Method for cylindricity error evaluation using Geometry Optimization Searching Algorithm

Measurement 44 (2011) 1556–1563

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Method for cylindricity error evaluation using Geometry Optimization Searching Algorithm Lei Xianqing ⇑, Song Hongwei, Xue Yujun, Li Jishun, Zhou Jia, Duan Mingde Henan University of Science and Technology, Luoyang, China

a r t i c l e

i n f o

Article history: Received 22 December 2010 Received in revised form 6 June 2011 Accepted 12 June 2011 Available online 24 June 2011 Keywords: Cylindricity Error evaluation Geometry searching algorithm Minimum zone

a b s t r a c t According to the geometrical characteristics of cylindricity error, a method for cylindricity error evaluation using Geometry Optimization Searching Algorithm (GOSA) has been presented. The optimization method and linearization method and uniform sampling could not adopt in the algorithm. The principle of the algorithm is that a hexagon are collocated based on the reference points in the starting and the end measured section respectively, the radius value of all the measured points are calculated by the line between the vertexes of the hexagon in the starting and the end measured section as the ideal axes, the cylindricity error value of corresponding evaluation method (include minimum zone cylinder method (MZC), minimum circumscribed cylinder method (MCC) and maximum inscribed cylinder method (MIC)) are obtained according to compare, judgment and arranged hexagon repeatedly. The principle and step of using the algorithm to solve the cylindricity error is detailed described and the mathematical formula and program flowchart are given. The experimental results show that the cylindricity error can be evaluated effectively and exactly using this algorithm. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The cylindricity error is the variation between measured cylindrical surface and its ideal cylindrical surface. It is an important quality index of precision of mechanical parts and assembled product, the value of cylindricity error has a great effect on wear resistance, rotation and assembly accuracy of the mechanic. In this case, searching and designing accurate geometric measurements and cylindricity error evaluating algorithm are useful in the practical production. Comparing with the evaluation of straightness error, planeness error and roundness error, the evaluation of cylindricity error is much more difficult and complex. The usual methods of evaluating cylindricity error includes least square cylinder method (LSC), minimum zone cylin⇑ Corresponding author. E-mail address: [email protected] (X.Q. Lei). 0263-2241/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.06.010

der method (MZC), minimum circumscribed cylinder method (MCC) and maximum inscribed cylinder method (MIC). The LSC method is used to minimize the sum of square errors for part profile evaluation and the error value is unique, because the LSC method is simple and operation easy, so it is now widely used. But the LSC method is in breach of the minimum conditions definition of cylindricity error and its principle is defectiveness in error evaluation, the cylindricity error value calculated by the LSC method usually is not minimal. It is very likely that the overestimated form geometric tolerance is artificially introduced and then used for part production. The MZC method is the error evaluation method for cylindricity conforms based on the ISO definition. The essential of the MZC method is to search two coaxial cylindrical surfaces containing all measured points, and the radius difference of the two coaxial cylindrical surfaces is min. As data processing is complexity in the MZC method, many approximative and relative accuracy methods, such as MCC

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method and MIC method, had been established. The objective functions of the MZC, MIC and MCC method are non-linear and many parameters need to be optimized [1,2]. To improve the error evaluation accuracy, the minimum zone method has received much attention in recent years. Mao et al. [3] and Cui et al. [4] proposed a method of evaluation cylindricity errors based on particle swarm optimization (PSO), their example showed that the method was suitable for the nonlinear optimization problems such as cylindricity error evaluations and the precision of PSO was better than the LSC method under minimum zone condition. Wen [5] believed the particle swarm optimization (PSO) not only has the advantages of algorithm simplicity and good flexibility, but also improves non-strict cylindrical parts error evaluation accuracy. And it is well suited for form error evaluation on CMM. Bei et al. [6] proposed the cylindricity error optimized model by genetic algorithm (GA). The mathematical model can work out the minimum zone solution of the cylindricity error with arbitrary position in space, and there are no special requirements in choosing measurement points. A test was given to prove that the optimal approximation solution to the cylinder axis’s vectors and minimum zone cylindricity can be worked out by the objective function. Li [7] presented DNA computing model to evaluate cylindricity error based on advantages of popular genetic algorithm, the method also has higher search efficiency, good overall optimal performance etc. It can made all form error evaluation algorithm in an algorithm frame by designing parameter genetic cipher table based on parameter range. Lai et al. [8] proposed a heuristic approach to model form errors for cylindricity evaluation based on genetic algorithms (GAS) and the theoretical basis for the cylindricity error evaluation algorithms. The performance of the method under various combinations of parameters and the precision improvement on the evaluation of cylindricity are carefully analyzed. The numerical results indicate that the GAS method does provide better accuracy on cylindricity evaluation. Venkaiah and Shunmugam [9] presented limaconcylinder algorithms for cylindricity evaluation by using computational geometric techniques, the algorithms developed in their study have been applied for minimum circumscribed, maximum inscribed and minimum zone evaluations of the data available in the literature. Cheraghi et al. [10] presented mathematical models and efficient methodologies for the evaluation of geometric characteristics that define form and function of cylindrical features; namely cylindricity and straightness of median line. Based on the methodologies, the cylindricity error evaluation (LSC, MIC, MCC and MZC) can be performed. Lao et al. [11] focus on cylindricity evaluation based on radial form measurements, and remove the shortcoming of the hyperboloid technique [12] by providing a simple procedure for appropriately initializing the data (axis estimation), which is axis estimation and the hyperboloid technique constitute an integrated methodology for cylindricity evaluation. Zhu and Ding [13] applied kinematical geometry to investigate the problem of determining the cylindricity error of a mechanical part by using measurement points obtained with a coordinate measuring machine. Four cylindricity evaluation methods commonly used are formulated as

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nonlinear constrained optimization problems and nonlinear least-squares problem, and efficient sequential approximation algorithms are developed to solve them. For enhancing the performance of the algorithms, the localization of cylinder and minimum variance fitting of cylinder is introduced to analytically provide proper initial solutions. By organizing all involved algorithms in a hierarchical structure, four complete cylindricity evaluation algorithms (LSC, MIC, MCC and MZC) are presented. From the references, we can know assessment method of cylindricity error is a very important application in metrology. The standard characterization of cylindricity is a spatial location between two coaxial cylinders with minimum radial difference, and the spatial location must contain all the measured points. Unfortunately, the construction of the zone cylinder is a very complex geometric problem, which can be formulated as a nonlinear optimization. In the process of the nonlinear optimization, the selection and application of algorithms are great importance. Meanwhile, convergence rate, precision of result and reliability of the algorithm directly affect the evaluating precision of cylindricity error. Although the methods were proposed by references can be understood by their writers and professionalism in the field of metrology, but the non-professional will not understand these ideas easily. Consequently, the simplified algorithms might not bring about accurate evaluation, the optimization algorithms are difficult in deciding initial center line and step length of optimization, and the computation process might be very complex. According to the definition of cylindricity error, an innovative and simple cylindricity error evaluation method, named as Geometry Optimization Searching Algorithm (GOSA) is presented, in which the cylindricity error can be gained by repeatedly call distance functions between point and space line through and simple judgments without the optimization method and linearization method.

2. The evaluation principle of the GOSA The core of cylindricity evaluation methods (LSC, MIC, MCC and MZC) is to resolve the parameters of the center line of cylindrical surface containing all real measuring points [1–13]. It is obvious that the ideal center line must be closed to the least square center line of measured cylinder. Now, a regular hexagon is allocated (the length of side is f, f is estimated value according to the machining accuracy of the measured cylinder) by the least square center as reference in initial and end measured section respectively, the 36 lines can be arranged connecting the hexagon each vertices in initial measured section to the hexagon each vertices in end measured section. If take one of the 36 lines as assumption ideal axis line of measured cylindrical surface respectively, by calculating the radiuses of all measurement points, the 36 cylindrical-tubular can be gained which contain all the data points. If the minimum of the radius difference of the 36 cylindrical-tubular (two coaxial cylinders surface) is less than the cylindricity error of the LSC, take the crossing points between the assumption ideal axis and initial and end measured section (i.e. the vertex of

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the pre-set hexagon in initial measured section and end measured section) as reference points, the re-establish new hexagon are re-established by using the 0.618  f as the side length. So, 36 new assumption ideal axis lines and corresponding cylindrical-tubular can be gained. If the minimum of the radius difference of the 36 cylindrical-tubular is not less than the cylindricity error of the LSC, the new hexagon are re-establish by using the 0.618  f as the side length and the least square center as reference. When the hexagon side length f is less than the set value d (normally, d = 0.0001 mm), it could be considered that the searched assumption ideal axis lines are getting close to the actual axis lines of the two coaxial cylinders which the minimum radius difference and contain all the data points, the search terminates. 3. The process and steps of the GOSA Assuming that the measuring points are expressed by Aij(xij, yij, zj) (j is serial number of the measured section and j = 1, 2, . . . , M, i is serial number of the measured point on the measured section and i = 1, 2, . . . , N). 3.1. Determine the initial reference It can be seen that from Refs. [1–13] and actual measurements, the ideal center line for evaluating cylindricity error must be closed the least square center line and the geometric axis of measured cylinder. Then, the selection of initial axis can be confirmed by following two methods. (1) While measured points are equally spaced sample in initial and end measured section, the initial axis can be established by the link between the least square center Od(Xd, Yd, Zd) of the starting measured section and the least square center Oe(Xe, Ye, ZM) of the end measured section. Their coordinates are calculated by the formula (1) or formula (2).

8 N N P P > 1 > xi1 ; Y d ¼ N1 yi1 ; > < Xd ¼ N i¼1

8 xsM ¼ ðyaM  ybM ÞðyaM ybM þ x2cM Þ > > > > > > þðybM  ycM ÞðybM ycM þ x2aM Þ > > > > > ðyaM  ycM ÞðyaM ycM þ x2bM Þ > > > > > > > xxM ¼ 2yaM ðxcM  xbM Þ > < 2y ðx  x Þ þ 2y ðx

cM bM aM1 cM bM  xaM Þ 2 > y ¼ ðx  x Þðx x þ y > aM aM bM bM sM cM Þ > > > 2 > þðx  x Þðx x þ y > cM cM bM bM2 aM Þ > > > 2 > ðx  x Þðx x þ y > aM cM aM1 cM bM Þ > > > > y ¼ 2x ðy  y Þ > aM xM cM bM > > : 2xbM ðyaM  ycM Þ þ 2xcM ðybM  yaM Þ

where the points (xa1, ya1), (xb1, yb1), (xc1, yc1) and points (xaM, yaM), (xbM, ybM), (xcM, ycM) are three points of moreor-less uniformly distributed in initial measured section (j = 1) and end measured section (j = M). 3.2. Construct assumption ideal axis Using center points Od(Xd, Yd, Z1), Oe(Xe, Ye, ZM) as reference points, a hexagon is set by the f (to avoid the area is too small and the ideal axis is missed, the f is more than four times estimation value of cylindricity error) as side length in initial measured section and end measured sec-

Z d ¼ z1

i¼1

N N > P P > > xiM ; Y e ¼ N1 yiM ; Z e ¼ zM : X e ¼ N1 i¼1

8 xs1 ¼ ðya1  yb1 Þðya1 yb1 þ x2c1 Þ þ ðyb1  yc1 Þðyb1 yc1 þ x2a1 Þ > > > > > þðya1  yc1 Þðya1 yc1 þ x2b1 Þ > > > < x ¼ 2y ðx  x Þ  2y ðx  x Þ þ 2y ðx  x Þ x1 c1 a1 b1 a1 c1 b1 a1 c1 b1 > ys1 ¼ ðya1  yb1 Þðya1 yb1 þ y2c1 Þ þ ðxb1  xc1 Þðxb1 xc1 þ y2a1 Þ > > > > > þðxa1  xc1 Þðxa1 xc1 þ y2b1 Þ > > : xx1 ¼ 2xa1 ðyc1  yb1 Þ  2xb1 ðya1  yc1 Þ þ 2xc1 ðyb1  ya1 Þ

ð1Þ

i¼1

where (xi1, yi1) and (xiM, yiM) are coordinates of measurement points in initial measured section (j = 1) and end measured section (j = M). (2) While measured points are unequally spaced sample in initial and end measured section, the center point Od(Xd, Yd, Zd) and Oe(Xe, Ye, ZM) are calculated based on the principle that the characters of three points can define a circle. The initial axis can be established by the link between the point Od(Xd, Yd, Zd) and the point Oe(Xe, Ye, ZM). The coordinates of the center point Od(Xd, Yd, Zd) and center point Oe(Xe, Ye, ZM) can be calculated by formula (2).

(

s1 s1 X d ¼ xxx1 ; Y d ¼ yyx1 ;

Xe ¼ where

xsM xxM

; Ye ¼

ysM yxM

Z d ¼ z1

; Z e ¼ zM

ð2Þ Fig. 1. The principle of geometry optimization searching evaluation for cylindricity errors.

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tion respectively (as shown in Fig. 1). Based on geometrical principle, the coordinates of the hexagon vertex dqm(xqm, yqm, zqm) and ezn(xzn, yzn, zzn) are:

8 p  > < xqm ¼ X d þ f cos  3 ðm  1Þ yqm ¼ Y d þ f sin p3 ðm  1Þ > : zqm ¼ z1 8 p  > < xzn ¼ X e þ f cos  3 ðn  1Þ yzn ¼ Y e þ f sin p3 ðn  1Þ > : zzn ¼ Z M

ðm ¼ 1; 2;    6Þ

ðn ¼ 1; 2;    6Þ

ð3Þ

ð4Þ

And then, the 36 lines can be constructed by connecting vertex of the hexagon in measured initial section with vertex of the hexagon in measured terminate section in turn, and If equation of the 36 lines are expressed as (x  a)/ P = (y  b)/Q = z, based on geometric principle, the direction numbers of the lines are:



Pmn ¼ ðxzn  xqm Þ=ðZ M  Z 1 Þ Q mn ¼ ðyzn  xqm Þ=ðZ M  Z 1 Þ

ð5Þ

3.3. Calculate the distances between all the measured points and the assumption ideal axis lines Using the following equation (formula (6)), the maximum distance, the minimum distance and the range of the distance between measured points and each presumptive ideal axes can be found by calculating the distances between all measured points Aij(xij, yij, zj) and the lines (presumptive ideal axes).

Rijlhmk ¼

jðxij  xqm ; yij  yqm ; zj Þ  ðPmn ; Q mn ; 1Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2mn þ Q 2mn þ 1

ð6Þ

At the distances between measured points and one line of the 36 presumptive ideal axes lines, there must has a max distance, a min distance and a distance range between the maximum and minimum distance. There are 36 presumptive ideal axes lines, therefore the 36 maximum distances Rmax, minimum distances Rmin and distance’s extreme difference DR can be gained. 3.4. Calculate the distances between all measured points and the initial axis line The maximum distance Dmax, minimum distance Dmin and distance’s extreme difference DD could be found by calculating the distances between all measured points Aij(xij, yij, zj) and the axis lines which established by reference points Od(Xd, Yd, Z1) and Oe(Xe, Ye, ZM).

Fig. 2. Flowchart of MCC for evaluating cylindricity errors.

initial and end measured section (i.e. The vertex of the hexagon in initial and end measured section, expressed as Odw(xdw, ydw, z1) and Oew(xew, yew, zM)), and the side is 0.618 times of the original side length (that is: f = 0.618f), repeat steps 3.2–3.5. When the hexagon side length f is less than the set value d (normally, d = 0.0001 mm), the search terminates. It is reputed that the searched assumption ideal axis lines are getting close to the actual axis lines of the minimum circumscribed cylinder. At this moment, the maximum and minimum cylindrical radius corresponding with this axis line are recorded and expressed as Rout and rout respectively, then, the cylindricity error of the MCC method is fmcc = Rout  rout. The program flowchart of the MCC is shown in Fig. 2.

3.5. The cylindricity error of the MCC method 3.6. The cylindricity error of the MIC method Among the 36 maximum distances Rmax which have been calculated in the step 3.3, if the minimum Rmax is more than the Dmax which has been calculated in the step 3.4 (that is: minRmax > Dmax), repeat steps 3.2–3.5, but f = 0.618f. Otherwise (that is: minRmax 6 Dmax), the reference points are displaced by the crossing points between the axis line corresponding with the minimum Rmax and

Among the 36 minimum distances Rmin which have been calculated in step 3.3, if the maximum Rmin is less than the Dmin which has been calculated in step 3.4 (that is: maxRmin < Dmin), repeat steps 3.2–3.4 and 3.6, but f = 0.618f. Otherwise (that is: maxRmin 6 Dmin), the reference points are displaced by the crossing points between

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3.2–3.4 and 3.7, but f = 0.618  f. Otherwise (that is: min DR 6 DD)), the reference points are displaced by the crossing points between the axis line corresponding with the minimum extreme range min DR and initial and end measured section (i.e. the vertex of the hexagon in initial and end measured section, expressed as Oda(xda, yda, z1) and Oea(xea, yea, zM)), and the side is 0.618 times of the original side length (that is: f = 0.618  f), repeat steps 3.2–3.4 and 2.7. When the hexagon side length f is less than the set value d (normally, d = 0.0001 mm), the search terminates. It is reputed that the searched assumption ideal axis lines is getting close to the actual axis lines of the two coaxial cylinders with minimum radius difference and containing all the data points. At this moment, the maximum and minimum cylindrical radius corresponding with this axis lines are recorded and expressed as Rarea and rarea respectively. Then, according to the definition of the minimum zone cylinder method, the cylindricity error of MZC method is fmzc = Rarea  rarea. The program flowchart of the MIC is shown in Fig. 4.

Fig. 3. Flowchart of MIC for evaluating cylindricity errors.

the axis line corresponding with the maximum Rmin and initial and end measured section (i.e. the vertex of the hexagon in initial and end measured section, expressed as Odn(xdn, ydn, z1) and Oen(xen, yen, zM)), and the side is 0.618 times of the original side length (that is: f = 0.618f), repeat steps 3.2–3.4 and 3.6. When the hexagon side length f is less than the set value d (normally, d = 0.0001 mm), the search terminates. It is reputed that the searched assumption ideal axis lines is getting close to the actual axis lines of the maximum inscribed cylinder. At this moment, the maximum and minimum cylindrical radius corresponding with this axis line are recorded and expressed as Rin and rin respectively, then, the cylindricity error of the MIC method is fmic = Rin  rin. The program flowchart of the MIC is shown in Fig. 3. 3.7. The cylindricity error of the MZC method Among the 36 extreme range DR which have been calculated in step 3.3, if the minimum extreme range min DR is more than the extreme range DD which has been calculated in step 3.4 (that is: (min DR > DD), repeat steps

Fig. 4. Flowchart of MCC for evaluating cylindricity errors.

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4. Experiments

4.3. Analysis of comparative experiment results

4.1. Cylindricity evaluation by CMM

The CMM is high precision measurement recognized by the persons all over the world, which is popularly used in the advanced manufacture technique and science research. The correctness of the proposed algorithm can be identified by comparing the data processing results between the proposed algorithm and the CMM. Comparing the cylindricity error in Tables 2 and 3, it could be seen that, though the initial reference point is different in the GOSA, the cylindricity errors obtained by the GOSA are in accord with the cylindricity errors by the CMM. It is seen from Table 3 that, the calculating results of the GOSA is slightly different with different starting point, the slightly different has no bearing on whether qualified of the measured parts for the common parts. For the sake of calculation, the least square center of the starting and end measured section as the initial reference point while measured points are equally spaced sampling, When Non-uniform sampling, the initial reference point may be calculated by formula (2). Moreover, when measuring the high precision parts, this slightly difference may decide on an erroneous verdict, the starting point can be selected in various ways, the minimum of calculated results is used

Put the measured cylinder on the worktable of the three coordinate measuring machines (Firm: Brown & Sharpe. Type: Global Status 574. Measuring range: 500 mm  700 mm  400 mm. Software: PC-DMIS. Measuring accuracy: 1 lm. Repeatable accuracy: 0.5 lm), measure five cross-sections and take 16 sampling points from each cross-section. The measured data and results are shown in Tables 1 and 2.

4.2. Cylindricity evaluation by the GOSA Using the proposed evaluation algorithm (select different starting reference point and the side length is f = 0.1 mm for initial hexagon) to process the data of the Table 1, the cylindricity errors and the corresponding axis of the three evaluation method are obtained when the final side of hexagon is less 0.0001 mm, the cylindricity errors and the corresponding axis parameters are listed as Table 3.

Table 1 Coordinates of measuring points (mm). No.

x

y

z

No.

x

y

z

No.

x

y

z

No.

x

y

z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7.169 14.324 25.509 33.683 34.167 25.051 7.1 8.071 19.797 28.945 34.891 34.168 28.525 18.113 7.971 6.778 11.322 20.262 24.016 31.014

34.21 31.887 23.877 9.283 7.378 24.377 34.238 34.034 28.86 19.647 2.595 7.484 20.224 29.908 34.044 34.290 33.073 28.475 25.389 16.098

30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 30.5 43.3 43.3 43.3 43.3

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

34.584 30.614 19.46 5.802 10.248 23.395 32.027 34.985 33.364 27.418 17.678 0.706 14.539 22.717 26.655 31.59 34.892 29.347 19.04 6.241

4.991 16.878 29.045 34.484 33.444 26.026 14.043 0.374 10.449 21.716 30.164 34.942 31.801 26.563 22.607 14.943 1.967 18.995 29.322 34.407

43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 43.3 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6.174 14.594 28.945 34.887 31.458 20.538 6.489 7.37 15.817 28.573 33.084 34.611 25.488 16.38 0.788 10.206 21.025 30.269 33.93 34.787

34.448 31.802 19.636 2.506 15.279 28.299 34.352 34.173 31.168 20.125 11.238 4.838 23.924 30.884 34.969 33.457 27.971 17.524 8.513 3.557

60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 81 81 81 81 81 81 81 81 81 81 81 81

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

31.626 18.362 3.4 5.318 8.484 18.607 26.163 33.973 34.49 29.015 17.564 3.842 15.389 29.428 34.054 34.931 33.204 29.818 21.451 15.791

14.895 29.755 34.794 34.548 33.909 29.583 23.175 8.156 5.71 19.502 30.231 34.765 31.424 18.935 8.042 2.026 11.008 18.273 27.629 31.216

81 81 81 81 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5 97.5

Table 2 Measurement results by CMM. Method

LSC

MZC

MIC

MCC

Axis parameter x0 (mm) y0 (mm) z0 (mm) i j k Cylindricity error (mm)

0.0125 0.0123 30.5 0.0000040172 0.000013612 1 0.0366

0.0118 0.0170 30.5 0.0000025775 0.00016886 1 0.0324

0.0121 0.0076 30.5 0.000012014 0.000074796 1 0.0367

0.0110 0.0165 30.5 0.000004866 0.00014041 1 0.0335

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Table 3 The results by Geometry Optimization Searching Algorithm. The initial reference points

Calculated by the least square axis parameter of the Table 2

Calculated by measuring points No. 1, No. 6 and No. 11.

Calculated by measuring points No. 65, No. 70 and No. 75 Calculated by measuring points No.1, No.6 and No.11.

Calculated by measuring points No. 70, No. 75 and No. 80. Calculated by measuring points No. 6, No. 11 and No. 16.

Calculated by measuring points No. 70, No. 75 and No. 80. Calculated by measuring points No. 6, No. 11 and No. 16.

Calculated by measuring points No. 65, No. 70 and No. 75

Method

8 < X d ¼ 0:0122 Od Y d ¼ 0:0132 : Z 1 ¼ 30:5 8 < X e ¼ 0:0125 Oe Y e ¼ 0:0132 : Z M ¼ 97:5 8 < X d ¼ 0:0405 Od Y d ¼ 0:0409 : Z 1 ¼ 30:5 8 < X e ¼ 0:0643 Oe Y e ¼ 0:0180 : Z M ¼ 97:5 8 < X d ¼ 0:0405 Od Y d ¼ 0:0409 : Z 1 ¼ 30:5 8 < X e ¼ 0:0227 Oe Y e ¼ 0:0232 : Z M ¼ 97:5 8 < X d ¼ 0:0231 Od Y d ¼ 0:0028 : Z 1 ¼ 30:5 8 < X e ¼ 0:0227 Oe Y e ¼ 0:0232 : Z M ¼ 97:5 8 < X d ¼ 0:0231 Od Y d ¼ 0:0028 : Z 1 ¼ 30:5 8 < X e ¼ 0:0643 Oe Y e ¼ 0:0180 : Z M ¼ 97:5

Axis parameter

Cylindricity error (mm)

x0 (mm)

y0 (mm)

z0 (mm)

i

j

k

MZC

0.0127

0.0169

30.5

1.5595e005

1.6156e004

1

0.0320

MIC MCC

0.0119 0.0152

0.0121 0.0170

30.5 30.5

1.1545e006 6.1470e005

1.8570e005 1.7490e004

1 1

0.0365 0.0328

MZC

0.0105

0.0145

30.5

1.9634e005

1.0904e004

1

0.0324

MIC MCC

0.0128 0.0080

0.0158 0.0170

30.5 30.5

1.5652e005 1.3544e004

8.4971e005 1.6255e004

1 1

0.0344 0.0349

MZC

0.0116

0.0160

30.5

2.9195e006

1.4113e004

1

0.0321

MIC MCC

0.0117 0.0047

0.0107 0.0166

30.5 30.5

1.6047e006 2.2725e004

3.8452e005 1.4364e004

1 1

0.0365 0.0369

MZC

0.0123

0.0170

30.5

1.0192e005

1.6365e004

1

0.0319

MIC MCC

0.0119 0.0088

0.0106 0.0157

30.5 30.5

5.7144e006 1.0316e004

1.0610e005 1.1819e004

1 1

0.0346 0.0334

MZC

0.0123

0.0170

30.5

7.9416e006

1.6305e004

1

0.0319

MIC MCC

0.0130 0.0052

0.0125 0.0150

30.5 30.5

2.2615e005 2.1227e004

3.4515e005 9.3533e005

1 1

0.0348 0.0355

as verification of conformity/inconformity of product with its geometrical specification. Experimental results show that this algorithm is effective and accurate. 5. Conclusions The GOSA method for cylindricity error presented in this paper is a new cylindricity evaluation algorithm. The algorithm is simple, intuitive and easy to program, and it has the commonality and better practicability. The cylindricity error of MZC, MIC and MCC can be gained at one time by using the GOSA. It is not necessary to require evenly distributed sampling interval and assume small error and deviation in this algorithm, the evaluating accuracy of the GOSA depends on the pre-set parameters d, the smaller the values of the d is, the more precise the evaluation is. Generally, d = 0.0001 mm can satisfy cylindricity error evaluating requirement. Acknowledgments The authors gratefully acknowledge National Natural Science Foundation of China (No. 50875076), the Ph.D. Ini-

tial Foundation of Henan University of Science & Technology, Scientific Research Innovation Ability Cultivation Foundation of Henan University of Science & Technology (No. 2010CZ0002) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province for financial support of this research work. References [1] Li Huifen, Jiang Xiangqian, Zhang Yu, et al., A practical algorithm for computing cylindricity error in the rectangular coordinates, Chinese Journal of Scientific Instrument 23 (2002) 424–426. [2] Chen Lijie, Zhang Lei, Zhang Yu, Mathematical models of cylindricality error with sampling points in rectangular spatial coordinates, Journal of Northeastern University (Natural Science) 26 (2005) 676–679. [3] Jian Mao, Huawen Zheng, Yanlong Cao, et al., Method for cylindricity errors evaluation using particle swarm optimization algorithm, Transactions of the Chinese Society for Agricultural Machinery 38 (2007) 146–149. [4] Changcai Cui, Fugui Huang, Rencheng Zhang, et al., Research on cylindricity evaluation based on the Particle Swarm Optimization (PSO), Optics and Precision Engineering 14 (2006) 256–260. [5] Xiulan Wen, Jiacai Huang, Danghong Sheng, et al., Conicity and cylindricity error evaluation using particle swarm optimization, Precision Engineering 34 (2010) 338–344. [6] Guangxia Bei, Peihuang Lou, Xiaoyong Wang, et al., Cylindricity error evaluation based on genetic algorithms, Journal of Shandong University (Engineering Science) 38 (2008) 33–36.

X.Q. Lei et al. / Measurement 44 (2011) 1556–1563 [7] Haolin Li, Application of DNA computing model to cylindricity error evaluation, Acta Metrologica Sinica 25 (2004) 107–110. [8] Hsinyi Lai, Wenyuh Jywe, Cha’OKuang Chen, et al., Precision modeling of form errors for cylindricity evaluation using genetic algorithms, Precision Engineering 24 (2000) 310–319. [9] N. Venkaiah, M.S. Shunmugam, Evaluation of form data using computational geometric techniques—Part II: Cylindricity error, International Journal of Machine Tools & Manufacture 47 (2007) 1237–1245. [10] S. Hossein Cheraghi, Guohua Jiang, Jamalsheikh Ahmad, Evaluating the geometric characteristics of cylindrical features, Precision Engineering 27 (2003) 195–204.

1563

[11] Y.Z. Lao, H.W. Leong, F.P. Preparata, et al., Accurate cylindricity evaluation with axis-estimation preprocessing, Precision Engineering 27 (2003) 429–437. [12] Devillers Olivier, Preparata Franco P. Evaluating the cylindricity of a nominally cylindrical point set, in: Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms, 2000, pp. 518–527. [13] Limin Zhu, Han Ding, Application of kinematics geometry to computational metrology: distance function based hierarchical algorithms for cylindricity evaluation, International Journal of Machine Tools and Manufacture 43 (2003) 203–215.