CONTEMPORARY multistation manufacturing systems,

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1 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER Stream-of-Variation (SoV)-Based Measurement Scheme Analysis in Multistation Machining Systems Dragan Djurdjanovic and Jun Ni Abstract Today, machining systems are complex multistation manufacturing systems that involve a large number of machining operations and several locating datum changes. Dimensional errors introduced at each machining operation get transformed and cause the occurrence of new errors as the workpiece propagates through the machining system. The appropriate choice of measurements in such a complex system is crucial for the subsequent successful identification of the root causes of machining errors hidden in dimensional measurements of the workpiece. In order to facilitate this measurement selection process, methods for quantitative characterization of measurement schemes must be developed. This problem of quantitative measurement characterization referred to as the measurement scheme analysis problem is dealt with in this paper. The measurement scheme analysis is accomplished through characterization of the maximal achievable accuracy of estimation of process-level parameters based on the measurements in a given measurement scheme. The stream of variation methodology is employed to establish a connection between the process-level parameters and measured product quality. Both the Bayesian and non-bayesian assumptions in the estimation are considered and several analytical properties are derived. The properties of the newly derived measurement scheme analysis methods are demonstrated in measurement scheme characterization in the multistation machining system used for machining of an automotive cylinder head. Note to Practitioners Techniques for the measurement scheme analysis presented in this paper enable one to formally and systematically evaluate the amount of information content a given combination of measurements carries about the process-level faults that cause quality problems in machining. Such techniques can be used to optimally select measurements that are the most informative about the machining process before a machining line is even built. Since the basis of the measurement scheme analysis methods presented in this paper is the model of the flow of machining errors referred to as the stream-of-variation (SoV) model, which can currently model only the influence of fixture and tool path parameter errors on the workpiece quality, the results presented in this paper only evaluate the amount of information about those error sources. However, as improvements to the SoV models are made in the future, the measurement scheme analysis methods and their properties demonstrated in this paper will remain valid as long as the newly obtained modes are in the linear state-space form possessed by the existing SoV models. Index Terms Measurement scheme analysis, multistation machining, nonexistent or uncertain statistical knowledge, stream of variation methodologies, worst-case maximum-likelihood (WCML) estimation. Manuscript received December 13, 2004; revised June 20, This work was supported by the Engineering Research Center for Reconfigurable Manufacturing Systems at the University of Michigan, Ann Arbor (NSF Grant EEC ). This paper was recommended for publication by Associate Editor Y. Narahari and Editor P. Ferreira upon evaluation of the reviewers comments. The authors are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI USA ( ddjurdja@umich.edu). Digital Object Identifier /TASE I. INTRODUCTION CONTEMPORARY multistation manufacturing systems, such as multistation machining or assembly lines, involve a large number of machining/assembly operations and locating datum changes. The identification and reduction of the root causes of dimensional errors in such processes are among the most influential problems during the ramp-up of new manufacturing lines [1] [4], as well as during full-scale production. It is therefore essential to devise methods to formally and quantitatively assess the amount of information that a given set of measurements carries about the root causes of dimensional errors, as different combinations of measurements in a manufacturing system carry different amounts of information on the parameters that are being estimated based on those measurements. These quantitative measurement characterization methods would, in turn, enable one to systematically choose combinations of measurements that carry the most information about the root causes of dimensional errors. The purpose of this paper is to develop formal methods for the characterization of any given combination of measurements in a multistation machining line through the quantitative description of how well those measurements mirror the root causes of dimensional errors in machining. In the remainder of this manuscript, any given combination of measurements taken in the machining line (not necessarily only at the end of the line) will be referred to as the measurement scheme, while the problem of quantitative characterization of measurement schemes will be referred to as the measurement scheme analysis problem. The introduction of the linear state-space models connecting dimensional machining errors with errors in the fixture and tool-path parameters in [4] [7] yielded a quantitative link connecting measurements taken in the multistation machining with the process-level parameters affecting those measurements. Following [8], where it was first suggested to explicitly model the flow and transformation of dimensional errors in assembly lines from one station to the next, thus yielding a model in the state-space form, the linear state-space models from [5] [7] will be referred to as the stream-of-variation (SoV) models of dimensional machining errors. The linear state-space form of models pursued in [5] [7] allowed the use of the wellestablished statistical and control theory tools to accomplish the estimation of statistical properties of process-level parameters based on the in-process and end-of-line measurements of the workpiece quality [9] [11]. This connection between process-level parameters and measured quality of the product can be used to allocate tolerances to variations in process parameters, such as fixture locator positions and in the case of machining also tool-path parameters [12], [13]. In addition, /$ IEEE

2 408 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 maximal achievable accuracy of the estimation of process-level parameters based on the in-process and end-of-line measurements can be utilized to describe how informative a given set of measurements is about the process parameters and, thus, quantitatively evaluate the goodness of that set of measurements. Measurement scheme analysis executed through the evaluation of the goodness of a set of measurements based on the corresponding achievable accuracy of process parameter estimation will be the topic of this paper. The rest of this paper is organized as follows. Section II gives a review of the literature addressing the characterization of measurements based on SoV modeling of the error flow in multistation manufacturing systems. Section III briefly describes the recently developed linear-state space modeling of dimensional errors in multistation machining processes, which serves as the basis for the measurement scheme analysis methods introduced in this paper. Section IV restates definitions and properties of the methods used in the Bayesian approach to the measurement scheme analysis in multistation machining systems. These methods were originally introduced in [14], and more details about the topic of Bayesian approach to measurement scheme analysis can be found in that paper. In Section V, methods for the non-bayesian approach to the measurement scheme analysis in multistation machining systems are introduced and their properties are discussed. Section VI gives a numerical demonstration of the non-bayesian measurement scheme analysis methods applied in the characterization of measurement schemes in the process used for machining of an automotive cylinder head. 1 The results presented in this paper are discussed in Section VII, while Section VIII offers the conclusions of the work presented in this paper and guidelines for future work. II. LITERATURE REVIEW The problem of quantitatively characterizing measurements in an assembly system was addressed in [15] [17]. The authors used the pairwise distinction between individual fault signatures to characterize each measurement scheme. Characterizing measurement schemes based on the pairwise distance between individual fault signatures is not the best approach for the characterization of measurements in the presence of multiple faults, since signatures from a group of linearly dependent faults can be pairwise very distant while, at the same time, one cannot uniquely decompose a given compound fault signature onto individual signatures within that group due to their linear dependence. Hence, this approach was applicable only to cases when single faults occur in the system. A more appropriate characterization of measurements can be accomplished using the linear state space model of errors in multistation manufacturing systems. In [18], the notion of measurement scheme diagnosability for multistation machining systems was introduced based on the rank of the regression matrix in the SoV-based linear model connecting the measured dimensional errors with their root causes. In [19] and [20], the concept of diagnosability of measurements in multistation manufacturing systems was studied more thoroughly and the notions 1 A numerical demonstration of the Bayesian measurement scheme analysis methods applied in the same machining process can be found in [14]. of within-station, between-station and overall process diagnosability [19], as well as the mean and variance diagnosability [20], were introduced. In addition, the process diagnosability index was introduced in [19] as the percentage of independent equations describing the measurements, versus the total number of possible faults in a multistation manufacturing process. This ad-hoc index is proposed as the quantitative measure describing the measurements in a given manufacturing system. Nevertheless, the rank-based criteria just convey the message of one s ability or inability to uniquely estimate the root causes of machining errors as a solution to the system of linear equations determined by a given set of measurements. In essence, the rank-based criteria are able to signal the existence or nonexistence of the root cause estimates based on a given set of measurements (and in the case of the diagnosability indices, also how close we are to having, or not having, the ability to uniquely estimate root causes of manufacturing errors), but are not able to quantify how good (accurate) those estimates are. For example, a set of measurements that is connected to the quality root causes through a system of a poorly conditioned system of linear equations of full rank is readily diagnosable, but would result in a highly uncertain (and, therefore, untrustworthy) estimate of the root causes, while diagnosability indices would not be able to signal such a poor selection of measurements. A more appropriate and more distinctive quantitative description of a measurement scheme can be based on its information content, which itself is mirrored in the amount of uncertainty related to the root cause estimation based on that measurement scheme. This approach allows a finer distinction among measurement schemes and is adopted in this paper. In [14] and [18], properties of the covariance matrix of the root-cause estimation error for the linear least-square estimator (LLSE) are used to quantitatively describe measurement schemes in multistation machining systems (for more on LLSE estimators, one may refer to any textbook on estimation theory, such as [21], or [22], and references therein). This Bayesian approach to measurement scheme analysis in multistation machining systems allows the integration of the a priori knowledge of statistical characteristics of the machining error root causes and sensor readings into root cause identification and quantitative measurement scheme characterization. Nevertheless, the Bayesian approach also requires this a priori statistical knowledge to exist in order for it to be applicable, which is one of the main reasons why it is often disputed among statisticians [21, p. 13]. Despite the series of recent standards on performance parameters assessment for computer numerically controlled (CNC) machining centers and lathes [23] [28], statistical properties of the root causes of dimensional machining errors may be available only in the latter stages of manufacturing, and are most likely not available during the ramp up and in the early stages of production. This is why one must acknowledge and address the need for measurement scheme analysis methods that are not based on the Bayesian approach in statistics. In this paper, we present the non-bayesian methods for measurement scheme analysis in multistation machining systems. First, measurement schemes analysis methods will be introduced to handle the case when only statistical characteristics of the measurement noise are known. These methods are based on

3 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 409 TABLE I REVIEW OF THE RESEARCH PERTAINING TO SYSTEMATIC CHARACTERIZATION OF MEASUREMENT SCHEMES BASED ON SoV MODEL OF THE FLOW OF DIMENSIONAL ERRORS IN MULTISTATION MANUFACTURING the well-known least-square estimation technique and characterization of the corresponding Fischer information matrix [29]. Second, the measurement scheme analysis methods dealing with inaccurate or possibly nonexistent knowledge of the statistical characteristics of the sensor noise and root causes of dimensional machining errors will be presented and discussed. These methods are based on the recently developed worst-case maximum-likelihood (WCML) estimation [30] [32]. Methods for quantitative assessment of how well different combinations of measurements mirror the process-level root causes of machining errors could ultimately be used to define the selection (synthesis) of measurements in a machining system as a formal, systematic optimization procedure, where the objective function for the optimization would be defined based on the measures of informativeness of any measurement scheme provided by the measurement scheme analysis tools. The topic of using the measurement scheme analysis tools to facilitate the optimal selection of measurements (measurement scheme synthesis) is out of the scope of this paper and is discussed thoroughly in [33]. Table I gives an overview of the work relevant to the systematic measurement scheme characterization based on the SoV model of the flow of dimensional errors in multistation manufacturing, and outlines the relative position of the problems dealt with in this paper relative to the prior work. III. STREAM OF VARIATION MODELING OF DIMENSIONAL MACHINING ERRORS Machining processes today are complex multistation processes, involving a large number of operations and several datum changes. In [5] [7], the propagation of dimensional errors in multistation machining systems was described through SoV models in the linear state-space form (1) where denotes the machining operation number, denotes the workpiece dimensional errors accumulated up to machining operation, describes dimensional errors introduced at operation, denotes the measured dimensional errors after operation, is the plant noise present due to the linearization errors and nonmodeled effects, and is the noise term present due to the linearization effects and sensor noise. The model of the linear state-space form (1) describes how machining errors are introduced, transformed, and accumulated as a workpiece is being machined in a multistation machining system. For each operation, matrix describes how machining errors accumulated up to and including operation are transformed and influence errors in machining operation, while matrix describes how new errors are introduced into the workpiece at operation. Matrix connects errors of the workpiece computer-aided-design (CAD) parameters to the measured errors, while matrix, which exists only in the case of on-machine measurements [5], [6], describes how machining errors introduced in machining operation directly influence the measured errors. In the existing SoV models of machining processes presented in [4] [7], each workpiece feature was described by its position, orientation, and a set of scalar features, such as the cylinder diameter, hole diameter and depth, slot width and depth, etc. The feature positions and orientations were expressed in a coordinate system determined by some workpiece measurement datum features, and the state vector from the model (1) comprised of errors in positions, orientations, and scalar parameters of each workpiece feature after machining operation.for each machining operation, the input vector contains errors in fixture parameters and errors in parameters of the newly machined surfaces at that machining operation. Such errors in orientation, position, and scalar parameters (diameters of holes, slot depths, etc.) of the newly machined surfaces could occur due to thermal effects, tool wear, or other error causes. Nevertheless, the existing SoV models of multistation machining are not able to represent the dependency of error parameters of the newly machined surfaces on other process parameters,

4 410 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 such as tool wear or thermal errors. Further identification of tool-path-related errors would have to be accomplished through other methods, such as thermal error, or tool wear monitoring. Following model (1), machining errors in a machining system represented using the linear state-space model (1) are introduced at each operation,, through nonzero elements of vectors. Now, the special form of model (1) can be used to estimate elements of vectors,, using measured errors, and, thus, accomplish identification of root causes of dimensional machining errors based on the measurements of the machined workpiece. Simple manipulations of (1) yield where and Matrices are the discretetime state transition matrices [34], satisfying Introducing the notation and viewing for for as the noise term, (2) becomes a linear model connecting the measured dimensional errors of the workpiece with the root causes of dimensional machining errors. Linear model (4) can now be used to estimate the elements of the root cause vector based on the measured errors of the machined workpiece. Furthermore, maximal achievable accuracy of that estimation can be utilized to characterize how much information about the root causes, (2) (3) (4) is reflected by any given combination of measurements, and, thus, evaluate the goodness of that combination of measurements. Evaluating this information content embedded in a given set of measurements is the main topic of this paper and will be discussed in the forthcoming sections. The notion of measurement scheme diagnosability should be mentioned prior to embarking into elaboration about measurement scheme analysis procedures. The diagnosability of a measurement scheme was previously introduced in [18] as the property that enables one to uniquely decompose each pattern of measured dimensional machining errors into root causes of those errors. Based on the model (4), one can formally define the notion of diagnosable measurement schemes. Definition 1: Measurements scheme is diagnosable if where is the number of root causes of dimensional machining errors modeled by the SoV model. 2 Measurement scheme diagnosability is a property that is analogous to the output reachability concept from the theory of discrete-time linear time-varying systems, as described in [34] (the concept that pertains to the relation between inputs and outputs of the system). Diagnosability was thoroughly studied in [19] and [20] and was utilized in [19] for quantitative characterization of measurements (i.e., for formal measurement scheme analysis). Nevertheless, as mentioned in the Introduction section, the regression-matrix rank-based concept of diagnosability is essentially a binary concept that indicates whether a system of linear equations can or cannot be uniquely solved. The following two sections of this paper deal with the problem of nonbinary characterization of measurements based on the error of estimation of process parameters from the measurements, rather than on the regression matrix rank as is the case with measurement diagnosability, even though diagnosability will have a profound effect on a number of properties of measurement scheme analysis methods that will be introduced. IV. BAYESIAN APPROACH TO MEASUREMENT SCHEME ANALYSIS In this section, the definitions and properties of the methods used in the Bayesian approach to measurement scheme analysis will be briefly reviewed. These methods have been introduced in [14] and more details on the Bayesian approach to measurement scheme analysis in multistation machining systems can be found in that paper. A. Methods for Measurement Scheme Analysis Based on the Bayesian Statistical Approach In order to facilitate the Bayesian approach to root cause identification and measurement scheme analysis in multistation machining systems, the following assumptions need to be made: Assumption 1. The noise process and root causes are independent. 2 The number of root causes of dimensional machining errors modeled by the SoV model is equal to the dimension of the vector U, or the number of rows in the regression matrix T of the linear model (4).

5 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 411 Assumption 2. The noise term is characterized as a zero mean process with the known covariance matrix. Assumption 3. Root causes are characterized by a random process with mean (expected) values given by vector and the known covariance matrix. Under these assumptions, the best LLSE of the root causes, given the measurements and the model (3), is described as [21] (5) while the covariance matrix of the corresponding estimation error is where denotes the mathematical expectation operator. The subscript emphasizes that is the covariance matrix of the root cause estimation error, for the LLSE estimation based on the measurement scheme and relationship (4). Assumption 1 that the estimated quantity and measurement noise processes are independent is a standard assumption in any estimation from noisy measurements. Assumption 2 states that statistical properties of the noise term in the linear model (4) are needed for the LLSE estimation, which is readily available from measurement noise characteristics (for example, accuracy of the coordinate measurement machine used for measurements). The LLSE estimation is essentially a Bayesian estimation method because of the need to have Assumption 3, stating the need to have a priori knowledge of the mean value vector and covariance matrix of the random vector that is being estimated [21, p. 13]. In the case of estimation considered in this paper, the estimated quantity is the vector of the root causes of dimensional machining errors, and the assumption that the mean and covariance matrix of the root causes are a priori known (Assumption 3) makes it possible to use an essentially Bayesian estimation method such as the LLSE estimation. The Bayesian character of the LLSE estimation method based on which measurement scheme characterization methods will be defined in this section is the reason why we refer to the resulting measurement scheme characterization approach as the Bayesian approach to measurement scheme analysis. Quantitative description of all measurement schemes will be introduced using the normalized covariance matrix of the LLSE root cause estimation error, given as (6) where denotes the total number of modeled root causes, denotes the unity matrix in the space, and and are such that ;. Matrices and exist because matrix is a covariance matrix. 3 3 It is assumed that all root causes are linearly independent and that therefore the matrix K is invertible. If linearly dependent root causes exist, one of them can be represented as the linear combination of the others and can be omitted, thus yielding a set of linearly independent root causes. Loosely speaking, larger elements in the matrix would signify more uncertainty in the root cause estimation and, hence, a less informative measurement scheme. Conversely, smaller elements in would signify less uncertainty in the root cause estimation and, hence, a more informative measurement scheme. It should be noted that prior knowledge of the mean values vector of the process describing machining error root causes does not participate in the formulation (6) of the normalized covariance matrix. Therefore, the change of the expected values of the machining error root causes, which could be caused by tool wear, thermal drift, or some other processes, will not alter the normalized covariance matrix and will not affect the measurement scheme analysis based on. The following definition allows more formal and mathematically rigid measurement scheme analysis and comparison. Definition 2: The trace-based LLSE root cause estimation uncertainty of a measurement scheme, labeled as,isdefined as the trace of the corresponding normalized LLSE root cause estimation error covariance matrix, given by (6). The norm-based LLSE root cause estimation uncertainty of a measurement scheme, labeled as,isdefined as the operator norm (maximum eigenvalue) of the corresponding normalized LLSE root cause estimation error covariance matrix, given by (6). The determinant-based LLSE root cause estimation uncertainty of a measurement scheme, labeled as,isdefined as the determinant of the corresponding normalized LLSE root cause estimation error covariance matrix, given by (6). Definition 3: Following Definition 2, two measurement schemes and can be compared according to the ordering defined as or ordering or ordering defined as defined as Superscripts (A), (E), and (D) are used in labeling the trace-based, norm-based, and determinant-based LLSE root cause estimation uncertainties of measurement schemes, respectively. This labeling is in accordance with [35], where the authors refer to optimizing the trace, norm, and determinant of the Fischer information matrix in the least-square estimation problems [29], as the A-optimality, E-optimality, and D-optimality, respectively. Since for each measurement scheme there exists a unique covariance matrix of the LLSE root cause

6 412 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 estimation error, relations,, and can be used to compare any two measurement schemes, with smaller root cause estimation uncertainty implying a better, (more informative ) measurement scheme. Furthermore, for any measurement scheme as, due to the positive semi-definiteness of the matrix, all of its eigenvalues are greater, or equal to zero. The following theorems summarize the properties of the measurement scheme characterization methods defined by definitions 2 and 3. Theorem 1: Smaller implies a smaller upper bound on all elements of the matrix. Furthermore Similarly, smaller implies a smaller upper bound on all of the elements of the matrix and Theorem 2: Let us observe measurement schemes and such that all of the measurements from the scheme are contained within the measurement scheme. Then Theorem 3: For each measurement scheme, we have that, where denotes the norm of a matrix. Furthermore, if the measurement scheme is not diagnosable, then. Theorem 1 means that smaller values of or imply less uncertainty in the estimation of the root causes of dimensional machining errors. Furthermore, if or, there is no uncertainty in the root cause estimation since the covariance matrix of the estimation error is zero. One cannot make similar statements about the determinant-based root cause estimation uncertainty, as it can be equal to zero even when (i.e., when there is uncertainty in the LLSE root cause estimation). A simple interpretation of Theorem 2 is that more measurements should convey more information about the root causes of dimensional machining errors, regardless of what property of the normalized covariance matrix of the LLSE root cause estimation error was used for measurement scheme analysis. This occurs because statistical properties of the measurement noise term are assumed to be known and the LLSE given by (5) ensures that estimation is not contaminated by the noise coming from additional measurements. Namely, measurements with higher noise content are being weighed by smaller weighing coefficients in the LLSE. Theorem 3 shows that all of the nondiagnosable measurement schemes have the same norm-based LLSE root cause estimation uncertainty. This property may not be desirable when one wishes to make a finer distinction among nondiagnosable measurement schemes. Trace-based and determinant-based LLSE root cause estimation uncertainties do not demonstrate this property. Proofs of the properties stated by Theorems 1, 2, and 3 can be found in [14]. B. Discussion on the Methods for Bayesian Approach to Measurement Scheme Analysis The analysis methods presented above are based on Assumptions 1 3 and allow the optimal integration of the a priori statistical knowledge into the root cause estimation and measurement scheme characterization. The normalized estimation error covariance matrix given by (6) describes how accurate root cause estimation based on a given set of measurements is and is used to describe the goodness of that set of measurements. The matrix property used to describe those matrices in measurement scheme synthesis should be able to give a good representation of the overall magnitude of all matrix elements (the smaller those elements are, the less root cause estimation uncertainty there is and, hence, the better the measurements scheme is). The normalized estimation error covariance matrix given by (6) is a covariance matrix and is therefore a positive definite matrix characterized by an ellipsoid with axes in the directions of the eigenvectors of that matrix and axis lengths equal to the corresponding matrix eigenvalues. The operator norm of such a matrix would be equal to the length of the largest axis of its ellipsoid, and even though other axes of the matrix ellipsoid could be small, its operator norm of the matrix will still be large (operator norm takes into account only the largest eigenvalue of the matrix and neglects all other eigenvalues [46]). On the other hand, the determinant takes into account lengths of all axes of the ellipsoid corresponding to the symmetric semidefinite matrix. The aggregate measure is created as a product of lengths of those axes (product of matrix eigenvalues), which is a measure of the ellipsoid volume. Such an aggregate measure enables one small ellipsoid axis to cover up the other potentially large axes (basically, if one of the eigenvalues is zero, which means that the length of one of the ellipsoid axes is zero, then the corresponding determinant will be zero, regardless of other eigenvalues [46]). Finally, the trace of a symmetric positive semidefinite matrix is the sum of lengths of all axes of its ellipsoid. Such an aggregate measure is equal to the sum of all matrix eigenvalues and can be seen as the Frobenius norm (sum of the squares of all matrix elements) of the matrix [46], offering a good overall representation of all elements of. This is the reason why the trace of the matrix is used in [33] to depict the overall magnitude of the matrix elements and, thus, assess the measurement

7 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 413 scheme fitness in an attempt to select combinations of measurements that are the most informative about the root causes of dimensional machining errors. This problem is referred to in [33] as the problem of the measurement scheme synthesis. Based on the elaboration given above, one may favor the use of the trace-based LLSE root cause estimation uncertainty for measurement scheme analysis. Furthermore, all analysis methods presented in this section are equally applicable to SoV models of both machining and auto-body assembly lines since, in addition to Assumptions 1 3, the analysis methods presented above only require the model of dimensional errors to be in a linear state-space form, such as that of the model presented in [36]. Nevertheless, as mentioned in the Introduction, prior knowledge of the statistical characteristics of the machining error root causes required by Assumption 3 may not be readily available for new machining lines. On the other hand, recent developments in international standards [23] [27] offer standard methods for [28] measurement of linear and angular positioning accuracies at specified locations; measurement of angular error motions of linear motion axes; measurement of critical machine alignments (i.e., squareness and parallelism); measurements of the error motions of machine tool spindles both statically (drift) and dynamically; measurements of the machine response to internal heat sources (e.g., spindles and drives). Due to these developments in standardization as well as the fact that the currently existing linear state-space models in machining only model the influence of the fixture and tool shift/path orientation errors on the product dimensional quality [4] [7], the third assumption that the nominal statistical properties of the machining error root causes are known, can be considered acceptable. Essentially, implementation of the aforementioned standards by machine tool builders could readily offer the user prior statistical information required by Assumption 3. However, with further refinement of the SoV models and subsequent modeling of more complex factors as root causes of dimensional errors, assumptions 1 3 may have to be relaxed. Furthermore, the lack of the a priori statistical knowledge required by assumptions 1 3 in the early stages of launch and ramp-up of manufacturing lines, necessitates the development of the non-bayesian measurement scheme analysis methods. Such an approach to the measurement scheme analysis will be the focal point of the next section. V. NON-BAYESIAN APPROACH TO MEASUREMENT SCHEME ANALYSIS The non-bayesian approach to the measurement scheme analysis in multistation machining systems will be introduced and discussed in this section. Section V-A presents the least-square estimation-based methods for measurement scheme analysis in the case when only statistical characteristics of the sensor noise term in the linear model (4) are known. Section V-B introduces an application of the recently developed worst-case maximum likelihood estimation techniques [30], [31], in solving the problem of measurement scheme analysis when the knowledge of statistical properties of the root causes in (4) is nonexistent, or inaccurate with bounded inaccuracy, and the knowledge of statistical characteristics of the sensor noise term in (4) is inaccurate with bounded inaccuracy. A. Least-Square Estimation-Based Measurement Scheme Analysis Let us assume that the noise term in the linear model (4) is characterized as a zero mean process with the known covariance matrix and that no a priori knowledge of the root causes exists. In this case, for a given measurement scheme, the well known least-square (LS) estimation can be used to estimate the root causes from the observation of dimensional part errors as where is the Fischer information matrix [29], and denotes the pseudoinverse of the matrix, [37]. If a measurement scheme is diagnosable, 4 then the pseudoinverse operator becomes a simple matrix inversion and the corresponding LS estimate given by (7) is an unbiased estimator of the root causes. It is easy to show that the variation of the LS estimate caused by the noise term is described by the matrix Thus, the matrix gives an overall description of the variations in the estimate caused by the noise term and can be used to quantitatively characterize the corresponding measurement scheme. Several properties of the Fischer information matrix can be used for measurement scheme analysis. Major criteria include the conditional number, trace, determinant, and operator norm (maximum eigenvalue) of, [35]. Larger trace, determinant, or operator norm of implies that the measurement scheme carries more information about the root causes. On the other hand, a smaller conditional number of implies a measurement scheme that is less sensitive to measurement noise and is therefore more desirable. For diagnosable measurement schemes, the conditional number of the corresponding Fischer information matrix indicates how close it is to being nondiagnosable. Namely, the larger is and the further away from 1 it is, the closer is to being nondiagnosable. For all nondiagnosable measurement schemes, their conditional number of the corresponding Fischer information matrix is infinite. Therefore, all nondiagnosable measurement schemes have the same conditional number of the corresponding Fischer information matrix, which could be seen as a potential drawback for its use in measurement scheme analysis. This is why in this paper, only the trace, determinant, and operator norm of the Fischer information matrix corresponding to a measurement scheme are considered for its characterization. 4 As defined by Definition 1 (using the rank of the regression matrix T ). (7)

8 414 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 Definition 4: The trace-based information content of a measurement scheme, labeled as, is the trace of the corresponding Fischer information matrix. The norm-based information content of a measurement scheme, labeled as, is the norm of the corresponding Fischer information matrix. The determinant-based information content of a measurement scheme, labeled as, is the determinant of the corresponding Fischer information matrix. It should be noted that the term information content is used to refer to the characterization criteria introduced in Definition 4 only because they are based on the commonly known Fisher information matrix [29]. These quantitative measures are not directly related to the Shannon s entropy-based information content [38], and should not be confused with this, also a well-known concept. From the definition of the operator norm, one has Furthermore, due to the positive semi-definiteness of Fischer information matrices, the trace-based and determinant-based information contents of a measurement scheme satisfy where denotes the determinant of a matrix. The following Lemma is crucial in proving the properties of,, and expressed in Theorem 4. Lemma 1: Let us observe measurement schemes and such that all of the measurements from the scheme are contained within the measurement scheme. Then is a positive semidefinite matrix. Proof of Lemma 1 is enclosed in the Appendix. Theorem 4: Let us observe measurement schemes and such that all of the measurements from the scheme are contained within the measurement scheme. Then 1) 2) 3) Proof: The proof of this Theorem is based on Lemma 1 stated above and Lemma 1 in [14]. Namely, based on these two lemmas, we have that for (8) where and denote ordered eigenvalues of and, respectively. Knowing that the trace of a symmetric matrix is equal to the sum of its eigenvalues, that the determinant of a matrix is equal to the product of its eigenvalues, and that the operator norm of a matrix is equal to its maximal eigenvalue, (8) readily gives statements 1), 2), and 3). Theorem 5: For a nondiagnosable measurement scheme Proof: Since is nondiagnosable, and, therefore,. This readily gives. Theorem 4 states properties analogous to those stated by Theorem 2 when the Bayesian approach was used in measurement scheme characterization. Similar to Theorem 2, the meaning of Theorem 4 is that regardless of whether one uses the tracebased, determinant-based, or norm-based information content of a measurement scheme for its characterization, the addition of measurements should result in a measurement scheme that is more informative about the root causes of dimensional machining errors. Again, as was the case with Theorem 2, the crucial element that facilitated the properties stated by Theorem 4 is the assumed knowledge of statistical properties of the measurement noise term, which, in turn, ensures that the LS estimation is not contaminated by the noise from the additional measurements. Theorem 5 states that all nondiagnosable measurement schemes have the same determinant-based information content. This property is analogous to Theorem 3 in the Bayesian approach to the measurement scheme analysis and it may not be desirable when one wishes to make a finer distinction among nondiagnosable measurement schemes. Such finer distinction could be made through the use of the trace, or the operator norm of the Fischer information matrix. 5 B. Worst-Case Maximum Likelihood Estimation-Based Measurement Scheme Analysis The LS estimation used in the previous subsection is based on the assumption that the knowledge of statistical properties of the noise term in the linear model (4) exists. In the case when one possesses an accurate SoV model connecting the dimensional machining errors with their root causes, the noise term is dominated by the sensor noise and the assumption used in the LS estimation-based measurement scheme analysis is realistic. However, one may not possess enough knowledge to postulate an accurate SoV model of the machining system, in (9) 5 As stated previously, all nondiagnosable measurement schemes have the infinite conditional number of the corresponding Fischer information matrix, which precludes the use of this Fischer information matrix property for finer distinction among nondiagnosable measurement schemes.

9 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 415 which case the unmodeled effects and nonlinearities could significantly contribute to the noise term, as can be seen from (3). Then, the knowledge of statistical characteristics of becomes inaccurate and this inaccuracy should be accounted for. Furthermore, the LS estimation used in the previous subsection does not take into account any statistical knowledge about the root causes in the linear model (4). If any such knowledge exists, one should be able to incorporate it into the root cause estimation and measurement scheme analysis. The recently developed WCML estimation technique [30] [32], can be used to tackle the two problems mentioned above. Let us assume that for a given measurement scheme, one can write the matrices,, and as where denotes matrices,, and It should be noted that if no a priori knowledge of the covariance matrix exists, one can simply write that is zero, 6 thus leaving the corresponding likelihood function only with the term associated with the measurement noise covariance matrix. With the problem of the SoV-based measurement scheme analysis in multistation machining systems, one deals only with the possibly uncertain knowledge of covariance matrices and, while the regression matrix in the linear model (4) can be found from the SoV model without any uncertainty. Following [31], this allows one to modify the WCML estimation scheme in order to obtain the robust linear unbiased (RLU) estimate of the root causes in the well-known innovations form (10) where matrices,,,,,,,,,,, and are known, and perturbation matrices,, and are unknown, but norm-bounded matrices ( denote the norm of matrices,, and ). Equation (10) expresses the inaccurate knowledge of matrices,, and in the linear fractional transformation (LFT) form, frequently used in robust control problems [39], [40], with matrices,,,,,,,, and describing the LFT structure of those inaccuracies. The meaning of the assumptions stated above is that one possesses inaccurate knowledge of the regression matrix in model (4), and inaccurate knowledge of the covariance matrices and, with bounded and structured inaccuracy, as given by (10). LFT representations are very general and are able to represent a wide range of uncertain perturbations in matrices,, and, including the cases when each element of the uncertain matrices is perturbed in a polynomial or rational manner [32]. Note that inaccurate knowledge of matrices and with bounded inaccuracies represented in the LFT form can be easily transformed into an LFT representation of matrices and, using the simple rules of LFT manipulations [39, pp ]. Let us also assume that the likelihood function corresponding to the linear model (4) can be represented as [31] (11) The likelihood function (11) corresponds to the case when root causes and noise term in the linear model (4) are independent zero-mean multivariate Gaussian processes with covariance matrices and, respectively. The WCML estimation gives the estimate of the root causes, based on the measured dimensional machining errors as the estimate that maximizes the likelihood function (11) regarding for the worst case of the likelihood function, regarding the bounded uncertainties,, and, that is [31] where (12) and is the a priori known mean of the root causes,ifitexists. If there exists no a priori knowledge of the mean of the root causes, one can simply use, [31]. The problem (12) can be efficiently solved in the form of a semidefinite program (SDP) (for more on SDP-s see [41] and references therein), and the variance of the RLU estimate is (13) which can be used to characterize the corresponding measurement scheme. If there is no a priori knowledge of the covariance matrix of the root causes, one can set in (13) and, thus, observe only part of the RLU estimate covariance matrix caused by the measurement noise term in the linear model (4). As in the previous elaboration, different properties of the matrix can be used to characterize the corresponding measurement scheme, such as its trace, operator norm, or determinant. A smaller trace, operator norm, or determinant of would imply a measurement scheme carrying more information about the root causes of dimensional machining errors. Definition 5: The trace-based WCML root cause estimation uncertainty of a measurement scheme, labeled as, is defined as the trace of the corresponding WCML estimate covariance matrix from (13). The norm-based WCML root cause estimation uncertainty of a measurement scheme, labeled as,isdefined as the operator norm of the corresponding WCML estimate covariance matrix from (13). 6 This, in turn, corresponds to K being infinite.

10 416 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 Fig. 1. Cylinder head used to demonstrate the SoV-based measurement scheme synthesis procedures introduced in this paper. The eight features taken into account by the SoV model are indicated in plot (a), while plot (b) shows the measurement datum features. Measurement datum axis z is pointed toward the cover face side of the cylinder head. The determinant-based WCML root cause estimation uncertainty of a measurement scheme, labeled as,isdefined as the determinant of the corresponding WCML estimate covariance matrix from (13). C. Discussion on the Non-Bayesian Approach to Measurement Scheme Analysis Two non-bayesian methods for the measurement scheme analysis have been proposed in this section, one based on the least-square (LS) estimation and the other based on the WCML estimation. In the case when one possesses an accurate SoV model of dimensional machining errors and accurate knowledge of the statistical characteristics of the measurement noise term in the linear model (4), the use of the LS-based measurement scheme analysis techniques is appropriate. Elaboration in Section V-A shows that the properties of the LS-based measurement scheme analysis methods to a large degree correspond to those displayed by the methods used in the Bayesian approach to the measurement scheme analysis. Similar remarks could not be made about the WCML-based measurement scheme analysis methods. Measurement scheme analysis methods based on the WCML estimation are able to handle cases when the a priori knowledge of the statistical properties of the root causes in the linear model (4) is inaccurate or nonexistent, and the knowledge of the statistical properties of the noise term in (4) is inaccurate. If no a priori knowledge of the root causes exists and there are no inaccuracies in the linear model (4), the WCML estimation becomes equivalent to the simple LS estimation discussed in Section V-A. On the other hand, if there exists accurate a priori knowledge of the statistical properties of the root causes and noise term in the linear model (4), the WCML estimation boils down to the well-known maximum-likelihood estimation, which, under the assumption of Gaussianity of and in (4), becomes equivalent to the LLSE estimation discussed in Section IV. Thus, one can see the WCML estimation-based measurement scheme analysis techniques presented in Section V-B as the methods that stand between the purely Bayesian LLSE estimation-based measurement scheme analysis methods that were presented in Section II, and the non-bayesian LS estimation-based measurement scheme analysis methods that were presented in Section V-B of this section. Finally, one should note that in order to accomplish characterization of more complex measurement schemes containing several hundreds or thousands of measurements, the use of the class of the so-called infeasible start methods for solving SDP problems is appropriate [43] [45]. These methods do not require a starting feasible primal and/or dual solutions and are able to solve SDP problems of several thousands of variables. VI. NUMERICAL ILLUSTRATIONS OF THE NON-BAYESIAN MEASUREMENT SCHEME ANALYSIS METHODS This section presents examples of the implementation of the measurement scheme analysis methods proposed in this paper. The methods are used to quantitatively characterize selected measurement schemes from the machining process used to produce an automotive cylinder head shown in Fig. 1. The process plan and locating datum features are identified in Table II. Each plane was described by a unit vector defining the plane orientation and one point in the plane defining its position. Cylindrical features were described by a unit vector defining the cylinder axis, one point on the cylinder axis defining its position, and by the cylinder diameter. The resulting state-space model for operations was of dimension 28 and the total number of inputs into the system was 37 [33] (only locating pin errors and geometric errors of newly machined surfaces were modeled for details on SoV modeling of dimensional

11 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 417 TABLE II PROCESS PLAN FOR MACHINING OF THE CYLINDER HEAD SHOWN IN FIG. 1 machining errors in this process, see [5] and [6]). After the last machining operation, all features were measured in the coordinate system defined by the machined joint face (primary measurement datum feature), datum Hole B (secondary measurement datum feature), and datum Hole C (tertiary measurement datum feature). 7 Such measurements could easily be taken using a coordinate measurement machine (CMM). As in [14], let us assume that we are just trying to estimate causes of dimensional errors in machining of the joint face feature in the second machining operation (Setup 2) and that other causes of dimensional machining errors are zero (hence, only vector is allowed to have nonzero elements, while all other vectors,, 3, 4, 5 are assumed to be zero). This assumption greatly simplifies the process at hand and allows for more intuitive understanding of the process to be used in reaching conclusions about ability or inability to diagnose root causes of machining errors based on a given set of measurements (diagnosability or lack of it for a given set of measurements). The goal of the elaboration below is to compare the intuitive conclusions with those obtained from theoretical considerations using methods introduced in this paper. During machining of the joint face, only fixture errors in the three locating pins under the cover face cause errors, while errors in the other three locating pins in the fixture used for this operation do not affect parameters of the machined joint face. The locating scheme used to hold the workpiece during machining operation 2 is shown in Fig. 2, while the fixture used in this operation is shown in Fig. 3. The tool path, which is the other source of errors during machining of the joint face, is modeled as a plane described by its orientation and position, with errors in the three orientation and three position parameters of this plane, resulting in errors of the machined joint face. However, the orientation of each plane is defined by a unit vector perpendicular to that plane [5], [6] and errors in the orientation of a plane in the direction perpendicular to the plane do not affect that particular plane. Hence, one should only consider two plane orientation error components in directions normal to the nominal plane orientation vector. Similarly, errors in the position of a plane perpendicular to the plane orientation vector do not affect that particular plane and one should only consider po- 7 This part coordinate system is also indicated in Fig. 1. Fig. 2. Milling of the joint face in the setup 2 (machining operation 2 from Table II). sition errors along the plane orientation vector, thus leaving us with three tool-path error parameters associated with machining of each plane [5]. It should be noted that the tool path errors could originate due to several factors, such as thermal errors, tool wear, or tool machine tool axis misalignment. However, the existing SoV models of multistation machining do not model the dependency of tool-path errors on other process parameters and further identification of tool-path-related errors would have to be accomplished through other methods, such as thermal error, or tool wear monitoring. In summary, considering that the existing SoV models of dimensional machining errors model the machining errors dependency only due to the fixture errors, or due to errors in the parameters of the tool path during machining of that feature [4] [7], there are only six root causes that need to be identified in this case: errors in the three fixture pads under the cover face, two orientational errors in machining of the joint face, and one positional error in machining of the cover face. Let us now observe the measurement scheme 1 from Table III, where measurements of only the cover face and Plane X created by the three rough casting datum points,, and relative to the joint face measurement datum are taken. This set of measurements could not yield information about whether errors in machining of the cover face occurred because of the errors in the fixture parameters or because of the errors in the tool path. For example, if the distance between the cover face and joint face is found to be too small, one cannot say that the reason for that was because the three fixture pins under the cover face were too high, or the cutting tool during milling of the joint face moved too deep, which is readily mirrored in the nondiagnosability of

12 418 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 Fig. 3. Cylinder head used to demonstrate the SoV-based measurement scheme synthesis procedures introduced in this paper. The eight features taken into account by the SoV model are indicated in plot (a), while plot (b) shows the measurement datum features. The measurement datum axis Z is pointed toward the cover face side of the cylinder head. TABLE III MEASUREMENT SCHEMES USED TO DEMONSTRATE THE MEASUREMENT SCHEME ANALYSIS METHODS the corresponding measurement scheme (the rank of the regression matrix associated with the measurement scheme 1 is 5 instead of 6). Four measurement schemes,,2,3,4 described in Table III are analyzed based on the SoV model postulated for the machining process at hand (see [5] and [6] for more details). Measurement scheme 1 consisted of measurements of the cover face plane and the plane X created by the three rough casting datum points,, and, relative to the joint face measurement datum. Such a measurement scheme is nondiagnosable because of the above-described inherent coupling of errors induced by the fixture and the cutting tool path. The inability of the measurement scheme to decouple sources of machining errors is readily mirrored in the nonfull rank of the regression matrix and subsequent nondiagnosability of the measurement scheme 1. Measurement scheme 2 also includes direct on-machine measurements of the fixture parameters, which allows one to directly identify errors in the fixture parameters and then use measurements of the cover face and X-plane (relative to the joint face, which is considered to be the measurement datum) to identify errors in the tool path during machining of the cover face. Therefore, the regression matrix is a full-rank matrix and the measurement scheme 2 is diagnosable. Measurement scheme 3 includes direct onmachine measurements of the tool-path parameters corresponding to machining of the joint face (could be measured by an onmachine probe measuring the newly machined joint face while the workpiece is still in the machine, in its fixture). Such a measurement could directly identify tool-path errors in machining of the joint face, while measurements of the cover face and X-plane relative to the joint face would identify errors in the locating pin parameters. Therefore, regression matrix is also a full-rank matrix and the diagnosable measurement scheme 3 can identify all six sources of errors in machining of the joint face. Finally, measurement scheme 4 encompasses all measurements in the schemes,, 2, 3 (i.e., measurements of the machined cover face and plane X relative to the joint face measurement datum, as well as on machine measurements of the fixture parameters and onmachine measurements of the tool-path parameters in machining of the cover face. As such, measurement scheme 4 is diagnosable with, but this measurement scheme involves the largest number of measurements and will therefore be most costly. The issue of achieving a tradeoff between informativeness of a measurement scheme about the root causes of dimensional machining errors and the cost of measurements is dealt with in [33]. In this paper, we focus on the examples of application of the non-bayesian measurement scheme analysis methods to the measurement schemes in Table III. Examples of applying the Bayesian measurement scheme analysis methods to the same measurement schemes can be found in [14]. For each of the four measurement schemes from Table III, we calculated their trace-based, determinant-based, and normbased information contents. Results of these calculations are given in Table IV. Table V shows results of the WCML estimation-based analysis of the measurement schemes described above. The WCML estimation was implemented using the SP version 1.1 Matlab toolbox for SDP [46]. A 50%

13 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 419 TABLE IV RESULTS OF THE LS ESTIMATION-BASED MEASUREMENT SCHEME ANALYSIS TABLE V RESULTS OF THE WCML ESTIMATION-BASED MEASUREMENT SCHEME ANALYSIS TABLE VI RESULTS OF THE LS ESTIMATION-BASED MEASUREMENT SCHEME ANALYSIS FOR MEASUREMENT SCHEMES A, B, AND C uncertainty is assumed in the covariance matrix of the noise term in the linear model (4). For each of the four measurement schemes,, 2, 3.4 we calculated the trace-based, determinant-based, and norm-based WCML root cause estimation uncertainties. The results shown above were obtained for a fairly simple situation involving only six possible root causes affecting the machining errors. Nevertheless, the methods proposed in this paper can readily be used for measurement scheme analysis without these assumptions and could uncover nondiagnosabilities in much more complicated and nonobvious cases. Let us observe the entire machining process for the cylinder head shown in Fig. 1 and described in Table IV. Measurement scheme A denotes the set of measurements of all orientations, positions, and diameters (if applicable) of the machined features in the part coordinate system. Measurement scheme B denotes the set of measurements obtained when onmachine measurements of fixtures in all three machining stations are added to the measurements from the measurement scheme A (note that all measurements from measurement scheme A are contained within the measurement scheme B). Finally, measurement scheme C will denote the set of measurements obtained when onmachine measurements of parameters of the newly machined features in all three machining stations are added to the measurements from measurement scheme B (all measurements from measurement schemes A and B are contained within measurement scheme C). Table VI shows results of the LS-estimation-based measurement scheme analysis applied on measurement schemes A, B, and C. VII. DISCUSSION Measurement scheme analysis, as defined in this paper, can be seen as a methodology to evaluate the amount of information of any given combination of measurements in a machining system carried about the root causes of errors in workpiece parameters. This informativeness of a given set of measurements is expressed through different measures (the trace, norm, and determinant) of the covariance matrix (or its inverse in the case of LS-based estimation) of estimation errors emanating from estimation of the root causes of dimensional machining errors, using the selected measurements as the basis for estimation. The previously developed SoV model in the linear state-space form (1) serves as the link between the dimensional machining errors and their root causes, enabling the aforementioned estimation and quantification of estimation uncertainty. In Sections III and IV, three different approaches of the characterization of measurement schemes have been presented and discussed. In the case when a priori knowledge of nominal statistical properties of the root causes of dimensional machining errors 8 exists, the Bayesian approach to root cause estimation and subsequent measurement scheme analysis is the appropriate method since it takes into account this a priori information and utilizes it to improve the root cause estimation. In the case when 8 The existing SoV models of dimensional machining errors model only errors in fixture parameters and errors in tool-path parameters (see Section II for more on the SoV models). For such models, this assumptions corresponds to having a priori knowledge about means and variances of errors in the fixture parameters as well as of linear and angular positioning errors of the machine tool in any machining operation.

14 420 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 3, NO. 4, OCTOBER 2006 Fig. 4. Relation between the three approaches to measurement scheme analysis. such a priori statistical knowledge does not exist, one must rely on the non-bayesian tools, such as the least-squares approach used in Section IV. Finally, when statistical knowledge about root causes exists, but is inaccurate (we only know the range within which mean and variance parameters of the vector are), the WCML estimation can take that information into account and the corresponding measurement scheme analysis tools can be used. Fig. 4 illustrates the relation between these three approaches to measurement scheme analysis. All matrices used in the measurement scheme characterization methods described in Section IV are covariance matrices, or their pseudoinverses. Such symmetric and positive semidefinite matrices are also characterized by an ellipsoid with axes in the directions of the eigenvectors of that matrix and axis lengths equal to the corresponding matrix eigenvalues, just like in the case of the normalized estimation error covariance matrix used for the Bayesian approach to measurement scheme analysis. Therefore, following the elaboration enclosed in Section III-B, the trace of a matrix is appropriate to depict the overall magnitude of the matrix elements and, thus, trace-based information content and trace-based WCML root-cause estimation uncertainties are suggested for measurement scheme analysis. Let us now observe results given in Section V of this manuscript. Note that all measurements present in the measurement scheme are also present in the measurement schemes,, 3.4. Furthermore, all measurement in schemes 2 and 3 are also present in the measurement scheme 4. Following Theorem 4, one would expect for,3,4 A comparison of numerical results in Tables IV and V suggests that the presence of uncertainty in the model and in the a priori knowledge about the system can significantly affect the measurement scheme characterization. Let us observe measurement schemes 2 and 3, which both contain onmachine measurements, both have the same number of measurements, and both are diagnosable. When the LS estimation is used as the basis for the measurement scheme analysis (i.e., when no uncertainty in the model is taken into account), different criteria for the corresponding Fischer information matrix give different answers to the question as to which of these two schemes is more informative about the root causes of dimensional machining errors (according to the trace-based and norm-based information contents, measurement scheme 2 is more informative than measurement scheme 3, while according to the determinant-based information content, measurement scheme 3 is more informative than measurement scheme 2). However, when one considers uncertainty in the covariance matrix of the noise term in the linear model, from Table V, one can see that measurement scheme 3 seems more informative than measurement scheme 2, regardless of the method used to characterize the covariance matrix of the WCML estimate. Therefore, if one had to choose between measurement schemes 2 and 3, the use of measurement scheme 3 may be more desirable. Nevertheless, the issue of systematically choosing the measurement scheme that is in some sense optimal is the problem of measurement scheme synthesis, rather than measurement scheme analysis, and is outside the scope of this paper and is discussed in detail in [33]. Examination of Table VI, offering results of the LS-estimation-based measurement scheme analysis for selected measurement schemes in the multistation cylinder-head machining process, indicates the same properties proven in Theorems 4 and 5 and illustrated in Table IV for the simplified, single machining station case. Namely, since all measurements in measurement scheme A are also in the measurement scheme B and in measurement scheme C, we have (following Theorem 4) which can be readily verified from Table VI. In addition, since all measurements in measurement scheme B are also in the measurement scheme C, Theorem 4 claims that and for,2,3, which is readily confirmed by inspecting Table IV. Furthermore, as predicted by Theorem 5, for the nondiagnosable measurement scheme, we have that which is also obvious from Table VI. Furthermore, since measurement scheme A is nondiagnosable, then according to Theorem 5 which is also depicted in Table VI.

15 DJURDJANOVIC AND NI: STREAM OF VARIATION (SoV)-BASED MEASUREMENT SCHEME ANALYSIS 421 Finally, it should be noted that the existing SoV models [4] [7] have been demonstrated only in linear state-space modeling of dimensional machining errors of prismatic parts, and future work is needed to demonstrate the SoV methodology in linear state-space modeling of nonprismatic workpieces, such as automotive crankshafts. Furthermore, the existing SoV models in machining only take into account kinematic effects affecting the dimensional machining errors, with each machined feature being modeled through a vector describing its orientation, a vector describing its position, and a vector of scalar parameters associated with that feature (diameter of a cylinder, width, and depth of a slot, etc.). Improvements of the existing SoV models through modeling of nondimensional machining errors, such as the workpiece form tolerances or surface quality as well as the inclusion of the influence of cutting forces, material properties, and dynamic effects into the SoV models, are both possible and necessary. Nevertheless, the SoV models in machining (as well as in assembly) are envisioned to be always in a linear state-space form (1) with the machining operation index playing the role of time in a discrete linear time-variant (LTV) system, states of the model describing the deviations of workpiece parameters away from nominal, inputs into the system representing errors introduced at each machining station, and outputs of the system representing measured deviations of the workpiece parameters. The methods introduced in this paper are applicable as long as the model of machining errors is in the linear state-space form (1) (i.e., in the SoV form). APPENDIX Proof of Lemma 1: Since measurements in the measurement scheme are within the measurement scheme, one can represent matrix as and matrix as where is the covariance matrix of the sensor noise terms present in the measurement scheme, but not in. Using the block-matrix identity [22, p. 39], shown in the equation at the top of the page, the matrix can be written as where (A.1) VIII. CONCLUSION AND FUTURE WORK This paper deals with the SoV-based approach to a formal and systematic measurement scheme analysis in multistation machining systems. Several methods of quantitative measurement scheme characterization based on both the Bayesian and non-bayesian approaches in statistics are presented and discussed. The recently developed WCML estimation techniques have been proposed as methods to deal with possible uncertainties in the knowledge about the machining system. Properties of the proposed measurement scheme analysis methods are derived and demonstrated using simple examples of selected measurement schemes in the machining system used to machine an automotive cylinder head. The methods proposed in this paper are in no way restricted only to the use in machining systems and can be applied in any manufacturing system in which dimensional errors are modeled in the linear state-space form. In the future, one needs to solve the issues of memory storage and robust identification of primal, or dual feasible points in the SDP-s underlying the WCML-based measurement scheme analysis methods. Furthermore, implementation of the newly introduced measurement scheme analysis techniques in order to facilitate a formal and systematic measurement scheme synthesis is a very important issue, but is outside the scope of this paper and will be presented in future publications. 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Amsterdam, The Netherlands: Elsevier Science, 1998, pp Dragan Djurdjanovic received the B.S. degree in mechanical engineering and applied mathematics in 1997 from the University of Nis, Serbia, in 1997, the M.Eng. degree in mechanical engineering from the Nanyang Technological University, Singapore, in 1999, the M.S. degree in electrical engineering (systems) in 2002, and the Ph.D. degree in mechanical engineering from the University of Michigan, Ann Arbor, in His research interests include quality control, intelligent proactive maintenance techniques, and applications of advanced signal processing in biomedical engineering. Dr. Djurdjanovic coauthored many journal and conference publications and is the recipient of several prizes and awards, including the 2006 Outstanding Young Manufacturing Engineer Award from the Society of Manufacturing Engineers (SME); Nomination for the Distinguished Ph.D. Thesis from the Department of Mechanical Engineering, University of Michigan in 2002; and The Outstanding Paper Award at the 2001 SME North American Manufacturing Research Conference. Jun Ni received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 1982, the M.S. degree in mechanical engineering from the University of Wisconsin, Madison, in 1984, and the Ph.D. degree in mechanical engineering from the University of Wisconsin, Madison, in Currently, he is a Professor in the Department of Mechanical Engineering, The University of Michigan, Ann Arbor, where he is the Director of the S. M. Wu Manufacturing Research Center. He also serves as the Deputy Director of a National Science Foundation-sponsored Engineering Research Center for Reconfigurable Machining Systems and is the Co-Director of a National Science Foundation-sponsored Industry/University Cooperative Research Center for Intelligent Maintenance Systems. His research and teaching interests are in the area of manufacturing science and engineering, with special focuses on precision machining, manufacturing process modeling and control, statistical quality design and improvement, micro/meso systems and manufacturing processes, and intelligent monitoring, maintenance, and service systems.