationship from voltage signal to spatial coordinates. Therefore, throughout the process, the values ​​of the 10 fitting parameters in the voltage-position calibration equation need to be continuously adjusted. Each adjustment involves calculating the global error, thus continuously adjusting the 10 fitting parameters to find the optimal fitting parameters.

[0115] Step 4 includes the following steps:

[0116] Step 41: Apply the voltage-position calibration equation and define the fitting coefficient k. j Domain U j , j represents the index of the fitting coefficient, j∈{1、2...10}, U j This represents the domain of the j-th fitting coefficient;

[0117] Step 42: Randomly select a value for each fitting coefficient within its domain to generate one particle x. i Randomly generate N particles to obtain the initial population;

[0118] , This represents the value of the j-th fitting parameter in the i-th particle;

[0119] ;

[0120] Step 43: Define a fitness function F(x) such that after substituting the particle swarm parameters into the calibration equation, the error between the calculated voltage and the measured voltage of each calibration sample is minimized;

[0121] Step 44: Calculate the fitness function value of each particle, and iterate based on the fitness function value of the population until the preset termination condition is reached.

[0122] This scheme employs a particle swarm optimization algorithm as the core optimization engine to drive the global optimal solution search for the multi-parameter calibration equation. The particle swarm explores in parallel within the solution space, dynamically adjusting the search direction based on individual historical optimal solutions and the group's optimal solution. This effectively overcomes the shortcomings of traditional gradient optimization, such as sensitivity to initial values ​​and susceptibility to local optima. Ultimately, it achieves high-precision global fitting of the 10-parameter strongly nonlinear calibration equation, providing a deterministic parameter solution set for sub-millimeter-level measurement of the spatial pose of a metal sphere.

[0123] In the particle swarm optimization (PSO) algorithm, the three most crucial parts are the fitness function setting, the particle update method, and the iteration conditions. These three parts are described in detail in step 44. Steps 41 to 43 provide the foundation for constructing the PSO algorithm.

[0124] The particle swarm update method in this scheme does not follow the traditional velocity + inertia update method of particle swarm optimization. Instead, it finds the optimal particle by randomly combining particles and updating them randomly. The reason for this design is that the parameter fitting in the current scheme is too complex, lacks a clear gradient direction, and introduces global optimal solutions, local optimal solutions, and parameters such as velocity and inertia, which would constrain the random perturbation of particles. Based on this, this application provides the following technical solution:

[0125] Step 44 includes the following steps:

[0126] Step 441: Calculate the fitness function value F(x) of all particles in the initial population N0. i This yields the probability that each particle is selected as part of the crossover population. , Represents particles The probability of being selected as a crossover population;

[0127] ;

[0128] ;

[0129] This represents the sum of the fitness function values ​​of all particles;

[0130] In step 441, the fitness function value F(x) i ) represents the mean squared error of all samples.

[0131] ;

[0132] Where q represents the sample index and R represents the total number of samples. This represents the voltage corresponding to the calibration point of the q-th sample. The voltage calculated for the voltage-position calibration equation.

[0133] Step 442: Based on the probability of each particle Each particle is mapped to a closed circle, and the area of ​​the sector mapped to each particle is... ; ; Represents the total area of ​​the circle;

[0134] Step 443: Randomly generate discrete points within the circle that have the same number of particles as in the population at the t-th iteration, extract the particles corresponding to the region of each discrete point, and obtain the crossover population.

[0135] During the iteration process, as the particle swarm continues to gather in the high-fitness fan-shaped region, the position update mechanism in step 442 will cause multiple particles to spatially overlap within the dominant solution domain, thereby triggering a dynamic reduction in the population size. This adaptive selection mechanism based on particle spatial distribution essentially achieves population elitism through natural convergence within the solution space, enabling the crossover population to automatically eliminate redundant particles in each iteration, simultaneously improving convergence efficiency and computational resource utilization.

[0136] Step 444: Randomly pair up the particles in the crossover population. For each pair of particles x... a x b Randomly generate the intersection point G;

[0137] ; Represents particle x a The j-th fitted parameter;

[0138] ; Represents particle x b The j-th fitted parameter;

[0139] Particle x a x b At the intersection point G, a new particle c is formed. a c b ;

[0140] ; c a The G-th fitted parameter;

[0141] ; c b The Gth fitted parameter.

[0142] The intersection point G defined in this scheme represents the interchange of fitting parameters between particles. A new solution is generated by exchanging the parameter segments of the two particles before and after point G. The selection of this intersection point directly affects the parameter recombination mode, thereby regulating the balance between the algorithm's global exploration and local exploitation capabilities. Therefore, this application adopts the following strategy:

[0143] In step 444, the random intersection point G is generated as follows:

[0144] Step 4441: Give all fitting parameters Generate intersection points and generate initial selection factors The initial selection factor gradually increases;

[0145] Step 4442: Update the selection factor for the previously selected intersection point G in each iteration. To obtain new selection factors ;

[0146] ;

[0147] in, This represents the number of particles in the population at the t-th iteration. The number of particles in the population at the next iteration. This represents the updated selection factor. Let G represent the selection factor before the update, and let G = {1, 2, ..., 10}.

[0148] Step 4443: Add the new selection factor p G Replace the selection factor of the corresponding fitting parameter, and randomly select a new fitting parameter as the crossover point based on the roulette algorithm;

[0149] The probability that the fitting parameter j is chosen as the intersection point is Pt. j ;

[0150] ; This represents the selection factor for the G-th fitting parameter in the t-th iteration.

[0151] For example, if the selection factors for the three fitting parameters in the current iteration are 0.2, 0.5, and 0.3 (summing up to 1.0), then the selection probability of parameter 1 is 20%, parameter 2 is 50%, and parameter 3 is 30%. When the system generates a random number 0.6 in the interval [0,1), since 0.6 falls within the cumulative probability interval [0.2,0.7) of parameter 2, parameter 2 is selected as the crossover point G in this round, realizing a directional control mechanism that prioritizes the selection of parameters with high selection factors. Of course, the selection probability does not guarantee that a crossover point will be selected; rather, an extraction operation is performed based on the probability to extract the probability of each fitting parameter, which is related to the selection probability.

[0152] This scheme dynamically adjusts the probability of each fitted parameter being selected as a crossover point by assigning an initial selection factor to each parameter. During the iteration process, the selection factor is continuously updated, so that the crossover point selection mechanism adaptively focuses on parameters that significantly contribute to the convergence of the system, thereby guiding the parameter recombination operation to evolve in a directional manner towards an efficient solution space.

[0153] Step 445: Fitting parameters for the new particles Based on the update probability up j Choose whether to perform an update; if not, fit the parameters. This is the original value. If an update is needed, the update formula is as follows:

[0154] ;

[0155] Represents the fitting parameters The updated value, L j U represents the lower bound of the j-th fitted parameter. j Indicates the first The upper limit of the fitted parameters, This represents a random number, ranging from 0 to 1.

[0156] This scheme dynamically adjusts the parameter update mode by updating the probability threshold: when the update probability is high, mutation operation is triggered first, causing more fitting parameters to undergo random perturbation to enhance global exploration; when the update probability is low, cross-recombination is preferred, and local fine development is achieved by parameter swapping, thereby adaptively balancing the breadth and depth of solution space search during the iteration process.

[0157] In step 445, the update probability is up. j Related to the number of iterations and the number of terms;

[0158] ;

[0159] Where t represents the iteration number, m represents the sum of the number of terms in the fitted parameters, e represents the natural constant, and f(t) represents the iteration function; the number of terms refers to the number of terms in the fitted parameters l and r, for example... The sum of the number of terms and their exponents is 2 and 3 respectively;

[0160] When t < 80 ;

[0161] When 120≥t≥80, ;

[0162] When T≥t>120 b0 is a constant factor.

[0163] Step 446: Repeat steps 442 to 445 until the termination condition is met.

[0164] Furthermore, the termination condition is that the number of iterations reaches a preset threshold or the gradient of the optimal direction of change of the fitness function value is less than a preset value.

[0165] Example 2: In Example 1, the fitness function is ;

[0166] ;

[0167] Where q represents the sample index and R represents the total number of samples. This represents the voltage corresponding to the calibration point of the q-th sample. The voltage calculated for the voltage-position calibration equation.

[0168] In general calibration schemes, the fitness function is relatively simple and can achieve millisecond-level response on conventional test platforms. However, in this scheme, increased calibration accuracy is required, thus necessitating a large number of calibration points (at least 500 were selected). Furthermore, the calibration equation itself is quite complex. Each calculation of the fitness function requires 500 calculations of the voltage-position calibration equation, resulting in significant computational time consumption. Therefore, to accelerate iteration efficiency, this application provides the following technical solution based on the example swarm algorithm provided in Example 1:

[0169] Furthermore, step 4 includes the following steps:

[0170] Step 41: Apply the voltage-position calibration equation and define the fitting coefficient k. j Domain U j , j represents the index of the fitting coefficient, j∈{1、2...10}, U j This represents the domain of the j-th fitting coefficient;

[0171] Step 42: Randomly select a value for each fitting coefficient within its domain to generate one particle x. i Randomly generate N particles to obtain the initial population;

[0172] , This represents the value of the j-th fitting parameter in the i-th particle;

[0173] ;

[0174] Step 43: Define a fitness function F(x) such that after substituting the particle swarm parameters into the calibration equation, the error between the calculated voltage and the measured voltage of each calibration sample is minimized;

[0175] The fitness function is ;

[0176] ;

[0177] Where q represents the sample index and R represents the total number of samples. This represents the voltage corresponding to the calibration point of the q-th sample. The voltage calculated for the voltage-position calibration equation.

[0178] Step 44: Calculate the fitness function value of each particle, and iterate based on the fitness function value of the population until the preset termination condition is reached.

[0179] In step 44 of Example 2, the update method for each particle is the same as that for particles in Example 1. However, the particle fitness function in Example 2 is calculated using a quadratic model.

[0180] Specifically, step 44 includes the following steps:

[0181] Step 441: Calculate the fitness function value F(x) of all particles in the initial population N0. i ), and each particle x i and the corresponding fitness value F(x) i ), and add to the database D. Based on the database D, generate a simple fitness function F(x) for each particle. i )`, calculate the fitness function value F(x) for all particles. i This yields the probability that each particle is selected as part of the crossover population. , Represents particles The probability of being selected as a crossover population;

[0182] ;

[0183] ;

[0184] ` represents the sum of the fitness function values ​​of all particles;

[0185] Step 442: Based on the probability of each particle Each particle is mapped to a closed circle, and the area of ​​the sector mapped to each particle is... ; ; Represents the total area of ​​the circle;

[0186] Step 443: Randomly generate discrete points within the circle that have the same number of particles as in the population at the t-th iteration, extract the particles corresponding to the region of each discrete point, and obtain the crossover population.

[0187] Step 444: Randomly pair up the particles in the crossover population. For each pair of particles x... a x b Randomly generate the intersection point G;

[0188] ; Represents particle x a The j-th fitted parameter;

[0189] ; Represents particle x b The j-th fitted parameter;

[0190] Particle x a x b At the intersection point G, a new particle c is formed. a c b ;

[0191] ; c a The G-th fitted parameter;

[0192] ; c b The G-th fitted parameter;

[0193] Step 445: Fitting parameters for the new particles Based on the update probability up j Choose whether to perform an update; if not, fit the parameters. This is the original value. If an update is needed, the update formula is as follows:

[0194] ;

[0195] Represents the fitting parameters The updated value, L j U represents the lower bound of the j-th fitted parameter. j Indicates the first The upper limit of the fitted parameters, rand(r) This represents a random number, ranging from 0 to 1.

[0196] Step 446: Repeat steps 442 to 445 until the termination condition is met.

[0197] Therefore, the difference between Example 2 and Example 1 is that Example 2 redesigns a fitness function F(x) for each particle after each initial iteration. i Using the new fitness function F(x)i This is used to control the selection of particles.

[0198] Specifically, step 441 includes the following steps:

[0199] Step 4411: Calculate the fitness function value F(x) of all particles in the initial population N0. i ), construct and initialize the shared database D;

[0200] ;

[0201] Set the trust region radius for each fitted parameter. ;

[0202] ;

[0203] in, , These represent the upper and lower bounds of the domain of the j-th fitting parameter, respectively;

[0204] Step 4412: Extract m0 geometric optimal points from the shared database D. Thus, the candidate subset P is obtained;

[0205] ;

[0206] ;

[0207] V is the Vandermonde matrix, used to describe the geometric distribution of the candidate subset P in the quadratic polynomial space. For V T The square root of the smallest eigenvalue of V, V T It is the transpose of V;

[0208] , The physical meaning is to select the m0 points that make the system most stable, that is, to maximize the 66-dimensional space volume spanned by the candidate subset P in 10-dimensional space.

[0209] ;

[0210] Denotes the basis function, p=66;

[0211] ;

[0212] express The element in row u and column q, The meaning of the row is the projection of the point in row u onto all basis functions. The row represents the values ​​of all points in the q-th column on the basis function;

[0213] , Represents the u-th particle in the candidate subset P;

[0214] ;

[0215] Where j and l are the indices of the fitting parameters, the input of the basis function is the particle, when q=1, the basis function outputs the constant term 1, when q=(2,...,11), the basis function outputs the value of the (j-1)th fitting parameter in the particle, when q=(12,...,21), the basis function outputs the square of the jth fitting parameter in the particle, and when q>21, the basis function outputs the product of the jth fitting parameter and the lth fitting parameter in the particle;

[0216] Choosing the geometric optimum instead of directly using the shared database D is to fundamentally address the numerical ill-conditioned problem in high-dimensional space: when interpolation points are poorly distributed in the parameter space (e.g., coplanar / collinear), the condition number of the Vandermonde matrix increases dramatically, amplifying the error in solving the quadratic model parameters by 10. 4 More than 10 times, which can cause prediction distortion and optimization divergence; while the geometric optimum, by actively selecting a spatially uniform point set, ensures stable model solution and positive definite Hessian matrix, reducing the prediction error of the local model at the boundary of the trust domain by 5-10 times.

[0217] Step 4413: Using the candidate subset P, construct a quadratic model using the least squares method. ;

[0218] ;

[0219] in, Let x be the geometric center of the candidate subset P, and H be the Hessian matrix used to describe the center point x. c Quadratic model The second-order partial derivatives of any two fitted parameters, c i Indicates center point c i Quadratic model of location The function value, g i Represents the center point x c The first derivative at x, where x is the input to the quadratic model. particles;

[0220] Step 4413 involves fitting a quadratic model using the points from the selected candidate subset P. Using this quadratic model To replace the actual fitness function.

[0221] Step 4414: Transform the quadratic model As a simple fitness function F(x) i )`, to update the particles.

[0222] Furthermore, in step 441 of Example 2, a simplified fitness function F(x) is provided. i The method for obtaining the position of a particle is as follows: Steps 442-445 in Example 1 allow each particle to iteratively produce a new particle (particles that were filtered out are not considered). After obtaining a new particle, it is necessary to verify whether its position exceeds the defined domain. Therefore, Example 2 omits the method for obtaining the position of the new particle, but provides a projection mechanism for the new particle.

[0223] Specifically, quadratic model The update method includes the following steps:

[0224] S1: Using a quadratic model Calculate the fitness function of the particles after iteration, select several better particles for position projection, and obtain new particles;

[0225] ;

[0226] in, Indicates a new particle, This represents the current globally optimal position. Indicates the radius of the trust domain. This indicates the final position of the new particle after projection. This represents the direction vector.

[0227] For example, in the t-th iteration, there are 500 particles. After the screening, crossover, and mutation processes in steps 441-445, 450 particles are obtained. These 450 particles are processed using a quadratic model. After calculation, the fitness function values ​​of each particle can be obtained. Then, the fitness function values ​​are used to sort the particles, and the top 50 particles closest to the optimal fitness function value direction are selected. Projection of these particles yields 50 new particles.

[0228] S2: Call the fitness function F(x) i )calculate fitness function value and update the shared database D;

[0229] For example, after obtaining 50 new particles in S1, the fitness function F(x) needs to be called. i )calculate fitness function value At this point, 50 new elements can be updated in the shared database D. Thus, in the next iteration, the quadratic model can be revised based on the updated shared database D.

[0230] S3: Update Trust Domain Radius ;

[0231] ;

[0232] in, This represents the radius of the trust region at time t+1. The initial trust region radius, where t represents the number of iterations. Represents the attenuation constant. Represents the natural exponential function;

[0233] After each iteration, the radius of the trust domain needs to be optimized.

[0234] The trust domain update mechanism in this application achieves a smooth transition from global exploration to local development through exponential decay, balancing the needs of search breadth and accuracy.

[0235] Thus, in Embodiment 2 provided in this application, only the actual fitness functions of a few particles extracted from "S1" need to be calculated in each iteration, while the fitness functions of the remaining particles can be replaced by a quadratic model. This minimizes the computational load during the iteration process.

[0236] Example 3: Example 3 provides a specific usage example of Example 1:

[0237] The small-diameter metal ball used in this embodiment has a diameter of 45mm, and the parameters of the eddy current sensor are shown in Table 1.

[0238] ;

[0239] Table 1: Parameter Table of Eddy Current Sensor;

[0240] To achieve accurate calibration, a high-precision coordinate measuring machine was selected as the calibration instrument for the eddy current sensor voltage measurement. A small-diameter metal ball was fixed to the moving end, and the eddy current sensor was fixed to the worktable to calibrate the voltage measurement of the eddy current sensor.

[0241] Calibration point and calibration trajectory design

[0242] According to the measurement requirements, the axial calibration range of the eddy current sensor is set to 0~0.5mm, and the radial calibration radius is set to 0~0.6mm. The resulting cylindrical space is used as the calibration space. The points on the coordinate axis are divided at intervals of 0.1mm to obtain calibration points, and the calibration trajectory is set in the order of each column.

[0243] Eddy current sensor calibration experiment

[0244] Along calibration equipment X C Y C Move the center of the small-diameter metal sphere in the direction of X. C Y C With all directional voltages at their lowest, the center of the small-diameter metal sphere is approximately located on the straight line of the eddy current sensor axis. Move the sphere center to the origin of the calibration coordinate system (at this point, the eddy current sensor voltage is 0.5). Control the calibration instrument to make the small-diameter metal sphere move along a specified trajectory for calibration. The eddy current sensor voltages are shown in Table 2.

[0245] ;

[0246] Table 2: Voltage meters of eddy current sensors at different calibration points;

[0247] Voltage preprocessing

[0248] First, the pre-acquired and recorded calibration voltages are read. After reading, the threshold is set to 0.00015 to obtain the accurate eddy current sensor voltages corresponding to each calibration point. The voltage preprocessing results are shown in Table 3.

[0249]

[0250] Table 3: Preprocessing results of eddy current sensor calibration voltage (partial) (unit: V);

[0251] Construction of voltage measurement calculation equations for eddy current sensors

[0252] The coordinate information of each calibration point in the calibration coordinate system, and the corresponding eddy current sensor calibration voltage, are input into the particle swarm optimization algorithm program for calculating the eddy current sensor measurement voltage. The equation for the eddy current sensor measurement voltage is as follows:

[0253] (3)

[0254] The calculated voltage was verified, as shown in Table 4, which indicates that the voltage calibration method for the eddy current sensor of small-diameter metal spheres is correct.

[0255]

[0256] Table 4: Comparison Table of Measured Voltage and Calculated Voltage (Unit: V);

[0257] After calibrating the eddy current sensor, three eddy current sensors are simultaneously arranged around a small metal sphere. The sphere is driven to move to multiple positions within the measurement space, and the sensor readings at each position are recorded synchronously. By solving the calibration equations simultaneously, the spatial attitude and position of the eddy current sensor coordinate system can be obtained. Once the spatial attitude of the eddy current sensor is obtained, the coordinates of the sphere's center point can be calculated using the sensor readings, expanding the application scenarios of the eddy current sensor. This section describes the specific application of the eddy current sensor after calibration. Solving the calibration equations simultaneously yields the spatial attitude and position of the eddy current sensor coordinate system. Mathematical calculations using mathematical tools can then be performed. The specific method for obtaining the sphere's center point coordinates using the three eddy current sensors is independent of the eddy current sensor calibration process; the fitting parameters must be calibrated first before the three eddy current sensors can be used to obtain the sphere's center point coordinates.

[0258] The above are merely preferred embodiments of this application and are not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A calibration method for an eddy current sensor for a small-diameter metal sphere, characterized in that, Includes the following steps: Step 1: Take the lowest point of the range on the central axis of the eddy current sensor as the origin O. C Construct a calibration coordinate system O C -X C Y C Z C X C Axis, Y C Axis, Z C The axis is parallel to the direction of motion of the degrees of freedom of the calibration instrument; The origin O is the point whose center coordinates within the range of the small-diameter metal sphere are closest to the probe plane of the eddy current sensor. S Construct a measurement coordinate system O S -X S Y S Z S And take the direction of the largest voltage gradient as Z. S axis; The relative positional relationship between the center of the small-diameter metal sphere and the eddy current sensor is used... l and r describe; in, l This indicates the relationship between the center of the small-diameter metal sphere and the X coordinate system. S O S Y S Distance between planes, r This indicates the relationship between the center of the small-diameter metal sphere and the Z-axis in the measurement coordinate system. S Distance between axes; Based on the positional relationship between the center of the small-diameter metal sphere and the center of the eddy current sensor, a voltage-position calibration equation is constructed. Step 2: Move the small-diameter metal ball along the preset calibration trajectory in the calibration coordinate system to obtain several calibration points and the voltage readings corresponding to the calibration points; Step 3: Take the calibration point and the voltage reading at the calibration point as a calibration sample, and obtain all the calibration samples to get the calibration sample set; Step 4: Use the particle swarm optimization algorithm to dynamically update the fitting coefficients in the voltage calibration equation to obtain an accurate voltage-position calibration equation; The voltage-position calibration equation is: ； k 1~ k 10 These are 10 fitting coefficients to be calibrated.

2. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 1, characterized in that, Step 2 includes the following steps: Step 21: In the X coordinate system of the calibration system C Axis, Y C Axis, Z C The shafts are respectively based on a fixed spacing Δx , Δy , Δz The effective range of the eddy current sensor is divided into calibration points, forming equally spaced calibration points. Step 22: Position the small-diameter metal ball at a Z-axis angle. C The axis starts with a column and passes through equally spaced calibration points in sequence. Step 23: Obtain the eddy current sensor readings of the small-diameter metal ball at each equally spaced calibration point.

3. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 2, characterized in that, In step 23, a voltage threshold is preset, and the readings of the small-diameter metal ball near the equally spaced calibration points are monitored in real time. When the reading of the eddy current sensor is less than the voltage threshold, the reading of the eddy current sensor at this time is taken as the effective calibration voltage of the calibration point, and the average value of the effective calibration voltage is taken as the calibration voltage of the calibration point.

4. The calibration method for an eddy current sensor for a small-diameter metal sphere according to any one of claims 1 to 3, characterized in that, Step 4 includes the following steps: Step 41: Apply the voltage-position calibration equation and define the fitting coefficients. kj domain Uj , j Indices representing the fit coefficients. j ∈{1、2...10}, Uj Indicates the first j The domain of each fitting coefficient; Step 42: Randomly select a value for each fitting coefficient within its domain to generate one particle. xi Randomly generated N Each particle is used to initialize the population. i Indices representing particles; , Indicates the first i The first particle j The values ​​of the fitting parameters; ； Step 43: Define a fitness function F(x) such that after substituting the particle swarm parameters into the calibration equation, the error between the calculated voltage and the measured voltage of each calibration sample is minimized; Step 44: Calculate the fitness function value of each particle, and iterate based on the fitness function value of the population until the preset termination condition is reached.

5. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 4, characterized in that, Step 44 includes the following steps: Step 441: Calculate the initial population N Fitness function values ​​of all particles in 0 F ( xi This yields the probability that each particle is selected as part of the crossover population. , Represents particles The probability of being selected as a crossover population; ； ； This represents the sum of the fitness function values ​​of all particles; Step 442: Based on the probability of each particle Each particle is mapped to a closed circle, and the area of ​​the sector mapped to each particle is... ; ; Represents the total area of ​​the circle; Step 443: Randomly generate a sequence within the circle that matches the first... t In the next iteration, for discrete points with an equal number of particles in the population, extract the particles corresponding to the region where each discrete point is located to obtain the crossover population. Step 444: Randomly pair up the particles in the crossover population. For each pair of particles... xa , xb Randomly generate intersection point G; ; Represents particles xa The Middle j One fitting parameter; ; Represents particles xb The Middle j One fitting parameter; particles xa , xb At the intersection point G, new particles are formed. ca , cb ; ; express ca The Middle G One fitting parameter; ; express cb The Middle G One fitting parameter; Step 445: Fitting parameters for the new particles Based on updated probability upj Choose whether to perform an update; if not, fit the parameters. This is the original value. If an update is needed, the update formula is as follows: ； Represents the fitting parameters The updated values, Lj Indicates the first j The lower bound of each fitted parameter. Uj Indicates the first The upper limit of the fitted parameters, This represents a random number, ranging from 0 to 1. Step 446: Repeat steps 442 to 445 until the termination condition is met.

6. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 5, characterized in that, In step 444, the random intersection point G is generated as follows: Step 4441: Give all fitting parameters k 1~ k 10 Generate intersection points and generate initial selection factors p 1~ p 10 The initial selection factor gradually increases; Step 4442: Update the selection factor for the previously selected intersection point G in each iteration. p G To obtain new selection factors p G `; ; in, This represents the number of particles in the population at the t-th iteration. The number of particles in the population at the next iteration. This represents the updated selection factor. Let G represent the selection factor before the update, and let G = {1, 2, ..., 10}. Step 4443: Add the new selection factor p G Replace the selection factor of the corresponding fitting parameter, and randomly select a new fitting parameter as the crossover point based on the roulette algorithm; Fitting parameters j The probability of being selected as an intersection point is Ptj ; ; Indicates the G-th fitted parameter at the th... t The selection factor in the next iteration.

7. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 6, characterized in that, In step 445, the probability is updated. upj Related to the number of iterations and the number of terms; ； in, t Indicates the number of iterations. l This represents the sum of the number and degree of the fitted parameters. e Represents the natural constant. f ( t () represents an iterative function; when t <80 hours, ; When 120≥ t When ≥80, ; when T ≥ t >120, ; b 0 is a constant factor.

8. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 7, characterized in that, Fitness function value in step 441 F ( xi ) represents the mean squared error of all samples; ； in, q Indicates the sample index. R Represents the total number of samples. Indicates the first q The voltage corresponding to the calibration point of each sample. The voltage calculated for the voltage-position calibration equation.

9. The method for calibrating an eddy current sensor for a small-diameter metal sphere according to claim 7, characterized in that, The termination condition is that the number of iterations reaches a preset threshold or the gradient of the optimal direction of change of the fitness function value is less than a preset value.

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