Measurement 44 (2011) 1556–1563

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Measurement

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Method for cylindricity error evaluation using Geometry OptimizationSearching Algorithm

Lei Xianqing ⇑, Song Hongwei, Xue Yujun, Li Jishun, Zhou Jia, Duan MingdeHenan University of Science and Technology, Luoyang, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 December 2010Received in revised form 6 June 2011Accepted 12 June 2011Available online 24 June 2011

Keywords:CylindricityError evaluationGeometry searching algorithmMinimum zone

0263-2241/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.measurement.2011.06.010

⇑ Corresponding author.E-mail address: ly-lxq@163.com (X.Q. Lei).

According to the geometrical characteristics of cylindricity error, a method for cylindricityerror evaluation using Geometry Optimization Searching Algorithm (GOSA) has been pre-sented. The optimization method and linearization method and uniform sampling couldnot adopt in the algorithm. The principle of the algorithm is that a hexagon are collocatedbased 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 vertexesof the hexagon in the starting and the end measured section as the ideal axes, the cylind-ricity error value of corresponding evaluation method (include minimum zone cylindermethod (MZC), minimum circumscribed cylinder method (MCC) and maximum inscribedcylinder method (MIC)) are obtained according to compare, judgment and arranged hexa-gon repeatedly. The principle and step of using the algorithm to solve the cylindricity erroris detailed described and the mathematical formula and program flowchart are given. Theexperimental results show that the cylindricity error can be evaluated effectively andexactly using this algorithm.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The cylindricity error is the variation between mea-sured cylindrical surface and its ideal cylindrical surface.It is an important quality index of precision of mechanicalparts and assembled product, the value of cylindricity errorhas a great effect on wear resistance, rotation and assem-bly accuracy of the mechanic. In this case, searching anddesigning accurate geometric measurements and cylind-ricity error evaluating algorithm are useful in the practicalproduction.

Comparing with the evaluation of straightness error,planeness error and roundness error, the evaluation ofcylindricity error is much more difficult and complex.The usual methods of evaluating cylindricity error includesleast square cylinder method (LSC), minimum zone cylin-

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der method (MZC), minimum circumscribed cylindermethod (MCC) and maximum inscribed cylinder method(MIC). The LSC method is used to minimize the sum ofsquare errors for part profile evaluation and the error valueis unique, because the LSC method is simple and operationeasy, so it is now widely used. But the LSC method is inbreach of the minimum conditions definition of cylindrici-ty error and its principle is defectiveness in error evalua-tion, the cylindricity error value calculated by the LSCmethod usually is not minimal. It is very likely that theoverestimated form geometric tolerance is artificiallyintroduced and then used for part production.

The MZC method is the error evaluation method for cyl-indricity conforms based on the ISO definition. The essentialof the MZC method is to search two coaxial cylindrical sur-faces containing all measured points, and the radius differ-ence of the two coaxial cylindrical surfaces is min. As dataprocessing is complexity in the MZC method, many approx-imative and relative accuracy methods, such as MCC

X.Q. Lei et al. / Measurement 44 (2011) 1556–1563 1557

method and MIC method, had been established. The objec-tive functions of the MZC, MIC and MCC method are non-lin-ear and many parameters need to be optimized [1,2]. Toimprove the error evaluation accuracy, the minimum zonemethod has received much attention in recent years.

Mao et al. [3] and Cui et al. [4] proposed a method ofevaluation cylindricity errors based on particle swarmoptimization (PSO), their example showed that the methodwas suitable for the nonlinear optimization problems suchas cylindricity error evaluations and the precision of PSOwas better than the LSC method under minimum zone con-dition. Wen [5] believed the particle swarm optimization(PSO) not only has the advantages of algorithm simplicityand good flexibility, but also improves non-strict cylindri-cal parts error evaluation accuracy. And it is well suited forform error evaluation on CMM. Bei et al. [6] proposed thecylindricity error optimized model by genetic algorithm(GA). The mathematical model can work out the minimumzone solution of the cylindricity error with arbitrary posi-tion in space, and there are no special requirements inchoosing measurement points. A test was given to provethat the optimal approximation solution to the cylinderaxis’s vectors and minimum zone cylindricity can beworked out by the objective function. Li [7] presentedDNA computing model to evaluate cylindricity error basedon advantages of popular genetic algorithm, the methodalso has higher search efficiency, good overall optimal per-formance etc. It can made all form error evaluation algo-rithm in an algorithm frame by designing parametergenetic cipher table based on parameter range. Lai et al.[8] proposed a heuristic approach to model form errorsfor cylindricity evaluation based on genetic algorithms(GAS) and the theoretical basis for the cylindricity errorevaluation algorithms. The performance of the method un-der various combinations of parameters and the precisionimprovement on the evaluation of cylindricity are carefullyanalyzed. The numerical results indicate that the GASmethod does provide better accuracy on cylindricity eval-uation. Venkaiah and Shunmugam [9] presented limacon-cylinder algorithms for cylindricity evaluation by usingcomputational geometric techniques, the algorithmsdeveloped in their study have been applied for minimumcircumscribed, maximum inscribed and minimum zoneevaluations of the data available in the literature. Cheraghiet al. [10] presented mathematical models and efficientmethodologies for the evaluation of geometric characteris-tics that define form and function of cylindrical features;namely cylindricity and straightness of median line. Basedon 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 formmeasurements, and remove the shortcoming of the hyper-boloid technique [12] by providing a simple procedure forappropriately initializing the data (axis estimation), whichis axis estimation and the hyperboloid technique consti-tute an integrated methodology for cylindricity evaluation.Zhu and Ding [13] applied kinematical geometry to inves-tigate the problem of determining the cylindricity error ofa mechanical part by using measurement points obtainedwith a coordinate measuring machine. Four cylindricityevaluation methods commonly used are formulated as

nonlinear constrained optimization problems and nonlin-ear least-squares problem, and efficient sequential approx-imation algorithms are developed to solve them. Forenhancing the performance of the algorithms, the localiza-tion of cylinder and minimum variance fitting of cylinder isintroduced to analytically provide proper initial solutions.By organizing all involved algorithms in a hierarchicalstructure, four complete cylindricity evaluation algorithms(LSC, MIC, MCC and MZC) are presented.

From the references, we can know assessment methodof cylindricity error is a very important application inmetrology. The standard characterization of cylindricity isa spatial location between two coaxial cylinders withminimum radial difference, and the spatial location mustcontain all the measured points. Unfortunately, the con-struction of the zone cylinder is a very complex geometricproblem, which can be formulated as a nonlinear optimiza-tion. In the process of the nonlinear optimization, theselection and application of algorithms are great impor-tance. Meanwhile, convergence rate, precision of resultand reliability of the algorithm directly affect the evaluat-ing precision of cylindricity error. Although the methodswere proposed by references can be understood by theirwriters and professionalism in the field of metrology, butthe non-professional will not understand these ideas eas-ily. Consequently, the simplified algorithms might notbring about accurate evaluation, the optimization algo-rithms are difficult in deciding initial center line and steplength of optimization, and the computation process mightbe very complex.

According to the definition of cylindricity error, an inno-vative and simple cylindricity error evaluation method,named as Geometry Optimization Searching Algorithm(GOSA) is presented, in which the cylindricity error canbe gained by repeatedly call distance functions betweenpoint and space line through and simple judgments with-out 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 lineof cylindrical surface containing all real measuring points[1–13]. It is obvious that the ideal center line must be closedto the least square center line of measured cylinder.

Now, a regular hexagon is allocated (the length of side isf, f is estimated value according to the machining accuracyof the measured cylinder) by the least square center as ref-erence in initial and end measured section respectively, the36 lines can be arranged connecting the hexagon each ver-tices in initial measured section to the hexagon each verti-ces in end measured section. If take one of the 36 lines asassumption ideal axis line of measured cylindrical surfacerespectively, by calculating the radiuses of all measure-ment points, the 36 cylindrical-tubular can be gainedwhich contain all the data points. If the minimum of theradius difference of the 36 cylindrical-tubular (two coaxialcylinders surface) is less than the cylindricity error of theLSC, take the crossing points between the assumption idealaxis and initial and end measured section (i.e. the vertex of

Fig. 1. The principle of geometry optimization searching evaluation forcylindricity errors.

1558 X.Q. Lei et al. / Measurement 44 (2011) 1556–1563

the pre-set hexagon in initial measured section and endmeasured section) as reference points, the re-establishnew hexagon are re-established by using the 0.618 � f asthe side length. So, 36 new assumption ideal axis linesand corresponding cylindrical-tubular can be gained. Ifthe minimum of the radius difference of the 36 cylindri-cal-tubular is not less than the cylindricity error of theLSC, the new hexagon are re-establish by using the 0.618� f as the side length and the least square center as refer-ence. When the hexagon side length f is less than the setvalue d (normally, d = 0.0001 mm), it could be consideredthat the searched assumption ideal axis lines are gettingclose to the actual axis lines of the two coaxial cylinderswhich the minimum radius difference and contain all thedata points, the search terminates.

3. The process and steps of the GOSA

Assuming that the measuring points are expressed byAij(xij, yij, zj) (j is serial number of the measured sectionand j = 1, 2, . . . , M, i is serial number of the measured pointon 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 mea-surements, the ideal center line for evaluating cylindricityerror must be closed the least square center line and thegeometric axis of measured cylinder. Then, the selectionof initial axis can be confirmed by following two methods.

(1) While measured points are equally spaced sample ininitial and end measured section, the initial axis canbe established by the link between the least squarecenter Od(Xd, Yd, Zd) of the starting measured sectionand the least square center Oe(Xe, Ye, ZM) of the endmeasured section. Their coordinates are calculatedby the formula (1) or formula (2).

Xd ¼ 1N

PNi¼1

xi1;Yd ¼ 1N

PNi¼1

yi1; Zd ¼ z1

Xe ¼ 1N

PNi¼1

xiM ;Ye ¼ 1N

PNi¼1

yiM; Ze ¼ zM

8>>><>>>:

ð1Þ

where (xi1, yi1) and (xiM, yiM) are coordinates of measure-ment points in initial measured section (j = 1) and endmeasured section (j = M).

(2) While measured points are unequally spaced samplein initial and end measured section, the center pointOd(Xd, Yd, Zd) and Oe(Xe, Ye, ZM) are calculated basedon the principle that the characters of three pointscan define a circle. The initial axis can be establishedby the link between the point Od(Xd, Yd, Zd) and thepoint Oe(Xe, Ye, ZM). The coordinates of the centerpoint Od(Xd, Yd, Zd) and center point Oe(Xe, Ye, ZM)can be calculated by formula (2).

Xd ¼ xs1xx1;Yd ¼ ys1

yx1; Zd ¼ z1

Xe ¼ xsMxxM;Ye ¼ ysM

yxM; Ze ¼ zM

(ð2Þ

where

xs1 ¼ ðya1 � yb1Þðya1yb1 þ x2c1Þ þ ðyb1 � yc1Þðyb1yc1 þ x2

a1Þþðya1 � yc1Þðya1yc1 þ x2

b1Þxx1 ¼ 2ya1ðxc1 � xb1Þ � 2yb1ðxa1 � xc1Þ þ 2yc1ðxb1 � xa1Þys1 ¼ ðya1 � yb1Þðya1yb1 þ y2

c1Þ þ ðxb1 � xc1Þðxb1xc1 þ y2a1Þ

þðxa1 � xc1Þðxa1xc1 þ y2b1Þ

xx1 ¼ 2xa1ðyc1 � yb1Þ � 2xb1ðya1 � yc1Þ þ 2xc1ðyb1 � ya1Þ

8>>>>>>>><>>>>>>>>:

xsM ¼ ðyaM � ybMÞðyaMybM þ x2cMÞ

þðybM � ycMÞðybMycM þ x2aMÞ

�ðyaM � ycMÞðyaMycM þ x2bMÞ

xxM ¼ 2yaMðxcM � xbMÞ�2ybMðxaM1 � xcMÞ þ 2ycMðxbM � xaMÞ

ysM ¼ ðxaM � xbMÞðxaMxbM þ y2cMÞ

þðxbM � xcMÞðxbM2xcM þ y2aMÞ

�ðxaM � xcMÞðxaM1xcM þ y2bMÞ

yxM ¼ 2xaMðycM � ybMÞ�2xbMðyaM � ycMÞ þ 2xcMðybM � yaMÞ

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:where the points (xa1, ya1), (xb1, yb1), (xc1, yc1) and points(xaM, yaM), (xbM, ybM), (xcM, ycM) are three points of more-or-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 refer-ence points, a hexagon is set by the f (to avoid the area istoo small and the ideal axis is missed, the f is more thanfour times estimation value of cylindricity error) as sidelength in initial measured section and end measured sec-

X.Q. Lei et al. / Measurement 44 (2011) 1556–1563 1559

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

xqm ¼ Xd þ f cos p3 ðm� 1Þ� �

yqm ¼ Yd þ f sin p3 ðm� 1Þ� �

zqm ¼ z1

8><>: ðm ¼ 1;2; � � �6Þ ð3Þ

xzn ¼ Xe þ f cos p3 ðn� 1Þ� �

yzn ¼ Ye þ f sin p3 ðn� 1Þ� �

zzn ¼ ZM

8><>: ðn ¼ 1;2; � � �6Þ ð4Þ

And then, the 36 lines can be constructed by connectingvertex of the hexagon in measured initial section with ver-tex 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 direc-tion numbers of the lines are:

Pmn ¼ ðxzn � xqmÞ=ðZM � Z1ÞQ mn ¼ ðyzn � xqmÞ=ðZM � Z1Þ

�ð5Þ

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

3.3. Calculate the distances between all the measured pointsand the assumption ideal axis lines

Using the following equation (formula (6)), the maxi-mum distance, the minimum distance and the range ofthe distance between measured points and each presump-tive ideal axes can be found by calculating the distancesbetween all measured points Aij(xij, yij, zj) and the lines(presumptive ideal axes).

Rijlhmk ¼jðxij � xqm; yij � yqm; zjÞ � ðPmn;Q mn;1Þjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P2mn þ Q 2

mn þ 1q ð6Þ

At the distances between measured points and one line ofthe 36 presumptive ideal axes lines, there must has a maxdistance, a min distance and a distance range between themaximum and minimum distance. There are 36 presump-tive ideal axes lines, therefore the 36 maximum distancesRmax, minimum distances Rmin and distance’s extreme dif-ference DR can be gained.

3.4. Calculate the distances between all measured points andthe initial axis line

The maximum distance Dmax, minimum distance Dmin

and distance’s extreme difference DD could be found bycalculating the distances between all measured pointsAij(xij, yij, zj) and the axis lines which established by refer-ence points Od(Xd, Yd, Z1) and Oe(Xe, Ye, ZM).

3.5. The cylindricity error of the MCC method

Among the 36 maximum distances Rmax which havebeen calculated in the step 3.3, if the minimum Rmax ismore than the Dmax which has been calculated in the step3.4 (that is: minRmax > Dmax), repeat steps 3.2–3.5, butf = 0.618f. Otherwise (that is: minRmax 6 Dmax), the refer-ence points are displaced by the crossing points betweenthe axis line corresponding with the minimum Rmax and

initial and end measured section (i.e. The vertex of thehexagon in initial and end measured section, expressedas Odw(xdw, ydw, z1) and Oew(xew, yew, zM)), and the side is0.618 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 va-lue d (normally, d = 0.0001 mm), the search terminates. It isreputed that the searched assumption ideal axis lines aregetting close to the actual axis lines of the minimum cir-cumscribed cylinder. At this moment, the maximum andminimum cylindrical radius corresponding with this axisline are recorded and expressed as Rout and rout respec-tively, then, the cylindricity error of the MCC method isfmcc = Rout � rout.

The program flowchart of the MCC is shown in Fig. 2.

3.6. The cylindricity error of the MIC method

Among the 36 minimum distances Rmin which havebeen calculated in step 3.3, if the maximum Rmin is lessthan the Dmin which has been calculated in step 3.4 (thatis: maxRmin < Dmin), repeat steps 3.2–3.4 and 3.6, butf = 0.618f. Otherwise (that is: maxRmin 6 Dmin), the refer-ence points are displaced by the crossing points between

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

1560 X.Q. Lei et al. / Measurement 44 (2011) 1556–1563

the axis line corresponding with the maximum Rmin andinitial and end measured section (i.e. the vertex of thehexagon in initial and end measured section, expressedas Odn(xdn, ydn, z1) and Oen(xen, yen, zM)), and the side is0.618 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 va-lue d (normally, d = 0.0001 mm), the search terminates. It isreputed that the searched assumption ideal axis lines isgetting close to the actual axis lines of the maximum in-scribed cylinder. At this moment, the maximum and min-imum cylindrical radius corresponding with this axis lineare 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.

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

3.7. The cylindricity error of the MZC method

Among the 36 extreme range DR which have been cal-culated in step 3.3, if the minimum extreme range minDR is more than the extreme range DD which has been cal-culated in step 3.4 (that is: (min DR > DD), repeat steps

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 thecrossing points between the axis line corresponding withthe minimum extreme range min DR and initial and endmeasured section (i.e. the vertex of the hexagon in initialand end measured section, expressed as Oda(xda, yda, z1)and Oea(xea, yea, zM)), and the side is 0.618 times of the ori-ginal side length (that is: f = 0.618 � f), repeat steps 3.2–3.4and 2.7.

When the hexagon side length f is less than the set va-lue d (normally, d = 0.0001 mm), the search terminates. It isreputed that the searched assumption ideal axis lines isgetting close to the actual axis lines of the two coaxial cyl-inders with minimum radius difference and containing allthe data points. At this moment, the maximum and mini-mum cylindrical radius corresponding with this axis linesare recorded and expressed as Rarea and rarea respectively.Then, according to the definition of the minimum zone cyl-inder method, the cylindricity error of MZC method isfmzc = Rarea � rarea.

The program flowchart of the MIC is shown in Fig. 4.

X.Q. Lei et al. / Measurement 44 (2011) 1556–1563 1561

4. Experiments

4.1. Cylindricity evaluation by CMM

Put the measured cylinder on the worktable of the threecoordinate measuring machines (Firm: Brown & Sharpe.Type: Global Status 574. Measuring range: 500 mm �700 mm � 400 mm. Software: PC-DMIS. Measuring accu-racy: 1 lm. Repeatable accuracy: 0.5 lm), measure fivecross-sections and take 16 sampling points from eachcross-section. The measured data and results are shownin Tables 1 and 2.

4.2. Cylindricity evaluation by the GOSA

Using the proposed evaluation algorithm (select differ-ent starting reference point and the side length isf = 0.1 mm for initial hexagon) to process the data of the Ta-ble 1, the cylindricity errors and the corresponding axis ofthe three evaluation method are obtained when the finalside of hexagon is less 0.0001 mm, the cylindricity errorsand the corresponding axis parameters are listed as Table 3.

Table 1Coordinates of measuring points (mm).

No. x y z No. x y z N

1 �7.169 �34.21 30.5 21 �34.584 �4.991 43.3 412 �14.324 �31.887 30.5 22 �30.614 16.878 43.3 423 �25.509 �23.877 30.5 23 �19.46 29.045 43.3 434 �33.683 �9.283 30.5 24 �5.802 34.484 43.3 445 �34.167 7.378 30.5 25 10.248 33.444 43.3 456 �25.051 24.377 30.5 26 23.395 26.026 43.3 467 �7.1 34.238 30.5 27 32.027 14.043 43.3 478 8.071 34.034 30.5 28 34.985 0.374 43.3 489 19.797 28.86 30.5 29 33.364 �10.449 43.3 49

10 28.945 19.647 30.5 30 27.418 �21.716 43.3 5011 34.891 2.595 30.5 31 17.678 �30.164 43.3 5112 34.168 �7.484 30.5 32 �0.706 �34.942 43.3 5213 28.525 �20.224 30.5 33 �14.539 �31.801 60.8 5314 18.113 �29.908 30.5 34 �22.717 �26.563 60.8 5415 7.971 �34.044 30.5 35 �26.655 �22.607 60.8 5516 �6.778 �34.290 30.5 36 �31.59 �14.943 60.8 5617 �11.322 �33.073 43.3 37 �34.892 1.967 60.8 5718 �20.262 �28.475 43.3 38 �29.347 18.995 60.8 5819 �24.016 �25.389 43.3 39 �19.04 29.322 60.8 5920 �31.014 �16.098 43.3 40 �6.241 34.407 60.8 60

Table 2Measurement results by CMM.

Method LSC MZC

Axis parameterx0 (mm) 0.0125 0.011y0 (mm) 0.0123 0.017z0 (mm) 30.5 30.5i �0.0000040172 �0.00j 0.000013612 �0.00k 1 1Cylindricity error (mm) 0.0366 0.032

4.3. Analysis of comparative experiment results

The CMM is high precision measurement recognized bythe persons all over the world, which is popularly used inthe advanced manufacture technique and science research.The correctness of the proposed algorithm can be identi-fied by comparing the data processing results betweenthe proposed algorithm and the CMM.

Comparing the cylindricity error in Tables 2 and 3, itcould be seen that, though the initial reference point is dif-ferent in the GOSA, the cylindricity errors obtained by theGOSA are in accord with the cylindricity errors by the CMM.

It is seen from Table 3 that, the calculating results of theGOSA is slightly different with different starting point, theslightly different has no bearing on whether qualified ofthe measured parts for the common parts. For the sake ofcalculation, the least square center of the starting andend measured section as the initial reference point whilemeasured points are equally spaced sampling, WhenNon-uniform sampling, the initial reference point may becalculated by formula (2). Moreover, when measuring thehigh precision parts, this slightly difference may decideon an erroneous verdict, the starting point can be selectedin various ways, the minimum of calculated results is used

o. x y z No. x y z

6.174 34.448 60.8 61 31.626 �14.895 8114.594 31.802 60.8 62 18.362 �29.755 8128.945 19.636 60.8 63 3.4 �34.794 8134.887 2.506 60.8 64 �5.318 �34.548 8131.458 �15.279 60.8 65 �8.484 �33.909 97.520.538 �28.299 60.8 66 �18.607 �29.583 97.5

6.489 �34.352 60.8 67 �26.163 �23.175 97.5�7.37 �34.173 60.8 68 �33.973 �8.156 97.5�15.817 �31.168 81 69 �34.49 5.71 97.5�28.573 �20.125 81 70 �29.015 19.502 97.5�33.084 �11.238 81 71 �17.564 30.231 97.5�34.611 4.838 81 72 �3.842 34.765 97.5�25.488 23.924 81 73 15.389 31.424 97.5�16.38 30.884 81 74 29.428 18.935 97.5�0.788 34.969 81 75 34.054 8.042 97.510.206 33.457 81 76 34.931 �2.026 97.521.025 27.971 81 77 33.204 �11.008 97.530.269 17.524 81 78 29.818 �18.273 97.533.93 8.513 81 79 21.451 �27.629 97.534.787 �3.557 81 80 15.791 �31.216 97.5

MIC MCC

8 0.0121 0.01100 0.0076 0.0165

30.5 30.500025775 �0.000012014 0.000004866016886 0.000074796 �0.00014041

1 14 0.0367 0.0335

Table 3The results by Geometry Optimization Searching Algorithm.

The initial reference points Method Axis parameter Cylindricityerror (mm)

x0

(mm)y0

(mm)z0

(mm)i j k

Calculated by the least squareaxis parameter of the Table 2 Od

Xd ¼ 0:0122Yd ¼ 0:0132Z1 ¼ 30:5

8<:

MZC 0.0127 0.0169 30.5 �1.5595e�005 �1.6156e�004 1 0.0320

MIC 0.0119 0.0121 30.5 �1.1545e�006 1.8570e�005 1 0.0365

Oe

Xe ¼ 0:0125Ye ¼ 0:0132ZM ¼ 97:5

8<:

MCC 0.0152 0.0170 30.5 �6.1470e�005 �1.7490e�004 1 0.0328

Calculated by measuring pointsNo. 1, No. 6 and No. 11. Od

Xd ¼ 0:0405Yd ¼ 0:0409

Z1 ¼ 30:5

8<:

MZC 0.0105 0.0145 30.5 1.9634e�005 �1.0904e�004 1 0.0324

MIC 0.0128 0.0158 30.5 �1.5652e�005 �8.4971e�005 1 0.0344Calculated by measuring points

No. 65, No. 70 and No. 75 Oe

Xe ¼ 0:0643Ye ¼ 0:0180ZM ¼ 97:5

8<:

MCC 0.0080 0.0170 30.5 1.3544e�004 �1.6255e�004 1 0.0349

Calculated by measuring pointsNo.1, No.6 and No.11. Od

Xd ¼ 0:0405Yd ¼ 0:0409

Z1 ¼ 30:5

8<:

MZC 0.0116 0.0160 30.5 2.9195e�006 �1.4113e�004 1 0.0321

MIC 0.0117 0.0107 30.5 1.6047e�006 3.8452e�005 1 0.0365Calculated by measuring points

No. 70, No. 75 and No. 80. Oe

Xe ¼ 0:0227Ye ¼ �0:0232

ZM ¼ 97:5

8<:

MCC 0.0047 0.0166 30.5 2.2725e�004 �1.4364e�004 1 0.0369

Calculated by measuring pointsNo. 6, No. 11 and No. 16. Od

Xd ¼ 0:0231Yd ¼ 0:0028

Z1 ¼ 30:5

8<:

MZC 0.0123 0.0170 30.5 �1.0192e�005 �1.6365e�004 1 0.0319

MIC 0.0119 0.0106 30.5 �5.7144e�006 �1.0610e�005 1 0.0346Calculated by measuring points

No. 70, No. 75 and No. 80. Oe

Xe ¼ 0:0227Ye ¼ �0:0232

ZM ¼ 97:5

8<:

MCC 0.0088 0.0157 30.5 1.0316e�004 �1.1819e�004 1 0.0334

Calculated by measuring pointsNo. 6, No. 11 and No. 16. Od

Xd ¼ 0:0231Yd ¼ 0:0028

Z1 ¼ 30:5

8<:

MZC 0.0123 0.0170 30.5 �7.9416e�006 �1.6305e�004 1 0.0319

MIC 0.0130 0.0125 30.5 �2.2615e�005 �3.4515e�005 1 0.0348Calculated by measuring points

No. 65, No. 70 and No. 75 Oe

Xe ¼ 0:0643Ye ¼ 0:0180ZM ¼ 97:5

8<:

MCC 0.0052 0.0150 30.5 2.1227e�004 �9.3533e�005 1 0.0355

1562 X.Q. Lei et al. / Measurement 44 (2011) 1556–1563

as verification of conformity/inconformity of product withits geometrical specification.

Experimental results show that this algorithm is effec-tive and accurate.

5. Conclusions

The GOSA method for cylindricity error presented inthis paper is a new cylindricity evaluation algorithm. Thealgorithm is simple, intuitive and easy to program, and ithas the commonality and better practicability. The cylind-ricity error of MZC, MIC and MCC can be gained at one timeby using the GOSA. It is not necessary to require evenly dis-tributed sampling interval and assume small error anddeviation in this algorithm, the evaluating accuracy ofthe GOSA depends on the pre-set parameters d, the smallerthe values of the d is, the more precise the evaluation is.Generally, d = 0.0001 mm can satisfy cylindricity errorevaluating requirement.

Acknowledgments

The authors gratefully acknowledge National NaturalScience Foundation of China (No. 50875076), the Ph.D. Ini-

tial Foundation of Henan University of Science & Technol-ogy, Scientific Research Innovation Ability CultivationFoundation of Henan University of Science & Technology(No. 2010CZ0002) and Innovation Scientists and Techni-cians Troop Construction Projects of Henan Province forfinancial support of this research work.

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