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Sampling uncertainty incoordinate measurementdata analysis

Woncheol Choi,* Thomas R. Kurfess,* and Jonathan Cagan†*School of Mechanical Engineering, Georgia Institute of Technology,Atlanta, GA; and †Department of Mechanical Engineering, Carnegie MellonUniversity, Pittsburgh, PA, USA

There are a number of important software related issues in coordinatemetrology. After measurement data are collected in the form of positionvectors, the data analysis software must derive the necessary geometricinformation from the point set, and uncertainty plays an important role inthe analysis. When extreme fit approaches (L2` norm estimation ap-proaches) are employed for form error evaluation, the uncertainty is closelyrelated to the sampling process used to gather the data. The measurementpoints are a subset of the true surface, and, consequently, the extreme fitresult differs from the true value. In this paper, we investigate the functionalrelationship between the uncertainty in an extreme fit and the number ofpoints measured. Two major issues are addressed in this paper. The firstaddresses and identifies the parameters that affect the functional relation-ship. The second develops a methodology to apply this relationship to thesampling of measurement points. © 1998 Elsevier Science Inc.

Keywords: metrology; coordinate measurement; geometry; uncertainty

Introduction

Three-dimensional (3-D) metrology has brought asignificant change in dimensional measurement.Compared to traditional comparison methods(e.g., comparing to a surface plate), 3-D measure-ment yields more comprehensive and, unfortu-nately complex, information about the real partgeometry. Three-dimensional metrology gener-ates surface coordinates of a measured featureinstead of measuring the geometric dimensions.Thus, the measurement output is a collection ofdigitized surface points. With these surface points,more details about surface variation can be ob-served, and various sections of different geometricfeatures can be measured in a single process. Co-ordinate measuring machines (CMMs) are pres-

ently widely employed as 3-D measurement de-vices.

After measurement data are collected by a3-D measuring machine, an independent numer-ical analysis must be performed. Because the mea-surement data are a set of points, the raw datamust be interpreted in a parametric form by anappropriate analysis procedure. This is necessaryto obtain geometric parameters of part variationor to make a tolerance conformance decision.Thus, the measurement data analysis for 3-D me-trology is a software procedure that is separate anddistinct form the measurement data-collectingprocedure, which is a hardware procedure. Thedata analysis function has become one of the keycomponents in 3-D metrology, and it is the focusof this paper.

For the evaluation of a form error, such asflatness as defined in ASME standard Y14.5,1,2 aminimum zone must be established from the mea-surement data. The minimum zone is a geometricvalue that defines the least possible height of the

Address reprint requests to Dr. T. R. Kurfess, Georgia Institute ofTechnology, GWW School of Mechanical Engineering, Manufac-turing Research Center, Room 435, Atlanta, GA 30332-0405,USA. E-mail: [email protected]

Precision Engineering 22:153–163, 1998© 1998 Elsevier Science Inc. All rights reserved. 0141-6359/98/$–see front matter655 Avenue of the Americas, New York, NY 10010 PII S0141-6359(98)00011-7

point set on a plane. Minimum zone evaluationalgorithms have been reported in various stud-ies.3–10 The problem is formulated as an extremefit evaluation that searches for the extreme pointsthat characterize the zone. The extreme fit refersto fitting using L2` norm estimation. The studiesshow that the extreme fit yields more accuratefitting results because: (1) it yields smaller zonevalue than the zone evaluated by a least-squares fit;and (2) it is more consistent with the standarddefinition and is a numerical realization of physi-cal fittings. Also, some concern has been ex-pressed regarding interpreting the results of theleast-squares fit, because it is a statistical estimationrather than an exact solution. Thus, the extremefit is becoming a more widely accepted fittingmethod. However, as indicated several stud-ies,11–13 such an extreme fit has limitations in ac-curacy. Part of the uncertainty is because of thenumerical instability in the actual implementationof the fit. Also, the plug-in nature of the extremefit itself causes uncertainty. A more detailed dis-cussion related to the uncertainty is addressedlater in this paper.

The important fact is that the uncertainty inthe extreme fit is closely related to the sampling ofmeasurement points. This leads to the issue ofuncertainty evaluation and its relationship to sam-pling. The sampling problem can be broken intotwo components, the location of the measurementpoints and the sample size. The measurementpoint allocation problem is related to the type oferror or deformation to be characterized by themeasurement. Typically, a uniform distributionover the measurement region is practiced; how-ever, a nonuniform distribution may be more ef-fective for characterizing specific systematic defor-mations. This measurement point allocationproblem is significant, but not the target of thisresearch. In this paper, we limit the scope of thesampling to the sample size only. For flatness eval-uation, where deviations from nominal are consid-ered to be a stochastic deformation, the probabil-ity that the error can occur for a unit area isconsidered uniform; thus, a uniform distributionof measurement points is typically preferred. It isobvious that an evaluation with more measure-ment points yields more accurate results. Unfortu-nately, this advantage is countered by the fact thatthe number of points is directly related to themeasurement time and the data-processing cost.Thus, to achieve efficient and accurate measure-ment results, guidelines for choosing the samplesize must be established.

The objective of this research is to investigatehow the sample size is related to uncertainty in the

extreme fit evaluation. We examine the functional(mathematical) relationship between the samplesize and the uncertainty. Also we examine keyparameters that affect to the uncertainty in theextreme fit evaluation. We also discuss using theresults of these analyses to develop a samplingstrategy in inspection planning stages. Further-more, a model for a generalized form error distri-bution is given along with two different ap-proaches to derive the functional relationship(analytic approximation model and numericalmodel). We employ order statistics for the analyticmodel and a neural network for the numericalmodel. The theoretical concepts developed in thispaper are then applied to a flatness example.

Related work

Recognition of the sampling problem can befound in Hocken et. al.12 and Caskey et. al.,14

where extensive numerical experiments on data-fitting algorithms were conducted to investigatesampling issues. They tested least-squares fit, min–max fit, and minimum zone evaluations on variousgeometric primitives, such as line, plane, circle,sphere, and cylinder. They generated simulatedmeasurement data embedded with a characteristicform error, such as tri-lobbing error for a circularfeature. Then, they ran multiple sampling andfitting procedures and determined the variation ofthe evaluated parameters. The results were re-ported for the cases of a line, plane, circle, sphere,and cylinder. With their results, they concludedthat variation is substantially large with the sam-pling size in current practice.

Weckenmann et. al.,15,16 also addressed sam-pling issues for coordinate metrology. They recog-nized that extreme fit is more suitable for func-tionality evaluation, but it yields variations that aresensitive to sampling. They investigated the effectof sample point location and sample size on acircular feature with tri-lobbing error and a linearfeature with an undulation form error. From theirnumerical experiments, they showed the uncer-tainty of the form error evaluation, represented asthe dispersion of the evaluated value, decreaseswith increasing sample size.

Mestre and Abou-Kandil17 approached theproblem from a different direction. Recognizingthe problem with current extreme fit evaluationsthat ignore the variation in unmeasured sites, theyproposed a new method for the flatness evalua-tion. Employing Bayesian prediction theory, theyderived the confidence interval of the surface vari-ation in unmeasured sites. Then, the minimumzone was determined with respect to the derived

Choi et al.: Sampling uncertainty in coordinate measurement data analysis

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confidence interval. In so doing, they could over-come the plug-in estimation error of extreme fitevaluation, and the ensuing estimated minimumzone was closer to the true flatness value.

As discussed above, the sampling uncertaintyproblem in coordinate metrology has been recog-nized in previous studies, but a formal evaluationmethod has not been reported, and the results arelimited to a specific case. Thus, it is impossible topredict the uncertainty for a given sampling con-dition or to predict the number of the samplesrequired for a given accuracy level. However, it isnecessary to provide a more formal analysis of theuncertainty so that more strategic measurementplanning can be achieved. To do so, the uncer-tainty must be precisely defined, and its mathemat-ical relationship with the measurement conditionmust be examined.

Extreme fit and uncertainty

See Equation (1). A typical form of the extremeproblem is represented as follows:

min M(u ) 5 max{distance[ pi, S(u )]} (1)

where S is the fitted surface model, u representsthe model parameters, and pi are measuredpoints. Because the function M(u) is nondifferen-tiable, it must be determined by an iterative searchalgorithm. If the fitting geometry is properly de-fined, there is a unique solution that satisfies the

optimality condition and is constrained by the ex-treme points. Thus, the fitting geometry is definedby a few extreme points in the measured point set.

The uncertainty associated with extreme fitcomes from the plug-in nature of the extreme fit.The plug-in estimation is the evaluation of a pa-rameter with a finite number of samples regardingthem as the entire population. The extreme fitgeometry is determined from a few extremepoints. However, the extreme points in the mea-surement data that contribute to the extreme fitgeometry are not necessarily the extreme points inthe true surface profile. As shown in Figure 1, thetrue flatness is defined as the minimum zone overthe true surface profile. However, the flatness isevaluated only from the measured points, and theextreme points in the measurement data are sam-ples of the true surface points. Thus, the trueflatness value and the flatness evaluated by mea-sured points are not equivalent, because the mea-surement data fail to capture the true extremepoints.

Depending upon the data sampling strategy,the evaluated extreme fit values will vary. Figure 2represents the flatness value evaluated from a sur-face with different test sets. Ninety-five test sets aretaken from a simulated surface. The surface hasnoise amplitude that is uniform over the area witha uniform noise distribution superimposed uponthe measurement. For each test set, the flatness is

Figure 1 True value versus esti-mated value by measured data

Figure 2 Variation of a minimum zone value with respect to different samples


155PRECISION ENGINEERING

evaluated by a minimum zone evaluation method,and the flatness value is plotted with respect to thetest set number. Although the samples are takenfrom the same surface, the evaluated flatness valuevaries with different test sets.

This variation is attributable to the plug-inevaluation of the minimum zone evaluation, and itis the sampling uncertainty. Thus, when an ex-treme fit value is evaluated, the value has the stan-dard deviation related to the sampling process.The standard deviation can be found only from aninfinite number of test sets and evaluations. Toalleviate this problem, we use standard error as anestimate of the standard deviation. The standarderror can be evaluated from the estimated stan-dard deviation of the finite number of test sets andevaluations. The standard error represents the un-certainty in the extreme fit evaluation. Therefore,we refer to the uncertainty as the “standard error”in this paper.

The next issue that must be addressed is therelationship between uncertainty and sample size.Clearly, as more datapoints are used for evaluatingthe fitting surface, the extreme fit surface becomescloser to the true value, because larger number ofmeasurement points yields a more comprehensivesampling of the true surface. Figure 3 shows therelation between the sample size and the uncer-tainty. The same flatness example is tested withdifferent numbers of measured points. From thesame surface, one hundred different test sets arerandomly taken with a given number of measuredpoints, and the flatness is evaluated for each testset. As tested in a previous example, the test setsyield different flatness values. Then, the standarderror of the flatness value is derived. The proce-dure has been repeated for the cases of different

numbers of measured points, and the results areshown in Figure 3. Intuitively, the standard errordecreases with respect to the sample size. How-ever, the metrologist requires more than intuition;rather, we must mathematically model such a re-lationship so that a prediction can be made. To doso, we also need to model the other parametersthat may affect to the uncertainty. One of theimportant parameters is the distribution of formerror, as discussed in the following section.

Distribution of form error

Form error is surface variation caused by imper-fect manufacturing. The form error of the surfaceis mapped into the measured points, and it can berepresented as the distribution of the probabilitydensity with respect to a fixed coordinate refer-ence frame (Figure 4). The probability density dis-tribution is referred to as form error distributionin this paper. To investigate the surface form error

Figure 3 The standarderror versus number ofsample

Figure 4 Form error distribution


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effect on uncertainty, the form error distributionmust be modeled. A normal distribution is a typi-cal assumption in a statistical situation. The ma-jority of random distributions with a large numberof data, that are encountered in the real world, areclose to the normal distribution. Even for engi-neering surfaces, form errors are often assumed tobe normally distributed, but the actual distribu-tion may seriously depart from normal distribu-tion (Figure 5). The normal distribution assump-tion is appropriate when form error is affected byvarious small-scale, random, and independent ef-fects. It is also assumed that mechanical propertiesmust be uniform and stable. Actual engineeringsurfaces undergo various nonuniform errorscaused by tool wear, variation of stiffness, heattreatment deformation, etc. It is difficult to modelsuch form errors with a normal distribution, be-cause the normal distribution is symmetric andhas only two degrees of freedom.

In this paper, a more appropriate model forthe form error distribution, the beta distribu-tion,18 is employed. The beta distribution has afinite range, a property that better represents realmaterial properties than the infinite tails of thenormal distribution. The beta distribution hasfour degrees of freedom and can be used to rep-resent a wide range of distributions from uniformdistribution to bell-shaped distributions similar tonormal distribution, as well as to model asymmet-ric distributions. Thus, the beta distribution is flex-ible enough to address the actual variations in real

surfaces. Furthermore, it has been shown that thebeta distribution is more appropriate than thenormal distribution for real surface form errors.19

See Equation (2), Equation (3), and Equation(4). The beta distribution can be formulated asfollows. It is defined in a finite range a, b. Theprobability density function of a generalized betadistribution is defined with the exponents h and las follows.

f( x, l, h, a, b)

51

(b 2 a) B(l, h) Sx 2 ab 2 aD

l21S1 2x 2 ab 2 aD

h21

0 , l, 0 , h, a # x # b (2)

where B(l, h) is the beta function defined as

B(l, h) 5 *0

1z21(1 2 z)h21dz (3)

Substituting a 5 0 and b 5 1 in Equation (3), theunit beta distribution can be defined as follows.

f( z, l, h) 51

B(l, h)zl21(1 2 z)h21 (4)

0 , l, 0 , h, 0 # z # 1

The function f represents the probability den-sity of the surface area with respect to the height z,which represents the normalized surface form er-ror level. The surface form error characteristic isdefined by the beta-distribution exponents l andh. When l and h values are unity, the distributionis equal to a uniform distribution, and as l and hvalues increase from unity, the distribution be-comes similar to the normal distribution (Figure6). l and h are the weight of the distribution ineach direction, respectively. As the value of l be-comes larger, the distribution skews toward 0, and

Figure 5 Typical non-normal engineering surface

Figure 6 Beta distribution



as h values become large, the distribution skewstoward 1. With the beta distribution, more generaltypes of form errors, including nonsymmetric dis-tribution, can be modeled.

Analytic approximation of theuncertainty function

The mathematical relationship between the un-certainty and the measured point characteristicscan be represented by the standard error functionin terms of the form error, sample size, and thegeometry. Because an extreme fit is a nondiffer-entiable function, it must be solved by numericaliteration. Thus, the standard error of the flatnessvalue can only be evaluated by a numerical simu-lation. An approximation is proposed and shownherein for the flatness example. The standard er-ror of flatness evaluation represents the varianceof the range of the maximum and the minimumpoints. We approximate it by the standard error ofthe range of the extreme points in a fixed geomet-ric domain. The approximation does not considerthe rigid body transformation; thus, the derivedstandard error may be smaller than the true error.The mathematical model for this approach is toadd height variations from a known distribution toa plane and approximate the flatness from theextreme values of this added variation. Therefore,a fit is not made to the points, and the referenceplane is fixed for all cases.

We employ order statistics to model the flat-ness evaluation. The order statistics provide statis-tical properties of ranked elements, including themaximum and the minimum. The extreme pointscan be modeled as the ranked elements in a cer-tain geometric domain. The standard error of anextreme fit feature depends upon the variance ofthe extreme points when sampled from an un-known true surface. The variance of the extremepoints is determined by the variance of order sta-tistics elements.

Suppose the statistic x is distributed by a prob-ability density function f(x). Let us denote thecumulative probability distribution as F(x). Fromthe distribution, a random sample of n data aretaken, and the sample data are denoted as xi, i 51, . . . n. The random sample is a set of measureddata in a certain geometric domain. Let us definexm as the minimum of xi, and xM as the maximumof the xi. See Equation (5). Then the range of xi isthe difference between the maximum and theminimum as

w 5 xM 2 xm, 0 , w , ` (5)

If xi is the normal directional variation for ithpoint of measurement data from a measured sur-

face, the range w is an approximation of the flat-ness value. The standard error of flatness evalua-tion can be derived from the variation of the rangew. See Equation (6). From order statistics,20 thejoint density for xm to be the minimum and for xMto be the maximum is:

g( xm, xM) 5 n(n 2 1)[F( xM)

2 F( xm)]n22f( xm) f( xM) (6)

See Equation (7). The probability density g can beexpressed as a function of the range w and theminimum xm as

g( xm, w) 5 n(n 2 1)[F( xm 1 w)

2 F( xm)]n22f( xm) f( xm 1 w)(7)

See Equation (8). The probability density for therange w can be derived as integrating Equation (7)with respect to xm over all possible ranges as

h(w) 5 n(n 2 1) *2`

`

[F( xm 1 w)

2 F( xm)]n22f( xm) f( xm 1 w)dxm

(8)

Equation (8) represents the probability density ofa variable w to be the range of the order statistics.Thus, it is the probability density of the flatnessvalue when n datapoints are taken from a surfacethat has the probability density f(x). See Equation(9) and Equation (10). Then, the mean of w is

mw 5 E[w] 5 *0

`

w z h(w)dw (9)

and the variance is

Var(w) 5 E[(w 2 mw)2]

5 *0

`

(w 2 mw)2 z h(w)dw (10)

From Equation (10), the standard error offlatness can be expressed as a function of thenumber of datapoints n. Let us consider the fol-lowing example. See Equation (11). Suppose asurface is modeled by the beta distribution withthe exponents h and l unity, then the probabilitydensity of the noise x is a simple uniform distribu-tion as

f( x) 5 H 1 0 # x # 10 elsewhere (11)

The functional relationship between the numberof measurement points and the standard error of


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the evaluation can be analytically modeled. SeeEquation (12). By Equation (8), the probabilitydensity of the range w can be determined as

h(w) 5 n(n 2 1) *0

12wwn22dxm

5 n(n 2 1)wn22(1 2 w), 0 , w , 1(12)

See Equation (13), Equation (14), and Equation(15). Substituting Equation (12) into Equation (9)gives the mean of the range as

mw 5 n(n 2 1) *0

1wn21(1 2 w)dw

5 n(n 2 1) S1n

21

n 1 1D5

n 2 1n 1 1

(13)

and the variance is given as

Var(w) 5 n(n 2 1) *0

1 Sw 2n 2 1n 1 1D

2

z wn22 z (1 2 w)dw

52(n 2 1)

(n 1 2)(n 1 1)2 (14)

Thus, the standard error of w can be expressed as

se(w) 5 ÎVar(w) 5 Î 2(n 2 1)(n 1 2)(n 1 1)2 (15)

Equation (15) represents the uncertainty of theflatness evaluation with respect to the sample sizen, when the distribution is uniform. According tothe formulation, for 100 points, the standard errorof the flatness evaluation is 0.0138. Because theform error distribution is uniform as Equation(11), the standard error is 1.38% of the evaluatedflatness value. If the standard error is to be smallerthan 1%, the number of samples must be morethan 140 points. This approximation yields opti-mistic results, because the rigid body transforma-tion of the measured points is not considered.

For the general case, the formulation of thestandard error is difficult to derive analytically.The terms in Equations (8), (9), and (10) are inintegral form that is difficult to analytically formu-late. For the general beta distribution as definedin Equation (4), analytical integration of Equa-tions (8), (9), and (10) is not possible. Thus, theanalytic formulation has limited applications. For

a more general case, a numerical approach mustbe investigated.

Numerical approach

For a given measurement condition, the func-tional relationship between the standard error as-sociated with evaluation and the measurement pa-rameters must be derived. The standard errorrepresents the uncertainty of the form error eval-uation. Because such a functional relationshipcannot be generally formulated as an analyticform, it must be estimated by an experimentalapproach. A neural network method is employedto model the functional relation. A neural networkis trained to learn the pattern from a set of knowndata. After the network is trained, it is able tooutput the standard error for a new set of inputdata. The network represents the functional rela-tionship between the input parameters and theoutput parameter. The advantage of this approachis that the explicit form of the function is notnecessary. If the function is approximated by amultivariate regression, the explicit form of thefunction, such as a polynomial and the order ofthe polynomial must known beforehand. A neuralnetwork can represent a function without knowingthe explicit form and is applicable to more generaltypes of functions.

To build a network that represents the func-tional relationship, a training dataset must be pro-vided. The training dataset includes various mea-surement conditions. In each case of themeasurement, the possible standard error of theevaluated value must be provided. The standarderror is evaluated by iterating the process of sam-pling from a given known surface and evaluatingthe extreme fit. Then, from repetitive evaluationsof the extreme fit, the standard error of the eval-uation is determined. The set of the measurementparameters, including the geometry, the form er-ror, the number of samples, and the evaluatedstandard error, are used for network input andoutput. The network identifies the functional re-lationship between the parameters from the dif-ferent training sets.

To demonstrate this approach, a flatnessmodel is tested. The model is an L by L flat squarerepresented by x–y plane (Figure 7). To simulatethe form error of a real surface, noise with a betadistribution is added to the surface. The beta dis-tribution is able to model a general type of sto-chastic error that occurs on engineering surfaces.The range of the form error is set as a 5 0, b 5 0.1of the beta distribution. Thus, the ideal flatnessvalue of the surface is 0.1. To generate training



datasets, flatness of the surface and the standarderror of the flatness must be evaluated. A set of npoints is randomly selected from the surface, thenflatness is evaluated from the point set. The spatialdistribution of the points on the surface is uni-form, and the specific value of the surface profileat the sampled point is represented by the betadistribution. (The uniform point distribution maynot be the best spatial distribution of points; how-ever, it is frequently used in practice.) Flatness ofthe model surface is evaluated by the minimumzone evaluation. When a different set of n pointsare taken, the evaluated flatness value changesslightly. The standard error of the flatness is de-termined as the samples are repetitively resa-mpled. The sampling uncertainty can then be rep-resented by the standard error of the evaluatedflatness value.

The parameters that define the model are thegeometric parameter, length L, and the noise pa-rameters h and l of the beta distribution. A feed-forward network with one hidden layer is used, asshown in Figure 8. The network consists of fourinputs and an output. Inputs are model parame-ters and the sample size. The output is the stan-dard error of flatness value.

The training data are generated, as shown in

Table 1. The ranges of the parameters for thetraining dataset are shown as the minimums andthe maximums of the parameter values. Differentparameter values, that increment from MIN toMAX, are tested. The beta distribution parameterspans from 1 to 5. With different l and h values,the different cases of the surface form error aremodeled. When l and h are both unity, the betadistribution is equivalent to uniform distribution.In each case of the beta distribution, differentnumber of measured data samples are taken. Thenumbers of samples are from 20 to 200, a typicalCMM measurement size. The flatness is then eval-uated. To determine the standard error of theevaluation, each case is iterated 100 times andfrom this iteration, the standard error of the flat-ness value is derived.

Results

The neural network was implemented using Stutt-gart Neural Network Simulator,21 version 4.1. Thenetwork was trained by the backpropagationmethod, using the momentum algorithm. A learn-ing rate of 0.3 and the momentum term m 5 0.4were used. The network was trained in 50,000cycles with the learning data. The square sum oferror defined in Equation (13) is 0.3535, and themean square error is 0.0026. Thus, the root-mean-square error is 0.0508. The approximated func-tion has about 5% error in the output. The erroris acceptable, considering the fact that the stan-dard error contains a certain level of randomnoise, because it has been evaluated from a finitenumber of samples. Note that if the network were

Figure 7 Test model

Figure 8 Network model

Table 1 Training dataset for neural network

Minvalue

Maxvalue

Number ofincrements

h 1. 5. 5l 1. 5. 51 10. 1000. 3Number of sample 20 200 4


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trained to fit all of the outputs exactly, it might beoverconstrained so that it generates unnecessaryfluctuations.

Because the function is in the form of a neu-ral network, the explicit mathematical form is notknown, and the result must be presented in anumerical form. Figure 9 is a plot of the functionwith respect to the sample size. The function hasfour input variables and an output variable, and,for the sake of visualization, four curves are drawnon Figure 9. The network function is able to yieldthe standard error value for any input variables. InFigure 9, the standard error monotonically de-creases with respect to the sample size. For thedistribution with l 5 1, h 5 1, a uniform distribu-tion with range 0.1, the standard error is below1 3 1023 with more than 170 points. Consideringthe magnitude of flatness (for l 5 1, h 5 1), 0.1,the standard error is less than 1%. Compared tothe analytic approximation, the experimental re-sults yield a 20% higher standard error for thesame number of points. This is mainly because ofthe simplification in the analytic approximationthat ignored the rigid body transformation of thesurface in the evaluation.

An interesting observation is that the stan-dard error increases as the pair h, l increases inbeta distribution. As h and l increase, the proba-bility distribution is more dense at the center asthe bell shape becomes steep, as shown in Figure 6.Thus, as h and l increase, although the variance ofthe form error decreases, the standard error offlatness increases. This is consistent with the ob-servation that the standard error of flatness evalu-ation from a surface with normally distributed

form error is larger than the standard error froma surface with uniformly distributed form error.

This phenomenon can be explained by thedistribution of the maximum and minimumpoints. Although the maximum and minimumpoints for initial noise distribution may not be theextreme points determined by the minimum zoneevaluation, the distribution of the maximum andthe minimum points is closely related to the mag-nitude of the minimum zone. As the noise distri-bution approaches a uniform distribution, themaximum and the minimum points from n sam-ples become fairly close to the limits. Thus, therange between the maximum and the minimum ismore consistent. If the noise distribution is denserat the center, the average range between the max-imum and the minimum is decreased but the vari-ation of the range is increased. Thus, as h, l in-creases, the standard error increases, as illustratedin Figure 9. As a result, the normal-like distributionyields larger standard error than the uniform dis-tribution. This tendency is reversed as the numberof samples decreases, as shown in Figure 9. If thenumber of samples is small, the samples are notfully distributed in the range. The discrepancybetween extreme points of the sample and thetrue surface is larger. As the distribution becomesmore uniform, the discrepancy increases. Forsteep bell-shaped distributions, the points aremostly distributed at the center, which has lesssensitivity to the number of points. Thus, com-pared to uniform distribution, the variation of therange is small.

Figure 10 shows the trend of the standard er-ror with respect to the skewness of the distribu-

Figure 9 Standard error with re-spect to different sizes of formerror versus number of samples



tion. As the coefficient of skewness a increases, thestandard error increases. Because the flatnessvalue is theoretically symmetric with respect to theskewness direction, either in the positive directionor the negative direction, the sign of skewnessshould not affect the standard error of the flatnessvalue. However, for actual measurement data, thesign of the skewness is important. A positive skew-ness value for a form error represents valleys in thesurface, as shown in Figure 5. For a contactingsurface, such as a bearing surface, the positive

skewness may be acceptable, because the valleys ofa surface may have a minimal effect on the func-tionality of the part. Also the high spatial fre-quency characteristics cannot be measured by arelatively large probe. On the contrary, a negativeskewness means spikes on the surface, which arecritical deformations. The spikes are more appar-ently measured, and the standard error of theevaluation increases.

An interesting observation is the relation be-tween the geometry of the model and the standard

Figure 10 Standarderror with respect toskewness a of distribu-tion versus number ofsamples

Figure 11 Standard errorand the size of measure-ment region (h 5 l 5 1.5)


162 JULY 1998 VOL 22 NO 3

error. As shown in Figure 11, there is very weakcorrelation between the standard error and thegeometry size. Throughout the range of thelength used, from 20 to 1000, Figure 11 showsalmost consistent standard error. This result indi-cates the density of the measured points has littleeffect on the uncertainty. For a 200 number pointcase, the densities varied from 0.5 points per unitarea to 2 3 1024 points per unit area, yet they yieldalmost similar standard errors. Thus, it is the num-ber of measurement points, not the density of thepoints, that is influential to the uncertainty.

Conclusions

The results show how the property of a measuredpoint set affects the uncertainty of an extreme fitevaluation. The established numerical model rep-resents how the standard error of the evaluationchanges with respect to the number of measuredpoints. The results indicate that the distribution ofthe form error is critical for the standard error aswell as the number of points. When the distribu-tion approaches a bell shape, which resembles anormal distribution, the standard error of the ex-treme fit evaluation tends to increase. When theskewness of the form error increases, the standarderror also tends to increase. However, the size ofmeasurement area has little effect on the standarderror. Thus, the density of the measurement datais not critical for the uncertainty.

One factor that has not been considered inthis paper is the distribution of measurementpoints on the measurement plane. The distribu-tion of measurement points is regarded as uni-form in this paper, because the flatness evaluationis a form error evaluation where the probability ofsuch a variation being observed is relatively uni-form compared to a systematic deformation char-acterization. Nonuniform distribution is more ef-fective for a nonuniform variation, which is asystematic deformation. If the evaluation is tocharacterize a systematic deformation, a nonuni-form distribution of measurement points shouldbe considered.

For practical measurement situations, thenumber of measured points must be determinedbefore the measurement. The results of this work,unfortunately, indicate that the uncertainty resultsdepend significantly upon the distribution of theform error. This does not lend itself well to thedetermination of the number of measurementpoints needed to achieve a specified level of un-certainty. Because the distribution of form error is

unknown before the measurement, it is difficult topredict the uncertainty of the evaluation basedupon the number of points. Thus, the knowledgeof the form distribution must be provided. Such afact, unfortunately, indicates that dimensional in-spection planning must be processed case by case.

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