A surface topography measuring device for large area workpieces and a method thereof

Abstract

The application provides a surface topography measuring device and method for large plane workpieces. The surface topography measuring method comprises the following steps: step 1, selecting several specific straight line paths as measuring paths on a required measuring plane of a workpiece, and measuring flatness curve data of each straight line path; step 2, screening effective data from the flatness curve data of the several measuring paths; step 3, sequentially processing the effective data by finding and filling outliers, deleting missing data, removing trends from the data, smoothing the data containing noise, and filling and scaling the data; step 4, performing surface interpolation fitting along the several specific paths among the several measuring paths; and step 5, drawing a full plane surface topography according to actual positions in a three-dimensional rectangular coordinate system. The application provides a high-precision simulation large plane surface topography measuring method, so that a surface topography with high precision of the whole plane is obtained.

Description

Technical Field

[0001] The present invention relates to the field of precision measurement and calculation methods. Background Art

[0002] Existing devices for measuring the surface topography of large planar workpieces are mainly linear guide flatness measuring instruments, which can only measure the surface topography of the workpiece along a certain straight line and cannot fully reflect the surface topography of larger workpieces. Summary of the Invention

[0003] In order to overcome the above technical deficiencies, a first aspect of the present invention provides a method for measuring the surface topography of a large planar workpiece, comprising:

[0004] Step S1: Selecting a number of specific straight paths on the required measurement plane of the workpiece as measurement paths, and measuring the flatness curve data of each straight path;

[0005] Step S2: Filtering out valid data from the flatness curve data of the plurality of measurement paths;

[0006] Step S3: performing the following processing on the valid data in sequence: discovering and filling outliers, deleting missing data, removing trends from the data, smoothing noisy data, and filling and scaling the data;

[0007] Step S4: performing surface interpolation fitting between the plurality of measurement paths and along the plurality of specific paths based on the processed data;

[0008] Step S5: Based on the fitted data, the full-plane surface topography is drawn in a three-dimensional rectangular coordinate system according to the actual position.

[0009] Furthermore, in step S1, a linear guide flatness measuring instrument is used to measure the original data sequence corresponding to each specific straight path.

[0010] Furthermore, the step S2 includes:

[0011] Step S2.1: Mark the data that satisfies both conditions 1 and 2 in the original data sequence corresponding to each specific straight line path as NaN, and filter the data other than NaN in the original data sequence as intermediate valid data.

[0012] Condition 1: 5 or more consecutive data show standard linear changes.

[0013] Condition 2: The position of the data at the beginning or end;

[0014] Step S2.2: Mark the data that meets both conditions 1 and 3 in the filtered intermediate valid data as NaN, and filter the data other than NaN in the intermediate valid data as valid data at both ends.

[0015] Condition 1: 5 or more consecutive data show standard linear changes.

[0016] Condition 3: The data is not at the beginning or end.

[0017] Furthermore, step S3 includes:

[0018] Step S3.1: Based on the filtered valid data, discover and fill in outliers. Step S3.1 includes steps S3.1.1 to S3.1.2:

[0019] Step S3.1.1: Find and remove outliers from the valid data corresponding to each specific straight line path: define all values ​​that meet condition 4 as outliers;

[0020] Condition 4: The difference from the median is more than three times the converted MAD value;

[0021] Convert MAD = c × median|(A - median(A))|,

[0022] in,

[0023] median(A) is the median of all elements of type A,

[0024]

[0025] erfcinv(erfc(x))=x,

[0026]

[0027] A is the valid data corresponding to each specific straight line path;

[0028] Step S3.1.2: Fill in outliers using a shape-preserving piecewise cubic spline interpolation polynomial. The polynomial P(n) satisfies conditions 5, 6, and 7 simultaneously.

[0029] Condition 5: In each subinterval n k ≤n≤n k +1, the polynomial P(n) is a cubic Hermite interpolation polynomial for the given data points, with the specified derivatives at the interpolation points;

[0030] Condition 6: P(n) performs y interpolation, that is, P(n j )=y j , and the first-order derivative is continuous; the second derivative May be discontinuous, so n j There may be jumps;

[0031] Condition 7: Cubic interpolation P(n) has a shape-preserving effect; j The slope at is chosen in a specific way so that P(n) preserves the shape of the data and the corresponding monotonicity;

[0032] Step S3.2: Delete missing data: Delete the data marked as NaN in step S2 from the data after filling outliers corresponding to each specific straight line path;

[0033] Step S3.3: removing the best straight-line fit line from the data after deleting missing data corresponding to each specific straight-line path, thereby obtaining the detrended data;

[0034] Step S3.4: Smoothing the detrended data corresponding to each specific straight line path;

[0035] Step S3.5: Fill and scale the smoothed data corresponding to each specific straight line path according to position.

[0036] Furthermore, the step S4 includes:

[0037] Step S4.1: extracting a number of data on any specific path;

[0038] Step S4.2: Fourier transform of the interpolation function value in the plurality of data;

[0039] Step S4.3: Perform the calculations in steps S4.1-S4.2 on all data on all specific straight line paths in sequence.

[0040] A second aspect of the present invention provides a surface topography measuring device for a large planar workpiece, comprising:

[0041] A straight path flatness measurement module, wherein the straight path flatness measurement module is used to select a number of specific straight paths on a required measurement plane of a workpiece as measurement paths and measure flatness curve data of each straight path;

[0042] a screening module, the screening module being used to screen out valid data from the flatness curve data of the plurality of measurement paths;

[0043] a data processing module, the data processing module being used to sequentially perform the following processing on the valid data: discover and fill outliers, delete missing data, remove trends from the data, smooth noisy data, and fill and scale the data;

[0044] a fitting module, configured to perform curved surface interpolation fitting along the several specific paths among the several measurement paths based on the processed data;

[0045] a drawing module, configured to draw the full plane surface topography according to the fitted data in a three-dimensional rectangular coordinate system according to the actual positions.

[0046] Further, the straight line path flatness measurement module uses a straight line guide rail type flatness measurement instrument to measure a corresponding original data sequence of each specific straight line path.

[0047] Further, the screening module is configured to:

[0048] (1) mark the data in the original data sequence of each specific straight line path that simultaneously satisfies condition 1 and condition 2 as NaN, and screen the data other than NaN in the original data sequence as intermediate effective data,

[0049] condition 1: the continuous 5 or more data are in standard linear variation,

[0050] condition 2: the data are at the beginning or end position;

[0051] (2) mark the data in the screened intermediate effective data that simultaneously satisfies condition 1 and condition 3 as NaN, and screen the data other than NaN in the intermediate effective data as two-end effective data,

[0052] condition 1: the continuous 5 or more data are in standard linear variation,

[0053] condition 3: the data are not at the beginning or end position.

[0054] Further, the data processing module is configured to:

[0055] (1) find and fill outliers based on the screened effective data, (1) including (1.1)-(1.2):

[0056] (1.1) find and remove outliers from the effective data corresponding to each specific straight line path: define any value that satisfies condition 4 as an outlier;

[0057] condition 4: the value is more than three times the conversion MAD different from the median; wherein,

[0058] conversion MAD = c x median|(A-median(A))|,

[0059] wherein,

[0060] median(A) is the median of all elements in A,

[0061]

[0062] erfcinv(erfc(x))=x,

[0063]

[0064] A is the valid data corresponding to each specific straight line path;

[0065] (1.2) Filling outliers: Filling is performed using a shape-preserving piecewise cubic spline interpolation polynomial, where the polynomial P(n) satisfies conditions 5, 6, and 7 simultaneously.

[0066] Condition 5: In each subinterval n k ≤n≤n k +1, the polynomial P(n) is a cubic Hermite interpolation polynomial for the given data points, with the specified derivatives at the interpolation points;

[0067] Condition 6: P(n) performs y interpolation, that is, P(n j )=y j , and the first-order derivative is continuous; the second derivative May be discontinuous, so n j There may be jumps;

[0068] Condition 7: Cubic interpolation P(n) has a shape-preserving effect; j The slope at is chosen in a specific way so that P(n) preserves the shape of the data and the corresponding monotonicity;

[0069] (2) Deleting missing data: The data marked as NaN in step S2 are deleted from the data after filling outliers corresponding to each specific straight path;

[0070] (3) removing the best straight-line fit line from the data after deleting missing data corresponding to each specific straight-line path to obtain the detrended data;

[0071] (4) Smoothing the detrended data corresponding to each specific straight line path;

[0072] (5) Fill and scale the smoothed data corresponding to each specific straight line path according to position.

[0073] Furthermore, the fitting module is used to:

[0074] (1) Extract some data on any specific path;

[0075] (2) Fourier transform of the interpolated function value in the plurality of data;

[0076] (3) Perform the calculations in (1)-(2) on all data on all specific straight line paths in sequence.

[0077] Compared with the existing technology, the above technical solution has the following beneficial effects:

[0078] This invention, based on a linear guide flatness measuring instrument, provides a high-precision method for measuring, predicting, calculating, and fitting large-plane surface topography, thereby obtaining a highly accurate surface topography for the entire plane. Specifically, it provides: 1) a method for measuring the surface topography of a large-plane workpiece; 2) a calculation method for identifying invalid measurement data; 3) a calculation method for predicting the true value of missing data; and 4) a method for predicting and fitting from points to lines and then to surfaces. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 A module diagram of a surface topography measurement device for a large planar workpiece according to the present application;

[0080] Figure 2 A flow chart of the surface topography measurement method of a large planar workpiece according to the present application;

[0081] Figure 3 A schematic diagram showing a marking of a specific straight line path on the upper surface of a circular component to be measured;

[0082] Figure 4 Schematic diagram of data processing for discovering and filling outliers;

[0083] Figure 5 Schematic diagram of data processing to remove missing data;

[0084] Figure 6 Schematic diagram of data processing to remove trends from data;

[0085] Figure 7 Schematic diagram of data processing for smoothing noisy data;

[0086] Figure 8 Schematic diagram of data processing for filling and scaling data;

[0087] Figure 9 Schematic diagram of the principle of surface interpolation fitting between measurement lines;

[0088] Figure 10 Schematic diagram of the principle of drawing the full-plane surface topography;

[0089] Figure 11 The high-precision large-plane surface morphology obtained by mapping. DETAILED DESCRIPTION

[0090] The advantages of the present invention are further described below with reference to the accompanying drawings and specific embodiments. Those skilled in the art should understand that the following specific description is illustrative rather than restrictive and should not be used to limit the scope of protection of the present invention.

[0091] like Figure 1 As shown, this embodiment provides a surface topography measuring device for a large planar workpiece, which includes: a straight path flatness measuring module, a screening module, a data processing module, a fitting module and a drawing module.

[0092] like Figure 2 As shown, the method for measuring the surface topography of a large planar workpiece using the surface topography measuring device includes steps S1 to S5:

[0093] Step S1: Select several specific straight paths on the required measurement plane of the workpiece as measurement paths, and measure the flatness curve data of each straight path.

[0094] In this step, a linear guide flatness measuring instrument is used to measure the original data sequence corresponding to each specific linear path.

[0095] For example, Figure 3 As shown in the figure, the upper surface of a circular component with a diameter of 1488 mm is used as the detection target. There is a missing circular hole with a diameter of 458 mm in the center, and there are small holes of varying sizes in some unspecified positions.

[0096] Step S1.1: Mark a certain position on the surface to be measured, which is marked as point a.

[0097] Step S1.2: Use a linear guide flatness measuring instrument to measure the straight line passing through point a and the center point of the circle. The obtained data is recorded as A, where A = {A1, A2...A n},n=1001,n∈Z, Z is an integer.

[0098] Step S1.3: Divide the component into eight equal parts based on line a, and mark the other three lines as b, c, and d.

[0099] Step S1.4: Measure along the straight lines passing through b, c, d and the center of the circle according to the method in step 1.2. The obtained data are recorded as:

[0100] B={B1,B2,…,B 1001},C={C1,C2,…,C 1001},D={D1,D2,…,D 1401}.

[0101] Step S2: Filter out valid data from the flatness curve data of the plurality of measurement paths.

[0102] Step S2.1: Mark the data in the original data sequence corresponding to each specific straight line path as NaN if the data satisfies both condition 1 and condition 2, and filter the data in the original data sequence except the NaN as intermediate valid data,

[0103] Condition 1: the data is linearly changed for 5 or more consecutive data,

[0104] Condition 2: the data is at the beginning or end of the data sequence.

[0105] In A, the data to be removed is: A 909 ~A 1001 The values are all 0, satisfying condition 1; and the data is at the end of the data sequence, satisfying condition 2.

[0106] In B, the data to be removed is: B 883 ~B 1001 The values are linearly changed, satisfying condition 1; and the data is at the end of the data sequence, satisfying condition 2.

[0107] Similarly, C and D are filtered.

[0108] The valid data obtained by filtering is: A = {A1,..., A 908}, B = {B1,..., B 882}, C, and D.

[0109] Step S2.2: Mark the data in the intermediate valid data as NaN if the data satisfies both condition 1 and condition 3, and filter the data in the intermediate valid data except the NaN as two-end valid data,

[0110] Condition 1: the data is linearly changed for 5 or more consecutive data,

[0111] Condition 3: the data is not at the beginning or end of the data sequence.

[0112] In A, the data to be removed is: A 309 ~A 566 The values are linearly changed, satisfying condition 1; and the data is not at the beginning or end of the data sequence, satisfying condition 3. The removed data is marked as NaN.

[0113] In B, the data to be removed is: B 309 ~B 564 The values are linearly changed, satisfying condition 1; and the data is not at the beginning or end of the data sequence, satisfying condition 3. The removed data is marked as NaN.

[0114] Similarly, C and D are filtered. The obtained data is marked as A, B, C, and D.

[0115] Step S3: The valid data is processed in the following order: outliers are found and filled, missing data are deleted, trends are removed from the data, noisy data are smoothed, and data are filled and scaled.

[0116] Step S3.1: Based on the filtered valid data, find and fill out the outliers, such as Figure 4 As shown. Step S3.1 includes steps S3.1.1-S3.1.2:

[0117] Step S3.1.1: Find and remove outliers from the valid data corresponding to each specific straight line path: define all values ​​that meet condition 4 as outliers;

[0118] Condition 4: The difference from the median is more than three times the converted MAD value;

[0119] Convert MAD = c × median|(A - median(A))|,

[0120] in,

[0121] median(A) is the median of all elements in A,

[0122]

[0123] erfcinv(erfc(x))=x,

[0124]

[0125] Specifically, in A, the condition 3 is satisfied if A 71 ~A 78 , and then remove these outliers.

[0126] Then perform step 3.1.1 for B, C, and D respectively.

[0127] Step S3.1.2: Fill in outliers using a shape-preserving piecewise cubic spline interpolation polynomial. The polynomial P(n) satisfies conditions 5, 6, and 7 simultaneously.

[0128] Condition 5: In each subinterval n k ≤n≤n k +1, the polynomial P(n) is a cubic Hermite interpolation polynomial for the given data points, with the specified derivatives at the interpolation points;

[0129] Condition 6: P(n) performs y interpolation, that is, P(n j )=y j , and the first-order derivative is continuous; the second derivative May be discontinuous, so n j There may be jumps;

[0130] Condition 7: Cubic interpolation P(n) has a shape-preserving effect; j The slope at is chosen in a specific way so that P(n) preserves the shape of the data and the corresponding monotonicity;

[0131] Step S3.2: Delete missing data: Delete the data marked as NaN in step S2 from the data after filling outliers corresponding to each specific straight line path, such as Figure 5 shown.

[0132] Specifically, the data that is NaN in A is: 309 ~A 566 , B in B 309 ~B 564 . Delete them, and then do the same for C and D.

[0133] Step S3.3: Remove the best straight line fit line from the data after deleting missing data corresponding to each specific straight line path to obtain the detrended data, such as Figure 6 shown.

[0134] Specifically, the best fitting straight line of A is Remove this line from A to obtain the detrended data.

[0135] Then, do the same for B, C, and D respectively.

[0136] Step S3.4: Smooth the detrended data corresponding to each specific straight line path, such as Figure 7 shown.

[0137] The smoothing method is a Gaussian weighted moving average within each window of A, with a window size factor of 0.25.

[0138] Specifically, A i The data of 0.25 times the number of data around is regarded as a window, and all Gaussian weighted moving averages in the window are calculated to obtain A ′ i , as A i Then perform this operation on all elements in A.

[0139] Then, do the same for B, C, and D.

[0140] Step S3.5: Fill and scale the smoothed data corresponding to each specific straight line path according to the position. Figure 8 shown.

[0141] The method is to fill in the larger positions of the component without measurement data, and then scale A, B, C, and D according to the actual size of the component.

[0142] In this example, the diameter of the circle left in the center is 458mm, which is proportional to the diameter of the component. From A in A N / 2 Start, insert There are vacancies.

[0143] Then, A i Scaled to the actual value at position i.

[0144] Furthermore, the same treatment method is adopted for B, C, and D.

[0145] Step S4: Based on the processed data, performing surface interpolation fitting between the plurality of measurement paths and along the plurality of specific paths.

[0146] like Figure 9 As shown, the method involves interpolating the Fourier transform of the function value between the data on any specific path. The number of interpolated data is determined based on the actual situation. In the figure, the hollow dots represent three cycles of data extracted in step S4.1, and the solid dots represent three cycles of data interpolated in step S4.2.

[0147] Step S4.1: extracting a number of data on any specific path;

[0148] For example, in this example, the data on the same radius of the polar coordinates are extracted. Extract A1, B1, C1, D1, A 1488 ,B 1488 ,C 1488 ,D 1488 There are 8 data in total, recorded as R1.

[0149] Step S4.2: Fourier transform of the interpolation function value in the plurality of data;

[0150] For example, in this example, the Fourier transform of the interpolated function value in R1 is 600. The interpolated data is recorded as R ′ 1, as a replacement for R1.

[0151] Step S4.3: Perform the calculations in steps S4.1-S4.2 on all data on all specific paths in sequence.

[0152] Perform the calculations in steps 4.1 and 4.2 for all data in A, B, C, and D in sequence.

[0153] Get {R1,…,R 744}.

[0154] Step S5: Based on the fitted data, the full-plane surface morphology is drawn in a three-dimensional rectangular coordinate system according to the actual position.

[0155] For example, Figure 10 As shown, the method is to set {R1,…,R 744 The data in} are plotted sequentially.

[0156] After implementing the above steps, the method can finally obtain a high-precision large-plane surface morphology, such as Figure 11 shown.

[0157] Data Appendix:

[0158] A＝{8,7,7,6,5,6,4,1,4,6,5,4,5,6,7,7,7,7,7,8,8,7,7,7,6,6,5,1,6,6,7,7,7,8,7,7,8,7,7,7,7,7,7,7,7,7,7,5,6,7,6,7,7,6,7,6,5,7,5,5,6,5,7,6,6,7,6,4,-358,-429,-482,-482,-482,-482,-482,-414,8,7,8,8,7,7,7,5,7,7,7,7,7,7,8,8,7,8,8,7,7,7,7,4,1,1,0,0,3,6,7,7,7,8,8,7,7,7,7,7,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,7,7,3,5,4,-1,1,1,2,5,5,6,6,6,7,7,7,7,7,7,6,2,5,-1,-1,1,-1,3,6,5,7,7,8,8,8,8,8,8,7,6,4,0,2,3,0,5,4,6,7,7,8,10,8,8,8,7,6,6,3,3,1,-1,0,0,0,6,6,6,7,7,7,7,5,5,4,0,-1,-1,-2,-1,0,0,2,5,5,6,7,7,7,7,7,7,6,6,5,2,3,4,0,1,2,-1,0,3,-1,0,2,-1,-1,-1,-2,-2,-2,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-3,-3,-3,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-5,-4,-7,-9,-7,-11,-10,-10,-11,-9,-6,-10,-7,-4,-9,-4,-7,-11,-5,-11,-11,-10,-12,-12,-12,-12,-12,-12,-13,-12,-13,-13,-12,-13,-13,-12,-14,-376,-376,-482,-482,-482,-482,-482,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-484,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-483,-484,-483,-484,-483,-483,-483,-483,-483,-484,-483,-484,-483,-483,-483,-483,-484,-483,-483,-484,-483,-484,-483,-483,-483,-483,-483,-484,-483,-483,-483,-483,-484,-483,-483,-484,-483,-483,-483,-483,-484,-483,-483,-484,-483,-484,-483,-483,-484,-484,-483,-484,-483,-483,-483,-483,-483,-483,-483,-483,-483,-484,-483,-484,-484,-483,-484,-483,-483,-483,-483,-484,-483,-483,-484,-483,-483,-484,-483,-483,-483,-484,-484,-483,-483,-484,-483,-483,-483,-484,-483,-483,-483,-484,-483,-483,-483,-484,-483,-483,-483,-484,-483,-471,-293,-13,-13,-13,-12,-12,-12,-11,-12,-12,-12,-12,-12,-12,-12,-11,-11,-11,-11,-8,-10,-8,-8,-7,-9,-4,-7,-3,-5,-6,-7,-6,-9,-6,-7,-6,-4,-5,-5,-4,-5,-4,-4,-4,-4,-4,-6,-7,-8,-5,-6,-5,-7,-5,-5,-3,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-3,-2,-2,-2,-2,-1,-2,-1,-1,-1,-1,-1,-1,0,-1,-1,-1,-1,-1,-2,-1,-1,-2,-1,-1,0,0,1,1,4,2,3,1,0,-1,-1,-2,-2,-1,-2,-2,-2,-2,-2,-1,-2,-1,-1,0,0,2,0,0,-1,-2,-2,-2,-2,-2,-2,-1,-1,2,4,7,4,7,6,6,6,5,2,4,-1,-1,-1,-2,-1,-1,-1,3,3,4,6,6,6,7,7,7,6,6,6,3,0,1,-2,-1,-1,-1,-1,1,4,6,7,7,7,7,6,5,0,0,-2,-2,-1,-1,-1,2,5,6,7,7,8,8,7,7,6,5,1,0,-1,-1,-2,-2,-1,-1,1,5,6,7,7,7,7,7,6,5,0,-1,-2,-2,-2,-2,-2,-1,0,0,5,7,7,7,8,7,7,7,4,2,-1,-2,-2,-1,0,4,6,7,8,8,8,8,8,7,7,7,5,0,-2,-2,-2,-2,-2,-1,0,6,7,8,8,7,8,7,6,6,-1,-1,-1,-2,-1,-1,-1,6,4,6,7,5,6,7,7,8,7,7,7,6,6,6,0,6,6,6,7,7,7,8,7,7,7,5,3,3,0,0,0,-1,0,0,0,2,4,5,7,7,7,8,8,8,8,7,7,7,6,6,4,4,4,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},

[0159] B＝{-3,-1,-4,-3.99,-1.99,-0.99,-0.99,0.01,-0.98,-0.98,0.02,-0.98,-0.98,-0.97,0.03,0.03,0.03,1.03,0.04,1.04,1.04,0.04,0.04,0.05,0.05,0.05,0.05,0.05,1.06,0.06,1.06,1.06,2.06,2.07,2.07,1.07,1.07,0.07,0.08,0.08,0.08,0.08,0.08,0.09,0.09,0.09,0.09,1.09,1.1,1.1,2.1,2.1,3.1,4.11,4.11,5.11,2.11,2.11,2.12,1.12,1.12,1.12,1.12,2.13,2.13,2.13,-248.87,-248.87,-248.86,-248.86,-248.86,-249.86,-249.86,-249.85,-70.85,8.15,8.15,5.15,1.16,1.16,0.16,0.16,1.16,1.17,4.17,6.17,4.17,9.17,10.18,10.18,10.18,10.18,10.18,9.19,8.19,8.19,8.19,5.19,8.2,7.2,7.2,9.2,8.2,9.21,10.21,9.21,10.21,10.21,10.22,11.22,10.22,10.22,10.22,10.23,10.23,10.23,10.23,11.23,10.24,10.24,11.24,10.24,10.24,10.25,10.25,10.25,10.25,10.25,10.26,10.26,10.26,11.26,11.26,11.27,11.27,11.27,11.27,11.27,11.28,11.28,11.28,10.28,10.28,10.29,10.29,10.29,10.29,10.29,11.3,11.3,11.3,11.3,11.3,11.31,11.31,11.31,11.31,11.31,11.32,11.32,11.32,11.32,11.32,11.33,11.33,11.33,10.33,10.33,11.34,11.34,11.34,11.34,11.34,11.35,11.35,11.35,11.35,11.35,11.36,11.36,12.36,11.36,11.36,12.37,11.37,11.37,12.37,11.37,12.38,12.38,12.38,12.38,12.38,12.39,14.39,12.39,12.39,14.39,11.4,13.4,12.4,11.4,12.4,12.41,11.41,12.41,11.41,11.41,12.42,11.42,12.42,12.42,11.42,11.43,12.43,11.43,12.43,11.43,11.44,11.44,11.44,11.44,11.44,11.45,11.45,11.45,11.45,11.45,17.46,11.46,12.46,11.46,12.46,11.47,11.47,10.47,10.47,10.47,11.48,11.48,11.48,11.48,11.48,11.49,11.49,12.49,11.49,11.49,11.5,12.5,11.5,11.5,11.5,11.51,10.51,11.51,10.51,10.51,10.52,10.52,10.52,10.52,10.52,11.53,10.53,11.53,11.53,11.53,11.54,11.54,10.54,11.54,10.54,11.55,10.55,10.55,10.55,10.55,10.56,10.56,10.56,10.56,10.56,10.57,10.57,10.57,9.57,9.57,9.58,9.58,9.58,9.58,8.58,9.59,9.59,9.59,9.59,9.59,8.6,8.6,7.6,8.6,8.6,6.61,6.61,3.61,-248.39,-248.39,-248.38,-248.38,-248.38,-248.38,-249.38,-248.37,-249.37,-249.37,-249.37,-249.37,-249.36,-249.36,-249.36,-249.36,-249.36,-249.35,-249.35,-249.35,-249.35,-249.35,-249.34,-249.34,-249.34,-249.34,-249.34,-249.33,-249.33,-249.33,-249.33,-249.33,-249.32,-249.32,-249.32,-249.32,-249.32,-249.31,-249.31,-249.31,-249.31,-249.31,-249.3,-249.3,-249.3,-249.3,-249.3,-249.29,-249.29,-249.29,-249.29,-249.29,-249.28,-249.28,-249.28,-249.28,-249.28,-249.27,-249.27,-249.27,-249.27,-249.27,-249.26,-249.26,-249.26,-249.26,-249.26,-249.25,-249.25,-249.25,-249.25,-249.25,-249.24,-249.24,-249.24,-249.24,-249.24,-249.23,-249.23,-249.23,-249.23,-249.23,-249.22,-249.22,-249.22,-249.22,-249.22,-249.21,-249.21,-249.21,-249.21,-249.21,-249.2,-249.2,-249.2,-249.2,-249.2,-249.19,-249.19,-249.19,-249.19,-249.19,-249.18,-249.18,-249.18,-249.18,-249.18,-249.17,-249.17,-249.17,-249.17,-249.17,-249.16,-249.16,-249.16,-250.16,-249.16,-249.15,-249.15,-249.15,-249.15,-249.15,-249.14,-249.14,-249.14,-249.14,-249.14,-249.13,-249.13,-250.13,-249.13,-249.13,-249.12,-249.12,-249.12,-249.12,-249.12,-249.11,-250.11,-250.11,-249.11,-249.11,-249.1,-249.1,-249.1,-250.1,-249.1,-249.09,-250.09,-249.09,-250.09,-250.09,-250.08,-249.08,-249.08,-250.08,-250.08,-249.07,-249.07,-250.07,-249.07,-249.07,-250.06,-249.06,-249.06,-249.06,-250.06,-249.05,-249.05,-249.05,-250.05,-250.05,-249.04,-249.04,-250.04,-250.04,-250.04,-250.03,-250.03,-250.03,-249.03,-250.03,-250.02,-250.02,-249.02,-250.02,-250.02,-250.01,-250.01,-249.01,-250.01,-250.01,-250,-250,-250,-250,-250,-249.99,-248.99,-249.99,-249.99,-249.99,-249.98,-249.98,-249.98,-249.98,-249.98,-249.97,-249.97,-249.97,-249.97,-249.97,-249.96,-248.96,-249.96,-249.96,-249.96,-249.95,-248.95,-249.95,-249.95,-249.95,-249.94,-249.94,-249.94,-249.94,-249.94,-249.93,-249.93,-249.93,-249.93,-249.93,-249.92,-249.92,-249.92,-249.92,-249.92,-249.91,-249.91,-249.91,-249.91,-249.91,-249.9,-249.9,-249.9,-249.9,-249.9,-249.89,-249.89,-249.89,-249.89,-249.89,-248.88,-249.88,-249.88,-249.88,4.12,5.13,5.13,6.13,5.13,5.13,5.14,5.14,6.14,4.14,5.14,7.15,8.15,9.15,10.15,11.15,11.16,11.16,11.16,11.16,11.16,11.17,11.17,11.17,10.17,10.17,10.18,10.18,10.18,10.18,11.18,11.19,11.19,11.19,12.19,12.19,12.2,12.2,11.2,11.2,11.2,11.21,11.21,11.21,11.21,10.21,10.22,11.22,11.22,11.22,12.22,12.23,12.23,13.23,12.23,12.23,13.24,12.24,12.24,12.24,12.24,12.25,11.25,11.25,12.25,12.25,12.26,12.26,12.26,12.26,12.26,13.27,13.27,13.27,13.27,13.27,12.28,12.28,12.28,12.28,12.28,12.29,12.29,12.29,12.29,13.29,13.3,13.3,14.3,16.3,15.3,14.31,15.31,13.31,12.31,13.31,12.32,12.32,13.32,13.32,12.32,12.33,12.33,12.33,13.33,13.33,14.34,16.34,13.34,13.34,13.34,13.35,12.35,12.35,12.35,11.35,12.36,11.36,12.36,12.36,14.36,-247.63,-247.63,-247.63,-247.63,-247.63,-247.62,-248.62,-112.62,19.38,22.38,13.39,13.39,12.39,12.39,12.39,11.4,11.4,11.4,12.4,12.4,12.41,13.41,14.41,14.41,15.41,13.42,13.42,13.42,12.42,11.42,11.43,11.43,11.43,11.43,11.43,12.44,12.44,12.44,13.44,13.44,15.45,14.45,13.45,14.45,13.45,12.46,12.46,11.46,10.46,13.46,10.47,10.47,11.47,11.47,12.47,13.48,13.48,13.48,13.48,12.48,12.49,12.49,10.49,11.49,9.49,5.5,9.5,6.5,4.5,10.5,7.51,7.51,11.51,10.51,11.51,12.52,11.52,11.52,11.52,10.52,10.53,10.53,7.53,7.53,4.53,3.54,7.54,8.54,10.54,11.54,7.55,10.55,11.55,11.55,11.55,10.56,7.56,7.56,4.56,3.56,3.57,2.57,2.57,2.57,2.57,3.58,4.58,6.58,10.58,11.58,11.59,11.59,11.59,10.59,8.59,2.6,1.6,1.6,0.6,11.6,0.61,0.61,1.61,1.61,3.61,7.62,10.62,11.62,11.62,9.62,10.63,9.63,5.63,3.63,2.63,1.64,1.64,1.64,0.64,1.64,0.65,0.65,2.65,3.65,7.65,11.66,10.66,9.66,11.66,7.66,6.67,4.67,1.67,1.67,-0.33,-0.32,0.68,0.68,-0.32,-3.32,0.69,1.69,1.69,2.69,4.69,3.7,5.7,3.7,2.7,2.7,1.71,0.71,0.71,-1.29,-3.29,-5.28,-4.28,-2.28,-0.28,0.72,0.73,1.73,1.73,0.73,1.73,26.74,26.74,26.74,25.74,25.74,24.75,24.75,24.75,23.75,25.75,25.76,26.76,27.76,27.76,1.76,1.77,1.77,1.77,1.77,1.77,1.78,1.78,1.78,1.78,1.78,1.79,1.79,1.79,1.79,1.79,1.8,1.8,1.8,1.8,1.8,1.81,1.81,1.81,1.81,1.81,1.82,1.82,1.82,1.82,1.82,1.83,1.83,1.83,1.83,1.83,1.84,1.84,1.84,1.84,1.84,1.85,1.85,1.85,1.85,1.85,1.86,1.86,1.86,1.86,1.86,1.87,1.87,1.87,1.87,1.87,1.88,1.88,1.88,1.88,1.88,1.89,1.89,1.89,1.89,1.89,1.9,1.9,1.9,1.9,1.9,1.91,1.91,1.91,1.91,1.91,1.92,1.92,1.92,1.92,1.92,1.93,1.93,1.93,1.93,1.93,1.94,1.94,1.94,1.94,1.94,1.95,1.95,1.95,1.95,1.95,1.96,1.96,1.96,1.96,1.96,1.97,1.97,1.97,1.97,1.97,1.98,1.98,1.98,1.98,1.98,1.99,1.99,1.99,1.99,1.99,2,2,2}.

[0160] C＝{2,5,3,8,5,7,10,7,6,3,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,4,6,10,10,10,10,11,11,11,11,11,11,10,10,9,6,2,2,2,2,3,4,8,10,10,10,11,10,9,10,4,3,8,2,4,7,6,8,7,9,10,6,7,9,6,8,9,4,3,2,1,1,1,1,4,4,7,10,9,10,10,8,9,9,3,6,5,2,4,6,5,9,9,10,11,11,11,11,11,11,11,10,11,10,10,10,8,8,10,9,10,11,11,11,11,11,11,11,11,11,10,11,11,10,10,10,10,10,11,11,11,12,13,14,12,12,12,11,11,11,10,10,10,9,11,11,11,12,12,13,12,12,12,12,11,11,11,10,9,8,6,6,7,9,10,11,11,12,13,13,16,14,12,12,11,11,10,9,9,9,9,9,10,10,11,11,11,12,12,12,12,11,11,11,10,10,10,10,9,10,10,10,11,11,11,11,11,11,11,11,11,10,10,9,8,8,9,9,10,10,10,11,11,11,11,11,11,11,11,10,10,9,7,7,7,7,8,9,10,10,10,11,11,11,11,11,11,10,10,9,6,4,3,3,3,4,5,7,7,5,8,9,8,9,9,9,8,6,4,3,2,1,1,1,1,2,2,3,4,5,7,9,6,3,3,1,1,0,0,0,0,-2,-2,-1,-2,-1,0,0,0,-350,-358,-369,-369,-369,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-370,-371,-370,-370,-370,-371,-370,-370,-370,-371,-370,-370,-371,-370,-370,-371,-370,-370,-370,-371,-370,-370,-371,-371,-370,-371,-371,-370,-370,-371,-371,-370,-370,-371,-370,-370,-371,-371,-370,-371,-371,-370,-371,-371,-370,-370,-371,-371,-370,-371,-371,-371,-371,-371,-371,-371,-370,-371,-371,-371,-370,-371,-371,-371,-370,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-370,-371,-371,-371,-371,-371,-371,-371,-370,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-371,-199,2,1,1,1,2,3,1,2,1,1,1,1,0,1,1,1,1,1,2,4,3,6,5,3,7,3,2,2,1,1,1,1,1,2,2,5,7,9,9,9,11,9,4,4,2,5,1,1,1,3,3,5,8,9,10,10,10,10,10,10,10,9,5,7,4,3,3,6,5,8,7,10,10,10,10,11,10,10,9,9,4,3,2,2,1,3,4,9,10,10,10,11,11,11,10,10,10,10,7,8,4,5,2,4,4,9,9,10,11,11,11,12,11,11,10,10,10,8,6,8,7,7,10,10,11,11,11,11,11,11,11,11,10,10,10,9,9,8,9,9,9,10,10,10,10,10,10,10,10,9,9,5,9,7,6,7,8,9,9,10,10,9,10,10,9,7,7,7,6,4,8,3,3,9,9,9,10,10,11,11,11,11,10,10,9,3,6,3,2,7,4,4,9,8,9,10,10,10,10,10,10,9,11,2,1,0,0,0,0,0,0,3,7,7,10,10,10,11,11,10,10,8,3,3,1,0,0,0,0,1,1,3,2,9,10,10,10,10,9,10,5,2,2,1,1,2,1,2,4,2,6,8,5,8,7,3,3,2,2,1,1,1,1,1,0,1,1,1,3,5,4,7,9,2,4,4,9,1,1,0,0,0,1,1,2,1,3,4,2,2,2,1,2,1,1,1,1,1,0,0,0,-1,0,0,35,37,37,37,38,37,36,36,34,35,34,29,34,33,33,36,35,35,36,36,36,37,37,38,40,37,37,36,36,36,34,32,32,31,32,33,35,35,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},

[0161] D＝{-6,-6,-3.99,-4.99,-3.99,3.02,4.02,5.03,13.03,15.03,14.04,17.04,22.04,15.05,14.05,14.05,12.06,4.06,2.06,0.07,-4.93,-5.93,-3.92,-4.92,2.09,4.09,10.09,12.1,14.1,17.1,21.11,22.11,23.11,22.12,18.12,14.13,12.13,4.13,3.14,2.14,2.14,3.15,3.15,4.15,6.16,12.16,14.16,14.17,14.17,16.18,20.18,21.18,22.19,22.19,22.19,15.2,14.2,13.2,7.21,3.21,1.21,-1.78,-1.78,0.23,2.23,5.23,14.24,14.24,17.24,23.25,22.25,22.25,21.26,16.26,14.26,13.27,12.27,5.28,4.28,4.28,3.29,4.29,3.29,4.3,10.3,13.3,13.31,15.31,17.31,15.32,15.32,16.33,14.33,13.33,13.34,6.34,4.34,5.35,4.35,3.35,9.36,12.36,12.36,14.37,19.37,15.38,19.38,23.38,23.39,22.39,23.39,23.4,17.4,15.4,14.41,7.41,4.41,3.42,3.42,2.43,3.43,4.43,6.44,13.44,15.44,19.45,23.45,23.45,22.46,19.46,15.46,14.47,11.47,5.48,4.48,4.48,3.49,5.49,7.49,11.5,14.5,15.5,15.51,20.51,22.51,19.52,16.52,15.53,14.53,13.53,12.54,9.54,10.54,5.55,5.55,5.55,6.56,9.56,14.56,14.57,14.57,15.58,19.58,22.58,20.59,22.59,22.59,18.6,15.6,14.6,13.61,12.61,12.61,9.62,5.62,9.63,13.63,13.63,14.64,18.64,21.64,22.65,22.65,22.65,17.66,15.66,14.66,14.67,12.67,12.68,8.68,6.68,11.69,13.69,13.69,14.7,16.7,19.7,19.71,22.71,22.71,21.72,21.72,19.73,15.73,14.73,13.74,10.74,6.74,9.75,8.75,11.75,13.76,14.76,15.76,17.77,21.77,23.78,23.78,22.78,21.79,15.79,14.79,13.8,10.8,4.8,4.81,4.81,3.81,4.82,4.82,6.83,12.83,14.83,16.84,18.84,21.84,22.85,22.85,20.85,15.86,14.86,13.86,5.87,4.87,4.88,4.88,4.88,7.89,13.89,14.89,16.9,22.9,15.9,-51.09,-86.09,-37.09,13.92,14.92,13.93,12.93,4.93,4.94,4.94,4.94,4.95,5.95,7.95,9.96,13.96,13.96,13.97,13.97,13.98,12.98,10.98,5.99,4.99,4.99,5,5,4,4.01,5.01,6.01,9.02,13.02,14.03,14.03,15.03,14.04,13.04,12.04,6.05,5.05,4.05,4.06,3.06,1.06,2.07,1.07,1.08,3.08,4.08,4.09,5.09,5.09,6.1,7.1,8.1,6.11,16.11,5.11,5.12,4.12,3.13,2.13,0.13,0.14,0.14,1.14,2.15,3.15,4.15,4.16,5.16,5.16,5.17,14.17,7.18,6.18,5.18,5.19,4.19,3.19,3.2,1.2,-1.8,-2.79,-4.79,-3.79,-3.78,-3.78,-2.78,-1.77,0.23,0.24,2.24,1.24,0.25,-1.75,-1.75,-3.74,-4.74,-4.74,-4.73,-4.73,-4.73,-4.72,-3.72,-3.71,-3.71,-2.71,-1.7,-3.7,-3.7,-3.69,-4.69,-245.69,-365.68,-424.68,-424.68,-424.67,-424.67,-425.66,-425.66,-425.66,-425.65,-425.65,-425.65,-425.64,-425.64,-425.64,-425.63,-425.63,-425.63,-425.62,-425.62,-425.61,-425.61,-425.61,-425.6,-425.6,-425.6,-425.59,-425.59,-425.59,-425.58,-425.58,-425.58,-425.57,-425.57,-425.56,-425.56,-425.56,-425.55,-425.55,-425.55,-425.54,-425.54,-425.54,-425.53,-425.53,-425.53,-425.52,-425.52,-425.51,-425.51,-425.51,-425.5,-425.5,-425.5,-425.49,-425.49,-425.49,-425.48,-425.48,-425.48,-425.47,-425.47,-425.46,-425.46,-425.46,-425.45,-425.45,-425.45,-425.44,-426.44,-426.44,-426.43,-426.43,-425.43,-425.42,-425.42,-425.41,-426.41,-425.41,-425.4,-426.4,-426.4,-425.39,-426.39,-426.39,-426.38,-425.38,-426.38,-425.37,-425.37,-425.36,-426.36,-425.36,-425.35,-425.35,-426.35,-426.34,-426.34,-425.34,-426.33,-426.33,-426.33,-426.32,-426.32,-425.31,-425.31,-426.31,-426.3,-426.3,-426.3,-426.29,-426.29,-426.29,-426.28,-425.28,-426.28,-426.27,-426.27,-426.26,-426.26,-426.26,-426.25,-426.25,-425.25,-425.24,-425.24,-426.24,-426.23,-426.23,-426.23,-426.22,-426.22,-426.21,-425.21,-425.21,-426.2,-426.2,-426.2,-426.19,-426.19,-426.19,-426.18,-425.18,-425.18,-426.17,-425.17,-426.16,-426.16,-426.16,-426.15,-426.15,-426.15,-426.14,-425.14,-426.14,-426.13,-426.13,-426.13,-426.12,-426.12,-426.11,-426.11,-425.11,-426.1,-426.1,-426.1,-426.09,-426.09,-426.09,-425.08,-426.08,-426.08,-426.07,-426.07,-426.06,-426.06,-426.06,-425.05,-426.05,-426.05,-426.04,-426.04,-426.04,-426.03,-426.03,-425.03,-425.02,-426.02,-426.01,-426.01,-426.01,-426,-426,-426,-425.99,-425.99,-425.99,-425.98,-425.98,-425.98,-425.97,-425.97,-425.96,-425.96,-425.96,-424.95,-425.95,-425.95,-425.94,-425.94,-424.94,-425.93,-425.93,-425.93,-425.92,-425.92,-425.91,-425.91,-424.91,-425.9,-425.9,-425.9,-425.89,-425.89,-425.89,-425.88,-425.88,-425.88,-425.87,-425.87,-425.86,-425.86,-425.86,-425.85,-425.85,-425.85,-425.84,-425.84,-424.84,-425.83,-425.83,-425.83,-425.82,-425.82,-425.81,-425.81,-425.81,-425.8,-425.8,-425.8,-425.79,-425.79,-425.79,-425.78,-425.78,-425.78,-425.77,-425.77,-424.76,-425.76,-425.76,-425.75,-425.75,-425.75,-425.74,-425.74,-424.74,-425.73,-425.73,-425.73,-425.72,-425.72,-425.71,-425.71,-425.71,-425.7,-425.7,-425.7,-425.69,-425.69,-425.69,-425.68,-425.68,-425.68,-425.67,-425.67,-424.66,-425.66,-425.66,-425.65,-425.65,-425.65,-425.64,-425.64,-425.64,-425.63,-425.63,-425.63,-425.62,-425.62,-425.61,-425.61,-425.61,-425.6,-425.6,-425.6,-425.59,-172.59,-3.59,-4.58,-4.58,-4.58,-4.57,-3.57,-3.56,-3.56,-2.56,-1.55,1.45,3.45,4.46,5.46,5.46,5.47,5.47,4.48,4.48,2.48,-1.51,-2.51,-2.51,-1.5,-1.5,0.5,4.51,5.51,5.51,6.52,6.52,6.53,6.53,6.53,6.54,5.54,5.54,5.55,5.55,4.55,4.56,4.56,4.56,4.57,4.57,4.58,5.58,6.58,6.59,7.59,10.59,8.6,9.6,13.6,12.61,7.61,6.61,6.62,6.62,5.63,5.63,5.63,6.64,6.64,6.64,7.65,11.65,14.65,12.66,11.66,9.66,8.67,6.67,6.68,6.68,6.68,6.69,6.69,7.69,7.7,7.7,9.7,10.71,9.71,7.71,8.72,7.72,7.73,6.73,6.73,6.74,6.74,6.74,7.75,9.75,8.75,11.76,15.76,15.76,15.77,15.77,16.78,15.78,14.78,15.79,16.79,15.79,15.8,14.8,15.8,14.81,10.81,11.81,12.82,13.82,12.83,13.83,14.83,15.84,14.84,14.84,14.85,14.85,13.85,12.86,12.86,13.86,13.87,13.87,15.88,15.88,16.88,16.89,38.89,16.89,15.9,15.9,15.9,14.91,8.91,8.91,8.92,8.92,8.93,12.93,14.93,15.94,15.94,16.94,16.95,16.95,16.95,16.96,16.96,16.96,16.97,16.97,15.98,15.98,15.98,15.99,14.99,14.99,15,15,16,16.01,16.01,16.01,17.02,17.02,17.03,18.03,17.03,16.04,14.04,8.04,7.05,6.05,5.05,6.06,6.06,8.06,15.07,17.07,18.08,22.08,21.08,23.09,19.09,17.09,14.1,8.1,6.1,2.11,-1.89,-2.89,-2.88,-1.88,0.13,5.13,6.13,8.14,15.14,16.14,17.15,17.15,18.15,17.16,17.16,27.16,15.17,8.17,7.18,6.18,3.18,1.19,1.19,-0.81,1.2,6.2,6.2,7.21,16.21,17.21,17.22,18.22,20.23,18.23,17.23,16.24,13.24,7.24,6.25,4.25,-0.75,-1.74,-0.74,1.26,3.27,5.27,7.28,10.28,15.28,17.29,18.29,18.29,19.3,18.3,17.3,16.31,9.31,7.31,5.32,2.32,-0.68,-1.67,-1.67,1.34,5.34,6.34,7.35,26.35,16.35,16.36,17.36,16.36,16.37,15.37,8.38,6.38,6.38,3.39,-0.61,-1.61,-1.6,-2.6,-2.6,-0.59,0.41,0.41,5.42,6.42,6.43,7.43,10.43,10.44,13.44,12.44,8.45,7.45,6.45,1.46,-2.54,-2.54,-2.53,-3.53,-2.53,-1.52,0.48,6.49,8.49,13.49,14.5,16.5,16.5,14.51,9.51,7.51,6.52,-0.48,-1.48,-1.47,-2.47,-1.46,1.54,4.54,6.55,7.55,9.55,11.56,14.56,15.56,12.57,12.57,11.58,7.58,4.58,-0.41,-1.41,-2.41,-3.4,-3.4,-2.4,-2.39,-1.39,4.61,5.62,7.62,15.63,15.63,16.63,16.64,14.64,7.64,6.65,4.65,-2.35,-3.34,-3.34,-3.34,-3.33,-1.33,3.68,6.68,12.68,17.69,17.69,17.69,17.7,16.7,10.7,7.71,2.71,-3.29,-6.28,-12.28,-13.28,-12.27,-4.27,-2.26,4.74,8.74,16.75,17.75,20.75,21.76,18.76,17.76,13.77,6.77,5.78,-0.22,-2.22,-3.21,-3.21,-4.21,-3.2,-2.2,-1.2,5.81,6.81,7.81,8.82,14.82,15.83,9.83,7.83,7.84,4.84,-1.16,-2.15,-3.15,-4.15,-4.14,-4.14,-3.14,-0.13,5.87,7.88,15.88,16.88,16.89,17.89,17.89,10.9,7.9,6.9,-1.09,-2.09,-3.09,-3.08,-4.08,-3.08,-2.07,-0.07,5.94,6.94,7.94,15.95,16.95,16.95,16.96,14.96,3.96,3.97,3.97,3.98,3.98,3.98,3.99,3.99,3.99,4,4,4,4.01,4.01,4.01,4.02,4.02,4.03,4.03,4.03,4.04,4.04,4.04,4.05,4.05,4.05,4.06,4.06,4.06,4.07,4.07,4.08,4.08,4.08,4.09,4.09,4.09,4.1,4.1,4.1,4.11,4.11,4.11,4.12,4.12,4.13,4.13,4.13,4.14,4.14,4.14,4.15,4.15,4.15,4.16,4.16,4.16,4.17,4.17,4.18,4.18,4.18,4.19,4.19,4.19,4.2,4.2,4.2,4.21,4.21,4.21,4.22,4.22,4.23,4.23,4.23,4.24,4.24,4.24,4.25,4.25,4.25,4.26,4.26,4.26,4.27,4.27,4.28,4.28,4.28,4.29,4.29,4.29,4.3,4.3,4.3,4.31,4.31,4.31,4.32,4.32,4.33,4.33,4.33,4.34,4.34,4.34,4.35,4.35,4.35,4.36,4.36,4.36,4.37,4.37,4.38,4.38,4.38,4.39,4.39,4.39,4.4,4.4,4.4,4.41,4.41,4.41,4.42,4.42,4.43,4.43,4.43,4.44,4.44,4.44,4.45,4.45,4.45,4.46,4.46,4.46,4.47,4.47,4.48,4.48,4.48,4.49,4.49,4.49,4.5,4.5,4.5,4.51,4.51,4.51,4.52,4.52,4.53,4.53,4.53,4.54,4.54,4.54,4.55,4.55,4.55,4.56,4.56,4.56,4.57,4.57,4.58,4.58,4.58,4.59,4.59,4.59,4.6,4.6,4.6,4.61,4.61,4.61,4.62,4.62,4.63,4.63,4.63,4.64,4.64,4.64,4.65,4.65,4.65,4.66,4.66,4.66,4.67,4.67,4.68,4.68,4.68,4.69,4.69,4.69,4.7,4.7,4.7,4.71,4.71,4.71,4.72,4.72,4.73,4.73,4.73,4.74,4.74,4.74,4.75,4.75,4.75,4.76,4.76,4.76,4.77,4.77,4.78,4.78,4.78,4.79,4.79,4.79,4.8,4.8,4.8,4.81,4.81,4.81,4.82,4.82,4.83,4.83,4.83,4.84,4.84,4.84,4.85,4.85,4.85,4.86,4.86,4.86,4.87,4.87,4.88,4.88,4.88,4.89,4.89,4.89,4.9,4.9,4.9,4.91,4.91,4.91,4.92,4.92,4.93,4.93,4.93,4.94,4.94,4.94,4.95,4.95,4.95,4.96,4.96,4.96,4.97,4.97,4.98,4.98,4.98,4.99,4.99,4.99,5,5}.

[0162] It should be noted that the embodiments of the present application have been described by way of illustration, and nothing therein should be taken as a limitation on the overall scope of the application. Any modification and equivalent arrangement made for the embodiments described by the specification, all of which are intended to come within the scope of the claims attached hereto, and in accordance with the sprit of the application, and all changes come within the meaning and range of equivalency of the claims.

Claims

1. A method for measuring the surface topography of a large planar workpiece, characterized in that: include: Step S1: Selecting a number of specific straight paths on the required measurement plane of the workpiece as measurement paths, and measuring the flatness curve data of each straight path; Step S2: Filtering out valid data from the flatness curve data of a plurality of measurement paths; Step S2 includes: Step S2.1: Mark the data that satisfies both conditions 1 and 2 in the original data sequence corresponding to each specific straight line path as NaN, and filter the data other than NaN in the original data sequence as intermediate valid data. Condition 1: 5 or more consecutive data show standard linear changes. Condition 2: The position of the data at the beginning or end; Step S2.2: Mark the data that meets both conditions 1 and 3 in the filtered intermediate valid data as NaN, and filter the data other than NaN in the intermediate valid data as valid data at both ends. Condition 1: 5 or more consecutive data show standard linear changes. Condition 3: The data is not at the beginning or end; Step S3: performing the following processing on the valid data in sequence: discovering and filling outliers, deleting missing data, removing trends from the data, smoothing noisy data, and filling and scaling the data; the step S3 also includes: step S3.1: discovering and filling outliers based on the filtered valid data; step S3.2: deleting missing data: deleting the data marked as NaN in step S2 from the data corresponding to each specific straight path after filling outliers; step S3.3: removing the best straight line fit line from the data corresponding to each specific straight path after deleting missing data, thereby obtaining detrended data; step S3.4: smoothing the detrended data corresponding to each specific straight path; step S3.5: filling and scaling the smoothed data corresponding to each specific straight path according to position; specifically as follows: supplementing larger positions without measurement data in the component, and scaling the flatness curve data of several measurement paths according to the actual size of the component; Step S4: performing surface interpolation fitting along a plurality of specific paths between the plurality of measurement paths based on the processed data; Step S5: Based on the fitted data, the full-plane surface topography is drawn in a three-dimensional rectangular coordinate system according to the actual position.

2. The surface topography measurement method of a large planar workpiece according to claim 1, wherein: In step S1 , a linear guide flatness measuring instrument is used to measure the original data sequence corresponding to each specific linear path.

3. The surface topography measurement method of a large planar workpiece according to claim 1, wherein: Step S3.1 includes steps S3.1.1 to S3.1.2: Step S3.1.1: Find and remove outliers from the valid data corresponding to each specific straight line path: define all values ​​that meet condition 4 as outliers; Condition 4: The difference from the median is more than three times the converted MAD value; ， in, is the median of all elements in A, , where The value of is determined by solving the following two equations, ， ， Valid data corresponding to each specific straight line path; Step S3.1.2: Fill out outliers using a shape-preserving piecewise cubic spline interpolation polynomial. At the same time, conditions 5, 6 and 7 are met. Condition 5: In each subinterval Above, polynomial is the cubic of a given data point interpolating polynomial with specified derivatives at the interpolation points; Condition 6: conduct Interpolation, that is , and the first-order derivative is continuous; the second derivative May not be continuous, so There may be jumps; Condition 7: Cubic interpolation Has a conformal effect; The slope at is chosen in a specific way so that Preserves the shape and corresponding monotonicity of the data.

4. The method for measuring the surface topography of a large planar workpiece according to any one of claims 1 to 3, wherein: The step S4 comprises: Step S4.1: extracting a number of data on any specific path; Step S4.2: Fourier transform of the interpolation function value in the plurality of data; Step S4.3: Perform the calculations in steps S4.1-S4.2 on all data on all specific straight line paths in sequence.

5. A surface topography measuring device for a large planar workpiece, characterized in that: include: A straight path flatness measurement module, wherein the straight path flatness measurement module is used to select a number of specific straight paths on a required measurement plane of a workpiece as measurement paths and measure flatness curve data of each straight path; A screening module is used to screen out valid data from the flatness curve data of a plurality of measurement paths; the screening module is used to: (1) Mark the data that satisfies both conditions 1 and 2 in the original data sequence corresponding to each specific straight line path as NaN, and filter the data other than NaN in the original data sequence as intermediate valid data. Condition 1: 5 or more consecutive data show standard linear changes. Condition 2: The position of the data at the beginning or end; (2) Mark the data that meets both conditions 1 and 3 in the filtered intermediate valid data as NaN, and filter the data other than NaN in the intermediate valid data as valid data at both ends. Condition 1: 5 or more consecutive data show standard linear changes. Condition 3: The data is not at the beginning or end; A data processing module is used to sequentially perform the following processing on the valid data: discover and fill outliers, delete missing data, remove trends from the data, smooth noisy data, and fill and scale the data; the data processing module is also used to: (1) discover and fill outliers based on the filtered valid data; (2) delete missing data: delete the data marked as NaN in the filtering module from the data corresponding to each specific straight line path after filling outliers; (3) removing the best straight-line fit line from the data after deleting missing data corresponding to each specific straight path, thereby obtaining detrended data; (4) smoothing the detrended data corresponding to each specific straight path; (5) filling and scaling the smoothed data corresponding to each specific straight path according to position; specifically, filling in the larger positions without measurement data in the component, and scaling the flatness curve data of several measurement paths according to the actual size of the component; A fitting module, configured to perform surface interpolation fitting along a plurality of specific paths between the plurality of measurement paths based on the processed data; A drawing module is used to draw the full-plane surface morphology according to the actual position in a three-dimensional rectangular coordinate system based on the fitted data.

6. The surface profile measuring device for a large planar workpiece according to claim 5, characterized in that: The linear path flatness measurement module uses a linear guide flatness measuring instrument to measure the original data sequence corresponding to each specific linear path.

7. The surface profile measuring device for a large planar workpiece according to claim 5, wherein: The data processing module is used for: (1) Based on the filtered valid data, discover and fill in outliers, (1) including (1.1)-(1.2): (1.1) Find and remove outliers from the valid data corresponding to each specific straight line path: define all values ​​that meet condition 4 as outliers; Condition 4: Conversion if the difference from the median is more than three times The value of ; where ， in, is the median of all elements in A, , where The value of is determined by solving the following two equations, ， ， Valid data corresponding to each specific straight line path; (1.2) Filling outliers: Use shape-preserving piecewise cubic spline interpolation polynomials for filling. At the same time, conditions 5, 6 and 7 are met. Condition 5: In each subinterval Above, polynomial is the cubic of a given data point interpolating polynomial with specified derivatives at the interpolation points; Condition 6: conduct Interpolation, that is , and the first-order derivative is continuous; the second derivative May not be continuous, so There may be jumps; Condition 7: Cubic interpolation Has a conformal effect; The slope at is chosen in a specific way so that Preserves the shape and corresponding monotonicity of the data.

8. The surface profile measuring device for a large planar workpiece according to any one of claims 5 to 7, characterized in that: The fitting module is used to: (1) Extracting some data on any specific path; (2) Fourier transform of the interpolated function value in the plurality of data; (3) Perform the calculations in the fitting modules (1)-(2) on all data on all specific straight line paths in sequence.

Images