An algorithm for extracting the center of a hinge shaft with additional constraint conditions

Abstract

The present invention discloses an algorithm for extracting the center of a hinge shaft with additional constraint conditions. The extraction algorithm includes four steps: S1. Use a plane fitting algorithm to process the data collected in the fieldwork to obtain the equation of the fitting plane P of the hinge shaft end face and the normal vector of the fitting plane P of the hinge shaft end face; S2. Use the indirect fitting method to process the data collected in the fieldwork and the normal vector of the fitting plane P of the hinge shaft end face obtained in step S1 to obtain the coordinates of a point on the axis L of the hinge shaft; S3. Use the fitting plane P of the hinge shaft end face and the axis L to obtain the center point of the hinge shaft end face; S4. Translate the center point of the end face to obtain the center point of the hinge shaft. The present invention does not require workers to obtain the center of the hinge shaft by placing a marker point at a high altitude, and has a higher solution accuracy, effectively solving the problems that the range where the marker point is placed is difficult to completely cover the circumference of the hinge and there are errors in the placed marker points, resulting in inaccurate solution of the center of the hinge shaft and potential safety hazards in high-altitude operations.

Description

Technical Field

[0001] The present invention relates to the field of installation and geometric tolerance detection of the hinge shaft of the radial gate in water conservancy projects, and particularly relates to an algorithm for scanning and extracting the center of the hinge shaft with additional constraint conditions. Background Art

[0002] The radial gate has the advantages of labor-saving opening and closing, reliable operation, a smaller thickness of the pier, no gate slot affecting the flow pattern of the water flow, good working conditions with a large amount of sediment, a smaller opening force, and good discharge conditions, etc. Therefore, the radial gate has become a commonly used water conservancy equipment. The hinge shaft is an important component of the radial gate, which bears the water pressure and part of the gravity of the arc gate, and has a great influence on the opening force and the reliability of the gate operation. The main parameters for its installation and detection include elevation deviation, mileage deviation, inclination, hinge shaft spacing, and coaxiality, etc. How to obtain the center of the hinge shaft is the basis for detecting these parameters. As shown in the attached drawing Figure 1 Currently, the measurement of the center of the hinge shaft is mostly based on a total station in cooperation with a special tooling, and the center of the circle is obtained by manually placing points at multiple positions around the edge and fitting the circle.

[0003] The existing method has the following disadvantages: (1) Placing at a high altitude, it is difficult to completely cover the circumference of the hinge with the range of the marked points, resulting in deviation in circle fitting and affecting the accuracy of solving the center of the hinge shaft; (2) Due to the chamfer on the end face of the hinge shaft and the human error in manual placement, there is an error in the placement of the marked points, and the error is generally about 1 mm; (3) Working at a high altitude, there are great potential safety hazards for personnel.

[0004] Because in the prior art, it is difficult to completely cover the circumference of the hinge with the range of the marked points and there are errors in the placed marked points, resulting in inaccurate solution of the center of the hinge shaft and potential safety hazards in working at a high altitude, so it has strong practical value to carry out research on scanning technology to obtain the center of the hinge shaft.

[0005] Therefore, the present invention provides an algorithm for scanning and extracting the center of the hinge shaft with additional constraint conditions to solve this problem. Summary of the Invention

[0006] In view of the deficiencies of the prior art, the purpose of this application is to provide a scanning and extraction algorithm for the center of a hinge shaft with additional constraint conditions, which processes the data collected in the field using a plane fitting algorithm and an indirect fitting method. The constraint conditions combine the accurate data of the hinge shaft design drawing with the collected data, thereby ensuring the accuracy of the obtained hinge shaft center data, solving the problem that the range where the marker points are placed is difficult to completely cover the hinge circumference and there are errors in the placed marker points, resulting in deviations in the obtained hinge shaft center. At the same time, there is no need to place marker points at high altitudes, solving the problem of potential safety hazards in high-altitude operations. Thus, it effectively solves the problems in the prior art that the range where the marker points are placed is difficult to completely cover the hinge circumference and there are errors in the placed marker points, resulting in inaccurate solution of the hinge shaft center and potential safety hazards in high-altitude operations.

[0007] A scanning and extraction algorithm for the center of a hinge shaft with additional constraint conditions, the extraction algorithm includes 4 steps:

[0008] S1. Use the plane fitting algorithm to process the data collected in the field to obtain the equation of the fitting plane P of the hinge shaft end face and the normal vector of the fitting plane P of the hinge shaft end face;

[0009] S2. Use the indirect fitting method to process the data collected in the field and the normal vector of the fitting plane P of the hinge shaft end face obtained in step S1 to obtain the coordinates of a point on the hinge shaft axis L;

[0010] The indirect fitting method is to use the cylindrical surface fitting algorithm, additional constraint conditions and parametric adjustment to process the data;

[0011] S3. Use the equation of the fitting plane P of the hinge shaft end face and the equation of the axis L to obtain the center point of the hinge shaft end face;

[0012] S4. Translate the center point of the end face to obtain the center point of the hinge shaft.

[0013] The beneficial effects achieved by the present invention:

[0014] This solution processes the data collected in the field using the plane fitting algorithm and the indirect fitting method. The constraint conditions fully combine the accurate data of the hinge shaft design drawing with the collected data, and the accuracy of the obtained hinge shaft center coordinate data is higher than that obtained by placing marker points. The method in this article directly detects the hinge shaft without the need for indirect measurement through a special tooling, eliminating human errors, and the staff does not need to perform high-altitude operations, significantly improving the safety of operations, solving the problems in the prior art that the range where the marker points are placed is difficult to completely cover the hinge circumference and there are errors in the placed marker points, resulting in inaccurate solution of the hinge shaft center and potential safety hazards in high-altitude operations. Description of the Drawings

[0015] Figure 1 It is a schematic diagram of the existing measurement method for the center of the hinge shaft.

[0016] Figure 2 It is a schematic diagram of the technical route for scanning to obtain the center of the hinge shaft.

[0017] Figure 3 It is a schematic diagram for determining the scanning area.

[0018] Figure 4 It is a schematic diagram of the fitting plane of the end face of the hinge shaft.

[0019] Figure 5 It is a schematic diagram of the fitting cylindrical surface of the hinge shaft.

[0020] Figure 6 It is a schematic diagram for determining the center of the circle of the end face of the hinge shaft.

[0021] Figure 7 It is a schematic diagram of the field measurement scenario 1 of the method in this article.

[0022] Figure 8 It is a schematic diagram of the field measurement scenario 2 of the method in this article. Specific implementation manner

[0023] The following combines specific embodiments to further elaborate on this solution. The embodiments are implemented on the premise of this solution and with reference to the accompanying drawings of the specification. The structural content mentioned in the following embodiments is all with reference to the accompanying drawings of the specification.

[0024] The following will refer to the accompanying drawings and describe in detail through embodiments a hinge shaft center scanning and extraction algorithm with additional constraint conditions provided by this solution.

[0025] A hinge shaft center scanning and extraction algorithm with additional constraint conditions, the extraction algorithm includes 4 steps:

[0026] S1. Use the plane fitting algorithm to process the data collected in the field to obtain the equation of the fitting plane P of the end face of the hinge shaft and the normal vector of the fitting plane P of the end face of the hinge shaft;

[0027] S2. Use the indirect fitting method to process the data collected in the field and the normal vector of the fitting plane P of the end face of the hinge shaft obtained in step S1 to obtain the coordinates of a point on the axis L of the hinge shaft;

[0028] The indirect fitting method is to use the cylindrical surface fitting algorithm, additional constraint conditions and parametric adjustment to process the data;

[0029] S3. Use the equation of the fitting plane P of the end face of the hinge shaft and the equation of the axis L to obtain the center point of the end face of the hinge shaft;

[0030] S4. Translate the center point of the end face to obtain the center point of the support hinge shaft.

[0031] The specific application technical route of an algorithm for scanning and extracting the center of a support hinge shaft with additional constraint conditions to obtain data related to the support hinge shaft is shown in the attached figure Figure 2 ;

[0032] Field data collection: After the scanner is set up, before scanning the support hinge shaft with the scanner in the field, it is necessary to first set the scanning parameters, set the scanning area and set the scanning resolution. Since the length of the part of the general support hinge shaft exposed from the hinge seat is relatively small, ranging from 50 mm to 60 mm, therefore, increasing the scanning resolution as much as possible to obtain more information about the cylindrical surface of the hinge shaft is the key to ensuring the fitting of the cylindrical surface of the support hinge shaft. While increasing the scanning resolution, as shown in the attached figure Figure 3 shown, refining the scanning area and reducing invalid fieldwork are effective methods to reduce the scanning time and improve the fieldwork efficiency. The specific measurement scenarios are shown in the attached figure Figure 3 、 Figure 4 shown.

[0033] The above-mentioned S1: Use the plane fitting algorithm to process the data collected in the field to obtain the equation of the fitting plane P of the end face of the support hinge shaft and the normal vector of the fitting plane P of the end face of the support hinge shaft. The specific content is as follows:

[0034] After scanning and obtaining the data of the end face of the support hinge shaft, as shown in the attached figure Figure 4 shown, use the plane fitting algorithm to fit and obtain the fitting plane P of the end face of the support hinge shaft and the normal vector T of the fitting plane P of the end face of the support hinge shaft;

[0035] Attached figure Figure 4 When performing plane fitting, as shown in Equation (1), it is necessary to first use the gradient filtering algorithm to obtain the point cloud data of the end face of the support hinge shaft, and then use the general equation of the plane for fitting, AX + BY + CZ + D = 0 (1). In formula (1), (X, Y, Z) are the three-dimensional coordinates of the scanned points. After using the least squares algorithm to fit and obtain the general equation of the fitting plane P of the end face of the support hinge shaft, the values of A, B, C, and D in the general equation can be obtained, so as to obtain the expression equation of the fitting plane P of the end face of the support hinge shaft and the normal vector T (A, B, C) of the fitting plane P of the end face of the support hinge shaft.

[0036] The above-mentioned S2: Use the indirect fitting method to process the data collected in the field and the normal vector of the fitting plane P of the end face of the support hinge shaft obtained in step S1 to obtain the coordinates of a point on the axis L of the support hinge shaft. The specific content is as follows:

[0037] The cylindrical surface of the support hinge shaft is fitted using the cylindrical surface fitting algorithm. Since the axis of the support hinge shaft is parallel to the normal vector of the end face of the support hinge shaft, and the normal vector of the end face of the support hinge shaft can be obtained through the fitting plane P of the end face of the support hinge shaft, when fitting the cylindrical surface, the constraint conditions regarding the axis of the support hinge can be added. At the same time, since the machining accuracy of the support hinge shaft is higher than the scanning accuracy, when fitting the cylinder, the constraint conditions regarding the radius of the support hinge shaft can be added according to the radius of the support hinge shaft. By adding the above-mentioned constraint conditions, the fitted cylindrical surface of the support hinge shaft is as shown in Figure 5 shown;

[0038] When performing cylindrical surface fitting, it is necessary to first determine the equation of the cylindrical surface. Theoretically, the cylindrical surface can be defined as the set of points in space that are equidistant from a certain straight line. Among them, this straight line is the axis L of the support hinge shaft, and this distance is the radius of the cylindrical surface of the support hinge shaft. The specific content of the cylindrical surface fitting algorithm is as follows:

[0039] S2.1 Set the scanning area before scanning and manually crop the scanned data to quickly obtain the point cloud data of the cylindrical surface and its coordinates (X1, Y1, Z1);

[0040] S2.2 As shown in Equation (2), the axis L of the support hinge shaft is expressed in the point-direction form:

[0041]

[0042] In formula (2), (x l , y l , z l ) is the three-dimensional coordinate of a point on the axis, denoted as P l , (a, b, c) is the direction vector T1 of the axis, and t is the distance from any point on the axis to the point P l (x l , y l , z l ). In formula (2), (X1, Y1, Z1) are the three-dimensional coordinates of the points on the axis of the support hinge shaft;

[0043] S2.3 Construct the distance equation from a point to a straight line. The straight line is the axis L of the cylindrical surface of the support hinge shaft constructed in S2.2. Substituting it in, the equation of the distance d can be obtained as:

[0044]

[0045] In formula (3), (x, y, z) are the three-dimensional coordinates of the points on the cylindrical surface of the support hinge shaft, (x l , y l , z l ) are the three-dimensional coordinates of a point on the axis, and (a, b, c) is the direction vector T1 of the axis;

[0046] S2.4. Solve for the coordinates of point P on the axis L of the hinge shaft using parametric adjustment with constraint conditions. The specific process is as follows: l as follows:

[0047] Through formula (3), the error equation for the fitting of the hinge shaft cylindrical surface can be established, as shown in formula (4):

[0048] v = d - r (4)

[0049] In formula (4), v is the fitting residual in parametric adjustment, and r is the radius of the cylindrical surface;

[0050] The additional constraint conditions are:

[0051]

[0052] In formula (5), (A, B, C) is the normal vector T of the fitting plane P of the hinge shaft end face, r0 is the actual radius of the hinge shaft, and the actual radius can be obtained from the design drawing of the hinge shaft. By combining formulas (3) - (5) and substituting the points on the hinge shaft cylindrical surface, the three-dimensional coordinates (x l , y l , z l ) of point P on the axis L are obtained. The obtained fitting cylindrical surface of the hinge shaft is shown in l . Since the area of the hinge shaft cylindrical surface that can be scanned in one setup is limited without relocating the station, the included angle range between the two sides of the limited scanning area and the center line of the hinge shaft is 100 degrees to 130 degrees. Therefore, by adding constraint conditions regarding the hinge shaft axis and the hinge shaft radius, the error of the deviation of the hinge shaft center and the normal scanning area can be reduced, thereby improving the solution accuracy of point P Figure 5 . l

[0053] S3: Use the fitting plane P of the hinge shaft end face and the axis L to obtain the center point of the hinge shaft end face. The specific content is as follows:

[0054] According to the three-dimensional coordinates (x l , y l , z l ) of point P on the hinge shaft axis L obtained in S2.5 l ​) and the normal vector T(A, B, C) of the fitting plane P of the end face of the hinge shaft obtained from S1.3. The normal vector T(A, B, C) of the fitting plane P of the end face of the hinge shaft is parallel to the direction vector T1 of the axis L of the hinge shaft. Use the normal vector T of the fitting plane P of the end face of the hinge shaft to replace the direction vector T1 of the axis L of the hinge shaft. Substitute the normal vector T of the fitting plane P of the end face of the hinge shaft into formula (2) in S2.2 to obtain the specific expression of the axis L of the hinge shaft. The axis L of the hinge shaft and the fitting plane P of the end face of the hinge shaft. Through the line-plane calculation formula, the intersection coordinates of the axis L of the hinge shaft and the fitting plane of the end face of the hinge shaft can be obtained. This intersection point is the center point of the end face of the hinge shaft, and the center point of the end face of the hinge shaft is also the center O1(x j , y j , z j ) of the end face of the hinge shaft. The line-plane calculation formula is as shown in formula (6):

[0055]

[0056] In formula (6), x j , y j , z j are the coordinates of the center O1 of the end face of the hinge shaft, and t is the distance from any point on the axis to the point P l (x l , y l , z l ).

[0057] S4: Translate the center point of the end face to obtain the coordinates of the center point of the hinge shaft. The specific content is as follows:

[0058] As shown in formula (7), according to the center O1 of the end face of the hinge shaft, the axis of the hinge shaft and the length of the hinge shaft itself, translate the center O1 of the end face of the hinge shaft to obtain the center coordinates O of the hinge shaft and the center O2 of the other end face;

[0059]

[0060] In formula (7), (x j , y j , z j ) are the three-dimensional coordinates of the center O1 of the end face of the hinge shaft, m is the distance from any point to the center O1 of the end face of the hinge shaft, (A, B, C) is the normal vector T of the fitting plane of the end face of the hinge shaft, and (X2, Y2, Z2) are the coordinates obtained by translating the center O1 of the end face of the hinge shaft;

[0061] When solving for the coordinates O of the center of the support hinge shaft, let the magnitude of m in formula (7) be half of the length of the hinge shaft, that is, the distance from the center of the support hinge shaft to the center of the circle at the end face of the support hinge shaft is half of the length of the support hinge shaft itself; when solving for the coordinates of the center of the circle at the other end face of the support hinge shaft, let the magnitude of m in formula (7) be the length of the hinge shaft itself, that is, the distance between the center of the circle at the other end face of the support hinge shaft and the center of the circle at the end face of the support hinge shaft is the length of the support hinge shaft itself.

[0062] After obtaining the coordinates of the center of the support hinge shaft, based on the center of the support hinge shaft, parameters such as the radius of curvature, coaxiality, mileage deviation of the support hinge shaft, and elevation deviation can be calculated.

[0063] In summary, the present invention improves the accuracy of solving the coordinates of the center of the support hinge shaft in the prior art. Taking the Leica Nova MS60 total station scanner as an example, the scanning accuracy in the method of this article is 0.6 mm, and the single-point measurement accuracy in the existing method is 0.5 mm. At present, the machining accuracy of the support hinge shaft is much higher than that of the detection instrument, but the existing method increases the marking point placement error by 1.0 mm - 2.0 mm. In comparison, the method of this article directly detects the support hinge shaft without indirectly measuring through a special tooling, reducing the accumulation of errors and having a high accuracy guarantee for the final result. It solves the problem in the prior art that the placement range of the marking points is difficult to completely cover the circumference of the support hinge and there are errors in the placed marking points, resulting in inaccurate solution of the center of the support hinge shaft; the measurement safety factor is higher. The original method requires personnel to climb to the support hinge through the support arm, with high-altitude operation and limited safety measures. The method of this article avoids personnel working at heights and significantly improves the safety of the operation, solving the problem of potential safety hazards in high-altitude operation; the work efficiency is higher. The original operation mode requires at least 2 people, and the operation time generally requires at least 40 minutes. The method of this article only requires 1 person, and the operation time only requires about 30 minutes, with higher economic benefits; the intelligent level is significantly improved. Compared with the manual operation method of the existing method, through scanning operation, the time and process of field operation are reduced, and the focus of work is placed on the later data processing, significantly improving the intelligent level of the detection process, which has a certain promoting effect on the high-level development of the industry.

[0064] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, and are not intended to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

[0065] For the sake of convenience in explanation, the above description has been made in connection with specific embodiments. However, the above exemplary discussion is not intended to be exhaustive or to limit the embodiments to the specific forms disclosed above. According to the above teachings, various modifications and variations can be obtained. The selection and description of the above embodiments are for the purpose of better explaining the principles and practical applications, so that those skilled in the art can better use the embodiments and various different modified embodiments suitable for specific use considerations.

Claims

1. An algorithm for extracting the center of a hinge shaft with additional constraint conditions, characterized in that The extraction algorithm includes four steps: S1. Use the plane fitting algorithm to process the data collected in the fieldwork to obtain the equation of the fitting plane P of the end face of the hinge shaft and the normal vector of the fitting plane P of the end face of the hinge shaft. S2. Use the indirect fitting method to process the data collected in the fieldwork and the normal vector of the fitting plane P of the end face of the hinge shaft obtained in step S1 to obtain the coordinates of a point on the axis L of the hinge shaft. The indirect fitting method is to use the cylindrical surface fitting algorithm, additional constraint conditions and parametric adjustment to process the data. S3. Use the fitting plane P of the end face of the hinge shaft and the axis L to obtain the center point of the end face of the hinge shaft. S4. Translate the center point of the end face to obtain the center point of the hinge shaft. The specific content of step S2 is as follows: S2.

1. Set the scanning area before scanning, and manually crop the data collected in the fieldwork to obtain the point cloud information on the cylindrical surface of the hinge shaft. S2.

2. Construct the point-direction equation of the axis L of the hinge shaft: In formula (2), (x l , y l , z l ) is the three-dimensional coordinates of a point on the axis, denoted as P l . (a, b, c) is the direction vector T1 of the axis, and t is the distance from any point on the axis to point P l (x l , y l , z l ). In formula (2), (X1, Y1, Z1) are the three-dimensional coordinates of the points on the axis of the support hinge shaft; S2.

3. Construct the distance equation from a point to a line. The line is the axis L of the cylindrical surface of the hinge shaft constructed in S2.

2. Substituting it in, the equation of the distance d can be obtained as: (x, y, z) in formula (3) are the three-dimensional coordinates of the points on the cylindrical surface of the hinge axis, (x l , y l , z l ) are the three-dimensional coordinates of a certain point on the axis, and (a, b, c) is the direction vector T1 of the axis; S2.

4. Solve for the coordinates of point P on the axis L of the support hinge shaft obtained by parametric adjustment with constraint conditions. The specific process is as follows: l The coordinates of point P on the axis L of the support hinge shaft obtained by parametric adjustment with constraint conditions are solved as follows: Through formula (3), the error equation for fitting the cylindrical surface of the hinge shaft can be established, as shown in formula (4): v = d - r (4) In formula (4), v is the fitting residual in parametric adjustment, and r is the radius of the cylindrical surface. The additional constraint conditions are: In formula (5), (A, B, C) is the normal vector T of the fitting plane of the end face of the support hinge shaft, r0 is the actual radius of the support hinge shaft, and the actual radius can be obtained from the design drawing of the support hinge shaft. After combining formulas (3) to (5) and substituting the points on the cylindrical surface of the support hinge shaft, the point P on the axis L is obtained. l The three-dimensional coordinates (x l ,y l , z l ).

2. The center scanning and extraction algorithm of the hinge shaft with additional constraint conditions according to claim 1, characterized in that The specific process of field data collection is as follows: After the scanner is set up, before the field personnel use the scanner to scan the hinge shaft, they need to set the scanning area and the scanning resolution; increase the scanning resolution to obtain the data on the cylindrical surface of the hinge shaft. While increasing the scanning resolution, refine the scanning area. After the scanner finishes scanning, the data collected in the fieldwork is obtained.

3. An additional constraint-based hinge axis center scanning and extraction algorithm according to claim 1, characterized in that The specific content of step S1 is as follows: S1.

1. Use the gradient filtering algorithm to obtain the point cloud data on the end face of the hinge shaft from the data collected by scanning. S1.

2. Construct the general equation of the fitting plane P of the end face of the hinge shaft: AX + BY + CZ + D = 0 (1). In formula (1), (X, Y, Z) are the three-dimensional coordinates of the points on the end face of the hinge shaft. S1.

3. Substitute the three-dimensional coordinates of the points on the end face of the hinge shaft into formula (1), and after fitting using the least squares algorithm, the values of A, B, C, and D in the general formula can be obtained, so as to obtain the expression equation of the fitting plane P of the end face of the hinge shaft and the normal vector T(A, B, C) of the fitting plane P of the end face of the hinge shaft.

4. An additional constraint-based hinge axis center scanning and extraction algorithm according to claim 3, characterized in that The specific content of step S3 is: according to the point P on the axis L of the support hinge shaft obtained in step S2 l The three-dimensional coordinates (x l ,y l , z l ) and the normal vector T(A, B, C) of the support hinge end face fitting plane P obtained by S1.

3. The normal vector T(A, B, C) of the support hinge end face fitting plane P is parallel to the direction vector T1 of the support hinge axis L. The normal vector T of the support hinge end face fitting plane P is used to replace the direction vector T1 of the support hinge axis L. The normal vector T of the support hinge end face fitting plane P is substituted into the formula (2) in S2.2 to obtain the specific expression of the support hinge axis L. The coordinates of the intersection of the support hinge axis L and the support hinge end face fitting plane P can be obtained by the line-surface calculation formula. The intersection is the center point of the support hinge end face, which is also the center point of the circle of the support hinge end face O1(x j ,y j , z j ), the line-surface calculation formula is shown in formula (6): In formula (6), (x j , y j , z j ) are the coordinates of the center O1 of the end face of the hinge shaft, and t is the distance from any point on the axis to point P l (x l , y l , z l ).

5. An additional constraint-based hinge axis center scanning and extraction algorithm according to claim 4, characterized in that The specific content of step S4: Translating the center point of the end face to obtain the center point coordinates of the hinge shaft is as follows: As shown in formula (7), according to the center O1 of the end face of the hinge shaft, the axis of the hinge shaft and the length of the hinge shaft itself, translate the center O1 of the end face of the hinge shaft to obtain the center coordinates O of the hinge shaft and the center O2 of the other end face. In formula (7), (x j , y j , z j ) is the three-dimensional coordinate of the center O1 of the end face of the hinge shaft, m is the distance from any point to the center O1 of the end face of the hinge shaft, (A, B, C) is the normal vector T of the fitting plane of the end face of the hinge shaft, and (X2, Y2, Z2) is the coordinate obtained by translating the center O1 of the end face of the hinge shaft; When solving the coordinates O of the center of the hinge shaft, let the magnitude of m in formula (7) be half of the length of the hinge shaft, that is, the distance from the center of the hinge shaft to the center of the end face of the hinge shaft is half of the length of the hinge shaft itself; when solving the coordinates of the center O2 of the other end face of the hinge shaft, let the magnitude of m in formula (7) be the length of the hinge shaft itself, that is, the distance between the center of the other end face of the hinge shaft and the center of the end face of the hinge shaft is the length of the hinge shaft itself.

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