Singularity Issues in Fixture Fault Diagnosis for Multi-Station Assembly Processes

Transcription

1 Yu Ding Abhishek Guta Deartment of Industrial Engineering, Texas A& University, TAU 33 College Station, TX Daniel W. Aley Deartment of Industrial Engineering and anagement Sciences, Northwestern University, Evanston, IL Singularity Issues in Fixture Fault Diagnosis for ulti-station Assembly Processes This aer resents a method of diagnosing variance comonents of rocess error sources in singular manufacturing systems. The singularity roblem is studied and the cause examined in the context of fixture error diagnosis in multi-station assembly rocesses. The singularity roblem results in nondiagnosable fixture errors when standard least-squares (LS) estimation methods are used. This aer suggests a reformulation of the original error roagation model into a covariance relation. The LS criterion is then alied directly to the samle covariance matrix to estimate the variance comonents. Diagnosability conditions for this variance LS estimator are derived, and it is demonstrated that certain singular systems that are not diagnosable using traditional LS methods become diagnosable with the variance LS estimator. odified versions that imrove the accuracy of the variance LS estimator are also resented. The various rocedures are thoroughly contrasted, in terms of accuracy and diagnosability. The results are illustrated with examles from anel assembly, although the alication of the aroach and the conclusions extend to more general discrete-art manufacturing rocesses where fixtures are used to ensure dimensional accuracy of the final roduct. DOI: 0.5/ Introduction Dimensional variation is a major roblem affecting roduct quality in discrete-art manufacturing. In automotive and aerosace industries, for examle, dimensional roblems contribute to roughly two-thirds of all quality-related roblems during new roduct launch,. Dimensional quality of the finished roduct in anel assembly deends largely on the accuracy of the fixtures used to hold arts. Fixture locators are used extensively in multistation assembly rocesses, such as automotive body assembly and aircraft fuselage assembly, in order to rovide art suort and dimensional reference within a given coordinate system, thereby determining the dimensional accuracy of the final assembly,3. Fixture locators may fail to rovide the desired ositioning reeatability relative to tolerances during roduction due to gradual deterioration of locators and/or catastrohic events such as broken locators. Considerable efforts have been made in recent years to diagnose fixture errors sometimes referred to as fixture faults based on roduct dimensional measurements 4 0. In these works, the effects of fixture errors on dimensional measurements are reresented via the linear diagnostic model ytdutvt, t,...,, () where y(t) is the vector of n measured roduct features, u(t) is the vector of error sources, v(t) is the additive noise vector e.g., sensor noise, Dd,d...,d is an n diagnostic matrix linking fixture errors to measurements, t is the observation number index, and is the samle size. Elements in u(t) are assumed indeendent random variables because fixture locators are assumed hysically indeendent. It is normally assumed that the sensor system is such that the elements of the noise vector are indeendent and have equal variance v. Thus, the covariance matrix of v(t) is v I. The matrix D can be determined from the relative ositions of fixture locators and sensors using standard kinematics analyses 4 9. Fixture errors manifest themselves as Contributed by the anufacturing Engineering Division for ublication in the JOURNAL OF ANUFACTURING SCIENCE AND ENGINEERING. anuscrit received Feb Associate Editor: S. Jack Hu. mean shifts and variance in the elements of u(t). Diagnosing fixture errors is equivalent to estimating the mean and variance comonents of u(t) based on the samle of measurement observations y(t) t. The focus of this aer is diagnosis of fixture error variance comonents, as oosed to mean shifts. ost aroaches are based on least-squares LS estimation, with a tyical rocedure as follows. The following rocedure has been slightly modified to accommodate a nonzero mean in y for the original resentation of this rocedure, lease refer to, e.g., Aley and Shi 5: DP Estimate û(t)(d T D) D T (y(t) ȳ), where the samle mean ȳ( t y(t))/; DP Estimate ˆ v (/()(n)) t vˆ(t) T vˆ(t), where vˆ(t)y(t)ȳ Dû(t); DP3 Estimate the variance comonents of u: ˆ i (/() t û i (t) ˆ v (D T D) i,i, where û i (t) reresents the ith element of û(t) and (D T D) i,i is the (i,i)th element of (D T D). The interretation is that we first estimate the random deviations û(t) t, and then use the samle variance of their elements to estimate the variance comonents of u. The quantity ˆ v (D T D) i,i is subtracted out in order to eliminate bias due to measurement noise. Because the deviations û(t) t are directly estimated via LS, this estimator will be referred to as a deviation LS estimator. In order to imlement rocedure DP to DP3 and roduce unique estimates, the following conditions are required: i D T D must be of full rank, or equivalently, the columns of D must be linearly indeendent; and ii n. These conditions are often satisfied for simle single-station assembly rocesses when a sufficient number of sensors are used to measure all degrees of freedom of each workiece. System singularity (D T D singular is often encountered in comlex multi-station assembly rocesses, however, where sensors can only be laced at a downstream station but variation sources are contributed from ustream stations. Singularity is also common in comliant-art assembly rocesses, where there are modes of rigid-body motion and comliant-art deformation. Section rovides an examle of singularity in multi-station assembly. LS modifications using generalized inverses and singular value decomosition,3 and artial least squares 4 have been 00 Õ Vol. 6, FEBRUARY 004 Coyright 004 by ASE Transactions of the ASE

2 develoed to accommodate singularity and ill-conditioning in arameter estimation 5,6. The basic idea behind these aroaches is to transform the columns of D into a smaller set of linearly indeendent basis vectors that san the column sace of D, and then use LS on the reduced-dimenaionality roblem. Although these are aroriate for many arameter estimation roblems, they may lead to erroneous conclusions in fixture diagnosis. The reason is that because the columns of D and the associated variance comonents reresent actual hysical henomena, any reduction in dimensionality and transformation of the columns of D will void its hysical meaning. Rong et al. roosed a artial solution to this roblem that they termed adjusted least squares. They artitioned DD D, where D consists of the columns of D that are linearly indeendent of all other columns. In other words, for any linear combination d d... d that equals zero, the coefficients associated with the columns of D must be zero. Assuming the set of linearly indeendent columns is nonemty, their method will rovide a unique estimate of the subset of variance comonents associated with the columns of D. As in standard least squares, the estimates of the other variance comonents associated with the columns of D ) are nonunique. This aer resents an aroach that can rovide unique estimates of all variance comonents in situations that satisfy certain diagnosability conditions, even if D T D is singular. We demonstrate that the deviation LS estimator ignores imortant information that can be utilized for this urose and derive a diagnosability condition for the new estimator that is more relaxed than the diagnosability condition for the deviation LS estimator. The relationshis between the various estimators are thoroughly discussed. The format of the remainder of the aer is as follows. Section reviews the modeling rocedure for fixture error roagation and exlains the cause of singularity in multi-station models. Section 3 introduces a new variance estimator and two modifications that imrove erformance. Section 3 also derives the diagnosability condition for the new estimator. Section 4 resents several examles of multi-station assembly rocesses corresonding to the situations discussed in Section 3. Section 5 concludes the aer. A Variation odel and Singularity in ulti-station Processes Previous work has develoed a fixture-error roagation model for general multi-station discrete-art manufacturing systems such as rigid-art assembly rocesses 7 9, comliant-art assembly rocesses 0, and machining rocesses,. In this section, we use a simlified two-station anel assembly rocess to illustrate the modeling rocedure, and exlain the cause of system singularity. Details on the modeling rocedure can be found in Jin and Shi 7 and Ding et al. 8. Fixtures in multi-station anel assembly rocesses generally use an n-- layout, consisting of two locating ins and n NC blocks to determine the art/subassembly location and orientation. A tyical 3-- i.e., n3) fixture is shown in Fig.. The two locating ins, P 4way and P way, constrain the three degrees of freedom of a art in the X-Z lane, where the 4-way in restricts art motion in both the X- and Z-directions, and the -way in restricts art motion in the Z-direction. The three NC blocks, NC i, i,, 3, constrain the remaining degrees of freedom of the workiece in the Y-direction. When a workiece is non-rigid, more than three NC blocks may be needed in order to reduce art deformation. For simlicity, this section illustrates with a D examle in the X-Z lane in which the art is rigid. ore general modeling examles that result in the same linear model structure of Eq. can be found in the aforementioned literature 7. In a multi-station rocess, 3-- fixtures are reeatedly used at every station to suort arts/subassemblies. To illustrate, we refer to the following examle throughout the aer. Figure shows a two-station rocess, which is a segment of the simlified automotive body assembly rocess from 0. Three workieces are welded together at Station I. The first workiece consists of two comonents and is a subassembly from the receding assembly oeration. After the welding oeration is finished, the whole assembly is transferred to a dedicated measurement station Station II for insection. This simle two-station segment involves all necessary assembly rocess oerations, including ositioning, joining, transferring, and insection. A full-scale assembly rocess will simly reeat these oerations when fabricating comlex roducts. In this rocess, each art or subassembly consisting of several arts is restrained by a 3-- fixture. Locators being used are marked P P 6 in Fig. note that P and P 6 are used to osition the whole subassembly in Station II. NC blocks are not shown since we are considering a D assembly rocess. The deviations of a 4-way locator in two directions or the deviation of a -way locator in the Z-direction could cause art deviation. They constitute otential fixture errors, numbered 9 on Station I and 0 on Station II, with arrows indicating their deviation directions. In such a D multi-station rocess, each art has three degrees of freedom. We use x i,k to denote the deviation state of art i at station k, x i,k X i,k Z i,k i,k T, () where is the erturbation oerator and is the orientation angle. Thus the state of the roduct, which consists of n (n 4 in this rocess arts, is reresented by T x k x,k T x n,k T, (3) where x i,k 0 if art i has not yet aeared at station k. Fig. Illustration of a 3-- fixture Journal of anufacturing Science and Engineering FEBRUARY 004, Vol. 6 Õ 0

3 Fig. A two-station assembly rocess units in mm Random fixture errors on station k are reresented by u k. Thus, we have u 9 T, and u 0 T, where j is the deviation associated with fixture error j. Nine coordinate sensors, denoted by m through m 9 in Fig. b, are installed in Station II. Each coordinate sensor measures the osition of a art feature e.g. a corner or hole in two orthogonal directions X and Z, so that the total number of measurements is n8. We use y to reresent the ositional deviations detected by sensors at roduct features. In the above rocess, since sensors are only available at Station II, we have y 0 and y m (X) m (Z) m 9 (X) m 9 (Z) T, where m j (XorZ) is the deviation detected at roduct feature j in the X(or Z) direction. For the two-station assembly rocess shown in Fig., the statesace reresentation 0 of the fixture error roagation model becomes x A 0 x 0 B u w x A x B u w (4) y C x v where x 0 reresents the art deviations resulting from the staming rocess rior to the assembly rocess, A x reresents the transformation of the roduct dimensional deviation from Station I to Station II, B k u k reresents the roduct deviations resulting from rocess variations at station k (k,), C characterizes the information regarding sensor locations at Station II, and w k (k,) reresents the higher order terms and other un-modeled rocess errors. Detailed exressions for the A, B, and C matrices in the above equation can be found in 0 and will not be reeated here. The numerical exression for A, which is needed in subsequent analyses, is rovided in Aendix A. In the aforementioned literature 7 on the multi-station error roagation model, the state sace reresentation is commonly adoted to model full-scale assembly and machining rocesses. Given a general N-station system as shown in Fig. 3, the state sace variation model takes the form x k A k x k B k u k w k and y k C k x k v k, k,...,n (5) where the notation corresonds to that in Eq. 4. Note that the subscrit reresents the station index and the observation index t is not exlicitly included. atrices A k, B k, and C k are determined by rocess design and sensor deloyment and C k 0 if no sensor is installed at station k e.g., C 0 in the above examle. The first equation in 5 is called the state transition equation and A k is accordingly called the state transition matrix because A k links x k to x k, the states of an assembly over two stations. In order to exress the state sace variation model in the same format as Eq., we reformulate Eq. 5 into an inut-outut linear model by eliminating all intermediate state variables x k. Assume that x 0 0 and sensors are laced at Station N. We have N N y N k C N N,k B k u k k C N N,k w k v N, (6) where N,k A N A k for Nk and k,k I. Further define and as C N N, B C N N, B C N B N and C N N, C N N, C N. (7) This inut-outut relationshi is of the same form as Eq., yduv, (8) where u T u T u T N w T w T N, D, and the subscrit N a station index is droed from y and v without causing any ambiguity. When the higher order terms and rocess background noise reresented by w k are negligible, i.e., u T u T u T N, Eq. 8 further simlifies to yuv. (9) In this aer, we focus on the model in Eq. 9. The aroaches develoed for Eq. 9 can be easily extended to the model in Eq. 8, because they share the same model structure. In our examle of the two-station assembly rocess, the measurement station Station II is in a well-controlled environment, and we only consider variation sources associated with locators in Station I i.e., fixture errors 9. Thus, u 0, and Eq. 4 becomes Fig. 3 Diagram of a multi-station discrete-art manufacturing rocess 0 Õ Vol. 6, FEBRUARY 004 Transactions of the ASE

4 Fig. 4 ultile ossibilities of fixture errors due to re-orientation y C A B C B u u v C A B u v, (0) where the diagnostic matrix DC A B. The numerical exression for for this two-station assembly rocess is rovided in Aendix A. It can be verified that the columns of are linearly deendent ( T is singular so that the deviation LS estimator is not alicable. The singularity roblem is quite common in multi-station systems, esecially when we include a comrehensive set of fixture errors in the model, and may be unavoidable regardless of how many sensors are used. The simle examle in Fig. 4 illustrates the reasons why. Suose the assembly deviation shown in Fig. 4a is observed at measurement Station k. Any of the Station k fixture error scenarios illustrated in Figs. 4b, 4c, and 4d could have resulted in the assembly deviation in Fig. 4a. Recall that the assembly deviation observed at measurement Station k is related to the fixture errors incurred at the revious station via the model y k C k A k B k u k k u k if the measurement station is free of fixture errors. The receding observation that any of the three fixture error scenarios could have resulted in the same observed assembly deviation means that, given y k, there is no unique solution for u k. athematically, this means that the columns of k are linearly deendent, so that T k k is singular. Because these conclusions clearly hold regardless of how many sensors we add at the measurement Station k, the only way to avoid a nonsingular system in this case is to add sensors to the assembly Station k. If this is not ossible, then the deviation LS estimator cannot be alied. This singularity roblem in multi-station assembly rocesses was also illustrated in the examles resented by Carlson et al. 8. With the fixture errors that are included in the error roagation models develoed in 7,8,0, the diagnostic matrices ( k s are all less than full rank. 3 Variation Diagnosis 3. Variance LS Estimator. Section resented an examle for multi-station assembly in which A k and T are both singular. Thus, the deviation LS estimator outlined in DP DP3 cannot be alied. This section develos an alternative aroach that circumvents this roblem. Taking the covariance matrix for both sides of Eq. 9 gives y u T v I, () where ( ) is the covariance matrix of a random vector. Since fixture errors associated with different fixture locators are assumed hysically indeendent, u diag is diagonal, where is the number of fixture errors included in the model. Define v T as the vector of variance comonents to be estimated. Equation can be written as y i i T i i v I () where i is the ith column vector of. In ractice, the oulation covariance y is estimated by the samle covariance matrix S y ytȳytȳ T y E, (3) t where E denotes the estimation error matrix. If we define V i j T j for i,...,, V I, and v, Eqs. and 3 become S y i V i i E. (4) In light of this, one aroach for estimating the variance comonents is to choose ˆ the symbol denotes an estimate to minimize the sum of the squares of the elements of the error matrix S y i V i ˆ. For square matrices A and B of comatible dimension, define the matrix inner roduct A,Btr(A T B) i and the associated matrix norm A A,A, which is exactly sum of the squares of the elements of A. Using standard results for least squares estimation in inner-roduct saces 3, the estimates in this case must satisfy the so-called normal equations ˆ b, (5) where the notation is as follows. is the Gram matrix, defined so that the ith-row, jth-column element is V i,v j for i, j. The ()-length column vector b is defined so that its ith element is V i,s y. For the articular inner roduct defined above, it can be verified that T T T ] ] T T T T T n and b T S y ] T. (6) S y trs y Journal of anufacturing Science and Engineering FEBRUARY 004, Vol. 6 Õ 03

5 Fig. 5 A three-anel two-station assembly rocess When is nonsingular, or equivalently when the matrices T,..., T, and I are linearly indeendent, ˆ b is a unique solution to Eq. 5. We refer to this aroach as the variance LS estimator. 3. Diagnosability of Deviation LS Estimator and Variance LS Estimator. The diagnosability condition required for a variance LS estimator is different from that required for a deviation LS estimator. For the variance LS estimator, the matrix must be of full rank in order for Eq. 5 to yield a unique solution. For the deviation LS estimator, the matrix T must be of full rank and also n in order to yield a unique solution. To more clearly illustrate the difference, consider the simlified examle shown in Fig. 5, in which each art has only one degree of freedom and can only translate no rotation in the Z-direction. Three locators are used to osition the three anels at Station I, and their instantaneous osition errors are denoted as u 3 T. After the joining oeration are finished, the three arts become one subassembly and it is transferred to Station II for measurement. The state vectors are x k Z,k Z,k Z 3,k T (k,) and the measurement vectors are y 0 and y m (Z) m (Z) m 3 (Z) T. At Station II, the locating hole on art is used to osition the whole assembly. The locator on Station II is assumed to be free of ositioning errors i.e., u 0). When this three-anel assembly is transferred to Station II, it undergoes a translation by the amount Z,, which can be reresented as 0 0 x x Ax. (7) 0 0x 0 x Because y x v at Station II, the state sace model becomes x u, x A x and y x v. The linear diagnostic model for this two-station rocess becomes y A u v. (8) Relating this to the model in Eq., we have DA, where A shown above is clearly singular. However, (9) 0 is of full rank. Consequently, the variance vector can be diagnosed using the variance LS estimator in Eq. 5, but not using the deviation LS estimator. An exlanation for diagnosability of variance vector using the variance LS estimator is aarent from the covariance matrix y v 0 0 v 0 (0) 0 3 v for this single degree of freedom assembly. The diagonal elements in y only rovide information regarding the summation of the fixture error variance comonents and 3, as well as the noise variance v. The non-zero off-diagonal element, which is the covariance between m (z) and m 3 (z), rovides extra information. In Station II, m (z) and m 3 (z) 3, so that cov(m (z),m 3 (z))var( ) recall that and 3 are assumed indeendent. This extra iece of information is utilized by the variance LS estimator so that variance comonents are diagnosable. The discussion so far did not include the fixture error u in Station II. A straightforward extension that includes u would result in the same conclusion. Another way of viewing the difference between these two estimators is the following: a variance LS estimator first calculates covariance matrices of u and y, and then alies the LS criterion on the samle covariance matrices, whereas a deviation LS estimator first calculates the LS estimates for the individual error vectors û(t) t, and then calculated the variances of u from û(t) t. Because estimating û(t) t requires more information than simly estimating its covariance matrix, it is not surrising that the deviation LS estimator requires a stronger diagnosability condition than the variance LS estimator. This is stated in the following theorem, the roof of which is included in Aendix A. Theorem. If T is of full rank and n, then is of full rank. The significance of Theorem is that a unique variance LS estimator exists whenever a unique deviation LS exists. The converse, however, is not necessarily true. As illustrated in the receding examle, there are situations where the variance LS estimator is unique but the deviation LS estimator is not. 3.3 Effect of System Structure odeled by on Variance Estimation. The erformance of the deviation LS estimator will deteriorate for ill-conditioned systems, even if T is not exactly singular. The most common criteria 4 used to quantify how ill-conditioned a system is include tr(( T ) ), cond( T ), and det(( T ) ), where cond and det are the condition number and the determinant of a matrix, resectively. These three measures are related to each other through the eigenvalues of T, which we denote i i. The relationshi is 04 Õ Vol. 6, FEBRUARY 004 Transactions of the ASE

6 tr(( T ) ) is small e.g., less than 0. We may conclude that the deviation LS estimator can outerform a variance LS estimator for a well-conditioned system. For an ill-conditioned system, however, the variance LS estimator will erform substantially better. In the following subsection, we resent a modified version of the variance LS estimator that erforms uniformly better than the deviation LS estimator. Although the erformances of these two estimators will generally differ, the estimators are actually equivalent in the secial case that all columns of are orthogonal, i.e., when i T j 0, i j. This obviously requires that T is of full rank. This is stated as Theorem, the roof of which is included in Aendix A3. Theorem. Ifn, and i T j 0, i j, the variance LS estimator in Eq. 5 is the same as the deviation LS estimator described in DP DP3. tr T i det T i i, i, and cond T max. () min The larger these measures are, the more ill-conditioned the system is. Since T is non-negative definite, all eigenvalues must be non-negative. The system is singular when these measures are infinite, or equivalently, when one or more eigenvalues are exactly zero. Throughout the remainder of the aer, we use tr(( T ) ) as the measure of how ill-conditioned a system is. The following simulations were used to investigate the extent to which both estimators are affected as the system changes from being well-conditioned to being ill-conditioned. The following arameters were used: n6, 3, 50, v 0.5, 3 j j,4,9, and T Fig. 6 SE vs tr T À a a , () where a was varied from to 0. so that the value of tr(( T ) ) changes accordingly. For each a, a onte Carlo simulation with K5,000 relicates was conducted for each estimator. The erformance measure is the mean square error SE of the estimates SE j K K k ˆ j,k j, (3) where ˆ j,k is the estimate of j for the kth relicate. Figure 6 shows the SEs of two estimators vs. the values of tr(( T ) )asais varied. From this, the following observations can be made: a The erformance of the deviation LS estimator deteriorates raidly the SE increases as tr(( T ) ) increases. In contrast, the SE of the variance LS estimator is largely indeendent of tr(( T ) ). Clearly, the variance LS estimator is less sensitive to linear deendencies in system structure. b Although the variance LS estimator erforms better for ill-conditioned systems, the deviation LS estimator has a smaller SE value when 3.4 odified Procedures to Enhance the Performance of the Variance LS Estimator. One observation in the revious section was that the variance LS estimator may erform worse than the deviation LS estimator for a well-conditioned system. This motivates the following algorithm for imroving the erformance of the variance LS estimator. A more general version of the algorithm was originally roosed by Anderson 5 as an aroximate maximum likelihood method. For the resent case, the exressions required in Ste of the algorithm simlify considerably to those shown below. The algorithm iterates over the following two stes until convergence. odified Procedure P Based on the estimate ˆ at the revious iteration, calculate the following estimate of the covariance matrix see Eq. ˆ y i i T i ˆ ˆ i v I Solve the equation *ˆ b* for the new estimate ˆ at the current iteration, where * T ˆ y T ˆ y T ˆ y ] ] T ˆ y T ˆ y T ˆ and y T ˆ y T ˆ y trˆ y b* T ˆ y S y ˆ y ] T ˆ y S y ˆ. y trˆ y S y At the initial iteration, we can use the estimate ˆ from the variance LS algorithm of Eq. 5. Convergence usually occurs after only one or two iterations. In all of the subsequent simulations, only a single iteration was used. The P algorithm bears a strong resemblance to the variance LS algorithm of Eq. 5. Suose we transform Eq. 4 by reand ost-multilying both sides by ˆ y /, which gives ˆ / y S y ˆ / y i ˆ / y V i ˆ / y i ˆ / y Eˆ / y. It is straightforward to verify that * is the Gram matrix of inner roducts for the transformed matrices ˆ y / V i ˆ y / :i,,..., and that the elements of b* are the inner roducts ˆ y / S y ˆ y /,ˆ y / V i ˆ y / for the transformed matrices. Journal of anufacturing Science and Engineering FEBRUARY 004, Vol. 6 Õ 05

7 Fig. 7 The erformance comarison of variance estimators At each iteration, the P algorithm is therefore the least squares solution that minimizes the norm of the transformed error matrix ˆ / y S y ˆ / y i ˆ / y V i ˆ / y ˆ. The transformation by ˆ i / y can be viewed as weighted least squares 6, since the elements of ˆ / y Eˆ / y are uncorrelated with equal variance. Note that the elements of the untransformed error matrix E are neither uncorrelated nor have equal variance. The P algorithm can substantially imrove the erformance over the variance LS estimator, esecially for a well-conditioned system. In fact, Anderson 5 has shown that the P estimator is asymtotically efficient. The imrovement is illustrated via simulation using the same as in Section 3.3 excet that we now use a larger samle size of 00). The simulation results are shown in Fig. 7a, which shows that the P estimator has uniformly smaller SE values than the other estimators. If ˆ y is ositive definite at each ste of the iteration, * will be full rank if and only if is full rank. Thus, the conditions required for the P algorithm to roduce a unique estimate are identical to the conditions required by the variance LS algorithm, rovided that ˆ y remains ositive definite at each iteration. This is not guaranteed, however, because elements of ˆ may take on negative values. This is more likely to occur when the samle size is relatively small and the true variance comonents are close to zero. The negativity of estimates is a roblem in almost all of the variance estimation algorithms the deviation LS estimator cannot avoid negative estimates, either. The most oular aroach to enforce non-negativity is to relace the negative elements in ˆ with zeros. As long as v 0, ˆ y is then guaranteed to be ositive definite. If v becomes negative and is relaced by 0, the seudoinverse of ˆ y can be used in ste. Rao and Kleffe develoed a different variance estimation aroach 7, Eq that avoids negative estimates. Their iterative algorithm will give ositive variance comonent estimates, rovided that the initial estimates are ositive. This rocedure is less intuitive and its develoment is rather mathematically involved. Consequently, we simly resent the final form of the algorithm for ractical use. Note that the form of the algorithm in Rao and Kleffe s Eq is much more comlex than the algorithm below. The reason is that for the articular model in Eq., the exressions simlify considerably to those shown below. odified Procedure P Select an initial ˆ 0 ˆ,0 ˆ,0 ˆ v,0 T with all ositive values. Calculate ˆ y,0 ˆ u,0 T ˆ v,0 I, where ˆ u,0 diagˆ,0 ˆ,0. 3 Solve the following set of linear equations for j0 and. ˆ i, j ˆ i, j trˆ y, j i T i trˆ y, j i T i "ˆ y, j S y, i,...,, and ˆ v, j ˆ v, j trˆ y, j trˆ y, j ˆ y, j S y. (4) The ˆ i, j i and ˆ v, j in P will remain ositive as long as the initial values of ˆ 0 are chosen ositive. The usual choice is to let ˆ i,0, for i,...,. The solution ˆ v,0 ˆ v, ˆ, ˆ, T is the final estimate. The results for the P estimator in the situation described in the receding simulation were also included in Fig. 7. For the relatively small samle size of 5, the P estimator outerforms the other estimators. For the more tyical samle size of 00, however, the P estimator erforms better than the P estimator. The reason is that the P estimator forces a bias in order to make ˆ ositive, and this bias does not disaear as samle size increases. In contrast, the P estimator is unbiased and consistent, meaning that its variance aroaches zero as samle size increases. Consequently, the P estimator is only recommended if samle size is very small. 4 Examles In this section, the estimators are alied to fixture error diagnosis in various automotive body assembly roblems. onte Carlo simulations with 5,000 relicates were conducted in a AT- LAB environment, and fixture errors were assumed to follow a normal distribution in all cases. For detailed descritions of the rocesses, the reader is referred to the various references cited below. For convenience, the diagnostic matrix is rovided below for each situation. 4. Assembly System With an Orthogonal Diagnostic atrix. The automotive assembly rocess was described in considerable detail in Aley and Shi 5. In Section 5 of their aer, they aly the deviation LS estimator to diagnosing errors in fixtures that locate the side frames of a car body. They assumed the linear structured model of Eq. 9 to reresent the effects of fixture errors on dimensional measurements. There were 4 measurements (n4) and two otential fixture errors (). The matrix which is the C matrix in their aer was through kinematics analysis determined to be 06 Õ Vol. 6, FEBRUARY 004 Transactions of the ASE

8 T (5) This matrix is of full column rank and n, suggesting that the variance LS estimator and the deviation LS estimator can both be alied. We also have tr(( T ) ).99, indicating that the system is well-conditioned. For this side-frame assembly system, both deviation LS estimator and variance LS estimator are used to estimate the variance comonents associated with fixture errors. Five different samle sizes were used (5,0,5,50,00) in the simulation. From the SE values shown in Fig. 8, it can be seen that the two estimators have almost identical erformance in this examle. The reason is that the two columns of are almost orthogonal ( T 0.08, T.005, and T ). This agrees with Theorem, which states that the two estimators are equivalent with the columns of are orthogonal. 4. Assembly System With a Non-Orthogonal Diagnostic atrix. any engineering systems do not result in an orthogonal matrix, in which case the erformance of the deviation LS and variance LS estimators will differ. For examle, the matrix used in Section 4 of Aley and Shi 5 is.093 T (6) , the columns of which are not orthogonal. In this case, n9, 3, and tr(( T ) )3.5, imlying the system is relatively well-conditioned. onte Carlo simulations were again conducted, but this time with a samle size of 5. A comarison of the deviation LS estimator, the variance LS estimator, and the P estimator due to the small samle size is shown in Table. The quantity ( i var(ˆ ))/() in the third row reresents the average i samle variance of the estimators for comarison with the SE. We found that the estimator from P demonstrates slightly more bias than the other two, but has smaller disersion. Based on the SE criterion, the P estimator erformed the best, followed by the deviation LS estimator. 4.3 Assembly System With a Singular Diagnostic atrix We next aly the variance estimators to the two-station examle introduced in Section.. The matrix for this model is given in Aendix A. Because the system is singular with tr(( T ) ), the deviation LS estimator cannot be used here. It can be verified that is full rank, so that the variance LS estimator and its modified versions are alicable. As discussed in Section, we only consider fixture errors associated with Station I. Hence 9 and n8. Simulations were carried out using a samle size of 00. The variance LS estimator of Eq. 5 and the P and P estimators were comared in this examle, and the results are shown in Table. In this examle, the variance LS estimator and P estimator erform comarably, although the latter has slightly smaller SE and disersion. This is consistent with the results shown in Fig. 7a as tr(( T ) ) increases. The P rocedure has the smallest SE and disersion among the three. But it also has quite noticeable bias. Although the P estimator outerformed the P estimator in this examle, our exerience indicates this is more an excetion than the norm. For examle, in Fig. 7a, the P has a smaller SE value. As another examle, suose we modify the twostation assembly examle considered in the receding aragrah so that we are now only interested in diagnosing the Z-direction fixture errors. The matrix in this case is the same as the matrix given in Aendix A, excet that we remove column, 4, and 7. It can be verified that that the new matrix is also less than full rank. Reeating the above simulations but with the new matrix, the SEs for the P and P estimators are.97 and.5. Thus, the P estimator is slightly more effective than the P estimator in this case. Table Comarison of three estimators for the linear system with as in Eq. 6 Deviation LS estimator ˆ Variance LS estimator P Fig. 8 SE for the linear system with as in Eq. 5 i varˆ i SE The true value of used in the simulation is,4,4,. Journal of anufacturing Science and Engineering FEBRUARY 004, Vol. 6 Õ 07

9 Table Comarison of three estimators for the linear system with a singular Variance LS estimator P P ˆ i varˆ i SE The true values of used in the simulation is Concluding Remarks Singularity is a common roblem in engineering systems, in which case the traditional least-squares estimation method cannot be alied effectively. This aer resents a new diagnosability condition and a variance LS estimator that takes into account the covariance between error terms and results in diagnosability for systems that are not diagnosable using traditional LS methods. Two modified versions of the algorithm were also resented to imrove the erformance of the variance LS estimator. We note that the resented methods tyically require a random samle of 5 50 units. For a dynamic rocess with tool wear, the rocess data are inherently autocorrelated. However, since 5 50 units tyically translates to roduction eriods of one hour or less, the samling eriod will generally be too small to observe any noticeable tool wear effects. Consequently, the methods should still be alicable to diagnosing other tyes of fixture errors in rocesses that also exerience relatively slow tool wear dynamics although other methods would be required to diagnose the tool wear itself. For rocesses with faster tool wear dynamics, recursive estimation methods would need to be develoed. We also oint out that the methods are for variance comonent estimation, as oosed to mean comonent estimation. Our exerience has been that the autobody industry views fixture error variance as more roblematic than mean shifts. A sustained, consistent deviation from nominal i.e., a mean shift can often be comensated quite easily by rocess engineers via shimming and other adjustments. In contrast, variation is much more difficult to comensate and requires either some form of on-line feedback control or the removal of the variation root cause. The methods resented in this aer are intended to be a tool to aid in detecting, identifying, and, ultimately, eliminating root causes of random variation. The examles in this study have been exclusively for fixture error diagnosis in multi-station assembly rocesses. However, all of the results and conclusions should also hold for other tyes of error sources and multi-station manufacturing rocesses, rovided that the linear structured model adequately reresents the effects of the error sources on the rocess and roduct measurements. Acknowledgment This research was artially suorted by the NSF grants DI and DI The authors also gratefully acknowledge the valuable comments and suggestions from the associate editor and referees. Aendices A Exression of atrices for Examle in Section m l A Õ Vol. 6, FEBRUARY 004 Transactions of the ASE

10 l m A Proof of Theorem. Suose T is of full rank and n, but that is singular. Because is the Gram matrix of T, T,..., T,I, its singularity imlies that T, T,..., T,I are linearly deendent. Thus, there exists a set of scalars,,...,,, not all zero, such that T T... T I. In order for this to hold, we must have 0. Otherwise, rank(i)n, whereas the summation of matrices on the left hand side can have at most rank n. It follows that T T... T 0, and at least one of the s say i ) is nonzero. Postmultilying the receding equation by i gives ( T i ) ( T i )... ( T i )0. Because at least one of the coefficients ( T i i i ) is nonzero, this imlies that the vectors,,..., are linearly deendent. Their Gram matrix T must therefore be singular, which contradicts the condition that T is full rank. A3 Proof of Theorem. Utilizing the fact that the columns in are orthogonal to each other, i.e., T i j 0, i j, we can re-write Eq. 5 as T 0 T ˆ ] ] ] ] ] 0 T T ˆ tr T S y ] T. S y T T n ˆ v trs y (a) The above equation is equivalent to i i T i ˆ T i i i ˆ v tr i T i S y, i,,..., T i i ˆ n ˆ i v trs y.. (a) We can solve ˆ i i in terms of ˆ v from the first equation and substitute it into the second equation. Then, we have nˆ v trs y i tr T i T i S y i i trs y tr S i y. i T i i i T (a3) Notice that T is a diagonal matrix with T i i as its (i,i)th element and i i i T i T, where diag and i, i,...,, is an arbitrary real number. Then, we have i T i T i T T. i i Given all these results, we can write Eq. a3 as (a4) ˆ v n tri S y. (a5) It can be further shown that this ˆ v is the same as the one in the deviation LS estimator. Substitute vˆ(t)y(t)ȳû(t) and û(t) (y(t)ȳ) into ˆ v ( t vˆ(t) T vˆ(t))/(()(n)). It turns out that ˆ v n ytȳ T I T I ytȳ t n tr ytȳ T I t ytȳ n T tr I ytȳytȳ t n tri S y. (a6) After obtaining the solution of ˆ v, we can substitute it into a to solve for ˆ as i Journal of anufacturing Science and Engineering FEBRUARY 004, Vol. 6 Õ 09

11 ˆ i T i i tr i T i S y ˆ v, i,,...,. (a7) T i i Recall that tr( i T i S y ) T i S y i and / T i i is the (i,i) element of ( T ). Then, T i / T i i is the ith row of ( T ) T. We can further write a7 as ˆ i T i S i T y ˆ i i T v i i T i S y i T ˆ v T i,i, i i i,,...,, (a8) where i is the ith row of ( T ) T. The second term ˆ v ( T ) i,i in the right hand side of the above equation is the same as the one in DP3 in Section. We shall show that t û i (t) /() is the same as i S y ( i ) T. In fact, û i (t) (( T ) T ) i (y(t)ȳ) i (y(t)ȳ). Then, t û i t i ytȳ i ytȳ T t i t i S y i T. This comletes the roof. References ytȳytȳ T i T (a9) Shalon, D., Gossard, D., Ulrich, K., and Fitzatrick, D., 99, Reresenting Geometric Variations in Comlex Structural Assemblies on CAD Systems, Proceedings of the 9th Annual ASE Advances in Design Automation Conference, DE-Vol. 44-,. 3. Ceglarek, D., and Shi, J., 995, Dimensional Variation Reduction for Automotive Body Assembly, anufacturing Review, 8, Cunningham, T. W., antriragada, R., Lee, D. J., Thornton, A. C., and Whitney D. E., 996, Definition, Analysis, and Planning of a Flexible Assembly Process, Proceedings of the 996 Jaan/USA Symosium on Flexible Automation, Vol., Ceglarek, D., and Shi, J., 996, Fixture Failure Diagnosis for the Autobody Assembly Using Pattern Recognition, ASE J. Ind., 8, Aley, D. W., and Shi, J., 998, Diagnosis of ultile Fixture Faults in Panel Assembly, ASE J. anuf. Sci. Eng., 0, Chang,., and Gossard, D. C., 998, Comutational ethod for Diagnosis of Variation-Related Assembly Problem, Int. J. Prod. Res., 36, Rong, Q., Ceglarek, D., and Shi, J., 000, Dimensional Fault Diagnosis for Comliant Beam Structure Assemblies, ASE J. anuf. Sci. Eng.,, Carlson, J. S., Lindkvist, L., and Soderberg, R., 000, ulti-fixture Assembly System Diagnosis Based on Part and Subassembly easurement Data, Proceedings of the 000 ASE Design Engineering Technical Conference, Setember 0 3, Baltimore, D. 9 Ding, Y., Ceglarek, D., and Shi, J., 00, Fault Diagnosis of ulti-station anufacturing Processes by Using State Sace Aroach, ASE J. anuf. Sci. Eng., 4, Ding, Y., Shi, J., and Ceglarek, D., 00, Diagnosability Analysis of ulti- Station anufacturing Processes, ASE J. Dyn. Syst., eas., Control, 4,. 3. Rong, Q., Shi, J., and Ceglarek, D., 00, Adjusted Least Squares Aroach for Diagnosis of Ill-Conditioned Comliant Assemblies, ASE J. anuf. Sci. Eng., 3, Schott, J. R., 997, atrix Analysis for Statistics, John Wiley & Sons, New York. 3 Golub, G. H., and Van Loan, C. F., 996, atrix Comutations, 3rd ed., The Johns Hokins University Press, Baltimore, D. 4 Geladi, P., and Kowalski, B. R., 986, Partial Least-Squares Regression: A Tutorial, Anal. Chim. Acta, 85, Beck, J. V., and Arnold, K. J., 977, Parameter Estimation in Engineering and Science, John Wiley & Sons, New York. 6 Hasan, W.., and Viloa, E., 997, Use of the Singular Value Decomosition ethod to Detect Ill-Conditioning of Structural Identification Problems, Comut. Struct., 63, Jin, J., and Shi, J., 999, State Sace odeling of Sheet etal Assembly for Dimensional Control, ASE J. anuf. Sci. Eng.,, Ding, Y., Ceglarek, D., and Shi, J., 000, odeling and Diagnosis of ulti- Station anufacturing Processes: Part I State Sace odel, Proceedings of the 000 Jaan/USA Symosium on Flexible Automation, July 3 6, Ann Arbor, I, 000JUSFA antriragada, R., and Whitney, D. E., 999, odeling and Controlling Variation Proagation in echanical Assemblies Using State Transition odels, IEEE Trans. Rob. Autom., 5, Camelio, A. J., Hu, S. J., and Ceglarek, D. J., 00, odeling Variation Proagation of ulti-station Assembly Systems With Comliant Parts, Proceedings of the 00 ASE Design Engineering Technical Conferences, Setember 9, Pittsburgh, PA. Djurdjanovic, D., and Ni, J., 00, Linear State Sace odeling of Dimensional achining Errors, Transactions of NARI/SE, XXIX, Zhou, S., Huang, Q., and Shi, J., 003, State Sace odeling for Dimensional onitoring of ultistage achining Process Using Differential otion Vector, IEEE Trans. Rob. Autom., 9, Luenberger, D. G., 968, Otimization by Vector Sace ethods, Wiley, New York. 4 Pukelsheim, F., 993, Otimal Design of Exeriments, John Wiley and Sons, New York. 5 Anderson, T. W., 973, Asymtotic Efficient Estimation of Covariance atrices With Linear Structure, Annals of Statistics,, Neter, J., Hutner,. H., Nachtsheim, C. J., and Wasserman, W., 996, Alied Linear Statistical odels, 4th ed., cgraw Hill-Irwin, Chicago, IL. 7 Rao, C. R., and Kleffe, J., 988, Estimation of Variance Comonents and Alications, North-Holland, Amsterdam, Õ Vol. 6, FEBRUARY 004 Transactions of the ASE