An optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy

Abstract

The present invention belongs to the technical field of bevel gear tooth top surface accuracy detection, and specifically is an optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy, comprising establishing a plurality of circular rings at equal intervals on the tooth top surface of the bevel gear, and taking the center of a circle of a bevel gear positioning inner hole as the origin, establishing a Cartesian coordinate system O-XYZ on the bottom surface of the large end of the bevel gear, wherein the O-XY plane is parallel to the bottom surface of the large end of the bevel gear, and the Z axis is perpendicular to the bottom surface of the large end of the bevel gear; the present invention calibrates the measurement system, obtains gear point cloud data, smoothes and denoises the point cloud data, aligns the point cloud, makes a gear point cloud model, and makes a measurement circular ring, and obtains measurement data, and only two steps of data processing are required, namely, smoothing and denoising the point cloud data and aligning the point cloud, and both steps are to remove abnormal noise points in the point cloud data and improve the measurement accuracy. Therefore, in the process of data processing, the present invention requires very few steps of data processing, reduces the error introduced by data processing, and shortens the data processing cycle.

Description

Technical Field

[0001] The invention belongs to the technical field of bevel gear tooth top surface accuracy detection, in particular to an optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy. Background Art

[0002] Bevel (bevel) gears are very important mechanical transmission components in mining, metallurgy, shipbuilding, automobile and other industries due to their advantages such as large overlap coefficient, large load-bearing capacity, stable transmission and low noise. The tooth geometry and processing and manufacturing accuracy of bevel gears have a great impact on their transmission performance, service life and reliability. Gear measurement and evaluation are the main ways to ensure gear processing quality.

[0003] In the past century, gear measurement technology has gone through four stages of development: from the perspective of measurement methods, it has gone through the traditional mechanical method, to the electromechanical combination method, to the optical-mechanical-electrical method, and finally to the machine vision measurement method that is now emerging; from the perspective of measurement methods, it has evolved from the contact method to the non-contact method;

[0004] Due to the rapid development of machine vision technology, this technology was quickly applied to the research and application of gear measurement methods. This measurement method is simple and fast, with high real-time performance, and can quickly and accurately feed back the measurement results to the processing equipment, thus turning the entire processing process into a closed-loop process, which can greatly improve the processing accuracy.

[0005] However, the existing visual measurement methods for bevel gears usually transform the bevel gear image data into the parameters of the bevel gear, and then judge whether it is qualified by comparing the error between the actual tooth top surface of the bevel gear and the theoretical tooth top surface. Among them, the traditional method of gear processing error measurement based on machine vision needs to go through measurement system calibration to obtain gear image - grayscale - noise reduction - binarization - morphological processing - Hough transform - edge extraction - obtain relevant measurement data; in the process of a total of six steps of image data processing from grayscale to edge extraction, each step will introduce errors, affecting the measurement accuracy of the tooth top surface of the bevel gear. Secondly, the data needs to undergo multiple transformations, which makes the measurement cycle of relevant parameters of the bevel gear longer.

[0006] In view of this, the present invention solves the above technical problems by proposing an optical coaxial multi-ring detection method for the accuracy of bevel gear tooth top surfaces. Summary of the invention

[0007] In order to make up for the deficiencies of the prior art and solve the technical problem that the existing anti-counterfeiting labels are mainly in the form of stickers, which makes the labels easy to be stolen, the present invention proposes an optical coaxial multi-ring detection method for the accuracy of the tooth top surface of bevel gears.

[0008] The technical solution adopted by the present invention to solve its technical problem is:

[0009] The optical coaxial multi-ring detection method for the tooth top surface accuracy of bevel gears described in the present invention has the following steps:

[0010] S1: First, clean the surface of the gear product, then randomly paste marking points on the key positions of the gear surface, and then use a high-precision surface structured light 3D scanner to measure the gear in full field to obtain complete 3D point cloud data of the gear;

[0011] S2: Create multiple circular rings at equal intervals on the tooth top surface of the bevel gear;

[0012] S3: With the center of the bevel gear positioning inner hole as the origin, a Cartesian coordinate system O-XYZ is established on the bottom surface of the large end of the bevel gear. The O-XY plane is parallel to the bottom surface of the large end of the bevel gear, and the Z axis is perpendicular to the bottom surface of the large end of the bevel gear.

[0013] S4: Establish intersection points between each circular ring and the edge lines on both sides of the tooth top of the gear tooth, and extract the coordinates of all intersection points between the circular ring and both sides of the same gear tooth;

[0014] S5: Based on the intersection coordinates extracted in S4, a plane is fitted using the eigenvalue method as the actual tooth top surface;

[0015] S6: measuring the maximum error between the actual tooth top surface and the theoretically designed tooth top surface by fitting the plane, determining whether the tooth top surface error of the bevel gear is within a predetermined value range, and then continuing to measure the tooth top surface errors of the bevel gear and other gear teeth by using this method, to determine whether the tooth top surface errors of all gear teeth of the bevel gear are less than a predetermined value;

[0016] S7: Measuring the surface profile deviation of bevel gears.

[0017] Preferably, in step S1, since the gear surface has strong reflectivity, a layer of developer is sprayed on the gear surface to improve the quality of acquiring the gear image data.

[0018] Preferably, in step S1, a new filter is designed to smooth and denoise the bevel gear point cloud data. The filter design can be realized by a combination of the intrinsic mode function and the residual obtained by EMD:

[0019]

[0020] Where: g' represents the input signal; η(k) is used to control different filter operations; D k is the intrinsic mode function obtained by decomposition, r N is the corresponding margin.

[0021] Preferably, in step S1, the three-dimensional point cloud data is registered by an iterative nearest point method based on point cloud boundary feature points.

[0022] Preferably, in the steps S3 and S4, the bevel gear teeth are numbered in clockwise order as follows: i∈(1, n), n∈R + , and name the intersection points of the ring and the gear teeth.

[0023] Preferably, in step S6, the actual tooth top surface is fitted with the theoretically designed tooth top surface by using the spatial plane equation: ax+by+cz=d.

[0024] Preferably, a, b, c in the spatial plane equation are unit normal vectors of the plane, that is: a 2 +b 2 +c 2 =1, d is the distance from the origin to the plane, d≥0, where to determine the plane features, four parameters a, b, c, d need to be determined.

[0025] Preferably, the fitting plane obtains the best effect, which needs to satisfy the following conditions when a2+b2+c2=1:

[0026]

[0027] And using the Lagrange multiplier method to find the extreme value of the function, the following function is formed:

[0028]

[0029] The function is then derived to determine the maximum error between the actual tooth top surface and the theoretically designed tooth top surface.

[0030] Preferably, in step S7, the surface profile error value is the minimum distance between two theoretical contour equidistant surfaces that contain the measured contour.

[0031] Preferably, in step S7, the surface profile evaluation principle is the least squares principle, that is, the sum of squares d of the distances from the measured point to the theoretical surface is minimized, so the objective function is constructed as:

[0032]

[0033] Where: d = distance (p i ,s), is the distance from the measuring point pi to the theoretical surface s, pi is the measuring data point, and r is the radius of the measuring point.

[0034] The beneficial effects of the present invention are as follows:

[0035] 1. The optical coaxial multi-ring detection method for the tooth top surface accuracy of bevel gears described in the present invention, through measurement system calibration - acquisition of gear point cloud data - point cloud data smoothing and denoising processing - point cloud registration - gear point cloud model - making measurement rings - acquisition of measurement data, only two steps of data processing are required, namely point cloud data smoothing and denoising and point cloud registration, and both steps are to remove abnormal noise points in the point cloud data to improve the measurement accuracy. Therefore, in the process of data processing, the present invention requires very few data processing steps, reduces the error introduced by data processing, and makes the data processing cycle shorter, so the measurement accuracy is minimally affected by the data processing process. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The present invention will be further described below in conjunction with the accompanying drawings.

[0037] Figure 1 It is a flow chart of the bevel gear coaxial multi-ring detection method of the present invention;

[0038] Figure 2 It is a three-dimensional projection view of the circular ring design of the coaxial multi-ring detection method of the present invention (taking five rings as an example);

[0039] Figure 3 It is a stereoscopic projection view of the gear model to be tested in the present invention (taking the model with the number of teeth z=30 as an example);

[0040] Figure 4 It is a schematic three-dimensional projection view of the position of the detection ring on the gear teeth of the present invention;

[0041] Figure 5 It is a plane diagram of the principle model of the coaxial multi-ring detection gear of the present invention;

[0042] Figure 6 yes Figure 5 A partial enlarged view of the position A shown;

[0043] Figure 7 It is a plane diagram of data points and point coordinates extracted from the coaxial multi-ring detection gear of the present invention;

[0044] Figure 8 is a plan view of the gear tooth top surface fitted by the present invention;

[0045] Fig. 9 It is a practical application principle diagram of the bevel gear coaxial multi-ring detection method of the present invention. DETAILED DESCRIPTION

[0046] In order to make the technical means, creative features, objectives and effects achieved by the present invention easy to understand, the present invention is further explained below in conjunction with specific implementation methods.

[0047] like Figures 1 to 9As shown, an optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy provided by an embodiment of the present invention has the following steps:

[0048] Step 1: Data Acquisition

[0049] (1) In order to improve the measurement accuracy of gears, the high-precision surface structured light 3D scanner needs to be calibrated;

[0050] (2) Clean the surface of the gear product;

[0051] (3) In order to obtain accurate data of key parts of the gear, randomly paste marking points on the surface of key positions of the gear;

[0052] (4) Since the gear surface has strong reflectivity, a layer of developer needs to be sprayed on the gear surface to improve the quality of gear image data acquisition;

[0053] (5) Use a high-precision surface structured light 3D scanner to perform full-field measurement of the gear and obtain complete 3D point cloud data of the gear.

[0054] Step 2: Gear point cloud data preprocessing (smoothing, denoising and point cloud registration)

[0055] Based on the empirical mode decomposition, point cloud data smoothing and denoising can be achieved through the design of new filters. For example, a low-pass filter can be used for smoothing, and a high-pass filter can be used for data enhancement.

[0056] 1. Filter design

[0057] Various applications of point cloud empirical mode decomposition (EMD) can be achieved by constructing different filters. Filter design can be achieved by combining the intrinsic mode function (IMF) obtained by EMD and the residual.

[0058]

[0059] Where: g' represents the input signal, η(k) is used to control different filter operations, and D k is the intrinsic mode function obtained by decomposition; r N is the corresponding margin.

[0060] 2. Data smoothing and denoising

[0061] Based on the EMD of the point cloud data, the point cloud data is smoothed by designing a filter. Since the noise is usually contained in the first few IMFs, a low-pass filter is set to filter out the high-frequency part of the signal and retain the low-frequency part, thereby removing the noise. For the smoothing of the point cloud data, the residual obtained by the feature-preserving point cloud EMD is used to reconstruct the point cloud data. In order to avoid the blurring of sharp edges and corners in the process of extracting details, the feature points in the point cloud data are extracted in advance using existing methods, and in order to better reconstruct the smooth point cloud model, the normal is updated using the normal weighted average.

[0062]

[0063] Where: N(i)' is the extreme point, g(p i ) is the corresponding signal value of the input, N(i) is P i The neighboring points of a point.

[0064] 3. Point cloud registration

[0065] The most classic application of point cloud registration is the iterative closest point method (ICP algorithm) based on the feature points of the point cloud boundary to perform point cloud data registration;

[0066] set up For two point cloud data sets with registration, the ICP algorithm first calculates each point p in the set P. i , search for its nearest point y on the point set X i As the corresponding point. The corresponding point set of C is the operation of finding corresponding points, that is

[0067] Y=C(P,X) (3)

[0068] The algorithm establishes an error measure:

[0069]

[0070] Then solve the rigid body transformation vector q=[q R |q T ] T , recorded as

[0071] (q,d)=Q(P,Y) (5)

[0072] Where: d is the corresponding mean square error, that is, d = f(q). The algorithm applies the solved cylinder transformation to the first point cloud data, denoted as q(P). The ICP algorithm iteratively performs this operation until a certain set convergence criterion is met.

[0073] The quaternion method is used to solve the rigid body transformation vector. First, a symmetric matrix Q(∑ px )

[0074]

[0075] Where: tr represents the trace of the matrix, I3 is the third-order unit matrix, Δ=[A 23 A 31 A 12 ] T , where ∑ px is the cross-covariance matrix between point set P and X, that is

[0076]

[0077] In the formula, μ p ,μ x Respectively represent the centroid of point sets P and X, that is

[0078]

[0079]

[0080] Matrix Q(∑ px )The unit eigenvector q corresponding to the maximum eigenvalue R =[q0 q1 q2 q3] T , which is the optimal rotation transformation represented by the unit quaternion, and the corresponding rotation matrix R(q R ) and the translation vector q T According to formula (10), we can get:

[0081]

[0082] q T =μ x -R(q R )μ p (11)

[0083] Step 3: If Figure 2 As shown, multiple rings are built on the bevel gear.

[0084] (1) Each tooth of the bevel gear is divided into five segments with equal spacing from the small end to the large end using a split point (in this case, five circular rings are established as an example);

[0085] (2) Make five circular rings with all the dividing points within a 360° range from the gear centerline and the gear centerline as the axis;

[0086] (3) The circular rings from the small end to the large end of the gear are named circle a, circle b, circle c, circle d, and circle e respectively.

[0087] Step 4: Get the intersection coordinates

[0088] (1) Establish a coordinate system with the center of the gear positioning inner hole as the origin, and establish a Cartesian coordinate system O-XYZ on the bottom surface of the big end of the bevel gear. The O-XY plane is parallel to the bottom surface of the big end of the bevel gear, and the Z axis is perpendicular to the bottom surface of the big end of the bevel gear;

[0089] (2) Number the gear teeth in clockwise order: i∈(1, n), n∈R + , (here we take the bevel gear with 30 teeth as an example);

[0090] (3) Figure 6 As shown in the figure, the intersection points of circle a, circle b, circle c, circle d, circle e and the left side line of the gear tooth are named as: P ai , P bi , P ci , P di , P ei The intersection points of circle a, circle b, circle c, circle d, circle e and the right side line of the gear tooth are named: P a'i , P b'i , P c'i , P d'i , P e'i ;

[0091] (4) Point P ai , P bi , P ci , P di , P ei The coordinates of the intersection point with the gear tooth numbered i are expressed as: (X ai ,Y ai , Z ai )、(X bi ,Y bi ,Z bi )、(X ci ,Y ci ,Z ci )、(X di ,Y di ,Z di )、(X ei ,Y ei ,Z ei );

[0092] (5) Point P a'i , P b'i , P c'i , P d'i , P e'i The coordinates of the intersection point with the gear tooth numbered i are expressed as: (X a'i,Y a'i ,Z a'i )、(X b'i ,Y b'i ,Z b'i )、(X c'i ,Y c'i ,Z c'i )、(X d'i ,Y d'i ,Z d'i )、(X e'i ,Y e'i ,Z e'i ).

[0093] Step 5: If Figure 8 As shown, the maximum error between the actual tooth top surface and the theoretically designed tooth top surface is measured by fitting the plane;

[0094] The plane equation of the space can generally be expressed as:

[0095] ax+by+cz=d (12)

[0096] Among them, a, b, c are the unit normal vectors of the plane. That is: a 2 +b 2 +c 2 =1, d is the distance from the origin to the plane, d≥0. To determine the plane features, only four parameters a, b, c, and d need to be determined;

[0097] According to the data point coordinates (X ai ,Y ai , Z ai )、(X bi ,Y bi ,Z bi )、(X ci ,Y ci ,Z ci )、(X di ,Y di ,Z di )、(X ei ,Y ei ,Z ei ) and (X a'i ,Y a'i ,Z a'i )、(X b'i ,Y b'i ,Z b'i )、(X c'i ,Y c'i ,Z c'i )、(X d'i ,Y d'i ,Z d'i )、(X e'i ,Y e'i,Z e'i ), using equation (12) to represent the plane equation, then any data point (x i ,y i ,z i ) to the plane is:

[0098] d i =|ax i +by i +cz i -d| (13)

[0099] To obtain the best fitting plane, the following conditions should be met under the condition a2+b2+c2=1:

[0100]

[0101] Using the Lagrange multiplier method to find the extreme value of a function, we can form the function

[0102]

[0103] Taking the derivative of (15) with respect to d and setting the derivative to zero, we obtain

[0104]

[0105] Then there is

[0106]

[0107] Therefore, equation (13) can be rewritten as

[0108]

[0109] in:

[0110]

[0111] Taking the derivative of equation (15) with respect to a and setting the derivative to zero, we can obtain

[0112]

[0113] in:

[0114]

[0115] Similarly, we can find the partial derivative of (15) with respect to b and c and set the derivative to zero, and we have

[0116]

[0117]

[0118] Formula (20), (22), and (23) form the eigenvalue equation, and we get

[0119]

[0120] From formula (24), we can see that by finding the matrix eigenvalues ​​and eigenvectors, we can solve the problem of a, b, c.

[0121]

[0122] x=(a,b,c) T (26)

[0123] Obviously, the matrix A is a 3×3 real symmetric matrix, and its eigenvalue can be obtained by the following formula (27):

[0124]

[0125] Where: (,) represents the dot product of two vectors;

[0126] Expand (27) and consider a 2 +b 2 +c 2 =1, that is, (x,x) = 1.

[0127]

[0128] therefore, The minimum value of is the minimum eigenvalue of A, and the eigenvector corresponding to the minimum eigenvalue is a, b, c.

[0129] The eigenvalues ​​of the real symmetric matrix A can be obtained using formula (28):

[0130] |A-λI|=0 (29)

[0131] The non-zero solution obtained is λ min The corresponding eigenvectors are the parameters a, b, and c of the plane equation.

[0132] Step 6: Compare the tooth top surface li fitted in the third step with the theoretical design tooth top surface l, and measure the maximum error dmax between the two planes;

[0133] Step 7: Surface profile deviation measurement

[0134] The surface profile error value is the minimum distance between two theoretical equidistant surfaces that contain the measured profile. The surface profile is a comprehensive tolerance that can control both the surface profile error and the line profile error of any cross-section profile on the surface.

[0135] Usually the principle of surface profile evaluation is the least square principle, that is, the sum of the squares of the distances from the measured point to the theoretical surface is the smallest, so the objective function is constructed as:

[0136]

[0137] Where: d = distance (p i ,s), is the distance from the measuring point pi to the theoretical surface s; pi is the measuring data point, r is the radius of the measuring point;

[0138] The free surface is matched by using the measured discrete data points, aiming to find the transformation matrix M between the measured coordinate points and the theoretical coordinate points, so that the actual surface is more closely matched with the theoretical design surface. The objective function (31) for matching the measured data with the theoretical surface can be constructed. It represents the sum of the squares of the distances between the two sets of data points. When the objective function reaches the minimum value, the match is achieved.

[0139]

[0140] In the formula, p i =[X i ,Y i ,Z i ,1] T , measured data point coordinates p′ i =[X′ i ,Y′ i ,Z′ i ,1] T , p′ i =Tp i , the coordinates of the measured point after transformation; Q i =[x i ,y i ,z i ,1] T , the coordinates of the theoretical data point, defined as the point on the theoretical surface closest to the Pi' point;

[0141] M is the transformation matrix:

[0142]

[0143] M contains the rotation matrix R and the translation matrix T, where α, β, γ are the rotation angles along the X, Y, and Z axes, and tx, ty, tz are the translations along the X, Y, and Z axes respectively;

[0144] Then the free surface matching is completed, assuming the theoretical model point p′ i,j (x′ i,j ,y′ i,j ,z′ i,j ) corresponds to the measuring point A and B are the number of measurement points in the x and y directions; therefore, q i,j and p′ i,j The distance between can be expressed as:

[0145]

[0146] if:

[0147]

[0148] The surface average profile error (St) is defined as

[0149]

[0150] The number of points on the measured surface is (n'×m').

[0151] Step 8: According to formula (35), if the maximum spacing error dmax between the actual tooth top surface and the theoretical tooth top surface of all gear teeth with corresponding numbers is within the predetermined value τ and the surface profile error St is within the corresponding predetermined value τt, then the gear is qualified; otherwise, it is unqualified;

[0152]

[0153] Where: dmax is the maximum distance between the actual tooth top surface and the designed tooth top surface, τ is the predetermined value of the gear product spacing error; St is the surface average profile error; τt is the expected value of the gear tooth surface profile deviation.

[0154] The measurement scheme of the present invention includes: measurement system calibration - acquisition of gear point cloud data - point cloud data smoothing and denoising - point cloud registration - gear point cloud model - making a measurement ring - acquisition of measurement data. There are only two steps of data processing, namely point cloud data smoothing and denoising and point cloud registration. Both steps are to remove abnormal noise points in the point cloud data, thereby improving the accuracy of bevel gear measurement.

[0155] Therefore, in the measurement scheme of the present invention, fewer data processing steps are required during the data processing process. Therefore, the data processing cycle will be shorter, and the measurement accuracy will be minimally affected by the data processing process. Therefore, the measurement scheme of the present invention has the advantages of short cycle and high accuracy.

[0156] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. An optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy, characterized in that: The detection steps are as follows: S1: First, clean the surface of the gear product, then randomly paste marking points on the key positions of the gear surface, and then use a high-precision surface structured light 3D scanner to measure the gear in full field to obtain complete 3D point cloud data of the gear; S2: Create multiple circular rings at equal intervals on the tooth top surface of the bevel gear; S3: With the center of the bevel gear positioning inner hole as the origin, a Cartesian coordinate system 0-XYZ is established on the bottom surface of the large end of the bevel gear. The 0-XY plane is parallel to the bottom surface of the large end of the bevel gear, and the Z axis is perpendicular to the bottom surface of the large end of the bevel gear. S4: Establish intersection points between each circular ring and the edge lines on both sides of the tooth top of the gear tooth, and extract the coordinates of all intersection points between the circular ring and both sides of the same gear tooth; S5: Based on the intersection coordinates extracted in S4, a plane is fitted using the eigenvalue method as the actual tooth top surface; S6: measuring the maximum error between the actual tooth top surface and the theoretically designed tooth top surface through the fitting plane, determining whether the tooth top surface error of the bevel gear is within a predetermined value range, and then continuing to measure the tooth top surface errors of the bevel gear and other gear teeth, and determining whether the tooth top surface errors of all gear teeth of the bevel gear are less than a predetermined value; S7: Measuring the surface profile deviation of bevel gears.

2. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In the step S1, since the gear surface has strong reflectivity, a layer of developer is sprayed on the gear surface to improve the quality of acquiring the gear image data.

3. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In step S1, a new filter is designed to smooth and denoise the bevel gear point cloud data. The filter design is implemented by combining the intrinsic modal function obtained by EMD and the residual: Where: represents the input signal, Used to control different filter operations, is the intrinsic mode function obtained by decomposition, is the corresponding margin.

4. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In the step S1, the three-dimensional point cloud data is registered by an iterative nearest point method based on point cloud boundary feature points.

5. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In the steps S3 and S4, the bevel gear teeth are numbered in clockwise order as follows: i∈(1, n), n∈R + , and name the intersection points of the ring and the gear teeth.

6. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In the step S6, the actual tooth top surface is fitted with the theoretically designed tooth top surface by using the spatial plane equation: ax+by+cz=d; a, b, c in the spatial plane equation are the unit normal vectors of the plane, that is: a 2 +b 2 +c 2 =1, d is the distance from the origin to the plane, d≥0, where to determine the plane features, four parameters a, b, c, d need to be determined.

7. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 6, characterized in that: To achieve the best effect of the fitting plane, it is necessary to satisfy the following when a2+b2+c2=1: And using the Lagrange multiplier method to find the extreme value of the function, the following function is formed: The function is then derived to determine the maximum error between the actual tooth top surface and the theoretically designed tooth top surface.

8. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In step S7, the surface profile error value is the minimum distance between two theoretical contour equidistant surfaces that contain the measured contour.

9. The optical coaxial multi-ring detection method for bevel gear tooth top surface accuracy according to claim 1, characterized in that: In step S7, the surface profile evaluation principle is the least squares principle, that is, the sum of the squares d of the distances from the measured point to the theoretical surface is minimized, so the objective function is constructed as: in: , is the measuring point The distance to the theoretical surface s, is the measurement data point, and r is the radius of the measurement point.

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