Method for measuring form and position error inside a rotating body part

Abstract

This invention relates to a method for measuring the internal form and position errors of rotating parts, belonging to the field of image processing technology. The method includes: inputting a three-dimensional point cloud model and calculating the normal vectors of the point cloud; extracting a set of points at the same latitude based on the geometric characteristics of the rotating part's point cloud; performing plane fitting on the set of points at the same latitude to obtain the initial value of the rotating part's axis; constructing an objective function based on the skew distances from the normal vectors on the rotating part's point cloud to the axis, and solving for the precise values ​​of the axis parameters; segmenting the point cloud based on the angle between the normal vectors of neighboring points and the normal vector of the seed point, and the curvature of the neighboring points; obtaining the initial values ​​of the geometric model's parameters based on the geometric dimensional relationships of cylinders and cones; setting the objective functions for cylinders and cones, and solving for the precise parameters of the geometric model; and calculating cylindricity, coaxiality, and perpendicularity according to the definition of form and position errors. This invention can accurately calculate the form and position errors of rotating parts, achieving higher accuracy in point cloud segmentation, surface reconstruction, and error calculation.

Description

Technical Field

[0001] This invention belongs to the field of image processing technology and relates to a method for measuring the internal form and position errors of rotating parts. Background Technology

[0002] With the continuous development of modern manufacturing technology, the structure of mechanical parts is becoming increasingly complex, and traditional measurement techniques can no longer meet the demands for high-precision measurement of precision parts. To address this issue, numerous emerging measurement technologies have emerged, such as machine vision measurement and laser scanning measurement. Among these, Industrial Computed Tomography (ICT) technology, with its ability to non-destructively inspect the internal morphology of enclosed cavities, has become one of the best non-destructive measurement methods, playing an indispensable role in the field of precision parts dimensional measurement. As the importance of precision parts in high-end manufacturing becomes increasingly prominent, achieving high-precision measurement has become the foundation for the manufacturing of various precision instruments and equipment, and a goal pursued by all industries. Therefore, researching high-precision measurement methods for precision parts based on industrial CT technology is extremely necessary.

[0003] Rotating parts are a common type of mechanical component, typically possessing complex structures such as gears, bearings, and transmission devices. They are widely used in various industries, including automotive manufacturing, aerospace, and machinery manufacturing. Due to their complex structure and demanding operating environments, such as high-speed rotation, high temperature, and high pressure, these parts generally require high precision and excellent dynamic performance to ensure stability. Therefore, error detection of rotating parts is crucial for guaranteeing their performance. However, due to the complexity of their structures, traditional measurement methods often cannot comprehensively and accurately obtain the dimensional and positional information of each component. This is especially true for enclosed cavities or concealed surfaces, where accurate measurement using traditional methods is even more challenging.

[0004] Point cloud segmentation is a crucial step in point cloud data processing, dividing the point cloud into blocks based on the geometric features, spatial structure, and surface characteristics of an object's surface. Region growing (RG) is a classic point cloud segmentation method, initially used for segmenting similar regions in 2D images, and also performs well in 3D point cloud segmentation. Region growing segmentation algorithms start with a selected initial seed point and continuously merge surrounding point clouds according to a given growth criterion to expand the segmentation region, thus achieving point cloud segmentation. In this type of algorithm, the selection of the seed point and the growth criterion are two key factors affecting the segmentation quality. The initial seed point is generally selected in the flattest region of the point cloud to avoid proximity to boundaries or edges, i.e., at the point of minimum curvature or quasi-planar residual value. The growth criterion judges the similarity between adjacent points, merging more similar points into the current region. Traditional region growing methods perform poorly in segmenting regions with indistinct surface transitions, often failing to accurately segment specific geometric shapes, such as smooth transition regions between cylinders and cones, and rounded corners of parts, resulting in low segmentation accuracy. At the same time, since the normal vector of this type of curved point cloud has a certain radial deflection, if the segmentation is still simply based on the normal vector, it will lead to incorrect segmentation, thus affecting the calculation of error.

[0005] For geometric model reconstruction, the least squares method or the Random Sample Consensus (RANSAC) method is generally used. RANSAC is robust and efficient in reconstructing geometric models, but its solution process involves randomness, requiring caution in high-precision scenarios. The least squares method is a classic approach. It obtains the model from prior knowledge and then iteratively solves the problem based on the linear relationship between observed data and unknown parameters. The solution obtained after iteration is the model parameter. For fitting complex models, the least squares method often gets stuck in local minima; therefore, appropriate initial values ​​are needed to ensure stable convergence to the global minimum.

[0006] In summary, the main challenges in measuring rotating parts lie in the uncertainty of the transition surfaces during point cloud segmentation and the accuracy of the initial values ​​for surface reconstruction. Therefore, this invention introduces axis calculation and the Rodrigues formula at the point cloud segmentation level to optimize traditional region growing methods and achieve accurate segmentation of rotating point clouds. At the surface reconstruction level, for common rotating surfaces such as cylinders and cones, after setting a reasonable objective function, the linear least squares method is used to solve for accurate initial iterative values, followed by the nonlinear least squares method Levenberg-Marquardt (LM) iterative solution for accurate parameters. Finally, the form and position errors in the 3D point cloud are defined, and the form and position errors of the point cloud are solved. Summary of the Invention

[0007] In view of this, the purpose of the present invention is to provide a method for measuring the internal form and position error of rotating parts, so as to meet the design conformity inspection requirements of rotating parts.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] A method for measuring the internal form and position errors of a rotating part, the method comprising the following steps:

[0010] S1: Input a 3D point cloud model and calculate the normal vector of the point cloud;

[0011] S2: Extract the point set at the same latitude based on the geometric characteristics of the point cloud of the body of revolution;

[0012] S3: Perform plane fitting using a set of points at the same latitude to obtain the initial value of the axis of revolution;

[0013] S4: Construct an objective function based on the skew distance from the normal vector on the point cloud of the body of revolution to the axis, substitute the initial value of the axis into it, and obtain the precise value of the axis parameter by nonlinear least squares solution;

[0014] S5: Rotate the normal vector of the neighboring point cloud to the seed point using the Rodrigues formula, and segment the point cloud according to the angle between the normal vector of the neighboring point and the normal vector of the seed point and the curvature of the neighboring point.

[0015] S6: Based on the geometric dimensional relationship between the cylinder and the cone, the initial values ​​of the geometric model parameters are obtained by solving using the least squares method;

[0016] S7: Set the objective functions for the cylinder and cone, substitute the initial values ​​of the geometric model obtained in S6, and obtain the precise parameters of the geometric model by nonlinear least squares solution;

[0017] S8: Calculate cylindricity, coaxiality, and perpendicularity according to the definition of form and position error;

[0018] In S2, the characteristic of the body of revolution is that two different points at the same latitude on the point cloud are equidistant from the intersection of the corresponding two normal vectors.

[0019] Furthermore, in S5, the Rodrigues formula for the 3D point cloud is:

[0020]

[0021] Where, vector v s For the target vector, the seed point p is actually... s Its projection point p on the axis vector s Let ' be the vector formed by two points, and let any point in the neighborhood set be ''. Rotation vector for The normal vector formed by v and its projection onto the axis is θ.s and The angle in the counterclockwise direction.

[0022] Furthermore, in step S6, the parameters to be solved for the cylinder are axis parameters, which are any points on the axis. and direction vector Radius r.

[0023] Furthermore, in S6, the geometric dimensional relationship of the cylinder is as follows: on the point cloud of the cylinder, there always exists any point... normal vector Vector perpendicular to the axis Establish the geometric distance expression and solve for the initial values.

[0024]

[0025] Furthermore, in S6, the geometric dimensions of the cylinder are as follows: the point cloud is divided into two point cloud clusters with q points each, and any point on the two point cloud clusters is p. j and p k The corresponding normal vector is n j n k Then the vector In n j n k Given that the projected lengths are equal, establish a geometric distance expression to solve for a point on the axis.

[0026]

[0027] Furthermore, in S6, the geometric dimensional relationship of the cylinder is: the cylinder radius r, i.e. Distance to the axis.

[0028] Furthermore, in step S6, the parameters required to solve for the cone are the axis parameters, which are any points on the axis. and direction vector Half-vertices α.

[0029] Furthermore, in S6, the geometric dimensional relationship of the cone is as follows: on the surface of the cone, there always exists any point... normal vector Perpendicular to With the vertex of the cone The vector m is used to establish a geometric distance expression for solving.

[0030]

[0031] Furthermore, in S6, the geometric dimensional relationship of the cone is as follows: calculate p0b Based on the distances to various points in the point cloud, the original point cloud is divided into multiple point cloud clusters. The point cloud cluster with the farthest distance is extracted, and a plane is fitted using this point cloud cluster to solve for the normal vector, which is the direction vector of the cone's axis.

[0032] Furthermore, in S6, the geometric dimensional relationship of the cylinder is: the geometric dimensional relationship of the cone is: α is the relationship between m and... The included angle.

[0033] The beneficial effects of this invention are as follows:

[0034] This invention optimizes the region growing method by solving for the axis of revolution and applying the Rodrigues formula to 3D point clouds, thereby achieving accurate segmentation of the point clouds of revolution and effectively improving measurement accuracy. Then, during surface reconstruction, a more stable LM fitting method is used to solve for the initial values ​​of the cylindrical and conical parameters. Unlike the RANSAC method, the LM method can effectively prevent randomness in the algorithm and avoid getting trapped in local minima during least squares solutions. Finally, the form and position errors are calculated according to the definition, thereby improving the accuracy of error measurement.

[0035] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0036] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0037] Figure 1 This is a flowchart of the present invention;

[0038] Figure 2 The distance between the normal vector and the axis is skew.

[0039] Figure 3 This is a schematic diagram of cylindricity error;

[0040] Figure 4 This is a schematic diagram of coaxiality error;

[0041] Figure 5 This is a schematic diagram of verticality error;

[0042] Figure 6 The point cloud segmentation results for each method;

[0043] Figure 7 The result of point cloud model segmentation for fuel nozzle A;

[0044] Figure 8 The result of segmenting the point cloud model of fuel nozzle B. Detailed Implementation

[0045] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0046] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0047] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0048] like Figure 1 As shown, a method for measuring the internal form and position error of a rotating part includes the following specific steps:

[0049] 1) Input a 3D point cloud model and calculate the normal vector of the point cloud;

[0050] 2) Extract the point set at the same latitude based on the geometric characteristics of the point cloud of the body of revolution;

[0051] 3) Perform plane fitting on the set of points at the same latitude to obtain the initial value of the axis of revolution;

[0052] 4) Construct an objective function based on the skew distance from the normal vector on the point cloud of the solid of revolution to the axis. Substitute the initial value of the axis obtained in step 3) into the function, and the precise value of the axis parameter can be obtained by nonlinear least squares. The skew distance from the normal vector to the axis is as follows: Figure 2 As shown, its expression is as follows:

[0053]

[0054] Where, n p Let d be the normal vector of any point in the point cloud. p The normal vector n of the point cloud of the volume of revolution p With axis n r Distance between skew planes, n op For o to point to p r The vector, vector n c Simultaneously perpendicular to n p With n r Therefore, n c is n p With n r The outer product, i.e., n c =n p ×n r .

[0055] 5) Based on the axis parameters obtained in step 4), rotate the normal vector of the neighboring point cloud to the seed point using the Rodrigues formula, and segment the point cloud according to the angle between the normal vector of the neighboring point and the normal vector of the seed point and the curvature of the neighboring point. The specific segmentation steps are as follows:

[0056] 5-1) Calculate the curvature of each point cloud and arrange the curvatures in ascending order to establish a point cloud sequence;

[0057] 5-2) If the point cloud sequence is empty, the process ends; otherwise, the point with the smallest curvature in the point cloud sequence is added to the seed point sequence.

[0058] 5-3) If the seed point sequence is empty, proceed to step 5-2); otherwise, proceed to step 5-4.

[0059] 5-4) Search the neighborhood of the seed point, calculate the rotation and translation matrix from the neighboring point to the seed point according to the Rodrigues formula, and calculate the target normal vector corresponding to the normal vector of the neighboring point.

[0060] 5-5) Project the target normal vector and the seed point normal vector onto the same plane, and calculate the angle between the target normal vector and the seed point normal vector;

[0061] 5-6) If the included angle is less than the normal included angle threshold, classify the point cloud into the same point cloud cluster and go to step 5-6); otherwise go to step 5-3.

[0062] 5-7) If the curvature of a neighboring point is less than the curvature threshold, then classify the neighboring point as a seed point and proceed to step 5-3.

[0063] 6) Based on the geometric dimensional relationships of the cylinder and cone, the initial values ​​of the geometric model parameters are obtained by solving using the least squares method;

[0064] 7) Set the objective functions for the cylinder and cone, substitute them with the initial values ​​of the geometric model obtained in step 6), and use nonlinear least squares to solve for the precise parameters of the geometric model;

[0065] 8) Based on the definition of geometrical errors, calculate cylindricity, coaxiality, and perpendicularity. The solution steps are as follows:

[0066] 8-1) Cylindricity error such as Figure 3 As shown, the calculation process is as follows: First, let the point cloud set P = {p1, p2, ..., p...} n The cylindrical surface f is obtained by surface reconstruction. v Iterate through the point set P and calculate the distance from each point to f. v The distances are calculated, and the resulting bidirectional distances form a set. in p represents the i-th point i to f v The directed distance, with the maximum and minimum positive distances being d, are respectively. max d min The corresponding enclosing cylindrical surfaces are f, respectively. max f min Finally, the cylindricity error E is obtained. Cyl for:

[0067] E Cyl =d max +d min

[0068] 8-2) Coaxiality error such as Figure 4 As shown, the calculation requires selecting a reference axis n. s The reference axis originates from the reference plane f. s The axis vector or surface normal vector. The set of point clouds to be measured, P = {p1, p2, ..., p...} n The cylindrical surface parameters and the corresponding fitting axis direction vector n are obtained through surface reconstruction. v Move each point in P to n v By projection, the corresponding projection points can be obtained, and the highest and lowest points among the projection points are p' and p', respectively. up and p' down The highest point in its corresponding point cloud is p. up and p down Finally, solve for p' separately. up and p' down to n s distance d up and ddown Coaxiality error value E Axi For d up and d down The larger value in, that is:

[0069] E Axi =max(d up d down )

[0070] 8-3) Verticality error such as Figure 5 As shown, the calculation requires selecting a reference plane f. s Given a point cloud set P = {p1, p2, ..., p...}, ... n}, f s The normal vector is n s The corresponding surface f is obtained by reconstructing the set P. v And f v The axis vector n v With n s Parallel, move each point in P towards n v By projection, the corresponding projection points can be obtained, and the highest and lowest points among the projection points are p' and p', respectively. up and p' down The highest point in its corresponding point cloud is p. up and p down Find p' respectively up and p' down to n s distance d up and d down Verticality error value E Ver For d up and d down The larger value, that is:

[0071] E Ver =max(d up ,d down )

[0072] Verification experiment:

[0073] To verify the application effect of this invention, experiments were conducted on three aspects: point cloud segmentation, surface reconstruction, and error measurement. The results were compared with those from multiple algorithms, the commercial measurement software VG Studio Max 3.0 (hereinafter referred to as VG), or coordinate measuring machines. The experimental software environment included Visual Studio 2019 and VG Studio Max 3.0; the experimental CT scanner was a CD-130BX / μCT micro-focus CT; the coordinate measuring machine was a LEITZ PMM-C with a measurement accuracy of 0.4μm.

[0074] To verify the effectiveness of the point cloud segmentation method of this invention, a fuel nozzle was selected as the experimental object, and the segmentation results were compared with those of traditional region growing algorithms and feature line extraction-based segmentation methods, as follows: Figure 6 As shown, surfaces a and c are conical surfaces, and surface b is a cylindrical surface. Figure 6 It can be seen that the present invention can better segment the curved surfaces a, b, and c.

[0075] The segmentation accuracy of this invention was verified by quantitative indicators. The experimental object was a conical surface c. The accuracy, precision, and segmentation quality were used as the indicators for the analysis. The comparison algorithms were the traditional region growing algorithm, random sampling consistency, and segmentation method based on feature line extraction. The segmentation results are shown in Table 1.

[0076] Table 1

[0077] method accuracy Accuracy Segmentation quality This invention 90.79％ 98.32％ 94.41％ Regional growth 70.47％ 78.57％ 73.33％ Random sampling consistency 84.67％ 94.45％ 89.29％ Feature extraction and segmentation 80.72％ 95.86％ 87.64％

[0078] To verify the accuracy of surface reconstruction in this invention, conical specimens were selected as experimental subjects. The included angles of the conical specimens were 60°, 90°, and 120°, respectively. The results were compared with the random sampling consistency method, and the comparison index was the error distance between the reconstructed model and the actual point cloud. The experimental results are shown in Table 2. As can be seen from Table 2, the LM method outperforms the RANSAC method in fitting the conical surfaces at different angles.

[0079] Table 2

[0080]

[0081] To further verify the accuracy of the surface reconstruction of this invention, a conical specimen was selected as the experimental object. The cone angle calculated by the surface reconstruction was compared with the actual three-coordinate measurement value. The results are shown in Table 3. As can be seen from Table 3, the difference between the calculated cone angle and the measured value is extremely small, all within 0.1°.

[0082] Table 3

[0083] Specimen cone angle This article's algorithm / ° Coordinate measuring machine value / ° Difference / ° 60° 60.00 60.05 0.05 90° 89.86 89.92 -0.06 120° 119.83 119.86 0.03

[0084] To verify the accuracy of this invention in calculating form and position errors, fuel nozzle A and fuel nozzle B were selected as experimental subjects. The two nozzles were divided, and the division results are as follows: Figure 7 , Figure 8As shown in Table 4, the experiment used high-precision coordinate measuring machine (CMM) measurements as nominal values ​​and compared them with the form and position error results measured by VG software. Table 4 shows that the algorithm of this invention has higher calculation accuracy compared to VG software. Specifically, cylindricity is improved by 0.0017 mm, coaxiality by 0.0039 mm, and perpendicularity by 0.0012 mm. Overall, the measurement accuracy is slightly better than VG software, meeting the measurement index requirements of industrial CT systems and can be used for high-precision measurement of precision parts.

[0085] Table 4

[0086] Test surface Reference surface Measurement items This article's algorithm / mm VG / mm Three coordinates / mm d - Cylindricity 0.0217 0.0200 0.0276 e f coaxiality 0.0224 0.0185 0.0233 h g Verticality 0.0279 0.0291 0.0112

[0087] In summary, this invention can better segment the point cloud of a rotating body, achieve higher precision surface reconstruction of the geometric model, and obtain more accurate form and position error values, thus effectively and better solving the problem of measuring internal errors of rotating body parts.

[0088] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can be implemented entirely as a software embodiment. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0089] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0090] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0091] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0092] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for measuring the internal form and position error of a rotating part, characterized in that: The method includes the following steps: S1: Input a 3D point cloud model and calculate the normal vector of the point cloud; S2: Extract the point set at the same latitude based on the geometric characteristics of the point cloud of the body of revolution; S3: Perform plane fitting using a set of points at the same latitude to obtain the initial value of the axis of revolution; S4: Construct an objective function based on the skew distance from the normal vector on the point cloud of the body of revolution to the axis, substitute the initial value of the axis into it, and obtain the precise value of the axis parameter by nonlinear least squares solution; S5: Rotate the normal vector of the neighboring point cloud to the seed point using the Rodrigues formula, and segment the point cloud according to the angle between the normal vector of the neighboring point and the normal vector of the seed point and the curvature of the neighboring point. The Rodrigues formula for 3D point clouds is: Where, vector The target vector has a seed point. Its projection point on the axis vector The vector formed by two points, where any point in the neighborhood set is... Rotation vector for The normal vector formed by the projection of itself onto the axis, for and The angle in the counterclockwise direction; S6: Based on the geometric dimensional relationship between the cylinder and the cone, the initial values ​​of the geometric model parameters are obtained by solving using the least squares method; S7: Set the objective functions for the cylinder and cone, substitute the initial values ​​of the geometric model obtained in S6, and obtain the precise parameters of the geometric model by nonlinear least squares solution; S8: Calculate cylindricity, coaxiality, and perpendicularity according to the definition of form and position error; In S2, the characteristic of the body of revolution is that two different points at the same latitude on the point cloud are equidistant from the intersection of the corresponding two normal vectors.

2. The method for measuring the internal form and position error of a rotating part according to claim 1, characterized in that: In S6, the parameters to be solved for the cylinder are the axis parameters, which are any points on the axis. and direction vector ,radius .

3. The method for measuring the internal form and position error of a rotating part according to claim 2, characterized in that: In S6, the geometric dimensional relationship of the cylinder is as follows: on the point cloud of the cylinder, there always exists any point... normal vector Vector perpendicular to the axis To solve for the initial value Establish the geometric distance expression: 。 4. The method for measuring the internal form and position error of a rotating part according to claim 3, characterized in that: In step S6, the geometric dimensions of the cylinder are as follows: the point cloud is divided into equal parts with a number of points. Two point cloud clusters, where any point on each cluster is... as well as The corresponding normal vector is , Then the vector , exist , The projected lengths are equal, so that a point on the axis can be solved. Establish the geometric distance expression: 。 5. The method for measuring the internal form and position error of a rotating part according to claim 4, characterized in that: In S6, the geometric dimensional relationship of the cylinder is as follows: cylinder radius Right now Distance to the axis.

6. The method for measuring the internal form and position error of a rotating part according to claim 5, characterized in that: In step S6, the parameters to be solved for the cone are the axis parameters, which are any points on the axis. and direction vector semi-apex angle .

7. The method for measuring the internal form and position error of a rotating part according to claim 6, characterized in that: In S6, the geometric dimensional relationship of the cone is as follows: on the surface of the cone, there always exists any point... normal vector Perpendicular to With the vertex of the cone The vector formed To solve Establish the geometric distance expression: 。 8. The method for measuring the internal form and position error of a rotating part according to claim 7, characterized in that: In S6, the geometric dimensional relationship of the cone is as follows: Calculate Based on the distances to various points in the point cloud, the original point cloud is divided into multiple point cloud clusters. The point cloud cluster with the farthest distance is extracted, and a plane is fitted using this point cloud cluster to solve for the normal vector, which is the direction vector of the cone's axis. .

9. The method for measuring the internal form and position error of a rotating part according to claim 8, characterized in that: In S6, the geometric dimensional relationships of the cylinder are as follows: the geometric dimensional relationships of the cone are as follows: half vertex angle for and The included angle.

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