Gear Integrated Error Determination Using the Gaussian Template Convolution-Facet Method

Abstract

A gear integrated error, a combination of individual and composite errors, carries richer information and has long been a key target of classic gear error measurement techniques. However, in the age of intelligent manufacturing, the classic methods for gear integrated error measurement are no longer able to meet the emerging requirements of large-scale gears and real-time online measurement. To address this gap, a novel approach to obtaining the gear integrated error based on GTC−Facet (Gaussian template convolution-Facet) is proposed. This method accurately pinpoints the sub-pixel contour of gears in images, enabling a quick derivation of the gear integrated error curve. From this curve, other individual and composite errors can be analyzed. The gear error information obtained through our method has higher measurement accuracy, achieving a positioning accuracy of 3.6 μm for the gear profile. Moreover, during the measurement process, the measured gear remains unclamped, and the entire measurement process can be completed within 0.35 s, which is much faster than classic methods. Our method meets the demands of online measurements and provides a new avenue for gear error measurement.

1. Introduction

Traditionally, gear accuracy measurement is classified into individual error measurement, composite error measurement, and integrated measurement [1]. Typical instruments include gear measurement centers, dual-flank mesh measuring instruments, one-sided mesh measuring instruments, and gear integrated error measuring instruments [2,3,4]. Nevertheless, conventional contact gear measuring instruments gradually reveal certain limitations [5]: the gear measurement center is unable to obtain composite errors; dual-flank mesh and one-sided mesh measuring instruments are incapable of obtaining individual errors [6].

Classic gear integrated error measuring instruments rely on single-tooth measuring devices, employing a specialized standard component known as the “special tooth-skipped master worm” to achieve “tooth-skipped one-sided meshing” measurements with a coincidence degree of less than 1, thereby obtaining the gear integrated error curve [7]. While these measurements yield results that encompass both individual and composite errors, the production of the special tooth-skipped master worm as a standard measuring element is challenging and lacks universal applicability. This limits the flexibility of classic gear integrated error measuring instruments to switch between different products, restricting their range of application [8]; the measurement efficiency cannot meet the online measurement needs of large-scale gears.

Unlike previous gear contact measurement technologies, gear vision measurement technology belongs to non-contact measurement with high measurement efficiency, making it well suited for online measurement and sorting [9,10]. In recent years, some scholars have applied sub-pixel detection algorithms to gear measurement, resulting in improved measurement accuracy. Chen, M.J. et al. [11] used the Zernike moment sub-pixel algorithm and morphological filtering method to extract and refine gear contours. A spur gear with a tip diameter of 49.751 mm and 23 teeth was selected as the measurement object, achieving a relative error of approximately 5% in the accumulated total deviation of the measured tooth pitch. Moru, D.K. et al. [12] measured the outer diameters of 12 spur gears with a nominal diameter of 62.014 mm using an interpolation-based sub-pixel operator provided by the Halcon algorithm library, with an error range between 2 μm and 30 μm. Duan, Z.Y. et al. [13] proposed a sub-pixel edge localization algorithm based on Gaussian integral surface fitting, which meets the profile deviation measurement requirements of five-level precision involute spur cylindrical gears with a tooth number of 60 and a modulus of 2.0 mm.

It is evident that while certain sub-pixel algorithms have been employed to enhance measurement accuracy, they are only able to accommodate gears of medium modulus, and there is a lack of research assessing the accuracy of fine modulus (modulus less than or equal to 1.0 mm) gears. Additionally, the current methods are limited to measuring individual errors, rather than gear integrated errors, leading to insufficient gear error information derived from images.

In this study, a gear integrated error measurement method based on GTC−Facet is proposed. This method utilizes the Canny operator to determine the gear edge transition area, and employs the GTC method to acquire the Hessian matrix. By matrix manipulation, the gray distribution Facet model is promptly obtained, enhancing the positioning speed and accuracy of the gear profile. Additionally, it integrates the theory of gear integrated error, enabling the instant acquisition of gear integrated error curves. This approach not only facilitates the decomposition of individual errors such as pitch deviation and profile deviation in gear analysis measurement, but also enables the decomposition of tangential composite deviation in gear functional measurement, significantly improving gear error analysis capabilities.

2. Theory of Gear Integrated Error

2.1. Gear Integrated Error Curve

The integrated error curve for gears serves as a comprehensive compilation of both individual and composite error data, providing a more extensive set of information compared to individual gear errors alone. As illustrated in Figure 1, the blue curve depictes the model of the integrated error curve within the gear section. The abscissa for each point on the curve corresponds to the angular coordinate $\phi$ (unit: °) associated with the respective point on the tooth face, while the ordinate represents the integrated error value denoted as GIE (unit: μm). Each segment of the curve signifies the integrated error of the left or right flank of each tooth.

The determination of pitch deviation, profile deviation, and other error parameters outlined in the international standard for gear accuracy is executed within a specific section of the tooth trace. Simultaneously, the calculation of the gear integrated error curve is conducted within the same tooth trace section. Consequently, the gear integrated error curve discussed in this article pertains specifically to the integrated error curve within the gear section, aligning with established gear accuracy standards.

The symbol meanings of the error term in Figure 1 are as follows: $f_{pt}$ represents the single-pitch deviation; $F_{pk}$ represents the sector pitch deviation; $F_p$ represents the cumulative pitch deviation; $f_i^{\prime }$ represents the tooth-to-tooth tangential composite deviation; $F_i^{\prime }$ represents the total tangential composite deviation; and $F_{\alpha }$ represents the total profile deviation. According to the integrated error curve, the profile form deviation, $f_{f\alpha }$, and profile slope deviation, $f_{H\alpha }$, can be further obtained; $f_{pbt}$ represents the base pitch deviation. The detailed definitions of these error items are given in the ISO-1328.1:2013 [14] and ISO/TR 10064-2:1996 [15] standards and will not be elaborated here.

Superimposing the gear integrated error curves of the left and right flanks on the same coordinate system can obtain a bidirectional integrated error curve in gear section (SJZ curve). Figure 2 is a segment on the SJZ curve, the pair of deviation values: $C_L$ and $C_R$ (or $C_L^{\prime }$ and $C_R^{\prime }$) representing the fixed chord position of the tooth, which should be located on the same angular coordinate $\phi$. The vertical coordinate distance between $C_L$/$C_R$ and $C_L^{\prime }$/$C_R^{\prime }$ is $L_C$ or $L_{\mathrm{C}}^{\prime }$, and its maximum variation is radial runout, defined as following:

$$F_r=\max (L_C,L_{\mathrm{C}}^{\prime })$$

(1)

2.2. Relationship between Integrated and Various Error Terms

The gear integrated error includes numerous gear error terms. The pitch deviation, profile deviation, and tangential comprehensive deviation can be separated from the integrated error curve in gear section. The radial runout can be separated from SJZ curve [16].

As shown in Figure 3, the AB segment is the overlapping meshing area between the tooth i and the tooth i − 1, the BC segment is the separate meshing area of the tooth i, and the CD segment is the overlapping meshing area between the tooth i + 1 and the tooth i. Assuming the number of sampling points for each tooth face is N, and if the overlap coefficient is ε, then the number of sampling points for the AC section is as follows:

$$N_3=INT(N/\epsilon )$$

(2)

In the formula, INT represents rounding. The number of sampling points for AB segment is as follows:

$$N_4=N-N_3$$

(3)

When $GI{E}_{0,j}=GI{E}_{z,j}$, z is the number of teeth, and $GI{E}_{i,j}$ is the integrated error value of the sampling point j on the tooth i, and then the longitudinal coordinate values of each point on the outer envelope of the gear integrated error curve are as follows:

$$L[(i-1)\times N+j]=\left\{\begin{array}{l}\max (GI{E}_{i-1,j+N_3},GI{E}_{i,j})\hspace{0.33em}1\le j\le N_4\hfill \\ GI{E}_{i,j}\hspace{0.33em}{N}_4\le N_3\hfill \end{array}\right\}1\le i\le z$$

(4)

The envelope of the gear integrated error curve is the total tangential composite deviation curve. The maximum variation of the total tangential composite deviation curve along the ordinate within the range of 0–360° is $F_i^{\prime }$, and the maximum variation of the ordinate within the range of 360°/z is $f_i^{\prime }$. The formulas are as follows:

$$F_i^{\prime }=\max L[k]-\min L[k]\hspace{0.33em}1\le k\le N_3\times z$$

(5)

$$f_{ik}^{\prime }=\max L[k\times j]-\min L[k\times j]\hspace{0.33em}1\le j\le N_3,1\le k\le z$$

(6)

$$f_i^{\prime }=\max (f_{ik}^{\prime })$$

(7)

According to the definition, the remaining errors can be obtained:

$$F_p=\max (F_{pi})-\min (F_{pi})\hspace{0.33em}1\le i\le z$$

(8)

$$\left.\begin{array}{c}{f}_{pt\max }=\max (f_{pti})\\ f_{pt\min }=\min (f_{pti})\end{array}\right\}\hspace{0.33em}1\le i\le z$$

(9)

where $F_{\alpha i}=\max (f_{fi,j})-\min (f_{fi,j})\hspace{0.33em}1\le j\le N$, then

$$F_{\alpha \max }=\max (F_{\alpha i})\hspace{0.33em}1\le i\le z$$

(10)

In the formula, $f_{fi,j}$ is the profile deviation of the sampling point j on the tooth i; $f_{pt\max }$ is the maximum value of a single-pitch deviation, and $f_{pt\min }$ is the minimum value of the single-pitch deviation. $F_{\alpha \max }$ is the maximum total profile deviation, where ${\upsilon }_i=\max \left|GI{E}_{i,j}-GI{E}_{i-1,j+N_3}\right|1\le j\le N_4$, then

$$f_{pbi}={\upsilon }_i\times \cos \alpha$$

(11)

$$\left.\begin{array}{c}{f}_{pb\max }=\max (f_{pbi})\\ f_{pb\min }=\min (f_{pbi})\end{array}\right\} 1\le i\le z$$

(12)

3. GTC−Facet Sub-Pixel Algorithm

The Facet model, introduced by Robert M. Haralick in 1984 [17], conceptualizes an image as a matrix comprised of numerous pixel units, allowing for its subdivision into several smaller matrices. Each region of these matrices conforms to a specific grayscale distribution, representing a Facet. The grayscale distribution function of each Facet is expressed as a polynomial function of the pixels within its corresponding region. However, directly employing polynomials for coefficient determination exhibits suboptimal resistance to interference, and computing the coefficients of binary cubic polynomials entails a considerable computational load.

This article addresses these challenges by transitioning from polynomial fitting of Facet to the utilization of Gaussian template convolution (GTC). The outlined steps are as follows:

By Taylor expansion of a unary function at $x_0$, we can obtain the following:

$$f(x_0+△x)=f(x_0)+△x·f^{\prime }(x_0)+\frac{1}{2!}△x^2·f^{″}(x_0)+o||△x^2||$$

(13)

The mapping of second-order derivatives to two-dimensional and multi-dimensional spaces is the Hessian matrix. In a two-dimensional image, $f(x,y)$ is a function that represents the grayscale value of the image, and expanding the $f(x+△x,y+△y)$ at $f(x_0,y_0)$, rewriting it in a matrix form, and rounding off the remaining term $o\left|\right|△x^2\left|\right|$ can obtain the following:

$$f(x+y)\approx f(x_0+y_0)+\left[△x,△y\right]\left[\begin{array}{c}{f}_x(x_0,y_0)\\ f_y(x_0,y_0)\end{array}\right]+\frac{1}{2!}\left[△x,△y\right]\left[\begin{array}{cc}{f}_{xx}(x_0,y_0)& f_{xy}(x_0,y_0)\\ f_{yx}(x_0,y_0)& f_{yy}(x_0,y_0)\end{array}\right]\left[\begin{array}{c}△x\\ △y\end{array}\right]$$

(14)

The second matrix in the third term to the right of the above equation, which is the Hessian matrix in two-dimensional space. The Hessian matrix is the second derivative at a point $(x,y)$ in two-dimensional space, written as a partial derivative matrix in the form of

$$H(x,y)=\left[\begin{array}{cc}{f}_{xx}(x,y)& f_{xy}(x,y)\\ f_{yx}(x,y)& f_{yy}(x,y)\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^{2}f(x,y)}{\partial x^2}& \frac{{\partial }^{2}f(x,y)}{\partial x\partial y}\\ \frac{{\partial }^{2}f(x,y)}{\partial x\partial y}& \frac{{\partial }^{2}f(x,y)}{\partial y^2}\end{array}\right]$$

(15)

The Hessian matrix simplifies the calculation of the grayscale distribution function by simply finding the elements in the matrix, which is actually the second derivative of the point. Because the data of a two-dimensional image are discrete, the general method for obtaining the second derivative of x at this point is to convert the derivative of a discrete function into a difference:

$$f_{xx}(x,y)=f(x,y)-f(x-△x,y)-(f(x-△x,y)-f(x+2△x,y))$$

(16)

However, this method only considers the information of the three pixels containing itself, and the amount of information is insufficient, so its robustness is poor. According to the linear scale space theory, taking the derivative of a function is equal to the convolution of the derivative of the function and Gaussian function:

$$\frac{\partial f(x,y)}{\partial x}=f(x,y)*\frac{\partial G(x,y)}{\partial x}$$

(17)

In the equation, * represents convolution, and $G(x,y)$ is the Gaussian function:

$$G(x,y)=\frac{1}{2\pi {\sigma }^2}{e}^{-(x^2+y^2)/(2{\sigma }^2)}$$

(18)

So, the elements in the Hessian matrix at $(x_0,y_0)$ can be solved by convolving the image grayscale function with the second-order derivative of the Gaussian function:

$$f_{xx}(x_0,y_0)=f(x_0,y_0)*\frac{{\partial }^{2}G(x,y)}{\partial x^2}$$

(19)

The Gaussian template is able to include information from all points within the n × n-neighborhood range of the point, which makes the results of the second derivative more accurate. When applying, it is necessary to convolute the complete image with the three partial derivative templates of the Gaussian function in order to calculate the Hessian matrix elements at the corresponding positions.

However, when we perform sub-pixel edge localization, truly useful information in the image only exists in the edge transition region. Therefore, we can first perform a coarse localization of the gear contour edge to determine the edge transition region, thereby reducing the image area that needs convolution. The rough positioning of gear contour edges is achieved using the Canny operator [18], and the edge transition region can be obtained through integer level pixel edges and corresponding gradients extracted by the Canny operator. Finally, an edge transition region with a width of 5–10 pixels is retained for convolution calculation.

According to the Steger algorithm [19], if there is a point $(x_0,y_0)$ on the integer level pixel edge of the gear contour, and $(n_x,n_y)$ is the absolute value of the maximum eigenvalue of the Hessian matrix at this point corresponds to the unit eigenvector, then $(n_x,n_y)$ is the unit normal vector (also as the unit gradient vector) of the gear edge contour. At $(x_0,y_0)$, referring to Equation (14), the grayscale of point $(x_0+t{n}_x,y_0+t{n}_y)$ in the normal direction can be expressed as follows:

$$f(x_0+t{n}_x,y_0+t{n}_y)=f(x_0,y_0)+[t{n}_x,{tn}_y]{[f_x(x_0,y_0),f_y(x_0,y_0)]}^T+\frac{1}{2}[t{n}_x,{tn}_y]H(x_0,y_0){[t{n}_x,{tn}_y]}^T$$

(20)

As shown in Figure 4, the gear profile conforms to the ramp type edge, so the sub-pixel coordinates should be located at the maximum grayscale value on the edge. Therefore, let $\frac{\partial f}{\partial t}=0$, then the solution is as follows:

$$t=-\frac{n_x{r}_x+n_y{r}_y}{n_x^2{r}_{xx}+2n_x{n}_y{r}_{xy}+n_y^2{r}_{yy}}$$

(21)

If $t{n}_x$ and ${tn}_y$ can simultaneously satisfy $t{n}_x\in [-\frac{1}{2},\frac{1}{2}]$ and ${tn}_y\in [-\frac{1}{2},\frac{1}{2}]$ and the sub-pixel coordinate points are located within the current pixel, then the sub-pixel coordinates of the gear contour are as follows:

$$\left\{\begin{array}{l}x=x_0+t{n}_x\\ y=y_0+t{n}_y\end{array}\right.$$

(22)

The specific steps of the gear sub-pixel contour extraction method based on the GTC−Facet model are depicted in Figure 5.

4. Obtaining Gear Integrated Error Based on GTC−Facet

The essence of the gear integrated error acquisition method proposed in this article, based on GTC−Facet, lies in the extraction of information from the image to synthesize the gear integrated error curve. The essential data for this synthesis comprise the gear end face center point, pitch deviation, and profile deviation.

4.1. Accurate Positioning of the Center Point of the Gear End Face

The center point of the gear end face is the basis for accurately calculating a series of gear accuracy detection items such as radial composite deviation, pitch deviation, and profile deviation, so it is necessary to first locate the gear center during measurement. The center of a gear can be fitted by the tip circle, root circle, or center hole circle. However, compared to the tip and root, the center hole is usually used as a fitting hole to be integrated with the output shaft or bearing. The size tolerance and cylindricity of the center hole reflect the performance of the gear, so the center hole is used to fit the gear center here.

The GTC−Facet sub-pixel fitting algorithm is used to extract the contour of the center hole, and the least squares method is used to fit the center of the circle. However, considering that using visual methods to measure and extract the gear contour is highly susceptible to dust interference and outliers, the least squares method is not reliable enough for outliers and may result in significant calculation errors. Therefore, this article adopts an iterative reweighted least squares method to improve the accuracy of circle fitting. Firstly, the coordinate formula for a circle is as follows:

$${(x-x_0)}^2+{(y-y_0)}^2=r^2$$

(23)

$(x_0,y_0)$ and r are the center and radius of the fitted circle, respectively. Assuming that there are n extracted center hole edge points, and the coordinates of each sub-pixel contour point are $(x_i,y_i)$, i = 1~n. The optimization objective function for the square of its error can be expressed as follows:

$$S={\displaystyle \sum _{i=1}^n(x_i^2-2x_0{x}_i+x_0^2+y_i^2-2y_0{y}_i+y_0^2-r^2}{)}^2$$

(24)

Where

$$b_1=-2x_0,b_2=-2y_0,b_3=x_0^2+y_0^2-r^2$$

(25)

After introducing distance weights $w_i$ for each sub-pixel contour point $(x_i,y_i)$, Equation (24) will be rewritten as follows:

$$S={\displaystyle \sum _{i=1}^n{w}_i(x_i^2+y_i^2+b_1{x}_i+b_2{y}_i+b_3}{)}^2$$

(26)

The ultimate goal of the iterative reweighted least squares method is to achieve a minimum value for the objective function S. At this point, simply take the partial derivatives of coefficient $b_1$, $b_2$, and $b_3$ separately, and make all partial derivatives zero to obtain the coefficients of the objective function. However, for Equation (26), the distance from each point to the center of the circle cannot be determined until the coefficients are fitted, so weights cannot be assigned. Therefore, iterative methods need to be used to solve the problem. In the first iteration, the standard least squares method is used to fit the circle, i.e., $w_i=1$. In the 2−n iterations, the weight function proposed by Huber is used:

$$w(\delta )=\left\{\begin{array}{l}1,\hspace{1em} \left|\delta \right|\le \gamma \\ \gamma /\left|\delta \right|, \left|\delta \right|>\gamma \end{array}\right.$$

(27)

In the equation, parameter $\gamma$ is the clipping function, which defines which points are outliers. The $\delta$ independent variable of the function represents the distance from a point to the center of a circle. After multiple iterations, calculate the optimal coefficients $b_1$, $b_2$, and $b_3$, and substitute them into Equation (26) to calculate the best fitting center and radius:

$$x_0=-\frac{b_1}{2},y_0=-\frac{b_2}{2},r=\frac{1}{2}\sqrt{b_1^2+b_2^2-4b_3}$$

(28)

Figure 6 shows the result of fitting the center and radius of the gear center hole using the iterative reweighted least squares method. Similarly, the tip circle and root circle can also be calculated using the method of fitting the center hole circle.

4.2. Calculation of Pitch Deviation and Profile Deviation

The key to calculating the gear integrated error based on visual images lies in the precise extraction of profiles. Classic edge detection algorithms can only extract pixel-level contours. By using the GTC−Facet sub-pixel positioning algorithm proposed in this article, the pixel-level edge contours can be further subdivided into sub-pixel levels. The effect is shown in Figure 7, and the blue arrow represents the gradient amplitude and direction of the gray level at the sub-pixel coordinate point. For ease of expression, draw an arrow every three coordinate points and reduce the amplitude of the arrow by ten.

Take the extracted center point of the gear end face as the center of the circle, measure the radius of the theoretical indexing circle as the radius, intersect with the profile, and sort the intersection points according to the serial number of the gear teeth. As it shown in Figure 8:

The actual angle ${\alpha }_i$ corresponding to the single left pitch can be calculated from the vectors $O{L}_i$ and $O{L}_{i+1}$ (the same to the single right pitch):

$${\alpha }_i=\arccos \frac{O{L}_i·O{L}_{i+1}}{\left|O{L}_i·O{L}_{i+1}\right|},i=1,2,\mathrm{\dots }z-1$$

(29)

This can further calculate the corresponding arc length between $L_i$ and $L_{i+1}$, which is the actual tooth pitch $P_i$:

$$P_i=d_M/2·{\alpha }_i=d/2·{\alpha }_i$$

(30)

The individual single-pitch deviation $f_{pti}$ is as follows:

$$f_{pti}=P_i-P$$

(31)

The theoretical tooth pitch $P=\pi d/z$ in the equation is calculated at the indexing circle, and the single-pitch deviation $f_{pt}$ is the maximum absolute value of $f_{pti}$:

$$f_{pt}=\max \left|f_{pti}\right|$$

(32)

Sector pitch deviation $F_{pk}$:

$$F_{pk}={\displaystyle \sum _{i=j}^{j+k}{f}_{pi}}$$

(33)

Cumulative pitch deviation $F_p$:

$$F_p=\max (F_{pk})-\min (F_{pk})$$

(34)

In the previous step, the sub-pixel coordinate set of the gear profile has been obtained. Before calculating the profile deviation, the theoretical involute corresponding to each profile needs to be generated. When the starting point $(r_b,0)$ of the involute is located, the involute equation is (the left and right profiles are the same) as follows:

$$x=r_b\cos \phi +r_b·\phi ·\sin \phi$$

(35)

$$y=r_b\sin \phi -r_b·\phi ·\cos \phi$$

(36)

In the formula, $r_b$ is the radius of the basic circle, $\phi$ is the rolling angle, and $\phi =\tan \alpha$; $\alpha$ is the pressure angle at any point on a segment profile, $\alpha =\arccos (r_b/r)$, and r is the distance from that point on the profile to the center of the circle.

Each profile has an intersection point with the basic circle, and to obtain the angle $\gamma$ at which each intersection rotates relative to $(r_b,0)$, the involute equation is as follows:

$$x=r_b·\cos (\phi +\gamma )+r_b·\phi ·\sin (\phi +\gamma )$$

(37)

$$y=r_b·\sin (\phi +\gamma )-r_b·\phi ·\cos (\phi +\gamma )$$

(38)

The position of the theoretical involute is shown in Figure 9.

As shown in Figure 10, place the coordinate origin at the center of the circle, where A$(x_i,y_i)$ is a point on the actual profile, the red curve represents the theoretical involute, and point A on the actual profile is perpendicular to the theoretical involute, with a perpendicular foot of P$(x,y)$. According to the properties of the involute, the line connecting PA is also tangent to the basic circle at point T. Assuming that the angle between the starting point of the involute and the tangent point T is $\theta$ and the length of OA is $\rho$, then

$$|TA{|}^2={\rho }^2-r_b^2=x_i^2+y_i^2-r_b^2$$

(39)

Write the coordinates of point A in polar coordinates, that is $(x_i,y_i)=(\rho ·\cos \alpha ,\rho ·\sin \alpha )$, and if the angle of the initial position of the involute is 0 degrees, then

$$\theta =\alpha +\arccos (r_b/\rho )$$

(40)

$$|AP|=r_b·\theta -|TA|=r_b·\theta -\sqrt{x_i^2+y_i^2-r_b^2}$$

(41)

The profile deviation at any point A on the profile evaluation segment is the length of segment AP, which needs to be judged as positive or negative. Generally, when the actual profile point is located outside the gear tooth area surrounded by the theoretical left and right profiles, it is positive and negative values are taken internally.

4.3. Synthesis of Gear Integrated Error Curve

The gear integrated error measurement method presented in this article necessitates only the acquisition of pitch and profile deviation, followed by the construction of the gear integrated error curve. Subsequently, additional analyses of other individual and composite errors can be conducted. Specifically, the profile deviation is superimposed onto a cross-sectional gear integrated error curve based on the actual graduation of each tooth, as determined by the pitch deviation.

However, the calculation direction of pitch deviation and profile deviation is different. When synthesizing, the component of profile deviation in the direction of pitch deviation should be considered. Based on Figure 10, the components of profile deviation at each point on the tooth face in the direction of pitch deviation (motion direction) should be considered. As shown in Figure 11, this component is marked in green, and the value of this component can be easily obtained, which is $f_{fi,j}/\cos \alpha$.

Therefore, the following synthesis formula should be used:

$$GI{E}_{i,j}=F_{Pi}+f_{fi,j}/\cos \alpha \hspace{0.33em}1\le i\le z,1\le j\le N$$

(42)

In the equation, $GI{E}_{i,j}$ is the integrated error value of the point j of the profile i on the gear integrated error curve, $F_{Pi}$ is the sector pitch deviation of the tooth i, and $f_{fi,j}$ is the profile deviation value of the point j of the tooth i; N is the number of sampling points on the profile.

Due to the fact that the values obtained in the image are in pixels, the final integrated error value needs to be converted to the actual unit of micrometers. Before taking the measurement, the gear vision measurement system needs to be calibrated. After calibration, a ratio coefficient $\lambda$ (pixel equivalent) between the image coordinate system and the world coordinate system can be obtained. The integrated error value obtained in the image can be multiplied by this coefficient to obtain the final integrated error value:

$${GIE}_{i,j}^{\prime }=GI{E}_{i,j}·\lambda \hspace{0.33em}1\le i\le z,1\le j\le N$$

(43)

In the formula, $GI{E}_{i,j}$ is the integrated error value of the tooth i and point j obtained from the image in pixels, and ${GIE}_{i,j}^{\prime }$ is the integrated error value of the tooth i and point j in micrometers.

5. Experiment and Discussion

5.1. GTC−Facet Algorithm Positioning Accuracy

Select master gears with an accuracy of JIS B1751−M00 from Japan Technomax Inc, Osaka, Japan. As shown in Figure 12a, the number of teeth is 38, the modulus is 1.0 mm, the pressure angle is 20 degrees, and the diameter of the center hole is labeled 12.0 mm. The positioning accuracy is compared by measuring the center hole of the gear. Figure 12b shows a comparison of positioning errors between the two algorithms, and the errors at each point represent the difference between the distance between the extracted center hole contour points and the nominal radius.

The experiment counted the difference between the distance from the contour point of 2200 holes to the center of the circle and the nominal radius, and the pixel equivalent of the gear image is 7.71 μm/pixel. The positioning error using a Canny pixel-level algorithm is relatively large, with a standard deviation of 28.77 μm. There is a significant issue of eccentricity. The standard deviation of positioning error obtained using the GTC−Facet algorithm is 3.58 μm. Using the standard deviation as the evaluation standard for positioning accuracy, its positioning accuracy is 3.58/7.71 = 0.46 pixel.

5.2. GVMS Integrated Measurement Experiment

This experiment requires building a visual measurement system and calibrating it.

5.2.1. Experimental Setup

The GVMS (Gear Visual Measurement System) has been built, including industrial cameras, lenses, LED backlight sources, visual brackets, and upper computers, as shown in Figure 13a. The system parameters of GVMS are shown in

Table 1. CVMS parameters.

Component	Parameters
Camera	CMOS, GigE interface, 5472 × 3648 px, 1″ target surface
Lens	Magnification: 0.5; distortion: <0.006%; resolution: 7.9 μm
Light source	LED surface light source
Pixel equivalent	4.76 μm/pixel
Gear parameters	Powder metallurgy spur gears; number of teeth: 17; modulus: 0.5 mm; pressure angle: 20°

, and the collected gear images are shown in Figure 13b.

Before taking the measurement, it is necessary to calibrate the GVMS using a 9 × 9 dot array calibration board. According to Zhengyou Zhang’s calibration method [20], collect eight calibration board images with different poses, perform sub-pixel extraction on each calibration board image, calculate the pixel distance between the dots, and convert it to the actual physical coordinate system. Finally, the actual physical size represented by one pixel is obtained using the calibration board as 4.76 μm. Therefore, the pixel equivalent is 4.76 μm/pixel.

5.2.2. Measurement Results

The integrated error curve in the gear section on the right/left flank obtained through GVMS is shown in Figure 14a and Figure 15a. Here, 1–17 teeth are sorted and distinguished by lines of different colors. There are 137–166 sampling points on each profile, with adjacent teeth spaced 360°/z.

The gear integrated error curve of the cross-section of the right flank takes the basic circle position of the first right flank as the initial position, where the rolling angle is 0°. The gear integrated error curve of the left flank section rotates 360°/z/2 more counterclockwise than the right flank, so the curve does not start from the rolling angle of 0°. The termination position of the gear integrated error curve of the left flank section is at the basic circle of the 17th left flank, and the angle at this position must be greater than 360°. This way, the integrated error curve in the gear section composed of the right and left flanks can fully contain the error information of the gear for one revolution.

ISO−1328.1:2013 [14] requires low-pass filtering of the original measurement data used for involute evaluation before analyzing and comparing tolerances. This article also applies Gaussian low−pass filtering to the integrated error curve in the gear section. Figure 14a and Figure 15a show the filtered integrated error curve in the gear section using the Cartesian coordinate system. Figure 14b and Figure 15b show the results of the extracted envelope from the error curve.

Figure 16 and Figure 17 are the polar coordinate system representations of the gear integrated error curve. The value of polar coordinates is calculated using the distance in the direction of the inscribed or circumscribed circle that contains the error curve. The use of the Cartesian coordinate system to represent the error curve is more in line with people’s habits of observation, while the use of polar coordinate representation can more intuitively reflect the integration of the gear corresponding to various angles on the circumference of the gear. Both methods can decompose the individual errors of gears from them.

5.3. Comparative Experiment between GVMS and P26

The error measurement of the same gear was carried out using Klinberg’s gear measurement center P26 and compared with the GVMS measurement results. The experimental site photos are shown in Figure 18.

5.3.1. Comparison of Integrated Measurement Results

The results of the pitch deviation, base pitch deviation, profile deviation, and tangential composite deviation measured by GVMS and P26 are summarized in

Table 2. Comparison of error results decomposed from the gear integrated error curve (unit: μm).

Items	Right-Hand Tooth Flank			Left-Hand Tooth Flank
	GVMS	P26	Error	GVMS	P26	Error
$F_i^{\prime }$	20.8	15.5	5.3	22.7	18.8	3.9
$f_i^{\prime }$	4.5	3.3	1.2	6.3	3.4	2.9
$F_{\alpha }$	5.5	7.3	−1.8	6.2	5.7	0.5
$f_{f\alpha }$	2.5	1.5	1.0	2.9	1.2	1.7
$f_{H\alpha }$	3.5	−7.3	−10.8	3.7	−5.8	9.5
$F_p$	15.1	12.9	2.2	15.2	14.9	0.3
$f_{pt\max }$	7.0	7.4	−0.4	9.5	11.1	−1.6
$f_{pt\min }$	0.2	0.1	0.1	0.3	0.1	0.2
$f_{pb\max }$	10.1	10.3	−0.2	11.5	10.6	0.9
$f_{pb\min }$	0.3	0.1	0.2	0.2	0.1	0.1

. The base pitch deviation and tangential composite deviation measured by P26 are decomposed from the gear integrated error curve.

5.3.2. Comparison of Pitch Deviation Results

The comparison results of the pitch deviation data decomposed from the gear integrated error curve are shown in Figure 19 and Figure 20.

According to ISO 1328−1:2013 [14], it can be seen from

Table 3. Comparison of pitch deviation (taking the absolute value of the maximum deviation).

Items	Tooth Flank	GVMS	P26	Error
$f_{p\max }$(μm)	left	9.5	11.1	−1.6
	right	7.0	7.4	−0.4
$F_p$(μm)	left	15.2	14.9	0.3
	right	15.1	12.9	2.2

that the pitch deviation of the gear meets level 8 accuracy, and the evaluation results of GVMS and P26 are consistent.

5.3.3. Comparison of Profile Deviation Results

As shown in Figure 21, the total profile deviation $F_{\alpha }$, the profile form deviation $f_{f\alpha }$, and the profile slope deviation $f_{H\alpha }$ can be decomposed from the gear integrated error curve. Taking the gear integrated error curve of the right flank of the third gear tooth as an example, it can be calculated that $F_{\alpha }$ = 1.9 μm, $f_{f\alpha }$ = 1.2 μm, and $f_{H\alpha }$ = 2.2 μm. $L_{\alpha }$ is the length of the profile measurement, $g_{\alpha }$ is the length of the meshing line, $C_f$ is the control point of the profile, $F_a$ is the forming point of the tip, and $N_f$ is the effective root point.

According to ISO 1328−1:2013 [14], it can be seen from the profile deviation data of the four teeth in

Table 4. Comparison of total profile deviation $F_{\alpha }$ (four teeth).

Tooth Number	Tooth Flank	GVMS	P26	Error
1	left	2.4	3.5	−1.1
	right	2.0	3.4	−1.4
6	left	3.2	5.7	−2.5
	right	3.0	7.3	−3.3
10	left	2.6	4.5	−1.9
	right	2.8	6.4	−3.6
14	left	3.2	3.5	−0.3
	right	2.1	1.2	0.9

,

Table 5. Comparison of profile form deviation $f_{f\alpha }$ (four teeth).

Tooth Number	Tooth Flank	GVMS	P26	Error
1	left	2.5	1.5	1.0
	right	2.3	1.5	0.8
6	left	3.1	1.2	1.9
	right	1.5	0.9	0.6
10	left	2.2	1.1	1.1
	right	0.9	1.2	−0.3
14	left	3.4	1.7	1.7
	right	1.3	1.2	0.1

and

Table 6. Comparison of profile slope deviation $f_{H\alpha }$ (four teeth).

Tooth Number	Tooth Flank	GVMS	P26	Error
1	left	−0.3	−3.1	2.8
	right	−2.2	−3.6	1.4
6	left	3.8	−5.8	2.0
	right	−3.4	−7.6	3.2
10	left	−1.2	−3.9	2.7
	right	−2.1	−6.0	3.9
14	left	−3.2	−2.1	−1.1
	right	2.5	−0.4	2.9

and Figure 22 and Figure 23 that the profile deviation of the gear meets level 7 accuracy, and the evaluation results of GVMS and P26 are basically consistent. The absolute value of the maximum error of $F_{\alpha }$ measured by both is 3.6 μm. The maximum error of $f_{f\alpha }$ is 1.9 μm. The maximum error of $f_{H\alpha }$ is 3.9 μm.

5.3.4. Measurement Speed Comparison

The average speed of measuring one gear using the method described in this article is about 0.35 s/piece. However, classic gear integrated error measurement methods use contact measurement, which is much slower. Wang, X.Y. [21] mentioned that measuring the one product gear with 28 teeth takes about 20 s, and the actual measurement also includes the clamping time, which takes even longer. Therefore, using the method described in this article to measure the gear integrated error has increased the measurement speed compared to classic methods, as shown in Figure 24.

5.4. Application of Methods

5.4.1. Detection of Tooth Face Burrs

The method proposed in this article can quickly and accurately identify burrs on the tooth’s face. As shown in Figure 25a, the burr is on the left flank of tooth 6. It is difficult to accurately identify and determine the position of the burr through conventional individual error analysis. However, by analyzing the integrated error curve in the gear section using the method described in this article, as shown in Figure 25b,c, it can be clearly observed that the burr’s radial length is approximately 27.2 μm within the range between 120° and 138° on the gear.

5.4.2. Detection of Gear Eccentricity

The method in this article only needs to analyze the eccentricity of the gear based on the SJZ curve. As shown in Figure 26, the gear to be tested has a certain eccentricity between the middle hole and the tip. The number of teeth is 23, and the modulus is 1.0 mm.

As shown in Figure 27, the SJZ curve is drawn, which can clearly reflect the eccentricity of the gear. The radial runout can be calculated from Equation (1), then it can be calculated that $F_r=35.1\mathsf{\mu }\mathrm{m}$.

5.4.3. Analysis of Profile Modification

By using the method presented in this article, various profile modifications can be analyzed, as shown in Figure 28. The integrated measurement of a spur gear with 23 teeth and 1.0 mm modules clearly demonstrates that the middle of the tooth width protrudes outward. Therefore, it can be concluded that the gear has undergone profile crowning modification.

6. Conclusions

Our method, based on the GTC−Facet algorithm, represents novel technology for obtaining gear integrated errors instantaneously. This approach affords several advantages, including fast measurement speed, high positioning accuracy, and comprehensive error information. Additionally, it holds the potential to enable online full inspection on large-scale and fine-module gear production lines in the future. As follows:

(1) A GTC−Facet sub-pixel algorithm is proposed to extract gear contours, achieving a positioning accuracy of 3.6 μm. This algorithm simplifies polynomial fitting to the Gaussian template convolution operation and utilizes the Canny operator to determine the gear edge transition region, effectively reducing the image convolution area and saving computational time.

(2) The iterative reweighted least squares method is employed to fit the center point of the gear end face. By calculating the pitch deviation and profile deviation, the gear integrated error curve can be synthesized. From the curve, both analytical and functional measurement results of the gears can be derived. Compared with the measurement results of P26, for gears with 23 teeth and module of 0.5 mm, the maximum error is 2.2 μm for $F_p$, −1.6 μm for $f_{p\max }$, 3.6 μm for the absolute value of $F_{\alpha }$, 1.9 μm for $f_{f\alpha }$, and 3.9 μm for $f_{H\alpha }$.

(3) The information about each tooth can be measured simultaneously without affecting the overall measurement speed. The average measurement speed of the method is approximately 0.35 s/piece, which is much faster than the classic contact measurement method (20 s/piece). There is no need for clamping during measurement, enhancing measurement speed while avoiding the introduction of clamping errors and gear damage. It also allows for the rapid identification and quantitative analysis of tooth face burrs and micro modifications of each profile.

At present, the method proposed in this article is only applicable to spur gear measurement, and it requires further improvement through the use of a multi-sensor fusion measurement scheme in future research.