SAMPLE SIZE PLANNING AND ESTIMATION OF FORM ERRORS USING A COORDINATE MEASURING MACHINE Sung Ho Chang Gary Herrin Shien-Ming Wu Department of Industrial & Operations Engineering University of Michigan Ann Arbor, MI 48109-2117 November 1990 Technical Report 90-35

Sample Size Planning and Estimation of Form Errors using a Coordinate Measuring Machine Sung Ho Chang Gary D. Herrin Shien-Ming Wu Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, Michigan 48109-2117 Technical Report 90-35 November 1990

ABSTRACT In this paper we have developed methods to statistically evaluate form errors using a coordinate measuring machine (CMM). The definitions of form errors in the current standards assume ideal inspection systems. However, there is no such ideal inspection systems actuality. Therefore, we establish practical mathematical definitions of form errors which can be applied for continuous or discrete measurements. They consider the characteristics of manufactured surfaces by assuming that the deviations from the nominal surface follow a Normal distribution. Importantly, these definitions are verified by measuring the real parts. Therefore, these definitions can serve as practical guideline for the inspection of real systems. In current CMM practice, there are no commonly accepted sample sizes for estimating form errors which have a statistical confidence. Practically, sample size planning is important for the geometric tolerance inspection using a CMM. We determine and validate appropriate sample sizes for form error estimation. Also, we develop form error estimation methods with certain confidence levels based on the obtained sample sizes in various form errors: straightness, flatness, circularity, and cylindricity. The determination of sample sizes use the new approach which is based on the maximum expectation of the straight prediction interval at a certain confidence level. The straight prediction interval is a new development which covers the variations of manufactured surfaces. This approach for estimating form errors, based on the proposed sample sizes, is superior to the current practice because it leads to better measurements 2

approximation. The proposed sample sizes and estimating method are verified by a simulation study and real part measurements. Furthermore, it considers the characteristics and functionality of manufactured parts. 3

1. Introduction High precision manufacturing, and thus increased new and improved high precision processes and machines, is in great demand today. This increasing demand is caused by the need: 1) to eliminate fitting problems and to promote assembly, especially in automatic assembly; 2) to improve interchangeability of components; 3) to improve quality control through higher machine accuracy capability which, in turn, will reduce scrap, rework and conventional inspection; and 4) to achieve further advances in technology. Many advanced (precision) technology products depend entirely on one or more components being manufactured to tolerances or dimensions in the micro- or even nanotechnology range. One of the tolerances considered in precision manufacturing is the form tolerance. Form tolerance is confirmed by evaluation of form errors. Evaluation and confirmation of form errors are executed by a computer controlled Coordinate Measuring Machine (CMM). It is one of the most widely used tools. Automobile companies alone are estimated to have over 300 CMMs. Generally CMMs are used for discrete measurements. Most CMMs use unique software programs, programs developed by their manufacturer, and, as a result, give different assessment of tolerances. These variations are due to the discrete measurements and mathematical definitions of tolerances built into the programs [Placek(1989) and Weill(1988)]. The tolerance we are dealing with in this paper is form errors. Formal definitions of form errors are given in current standards (ISO 1101 and ANSI Y14.5). Form errors are the linear distance between 4

two parallel geometrical curves or surfaces. These surfaces contain all the elements of the manufactured object surface. The American National Standards Institute in conjunction with the American Society of Mechanical Engineering offers the following definition of form error: "the error of form is considered as being that deviation from the nominal surface which is not included in surface texture (ANSI/ASME B46.1-1985)". These definitions assume perfect (continuous) measurements. However, continuous measurements are impossible as we can never measure actual maximum (peak) and minimum (valley) points, the points which theoretically contain all the elements of the surface. Therefore, this definition is limited to ideal measurements and does not lead to mathematical definitions for discrete measurements. To compensate for this limitation in the standards, it is current practice to estimate form errors as the sum of the algebraic maximum and minimum deviations from discrete measurements. These deviations are obtained from estimated surfaces. These surfaces are estimated by the various methods [ElMaraghy, Wu and ElMaraghy (1989), Shunmugam(1987, 1986), Fukuda and Shimokobe (1984), Murthy (1982), Murthy and Abdin(1980), Kakino and Kitazawa (1978), Gota and Lizuka (1977)]. These estimated surfaces vary or change depending on the number of discrete (sample) measurements. Consequently, the estimated form errors also vary. The evaluation of form errors (e.g. straightness, flatness, circularity, and cylindericity) using a coordinate measuring machine (CMM) relies on discrete measurements. However, definitions of form errors in the current standards (ISO 1101, ANSI Y14.5) assume perfect (continuous) 5

measurements, not discrete measurements. Therefore, there is no commonly accepted method for calculating form errors using discrete measurements; it is current practice to satisfy the definitions of the standards using discretely measured points. However, current practice does not consider the uncertainty of manufactured surfaces. As a result, it is not possible to give statistical confidence to the estimated form errors or to suggest statistically reliable minimum sample points. At the same time, the number of measured points needed to be large enough to provide reliable results. Theoretically, the minimum number of points to calculate form errors are straight forward. As an example, a minimum of three points are necessary to get a straightness error. Two points are used to estimate a straight line and one point is used to get the information about the uncertainty of the estimated straight line. If all three measured points lie on the perfect straight line, then there is no straightness error because the third point does not give any information. If they are not on a straight line, the third point gives information about the straightness error. However, there are no surfaces or curves whose uncertainty information can be explained by one point. Therefore, the theoretical minimum number of three points are not enough to obtain information about form errors. Additional measurements are needed to get statistically reliable information. By establishing a statistically reliable minimum, the manufacturer does not have to measure an inordinate number of points. In order to overcome the problem of inconsistency or change, we propose new definitions of form errors which can be used for discrete measurements. Also, these new definitions can be applied to the 6

continuous measurements. In other words, these definitions have the ability to represent the continuous measurements by the discrete measurements. One of the methods used to approximate continuous events by discrete events is probability distribution. It has often been assumed that there is a Normal (Gaussian) distribution [Greenwood and Williamson (1966)] of the deviations from the general surface shape of all manufactured parts. However, variations [Weckenmann and Heinrichowski (1985), Bourdet, Clement and Weill (1984)] in manufactured surface shape characteristics are significant due to various types of manufacturing processes. We take these effects into account with probability distribution. Then, we classify form errors into two cases depending on the surface shape characteristics; without systematic variation and with systematic variation. Form error without systematic variation occurs when the manufactured surface shape is the same as the specified shape. Here, form error depends on the deviations from the desired surface shape. Form error with systematic variation is the case that when manufactured surface shape is different from the specified shape. Here form error is more affected by the varied shape than by the deviations from the varied shape. In both cases, deviations are assumed to follow normal distribution. 2. Proposed Definition without Systematic Variation If it is assumed that manufacturing process is noisy, then the deviations of the product surface from the nominal (designed) product surface can be expected to follow a normal distribution. A nominal surface 7

is the intended surface contour which is usually shown and dimensioned on a drawing or descriptive specification. Theoretically, normal distributions have no finite minimum or maximum values. However, such large values are not found among the deviations. Practically, most of the data values lie within ~3a (standard deviation) of the nominal surface and these are the range from +3a to -3a, called the range of natural tolerance limits. If we define a form error as 60, then we can say that this range contains all the elements of a manufactured surface from a practical point of view. Therefore, this range satisfies the definition of the standard (ISO 1101). Accordingly we propose the following definition which can be applied to any kind of form error. Form error is 60 or the range of natural tolerance limits when the deviations from a nominal surface follow a Normal distribution Because we usually do not know the exact value of the standard deviation, we estimate the standard deviation from a sample. When the sample size is large enough to be considered a continuous measurement, we can use the estimated standard deviation to obtain form error. The estimated standard deviation is, however, a random variable. When the sample size is small, the estimated form error could vary depending on sample size and the form error can be overestimated. We will consider that problem in the later section. 8

3 Proposed Definition with Systematic Variation Because the characteristics of manufactured surface vary due to types and noises of manufacturing processes, we need additional definitions of form error to identify those surfaces. Surface characteristics are influenced by clamping setup, residual stresses and tool wear. Even though various surface characteristics have been described by Weckenmann and Heinrichowski [1985], and Bourdet, Clement and Weill [1984], and are shown in Fig. 1, we consider the second order polynomial curve for straightness and the special second order surface for flatness. These are combined with Gaussian case in this section. When the second order polynomial represents the straight manufactured surface, we can define the straightness error in a new way while still satisfying the definition given in ISO. When the surface shape is perfectly fit to the second order polynomial, the distance, between the line which passes through two end points and another line which is parallel to the previous line and which is tangential to the second order polynomial, is defined as a straightness error (See Fig. 2). However, no surface can be perfectly fit to the second order polynomial function because of manufacturing noises. Therefore, the straightness error is estimated by the variation of the shape of second order polynomial model (See Fig. 3). 9

(a) Spiral 'VWf\V (b) Convex or Concave (c) Sinusoidal (d) Gaussian Note:Each of (a), (b) and (c) can be combined with case (d) Figure 1 Possible Surface Characteristics 10

Workpiece magnified surface Workpiece magnified surface Measured Value A B M~~er~ur~d ~cation A,B: Two end points of measured workpiece Figure 2 Straightness Error in Second Order Polynomial Fitting Possible Measured \\ Straightness Value EnIr Measured otion Figure 3 Possible Straightness error in Second Order Polynomial Fitting 11

The new definition of a straightness error for the second order polynomial can be described in a mathematical expression because the two end points are decided by a specified straightness error given in the specification. A specified straightness error could be for a whole length (total straightness) or a partial length (straightness per unit length). Let the curve shape be a second order polynomial form, then its real function, its estimated function and its two assessment lines are y = 3P + Px + P2x2 + e e - N(0,o2) y = bo + blx + b2x2 Yl = ao + alx = co + alx (1) respectively. Xmax and Xmin are two end points of the given straightness range. Then two expected end points will be (Xmin, b + mi+ b + b2xmn), (Xmax, bo + blxmax + b2xm2x) The slope of two assessment lines, al, can be calculated b2xm2x + blxmax- b2xmn - blmin al = Xmax - Xmin = bI + b2 (max + Xmin) - (2) When y1 = as + alx is passing through two expected end points, ao can be calculated as 12

of confidence based on the confidence interval of the true coefficient y. (1 - a)*100 per cent confidence interval of coefficient y is C - t(1-a/2; n-4) s(C) < y < C + t(1-a/2; n-4) s(C) where n: number of measurements s(C): estimated standard deviation of y 1-a: confidence level. We can calculate the possible flatness error using the upper or lower bound value depending on the surface characteristic (lower bound for concave form and upper bound for convex form). The appropriate sample size problem will be discussed in the later section. 4. Various Surface Shapes in Real Parts We conducted real parts measurement using a CMM to observe the surface shapes and the deviations. This experiment was performed using the Sheffield Cordax RS-30 DCC CMM at the CMM Lab in the University of Michigan. A 165mm long bar (Fig. 4), a 200mm long bar (Fig. 5), and a 60x30mm rectangular bar (Fig. 6) were measured every 1mm. Also a 100mm long cylinderical bar was measured along the cylinder axis every 1mm (Fig. 7). From these measurements we can say that the 165mm bar and 100mm cylinderical bar surfaces satisfy the condition. The condition is that the deviations from the nominal surface follow the normal distribution. The 200mm bar and the 60x30mm bar surface satisfy the condition. The condition is that the deviations from the characteristic surface shape follow the normal distribution. The characteristic surface 16

shape depends on the manufacturing process characteristics. Therefore, we can say that our proposed definitions are appropriate for real parts. 17

-.708 -.709 E i -.71 0 ' -.711 -.712 - -.713 3 o.-714 -.715 -.716 -20 0 20 40 60 80 100 120 140 160 180 Measured Location (mm) Figure 4 Surface Measurements of 165mm long Bar -3.605, -3.61 E F3.615 3 > -3.62 <3.625.c ~ -3.63 13.635 -3.64 -25 0 25 50 75 100 125 Measured Location (mm) 150 175 200 225 Figure 5 Surface Measurements of 200mm long Bar 18

60(mm 30 (mm) 60 (mm) X Figure 6 Surface Measurements of 60x30 mm area -12.65 _-12.7 E E (''12.75 it 8 -12.8 2.85 cy12.85 I -12.9 -12.95 -13 Figure 7 Surface Measurements of 100mm long Cylinderical Bar along its axis 19

5. Proposed Approach We define form error as 6a or the function of the surface shape parameter in previous sections. We usually do not know the real values of a standard deviation or a surface shape parameter. They are estimated from a estimated nominal surface. The nominal surface is estimated from sample measurements. It is estimated because we do not know the exact location of real nominal surface (Fig. 8). Therefore, there are two variations in estimating form error: 1) variation in possible location of the nominal surface, and 2) variation within the estimation of standard deviation or surface shape parameter which involves probability distribution. We use the linear regression method to estimate the nominal surface. To use the linear regression method, we make assumptions based on the fact that machining processes are always disturbed by various noises which are independent of the form of the surface. Hence the cumulative effect of these noises is subject to the central limit theorem and is governed by a Normal distribution [Greenwood and Williamson (1966)]. Under these assumptions, we can make basic assertions that involve probability distributions. Let our manufactured surface be represented by the functional form Zi = f(Xi,Yi) + i where f(Xi,Yi): function of manufactured surface f(Xi,Yi) = po + P1xi (for simple straightness case) ei: combined noise 20

1. ei is a normal random variable with mean zero and variance a2 (unknown), that is, Ei - N (0, a2), E(ej)=0, V(ei)=a2. 2. ei and ej are uncorrelated, i * j, so that Cov(ei, ej) = 0 and Zi and Zj, i * j, are uncorrelated. Thus E(Zi) = fXi,Yi), V(Zi) = o2 Based on these assumptions, each observation comes from a normal distribution centered vertically at the level implied by the proposed model. The variance of each normal distribution is assumed to be the same. Reference Measuring Axes Y Measured Surface M u / Pr EstiatedNomiaSr Measuring Part Estimated Nominal Surface Figure 8 Reference measuring axes in a CMM and estimated nominal surface 21

We can simply use these estimated standard deviation or surface shape parameter values to obtain form errors without considering statistical confidence. We will not know much about the variations of the real nominal surface and the probability of parameter estimations. Therefore, we use a prediction interval length that considers the variation of real nominal surface and the variation of parameter estimation. The prediction interval length (PI) can be represented by the function of the sample size, a specified point and an estimated standard deviation in general linear regression analysis. PI = 2*t(n-p,l-a)*{1 + f(n,Po))*MSE where t(n-p,l-a): upper (1-a) percentage point of t-distribution with (n-p) degrees of freedom f(n,Po): function of sample size n and a specified point Po MSE: estimated variance When a certain PI, with given sample size n, confidence level (1-a) and MSE, is approximately equal to 6a, we can say that it is an estimated form error. The bands of the PI, however, are curvilinear and our objective is to find linear bands which cover the maximum variations of the nominal surface and its estimation. The maximum PI is chosen at a given sample size (Fig. 9). However, the interval estimate of the PI is a random variable because the sample standard deviation (or JMSE) is a random variable. The expected prediction interval is compared to 6a. We can say that PI at that sample size is an estimation of form error which has no systematic error

when the expected length of the maximum prediction interval, at a certain sample size with a certain confidence level, is approximately equal to 60. The upper or lower confidence limits of the surface shape parameter at that sample size can be used to estimate form error. This has a systematic error with the same confidence level because the confidence interval of the surface shape parameter is narrower than the PI with the same sample size. Y i Prediction Interval Bands i P,, " Estimated Line.Ioi'l I o — ' "~ Straight Prediction Xi/,, '"""" Interval Bands I I"! 1X xl x2 Figure 9 Illustration of Maximum Straight Prediction Interval Bands The statements above can be represented in mathematical terms as follow; PI(Po) = 2 * tn-pla2 { 1 + Po' (PP)-1 P0o 2 /MSE (12) 23

where PI(Po): length of prediction interval at Po PO: specified column vector of P P: observation design matrix MSE: estimated variance of least square residuals p: # of parameters estimated a: confidence level. Since the residual error variance (MSE) follows a Chi-square distribution (n-p)MSE 2 2 Xn-P the expected length of the prediction interval (E[PI]) can be represented as follows E[PI] = 2 tn p,1a/2 h(n,P0) j-E[Cn-p] (13) where h(nPo) = (1 + Po' (pp)-1 PO 1/2 Xn [(n-p)/2] r(n) = Gamma function 00 = xn1 ex dx. If E[PI] = 6a, we can determine the appropriate sample size needed to estimate the form error when the confidence level is given. Or, we can determine the appropriate confidence level when the sample size is given which satisfies 24

E[Xn-p] 3. tn pla/2 ) h(n,PO) n-p=3 (14) In this section we are only considering sample size determination when the confidence level is given for straightness, flatness, circularity, and cylindericity errors. The proposed new approach for determining the appropriate sample size and for estimating form errors not only satisfies the proposed definitions but also accounts for the possible variations in the estimating procedure. 6. Sample Size Determination In this section, we explain the procedure for determining sample size using the prediction interval approach for various functional forms. Observation matrix P is constructed with the assumption that each measurement is equi-distance in every dimension. 6.1 Simple Straight Tine Function The general simple straight line regression function from sample size n is represented by Y= bo + blX.

In the simple straight line case, the maximum value of Po'(P'P)'Po can be 1 obtained at one of two end points. As an example, we have (3 + 0.5) when n=3 as follows: 1/3 0 P= 1 0o (p)-l=[ 12] Po=(1, -1) or (1, 1) 1 PO'(PP)'P0 = + 0.5). In the same way we can get the maximum value of Po'(P'P) 1Po for different sample sizes. Then, we can find the appropriate sample size which satisfies the following condition: 2 tn.2,1./2 (Po'(P'P)'lpo)12 i } 6 (15) Nn-2 where l-a: confidence coefficient of prediction interval. The appropriate sample sizes with 95% (a=0.05) and 99% (a=0.01) confidence for the simple straight line function are 7 and 24, and are shown in Tables 1 and 2 respectively. Numerical values for E[Xnp] are in Appendix B. 26

Sample Size tn-2,1-0.05/2 E[Xn-21/nI2 I+Xo'(X'X)-lX0 E[PII/ 3 12.706.7979 1+3+ 0.50 27.454 4 4.303.8862 1++0.45 9944 5 3.182.9213 ++ 0.40 7.416 6 2.776.9400 + 0.36 6.448 7 2.571.9513 1+ + 0.3 5.916 8 2.447.9594 1+ 29 5.585 Table 1 Sample Size for Simple Straight Line Function with a=0.05 Sample Size tn.2,1-0.01/2 E[Xn. n-2 1 i +XO'(X'X)-1X E[PII/ 22 2.845.9876 1++22 6.066 23 2.831.9882 1+3 6.030 24 2.819.9887 1+24+ 6.008 25 2.807.9892 1+5 + 0.11 5.955 Table 3.2 Sample Size for Simple Straight Line Function with a=0.01 62 Second Order Poynomial Curve Function The general second order polynomial regression function from sample size n is represented by 27

Y= bo +blX+b2X2. As an example, the maximum value of Po'(P'P)Po will be (+ 0.7) when n=4 as follows: b X X2 1 1 P= 1 V13 19 1 -1/3 1/9 41/64 0 -45/641 (PP)-l = 0 9/20 0 L-45/64 0 81/64j Po0' =(1, 1, 1) or (1, -1, 1) Po'(P'P)1Po 1 + 0.7). In the same way we can get the maximum value of Po'(P'P)Y'Po for different sample sizes. The appropriate sample sizes with 95% and 99% confidence for the second order polynomial curve function are 9 and 36. 6.3 Simple Plane Function The general simple plane regression function from sample size n is represented by Z = bo + blX + b2Y. As an example, the maximum value of P'(P'P)'Po will be (+ 0.5) when n=4 as follows:

-b X 1 400 P= 1 1-1 (pp)- =04 0 1 -11 004 _1 -1 -1J PO' = (1, 1, 1) or (1, -1, -1) P0'(P'P)'Po = ( + 0.5). In the same way, we can get the maximum value of Po'(P'P)-1Po for different sample sizes. The result is similar to the simple straight line function except for the number of parameters to be estimated. The appropriate sample sizes with 95% and 99% confidence for the simple plane function are 8 and 25. 6.4 Second Order Surface Function In this case we consider only the specific form of a surface Z = bo + b + b2 + b3Y. As an example, the maximum value of Po'(P'P)1Po will be + 0.8) when n=5 as follows: b bXY X2 1-1-1 1 p= 1 1-1 1 100 0 1-1 1 1 111 1 1 0 0 -1 (p1p)-= 0 1/4 0 0 0 0 1/4 0 -- 0 0 5/4

PO' =(1, 1, 1, 1) or (1, -1, -1, 1) Po'(P'P)'Po = (1 + 0.8). In the same way, we can get the maximum value of Po'(P'P)'1Po for different sample sizes. The result is similar to the simple straight line function except for the number of parameters. The appropriate sample sizes with 95% and 99% confidence for the simple plane function are 9 and 36. 6.5 Circular Function The linearized deviation (Fig. 10) is used [Shunmugam (1986)] to estimate the circle from n observations which are represented by polar coordinates (ri, Oi): ei = ri - (Ro + xoCos0i + yoSin0i) (16) where Ro = radius of the estimated circle xo, yo = coordinates of origin of the estimated circle. Then, the desired regression function can be written as follows: ri = Ro + xoCos0i + yoSin0i (17) 30

Y Figure 10 Linearized Deviation from Circle and if Y = ri, bo = Ro, b1 = x0, b2 = yo, X1 = CosOi and X2 = Sin0i, then Y = bo + blX1 + b2X2. (18) Because CosO cannot be represented by the linear combination of Sine and there are three parameters to be estimated, Eq.(18) is exactly same as the simple plane function. Therefore, the appropriate sample sizes with 95% and 99% confidence for circular function are 8 and 25. 6.6 Cylindrical Function The linearized deviation (Fig. 11) is used Shunmugam (1986)] to estimate the cylinder from n observations which are represented by cylindrical coordinates (ri, i, zi): 31

ei = ri - [Ro + (xo + lozi)Cos0i + (yo + mozi)Sin0i] (19) where Ro = radius of estimated cylinder x0 = x coordinate of origin of estimated cylinder YO = y coordinate of origin of estimated cylinder lo, mo = slopes of estimated cylinder axis. Axis of cylinder Xo+ 10 Zi YO + mozi z y Pi i^zi) e. xEstimated Cylind Estimated Cylinder x 0+l0z x Figure 11 Linearized Deviation from Cylinder Then, the desired regression function can be written as follows:

ri = Ro + xoCos0i + yoSinei + loziCos0i + moziSin0i (20) and if Y = ri, bo = Ro, b1 = x0, b2 = yo, b3 = lo, b4 = mo, X1 = CosOi, X2 = SinOi, X3 = ziCos0i and X4 = ziSinOi, then Y = bo + blX1 + b2X2 + b3X3 + b4X4. (21) As an example, the maximum value of Po'(P'P)-'Po will be (+ 0.82) when n=6 as follows: 1 1 0 -1 0 1 3 3 3~3 1 2 '2 10 ' 10 1 3 1 3 i-6 '10 1 -2 2 10 -10 P= 2 1 - 10 1 X3 _ 3 3N3 1 - ~2 2 10 10 1 X3 5 5^3 1 2 -2 10 '10 3 2 243 5 5N3 2 3 9 '3 3 2 71 323 19 17__3 3 106 315 21 21 (pp)-I= 23 323 149 133 19 _ 9 315 315 63 21 5 19 1313 55 4041l 3 21 ' 63 -21 21 5N3 17N/3 19 40N 45 3 21 21 21 7

Po' =(1, 1, 0, -1, 0) or (1, 2,, 10' -10) PO'(P'P)Po = + 0.82). In the same way we can get the maximum value of Po'(P'P)-1Po for different sample sizes. The appropriate sample sizes with 95% and 99% confidence for the cylindrical function are 13 and 55. 7. Testing the Aptness of the Proposed Sample Sizes Appropriate sample sizes were obtained based on the expected length of the prediction interval. These sample sizes require a minimum number at a certain confidence level. Estimated form errors are the length of the maximum prediction interval at the same confidence level. To verify the aptness of this proposal, a simulation study was conducted and real parts measurements were made. 7.1 Simulation In the previous section, we obtained the appropriate sample sizes for various functions with two different confidence levels. To test the aptness of these sample sizes, we conducted a simulation study for the simple straight line function. We expect that similar results will be obtained in other functions. We generated 1000 normal random number sets, each of which consists of 1000 numbers, with a mean of zero and a standard deviation of 34

0.001, to test the aptness of the sample sizes. We assume that each data set represents all the surface elements of a measured part. By observing each data set, we can say that straightness error of each set is 0.006 (6a). We collected sample sizes from 3 through 25 numbers from each set with equi-intervals. In other words, we assigned the number 1 to 1000 for each number in each data set assuming that each number represented a serial measurement location. As an example, with sample size 7, we collected values which have assigned numbers of 1, 167, 333, 500, 667, 833 and 1000. Next, we estimated a straight line using each sample based on the least squares estimation. From each straight line estimation, we collected square roots of the mean square error (MSE) values and calculated estimated straightness error, and calculates mean and standard deviation of those values (Table 3 and Fig. 12). The constants multiplied by tFMSiEto estimate straightness error are different depending on the sample size and the confidence level. Exact values of these constants are obtained using Eq.(13). Even though the constant values at sample size 7 and 24 are not exactly 6 (Table 4), the estimated straightness errors are not significantly different from that of multiplied by 6 because the IVMSE value is very small in practical form error measurements. The same mean value of 0.006 was the estimated straightness error of the sample sizes 7 and 24 for 95% and 99% confidence level, respectively. Based on these results, we can say that our sample sizes were appropriate for each confidence level. Also, the mean value of the estimated straightness error was calculated by simply multiplying 6 by VMSE for the practical purpose (Table 5). We can observe that our proposed definition is appropriate. This means that if we do not have to worry about

the confidence level, then simply estimated straightness error can be a reasonable estimation. We can determine the confidence level at each sample size using Eq.(14). However, we did not calculate these confidence levels because our objective is to verify the sample size at a certain confidence level (95% and 99%). 36

I Mean of 95% Stdof 95% Mean of 99% Std. of 95% Sample Size ConfideneSE CnfednceSE Cofidence SE Confiednce SE 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0.0337 0.0102 0.0075 0.0066 0.0060 0.0055 0.0054 0.0051 0.0050 0.0049 0.0048 0.0048 0.0047 0.0047 0.0047 0.0046 0.0045 0.0045 0.0045 0.0044 0.0044 0.0044 0.0044 0.0190 0.0049 0.0030 0.0023 0.0018 0.0015 0.0014 0.0013 0.0011 0.0011 0.0010 0.0010 0.0009 0.0009 0.0008 0.0008 0.0008 0.0007 0.0007 0.0007 0.0007 0.0006 0.0007 0.1686 0.0236 0.0138 0.0109 0.0094 0.0084 0.0080 0.0075 0.0072 0.0070 0.0068 0.0067 0.0066 0.0065 0.0064 0.0063 0.0062 0.0062 0.0061 0.0061 0.0060 0.0060 0.0060 0.0953 0.0113 0.0055 0.0037 0.0029 0.0023 0.0021 0.0019 0.0016 0.0015 0.0014 0.0013 0.0013 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 0.0009 0.0009 0.0009 0.0009 L I I I Note: SE: Straightness Error Std.: Standard Deviation Real SE = 0.006 Table 3 Mean and Standard Deviation of estimated straightness error with 95% and 99% confidence for different sample sizes 37

0.015 | 0.010 5 I, r./'2,O 0.005 0.000 0.015, 0 10 20 Sample Size (a) 95% Confidence 30 0 t 04 E 1-3 Cd I Po 0.010 0.005 0.000 0 10 20 30 Sample Size (b) 99% Confidence Note: Real Straightness Error = 0.006 Figure 12 Mean and Standard Deviation of Estimated Straightness Error with (a) 95% and (b) 99%o confidence 38

Constant for 95% Constant for 99% Sample Size Confidence Confidence I 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 21 22 23 24 25 34.408 11.221 8.049 6.860 6.219 5.821 5.559 5.359 5.200 5.068 4.974 4.895 4.824 4.773 4.706 4.655 4.608 4.584 4.543 4.504 4.487 4.471 4.439 I 172.384 25.881 14.777 11.378 9.753 8.819 8.224 7.796 7.471 7.208 7.019 6.862 6.726 6.610 6.507 6.414 6.329 6.279 6.209 6.142 6.102 6.077 6.020 Table 4 Constants multiplied by 4MSE for estimating Straightness Error 39

Sample Size Straightness Error (64MSE ) Standard Deviation I I~~~~~~~~~~ 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25.0059.0055.0056.0058.0058.0057.0058.0058.0058.0058.0058.0058.0059.0059.0059.0059.0059.0059.0059.0059.0059.0060.0060 I.0033.0026.0022.0020.0018.0016.0016.0014.0013.0013.0012.0012.0012.0011.0010.0010.0010.0009.0009.0009.0009.0009.0009 Note: Real Straightness Error = 0.006 Table 5 Mean value and Standard Deviation of Simply Estimated Straightness Error without considering Confidence Level 40

7.2 Type I and II errors of the proposed approach The purpose for estimating form error is to evaluate the conformance of a product to its tolerance. However, we cannot guarantee 100% assurance unless we measure all the surface points of the product. Therefore, we determined the minimum sample size with a specified confidence level. Now we want to see the tolerance conformity in terms of Type I and Type II errors. We estimate a form error using the length of the prediction interval at a specific sample size. However, it is a random variable because of INMSE. This interval has its own confidence interval with a given specified confidence level (1-y). When we assume that the manufacturing process is in-control, we are interested in determining the out-of-control state or the increment of variance (Type I error). A Type I error is (y*100)% assuming that this specification is the upper confidence limit of the estimated form error with (1-y) confidence. Therefore, when the specification is given, the Type I error can be obtained as follows: Ho (Null Hypothesis): a2 < 20 Ha (Alternative): o2 > c0 (22) where o2: estimated variance 20: desired (or tolerable) variance PI(Po) = 2 * tnp,.2 { 1 + Po0 (PP)-1 PO }12 VMSE and we already know (n-p)MSE 2 and we already know on- M XSna2 Xn. 41

we want to find the probability to reject the Null Hypothesis Pr(n-p) MSE 2 ) an 02 > X (-p, Y Pr( iMSE> ao. c n-p ) Pr( PI> o n-p )=Y (23) 4"MSE %Jo n-P Therefore, we can find Type I error y by solving the equation when the specification is represented by the multiplication of the standard deviation (C*ao): C = 2 * tnl2 ( 1 + P0 (P'p)- Po }1/ 2 n (24) n-p When we assume that the manufacturing process is out-of-control, we are interested in determining the in-control state. This decision is a Type II error. A Type II error can be obtained when the state of out-ofcontrol is given. We assume that the state of out-of-control is the increment of variance by the multiplication of the standard deviation. It also can be expressed as follows: Ho (Null Hypothesis): (Ka)2 > (Cao)2 Ha (Alternative): (KO)2 < (Coo)2 (25) where (Ka)2: estimated variance Cao: specification K: out-of-control state 42

we want to find the probability to reject the Null Hypothesis Pr(n-p) MSE 2 = 1 r (CoO)2 < n-p,l- ) 1 -_ /MSE ] (Xn%, 1-) Pr( K < Cao - ) =K o n-p Pr( PI S P C Xn )=-p (26) Then, we can obtain the Type II error P by solving the equation K= 2 * tnp,lc/2 { 1 + P0' (p 'p)1 Po } 2 (np, ~K *" - t~-pl-a/211 +.O ^0n-p (27) In order to test the procedure for obtaining Type I and II errors, we generated 5000 normal random number sets with a mean of zero, and a standard deviation of 0.001. Each of them had 1000 numbers. We collected samples of size 7 and 24 with equi-intervals from each set for 95% and 99% confidence, respectively. Then, estimated a straightness error for each sample size. For both sample sizes, we obtained a mean value of 0.006 which is the same as the estimated straightness error. However, their variation (Fig. 14) is different because of the difference of the sample sizes. When we assumed that the specification is 9a or 0.009, the Type I error for sample size 7 was a little bit greater than 0.05 and for sample size 24 was much less than 0.005. It was obtained by following procedures. 43

n-p C = 2 * tp 11a/2 l + PO (P'P)'1 PO } V ^ from Eq.(3.13) where C = 9, p =2 For n=7, a= 0.1 2 * tn-p,la/2 ( 1 + Po' (P'P)-1 PO }2 = 6.22 For n= 24, a=0.02 2* tn.p, 1 + PO (PP)-' Po /2 = 6.05 2,0.05 =11.07 220.00 =42.80 X = 10.47 2 = 48.69 X5,y 22,= 0.1 = 9.24, = 40.29 Then, yis a little bit greater than 0.05 for n=7, a= 0.1 y is much less than 0.005 for n= 24, a=0.02. To test for the Type II error, we did the same procedure except for the standard deviation of 0.002. The mean value of the estimated straightness error was 0.012 as expected. Their distributions are shown in Fig. 15. We can obtain the Type II error for sample size 7 as a little bit greater than 0.1 and for sample size 24 as a little bit less than 0.05 because we increased the standard deviation two times. It can be also obtained by following procedure. = 2 * t-p,2 { 1 + Pod (PP)'1 PO )12 A o) from Eq.(3.16) n-p 44

where C=9, K=2, p=2 251. = 4.35 X21- =2.62 x25-0. = 1.61 x22,1o = 12.34 22,1- 0.05 22,1- -= 12.17 5,1-0.025 = 1098 Then, 3 is a little bit greater than 0.1 for n=7, a= 0.1 p is a little bit less than 0.05 for n= 24, a=0.02. 45

1200 1000. 800. o 600. 400. 200. a _ _ ~ ~ ----— I ~ — ~- n. a r —4 I +LL~ 0.002 I. * I *I I. ~.004.006.008.01.012 Estimated Straightness Error with sample size 7 014.016 1400 1200 1000.40 0 800 600 400 200.002.004.006.008.01.012.014 Estimated straightness error with sample size 24.016 Figure 14 Distribution of Estimated Straightness Error 46

1000 600 o 500 400 300 200 100 0. -..-. 0.005.01.01 5.0 2.025.03 Estimated Straightness Error with sample size 7 1200,- 1000 800 o 600 400 200 0!.006.008.01.012.014.016.018.02.022 Estimated Straightness Error with sample size 24 Figure 15 Distribution of Estimated Straightness Error when standard deviation is doubled 47

7.3 Experiments with Real Data In addition to the simulation, we conducted real part measurements using the CMM to assess the straightness error. This experiment was performed using Sheffield Cordax RS-30 DCC CMM at the CMM Lab in the University of Michigan. A 165mm long rectangular bar was measured in increment of 1mm (Fig. 4). Based on these measurements we concluded that its straightness error is approximately 0.006mm. We collected sample sizes 3 to 25 from these data with almost equi-interval. For each sample size, we estimate 95% and 99% confidence straightness errors (Table 6). A 200mm long bar was measured in the same way (Fig. 5) and the estimated real straightness error was 0.017mm. Sample sizes 4 to 15 and 36, with almost equi-interval, were collected. These Estimated straightness errors, using Eq.(9), are shown in Table 7 and Figure 17. The estimated straightness errors have a tendency to decrease when the sample size is increased even though there are some fluctuations (Figs. 16 and 17). We expected those fluctuations because our form error estimation approach uses 4iMSE,. Even though we did not get the exact value of a straightness error at the desired sample size, we got reasonably estimated values around the desired sample size. 48

Smple Size 95% Confidence (mm) 99% Confidence (mm) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 24 25 0.014 0.007 0.008 0.009 0.005 0.006 0.005 0.005 0.008 0.005 0.006 0.004 0.008 0.005 0.007 0.006 0.005 0.005 0.007 0.005 0.005 0.007 0.005 0.070 0.016 0.015 0.015 0.007 0.008 0.007 0.007 0.011 0.008 0.008 0.005 0.011 0.007 0.010 0.009 0.007 0.007 0.010 0.007 0.006 0.009 0.006 Table 6 Estimated Straightness Error for 165mm long bar 49

Sample Si 95% Confidence (mm) 99% Confidence (mm) 4 0.050 0.191 5 0.026 0.041 6 0.018 0.021 7 0.019 0.022 8 0.019 0.021 9 0.019 0.021 10 0.019 0.021 11 0.017 0.018 12 0.020 0.022 13 0.017 0.018 14 0.018 0.019 15 0.017 0.019 36 0.015 0.016 Table 7 Estimated Straightness Error for 200mm long bar 50

(mm 0.020 | 0.015 0.010 0.005 0.000 ) A.. \ [ 1 1,w H 1 I 'ur................ II: _ ^_______ --- —--------- - -------------- ------------ '~\A/~^^ ^~~~~~~~~~~~~~~~~~~~~~~~~ --- —--- ^ -^K ^_~~~~~~~~~~~~~~~~~~~ --- — --- ---- 95% —. -- 99% 0 5 10 15 Sample Size 20 25 30 Figure 16 Variation of Estimated Straightness Error according to the sample size for 165mm long bar 51

(mm) 0.030 - 0.025 * 0.020 0.015 0.010 X --- 95%:*- 99% 0 5 10 15 20 25 30 35 40 Sample Size Figure 17 Variation of Estimated Straightness Error according to the sample size for 200mm long bar 52

& Comparison to other Techniques using Simulation We proposed an alternate approach for estimating form errors in previous sections. We can compare the results of this proposal with those of current approaches in order to test its aptness. The Least Squares (LS) and the Minimum Deviation (MD) approaches are used for comparison purpose. We only considered straightness for simplicity because other form error cases will give the similar results,. We used normal random numbers to test the various approaches because it has often been assumed that there is a Gaussian distribution [Thomas (1982)] to all manufactured parts that are being measured. However, whatever approach is applied to estimate the straightness error, the estimated value is a random variable because it is estimated from the combination of random numbers. We compared the expected values as well as several example cases. We generated 5 sets of 1000 normal random numbers with a mean of zero and a standard deviation of 0.001 to represent 5 different surfaces. From these numbers, we said that the real straightness error is approximately 0.006 for each surface. We collected samples of size 7 and 24 from each set. We collected 1st, 167th, 333rd, 500th, 667th, 833rd and 1000th numbers at equi-distances of sample size 7. The same procedure was applied to sample size 24. By applying three different approaches - prediction interval, least squares and minimum deviation -- we estimated straightness errors for each simulated surface. Actually we did not calculate the results of the 53

minimum deviation approach because they are always less than or equal to those of the least squares approach. The estimated straightness errors of each approach are given in Table 8 for sample sizes 7 and 24. Case Real PIMethod ISMethod MD Method Sample Size _7 21 7 24 7 24 Surface 1 0.006 0.005 0.006 0.002 0.004 0.002 0.004 Surface 2 0.006 0.004 0.008 0.002 0.005 < 0.002 < 0.005 Surface 3 0.006 0.007 0.005 0.003 0.003 <0.003 <0.003 Surface 4 0.006 0.005 0.006 0.002 0.005 < 0.002 < 0.005 Surface 5 0.006 0.004 0.006 0.002 0.004 ~ 0.002 <0.004 where PI: Prediction Interval LS: Least Squares MD: Minimum Deviation Table 8 Estimated Straightness Errors using Different Approaches We compared the expected values of each approaches to show the generality even though the prediction interval approach gives results close to the real value in the cases above. The expected value of the prediction interval approach was given in previous section as follows: E[PI] = 2 tn.p,1./2 h(n,Po) -w(n-p) where h(n,Po) = { 1 + Po' (P'P)-1 PO )2 (28) 54

w(n-p) = E[Xn-p] [(n-p+)/2] UF(n-p)/2] r(n) = Gamma function 00 = xn-1 e- dx For the least squares approach we used the expected values of the range, X(n) - X(1, in the order statistics [Sarhan and Greenberg (1962), Harter (1969)]. We did not use the residual because the difference between maximum and minimum residuals is always less than or equal to the difference between maximum and minimum values in certain sample size from order statistics. If we have a sample of n observations, X1, X2,..., Xn, and rearrange them in ascending order of magnitude as X(1) < X(2) -<. < X(n), we call X(r) is the rth order statistic. Xi are assumed to be statistically independent and identicaly distributred. The expected value of the kth largest observation, in a sample of size n from a standard normal population (g=0, a2=1), is given by 00 n! ( i I 1 ) E(X(k)) = (n-k)! (k-l)! [ (X)] [ +X)]-k (X) dX (29) 00 1 X2 where (X) = exp(- 2 v^ z 55

x <>(X) = J(X)dX Utilizing Eqs.(28) and (29), we obtained the expected values of the estimated straightness error for the different sample sizes as given in Table 9. Case Real PI Method LS Method MD Method Sample Size _7 2a 7 2a 7 2X Straightness 0.006 0.006 0.006 0.003 0.004 < 0.003 0.004 * We assume that the population is a normal distribution with 1=0 and a=0.001. Table 9 Expected Values of Estimated Straightness Error in different Approaches 9. Conclusion This paper has presented two definitions for various form errors which can be operationalized and represented in mathematical terms even with discrete measurements. This chapter has also presented a procedure for determining the appropriate sample size and a formulation for evaluating form errors using the CMM. These new definitions have the following characteristics. 1) They consider the characteristics of manufactured surfaces. 2) They can be represented in mathematical terms. 56

3) They can be used to determine appropriate sample sizes with a certain confidence level. 4) They were carefully tested by measuring the real surfaces. The approach for sample size determination have the following charaterisitcs. 1) It determines the sample size with a new criterion which is applied to the expectation of prediction interval with various confidence levels (95% and 99%). 2) It can be used to determine the confidence level when the sample size is given. 3) It uses the least squares criterion to estimate the desired feature in functional form. 4) It can be used to calculate Type I and II errors when the specification is given because it is statistically well defined. The results of testing and verifying of this new sample size determination approach are as follows. 1) It was carefully tested for determining the sample size for straightness, flatness, circularity and cylindericity. 2) The formulation was carefully tested for determining the straightness and flatness errors from simulated data and real measurement data. 3) Finally and most importantly, the results were tested and shown to be successful and satisfactory. 57

The approach proposed in this paper can provide a useful basis for further research for estimating form errors using the CMM. The formulations developed for straightness and flatness errors can be extended to a higher order of dimensional geometric tolerances. Consequently, the formulation can be established to estimate true geometric errors using the CMM. 58

Appendix A. Expectation of k The probability density function of X2 distribution is k k x I1 1 2 2 2 f(x)= () x e rF(\ The expectation of Xk can be calculated by substituting Fx into x in applying the definition of expectation, 00 k k x 2 e dx E[4x] = - (12 x e dx 00 r k k 1 x 1 122 2 2 = '" x e dx O0 00 k k+1 x f I 1) 2 2 ' 12 ( j) x e dx

J jk+l k+l x rk+l 1 1) 2 2 2 2 -2 k lQ2' x e k x =_k+l 2 2 ' k+l 1 k+l k+l x 2I 12 2 2 1 -2 =- k k+2) x e dx r() F(2) k+l 1 2 1 2 R2k 2 F (2) - ()2 60

B. Table for Numerical Values of E[xn.p/Inn-p E[Xn.p]/n-p__ 1 0.7979 2 0.8862 3 0.9213 4 0.9400 5 0.9513 6 0.9594 7 0.9650 8 0.9693 9 0.9727 10 0.9754 11 0.9776 12 0.9794 13 0.9810 14 0.9823 15 0.9835 16 0.9845 17 0.9854 18 0.9862 19 0.9869 20 0.9876 21 0.9882 22 0.9887 23 0.9892 24 0.9896 25 0.9901 26 0.9904 27 0.9908 28 0.9911 29 0.9914 61

American National Standard Institutes (ANSI Y14.5M-1982) Dimensioning and Tolerancing. American National Standard Institutes (ANSI/ASME B46.1-1985) Surface Texture (Surface RoughnessnWaviness, and Lay) American National Standard Institutes (ANSI/ASME B89.1.12M-1985) Methods for Performance Evaluation of Coordinate Measuring Machines. Bourdet, P., A. Clement and R. Weill [1984], "Methodology and Comparative Study of Optimal Identification Processes for Geometrically Defined Surfaces", Proceedings of the International Symposium on Metrology for Quality Control in Production, Tokyo. Box, G. E. P. and N. R. Draper [1987], Emprical Model-Building and Response Surfaces, John Wiley & Sons, Inc. Draper, N. and H. Smith [1981], Applied Regression Analysis, 2nd. Ed., John Wiley & Sons, Inc. ElMaraghy, W. H., Z. Wu and H. A. ElMaraghy [1989], "Evaluation of Actual Geometric Tolerances Using Coordinate Measuring Machine Data", ASME Design Technical Conferences, vol. 19-1. Fukuda, M and A. Shimokohbe [1984], "Algorithms for form error evaluation - methods of the minimum zone and least squares", Proc. Int. Sym. Metrology Quality Control Production, Tokyo, Japan, pp 197-202. Gota, M. and K. Lizuka [1977], "An analysis of the relationship between minimum zone deviation in circularity and cylindericity", Proc. Int. Conf. Prod. Eng., New Delhi, India, pp x61-x70. Greenwood, J. A. and J. B. P. Williamson [1966], "Contact of Nominally Flat Surfaces", Proc. Roy. Soc., London, A295, pp 300-319. Harter, H. L. [1969], Order Statistics and their use in Testing and Estimation, Aerospace Research Laboratories, Office of Aerospace Research, vol. 2, U.S.A. International Standard Organization (ISO 1101-1983(E)) Technical drawings: geometrical tolerancing - tolerancing of form, orientation, location and runout - generalities, definitions, symbols, indications on drawing.

International Standard Organization (ISO 1101/II-1974(E)) Technical Drawings - Tolerances of form and of position - Part II: Maximum material principle. Kakino, Y. and J. Kitazawa [1978], "In situ measurement of cylindricity", Ann. C.I.R.P., Vol. 27, No. 1, pp 371-375. Lowell W. F. [1982], Modern Geometric Dimensioning and Tolerancing, National Tooling and Machining Association. Murthy, T. S. R. [1982], "A comparison of different algorithms for cylindricity evaluation", Int. J. Mach. Tool Des. Res., Vol. 22, No. 4, pp 283-292. Murthy, T. S. R., and S. Z. Abdin [1980], "Minimum zone evaluation of surfaces", Int. J. Mach. Tool Des. Res., Vol. 20, No. 2, pp 123-136. Murthy, T. S. R., and S. Z. Abdin [1980], "Minimum zone evaluation of surfaces", Int. J. Mach. Tool Des. Res., Vol. 20, No. 2, pp 123-136. Neter, J., W. Wasserman and M. H. Kutner [1985], Applied Linear Statistical Models, 2nd. Ed., Richard D. Irwin, Inc. Placek, C. [1989], "Mathematizing Y14.5", Quality, December, pp. Q10-Q13. Sarhan, A. E. and B. G. Greenberg, eds. [1962], Contributions to Order Statistics, Wiley, New York. Shunmugam, M. S. [1986], "On assessment of geometric errors", Int. J. Prod. Res., Vol. 24, No. 2, pp 413-425. Shunmugam, M. S. [1987], "Comparison between linear and normal deviations of forms of engineering surfaces", Precision Engineering, Vol. 9, No. 2, pp 96-102. Shunmugam, M. S. [1987], "New approach for evaluating form errors of engineering surfaces", Computer Aided Design, Vol. 19, No. 7, pp 368-374. Shunmugam, R. C. and D. J. Whitehouse [1971], "A unified approach to surface metrology", Proc. Inst. Mech. Eng., London, UK, Vol. 185, pp 697-707. Shunmugam, R. C. and D. J. Whitehouse [1971], "A unified approach to surface metrology", Proc. Inst. Mech. Eng., London, UK, Vol. 185, pp 697-707. Society of Manufacturing Engineers [1988], Dimensional Metrology and Geometric Conformance. 63

Thomas, T. R. [1982], Rough Surfaces, Longman. Weckenmann, A. and M. Heinrichowski [1985], "Problems with software for running coordinate measuring machines", Precision Engineering, Vol. 7, No. 2, pp. 87-91. Weill, R. [1988], "Tolerancing for Function", Ann. C.I.R.P., Vol. 37, No. 2.

B. Table for Numerical Values of E[Xn.pl]/n-p L n-p __ E[n.p]/n-p_ I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.7979 0.8862 0.9213 0.9400 0.9513 0.9594 0.9650 0.9693 0.9727 0.9754 0.9776 0.9794 0.9810 0.9823 0.9835 0.9845 0.9854 0.9862 0.9869 0.9876 0.9882 0.9887 0.9892 0.9896 0.9901 0.9904 0.9908 0.9911 0.9914 0.9917 - 61

References American National Standard Institutes (ANSI Y14.5M-1982) Dimensioning and Tolerancing. American National Standard Institutes (ANSI/ASME B46.1-1985) Surface Texture (Surface RoughnessnWaviness, and Lay) American National Standard Institutes (ANSI/ASME B89.1.12M-1985) Methods for Performance Evaluation of Coordinate Measuring Machines. Bourdet, P., A. Clement and R. Weill [1984], "Methodology and Comparative Study of Optimal Identification Processes for Geometrically Defined Surfaces", Proceedings of the International Symposium on Metrology for Quality Control in Production, Tokyo. Box, G. E. P. and N. R. Draper [1987], Emprical Model-Building and Response Surfaces, John Wiley & Sons, Inc. Draper, N. and H. Smith [1981], Applied Regression Analysis, 2nd. Ed., John Wiley & Sons, Inc. ElMaraghy, W. H., Z. Wu and H. A. ElMaraghy [1989], "Evaluation of Actual Geometric Tolerances Using Coordinate Measuring Machine Data", ASME Design Technical Conferences, vol. 19-1. Fukuda, M and A. Shimokohbe [1984], "Algorithms for form error evaluation - methods of the minimum zone and least squares", Proc. Int. Sym. Metrology Quality Control Production, Tokyo, Japan, pp 197-202. Gota, M. and K. Lizuka [1977], "An analysis of the relationship between minimum zone deviation in circularity and cylindericity", Proc. Int. Conf. Prod. Eng., New Delhi, India, pp x61-x70. Greenwood, J. A. and J. B. P. Williamson [1966], "Contact of Nominally Flat Surfaces", Proc. Roy. Soc., London, A295, pp 300-319. Harter, H. L. [1969], Order Statistics and their use in Testing and Estimation, Aerospace Research Laboratories, Office of Aerospace Research, vol. 2, U.S.A. International Standard Organization (ISO 1101-1983(E)) Technical drawings: geometrical tolerancing - tolerancing of form, orientation, location and runout - generalities, definitions, symbols, indications on drawing.

UNIVERSITY OF MICHIGAN 3 9015 04733 7921 Thomas, T. R. [1982], Rough Surfaces, Longman. Weckenmann, A. and M. Heinrichowski [1985], "Problems with software for running coordinate measuring machines", Precision Engineering, Vol. 7, No. 2, pp. 87-91. Weill, R. [1988], "Tolerancing for Function", Ann. C.I.R.P., Vol. 37, No. 2. 64