A method for fast calculating surface error of workpiece machining

Abstract

The present application relates to a kind of workpiece machining curved surface error fast calculation method, adopt the way of combining triangular sheet grid and space index structure, improve the calculation efficiency and accuracy;Through the grid processing to CAD model, construct space index structure, then based on the model after gridding constructs space index structure to accelerate the query process, then based on the vertex of measurement model constructs containment box, and quickly searches the triangular sheet grid intersected with containment box in space index structure, and then accurately calculates the error of workpiece machining curved surface.This method is suitable for complex curved surface and large-scale point cloud data, can quickly evaluate machining error, widely used in error detection and optimization in numerical control machining process, meet the demand of high-precision machining.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of numerical control machining and optimization, and particularly relates to a workpiece machining curved surface error rapid calculation method. BACKGROUND

[0002] Influenced by factors such as machine tool precision, tool wear, material properties and machining process, there is a deviation between the actual machined curved surface and the designed curved surface in the machining process. In the modern manufacturing field, the evaluation of high-precision curved surface machining quality has an important influence on the precision and use performance of the product, and the rapid and accurate calculation of the workpiece machining curved surface error has become one of the core problems of precision manufacturing and quality control. The curved surface machining quality of the workpiece directly affects the performance, service life and assembly precision of the product. The traditional workpiece machining curved surface error algorithm mainly compares and evaluates the designed curved surface of the workpiece and the actual curved surface after machining, calculates the error size and optimizes the machining precision. However, the traditional method still has some limitations in practical application, especially in terms of measurement accuracy, calculation efficiency and applicability, and a more efficient algorithm is needed to meet the needs of modern manufacturing.

[0003] The traditional curved surface error calculation method is usually based on the point cloud data collected by the coordinate measuring device (Coordinate Measuring Machines, CMM), and the point coordinates on the actual machined surface are matched with the theoretical curved surface in the designed model (such as CAD model) of the workpiece. Such method has low calculation efficiency, especially in the case of large data volume or high measurement density, the error calculation time is long, which is difficult to meet the needs of efficient machining and real-time quality control. The traditional curved surface error calculation method mainly includes the following steps: 1. Data acquisition: use CMM, laser scanner and other devices to obtain point cloud data on the surface of the workpiece. These data usually represent the actual machined surface of the workpiece in the form of discrete points. The accuracy and density of data acquisition will directly affect the results of error calculation. 2. Point cloud data processing: the collected point cloud data often contains noise, redundant points, abnormal points, etc. In order to improve the calculation accuracy, the point cloud data must be preprocessed, including noise filtering, downsampling and smoothing processing, etc. 3. Error matching and calculation: compare the processed actual point cloud data with the theoretical curved surface model. Common error calculation methods include least squares fitting method, minimum bounding box method, Hausdorff distance, etc. However, the traditional method has significantly increased calculation amount and time when dealing with complex curved surfaces or free curved surfaces due to their irregular geometric characteristics and large data volume. 4. Error analysis and feedback: through error analysis, the size and distribution of machining error are obtained, so as to analyze the error in order to achieve the purpose of further optimization. This process provides an important basis for error compensation and process improvement.

[0004] Traditional error calculation methods offer certain advantages in terms of computational accuracy, but they also have significant limitations. First, traditional methods often rely on point-by-point calculations, resulting in low computational efficiency in scenarios with large amounts of data. This is especially true when the surface model is a complex or free-form surface, with variable shapes and uneven curvature; point-by-point calculations require substantial computational resources, making it difficult to meet the high efficiency and real-time requirements of modern manufacturing. Second, traditional methods depend on direct comparisons between measurement points and theoretical models, making them insensitive to error distribution in local areas and potentially overlooking deviations in some local features. Therefore, traditional methods are insufficient for rapid error calculation and real-time error assessment of large-scale point cloud data. Summary of the Invention

[0005] The main objective of this invention is to provide a method for rapidly calculating the surface error of a workpiece during machining.

[0006] To achieve the above objectives, the present invention provides a method for rapid calculation of workpiece machining surface errors, comprising the following steps:

[0007] The CAD model is meshed to generate the corresponding mesh model;

[0008] Extract the vertex set from the actual measurement data of the workpiece as a reference point for error calculation;

[0009] A spatial index structure is constructed based on the meshed CAD model;

[0010] A bounding box is generated for each vertex to enclose a local region of the vertex.

[0011] By searching the spatial index structure, the triangular mesh pieces intersecting with the containing box can be quickly located to obtain the geometric data related to the workpiece surface error.

[0012] Based on the projection direction, distance, and intersecting mesh information of each vertex, a corresponding error model is constructed, and the error data of all vertices are integrated into the final error model output.

[0013] Optionally, the CAD model can be meshed to obtain...

[0014] ;

[0015] Among them, Represents the set of vertices. Describe the set of edges. This represents the set of triangular pieces.

[0016] Optionally, the constructed spatial index structure can be one of a K-dimensional tree, an octree, or a directed bounding box tree.

[0017] Optionally, the spatial index structure is a K-dimensional tree.

[0018] The K-dimensional tree construction process is performed according to the following steps:

[0019] Select a division axis to determine the hyperplane of spatial division;

[0020] Select a division point to determine the node to search the current data in the division axis dimension to find the median data;

[0021] Divide the space into two parts through the selected division point after the division point is selected;

[0022] Recursively construct a sub-tree.

[0023] Optionally, the inclusive box is one of an axis-aligned inclusive box, a directed inclusive box, and a spherical inclusive box.

[0024] Optionally, the inclusive box is a spherical inclusive box; and the generating an inclusive box for each point based on the vertex to enclose the local area of the vertex includes the following steps:

[0025] Taking the vertex as the center and the domain range as the radius, a spherical inclusive box of the point is constructed, and then all the triangle patches intersecting with the spherical inclusive box are searched in the constructed K-dimensional tree to obtain a set of all the triangle patches

[0026] wherein is a set composed of the indices of all the intersecting triangle patches, represents the th element of the set ;

[0027] The vertex normal vector

[0028] is calculated.

[0029] wherein represents the area of the triangle patch , represents the normal vector of the triangle patch , represents the modulus operation on the vector;

[0030] The vertex projection direction and the projection distance are calculated to construct a vertex error model .

[0031] All the triangle patches are traversed to calculate the point to the triangle constituted plane of the projection point , determine whether the projection point is located inside the triangle , if inside the triangle, calculate the projection distance and the projection vector ;

[0032] .

[0033] Optionally, further comprising determining the direction of the vertex normal vector , if the directions are opposite, the projection distance and the projection vector need to be reversed, that is

[0034] ;

[0035] ;

[0036] ;

[0037] If > , that is, the maximum value of the angle constraint, the value of is modified to , that is, the maximum value of the custom error.

[0038] Optionally, if the projection point is not inside the triangle, calculate the projection of the point to the edge of the triangle, calculate the projection of the vertex to the three edges, if the projection point is located on the vertex of the triangle, take the minimum edge projection point in the angle constraint range as the result, get the projection distance and the projection vector , then determine the direction of the vertex normal vector, that is

[0039]

[0040] .

[0041] Optionally, if there is no projection point on the three edges, calculate the projection of the vertex to the vertex of the triangle, that is, the distance to the three vertices, take the shortest distance to the vertex of the triangle in the angle constraint range as , that is, the projection point, calculate the projection distance and the projection vector , and then adjust through the normal vector and i.e.

[0042] ;

[0043] ;

[0044] traverse each element in the mesh, and after traversing all the triangles in the mesh, calculate the minimum projection distance and triangle index

[0045] ;

[0046] if exists, output and as the projection distance and projection direction vector, that is, the error distance and error vector, and then construct the error model of the point based on and . Determine whether the value of i is , if so, integrate the error data of all the vertices into the final error model output, otherwise calculate the next vertex .

[0047] Compared with the prior art, the algorithm proposed by the present application has the following advantages:

[0048] Efficient error calculation method: by constructing the spatial index structure of the triangular mesh grid (such as KDTree, OCtree, OBBTree, etc.), the present application greatly improves the efficiency of finding intersecting grids in large-scale three-dimensional models, and optimizes the error calculation process. Compared with the traditional method, invalid calculation is reduced, and only the part intersecting with the containing box needs to be accurately calculated, so that the key area where the error occurs can be quickly determined, the overall error calculation efficiency is improved, and the calculation time is significantly reduced.

[0049] 2. High-precision error calculation capability: the algorithm accurately matches the workpiece surface and the design surface, and ensures the accuracy of error evaluation through the calculation of error vectors. The error not only reflects in the numerical value, but also provides detailed error direction information, which is convenient for subsequent processing correction and quality control.

[0050] ​​​​3. Superior complex surface model applicability: The algorithm proposed in the present application can efficiently process non-uniformly distributed data and complex surface structures, especially in evaluating the surface error of complex industrial workpieces. By meshing the complex surface model into a simplified grid model and flexibly selecting different spatial index tree structures to adapt to various complexities and data distributions, the algorithm ensures the accuracy and robustness of error calculation while being applicable to all gridable complex surfaces.

[0051] 4. Excellent scalability and versatility: The algorithm of the present application can be extended to different surface error calculation scenarios, including various workpiece surfaces commonly encountered in industrial processing. At the same time, the algorithm has good scalability and is applicable to various spatial index structures and geometric models of different complexities, providing a technical foundation for diversified industrial applications. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1 is a flowchart of the workpiece processing surface error fast calculation method of the present application;

[0053] Figure 2 is a field map for a vertex;

[0054] Figure 3 is a point-to-triangle projection map. DETAILED DESCRIPTION

[0055] The technical solutions in the embodiments of the present application will be described in detail below with reference to the accompanying drawings, and the described embodiments are only possible technical implementations of the present application, not all possible implementations. Those skilled in the art can easily obtain other embodiments by combining the embodiments of the present application without creative labor, and these embodiments are also within the protection scope of the present application.

[0056] Embodiment One

[0057] Referring to Figure 1 , the present embodiment provides a workpiece processing surface error fast calculation method, which first acquires a CAD model and measurement data, and performs meshing processing on the CAD model to generate a triangle mesh model; then constructs a spatial index structure (KDTree, OCtree, OBBTree, etc.) based on the meshed model to accelerate the query and detection of objects in three-dimensional space. Subsequently, a containing box is constructed based on the vertices of the measurement model as the center, and the triangle pieces intersecting with the containing box are searched in the constructed spatial index structure. Then the normal vector of each vertex is calculated to determine its projection point and error vector to the triangle piece, and finally the overall error model of the workpiece surface is constructed according to the error vectors of all vertices.

[0058] Specifically, it is divided into the following four steps:

[0059] S1 preprocessing

[0060] In the present invention, we first mesh the CAD model

[0061]

[0062] i.e. the meshed , where represents the set of vertices, represents the set of edges, represents the set of triangles.

[0063] Based on the measured data N, we get the set of all vertices

[0064]

[0065] where represents the maximum number of vertices.

[0066] S2 Constructing the spatial Tree structure of the triangle mesh

[0067] The trees (Tree) that can be used to construct the triangle network in the present invention mainly include K-dimensional tree (KDTree), octree (Octree), and oriented bounding box tree (OBBTree), etc. These are several commonly used spatial index structures, which are used to accelerate the query and detection of objects in three-dimensional space. Taking KDTree as an example, KDTree is a tree structure based on spatial partitioning, which realizes efficient nearest neighbor query by recursively dividing the space into hyperplanes, and is suitable for fast searching of point cloud data. The construction of KDTree is to recursively divide the space into multiple regions, each node represents a hyperplane, and each node points to a set of data points or triangles in the space. Based on the in S1, we construct the KDTree about the triangles.

[0068] S3 Traverse the vertices Construct the bounding box, construct the error model of the point

[0069] According to the shape of the object, different types of bounding boxes can be used, such as axis-aligned bounding box (Axis-Aligned Bounding Box, AABB), oriented bounding box (Oriented Bounding Box, OBB), and spherical bounding box. In the present invention, spherical bounding box is usually used as the basic form of constructing bounding box, because it is simple, computationally efficient, and suitable for boundary calculation of most three-dimensional objects.

[0070] S31 Traverse the vertices​​​ center, field range radius, build point sphere containing box Then search in the built KDTree to get all the triangle patches intersecting with , where is the set of all intersecting triangle patches, is the set of all the indices of the intersecting triangle patches, ) represents the th element of the set .

[0071] S32 Calculate vertex normal vector

[0072] ;

[0073] where represents the area of triangle patch , represents the normal vector of triangle patch , represents the modulo operation on the vector. S33 Calculate vertex projection direction and projection distance, build vertex error model . Traverse all the triangle patches

[0074] , calculate the projection point of point to the plane composed of triangle patches , judge whether the projection point is located inside the triangle patch , if it is inside the triangle, calculate the projection distance and the projection vector ; if it is not, calculate the projection of the point to the edges of the triangle patch, i.e. the projection of vertex to the three edges, if the projection point is located at the vertex of the triangle patch, take the edge projection point with the smallest projection distance within the angle constraint range as the result, get the projection distance and the projection vector ; if there is no projection point on the three edges, calculate the projection of vertex to the vertices of the triangle patch, i.e. the distance to the three vertices, take the vertex of the triangle patch with the shortest distance within the angle constraint range as , i.e. the projection point, calculate the projection distance and the projection vector , then adjust and through the normal vector, i.e.

[0075] ​​ ;

[0076] ;

[0077] Traverse one by one Elements, traversed After finding all the triangles, calculate the minimum projection distance and the triangle index.

[0078] ;

[0079] If it exists, output it. and As the projection distance and projection direction vectors, i.e., the error distance and error vector, then based on and Build Point error model Determine if the value of i is... If yes, proceed to S4; otherwise, return to S31 to calculate the next vertex. .

[0080] S4: Constructing the error model and output

[0081] Error model integrating all vertices Constructing an error model It outputs the result, enabling rapid calculation of workpiece machining surface errors.

[0082] Example 2

[0083] As another embodiment of the present invention, combined with Figures 1-3 As shown, the rapid calculation method for workpiece machining surface error in this embodiment mainly includes the following steps:

[0084] Firstly, input the CAD model and the measured network model of the workpiece. Mesh the CAD model to generate the corresponding mesh model, and extract the vertex set from the measurement data as the reference point for error calculation. Then, based on the meshed CAD model, construct a spatial index structure (such as KDTree), which can accelerate the search and matching operations in subsequent error calculation. Subsequently, process each measured vertex, calculate its projection direction and projection distance from the theoretical model. Based on these vertices, generate an inclusive box for each point, which encloses the local area of the vertex. Then, search the spatial index structure to quickly find the triangular mesh pieces intersected with these inclusive boxes and obtain the geometric data related to the workpiece surface error. Finally, based on the projection direction, distance and intersecting mesh information of each vertex, construct the corresponding error model, and integrate the error data of all vertices into the final error model output. This algorithm flow realizes fast and efficient calculation of complex workpiece surface error, and can be widely applied to real-time error evaluation and quality control in industrial field. The specific implementation steps are as follows:

[0085] S1: preprocessing

[0086] Get CAD model and measurement data , mesh to

[0087] ;

[0088] i.e. the meshed , where represents the set of vertices, represents the set of edges, represents the set of triangular pieces. Based on N, get the set of all vertices .

[0089] S2: construct spatial Tree structure of triangular piece mesh

[0090] Based on the triangular piece information of S1 , each dimension coordinate axis is used as the division axis to establish the Tree. Several commonly used spatial index structures include KDTree, OCTree and OBBTree. Among them, OCtree is an octree structure that recursively divides the three-dimensional space into 8 equal volume subspaces, suitable for processing non-uniformly distributed three-dimensional data, especially for complex curved surface models. OBBTree is a binary tree structure that uses the directional inclusive box (OBB) feature, which divides the space along the longest axis of the OBB to recursively construct a tight and efficient spatial hierarchical structure.

[0091] While other spatial indexing structures (OCtree, OBBTree, etc.) have advantages in handling specific types of spatial data, KDTree's efficiency is particularly outstanding when fast querying and processing of large-scale, dense data is required. Therefore, this invention chooses KDTree as an example for detailed explanation, but this invention is not limited to KDTree. Specifically, the tree construction process follows these steps:

[0092] 1): Select the dividing axis to determine the hyperplane for spatial division;

[0093] 2): Select the dividing point and determine the node. Search the current data according to the dividing axis dimension to find the median data (dividing point);

[0094] 3): Divide the space. After selecting a dividing point, the space is divided into two parts through that point;

[0095] 4): Recursively construct subtrees.

[0096] When choosing a hyperplane, taking point clouds as an example, there are two approaches. First, for the dimension where the point cloud distribution is not concentrated, the variance is relatively large. The maximum variance method can be used to select the dimension for each classification. That is, before each new division, we determine which dimension to divide on by judging the variance. Second, the coordinate axes can be used alternately as the dividing axis. For example, the first layer uses the X-axis, the second layer uses the Y-axis, the third layer uses the X-axis again, and so on.

[0097] When constructing a KDTree using triangles as objects, steps 1) and 2) are consistent with the KDTree for point clouds. However, there is a slight difference in S23. When a triangle intersects with a hyperplane, the processing strategy is to assign it to the right branch, i.e., divide it into the left branch and branches not on the left; alternatively, it can be assigned to the left branch, i.e., divided into the right branch and branches not on the right. The construction methods for other trees are similar to those for KDTrees, but there are differences in steps 1) and 2).

[0098] S3: Traversing Vertices Build an inclusive box, build Point error model .

[0099] S31: The range of the area calculated by referencing the workpiece error estimate. by Establish the radius about the vertex spherical containment box Searching for all triangles in the KDTree that intersect with the containing box yields the set of triangles. For any They all

[0100]

[0101]

[0102]

[0103]

[0104] is established, where is a set consisting of subscripts of all intersected triangles, represents the th element of the set , are three vertices of the triangle , is the unit normal of the triangle , and are the projection distance and the projection point of the vertex to the triangle , respectively.

[0105] S32: Calculate the normal vector of the vertex as shown in Figure 2 , and the vertex is associated with a triangle set ,

[0106] ;

[0107] represents the normal vector of the triangle . Use the average of each triangle normal vector as the normal vector of the vertex, that is

[0108] ;

[0109] where represents the area of the triangle , represents the normal vector of the triangle , represents the modulus operation on the vector.

[0110] S33: Calculate the vertex projection direction and projection distance, and construct the vertex error model.

[0111] (1): Traverse the triangle , calculate the projection of the vertex to the triangle , as shown in Figure 3 , calculate the projection point of the point to the plane composed of triangles, and judge whether the projection point is located in the triangle If inside the triangle, then calculate the projection distance and the projection vector ,

[0112] ;

[0113] Then judge the direction of and the vertex normal vector, if the directions are opposite, then the projection distance and the projection vector need to be reversed, that is

[0114] ;

[0115] ;

[0116] ;

[0117] If > , that is, the maximum value of the angle constraint, then modify the value to , that is, the maximum value of the custom error.

[0118] If the projection point is not inside the triangle, then calculate the projection of the point to the edges of the triangle, calculate the projection of the vertex to the three edges, if the projection point is located on the vertex of the triangle, then take the edge projection point with the minimum projection distance within the angle constraint range as the result, obtain the projection distance and the projection vector , then judge the direction of and the vertex normal vector, that is

[0119] ;

[0120] ;

[0121] If there is no projection point on the three edges, then calculate the projection of the vertex to the vertices of the triangle, that is, the distance to the three vertices, take the vertex with the shortest distance within the angle constraint range as , that is, the projection point, calculate the projection distance and the projection vector ;

[0122]

[0123] =

[0124] ​Iterate operation (1) until all the triangles are traversed, then proceed to (2).

[0125] (2): Select the minimum projection distance of all the triangles in the middle

[0126] ;

[0127] If it exists, output the projection distance and the projection direction vector as the projection distance and the projection direction vector, and proceed to (4). If it does not exist, proceed to (3). (3): Calculate the angle between the projection direction and the surface normal vector

[0128]

[0129] ;

[0130] If it exists, output the projection distance and the projection direction vector, and proceed to (4). If it does not exist, the projection cannot be made, and it is determined whether the value of i is n. If yes, proceed to S4. If no, return to S31 to calculate the next vertex .

[0131] (4): Construct an error model of the vertex based on the projection distance and the projection direction

[0132] S4: Construct an error model

[0133] Integrate all the error models , construct an error model , and output, to realize fast calculation of the error of the workpiece machining surface.

[0134] ​​​​​​​​​​​​​​​​The present application mainly solves the problems of low calculation efficiency and inaccurate error evaluation in the existing workpiece machining curved surface error calculation method. The core idea of the present application is to grid the CAD model of the workpiece, and accelerate the search of the grid by constructing a spatial index structure (such as KDTree, OBBTree, etc.). First, the inclusive box of the actual measurement point of the workpiece is constructed, and the intersecting triangular mesh is quickly found in the spatial index structure, and then the error value between the actual machining curved surface and the theoretical model is calculated. This method not only significantly improves the error calculation speed, but also guarantees the accuracy of error evaluation, and is especially suitable for rapid detection of complex curved surfaces.

[0135] It should be noted that in this paper, the term "includes", "contains" or any other variant thereof is intended to cover non-exclusive inclusion, so that the process, device, article or method including a series of elements not only includes those elements, but also includes other elements not explicitly listed, or includes elements inherent to such process, device, article or method. Without more limitations, the element defined by the statement "includes one" does not exclude the existence of other identical elements in the process, device, article or method including the element.

[0136] The above is only the preferred embodiment of the present application, and does not limit the patent scope of the present application, and any equivalent structure or equivalent process transformation using the content of the present application specification and drawings, or direct or indirect application in other related technical fields, are also included in the patent protection scope of the present application.

Claims

1. A method for fast computation of surface error in workpiece machining, characterized in that, The method comprises the following steps: The CAD model is meshed to obtain Generate the corresponding mesh model; where, where Represents the set of vertices. Describe the set of edges. Represents a set of triangular pieces; extracting a vertex set from actual measurement data of a workpiece as reference points for error calculation; constructing a space index structure based on a gridded CAD model; generating an inclusive box for each vertex based on the vertex to enclose a local area of the vertex; the inclusive box is one of an axis-aligned inclusive box, a directed inclusive box, and a spherical inclusive box; the inclusive box is a spherical inclusive box; the step of generating an inclusive box for each vertex based on the vertex to enclose a local area of the vertex specifically comprises the following steps: With vertex Centered on the scope of the field Construct points with radius . spherical containment box Then in the construction K Searching in the tree yields results related to the spherical containment box. The set of all intersecting triangles ; wherein is the set of all intersecting subscripts of the triangles, ) denotes the th element of the set . calculating a vertex normal vector wherein denotes the area of a triangle denotes the normal vector of a triangle denotes a modulo operation on a vector;​​ Calculate the vertex projection direction and projection distance, and construct a vertex error model ; traversing all the triangles , calculating the projection point of the point to the plane composed of the triangle , judging whether the projection point is located inside the triangle , if inside the triangle, calculating the projection distance and the projection vector ;​ ； quickly searching for a triangular mesh piece intersecting the inclusive box by searching the space index structure to obtain geometric data related to surface error of the workpiece; constructing a corresponding error model based on projection direction, distance, and intersecting mesh information of each vertex, and integrating error data of all vertices into a final error model output.

2. The computational method of claim 1, wherein, The constructed space index structure is one of a K-dimensional tree, an octree, and a directed bounding volume tree.

3. The computational method of claim 2, wherein, The space index structure is a K-dimensional tree. The K-dimensional tree construction process is performed according to the following steps: selecting a division axis to determine a hyperplane for space division; selecting a division point to determine a median data for current data searching in the division axis dimension; dividing the space into two parts after selecting the division point by dividing the point; recursively constructing a sub-tree.

4. The computational method of claim 3, wherein, Also included is determining a projection vector with the vertex normal vector direction, if the directions are opposite, then the projection distance and the projection vector need to be reversed, i.e. ； ； ； If , the maximum value of the angle constraint, the projection distance value is modified to , i.e. the custom maximum error value.​ 5. The computational method of claim 4, wherein, If the projection point is not inside the triangle, then the projection of the point to the edges of the triangle is calculated, the projection of the vertices to the three edges is calculated, if the projection point is located at the vertex of the triangle, then the projection point with the smallest projection distance in the angle constraint range is taken as the result, the projection distance and the projection vector are obtained, then the direction of the projection vector and the vertex normal vector is judged, that is ； 。 6. The computational method of claim 5, wherein, If there are no projection points on any of the three edges, then calculate the vertex. The projection to the vertices of the triangle is the distance to each of the three vertices. The vertex of the triangle with the shortest distance within the angular constraint range is selected as the projection. That is, the projection point, and the calculation of the projection distance. and projection vector Then through the normal vector Adjust the projection distance and projection vector ,Right now ； ； traverse one by one middle element, traverse all the triangles calculate the minimum projection distance and triangle index after traversing all the triangles ； Output if present and as the projection distance and projection direction vector, i.e. the error distance and error vector, then based on and construct an error model for the point ; determining i whether the value is , and if so, integrating error data for all of the vertices into a final error model output, and if not, calculating a next vertex .

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