Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 1: geometric error decomposition

International Journal of Machine Tools & Manufacture 39 (1999) 1199–1217

Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 1: geometric error decomposition Zhongcheng Yan, Been-Der Yang, Chia-Hsiang Menq* Coordinate Metrology and Measurement Laboratory, Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA Received 7 January 1998; received in revised form 27 November 1998

Abstract In this paper our study focuses on the uncertainty analysis and variation reduction of coordinate system estimation using discrete measurement data and is associated with the applications that deal with parts produced by end-milling processes and having complex geometry. This paper consists of three parts. Since the uncertainty of the estimated coordinate transformation arises from the geometric errors on a part surface, Part 1 is devoted to the study of surface geometric errors. In this study, according to the characteristics of end-milling processes the sampled geometric error is divided into two components, and a decomposition procedure is developed for geometric error decomposition. The results of surface geometric error decomposition will be used in Part 2 for uncertainty analysis and in Part 3 for variation reduction.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Coordinate metrology; Geometric error decomposition; Uncertainty analysis

1. Introduction The methods divergence problem in coordinate metrology is a well-known phenomenon [1,2] when dealing with discrete measurement data. The problem can be divided into two categories: (a) different data analysis algorithms give different inspection results when using the same set of measurement data; (b) different sampling schemes produce different inspection results for the * Corresponding author. Tel.: ⫹ 1-614-292-4232; fax: ⫹ 1-614-292-3163. 0890-6955/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 8 9 - 3

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same part, even when the same data analysis algorithm is used. The first category of the methods divergence problem arises from the lack of a mathematical definition for geometric dimensioning and tolerancing in the current ANSI Y14.5 standard [3], which was originally designed for hard gage based inspection. The lack of a mathematical definition of the current standard results in different interpretations of the measurement data that are used to evaluate the dimensional quality of a part. Currently, the mathematization of the Y14.5 standard is being studied by a Y14.5 subcommittee in an effort to standardize the interpretation of measurement data. The second category of the methods divergence problem is due to the fact that any sampling scheme in coordinate metrology measures only a finite number of discrete points from the surface of a part. However, the surface of a real part consists of an infinite number of points, and at any point geometric deviation exists. Since the geometric error varies from one point to another, the inspection results will depend on the number and location of the discrete points being used. In other words, as long as geometric errors exist on the part surface and discrete data points are used for dimensional inspection, variation of the inspection result is inevitable. Nevertheless, for many applications that require high precision coordinate metrology it is important to be able to control this variation. In order to control the variation in dimensional inspection, it is necessary to characterize the coordinate variations resulted from discrete sampling. Since coordinate variation associated with changing sampling scheme in dimensional inspection is closely related to the uncertainty of coordinate system estimation [4–7], in this paper, we investigate the uncertainty of coordinate system estimation using discrete measurement data. In this study, it is assumed that the coordinate transformation between the part design coordinate system and the measuring machine coordinate system is estimated by fitting the measurement data, sampled from the part surface, to the part nominal model using the nonlinear least squares best-fit method [8–10]. Traditionally, the coordinate transformation is determined by using the 32-1 approach, in which standard features, such as planar surfaces, spherical surfaces, or cylindrical surfaces, are calculated to establish a datum reference frame. Using the nonlinear least squares best-fit method, the estimation of coordinate transformation becomes an optimization process. Using the best-fit method, all the measurement points contribute to the result of coordinate estimation, thus a more robust coordinate estimation result can be obtained. Another advantage of the best-fit method is that it can be used for the evaluation of free-form surfaces. For cases where datum surfaces are used, the best-fit method can be used to establish each datum surface, and the part coordinate system can be synthesized from the datum surfaces accordingly. Therefore, in this paper our study focuses on the uncertainty analysis and variation reduction of coordinate system estimation and is associated with the applications that deal with parts produced by end-milling processes and having complex geometry. Applications that deal with geometric parameter estimation for standard features, such as cylindrical or conical surfaces, can be found in [11,12]. This paper consists of three parts. Since the uncertainty of the estimated coordinate transformation arises from the geometric errors on a part surface, Part 1 is devoted to the study of surface geometric errors. In this study, according to the characteristics of end-milling processes the sampled geometric error is divided into two components, and a decomposition procedure is developed for geometric error decomposition. The results of surface geometric error decomposition will be used in Part 2 for uncertainty analysis and in Part 3 for variation reduction.

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2. Estimation of coordinate transformation The nominal shape of each surface patch of a designed part can be represented by its shape function Ni(s˜), where s˜ is called the feature variables [13]. After the part is produced, the actual geometric form of the surface patch can be expressed as follows: Mi(s˜) ⫽ Ni(s˜) ⫹ ⑀i(s˜)

(1)

where ⑀i(s˜) represents the geometric deviations of the manufactured surface patch. In coordinate metrology, the coordinates of the measurement points sampled from the real part surface are given in the machine coordinate system of the measuring device. The sampled coordinates of the jth measurement point on the ith surface patch can be expressed as follows: Xij ⫽ T(t˜)Mi(s˜ij ) ⫹ ⑀m

(2)

where ⑀m is the measurement error; T(t˜) are the transformation matrix between the part coordinate system and the machine coordinate system of the coordinate measuring device, and ˜t are the transformation parameters. If the measurement error is small as compared to the geometric errors, the sampled geometric errors at the measurement points can be computed by the following formula:

⑀ij (s˜ij ,t˜) ⫽ T

⫺1

(t˜)Xij ⫺ Ni(s˜ij )

(3)

The calculated geometric deviations ⑀ij, are then used for the purpose of tolerance evaluation. However, before Eq. (3) can be used to calculate the sampled geometric errors, the transformation matrix T(t˜) must be determined. Using the nonlinear least squares best-fit method, the estimation of coordinate transformation becomes an optimization process. The objective function to be minimized is formulated as follows [8]:

冘冘 n

F⫽

mi

i⫽1 j⫽1

冘冘 n

⑀ (s˜ij ,t˜) ⫽ 2 ij

mi

兩T

⫺1

(t˜)Xij ⫺ Ni(s˜ij )兩2

(4)

i⫽1 j⫽1

where n is the number of surface patches of the part surface; mi is the number of measurement points sampled from the ith surface patch; Ni(s˜ij ) is the nearest point on the nominal surface corresponding to the transformed measurement point T ⫺ 1(t˜)Xij . Using the best-fit method, all the measurement points contribute to the result of coordinate estimation, thus more robust coordinate estimation result can be obtained. Another advantage of the best-fit method is that it can be used for the evaluation of free-form surfaces. For cases where datum surfaces are used, the best-fit method can be used to establish each datum surface, and the part coordinate system can be synthesized from the datum surfaces accordingly. However, even if the best-fit method is used to estimate the coordinate transformation, variation of the resulting coordinate transformation is still inevitable when changing the number and/or location of the discrete measurement points. Consequently, variation will be associated with the computed geometric deviations at the measurement points when different measurement points are used in best-fit. As an example, Fig. 1 shows the variation of the computed geometric errors at the measurement points sampled from a turbine blade. The turbine blade consists of two B-spline surfaces. The nominal design model and the 2671 actual measurement data scanned from seven sections of the turbine blade are shown in Fig. 1(a). From the scanned data, three sets of measure-

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Fig. 1. Geometric deviations of a turbine blade. (a) Turbine blade model and scanned data before best-fit. (b) Geometric deviations after best-fitting 35 points. (c) Geometric deviations after best-fitting 136 points. (d) Geometric deviations after best-fitting 2035 points.

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ment points having different number and locations are selected to best-fit with the nominal model to estimate the coordinate transformation. After best-fitting each set of the selected points with the nominal model, the sampled geometric deviations at the measurement points are computed and graphically shown in Fig. 1(b–d) after being magnified by 10 times for visualization purposes. Among Fig. 1(b–d), the variation of the calculated geometric deviations can be clearly observed. Since the uncertainty of coordinate estimation arises from the geometric errors on the manufactured part surface, the geometric errors on the part surface play an important role in the uncertainty analysis of coordinate estimation. In light of this, the characteristics of surface geometric errors is investigated in this paper. 3. Characterization of sampled geometric errors The geometric errors on a part surface are attributed to many factors. Different type of error sources in the manufacturing system leave different signatures on the part surface, and the geometric errors on the part surface are the result of the joint effect of all the error sources in the manufacturing system. For example, the geometric errors on a surface produced by end-milling processes can be categorized into at least two distinct components: waviness1 and roughness. The waviness component of the surface geometric errors is the widely spaced surface irregularity which forms a smooth error trend imposed on the nominal surface. The roughness component of the surface geometric errors is the closely spaced surface irregularity which has high frequency variation. The actual part surface can be considered as a surface resulted from the superposition of the waviness trend and the roughness component on the nominal surface. In coordinate metrology, only a finite number of measurement points are sampled from the part surface. The purpose is to capture the smooth error trend imposed on the nominal surface. However, in the coordinate sampling process, the roughness component of the geometric error aliases with the waviness component. Consequently, the sampled geometric errors at the individual measurement points consist of two distinct components. Among the sampled geometric errors, the component corresponding to the waviness of the surface geometric errors represents the smooth error trend and is spatially correlated. It is called the deterministic component of the sampled geometric errors. On the other hand, the discontinuous roughness component of the sampled geometric errors is weakly correlated and can be considered to be spatially random. Due to the different characteristics between the deterministic error component and the random error component of the sampled geometric errors, different effects of the two error components on the result of coordinate estimation are expected. Therefore, for the purpose of uncertainty analysis, it is necessary to separate these two distinct error components from the measurement data and treat them separately in the uncertainty analysis of coordinate estimation. 4. Decomposition of geometric errors Since the number of measurement points used in coordinate metrology is often not large enough to recover the closely spaced surface irregularity, frequency domain approaches such as Fast 1

In this paper, the form errors of the part surface are included in the waviness component.

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Fourier Transform (FFT) may not be applicable for the purpose of decomposition. Therefore, a unique approach needs be developed for geometric error decomposition. In this paper, the proposed approach is to capture the smooth error trend of the machined surface. 4.1. Construction of a deterministic surface Since the waviness trend of the surface geometric errors is a smooth error trend, the superposition of the waviness trend on the nominal surface will form another smooth surface. This smooth surface is referred to as the deterministic surface. The actual part surface can be considered as the surface resulted from imposing the roughness component on the deterministic surface. After coordinate sampling, the actual part surface is digitized into discrete measurement data which consists of the deterministic surface and the discretized roughness portion of the surface profile. If the number of points measured from the part surface is large enough, the measurement data should contain sufficient information that allows the deterministic surface be re-constructed. On the other hand, the high frequency roughness error component becomes additional error component at the measurement points, and these additional errors are of spatially statistical independence. From the statistical characteristic of the additional error component at the measurement points, a statistical model for the sampled surface geometry can be proposed as follows [14]: X ⫽ S(u,v) ⫹ ⑀r

(5)

where S(u,v) is a surface regression model, and ⑀r is a random variable of spatially statistical independence with its mean E{⑀r} ⫽ 0 and its variance Var{⑀r} ⫽ ␴2⑀r. The surface regression model is used to describe the form of the deterministic surface. The random variable ⑀r is employed to represent the additional error component of the sampled geometric errors. The separation of the additional error component can be achieved by finding an adequate surface regression model such that the normal deviations from the measurement points to the established deterministic surface satisfy the condition of spatial randomness. However, finding an adequate surface regression model is a very complicated problem which is closely related to the complexity of the deterministic surface form. The more complicated the deterministic surface is, the more measurement points have to be sampled from the part surface to represent the complexity of the deterministic surface, and the more patches will be needed. Therefore, in this paper the proposed approach consists of two iterative loops. The purpose of the inner loop is to determine an adequate surface regression model for a set of discrete measurement data by increasing the number of surface patches. The adequacy of the selected surface regression model is examined by judging the spatial randomness of the residual normal deviations. The purpose of the outer loop is to determine an adequate surface regression model for the existing deterministic surface by increasing the number of measurement points. 4.1.1. Deterministic surface model The deterministic surface is a smooth surface and can be modeled by a bicubic B-spline surface [15–17] as follows:

冘冘

h⫹3 l⫹3

S(u,v) ⫽

i⫽1 j⫽1

Ui(u)Vj (v)Cij

(6)

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where Ui(u) and Vj(v) are the cubic B-spline basis functions, and Cij are the control points. The patch number of the surface is h ⫻ l. The surface model is actually a linear combination of the control points with the tensor products of the basis functions as the coefficients. Therefore, by rearrangement, Eq. (6) can be rewritten into the following form:

冘 N

S(u,v) ⫽

gkCk

(7)

k⫽1

where gk are the tensor products of the B-spline basis functions, Ck ⫽ [xck,yck,zck]T are the corresponding control points, and N ⫽ (h ⫹ 3) ⫻ (l ⫹ 3) is the number of control points. 4.1.2. Regression surface model Suppose n points have been measured from the actual part surface, and the coordinates of the measurement points are Pi ⫽ [xi,yi,zi]T (i ⫽ 1,2,…,n). If the knot sequences of the B-splines in the u and v directions are specified and the number of patches of the deterministic surface model and the uv parameters of each measurement point are determined, the tensor product terms in Eq. (7) can be computed. Therefore, a linear regression equation for the control points of the deterministic surface can be derived as follows:  PT1 

 g11 g12 $ g1N   CT1 

P  g 

T 2

⯗

⫽ 

21

⯗

P  g 

T n





n1

g22 $ g2N

C 

⯗ 哻 ⯗

⯗





T 2

 or P ⫽ GC



(8)

gn2 $ gnN   CTN 

Since both Pi and Ci in Eq. (8) have x,y,z components, Eq. (8) can be further rewritten into the following three sub-equations:

冦

Xp ⫽ GXc Yp ⫽ GYc

(9)

Zp ⫽ GZc

where Xp, Yp, and Zp are the three vectors consisting of the x, y and z coordinates of the measurement data, respectively; Xc, Yc, and Zc are the control points. From Eq. (9), the control points of the deterministic surface can be obtained by regression as follows [18]:

冦

Xc ⫽ (GTG)−1GTXp Yc ⫽ (GTG)−1GTYp

(10)

Zc ⫽ (GTG)−1GTZp

The entities of the G matrix in Eq. (8) are the tensor products corresponding to each measure-

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ment point. To compute the G matrix, the number of patches of the deterministic surface model, the knot sequences of the B-splines in the u and v directions, and the uv parameters of each measurement point need be specified first. The determination of the knot sequences for the Bsplines is not very critical in this study. Uniform knot sequences are used in this paper. In assigning the uv parameters for the measurement points, it can be assumed that the uv parameters of the measured points on the part surface are close to the uv parameters of its corresponding target points used for coordinate sampling. Although the deterministic surface is usually of higher complexity than the nominal design surface, the above assumption is quite accurate since the magnitude of the geometric deviations on the part surface is usually quite small as compared to the dimension of the part surface.

4.1.3. Under-fit and over-fit prevention For a B-spline surface representation, the complexity of the surface regression model is related to its patch number. If the surface regression model is simpler than the complexity of the deterministic surface, then under-fit problem will occur. On the other hand, if the surface regression model is of higher complexity than the form of the deterministic surface, then over-fit problem will arise. The under-fit problem can be avoided by checking the spatially statistical independence of the residual normal deviations. From Eq. (5), it is known that the residual normal deviations should have spatially statistical independence. A strong autocorrelation among the residual normal deviations indicates the existence of under-fit and implies that more patches are required to describe the deterministic surface. On the other hand, the over-fit problem can be prevented by adopting an iterative procedure in which the number of patches of the regression model is gradually increased in each iteration until the randomness test for residual normal deviations is passed. Since the deterministic surface often has higher complexity than the nominal model of the surface, the patch number of the surface nominal model can be used as the starting point in the determination of the patch number for the regression model of the deterministic surface. Figs. 2–4 show the example of fitting a set of 300 points measured from a ball-end milled sculptured surface. Fig. 2 is the nominal design model of the surface which has 2 ⫻ 2 patches. Two regression models with different patch numbers are used to fit the 300 measurement data, and the results are shown in Fig. 3 and Fig. 4. Fig. 3 is the result using a regression model with 2 ⫻ 2 patches, and Fig. 4 is the result using a regression model with 4 ⫻ 5 patches. Fig. 3a and Fig. 4a show the fitted surfaces and the residual normal deviations. A magnification factor of 50 is used in both figures to make the residual normal deviations graphically visible. Fig. 3b and Fig. 4b show the signs of the residual normal deviations in the uv space. The plus signs represent the normal deviations on one side of the fitted surface, and the diamond signs indicate the normal deviations on the other side of the fitted surface. The aggregations of the same signs in some regions, as shown in Fig. 3b, indicate the strong autocorrelation among the normal deviations in those regions. By increasing the patch number, the regression model becomes more suitable for the description of the deterministic surface. The scattered distribution pattern of the normal deviation signs, as shown in Fig. 4b, indicate that the spatially statistical dependence among the residual normal deviations has been significantly reduced.

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Fig. 2. Nominal model of the ball-end milled surface.

4.1.4. Spatial independence In addition to graphical examination, there are a number of statistics that can be used to test the spatial autocorrelation among some spatially distributed data. Among the statistics for detecting the presence of spatial autocorrelation with interval scaled data, Moran’s I is preferred by researchers [19,20]. Let ⑀i be the normal deviation from the ith measurement point to the fitted deterministic surface, and ⑀¯ the mean of {⑀i} over the n measurement points. Moran’s I is defined as follows:

冘冘 n

n Moran⬘s I ⫽ S0

n

wij (⑀i ⫺ ⑀¯ )(⑀j ⫺ ⑀¯ )

i⫽1 j⫽1

冘⑀

(11)

n

( i ⫺ ⑀¯ )

2

i⫽1

冘冘 n

where S0 ⫽

n

wij , and wij is the weight which represents a measure of spatial interrelation

i⫽1 j⫽1

between ⑀i and ⑀j. The selection of weights is ‘subjective’; nevertheless, there is a tendency to follow, i.e., the interaction between two points decreases as the distance between the two points increases. In this study, the following scheme for the weight calculation is used: wij ⫽

d ij⫺ k , i⫽j Li

wij ⫽ 0, i ⫽ j

(12)

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Fig. 3. Fitting measurement data using 2 ⫻ 2 patches. (a) Fitted deterministic surface and residual normal deviations. (b) Sign distribution of residual normal deviations.

where dij is the distance between the ith point and the jth point, k is a constant (k ⱖ 1) and

冘 n

Li ⫽

d ij⫺ k. The determination of k value is not very crucial. In this study, k ⫽ 4 is used.

j ⫽ 1,j⫽i

According to Cliff and Ord, the distribution of Moran’s I is asymptotically normal. In practical

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Fig. 4. Fitting measurement data using 4 ⫻ 5 patches. (a) Fitted deterministic surface and residual normal deviations. (b) Sign distribution of residual normal deviations.

applications, a normal approximation may be assumed for the distribution of Moran’s I for moderate sample sizes. Therefore, the hypothesis test for the spatial independence among the residual normal deviations {⑀i} can be formulated as follows: Step 1. The hypothesis is stated. H0: Normal residual deviations are of spatially statistical independence. Ha: Normal residual deviations are of spatially statistical dependence.

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Step 2. A level of significance is chosen. The level of significance is the probability of rejecting the null hypothesis when it is true. The most common level of significance used in applications is 0.01. In this study, a significance level of 0.01 is adopted. Step 3. A test statistics is chosen. The test statistics is expressed as Z⫽

(Moran⬘s I) ⫺ ␮M , ␴M

where

␮M ⫽ ⫺ ␴2M ⫽

1 , n⫺1

n{(n2 ⫺ 3n ⫹ 3)S1 ⫺ nS2 ⫹ 3S 20} ⫺ r{n(n ⫺ 1)S1 ⫺ 2nS2 ⫹ 6S 20} 1 ⫺ , 2 (n ⫺ 1)(n ⫺ 2)(n ⫺ 3)S 0 (n ⫺ 1)2

冘冘 冘

1 S1 ⫽ 2

n

n

(wij ⫹ wji) , S2 ⫽

i⫽1 j⫽1

n

w.i ⫽

wji, r ⫽

j⫽1

冘 n

2

i⫽1

冘 n

(wi. ⫹ w.i), wi. ⫽

wij ,

j⫽1

m4 , m22

and mr ⫽

1 n

冘⑀ n

( i ⫺ ⑀¯ )r.

i⫽1

It has been proven that the selected test statistics Z follows the standard normal distribution. Step 4. A decision rule is set up. For the selected significance level of 0.01, the critical value for the test statistics Z is Z0.01 ⫽ 2.33. If Z ⬍ Z0.01, the null hypothesis is accepted. Otherwise, the alternative hypothesis is accepted. The acceptance of the null hypothesis indicates that the residual normal deviations from the measurement points to the fitted surface is of spatial independence. The rejection of the null hypothesis suggests that the number of patches of the regression model for the deterministic surface should be increased. The values of the Z statistics for Figs. 3 and 4 are 11.895 and 0.041, respectively. Based on the above hypothesis test, it can be seen that the number of patches of the regression model used in Fig. 3 should be increased, and the adequacy of the regression model used in Fig. 4 is justified. 4.2. Number of measurement points Since the extraction of the deterministic surface is based on the discrete points sampled from the actual part surface, the sampled discrete points must contain sufficient information that allows the deterministic surface to be re-constructed. Therefore, determining the number of measurement points that are required to represent the form of the deterministic surface becomes an important issue in geometric error decomposition.

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Since the error formation process is often very complicated, the geometric errors on a real part surface are usually very complex. One experiment was conducted to illustrate the possible geometric errors produced by end-milling processes. In the experiment, two identical sculptured surfaces with their nominal model shown in Fig. 2 were machined using ball-end milling process. All the cutting conditions were identical except the cutting paths: one using the one-way cutting path, the other using the zigzag cutting path. After machining, 400 measurement data were collected from each of the machined surfaces using a Coordinate Measuring Machine (CMM). The measurement data were best-fitted with the nominal model of the surface, and the geometric deviations for these two machined surfaces are shown in Fig. 5 in the uv parameter space. It can be seen that the resulting geometric errors vary over the surface and are much more complex than the original design geometry of the surface. In addition, the variation of the geometric errors resulting from the one-way cutting pattern is quite different from that of the zigzag cutting pattern. In practice, it is difficult to have much information about the formation of the geometric errors before any measurement data are sampled from the surface of the manufactured part. To ensure that the sampled points properly represent the characteristics of the deterministic surface, the number of measurement points has to be determined in an iterative fashion. In each iteration, the density of measurement points is increased and more measurement data are sampled from the

Fig. 5. Resulting surface geometric errors in ball-end milling process. (a) One-way cutting. (b) Zigzag cutting.

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part surface. Based on the previous discussion in Section 4.1, a proper surface model can be constructed based on the collected measurement data. The iteration is carried out until a selected criterion is satisfied. Generally speaking, if the density of sampling points is not high enough to catch the form of the deterministic surface, when the sampling density is increased, the number of surface patches needed to fit the deterministic surface will also have to be increased in order to pass the randomness test for the residual normal errors. When the sampling density is high enough to represent the form of the deterministic surface, the patch number will tend to saturate if the sampling density is further increased. Fig. 6 shows the experimental result about the required number of patches versus number of measurement points. The ball-end milled surface of Fig. 2 is used in the experiment, and the measurement data were sampled from the surface with a random distribution pattern. From the experiment result, it can be seen that the patch number becomes saturated when the number of measurement points is increased to 300 points. Therefore, it can be considered that at least 300 points must be sampled from this ball-end milled surface to ensure the accuracy of the fitted deterministic surface. In practice, the existence of the deterministic error component in the sampled geometric errors can be often observed. However, it is still possible in precision machining that the deterministic error component is negligibly small compared to the random error component on the part surface. If the patch number of the deterministic surface saturates to the patch number of the nominal surface model, then it is very likely that the random error component is dominant. Otherwise, the surface should be considered to have both the deterministic and random error component. 4.3. Deterministic error component After constructing the deterministic surface, a best-fit procedure is needed to determine the deterministic error component of the machined surface. However, as discussed in Section 2, the current approach can only be used to best-fit discrete measurement points to the nominal surface.

Fig. 6. Patch number versus number of measurement points.

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It can not be directly used to best-fit the continuous deterministic surface to the nominal surface. In order to best-fit the deterministic surface with the nominal surface, approximation has to be made. The constructed continuous deterministic surface can be approximated by a large number of discrete points, and the best-fit method can then be used to minimize the sum of the normal distances from the selected discrete points to the nominal surface model. After best-fitting the deterministic surface with the nominal surface, the deterministic error component can be computed. 4.4. Summary In summary, the geometric error decomposition can be carried out following the iterative procedure outlined in Fig. 7. It can be seen that the first part of the iterative procedure has two iterative loops. The inner loop is to increase the number of surface patches, while the outer loop is to increase the number of sampling points. The second part of the procedure is to extract the

Fig. 7.

Procedure for geometric error decomposition.

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deterministic error components from the constructed deterministic surface. Fig. 8 gives an example of the error decomposition results for the ball-end milled surface. Again, 300 points are measured from the ball-end milled surface. The deterministic surface extracted from these 300 measurement points is shown in Fig. 3a, and the random error component is depicted in Fig. 8(a). Fig. 8(b) shows the deterministic error component in the uv parameter space. It can be seen that the deterministic error component of the ball-end milled surface is much more complicated than the nominal surface model.

Fig. 8. Geometric error decomposition result for the ball-end milled surface. (a) Random error component. (b) Deterministic error component.

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5. Classification of uncertainty analysis Through the process of geometric error decomposition, the information about the random error component, the deterministic error component, and the position and orientation of the surface is obtained. The uncertainty of coordinate estimation can then be studied based on the obtained information. Since the mechanisms through which the two different error components affect the variations of coordinate system estimation are different, the two error components need to be considered separately in the process of uncertainty analysis. In order to facilitate the subsequent discussion, we classify the surface geometric error into two types: Type I Geometric Error: when the random error component is the dominant error component on a part surface and the deterministic error component is negligible as compared with the random error component, the geometric error on the part surface is classified as Type I Geometric Error. The problem involving datum surfaces which are carefully finished often falls into this category. The uncertainty problem which involves the Type I Geometric Error is called the Type I Uncertainty Problem. Type II Geometric Error: when the deterministic error component of the geometric error on a part surface is no longer negligible as compared with the random error component, the geometric error on the part surface is classified as Type II Geometric Error. The uncertainty problem which involves the Type II Geometric Error is called the Type II Uncertainty Problem.

6. Conclusions In this paper, the surface geometric errors of parts produced by end-milling processes are characterized, and an iterative procedure for geometric error decomposition is developed. The resulting surface geometric errors of a part produced by end-milling processes can be categorized into two distinct components: waviness and roughness. After coordinate sampling, the sampled geometric errors of the part surface also consist of two different components: a deterministic error component and a random error component. It is necessary to separate these two different error components for the purpose of dimensional inspection and uncertainty analysis. The proposed geometric error decomposition procedure consists of two steps. In the first step, the random error component is separated from the deterministic component by constructing a deterministic surface model using an iterative approach. The approach consists of two iterative loops. The inner loop is to increase the number of surface patches, while the outer loop is to increase the number of sampling points. The proposed iterative approach is used to find an adequate surface regression model such that the normal deviations from the measurement points to the established deterministic surface satisfy the condition of spatial randomness. The second step of the procedure is to extract the deterministic error components from the constructed deterministic surface. By bestfitting the constructed deterministic surface with the surface nominal model, the coordinate transformation between the machine coordinate system in which the measurement data are sampled and the design coordinate system of the part can be obtained. Based on the estimated coordinate transformation, the deterministic error component which is the normal deviation from the deter-

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ministic surface to the nominal surface model can be calculated. The results of surface geometric error decomposition will be used in Part 2 for uncertainty analysis and in Part 3 for variation reduction.

Acknowledgements This material is based on work supported by the National Science Foundation under grant no. DDM-9215600 and DMI-9500025. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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