Sphericity error evaluation method based on cone search algorithm
By combining the cone search algorithm with the minimum containment region method, auxiliary sphere and cone models are constructed, which solves the problem of insufficient accuracy of existing sphericity error assessment methods in precision manufacturing and achieves more efficient and higher accuracy sphericity error assessment.
Patent Information
- Application Number
- CN202510574269.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-06
- Publication Date
- 2026-01-30
AI Technical Summary
Existing methods for assessing sphericity error are insufficient to meet the increasingly stringent precision requirements in the precision manufacturing industry, especially in fields such as ultra-precision machining and aerospace. More efficient and higher-precision assessment methods are needed.
A sphericity error assessment method based on the cone search algorithm is adopted. By constructing auxiliary sphere and cone models, and combining the least squares sphere center coordinates and the minimum bounding region method, a fine search is performed to improve the search accuracy and efficiency, and finally output accurate sphericity error assessment results.
It significantly improves the accuracy and efficiency of sphericity error assessment, has better stability, and the calculation results are superior to traditional methods, meeting the high-precision requirements of the precision manufacturing industry.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of error detection for spherical parts, and specifically to a method for evaluating sphericity error based on a conic search algorithm. Background Technology
[0002] Due to their unique geometric shape, spherical components are increasingly widely used in various industries, such as air bearings in ultra-precision machining and gyroscopes in aerospace. Spherical components play an important role in these applications. Furthermore, in the field of micro-nano coordinate measuring machines, most measuring machine probes use spherical probes. Therefore, research on spherical error assessment has significant practical implications and promotes the development of the precision manufacturing industry.
[0003] Current research on sphericity error assessment mainly focuses on the least squares method, the maximum inscribed method, the minimum bounding method, and the minimum containment region method. Both the establishment of assessment models and the research on search methods are based on these four principles. Among them, the sphericity error assessment based on the minimum containment region principle conforms to the definition of sphericity, meets international standards, and provides better assessment results, making it a current research focus. However, with the continuous development of the precision manufacturing industry, the requirements for sphericity error are becoming increasingly stringent, necessitating further research into more accurate sphericity error assessment methods. Summary of the Invention
[0004] The present invention aims to provide a sphericity error assessment method based on the conic search algorithm, which has the characteristics of high assessment efficiency and high accuracy.
[0005] The technical solution of the present invention is as follows:
[0006] The sphericity error assessment method based on the conic search algorithm includes the following steps:
[0007] A. Obtain the coordinates p of the sampling point. i (x i ,y i ,z i ), i = 1, 2, ..., n; calculate the coordinates of the center of the least squares sphere based on the coordinates of the collection points, and then calculate the least squares sphere error F;
[0008] B. Using the coordinates of the least squares sphere center as the coordinates of the auxiliary sphere center, construct an auxiliary sphere with radius F. Take the 6 intersection points of the auxiliary sphere with the x, y, and z coordinate axes, and the center of the least squares sphere, for a total of 7 points, as the first series of search points. The coordinates of the first series of search points are as follows: x a0 (a,b,c), x a1 (a+F,b,c), x a2 (aF,b,c), x a3 (a,b+F,c), xa4 (a,bF,c), x a5 (a,b,c+F), x a6 (a,b,cF);
[0009] Calculate p for each collection point i Calculate the distances to each of the seven search points; obtain the maximum and minimum distances from each search point to all collection points, and calculate the range between the maximum and minimum distances to obtain seven range values.
[0010] Find the coordinates of the search points corresponding to the minimum and second minimum values of the range, and let the coordinates of the search point corresponding to the minimum range be N = (x n ,y n ,z n The second minimum range corresponds to the search point coordinates M = (x m ,y m ,z m );
[0011] C. Based on the coordinates M and N obtained in step B, construct a cone model with coordinate M as the center of the cone base and coordinate N as the cone vertex, and F as the radius of the cone base.
[0012] Using a plane parallel to the base of the cone, the cone model is cut at equal intervals to obtain g cut surfaces, where the distance d between adjacent cut surfaces is the distance between points M and N divided by g;
[0013] Take m points evenly along the outer circumference of each cut surface as cone search points, and calculate the coordinates (x, y) of each cone search point. sq ,y sq ,z sq );
[0014] Using each cone search point and points M and N as the second series of search points, calculate the distance from the second series of search points to each acquisition point p. i (x i ,y i ,z i Given n points (i = 1, 2, ..., n), calculate the distance from each point in the second series of search points to these n collection points, obtaining n distances. Find the range between the maximum and minimum distances, obtaining n range values. Calculate the difference σ1 between the minimum and second minimum range values. If σ1 ≤ σ, where σ is a preset value, then the minimum and second minimum values are considered close. The cone search point corresponding to the minimum range value is then considered the optimal point, and this point is output as the quasi-sphere center O. e (x e ,y e ,z eIf σ1 > σ, update the cone search point corresponding to the second minimum range value to the new cone base center M, update the cone search point corresponding to the minimum range value to the new cone vertex N, reduce F to half its original value, keep other parameters unchanged, repeat this step, loop until the iteration termination condition is met, and output the cone search point corresponding to the minimum range value at this time as the quasi-sphere center O. e (x e ,y e ,z e );
[0015] D. Calculate p for each data collection point. i (x i ,y i ,z i ) to the center of the ball O e (x e ,y e ,z e Find the coordinates of the collection points corresponding to the maximum, second-maximum, second-second-maximum, second-minimum, and minimum distances, and denote them as D. 3+2 (x D ,y D ,z D W 3+2 (x W ,y W ,z W Z 3+2 (x Z ,y Z ,z Z ), I 3+2 (x I ,y I ,z I ), U 3+2 (x U ,y U ,z U ); with D 3+2 (x D ,y D ,z D W 3+2 (x W ,y W ,z W Z 3+2 (x Z ,y Z ,z Z ) as three control points on the outer sphere of the bounding sphere in the minimum bounding region method, with I 3+2 (x I ,y I ,z I ), U 3+2 (x U ,yU ,z U As two control points on the sphere within the bounding sphere using the minimum bounding region method, the value of f1 and the coordinates of the sphere's center O are calculated based on the following formula (1). od (x od ,y od ,z od ):
[0016]
[0017] E. Calculate p for each data collection point. i (x i ,y i ,z i ) to the center of the ball O e (x e ,y e ,z e Find the coordinates of the collection points corresponding to the maximum, second-maximum, second-second-minimum, second-minimum, and minimum distances, and denote them as t. 2+3 (x t ,y t ,z t ), k 2+3 (x k ,y k ,z k ), s 2+3 (x s ,y s ,z s ), c 2+3 (x c ,y c ,z c b 2+3 (x b ,y b ,z b ); with t 2+3 (x t ,y t ,z t ), k 2+3 (x k ,y k ,z k ) as two control points on the outer sphere of the bounding sphere in the minimum bounding region method, with s 2+3 (x s ,y s ,z s ), c 2+3 (x c ,y c ,z c b 2+3 (x b ,y b ,zb As three control points on the sphere within the bounded sphere using the minimum bounded area method, the value of f2 and the coordinates of the sphere's center O are calculated based on the following formula (2). ow (x ow ,y ow ,z ow ):
[0018]
[0019] F. Compare the values of f1 and f2, and output the minimum value as the sphericity error assessment result.
[0020] In step A, the coordinates of the center of the least squares sphere are calculated using the Levenberg-Marquardt nonlinear optimization method, and the calculation process is as follows:
[0021] Let the error function be e i (pi,h,r)=(x i -a) 2 +(y i -b) 2 +(z i -c) 2 -r 2 Where h = (a, b, c) are the coordinates of the sphere's center, and r is the radius of the sphere; the error vector e = [e1, e2, ..., e n ] T Regarding the parameter vector p = [a, b, c, r] T The Jacobian matrix J is an n×4 matrix whose elements
[0022] The iteration begins, given the initial estimate of the parameter vector p: p0 = [a0, b0, c0, r0]. T The centroid (a0, b0, c0) of the point cloud is calculated as a reference for the initial estimate of the sphere center. Then, the sphere radius is initially calculated by considering the degree of discretization of the point cloud.
[0023] In the k-th iteration:
[0024] Based on parameter vector p k =[a k ,b k ,c k ,r k ] T Calculate the Jacobian matrix J k .
[0025] Solve the system of linear equations Obtain the parameter update amount Δp k , where λ k It is the damping factor (take λ).k =0.01), I is a 4×4 identity matrix, e k Based on p k The calculated error vector;
[0026] Update parameter vector: p k+1 =p k -Δp k ;
[0027] Calculate Δp k Euclidean norm ||Δp k ‖, when ‖Δp k ‖<∈(∈ is a preset threshold, taken as ∈=1×10 -10 When p = [a, b, c, r], the iteration converges. T (a, b, c) in the equation are the coordinates of the sphere's center.
[0028] In step A, after obtaining the coordinates of the sphere center (a, b, c), the least squares sphere error F is calculated using the following formula:
[0029]
[0030] The formula for calculating the range is as follows:
[0031] ΔR iz =max{R iz}-min{R iz} (4)
[0032] Where R iz Represents the distance from the collection point to the search point, max{R iz} represents the maximum distance, min{R} iz} represents the minimum distance; i = 1, 2, 3, ..., n, z = 0, 1, 2, 3, 4, 5, 6.
[0033] Cone search point (x) sq ,y sq ,z sq The formula for calculating coordinates is:
[0034]
[0035] in, u and v are orthogonal unit vectors; R s =F(gs-1) / g, C=M+(gs-1)(NM) / g, s=1,2,…,g, q=1,2,…,m.
[0036] The calculation process for orthogonal unit vectors u and v is as follows:
[0037] Let the direction vector of the cone's axis be w = (x n -x m ,y n -y m ,z n -z m Let t = (1,0,0) be the unit vector in the positive x-axis direction, and calculate u = w × t; if the calculated result of u is not a zero vector, then calculate v = w × u; if the calculated result of u is a zero vector, then replace t with b = (1,0,0), which is the unit vector in the positive y-axis direction, recalculate u = w × b, and then calculate v = w × u.
[0038] The beneficial effects of this invention are as follows:
[0039] This invention constructs a unique conic search algorithm. Through an innovative search strategy combining an auxiliary sphere and a conic model, the search point is controlled to be near the optimal point, effectively improving search accuracy and efficiency. Simultaneously, by combining the minimum encompassing region method structure, accurate sphericity error is calculated, further enhancing the stability and accuracy of the assessment. This invention establishes a new method for assessing sphericity error using a minimum region, which can further improve the accuracy of sphericity error assessment and has promising application prospects. Attached Figure Description
[0040] Figure 1 A schematic diagram of the auxiliary ball position established for Example 1;
[0041] Figure 2 A schematic diagram of the distribution of auxiliary ball search points established in Example 1;
[0042] Figure 3 This is a schematic diagram of the conical cross-section established in this invention;
[0043] Figure 4 This is a schematic diagram of the cone search point distribution established in this invention. Detailed Implementation
[0044] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.
[0045] Example 1
[0046] The sphericity error assessment method based on the conic search algorithm provided in this embodiment, taking the data in Reference 1 as an example, as shown in Table 1, includes the following steps:
[0047] Table 1 Data from Reference 1
[0048]
[0049]
[0050] Step 1: Calculate the coordinates of the center of the least squares sphere and the least squares sphere error F. Let the coordinates of the acquisition point be p. i (x i ,y i ,z i (i = 1, 2, ..., 25). The coordinates of the center of the least squares sphere are calculated using the Levenberg-Marquardt nonlinear optimization method. Before the iteration begins, the collected points are preprocessed to calculate the centroid of the point cloud: Preliminary estimate of sphere radius
[0051] Let the error function be e i (pi,h,r)=(x i -a) 2 +(y i -b) 2 +(z i -c) 2 -r 2 Where h = (a, b, c) are the coordinates of the sphere's center, and r is the sphere's radius. Error vector e = [e1, e2, ..., e 25 ] T Regarding the parameter vector p = [a, b, c, r] T The Jacobian matrix J is a 25×4 matrix, and its elements are...
[0052] In the k-th iteration:
[0053] Based on parameter vector p k =[a k ,b k ,c k ,r k ] T Calculate the Jacobian matrix J k .
[0054] Solve the system of linear equations Obtain the parameter update amount Δp k , where λ k It is the damping factor (take λ). k =0.01), I is a 4×4 identity matrix, e k Based on p k The calculated error vector.
[0055] Update parameter vector: p k+1 =p k -Δp k .
[0056] Calculate Δp k Euclidean norm ||Δp k ‖, when ‖Δp k‖<∈(∈ is a preset threshold, taken as ∈=1×10 -10 When p = [a, b, c, r], the iteration converges. T (a, b, c) in the equation are the coordinates of the sphere's center.
[0057] The least-squares sphere center (a, b, c) obtained through the above iterations is (-0.25759679, -0.66103733, -0.01354323). After obtaining the sphere center coordinates, the least-squares sphere error F is calculated using the following formula:
[0058]
[0059] The least squares ball error F is obtained as 3.6670783 mm.
[0060] Step 2: Construct an auxiliary sphere and perform a coarse search. Using the least-squares center (-0.25759679, -0.66103733, -0.01354323) as the center of the auxiliary sphere, and a radius of F = 3.6670783 mm, construct an auxiliary sphere. The positions of the auxiliary spheres are as follows: Figure 1 As shown, the six intersection points of the auxiliary sphere with the x, y, and z coordinate axes, and the center of the least squares sphere, totaling seven points, are taken as the first series of search points. The coordinates of the first series of search points are x, y, and z respectively. a0 (a,b,c), x a1 (a+F,b,c), x a2 (aF,b,c), x a3 (a,b+F,c), x a4 (a,bF,c), x a5 (a,b,c+F), x a6 The distribution of search points on the auxiliary sphere is as follows: (a, b, cF) Figure 2 As shown.
[0061] Calculate the collection point p i The distance R to these 7 search points iz And find the maximum distance max{R}. iz} and minimum distance min{R iz The range ΔR iz The calculation formula is as follows:
[0062] ΔR iz =max{R iz}-min{R iz} (4)
[0063] Where i = 1, 2, 3, ..., 25, z = 0, 1, 2, 3, 4, 5, 6.
[0064] Find ΔR izThe minimum and second minimum values correspond to the coordinates of the search points. Let ΔR be the coordinates of these points. iz The minimum value corresponds to the search point coordinates N = (x n ,y n ,z n ), ΔR iz The search point coordinates corresponding to the second minimum value are M = (x m ,y m ,z m ).
[0065] Step 3: Construct the cone model and perform a fine search, using the M = (x) obtained in Step 2. m ,y m ,z m ), N = (x n ,y n ,z n Let F be the center of the cone's base, M, and the vertex, N, respectively. The radius of the cone's base is F = 3.6670783 mm. Set the number of sections g = 5, and the number of search points per section m = 5. The intercepted distance d is the distance between points M and N divided by the number of sections g. The cross-sections of the cone model are as follows... Figure 3 As shown, the search point for the conical section is as follows: Figure 4 As shown, parameters are introduced. R s =F(gs-1) / g, C = M + (gs-1)(NM) / g, where s = 1, 2, ..., g, q = 1, 2, ..., m. Cone search point (x sq ,y sq ,z sq The formula for calculating coordinates is:
[0066]
[0067] Let the direction vector of the cone's axis be w = (x n -x m ,y n -y m ,z n -z m Let t = (1,0,0) be the unit vector in the positive x-axis direction, and calculate u = w × t. If the calculated result of u is not a zero vector, then calculate v = w × u; if the calculated result of u is a zero vector, then replace t with b = (1,0,0) (the unit vector in the positive y-axis direction), recalculate u = w × b, and then calculate v = w × u. Normalize u and v:
[0068] Each time, the conical search point is added to the search points M and N corresponding to the previous minimum and second minimum values, resulting in a total of mg+2 points, which are used as the second series of search points; the second series of search points are then compared with all collected points p. i (xi ,y i ,z i The maximum and minimum distances between (i = 1, 2, ..., 41) and the range of the maximum and minimum distances calculated based on formula (4) are used to calculate the difference σ1 between the current minimum and the second minimum range. If σ1 ≤ σ (σ takes the value of 0.000000001mm), then the minimum and the second minimum are considered to be close, and the search is completely converged. At this time, the cone search point corresponding to the minimum range is the optimal point, and this point is output as the quasi-sphere center O. e (x e ,y e ,z e If σ1 > σ, update the cone search point corresponding to the second minimum range value to the new cone base center M, update the cone search point corresponding to the minimum range value to the new cone vertex N, reduce F to half its original value, keep other parameters unchanged, and repeat step 3 until the iteration termination condition is met. Output the cone search point corresponding to the minimum range value at this time as the quasi-sphere center O. e (x e ,y e ,z e ).
[0069] The coordinates of the quasi-sphere center (x) were obtained through calculation. e ,y e ,z e The values are (-0.3881881479, -0.3581302075, -0.2988156113).
[0070] Step 4: Calculate the collection point p i (x i ,y i ,z i ) to the center of the ball O e Find the coordinates of the collection points corresponding to the maximum, second-maximum, second-second-maximum, second-minimum, and minimum distances, and denote them as D. 3+2 (x D ,y D ,z D W 3+2 (x W ,y W ,z W Z 3+2 (x Z ,y Z ,z Z ), I 3+2 (x I ,y I ,z I ), U 3+2 (x U ,yU ,z U Calculate the value of f1 and the coordinates of the sphere's center O. od (x od ,y od ,z od ), collect point p i Substituting into the following formula, we obtain:
[0071]
[0072] The coordinates of the sphere's center (x) are calculated. od ,y od ,z od The range is (-0.4793784466, 0.0873316049, -0.4794599725), and the minimum value of f1 is 3.7057458235 mm.
[0073] Step 5: Calculate the collection point p i To the center of the ball O e Find the coordinates of the collection points corresponding to the maximum, second-maximum, second-second-minimum, second-minimum, and minimum distances, and denote them as t. 2+3 (x t ,y t ,z t ), k 2+3 (x k ,y k ,z k ), s 2+3 (x s ,y s ,z s ), c 2+3 (x c ,y c ,z c b 2+3 (x b ,y b ,z b Calculate the value of f2 and the coordinates of the sphere's center O. ow (x ow ,y ow ,z ow ), collect point p i The following formula is used to obtain it:
[0074]
[0075] The coordinates of the sphere's center (x) are calculated. ow ,y ow ,z owThe range is (-0.3887296494, -0.3554883447, -0.2998875923), and the minimum value of f2 is 3.3325176161635426mm.
[0076] Step 6: Determine the magnitudes of f1 and f2, and output the minimum value of the two, 3.3325176161635426mm, as the sphericity error assessment result.
[0077] As shown in Table 2, the sphericity error assessment results calculated by the method of the present invention are significantly better than those of the least squares method, Reference 1, Reference 2 and Reference 3, which confirms the effectiveness of the present method.
[0078] Table 2 Data Processing Results
[0079]
[0080]
[0081] The data in this implementation case references data from Reference 1, "Sphericity Error Assessment Based on Genetic Algorithm," Cui Changcai, Che Rensheng, Ye Dong, et al., Optics and Precision Engineering, Vol. 0, No. 04, pp. 333-339.
[0082] The comparison data for this implementation case are cited from references 2 and 3. Reference 2: "A new algorithm and program for sphericity error, Lin Xiang, Journal of Fujian Commercial College, Vol. 0, No. 5, pp. 100-102"; Reference 3: "Sphericity error evaluation: theoretical derivation and algorithm development, Wang M, Cheraghi HS, Masud SA, IIE Transactions, Vol. 33, No. 4, pp. 281-292".
Claims
1. A method for sphericity error evaluation based on a conic search algorithm, characterized in that, The method comprises the following steps: A, sampling to obtain the collection point coordinates p i (x i ,y i ,z i ), i = 1, 2, …, n; based on the collection point coordinates, the least squares sphere center coordinates are calculated, and then the least squares sphere error F is calculated; B, with the least square sphere center coordinates as the auxiliary sphere center coordinates, an auxiliary sphere is constructed with F as the radius, and the 6 intersection points of the auxiliary sphere and the x, y, z coordinate axes and the least square sphere center are taken as 7 first series search points, and the first series search point coordinates are as follows: x a0 (a,b,c), x a1 (a+F,b,c), x a2 (a-F,b,c), x a3 (a,b+F,c), x a4 (a,b-F,c), x a5 (a,b,c+F), x a6 (a,b,c-F); Calculating the respective acquisition points p i the distances to the 7 search points respectively; Obtaining the maximum distance and the minimum distance from each search point to all acquisition points, calculating the range of the maximum distance and the minimum distance, and obtaining 7 range values; Find the minimum value and the second minimum value of the range, and the corresponding search point coordinates, let the minimum value of the range correspond to the search point coordinates N=(x n ,y n ,z n ), and the second minimum value of the range correspond to the search point coordinates M=(x m ,y m ,z m ); C. Based on the coordinates M and N calculated in step B, a conical model is constructed, with the coordinate M as the center of the conical base, the coordinate N as the apex of the cone, and F as the radius of the conical base; Using a plane parallel to the conical base, the conical model is cut at equal distances to obtain g cut surfaces, wherein the distance d between adjacent cut surfaces is the distance between points M and N divided by g; Take m points evenly along the outer circumference of each cut surface as cone search points, and calculate the coordinates (x, y) of each cone search point. sq ,y sq ,z sq ); Take each conic search point and points M, N as the second series of search points, calculate the distance between the second series of search points to each acquisition point p i (x i ,y i ,z i )(i = 1, 2, …, n) distance, calculate the distance between each point in the second series of search points and the n acquisition points, get n distances, find the maximum distance and the minimum distance, get n range values, calculate the difference between the minimum value and the second minimum value σ1, if σ1≤σ, σ is a preset value, then the minimum value and the second minimum value are close, at this time the minimum range value corresponding to the conic search point is the optimal point, output this point as the quasi-sphere center O e (x e ,y e ,z e ); if σ1>σ, update the conic search point corresponding to the second minimum range value as the new conic base circle center M, update the conic search point corresponding to the minimum range value as the new conic apex N, reduce F to half of the original, other parameters remain unchanged, repeat the step, and cycle until the iteration termination condition is met, output the conic search point corresponding to the minimum range value at this time as the quasi-sphere center O e (x e ,y e ,z e ); D, calculate the distance of each collection point p i (x i ,y i ,z i ) to the quasi-sphere center O e (x e ,y e ,z e ), find the maximum distance, the second maximum distance, the third maximum distance, the second minimum distance and the minimum distance corresponding to the collection point coordinates, respectively recorded as D 3+2 (x D ,y D ,z D ), W 3+2 (x W ,y W ,z W ), Z 3+2 (x Z ,y Z ,z Z ), I 3+2 (x I ,y I ,z I ), U 3+2 (x U ,y U ,z U ); D 3+2 (x D ,y D ,z D ), W 3+2 (x W ,y W ,z W ), Z 3+2 (x Z ,y Z ,z Z ) as the three control points on the outside of the sphere, I 3+2 (x I ,y I ,z I ), U 3+2 (x U ,y U ,z U ) as the two control points on the inside of the sphere, based on the following formula (1) to calculate the value of f1 and the sphere center coordinates O od (x od ,y od ,z od ): E、Calculate the distance of each acquisition point p i (x i ,y i ,z i ) to the center of the ball O e (x e ,y e ,z e ), find the maximum distance, the second maximum distance, the third minimum distance, the second minimum distance and the minimum distance corresponding to the acquisition point coordinates, respectively recorded as t 2+3 (x t ,y t ,z t ), k 2+3 (x k ,y k ,z k ), s 2+3 (x s ,y s ,z s ), c 2+3 (x c ,y c ,z c ), b 2+3 (x b ,y b ,z b ); t 2+3 (x t ,y t ,z t ), k 2+3 (x k ,y k ,z k ) as the minimum containing area method containing two control points on the outside of the ball, s 2+3 (x s ,y s ,z s ), c 2+3 (x c ,y c ,z c ), b 2+3 (x b ,y b ,z b ) as the minimum containing area method containing three control points on the inside of the ball, calculate the f2 value and the center coordinates O ow (x ow ,y ow ,z ow ) based on the following formula (2): F. Comparing the size of f1 and f2, and outputting the minimum value as the sphericity error evaluation result.
2. The sphericity error evaluation method based on the conical search algorithm according to claim 1, wherein: In step A, the least square sphere center coordinates are calculated by the Levenberg-Marquardt nonlinear optimization method, and the calculation process is as follows: Let the error function e i (pi, h, r) = (x i -a) 2 +(y i -b) 2 +(z i -c) 2 -r 2 where h = (a, b, c) is the center coordinate of the sphere and r is the radius of the sphere; the error vector e = [el, e2,..., en]Tis the vector of the errors of the n points pi, i = 1, 2,..., n. n T The Jacobian matrix J with respect to the parameter vector p = [a, b, c, r] T is an n x 4 matrix whose elements The iteration starts with an initial estimate of the parameter vector p, p0= [a0, b0, c0, r0] T The center of mass (a0, b0, c0) of the point cloud is computed as a reference for the initial estimate of the sphere center The sphere radius is then initially computed in combination with the dispersion of the point cloud In the kth iteration: According to the parameter vector p k = [a k , b k , c k , r k ] T , the Jacobian matrix J k is calculated. Solving linear equation systems Obtaining parameter updates Δp k where λ k is a damping factor (take λ k = 0.01), I is a 4x4 identity matrix, e k is the error vector based on p k calculation; Update parameter vector: p k+1 = p k - Δp k ; Compute Δp k Euclidean norm of Δp k ‖, when ‖Δp k ‖ < ∈ ( ∈ is a preset threshold, take ∈ = 1 × 10 -10 ), iteration converges, at this time p = [a, b, c, r] T (a, b, c) in (a, b, c, r) is the spherical center coordinate.
3. The sphericity error evaluation method based on the conical search algorithm according to claim 2, wherein: In step A, after obtaining the sphere center coordinates (a, b, c), the least square sphere error F is calculated according to the following formula:
4. The sphericity error evaluation method based on the conical search algorithm according to claim 1, wherein: The calculation formula of the range is as follows: ΔR iz = max{R iz} - min{R iz} (4) where R iz represents the distance from the collection point to the search point, max{R iz} represents the maximum distance, and min{R iz} represents the minimum distance; i = 1, 2, 3, …, n, and z = 0, 1, 2, 3, 4, 5, 6.
5. The sphericity error evaluation method based on the conical search algorithm according to claim 1, wherein: The conic search point (x sq ,y sq ,z sq ) coordinate calculation formula is: wherein u and v are orthogonal unit vectors; R s = F(g-s-1) / g, C = M + (g-s-1)(N-M) / g, s = 1,2,..., g, q = 1,2,..., m.
6. The sphericity error evaluation method based on the conical search algorithm according to claim 5, wherein: The calculation process of the orthogonal unit vectors u and v is as follows: Let the conic axis direction vector w = (x n - x m , y n - y m , z n - z m ), take t = (1, 0, 0) as the unit vector of the positive direction of the x axis, calculate u = w x t; if the calculation result of u is not a zero vector, then calculate v = w x u; if the calculation result of u is a zero vector, then replace t with b = (1, 0, 0), that is, the unit vector of the positive direction of the y axis, recalculate u = w x b, and then calculate v = w x u.