Eddy current sensor error correction method suitable for displacement measurement of plane with holes

Abstract

The invention belongs to the technical field of precision measurement and intelligent manufacturing, and discloses an eddy current sensor error correction method suitable for hole-containing plane displacement measurement. The method comprises the following steps: firstly, constructing an equivalent eddy current field and carrying out theoretical derivation to finish accurate physical description of a quantitative relation between geometric parameters of a plane with holes and inductance change of a sensor probe; secondly, a virtual sensor model capable of simulating displacement-inductance-voltage full-link response of a real sensor is constructed by combining an inductance theoretical model and transmission characteristics of an internal signal processing circuit of the sensor; then, an optimization strategy combining global search and local fine tuning is adopted, a small amount of measured data is fused, and efficient identification and calibration of the key parameters of the virtual model are achieved; and finally, parameterized simulation is carried out by using the optimized virtual model to generate high-density calibration data, a corrected high-precision actual displacement value is inversely solved, and high-precision error correction of the real sensor in any hole-containing scene is completed.

Description

Technical Field

[0001] This invention relates to the fields of precision measurement and intelligent manufacturing technology, and in particular to a method for error correction and accuracy assurance when using an eddy current sensor to measure displacement on a surface with holes and discontinuities in an intelligent manufacturing scenario. Background Technology

[0002] Eddy current sensors, as one of the core sensing elements in modern intelligent manufacturing and precision measurement, have been widely used due to their significant advantages such as high measurement accuracy, non-contact operation, fast response speed, strong resistance to oil and dust interference, and ease of embedding and integration. However, in actual industrial scenarios, many key components have various perforated structures, such as the positioning pin holes of assembly tooling positioners, the bolt holes of internal combustion engine cylinder blocks, the cooling holes of aero-engine blades, and the weight reduction holes of robot end effectors. The perforated structure severely disrupts the stable eddy current field, directly leading to significant nonlinear errors between the sensor's output signal and the actual displacement, resulting in severely inaccurate measurement results. This technical bottleneck has seriously restricted the depth and breadth of its application in high-end manufacturing. Therefore, high-precision, high-efficiency, and low-cost calibration of eddy current sensors for perforated planes has become one of the urgent problems to be solved in the fields of precision measurement and intelligent manufacturing.

[0003] The nonlinear relationship between the input and output of eddy current sensors is the primary source of measurement error. To improve measurement accuracy, scholars both domestically and internationally have conducted extensive research on calibrating the output characteristics of sensors, primarily through three approaches: physical model-driven, data-driven, and hybrid-driven methods. In terms of model-driven approaches, Liu Wei et al., in their patent CN119164277A published in 2024, titled "A Nonlinear Error Correction Method for Eddy Current Displacement Sensors Based on Error Variation Trend Partition Calibration," proposed a full-range calibration model for eddy current sensors, achieving significant correction of the nonlinear error of eddy current sensors. In terms of data-driven approaches, Ma Lu et al., in their patent CN119783508A published in 2024, titled "Eddy Current Sensor Coil Parameter Optimization Method," proposed an eddy current sensor parameter optimization method that constructs a BP neural network and uses a particle swarm optimization algorithm, achieving high-precision linear measurement of eddy current sensors within the 2-15mm range. In terms of hybrid-driven approaches, Miao Enming et al., in their patent CN101793493B published in 2010, titled "Eddy Current Sensor Accuracy Improvement Calibration Method," proposed a calibration method that integrates high-precision measurement data with the output characteristic model of eddy current sensors, achieving improved measurement accuracy of eddy current sensors.

[0004] In summary, while domestic and international scholars have conducted extensive research on the calibration of eddy current sensor output characteristics, there is currently no effective method to reduce the measurement errors caused by irregular eddy current fields in a plane with holes, limiting the application of eddy current sensors in industrial measurement. Therefore, it is urgent to develop an error correction method for eddy current sensors suitable for displacement measurement in planes with holes. Summary of the Invention

[0005] This invention proposes an error correction method for eddy current sensors that combines a coupled physical model with data-driven error correction, suitable for displacement measurement of surfaces with holes. The aim is to correct the error of eddy current sensors for displacement measurement of surfaces with holes by incorporating a portion of the physical model and a small amount of high-precision measurement results, without relying on a large amount of measured data.

[0006] The technical solution of the present invention:

[0007] An error correction method for eddy current sensors suitable for displacement measurement of planes with holes is proposed. First, by constructing an equivalent eddy current field and performing theoretical derivation, a precise physical description of the quantitative relationship between the geometric parameters of the plane with holes and the inductance change of the sensor probe is obtained. Second, combining the inductance theoretical model with the transmission characteristics of the sensor's internal signal processing circuit, a virtual sensor model capable of simulating the entire "displacement-inductance-voltage" response of a real sensor is constructed. Then, an optimization strategy combining global search and local fine-tuning is employed, and a small amount of measured data is integrated to achieve efficient identification and calibration of key parameters of the virtual model, enabling it to accurately reproduce the sensor's output characteristics. Finally, parametric simulation is performed using the optimized virtual model to generate high-density calibration data, completing the calibration of the output characteristics of the real sensor in a hole-containing scenario, thereby achieving accurate correction of measurement errors. The specific steps are as follows:

[0008] Step 1: Theoretical modeling of inductance variation on a surface with holes;

[0009] First, the equivalent eddy current field on the surface of the conductor under test within the influence range of the eddy current sensor's magnetic field is discretized into multiple mutually insulated cross-sectional rings. The thickness of the cross-sectional rings is then calculated, which is the skin depth of the current at the corresponding frequency.

[0010] (1)

[0011] Where μ0 is the vacuum permeability μ r The relative permeability of the material, Let f be the electrical conductivity of the material, f be the frequency of the AC current in the coil, and H be the electrical conductivity of the material. e The skin depth of the current at the corresponding frequency;

[0012] Subsequently, considering the single-turn case of the eddy current sensor probe coil, with an equivalent number of eddy current loops of n, and applying Kirchhoff's voltage law to the probe coil circuit, based on the fact that the voltage of the voltage source is equal to the sum of the voltage drops of all components in the probe coil circuit, the following matrix equation is obtained:

[0013] (2)

[0014] in, and These represent the voltage and current of the probe coil, respectively. The impedance of the probe itself; The complex frequency of the circuit is represented by j, where j is the imaginary unit and ω is the angular frequency. and The first Current and impedance of an equivalent coil; For the first The and the first Mutual inductance coefficient between equivalent coils;

[0015] When a circular hole exists on the surface of the conductor being measured, its effect is equivalent to the change in inductance caused by the disappearance of the original eddy currents within the hole region; the magnetic field generated by the original eddy currents at the probe coil is calculated analytically. Then, by integrating according to the definition of inductance, the change in probe inductance caused by the circular hole is obtained. :

[0016] (3)

[0017] Where I1 is the current of the first equivalent coil, and B e The magnetic field generated by the existing eddy current at the probe coil. Let A be the change in probe inductance caused by the aperture, and let A be the effective area perpendicular to the magnetic field direction.

[0018] Since the magnetic field is calculated only for magnetic field lines that pass through the plane of the probe coil perpendicularly, only... The eddy current magnetic flux density component in the direction causes a change in inductance, hence:

[0019] (4)

[0020] Establish a global coordinate system: The origin O is the perpendicular projection of the center point of the probe coil onto the surface of the conductor being measured. The plane containing the conductor surface is the xy-plane, and the direction perpendicular to the xy-plane is the z-axis; where r is the effective radius of the eddy current, and H... e The effective depth of the eddy current;

[0021] Establish a local cylindrical coordinate system: with the center of the circular hole as the origin. d represents the coordinates of the current integration point in the local cylindrical coordinate system; d is the distance from the central axis of the circular hole on the surface of the conductor to the origin O of the coordinate system; J e denoted as , where is the current density distribution function of the original eddy current that disappears in the region due to the presence of the circular hole; z is the perpendicular distance between the probe coil plane and the surface of the conductor being measured.

[0022] Step 2: Construction of the virtual eddy current sensor model;

[0023] Establish a mathematical model to simulate the entire process of a real eddy current sensor from physical induction to signal output, that is, construct a virtual eddy current sensor model;

[0024] The change in probe inductance caused by the circular hole calculated in step 1 As a fundamental concept, after the probe of an eddy current sensor interacts electromagnetically with the conductor being measured at a distance x, an inductance change related to the distance x is generated in the probe coil. To convert this inductance change into a voltage signal, the signal processing circuit inside the eddy current sensor performs conversion and conditioning: first, the inductance change is initially converted into a voltage signal using a bridge circuit, followed by demodulation and amplification; to achieve linearization of the output, a logarithmic amplifier is used in the signal processing circuit inside the eddy current sensor to make the actual output voltage U of the eddy current sensor... out With the coil normalized inductance L norm The two relationships exhibit a stable and predictable mapping function relationship; therefore, this mapping function is introduced:

[0025] (3)

[0026] Among them, L norm =L / L0, where L is the equivalent inductance of the probe coil at the current measurement displacement, L0 is the inherent inductance of the probe coil when it is far away from the conductor being measured, and a, b, and c are all positive constants.

[0027] Curve fitting was performed using finite element simulation data, fitting only the mapping function within the displacement range of 0–4 mm; the final fitting equation is as follows:

[0028] (4)

[0029] Step 3: Global-local optimization of key parameters of the virtual sensor;

[0030] Based on the particle swarm optimization algorithm, automatic parameter optimization is achieved by matching data measured by real eddy current sensors with data from virtual eddy current sensors.

[0031] The key parameter set to be optimized for the virtual eddy current sensor model is Including the outer diameter of the coil Coil inner diameter Coil thickness and number of coil turns ;

[0032] Considering calibration requirements in all scenarios, a full-scenario error function for the virtual eddy current sensor is constructed. The key parameter identification problem is transformed into an optimization problem, with the optimization objective being to minimize the error function across the entire scene.

[0033] (5)

[0034] Where i is the index of the key parameter combination, w i Let be the weight coefficient of the i-th key parameter group, satisfying The weighting coefficients are allocated based on the importance of the scenario; For the error function in a single scenario, and These represent the measurement results of the virtual eddy current sensor and the real eddy current sensor under the i-th set of key parameters, respectively.

[0035] The optimization process for key parameters adopts a combined global-local strategy. First, the particle swarm optimization algorithm is used to perform a global search across the entire parameter space to locate the region of optimal parameters. The iterative update rule is as follows:

[0036] (6)

[0037] (7)

[0038] Where j is the dimensional index of the key parameter set of the virtual eddy current sensor. and These represent the j-th dimension components of the i-th scene of the virtual eddy current sensor in the i-th dimension. Velocity and position at the next iteration and These represent the individual optimal solution and the global optimal solution for the key parameters of the virtual eddy current sensor, respectively. and It is the optimization learning factor of the particle swarm optimization algorithm. and yes Random numbers within a range, i.e. w is the inertia weight factor in the particle swarm optimization algorithm, which is taken as a non-negative constant in this embodiment;

[0039] After the global optimal search for parameters, local fine-tuning is performed; a surrogate model for local fine-tuning is constructed using Gaussian process regression.

[0040] (8)

[0041] In the formula, and For two different sets of parameters, hyperparameter sets The behavior and performance of the locally refined optimization surrogate model are controlled by the length scale *l*, which determines the smoothness of the function variation and the signal variance. Control the amplitude and noise variance of this function. Random noise in the modeling data, For the Kronecker delta function, when The value is 1 if it is true, and 0 otherwise.

[0042] The parameter layout was optimized and adjusted using a local optimization algorithm. The convergence process of the coil thickness, number of coil turns, outer diameter, and inner diameter was observed. It was found that the objective function converged well, and the iteration stopping condition is as follows:

[0043] (11)

[0044] Step 4: Output characteristic calibration and error correction based on interpolation algorithm;

[0045] Using the optimized virtual eddy current sensor model from step 3, high-density "displacement-voltage" data points covering the entire measurement range in the target measurement scenario are efficiently generated in the simulation environment through parameter scanning. Based on these data points, an interpolation algorithm is used to construct a smooth and accurate output characteristic calibration curve. In actual measurement, the voltage value measured by the real eddy current sensor is input into the calibration curve, which is used to solve for the corrected actual displacement value and complete the error correction.

[0046] The beneficial effects of this invention are as follows: Compared with existing error correction methods that rely on numerous physical experiments for calibration, this invention effectively combines well-understood electromagnetic field theory modeling with efficient and flexible data-driven optimization strategies. It avoids the need for repetitive, time-consuming, and costly calibration experiments required to address holes of different sizes and locations, resolving the contradiction between insufficient prediction accuracy of purely physical models and excessively high calibration costs of purely data-driven methods. By constructing a high-fidelity virtual sensor model and using a small amount of measured data for precise parameter identification, this method can achieve high-precision error correction for sensors in hole-containing surface measurement scenarios at extremely low cost, generally reducing the absolute measurement error to below 20 micrometers, with an accuracy improvement exceeding 76.5%. This method has strong generalization capabilities; when measurement conditions change, new calibration relationships can be quickly generated simply by parametric simulation in the virtual model, providing an efficient, accurate, and economical solution to the challenge of high-precision displacement measurement of complex surfaces in mechatronic systems. Attached Figure Description

[0047] Figure 1 This is a flowchart of the error correction method for eddy current sensors.

[0048] Figure 2 The diagram shows the equivalent calculation model of the eddy current ring segmented within the conductor under test.

[0049] Figure 3 This is a distribution diagram of measurement sampling points for circular hole-type defects.

[0050] Figure 4 This is a platform for verifying the accuracy of the output characteristic relationship of eddy currents.

[0051] In the diagram: 1. Magnetic clamp, 2. Eddy current sensor, 3. Calibration plane, 4. Six-degree-of-freedom displacement stage, 5. Computer. Detailed Implementation

[0052] The specific embodiments of the present invention are described in detail below with reference to the technical solutions and accompanying drawings.

[0053] In this embodiment, the eddy current displacement sensor used has a displacement measurement error of less than 15 µm within a 4 mm measurement range. The displacement stage used for experimental calibration is a six-degree-of-freedom stage with a vertical repeatability better than ±0.06 µm to ensure the accuracy of calibration data acquisition. Accuracy verification experiments of this method were conducted on circular holes of three different sizes—6.2 mm, 7.5 mm, and 8.7 mm—on the surface of the positioner. A flowchart of the detailed implementation method is shown below. Figure 1 As shown.

[0054] Step 1: Theoretical modeling of inductance variation on a surface with holes;

[0055] This example first delves into the measurement principle of an eddy current sensor. A high-frequency alternating current flowing through the probe coil induces an eddy current field on the adjacent metal surface being measured. This eddy current field generates a secondary magnetic field opposite in direction to the original magnetic field, thus weakening the total magnetic flux of the coil, ultimately manifesting as a change in the probe's equivalent impedance (mainly inductance L). When the distance x between the probe and the measured surface changes, the electromagnetic coupling strength between them changes accordingly, causing a regular change in inductance L. This is the physical basis for displacement measurement.

[0056] To accurately describe the effect of the aperture on this process, this invention employs an equivalent eddy current ring model, discretizing the continuous eddy current field on the surface of the conductor under test into a series of concentric and mutually insulated virtual current rings. Through this method, the complex electromagnetic field problem is transformed into a more easily solvable multi-coil coupled circuit problem. Based on the Biot-Savart law and Kirchhoff's voltage law, precise mathematical equations describing the electromagnetic coupling relationship between the probe coil and each eddy current ring can be established (the mathematical equations are established based on the equivalent calculation model of the eddy current ring, such as...). Figure 2The existence of the aperture is equivalent to the disappearance of the current in all eddy current rings in the region. By solving this equation, a quantitative physical relationship can be established between the aperture's geometric dimensions, relative position, and other parameters and the change in probe inductance ΔL, providing a solid theoretical foundation for the subsequent construction of a virtual model.

[0057] Step 2: Construction of the virtual eddy current displacement sensor model;

[0058] The construction process of the virtual sensor integrates theoretical models and signal processing circuit characteristics. First, the displacement x to be measured and the orifice parameters are input, and the corresponding coil inductance L is calculated using the theoretical model from step 1. Next, the signal processing is simulated. To convert the inductance change into a voltage signal, the signal processing circuit inside the eddy current sensor performs conversion and conditioning: first, the inductance change is initially converted into a voltage signal using a bridge circuit, followed by demodulation and amplification; to achieve linearization of the output, a logarithmic amplifier is used in the signal processing circuit inside the eddy current sensor to make the actual output voltage U of the eddy current sensor... out With the coil normalized inductance L norm There is a stable and predictable mapping function relationship between them, as follows:

[0059] (12)

[0060] Among them, L norm =L / L0, where L is the equivalent inductance of the probe coil at the current measurement displacement, L0 is the inherent inductance of the probe coil when it is far from the conductor being measured, and a, b, c are key parameters characterizing the sensor. A series of (L) values ​​are obtained through finite element simulation. norm U out ) data points and perform curve fitting (R) 2 >0.998), thus obtaining the initial values ​​for this set of parameters. At this point, a complete virtual sensor model is established, forming the following relationship between the measured displacement x and the normalized inductance L of the coil. norm Mapping function between:

[0061] (13)

[0062] Step 3: Global-local optimization of key parameters of the virtual sensor;

[0063] To ensure that the virtual model is accurately matched to the specific sensor used (Kaman KD2306 in this example), the set of sensitive parameters in the virtual model needs to be optimized. (including the equivalent outer diameter of the coil) , inner diameter ,thickness Number of turns And optimize the formula for a, b, and c). The objective function for optimization is... Defined as the virtual sensor prediction value y under M preset measurement scenarios (different hole sizes and positions). sim The actual sensor measurement values ​​y at N sampling points real The weighted root mean square error is minimized, and the sampling point distribution is as follows: Figure 3 The optimization process consists of two steps:

[0064] Global Search: Due to the high dimensionality and strong nonlinearity of the parameter space, the Particle Swarm Optimization (PSO) algorithm is first employed. Utilizing sparse simulation data obtained under different parameters, global optimization is performed across the entire parameter space to quickly identify a smaller region containing the optimal solution, providing an ideal initial value for subsequent fine-tuning.

[0065] Local fine-tuning: Within the globally optimal region, a small number of samples are collected (e.g., in...). Figure 4 (5-10 points are selected from the sampling points shown) Actual measurement data are used. To avoid directly calculating gradients from the actual data, Gaussian process regression (GPR) is first used to establish a connection parameter... A continuously differentiable surrogate model with respect to the output deviation is used. Then, based on this surrogate model, the Levenberg-Marquardt (LM) algorithm is employed for efficient local optimization, resulting in final fine-tuning of the parameters. The convergence of parameters and objective function values ​​during optimization is good, indicating that the optimization strategy is effective and can quickly obtain accurate parameter solutions.

[0066] Step 4: Output characteristic calibration and error correction based on the optimization model;

[0067] This step is the final application and verification of the method. The experiment was conducted on a high-precision optical platform that integrated a calibration board (containing the necessary components for constructing a high-precision experimental dataset). (Circular hole), eddy current sensor, six-degree-of-freedom displacement stage, fixture and computer.

[0068] To fully verify the effectiveness and universality of the method of this invention, a complete testing and calibration process was performed on all the circular holes on the calibration plate. First, the optimized final parameters from step 3 were substituted into the virtual sensor model. For each diameter of circular hole, three typical working conditions were simulated in the simulation environment, with horizontal distances between the sensor probe and its central axis of 0mm, 3mm, and 6mm, respectively. Through parameter scanning, high-density, high-precision "displacement-voltage" calibration curves were generated for the corresponding working conditions within the vertical displacement range of 0 to 4mm. The eddy current output characteristic relationship calibration accuracy verification platform is shown below. Figure 4The eddy current sensor is fixed by a magnetic clamp and placed above the calibration plane. Then, a six-free platform is used to control the relative position between the calibration plane and the eddy current sensor. Finally, the data of the eddy current sensor is transmitted to the computer for data processing.

[0069] In the experimental verification, a high-precision displacement stage was used to position the real sensor under all the aforementioned test conditions, and the voltage reading of the real sensor was collected every 0.2 mm. Then, these readings, which contained significant errors, were substituted into the corresponding calibration curves generated by the virtual model, and the corrected displacement values ​​were obtained through interpolation and inverse kinematics. Finally, the error correction of the eddy current displacement sensor in the plane containing the hole can be achieved.

Claims

1. A method of error correction for an eddy current sensor suitable for bore containing planar displacement measurements, characterised in that, The steps include the following: Step 1: Theoretical modeling of the inductance change of a surface containing holes; Step 2: Construction of a virtual eddy current sensor model; A mathematical model simulating the whole process from physical sensing to signal output of a real eddy current sensor is established, that is, a virtual eddy current sensor model is constructed; Step 3: Global-local optimization of key parameters of the virtual sensor; Step 4: Output characteristic calibration and error correction based on an interpolation algorithm; Using the optimized virtual eddy current sensor model in step 3, high-density data points of "displacement-voltage" covering the entire range are efficiently generated in the simulation environment in the target measurement scenario through parameter scanning; based on these data points, a smooth and accurate output characteristic calibration curve is constructed using an interpolation algorithm; in actual measurement, the voltage value measured by the real eddy current sensor is input into the calibration curve, and the corrected actual displacement value is inversely solved, completing error correction.

2. The eddy current sensor error correction method suitable for hole-containing plane displacement measurement according to claim 1, characterized in that: The specific implementation process of step 1 is as follows: First, the equivalent eddy current field of the conductor surface to be measured within the influence range of the eddy current sensor magnetic field is discretized into multiple mutually insulated cross-section circular rings, and the thickness of the cross-section circular ring, that is, the skin depth of the current at the corresponding frequency, is calculated: (1) wherein Subsequently, taking the probe coil of the eddy current sensor as a single turn and the number of equivalent eddy current rings as n, the Kirchhoff voltage law for the probe coil loop is as follows: 0 is the vacuum permeability , 3. The eddy current sensor error correction method suitable for hole-containing plane displacement measurement according to claim 2, characterized in that: r is the relative permeability of the material, is the electrical conductivity of the material, f is the coil AC frequency, H e is the skin depth of the current at the corresponding frequency; The specific implementation process of step 2 is as follows: (2) in, and These represent the voltage and current of the probe coil, respectively. The impedance of the probe itself; Represents the complex frequency of the circuit. j The imaginary unit, Using finite element simulation data for curve fitting, only the mapping function in the 0~4 mm displacement range is fitted; finally, the fitting equation is as follows: Angular frequency; and The first Current and impedance of an equivalent coil; For the first The and the first Mutual inductance coefficient between equivalent coils; When there is a circular hole on the surface of the measured conductor, the influence is equivalent to the inductance change caused by the disappearance of the original vortex in the hole area; the magnetic field generated by the original vortex at the probe coil is calculated analytically , and the inductance definition is integrated, that is, the inductance change of the probe caused by the circular hole is obtained : (3) wherein, I 1 is the current of the first equivalent coil, B e B0is the magnetic field generated by the original eddy current at the probe coil, ΔL is the amount of change in the inductance of the probe caused by the hole, A A is the effective area perpendicular to the direction of the magnetic field; Since the magnetic field is calculated only for the magnetic flux lines passing through the probe coil plane perpendicularly, there is only The component of the eddy current magnetic induction in the direction results in a change of the inductance, so that (4) Establish a global coordinate system: the origin is the vertical projection of the center point of the probe coil onto the surface of the conductor being measured. O The plane containing the surface of the conductor is 4. The eddy current sensor error correction method suitable for hole-containing plane displacement measurement according to claim 3, characterized in that: plane, perpendicular to The specific implementation process of step 3 is as follows: The direction of the plane is z Axis; where, r The effective radius of the eddy current. H e The effective depth of the eddy current; Establish local cylindrical coordinate system: take the center of the hole as the origin, is the coordinate of the current integration point in the local cylindrical coordinate system; d is the distance from the center axis of the hole on the surface of the measured conductor to the origin of the coordinate system O ; J e is the current density distribution function of the original vortex flow that disappears in this area due to the existence of the hole; z is the vertical distance of the probe coil plane relative to the surface of the measured conductor. Based on the particle swarm optimization algorithm, the matching of the data measured by the real eddy current sensor and the virtual eddy current sensor data is realized to automatically optimize the parameters; The optimization process of the key parameters adopts a global-local combination strategy; first, the particle swarm optimization algorithm is used to perform global search in the entire parameter space to locate the region of the optimal parameters, and the iterative update rule is as follows: The amount of change in the probe inductance caused by the circular hole calculated in step 1 As a basis, the probe of the eddy current sensor produces an electromagnetic interaction with the measured conductor at a distance of x The inductance change in the probe coil is related to the distance x In order to convert the inductance change into a voltage signal, the signal processing circuit inside the eddy current sensor performs conversion and conditioning: first, the inductance change is preliminarily converted into a voltage signal through a bridge circuit, and then demodulation and amplification are performed; In order to realize the linearization of the output, the logarithmic amplifier is used in the signal processing circuit inside the eddy current sensor, so that the actual output voltage of the eddy current sensor U out and the normalized inductance of the coil L norm present a stable and predictable mapping function relationship; Therefore, the mapping function is introduced: (3) wherein L norm = After the global optimization of the parameters, local fine optimization is performed; a local fine optimization surrogate model is constructed by Gaussian process regression: 0 , L Lp is the equivalent inductance value of the probe coil at the current measured displacement, L 0 L0 is the intrinsic inductance value of the probe coil when far from the measured conductor, Through parameter layout optimization adjustment by the local optimization algorithm, the convergence process of the coil thickness, the number of coil turns, the coil outer diameter, and the coil inner diameter is observed, and it is found that the target function converges well, and the iteration stopping condition is as follows: are all normal numbers; ​ (4)。 ​ ​ ​ The key parameter set to be optimized for the virtual eddy current sensor model is , including the coil outer diameter , the coil inner diameter , the coil thickness , and the coil turns ; Considering the calibration requirements in all scenarios, a virtual eddy current sensor full-scene error function is constructed The key parameter identification problem is converted into an optimization problem, and the optimization objective is to minimize the full-scene error function: (5) wherein, i is an index of the key parameter combination, w i is a weight coefficient of the i th key parameter, satisfying , the weight coefficient is assigned according to the scene importance; is an error function under a single scene, and represent the measurement results of the virtual eddy current sensor and the real eddy current sensor under the i th key parameter, respectively; ​ (6) (7) wherein, j is a dimension index of the virtual eddy current sensor key parameter set, and respectively represent the velocity and position of the virtual eddy current sensor at the i-th iteration of the j-th component of the k-th scenario, i j respectively represent the individual optimal solution and the global optimal solution of the virtual eddy current sensor key parameter, is a learning factor of the particle swarm optimization algorithm, is a random number in the range of [0, 1], that is, ; w is an inertia weight factor in the particle swarm algorithm, which is taken as a non-negative constant in the embodiment.​​​​​​​ ​ (8) wherein, and are two different sets of parameters, the hyperparameter set controls the behavior and performance of the local fine-tuned surrogate model, where the length scale l determines the smoothness of the local fine-tuned surrogate model function variation, the signal variance controls the amplitude of the function, the noise variance models the random noise in the data, is the Kronecker delta function, which is 1 when and 0 otherwise. ​ (11)。

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