Optical system imaging quality prediction method based on assembly error

Abstract

The invention relates to the field of mechanical engineering, in particular to an assembly error-based optical system imaging quality prediction method, which comprises the following steps of: firstly, constructing an accurate optical system finite element model, simulating a bolt tightening process, obtaining mirror surface deformation under the action of assembly stress, fitting a mirror surface shape by utilizing a Zernike polynomial, taking the mirror surface shape as an intermediate medium, and predicting the imaging quality of an optical system. The Zemax is adopted to simulate the imaging of the optical system under the conditions of mirror surface deformation and assembly poses, and the influence rule of various assembly errors on the imaging quality of the optical system is explored. Thirdly, an assembly data set is constructed through an integrated simulation method, an MLP-XGBoost combined neural network agent model is established, model training is carried out, and compared with a single MLP model, a single XGBoost model and a traditional regression model, the method has higher superiority and reliability in the prediction precision aspect. And finally, a stochastic gradient descent method is adopted to optimize the pre-tightening force under different primary and secondary mirror assembly pose conditions, the energy concentration ratio is averagely improved by 11.34% compared with the energy concentration ratio before optimization, and the assembly and adjustment efficiency and precision of the double-mirror optical system are improved.

Description

Technical Field

[0001] The invention relates to the technical field of mechanical engineering, and in particular to a method for predicting the imaging quality of an optical system based on assembly errors. Background Art

[0002] The coaxial double-reflective optical system is mainly composed of a primary mirror, a secondary mirror, and a mirror frame. It has the advantages of no chromatic aberration, strong light-gathering ability, easy processing, and can reach the diffraction limit. It is widely used in astronomical observation, space remote sensing, precision guidance, chip lithography and other fields. With the improvement of product performance requirements and the complex and harsh service environment, the requirements for the imaging quality of coaxial double-reflective optical systems are also getting higher and higher. The imaging quality of the double-reflective optical system not only depends on the optical-mechanical structure design and processing accuracy, but also the adjustment posture and connection and tightening mechanical state during the assembly process. It also significantly affects the final performance of the precision optical system. How to accurately evaluate the influence of various external loads on the deformation of the optical system mirror during the assembly process, and to achieve the prediction of the imaging quality of the optical system has become a key issue that needs to be solved urgently. Summary of the invention

[0003] In view of the above problems, an embodiment of the present invention provides an optical system imaging quality prediction method based on assembly error.

[0004] An embodiment of the present invention provides a method for predicting the imaging quality of an optical system based on assembly errors, comprising:

[0005] The finite element model of the double-reflective optical system was established, and the tightening process of the bolts was simulated to obtain the mirror deformation under different preload conditions;

[0006] The Zernike polynomials are used to fit the mirror surface shape in the double-reflection optical system to obtain the Zernike polynomial coefficients used to characterize the mirror surface shape error.

[0007] The mirror deformation is imported into the optical-mechanical-thermal coupling analysis module and the mirror surface shape is fitted using the Zernike polynomial coefficients as the basis function to obtain a macro file including the mirror surface shape deviation and the primary and secondary mirror assembly posture deviation;

[0008] The obtained macro file is loaded into the optical simulation module to perform light path imaging simulation to obtain the energy concentration used to characterize the imaging quality;

[0009] A data set was constructed based on the preload force, primary and secondary mirror assembly posture deviation, and energy concentration, and the constructed data set was used to train the MLP model and XGBoost model.

[0010] The current preload force and the primary and secondary mirror assembly posture deviations of the optical system to be predicted are obtained, and the obtained preload force and the primary and secondary mirror assembly posture deviations are respectively input into the trained MLP model and the XGBoost model to obtain the first energy concentration and the second energy concentration;

[0011] The first weight matrix and the second weight matrix of the MLP model and the XGBoost model are obtained, and the first energy concentration and the second energy concentration are weightedly calculated using the determined first weight matrix and the second weight matrix to obtain the final energy concentration as a prediction result.

[0012] Optionally, also include:

[0013] The assembly posture deviation of the primary and secondary mirrors is taken as the input value and set as a constant according to the actual assembly situation. The preload force is taken as the input value and set as the optimization variable. The optimization goal is to maximize the predicted result, and the stochastic gradient descent algorithm is used to iteratively calculate the optimal preload force. The actual assembly process is guided by the calculated optimal preload force.

[0014] Optionally, the process of establishing a finite element model of a double-reflective optical system includes:

[0015] Establish a three-dimensional geometric model of the double-mirror optical system, including the primary mirror, secondary mirror and mirror frame. The primary mirror is fixed to the mirror frame through three sets of bolts and nuts, and the secondary mirror is assembled with the cylindrical hole on the mirror frame through a pressure ring.

[0016] The finite element model is established by meshing the three-dimensional geometric model, wherein the bolts and nuts are meshed with hexahedron, and the primary mirror, secondary mirror and mirror frame are meshed with tetrahedron.

[0017] Optionally, the process of loading the obtained macro file into the optical simulation module for light path imaging simulation includes:

[0018] Establish the reflection surfaces corresponding to the primary mirror and the secondary mirror, where the primary mirror is a rotating parabola and the secondary mirror is an even-order hyperboloid, and set the size of the opening at the center of the primary mirror and the distance between the primary and secondary mirrors;

[0019] The mirror surface deviation is inserted into the mirror coordinate discontinuity surface to achieve the setting of the primary mirror and the secondary mirror eccentricity and tilt;

[0020] The assembly posture deviations of the primary and secondary mirrors are superimposed on the reflective surface to generate a composite surface, thereby fitting the mirror deformation to the mirror surface.

[0021] Optionally, the process of constructing a data set based on the preload force, the primary and secondary mirror assembly posture deviations, and the energy concentration includes:

[0022] The preload force and the primary and secondary mirror assembly posture deviation are used as feature variables, and the energy concentration is used as the target variable to construct a data set. The primary and secondary mirror assembly posture deviation includes the eccentricity error of the primary and secondary mirrors along the X-axis and Y-axis, as well as the skew error around the X-axis and Y-axis.

[0023] The feature variables and target variables in the constructed dataset are randomly sorted and normalized, and finally the dataset is divided into training set and test set according to the set ratio.

[0024] Optionally, the process of training the MLP model and XGBoost model using the constructed dataset includes:

[0025] Use the training set to train the MLP model and XGBoost model separately;

[0026] The MLP model and the XGBoost model were tested using the test set to obtain the energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set;

[0027] Calculate a first weight matrix and a second weight matrix according to the obtained energy concentration prediction value under the test set and the true value of the energy concentration under the test set;

[0028] The energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set are weightedly calculated using the first weight matrix and the second weight matrix to obtain the prediction value of the MLP-XGBoost combination model;

[0029] The mean absolute error, mean square error, and mean error percentage are calculated based on the number of samples in the test set, the predicted value of the combined model, and the true value of the energy concentration;

[0030] When the mean absolute error, mean square error, and average error percentage meet expectations, the MLP-XGBoost combined model is used as the final model for predicting the imaging quality of the optical system; if they do not meet expectations, the training set and test set are repartitioned and the step of using the training set to train the MLP model and the XGBoost model separately is returned.

[0031] Optionally, the first weight matrix and the second weight matrix are determined using the following expressions:

[0032]

[0033] Among them, W * The weight matrix for combining a single model, matrix I1 = [1,1,…,1] T , S is the prediction error information matrix of a single model, S = e T e, e is the prediction error matrix, e=[E1,E2,…,Ei ] T , where E i is the prediction error vector of the i-th model:

[0034] E i =Z i -Y i =[e i1 ,e i2 ,…,e iN ] T

[0035] Among them, Z i =[z1,z2,…,z j ,…,z N ] T is the true value of energy concentration, Y i =[y i1 ,y i2 ,…,y ij ,…,y iN ] T is the predicted value of energy concentration.

[0036] Optionally, the prediction result is expressed as:

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] 1) Through the optical-mechanical integration simulation method of the optical system, the influence of different assembly errors on the imaging quality of the optical system is analyzed. It is found that when there is a posture deviation between the primary and secondary mirrors, the influence of different bolt preloads is different. It is necessary to adjust the appropriate preload size to correct the degradation of imaging quality caused by the posture deviation. The segregation and skew errors of the primary and secondary mirrors along the X and Y axes have a significant impact on the imaging quality, while the segregation and skew errors in the Z axis direction (optical axis direction) have almost no effect.

[0039] 2) Considering the mirror deformation and optical element posture deviation caused by the preload during assembly, a combined prediction model for the energy concentration of the double-mirror optical system based on MLP-XGBoost was constructed. The average error of the training set was only 4.70%, which was significantly better than the single MLP and XGBoost models, as well as the traditional regression models (BP and SVR);

[0040] 3) A preload force optimization method based on the MLP-XGBoost model and the stochastic gradient descent algorithm was proposed. Under different assembly posture conditions, the energy concentration of the optical system was increased by an average of 11.34% compared with before optimization, which can be used to guide the actual assembly process of the optical system. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The drawings described herein are used to provide a further understanding of the present invention, constitute a part of the present invention, and do not constitute a limitation of the present invention. In the drawings:

[0042] Figure 1 It is a three-dimensional geometric model of a double-reflective optical system;

[0043] Figure 2 It is the finite element model of the double-reflective optical system;

[0044] Figure 3 This is the mirror deformation result diagram;

[0045] Figure 4 Torsion-tension relationship diagram obtained for simulation and theoretical model;

[0046] Figure 5 It is the residual diagram of the fitting between the sag plane and the mirror deformation;

[0047] Figure 6 It is the optical path diagram of the double-reflection optical system;

[0048] Figure 7 To circle the energy distribution diagram;

[0049] Figure 8 This is the influence diagram of bolt preload on energy concentration;

[0050] Fig. 9 This is the law diagram of the influence of posture deviation on energy concentration;

[0051] Fig.10 It is the dot diagram under the influence of assembly error;

[0052] Fig.11 This is the schematic diagram of the MLP neural network;

[0053] Fig.12 This is a comparison chart of energy concentration before and after optimization;

[0054] Fig.13 A flowchart for predicting imaging quality and guiding the actual assembly process. DETAILED DESCRIPTION

[0055] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments and the accompanying drawings. Here, the illustrative embodiments of the present invention and their descriptions are used to explain the present invention, but are not intended to limit the present invention.

[0056] An embodiment of the present invention provides a method for predicting the imaging quality of an optical system based on assembly errors, comprising:

[0057] Step 1: Establish a finite element model of the double-reflective optical system and simulate the tightening process of the bolts to obtain the mirror deformation under different preload conditions.

[0058] In the implementation, a three-dimensional geometric model of the double-reflection optical system is established, such as Figure 1 As shown in the figure, it includes a primary mirror, a secondary mirror and a mirror frame. The primary mirror is fixed to the mirror frame by three sets of bolts and nuts, and the secondary mirror is assembled with the cylindrical hole on the mirror frame by a pressure ring. In practice, light is incident parallel to the axis direction, and is converged into an image after being reflected twice by the primary mirror and the secondary mirror. During the assembly process, the mirror deformation caused by the pre-tightening force of the bolts and nuts and the assembly posture deviation of the primary and secondary mirrors will significantly affect the optical path and imaging quality of the assembled optical system.

[0059] The commercial pre-processing software Hypermesh 12.0 can be used to mesh the three-dimensional geometric model and establish a finite element model. In order to realize the simulation of the tightening process of bolts and nuts, the existing threaded mesh precise modeling method can be used to perform hexahedral mesh division on the bolts and nuts; the automatic meshing method can be used to perform tetrahedral mesh division on the primary mirror, secondary mirror and mirror frame. Figure 2 The partitioned finite element model of the double-reflective optical system is shown, which includes 152,937 tetrahedral meshes and 251,136 hexahedral meshes.

[0060] Then the parameters of the finite element model are set. All meshes use Solid185 units. This type of mesh is mainly used for three-dimensional solid structures and is sensitive to deformation and stress. In terms of material properties, the bolts and mirror frame are made of 1045 steel, and the primary and secondary mirrors are made of aluminum alloy. The specific parameters are shown in Table 1.

[0061] Table 1 Material parameters of double-reflective optical system

[0062]

[0063] The finite element model of the double-mirror optical system contains a total of 11 pairs of contact interfaces: thread contact interface, nut and primary mirror contact interface, primary mirror and frame contact interface, frame and secondary mirror contact interface, etc. During simulation, the contact interface is divided into contact surface and target surface, and meshing is performed using CONTA173 unit and TARGE170 unit respectively. The friction coefficient of the contact interface is set to 0.15. In terms of boundary conditions, the bottom of the fixed bolt and the frame is constrained to constrain the translation of the primary mirror base to simulate the real assembly state; in terms of load loading, the same rotation angle is applied to the nut surface nodes in sequence to simulate the bolt tightening process. The bolt specification is M3, and the actual preload generated in the project is about 1kN. Therefore, the nut rotation angle is about 3°.

[0064] ANASYS Workbench 19.0 is used to simulate the assembly process of the double-reflective optical system. The mirror deformation obtained by simulation is as follows: Figure 3 As shown, Figure a shows the deformation of the primary mirror, with a maximum deformation of 6.1935e-4mm, and Figure b shows the deformation of the secondary mirror, with a maximum deformation of 1.975e-8mm. It can be seen that the deformation of the primary mirror is much greater than that of the secondary mirror during the assembly process. From the magnitude of the deformation, the deformation of the secondary mirror can be ignored.

[0065] In order to ensure the reliability and accuracy of the simulation results, the relationship curve between the torque and preload obtained by simulation is compared with the theoretical results. Specifically, the theoretical formula for the torsion-tension relationship of the bolt connection is:

[0066]

[0067] Among them, μ b is the end surface friction coefficient, r b is the effective radius of the end face, P is the pitch, μ t is the thread friction coefficient, r t is the effective radius of the thread, and α is the tooth angle. It can be seen that the tightening torque T and the preload F are linearly related. For the bolt connection structure with a specification of M3, when the above parameters are known, a quantitative relationship curve between the tightening torque and the preload can be obtained.

[0068] Extract the tightening force and pre-tightening force from the finite element simulation results and obtain discrete data points at different tightening times. Figure 4 The torsion-tension relationship discrete data points obtained by finite element simulation and the torsion-tension relationship curve obtained by the torsion-tension relationship theoretical formula of bolt connection are shown. It can be seen from the figure that the discrete data points obtained by simulation basically fall on the torsion-tension relationship curve obtained by the theoretical formula, and the two are basically consistent. Further linear fitting is performed on the discrete data points, and the slope of the straight line calculated is 0.881, and the slope of the torsion-tension relationship obtained by the theoretical formula is 0.874. The error between the two is less than 1%. Therefore, it can be determined that the finite element model constructed by the present invention is accurate and reliable.

[0069] Step 2: Use Zernike polynomials to fit the mirror surface shape in the double-reflecting optical system to obtain Zernike polynomial coefficients for characterizing the mirror surface shape error.

[0070] In practice, Zernike polynomials can accurately fit the optical mirror surface shape, which can be described by mathematical polar coordinates as follows:

[0071]

[0072] Among them, P nm , Q mmis the Zernike coefficient, the angle is the angle measured counterclockwise from the local +x axis, ρ is the normalized radial coordinate, ρ∈[0,1].

[0073] The Zernike polynomial radial function is as follows:

[0074]

[0075] Where m and n are both integer values, representing the annular and radial wave numbers, respectively. nm must be an even number, and n ≥ m. s is The free variable of the interval.

[0076] Zernike polynomials can accurately fit the mirror surface shape. Each of its terms has a clear material meaning and is easy to correspond to the Seidel aberration terms of optical design. The coefficients of the first 9 terms of the Zernike polynomials and their corresponding Seidel aberrations are shown in Table 2.

[0077] Table 2 Zernike polynomial coefficients and their corresponding Seidel aberrations

[0078]

[0079] Step 3: Import the mirror deformation into the optical-mechanical-thermal coupling analysis module and use the Zernike polynomial coefficients as the basis function to fit the mirror surface shape, and obtain a macro file including the mirror surface shape deviation and the primary and secondary mirror assembly posture deviation.

[0080] In the implementation, the optical-mechanical-thermal coupling analysis module uses Sigfit software, which can convert the mirror deformation into a form that can be read by the optical simulation software to achieve accurate fitting of the mirror deformation; and through polynomial fitting, an optical macro file with Zernike coefficients can be generated. The macro file can be used to directly load the sagittal height surface obtained by fitting into the optical simulation software. The fitting residual between the sagittal height surface obtained by polynomial fitting and the input mirror deformation is shown in Figure 5 As shown in the figure, the unit waves = 1e-9m, the fitting residual distribution is relatively uniform, and most of the residuals are less than 1e-6mm, the maximum residual is 4.488e-6mm, and the maximum deformation obtained by finite element simulation is 6.1935e-4mm, which is much larger than the fitting residual, proving that the fitting error is very small and the fitting effect is good.

[0081] Step 4: Load the obtained macro file into the optical simulation module to perform light path imaging simulation to obtain energy concentration used to characterize imaging quality.

[0082] In the implementation, Zemax simulation software was used for the optical simulation module. In the optical simulation process, the double-reflective optical system needs to be modeled. In the specific modeling, the reflection surfaces corresponding to the primary mirror and the secondary mirror are established according to Table 3, where the primary mirror is a rotating parabola and the secondary mirror is an even-order hyperboloid. At the same time, the size of the opening at the center of the primary mirror is set, the center opening is 16.2 mm, and the distance between the primary and secondary mirrors is 38 mm.

[0083] Table 3 Optical system structural parameters

[0084]

[0085]

[0086] Then load the macro file into the Zemax simulation software to realize the setting of the eccentricity and tilt of the primary and secondary mirrors by inserting the mirror coordinate discontinuity surface to adjust the mirror surface shape deviation; superimpose the primary and secondary mirror assembly posture deviations with the reflection surface to generate a composite surface, so as to fit the mirror deformation to the mirror surface. In terms of light source, set the parallel light with an entrance pupil diameter of 84mm. The light is incident parallel to the axis direction and converges into an image after being reflected twice by the primary mirror and the secondary mirror. The ideal optical path is as follows Figure 6 As shown, the focal length of this system is 120mm and the wavelength of the incident light is 3.7-5.3μm.

[0087] In terms of imaging quality evaluation, since this optical system is mainly used for point target imaging detection, that is, energy detection, the imaging quality of the optical system is evaluated mainly from the perspective of energy. The optical system is an energy collection system. The energy concentration can directly reflect the energy collection ability of the optical system and can also comprehensively describe the imaging quality of the system. Therefore, energy concentration is selected as a quantitative evaluation indicator of the imaging quality of the optical system. In engineering, the diameter of the circle containing 83% of the energy is generally used to judge the energy concentration. Therefore, the energy concentration of the optical system is evaluated by the percentage of the enclosed energy in an Airy disk to the total energy. Optical simulation with Zemax software can obtain the Airy disk radius of the optical system and the fractional curve of the enclosed energy, such as Figure 7 As shown in FIG. 1 , the circled energy fraction curve represents the distribution of the spot energy with the spot diameter. In the obtained circled energy fraction curve, the circled energy fraction corresponding to the Airy disk radius is the energy concentration of the analyzed optical system.

[0088] Step 5: Construct a data set based on the preload force, primary and secondary mirror assembly posture deviation, and energy concentration, and use the constructed data set to train the MLP model and XGBoost model.

[0089] In practice, there are many factors that affect the imaging quality of the twin-mirror optical system. Usually, the mirror deformation error caused by the bolt preload is 3-4 orders of magnitude higher than the processing error of the mirror itself. Therefore, two aspects of assembly errors are mainly considered during the research process: the mirror deformation error caused by the bolt preload and the position deviation of the primary and secondary mirrors.

[0090] Influence of bolt preload: A set of primary and secondary mirror position deviations are randomly generated. The double-mirror optical system will have a certain initial deflection and eccentricity error. On this basis, the influence of different preloads on the imaging quality of the optical system is studied. For different bolt preloads, the nut rotation angle can be adjusted for characterization. Figure 8 The preload forces corresponding to the bolt connections No. 1, 2, and 3 shown in the figure are F1, F2, and F3, respectively. The single-factor control variable concept is adopted. When analyzing the preload force of one bolt, the preload forces of the other bolts are all set to 1000N. The law of energy concentration changing with preload force is shown in Figure 8 As shown in the figure, it can be seen that as the preload of bolt No. 1 increases, the energy concentration gradually increases and the imaging quality gradually improves; for bolt No. 2, as the preload increases, the energy concentration gradually decreases; for bolt No. 3, the energy concentration decreases as F3 increases, and when the preload exceeds 1000N, the energy concentration fluctuates and tends to be stable. In summary, considering the influence of the posture deviation of the primary and secondary mirrors, the influence of different bolt preloads on the energy concentration is different, and the preload size needs to be precisely controlled to ensure the best imaging quality of the optical system.

[0091] The influence of the primary and secondary mirror assembly posture deviation: For the posture deviation of the primary and secondary mirrors, the eccentricity and tilt of the mirror surface can be adjusted in the Zemax software for characterization. In the Cartesian coordinate system, the six degrees of freedom of the primary and secondary mirrors are the eccentricity errors (D x , D y and D z ), and the skew errors around these three axes (T x , T y and T z ). Applying a set of random bolt preloads will cause a certain deformation of the mirror surface, and explore the influence of posture deviation on the imaging quality of the optical system. Considering the posture deviation of the primary and secondary mirrors with 12 degrees of freedom: the eccentricity error D of the primary mirror x1 , D y1 and D z1 ; Primary mirror tilt error T x1 , T y1 and T z1 ; Secondary mirror eccentricity error D x2 , D y2 and D z2 ; Secondary mirror tilt error T x2 , T y2 and Tz2 Using the single factor control variable idea, when analyzing the posture deviation of one degree of freedom, the other posture deviations are set to 0. The change law of energy concentration with posture deviation is as follows: Fig. 9 As shown in the figure, it can be seen that the eccentricity and deflection errors around the Z axis have little effect on the energy concentration of the optical system, but the other 8 posture deviation parameters have a significant effect on the energy concentration. When the eccentricity and deflection are within ±0.01mm, the energy concentration does not change much, but when the eccentricity and deflection are greater than 0.01mm, the energy concentration drops rapidly, especially for the primary mirror. When the eccentricity and deflection are ±0.03, the energy concentration of the primary mirror drops by nearly 60% and 50% respectively.

[0092] In order to intuitively show the impact of overall assembly error on the energy concentration of the optical system, the concept of spot diagram is introduced: for the actual optical system, the light is not an ideal straight line, and the image is a diffuse light spot. The light spot image formed is called a spot diagram. The distribution of points in the spot diagram approximately represents the energy distribution of the image point. According to the definition of energy concentration, if all points in the spot diagram fall into the Airy disk, it means that the energy concentration has reached the maximum value. Fig.10 It is a point diagram of the optical system under different conditions. The black circle in the figure is the Airy disk of the optical system. Figure a corresponds to the imaging result of the ideal optical path, Figure b and Figure c correspond to the imaging results under the influence of mirror deformation and posture deviation respectively, and Figure d is the imaging result under the combined effect of mirror deformation and posture deviation. The energy concentration of Figure a and Figure b is 83.8%, the energy concentration of Figure c is 52.4%, and the energy concentration of Figure d is 47.5%. It can be seen from the figure that the energy concentration may be affected by the combined effect of mirror deformation caused by preload and posture deviation, especially posture deviation, which can easily cause the light spot to exceed the circle range of the Airy disk, resulting in a significant reduction in imaging quality.

[0093] from Fig.10 It can be seen that within the allowable error range, the influence of the eccentricity and skew errors of the primary and secondary mirrors in the Z-axis direction on the imaging quality can be ignored. Therefore, the mirror posture deviation only considers the eccentricity errors of the primary and secondary mirrors along the X-axis and Y-axis, as well as the skew errors around the X-axis and Y-axis. The sample format of the data set is shown in Table 4.

[0094] Table 4 Datasets of assembly error and imaging quality

[0095]

[0096] After that, the feature variables and target variables in the constructed data set are randomly sorted and normalized, and finally the data set is divided into training set and test set according to the set ratio.

[0097] In practice, a single model is often limited to a specific data distribution or pattern and may not perform well when faced with new data. Therefore, the present invention introduces the idea of ​​a combined model, combines multiple different models, fully utilizes the advantages of a single model algorithm, obtains information about the prediction target from multiple angles, and improves the generalization ability and prediction accuracy of the overall model. In one implementation, an MLP-XGBoost model can be constructed, which is a combined prediction model that combines a multi-layer perceptron (MLP) with a gradient boosted tree (XGBoost).

[0098] (1) MLP model

[0099] The MLP multilayer perceptron neural network is a multilayer feedforward neural network that can achieve nonlinear mapping between input and output and is widely used in machine learning, pattern recognition and other fields. Fig.11 As shown, the entire neural network consists of an input layer, a hidden layer, and an output layer, where x = [x1, x2, x3, …, x i ] is the input variable, k=[k1,k2,k3,…,k j ] is the hidden layer output variable, y is the final output variable, and the relationship between y and k is as follows:

[0100]

[0101] In the formula, w j is the weight of the connection between the jth neuron and y, θ y is the bias of y, k j is the output of the jth hidden neuron, and the relationship between k and x is as follows:

[0102]

[0103] In the formula, v ij is the weight of the connection between the ith input variable x and the jth neuron, x i is the ith input variable, is the bias of the jth neuron in the kth hidden layer.

[0104] (2) XGBoost Model

[0105] XGBoost is a gradient boosting ensemble machine learning model that helps avoid overfitting. It mainly uses an iterative regression tree algorithm to construct multiple regression trees in a serial manner, where the goal of the newly generated regression tree is to fit the residual of the previous tree. In each round of iteration, the new model will contribute to reducing the total loss of all previous models. In general, the goal of XGBoost is to learn a prediction function To minimize a given loss function, the loss function can be expressed as:

[0106]

[0107] In the formula, x i is the feature vector of the i-th sample, y i is the corresponding output, l is a differentiable convex loss function used to measure the prediction and the true value y i The gap between k is the kth weak learner, Ω represents the complexity of the model, which is usually in the form of Where T is the number of leaf nodes in the tree, w is the vector of leaf node weights, and γ and λ are regularization parameters.

[0108] In each iteration t, XGBoost adds a new weak learner f t , used to predict the residual or error The prediction function is updated as: In order to find the best f t , will define a t The second-order approximate loss function is:

[0109]

[0110] where g i is about The first derivative of i is the second-order derivative. In each round of training, XGBoost uses a greedy algorithm to traverse all possible split points and selects the split point with the smallest loss function to split the tree node. This method can effectively handle discrete data and missing value problems.

[0111] In practice, the main advantage of MLP is that the model structure is flexible, with enough hidden layers and neurons, and it can approximate any complex function. However, it relies on a large number of parameter adjustments and requires sufficient data to reduce the risk of overfitting, and has low computational efficiency; XGBoost performs well in capturing the structured features of data sets, but may not be comparable to neural networks when processing complex nonlinear relationships of input data. Therefore, the present invention combines the MLP and XGBoost models to give full play to the advantages of both and make up for each other's shortcomings. MLP can extract highly abstract features through its deep structure, while XGBoost can effectively integrate these features and has better coordination with outliers and unbalanced data. The combination of these two models to predict the imaging quality of the double-reflection optical system can achieve more accurate and stable prediction results.

[0112] During the specific training process, the MLP model and XGBoost model are trained separately using the training set;

[0113] Then, the MLP model and XGBoost model were tested using the test set to obtain the energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set;

[0114] Calculate a first weight matrix and a second weight matrix according to the obtained energy concentration prediction value under the test set and the true value of the energy concentration under the test set;

[0115] The energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set are weightedly calculated using the first weight matrix and the second weight matrix to obtain the prediction value of the MLP-XGBoost combination model;

[0116] The mean absolute error, mean square error, and mean error percentage are calculated based on the number of samples in the test set, the predicted value of the combined model, and the true value of the energy concentration;

[0117] When the mean absolute error, mean square error, and average error percentage meet expectations, the MLP-XGBoost combined model is used as the final model for predicting the imaging quality of the optical system; if they do not meet expectations, the training set and test set are repartitioned and the step of using the training set to train the MLP model and the XGBoost model separately is returned.

[0118] In order to evaluate the prediction accuracy of the MLP-XGBoost combination model, MLP, XGBoost models and two common prediction models: BP neural network and SVR regression model were selected as the control group, and the data set construction and division methods, model training and error evaluation methods were consistent. Table 5 shows the error evaluation results of the above models in the test set. It can be seen from the table that the average prediction errors of the single models MLP and XGBoost are 4.83% and 7.85% respectively, which are higher than the proposed MLP-XGBoost prediction model (4.70%), and the MAE and MSE indicators are also smaller than the corresponding indicators of the proposed MLP-XGBoost, which shows that the prediction accuracy of the combination model is better than the two single models, and the proposed MLP and XGBoost combination model is reliable. In addition, compared with the other two regression prediction models, the average prediction errors of the BP neural network and SVR regression models reached 6.07% and 24.96% respectively, which are much larger than the prediction error of MLP-XGBoost, and the MAE and MSE indicators of MLP-XGBoost are also the lowest. Therefore, the MLP-XGBoost combination model proposed in the present invention shows the best optical imaging quality prediction accuracy.

[0119] Table 5 Comparison of prediction model error evaluation indicators

[0120] Prediction Model MAE / mm MSE / mm MPE MLP-XGBoost 0.0320 0.0416 4.70％ MLP 0.0335 0.0430 4.83％ XGBoost 0.0551 0.0707 7.85％ BP 0.0379 0.0615 6.07％ SVR 0.1204 0.1729 24.96％

[0121] Step 6: Obtain the current preload force and primary and secondary mirror assembly posture deviations of the optical system to be predicted, and input the obtained preload force and primary and secondary mirror assembly posture deviations into the trained MLP model and XGBoost model respectively to obtain the first energy concentration and the second energy concentration.

[0122] Step 7: Obtain the first weight matrix and the second weight matrix of the MLP model and the XGBoost model, and use the determined first weight matrix and the second weight matrix to perform weighted calculation on the first energy concentration and the second energy concentration to obtain the final energy concentration as a prediction result.

[0123] In implementation, the first weight matrix and the second weight matrix may be determined using the following expressions:

[0124]

[0125] Among them, W * The weight matrix for combining a single model, matrix I1 = [1,1,…,1] T , S is the prediction error information matrix of a single model, S = e T e, e is the prediction error matrix, e=[E1,E2,…,E i ] T , where E i is the prediction error vector of the i-th model:

[0126] E i =Z i -Y i =[e i1 ,e i2 ,…,e iN ] T

[0127] Among them, Z i =[z1,z2,…,z j ,…,z N ] T is the true value of energy concentration, Y i =[y i1 ,y i2 ,…,y ij ,…,y iN ] T is the predicted value of energy concentration.

[0128] Correspondingly, the predicted value of the MLP-XGBoost combination model is expressed as:

[0129]

[0130] In a specific implementation, during the actual assembly and adjustment of the double-reflective optical system, the posture deviation of the primary and secondary mirrors is inevitable and difficult to completely eliminate. It can only be controlled within a certain range. It is usually necessary to adjust the bolt preload to improve the imaging quality of the double-reflective optical system. For the assembly of primary and secondary mirrors with a certain posture deviation, the preload can be optimized through the random gradient descent algorithm to maximize the imaging quality of the optical system and guide the actual assembly and adjustment process of the bolt connection.

[0131] The core idea of ​​the stochastic gradient descent algorithm is to randomly select a sample in each iteration, and iteratively update the parameters through the gradient of the sample to minimize the loss function. This method can improve the optimization efficiency while avoiding falling into the local minimum, making the training results more accurate. Specifically, for the loss function L(θ), θ is the parameter vector, and the update rule of stochastic gradient descent is as follows:

[0132]

[0133] In the formula, θ t is the parameter value after the tth iteration, η is the learning rate, is the loss function at the current parameter value θ t About the sample (x i ,y i )’s gradient.

[0134] Based on this, after obtaining the energy concentration predicted by the MLP-XGBoost combined model, the assembly posture deviation of the primary and secondary mirrors can be used as input values ​​and set as constants according to the actual assembly situation; the preload force can be used as an input value and set as an optimization variable; the maximization of the predicted energy concentration result is used as the optimization goal, and the stochastic gradient descent algorithm is used to iteratively calculate the optimal preload force, and the actual assembly process is guided by the calculated optimal preload force.

[0135] Effect verification: 10 groups of primary and secondary mirror posture errors and initial bolt preload forces were randomly generated. The energy concentration was calculated by combining the model. Then, the initial preload was optimized using the stochastic gradient descent algorithm to obtain the optimized energy concentration. Fig.12 As shown in the figure, the energy concentration of the 10 groups of experiments increased by an average of 11.34% compared with that before optimization, proving that the optimization algorithm used is effective. The preload optimization using this method can significantly improve the energy concentration of the optical system and can be used to guide the actual assembly process.

[0136] See also Fig.13The overall design idea of ​​the present invention includes: constructing an accurate finite element model of a double-reflective optical system, simulating the bolt tightening process, obtaining the mirror deformation under the action of assembly stress, fitting the mirror surface shape using Zernike polynomials, and using this as an intermediate medium. Zemax is used to simulate the imaging of the optical system under the conditions of mirror deformation and assembly posture, and the influence of various relevant assembly parameters on the imaging quality of the optical system is explored, and then a data set is constructed; the data set is used to train the established MLP-XGBoost combination model, and then the trained MLP-XGBoost combination model can be used to predict the imaging quality according to the relevant assembly parameters, so as to evaluate the assembly quality; in addition, the maximization of the predicted energy concentration result can be used as the optimization goal, and the stochastic gradient descent algorithm is used to iteratively calculate the optimal preload, and the actual assembly process is guided according to the calculated optimal preload.

[0137] The imaging quality of the optical system under different assembly error conditions is accurately and quickly predicted, with a prediction accuracy of up to 95%. Finally, considering the different primary and secondary mirror posture deviations, the bolt preload is optimized using the stochastic gradient descent method, and the following conclusions are obtained:

[0138] 1) Through the optical-mechanical integration simulation method of the optical system, the influence of different assembly errors on the imaging quality of the optical system is analyzed. It is found that when there is a posture deviation between the primary and secondary mirrors, the influence of different bolt preloads is different. It is necessary to adjust the appropriate preload size to correct the degradation of imaging quality caused by the posture deviation; the eccentricity and skew errors of the primary and secondary mirrors along the X and Y axes have a significant impact on the imaging quality, while the segregation and skew errors in the Z axis direction (optical axis direction) have almost no effect.

[0139] 2) Considering the mirror deformation and optical element posture deviation caused by the preload during the assembly process, a combined prediction model for the energy concentration of the double-mirror optical system based on MLP-XGBoost was constructed. The average error of the training set was only 4.70%, which is significantly better than the single MLP and XGBoost models, as well as the traditional regression models (BP and SVR).

[0140] 3) A preload force optimization method based on the MLP-XGBoost model and the stochastic gradient descent algorithm was proposed. Under different assembly posture conditions, the energy concentration of the optical system was increased by an average of 11.34% compared with before optimization, which can be used to guide the actual assembly process of the optical system.

[0141] The above description is only a preferred embodiment of the present invention and is not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.

Claims

1. A method for predicting the imaging quality of an optical system based on assembly error, characterized in that: include: The finite element model of the double-reflective optical system was established, and the tightening process of the bolts was simulated to obtain the mirror deformation under different preload conditions; The Zernike polynomials are used to fit the mirror surface shape in the double-reflection optical system to obtain the Zernike polynomial coefficients used to characterize the mirror surface shape error. The mirror deformation is imported into the optical-mechanical-thermal coupling analysis module and the mirror surface shape is fitted using the Zernike polynomial coefficients as the basis function to obtain a macro file including the mirror surface shape deviation and the primary and secondary mirror assembly posture deviation; The obtained macro file is loaded into the optical simulation module to perform light path imaging simulation to obtain the energy concentration used to characterize the imaging quality; A data set was constructed based on the preload force, primary and secondary mirror assembly posture deviation, and energy concentration, and the constructed data set was used to train the MLP model and XGBoost model. The current preload force and the primary and secondary mirror assembly posture deviations of the optical system to be predicted are obtained, and the obtained preload force and the primary and secondary mirror assembly posture deviations are respectively input into the trained MLP model and the XGBoost model to obtain the first energy concentration and the second energy concentration; The first weight matrix and the second weight matrix of the MLP model and the XGBoost model are obtained, and the first energy concentration and the second energy concentration are weightedly calculated using the determined first weight matrix and the second weight matrix to obtain the final energy concentration as a prediction result.

2. The optical system imaging quality prediction method based on assembly error according to claim 1, characterized in that: Also includes: The assembly posture deviation of the primary and secondary mirrors is taken as an input value and set as a constant according to the actual assembly situation; the preload force is taken as an input value and set as an optimization variable; Taking the maximization of the prediction result as the optimization goal, the stochastic gradient descent algorithm is used to iteratively calculate the optimal preload force, and the actual assembly process is guided according to the calculated optimal preload force.

3. The optical system imaging quality prediction method based on assembly error according to claim 1, characterized in that: The process of establishing the finite element model of the double-reflective optical system includes: Establish a three-dimensional geometric model of the double-mirror optical system, including the primary mirror, secondary mirror and mirror frame. The primary mirror is fixed to the mirror frame through three sets of bolts and nuts, and the secondary mirror is assembled with the cylindrical hole on the mirror frame through a pressure ring. The finite element model is established by meshing the three-dimensional geometric model, wherein the bolts and nuts are meshed with hexahedron, and the primary mirror, secondary mirror and mirror frame are meshed with tetrahedron.

4. The optical system imaging quality prediction method based on assembly error according to claim 1, characterized in that: The process of loading the obtained macro file into the optical simulation module for light path imaging simulation includes: Establish the reflection surfaces corresponding to the primary mirror and the secondary mirror, where the primary mirror is a rotating parabola and the secondary mirror is an even-order hyperboloid, and set the size of the opening at the center of the primary mirror and the distance between the primary and secondary mirrors; The mirror surface deviation is inserted into the mirror coordinate discontinuity surface to achieve the setting of the primary mirror and the secondary mirror eccentricity and tilt; The assembly posture deviations of the primary and secondary mirrors are superimposed on the reflective surface to generate a composite surface, thereby fitting the mirror deformation to the mirror surface.

5. The optical system imaging quality prediction method based on assembly error according to claim 1, characterized in that: The process of constructing a data set based on preload, primary and secondary mirror assembly posture deviation, and energy concentration includes: The preload force and the primary and secondary mirror assembly posture deviation are used as feature variables, and the energy concentration is used as the target variable to construct a data set. The primary and secondary mirror assembly posture deviation includes the eccentricity error of the primary and secondary mirrors along the X-axis and Y-axis, as well as the skew error around the X-axis and Y-axis. The feature variables and target variables in the constructed dataset are randomly sorted and normalized, and finally the dataset is divided into training set and test set according to the set ratio.

6. The optical system imaging quality prediction method based on assembly error according to claim 5, characterized in that: The process of training the MLP model and XGBoost model using the constructed dataset includes: Use the training set to train the MLP model and XGBoost model separately; The MLP model and the XGBoost model were tested using the test set to obtain the energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set; Calculate a first weight matrix and a second weight matrix according to the obtained energy concentration prediction value under the test set and the true value of the energy concentration under the test set; The energy concentration prediction values ​​of the MLP model and the XGBoost model under the test set are weightedly calculated using the first weight matrix and the second weight matrix to obtain the prediction value of the MLP-XGBoost combination model; The mean absolute error, mean square error, and mean error percentage are calculated based on the number of samples in the test set, the predicted value of the combined model, and the true value of the energy concentration; When the mean absolute error, mean square error, and average error percentage meet expectations, the MLP-XGBoost combined model is used as the final model for predicting the imaging quality of the optical system; if they do not meet expectations, the training set and test set are repartitioned and the step of using the training set to train the MLP model and the XGBoost model separately is returned.

7. The optical system imaging quality prediction method based on assembly error according to claim 6, characterized in that: The first weight matrix and the second weight matrix are determined using the following expressions: Among them, W * The weight matrix for combining a single model, matrix I1 = [1,1,…,1] T , S is the prediction error information matrix of a single model, S = e T e, e is the prediction error matrix, e=[E1,E2,…,E i ] T , where E i is the prediction error vector of the i-th model: E i =Z i -Y i =[e i1 ,e i2 ,…,e iN ] T Among them, Z i =[z1,z2,…,z j ,…,z N ] T is the true value of energy concentration, Y i =[y i1 ,y i2 ,…,y ij ,…,y iN ] T is the predicted value of energy concentration.

8. The optical system imaging quality prediction method based on assembly error according to claim 7, characterized in that: The prediction results are expressed as:

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