An aircraft skin deformation analysis and spraying path conforming optimization method

Abstract

The present application relates to a kind of aircraft skin deformation analysis and spraying path conforming optimization method.This method includes: accurate registration is carried out to actual point cloud data and mathematical model;Actual point cloud data and mathematical model are divided into regions and corresponding region matching is carried out;Actual point cloud data and mathematical model corresponding region and overall deformation parameter are calculated and analyzed;Actual point cloud deformation parameter obtained in the step is matched and integrated in local region, respectively, additional to the theoretical spraying path based on mathematical model is generated in advance, and the path is optimized and corrected conforming, and the actual spraying path conforming actual point cloud shape is obtained;Actual spraying path obtained after conforming optimization is overall processed.This method is analyzed by corresponding block deformation of point cloud data and theoretical mathematical model, and the theoretical spraying path is optimized, and the robot is guided to complete conforming spraying movement task, and the processing process adaptability is improved, and the path planning efficiency is improved.

Description

Technical Field

[0001] This invention belongs to the technical field of aircraft skin spraying path planning methods, and particularly relates to an aircraft skin deformation analysis and spraying path conformal optimization method. Background Technology

[0002] In aircraft skin painting, the general requirements for the paint surface are that the paint film uniformly and smoothly covers the substrate material, free from defects such as pinholes, bubbles, pores, runs, flow marks, ripples, dust, fuzz, dry spraying, overspraying, debris, and scratches. Without magnified visual inspection, the applied primer should provide continuous coverage without gaps, and be free from visible solid particles, contaminants, fisheyes, pinholes, bubbles, overspray, and other defects. At a distance of 2-3 meters from the coating surface without magnified visual inspection, there should be no visible unevenness, runs, lines, runs, or orange peel texture. Furthermore, depending on the type of paint used, the dry film thickness is generally required to be between 10-50 μm. With the aircraft manufacturing industry's increasing demands for aircraft appearance and performance, the requirements for the dry film thickness of the paint are also correspondingly rising.

[0003] During aircraft skin painting, the skin is installed onto specific fixtures. The painting path is designed and planned based on manual observation or a mathematical model of the skin, combined with the painting process requirements. Due to the thinness of the skin and its often complex hyperboloid shape, uneven stress occurs when it is installed on the fixture, easily causing deformation. Current methods of manual painting and generating painting paths based on mathematical models lack the perception and quantification of actual skin deformation. They can only adjust the process or calculate the actual deformation based on real-time data to ensure the paint film quality meets process requirements as much as possible. Currently, the path optimization method that considers deformation involves collecting point cloud data each time a skin is placed, slicing and fitting the point cloud data, and calculating the actual painting path. However, collecting data for a single skin and calculating the path based on point cloud data involves a large amount of data, cumbersome data processing, long processing time, and high requirements for computer storage space and processing speed, resulting in low path planning efficiency. Therefore, in order to ensure the quality of skin coating, improve processing efficiency, and meet the design and manufacturing standards of the next generation of aircraft, developing efficient, automated, and intelligent methods for skin coating path planning has become an important issue that technical personnel need to address. Summary of the Invention

[0004] To address the problems of existing aircraft skin coating path optimization methods, which require data acquisition for individual skin sheets and path calculation based on point cloud data, resulting in large data volumes, cumbersome data processing, long processing times, high requirements for computer storage space and operating speed, and low path planning efficiency, this invention proposes a new method for aircraft skin deformation analysis and conformal optimization of coating paths.

[0005] This method targets the skin frame structure of aircraft. Using a KUKA industrial robot as a carrier, it utilizes a 3D surface laser sensor to acquire skin shape information, providing point cloud data that conforms to the actual skin shape. Then, by analyzing the deformation of corresponding blocks in the point cloud data and the theoretical mathematical model, the theoretical spraying path is optimized, guiding the robot to complete the conformal spraying motion task and improving the adaptability of the processing.

[0006] The implementation system of this method is primarily software-based, with a hardware platform built for verification. The software system mainly includes a data analysis and processing system located on a host computer; the hardware platform mainly includes a 3D surface laser sensor, a KUKA robot, a spraying module, a laser tracker, and the structural components necessary for system assembly and functional implementation. Through the combination of the hardware platform and the software system, the implementation system of this method can achieve accurate extraction of processing references and real-time planning of spraying paths in both online and offline modes.

[0007] Specifically, this invention provides a method for aircraft skin deformation analysis and conformal optimization of coating path, the method comprising:

[0008] S1. Perform precise registration between actual point cloud data and mathematical models;

[0009] S2. Divide the actual point cloud data and mathematical model into regions and match the corresponding regions;

[0010] S3. Calculate and analyze the deformation parameters of the corresponding regions and the overall deformation of the actual point cloud data and the mathematical model;

[0011] S4. The actual point cloud deformation parameters obtained in the previous step are matched locally and integrated as a whole, and then attached to the theoretical spraying path pre-generated based on the mathematical model. The path is then optimized and corrected to obtain the actual spraying path that conforms to the actual point cloud shape.

[0012] S5. Perform overall processing on the actual spraying path obtained after conformal optimization.

[0013] Furthermore, in step S1 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention, the precise registration of the actual point cloud data and the mathematical model is achieved by calculating the rotation and translation matrix between the two point clouds, transforming the source point cloud or mathematical model to the same coordinate system as the target point cloud or mathematical model. This can be expressed as the following equation:

[0014] p t =R·p s +T

[0015] Where, p t and p s It is a pair of corresponding points, where R is the rotation matrix between the point cloud and the mathematical model, and T is the translation vector.

[0016] Furthermore, the precise registration of actual point cloud data and mathematical model in step S1 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention includes:

[0017] S11. First, perform coarse registration to ensure that the actual point cloud data and the mathematical model coincide in the principal direction. The principal direction of the point cloud data is obtained by calculating the feature vectors of all points or by calculating the bounding box. The principal direction of the mathematical model is obtained by calculating the feature vectors of the vertices of the triangular facets. In addition to the principal direction, two secondary directions perpendicular to the principal direction are further obtained, thereby establishing a reference coordinate system with the centroid as the origin and the principal and secondary directions as the coordinate axes. Adjusting the reference coordinate system of the actual point cloud data and the mathematical model to be consistent achieves coarse registration between the actual point cloud data and the mathematical model.

[0018] S12. After completing the coarse registration of the actual point cloud data and the mathematical model, further fine registration is performed;

[0019] The target point set A corresponds to the reference point set S, and the corresponding point set should satisfy the following conditions:

[0020] (1) The number of points in A is equal to the number of points in S;

[0021] (2) For each point a in A i They should all correspond to points s in S with the same subscript. i ;

[0022] Let the rotation transformation vector be the unit quaternion q. R =[q0q1q2q3] t Where q0≥0, and q0 2 +q1 2 +q2 2 +q3 2 =1, thus obtaining a 3×3 rotation matrix R(q) R );

[0023] Let the translation transformation vector be q. T =[q4q5q6] t We can obtain the complete coordinate transformation vector q = [q R |q T ] t ;

[0024] The problem of finding the optimal coordinate transformation vector between corresponding point sets is transformed into the problem of finding q that minimizes the function:

[0025] The algorithm is as follows:

[0026] (1) Obtain the corresponding point cloud dataset A and mathematical model set S;

[0027] (2) Calculate the centroids of A and S respectively:

[0028]

[0029] (3) Construct a 3×3 covariance matrix from two point sets:

[0030]

[0031] (4) Construct a 4×4 symmetric matrix from the covariance matrix:

[0032]

[0033] Where, tr(∑ as ) is a matrix ∑ as The trace, Δ=[P 23 P 31 P 12 ] T , I3 is a 3×3 identity matrix;

[0034] (5) Calculate Q(∑ as The eigenvalues ​​and eigenvectors, the eigenvector corresponding to the largest eigenvalue is the optimal rotation vector q. R =[q0q1q2q3] t ;

[0035] (6) Calculate the optimal translation vector:

[0036] q T =μ s -R(q R )μ a

[0037] in,

[0038]

[0039] (7) Perform coordinate transformation on A based on the rotation and translation vectors;

[0040] (8) Calculate the error to determine whether iteration is still needed. If the error is less than the set value, stop the iteration; otherwise, continue the iteration.

[0041] (9) Complete the fine registration.

[0042] Furthermore, the coarse registration described in step S11 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention includes: setting V p V is the bounding box volume of the point cloud data. s V is the bounding box volume of the mathematical model. iLet λ be the volume of the bounding box obtained by intersecting the bounding box of the point cloud data and the bounding box of the mathematical model. Then the bounding box coincidence coefficient λ is:

[0043]

[0044] ε is a set value. If λ>ε, the point cloud data and the mathematical model are close to coincide. Otherwise, they are not close to coincide and still need to be optimized.

[0045] Furthermore, step S2 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention, which describes the region division and corresponding region matching of actual point cloud data and mathematical model, includes: pre-generating a theoretical coating path that conforms to the shape characteristics and process requirements of the mathematical model based on the mathematical model; calculating the region division method of the mathematical model in reverse according to the theoretical coating path and process parameters; generating region boundary lines or dividing lines; dividing the registered actual point cloud into multiple regions, so that the divided regions can accurately express the shape information of the actual point cloud and make the actual point cloud of each region accurately correspond to the mathematical model.

[0046] Furthermore, in step S3 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention, the calculation and analysis of the deformation parameters of the corresponding region and the overall deformation of the actual point cloud data and the mathematical model includes, according to the corresponding region divided in step S2, performing surface fitting on the actual point cloud data in each region, calculating its center point coordinates, regional point cloud curvature and surface normal vector parameters, thereby analyzing the local deformation and overall deformation of the actual point cloud relative to the mathematical model.

[0047] Furthermore, the regional point cloud curvature and surface normal vector in the aircraft skin deformation analysis and coating path conformal optimization method of the present invention are obtained by calculation using either of the following two methods:

[0048] (I) Using quadratic surface fitting to determine the curvature of point clouds

[0049] Take a data point P from the scattered point cloud. i Then with P i Take n points uniformly around the center of the point cloud, ensuring these n points cover the entire point cloud as much as possible; fit the quadratic surface z(x, y) = ax using the least squares method. 2 +bxy+cy 2 After obtaining the coefficients, calculate the Gaussian curvature and mean curvature of the data points based on the properties of the spatial surface curve;

[0050] For the m-neighborhood of any data point within the measurement point cloud, according to the least squares principle, the following expression should be minimized:

[0051]

[0052] In the formula, xj y j , z j Let a, b, and c be points in the neighborhood, and a, b, and c be the coefficients of the quadratic surface equation. Taking the derivative of the above equation with respect to the coefficients and setting them to zero, we obtain the following equation:

[0053]

[0054] Therefore, the coefficients of the quadratic surface can be solved; the equation of the curve can then be written in the form of a surface parametric equation:

[0055]

[0056] If there exists a curve r on the surface, where t is time and r is a spatial vector, then the expression for r is:

[0057] r = r(x(t), y(t));

[0058] If s represents the arc length of curve r, then the differential formula for arc length can be obtained from the chain rule:

[0059]

[0060] From the first fundamental formula of a surface, we can obtain:

[0061] (ds) 2 =I=E(dx) 2 +2Fdxdy+G(dy) 2 ;

[0062] Where E, F, and G are the fundamental quantities of the first fundamental equation of the surface; if P is a point on the curve r, and t and n represent the unit tangent vector and unit normal vector of point P respectively, then the curvature vector can be decomposed as:

[0063]

[0064] The unit normal vector n of the curve is expressed as:

[0065]

[0066] The second fundamental formula for surfaces can be obtained:

[0067]

[0068] Normal curvature can be expressed as:

[0069]

[0070] By processing the variable λ, we can obtain k. n The two fundamental roots k1 and k2 can be obtained from the curvature characteristics:

[0071] Gaussian curvature:

[0072]

[0073] Mean curvature:

[0074]

[0075] (ii) Calculate the curvature of the point cloud by combining the normal vectors of all points within the adjacent range of a certain point.

[0076] To estimate the curvature of a point, we only consider the contribution of a neighboring point and convert it into the construction of a positive cross-section curve;

[0077] Construct a normal cross-section circle and estimate the normal curvature based on the position and normal vector of the target point and its adjacent points;

[0078] The normal curvature is calculated using a local fitting method. For each point p in the point cloud, let N be the unit normal vector of point p. The method for estimating the normal curvature at point p using the point coordinates and the normal vector is as follows:

[0079] Assume that point p has m nearest neighbors, q i Let q be the i-th nearest neighbor of point p. i The normal vector is M i Let p, X, Y, z be an orthogonal coordinate system, and let L be a local coordinate system of point p. Let N denote the normal vector of point p, and let X and Y be orthogonal unit vectors. In L, p, q i M i The coordinates are p(0, 0, 0) and q respectively. i (x i y i , z i M i (n x，i n y，i n z，i The normal curvature of point p is estimated using a osculating circle passing through it.

[0080] Then p relative to q i The normal curvature is estimated as follows:

[0081]

[0082] In the formula, α represents vectors -N and pq. i The angle between them, β is the angle between vectors N and M i The angle between them.

[0083] Furthermore, step S3 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention, which involves calculating and analyzing the deformation parameters of the corresponding region and the overall deformation parameters of the actual point cloud data and the mathematical model, also includes:

[0084] Calculating the actual point cloud normal vector within the corresponding region transforms the problem of determining the normal of a point on the surface into a least-squares plane fitting problem. Therefore, the solution for estimating the surface normal becomes analyzing the eigenvectors and eigenvalues ​​of a covariance matrix, which is created from the nearest neighbors of the query point. The maximum principal component is the direction with the largest variance after projection onto a certain direction, while the normal vector is the direction with the smallest information after projection onto a certain direction. Therefore, principal component analysis (PCA) transformation is required to project the point cloud onto the direction with the smallest eigenvalue.

[0085] Steps to solve for the normal vector:

[0086] The target plane consists of a point x and a normal vector. Let P represent a point P. i ∈P k The distance to the plane is: That is, take a point on the plane, and the distance from a point outside the plane to the plane is the projection of the vector formed by these two points onto the normal vector; if the distance from the point to the plane is 0, that is, d i =0, so the normal vector of the plane can be found; therefore, the following steps are followed:

[0087] (1) Find the centroid P of all points. k :

[0088]

[0089] (2) Calculate the centroid P k The covariance matrix C∈R 3×3 , and its eigenvectors and eigenvalues:

[0090]

[0091] Where ∈ is 1, C is a symmetric and positive semi-definite matrix and its eigenvalue is the real number λ. j ∈R, eigenvectors An orthogonal coordinate system is formed, corresponding to the centroid P. k Principal components; if 0 ≤ λ0 ≤ λ1 ≤ λ2, the eigenvectors with the smallest eigenvalue λ0. It is the normal vector or Approximation, or Can be used in spherical coordinates express:

[0092]

[0093] Principal component analysis (PCA) is used to calculate the orientation, but it exhibits bivariate properties, making it impossible to consistently orient the normal direction of the entire point cloud dataset. A solution to this problem is to use viewpoint V. P For all normals Orientation simply requires aligning them all towards the viewpoint, satisfying the following equation:

[0094]

[0095] For any point p i The covariance matrix is ​​calculated from the following formula:

[0096]

[0097] Where k is p i The number of nearest neighbors of a point Let λ represent the three-dimensional centroid of the nearest neighbor point. j V represents the j-th eigenvalue of the covariance matrix. j Let represent the j-th eigenvector.

[0098] Furthermore, the step S5 of the aircraft skin deformation analysis and coating path conformal optimization method of the present invention, which involves overall processing of the actual coating path obtained after conformal optimization, includes:

[0099] Perform smooth optimization fitting on the path;

[0100] Based on the robot's operating speed requirements, the path is fitted and interpolated.

[0101] Adaptive intelligent planning is performed on multiple non-connected paths, and multiple paths are connected according to the shortest distance and robot limit rules;

[0102] Plan the timing of the end-spraying switch and the timing of the paint bucket replacement.

[0103] On the other hand, the present invention also provides an aircraft skin deformation analysis and spraying path conformal optimization system. This system targets the aircraft skin frame structure, uses a KUKA industrial robot as a carrier, and uses a 3D surface laser sensor to obtain skin shape information, providing point cloud data that conforms to the actual skin shape. Then, by analyzing the deformation of corresponding blocks of the point cloud data and the theoretical mathematical model, the theoretical spraying path is optimized, guiding the robot to complete the conformal spraying motion task and improving the adaptability of the processing process.

[0104] This system is primarily software-based, with a hardware platform built to validate it. The software system mainly includes a data analysis and processing system located on a host computer. The hardware platform mainly includes a 3D surface laser sensor, a KUKA robot, a spraying module, a laser tracker, and structural components necessary for system assembly and functional implementation. Through the combination of the hardware platform and the software system, this system can achieve functions such as accurate extraction of processing benchmarks and real-time planning of spraying paths in both online and offline modes.

[0105] Furthermore, this invention also relates to the application of the aforementioned aircraft skin deformation analysis and coating path conformal optimization method and system in the aircraft assembly industry.

[0106] In summary, the aircraft skin deformation analysis and coating path conformal optimization method and system of the present invention have the following characteristics:

[0107] 1) Precise registration of actual point cloud data with mathematical models

[0108] In addition to common registration methods, this scheme also introduces techniques such as bounding box registration, local feature registration, and deformation data registration to comprehensively register point cloud data with mathematical models. By combining multiple techniques, the registration problem can be solved from multiple perspectives, thereby reducing computational load and improving accuracy in practical applications.

[0109] 2) Regional division and corresponding region matching of point cloud data and mathematical models

[0110] Based on the mathematical model, a theoretical spraying path that conforms to the shape characteristics and process requirements of the mathematical model is generated in advance. The region division method of the mathematical model is calculated by reverse calculation according to the theoretical spraying path and process parameters, and the region boundary line or dividing line is generated. The registered actual point cloud is divided into multiple regions, so that the divided regions can accurately express the shape and other information of the actual point cloud, and the actual point cloud of each region corresponds accurately with the mathematical model.

[0111] 3) Calculation and analysis of deformation parameters of corresponding regions and the whole based on point cloud data and mathematical models.

[0112] Based on the corresponding regions after division, parameters such as center, curvature, and normal vector are calculated to analyze the local and overall deformation of the actual point cloud relative to the mathematical model.

[0113] 4) Path conformal optimization based on point cloud deformation parameters

[0114] The calculated local deformation parameters and overall deformation parameters are matched locally and integrated globally, and then attached to the theoretical path pre-generated based on the mathematical model. The path is then optimized and corrected to obtain the actual spraying path that conforms to the actual point cloud shape.

[0115] 5) Overall processing of the path after contour optimization

[0116] The actual spraying path after conformal optimization is processed as a whole, and the path is smoothed and fitted. According to the requirements of robot running speed, the path is fitted and interpolated. Adaptive intelligent planning is performed on multiple non-connected paths. Multiple paths are connected according to the shortest distance, robot limit and other rules. The timing of end spraying switch and paint bucket replacement is planned to make reasonable use of paint, improve spraying efficiency and quality, avoid paint waste and reduce environmental pollution. Attached Figure Description

[0117] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments of the present invention will be briefly introduced below. Obviously, the following drawings are only some embodiments recorded in the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0118] Figure 1 This is a flowchart illustrating the implementation of the method of the present invention.

[0119] Figure 2 This is a schematic diagram illustrating the registration of actual point cloud data with a mathematical model in an embodiment of the method of the present invention.

[0120] Figure 3 This is a schematic diagram of the region division method of the mathematical model in the embodiment of the method of the present invention.

[0121] Figure 4 This is a schematic diagram of a mathematical model (top) and an actual point cloud (bottom) for a certain local area in an embodiment of the method of the present invention.

[0122] Figure 5 This is a schematic diagram of the triangle (left) and local coordinate system (right) defined by the circumscribed circle, neighboring points, and normal vector in an embodiment of the method of the present invention.

[0123] Figure 6 This is a schematic diagram showing the theoretical path of the mathematical model (left) and the surface center (right) of the corresponding region of the actual point cloud in an embodiment of the method of the present invention.

[0124] Figure 7 This is a schematic diagram comparing the registered mathematical model with the actual point cloud in an embodiment of the method of the present invention.

[0125] Figure 8 This is a schematic diagram of the mathematical model and spraying path (left) and the actual point cloud and spraying path (right) in an embodiment of the method of the present invention.

[0126] Figure 9 This is a schematic diagram comparing the theoretical spraying path and the actual spraying path in an embodiment of the method of the present invention. Detailed Implementation

[0127] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. This invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of this invention.

[0128] At the same time, it should be understood that the scope of protection of the present invention is not limited to the specific embodiments described below; it should also be understood that the terminology used in the embodiments of the present invention is for describing specific embodiments and not for limiting the scope of protection of the present invention.

[0129] Example: A method for aircraft skin deformation analysis and conformal optimization of spraying path

[0130] This method targets the skin frame structure of aircraft. Using a KUKA industrial robot as a carrier, it utilizes a 3D surface laser sensor to acquire skin shape information, providing point cloud data that conforms to the actual skin shape. Then, by analyzing the deformation of corresponding blocks in the point cloud data and the theoretical mathematical model, the theoretical spraying path is optimized, guiding the robot to complete the conformal spraying motion task and improving the adaptability of the processing.

[0131] The implementation system of this method is primarily software-based, with a hardware platform built for verification. The software system mainly includes a data analysis and processing system located on a host computer; the hardware platform mainly includes a 3D surface laser sensor, a KUKA robot, a spraying module, a laser tracker, and the structural components necessary for system assembly and functional implementation. Through the combination of the hardware platform and the software system, the implementation system of this method can achieve accurate extraction of processing references and real-time planning of spraying paths in both online and offline modes.

[0132] The measurement benchmark of this method system consists of a theoretical mathematical model and actual point cloud data, where the feature structure and its changes serve as a reference for the conformal coating process. Therefore, it is necessary to align / register the collected point cloud and the mathematical model before dividing the corresponding regions.

[0133] Overall, such as Figure 1 As shown, this method includes the following steps:

[0134] (I) Precise registration of actual point cloud data with mathematical models

[0135] In addition to commonly used registration methods, this scheme also introduces techniques such as bounding box registration, local feature registration, and deformation data registration to comprehensively register multi-source point cloud data with mathematical models. By combining multiple techniques for registration, the registration problem can be solved from multiple perspectives, thereby reducing computational load and improving accuracy in practical applications.

[0136] like Figure 2 As shown, the registration of actual point cloud data with a mathematical model is similar to the registration between point clouds. Both involve calculating the rotation and translation matrix between two point clouds to transform the source point cloud or mathematical model into the same coordinate system as the target point cloud or mathematical model. This can be expressed as the following equation:

[0137] p t =R·p s +T

[0138] Where, p t and p s It is a pair of corresponding points, where R is the rotation matrix between the point cloud and the mathematical model, and T is the translation vector.

[0139] To improve the registration accuracy between point cloud data and the mathematical model, coarse registration is required first, ensuring they coincide in the principal direction. The principal direction of the point cloud data is obtained by calculating the eigenvectors of all points, or by calculating bounding boxes. The principal direction of the mathematical model is obtained by calculating the eigenvectors of the vertices of the triangular facets. In addition to the principal direction, two secondary directions perpendicular to the principal direction can be obtained, allowing the establishment of a reference coordinate system with the centroid as the origin and the principal and secondary directions as the coordinate axes. Aligning the actual point cloud data with the reference coordinate system of the mathematical model achieves coarse registration between the point cloud data and the mathematical model. Let V... p V is the bounding box volume of the point cloud data. s V is the bounding box volume of the mathematical model. i Let λ be the volume of the bounding box obtained by intersecting the bounding box of the point cloud data and the bounding box of the mathematical model. Then the bounding box coincidence coefficient λ is:

[0140]

[0141] ε is a set value. If λ>ε, the point cloud data and the mathematical model are close to coincide. Otherwise, they are not close to coincide and still need to be optimized.

[0142] After coarse registration of point cloud data and mathematical model, fine registration is required. Assuming the target point set A corresponds to the reference point set S, the corresponding point sets should satisfy the following conditions: (1) the number of points in A is equal to the number of points in S; (2) for each point a in A... i They should all correspond to points s in S with the same subscript. i Let the rotation transformation vector be the unit quaternion q. R=[q0q1q2q3] t Where q0≥0, and q0 2 +q1 2 +q2 2 +q3 2 =1, thus obtaining a 3×3 rotation matrix R(q) R Let the translation transformation vector be q. T =[q4q5q6] t We can obtain the complete coordinate transformation vector q = [q R |q T ] t The problem of finding the optimal coordinate transformation vector between corresponding point sets is transformed into the problem of finding q that minimizes the function.

[0143] The algorithm is as follows:

[0144] (1) Obtain the corresponding point set A (actual point cloud data) and S (mathematical model);

[0145] (2) Calculate the centroids of A and S respectively:

[0146]

[0147] (3) Construct a 3×3 covariance matrix from two point sets:

[0148]

[0149] (4) Construct a 4×4 symmetric matrix from the covariance matrix:

[0150]

[0151] Where, tr(∑ as ) is a matrix ∑ as The trace, Δ=[P 23 P 31 P 12 ] T , I3 is a 3×3 identity matrix;

[0152] (5) Calculate Q(∑ as The eigenvalues ​​and eigenvectors, the eigenvector corresponding to the largest eigenvalue is the optimal rotation vector q. R =[q0q1q2q3] t ;

[0153] (6) Calculate the optimal translation vector:

[0154] q r =μ s -R(q R )μ a

[0155] in,

[0156]

[0157] (7) Perform coordinate transformation on A based on the rotation and translation vectors;

[0158] (8) Calculate the error to determine whether iteration is still needed. If the error is less than the set value, stop the iteration; otherwise, continue the iteration.

[0159] (9) Complete the fine registration.

[0160] (II) Regional division and corresponding region matching of point cloud data and mathematical models

[0161] Based on the mathematical model, a theoretical spraying path that conforms to the shape characteristics and process requirements of the mathematical model is pre-generated. Then, the region division method of the mathematical model is calculated by reverse calculation according to the theoretical spraying path and process parameters, such as... Figure 3 As shown, the generated region boundary lines or dividing lines divide the registered actual point cloud into multiple regions, so that the divided regions can accurately represent the shape and other information of the actual point cloud, and the actual point cloud of each region corresponds accurately with the mathematical model.

[0162] Since the mathematical model and the actual point cloud almost overlap in space after alignment / registration, the actual point cloud can be divided based on the region division boundary of the mathematical model. Figure 4 The display shows a mathematical model and actual point cloud of a certain local area. In actual calculations, data segmentation is not performed; the data used for calculation is only selected based on the results of the region division.

[0163] like Figure 4 The mathematical model and actual point cloud of a portion of the displayed area are divided into 7 pairs of grids, with each grid area divided into 500mm sections from left to right. The actual point cloud data within each grid is then fitted to a surface, and parameters such as the center point coordinates, area curvature, and surface normal vector are calculated.

[0164] (III) Calculation and analysis of deformation parameters of corresponding regions and the whole based on point cloud data and mathematical models

[0165] Based on the corresponding regions divided in step (II), calculate parameters such as center, curvature, and normal vector to analyze the local and overall deformation of the actual point cloud relative to the mathematical model.

[0166] Curvature is a measure of the degree of curvature of a curve. Mean curvature, as an "external" standard of curvature measurement in differential geometry, provides a local description of the curvature of a surface embedded in its surrounding space (e.g., a two-dimensional surface embedded in three-dimensional Euclidean space). Gaussian curvature, as a quantity describing the concavity and convexity of a surface, indicates that a large variation in this quantity also indicates a large variation within the surface, suggesting a lower degree of smoothness. For any point in a point cloud, there exists a surface z = r(x, y) that approximates the neighboring point cloud. The curvature at a point can be characterized by the curvature of the local surface fitted to that point and its neighbors. Through least-squares fitting, a quadratic surface can be used to characterize the local region, calculating the principal curvatures k1, k2, mean curvature H, and Gaussian curvature K at each point:

[0167]

[0168]

[0169]

[0170] Where L = r xx n;N=r yy n;E=r x r x F = r x r y G = r y r y ;r x r y r xx r yy r xy These are the partial differentials of the surface. E, F, and G are called the first fundamental invariants of the surface, and L, M, and N are called the second fundamental invariants of the surface.

[0171] Principal curvature, mean curvature, and Gaussian curvature are curvature concepts in mathematics. Surface curvature, on the other hand, is a concept that describes the degree of change of the surface of a point cloud using characteristic values ​​of the point cloud data, and it is different from curvature in the mathematical sense.

[0172] This method uses two curvature calculation methods to calculate the curvature of the regional point cloud, thereby obtaining parameters such as its surface normal vector. The first method uses quadratic surface fitting to calculate the point cloud curvature, and the second method uses the normal vectors of all points within a certain range of a point to calculate the point cloud curvature.

[0173] The method for finding the curvature of a point cloud by fitting a quadratic surface is to take a data point P in the scattered point cloud. i Then with P i Centered on a point cloud, select n points evenly distributed across the point cloud, ensuring these n points cover the entire point cloud as much as possible. Using these n points, fit a quadratic surface z(x, y) = ax using the least squares method.2 +bxy+cy 2 After obtaining the coefficients, the Gaussian curvature and mean curvature of the data points are calculated based on the properties of the spatial surface curve. For the m-neighborhood of any data point within the measurement point cloud, according to the least squares principle, the following formula needs to be minimized:

[0174]

[0175] In the formula, x j y j , z j Let a, b, and c be points in the neighborhood, and a, b, and c be the coefficients of the quadratic surface equation. Taking the derivative of the above equation with respect to the coefficients and setting them to zero, we obtain the following equation:

[0176]

[0177] Therefore, the coefficients of the quadratic surface can be solved. The equation of the curve can then be written in the form of a surface parametric equation:

[0178]

[0179] If there exists a curve r on the surface, where t is time and r is a spatial vector, then the expression for r is:

[0180] r = r(x(t), y(t))

[0181] If s represents the arc length of curve r, then the differential formula for arc length can be obtained from the chain rule:

[0182]

[0183] From the first fundamental formula of a surface, we can obtain:

[0184] (ds) 2 =I=E(dx) 2 +2Fdxdy+G(dy) 2

[0185] Where E, F, and G are the fundamental quantities of the first fundamental equation of the surface; if P is a point on the curve r, and t and n represent the unit tangent vector and unit normal vector of point P respectively, then the curvature vector can be decomposed as:

[0186]

[0187] The unit normal vector n of the curve is expressed as:

[0188]

[0189] The second fundamental formula for surfaces can be obtained:

[0190]

[0191] L, M, and N are the second kind of fundamental quantities of the surface; the normal curvature can be expressed as:

[0192]

[0193] By processing the variable λ, we can obtain k. n The two fundamental roots k1 and k2 can be obtained from the curvature characteristics:

[0194] Gaussian curvature:

[0195]

[0196] Mean curvature:

[0197]

[0198] Another approach is to calculate the curvature of a point using the normal vectors of neighboring points. The local shape of a point on a point cloud surface is determined by all its neighboring points. Estimating curvature through surface fitting can introduce significant errors. Therefore, the contribution of the normal vector should be considered. To estimate the curvature of a point, we will consider the contribution of only one neighboring point, which is transformed into the construction of a normal cross-section curve. We will construct a normal cross-section circle and estimate the normal curvature based on the positions and normal vectors of two points (the target point and one of its neighbors).

[0199] The normal curvature can be calculated using a local fitting method. For each point p in the point cloud, let N be the unit normal vector at point p. The method for estimating the normal curvature at point p using the point coordinates and the normal vector is as follows:

[0200] Assume that point p has m nearest neighbors, q i Let q be the i-th nearest neighbor of point p. i The normal vector is M i Let p, X, Y, z be an orthogonal coordinate system, and let L be a local coordinate system of point p. Let N denote the normal vector of point p, and let X and Y be orthogonal unit vectors. In L, p, q i M i The coordinates can be p(0, 0, 0) and q respectively. i (x i y i , z i M i (n x，i n y，i n z，i Then we can estimate the normal curvature of point p using a osculating circle passing through point p. Figure 5 The geometric relationships between these variables are shown.

[0201] Then p relative to q i The normal curvature is estimated as follows:

[0202]

[0203] In the formula, α represents vectors -N and pq. i The angle between them, β is the angle between vectors N and M i The angle between them.

[0204] Calculating the actual point cloud normal vector within the corresponding region and determining the normal of a point on the surface is approximately equivalent to estimating the normal of a tangent plane fitted to the surface by nearby points. Therefore, after transformation, it becomes a least-squares plane fitting problem. Thus, the solution for estimating the surface normal becomes analyzing the eigenvectors and eigenvalues ​​of a covariance matrix (or using Principal Component Analysis (PCA)). This covariance matrix is ​​created from the nearest neighbors of the query point. The largest principal component is the direction with the largest variance (the direction with the most information) after projection onto a certain direction; while the normal vector is the direction with the smallest information after projection onto a certain direction. Therefore, PCA transformation is needed to project the point cloud onto the direction with the smallest eigenvalue. Three important concepts are involved: the definition of a plane and the distance from a point to a line, least squares method and principal component analysis (PCA), and the covariance matrix, its eigenvectors, and eigenvalues.

[0205] Steps to solve for the normal vector:

[0206] The target plane can be defined by a point x and a normal vector. Let P represent a point P. i ∈P k The distance to the plane is: That is, take a point on the plane, and the distance from a point outside the plane to the plane; this distance is the projection of the vector formed by these two points onto the normal vector. If the distance from the point to the plane is 0, i.e., d... i If the value is 0, the normal vector of the plane can be found. Therefore, the following steps can be taken:

[0207] (1) Find the centroid P of all points. k

[0208]

[0209] (2) Calculate the centroid P k The covariance matrix C∈R 3×3 , and its eigenvectors and eigenvalues

[0210]

[0211] Where ∈ is usually 1, C is a symmetric and positive semi-definite matrix and its eigenvalues ​​are real numbers λ. j ∈R, eigenvectors An orthogonal coordinate system is formed, corresponding to the centroid P. k Principal components. If 0 ≤ λ0 ≤ λ1 ≤ λ2, the eigenvector with the smallest eigenvalue λ0 is... It is the normal vector or Approximation, or Can be used in spherical coordinates express:

[0212]

[0213] Principal component analysis (PCA) is used to calculate the orientation, but it exhibits bivariate properties, making it impossible to consistently orient the normal directions across the entire point cloud dataset. A solution to this problem is to use viewpoint V. P For all normals Orientation simply requires aligning them all towards the viewpoint, satisfying the following equation:

[0214]

[0215] For any point p i We calculate the covariance matrix from the following formula:

[0216]

[0217] Where k is p i The number of nearest neighbors of a point Let λ represent the three-dimensional centroid of the nearest neighbor point. j V represents the j-th eigenvalue of the covariance matrix. j Let represent the j-th eigenvector.

[0218] Generally, since there is no mathematical method to solve for the sign of the normal, the direction of the normal calculated by principal component analysis is ambiguous and inconsistent across the entire point cloud dataset. Figure 6 This figure shows the effects of these effects on two parts of a larger dataset representing a portion of a kitchen environment. The figure displays an Extended Gaussian Image (EGI), also known as a normal sphere, which describes the orientation of all normals in the point cloud. Since the dataset is 2.5-dimensional and thus acquired from a single viewpoint, the normals should only appear on more than half of the sphere in the EGI; however, due to inconsistent orientation, they are distributed across the entire sphere.

[0219] If the viewpoint Vp is actually known, to make all normal vectors n i All of them are oriented towards the viewpoint and must satisfy the following equation:

[0220]

[0221] Estimating the normal vector of a point on a surface requires finding its nearest neighbors. The details of the nearest neighbor estimation problem raise the question of the correct scaling factor: given a point cloud dataset, choosing an appropriate nearest neighbor value (k) is crucial and constitutes a constraint in the automatic estimation of point feature representations (i.e., without a user-given threshold). To better illustrate this problem, Figure 6 It demonstrates the effect of choosing a small scale to compare with a large scale.

[0222] The estimated surface normals are approximately perpendicular to both planes, and small edges are visible across the entire table. However, if the scale factor is too large (right side), and therefore the set of nearest neighbors is larger than the coverage points of adjacent surfaces, the estimated point feature representation will be distorted, rotating the surface normals at the edges of the two planar surfaces and smearing the edges and suppressing fine details. Without delving too much into detail, let's assume for now that it's sufficient to choose a scale for determining the neighborhood of points based on the level of detail required by the application. Simply put, if the curvature of the edge between the cup handle and the cylindrical section is important, the scale factor needs to be small enough to capture those details; otherwise, it needs to be large enough. The output surface curvature estimate is the relationship between the eigenvalues ​​of the covariance matrix:

[0223]

[0224] Using the methods described above, the midpoint, curvature, and normal vector, among other relevant parameters, are calculated for each pair of mathematical models and actual point cloud meshes. Figure 7 The image shows the center point of each pair of mathematical models and actual point clouds. It can be seen that due to the shape difference between the mathematical model and the actual point cloud, i.e., the skin has been deformed, the center point obtained by the mathematical model and the actual point cloud at the corresponding position differs in the Z-axis direction (distance / depth direction) of the sensor coordinate system.

[0225] (iv) Path conformal optimization based on point cloud deformation parameters

[0226] The local deformation parameters and overall deformation parameters calculated in step (iii) are matched locally and integrated globally, and then attached to the theoretical path pre-generated based on the mathematical model. The path is then optimized and corrected to obtain the actual spraying path that conforms to the actual point cloud shape.

[0227] Depend on Figure 7 As can be seen from left to right, the shape, mean, and other features of the point cloud data and mathematical model within the corresponding grids are all changing, and the difference between the two is not fixed, meaning the deformation is not constant. The multivariate changes of each grid in the entire skin are used as parameters to guide the deformation optimization path. These parameters are then appended to the theoretical spraying path pre-generated based on the mathematical model, resulting in an actual spraying path that better matches the actual shape of the skin after deformation optimization.

[0228] like Figure 8 As shown, the left side displays the mathematical model and the theoretical spraying path pre-generated based on the mathematical model, while the right side displays the actual collected point cloud data of the deformed skin, and the actual spraying path after the mathematical model and actual point cloud deformation parameters are appended to the theoretical spraying path to complete the conformal path optimization. It can be seen that the actual spraying path optimized based on the deformation parameters strictly follows the shape of the actual point cloud data, and the spatial relationships such as the spacing between points on the path and their distance from the actual point cloud are similar to the spatial relationships of the mathematical model and its theoretical path.

[0229] like Figure 9 As shown, by comparing the actual point cloud with the mathematical model and their respective paths, it can be seen that the actual spraying path accurately responds to the skin deformation measured by the actual point cloud based on the theoretical spraying path, and obtains an actual spraying path that fits the actual shape.

[0230] While the steps, formulas, and parameter types for deformation analysis and parameter calculation of actual point clouds and mathematical models can be designed and optimized manually according to actual needs, the actual calculation and analysis process is automated. It can perform adaptive analysis and calculations with high accuracy for various situations, including different positions, poses, shapes, sizes, and deformations. Furthermore, after automated deformation calculation and analysis, automated actual path optimization is performed, automatically optimizing the path according to the deformation parameters of the corresponding mesh, thereby automatically generating the actual path. This process requires no manual parameter setting, robustly adapts to different deformation conditions, and significantly reduces manual costs.

[0231] (V) Overall processing of the path after contour optimization

[0232] The actual spraying path after conformal optimization is processed as a whole, and the path is smoothed and fitted. According to the requirements of robot running speed, the path is fitted and interpolated. Adaptive intelligent planning is performed on multiple non-connected paths. Multiple paths are connected according to the shortest distance, robot limit and other rules. The timing of end spraying switch and paint bucket replacement is planned to make reasonable use of paint, improve spraying efficiency and quality, avoid paint waste and reduce environmental pollution.

[0233] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to obtain equivalent embodiments without departing from the scope of the technical solution of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the technical solution of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for aircraft skin deformation analysis and conformal optimization of spraying path, characterized in that, The method includes: S1. Accurately register the actual point cloud data with the mathematical model, including: S11. First, perform coarse registration to ensure that the actual point cloud data and the mathematical model coincide in the principal direction; the coarse registration includes: setting Let the bounding box volume of the point cloud data be . Let V be the bounding box volume of the mathematical model. Let the bounding box obtained by intersecting the bounding box of the point cloud data and the bounding box of the mathematical model be the bounding box overlap coefficient. for: ， For the set value, if If the point cloud data is close to the mathematical model, then the point cloud data is close to the model; otherwise, it is not close to the model and still needs to be optimized. S12. After completing the coarse registration of the actual point cloud data and the mathematical model, further fine registration is performed; The target point set A corresponds to the reference point set S, and the corresponding point set should satisfy the following conditions: (1) The number of points in A is equal to the number of points in S; (2) For each point in A They should all correspond to points in S with the same subscript. ; Let the rotation transformation vector be a unit quaternion. ,in, ,and We can obtain a 3×3 rotation matrix. ; Let the translation transformation vector be... The complete coordinate transformation vector can be obtained. ; The problem of finding the optimal coordinate transformation vector between corresponding point sets is transformed into finding... The problem of minimizing the function; S2. Perform region division and corresponding region matching on the actual point cloud data and mathematical model, including: pre-generating a theoretical spraying path that conforms to the shape characteristics and process requirements of the mathematical model based on the mathematical model; calculating the region division method of the mathematical model in reverse according to the theoretical spraying path and process parameters; generating region boundary lines or dividing lines; dividing the registered actual point cloud into multiple regions; making the divided regions accurately express the shape information of the actual point cloud; and making the actual point cloud of each region accurately correspond to the mathematical model. S3. Calculate and analyze the deformation parameters of the corresponding regions and the overall model based on the actual point cloud data; S4. The actual point cloud deformation parameters obtained in the previous step are matched locally and integrated as a whole, and then attached to the theoretical spraying path pre-generated based on the mathematical model. The path is then optimized and corrected to obtain the actual spraying path that conforms to the actual point cloud shape. S5. Perform overall processing on the actual spraying path obtained after conformal optimization.

2. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 1, characterized in that, The precise registration of the actual point cloud data and the mathematical model described in step S1 is achieved by calculating the rotation and translation matrix between the two point clouds, transforming the source point cloud or mathematical model to the same coordinate system as the target point cloud or mathematical model. This can be expressed as the following equation: ， in, and It is a pair of corresponding points, where R is the rotation matrix between the point cloud and the mathematical model, and T is the translation vector.

3. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 1, characterized in that, Step S11 specifically includes: the main direction of the point cloud data is obtained by calculating the feature vectors of all points or by calculating the bounding box; the main direction of the mathematical model is obtained by calculating the feature vectors of the vertices of the triangular facets; in addition to the main direction, two secondary directions perpendicular to the main direction are further obtained, thereby establishing a reference coordinate system with the centroid as the origin and the main and secondary directions as the coordinate axes; by adjusting the reference coordinate system of the actual point cloud data and the mathematical model to be consistent, coarse registration between the actual point cloud data and the mathematical model can be achieved. The specific algorithm in step S12 is as follows: (1) Obtain the actual point cloud dataset A and the mathematical model set S corresponding to the point set; (2) Calculate the centroids of A and S respectively: ， (3) Construct a 3×3 covariance matrix from two point sets: ， (4) Construct a 4×4 symmetric matrix from the covariance matrix: ， in, It is a matrix traces, , , It is a 3×3 identity matrix; (5) Calculation Eigenvalues ​​and eigenvectors, the eigenvector corresponding to the largest eigenvalue is the optimal rotation vector. ; (6) Calculate the optimal translation vector: ， in, ， (7) Perform coordinate transformation on A based on the rotation and translation vectors; (8) Calculate the error to determine whether iteration is still needed. If the error is less than the set value, stop the iteration; otherwise, continue. Continued iteration; (9) Complete the fine registration.

4. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 1, characterized in that, The calculation and analysis of the actual point cloud data and the corresponding regions and overall deformation parameters of the mathematical model in step S3 includes performing surface fitting on the actual point cloud data in each region according to the corresponding regions divided in step S2, calculating the coordinates of its center point, the curvature of the regional point cloud and the surface normal vector parameters, so as to analyze the local deformation and overall deformation of the actual point cloud relative to the mathematical model.

5. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 4, characterized in that, The curvature of the point cloud and the surface normal vector of the region are obtained by either of the following two methods: (a) Using quadratic surface fitting to determine the curvature of point clouds Take a data point from the scattered point cloud. , and then with Take uniform samples from the point cloud with the center as the center. This point, Each point should cover the entire point cloud as much as possible; a quadratic surface is fitted using the least squares method. After obtaining the coefficients, calculate the Gaussian curvature and mean curvature of the data points based on the properties of the spatial surface curve; For any data point within the measurement point cloud According to the least squares principle, the following expression should be minimized in the neighborhood: ， ； In the formula, Let a, b, and c be points in the neighborhood, and a, b, and c be the coefficients of the quadratic surface equation. Taking the derivative of the above equation with respect to the coefficients and setting them to zero, we obtain the following equation: ； Therefore, the coefficients of the quadratic surface can be solved; the equation of the curve can then be written in the form of a surface parametric equation: ； If there exists a curve on the surface It is time. If it is a spatial vector, then The expression is: ； If we take Represents curve Given the arc length, the differential formula for the arc length can be obtained using the chain rule for the chain rule: ； From the first fundamental formula of a surface, we can obtain: ； Where E, F, and G are the fundamental quantities of the first fundamental form of the surface; if It is a curve A little bit above, using and They represent Given the unit tangent vector and unit normal vector of a point, the curvature vector can be decomposed into: ； unit normal vector of the curve Represented as: ； The second fundamental formula for surfaces can be obtained: ； L, M, N are the second kind of fundamental quantities of the surface; the normal curvature can be expressed as: ； Processing variables available Two basal roots Therefore, based on the curvature characteristics, we can obtain: Gaussian curvature: ； Mean curvature: ； (ii) Calculate the curvature of the point cloud by combining the normal vectors of all points within the adjacent range of a certain point. To estimate the curvature of a point, we only consider the contribution of a neighboring point and convert it into the construction of a positive cross-section curve; Construct a normal cross-section circle and estimate the normal curvature based on the position and normal vector of the target point and its adjacent points; Calculated using a local fit of the normal curvature; for each point in the point cloud. ,set up The unit normal vector of the point is Use point coordinates and normal vector to estimate points The method for determining the normal curvature at a given point is as follows: Assumption There are points nearby Nearest neighbor points, For point The Nearest neighbor points, The normal vector is Let an orthogonal coordinate system be established. for Local coordinate system of a point express The normal vector of a point and They are orthogonal unit vectors; in middle, The coordinates are respectively , , ; using a point To estimate the point using the close circle normal curvature ; but Compared to The normal curvature is estimated as follows: ； In the formula, It is a vector and The angle between them It is a vector and The angle between them.

6. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 5, characterized in that, Step S3, which involves calculating and analyzing the deformation parameters of the corresponding regions and the overall model based on the actual point cloud data, further includes: Calculating the actual point cloud normal vector within the corresponding region transforms the problem of determining the normal of a point on the surface into a least-squares plane fitting problem. Therefore, the solution for estimating the surface normal becomes analyzing the eigenvectors and eigenvalues ​​of a covariance matrix, which is created from the nearest neighbors of the query point. The maximum principal component is the direction with the largest variance after projection onto a certain direction, while the normal vector is the direction with the smallest information after projection onto a certain direction. Therefore, principal component analysis (PCA) transformation is required to project the point cloud onto the direction with the smallest eigenvalue. Steps to solve for the normal vector: The target plane consists of a point and a normal vector To represent, and a point The distance to the plane is: That is, take a point on the plane, and the distance from a point outside the plane to the plane is the projection of the vector formed by these two points onto the normal vector; if the distance from the point to the plane is 0, that is... Then, the normal vector of the plane can be found; therefore, the following steps are taken: (1) Find the centroid of all points. : ； (2) Calculate the centroid covariance matrix , and its eigenvectors and eigenvalues: ； in, =1, It is a symmetric and positive semi-definite matrix and its eigenvalues ​​are real numbers. eigenvectors An orthogonal coordinate system is formed, corresponding to the center of mass. Principal components; if Minimum eigenvalue eigenvectors It is the normal vector or Approximation, or In spherical coordinates express: ； Principal component analysis (PCA) is used to calculate the orientation, but it exhibits bivariate properties, making it impossible to consistently orient the normal direction of the entire point cloud dataset. A solution to this problem is to use viewpoints. For all normals Orientation simply requires aligning them all towards the viewpoint, satisfying the following equation: ； For any point The covariance matrix is ​​calculated from the following formula: ； in, yes The number of nearest neighbors of a point Represents the three-dimensional centroid of the nearest neighbor points. Represents the covariance moments The j-th eigenvalue of the matrix, Let represent the j-th eigenvector.

7. The method for aircraft skin deformation analysis and conformal optimization of spraying path according to claim 1, characterized in that, Step S5 involves performing overall processing on the actual spraying path obtained after conformal optimization, including: Perform smooth optimization fitting on the path; Based on the robot's operating speed requirements, the path is fitted and interpolated. Adaptive intelligent planning is performed on multiple non-connected paths, and multiple paths are connected according to the shortest distance and robot limit rules; Plan the timing of the end-spraying switch and the timing of the paint bucket replacement.

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