Sampling uncertainty in coordinate measurement data analysis

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<ul><li><p>Sampling uncertainty incoordinate measurementdata analysis</p><p>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</p><p>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.</p><p>Keywords: metrology; coordinate measurement; geometry; uncertainty</p><p>Introduction</p><p>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-</p><p>ently widely employed as 3-D measurement de-vices.</p><p>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.</p><p>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</p><p>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: Tom.kurfess@me.gatech.edu</p><p>Precision Engineering 22:153163, 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</p></li><li><p>point set on a plane. Minimum zone evaluationalgorithms have been reported in various stud-ies.310 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,1113 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.</p><p>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.</p><p>The objective of this research is to investigatehow the sample size is related to uncertainty in the</p><p>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.</p><p>Related work</p><p>Recognition of the sampling problem can befound in Hocken et. al.12 and Caskey et. al.,14where extensive numerical experiments on data-fitting algorithms were conducted to investigatesampling issues. They tested least-squares fit, minmax 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.</p><p>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.</p><p>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</p><p>Choi et al.: Sampling uncertainty in coordinate measurement data analysis</p><p>154 JULY 1998 VOL 22 NO 3</p></li><li><p>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.</p><p>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.</p><p>Extreme fit and uncertainty</p><p>See Equation (1). A typical form of the extremeproblem is represented as follows:</p><p>min M(u ) 5 max{distance[ pi, S(u )]} (1)</p><p>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</p><p>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.</p><p>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.</p><p>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</p><p>Figure 1 True value versus esti-mated value by measured data</p><p>Figure 2 Variation of a minimum zone value with respect to different samples</p><p>Choi et al.: Sampling uncertainty in coordinate measurement data analysis</p><p>155PRECISION ENGINEERING</p></li><li><p>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.</p><p>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 errorin this paper.</p><p>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</p><p>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.</p><p>Distribution of form error</p><p>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</p><p>Figure 3 The standarderror versus number ofsample</p><p>Figure 4 Form error distribution</p><p>Choi et al.: Sampling uncertainty in coordinate measurement data analysis</p><p>156 JULY 1998 VOL 22 NO 3</p></li><li><p>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 appropr...</p></li></ul>