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Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 3: variation reduction
International Journal of Machine Tools & Manufacture 39 (1999) 1239–1261
Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 3: variation reduction Zhongcheng Yan, 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, the study focuses on the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component. Two different approaches are proposed for the estimation of coordinate transformation. In the onestep approach, the best-fit method is applied to directly fit the measurement data to the nominal surface. In the two-step approach, a deterministic surface is first constructed from the measurement data and the fitted deterministic surface is then best-fitted with the nominal surface to estimate the coordinate transformation. The computation needed in the two-step coordinate estimation approach is more expensive than that required by the one-step approach. However, by estimating the deterministic error component of the surface geometric error, the two-step approach can effectively reduce the influence of the deterministic error component on the result of coordinate estimation. Therefore, with the same measurement data, the two-step approach gives a much more accurate coordinate estimation result than the one-step approach. In addition, the variation range of the obtained coordinate transformation parameters is much reduced. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Coordinate metrology; Uncertainty analysis; Variation reduction; Deterministic error
1. Introduction The variation of the estimated coordinate transformation arises from the geometric errors on the part surface. In Part 2 of the paper, the uncertainty analysis for the case in which the sampled * 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 9 1 - 1
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geometric errors are dominated by the random component is investigated. In practice, the uncertainty analysis for coordinate estimation using high precision datum surfaces often falls into this type. In this part of the paper, the study focuses on the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component [1]. Due to the distinct characteristics of the two different error components, their influences on the result of coordinate estimation are also different. In current practice of coordinate estimation, however, the two different error components are not distinguished in the process of coordinate estimation. In other words, in both the Type I and the Type II problem, the coordinate transformation is estimated by directly best-fitting the measurement data with the nominal surface model. We call this approach the One-step Coordinate Estimation Approach. In addition to this one-step approach, we propose a different approach which is called the Two-step Coordinate Estimation Approach. In the proposed two-step coordinate estimation approach, the estimation of coordinate transformation is carried out in two steps. First, a deterministic surface is constructed from the measurement data using the surface fitting techniques discussed in Part 1 of the paper. Then, the deterministic surface is best-fitted with the surface nominal model to estimate the coordinate transformation. The best-fit of the deterministic surface with the nominal model is performed by using the non-linear least squares method to fit a large number of discrete points selected from the deterministic surface with the nominal surface. It is evident that the coordinate estimation results from the two different approaches will be different. In this paper, uncertainty analysis for the two different approaches will be investigated and the merits and drawbacks of the two approaches will be compared.
2. One-step approach In the one-step approach, the best-fit algorithm is applied directly to fit the measurement data with the nominal surface model [2–4]. Since both the random and the deterministic error components exist, the variation of the estimated coordinate transformation is the result of the joint effect of the two different geometric error components. The influences of both the random and the deterministic error components on the coordinate transformation estimated by using the one-step approach is discussed in this section. 2.1. Uncertainty of coordinate estimation To study the influences of the random and the deterministic error components on the result of coordinate estimation by the one-step approach, the relationship between the uncertainty parameters and the sampled geometric errors must be established. The sensitivity matrix of coordinate estimation derived in Part 2 can again be employed here for this purpose. A␦˜t ⫽ ⑀˜
(1)
where A is the sensitivity matrix of coordinate estimation, and ⑀˜ is the geometric deviations at the measurement points. As shown in Fig. 1, suppose that the corresponding normal projection
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Fig. 1. Geometric error components.
point for measurement point Pi⬘ is Pi. Then the sampled geometric error at point Pi can be expressed as follows:
⑀i ⫽ ⑀di ⫹ ⑀ri
(2)
where ⑀di and ⑀ri are the deterministic and the random error components at point Pi, respectively. For a total of N measurement points, the geometric error vector will be
⑀˜ ⫽ ⑀˜ d ⫹ ⑀˜ r
(3)
where ⑀˜ d ⫽ [⑀d1,⑀d2,$,⑀dN]T is the deterministic error vector, and ⑀˜ r ⫽ [⑀r1,⑀r2,$,⑀rN]T is the random error vector. Substituting Eq. (3) into Eq. (1) yields A␦˜t ⫽ ⑀˜ d ⫹ ⑀˜ r
(4)
From Eq. (4), the inverse mapping from the geometric error vector space to the uncertainty parameter vector space can be determined as follows:
冋冘 册 冋冘 册 6
␦˜t ⫽
k⫽1
6
vikujk (⑀˜ d ⫹ ⑀˜ r) ⫽ k
冋冘 册 冋冘 册 6
k⫽1
vikujk ⑀˜ ⫹ k d
6
k⫽1
vikujk ⑀˜ ⬅ ␦˜td ⫹ ␦˜tr k r
(5)
vikujk is the inverse mapping matrix, vik and ujk are the entities of the left and right k k⫽1 singular matrices of the sensitivity matrix A respectively, and k are the singular values of A. It can be seen from Eq. (5) that the coordinate deviation ␦˜t consists of two parts: ␦˜td and ␦˜tr; ␦˜td is caused by the deterministic error component, and ␦˜tr comes from the effect of the random error component. Since the real coordinate transformation is defined as the limit of the estimated coordinate transformation when the number of measurement points approaches infinity, both ␦˜td and ␦˜tr will converge to zero as the number of sampling points approaches to infinity [1]. However, when using a finite number of measurement points, since the magnitude of the deterministic error component differs from one point to another, ␦˜td will change when changing the number and location of measurement points. Since the deterministic error trend on the machined surface is not estimated in the one-step coordinate estimation approach, it is impossible to assess where
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the magnitude of ␦˜td. Without assessing the magnitude of ␦˜td, the accuracy of the coordinate estimation result will deteriorate. This is the drawback of the one-step coordinate estimation approach in estimating coordinate transformation for the Type II problem. The only way to improve coordinate estimation accuracy is to increase the sampling density so as to make ␦˜td as small as possible. From Eq. (5), the mean and the variance of the uncertainty parameter vector ␦˜t can be calculated. The mean of ␦˜t is given by the following formula:
冋冘 册 6
vikujk ␦˜t ⫽ ␦˜td ⫹ ␦˜tr ⫽ E(⑀˜ d) ⫹ k k⫽1
冋冘 册 6
vikujk E(⑀˜ r) k k⫽1
(6)
When the number and location of the measurement points are determined, the deterministic error vector ⑀˜ d is the same for every surface machined under the same process conditions. Therefore, we have E(⑀˜ d) ⫽ ⑀˜ d. Since the random error vector ⑀˜ r has zero mean, i.e. E(⑀˜ r) ⫽ 0, Eq. (6) can be simplified as follows:
冋冘 册 6
␦˜t ⫽ ␦˜td ⫽ ␦˜td ⫽
k⫽1
vikujk ⑀˜ k d
(7)
It is clear that ␦˜td is a bias in the estimated coordinate parameters. For a specified sampling scheme, ␦˜td represents a constant shift of ␦˜t from the origin of the uncertainty parameter space and ␦˜tr, which is the variation of ␦˜t, will be added to this constant shift ␦˜td. Since the deterministic error vector ⑀˜ d is a constant vector for a selected sampling plan, it can be proven that Cov(⑀˜ ) ⫽ Cov(⑀˜ d ⫹ ⑀˜ r) ⫽ Cov(⑀˜ r). Therefore, the covariance of ␦˜t can be obtained from Eq. (5) as follows:
冋冘 册 6
vikujk Cov(⑀˜ r) Cov(␦˜t) ⫽ k k⫽1
冋冘 册 6
k⫽1
vikujk k
T
⫽ Cov(␦˜tr)
(8)
From the theory of statistics [5], we know that ␦˜t will follow a normal distribution with ␦˜td as its mean. The joint probability density function of ␦˜t is given by Eq. (9), and the hyper-ellipsoid shaped probability envelope is illustrated in Fig. 2. f (␦˜t) ⫽
1 exp{ ⫺ 0.5·␦˜tTr[Cov(␦˜tr)]−1␦˜tr} 83兩Cov(␦˜tr)兩1/2
(9)
If the statistics of the random errors on the surfaces are available, then Cov(␦˜t) can be obtained from Eq. (8). Thus the normal distribution shape of ␦˜t is known. However, as mentioned above, the magnitude of the bias component ␦˜td in the estimated coordinate parameters can not be obtained in the one-step coordinate estimation approach. This means that it is impossible to know the center of the normal distribution which is shifted from the origin of the uncertainty parameter
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Fig. 2. Illustration of probability envelope of one-step approach.
vector space. As for the variation of the uncertainty parameters, it can be proven that the sum of the variances of the uncertainty parameters follows the following equation:
冘 6
i⫽1
冘 6
2␦ti ⫽
2 k
(10)
k⫽1
where 2k are the singular values of the covariance matrix Cov(␦˜t). 2.2. Computer simulations To verify the above discussion about the one-step coordinate estimation approach, computer simulations were conducted. In this study, the B-spline surface shown in Fig. 3 is used as the nominal surface. A deterministic error trend ⑀d(u,v) with 3⫻3 patches as shown in Fig. 4 is added to the nominal surface along the normal direction of the nominal surface. Random errors are then added to a 101⫻101 grid on the deterministic surface to simulate a surface having both deterministic error trend and random errors. A total of 50 surfaces having the same deterministic error trend but different random errors are created. Best-fit is applied to fit the 10 201 data points of each surface with the nominal surface model, and the obtained coordinate transformation parameters serve as the real coordinate transformation parameters ˜treal for each surface. A total of 20 sets of target points with the number ranging from 50 to 1000 are first determined in such a way that the (i⫹1)th set is obtained by adding 50 more randomly picked points to the ith set. In each set of the target points, any single point is only selected once and there is no repeated selection of the same point. Based on each set of the selected target points, measurement data are picked from the 10 201 data points on each of the 50 surfaces. The measurement data taken from each
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Fig. 3.
Fig. 4.
Nominal B-spline surface model for simulation study of coordinate uncertainty (1⫻1 patch).
Deterministic error trend for simulation study of the Type II uncertainty problem (3⫻3 patches).
surface are best-fitted with the nominal surface model to get the estimated coordinate transformation parameters ˜test. Consequently, ␦˜t which is the deviation of ˜test from ˜treal can be calculated for each surface. The simulation results are presented in Figs. 5–7. Fig. 5 shows the norm of ␦˜t against 1/Unorm, where Unorm is defined in Part 2 of the paper. For each set of the selected target points, Unorm is computed from the covariance matrix Cov(␦˜t) in Eq. (8). It can be seen from Fig. 5 that, for the Type II uncertainty problem, the norm of ␦˜t resulted from the one-step approach is no longer bounded by the maximum principle axis length of the 99% probability envelope. It is evident that this is due to the non-zero mean ␦˜t ⫽ ␦˜td of the uncertainty parameters that is caused by the deterministic error component of the surface geometric errors. The mean values of the uncertainty parameters for different sampling schemes are computed from the simulation results and plotted in Fig. 6. It can be seen from Fig. 6 that the mean value of each uncertainty parameter has no clear convergence trend associated with the increase of measurement points. The fluctuation of the mean value of each uncertainty parameter associated with the increase of measurement points is caused by the changing magnitude of the deterministic
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Fig. 5. 兩␦˜t兩 vs 1/Unorm (one-step approach, simulation results).
error component at different sampling points. Due to the influence of the deterministic error component, the increase of measurement points does not necessary reduce the mean value of each uncertainty parameter. On the other hand, the variances of the uncertainty parameters are mainly due to the influences of the random errors. More measurement points usually result in smaller variation of ␦˜t. This can be verified from Fig. 7. In Fig. 7, the predicted variances for the uncertainty parameters are computed from Eq. (8). The predicted variances and the variances computed from simulation results are in good agreement. The small discrepancies are possibly due to the weak correlation among the random errors used in the simulations and the fact that the best-fit is a nonlinear process and the sensitivity matrix used is a linear approximation. 2.3. Experimental verification In addition to the computer simulations, experiments were also carried out for the verification of the uncertainty analysis for the one-step coordinate estimation approach. The nominal model of the ball-end milled surface used in the experiments is shown in Fig. 2 of Part 2. It is a freeform surface with 2⫻2 patches. This ball-end milled surface has a deterministic error trend as shown in Fig. 8 of Part 1. The distribution of the random error on the surface is shown in Fig. 11 of Part 2, and the standard deviation of the random errors over the surface is ⫽ 0.0155. From the surface, 800 points are sampled by using a CMM. In each experiment, a group of points are arbitrarily picked from the 800 measurement points. The best-fit method is then used to directly fit the picked data points to the nominal model of the ball-end milled surface. The obtained coordinate transformation parameters are compared with the real coordinate transformation parameters to get the coordinate deviation ␦˜t. With the presence of the deterministic error component in this case, 800 points is not enough to get a close approximation of the real coordinate transformation. For this reason, the coordinate transformation estimated by the two-step coordinate estimation approach (discussed in the next section) using the sampled 800 points is used as the approximation of the real coordinate transformation. A total of 793 experiments were carried out.
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Fig. 6. Mean of uncertainty parameters (one-step approach, simulation results). (a) Mean of dx, dy, dz. (b) Mean of ␦x, ␦y, ␦z. (c) Norm of 兩␦˜t兩.
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Fig. 7. Variances of uncertainty parameters (one-step approach, simulation results). (a) Variances of dx, dy, dz. (b) Variances of ␦x, ␦y, ␦z. (c) Sum of variances of uncertainty parameters.
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The number of points used in the experiments ranges from 10 to 650 points. The norm of the coordinate deviation ␦˜t obtained from the experiments is shown in Fig. 8. Shown in Fig. 8(a) are the results with the number of points from 10 to 250, and those presented in Fig. 8(b) are the results with the number of points from 300 to 650. The robustness measure Rnorm is the same as defined in Part 2. Due to the influence of the deterministic error component which causes a bias shift of ␦˜t, the norm of ␦˜t is no longer bounded by the maximum principle axis length of the 99% probability envelope. This points out that the coordinate deviation ␦˜t in the Type II uncertainty problem is not predictable if the coordinate transformation is estimated by using the onestep approach.
Fig. 8. 兩␦˜t兩 vs Rnorm (one-step approach, experimental results). (a) 兩␦˜t兩 vs Rnorm (100–250 points). (b) 兩␦˜t兩 vs Rnorm (300– 650 points).
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3. Two-step approach In the two-step approach, a deterministic surface is first constructed from the measurement data using the procedure discussed in Part 1. After the construction of the deterministic surface, coordinate transformation is estimated by best-fitting the deterministic surface with the nominal surface. The best-fit of the deterministic surface with the nominal surface is performed by selecting a large number of discrete points from the deterministic surface and applying the non-linear least squares method to fit the selected points with the nominal surface. After best-fitting the deterministic surface with the nominal surface model, the deterministic error component can be obtained by computing the normal deviation from the deterministic surface to the nominal surface. 3.1. Uncertainty of the fitted deterministic surface Due to the variation of the random errors on the machined surface, variation of the deterministic surface fitted from the measurement data is unavoidable. In other words, uncertainty will be always associated with the fitted deterministic surface. As discussed in Part 1, fitting the deterministic surface from the measurement data is to determine the control points of the deterministic surface by regression. Therefore, the uncertainty of the fitted deterministic surface will be reflected by the variation of the obtained control points. To study the influence of the random errors on the uncertainty of the fitted deterministic surface, it is necessary to establish the relationship between the variation of the control points and the random errors on the part surface. This relationship can be derived by using the perturbation method. From the definition of B-spline surface, we know that the coordinates of an arbitrary point Pi on the deterministic surface is a linear combination of the control points of the B-spline representation, i.e.
gkxck
k⫽1 N
yi ⫽
冘 冘 冘 N
xi ⫽
gkyck
(11)
k⫽1 N
zi ⫽
gkzck
k⫽1
where [xi,yi,zi]T are the coordinates of point Pi, gk are the tensor products of the B-spline basis functions, and [xck,yck,zck]T (k ⫽ 1,2,…,N) are the coordinates of the B-spline surface control points. If small perturbations ⌬Ck ⫽ [dxck,dyck,dzck]T (k ⫽ 1,2,…,N) are applied to the N control points of the deterministic surface, the perturbation induced deviation ⌬Pi ⫽ [dxi,dyi,dzi]T at point Pi, will be given by the following formula:
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冘 冘 冘 N
dxi ⫽
N
dyi ⫽
gkdxck
k⫽1
gkdyck
(12)
k⫽1 N
dzi ⫽
gkdzck
k⫽1
If a total of n points on the deterministic surface are considered, then the perturbation induced deviations at these n points can be written into the following vector form: ⌬Xp ⫽ G⌬Xc ⌬Yp ⫽ G⌬Yc ⌬Zp ⫽ G⌬Zc
(13)
where ⌬Xp ⫽ [dx1,dx2,$,dxn]T, ⌬Yp ⫽ [dy1,dy2,$,dyn]T, ⌬Zp ⫽ [dz1,dz2,$,dzn]T, ⌬Xc ⫽ [dxc1,dxc2,$,dxcN]T, ⌬Yc ⫽ [dyc1,dyc2,$,dycN]T, ⌬Zc ⫽ [dzc1,dzc2,$,dzcN]T, and G is the following matrix consisting of the tensor products of the B-spline basis functions corresponding to each of the n points on the deterministic surface: g11 g12 $ g1N
g
21
G⫽
⯗
⯗ 哻 ⯗
g
g22 $ g2N
n1
(14)
gn2 $ gnN
Eq. (13) describes the relationship between the perturbation of the control points of the deterministic surface and the induced deviations at the selected points on the deterministic surface. Similarly, if a deterministic surface is fitted from a group of measurement points, the random errors at the measurement points will affect the control points of the fitted deterministic surface. Considering the regression process of fitting the deterministic surface from measurement data, the reverse mapping of Eq. (13) can be expressed as follows ⌬Xc ⫽ G+⌬Xp ⌬Yc ⫽ G+⌬Yp + ⌬Zc ⫽ G ⌬Zp
(15)
where G+ is the following pseudo-inverse matrix of the G matrix: g+11 g+12 $ g+1n
g
G+ ⫽
⯗
g
+ 21
+ N1
g+22 $ g+2n
⯗ 哻 ⯗
g+N2 $ g+Nn
(16)
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The three deviation vectors ⌬Xp, ⌬Yp and ⌬Zp in Eq. (15) are determined by the projection of the random errors on the x–y–z coordinate axes. As shown in Fig. 1, if the random error at target point Pi is ⑀ri and the corresponding surface normal at Pi is nˆi ⫽ [nxi,nyi,nzi]T, then ⌬Xp, ⌬Yp and ⌬Zp can be computed as follows: ⌬Xp ⫽ [nx1⑀r1,nx2⑀r2,$,nxn⑀rn]T ⌬Yp ⫽ [ny1⑀r1,ny2⑀r2,$,nyn⑀rn]T T ⌬Zp ⫽ [nz1⑀r1,nz2⑀r2,$,nzn⑀rn]
(17)
By substituting ⌬Xp, ⌬Yp and ⌬Zp into Eq. (15) and rearranging ⌬Xc, ⌬Yc and ⌬Zc, the following matrix equation which links the random errors at the measurement points and the variation of the fitted deterministic surface can be obtained:
␦C˜ ⫽ B⑀˜ r (18) T where ␦C˜ ⫽ [dxc1,dyc1,dzc1,$,dxcN,dycN,dzcN] is the 3N ⫻ 1 control point deviation vector, ⑀˜ r ⫽ [⑀r1,⑀r2,$,⑀rn]T is the n ⫻ 1 random error vector, and B is the following 3N ⫻ n matrix which is referred to as the sensitivity matrix of surface fitting. g+11nˆ1 g+12nˆ2 $ g+1nnˆn
g
B⫽
g
+ 21 1
nˆ g+22nˆ2 $ g+2nnˆn
⯗
⯗
哻
+ N1 1
⯗
(19)
nˆ g+N2nˆ2 $ g+Nnnˆn
Eq. (18) can be used to estimate the variation of the obtained control points when the statistics of the random errors on the machined surface are available. Since the mean of the random error vector is E(⑀˜ r) ⫽ 0, the mean of the control point deviation vector will be E(␦C˜) ⫽ BE(⑀˜ r) ⫽ 0. Due to the fact that the random errors at the measurement points are of spatial independence, the covariance matrix of the random error vector ⑀˜ r is a diagonal matrix, i.e. Cov(⑀˜ r) ⫽ diag{2⑀r1,2⑀r2,$,2⑀rn}
(20)
where 2⑀ri is the variance of ⑀ri which is the random error at the ith measurement point. Therefore, the covariance matrix of ␦C˜ can be obtained from Eq. (18) as follows: Cov(␦C˜) ⫽ BCov(⑀˜ r)BT
(21)
The covariance matrix Cov(␦C˜) is the numerical representation of the uncertainty of the fitted deterministic surface. Eq. (21) indicates that Cov(␦C˜) is of the same format as Cov(␦˜t) in Eq. (6) of Part 2. From this analogy, the uncertainty analysis for the fitted deterministic surface, including the study of the distribution of ␦C˜, the theoretical bounds of ␦C˜, and the uncertainty and robustness measures for surface fitting, can be performed in the same manner as that discussed for ␦˜t in Part 2. Therefore, when the statistics of the random errors on the machined surface are available, the accuracy of the deterministic surface fitted from a set of measurement data can be estimated.
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3.2. Uncertainty of coordinate estimation The second step of the two-step approach is to best-fit the deterministic surface with the nominal surface to estimate the coordinate transformation. The best-fit of the deterministic surface with the nominal surface is performed in an approximation fashion. First, a large number of discrete points from the obtained deterministic surface are selected to approximately represent the deterministic surface. Then, the non-linear least squares method is applied to fit these discrete points with the nominal surface to estimate the coordinate transformation. Since uncertainty is associated with the extracted deterministic surface, as discussed in the previous section, each of the selected discrete points on the fitted deterministic surface will have a variation. The variation of these selected discrete points will propagate into the result of coordinate estimation. To study the uncertainty of the coordinate transformation estimated by the two-step approach, the variation of the estimated deterministic error component at the discrete points used in coordinate estimation needs to be examined. As shown in Fig. 9, the deterministic error ⑀d is the distance from the deterministic surface to ˆ . ⑀d can be divided into an invariant part the nominal surface along the surface normal vector m ⑀f and a variant part ⑀v, i.e.
⑀d ⫽ ⑀f ⫹ ⑀v
(22)
The invariant part ⑀f of the deterministic error is the mean value of the deterministic error at point P. The variant part ⑀v of the deterministic error comes from the propagation of the uncertainty of the fitted deterministic surface, which is due to the influences of the random errors on the part surface and represented by the variation of the control points of the fitted deterministic surface. If the deviation of the control points is ␦C˜, its propagation will cause a deviation ⌬Pd at point Pd, as shown in Fig. 9. By using Eq. (11), ⌬Pd can be written into the following form: (23) ⌬Pd ⫽ [g1 g2$gN]␦C˜ where gi are the tensor products of the B-spline basis functions at point Pd on the estimated deterministic surface. The variant part ⑀v of the estimated deterministic error ⑀d can be approxiˆ of the nominal surface: mated by the projection of ⌬Pd in the normal direction m
Fig. 9. Deterministic error ⑀d and its variation.
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ˆ ⫽ [g1m ˆ T g 2m ˆ T$gNm ˆ T]␦C˜ ⑀v ⫽ ⌬Pd·m
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(24)
From the discussion in Section 3.1, we have E(␦C˜) ⫽ 0. Therefore, ⑀v will also have zero mean value. This confirms that the invariant part ⑀f of the estimated deterministic error is the mean value of the deterministic error at point P. For a total of M points selected from the extracted deterministic surface, the deterministic error vector will be
⑀˜ d ⫽ ⑀˜ f ⫹ ⑀˜ v
(25)
By using Eq. (24), ⑀˜ v can be written into the following form: g11m ˆ T1
g
⑀˜ v ⫽
g
ˆ T1 $ g1Nm ˆ T1 g12m
ˆ T2 g22m ˆ T2 $ g2Nm ˆ T2 m
21
⯗
⯗
哻
⯗
˜ ⬅ B1␦C˜ ␦C
(26)
ˆ TM gM2m ˆ TM $ gMNm ˆ TM m where gij are the tensor products of the basis functions of the estimated deterministic surface, ˆ i is the normal vector of the nominal surface at point Pi. From Eq. (26), the covariance matrix m of the deterministic error vector ⑀˜ d can be obtained: (27) Cov(⑀˜ d) ⫽ Cov(⑀˜ v) ⫽ B1Cov(␦C˜)BT1
M1
In terms of fitting discrete points with the nominal surface model to estimate the coordinate transformation, the second step of the two-step approach is similar to the coordinate estimation of the one-step approach. Therefore, the sensitivity matrix for coordinate estimation derived in Part 2 can be easily adopted to link the coordinate deviation ␦˜t with the deterministic error vector ⑀˜ d as follows: A␦˜t ⫽ ⑀˜ d ⫽ ⑀˜ f ⫹ ⑀˜ v
(28)
where A is the following sensitivity matrix which is related to the number and location of the discrete points used in coordinate estimation. m ˆ T1 (X1 ⫻ m ˆ 1)T
mˆ
A⫽
mˆ
T 2
$
T M
ˆ 2)T (X2 ⫻ m
$
(29)
ˆ M)T (XM ⫻ m
From Eq. (29), the deviation of the coordinate transformation estimated by the two-step approach can be estimated by the following formula:
冋冘 册 冋冘 册 冋冘 册 6
␦˜t ⫽
k⫽1
vikujk ⑀˜ ⫽ k d
6
k⫽1
vikujk ⑀˜ ⫹ k f
6
k⫽1
vikujk ⑀˜ ⬅ ␦˜tf ⫹ ␦˜tv k v
(30)
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冋冘 册 6
vikujk is the inverse mapping matrix from the geometric error space to the uncertainty k k⫽1 parameter space, vik and ujk are the entities of the left and right singular matrices of the sensitivity matrix A, respectively, and k are the singular values of A. When the number and location of the discrete points are determined, the mean and the variance of the coordinate deviation ␦˜t can be derived from Eq. (30). The mean of ␦˜t is given by the following equation:
where
冋冘 册 6
␦˜t ⫽ ␦˜tf ⫹ ␦˜tv ⫽
k⫽1
vikujk E(⑀˜ f) ⫹ k
冋冘 册 6
k⫽1
vikujk E(⑀˜ v) k
(31)
From earlier discussion, we have E(⑀˜ v) ⫽ 0 and E(⑀˜ d) ⫽ E(⑀˜ f) ⫽ ⑀˜ f. Therefore, Eq. (31) becomes
冋 冘 册⑀ 6
␦˜t ⫽ ␦˜tf ⫽ ␦˜tf ⫽
k⫽1
vikujk
(32)
˜f
k
Eq. (32) indicates that ␦˜tf is a constant component of the coordinate deviation in the two-step coordinate estimation approach. Since the estimated deterministic surface is a continuous surface, theoretically an infinite number of discrete points can be selected from the estimated deterministic surface. Therefore, by increasing the number of the discrete points selected from the estimated deterministic surface for coordinate estimation, ␦˜tf can be reduced to a negligible level compared to the required accuracy. As for the variation of ␦˜t, it is intuitive that even the number of the discrete points selected from the fitted deterministic surface for coordinate estimation is increased to infinity, due to the uncertainty of the constructed deterministic surface, variation will still exist in the result of coordinate estimation. From Eq. (30), the covariance matrix of ␦˜t can be obtained as follows:
冋冘 册 6
Cov(␦˜t) ⫽
vikujk Cov(⑀˜ v) k k⫽1
冋冘 册 6
k⫽1
vikujk k
T
(33)
Substituting Eq. (21) and Eq. (27) into Eq. (33), Cov(␦˜t) becomes Cov(␦˜t) ⫽ DBCov(⑀˜ r)BTDT
(34)
where D is the following uncertainty propagation matrix which maps the uncertainty of the fitted deterministic surface into the result of coordinate transformation through the second step of the two-step coordinate estimation approach:
冋冘 册 6
D⫽
k⫽1
vikujk B1 k
(35)
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Having obtained the covariance matrix Cov(␦˜t), the distribution of ␦˜t, the theoretical bounds of ␦˜t, and the uncertainty and robustness measures for coordinate estimation can be studied by applying the techniques developed in Part 2. 3.3. Computer simulations For the verification of the above analysis for the proposed two-step approach, computer simulations are conducted. In the computer simulations, the same data set used in Section 2.2 for the one-step approach are used. The data set includes a total of 50 simulated surfaces having the same deterministic error but different random errors. Each surface has 10 201 data points on a 101 ⫻ 101 grid evenly distributed in the uv-space. To obtain measurement data from the simulated surfaces, 20 different sampling schemes with the number of points ranging from 50 to 1000 are first determined. Based on each sampling scheme, measurement points are taken from the 10 201 data points on each of the 50 surfaces. A total of 1000 sets of simulated measurement data are used in the simulations. For each set of the simulated measurement data, a deterministic surface is fitted from the data points in the first step of the two-step approach. The number of measurement data needed to capture the form of the deterministic surface is about 400 points (Fig. 10). In the second step of the two-step approach, 10 000 discrete points on a 100 ⫻ 100 grid evenly spaced in the uv-space are selected from the extracted deterministic surface. The selected 10 000 discrete points are then best-fitted with the nominal surface model to get the estimated coordinate transformation parameters ˜test. By comparing ˜test with ˜treal, which is the real coordinate transformation parameters obtained from the best-fit of the 10 201 data points on each surface with the nominal surface model, coordinate deviation ␦˜t can be obtained for each surface. The simulation results are shown in Figs. 11–13. Fig. 11 shows the norm of ␦˜t against 1/Unorm, where Unorm is the uncertainty measure computed from Cov(␦˜t) in Eq. (34). In the figure, the norms of the coordinate deviation ␦˜t obtained from each set of the simulated measurement data all fall below the maximum principle axis length of
Fig. 10.
Number of patches vs number of points.
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Fig. 11. 兩␦˜t兩 vs 1/Unorm (two-step approach, simulation results).
the 99% probability upper bound hyper-ellipsoid. Therefore, for the two-step coordinate estimation approach, the variation range of 兩␦˜t兩 is predictable. By comparing Fig. 11 with Fig. 5, which shows the norm of ␦˜t obtained from the one-step approach, one can clearly see the effectiveness of the two-step approach in eliminating the influence of the deterministic error component on the result of coordinate estimation. Shown in Fig. 12 are the mean values of the uncertainty parameters with different sampling schemes. As the number of measurement points increases, the convergence trend of the mean values of the uncertainty parameters can be observed in Fig. 12(a and b). This is an important advantage of the two-step approach over the one-step approach. When the number of measurement points increases from 50 points to 100 points, the mean values of the uncertainty parameters quickly decrease to an acceptable level (dx ⫽ ⫺ 10.3m, dy ⫽ 11.7m, dz ⫽ 2.7m, ␦x ⫽ ⫺ 0.009°, ␦y ⫽ ⫺ 0.007°, ␦z ⫽ 0.025°). After the number of measurement points reaches 100 points, the mean values of the uncertainty parameters continue to decrease, but the convergence speed slows down. When the number of measurement points increases to 400, which is about the number of points required for capturing the deterministic surface form, the mean values decrease to dx ⫽ ⫺ 4.2m, dy ⫽ 8.1m, dz ⫽ 0.3m, ␦x ⫽ ⫺ 0.002°, ␦y ⫽ ⫺ 0.003° and ␦z ⫽ 0.003°. The variances of the uncertainty parameters obtained from the simulation results and the variances predicted by Eq. (34) are plotted in Fig. 13. It can be seen that the predicted variances and the variances obtained from computer simulations are in good agreement. By comparing Fig. 13 with Fig. 7, one can see that the magnitude of the variances of the uncertainty parameters resulted from the one-step approach and that obtained from the two-step approach are very close. This is due to the fact that the variances of the uncertainty parameters are determined by the random errors at the sampling points. Although the two-step approach can effectively eliminate the influences of the deterministic error component which causes a constant coordinate deviation from the real coordinate transformation, it can not reduce the influences of the random errors which cause the variation in the result of coordinate estimation.
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Fig. 12. Mean of uncertainty parameters (two-step approach, simulation results). (a) Mean of dx, dy, dz. (b) Mean of ␦x, ␦y, ␦z. (c) Norm of 兩␦˜t兩.
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Fig. 13. Variances of uncertainty parameters (two step approach, simulation results). (a) Variances of dx, dy, dz. (b) Variances of ␦x, ␦y, ␦z. (c) Sum of variances of uncertainty parameters.
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3.4. Experimental verification The same measurement data used in the experimental study of the one-step approach are used here for the study of the two-step approach. However, from Fig. 6 of Part 1, we know that the ball-end milled surface used in the experiments has a deterministic surface of 4 ⫻ 5 patches. Therefore, the number of control points of the deterministic surface is 56. In order to use the two-step approach, the number of measurement points must be larger than 56. Fig. 6 of Part 1 also points out that about 300 points are required to capture the deterministic surface form of the ball-end milled surface. For these reasons, out of the 793 sets of measurement data used in the experimental study of the one-step approach, only 343 sets of measurement data with number of points ranging from 100 to 650 are used in the experimental study of the two-step approach. The norm of coordinate deviation ␦˜t obtained from the experiments are plotted against the robustness measure Rnorm in Fig. 14. Fig. 14(a) shows the results with number of points from 100 to 250,
Fig. 14. 兩␦˜t兩 vs Rnorm (two-step approach, experimental results). (a) 兩␦˜t兩 vs Rnorm (100–250 points). (b) 兩␦˜t兩 vs Rnorm (300–650 points).
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and the results with number of points from 300 to 650 are shown in Fig. 14(b). The experimental results confirm that the norm of ␦˜t resulted from the proposed two-step approach are bounded by the maximum principle axis length of the 90% probability envelope. It is worth noting from Fig. 14(a) that when the number of measurement points is less than 300 points, which is the number of points required for the capture of the deterministic surface form, the robustness measure Rnorm could be quite small. Consequently, the variation range of 兩␦˜t兩 can be quite large. For example, by comparing Fig. 14(a) with Fig. 8(a), one can see that the variation range of 兩␦˜t兩 obtained from 100 points by the two-step approach is even larger than the variation range of 兩␦˜t兩 obtained from 10 points by the one-step approach. However, when the number of measurement points is more than 300 points, Fig. 14(b) shows that the variation range of 兩␦˜t兩 of the two-step approach is much smaller than the results of the one-step approach shown in Fig. 8(b). This result implies that the number of measurement points must be large enough to capture the form of the deterministic surface in order to take the advantage of the two-step approach. 4. Conclusions In this part of the paper, we investigate the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component. Two different approaches are proposed for the estimation of coordinate transformation. The one-step approach directly fits the measurement data with the part model to estimate the coordinate transformation. The two-step approach first uses surface fitting techniques to extract the deterministic surface from the measurement data, and then best-fit the deterministic surface with the nominal part model to estimate the coordinate transformation. Although the computation needed in the two-step coordinate estimation approach is more expensive than that required by the one-step approach, theoretical analysis, computer simulation and experimental results show that the two-step approach can effectively reduce the influence of the deterministic error component on the result of coordinate estimation. Therefore, with the same measurement data, the two-step approach gives a much more accurate coordinate estimation result than the one-step approach. In addition, the variation range of the coordinate transformation parameters obtained by the two-step approach is much reduced. 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. References [1] Z.C. Yan, Geometric tolerance evaluation and uncertainty analysis for coordinate metrology, Ph.D. dissertation, The Ohio State University, Columbus, Ohio, 1994.
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[2] K.C. Sahoo, C.H. Menq, Localization of 3D objects using surface representation and tactile sensing, in: Proceedings of the Symposium on Computer Aided Design and Manufacturing of Dies and Molds, ASME Winter Annual Meeting, Chicago, 28 Nov.–2 Dec., 1988, pp. 105–118. [3] A.B. Forbes, Least-squares best-fit geometric elements, Report DITC 140/89, National Physical Laboratory, UK, 1989. [4] C.H. Menq, H.T. Yau, G.Y. Lai, Automated precision measurement of surface profile in CAD-directed inspection, IEEE Trans. on Robotics and Automation 8 (2) (1992) 268–278. [5] D.F. Morrison, Multivariate Statistical Methods, McGraw-Hill, New York, 1976.
Uncertainty analysis and variation reduction of three dimensional coordinate metrology. Part 3: variation reduction Zhongcheng Yan, 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, the study focuses on the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component. Two different approaches are proposed for the estimation of coordinate transformation. In the onestep approach, the best-fit method is applied to directly fit the measurement data to the nominal surface. In the two-step approach, a deterministic surface is first constructed from the measurement data and the fitted deterministic surface is then best-fitted with the nominal surface to estimate the coordinate transformation. The computation needed in the two-step coordinate estimation approach is more expensive than that required by the one-step approach. However, by estimating the deterministic error component of the surface geometric error, the two-step approach can effectively reduce the influence of the deterministic error component on the result of coordinate estimation. Therefore, with the same measurement data, the two-step approach gives a much more accurate coordinate estimation result than the one-step approach. In addition, the variation range of the obtained coordinate transformation parameters is much reduced. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Coordinate metrology; Uncertainty analysis; Variation reduction; Deterministic error
1. Introduction The variation of the estimated coordinate transformation arises from the geometric errors on the part surface. In Part 2 of the paper, the uncertainty analysis for the case in which the sampled * 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 9 1 - 1
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geometric errors are dominated by the random component is investigated. In practice, the uncertainty analysis for coordinate estimation using high precision datum surfaces often falls into this type. In this part of the paper, the study focuses on the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component [1]. Due to the distinct characteristics of the two different error components, their influences on the result of coordinate estimation are also different. In current practice of coordinate estimation, however, the two different error components are not distinguished in the process of coordinate estimation. In other words, in both the Type I and the Type II problem, the coordinate transformation is estimated by directly best-fitting the measurement data with the nominal surface model. We call this approach the One-step Coordinate Estimation Approach. In addition to this one-step approach, we propose a different approach which is called the Two-step Coordinate Estimation Approach. In the proposed two-step coordinate estimation approach, the estimation of coordinate transformation is carried out in two steps. First, a deterministic surface is constructed from the measurement data using the surface fitting techniques discussed in Part 1 of the paper. Then, the deterministic surface is best-fitted with the surface nominal model to estimate the coordinate transformation. The best-fit of the deterministic surface with the nominal model is performed by using the non-linear least squares method to fit a large number of discrete points selected from the deterministic surface with the nominal surface. It is evident that the coordinate estimation results from the two different approaches will be different. In this paper, uncertainty analysis for the two different approaches will be investigated and the merits and drawbacks of the two approaches will be compared.
2. One-step approach In the one-step approach, the best-fit algorithm is applied directly to fit the measurement data with the nominal surface model [2–4]. Since both the random and the deterministic error components exist, the variation of the estimated coordinate transformation is the result of the joint effect of the two different geometric error components. The influences of both the random and the deterministic error components on the coordinate transformation estimated by using the one-step approach is discussed in this section. 2.1. Uncertainty of coordinate estimation To study the influences of the random and the deterministic error components on the result of coordinate estimation by the one-step approach, the relationship between the uncertainty parameters and the sampled geometric errors must be established. The sensitivity matrix of coordinate estimation derived in Part 2 can again be employed here for this purpose. A␦˜t ⫽ ⑀˜
(1)
where A is the sensitivity matrix of coordinate estimation, and ⑀˜ is the geometric deviations at the measurement points. As shown in Fig. 1, suppose that the corresponding normal projection
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Fig. 1. Geometric error components.
point for measurement point Pi⬘ is Pi. Then the sampled geometric error at point Pi can be expressed as follows:
⑀i ⫽ ⑀di ⫹ ⑀ri
(2)
where ⑀di and ⑀ri are the deterministic and the random error components at point Pi, respectively. For a total of N measurement points, the geometric error vector will be
⑀˜ ⫽ ⑀˜ d ⫹ ⑀˜ r
(3)
where ⑀˜ d ⫽ [⑀d1,⑀d2,$,⑀dN]T is the deterministic error vector, and ⑀˜ r ⫽ [⑀r1,⑀r2,$,⑀rN]T is the random error vector. Substituting Eq. (3) into Eq. (1) yields A␦˜t ⫽ ⑀˜ d ⫹ ⑀˜ r
(4)
From Eq. (4), the inverse mapping from the geometric error vector space to the uncertainty parameter vector space can be determined as follows:
冋冘 册 冋冘 册 6
␦˜t ⫽
k⫽1
6
vikujk (⑀˜ d ⫹ ⑀˜ r) ⫽ k
冋冘 册 冋冘 册 6
k⫽1
vikujk ⑀˜ ⫹ k d
6
k⫽1
vikujk ⑀˜ ⬅ ␦˜td ⫹ ␦˜tr k r
(5)
vikujk is the inverse mapping matrix, vik and ujk are the entities of the left and right k k⫽1 singular matrices of the sensitivity matrix A respectively, and k are the singular values of A. It can be seen from Eq. (5) that the coordinate deviation ␦˜t consists of two parts: ␦˜td and ␦˜tr; ␦˜td is caused by the deterministic error component, and ␦˜tr comes from the effect of the random error component. Since the real coordinate transformation is defined as the limit of the estimated coordinate transformation when the number of measurement points approaches infinity, both ␦˜td and ␦˜tr will converge to zero as the number of sampling points approaches to infinity [1]. However, when using a finite number of measurement points, since the magnitude of the deterministic error component differs from one point to another, ␦˜td will change when changing the number and location of measurement points. Since the deterministic error trend on the machined surface is not estimated in the one-step coordinate estimation approach, it is impossible to assess where
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the magnitude of ␦˜td. Without assessing the magnitude of ␦˜td, the accuracy of the coordinate estimation result will deteriorate. This is the drawback of the one-step coordinate estimation approach in estimating coordinate transformation for the Type II problem. The only way to improve coordinate estimation accuracy is to increase the sampling density so as to make ␦˜td as small as possible. From Eq. (5), the mean and the variance of the uncertainty parameter vector ␦˜t can be calculated. The mean of ␦˜t is given by the following formula:
冋冘 册 6
vikujk ␦˜t ⫽ ␦˜td ⫹ ␦˜tr ⫽ E(⑀˜ d) ⫹ k k⫽1
冋冘 册 6
vikujk E(⑀˜ r) k k⫽1
(6)
When the number and location of the measurement points are determined, the deterministic error vector ⑀˜ d is the same for every surface machined under the same process conditions. Therefore, we have E(⑀˜ d) ⫽ ⑀˜ d. Since the random error vector ⑀˜ r has zero mean, i.e. E(⑀˜ r) ⫽ 0, Eq. (6) can be simplified as follows:
冋冘 册 6
␦˜t ⫽ ␦˜td ⫽ ␦˜td ⫽
k⫽1
vikujk ⑀˜ k d
(7)
It is clear that ␦˜td is a bias in the estimated coordinate parameters. For a specified sampling scheme, ␦˜td represents a constant shift of ␦˜t from the origin of the uncertainty parameter space and ␦˜tr, which is the variation of ␦˜t, will be added to this constant shift ␦˜td. Since the deterministic error vector ⑀˜ d is a constant vector for a selected sampling plan, it can be proven that Cov(⑀˜ ) ⫽ Cov(⑀˜ d ⫹ ⑀˜ r) ⫽ Cov(⑀˜ r). Therefore, the covariance of ␦˜t can be obtained from Eq. (5) as follows:
冋冘 册 6
vikujk Cov(⑀˜ r) Cov(␦˜t) ⫽ k k⫽1
冋冘 册 6
k⫽1
vikujk k
T
⫽ Cov(␦˜tr)
(8)
From the theory of statistics [5], we know that ␦˜t will follow a normal distribution with ␦˜td as its mean. The joint probability density function of ␦˜t is given by Eq. (9), and the hyper-ellipsoid shaped probability envelope is illustrated in Fig. 2. f (␦˜t) ⫽
1 exp{ ⫺ 0.5·␦˜tTr[Cov(␦˜tr)]−1␦˜tr} 83兩Cov(␦˜tr)兩1/2
(9)
If the statistics of the random errors on the surfaces are available, then Cov(␦˜t) can be obtained from Eq. (8). Thus the normal distribution shape of ␦˜t is known. However, as mentioned above, the magnitude of the bias component ␦˜td in the estimated coordinate parameters can not be obtained in the one-step coordinate estimation approach. This means that it is impossible to know the center of the normal distribution which is shifted from the origin of the uncertainty parameter
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Fig. 2. Illustration of probability envelope of one-step approach.
vector space. As for the variation of the uncertainty parameters, it can be proven that the sum of the variances of the uncertainty parameters follows the following equation:
冘 6
i⫽1
冘 6
2␦ti ⫽
2 k
(10)
k⫽1
where 2k are the singular values of the covariance matrix Cov(␦˜t). 2.2. Computer simulations To verify the above discussion about the one-step coordinate estimation approach, computer simulations were conducted. In this study, the B-spline surface shown in Fig. 3 is used as the nominal surface. A deterministic error trend ⑀d(u,v) with 3⫻3 patches as shown in Fig. 4 is added to the nominal surface along the normal direction of the nominal surface. Random errors are then added to a 101⫻101 grid on the deterministic surface to simulate a surface having both deterministic error trend and random errors. A total of 50 surfaces having the same deterministic error trend but different random errors are created. Best-fit is applied to fit the 10 201 data points of each surface with the nominal surface model, and the obtained coordinate transformation parameters serve as the real coordinate transformation parameters ˜treal for each surface. A total of 20 sets of target points with the number ranging from 50 to 1000 are first determined in such a way that the (i⫹1)th set is obtained by adding 50 more randomly picked points to the ith set. In each set of the target points, any single point is only selected once and there is no repeated selection of the same point. Based on each set of the selected target points, measurement data are picked from the 10 201 data points on each of the 50 surfaces. The measurement data taken from each
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Fig. 3.
Fig. 4.
Nominal B-spline surface model for simulation study of coordinate uncertainty (1⫻1 patch).
Deterministic error trend for simulation study of the Type II uncertainty problem (3⫻3 patches).
surface are best-fitted with the nominal surface model to get the estimated coordinate transformation parameters ˜test. Consequently, ␦˜t which is the deviation of ˜test from ˜treal can be calculated for each surface. The simulation results are presented in Figs. 5–7. Fig. 5 shows the norm of ␦˜t against 1/Unorm, where Unorm is defined in Part 2 of the paper. For each set of the selected target points, Unorm is computed from the covariance matrix Cov(␦˜t) in Eq. (8). It can be seen from Fig. 5 that, for the Type II uncertainty problem, the norm of ␦˜t resulted from the one-step approach is no longer bounded by the maximum principle axis length of the 99% probability envelope. It is evident that this is due to the non-zero mean ␦˜t ⫽ ␦˜td of the uncertainty parameters that is caused by the deterministic error component of the surface geometric errors. The mean values of the uncertainty parameters for different sampling schemes are computed from the simulation results and plotted in Fig. 6. It can be seen from Fig. 6 that the mean value of each uncertainty parameter has no clear convergence trend associated with the increase of measurement points. The fluctuation of the mean value of each uncertainty parameter associated with the increase of measurement points is caused by the changing magnitude of the deterministic
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Fig. 5. 兩␦˜t兩 vs 1/Unorm (one-step approach, simulation results).
error component at different sampling points. Due to the influence of the deterministic error component, the increase of measurement points does not necessary reduce the mean value of each uncertainty parameter. On the other hand, the variances of the uncertainty parameters are mainly due to the influences of the random errors. More measurement points usually result in smaller variation of ␦˜t. This can be verified from Fig. 7. In Fig. 7, the predicted variances for the uncertainty parameters are computed from Eq. (8). The predicted variances and the variances computed from simulation results are in good agreement. The small discrepancies are possibly due to the weak correlation among the random errors used in the simulations and the fact that the best-fit is a nonlinear process and the sensitivity matrix used is a linear approximation. 2.3. Experimental verification In addition to the computer simulations, experiments were also carried out for the verification of the uncertainty analysis for the one-step coordinate estimation approach. The nominal model of the ball-end milled surface used in the experiments is shown in Fig. 2 of Part 2. It is a freeform surface with 2⫻2 patches. This ball-end milled surface has a deterministic error trend as shown in Fig. 8 of Part 1. The distribution of the random error on the surface is shown in Fig. 11 of Part 2, and the standard deviation of the random errors over the surface is ⫽ 0.0155. From the surface, 800 points are sampled by using a CMM. In each experiment, a group of points are arbitrarily picked from the 800 measurement points. The best-fit method is then used to directly fit the picked data points to the nominal model of the ball-end milled surface. The obtained coordinate transformation parameters are compared with the real coordinate transformation parameters to get the coordinate deviation ␦˜t. With the presence of the deterministic error component in this case, 800 points is not enough to get a close approximation of the real coordinate transformation. For this reason, the coordinate transformation estimated by the two-step coordinate estimation approach (discussed in the next section) using the sampled 800 points is used as the approximation of the real coordinate transformation. A total of 793 experiments were carried out.
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Fig. 6. Mean of uncertainty parameters (one-step approach, simulation results). (a) Mean of dx, dy, dz. (b) Mean of ␦x, ␦y, ␦z. (c) Norm of 兩␦˜t兩.
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Fig. 7. Variances of uncertainty parameters (one-step approach, simulation results). (a) Variances of dx, dy, dz. (b) Variances of ␦x, ␦y, ␦z. (c) Sum of variances of uncertainty parameters.
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The number of points used in the experiments ranges from 10 to 650 points. The norm of the coordinate deviation ␦˜t obtained from the experiments is shown in Fig. 8. Shown in Fig. 8(a) are the results with the number of points from 10 to 250, and those presented in Fig. 8(b) are the results with the number of points from 300 to 650. The robustness measure Rnorm is the same as defined in Part 2. Due to the influence of the deterministic error component which causes a bias shift of ␦˜t, the norm of ␦˜t is no longer bounded by the maximum principle axis length of the 99% probability envelope. This points out that the coordinate deviation ␦˜t in the Type II uncertainty problem is not predictable if the coordinate transformation is estimated by using the onestep approach.
Fig. 8. 兩␦˜t兩 vs Rnorm (one-step approach, experimental results). (a) 兩␦˜t兩 vs Rnorm (100–250 points). (b) 兩␦˜t兩 vs Rnorm (300– 650 points).
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3. Two-step approach In the two-step approach, a deterministic surface is first constructed from the measurement data using the procedure discussed in Part 1. After the construction of the deterministic surface, coordinate transformation is estimated by best-fitting the deterministic surface with the nominal surface. The best-fit of the deterministic surface with the nominal surface is performed by selecting a large number of discrete points from the deterministic surface and applying the non-linear least squares method to fit the selected points with the nominal surface. After best-fitting the deterministic surface with the nominal surface model, the deterministic error component can be obtained by computing the normal deviation from the deterministic surface to the nominal surface. 3.1. Uncertainty of the fitted deterministic surface Due to the variation of the random errors on the machined surface, variation of the deterministic surface fitted from the measurement data is unavoidable. In other words, uncertainty will be always associated with the fitted deterministic surface. As discussed in Part 1, fitting the deterministic surface from the measurement data is to determine the control points of the deterministic surface by regression. Therefore, the uncertainty of the fitted deterministic surface will be reflected by the variation of the obtained control points. To study the influence of the random errors on the uncertainty of the fitted deterministic surface, it is necessary to establish the relationship between the variation of the control points and the random errors on the part surface. This relationship can be derived by using the perturbation method. From the definition of B-spline surface, we know that the coordinates of an arbitrary point Pi on the deterministic surface is a linear combination of the control points of the B-spline representation, i.e.
gkxck
k⫽1 N
yi ⫽
冘 冘 冘 N
xi ⫽
gkyck
(11)
k⫽1 N
zi ⫽
gkzck
k⫽1
where [xi,yi,zi]T are the coordinates of point Pi, gk are the tensor products of the B-spline basis functions, and [xck,yck,zck]T (k ⫽ 1,2,…,N) are the coordinates of the B-spline surface control points. If small perturbations ⌬Ck ⫽ [dxck,dyck,dzck]T (k ⫽ 1,2,…,N) are applied to the N control points of the deterministic surface, the perturbation induced deviation ⌬Pi ⫽ [dxi,dyi,dzi]T at point Pi, will be given by the following formula:
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冘 冘 冘 N
dxi ⫽
N
dyi ⫽
gkdxck
k⫽1
gkdyck
(12)
k⫽1 N
dzi ⫽
gkdzck
k⫽1
If a total of n points on the deterministic surface are considered, then the perturbation induced deviations at these n points can be written into the following vector form: ⌬Xp ⫽ G⌬Xc ⌬Yp ⫽ G⌬Yc ⌬Zp ⫽ G⌬Zc
(13)
where ⌬Xp ⫽ [dx1,dx2,$,dxn]T, ⌬Yp ⫽ [dy1,dy2,$,dyn]T, ⌬Zp ⫽ [dz1,dz2,$,dzn]T, ⌬Xc ⫽ [dxc1,dxc2,$,dxcN]T, ⌬Yc ⫽ [dyc1,dyc2,$,dycN]T, ⌬Zc ⫽ [dzc1,dzc2,$,dzcN]T, and G is the following matrix consisting of the tensor products of the B-spline basis functions corresponding to each of the n points on the deterministic surface: g11 g12 $ g1N
g
21
G⫽
⯗
⯗ 哻 ⯗
g
g22 $ g2N
n1
(14)
gn2 $ gnN
Eq. (13) describes the relationship between the perturbation of the control points of the deterministic surface and the induced deviations at the selected points on the deterministic surface. Similarly, if a deterministic surface is fitted from a group of measurement points, the random errors at the measurement points will affect the control points of the fitted deterministic surface. Considering the regression process of fitting the deterministic surface from measurement data, the reverse mapping of Eq. (13) can be expressed as follows ⌬Xc ⫽ G+⌬Xp ⌬Yc ⫽ G+⌬Yp + ⌬Zc ⫽ G ⌬Zp
(15)
where G+ is the following pseudo-inverse matrix of the G matrix: g+11 g+12 $ g+1n
g
G+ ⫽
⯗
g
+ 21
+ N1
g+22 $ g+2n
⯗ 哻 ⯗
g+N2 $ g+Nn
(16)
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The three deviation vectors ⌬Xp, ⌬Yp and ⌬Zp in Eq. (15) are determined by the projection of the random errors on the x–y–z coordinate axes. As shown in Fig. 1, if the random error at target point Pi is ⑀ri and the corresponding surface normal at Pi is nˆi ⫽ [nxi,nyi,nzi]T, then ⌬Xp, ⌬Yp and ⌬Zp can be computed as follows: ⌬Xp ⫽ [nx1⑀r1,nx2⑀r2,$,nxn⑀rn]T ⌬Yp ⫽ [ny1⑀r1,ny2⑀r2,$,nyn⑀rn]T T ⌬Zp ⫽ [nz1⑀r1,nz2⑀r2,$,nzn⑀rn]
(17)
By substituting ⌬Xp, ⌬Yp and ⌬Zp into Eq. (15) and rearranging ⌬Xc, ⌬Yc and ⌬Zc, the following matrix equation which links the random errors at the measurement points and the variation of the fitted deterministic surface can be obtained:
␦C˜ ⫽ B⑀˜ r (18) T where ␦C˜ ⫽ [dxc1,dyc1,dzc1,$,dxcN,dycN,dzcN] is the 3N ⫻ 1 control point deviation vector, ⑀˜ r ⫽ [⑀r1,⑀r2,$,⑀rn]T is the n ⫻ 1 random error vector, and B is the following 3N ⫻ n matrix which is referred to as the sensitivity matrix of surface fitting. g+11nˆ1 g+12nˆ2 $ g+1nnˆn
g
B⫽
g
+ 21 1
nˆ g+22nˆ2 $ g+2nnˆn
⯗
⯗
哻
+ N1 1
⯗
(19)
nˆ g+N2nˆ2 $ g+Nnnˆn
Eq. (18) can be used to estimate the variation of the obtained control points when the statistics of the random errors on the machined surface are available. Since the mean of the random error vector is E(⑀˜ r) ⫽ 0, the mean of the control point deviation vector will be E(␦C˜) ⫽ BE(⑀˜ r) ⫽ 0. Due to the fact that the random errors at the measurement points are of spatial independence, the covariance matrix of the random error vector ⑀˜ r is a diagonal matrix, i.e. Cov(⑀˜ r) ⫽ diag{2⑀r1,2⑀r2,$,2⑀rn}
(20)
where 2⑀ri is the variance of ⑀ri which is the random error at the ith measurement point. Therefore, the covariance matrix of ␦C˜ can be obtained from Eq. (18) as follows: Cov(␦C˜) ⫽ BCov(⑀˜ r)BT
(21)
The covariance matrix Cov(␦C˜) is the numerical representation of the uncertainty of the fitted deterministic surface. Eq. (21) indicates that Cov(␦C˜) is of the same format as Cov(␦˜t) in Eq. (6) of Part 2. From this analogy, the uncertainty analysis for the fitted deterministic surface, including the study of the distribution of ␦C˜, the theoretical bounds of ␦C˜, and the uncertainty and robustness measures for surface fitting, can be performed in the same manner as that discussed for ␦˜t in Part 2. Therefore, when the statistics of the random errors on the machined surface are available, the accuracy of the deterministic surface fitted from a set of measurement data can be estimated.
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3.2. Uncertainty of coordinate estimation The second step of the two-step approach is to best-fit the deterministic surface with the nominal surface to estimate the coordinate transformation. The best-fit of the deterministic surface with the nominal surface is performed in an approximation fashion. First, a large number of discrete points from the obtained deterministic surface are selected to approximately represent the deterministic surface. Then, the non-linear least squares method is applied to fit these discrete points with the nominal surface to estimate the coordinate transformation. Since uncertainty is associated with the extracted deterministic surface, as discussed in the previous section, each of the selected discrete points on the fitted deterministic surface will have a variation. The variation of these selected discrete points will propagate into the result of coordinate estimation. To study the uncertainty of the coordinate transformation estimated by the two-step approach, the variation of the estimated deterministic error component at the discrete points used in coordinate estimation needs to be examined. As shown in Fig. 9, the deterministic error ⑀d is the distance from the deterministic surface to ˆ . ⑀d can be divided into an invariant part the nominal surface along the surface normal vector m ⑀f and a variant part ⑀v, i.e.
⑀d ⫽ ⑀f ⫹ ⑀v
(22)
The invariant part ⑀f of the deterministic error is the mean value of the deterministic error at point P. The variant part ⑀v of the deterministic error comes from the propagation of the uncertainty of the fitted deterministic surface, which is due to the influences of the random errors on the part surface and represented by the variation of the control points of the fitted deterministic surface. If the deviation of the control points is ␦C˜, its propagation will cause a deviation ⌬Pd at point Pd, as shown in Fig. 9. By using Eq. (11), ⌬Pd can be written into the following form: (23) ⌬Pd ⫽ [g1 g2$gN]␦C˜ where gi are the tensor products of the B-spline basis functions at point Pd on the estimated deterministic surface. The variant part ⑀v of the estimated deterministic error ⑀d can be approxiˆ of the nominal surface: mated by the projection of ⌬Pd in the normal direction m
Fig. 9. Deterministic error ⑀d and its variation.
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ˆ ⫽ [g1m ˆ T g 2m ˆ T$gNm ˆ T]␦C˜ ⑀v ⫽ ⌬Pd·m
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(24)
From the discussion in Section 3.1, we have E(␦C˜) ⫽ 0. Therefore, ⑀v will also have zero mean value. This confirms that the invariant part ⑀f of the estimated deterministic error is the mean value of the deterministic error at point P. For a total of M points selected from the extracted deterministic surface, the deterministic error vector will be
⑀˜ d ⫽ ⑀˜ f ⫹ ⑀˜ v
(25)
By using Eq. (24), ⑀˜ v can be written into the following form: g11m ˆ T1
g
⑀˜ v ⫽
g
ˆ T1 $ g1Nm ˆ T1 g12m
ˆ T2 g22m ˆ T2 $ g2Nm ˆ T2 m
21
⯗
⯗
哻
⯗
˜ ⬅ B1␦C˜ ␦C
(26)
ˆ TM gM2m ˆ TM $ gMNm ˆ TM m where gij are the tensor products of the basis functions of the estimated deterministic surface, ˆ i is the normal vector of the nominal surface at point Pi. From Eq. (26), the covariance matrix m of the deterministic error vector ⑀˜ d can be obtained: (27) Cov(⑀˜ d) ⫽ Cov(⑀˜ v) ⫽ B1Cov(␦C˜)BT1
M1
In terms of fitting discrete points with the nominal surface model to estimate the coordinate transformation, the second step of the two-step approach is similar to the coordinate estimation of the one-step approach. Therefore, the sensitivity matrix for coordinate estimation derived in Part 2 can be easily adopted to link the coordinate deviation ␦˜t with the deterministic error vector ⑀˜ d as follows: A␦˜t ⫽ ⑀˜ d ⫽ ⑀˜ f ⫹ ⑀˜ v
(28)
where A is the following sensitivity matrix which is related to the number and location of the discrete points used in coordinate estimation. m ˆ T1 (X1 ⫻ m ˆ 1)T
mˆ
A⫽
mˆ
T 2
$
T M
ˆ 2)T (X2 ⫻ m
$
(29)
ˆ M)T (XM ⫻ m
From Eq. (29), the deviation of the coordinate transformation estimated by the two-step approach can be estimated by the following formula:
冋冘 册 冋冘 册 冋冘 册 6
␦˜t ⫽
k⫽1
vikujk ⑀˜ ⫽ k d
6
k⫽1
vikujk ⑀˜ ⫹ k f
6
k⫽1
vikujk ⑀˜ ⬅ ␦˜tf ⫹ ␦˜tv k v
(30)
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冋冘 册 6
vikujk is the inverse mapping matrix from the geometric error space to the uncertainty k k⫽1 parameter space, vik and ujk are the entities of the left and right singular matrices of the sensitivity matrix A, respectively, and k are the singular values of A. When the number and location of the discrete points are determined, the mean and the variance of the coordinate deviation ␦˜t can be derived from Eq. (30). The mean of ␦˜t is given by the following equation:
where
冋冘 册 6
␦˜t ⫽ ␦˜tf ⫹ ␦˜tv ⫽
k⫽1
vikujk E(⑀˜ f) ⫹ k
冋冘 册 6
k⫽1
vikujk E(⑀˜ v) k
(31)
From earlier discussion, we have E(⑀˜ v) ⫽ 0 and E(⑀˜ d) ⫽ E(⑀˜ f) ⫽ ⑀˜ f. Therefore, Eq. (31) becomes
冋 冘 册⑀ 6
␦˜t ⫽ ␦˜tf ⫽ ␦˜tf ⫽
k⫽1
vikujk
(32)
˜f
k
Eq. (32) indicates that ␦˜tf is a constant component of the coordinate deviation in the two-step coordinate estimation approach. Since the estimated deterministic surface is a continuous surface, theoretically an infinite number of discrete points can be selected from the estimated deterministic surface. Therefore, by increasing the number of the discrete points selected from the estimated deterministic surface for coordinate estimation, ␦˜tf can be reduced to a negligible level compared to the required accuracy. As for the variation of ␦˜t, it is intuitive that even the number of the discrete points selected from the fitted deterministic surface for coordinate estimation is increased to infinity, due to the uncertainty of the constructed deterministic surface, variation will still exist in the result of coordinate estimation. From Eq. (30), the covariance matrix of ␦˜t can be obtained as follows:
冋冘 册 6
Cov(␦˜t) ⫽
vikujk Cov(⑀˜ v) k k⫽1
冋冘 册 6
k⫽1
vikujk k
T
(33)
Substituting Eq. (21) and Eq. (27) into Eq. (33), Cov(␦˜t) becomes Cov(␦˜t) ⫽ DBCov(⑀˜ r)BTDT
(34)
where D is the following uncertainty propagation matrix which maps the uncertainty of the fitted deterministic surface into the result of coordinate transformation through the second step of the two-step coordinate estimation approach:
冋冘 册 6
D⫽
k⫽1
vikujk B1 k
(35)
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Having obtained the covariance matrix Cov(␦˜t), the distribution of ␦˜t, the theoretical bounds of ␦˜t, and the uncertainty and robustness measures for coordinate estimation can be studied by applying the techniques developed in Part 2. 3.3. Computer simulations For the verification of the above analysis for the proposed two-step approach, computer simulations are conducted. In the computer simulations, the same data set used in Section 2.2 for the one-step approach are used. The data set includes a total of 50 simulated surfaces having the same deterministic error but different random errors. Each surface has 10 201 data points on a 101 ⫻ 101 grid evenly distributed in the uv-space. To obtain measurement data from the simulated surfaces, 20 different sampling schemes with the number of points ranging from 50 to 1000 are first determined. Based on each sampling scheme, measurement points are taken from the 10 201 data points on each of the 50 surfaces. A total of 1000 sets of simulated measurement data are used in the simulations. For each set of the simulated measurement data, a deterministic surface is fitted from the data points in the first step of the two-step approach. The number of measurement data needed to capture the form of the deterministic surface is about 400 points (Fig. 10). In the second step of the two-step approach, 10 000 discrete points on a 100 ⫻ 100 grid evenly spaced in the uv-space are selected from the extracted deterministic surface. The selected 10 000 discrete points are then best-fitted with the nominal surface model to get the estimated coordinate transformation parameters ˜test. By comparing ˜test with ˜treal, which is the real coordinate transformation parameters obtained from the best-fit of the 10 201 data points on each surface with the nominal surface model, coordinate deviation ␦˜t can be obtained for each surface. The simulation results are shown in Figs. 11–13. Fig. 11 shows the norm of ␦˜t against 1/Unorm, where Unorm is the uncertainty measure computed from Cov(␦˜t) in Eq. (34). In the figure, the norms of the coordinate deviation ␦˜t obtained from each set of the simulated measurement data all fall below the maximum principle axis length of
Fig. 10.
Number of patches vs number of points.
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Fig. 11. 兩␦˜t兩 vs 1/Unorm (two-step approach, simulation results).
the 99% probability upper bound hyper-ellipsoid. Therefore, for the two-step coordinate estimation approach, the variation range of 兩␦˜t兩 is predictable. By comparing Fig. 11 with Fig. 5, which shows the norm of ␦˜t obtained from the one-step approach, one can clearly see the effectiveness of the two-step approach in eliminating the influence of the deterministic error component on the result of coordinate estimation. Shown in Fig. 12 are the mean values of the uncertainty parameters with different sampling schemes. As the number of measurement points increases, the convergence trend of the mean values of the uncertainty parameters can be observed in Fig. 12(a and b). This is an important advantage of the two-step approach over the one-step approach. When the number of measurement points increases from 50 points to 100 points, the mean values of the uncertainty parameters quickly decrease to an acceptable level (dx ⫽ ⫺ 10.3m, dy ⫽ 11.7m, dz ⫽ 2.7m, ␦x ⫽ ⫺ 0.009°, ␦y ⫽ ⫺ 0.007°, ␦z ⫽ 0.025°). After the number of measurement points reaches 100 points, the mean values of the uncertainty parameters continue to decrease, but the convergence speed slows down. When the number of measurement points increases to 400, which is about the number of points required for capturing the deterministic surface form, the mean values decrease to dx ⫽ ⫺ 4.2m, dy ⫽ 8.1m, dz ⫽ 0.3m, ␦x ⫽ ⫺ 0.002°, ␦y ⫽ ⫺ 0.003° and ␦z ⫽ 0.003°. The variances of the uncertainty parameters obtained from the simulation results and the variances predicted by Eq. (34) are plotted in Fig. 13. It can be seen that the predicted variances and the variances obtained from computer simulations are in good agreement. By comparing Fig. 13 with Fig. 7, one can see that the magnitude of the variances of the uncertainty parameters resulted from the one-step approach and that obtained from the two-step approach are very close. This is due to the fact that the variances of the uncertainty parameters are determined by the random errors at the sampling points. Although the two-step approach can effectively eliminate the influences of the deterministic error component which causes a constant coordinate deviation from the real coordinate transformation, it can not reduce the influences of the random errors which cause the variation in the result of coordinate estimation.
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Fig. 12. Mean of uncertainty parameters (two-step approach, simulation results). (a) Mean of dx, dy, dz. (b) Mean of ␦x, ␦y, ␦z. (c) Norm of 兩␦˜t兩.
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Fig. 13. Variances of uncertainty parameters (two step approach, simulation results). (a) Variances of dx, dy, dz. (b) Variances of ␦x, ␦y, ␦z. (c) Sum of variances of uncertainty parameters.
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3.4. Experimental verification The same measurement data used in the experimental study of the one-step approach are used here for the study of the two-step approach. However, from Fig. 6 of Part 1, we know that the ball-end milled surface used in the experiments has a deterministic surface of 4 ⫻ 5 patches. Therefore, the number of control points of the deterministic surface is 56. In order to use the two-step approach, the number of measurement points must be larger than 56. Fig. 6 of Part 1 also points out that about 300 points are required to capture the deterministic surface form of the ball-end milled surface. For these reasons, out of the 793 sets of measurement data used in the experimental study of the one-step approach, only 343 sets of measurement data with number of points ranging from 100 to 650 are used in the experimental study of the two-step approach. The norm of coordinate deviation ␦˜t obtained from the experiments are plotted against the robustness measure Rnorm in Fig. 14. Fig. 14(a) shows the results with number of points from 100 to 250,
Fig. 14. 兩␦˜t兩 vs Rnorm (two-step approach, experimental results). (a) 兩␦˜t兩 vs Rnorm (100–250 points). (b) 兩␦˜t兩 vs Rnorm (300–650 points).
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and the results with number of points from 300 to 650 are shown in Fig. 14(b). The experimental results confirm that the norm of ␦˜t resulted from the proposed two-step approach are bounded by the maximum principle axis length of the 90% probability envelope. It is worth noting from Fig. 14(a) that when the number of measurement points is less than 300 points, which is the number of points required for the capture of the deterministic surface form, the robustness measure Rnorm could be quite small. Consequently, the variation range of 兩␦˜t兩 can be quite large. For example, by comparing Fig. 14(a) with Fig. 8(a), one can see that the variation range of 兩␦˜t兩 obtained from 100 points by the two-step approach is even larger than the variation range of 兩␦˜t兩 obtained from 10 points by the one-step approach. However, when the number of measurement points is more than 300 points, Fig. 14(b) shows that the variation range of 兩␦˜t兩 of the two-step approach is much smaller than the results of the one-step approach shown in Fig. 8(b). This result implies that the number of measurement points must be large enough to capture the form of the deterministic surface in order to take the advantage of the two-step approach. 4. Conclusions In this part of the paper, we investigate the coordinate variations for the case in which the deterministic error component of the sampled geometric errors is no longer negligible compared to the random error component. Two different approaches are proposed for the estimation of coordinate transformation. The one-step approach directly fits the measurement data with the part model to estimate the coordinate transformation. The two-step approach first uses surface fitting techniques to extract the deterministic surface from the measurement data, and then best-fit the deterministic surface with the nominal part model to estimate the coordinate transformation. Although the computation needed in the two-step coordinate estimation approach is more expensive than that required by the one-step approach, theoretical analysis, computer simulation and experimental results show that the two-step approach can effectively reduce the influence of the deterministic error component on the result of coordinate estimation. Therefore, with the same measurement data, the two-step approach gives a much more accurate coordinate estimation result than the one-step approach. In addition, the variation range of the coordinate transformation parameters obtained by the two-step approach is much reduced. 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. References [1] Z.C. Yan, Geometric tolerance evaluation and uncertainty analysis for coordinate metrology, Ph.D. dissertation, The Ohio State University, Columbus, Ohio, 1994.
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[2] K.C. Sahoo, C.H. Menq, Localization of 3D objects using surface representation and tactile sensing, in: Proceedings of the Symposium on Computer Aided Design and Manufacturing of Dies and Molds, ASME Winter Annual Meeting, Chicago, 28 Nov.–2 Dec., 1988, pp. 105–118. [3] A.B. Forbes, Least-squares best-fit geometric elements, Report DITC 140/89, National Physical Laboratory, UK, 1989. [4] C.H. Menq, H.T. Yau, G.Y. Lai, Automated precision measurement of surface profile in CAD-directed inspection, IEEE Trans. on Robotics and Automation 8 (2) (1992) 268–278. [5] D.F. Morrison, Multivariate Statistical Methods, McGraw-Hill, New York, 1976.