An error compensation method for grating interferometer measurements

Abstract

This application belongs to the field of grating measurement and provides an error compensation method for grating interferometer measurement. The method includes: constructing an angle error function based on the angle error measured by the grating interferometer; constructing a relative displacement error function based on the relative displacement error measured by the grating interferometer; constructing an explicit compensation model based on the angle error function and the relative displacement error function; constructing an implicit compensation model based on the dataset formed by the angle error and the relative displacement error; obtaining the explicit accuracy mean based on the explicit compensation model and the implicit accuracy mean based on the implicit compensation model; when the implicit accuracy mean is not less than the explicit accuracy mean, the explicit compensation model is used to compensate for the error in the grating interferometer measurement. This method ensures the reliability of the selected model by analyzing the consistency between the internal mapping of the model and physical laws, thereby achieving high-precision, high-robustness, and highly interpretable displacement measurement compensation.

Description

Technical Field

[0001] This application relates to the field of grating measurement, and more specifically, to an error compensation method for grating interferometer measurement. Background Technology

[0002] Grating interferometers are widely used for nanometer-level displacement measurement in high-end equipment manufacturing, integrated circuit testing, and precision metrology due to their advantages such as high resolution and large measurement range. However, under actual dynamic measurement conditions, their theoretical measurement accuracy is often severely limited by a variety of complex error sources. The main error sources include: Cosine error: Due to factors such as the parasitic motion of the multi-degree-of-freedom motion of the motion platform (e.g., pitch, yaw, and roll), the angular deviation between the measurement coordinate system and the motion coordinate system will introduce nonlinear measurement error through the cosine error principle.

[0003] Vibration error: Due to the inconsistency of motion caused by the vibration of the mechanical transmission chain, relative sliding occurs between the reading head of the grating interferometer and the target. This relative displacement will be directly coupled into the measurement signal.

[0004] To address the aforementioned errors and improve measurement accuracy, existing technologies primarily employ the following two types of compensation methods: (1) Compensation method based on physical model These methods predict and compensate for errors by establishing a physical model of the measurement system (such as a mathematical model based on geometric optics or rigid body kinematics). Their advantages include a model with clear physical meaning, strong interpretability, and stability and reliability under accurate operating conditions.

[0005] However, this method has significant limitations: First, accurate physical models often rely on a perfect understanding of system parameters and boundary conditions. In complex real-world working environments, there are numerous unmodeled dynamic and nonlinear factors (such as minute deformations, thermal effects, and nonlinear friction), leading to model mismatch and reduced compensation effectiveness. Second, to simplify calculations, physical models often employ small-angle linearization approximations, which limit compensation accuracy when angular deviations are large or higher-order nonlinear effects exist.

[0006] (2) Data-driven compensation method This type of method does not rely on a specific physical model, but treats the error as a "black box," training the mapping relationship between inputs (such as sensor signals) and outputs (measurement errors) through a large amount of experimental data. Its advantages lie in its ability to adaptively fit complex nonlinear relationships, its ability to capture unmodeled dynamics, and its potential to achieve high compensation accuracy within the range of training data.

[0007] However, this method also has significant drawbacks: First, the model is a "black box" lacking physical interpretability, with opaque internal decision-making logic, making it difficult to determine whether its predictions are based on reasonable physical laws. When operating conditions exceed the distribution of training data, its behavior becomes unpredictable, raising doubts about its generalization reliability. Second, its performance heavily depends on the quantity, quality, and representativeness of the training data, resulting in high data acquisition costs and the inability to utilize known prior physical knowledge, potentially leading to "data waste."

[0008] In summary, the two existing mainstream technical approaches each have their inherent drawbacks: physical models offer strong interpretability but are limited in accuracy and adaptability; data models have high potential but suffer from poor reliability and weak interpretability. Currently, there is a lack of a solution that can effectively integrate the advantages of both. Summary of the Invention

[0009] The purpose of this application is to provide an error compensation method for grating interferometer measurements, which can solve at least one of the aforementioned technical problems. The specific solution is as follows: According to a specific embodiment of this application, this application proposes an error compensation method for grating interferometer measurement, comprising: Based on the angular error x1{j} measured by the grating interferometer, an angular error function y1{j} is constructed, which satisfies: , where a is the first angle gain coefficient; Based on the relative displacement error x2{j} measured by the grating interferometer, a relative displacement error function y2{j} is constructed, satisfying: Where b is the first displacement gain coefficient; An explicit compensation model y{j} is constructed based on the angle error function and the relative displacement error function, where, ; An implicit compensation model is constructed based on the dataset formed by the angle error x1{j} and the relative displacement error x2{j}, where j is a natural number (1≤j≤n), representing the experiment number, and n is the total number of experiments, which is a natural number. The explicit accuracy range A is obtained based on the explicit compensation model. exp (j), and the implicit accuracy range A is obtained based on the implicit compensation model. imp (j), where, , in, , Indicates explicit accuracy. Indicates implicit accuracy. This represents the uncompensated displacement value of the grating interferometer. yexp {j} represents the displacement value obtained using the explicit compensation model. y imp {j} represents the displacement value calculated using the implicit compensation model; y 真 {j} represents the true value of the displacement in the j-th experiment; Based on explicit accuracy range A exp (j) Obtain the explicit accuracy mean A exp And based on the implicit accuracy range A imp (j) Obtain the implicit accuracy mean A imp ;in When the implicit accuracy mean A imp Not less than the explicit accuracy mean A exp In this case, an explicit compensation model is used to compensate for the error in the grating interferometer measurements.

[0010] In some embodiments, it also includes: When the implicit accuracy mean A imp Less than the explicit accuracy mean A exp In each experiment, implicit accuracy is used. y imp {j} and explicit accuracy y exp Calculate the difference sequence d{j} for {j}, where the difference sequence d{j} satisfies the following relationship: ; Calculate the first Pearson correlation coefficient ρ in the j-th experiment based on the difference sequence d{j}. d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j); First Pearson correlation coefficient ρ d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j) satisfies the following relation: in, Let be the mean of a single difference sequence d{j}. Let x1{j} be the mean of the angle error sequence in the j-th experiment. Let x2{j} be the mean of the angle error sequence in the j-th experiment; ρ for n experiments d,x1 (j) and ρ d,x2 (j) Calculate the average to obtain the mean ρ of the first Pearson correlation coefficient. d,x1 Mean of the second Pearson correlation coefficient ρ d,x2 : When the condition |ρ is satisfied d,x1 |<0.2 and |ρ d,x2 When | < 0.2, an explicit compensation model is used to compensate for the error in the grating interferometer measurements.

[0011] In some embodiments, it also includes: When the condition |ρ is not satisfied d,x1 |<0.2 and |ρ d,x2 When |<0.2, the mean accuracy A of the ridge regression model is constructed based on the ridge regression model. ridge The ridge regression model satisfies: f{j}=a1⋅x1{j}+b1⋅x2{j}, where a1 is the second angle gain coefficient and b2 is the second displacement gain coefficient. If A imp ridge Then calculate the angle error gain coefficient A and the displacement error gain coefficient B of the implicit model; When the angle error gain coefficient A is of the same order of magnitude as the first angle gain coefficient a, and the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0012] In some embodiments, it also includes: When the angle error gain coefficient A is not of the same order of magnitude as the first angle gain coefficient a, or when the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0013] In some embodiments, it also includes: If A ridge ≤A imp Then compare the second angle gain coefficient a1 and the first angle gain coefficient a of the ridge regression model, and the second displacement gain coefficient b2 and the first displacement gain coefficient b. When the second angle gain coefficient a1 and the first angle gain coefficient a are of the same order of magnitude, and the second displacement gain coefficient b2 and the first displacement gain coefficient b are of the same order of magnitude, the ridge regression model is used to compensate for the error in the grating interferometer measurement.

[0014] In some embodiments, it also includes: When the second angle gain coefficient a1 and the first angle gain coefficient a are not of the same magnitude, or the second displacement gain coefficient b2 and the first displacement gain coefficient b are not of the same magnitude, the explicit compensation model or the implicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0015] ​In some embodiments, the explicit compensation model or implicit compensation model is used to compensate for errors in the grating interferometer measurements, including: When the angle error gain coefficient A is of the same order of magnitude as the first angle gain coefficient a, and the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0016] In some embodiments, it also includes: When the angle error gain coefficient A is not of the same order of magnitude as the first angle gain coefficient a, or when the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0017] In some embodiments, an implicit compensation model is constructed based on the datasets of the angle error x1{j} and the relative displacement error x2{j}, including: A pre-experiment training set is constructed based on the angle error x1{j} and the relative displacement error x2{j}, the pre-experiment training set including a training set, a validation set and a test set; The layered recurrent neural network is trained based on the training set, validated based on the validation set, and tested based on the test set to obtain the implicit compensation model.

[0018] In some embodiments, 60% of the pre-experiment training set data is the training set, 20% is the validation set, and 20% is the test set.

[0019] Compared with the prior art, the above-described solutions of this application have at least the following beneficial effects: This application proposes an error compensation method for grating interferometer measurements. The method aims to establish a hybrid modeling framework integrating physical mechanisms and data-driven approaches, and designs a systematic model decision-making mechanism. This mechanism not only automatically selects the optimal compensation model based on actual data, but also ensures the reliability of the selected model by analyzing the consistency between the model's internal mapping and physical laws. Thus, it simultaneously achieves high-precision, high-robustness, and highly interpretable displacement measurement compensation under complex multi-source error environments. Attached Figure Description

[0020] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. In the drawings: Figure 1 A flowchart illustrating an error compensation method for grating interferometer measurement provided in an embodiment of this application.

[0021] Figure 2 This is a schematic diagram of the measurement structure for an error compensation method for grating interferometer measurement provided in an embodiment of this application.

[0022] Figure 3 This is a schematic diagram of the measurement optical path structure for the error compensation method for grating interferometer measurement provided in the embodiments of this application.

[0023] Figure 4 This is an overall flowchart of the error compensation method for grating interferometer measurement provided in the embodiments of this application.

[0024] Explanation of reference numerals in the attached figures: Connecting component 101, measuring module 102, target component 103, grating 104, reference interferometer 105; First reading head 201, second reading head 202, third reading head 203, fourth reading head 204, fifth reading head 205, sixth reading head 206. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this application clearer, the application will be further described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0026] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that an article or device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such an article or device. Without further limitation, an element defined by the phrase "comprising one" does not exclude the presence of other identical elements in the article or device that includes said element.

[0027] The following is in conjunction with the appendix Figures 1-4 Detailed description of optional embodiments of the present invention.

[0028] like Figure 1 As shown, this application proposes an error compensation method for grating interferometer measurements, comprising the following steps: Step S102: Based on the angle error x1{j} measured by the grating interferometer, construct the angle error function y1{j}, which satisfies: , where a is the first angle gain coefficient; Step S104: Based on the relative displacement error x2{j} measured by the grating interferometer, construct the relative displacement error function y2{j}, which satisfies: Where b is the first displacement gain coefficient; Step S106: Construct an explicit compensation model y{j} based on the angle error function and the relative displacement error function, where, ; Step S108: Construct an implicit compensation model based on the dataset formed by the angle error x1{j} and the relative displacement error x2{j}, where j is a natural number (1≤j≤n), representing the experiment number, n is the total number of experiments, and n is a natural number; Step S110: Obtain the explicit accuracy range A based on the explicit compensation model. exp (j), and the implicit accuracy range A is obtained based on the implicit compensation model. imp (j), where, , in, , Indicates explicit accuracy. Indicates implicit accuracy. This represents the uncompensated displacement value of the grating interferometer. y exp {j} represents the displacement value obtained using the explicit compensation model. y imp {j} represents the displacement value calculated using the implicit compensation model; y 真 {j} represents the true value of the displacement in the j-th experiment; Step S112: Based on the explicit accuracy range A exp (j) Obtain the explicit accuracy mean A exp And based on the implicit accuracy range A imp (j) Obtain the implicit accuracy mean A imp ;in Step S114: When the implicit accuracy mean A imp Not less than the explicit accuracy mean A exp In this case, an explicit compensation model is used to compensate for the error in the grating interferometer measurements.

[0029] Measurement diagram as follows Figure 2As shown, this design aims to achieve precise correction of the target displacement. The connecting component 101 mechanically connects the grating interferometer measurement module 102 to the target component 103 to transmit motion or force. The grating 104 remains independently stationary and is mounted on a common reference externally with the connecting component. The grating 104 cooperates with the measurement module 102 to measure the displacement of the target component 103. During operation, the connecting component 101 drives the grating interferometer measurement module 102 to move synchronously with the target component 103. Ideally, the displacement output by the measurement module 102 should equal the actual displacement of the target component 103. However, in actual operation, there are two main sources of error: first, the measurement module 102 may experience attitude changes relative to the target component 103, resulting in a cosine error; second, the two components may slip relative to each other due to the vibration of the connecting component 101, resulting in a relative displacement and vibration error. In practice, the displacement value output by the measurement module 102 is referred to as the uncompensated displacement value of the grating interferometer.

[0030] During the j-th measurement experiment, the change in attitude angle x1{j} of the target component 103 during the measurement process is collected by the measurement module 102. The application scenario of this application is linear displacement measurement, where the linear displacement direction is defined as the x-axis, the grating direction is defined as the z-axis, and the y-axis is defined according to the right-hand rule. The optical path arrangement is as follows... Figure 3 As shown, the measurement module 102 includes a first reading head 201, a second reading head 202, a third reading head 203, a fourth reading head 204, a fifth reading head 205, and a sixth reading head 206. The first and second reading heads 201 and 202 are symmetrical Littoral incident grating interferometers, the fourth, fifth, and sixth reading heads 204 and 205 are perpendicular incident grating interferometers, and the third reading head 203 is a relative displacement measurement reading head (laser interferometer). Three rotation angles are calculated using differential calculations: the rotation angle about the x-axis (roll). x The rotation angle (yaw) around the y-axis is calculated from the difference in readings between the fourth reading head 204 and the sixth reading head 206. y The rotation angle (pitch) around the z-axis is calculated from the difference in readings from the first reading head 201 and the second reading head 202. z Calculated from the difference in readings between the fourth reading head 204 and the fifth reading head 205.

[0031] Simultaneously, the relative displacement x2{j} between the target component 103 and the target component 103 is measured by the measurement module 102. Multiple sets of attitude angle changes x1{j} and relative displacements x2{j} can be obtained through repeated measurements. Meanwhile, the displacement of the target component 103 is measured on the other side of the target component 103 using a reference interferometer 105, and this displacement is used as the true value y. 真 {j}.

[0032] Explicit model construction: Based on the analysis of the physical structure and working principle of the grating interferometer, a compensation model with parameters having clear physical meaning is established.

[0033] First, the mapping relationship between attitude angle change and measurement error is established. Since the application scenario of this application is linear displacement measurement, grating 104 is a one-dimensional grating, and the grating line direction is perpendicular to the linear displacement direction. x Angular deviation does not affect the measurement results; only angular deviation needs to be considered. y and z Angle deviation.

[0034] Next, the uncompensated displacement value x 0 The optical path plane of these two reading heads is the XOY plane, obtained by averaging the measurements from the first reading head 201 and the second reading head 202. This plane is spatially symmetrical with the measurement axis of the third reading head 203 (parallel to the x-axis). Under this optical path arrangement, y The influence of the phase measurement values ​​of the first reading head 201 and the second reading head 202 on the phase values ​​has opposite signs, therefore, this averaging calculation can not only eliminate the influence to a certain extent... y It can also reduce the synchronous measurement error of the first reading head 201 and the second reading head 202.

[0035] From the above two steps, only need to consider z The influence of this error is referred to in this application as the angular deviation x1{j}. After the measurement structure is established, the direction of each ray and its intersection with the optical device are determined. Based on the ray tracing method, a mapping model is established between the angular deviation x1{j} and the final measurement error y1{j}. Within a small angular range (e.g., within 10 arcseconds), this mapping relationship can be approximated as linear: y1{j}=a⋅x1{j} Where 'a' is the first angle gain coefficient.

[0036] Then, a mapping relationship between relative displacement and vibration error is constructed based on the rigid body assumption. Consider the relative displacement error x2{j} between the grating and the reading head. Based on a simplified rigid body kinematics assumption: when the connecting component 101 vibrates in the x-direction, since the measuring module 102 and the target component 103 are rigidly connected in a plane perpendicular to the direction of motion, the vibration will be transmitted to the reading head position with the same amplitude and phase. Based on this physical assumption, a linear mapping relationship is constructed between the relative displacement error x2{j} and the relative displacement error function y2{j} caused by it: y2{j}=b⋅x2{j} Where b is the first displacement gain coefficient.

[0037] An explicit linear model is constructed based on physical rules. Since the measurement errors caused by angular deviation and relative displacement are independent, and the total system error is the result of the linear superposition of the two, an explicit compensation model y{j} is constructed based on the angular error function and the relative displacement error function: y{j}=y1{j}+y2{j}=a⋅x1{j}+b⋅x2{j}.

[0038] Where j is a natural number (1≤j≤n), representing the experiment number, and n is the total number of experiments, where n is a natural number. Taking x1{1} as an example, it represents the angle error collected during the entire experimental time in the first experiment, which is a curve. The following "{}" represents a vector, and "()" represents a numerical value. Furthermore, the contribution of measurement errors caused by angular deviation and relative displacement can be analyzed.

[0039] To evaluate the relative contribution of the two error sources to the total error under a specific measurement task or dataset, the formula for calculating the component percentage R is as follows: using the fitted first angle gain coefficient a, the first displacement gain coefficient b, and the input angle error x1 and relative displacement error x2 of the entire dataset. in, In the formula, j represents the experiment number, and n represents the total number of experiments. halfrange1(j) represents half of the peak-to-peak value of the cosine error in the j-th experiment (max represents the maximum value, and min represents the minimum value), halfrange2(j) represents half of the peak-to-peak value of the vibration error in the j-th experiment, R(j) represents the proportion of the cosine error component in the j-th experiment, and R represents the average of the proportions of the cosine error component in the total n experiments.

[0040] If R > 50%, it indicates that the angle error is dominant; if R < 50%, it indicates that the displacement error is dominant; if R ≈ 50%, the two contribute equally.

[0041] Implicit model construction: based on the angle error x1 and the relative displacement error x2 The dataset is used to construct an implicit compensation model, including: based on the angle error x1 and the relative displacement error x2 A pre-experiment training set is constructed, which includes a training set, a validation set, and a test set. The layer recurrent neural network is trained based on the training set, the trained layer recurrent neural network is validated based on the validation set, and the layer recurrent neural network is tested based on the test set to obtain an implicit compensation model.

[0042] The pre-experiment training set data consists of 60% training set, 20% validation set, and 20% test set.

[0043] First, a dataset is constructed, consisting of a pre-experiment training set and a formal experiment set, both of which are identical. The pre-experiment training set comprises data from multiple experiments (angle error x1). and relative displacement error x2 As input features, the dataset is divided into training, validation, and test sets according to a ratio of 60%:20%:20% of the number of experiments.

[0044] Then, the neural network model was trained using a pre-experiment training set, with a layered recurrent neural network (LRNN) chosen as the implicit model. This network structure can effectively handle time-series or dynamic data. The network input is the angle error x1 at the current and possible historical moments. and relative displacement error x2 The network is trained on the training set, and hyperparameters are tuned and early stopping is implemented using the validation set to prevent overfitting. Finally, its accuracy is evaluated on the test set (using metrics such as root mean square error (RMSE) and coefficient of determination (R²). The successfully trained neural network is the implicit model.

[0045] Furthermore, the explicit accuracy range Aexp(j) and implicit accuracy range Aimp(j) of the j-th experimental data are calculated, where, under the explicit model, the output value of the j-th experimental data and the true displacement y of the target 103 measured by the reference interferometer 105 are... 真 The comparison result of {j} is the explicit accuracy. y exp {j}, where half of its peak-to-peak value is the explicit accuracy range Aexp(j), calculated as follows: , max represents the maximum value, min represents the minimum value; under the implicit model, the output value of the j-th experiment data and the true value y of the displacement of the target 103 measured by the reference interferometer 105 are compared. 真 The comparison result of {j} is the implicit accuracy. y imp {j}, where half of its peak-to-peak value is the implicit accuracy range Aimp(j), calculated as follows: The comparison of explicit and implicit accuracy ranges can be made as percentages, differences, etc., and there are no restrictions on this. By averaging the data from multiple (e.g., n) experiments, the explicit compensation model can be used to obtain the explicit accuracy mean A. exp And the implicit compensation model yields the implicit accuracy mean A. imp .

[0046] like Figure 4 As shown, in some embodiments, the method further includes the following step: when the implicit accuracy mean A imp Less than the explicit accuracy mean A exp In each experiment, implicit accuracy is used. y imp {j} and explicit accuracy Calculate the difference sequence d{j}, which satisfies the following relationship: ; Calculate the first Pearson correlation coefficient ρ in the j-th experiment based on the difference sequence d{j}. d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j); First Pearson correlation coefficient ρ d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j) satisfies the following relation: in, Let be the mean of a single difference sequence d{j}. Let x1{j} be the mean of the angle error sequence in the j-th experiment.

[0047] n is a natural number, and ρ for n experiments d,x1 (j) and ρ d,x2 (j) Calculate the average to obtain ρ d,x1 and ρ d,x2 : When the condition |ρ is satisfied d,x1 |<0.2 and |ρ d,x2 When |<0.2, an explicit compensation model is used to compensate for the error in the grating interferometer measurements. If |ρ d,x1 |<0.2 and |ρ d,x2 If | < 0.2, it indicates that the difference between the two models has no obvious linear relationship with the input variables. The implicit model does not systematically correct errors directly related to the input and may only fit noise. In this case, the explicit model, which is more physically interpretable, should be selected first, and the process terminates.

[0048] In some embodiments, the method further includes: when the condition |ρ is not met d,x1 |<0.2 and |ρ d,x2 When |<0.2, the mean accuracy A of the ridge regression model is constructed based on the ridge regression model. ridge The ridge regression model satisfies: f{j}=a1⋅x1{j}+b1⋅x2{j}, where a1 is the second angle gain coefficient and b2 is the second displacement gain coefficient; the mean accuracy A of the ridge regression is... ridge The calculation process is as follows: Explicit accuracy mean A exp or implicit accuracy mean A imp This will not be elaborated upon here.

[0049] If the correlation coefficient does not meet the weak correlation condition mentioned above, a ridge regression model is introduced as a third comparative model. The ridge regression model is a linear model with L2 regularization, which can provide a more stable solution when collinearity exists. Using the angle error x1{j} and relative displacement error x2{j} as inputs, and the measurement error f{j} of the ridge regression model as output, the ridge regression model is trained on a pre-experiment training set, and the optimal regularization coefficient is selected through a validation set. Finally, the trained ridge regression model is output, and its mean accuracy A is evaluated on the formal experimental set. ridge .

[0050] Furthermore, if A imp ridge Then, the angular error gain coefficient A and the displacement error gain coefficient B of the implicit model are calculated; when the mean value of the angular error gain coefficient A is of the same order of magnitude as the first angular gain coefficient a, and the mean value of the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then the implicit compensation model is used to compensate for the error of the grating interferometer measurement value.

[0051] Specifically, if A imp ridge If the neural network is superior, then a separate input contribution analysis is performed on the implicit model. A separate input sensitivity analysis is then performed using the trained neural network. In the j-th experiment, the relative displacement error x2{j} is fixed at zero, and only the angle error x1{j} is varied; the predicted value Δy of the neural network is observed. 1预测 {j},Δy 1预测 Half of the peak value of {j} is the predicted range of range_Δy for only the angle error. 1预测 (j), the calculation method is as follows The angle error range range_x1(j) is half the peak-to-peak value of the angle error x1{j}, and is calculated as follows: The predicted range of values ​​for only varying angle errors is range_Δy. 1预测 ​​The approximate ratio of (j) to the angle error range range_x1(j) determines the angle error gain coefficient, denoted as A(j).

[0052] With the fixed angular error x1{j} at zero, and only the relative displacement error x2{j} varying, observe the predicted value Δy of the neural network. 2预测 {j},Δy 2预测 Half of the peak value of {j} is the predicted range of range_Δy, which only varies with the relative displacement error. 2预测 (j), the calculation method is as follows The relative displacement error range range_x2(j) is half the peak-to-peak value of the relative displacement error x2{j}, and is calculated as follows: The predicted range of values ​​for only varying relative displacement error is range_Δy. 2预测 The approximate ratio of (j) to the relative displacement error range range_x2(j) determines the displacement error gain coefficient, denoted as B(j).

[0053] The mean value of the angle error gain coefficient A and the mean value of the displacement error gain coefficient B are obtained by averaging A(j) and B(j) for a total of n experiments.

[0054] Then, the coefficients are compared, and the mean value of the angle error gain coefficient A and the mean value of the displacement error gain coefficient B are compared with the first angle gain coefficient a and the first displacement gain coefficient b, respectively.

[0055] When the mean value of the angle error gain coefficient A is on the same order of magnitude as the first angle gain coefficient a (e.g., within one order of magnitude), and the mean value of the displacement error gain coefficient B is on the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurements. This indicates that the mapping relationship obtained by the neural network training is basically consistent with the physical laws, but its nonlinear mapping capability is stronger; therefore, an implicit model (i.e., a neural network) is selected.

[0056] When the angular error gain coefficient A is not of the same order of magnitude as the first angular gain coefficient a, or when the mean value of the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurements. In this case, it indicates that the neural network training has obtained a mapping that differs significantly from physical laws, and its generalization ability may be questionable; therefore, an explicit model is selected.

[0057] In some embodiments, the method further includes: If A ridge ≤A imp Then compare the second angle gain coefficient a1 and the first angle gain coefficient a of the ridge regression model, and the second displacement gain coefficient b2 and the first displacement gain coefficient b. When the second angle gain coefficient a1 and the first angle gain coefficient a are of the same order of magnitude, and the second displacement gain coefficient b2 and the first displacement gain coefficient b are of the same order of magnitude, it indicates that the linear relationship supported by the data is in good agreement with the physical model. Therefore, the ridge regression model is used to compensate for the error in the grating interferometer measurement.

[0058] Furthermore, the method also includes: when the second angular gain coefficient a1 and the first angular gain coefficient a are not of the same magnitude, or the second displacement gain coefficient b2 and the first displacement gain coefficient b are not of the same magnitude, it is still necessary to compare the relationship between the neural network equivalent gain and the physical coefficients, and use the explicit compensation model or the implicit compensation model to compensate for the error of the grating interferometer measurement value according to the comparison result.

[0059] Specifically, the explicit or implicit compensation model is used to compensate for errors in the grating interferometer measurements, including: When the angle error gain coefficient A is of the same order of magnitude as the first angle gain coefficient a, and the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0060] When the angle error gain coefficient A is not of the same order of magnitude as the first angle gain coefficient a, or when the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurement.

[0061] Finally, it should be noted that the various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems or apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to in the method section. This embodiment only describes an electromagnetic field structure designed using a Helmholtz coil and DC high voltage; other methods that utilize electromagnetic fields to confine plasma in a discharge region are within the scope of protection of this patent.

[0062] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. An error compensation method for grating interferometer measurements, characterized in that, include: Based on the angular error x1{j} measured by the grating interferometer, an angular error function y1{j} is constructed, which satisfies: , where a is the first angle gain coefficient; Based on the relative displacement error x2{j} measured by the grating interferometer, a relative displacement error function y2{j} is constructed, which satisfies: Where b is the first displacement gain coefficient; An explicit compensation model y{j} is constructed based on the angle error function and the relative displacement error function, where, ; An implicit compensation model is constructed based on the dataset formed by the angle error x1{j} and the relative displacement error x2{j}, where j is a natural number (1≤j≤n), representing the experiment number, and n is the total number of experiments, which is a natural number. The explicit accuracy range A is obtained based on the explicit compensation model. exp (j), and the implicit accuracy range A is obtained based on the implicit compensation model. imp (j), where, , in, , Indicates explicit accuracy. Indicates implicit accuracy. This represents the uncompensated displacement value of the grating interferometer. y exp {j} represents the displacement value obtained using the explicit compensation model. y imp {j} represents the displacement value calculated using the implicit compensation model; y 真 {j} represents the true value of the displacement in the j-th experiment; Based on explicit accuracy range A exp (j) Obtain the explicit accuracy mean A exp And based on the implicit accuracy range A imp (j) Obtain the implicit accuracy mean A imp ;in When the implicit accuracy mean A imp Not less than the explicit accuracy mean A exp In this case, an explicit compensation model is used to compensate for the error in the grating interferometer measurements.

2. The method according to claim 1, characterized in that, Also includes: When the implicit accuracy mean A imp Less than the explicit accuracy mean A exp In each experiment, implicit accuracy is used. and explicit accuracy Calculate the difference sequence d{j}, which satisfies the following relationship: ; Calculate the first Pearson correlation coefficient ρ in the j-th experiment based on the difference sequence d{j}. d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j); First Pearson correlation coefficient ρ d,x1 (j) and the second Pearson correlation coefficient ρ d,x2 (j) satisfies the following relation: in, Let be the mean of a single difference sequence d{j}. Let x1{j} be the mean of the angle error sequence in the j-th experiment. Let x2{j} be the mean of the angle error sequence in the j-th experiment; ρ for n experiments d,x1 (j) and ρ d,x2 (j) Calculate the average to obtain the mean ρ of the first Pearson correlation coefficient. d,x1 Mean of the second Pearson correlation coefficient ρ d,x2 : When the condition |ρ is satisfied d,x1 |<0.2 and |ρ d,x2 When | < 0.2, an explicit compensation model is used to compensate for the error in the grating interferometer measurements.

3. The method according to claim 2, characterized in that, Also includes: When the condition |ρ is not satisfied d,x1 |<0.2 and |ρ d,x2 When |<0.2, the mean accuracy A of the ridge regression model is constructed based on the ridge regression model. ridge The ridge regression model satisfies: f{j}=a1⋅x1{j}+b1⋅x2{j}, where a1 is the second angle gain coefficient and b2 is the second displacement gain coefficient. If A imp ridge Then calculate the angle error gain coefficient A and the displacement error gain coefficient B of the implicit model;​ When the angle error gain coefficient A is of the same order of magnitude as the first angle gain coefficient a, and the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurement.

4. The method according to claim 3, characterized in that, Also includes: When the angle error gain coefficient A is not of the same order of magnitude as the first angle gain coefficient a, or when the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurement.

5. The method according to claim 3, characterized in that, Also includes: If A ridge ≤A imp Then compare the second angle gain coefficient a1 and the first angle gain coefficient a of the ridge regression model, and the second displacement gain coefficient b2 and the first displacement gain coefficient b. When the second angle gain coefficient a1 and the first angle gain coefficient a are of the same order of magnitude, and the second displacement gain coefficient b2 and the first displacement gain coefficient b are of the same order of magnitude, the ridge regression model is used to compensate for the error in the grating interferometer measurement.

6. The method according to claim 5, characterized in that, Also includes: When the second angle gain coefficient a1 and the first angle gain coefficient a are not of the same magnitude, or the second displacement gain coefficient b2 and the first displacement gain coefficient b are not of the same magnitude, the explicit compensation model or the implicit compensation model is used to compensate for the error in the grating interferometer measurement.

7. The method according to claim 6, characterized in that, The explicit or implicit compensation model is then used to compensate for errors in the grating interferometer measurements, including: When the angle error gain coefficient A is of the same order of magnitude as the first angle gain coefficient a, and the displacement error gain coefficient B is of the same order of magnitude as the first displacement gain coefficient b, then an implicit compensation model is used to compensate for the error in the grating interferometer measurement.

8. The method according to claim 7, characterized in that, Also includes: When the angle error gain coefficient A is not of the same order of magnitude as the first angle gain coefficient a, or when the displacement error gain coefficient B is not of the same order of magnitude as the first displacement gain coefficient b, an explicit compensation model is used to compensate for the error in the grating interferometer measurement.

9. The method according to claim 1, characterized in that, An implicit compensation model is constructed based on the datasets of the angle error x1{j} and the relative displacement error x2{j}, including: A pre-experiment training set is constructed based on the angle error x1{j} and the relative displacement error x2{j}, the pre-experiment training set including a training set, a validation set and a test set; The layered recurrent neural network is trained based on the training set, validated based on the validation set, and tested based on the test set to obtain the implicit compensation model.

10. The method according to claim 9, characterized in that, The pre-experiment training set data consists of 60% training set, 20% validation set, and 20% test set.

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