Method for determining optical processing parameters by allocating component manufacturing tolerances

Abstract

The present invention relates to the field of optical processing technology, and in particular to a method for determining optical processing parameters by allocating component manufacturing tolerances. The method characterizes the component surface processing errors generated under the process parameters in the form of characteristic spectra, forms a set of error characteristic spectra of different process parameters, and performs tolerance analysis on the error characteristic spectra together with the component standard surface in optical design software to obtain a characteristic spectrum that requires strict control. The strictly controlled characteristic spectrum is compared with the characteristic spectrum set, and the process parameters corresponding to the optimal error characteristic spectrum are obtained, which are the optimal process parameters. The present invention effectively solves the technical problem that the existing method cannot determine a processing method that meets the tolerance, and provides a theoretical basis for subsequent processing.

Description

Technical Field

[0001] The present invention belongs to the field of optical processing technology, and in particular relates to a method for determining optical processing process parameters by allocating component manufacturing tolerances. Background Art

[0002] Early approaches to surface tolerances in optical systems, such as the paper "Perturbations of optical systems" published in the Journal of the Optical Society of America A, first proposed an optical system perturbation model. This model used analytical methods to analyze the perturbations of light by various order errors and determined tolerance indicators. Modern approaches have proposed establishing tolerance indicators using the tracing index method and the polynomial index method, and finally using Monte Carlo analysis to determine tolerance ranges and complete tolerance analysis. Among them, the tracing index method uses the ray tracing method to give the surface geometric error tolerance index. The paper "Slope sensitivities for optical surfaces" published in "SPIE:Optical Engineering + Applications Proceedings" analyzes the sensitivity index of light to surface gradients. The paper "Local tolerance and quality evaluation for optical surfaces" published in "Optica" gives local tolerance index for figure error. The polynomial index method performs polynomial fitting on the error and uses its polynomial coefficient as the surface error tolerance index. The paper "Optical design and tolerancing of freeform surfaces using anisotropic radialbasis functions" published in "Optical Engineering" uses radial basis function to characterize the error and then analyze the error tolerance index. The paper "Description and tolerance analysis of freeform surface figure error using specific probability distributed Gaussian radial basis functions" published in "Optics Express" upgrades the radial basis function to a two-dimensional analysis tolerance index. The paper "Surface variation analysis of freeform optical systems over surface frequency bands for prescribed wavefront" published on the "Optics&Laser Technology" platform errors》Analyze the error tolerance index after using Zernike polynomials to characterize the errors.Finally, the tolerance range is given through Monte Carlo analysis.

[0003] The papers "Desensitization design method of a freeform optical system based on local curve control" published in the journal "Optics Letters" and "Desensitization to mid-spatial-frequency surface errors" published in "Optics Express" proposed that another indirect treatment method is error desensitization design, which uses a more ideal design to relax the tolerance range of surface errors.

[0004] However, after analyzing the error tolerance, there is no specific and clear method to find a processing method that meets the tolerance. Summary of the Invention

[0005] In view of this, the present invention aims to provide a method for determining optical processing parameters by allocating component manufacturing tolerances, characterizing the component surface processing errors generated under the process parameters in the form of characteristic spectra, forming a set of error characteristic spectra of different process parameters, and performing tolerance analysis on the error characteristic spectra together with the component standard surface in optical design software to obtain the characteristic spectra that need to be strictly controlled, and comparing the strictly controlled characteristic spectra with the characteristic spectrum set to obtain the process parameters corresponding to the optimal characteristic spectra, which are the optimal process parameters.

[0006] To achieve the above object, the technical solution created by the present invention is implemented as follows:

[0007] A method for determining optical processing parameters by allocating component manufacturing tolerances, comprising:

[0008] S1: Determine the structure of the optical system, determine the optical elements that need to control processing errors in the optical system, and determine the process parameter space when processing the optical elements;

[0009] S2: Obtaining a corresponding error characteristic spectrum generated when processing the optical element according to the process parameters in the process parameter space of step S1;

[0010] S3: After adding the error characteristic spectrum obtained in step S2 to the surface of the optical element in step S1, tolerance analysis is performed on the current optical system to obtain a set of main influencing items;

[0011] S4: According to the set of main influencing items obtained in step S3, the optimal error characteristic spectrum is determined from the error characteristic spectrum, and the process parameters corresponding to the optimal error characteristic spectrum are the optimal process parameters.

[0012] Furthermore, in step S2:

[0013] The error characteristic diagram corresponding to the process parameters is obtained by the following formula:

[0014] E i (x,y)=Fig(para i ), para i ∈Ω;

[0015] Among them, Ω represents the process parameter space, para i represents the i-th process parameter, Fig represents the function of generating the error characteristic diagram from the process parameter, E i (x,y) represents the error characteristic diagram corresponding to the i-th process parameter;

[0016] The error characteristic graph is expanded into the following polynomial form:

[0017] ;

[0018] Among them, K represents the total number of terms in the polynomial expansion, L i (k) represents the kth error characteristic spectrum corresponding to the i-th process parameter, P k represents the k-th characteristic polynomial.

[0019] Furthermore, step S3 includes:

[0020] From the error characteristic spectrum, randomly select the error characteristic spectrum corresponding to one of the process parameters;

[0021] Adding the selected error characteristic spectrum as a formal error characteristic spectrum to the optical element to obtain the current optical system;

[0022] Set the main tolerance index range;

[0023] Conduct Monte Carlo tolerance analysis on the current optical system according to the main tolerance index range;

[0024] According to the main tolerance index range and formal error characteristic spectrum, the set of main influencing items is determined by the following formula:

[0025] ;

[0026] Among them, σ represents the set of main influencing items, L f (k) represents the k-th formal error characteristic spectrum, δ(k) represents the main tolerance index range corresponding to the k-th formal error characteristic spectrum, MC(·) represents the Monte Carlo analysis function, It represents the number of items in the η formal error characteristic spectra corresponding to the first η maximum values.

[0027] Further, in the process of adding the form error characteristic spectrum to the optical element;

[0028] Combined with the characteristic polynomial, the formal error characteristic graph is obtained by the following formula:

[0029] ;

[0030] Among them, E f Representation error characteristic map, L f (k) represents the k-th formal error characteristic spectrum;

[0031] Superimpose a form error signature onto the surface of the optical element.

[0032] Furthermore, in step S4, the process of determining the optimal error characteristic spectrum from the error characteristic spectrum includes: obtaining the optimal error characteristic spectrum by the following formula:

[0033] ;

[0034] Among them, L0(k) represents the optimal error characteristic spectrum, L i (k) represents the kth error characteristic spectrum corresponding to the i-th process parameter, N represents the total number of process parameters in the process parameter space, Indicates the error characteristic spectrum L corresponding to the minimum value i (k), represents the two-norm.

[0035] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0036] In the method for determining optical processing parameters by allocating component manufacturing tolerances created by the present invention, the component surface processing errors generated under the process parameters are characterized in the form of characteristic spectra, and the error characteristic spectra of different process parameters are superimposed on the standard surface of the component for tolerance analysis to determine the error characteristic spectrum that needs to be strictly controlled; the strictly controlled error characteristic spectrum is compared with the set main tolerance index range, and the process parameters corresponding to the optimal error characteristic spectrum are the optimal process parameters, which successfully solves the technical problem that the existing method cannot find a processing method that meets the tolerance. The optimal process parameters are calculated using this method without repeated experiments, which can reduce the time cost, material cost, and labor cost of process trial and error in the manufacturing process, provide a theoretical basis and processing guidance for subsequent processing, and further improve the subsequent processing accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The accompanying drawings, which constitute part of the present invention, are intended to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are intended to explain the present invention and do not constitute an undue limitation of the present invention. In the accompanying drawings:

[0038] Figure 1 A schematic flow chart of a method for determining optical processing parameters by allocating component manufacturing tolerances according to an embodiment of the present invention;

[0039] Figure 2 A schematic diagram of the Cooke three-separation objective lens according to an embodiment of the present invention;

[0040] Figure 3 A schematic diagram of a removal function according to an embodiment of the present invention;

[0041] Figure 4 To create the error characteristic diagram described in the embodiment of the present invention;

[0042] Figure 5 The error characteristic curve diagram described in the embodiment of the present invention is created;

[0043] Figure 6 A comparison chart of the machining gauge and characteristic spectrum described in the embodiment of the present invention;

[0044] Figure 7 A surface data map of a concave lens in the form of a dynamic link library file according to an embodiment of the present invention;

[0045] Figure 8 This is a tolerance analysis result diagram for the embodiment of the present invention.

[0046] Reference numerals:

[0047] 1. Concave lens; 2. Surface to be analyzed. DETAILED DESCRIPTION

[0048] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation of the present invention.

[0049] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

[0050] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention. In addition, the terms "first", "second" and the like are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, features defined as "first", "second" and the like may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0051] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "installed," "connected," and "connected" should be understood in a broad sense. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to mechanical connections or electrical connections; they can refer to direct connections or indirect connections through an intermediate medium; and they can refer to internal connections between two components. Those skilled in the art can understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0052] The present invention will be described in detail below with reference to the accompanying drawings and in combination with embodiments.

[0053] like Figure 1 As shown, the method for determining optical processing parameters by allocating component manufacturing tolerances according to an embodiment of the present invention includes:

[0054] S1: Determine the structure of the optical system, determine the optical elements that need to control processing errors in the optical system, and determine the process parameter space when processing the optical elements.

[0055] The optical system is selected based on the actual situation. It can be a Cooke three-separation objective, a Gaussian objective, a coaxial two-mirror objective, etc. After the optical system is determined, the optical elements that need to control the processing error are determined based on the actual situation or research experience. For example, in the off-axis three-mirror optical system, the main consideration is the error of the primary mirror. It should be noted that the optical elements in the structure of the optical system at this time are in a state without any error. When the surface of the optical element is combined with the error characteristic diagram generated under different process parameters, an optical element with processing error can be obtained. The process parameter space is composed of multiple process parameters, and each process parameter includes no less than one specific processing parameter, such as the processing gauge h, the removal function RF, the processing angle theta and the processing residence time dt. Correspondingly, the process parameter can be expressed as para=(RF,h,theta,dt), where para represents the process parameter.

[0056] S2: Obtaining a corresponding error characteristic spectrum generated when processing the optical element according to the process parameters in the process parameter space of step S1.

[0057] In some embodiments, the error characteristic graph is obtained by the following formula:

[0058] E i (x,y)=Fig(para i ), para i ∈Ω;

[0059] Among them, Ω represents the process parameter space, para i represents the i-th process parameter in the process parameter space, Fig represents the function of generating the error characteristic diagram from the process parameter, E i (x,y) represents the error characteristic diagram corresponding to the i-th process parameter.

[0060] The error characteristic graph is expanded into the following polynomial form:

[0061] ;

[0062] Among them, K represents the total number of terms in the polynomial expansion, L i (k) represents the kth error characteristic spectrum corresponding to the i-th process parameter, P k represents the k-th characteristic polynomial.

[0063] The specific polynomial expansion form of the error characteristic diagram is selected according to the characteristics of the optical element whose processing error needs to be controlled. For example, the annular error generated by the spiral processing trajectory can be expanded into a Zernike polynomial, and the random error generated by the random trajectory can be expanded into a B-spline polynomial.

[0064] S3: After adding the error characteristic spectrum obtained in step S2 to the surface of the optical element in step S1, the current optical system is analyzed to obtain a set of main influencing items.

[0065] In some embodiments, step S3 includes:

[0066] From the error characteristic spectrum, randomly select the error characteristic spectrum corresponding to one of the process parameters;

[0067] The selected error characteristic spectrum is added to the optical element as the formal error characteristic spectrum to obtain the current optical system; it can be understood that the currently selected process parameters also correspond to the formal error characteristic spectrum;

[0068] Set the main tolerance index range, which is adaptively adjusted according to actual conditions and experimental experience;

[0069] Conduct Monte Carlo tolerance analysis on the current optical system according to the main tolerance index range;

[0070] The set of items corresponding to the formal error characteristic spectrum that plays a major role within the main tolerance index range is the set of main influencing items. In the process of Monte Carlo tolerance analysis, by adding random perturbations to the formal error characteristic spectrum, the perturbation value is within the main tolerance index range, and the current optical system with the perturbation is obtained for optical analysis; by continuously changing the perturbation, it is determined which formal error characteristic spectrum under the current process parameters plays a major role, thereby determining the set of main influencing items. Specifically, according to the main tolerance index range and the formal error characteristic spectrum, the set of main influencing items is determined by the following formula:

[0071] ;

[0072] Among them, σ represents the set of main influencing items, L f (k) represents the k-th formal error characteristic spectrum, δ(k) represents the main tolerance index range corresponding to the k-th formal error characteristic spectrum, MC(·) represents the Monte Carlo analysis function, It represents the number of items in the η formal error characteristic spectra corresponding to the first η maximum values.

[0073] In some embodiments, when adding a form error characteristic spectrum to an optical element, combined with a characteristic polynomial, a form error characteristic map is obtained by the following formula:

[0074] ;

[0075] Among them, E f Representation error characteristic map, L f (k) represents the kth form error characteristic spectrum. The form error characteristic map is superimposed on the surface of the optical element.

[0076] In one embodiment, the form error characteristic map is added to the corresponding elements of the surface of the optical element, the result of the addition is written into a dynamic link library file, the surface type is selected as user-defined, and the dynamic link library file is loaded into the optical system as the element surface.

[0077] S4: According to the set of main influencing items obtained in step S3, the optimal error characteristic spectrum is determined from the error characteristic spectrum, and the process parameters corresponding to the optimal error characteristic spectrum are the optimal process parameters.

[0078] In some embodiments, in step S4, the process of determining the optimal error characteristic spectrum from the error characteristic spectrum includes:

[0079] The optimal error characteristic spectrum is obtained by the following formula:

[0080] ;

[0081] Where L0(k) represents the optimal error characteristic spectrum, N represents the total number of process parameters in the process parameter space, Indicates the error characteristic spectrum L corresponding to the minimum value i (k), represents the two-norm.

[0082] In order to clearly illustrate the method for determining optical processing parameters by allocating component manufacturing tolerances according to an embodiment of the present invention, an embodiment is provided.

[0083] Example: Method for determining optical processing parameters by allocating component manufacturing tolerances based on Zemax software, combined with Figure 1 , the method provided in this embodiment specifically includes:

[0084] S1: Determine the structure of the optical system, determine the optical elements that need to control processing errors in the optical system, and determine the process parameter space when processing the optical elements.

[0085] In this embodiment, the optical system to be analyzed is determined as follows Figure 2 In the Cooke three-separation objective lens shown, the optical element whose processing error needs to be controlled is the concave lens 1. The surface equation Z(x, y) of the surface to be analyzed 2 in the concave lens 1 is:

[0086] ;

[0087] Wherein, R represents the curvature radius of the surface 2 to be analyzed, where R=20.2919 mm.

[0088] In this embodiment, the process parameters that need to be determined include the removal function RF and the processing track gauge h. Specifically, the concave lens 1 is processed by the magnetorheological polishing process. The corresponding removal function RF is as follows: Figure 3 As shown, the process parameters can be expressed as para i =(RF,h i ), we can further determine that the process parameter space Ω can be expressed as Ω={para1,para2,...,para i ,...,para N}, that is, the process parameter space Ω can be expressed as Ω={(RF,h1),(RF,h2),...,(RF,h i ),...,(RF,h N In this embodiment, the process parameter space Ω includes three groups of process parameters, that is, N=3, and the machining gauges h in the three groups of process parameters are 1, 0.8, and 2.1, respectively, that is: Ω={(RF,1),(RF,0.8),(RF,2.1)}.

[0089] S2: Obtaining a corresponding error characteristic spectrum generated when processing the optical element according to the process parameters in the process parameter space of step S1.

[0090] In this embodiment, the continuous tool influence function method disclosed in the invention patent application with Chinese patent publication number CN116679622A, publication date October 3, 2023, and patent name “Surface shape tool mark error prediction method based on continuous tool function” is introduced to obtain the following Figure 4 The error characteristic diagram shown is:

[0091] E i (x,y)=Fig(para i ), para i ∈Ω;

[0092] The Fig here represents the continuous tool influence function method.

[0093] In this embodiment, since the error is consistent in the x-direction, that is, the intersection line in the y-direction can fully characterize the processing error, the error characteristic diagram E i Error characteristic curve E i '(y) is expanded according to Legendre polynomials as follows:

[0094] ;

[0095] .

[0096] In this embodiment, the error characteristic curve E i '(y) Figure 5 As shown, it is from the error characteristic diagram E i Take the y-direction intercept (i.e. Figure 4 The black line in the figure is obtained. Figure 5 The X coordinate is the normalized coordinate of the aperture of concave lens 1, and the Y coordinate is the normalized value of the processing amplitude. This operation is to ensure that the Legendre polynomial fitting can be performed normally (that is, the fitting requires the interval to be [-1, 1] and the amplitude to be within [-1, 1]).

[0097] S3: After adding the error characteristic spectrum obtained in step S2 to the surface of the optical element in step S1, the current optical system is analyzed to obtain a set of main influencing items.

[0098] In this embodiment, step S3 includes:

[0099] S31: Randomly select an error characteristic spectrum corresponding to one of the process parameters from the error characteristic spectrum.

[0100] S32: Add the selected error characteristic spectrum as the formal error characteristic spectrum to the surface to be analyzed 2 to obtain the current optical system. In this embodiment, step S32 includes:

[0101] Combined with the characteristic polynomial, the formal error characteristic graph is obtained by the following formula:

[0102] ;

[0103] The formal error characteristic map is superimposed on the surface of the surface to be analyzed 2, that is:

[0104] Z'=Z+E f ;

[0105] Where Z' represents the surface equation of the surface to be analyzed 2 after superimposing the formal error characteristic map. Write the surface equation Z' into a dynamic link library file, select the surface type as user-defined, and use the dynamic link library file as the surface to be analyzed 2. The results of the analysis of the current surface to be analyzed 2 using Zemax software are as follows Figure 7 As shown in the figure, the tolerance analysis results are as follows: Figure 8 As shown, the form error characteristic spectrum L f (1) Correspondence Figure 7 The "1st Order Term" form error characteristic spectrum L f (2) Correspondence Figure 7 The "2nd Order Term" in Figure 8In the figure, "TPAR(3,2)" and "TPAR(a,b)" represent the b-th process parameter of surface a in the current optical system. It can be understood that this embodiment is for surface 2 to be analyzed, that is, the third surface in the optical system, so it corresponds to "TPAR(3,2)".

[0106] S33: Set the main tolerance index range to δ(k)=[-4,4];

[0107] S34: According to the main tolerance index range, the current optical system is subjected to Monte Carlo tolerance analysis. The set of items corresponding to the formal error characteristic spectrum that plays a major role within the main tolerance index range is the set of main influencing items. Specifically, according to the main tolerance index range δ(k) and the formal error characteristic spectrum L f (k), let η = 1, and determine the set of main influencing items σ by the following formula:

[0108] .

[0109] The above formula shows that the second characteristic spectrum needs to be strictly controlled. As is well known to optical designers, aspheric design cannot contain quadratic terms, and the polynomial corresponding to the second characteristic spectrum is P2=1.5x 2 -0.5 contains a quadratic term and needs to be strictly controlled, which is consistent with the analysis results.

[0110] S4: According to the set of main influencing items obtained in step S3, the optimal error characteristic spectrum is determined from the error characteristic spectrum, and the process parameters corresponding to the optimal error characteristic spectrum are the optimal process parameters.

[0111] In this embodiment, step S4 includes:

[0112] The optimal error characteristic spectrum is obtained by the following formula:

[0113] ;

[0114] It is obtained that L0(2)=L2(2), indicating that L2(2) is the optimal error characteristic spectrum, that is, for the current three processing parameters, the track gauge h=0.8mm is the optimal processing parameter.

[0115] It should be understood that the various forms of the processes shown above can be used to reorder, add, or delete steps. For example, the steps described in the present disclosure can be performed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in the present disclosure can be achieved. This is not limited herein.

[0116] The above specific embodiments do not limit the scope of protection of the present invention. Those skilled in the art will appreciate that various modifications, combinations, sub-combinations, and substitutions may be made based on design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention are intended to be included within the scope of protection of the present invention.

Claims

1. A method for determining optical processing parameters by allocating component manufacturing tolerances, characterized in that: include: S1: Determine the structure of the optical system, determine the optical element whose processing error needs to be controlled in the optical system, and determine the process parameter space when processing the optical element; S2: Obtaining a corresponding error characteristic graph according to the process parameters in the process parameter space of step S1; performing a polynomial expansion on the error characteristic graph to obtain a corresponding error characteristic spectrum generated when processing the optical element; S3: After adding the error characteristic spectrum obtained in step S2 to the surface of the optical element in step S1, tolerance analysis is performed on the current optical system. The set of the error characteristic spectrum after tolerance analysis and the maximum value of the set main tolerance index range is the set of main influencing items; Step S3 includes: randomly selecting an error characteristic spectrum corresponding to one of the process parameters from the error characteristic spectrum; adding the selected error characteristic spectrum to the optical element as a formal error characteristic spectrum to obtain a current optical system; setting a main tolerance index range; performing a Monte Carlo tolerance analysis on the current optical system based on the main tolerance index range: determining the set of main influencing items based on the main tolerance index range and the formal error characteristic spectrum by the following formula: ； Wherein, σ represents the set of main influencing items, L f (k) represents the k-th formal error characteristic spectrum, δ(k) represents the main tolerance index range corresponding to the k-th formal error characteristic spectrum, MC(·) represents the Monte Carlo analysis function, represents the number of terms in the η formal error characteristic spectra corresponding to the first η maximum values; S4: According to the set of main influencing items obtained in step S3, the minimum value in the error characteristic spectrum is the optimal error characteristic spectrum, and the process parameters corresponding to the optimal error characteristic spectrum are the optimal process parameters.

2. The method for determining optical processing parameters by allocating component manufacturing tolerances according to claim 1, characterized in that: In step S2: The error characteristic diagram corresponding to the process parameters is obtained by the following formula: AND i (x,y)=Fig(for i ),to i ∈Ω; Wherein, Ω represents the process parameter space, para i represents the i-th process parameter, Fig represents the function that generates the error characteristic diagram from the process parameter, which is the continuous tool influence function method, E i (x,y) represents the error characteristic diagram corresponding to the i-th process parameter; The error characteristic graph is expanded into a polynomial as follows: ； Among them, K represents the total number of terms in the polynomial expansion, L i (k) represents the kth error characteristic spectrum corresponding to the i-th process parameter, P k represents the k-th characteristic polynomial.

3. The method for determining optical processing parameters by allocating component manufacturing tolerances according to claim 1, characterized in that: in the process of adding the form error signature to the optical element; Combined with the characteristic polynomial, the formal error characteristic graph is obtained by the following formula: ； Among them, E f Represents the formal error characteristic diagram, L f (k) represents the k-th formal error characteristic spectrum; The form error characteristic map is superimposed on the surface of the optical element.

4. The method for determining optical processing parameters by allocating component manufacturing tolerances according to claim 1, characterized in that: In step S4, the process of determining the optimal error characteristic spectrum from the error characteristic spectrum includes: The optimal error characteristic spectrum is obtained by the following formula: ； Wherein, L0(k) represents the optimal error characteristic spectrum, L i (k) represents the kth error characteristic spectrum corresponding to the i-th process parameter, N represents the total number of process parameters in the process parameter space, Indicates the error characteristic spectrum L corresponding to the minimum value i (k), represents the two-norm.

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