A rectangular aperture curved surface figure error compensation method and system

Abstract

The application provides a rectangular aperture curved surface form error compensation method and system, and belongs to the field of ultra-precision machining of free-form optical elements. The method comprises the following steps: based on the optimal moving path, the workpiece surface is machined, the coordinates and actual height values of each point on the curved surface form are collected, and the machining error is obtained; the machining error curved surface is normalized by using the Zernike polynomial to fit the predicted error curved surface; the optimal moving path in the machining process is corrected by taking the negative of the predicted error curved surface, and the workpiece surface is re-machined until the machining error meets the machining precision. The application strictly controls the precision and range of error compensation, and ensures that the compensated curved surface can meet the design requirements.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of ultra-precision machining of free-form optical elements, and more particularly relates to a rectangular aperture curved surface shape error compensation method and system. BACKGROUND

[0002] Free-form optical elements play an important role in optical systems. Their high curved surface design freedom can effectively reduce the influence of optical aberration and chromatic aberration, improve imaging quality, meet various complex optical requirements, and are widely used in fields such as lithography machines, laser nuclear fusion devices, space telescopes, and medical equipment. High-precision free-form optical elements play a key role in ensuring the performance of optical systems. However, during the machining of free-form surfaces, various factors such as positioning and following errors of machining equipment, thermal deformation, tool wear, and material inhomogeneity can affect the machining process, resulting in surface shape errors in the machined free-form surface, which seriously affects the performance of the optical system. Therefore, a high-precision compensation method is a key technology for the convergence of free-form surface shape error machining. At the same time, in many current optical elements, such as optical filters, laser resonant cavities, and spectrometers, rectangular apertures are commonly used. However, current compensation algorithms are mainly designed for circular apertures, and when applied to rectangular apertures, they may have insufficient fitting accuracy and low convergence efficiency, which affects the performance and production efficiency of optical elements. Therefore, in order to meet the needs of rectangular aperture optical free-form surfaces, it is urgent to develop a more advanced and effective surface shape error compensation method to improve fitting accuracy and convergence efficiency, achieve efficient and accurate error compensation, and thus improve the surface shape accuracy and optical performance of free-form optical elements. SUMMARY

[0003] In view of the defects of the prior art, the purpose of the present application is to provide a rectangular aperture curved surface shape error compensation method and system, which aims to solve the problem of insufficient fitting accuracy and low convergence efficiency when the existing compensation algorithm for free-form surface shape is applied to rectangular apertures.

[0004] To achieve the above-mentioned purpose, the present application provides a rectangular aperture curved surface shape error compensation method, comprising the following steps:

[0005] Step S1: Based on the parameters and characteristics of the curved surface to be machined, a machining program is constructed to obtain the best movement path during machining;

[0006] Step S2: Based on the best movement path obtained in step S1, the workpiece surface is machined, the coordinates and actual height values of each point on the curved surface shape are collected, and the machining error is obtained;

[0007] Step S3: judging whether the machining error meets the machining precision, if yes, terminating the execution of subsequent steps, and taking the workpiece processed in step S2 as the final workpiece; if not, turning to step S4;

[0008] Step S4: normalizing the machining error surface in a unit square and fitting the predicted error surface by using the Zernike polynomial;

[0009] Step S5: adding the predicted error surface to the machining program in step S1 in reverse, correcting the optimal moving path in the machining process, and turning to step S2 until the machining error meets the machining precision; wherein the surface shape of the workpiece is a rectangular aperture.

[0010] Further preferably, step S4 specifically comprises the following steps:

[0011] Step S4.1: constructing a machining error surface coordinate system based on the machining error obtained in step S2, and mapping the rectangular aperture surface area to a unit square for normalization processing;

[0012] Step S4.2: taking the center of the unit square as the origin, calculating the relative distance and angle of each point to the center, and mapping the points of the rectangular aperture surface area to the polar coordinate system of a virtual unit circle;

[0013] Step S4.3: determining the order of the Zernike polynomial based on the radius and angle parameters in the virtual unit circle, and initializing the coefficients of the Zernike polynomial in the polar coordinate system;

[0014] Step S4.4: multiplying each term of the Zernike polynomial by the corresponding coefficient of the Zernike polynomial in the polar coordinate system and accumulating to obtain the predicted height value of each point of the surface shape;

[0015] Step S4.5: fitting the target function by using the least squares method to obtain the predicted error, adjusting the coefficients of the Zernike polynomial, and turning to step S4.4 until the minimum predicted error is obtained; wherein the error between the predicted height value and the actual height value is the predicted error, and the minimum predicted error is the target function;

[0016] Step S4.6: judging whether the minimum predicted error obtained in step S4.5 meets the preset condition, if yes, obtaining the predicted error surface, if not, adjusting the order of the Zernike polynomial and the step size of the least squares method, and turning to step S4.4.

[0017] Further preferably, the construction method of the machining error surface coordinate system is to keep the horizontal and vertical coordinates of the rectangular aperture surface unchanged, and to take the machining error as the vertical coordinate axis.

[0018] Further preferably, when the predicted error does not meet the preset condition, the order of the Zernike polynomial is increased or the step size of the least squares method is decreased.

[0019] Further preferably, when the prediction error surface is taken inversely into the machining program, the cutting depth, speed and direction of the tool at each point position are adjusted.

[0020] Further preferably, the parameters and characteristics of the surface to be machined include process parameters of machining, precision parameters of the machined surface and material properties of the workpiece.

[0021] In a second aspect, the application provides a rectangular aperture curved surface error compensation system, comprising:

[0022] A machining path acquisition module is configured to construct a machining program based on the parameters and characteristics of the surface to be machined, so as to obtain an optimal movement path in the machining process.

[0023] A tool is configured to machine the surface of the workpiece according to the optimal movement path.

[0024] A profilometer measurement probe is configured to collect the coordinates and actual height values of each point on the curved surface.

[0025] A machining error calculation module is configured to calculate the machining error of each point based on the actual height values of each point on the curved surface and the preset height values.

[0026] A first determination module is configured to determine whether the machining error meets the machining precision. If yes, the workpiece after the current machining is taken as the final workpiece. Otherwise, the prediction error surface fitting module is driven to run.

[0027] The prediction error surface fitting module is configured to normalize the machining error surface using Zernike polynomials to fit the prediction error surface.

[0028] A movement path compensation module is configured to take the prediction error surface inversely into the machining program in the machining path acquisition module to correct the optimal movement path in the machining process.

[0029] The curved surface of the workpiece is of a rectangular aperture.

[0030] Further preferably, the prediction error surface fitting module comprises:

[0031] A rectangular aperture curved surface normalization unit is configured to construct a machining error surface coordinate system based on the machining error, and map the rectangular aperture curved surface region to a unit square for normalization processing.

[0032] A coordinate conversion unit is configured to take the center of the unit square as the origin, calculate the relative distance and angle of each point to the center, and map the points of the rectangular aperture curved surface region to the polar coordinate system of a virtual unit circle.

[0033] The data initialization unit is configured to determine the order of the Zernike polynomial based on the radius and angle parameters in the virtual unit circle, and initialize the coefficients of the Zernike polynomial in the polar coordinate system.

[0034] The height prediction unit is configured to accumulate the Zernike polynomial terms multiplied by the corresponding coefficients of the Zernike polynomial in the polar coordinate system to obtain the predicted height value of each point on the surface profile.

[0035] The prediction error solving unit is configured to fit a target function by using the least square method to obtain the prediction error.

[0036] The second determination unit is configured to determine whether the minimum prediction error meets a preset condition.

[0037] The minimum prediction error finding unit is configured to adjust the coefficients of the Zernike polynomial, or adjust the order of the Zernike polynomial and the step size of the least square method based on the prediction error.

[0038] The error between the predicted height value and the actual height value is taken as the prediction error, and the minimum prediction error is taken as the target function.

[0039] Further preferably, the horizontal and vertical coordinates of the machining error surface coordinate system in the rectangular aperture surface normalization unit are the horizontal and vertical coordinates of the rectangular aperture surface, and the vertical coordinate is the machining error.

[0040] Further preferably, in the minimum prediction error finding unit, when the prediction error does not meet the preset condition, the order of the Zernike polynomial is increased, or the step size of the least square method is reduced.

[0041] Further preferably, in the movement path compensation module, when the prediction error surface is added to the machining program, the cutting depth, speed and direction of the tool at each point are adjusted.

[0042] Further preferably, in the machining path obtaining module, the parameters and characteristics of the to-be-machined surface include the machining process parameters, the precision parameters of the machining surface and the material properties of the workpiece.

[0043] Overall, the above technical solutions conceived by the present application have the following beneficial effects compared with the prior art:

[0044] In view of the characteristics of the rectangular aperture free-form surface, the rectangular aperture free-form surface shape error compensation method is provided to improve the precision and efficiency of compensation; the Zernike polynomial is used for unit square normalization operation of the curved surface, and since the Zernike polynomial is orthogonal in the circular area, when directly converted and used in the rectangular aperture curved surface, the orthogonal condition is no longer met, therefore, orthogonalization processing is needed; through calculation and adjustment, the Zernike polynomial fitting curved surface can better adapt to the shape characteristics of the rectangular aperture, so as to realize the normalization of the unit square; after the normalization is completed, the amount of compensation is calculated according to the normalized Zernike curved surface fitting result; then, through the corresponding processing or adjustment measures, the error of the curved surface is accurately compensated, and in the compensation process, the precision and range of compensation are strictly controlled to ensure that the compensated curved surface can meet the design requirements.

[0045] The method can comprehensively and effectively solve the problem of rectangular aperture free-form surface shape error compensation by using Zernike polynomial to fit the overall three-dimensional surface error, and significantly improve the machining precision and quality of the rectangular aperture curved surface.

[0046] More specifically, after the rectangular aperture curved surface is normalized by the Zernike curved surface, the fitting compensation is performed, which significantly improves the fitting precision; through accurate modeling and analysis of the curved surface shape, more subtle surface features can be captured, so that the deviation between the fitting result and the actual demand is greatly reduced; compared with the traditional method, the processing of complex curved surface and small error is more accurate, thereby effectively improving the machining quality of the rectangular aperture curved surface.

[0047] The rectangular aperture curved surface shape error compensation method provided by the application greatly improves the iteration efficiency; compared with the prior art, the application firstly uses Zernike polynomial to fit the error curved surface and performs unit square normalization operation on the error curved surface for the specific object of the rectangular aperture curved surface, so that the originally complex rectangular aperture curved surface problem can be converted into a relatively standard unit square problem for processing; in terms of calculation process, the rectangular region is mapped to a virtual unit circle polar coordinate system through region transformation, and the characteristics of Zernike polynomial in the unit circle are used to reduce the calculation steps and complexity. In terms of algorithm, an optimization algorithm is used to adjust the Zernike polynomial coefficients during the fitting process, so that the constructed curved surface is close to the actual machining error curved surface, the error is calculated according to the objective function, and the coefficients are adjusted to realize fast convergence. In terms of data processing and error evaluation, the data is arranged to form an ordered matrix, the fitting result is evaluated and compared with the original data, the parameters and error algorithm are adjusted, and the iteration efficiency is improved. Through a series of means and optimized calculation process and algorithm, unnecessary repeated calculation and parameter adjustment are reduced, so that a relatively ideal compensation result can be obtained in a short time, the iteration efficiency of the rectangular aperture curved surface machining is improved, the entire machining cycle is greatly shortened, and a strong guarantee is provided for batch production.

[0048] The rectangular aperture curved surface high-precision surface error compensation method provided by the application significantly reduces the surface error. First, the method is innovative in using a fitting strategy based on Zernike polynomials for the specific object of rectangular aperture free-form surfaces. Compared with the prior art, the traditional compensation method often has the problem of insufficient fitting accuracy when facing rectangular aperture curved surfaces, while the method of the application can more accurately describe the shape characteristics of rectangular aperture curved surfaces by using Zernike polynomials to normalize the unit square and then fitting and compensating. In the compensation process, the mathematical properties of Zernike polynomials are used to fit the error surface in the unit square coordinate system. In the fitting process, the coefficients of the polynomials are constantly adjusted so that the surface constructed by the polynomials is as close as possible to the actual machining error surface. Through repeated calculation and optimization, the Zernike polynomial expression that best describes the characteristics of the machining error surface is found, thereby achieving high-precision fitting of the machining error surface. In addition, the method also includes steps such as adding the original machining program to the machining error surface after taking the inverse, and performing secondary machining after correcting the tool path. By repeating these steps, the machining process and parameters are continuously improved, gradually reducing the machining error. Compared with the prior art, this iterative compensation method can more effectively eliminate surface errors and improve the machining accuracy of rectangular aperture curved surfaces.

[0049] In summary, the rectangular aperture curved surface surface error compensation method provided by the application significantly reduces surface errors through innovative fitting strategies and iterative compensation steps, compared with the prior art, providing a strong guarantee for reducing subsequent processing and detection costs. Due to the improvement in compensation accuracy, the time for additional finishing and fine polishing processes is reduced, saving costs. At the same time, the improved production efficiency allows more machining tasks to be completed in the same time, increasing output and further improving economic benefits. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 is a rectangular aperture curved surface surface error compensation method flowchart provided by the application;

[0051] Figure 2 is a rectangular aperture curved surface surface error compensation method schematic diagram provided by the application. DETAILED DESCRIPTION

[0052] In order to make the purpose, technical scheme and advantages of the application more clear and explicit, the application will be further described in detail below in combination with the drawings and examples. It should be understood that the specific examples described herein are only used to explain the application and do not limit the application.

[0053] In the embodiments of this application, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of this application should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0054] The technical solutions provided in the embodiments of this application are introduced.

[0055] In response to the demand for surface error compensation of rectangular aperture free-form surfaces, the present application provides a surface error compensation method for rectangular aperture free-form surfaces based on Zernike polynomial fitting, aiming to significantly improve the machining accuracy of rectangular aperture free-form surfaces, effectively solve the limitations of existing compensation methods in the application of rectangular aperture surfaces, greatly improve the accuracy and stability of compensation, reduce the adverse effects of machining errors on optical performance, and meet the strict requirements of high-precision optical systems for rectangular aperture free-form surfaces.

[0056] Example 1

[0057] like Figure 1 and Figure 2 As shown, the present application provides a surface error compensation method for a rectangular aperture free-form surface based on Zernike polynomial fitting, comprising the following steps:

[0058] Step S1: Design a machining program based on the ideal machining surface, plan the tool path, and perform the first machining;

[0059] More specifically, step S1 includes analyzing various parameters and characteristics of the ideal machined surface, fully considering factors such as machining process requirements, precision standards, and material properties, designing a suitable machining program based on the characteristics and performance of the machining equipment, and planning the optimal movement path of the tool during the machining process to ensure that the tool can efficiently and accurately cut the workpiece to achieve the ideal surface shape; starting the machining equipment to perform the first machining operation according to the designed machining program and tool path to initially form a surface close to the ideal shape;

[0060] Step S2: Using a profilometer to measure the actual machined surface shape and obtain the machining error;

[0061] More specifically, step S2 is: scanning the workpiece surface point by point and collecting data using a profilometer measuring probe, recording information such as the coordinates and height of each point on the actual machined surface, and then comparing and analyzing the measured data with the design data of the ideal machined surface to calculate the machining error generated during the actual machining process;

[0062] Step S3: Determine whether the machining error meets the machining accuracy. If so, terminate the execution of the following steps and obtain the final machined workpiece; if not, go to step S4;

[0063] Step S4: using Zernike polynomials to normalize the machining error surface to a unit square and then fitting the predicted error surface;

[0064] More specifically, S4 includes the following steps:

[0065] Construct a machining error surface coordinate system, map the rectangular area to a relative unit square for normalization; take the center of the unit square as the origin, calculate the relative distance and angle of each point to the center, and map the points in the rectangular area to a polar coordinate system of a virtual unit circle;

[0066] The order of the Zernike polynomial is determined based on the "radius" and "angle" parameters in the transformed area, and the coefficients of the Zernike polynomials in the polar coordinate system are initialized. In each iteration, the predicted height value is calculated by multiplying each Zernike polynomial term by the corresponding Zernike polynomial coefficient on the virtual unit circle according to the current Zernike polynomial coefficients, and then accumulating the results. The objective function is optimized by fitting using least squares, etc., and the prediction error is calculated based on the objective function. The coefficients of the Zernike polynomials are adjusted to gradually reduce the error. The objective function is to minimize the error between the predicted height value and the actual measured height value.

[0067] More specifically, the fitting result evaluation and correction method is as follows: if the minimum prediction error does not meet the requirements, the order of the Zernike polynomial can be increased, the step size of the least squares optimization algorithm can be adjusted, and the processed data can be preprocessed by filtering, etc., and step S4 can be repeated until the obtained prediction error is less than the preset error value, the predicted height value corresponding to the minimum prediction error is obtained, and the prediction error surface is constructed;

[0068] Step S5: After inverting the predicted error surface, add it to the original machining program, and perform secondary machining after correcting the tool path;

[0069] More specifically, S5 is as follows: performing an inversion operation on the prediction error surface obtained by analysis in step S4, adding the inverted prediction error data to the originally designed machining program, correcting and supplementing the original machining program, replanning the tool path, adjusting the cutting depth, speed, direction and other parameters of the tool at each position, and starting the machining equipment to perform secondary machining according to the corrected tool path;

[0070] Step S6: Repeat steps S1 to S5, iterate the machining process and parameters, and gradually reduce the machining error until the machining accuracy meets the requirements.

[0071] Example 2

[0072] In a second aspect, the application provides a rectangular aperture curved surface form error compensation system, comprising:

[0073] A machining path acquisition module, configured to construct a machining program based on parameters and characteristics of the curved surface to be machined, so as to obtain an optimal movement path in the machining process;

[0074] A tool, configured to machine the workpiece surface according to the optimal movement path;

[0075] A profilometer measurement probe, configured to collect coordinates and actual height values of each point on the curved surface form;

[0076] A machining error calculation module, configured to calculate the machining error of each point based on the actual height value of each point on the curved surface form and a preset height value;

[0077] A first determination module, configured to determine whether the machining error meets the machining accuracy, if yes, the workpiece after the current machining is taken as the final workpiece, otherwise, the prediction error surface fitting module is driven to run;

[0078] A prediction error surface fitting module, configured to fit the prediction error surface after the machining error surface is normalized by using the Zernike polynomial;

[0079] A movement path compensation module, configured to add the prediction error surface to the machining program in the machining path acquisition module in reverse, so as to correct the optimal movement path in the machining process;

[0080] Preferably, the curved surface form of the workpiece is a rectangular aperture.

[0081] Further preferably, the prediction error surface fitting module comprises:

[0082] A rectangular aperture curved surface normalization unit, configured to construct a machining error curved surface coordinate system based on the machining error, and map the rectangular aperture curved surface region to a unit square for normalization processing;

[0083] A coordinate conversion unit, configured to take the center of the unit square as the origin, calculate the relative distance and angle of each point to the center, and map the points of the rectangular aperture curved surface region to the polar coordinate system of a virtual unit circle;

[0084] More specifically, the Zernike polynomial is usually defined in the polar coordinate system, has good orthogonality, and performs well in the unit circle; for any point (r, θ) in the unit circle, the Zernike polynomial can be expressed as the product of a radial polynomial and an angle function; the basic form is as follows:

[0085]

[0086] wherein: n is the order of the polynomial (a non-negative integer); m is the order related to the angle, satisfying |m|≤n and n-|m| is even; p is the normalized radial distance; θ is the polar angle;

[0087] Radial polynomial is defined as follows:

[0088]

[0089] For practical applications, especially in optics, the Zernike polynomials are usually used in real-valued form. The real-valued form can be expressed as:

[0090]

[0091] For a rectangular region, a bilinear interpolation nonlinear mapping technique is used to convert the points in the rectangular region to the points in the unit circle; the side length of the rectangular region is set as L, and the center point is the origin, and the coordinates (x, y) of the rectangular region can be converted into polar coordinate form (r', θ'), wherein:

[0092]

[0093] This conversion maps each point in the rectangular region to each point in the unit circle;

[0094] The data initialization unit is configured to determine the order of the Zernike polynomial based on the radius and angle parameters in the virtual unit circle, and initialize the coefficients of the Zernike polynomial in the polar coordinate system;

[0095] The height prediction unit is configured to multiply each term of the Zernike polynomial by the corresponding coefficient of the Zernike polynomial in the polar coordinate system, and then accumulate to obtain the predicted height value of each point on the curved surface;

[0096] The prediction error solving unit is configured to use the least squares method to fit the objective function and obtain the prediction error;

[0097] The second determination unit is configured to determine whether the minimum prediction error meets a preset condition;

[0098] The minimum prediction error finding unit is configured to adjust the coefficients of the Zernike polynomial, or adjust the order of the Zernike polynomial and the step size of the least squares method based on the prediction error;

[0099] Wherein, the error between the predicted height value and the actual height value is the prediction error, and the minimum prediction error is the objective function.

[0100] Further preferably, the horizontal and vertical coordinates of the machining error curved surface coordinate system in the rectangular aperture curved surface normalization unit are the horizontal and vertical coordinates of the rectangular aperture curved surface, and the vertical coordinate is the machining error.

[0101] Further preferably, in the minimum prediction error searching unit, when the prediction error does not satisfy the preset condition, the order of the Zernike polynomial is increased or the step of the least square method is reduced.

[0102] Further preferably, in the moving path compensation module, when the prediction error surface is added to the machining program in reverse, the cutting depth, speed and direction of the tool at each point position are adjusted.

[0103] Further preferably, in the machining path acquisition module, the parameters and characteristics of the surface to be machined include process parameters of machining, precision parameters of the machining surface and material characteristics of the workpiece.

[0104] Compared with the prior art, the present application has the following advantages:

[0105] In view of the characteristics of the rectangular aperture free-form surface, the present application provides a rectangular aperture surface shape error compensation method to improve the accuracy and efficiency of compensation. The Zernike polynomial is used for unit square normalization of the surface. Since the Zernike polynomial is orthogonal in the circular region, it no longer satisfies the orthogonal condition when directly converted for use in the rectangular aperture surface, and therefore needs to be orthogonalized. Through calculation and adjustment, the Zernike polynomial fitting surface can better adapt to the shape characteristics of the rectangular aperture, thereby realizing unit square normalization. After normalization, the amount of compensation is calculated according to the normalized Zernike surface fitting result. Then, the error of the surface is accurately compensated through corresponding machining or adjustment measures. In the compensation process, the accuracy and range of compensation are strictly controlled to ensure that the compensated surface can meet the design requirements.

[0106] For rectangular aperture free-form surfaces, the method uses Zernike polynomials to fit the overall three-dimensional surface error, which can comprehensively and effectively solve the problem of rectangular aperture surface shape error compensation and significantly improve the machining accuracy and quality of rectangular aperture surfaces.

[0107] More specifically, after the rectangular aperture surface is normalized by the Zernike surface, the fitting compensation is performed, which significantly improves the fitting accuracy. Through accurate modeling and analysis of the surface shape, finer surface features can be captured, and the deviation between the fitting result and the actual demand is greatly reduced. Compared with traditional methods, the method performs more accurately in the processing of complex surfaces and small errors, thereby effectively improving the machining quality of rectangular aperture surfaces.

[0108] The rectangular aperture surface shape error compensation method provided by the present application greatly improves the iteration efficiency. In the compensation process, the calculation process and algorithm are optimized, unnecessary repeated calculations and parameter adjustments are reduced, and the characteristics of fast convergence enable ideal compensation results to be obtained in a short time, greatly shortening the entire machining cycle and providing a strong guarantee for batch production.

[0109] This application provides a method for compensating surface errors of rectangular aperture curved surfaces, significantly reducing surface errors and thus lowering the costs of subsequent processing and testing. The improved compensation accuracy reduces the time required for additional finishing and fine grinding processes, saving costs. Furthermore, the improved production efficiency enables more processing tasks to be completed within the same timeframe, increasing output and further improving economic benefits.

[0110] It will be understood that the various numerical numbers involved in the embodiments of the present application are merely distinctions for the convenience of description and are not intended to limit the scope of the embodiments of the present application.

[0111] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application should be included in the scope of protection of the present application.

Claims

1. A method for compensating for surface errors of rectangular aperture curved surfaces, characterized in that: The following steps are involved: Step S1: Based on the parameters and features of the surface to be processed, a machining program is constructed to obtain the optimal movement path during the machining process; Step S2: Based on the optimal moving path obtained in step S1, the workpiece surface is processed, the coordinates and actual height values ​​of each point on the curved surface are collected, and the processing error is obtained; Step S3: Determine whether the machining error meets the machining accuracy. If so, terminate the subsequent steps and use the workpiece processed in step S2 as the final workpiece. If not, proceed to step S4. Step S4: using Zernike polynomials to normalize the machining error surface to a unit square and then fitting the predicted error surface; Step S5: Invert the predicted error surface and add it to the machining program in step S1 to correct the optimal movement path during the machining process, and go to step S2 until the machining error meets the machining accuracy; wherein the curved surface shape of the workpiece is a rectangular aperture; Step S4 specifically includes the following steps: Step S4.1: Construct a machining error surface coordinate system based on the machining error obtained in step S2, and map the rectangular aperture surface area to a unit square for normalization; Step S4.2: Taking the center of the unit square as the origin, calculate the relative distance and angle of each point to the center, and map the points of the rectangular aperture surface area to the polar coordinate system of the virtual unit circle; Step S4.3: Determine the order of the Zernike polynomial based on the radius and angle parameters within the virtual unit circle, and initialize the coefficients of the Zernike polynomial in the polar coordinate system; Step S4.4: In the polar coordinate system, multiply each term of the Zernike polynomial by the corresponding coefficient of the Zernike polynomial and add them together to obtain the predicted height value of each point on the surface; Step S4.5: Use the least squares method to fit the objective function, obtain the prediction error, adjust the coefficients of the Zernike polynomials, and go to step S4.4 until the minimum prediction error is obtained; the error between the predicted altitude value and the actual altitude value is the prediction error, and minimizing the prediction error is the objective function; Step S4.6: Determine whether the minimum prediction error obtained in step S4.5 meets the preset conditions. If so, obtain the prediction error surface. If not, adjust the order of the Zernike polynomial and the step size of the least squares method, and go to step S4.

4.

2. The rectangular aperture curved surface shape error compensation method according to claim 1, characterized in that: The method for constructing the machining error surface coordinate system is to keep the horizontal and vertical coordinates of the rectangular aperture surface unchanged and use the machining error as the vertical coordinate axis.

3. The method for compensating for rectangular aperture curved surface shape errors according to claim 1, wherein: When the prediction error does not meet the preset conditions, the order of the Zernike polynomial is increased, or the step size of the least squares method is reduced.

4. The method for compensating for rectangular aperture curved surface shape errors according to any one of claims 1 to 3, characterized in that: When the predicted error surface is inverted and added to the machining program, the cutting depth, speed and direction of the tool at each point are adjusted.

5. The method for compensating for rectangular aperture curved surface shape error according to claim 1, characterized in that: The parameters and characteristics of the surface to be machined include: machining process parameters, precision parameters of the machined surface and material properties of the workpiece.

6. A rectangular aperture curved surface shape error compensation system, characterized in that: include: The machining path acquisition module is used to construct a machining program based on the parameters and features of the surface to be machined to obtain the optimal movement path during the machining process; The tool is used to process the workpiece surface according to the optimal movement path; Profilometer measuring probe, used to collect the coordinates and actual height values ​​of each point on the curved surface; A machining error calculation module is used to calculate the machining error of each point based on the actual height value of each point on the curved surface and the preset height value; The first judgment module is used to judge whether the machining error meets the machining accuracy. If so, the currently machined workpiece is used as the final workpiece. Otherwise, the prediction error surface fitting module is driven to run. A prediction error surface fitting module is used to fit the prediction error surface after normalizing the machining error surface to a unit square using Zernike polynomials; The moving path compensation module is used to invert the predicted error surface and add it to the machining program in the machining path acquisition module to correct the optimal moving path during the machining process; Among them, the curved surface of the workpiece is a rectangular aperture; The prediction error surface fitting module includes: A rectangular aperture surface normalization unit is used to construct a machining error surface coordinate system based on the machining error, and to map the rectangular aperture surface area into a unit square for normalization processing; A coordinate conversion unit is used to calculate the relative distance and angle between each point and the center of the unit square, and map the points of the rectangular aperture surface area to the polar coordinate system of the virtual unit circle; A data initialization unit, configured to determine the order of the Zernike polynomial based on the radius and angle parameters within the virtual unit circle, and to initialize the coefficients of each Zernike polynomial in the polar coordinate system; A height prediction unit is used to multiply each Zernike polynomial by the corresponding Zernike polynomial coefficient in a polar coordinate system and then accumulate the results to obtain a predicted height value for each point on the surface; A prediction error solving unit is used to fit the objective function using the least square method to obtain the prediction error; A second determination unit is used to determine whether the minimum prediction error meets a preset condition; A minimum prediction error search unit, configured to adjust the coefficients of each Zernike polynomial based on the prediction error; or adjust the order of the Zernike polynomial and the step size of the least squares method; The error between the predicted height value and the actual height value is the prediction error, and the objective function is to minimize the prediction error.

7. The rectangular aperture curved surface shape error compensation system according to claim 6, characterized in that: When the prediction error in the minimum prediction error search unit does not meet the preset conditions, the order of the Zernike polynomial is increased, or the step size of the least squares method is reduced.

8. The rectangular aperture curved surface shape error compensation system according to claim 6, characterized in that: In the moving path compensation module, when the predicted error surface is inverted and added to the machining program, the cutting depth, speed and direction of the tool at each point are adjusted.

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