Research on Visual Measurement of Aperture and Hole Group Center Distance Based on Geometric Constraint

Abstract

Hole is the most common symmetrical geometric structure of mechanical parts, and aperture and hole center distance are important measured dimensions in machining. However, existing visual measurement methods often require high equipment installation accuracy and low measurement accuracy. To solve the problem of projection deformation of the measured hole: Firstly, a local coordinate system is created, and the coordinate plane of the coordinate is parallel to the end face of the measured hole. Secondly, a cone is established, the end face of the measured hole is alike to the bottom face of the cone, and the optical center of the camera is the vertex. The space equation of the cone is represented by the coordinates of the center point of the hole. Finally, a cutting plane is established near the imaging plane of the camera, and the intersection line equation between the plane and the cone is obtained, and the diameter of the measured hole and the coordinates of the center of the circle are obtained through the parameters of the intersection line equation. In order to solve the problem of errors in edge points of the hole, geometric constraints based on the center of the circle are used to improve measurement accuracy. This experiment takes the automobile insert mold as the measured part and uses the measurement model of this paper to measure the aperture and hole center distance of the mold. The measurement results show that the measurement accuracy of the aperture is 0.018 mm, the measurement accuracy of the center distance of the hole is 0.05 mm, and the maximum relative error is 2.2%.

1. Introduction

A hole and a hole group composed of multiple holes according to a certain positional relationship are some of the most basic geometric elements in mechanical structure [1,2]. Aperture and center distance are important dimensions of mechanical parts.

The existing hole diameter and hole group center distance measurement methods are mainly based on contact measurement [3,4,5]. Inner hole plug gauges or special position gauges are used to complete the measurement in mass production, and inner diameter micrometers are often used to measure holes in small batch production. For parts with high precision requirements, the three-coordinate measuring instrument is used to obtain the position size between the hole groups. Since contact measurement requires human participation and cannot achieve online detection, it is not suitable for intelligent manufacturing.

Non-contact measurement uses a variety of sensors to measure the apertures and the center distance of the hole group without contacting the holes [6,7,8,9,10,11]. The vision measurement is an important measurement method and is widely used in hole size measurement [12,13,14,15,16,17,18,19].

Gong et al. [20] introduced the methods of obtaining aperture by different measurement methods, and these methods included contact probes, inductive sensor, laser sensor, and the machine vision method. Compared with other methods, the visual method was easily affected by environment lighting. Li et al. [21] proposed a method for measuring the position and size of hole groups by using a vision system consisting of two cameras and a cross-laser. The method obtains the center line of the light stripe through the improved direction template, and the position error of the hole group can be solved by the center line of the structured light. The experimental results show that the aperture measurement accuracy of this method is 0.001 mm, and the standard deviation of measurement results is 0.001 mm. In the measurement of hole center distance, the measurement error of this method is 0.02 mm. The accuracy is high, but the equipment is relatively complex. To detect the position of the brake disc hole group, Guo et al. [22] designed a measurement scheme combining a CCD image processing system and a contact displacement sensor. A CCD image processing system is used to obtain the diameter of each mounting hole and the coordinates of its center through an edge extraction algorithm and least square fitting. The difference between the diameter of the measured hole by this method and a coordinate measuring machine is 0.012 mm. For the detection of the center hole, a contact displacement sensor is used, and the measurement error of the sensor is 0.0005 mm. This method adopts a combination of a contact displacement sensor and a CCD and has high precision, but it is a compromised method for solving the insufficient precision of non-contact measurement. Jin et al. [23] proposed a method for obtaining hole diameter by a visual system which consists of dual interferometers and cameras. The measured hole dimension could be obtained by the reconstructed three-dimensional shapes. This paper uses the thickness of standard gauge blocks and the diameters of holes as the measured objects. In the measurement of gauge block thickness, the measurement accuracy is 0.04 mm. In the measurement of hole diameter, the measurement accuracy is 0.1 mm. The vision system of this method is complex and expensive, and the method is not suitable for the processing site. There are many similarities between hole diameter measurement and shaft diameter measurement based on machine vision. Both the hole diameter and the shaft diameter are obtained by ellipse fitting. Tan et al. [24] proposed a shaft diameter measurement algorithm based on linear structured light vision, and the accuracy of the algorithm was verified in static and dynamic experiments, respectively. The experimental results show that the accuracy of this algorithm is 0.019 mm. In this method, the error points in the data points were not removed, and with all data points involved in ellipse fitting, the fitting accuracy might be affected.

There are two common problems in the existing measurement of aperture and center distance based on vision algorithms: First, the existing methods have higher requirements on the position between the camera and the surface of the measured hole in high-precision measurement. When the imaging plane of the camera is substantially parallel to the plane of the measured hole, measurement accuracy can be ensured. The installation condition reduces the applicability of the measurement method. Second, the aperture and hole center distance must be obtained by using the coordinates of the hole edge points through ellipse fitting. However, the coordinates of these points obtained by the visual system generally have errors, and high measurement accuracy cannot be obtained by simple algebraic fitting or geometric fitting.

To solve these two problems, this paper proposes a visual measurement method based on the geometric constraints of the hole center for aperture and center distance. First, a local coordinate system on the surface of the measured hole is established, and the rotation matrix of the local coordinate system and the camera coordinate system can be solved by the calibration plate. Second, the expressions of the circle and conical surface corresponding to the measured hole are established in the world coordinate system, and the optical center is a conic node. Thirdly, a projection plane parallel to the surface of the measured hole is established near the camera image plane, and points on the edge of the hole onto the image plane are projected to the projection plane. The projection is the intersection circle of the projection plane truncating the conical surface. Finally, the coordinates of the center and radius of the circle are determined by the least square fitting of the intersecting circle. The optimization objective function is established according to the geometric characteristics of the circle, and these fitting results are used as initial values for optimization. The final measurement result is obtained by a nonlinear optimization algorithm.

2. Establishment of Local Coordinate System and Solution of Coordinate Transformation Matrix

According to the camera projection law, the contour formed by the edge points of the measured hole is an ellipse on the imaging plane. To facilitate the solution of aperture and center point coordinates, a local coordinate system (O_E-X_EY_EZ_E) is established in this paper. The coordinate plane (X_EO_EY_E) of this coordinate system is parallel to the surface of the measured hole group, and the direction vector of O_EZ_E is the direction vector of the hole group surface. The origin of the coordinate system coincides with the origin of the camera coordinate system. In the vision system, O_C-X_CY_CZ_C is the camera coordinate system, the origin of the coordinate system is the optical center of the camera, and the coordinate plane (X_CO_CY_C) is parallel to the imaging plane of the camera. The relationship between the coordinate systems in the vision model is shown in Figure 1.

O_C-X_CY_CZ_C is the camera coordinate system; O_E-X_EY_EZ_E is the local coordinate system.

In the camera coordinate system, the plane equation of the measured hole surface is:

$$A_1{X}_C+B_1{Y}_C+C_1{Z}_C+1=0$$

(1)

The direction vector of O_EZ_E in local coordinates can be obtained by the surface expression of the hole group, and the direction vector is (A₁, B₁, C₁). As shown in Figure 1, to obtain the expression of the surface of the hole group, this paper attaches the lower surface of the checkerboard calibration plate to the surface of the measured hole to ensure that the surface of the calibration plate is parallel to the surface of the hole. According to the two-step calibration method, the camera coordinates of each corner on the calibration plate can be obtained, the coordinates of the corners are substituted into Equation (1), and the direction vector of the calibration plate can be solved by least squares fitting. Since the surface of the calibration plate is parallel to the surface of the measured hole, the direction vector of the calibration plate is the direction vector of O_EZ_E. According to the direction vector of O_EZ_E, the direction cosine of the local coordinate axis in the camera coordinate system can be obtained:

$$e_{13}=\frac{A_1}{\sqrt{A_1^2+B_1^2+C_1^2}},e_{23}=\frac{B_1}{\sqrt{A_1^2+B_1^2+C_1^2}},e_{33}=\frac{C_1}{\sqrt{A_1^2+B_1^2+C_1^2}}$$

(2)

As shown in Figure 1, the straight line L is the connection line between any two corner points on the calibration board, and the coordinates of the corner points are (X_C1, Y_C1, Z_C1), (X_C2, Y_C2, Z_C2). According to the geometric relationship, the straight line L is parallel to the local coordinate system of O_EX_E, and the direction vector of the O_EX_E in the camera coordinate system is (n_x, n_y, n_z), and the vector could be Equations (3) and (4).

$$n_x=\frac{X_{c2}-X_{c1}}{L_{1-2}},n_y=\frac{Y_{c2}-Y_{c1}}{L_{1-2}},n_z=\frac{Z_{c2}-Z_{c1}}{L_{1-2}}$$

(3)

$$L_{1-2}=\sqrt{(X_{c2}-X_{c1}{)}^2+(Y_{c2}-Y_{c1}{)}^2+(Z_{c2}-Z_{c1}{)}^2}$$

(4)

Through the direction vector of the straight line L, the direction cosine of the O_EX_E axis in the camera coordinate system can be obtained:

$$e_{11}=\frac{n_x}{\sqrt{n_x^2+n_y^2+n_z^2}},e_{21}=\frac{n_y}{\sqrt{n_x^2+n_y^2+n_z^2}},e_{31}=\frac{n_z}{\sqrt{n_x^2+n_y^2+n_z^2}}$$

(5)

According to the coordinate axis relationship of the spatial rectangular coordinate system, the direction cosine of the O_EY_E can be solved by the direction cosine of the O_EX_E and the O_EZ_E camera coordinate system, and this is shown in Equations (6) and (7).

$$\mathit{J}=\mathit{K}\times \mathit{I}=\left|\begin{array}{ccc}i& j& k\\ e_{13}& e_{23}& e_{33}\\ e_{11}& e_{21}& e_{31}\end{array}\right|=e_{y1}i+e_{y2}j+e_{y3}k$$

(6)

$$e_{y1}=(e_{23}{e}_{31}-e_{33}{e}_{21}),e_{y2}=(e_{33}{e}_{11}-e_{13}{e}_{31}),e_{y3}=(e_{13}{e}_{21}-e_{23}{e}_{11})$$

(7)

The direction cosine of the O_EY_E axis is:

$$e_{12}=\frac{e_{y1}}{\sqrt{e_{y1}^2+e_{y2}^2+e_{y3}^2}},e_{22}=\frac{e_{y2}}{\sqrt{e_{y1}^2+e_{y2}^2+e_{y3}^2}},e_{32}=\frac{e_{y3}}{\sqrt{e_{y1}^2+e_{y2}^2+e_{y3}^2}}$$

(8)

The transformation relationship between the local coordinate system and the camera coordinate system is obtained from Equations (2), (5) and (8).

$$\left[\begin{array}{c}{X}_C\\ Y_C\\ Z_C\end{array}\right]=\left[\begin{array}{ccc}{e}_{11}& e_{12}& e_{13}\\ e_{21}& e_{22}& e_{23}\\ e_{31}& e_{32}& e_{33}\end{array}\right]\left[\begin{array}{c}{X}_E\\ Y_E\\ Z_E\end{array}\right]+\mathit{T}$$

(9)

In Equation (9), [X_C, Y_C, Z_C] is the camera coordinates, and [X_E, Y_E, Z_E] is the coordinates in the local coordinate system. According to the method of establishing the local coordinate system, the origin of the local coordinate system coincides with the origin of the camera coordinate system, so the translation matrix T is a zero matrix.

3. A Visual Measurement Model of Hole Group Size Based on Circle Center Constraints

The spatial moment edge extraction algorithm has a simple calculation process and a high detection speed [25], so the paper uses the Zernike moment algorithm to obtain pixel coordinates of the edge of the measured hole [26]. In Figure 1, P is any edge point of the hole to be measured, P’ is the projection point of P on the camera image plane. The camera calibration of P’ can be solved by the internal parameters and distortion coefficients, and the camera calibration of the point is (x_ui, y_ui, 1). The local coordinates of P’ can be obtained by Equation (10), and the local coordinate of P’ is [x_Ei, y_Ei, z_Ei].

$$\left[\begin{array}{c}{x}_{Ei}\\ y_{Ei}\\ z_{Ei}\end{array}\right]={\left[\begin{array}{ccc}{e}_{11}& e_{12}& e_{13}\\ e_{21}& e_{22}& e_{23}\\ e_{31}& e_{32}& e_{33}\end{array}\right]}^{-1}\left[\begin{array}{c}{x}_{ui}\\ y_{ui}\\ 1\end{array}\right]$$

(10)

As shown in Figure 1, the surface of the measured hole, the surface of the checkerboard calibration plate, and the coordinate plane (O_EX_EY_E) are parallel to each other according to establishment process of the local coordinate system.

$$Z_B=e_{13}{X}_{C1}+e_{23}{Y}_{C1}+e_{33}{Z}_{C1}$$

(11)

In Equation (11), Z_B is the Z-axis coordinate corresponding to the surface of the calibration plate. The Z-axis coordinate corresponding to the surface of the measured hole (Z₀) is solved by Equation (12), and D is the thickness of the checkerboard calibration plate.

$$Z_0=Z_B+D$$

(12)

In local coordinates, the coordinates of the center point in any group of holes measured are (X₀, Y₀, Z₀), and the radius is r. The hole edge equation is:

$$\left\{\begin{array}{l}(X_E-X_0{)}^2+(Y_E-Y_0{)}^2=r_{}^2\\ Z_E=Z_0\end{array}\right.$$

(13)

As shown in Figure 2, a cone is constructed with the measured hole as the bottom surface and the camera optical center as the cone top. In the local coordinate system, the cone surface equation can be expressed as:

$$(X_E-\frac{X_0}{Z_0}{Z}_E{)}^2+(Y_E-\frac{Y_0}{Z_0}{Z}_E{)}^2=(\frac{Z_E}{Z_0}{r}_{}{)}^2$$

(14)

A cutting plane π is established near the image plane, which is parallel to the surface of the measured hole (i.e., the bottom surface of the cone), and the Z-axis coordinate corresponding to the cutting plane π is as Z₁. According to the geometric relationship, the intersection line of the plane and the cone is also a circle, and the equation of the intersection line corresponding to any hole in the hole group can be obtained according to Equation (13).

$$(X_E-\frac{X_0}{Z_0}{Z}_1{)}^2+(Y_E-\frac{Y_0}{Z_0}{Z}_1{)}^2=(\frac{Z_1}{Z_0}{r}_{}{)}^2$$

(15)

As shown in Figure 2, the edge point of the measured hole on the image plane is projected to the cutting plane π, and the coordinates of the edge points are [x_Ei, y_Ei, z_Ei] in the local coordinate system. The equation of the projected ray L can be obtained according to the coordinates of the projected point on the image plane and the direction vector of the O_EX_E axis.

$$\frac{(X_E-x_{Ei})}{A_1}=\frac{(Y_E-y_{Ei})}{B_1}=\frac{(Z_E-z_{Ei})}{C_1}$$

(16)

Through Equation (16) and the equation of the cutting plane π, the coordinates of the projection point, which is the edge point of the measured hole on the image plane, could be obtained, and the coordinates of the projection point are set as (x_i, y_i, z_i).

$$\left\{\begin{array}{c}\frac{(X_E-x_{Ei})}{A_1}=\frac{(Y_E-y_{Ei})}{B_1}=\frac{(Z_E-z_{Ei})}{C_1}\\ Z=Z_1\end{array}\right.$$

(17)

According to the camera projection principle and the geometric relationship shown in Figure 2, all projection points on the cutting plane are theoretically on the circle represented by Equation (14). Equation (14) is rewritten as a general expression for a circle and is shown in Equation (18)

$$x^2+y^2+A_{2}x+B_{2}y+C_2=0$$

(18)

The coefficients A₂, B₂, and C₂ of the general equation of the circle are:

$$\left\{\begin{array}{c}{A}_2=-2X_0\frac{Z_1}{Z_0}\\ B_2=-2Y_0\frac{Z_1}{Z_0}\\ C_2=(X_0\frac{Z_1}{Z_0}{)}^2+(Y_0\frac{Z_1}{Z_0}{)}^2-(r_{}\frac{Z_1}{Z_0}{)}^2\end{array}\right.$$

(19)

All projection points corresponding to the measured hole could be substituted into Equation (18), and the coordinates of the center and radius r_i could be solved by least squares fitting. However, there must be errors in the process of using the above method to obtain the measured results. The main reasons for the errors are as follows.

First, because there are errors in both the edge detection algorithm and the camera calibration, the three-dimensional coordinates of the hole edge points are inaccurate, and the hole edge points are the fitting points. However, the error points in the data cannot be eliminated in least squares fitting, which causes measurement error.

Second, it is difficult to ensure that the surface of the hole is parallel to the imaging plane of the camera. The edges of the holes on two different surfaces are captured in the image simultaneously. It leads to only the edge points of a part of the hole participating in the calculation as fitting points, and these points can’t represent the complete measured hole information. As shown in Figure 3, the edge of a hole in the mold is detected by the edge detection algorithm. The green point is the edge point of the front surface of the tested hole, which is the data point that can be used for fitting. The red point contains the edge points of the front and rear hole surfaces, and the two types of points are difficult to distinguish through the algorithm, and the edge points on the rear surface of the measured hole cannot participate in the circle fitting calculation. Therefore, the points participating in the fitting can only express part of the geometric information of the measured hole, resulting in the fitting error.

To solve this problem, the paper proposes an optimization model of hole group measurement based on center constraint, which optimizes the center point coordinates and aperture obtained by the least squares fitting algorithm to improve the measurement accuracy. As shown in Figure 4, let LL be a chord in the circle and point F be the midpoint of the chord. According to the characteristics of the circle, OF is perpendicular to the chord LL, that is $OF\cdot QQ=0$. According to this geometric relationship, the objective function of optimization can be established:

$$F(x_0,y_0)=\mathit{min}\sum _{i=1}^{N/2}[(x_0-\frac{x_i+x_{N/2+i}}{2})(x_i-x_{N/2+i})+{(y}_0-\frac{y_i+y_{N/2+i}}{2})(y_i-y_{N/2+i})]$$

(20)

In the equation, N is the number of participating points. When the number of fitting points is odd, remove the last fitting point and change the number of fitting points to an even number. (x_i, y_i) are the coordinates of the projection point. (x₀, y₀) are the coordinates of circle center. The objective function is optimized by the Levenberg–Marquardt algorithm, and the initial value of the optimization is obtained by the least squares fitting method. The diameter of the measured hole can be obtained by substituting the optimized coordinates of the center into Equation (21).

$$d_{}=2\times \frac{Z_0}{Z_1}\times \frac{{\sum }_{i=1}^N\sqrt{(x_i-X_0{)}^2+(y_i-{(Y}_0{)}^2}}{N}$$

(21)

Among them, Z₀ is the Z-axis local coordinate corresponding to the surface of the measured hole, and Z₁ is the Z-axis local coordinate corresponding to the section plane. x_i and y_i are the local coordinates of the fitting point on the measured hole. (X₀, Y₀) is the optimized coordinates of circle center by Equation (20). d is the diameter of the measured hole.

4. Results of Hole Group Measurement Experiment and Result Analysis Based on Vision System

In the experiment, the insert mold with higher requirements on the center distance of the hole group was used as the measured object, and the measured object is shown in Figure 5. The paper adopts a single camera vision system, the hardware parameters of the system are shown in

Table 1. Device hardware parameters.

Device Name	Device Model	The Main Parameters
Camera	MER-125-30UM	Resolution: 1292 × 964 pixel
Lens	Computar M2514-MP	Focal length: 25 mm
Light source	CCS LFL-200	Light-emitting area: 200 × 180 mm
Checkerboard Calibration Board	NANO 25 mm-2.0	Precision: ±1.0 μm

, the camera calibration experiment site is shown in Figure 6, and the measurement site is shown in Figure 7.

In this experiment, the camera internal parameters and distortion coefficients were obtained by the two-step calibration method [27], and the calibration results are shown in

Table 2. Camera internal parameters and distortion coefficient.

α	β	γ	u0	v0	k1	k2	p1	p2
6788.80	6790.21	−2.178	576.937	420.698	0.098	0.677	−0.002	−0.002

. Based on the second part of the paper, the transformation matrix of the local coordinate system, and the camera coordinate system, the normal vector of the hole surface was obtained. To ensure the measurement accuracy of the vision system, the thickness of the calibration plate was obtained with a micrometer, and the measurement results are shown in

Table 3. Normal vector transformation matrix and calibration plate thickness corresponding to the surface of the hole group.

	Hole Surface Normal Vector	Coordinate System Transformation Matrix
Insert mold 1	$\left[0.0233,0.0229,-1.3347\right]$	$\left[\begin{array}{ccc}-0.3433& -0.9391& 0.0175\\ -0.9390& -0.3436& 0.0171\\ -0.0221& -0.0105& -0.9997\end{array}\right]$
Insert mold 1	$\left[-0.016,0.466,-1.3206\right]$	$\left[\begin{array}{ccc}-0.3439& -0.9389& -0.0121\\ -0.9386& 0.3433& 0.0353\\ -0.0289& 0.0235& -0.9993\end{array}\right]$
Calibration plate thickness	2.003 mm

.

To verify the accuracy of the hole center distance and the hole diameter measurement model proposed in this paper, the distance between the center of each hole and the hole diameter was obtained by a three-coordinate measuring instrument. The coordinate measuring machine used in this paper is made by ZEISS company, and the model is Spectrum ll 7/10/6. The measurement accuracy of the device is 2 μm. The measurement results are shown in

Table 4. Measured size value of the insert mold (three-coordinate measurement value)/mm.

	The Diameters of Holes
	d1	d2	d3	d4
Mold 1	20.05	20.058	7.013	7.019
Mold 2	20.055	20.063	7.019	7.005
	Center distances of holes
	D1	D2	D3	D4	D5
Mold 1	100.056	40.025	60.040	20.026	20.016
Mold 2	100.048	40.038	60.025	20.028	20.028

, and the dimensions codes in this table are shown in Figure 8.

Using the edge detection algorithm of Zernike moment calculation, the pixel coordinates of the fitting data points on the insert molds 1 and 2 were obtained, and the detection results of the edge points are shown in Figure 9. Using the method proposed in this paper, all coordinates of centers and apertures could first be solved by least squares fitting, the final measurement results were obtained by the optimization method based on the center constraint, and the measurement results are shown in

Table 5. Measured size value on the insert mold (three-coordinate measurement value)/m.

	d1	d2	d3	d4
Mold 1	20.039	20.048	6.995	7.002
Mold 2	20.041	20.058	7.005	6.990
	D1	D2	D3	D4	D5
Mold 1	99.988	39.982	59.984	19.983	19.977
Mold 2	99.989	39.98	59.974	19.983	19.993

.

The measurement results on the paper were compared with the measurement results obtained by the three-coordinate measuring instrument. Measurement errors for each dimension are shown in

Table 6. Aperture measurement errors based on vision measurement model/mm.

	d1	d2	d3	d4	Mean
Mold 1	0.011	0.01	0.018	0.017	0.014
Mold 2	0.014	0.005	0.014	0.015	0.012

and

Table 7. Comparison results of visual measurement model and three-coordinate measurement/mm.

	D1	D2	D3	D4	D5	Mean
Mold 1	0.068	0.043	0.056	0.043	0.039	0.05
Mold 2	0.059	0.058	0.051	0.045	0.035	0.05

.

According to the measurement comparison results, the average error of the diameter measurement is 0.013 mm, the maximum error is less than 0.018 mm, and the variance of the aperture measurement error is 1.77 × 10⁻⁵. To further analyze the experimental results, the relationship between the hole diameter and the corresponding measurement error is plotted, as shown in Figure 10 and Figure 11. The aperture measurement error corresponding to smaller diameter holes is higher than the measurement error corresponding to larger diameter holes. The reason for this trend is that a smaller diameter hole contains fewer data points. The error in the data points has a greater impact on the accuracy of the aperture measurement.

In the center distance measurement experiment, the error of each hole center distance is shown in Table 7. The maximum error is 68 μm, and the average error is 50 μm. According to the experimental results, the larger the hole center distance between the two holes, the greater the measurement error of the hole center distance. Therefore, the distance between the two holes is an important factor affecting measurement accuracy. To better analyze the measurement accuracy of the algorithm proposed in the paper, the relative error is used to evaluate the accuracy of the measurement algorithm, and the relative error of the hole center distance is shown in

Table 8. Relative errors of the hole center distance.

	D1	D2	D3	D4	D5	Mean
Mold 1	0.7%	1.1%	0.9%	2.2%	2.0%	1.4%
Mold 2	0.6%	1.5%	0.9%	2.3%	1.8%	1.4%

. Based on the relative error measurement results, the maximum relative error of the hole center distance is 2.2%, and the mean value of the error is 1.4%, the variance of the measurement error distribution is 3.91 × 10⁻⁹.

According to the measurement comparison results, the average error of the diameter measurement is 0.013 mm, and the maximum error is less than 0.018 mm. In the center distance measurement experiment, the average measurement error of this paper relative to the three-coordinate measuring instrument is 0.05 mm, the maximum error was less than 0.07 mm, and the maximum relative error is 2.2%.

5. Summary

The paper proposes a method for measuring aperture and hole group center distance based on machine vision. A local coordinate system is established, and the transformation matrix between the coordinate system and the camera coordinate system can be obtained through a checkerboard calibration board. Then a cutting plane is established near the imaging plane of the camera, and the intersection line equation between the plane and the cone is obtained. The cone is composed of the end face of the measured hole and the apex of the cone. Through spatial geometric relationships, the diameter and center point of the measured hole can be obtained by the intersection arc on the cutting plane. In order to further improve measurement accuracy, this paper adopts a measurement optimization model based on center constraint. In this experiment, the automobile insert mold is used as the measurement object, the aperture measurement accuracy is 0.013 mm, and the maximum measurement error is 0.018 mm. In the hole center distance measurement, the evaluation error of the hole center distance measurement is 0.05 mm, the average relative error of the hole center distance is 1.4%, and the maximum relative measurement error is 2.2%.