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record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam Ka controlfield tag 001 001967169 003 fts 005 20081024114055.0 006 med 007 cr mnuuuuuu 008 081024s2007 flu sbm 000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0002252 035 (OCoLC)263421613 040 FHM c FHM 049 FHMM 090 T56 (ONLINE) 1 100 Wang, Hui. 0 245 Error equivalence theory for manufacturing process control h [electronic resource] / by Hui Wang. 260 [Tampa, Fla.] : b University of South Florida, 2007. 3 520 ABSTRACT: Due to uncertainty in manufacturing processes, applied probability and statistics have been widely applied for quality and productivity improvement. In spite of significant achievements made in causality modeling for control of process variations, there exists a lack of understanding on error equivalence phenomenon, which concerns the mechanism that different error sources result in identical variation patterns on part features. This so called error equivalence phenomenon could have dual effects on dimensional control: significantly increasing the complexity of root cause identification, and providing an opportunity to use one error source to counteract or compensate the others. Most of previous research has focused on analyses of individual errors, process modeling of variation propagation, process diagnosis, reduction of sensing noise, and error compensation for machine tool.^ This dissertation presents a mathematical formulation of the error equivalence to achieve a better, insightful understanding, and control of manufacturing process. The first issue to be studied is mathematical modeling of the error equivalence phenomenon in manufacturing to predict product variation. Using kinematic analysis and analytical geometry, the research derives an error equivalence model that can transform different types of errors to the equivalent amount of one base error. A causal process model is then developed to predict the joint impact of multiple process errors on product features. Second, error equivalence analysis is conducted for root cause identification. Based on the error equivalence modeling, this study proposes a sequential root cause identification procedure to detect and pinpoint the error sources. Comparing with the conventional measurement strategy, the proposed sequential procedure identifies the potential error sources more effectively.^ ^Finally, an errorcancelingerror compensation strategy with integration of statistical quality control is proposed. A novel error compensation approach has been proposed to compensate for process errors by controlling the base error. The adjustment process and product quality will be monitored by quality control charts. Based on the monitoring results, an updating scheme is developed to enhance the stability and sensitivity of the compensation algorithm. These aspects constitute the "Error Equivalence Theory". The research will lead to new analytical tools and algorithms for continuous variation reduction and quality improvement in manufacturing. 502 Dissertation (Ph.D.)University of South Florida, 2007. 504 Includes bibliographical references. 516 Text (Electronic dissertation) in PDF format. 538 System requirements: World Wide Web browser and PDF reader. Mode of access: World Wide Web. 500 Title from PDF of title page. Document formatted into pages; contains 122 pages. Includes vita. 590 Adviser: Qiang Huang, Ph.D. 653 Error cancellation. Process modeling. Root cause identification. Automatic process adjustment. Statistical quality control. 690 Dissertations, Academic z USF x Industrial Engineering Doctoral. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.2252 PAGE 1 Error Equivalence Theory for Manufacturing Process Control by Hui Wang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida Major Professor: Qiang Huang, Ph.D. Shekhar Bhansali, Ph.D. Tapas K. Das, Ph.D. Yuncheng You, Ph.D. Jos L. ZayasCastro, Ph.D. Date of Approval: April 17, 2007 Keywords: error cancellation, process mode ling, root cause identification, automatic process adjustment, stat istical quality control Copyright 2007, Hui Wang PAGE 2 Dedication To my parents. PAGE 3 Acknowledgements I would like to express my sincere gratitu de to Prof. Qiang Huang, my advisor, for sharing his insightful ideas in researc h, for providing continual encouragement and critical mentoring. Prof. Huangs innovative ideas, broad and indepth knowledge in manufacturing and applied statis tics have been a great inspir ation to me. Without him, I would not be able to accomplish what I have accomplished. Throughout my fouryear PhD studies, he has always been wholeheart edly supporting me to steer through countless difficulties in all aspects of my life. I want to thank my dissertation committ ee members, Prof. Tapas K. Das, Prof. Jos ZayasCastro, Prof. Shekhar Bhansali, and Prof. Yuncheng You for their valuable suggestions and assistance. I wish to tha nk Prof. Louis MartinVega and Prof. A.N.V. Rao for their constructive suggestions wh en they were my dissertation committee members. I am also grateful to Dr. Reuven Katz from the University of Michigan for his advice on clarifying several vague concepts in Chapter Two. Special thanks are given to all other f aculty members, Ms. Gloria Hanshaw, Ms. Jackie Stephens, and Mr. Chris Paulus in th e IMSE department for their kind help during my Ph.D. studies. In addition, I would like to express my sincere appreci ation to my friends, Mr. Shaoqiang Chen, Mr. Xi Zhang, Ms. Diana Pr ieto, Mr. Yang Tan, and other fellow IMSE graduate students. Finally, I am forever indebted to my wonderful parents Zulin Chen and Yongkang Wang. I would have never fini shed this dissertation w ithout their endless love, encouragement and unconditional support. I owe them so much. PAGE 4 i Table of Contents List of Tables iii List of Figures iv Abstract vi Chapter 1 Introduction 1 1.1 Phenomena of Error Equivale nce in Manufacturing Processes 2 1.2 Related Work and the State of the Arts 4 1.2.1 Research Review for Modeling Process Errors 4 1.2.2 Research Review for Process Root Cause Diagnosis 8 1.2.3 Research Review for Process Control 9 1.2.4 Summary of Literature Review 12 1.3 Dissertation Outline 13 Chapter 2 Error Equivalence Modeling and Variation Propagation Modeling Based on Error Equivalence 15 2.1 Preliminaries and Notations 16 2.2 Mathematical Modeling of the Error Equivalence Phenomenon in Manufacturing 20 2.3 Error Equivalence Modeli ng for Machining Processes 22 2.3.1 Concept of Equivalent Fixture Error 22 2.3.2 Derivation of EFE Model 24 2.4 Variation Propagation Modeling Ba sed on Error Equivalence for MultiOperation Machining Process 28 2.4.1 Background Review for Mu ltiOperational Manufacturing Process 28 2.4.2 Variation Propagation Model Derivation 30 2.4.3 Discussion for Error Grouping in Machining Processes 35 2.5 EFE Validation and Modeling Demonstration 36 2.5.1 Experimental Validation of EFE 37 2.5.2 MultiOperational Variat ion Propagation Modeling with Grouped EFEs 38 2.6 Summary 45 Chapter 3 Error Cancellation Modeling a nd Its Application in Process Control 47 3.1 Error Cancellation and It s Theoretical Implications 48 3.1.1 Diagnosability Analysis of Manufacturing Pro cess with Error Equivalence 49 PAGE 5 ii 3.1.2 Sequential Root Cause Identification 50 3.1.3 ErrorCancelingError Compensation Strategy 52 3.2 Applications of Error Can cellation in a Mi lling Process 55 3.2.1 Diagnosis Based on Error Equivalence 55 3.2.2 Error Compensation Simulation 60 3.3 Summary 62 Chapter 4 Dynamic Error Equivalence Modeling and InLine Monitoring of Dynamic Equivalent Fixture Errors 64 4.1 Introduction to Modeling of Dynamic Errors 66 4.2 Latent Variable Modeling of Machine Tool Dynamic Errors 67 4.2.1 Description of Data 67 4.2.2 Latent Variable Modeling of Machine Tool Dynamic Error 71 4.3 InLine Monitoring of Dynamic Equi valent Errors of Machine Tool 78 4.4 Isolation of Lagged Variables and Sensors Responsible for the OutofControl Signal 82 4.5 Summary 85 Chapter 5 Error Compensation Based on Dynamic Error Equivalence for Reducing Dimensional Variation in Discrete Machining Processes 87 5.1 Automatic Process Adjustment Ba sed on Error Equiva lence Mechanism 88 5.2 SPC Integrated Process Adjust ment Based on Error Equivalence 91 5.3 Simulation of Error Equivalence Process Adjustment 94 5.4 Adjustment Algorithm Evaluation 98 5.5 Summary 102 Chapter 6 Conclusions and Future Work 104 6.1 Conclusions 104 6.2 Future Work 106 References 107 Appendices 115 Appendix A: Infinitesimal An alysis of Workpiece Deviation Due to Fixture Errors 116 Appendix B: Proof for Proposition in Chapter 2 117 Appendix C: Proof for Corollary in Chapter 2 118 Appendix D: Determine Difference Order for D( q ) 119 Appendix E: Screened Variables 120 Appendix F: Results of Partial Least Square Estimation 121 About the Author End Page PAGE 6 iii List of Tables Table 2.1 Measurement Results (Under PCS0) 38 Table 2.2 Machined Features Specification 39 Table 2.3 Coordinates of Locating Points on the Primary Datum Surfaces (Unit: mm) 39 Table 3.1 Measured Features (mm) 59 Table 3.2 Estimation of u for 5 Replicates (mm) 59 Table 3.3 Error Decomposition (mm) 60 Table A.1 First Order Difference 119 Table A.2 Second Order Difference 119 Table A.3 Screened Vari ables With Autoregressive Terms 120 Table A.4 Screened Vari ables without Autoregressive Terms 120 Table A.5 Percentage of Va riance Explained by Latent Variables 121 Table A.6 Regression Coefficient B 121 Table A.7 Matrix W(PTW)1 122 Table A.8 Scores for Points 10, 33, and 56 122 PAGE 7 iv List of Figures Figure 1.1 Error Equivalence in Machining 3 Figure 1.2 Error Equivalence in Assembly 3 Figure 1.3 The Framework of Error Equivalence Theory 14 Figure 2.1 Modeling of Part Feature Deviation 17 Figure 2.2 General 321 Locating Scheme and FCS0 18 Figure 2.3 Modeling of Workpiece Positioning Error 20 Figure 2.4 Mathematical Mode ling of Error Equivalence 21 Figure 2.5 Equivalent Fixture Error 23 Figure 2.6 EFE Derivation 25 Figure 2.7 NonPlanar Datum Surfaces 26 Figure 2.8 PinHole Locating Scheme 27 Figure 2.9 Model Derivation 30 Figure 2.10 Raw Workpiece and Locating Scheme (Unit: mm) 37 Figure 2.11 Workpiece and Locating 39 Figure 2.12 Two Cutting Operations 40 Figure 3.1 ErrorC ancelingError Strategy 53 Figure 3.2 Process Adjustment Using EFE Concept 53 Figure 3.3 Sequential Root Cause Identification Procedures 58 Figure 3.4 Error Compensation for Each Locator 60 Figure 3.5 Mean and Standa rd Deviation of Two Features 61 PAGE 8 v Figure 4.1 Thermal Sensor Locations on a Machine Tool 68 Figure 4.2 Machine Tool Te mperature and Thermal Error Data 69 Figure 4.3 Stationarity Treatment 70 Figure 4.4 Equivalent Fixture Error of Fig. 4.2 75 Figure 4.5 Model Prediction and Residuals 78 Figure 4.6 Ellipse Format Chart 80 Figure 4.7 Control Ellipse for Future Observations 81 Figure 4.8 Standardized Scores in Points 10, 33, and 56 83 Figure 4.9 Lagged Variable C ontributions to Score Component 2 84 Figure 4.10 Sensor Contribu tions to Score Component 2 85 Figure 5.1 Adjustment Based on Error Equivalence 94 Figure 5.2(a) Machine Tool Temperature and Error Data 95 Figure 5.2(b) Thermal Error Measurements 95 Figure 5.3 EFE Adjustment 97 Figure 5.4 Monitoring Thickness a nd Standard Deviation of Edge Length 98 Figure 5.5 Effect of Parameters Change in Process Adjustment Algorithm 101 PAGE 9 vi Error Equivalence Theory For Manufacturing Process Control Hui Wang ABSTRACT Due to uncertainty in manufacturing proce sses, applied probability and statistics have been widely applied for quality and productivity improvement. In spite of significant achievements made in causality modeling for control of process variations, there exists a lack of un derstanding on error equivale nce phenomenon, which concerns the mechanism that different error sources resu lt in identical variation patterns on part features. This so called error equivalence phenomenon could have dual effects on dimensional control: significantly increasing th e complexity of root cause identification, and providing an opportunity to use one erro r source to counteract or compensate the others. Most of previous research has focused on analyses of individual errors, process modeling of variation propagation, process di agnosis, reduction of sensing noise, and error compensation for machine tool. This dissertation presents a mathematical formulation of the error equivalence to ach ieve a better, insightful understanding, and control of manufacturing process. The first issue to be studied is mathem atical modeling of the error equivalence phenomenon in manufacturing to predict produc t variation. Using kinematic analysis and analytical geometry, the research derives an error equivalence mode l that can transform PAGE 10 vii different types of errors to the equivalent amount of one base error. A causal process model is then developed to predict the joint impact of multiple process errors on product features. Second, error equivalence analysis is conducted for root cause identification. Based on the error equivalence modeling, this study proposes a sequential root cause identification procedure to detect and pinpoi nt the error sources. Comparing with the conventional measurement strategy, the propos ed sequential procedure identifies the potential error sources more effectively. Finally, an errorcancelingerror comp ensation strategy with integration of statistical quality control is proposed. A novel error compensation approach has been proposed to compensate for process errors by controlling the base error. The adjustment process and product quality will be monito red by quality control charts. Based on the monitoring results, an updating scheme is developed to enhance the stability and sensitivity of the compensation algorith m. These aspects constitute the Error Equivalence Theory. The research will lead to new analytical tools and algorithms for continuous variation reduction and qu ality improvement in manufacturing. PAGE 11 1 Chapter 1 Introduction The intense global competition has b een driving the manufacturers to continuously improve quality in the life cycle of product de sign and manufacturing. Vital to the competition success is the product variation reduction to achieve the continuous manufacturing process improvement. Howeve r, variation reduction for the process improvement has been an extremely challengi ng issue because of the following reasons: Prediction of quality perform ance with process variation. Due to the uncertain nature of the manufacturing process, probabilistic models and statistics have been widely applied to depict the process variation. However, there exis ts a lack of understanding on error equivalence, an engineering phenomenon concerning the mechanism that multiple error sources result in the identic al variation pattern. This fact impacts almost every stage of variation control (e .g., process root cause diagnosis and error compensation). Therefore, to better pr edict the process performance, error equivalence has to be quantita tively modeled and analyzed. Control of a varying process. Variation control strategies mu st be incorporated in the early stage of manufacturi ng process design. The control strategy involves statistical quality control (SQC), root cause iden tification and automatic process error compensation to reduce potential large variations. The dual effects of error equivalence on process control have not been well studied. For instance, the phenomenon of error equivalence could conceal the information of multiple errors PAGE 12 2 and thus significantly increase the complexity of root cause iden tification (diagnosis). It may provide an opportunity to purposel y use one error source to counteract the others and thereby reduce overall process va riations. Hence, the inclusion of error equivalence mechanism into quality contro l may create a new c ontrol paradigm of manufacturing process, i.e., information collection in support of process diagnosis, root cause identification, and SPC (statistical process contro l) integrated process error compensation. Therefore, the aforementioned issues en tail an essential analysis of error equivalence for process improvement. The goa l of this work is to model the error equivalence in traditional di screte manufacturing to achieve an insightful understanding of process variation and a better process control. 1.1 Phenomena of Error Equivale nce in Manufacturing Processes In a manufacturing process, product quali ty can be affected by multiple error sources. For example, the dominant root cau se of quality problems in a machining process includes fixture, datum, and machine t ool errors. A fixture is a device used to locate, clamp, and support a workpiece dur ing machining, assembly, or inspection. Fixture error is considered to be a signifi cant fixture deviation of a locator from its specified position. Machining datum surfaces are those part features that are in direct contact with the fixture locator s. Datum error is deemed to be the significant deviation of datum surfaces and is mainly induced by im perfections in raw workpieces or faulty operations in the previous stages. Together the fixture and datum surfaces provide a reference system for accurate cutting operations using machine tools. Machine tool error PAGE 13 3 is modeled in terms of significant tool path deviations from its intended route. This dissertation mainly focuses on kinematic aspects of these three error types. A widely observed engineering phenomenon is that the individual error sources can result in the identical variation patterns on product features in manufacturing process. For instance, in a machining process, all af orementioned process deviations can generate the same amount of feature deviation x as shown in Fig. 1.1 (Wang, Huang, and Katz, 2005; and Wang and Huang, 2006). This error e quivalence phenomenon is also observed in many other manufacturing processes, e.g., the automotive body assembly process (Fig. 1.2, Ding, et al. 2005). Deviated tool path Nominal tool path (b) Machine process with machine tool error (c) Machining process with datum error (a) Machine process with fixture error Nominal tool path Deviated datum surface Fixture locator deviation s Deviated tool path Nominal tool path (b) Machine process with machine tool error (c) Machining process with datum error (a) Machine process with fixture error Nominal tool path Deviated datum surface Fixture locator deviation s x x x x x x f m d Deviated tool path Nominal tool path (b) Machine process with machine tool error (c) Machining process with datum error (a) Machine process with fixture error Nominal tool path Deviated datum surface Fixture locator deviation s Deviated tool path Nominal tool path (b) Machine process with machine tool error (c) Machining process with datum error (a) Machine process with fixture error Nominal tool path Deviated datum surface Fixture locator deviation s x x x x x x f m d Part 1 Part 2 Part 1 Part 2 Part 1 Part 2 Fixture deviation Part 1 Part 2 Part 1 Part 2 Fixture deviation Workpiece deviation or reorientation error (a) (b) Figure 1.1 Error Equivalence in Machining Figure 1.2 Error Equivalence in Assembly The impact of such an error equiva lence phenomenon on manufacturing process control is twofold. On the one hand, it signifi cantly increases the complexity of variation control. As an example, identifying the root causes becomes extremely challenging when different error sources are able to produce the identical dimensional variations. On the other hand, the error equivalence phenomenon provides an opportunity to purposely use PAGE 14 4 one error source to coun teract another in order to reduce process variation. In both cases, a fundamental understanding of this comple x engineering phenomenon will assist to improve manufacturing process control. 1.2 Related Work and the State of the Arts The study on error equivalence is, howeve r, very limited. Most related research on process error modeling has been focused on the analysis of th e individual error sources, e.g., the fixture errors and machin e tool errors, how these errors impact the product quality, and thereby how to diagnose the errors and reduce variation by process control. This section review s the related research on pro cess errors modeling, diagnosis and control. 1.2.1 Research Review for Modeling Process Errors Fixture error. Fixture error has been c onsidered as one of crucia l factors in the optimal fixture design and analysis. Shawki and A bdelAal (1965) experime ntally studied the impact of fixture wear on the positional accuracy of the workpiece. Asada and By (1985) proposed kinematic modeling, analysis, and characterization of adaptable fixturing. Screw theory has been developed to estim ate the locating accuracy under the rigid body assumption (Ohwovoriole, 1981). Weil, Darel, and Laloum (1991) then developed several optimization approaches to mini mize the workpiece positioning errors. A robust fixture design was proposed by Cai, Hu, a nd Yuan (1997) to minimize the positional error. Marin and Ferreira (2003) analyzed the influe nce of dimensional locator errors on tolerance allocation problem. Researchers also considered the geometry of datum surface for the fixture design. Optimization of locating setup proposed by Weil, et al. (1991) was PAGE 15 5 based on the locally linearized part geometry. Choudhuri a nd De Meter (1999) considered the contact geometry between the locators and workpiece to investigate the impact of fixture locator tolerance sche me on geometric error of the feature. Machine tool error. Machine tool error can be due to thermal effect, cutting force, and geometric error of machine tool. Various approaches have been proposed for the machine tool error modeling and compensation. The cu tting process modeling has been focused on the understanding of cutting forces, dynamics of machine tool structure, and surface profile generation (Smith and Tlusty, 1991; Ehmann, et al. 1991; Kline, Devor, and Shareef, 1982; Wu and Liu, 1985; Sutherland and DeVor, 1986; Altintas and Lee, 1998; Kapoor, et al. 1998; Huang and Liang, 2005; Mann, et al. 2005; Li and Shin, 2006; and Liu, et al. 2006). Machine volumetric error modeli ng studies the erro r of the relative movement between the cutting tool and the ideal workpiece for error compensation or machine design (Schultschik, 1977; Ferreira and Liu, 1986; Donmez, et al. 1986; Anjanappa, et al. 1988; Bryan, 1990; Kurtoglu, 1990; S oons, Theuws, and Schellekens, 1992; Chen, et al. 1993; and Frey, Otto, and Pflager, 1997). A volumetric error model of a 3axis jig boring machine is developed by Schultschik (1977) us ing a vector chain expression. Ferreira and Liu ( 1986) developed a model studyi ng the geometric error of a 3axis machine using homogeneous coordi nate transformation. A general methodology for modeling the multiaxis machine was developed by Soons, Theuws, and Schellekens (1992). The volumetric error model combin ing geometric and thermal errors was proposed to compensate for time va rying error in real time (Chen, et al. 1993). Other approaches, including empiri cal, trigonometric, and e rror matrix methods were summarized by Ferreira and Liu (1986). PAGE 16 6 Machine tool thermal error. With the increasing demand for improved machining accuracy in recent years, the problem of ther mal deformation of machine tool structures is becoming more critical than ever. In or der to maintain part quality under various thermal conditions, two approaches have been studied extensively over the past decades: error avoidance approach and error comp ensation approach (Bryan, 1990). Thermal errors could be reduced with structural improvement of machine tools through careful design and manufacturing technology. This is known as the error avoidance approach. However, there are, in many cases, cost or physical limitations to accuracy improvement that cannot be overcome solely by production and design techniques. Recently, due to the development of sensing, modeling, and comput er techniques, the th ermal error reduction through real time machine tool error compensation has been increasingly considered, in which the thermal error is modeled as a func tion of machine temperatures collected by thermal sensors (Chen, et al. 1993). For most thermal error compensation systems, the thermal errors are predicted with temperatureerror models. The effectiv eness of thermal error compensation largely relies on the accuracy of prediction of tim e varying thermal errors during machining. Various thermal error modeling schemes have been reported in literature, which can be classified into two categories: time inde pendent static modeling and time dependent dynamic modeling. The first category of st udies, time independent static modeling, assumes that thermal errors can be uni quely described by current machine tool temperature measurements (Chen, et al. 1993; and Kurtoglu, 1990). It only considers the statistical relationship between temperatur e measurements and thermal deformations, while neglects the dynamic characteristic s of machine thermoelastic systems. PAGE 17 7 Nevertheless, the information contained in the discrete temperature measurements, which only catches a subset of the whole machine to ol temperature field (Venugopal and Barash, 1986), is incomplete and therefore the problem is not uniquely defined. This motivates the second category of studies for modeling the dynamic effects of thermal errors (Moriwaki, et al. 1998) and the recent pr ogress is to apply syst em identification (SI) theory to thermal error modeling (Yang a nd Ni, 2003). Both these two categories of studies reveal that the number of sensors, sensor location, temperature history, and lagged variable selection are criti cal to achieve high model prediction accuracy and model robustness to different working conditions. As a summary, the studies of process erro rs have been focused on the modeling of individual error sources, process variation monitoring, and variation reduction. Equivalence relationship between multiple errors has not been sufficiently addressed. Causality modeling Models of predic ting surface quality are often deterministic and used for a single machining station (Li and Shin, 2006). In the recent decade, more research can be found to investigate th e causal relationship between part features and errors, especially in a complex manufacturing syst em. The available model formulation includes time series model (Lawless, Mackay, and Robinson, 1999), st ate space models (Jin and Shi, 1999; Ding, Ceglarek, and Shi, 2000; Hua ng, Shi, and Yuan, 2003; Djurdjanovic and Ni, 2001; Zhou, Huang, and Shi, 2003; and Hua ng and Shi, 2004), and state transition model (Mantripragada and Whitney, 1999). The results of the process error model can be summarized as follows. Denote by x the dimensional deviation of a workpiece of N operations and by u =( u1, u2, , up)T the multiple error sources from all operations. The relationship between x and u can be represented by PAGE 18 8 x = =1 +=+,p iii u u (1.1) where is are sensitivity matrices determ ined by process and product design and = 12 p is the noise term. This line of research (Hu, 1997; Jin and Shi, 1999; Mantripragada and Whitney, 1999; Djur djanovic and Ni, 2001; Camelio, Hu, and Ceglarek, 2003; Agapiou, et al., 2003; Agapiou, et al., 2005; Zhou, et al., 2003; Huang, Zhou, and Shi, 2002; Zhou, Huang, and Shi, 2003; Huang, Shi, and Yuan, 2003; and Huang and Shi, 2004) provides a solid foundatio n for conducting further analysis of the error equivalence. 1.2.2 Research Review for Process Root Cause Diagnosis The approaches developed for root caus e diagnosis include variation pattern mapping (Ceglarek and Shi, 1996) variation estimation based on physical models (Apley and Shi, 1998; Chang and Gossard, 1998; Ding, Ceglarek, and Shi, 2002; Zhou, et al., 2003; Camelio and Hu, 2004; Carlson and Sde rberg, 2003; Huang, Zhou, and Shi, 2002; Huang and Shi, 2004; and Li and Zhou, 2006) and variation pattern extraction from measurement data. Ceglarek, Shi, and Wu (1994) develope d root cause diagnostic algorithm for autobody assembly line where fixture errors are dominant process faults. Principal component analysis (PCA) has been applied to fixture error diagnosis by Hu and Wu (1992), who make a physical interpretation of the principal components and thereby get insightful understanding of r oot causes of process variati on. Ceglarek and Shi (1996) integrated PCA, fixture design, and pattern recognition and have achieved considerable success in identifying problems re sulting from worn, loose, or broken fixture elements in PAGE 19 9 the assembly process. However, this method cannot detect multiple fixture errors. A PCA based diagnostic algorithm has also been proposed by Rong, Ceglarek, and Shi (2000). Apley and Shi (1998) developed a diagnostic al gorithm that is able to detect multiple fixture faults occurring simultaneously. Their continuing work in 2001 presented a statistical technique to diagnose root cause s of process variability by using a causality model. Ding, Ceglarek, and Shi (2002) deri ved a PCA based diagnostics from the state space model. However, the number of the simultaneous error patterns may grow significantly as more manufacturing operations are involv ed. The multiple error patterns are rarely orthogonal and they are difficult to distin guish from each other. Therefore, the manufacturing process may not be diagnosable Ding, Shi, and Cegl arek (2002) analyzed the diagnosability of multistage manufacturing processes and applied the results to the evaluation of sensor dist ribution strategy. Zhou, et al. (2003) developed a more general framework for diagnosability analysis by co nsidering aliasing faulty structures for coupled errors in a partially diagnosable pro cess. Further studies are needed on the fault diagnosis for a general machining process where multiple types of errors occur. 1.2.3 Research Review for Process Control The objective of process control is to keep the output as close as possible to the target all the time. Other than the trad itional SPC where Shewhart, EWMA, and CUSUM control charts are the common techniques, automatic process control (APC) and its integration with SPC have gained mo re attention in recent decades. PAGE 20 10Automatic process control. APC uses feedback or feedfo rward control to counteract the effects of root causes and reduce the process variation. Although SPC achieved great success in discrete manufacturing, APC is more likely to be used in continuous process industries where the process out put has a tendency to drift away. The early research on APC can be tracked back to Boxs early research (Box, 1957; Box and Jenkins 1963, 1970; Box and Draper, 1969; 1970; and Box and Kramer, 1992). In APC, the most theoretically discussed control rule is the minimum mean squared error (MMSE) control. It is based on the stochastic control theory (strm, 1970) to find out the optimal control rule to minimize the mean square error of the process output. However, since MMSE control has unstable modes (strm and Wittenmark, 1990; and Tsung, 2000), in some occasions, it causes the process to adapt to the disturbance changes and causes larger output response. In industries, proportionalinte gralderivative (PID) control tuning is the most common control technique (strm, 1988). Its purpose is to reduce the output variance as much as possible based on the PID controller. Compared with many MMSE controllers, PID controller is more robust in varying environments. Integration of APC and SPC. More recently, more research efforts are directed towards the approach combining SPC and APC to secure both the process optimization and quality improvement. MacGregor (1988) was am ong the first to suggest SPC charts to monitor the controlled process. The similari ties and overlap betw een SPC and APC were described. The integration of APC and SPC has been reviewed by Box and Kramer (1992). In these early disserta tions, a minimum cost strategy is suggested to adjust the process and SPC chart is used as dead bands or filtering device (English and Case, 1990) for feedback controlled process. This dead band concept was extended for multivariate PAGE 21 11 problems by Del Castillo (1996). Vander Wiel, et al. (1992) proposed an algorithmic statistical process control (ASPC), which re duces the process vari ation by APC and then monitors the process to detect and remove root cause of variation using SPC. Tucker, et al. (1993) elaborated on the ASPC by gi ving an overall philosophy, guidelines, justification, and indicating related research issues. Parallel to the integration work, rese arch (MacGregor and Harris, 1997; Harris and Ross, 1991) has been implemented for correc ting SPC procedures due to the effect of correlation and applying these procedures for monitoring a controlled process. Tsung (2000) proposed an integrat ed approach to simultane ously monitor and diagnose controlled process using dynamic principal component analysis and minimax distance classifier. In the early research of integrating AP C and SPC, the only monitored variable is the controlled output. Output monitoring alone cannot provide suffic ient information on the process change because it has been comp ensated for by controllers. MacGregor (1991) suggested monitoring the output of the controller. Messina, et al. (1996) then considered the monitoring controller output under an autoregressive moving average disturbance process and proposed jointly monitoring for process output and cont rolled signal. Tsung, et al. (1999) proposed a procedure for jointly monitoring the PID controlled output and controlled signal using bivariate SPC. The SPC robustness was also investigated. In addition, researchers also applied APC and SPC to runtorun (RTR ) process control, which refers to performing control action be tween runs instead of during a run (Del Castillo, 1996; Butler and Stefani, 1994; Mozumder, et al., 1994; Sachs, et al., 1995; and PAGE 22 12 Tsung and Shi, 1999). Del Castillo and Hurwitz (1997) reviewed research work on RTR control. Most of SPC integrated APC approaches have been mainly applied to continuous process. The adjustment in discrete pro cess relies on the control of servo motor, interpolator and adaptive loop in th e machine tools (strm, 1970, 1990) or compensation of individual error sources. Litt le work discussed the potential application of APC in a discrete manufacturing process where the dominant control strategy is to construct control chart to id entify the assignable cause. There is a lack of methodology that can compensate for the joint effect of multiple error sources. 1.2.4 Summary of Literature Review Process modeling. Previous research has been focu sed on the analyses of individual errors and causality modeling in manufact uring processes. The research on the variation reduction and process control has not studied the error equivalence phenomenon in manufacturing processes. There is a lack of physical model to describe the error equivalence so as to study its impact on process control. Model based root cause diagnosis. Previous research has extensively studied the process sensing strategy, statistical proce ss monitoring, diagnosability analysis, and diagnostic algorithms. Those studies did not address the challenges the error equivalence brings to the root cause diagnos is of manufacturing process with multiple error sources. Error compensation. Previous research widely studied the SPC integrated automatic process adjustment in continuous manufact uring processes. Th e traditional error PAGE 23 13 compensation strategy for a disc rete manufacturing process is to offset the process errors individually and may not be cost e ffective. Hence it is desirable to study the impact of the error equivalence mechanism on the error compensation. 1.3 Dissertation Outline The insightful understanding and full util ization of the error equivalence require advances in: mathematical modeling of the error equivalence phenomenon in manufacturing, error equivalence analysis for root cause identification, and error equivalence analysis for automatic process error compensation with integration of SPC. These research aspects constitute the error equivalence theory. The challenge for these research advances is the fusion of engineering science and statistics into the modeling of error equivale nce and the life cycle of controlling process variations. The overall framewor k of error equivalence theo ry is shown in Fig 1.1. Chapter 1 describes phenomenon of error equivalence and reviews the related work for process modeling, diagnosis, and process control. Chapter 2 presents a tentative mathemati cal definition of error equivalence and models the error equivalence phenomenon thr ough a kinematic analysis of workpiece and errors. The error equivalence model has been verified by a real milling process. In addition, a state space model based on error e quivalence is derived to study the variation stackup in the multistage manufacturing proce ss. The procedure of variation propagation model based on error equi valence has been demons trated via a case study. Chapter 3 intends to further explore the error equivalence mechanism and discusses its theoretical implication in root cause identification as well as automatic PAGE 24 14 process adjustment for time invariant errors. A sequential root cause identification procedure has been proposed to distinguish multiple types of errors in the machining processes. The diagnostic algorithm is experi mentally validated by a milling process. The process adjustment based on error equivalence is illustrated with a simulation. Chapter 4 builds a dynamic model of pr ocess errors to study the dynamic error equivalence. In addition, stat istical process control is in troduced to monitor the dynamic equivalent errors. Based on the conclusion of Chapter 4, an automatic process adjustment algorithm using error equivalence is de rived to compensate for dynamic errors in a discrete manufacturing process in Chapter 5. The perf ormance of the adjust ment rule, including stability and sensitivity has been evaluated. Furthermore, the adjustment algorithm is integrated with SPC so that changes in both adjustment algorith m and manufacturing can be detected. Chapter 6 concludes the dissertation. Pr ospects of future research are also discussed. Error equivalence methodology Chapter 2 Error equivalence modeling to predict quality Chapter 3 Error cancellation and its implication Chapter 4 Dynamic equivalent error modeling and inline monitoring Chapter 5 Errorcancelingerror strategy with integration of SPC Design phase Manufacturing phase Engineering science Kinematics Product design Manufacturing Statistics Probabilistic modeling of process uncertainty Dynamic errors Sensing constraints Regression analysis System identification Error sources Variation patterns Sensing constraints Information collection Hypothesis testing Variance decomposition Statistical process control Error sources Adjustment & evaluation Engineering constraints Error equivalence methodology Chapter 2 Error equivalence modeling to predict quality Chapter 3 Error cancellation and its implication Chapter 4 Dynamic equivalent error modeling and inline monitoring Chapter 5 Errorcancelingerror strategy with integration of SPC Design phase Manufacturing phase Engineering science Kinematics Product design Manufacturing Statistics Probabilistic modeling of process uncertainty Dynamic errors Sensing constraints Regression analysis System identification Error sources Variation patterns Sensing constraints Information collection Hypothesis testing Variance decomposition Statistical process control Error sources Adjustment & evaluation Engineering constraints Figure 1.3 The Framework of Error Equivalence Theory PAGE 25 15 Chapter 2 Error Equivalence Modeling and Variati on Propagation Modeling Based on Error Equivalence* This chapter models the phenomenon of the error equivalence in the machining processes by considering how multiple errors (including fixture, and datum, and machine tool) generate the same pattern on part feat ures. The equivalent tr ansformations between multiple errors are derived through a kinematic analysis of process errors. As a result, error sources can be grouped so that root cause identification can be conducted in a sequential manner, which generally require s fewer feature measurements than the previous approaches. The case study demons trates the model validity through a real cutting experiment. The chapter is organized as follows. S ection 2.1 introduces some preliminaries and notations. Section 2.2 defines the erro r equivalence and over views the methodology. Error equivalence model in machining processes is derived in Section 2.3. As an example of applying the error equivalence model, Sect ion 2.4 presents a ne w variation propagation model for multioperational machining proce sses. The case studies have been conducted in Section 2.5. Conclusions and future rese arch work are discussed in Section 2.6. *The work in this chapter ha s appeared in Wang, H., Huang, Q., and Katz, R., 2005, MultiOperationa l Machining Processes Modeling for Sequential Root Cause Iden tification and Measurement Reduction, ASME Transactions, Journal of Manufacturing Science and Engineering 127, pp. 51252. PAGE 26 16 2.1 Preliminaries and Notations This section introduces kinematic analysis of machining process, including representations of surface and its spatial tr ansformation caused by process errors in a manufacturing process. The results will be used to derive error equivalence transformation. By vectorial surface model (Martinsen, 1993; and Huang, Shi, and Yuan, 2003), an Msurface part X is represented as a vector in the part coordinate system (PCS) X = 1 T TTT jMXXX j=1, , M, (2.1) where Xj denotes the jth surface and it is represented as Xj =T TT jjjrvp= (vjx vjy vjz pjx pjy pjz rj)T (2.2) where vj=(vjx vjy vjz)T, pj=( pjx pjy pjz)T, and rj are orientation, location and size of surface j, respectively. Subscripts x, y, and z denote orthogonal directi ons in the coordinate system. M is determined by product design and pro cess planning. The size of cylindrical hole can be represented by the radius of the hole and size of plane is zero. The nominal surface j and part are denoted as 0 jX and X0, respectively. The deviation of Xj is denoted as xj=Xj0 jX= T TT jjjr vpas shown in Fig. 2.1, where Euler parameters and matrix H will be described in Eq. (2 .4). Accordingly, the part deviation is denoted as x= 1 T TTT jMxxx. The feature deviation x of a workpiece can be represented as a function of multiple errors sources (u1, u2, , up)T, x = =1 ()+,p iiifu (2.3) where fi(.)s are functions determined by process and product design. is the noise term. Process errors {ui} involved in machining mainly incl ude those during setup and cutting PAGE 27 17 operations. Since the part is modeled as a vector, operations and their errors can be viewed as vector transformations. Therefor e, homogeneous transformation matrix (HTM) is generally applied to model both operati ons and operational errors. For instance, HTM FHP is used to model the nominal setup at operation k. It transforms 0 jX from the nominal PCS (denoted as PCS0) to the nominal fixture coordinate system (FCS0). Since setup error could be induced by fixture er ror and datum error, we use HTMs Hf and Hd to denote the additional transformation of 0 jX in the FCS0 caused by fixture error and datum error, respectively. y x zn0p0o D0Nominal Feature p1n1D1Machined Feature 32 31 21122 212 ()= 221 0001 eex eey eez Hq y x zn0p0o D0Nominal Feature p1n1D1Machined Feature 32 31 21122 212 ()= 221 0001 eex eey eez Hq Figure 2.1 Modeling of Pa rt Feature Deviation To describe fixture error, the common 321 fixture locating scheme is adopted (Fig. 2.2). The fixture is represented by the positions of 6 locators in the FCS, i.e., (fix fiy fiz)T, i=1, , 6. Not losing generality, the FCS0 is established with f1 z=f2 z=f3 z =f4 y=f5 y=f6 x0. The fixture error is denoted as deviations of locators, i.e., f=( f1 z f2 z f3 z f4 y f5 y f6 x)T .. Cai, Hu, and Yuan (1997) nicely presented the relationship between f and Hf. Their key results are summa rized in Appendix A. PAGE 28 18 Block workpiece Figure 2.2 General 321 Locating Scheme and FCS0 The datum error is included in the incoming workpiece x. For the surfaces used as the primary, secondary, and tertiary datum, their errors are denoted as xI, xII, and xIII, respectively. Datum error is then DIIIIII()TTTTxxxx. The relationship between datum error and Hd will be derived in Section 2.2 using th e concept of equivalent fixture error. The datum error is first converted to the e quivalent amount of fi xture locator errors (denoted as d). Then the results in Cai Hu, and Yu an (1997) can be directly applied to find Hd through d. The nominal cutting operation or th e tool path can be modeled as MHF FHP0 jX, where MHF transforms a part surface from the FCS0 to the nominal machine tool coordinate system (MCS0). (When deriving the results, we choose the MCS0 to be the same as the FCS0, i.e., MHF= I88. Discussion is given in Section 2.4.3 when MHF is not identity matrix.) We use Hm to represent the transformation of tool path (from nominal to the real one) caused by machine tool error. On ly geometric errors of machine tool are considered in this work. Fig. 2.3 shows the transformation due to process errors. As an example to show the form of HTM, Hm is given as PAGE 29 19 0 01m m m mm m mx y z Rot000 H= 0Rot0 00 00 (2.4) where rotation matrix Rotm has the following form under small deviation assumption (Huang and Shi, 2003), 32 31 21122 =212 221mm mmm mmee ee ee Rot and ( e1 m e2 m e3 m)T are deviations of Euler paramete rs, representing deviation of tool path orientation. Rotm on the upper left corner of Eq. (2 .4) transforms the orientation of surface, while the second Rotm transforms the surface position. (xm ym zm)T represents deviation of tool path in position. m is the ratio of actual a nd ideal surface size. When m=1, there is no size deviation due to the machin e tool error. Accordingly, we define the machine tool error as (T m q m1)T, where qm=(xm ym zm e1 m e2 m e3 m)T. The equivalent fixture error due to machine tool is denoted as m. Notations qd and qf can also be introduced for the parameters in Hd and Hf in a similar way. Since datum and fixture errors have no impact on the surface size, we have d= f=1. PAGE 30 20 H ( q ) y x z o y x z oLocator 1 Locator m Locator 2 H ( q ) 32 31 21122 212 ()= 221 0001 eex eey eez Hq Figure 2.3 Modeling of Workpiece Positioning Error 2.2 Mathematical Modeling of the Error Equivalence Phenomenon in Manufacturing Suppose p random error sources uis lead to dimensional deviation x as x = fi(ui)+i, i=1,2, ,p. uis are assumed to be independent from one another and the noise term has mean E(i)=0, and covariance Cov(i)=2i I, where I is an identity matrix. A tentative definition of error e quivalence is given as follows. Definition: Two error sources ui and uj are equivalent if expectation E[fi(ui)]= E[fj(uj)]. That is, the equivalence among random errors is evaluated by the resultant mean shift patterns in product features. It should be noted that errors might not be equivalent under all situations. For instance, the surface profile deviation caused by a machine tool might not be reproduced by a fixture. This study only focuses on the situations that error equivalence holds. If error sources ui and uj are equivalent, it is feasible to transform ui into equivalent amount of error in terms of uj without affecting the analysis of feature deviation x. This fact prompts error equivalen ce transformation to derive the error equivalence model. PAGE 31 21 Positioning errors of workpiece and HTM matrices Process errors *=ix q1 q2 qp H ( q1) H ( q2) H ( qp) Equivalent errors to x1Feature deviation Kiui ** =1= +p iix u 1 2 pu u u 1 2 pu u u Positioning errors of workpiece and HTM matrices Process errors *=ix q1 q2 qp H ( q1) H ( q2) H ( qp) Equivalent errors to x1Feature deviation Kiui ** =1= +p iix u 1 2 pu u u 1 2 pu u u 1 2 pu u u Figure 2.4 Mathematical Mode ling of Error Equivalence Fig. 2.4 outlines the basic idea of mathem atical modeling of the error equivalence phenomenon. If p process errors uis are equivalent, the first step of modeling is to transform uis into a base type error u1 through *=iiiuKu. A significant advantage of this equivalent transformation is that the causal relationship between base error u1 and feature deviation, i.e., x=f1(u1), can be generally applied to ot her types of error sources. The manufacturing operation (e.g., cutting or setup operation) can be represented by a HTM matrix H(q), where the deviation of Euler parameters (q) (see Fig. 2.3) are related to the operational error. The remaining modeli ng steps can therefore be focused on the causal model x=f1(u1) because the transformed errors ius are to be grouped together into =1p iiu with 1u = u1. The process model presented by Eq. (2.3) can be rewritten as x = =1 ()p iiifu Since ius are treated as base error u1, the process model based on error equivalence modeling thus becomes x* 12=11=(,,...,)+= ()p piifuuu fu + (2.5) If function f1 could be approximated by a linear function *, the model becomes x =* u + with u=* =1p iiu. (2.6) PAGE 32 22 The definition also shows the way to check th e error equivalence cond ition. We can first estimate E(*iu) and Cov(*iu) from measurement data using maximum likelihood estimation (MLE) method. Then the definition of error equivalence can be directly applied. Transforming error sources into a base error implies the transformation of manufacturing operations into a base operation, i.e., the operation with base error only. Operations with other types of errors become flawless because all the process errors have been transferred to the base operation. The derivation of this dissertation is under the linearity assumption, under which equivalence transformation and quality pr ediction model assume linear form. The nonlinear deformation of products is not considered in this study. 2.3 Error Equivalence Modeli ng for Machining Processes We first introduce the concept of equivale nt fixture error, by which a variation propagation model is develope d by grouping fixture, datum, and machine tool errors. Condition of error grouping is also discussed in this section. 2.3.1 Concept of Equivalent Fixture Error In a general machining process, three ma jor error sources are considered: fixture error f, machine tool error qm, and datum surface error xD. The fixture error is chosen as the base error because of the following reasons: Fixture error is simply repres ented by the deviation of fi xture locators, while machine tool error is relatively complicated. The da tum error is usually caused by fixture or machine tool errors. PAGE 33 23 Fixture error has been well st udied. Methods are readily avai lable for the analysis of workpiece positioning error (Weill, Darel, and Laloum, 1991; Rong and Bai, 1996; Cai, Hu, and Yuan, 1997; Wang, 2000; and Marin and Ferreira, 2003), the resultant feature deviation, and fixtur e error diagnosis (Hu and Wu, 1992; Apley and Shi, 1998, 2001; and Ceglarek and Shi, 1996). Flexible fixtures have been availa ble whose locators are adjustable for accommodating a product family. It is possible to adjust the locator lengths for the purpose of error compensation. The base error in terms of fixture error is called equivalent fixture error (EFE), which can be illustrated with a 2D block workpiece (Fig. 2.5). Equivalent locator error Machine tool error Actual tool path Nominal tool path m1 m2 m3Equivalent locator error Machine tool error Actual tool path Nominal tool path m1 m2 m3 (a) (b) Figure 2.5 Equivalent Fixture Error In Fig. 2.5(a), the dash line block with surfaces (0 1X 0 2X 0 3X 0 4X) is in its nominal setup position. Due to datum error occurring on surface X1, the block has to be transformed to position (X1 X2 X3 X4) (the solid line block) around the locating point f3. The workpiece position transf ormation is described by HTM Hd. The EFE due to datum error, denoted by d=( d1 z d2 z d3 z d4 y d5 y d6 x)T, can be derived by finding the difference between actual (Hd FHP (0T jX 1)T) and nominal datum surfaces (FHP (0T jX 1)T), where {j}{I, II, III}. In Fig. 2.5(a), the equivalent fixtur e deviation is d1 and d2. In Fig. 2.5(b), EFE due to machine tool error can be derived in a similar way. The left panel X3 X2 X1 X4 f1 f2 f3 0 4X0 1X0 2X d1 d2 0 3X PAGE 34 24 shows that the machined surface X3 deviates from designed position 0 3X due to machine tool errors. The EFE transforms the workpiece from nominal position (0 1X 0 2X 0 3X 0 4X) to dash line position shown in right panel. A nominal cutting operation can yield the same surface deviation as machine tool error does in the left panel. Therefore, the inverse of Hm transforms X3 to its nominal position 0 3X in the FCS. The EFE due to machine tool error, denoted by m=( m1 z m2 z m3 z m4 y m5 y m6 x)T, can be uniquely determined by the difference between 1mHFHP (0T jX 1)T and FHP (0T jX 1)T at the locating point, where {j}{I, II, III}. In this example, the equivalent fixture locator deviation m1 and m2 is determined by difference between surfaces 0 1X and X1 at locating point 1 and 2. m3 can be computed by the difference between surfaces 0 2X and X2 at locating point 3. 2.3.2 Derivation of EFE Model The equivalent locator deviation caused by either datum error or machine tool error can be computed by the distance be tween two points where locators intersect the nominal datum 0jX=000000( )T jxjyjzjxjyjzvvvppp and deviated datum surfaces Xj=(vjx vjy vjz pjx pjy pjz) (Fig. 2.6), where j=I, II, III represents three datum surfaces. nj is the normal vector of datum surface and it is equal to orientation vector vj when datum surface is planar. Let d=( d1 z d2 z d3 z d4 y d5 y d6 x)T and m=( m1 z m2 z m3 z m4 y m5 y m6 x)T represent EFEs caused by datum and machine tool errors, respectively. Using analytical geometry, EFEs can be derived as PAGE 35 25IIIIII IIIIIIIIIIII IIIIIIIIIIIIIIIIII()[()()]/,1,2,3, ()[()()]/,4,5, ()[()()]/,6.izizxixxyiyyzziz iyiyxixxzizzyyiy ixixyiyyzizzxxixdormnfpnfpnpfi dormnfpnfpnpfi dormnfpnfpnpfi (2.7) The orientation vector nj and position pj of the plane Xj can be further expanded by datum error xj or machine tool error qm. Figure 2.6 EFE Derivation When computing d, deviated surface Xj can be determined by datum error plus the nominal, i.e., 0jjjXXx. Eq. (2.7) is then linearized as: III IIIIII IIIIIIIII,1,2,3, ,4,5, ,6.izixxiyyz iyixxizzy ixiyyizzxdfvfvpi dfvfvpi dfvfvpi or I 2II III. x dKx x (2.8) The mapping matrix relating datum error to d is 1 22 3 G00 K0G0 00G where 11 122 330001 0001 0001xy xx xxff ff ff G, 44 2 550010 0010xz xzff ff G, and 3660100yzff G. When deriving m, we use the relationship between Xj and machine tool error qm. Linearization of Eq. (2.7) using the firs t order of Tayler e xpansion then yields d (or m) 0 j X Xj Deviated datum surface Locators Nominal datum surface nj 0 j n PAGE 36 263 mKqm, and 11 22 33 3 44 55 66001220 001220 001220 010202 010202 100022yx yx yx z x z x zyff ff ff f f f f ff K. Figure 2.7 NonPlanar Datum Surfaces This modeling is applicable for the case where datum surfaces are all planes. When the surface is not planar, we should use tangential plane of su rface at each locating point as datum surface. Fig. 2.7 shows the setup of a 2D part with nonplanar datum surfaces. The datum surfaces are tangential planes T1, T2, and T3. The corresponding normal vectors are n1, n2, and n3, respectively. If the implic it form surface equation is represented by fj(xj, yj, zj)=0, nj and pj are determined by ,(,,)0T jjj jjjxjyjz jjjfff fppp xyz n, j=I,II, ,VI. (2.10) The following is for a brief derivation on orientation v and position vector p of three datum surfaces. If the features j1, j2 and j3 that are selected as the first, second and tertiary datum surface are planar, orientati on vector of three datum surfaces can be vI=vj 1, vII=vj 2, and vIII=vj 3. However, if j2 and j3 are cylindrical holes where round pin and diamond pin reside respectively, such locating scheme is equi valent to a simplified 321 T1 T2 n2 n1 n3 T3 (2.9) PAGE 37 27 fixture locating scheme as shown in Fig. 2.8. We can set the origin of fixture coordinate system at the point of pj2, and f1 z=f2 z=f3 z=f4 x=f4 y=f4 z=f5 y=f5 z=f6 x=f6 y=f6 z=0. z y vIvIII j2j3 f5f1f2f f3f6vj 3vj 2pj 3pj Second datum surface II Primary datum surface I Tertiary datum surface III Locating point z y vIvIII j2j3 f5f1f2f f3f6vj 3vj 2pj 3pj Second datum surface II Primary datum surface I Tertiary datum surface III Locating point Figure 2.8 PinHole Locating Scheme The orientation vector for second datum surface is defined to be 223 323233323332333233323332II() = ,jjj jyjzjyjzjzjyjzjyjzjxjzjxjxjzjxjzjxjyjxjyjyjxjyjxvpvpvpvpvpvpvpvpvpvpvpvp vvpp (2.11) where orientation vj and position pj of holes are parameters that vary within infinitesimal range. The normal vector for th e tertiary datum surface is 23232323IIIjjjxjxjyjyjzjz p ppppp npp. (2.12) Deviation of normal vector is determin ed by differentiation and linearization of vI, vII, and vIII. The results are given as follows: If three datum surfaces are planar: nI= vj1, nII= vj2, nIII= vj3 and (2.13) pI= pj1, pII= pj2, pIII= pj3. PAGE 38 28 If j1 is plane, j2 and j3 are cylindrical hole: n1= vj 1, 2333232 32332 323233233II II II(), (), ()()(),xjyjyjzjyjyjyjy yjxjxjzjxjx zjyjyjxjxjxjzjxjxjznppvpppp nppvpp nppvppvppv 23 23 233 3 3, x jxjx yjyjy zjzjznpp npp npp 2.4 Variation Propagation M odeling Based on Error Equiva lence for MultiOperation Machining Process 2.4.1 Background Review for MultiOp erational Manufacturing Process Due to the increasing complexity of pr oducts and the requirements of quick response and flexibility, manufacturing pro cess has evolved into complex systems consisting of many stages, where the vari ation can be accumulated through multiple stages onto the final product. Such variation transmission ha s been widely investigated. Variation propagation modeling has been proved to be an effective way for variation reduction and design synthesis in multioperational manufacturing processes. A brief review is given to the prev iously developed state space model. For an Noperation manufacturing pr ocess, the state of the kth operation x(k) is described as a linear combination of the previous state x(k1), process input u(k), and natural process variation (k). Quality characteristic y(k) is a linear transformation of state x(k) plus measurement noise (k). Under small deviation assumption, the model has the following form (2.14) PAGE 39 29x(k)=A(k1)x(k1)+B(k)u(k)+(k), k=1, 2, , N, (2.15) y(k)=C(k)x(k)+(k), {k}{1, 2, , N}. For machining processes, state vector x(k) represents the deviations of part features. The process deviation u(k) includes fixture and machin e tool deviations, while the datum deviation is contained in x(k1). State transition matrix A(k1) and input coefficient matrix B(k) are constant matrices determin ed by product and process design. The matrix C(k) is determined by measurement design. Denote by y the quality characteristics of N operations and by u the process deviations from all operations. The relationship between y and u can be obtained by solving Eq. (2.15), which ends up with a linear model in the form y= u+ or x= u+. Diagnosis and measur ement synthesis can be performed by analyzing the rank of matrix (Ding, et al., 2003; and Zhou, Huang, and Shi, 2003). The problem en countered, however, is that is often not full rank for machining processes. One natural thought is to increase the di mension of quality characteristics y to increase the rank of matrix. Nevertheless, this strategy cannot guarantee the full rank of because datum, fixture, and machine tool errors could generate the same error patterns on part feat ures. Previously developed approaches for machining processes (Huang, Shi, and Yuan 2003; Djurdjanovic a nd Ni, 2001; and Zhou, Huang, and Shi, 2003), however, did not model th e error equivalence. Consequently, it is difficult to distinguish error sources at each operation (Huang and Shi, 2004). The strategy proposed in th is chapter is to formulat e the variation propagation model using the proposed EFE concept. With th is concept, datum e rror and machine tool error are transformed to equiva lent fixture locator errors at each operation. As a result, the dimension of u can be reduced by properly grouping three types of errors together. PAGE 40 30 The rationale of the proposed methodology is to conduct measurement in a sequential manner for root cause identification. First, only necessary inform ation is provided to identify whether there is any error in the process. If not, additional measurement is deemed as waste of resources. Second, if any error is identified, further measurement will be conducted to distinguish three types of errors. This methodology generally requires less feature measurements than the previous approaches. A detailed diagnostic algorithm will be presented in Chapter 3. 2.4.2 Variation Propagation Model Derivation This section shows the derivation procedure for the surface deviation xj(k) (Fig. 2.9). It can be easily ex tended for part deviation x(k) and establishing state space model. Figure 2.9 Model Derivation Step 1 models how feature quality is affected by faulty setup and cutting operation at the kth stage. Parameters qd(k), qf(k), and qm(k) in HTMs are intermediate variables linking f(k) d(k), and m(k) with feature deviation xj(k). Step 2 derives how fixture error f(k), EFE d(k) and m(k) affect qd(k), qf(k), and qm(k), respectively. Step 3 describes how errors from previous ope ration (datum error) affect d(k). f(k) m(k) qm(k) d ( k ) qd(k) xj(k) qf(k) Step 1 Step 3 Ste p 2 xI(k1) xII(k1) xIII(k1) PAGE 41 31Step 1. After setup operation, the part surface can be represented by FHP(k)(0()T jk X 1)T. The machined surface j is represented as Hm(k)FHP(k)(0()T jkX 1)T in the FCS0. After transforming the surface to the PCS0 (Huang and Shi, 2003), the actual surface Xj(k) is: 1110()1()()()()()()1,T T TFFT jPdfmPjkkkkkkk X=HHHHHX (2.16) where 0000000()(()()()()()())T jjxjyjzjxjyjzkvkvkvkpkpkpk X. By substituting Eq. (2.4) into Eq. (2.16), we can com pute the actual machined surface Xj(k). After ignoring higher order error terms, Eq. (2.16) can be rewritten as: 61 0 118()()() ()()() ()jdjfjm j j jkkk kkk rk AAA0 xq 0, and 00 00 00 00 00 00000022 000202 000220 ()()() 100022 010202 001220jzjy jzjx jyjx jdjfjm jzjy jzjx jyjxvv vv vv kkk pp pp pp AAA if FHP(k)=I, where rank(Ajd) 5 and ()=()()()()1T TTT dfmmkkkkk qqqq. (k) is the modeling error for operation k. Index k is omitted within matrices Ajd(k), Ajf(k) and Ajm(k). The q(k) can be grouped because of Ajd(k)=Ajf(k)=Ajm(k). Eq. (2.17) is 61 T 0 16() ()(()0)(()())()1() ()T jd TT jdfmm jk kkkkkk rk A0 xqqq 0 (2.18) where the dimension of q ( k ) is reduced from 19 to 7. The expression for Ajd( k ), Ajf( k ) and Ajm( k ) in Eq. (2.17) is only given under the condition of FHP( k )= I In Section 2.4.3, we will show that Ajd( k )= Ajf( k )=Ajm( k ) and error grouping still hold if FHP( k ) I (2.17) PAGE 42 32 Step 2 Relationship between qf( k ) and f has been given as 1fqJ Ef by Cai, et al. (1997) (refer to Appendix A for a brief su mmary of the result). By the concept of EFE, d and m are equivalent to f Therefore, qd( k ) and qm( k ) can be determined accordingly by the same approach, i.e., 1dqJ Ed and (2.19) 11( ) ,mqJ EmJ Em (2.20) where matrix E is an 186 matrix (see Appendix A). Since 1 m H (not Hm) transforms the workpiece from nominal position to its real position in the FCS (refer to Fig. 2.5(b)), we add minus sign before 1J Em in Eq. (2.20). It turns out that Jacobian matrix J and orientation matrix in Eqs. (2.19) and (2.20) are the same as those in Eq. (A.1). Therefore, we still can group e rrors after substituting Eqs. (A.1), (2.19), and (2.20) into Eq. (2.18), ()() ()0()()()T T jjjjkkkkkk xBdBu (2.21) where 1 61 0 16()()() () ()jd j jkkk k rk AJ E0 B 0 is the input coefficien t matrix linking errors at the current operation with feature deviation, rank (1()()()jdkkkAJ E) 5, and u ( k )=(( f ( k ) + m ( k )) T, m( k )1) T. Step 3 EFE d ( k ) in Eq. (2.18) becomes I II 2222 III 7222222 221(1) (1) () (1) 111 1 k k k k x x d 0G0 H x 00 (2.22) PAGE 43 33 where matrix H transforms deviations of th ree datum surfaces from PCS0 to FCS0. It is defined as 13 13 13 121(0) (0) (0) 1FFF T PPP FFF FT PPPP FFF T PPPxyz xyz xyz 0 R0 0 0; where FRP= diag(FRotP FRotP m FRotP FRotP m FRotP FRotP m). FRotP is the rotational block matrix in FHP. (FxP FyP FzP)T are translation parameters. Matrix 1 2 3= 00 0 0 00 maps the deviation of workpiece to the EFE with 11 122 3300010 00010 00010xy xy xyff ff ff 44 2 5500100 00100xz xzff ff and 36601000yzff Matrix G is introduced for computing deviation of orientation vector of datum surface under two conditions: If all datum surf aces are planar: G = I ; If XI is plane, XII and XIII are cylindrical holes, G can be obtained by differentiating IIIIIII() vpp and pIIpIII. Considering the results in Eq. (2.14), we have 77 1112 434447 2122 434447 00 I GG G0 0I0 GG 0 0I0, where 23 32110000100 001000 000000jxjx jxjxpp pp G 120000100 0001000 0000000 G, 210001000 0000100 0000010 G, 220001000 0000100 0000010 G. PAGE 44 34 Substituting Eq. (2.22) into Eq. (2.21), state transition matrix Aj( k 1) can be obtained and we derive the variatio n propagation model for the surface j at operation k If we assemble the model for all the features and datum surfaces, the equation in the form of the state space model can be obtained. The dimension of input vector u ( k ) is reduced from 13 to 7 because of error grouping. Thus the order of matrix T* is greatly reduced. The dimension of output vector x ( k ) required to make T* full rank is reduced as well. When FCS, PCS, and MCS coincide, and the orie ntation vectors of datum surfaces are (0 0 1 0 0 0 0)T, (0 1 0 0 0 0 0)T, and (1 0 0 0 0 0 0)T in the FCS, we get input matrix *j corresponding to the machined surface j 0000000T xyzxyzvvvpppas *1jjd AJ E, (2.23) which yields *j matrix, i.e., PAGE 45 35 j f 3x f 4z f 5z v y0 f 2x f 4z f 5z v y0 f 4x f 5x f 2y f 3y v z0 f4xf5xf3xf1yf2yf2xf1yf3yfxf2yf3y f 3x f 4z f 5z v y0 f 1x f 4z f 5z v y0 f 4x f 5x f 1y f 3y v z0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3y f2xf3xf4zvx0f5zvx0f4xf5xvz0 f4xf5xf3xf1yf2yf2xf1yf3yfxf2yf3y fxf3xf4zvx0f5zvx0f4xf5xvz0 f4xf5xf3xf1yf2yf1xf2yf3yf2xf1yf3yf2yvx0f3yvx0f2xf3xvy0 f3xf1yf2yf1xf2yf3yf2xf1yf3yf1yv1x0f3yvx0f1xf3xvy0 f3xf1yf2yf1xf2yf3yf2xf1yf3yf2xf4zf5zf6ypy0f3xf4zf5zf6ypy0f4xf5xf2yf3yf6zpz0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3yf1xf4zf5zf6ypy0f3xf4zf5zf6ypy0f4xf5xf1yf3yf6zpz0 f4xf5xf3xf1yf2yf1xf2yf3yf2xf1yf3y f2xf3xf4zf5zpx0f5xf4zpz0f4xf5zpz0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3y f1xf3xf4zf5zpx0f5xf4zpz0f4xf5zpz0 f4xf5xf3xf1yf2yf1xf2yf3yf2xf1yf3y f2yf3ypx0f3xf2ypy0f2xf3ypy0 f3xf1yf2yf1xf2yf3yf2xf1yf3y f1yf3ypx0f3xf1ypy0f1xf3ypy0 f3xf1yf2yfxf2yf3yf2xf1yf3y f 2x f 4z f 5z v y0 f 1x f 4z f 5z v y0 f 4x f 5x f 1y f 2y v z0 f4xf5xf3xf1yf2yf2xf1yf3yfxf2yf3y v y0 f4xf5x v y0 f4xf5x0fxf2xf4zvx0f5zvx0f4xf5xvz0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3yvx0 f4xf5xvx0 f4xf5x0f1yvx0f2yvx0f1xf2xvy0 f3xf1yf2yf1xf2yf3yf2xf1yf3y000f1xf4zf5zf6ypy0f2xf4zf5zf6ypy0f4xf5xf1yf2yf6zpz0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3yf6ypy0 f4xf5xf6ypy0 f4xf5x1f1xf2xf4zf5zpx0f5xf4zpz0f4xf5zpz0 f4xf5xf3xf1yf2yf2xf1yf3yf1xf2yf3yf5xpx0 f4xf5xf4xpx0 f4xf5x0fyf2ypx0f2xf1ypy0fxf2ypy0 f3xf1yf2yf1xf2yf3yf2xf1yf3y000 where we can see that matrices *j corresponding to three EFEs are the same. The structure of Eq. (2.17) proves our prev ious claim that it is hard to conduct root cause identification using previously developed models. It also reveals that fixture and machine tool cannot be distinguished without inprocess measurements on either fixture locators or the machin e tool at each operation. 2.4.3 Discussion for Error Grouping in Machining Processes In Section 2.4.2, the model derivation is based on the assumption that transformation matrix FHM( k ) is identity. In addition, the expression of Ajd ( k ), Ajf( k ) and Ajm( k ) are given under the condition of FHM( k )= I In this section, a necessary and sufficient condition for erro r grouping is discussed. PAGE 46 36 Proposition 2.1 (Condition on grouping variables) The linear equation 1212==,TT nmxxxuuu x (2.24) where ={ gij}nm, i =1,2, , n ; j =1, 2, , m ; x1, x2, ..., xn and u1, u2, ..., um are variables, can be grouped into the following form 12T n p ppu x with 1122...mmukukuku (2.25) where pi and kj are certain coefficients, if and only if the rank of matrix is one or zero. In our study, the coefficient matrices of d f and m are the same, (see Eqs. (A.1), (2.19), and (2.20)), which satisf ies the sufficient condition for grouping. In the above discussion, we a ssume the transformation matrix FHP and FHM to be identities. If three coordinate systems do not coincide with each other, the coefficient matrices for d f and m are still the same when FHP I88 and FHM = I88. However, this is not true when FHM I88. We have the following conclusion. Corollary MCS0 and FCS0 must coincide to perform error grouping in the proposed model. However, this requirement can be easily satisfied in modeling stage. The proofs of the proposition and corollary ar e listed in Appendix B. 2.5 EFE Validation and Modeling Demonstration This section validates the EFE with a milling process and demonstrates the modeling procedures for a multioperational machining process. PAGE 47 37 2.5.1 Experimental Validation of EFE We machine 6 blocks to validate EFE mode l. The first three parts are cut with only datum error, while the rest are cut with only machine tool error. The datum error and machine tool error are set in such a way that d = m =(1.105 0 0 0 0 0)T, i.e., their EFEs are the same based on Eqs. (2.8) and (2.9). Then we can measure the machined surface and compare the surface orientation and position. Fig. 2.10 shows the specification of ra w workpiece and fixture layout. Only top surface X is machined and its specification is X0= (0 0 1 0 0 20.32 0) T. Using Eq. (2.21), the deviated surface X is predicted as (0 0.0175 0.9998 0 0 18.88)T. Figure 2.10 Raw Workpiece and Locating Scheme (Unit: mm) Table 2.1 shows the measurement of the machined surface. As can be seen, the discrepancies between two samples are very small. The measurement data are also comparable with the predicte d results. Therefore, the e xperiment supports EFE model. Locating Point f1 f2 f3 f4 f6 z y ( a ) ( b ) 22.860.01 101.60.1 76.20.1 19.20.01 570.01 190.01 63.30.01 50.80.01 80.01 76.20.1 X f5 y x z x x Grooves 380.01 190.01 PAGE 48 38 Table 2.1 Measurement Results (Under PCS0) X v x v y v z p x p y p z Sample 1 (Datum Error) 0 0.0174 0.9998 0 0 18.880 0.0001 0.0174 0.9998 0 0 18.882 0 0.0174 0.9998 0 0 18.881 Sample 2 (Machine Tool Error) 0 0.0172 0.9999 0 0 18.880 0.0001 0.0173 0.9999 0 0 18.884 0 0.0163 0.9999 0 0 18.887 2.5.2 MultiOperational Variation Pr opagation Modeling With Grouped EFEs A machining process for V8 cylinder head is employed to illustrate modeling procedure and the advantage of the modeling approach. Th e drawing of workpiece and the locating points are shown in Fig. 2.11. The surfaces chosen are marked as X1X8. X1 is the exhaust face, while X2 and X3 are two cup plug holes on the X1. X4 is spark plug tube hole and X5 is a hole for the exhaust lash adjuster. X4 and X5 are two angle holes and the specifications are given in section plots S1S1 and S2S2. Center of X7 is set to be the origin of nominal part coordinate system. Based on the dimensions shown in Fig. 2.11, the specification of each machined surface is listed in Table 2.2. PAGE 49 39 x X3p2X8 s1 X6 p3 X4 p2 s1 400 350X1p3 300 X4 s1s1 X7 y p1 p1 52.69X6 X7 X6 64.5 X5 s2 19.25 X2 X5 s2s2115.09 44.41 s250 X7 X6 74 Figure 2.11 Workpiece and Locating The workpiece goes through two operations (Fig. 2.12): the first operation mills X1 and drills X2 and X3 using datum surfaces X6, X7 and X8; and the second operation drills X4 and X5 using datum surface X1, X2 and X3. The locator positions on the primary datum planes are given in Table 2.3. Table 2.2 Machined Features Specification Feature Component Part Features (In the PCS0, Unit: mm) X1 X2 X3 X4 X5 X6 X7 X8 vx( k ) vy( k ) vz( k ) px( k ) py( k ) pz( k ) r ( k ) 0 1 0 0 131 0 0 0 1 0 19.25 131 81.25 7.5 0 1 0 319.25 131 81.25 7.5 0 0.43 0.90 350 52.69 0 4.6 0 0.28 0.96 50 44.41 115.09 16.92 0 0 1 0 0 0 0 0 0 1 0 0 0 5 0 0 1 400 0 0 5 Table 2.3 Coordinates of Locating Points on the Primary Datum Surfaces (Unit: mm) p1 p2 p3 Operation 1 (7, 109, 0) (407, 109, 0) (200, 11, 0) Operation 2 (19.25, 131, 61.25)(319. 25, 131, 61.25) (169.25, 131, 11.25) PAGE 50 40 X6 X1 X6 X1 Operation 1 Operation 2 Figure 2.12 Two Cutti ng Operations The state vector is x ( k )= 12345()()()()()T TTTTTkkkkkxxxxx. Since diagnosis of feature size is relatively straightforward, we do not consider effect of size. In this case study, we also assume that the workpiece is perfect, i.e., (0)j x0, j =1,2, ,5. As a comparison, before using the proposed methodology, we can check the number of necessary measurements for identif ying errors via previously proposed model (Zhou, Huang, and Shi, 2003; Huang and Shi, 2004). It can be observe d that there are 12 error components (6 fixture a nd 6 machine tool error compone nts) as input to the model for each operation and therefore, total 24 inputs entails 24 components in quality characteristic for root cause identification. Since each feat ure contains 6 components, at least 12/6=2 features are requi red for each operation. However, we have shown in Eq. (2.21) that the rank of block matrix 1()()jdkkAJ E in Bj( k ) does not exceed 5. More features information is needed to identify all the errors. Therefore, the number of features identifying errors for each operation should be no less than 3. In this case study where only two operations are considered, total amount of measured features should not be less than 32=6 even if the purpose is to identify wh ether errors occur in the process. Using Eqs. (2.21) and (2.22), we calculate Aj( k ) and Bj( k ), based on which the model in the grouped form is formulated as follows. PAGE 51 41 Operation 1: Because the first operation only mills X1 and drills X2 and X3, input matrices for features 4 and 5 are zero. The results are: 61 1 160000.00250.00250 000000 0.00420.00420.0083000 0000.32750.32751 (1), 000100 1.07480.10860.1833000 0 0 B 0 61 2 160000.00250.00250 000000 0.00420.00420.0083000 0.19630.196300.32750.32751 (1), 0.33850.33850.67710.95190.04810 1.0283015510.1833000 7.5 0 B 0 61 66 3 45 160000.00250.00250 000000 0.00420.00420.0083000 0.19630.196300.32750.32751 (1),(1),and(1) 4.6 0.33850.33850.67710.20190.79810 0.30360.87970.1833000 7.5 0 0 BBB 066. 16.92 0 The state equation for operation k can be assembled as: 1 77 1 77 2 77 3 77 4 77 5 ((0)1) ((0)1) ((1)1)(1) (1) (1) (1) (1) (1) 11 1TT TTdiag x xB 000000 x B 0 00000 x 00 0000 x 000 000 x 0000 00 x 00000 11 22 2 33 3351 44 4 55 5 66 135 21 ((1)0)(1)(1) (1)(1) (1) (1)(1) (1) (1) (1)(1) (1) 0 (1)(1) (1) (1)(1) 0zz zz zz yy yy xx diagfm fm fm fm fm fm BB0 B B 0 0 where identity block matrix in A (0) represents that the corresponding features have not been machined. Since HTM is used to derive d ( k ) as shown in Eq. (2.22), dimension of state vector has to be increased by us ing as the last entry, i.e., ( xT(k) 1)T. ( k ) are the stackup of j( k ), where j =1, 2,,5. Zeros in the last ro w of the model are introduced to make the matrix dimension consistent. Operation 2: Since FHP(2) I expression of Ajd( k ) presented in Eq. (2.17) does not apply for the second operation. However, accordi ng to the corollary in Section 2.4.3, we PAGE 52 42 can still derive A (1) and B (2) by substituting nonidentity matrix FHP in (2.16), followed by the same procedure for deriving Eqs. (2.17), (2.18), and (2.21). 61 4 160.0010.00100.00230.00230 0.00380.00380.0075000 0.00180.00180.0036000 0.18920.189200.2025020251 (2), 0.60810.024815833000 0.32630.32630.6526017250.82750 4.6 0 B 0 61 5 160.0070.00700.00240.00240 0.0040.0040.008000 0.00110.00110.0023000 0.20920.209200.08520.08521 (2), 0.59610.22030.6242000 0.36080.36080.72160.92250.07750 1692 0 B 0 B1(2)= 077, and B2(2)= B3(2)=diag( 066, 7.5). Since datum error is generated by the firs t operation, state transition matrix must be calculated. By Eqs. (2.21) and (2.22), rotational deviation of the surface caused by datum errors can be expressed by 1 1 2 3(1) ()()()(1) (1)F jdPkkk x AJ E RGx x, where j =4, 5. For the convenience of displaying results, we can denote 1 123()()()F jjjjdPkkk AJ E RG. The results are 41 42 17 170.430500000000000902600 000.902600000000000 000.4305 00000000000 78.3100000000018100 ,, 33108101000000000 0078.310000000033110 00 43 1700000.902600 0000000 0000000 00008100 0000000 000033100 0 51 52 17 170.275600000000000.961300 000.961300000000000 000.275600000000000 86.59000000000134.0900 ,, 31034.0901000000000 0086.59000000003110 00 53 1700000.961300 0000000 0000000 000034.0900 0000000 00003100 0 PAGE 53 43 The translational deviation of surface can be calculated by 1 131414()()()(0) F FFFFFFFFT jdPPPPPPPPPkkkxyzxyzxyz AJ E 000. We denote this expression as a column vector 4j The calculation results are 54= 44=( 013 19.25 131 81.25 0)T. The state equation can be assembled as 7771 11 7771 22 7771 3 414243777744 4 515253777754 5 ((1)1) ((2)1)(2)(1) (2)(1) (2) (2) (2) 1 1TTdiag A xI00000 xx 0I0000 xx 00I000 xx 00 x 00 x 00000 11 1 22 2 33 3351 3 44 4 4 55 5 5 135 ((2)1) ((1)1)(2)(2) (2) (2)(2) (2) (2)(2) (2) (1) (2)(2) (2) (1) (2)(2) (2) (1) 0 1TTzz zz zz yy yy diagfm fm fm fm fm B xB B B0 B x B x 0 66 21(2) 0 (2)(2)xxfm 0 Solving the state equation for two operations, the model for root cause identification is (1)0(1)0(1)0 (1) givenby (1)(1)(1) (2)0(2)0(2)0TTT TTT TTT TTT yu B0 y=C ABB yu Output matrix C is determined by the selection of measured features. An optimized selection of measured features for root cause identification must maximize the rank of matrix while minimize number of rows in matrix C i.e., the minimum number of components in vector y In this example, the number of errors to be determined is 12 and the minimum number of feature components to be measured should be 12. Each feature component is selected as one entry in vector xj( k ), e.g., vj( k ) in xj( k ) can be chosen as a feature component. Thus, th e entry 1 appears at most on ce in each row of feature selection matrix C The position of 1 is dete rmined by nonzero entry in (1) (1)(1)(2) B0 ABB. For this case study, 4 features are selected and the output matrix C is PAGE 54 44 chosen as: 12 1272 34 0CC0000000 C 00000000CC, where 14 141414461 1410 00 0 C00I0 0, 55 2611 0 C0, 14 341414461 1410 00 0 C00I0 0, and 55 4621 0 C0. The size of other zero block matrices in C is 77. After removing the zero rows in u and corresponding columns in (1) (1)(1)(2) B0 C ABB, we obtain the equation (1) (1) (1) (2) (1) (1) yfm y= yfm for diagnosis of errors that oc cur at each operation, where is the noise term composing of (1) and (2) in the first and second operations and *0000.00250.00250000000 0.00830.00420.0042000000000 00.19630.19630.32750.32751000000 0.67710.33850.33850.95190.04810000000 0.18331.02830.1551000000000 0.18330.30360.8797000000000 0000.6780.6 78000.0010.0010.00230.00230 0.00360.00180.00180000.00360.00180.0018000 00.19630.196360.618360.6183100.18920.18920.20250.20251 0.6750.33750.33750.17250.827501.58330.68010.0248000 0.46920 .7020.1712248.25248.2500.65260.32630.32630.17250.82750 0.53830.66750.205723.2523.2500.72160.36080.36080.92250.07750 It can be observed that the rank of is 12. The least square estimation can thus be performed. Therefore, measuring 4 featur es makes it possible to identify 12 error components. Only 12 components in quality ch aracteristic y are needed for identifying if there are errors. The proposed approach iden tifies location of root cause without having to find out every potential error. Compared with quality characte ristic components (at PAGE 55 45 least 24) and 6 features measured based on th e previous model, reduction on the model dimension and measurements by the proposed ap proach is significan t. If fixture and machine tool errors should be further disti nguished, the strategy of sequential root cause identification suggests that a dditional inprocess measurement only needs to be taken on the faulty (equivalent) locator(s). Therefore, the proposed strategy generally requires less features and inprocess measurement for root cause identification. 2.6 Summary This chapter presents a mathematical formulation of error equivalence and prediction of process variations. The error equivalence formulation, based on a novel concept of error equivalence transforma tion, helps to understand and model the mechanism that different error sources result in the identical variation pattern on part features. The derived qualit y prediction model (causal m odel) embedded with error equivalence mechanism can reveal more physic al insights into the process variation. As an application of error equivalence model in a multioperational machining processes, this chapter presents a variation propagation modeling that facilitates root cause identification and measurement strate gy. The benefit of introducing equivalent errors in the process modeling is that the process errors can be grouped with the base error (in the machining process, datum error and machine tool error can be grouped with fixture error). As a result, the dimension of model inputs is significantly reduced compared with previous modeling methodologies. The feasibility of error grouping is discussed It is shown that the symmetry of HTM in the infinitesimal analys is is the key factor for erro r grouping since the coordinate PAGE 56 46 transformation may possibly violate the symm etry in HTM multiplication. The modeling results indicate that HTM between the PCS a nd the FCS does not affect the symmetry in HTM multiplication. This grouping approach requires merging the MCS and the FCS during modeling to satisfy the condition of grouping. The requirement can easily be satisfied in the modeling stage. The case studies demonstrated the valid ity of error equivalence model in the machining process, modeling procedure, and its implementation in measurement reduction. The modeling work presented in th is chapter establishes the basis for root cause identification of multiple error source s and errorcancelingerror automatic process adjustment. PAGE 57 47 Chapter 3 Error Cancellation Modeling and Its Application in Process Control* Due to the error equivalence mechanism, the impacts of errors on part features may cancel one another. Error cancellation may hinder the error information from being identified and therefore increase the complexity of root cause identification. However, we can maneuver one error to cancel other errors and reduce process variation. By considering such dual effects of error canc ellation, this chapter intends to study the implications of error cancellation based on the derived error e quivalence model. Section 3.1 analyzes error cancellation a nd its theoretical implications from the perspectives of process mon itoring and control, including root cause diagnosis and error compensation. Using error cancellation, a sequential root cause identification procedure and errorcancelingerror methodology are devel oped to reduce the time invariant process errors. In Section 3.2, the proposed diagnosti c procedure is demonstrated by a machining experiment and the error compensation is al so illustrated with a simulation study. A summary is given in Section 3.3. *The work in this chapter has appeared in Wang, H. and Hu ang, Q, 2006, Error Cancellation Modeling and Its Application in Machining Process Control, IIE Transactions on Quality and Reliability, 38, pp.379388. PAGE 58 48 3.1 Error Cancellation and It s Theoretical Implications It has been widely noted that the impact of multiple error sources on product features may cancel out one anot her. This phenomenon may have the drawback that it is possible for it to conceal the fact that multip le errors have occurred in the process, however, there is the op portunity for us to purposely use one type of error to counteract or compensate another error and thereby redu ce variation. Error equivalence can model the error ca ncellation and the im pact of errors on feature deviation. By Eq. (2.6), we have E( x )= E( u*)= *E(* =1p iiu )= *E(=1p iiiKu ), and (3.1a) Cov( x )= *Cov( u*) T+Cov() = *Cov[=1 Cov()pT iiiiKuK]* T+Cov(), (3.1b) where E(.) and Cov(.) repres ent expectation and variancecovariance matrix of random variables in the parentheses. Eq. (3.1a) indicates that the cancellation e ffect of three types of errors can be modeled as a linear combination of mean shift of equivalent amount of base errors, i.e., =1( )p iiEu,. Their impacts on feature deviati on are described by mapping matrix in Eq. (2.6). For a special case that three types of errors completely cancel each other, i.e., =1( )p iiEu is statistically insignificant, the mean of process output is within control. It should be noticed that the va riances caused by three types of errors cannot be cancelled (see Eq. (3.1b)). In the machining process, Eq. (2.6) becomes x= ***+T TTT dfm =*( d+ f+ m)+ (3.2) and error cancellation is modeled by E( d+ f+ m). PAGE 59 49 Modeling of error cancellation has many theoretical implications on machining process control. This section discusses the implications on three issues: diagnosability analysis, root cause identification, and error compensation. 3.1.1 Diagnosability Analysis of Manufact uring Process with Error Equivalence This chapter studies the diagnosability of the process that is governed by a general linear causal model as follows, which relates the errors to the feature deviation x, x 12.T TTT p u uuu (3.3a) where matrix is determined by the part speci fication. Its relationship with will be discussed in Proposition 3.1. In the machining process, the model becomes x= D.T TTT m xfq (3.3b) Under a certain measurement strategy, diagnosability study aims to determine whether all the process errors uis are estimable. If the process is diagnosable, the least square estimation (LSE) can be performed, i.e., 12T TTT puuu=(T )1 x. (3.4) The diagnosability depends on the rank of (Zhou, et al., 2003). We can see that Eq. (3.4) requires T to be full rank, or equivalently, all the columns in to be independent. Proposition 1 addresses the structure of for a machining process. Proposition 3.1. If error equivalence hol ds for process errors 12T TTT puuu, the process will not be diagnosable with measurement of quality characteristic x. In the machining process, block matrices in matrix (see Eq. (3.4)) corresponding to three PAGE 60 50 types of errors are dependent and matrix T is always not full rank, i.e., fixture, datum, and machine tool errors cannot be distinguish ed by measuring the part features only. Proof. If we use transformation matrices Ki to transform errors ui to base error u1, Eq. (2.6) becomes *** 212=.T TTT pp x K Kuuu (3.5a) In the machining process, matrices K2 (from Eq. (2.8)) and K3 (Eq. (2.9)) transform datum error Dx to d and machine tool error qm to m, respectively. Eq. (2.6) becomes *** 23D=[]+.T TTT m x K Kxfq (3.5b) Comparing Eq. (3.5a) with Eq. (3.4), we obtain matrix =*** 2p K K. However, the columns corresponding to fixt ure and machine tool errors in matrix are dependent because columns of *Kis are the linear combination of columns of *. Therefore, rank of equals the rank of *. This also implies that the system is not diagnosable. An implication of this proposition is that LSE of 12T TTT puuu in Eq. (3.4) cannot be obtained. However, the causal mode l (3.2) with error grouped eliminates the linearly dependent columns in matrix and therefore can be full rank. This fact leads to sequential root cause identification in Section 3.1.2. 3.1.2 Sequential Root Cause Identification Using Eq. (3.2), the grouped errors u can be estimated as *()*()**1*() 1 ()p nnTTn i i uu x, n=1, 2, , N, (3.6a) PAGE 61 51 where *()n iu is the LSE of ui for the nth replicate of measurement. In a machining process, *() 1p n i iu becomes ()()() nnn dfm. Each row of corresponds to output feature while each column of corresponds to component of e rror vectors. Hence, the number of rows of must be larger than the number of its columns to ensure that sufficient features are measured for LSE. The mean a nd variancecovariance of the detected errors are 1 E()E()p ii iuKu and 1Cov()Cov().p T iii i uKuK (3.6b) Proposition 3.1 indicates that measuremen t other than quality characteristics x is necessary to distinguis h error sources. However, it will not be economical to take the additional measurement if no process error occurs. A sequential procedure is thus proposed for root cause identification: Necessary error information (e.g., offline measurement on workpieces) is collected first to identify the occurrence of error sources using Eq. (3.6a). The process error information can be analyzed by conducting hypothesis test on () 1 {}nN nu. Since the estimated u is a mixture of noise and errors, a proper test statistic should be developed to detect the erro rs from process noise. Hypot hesis testing for mean and variance can then be used to find out if the errors are mean shift or large variance. Additional measurement (e.g., inline m easurement on process errors) is then conducted to distinguish di fferent types of errors 12T TTT puuu E( u) and Cov( u) will be estimated with the inprocess measurement of (p1) error sources. By Eq. (3.6b) the remaining unmeasured errors can be obtained. The detailed procedures for the machining process will be given in the Section 3.2.1. PAGE 62 52 3.1.3 ErrorCancelingError Compensation Strategy We can use the effect of error cancellation to compen sate process errors. An adjustment algorithm based on error equivalen ce mechanism can be designed to adjust the base error u1 to compensate the other process errors {ui}2 p i With the development of adjustable fixture whose locator length is cha ngeable, it is feasible to compensate errors only by changing the length of locators. We use index i to represent the ith adjustment period. During period i, N part feature deviations {x( i ), ( n )}1N n are measured to determine the amount of locator adjustment. Such compensation is only implemented at the beginning of the period. Denote c( i ) as the accumulative amount of locator length adjusted after the ith period and the beginning of period i+1. The compensation procedure can be illustrated with Fig. 3.1. One can see that a nominal machining process is disturbed by errors d, f and m, and the observation noise Error sources, noi se, and machining process constitute a disturbed process, as marked in the dash line block. Using the feature deviation x( i ) for the ith period as input (x( i ) can be estimated by the average of N measured parts in the period i, i.e., ()()ii xx), an adjustment algorithm is introduced to generate signal c( i ) to manipulate adjustable fixture locators to counteract the errors for the (i+1)th machining period. The amou nt of compensation at period i+1 should be c( i )c( i 1). The error compensation model can then be x( i +1)=S( i +1)+*c( i ) and S( i +1)=*u( i +1)+( i +1), (3.7) where S( i +1) is the output of the di sturbed process for time i+1. This term represents the feature deviation measured wit hout any compensation being made. PAGE 63 53 Figure 3.1 ErrorCancelingError Strategy The adjustment using equivalent errors can be illustrated with an example in Fig. 3.2, where a prismatic part is set up in a fixture with locators f1, f2, and f3. We expect to perform a parallel cutting on the top plane of th e part. If the tool path tilts due to thermal effect, the yielded top plane w ill also tilt the same angle. However, under the adjustable fixture where the length of locator pin is adjustable, we may find out the adjustment amount (black bar in right panel of Fig. 3.2) for f1, f2, and f3 such that the part tilts the same angle as the deviated tool path. Obviously, a conforming part can still be obtained. Similarly, we can also adjust fixture locator s to compensate the datum error. The amount of adjustment can be determined by EFE using Eq. (2.9). With this concept, the feature deviation caused by machine tool thermal error (tilted tool path) can also be generated by EFE ( m1 m2 m3) alone. In order to compensate th is error, we must apply the amount of adjustment (m1 m2 m3) to these locating pins. Figure 3.2 Process Adjustment Using EFE Concept f1 f2 f3 Deviated tool path Deviated tool f1 m1m2 m3 f2 f3 x Adjustment c u Nominal Machining ( ) PAGE 64 54 In this chapter, the compensation focuses on time invariant error because they account for the majority of overall machini ng errors (Zhou, Huang, and Shi, 2003). The negative value of predicted equivalent errors can be used to adjust locators. From Eq. (3.7), it is clear that if we set *c( i )=*u( i +1), then the adjustment can cancel out the process errors and deviation is x( i +1)= ( i +1). The adjustment c( i ) can be the LSE of u( i +1), i.e., c( i )=(1)i u=[**1*()()TTi xc( i 1)], and c(1)=**1*(1)()TT x. (3.8) By solving the recursive Eq. (3.8), we derive an integral adjustment that can minimize mean square error (MSE) of the feature deviation, i.e., c( i )=**1*()()()() 11 ()()ii TTtttt tt xdfm. (3.9) Eq. (3.9) shows that the accumulative amount of compensation for the next period is equal to the sum of the LSE of EFE of all current and previous time periods of machining. The accumulative compensation c( i ) is helpful for evaluation of adjustment performance such as stability and robustn ess analysis. The amount of compensation for the i+1th period is c( i )c( i 1), ()(1)**1*()()iiTTi cc x. (3.10) The compensation accuracy can be estimated by x( i )*(* T*)1* T x( i ), i.e., the difference between x( i ) and its LSE. Denote range space of as R(*) and null space of *T as N(*T). Spaces R(*) and N(*T) are orthogonal and constitute the whole vector space Rq1, where q is the number of rows in x( i ) (or *). By the property of LSE, we know that the estimation error vector x( i )*(*T*)1*T x( i ) is orthogonal to R(*). Therefore, the compensation accuracy of Eq (3.9) can be estimated by projection of PAGE 65 55 observation (feature deviation) vector x( i ) onto N(*T). This conclusion also shows the components of observation that can be compen sated. The projection of observation vector x( i ) onto space R(*) can be fully compensated with Eq (3.9) whereas the projection onto N(* T) cannot be compensated. In practice, the accuracy that the adjustable locator can achieve must be considered. Suppose the standard devi ation of locators movement is f. We can set the stopping region for applying error comp ensation with 99.73% confidence 3 fc( i )c( i 1) 3 f. (3.11) It should be noted that the errorcancelingerror strategy in Eq. (3.9) is valid for compensation of time invariant process erro rs. Compensation strategy for dynamic errors will be studied in Chapter 5. 3.2 Applications of Error Can cellation in a Milling Process Discussion in Section 3.1 implies the appli cation of equivalent errors in sequential root cause identification and error compensation. The dia gnostic algorithms are proposed in this section and demonstrated with a machining experiment. EFE compensation for process control is illust rated with a simulation. 3.2.1 Diagnosis Based on Error Equivalence There are several diagnostic approaches (Ceglarek and Shi, 1996; Apley and Shi, 1998; and Rong, Shi, and Ceglarek, 2001) that have achieved consid erable success in fixture errors detection. The approach pr oposed by Apley and Shi (1998) can effectively identify multiple fixture errors. By extending this approach, we use it for sequential root cause identification: PAGE 66 56Step 1: Conduct measurement on features and datum surfaces of raw workpiece to estimate error sources *()nu for the nth replicate by Eq. (3.6). The grouped error can be estimated by the average of ()nu over N measured workpieces, i.e., **() 11 N n nNuu n =1,2, N As mentioned in Secti on 3.2, the error vector ()nu is the mixture of error sources and process noise. Step 2 : To detect the errors from the process noise, we can use F test statistic introduced by Apley and Shi (1998): 2 **12 , [()]i i T iiS F S i =1, 2, , 6, (3.12) where 2()2 11 []N n ii nSu N, and ()n iu represents the i th component in vector u( n ). **1 ,()T ii is the i th diagonal entry of matrix **1()T The estimator for variance of noise is 2()() 11 (6)N nTn nS Nq, and ()()*()nnn x u is for noise terms. When Fi> F1( N N ( q 6)), we conclude that the i th error significantly occu rs with confidence of 100(1)%. By investigating {()n iu }1 N n for mean ui (H0: ui=0 vs. H1: ui 0), and variance 2 ui (H0: 2 ui 2 0 vs. H1: 2 ui> 2 0), one can determine whether the pattern of the errors is mean shift or variance. 2 0 is a small value. In the case study, we choose 2 0= 0.1mm2. Under the normality assumption of EFEs ( d, f, and m), we can use the T test statistic ()2 11 /() (1)N n iii nTuuu NN and compare it with t1/2( n 1) to test mean shift. 2()22 0 1()/N n ii nuu is used and compared with 2 1 ( n 1) to test variance. is the PAGE 67 57 significance level. If Fi< F1( N N ( q 6)), no errors occur at the i th locator, or the errors cannot be distinguished from process noise. Step 3 : Apply the additional measurement on locat ors and datum surfaces to distinguish errors whenever errors are identified. Denote f() n i, d() n i, and m() n i as the i th component in vector f( n ), d( n ), and m( n ), respectively. Locator deviation { f() n i}1 N n and datum surfaces {X() n j}1 N n are measured. The EFE { d() n i}1 N n caused by datum error can be calculated by Eq. (2.8). The mean shif t of the errors can be estimated using the sample mean of d() n i, f() n i, and m() n i= u() n id() n if() n i. The variance can then be estimated by the sample variance for d() n i, f() n i, and m() n i. If the errors turn out to be the mean shift ( ui 0 for certain i ), machine tool error in terms of EFE is im= iu difi, where di and fi are the average EFE over all N parts. Machine tool error qm is then determined by the inverse of Eq. (2.9) qm1 3. Km (3.13) The variance of grouped error ( 2 ui) can then be decomposed as 2222 uidifimi (3.14) If 2 ui> 2 0, variances caused by three types of errors 2 di 2 f i and 2 mi can be estimated by the sample variance of { d() n i}1 N n { f() n i}1 N n and { ()n im }1 N n The 100(12 )% confidence interval (CI) of m is ( mL), where z1follows the cumulative standard normal distribution such that 2 1/21 1 2z uedu and 11 11,116,6 61 (()...())TTT uuuuzz L The corresponding CI vector for qm is PAGE 68 58 (11 33 KmKL). The CI for d and f can be obtained by ( di 1/2(1)/diStnn) and ( fi 1/2(1)/fiStnn), where Sdi and Sfi are the sample variance for { d() n i}1 N n and { f() n i}1 N n. This approach can effectively identify the machine tool errors. Identification of error occurrence Decisionmaking on taking inprocess measurement on certain Error decomposition and individual error identification Diagnosability analysis *()**1*() =()nTTnu x 2 **12 [()]i i T iiS F S * 2[]p K K () 1 E()p n ii i uKu 1 Cov()Cov()p T iii i uKuK iuIdentification of error occurrence Decisionmaking on taking inprocess measurement on certain Error decomposition and individual error identification Diagnosability analysis *()**1*() =()nTTnu x 2 **12 [()]i i T iiS F S * 2[]p K K () 1 E()p n ii i uKu 1 Cov()Cov()p T iii i uKuK iu Sequential Root Cause Identification Identification of error occurrence Decisionmaking on taking inprocess measurement on certain Error decomposition and individual error identification Diagnosability analysis *()**1*() =()nTTnu x 2 **12 [()]i i T iiS F S * 2[]p K K () 1 E()p n ii i uKu 1 Cov()Cov()p T iii i uKuK iuIdentification of error occurrence Decisionmaking on taking inprocess measurement on certain Error decomposition and individual error identification Diagnosability analysis *()**1*() =()nTTnu x 2 **12 [()]i i T iiS F S * 2[]p K K () 1 E()p n ii i uKu 1 Cov()Cov()p T iii i uKuK iu Sequential Root Cause Identification Figure 3.3 Sequential Root Caus e Identification Procedures Fig. 3.3 shows the sequential diagnos tic methodology under the error equivalence mechanism. It can be seen that the sequential diagnostic methodology includes diagnosability analysis (Proposition 3.1) and sequential root cau se identification. To demonstrate the model and the dia gnostic procedure, we intentionally introduced datum and machine tool errors to mill 5 block workpieces. We use the same setup, raw workpiece, and fixturing sche me as Fig. 2.10. Coordinate system xyz fixed with nominal fixture is also introduce d to represent the plane. Top plane X1 and side plane X2 are to be milled. All 8 vertices are marked as 1~8 and their coordinates in the coordinate system xyz are measured to help to determine X1 and X2. In this chapter, the unit is mm for the length and radian for the angl e. Under the coordinate system in the Fig. 2.9, surface specifications are X1=(0 0 1 0 0 15.24)T, and X2=(0 1 0 0 96.5 0)T. From model (3.2) and Eq. (2.23), we get 1 2(),iiii i xdfm where PAGE 69 59100.02630.0263000 0.01580.00790.0079000 000000 00.13790.137913368133681 0.08280.04140.04141.50.50 130330.84831.1517000 20000.02630.02630 000000 0.01580.00790.0079000 00.26320.26321.20261.20261 0.1580.0790.0791.5050 0.22121.61060.3894000 The number of rows q in is 12. We set fixture error to be zero ( f=0). The primary datum plane I is premachined to be XI=(0 0.018 0.998 0 0.207 1.486)T and its corresponding EFE is d=(1.105 0 0 0 0 0)Tmm. The machine tool error is set to be qm=(0 0.175 1.44 0.0175 0 0)T by adjusting the orientation and position of tool path. Based on coordinates of the vertices 1~8 measured, the feat ure deviations are given in Table 3.1. Since X1 and X2 are all planes, the deviations rjs of surface size are all zero. Following steps 13, the identified EFEs are given in Tables 3.2 and 3.3. Table 3.1 Measured Features (mm) X1 X2 n 1 2 3 4 5 1 2 3 4 5 vx vy vz px py pz 0.001 0.033 0.000 0.000 0.145 3.877 0.000 0.034 0.000 0.000 0.163 2.749 0.000 0.039 0.000 0.000 0.119 2.329 0.000 0.034 0.000 0.000 0.185 3.509 0.001 0.035 0.000 0.000 0.153 2.459 0.000 0.000 0.032 0.000 0.347 0.579 0.000 0.000 0.034 0.000 0.379 0.358 0.000 0.000 0.032 0.000 0.253 0.479 0.000 0.000 0.036 0.000 0.307 0.539 0.000 0.000 0.035 0.000 0.268 0.429 Table 3.2 Estimation of u for 5 Replicates (mm) u(1) u(2) u(3) u(4) u(5) u T 2 2.937 0.050 0.002 0.055 0.047 0.004 2.133 0.090 0.090 0.031 0.031 0.000 1.775 0.064 0.0562 0.003 0.004 0.001 2.697 0.057 0.057 0.039 0.039 0.000 1.902 0.002 0.020 0.015 0.018 0.001 2.289 0.027 0.023 0.016 0.015 0.001 10.119 10.247 We choose to be 0.01. The threshold value F0.99(5,5(126))= F0.99 (5,30) =3.699. In Table 3.2, we can see that F1>3.699, which indicates that error occurs at locator 1. Using the data in the first row of Table 3.2 to conduct T and 2 tests for mean and PAGE 70 60 variance, respectively, we find that T > t10.01/2 (51)= t0.995(4)=4.604 and 2< 2 1001(4)=13.277. Hence, we conclude that th ere is significant m ean shift while the variance is not large. If we make the a dditional measurement, by Eq. (3.13), the 98% confidence interval for the detected m ean shift of machine tool error is qm=(0.006 0.167 1.540 0.018 0.000 0.000)T(0.008 0.001 0.000 0.000 0.001 0.000)T, which is consistent with the preintroduced errors. The EFE causal model and dia gnostic algorithm is experimentally validated. Table 3.3 Error Decomposition (mm) Locators uFi f d m 1 2 3 4 5 6 2.289 0.027 0.023 0.016 0.015 0.001 19.525 0.051 0.005 0.613 0.073 0.002 0 0 0 0 0 0 1.105 0 0 0 0 0 1.184 0.027 0.023 0.016 0.015 0.003 Figure 3.4 Error Compensation for Each Locator 3.2.2 Error Compensation Simulation Using the same machining process as in Section 3.2.1, we can simulate error compensation for 5 adjustment periods. Total 5 parts are sampled during each period. The fixture error is set to be f=(0.276 0 0 0.276 0 0)Tmm. The machine tool error is set to be PAGE 71 61 qm=(0.075 0.023 0.329 0.0023 0.0075 0)T and its EFE is m=(0 0 0.286 0 0 0)Tmm. We assume the measurement noise to follow N (0, (0.002mm)2) for displacement and N (0, (0.001rad)2) for orientation. The compensation values can be calculated by Eqs. (3.9) and (3.10). In this case study, the accuracy of the locator movement is assumed to be f=0.01mm and the criterion for st opping the compensation is 0.03 c(i)c(i1) 0.03mm (see Eq. (3.11)). Fig. 3.4 shows the compensation (c(i)c(i1)) for locators f1~ f4. The values of adjustment periods 2~5 are given by the solid line in the figure. The dash dot line represents the value of 3 f. The adjustments for locators f5 and f6 are all zero and not shown in the figure. One can see that the e ffect of compensation in the second period is dominant. The compensation for the subsequent periods is relatively small because no significant error sources are introduced for these periods. 1 2 3 4 5 15 15.1 15.2 15.3 15.4 Adjustment Period l z ( mm ) 1 2 3 4 5 96.3 96.35 96.4 96.45 96.5 96.55 Adjustment Periodly (mm) 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Adjustment PeriodStandard deviation of lz(mm) 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Adjustment Period Standard deviation of ly (mm) Figure 3.5 Mean and Standard Deviation of Two Features The effect of error compensation can be illustrated with the quality improvement of two features, the plane distance along z axis ( lz) and y axis ( ly) as shown in Fig. 3.5. lz can be estimated by the mean and sta ndard deviation of length of edges l15, l 26, l 37 and l 48 and ly can be estimated by l14, l23, l67 and l58 for each machining period, where lmn is the PAGE 72 62 distance between the vertices m and n and is estimated by the edge length of 5 parts in each period. Milling of planes X1 and X2 impacts the plane distance along z and y axes. The nominal part should have the same length of edges along z and y directions (15.24 and 96.5mm, see the dash line in Fig. 3.5), re spectively. However, in the first adjustment period ( i =1) without erro r compensation, the errors of edge lengths are beyond specified tolerance. In the periods 2~5 when compen sation algorithm has been applied, deviation of lz and ly is significantly reduced. 3.3 Summary This chapter investigates error cancellati on among multiple errors (datum, fixture, and machine tool errors) for improving quality control in machining processes. As a summary, the implications of studyi ng error cancellation are as follows: First, process errors may cancel one anothe r and conceal the error information. It has been proved that a machining process with datum, fixture, and machine tool errors cannot be diagnosable by only measuring the pa rt features. To overcome this problem, a sequential procedure is therefore proposed, i. e., first identify error occurrence based on measurement of product deviation x and an F test statistics, and then discriminate error sources using inprocess measurements (not pr oduct features) and hypot hesis test only if process error is detected. This procedure can de tect the mean shift as well as the variance of process errors from the process noise. A case study for a milling process of block parts has shown that the proposed approach can effectively identify the error sources. Second, an errorcancelinge rror process adjustment st rategy can be developed. Study of error cancellation also suggests that erro rs (machine tool and datum errors) can PAGE 73 63 be compensated by adjusting the base error (the length of fixt ure locators). An integral adjustment algorithm is presented in this chap ter for compensation of time invariant error. It has been shown that the accumulative amount of compensation is equal to the sum of the LSE of EFE of all previous time periods of machining. The procedure has been demonstrated with a simulation study. PAGE 74 64 Chapter 4 Dynamic Error Equivalence Modeling and In Line Monitoring of Dynamic Equivalent Fixture Errors* This chapter studies the error equiva lence of dynamic errors and thereby establishes a process model for the purpose of APC based on the dynamic equivalent errors. Considering process monitoring and data collection, this chapter presents a new concept, in addition to the widely recognized error avoi dance and error compensation approaches, to control the e ffects of dynamic errors by in line monitoring of process dynamic errors. This chapter selects the therma l effect of machine tool errors as an example to demonstrate the modeling and m onitoring of dynamic equivalent errors. The remainder of the chapter includes 4 sections. Section 4.1 introduces the problems in dynamic process modeling. In Sect ion 4.2, based on an experiment, latent variable modeling (LVM) method is applied to build an ARX model for dynamic errors (thermal errors). Variable selection strategy is also discussed for the situation that high accuracy is required for model prediction. Usin g the latent variable model, the inline monitoring of thermal error and control chart design are presented in Section 4.3. Section 4.4 discusses the isolation of lagged variables and sensors responsible for outofcontrol signals. A summary is given in Section 4.5. *This work will appear in Wang, H., Huang, Q., and Yang, H., 2007, Latent Variable Modeling and InLine Monitoring of Machine Tool Thermal Errors, accepted by Journal of Manufacturing System. PAGE 75 65 Nomenclature N number of observations A number of latent variables p number of part characterist ics required by design specification c number of thermal sensors mounted onto a machine tool t time index l time lag l ZT transpose of matrix Z p1 (n) thermal errors at time period n Np thermal error history from t = 1 to t = N [(1) (2) (N)]T sc1 (n) readings of c thermal sensors at time t [ s1 (n) s2 (n) sc(n)]T SNc temperature history of machine tool from t = 1 to t = N [s(1) s(2) s(N)]T XNk descriptor data YNm response data LV latent variables of X and Y, [ LV1 LVA]T ( A k + m ) TNA common scores of X and Y SLV sample covariance matrix of latent variables LV PkA X loadings QmA Y loadings PAGE 76 66WkA weights of variables in X ENk X residuals FNm Y residuals q back shift operator D( q ) difference operator with dj = 1 or 2 ( j =1,2, , c or p ), Diag [(1q1)d1, (1q1)d2, ] S( i j ) a matrix [D( q )s(i) D( q )s(i+1) D( q )s(j)]T ( i j ) a matrix [(D( q )(i) D( q )(i+1) , D( q )(j)]T 4.1 Introduction to Modeling of Dynamic Errors Since dynamic errors may have great impact on part quality, inline monitoring of dynamic errors is a very important issue for quality improvement. For example, thermally induced errors account for a large percentage of machine tool errors and hence inline monitoring and compensation of thermal errors are critical to reduce process variations. Although as shown in Chapter 1, the SI based dynamic modeling methodology shows significant advantages over the sta tic one in terms of model accuracy and robustness, several barriers still remain when applying SI theory to the thermal error modeling: how to determine the number of temperature measurements that is sufficient to build an adequate SI model, in order to avoid excess amount of sensors to be mounted onto a machine tool. how to select appropriate lagged variables when a larg e number of thermal sensors are available. The stepwise regression is commonly applied for variable selection PAGE 77 67 (Chen, et al. 1993). However, this method has certa in limitations when being applied to model strongly correlated historical data. Thus a systematic approach for the selection of appropriate lagge d variables is necessary for determining the structure of the dynamic model. how to effectively estimate machine thermal status and predict machine performance when the sensing resource is limited. The purpose of this chapter is to overcome aforementioned difficulties. It presents a new concept of controlling machining ther mal effects by inline monitoring of machine thermal status based on SPC. Limited number of thermal sensors are employed to track the temperature distribution of machine t ools and to detect out ofcontrol machine thermal status as the results of environmen t change, machine degradation, or process parameters change. The recently devel oped LVM method (Shi and MacGregor, 2000) provides a powerful tool for variable sele ction and model order determination. This method will be employed in both dynamic modeling and inline monitoring. 4.2 Latent Variable Modeling of Machine Tool Dynamic Errors 4.2.1 Description of Data In thermal error modeling, the collected information includes machine tool temperature and thermal error (provide d by inline probing system). Suppose c thermal sensors are mounted on a machine tool. Let sc1 (n) = [ s1 (n) s2 (n) sc(n)]T denote the c sensor readings at time t and SNc = [s(1) s(2) s(N)]T denote temperature history of machine tool from time t = 1 to t = N Suppose p characteristics are probed and the measured readings for thermal error qm(n) are denoted by p1 (n). Then Np = [(1) PAGE 78 68(2) (N)]T represents thermal error history. SNc and Np are nonstationary multivariate time series data. This fact can be illustrated by an example. As shown in Fig. 4.1, 11 sensors are mounted onto a CNC machine tool, where S # i denotes sensor i ( i =1, 2, , 11). Under certain working conditions, the sensor readings over time index are shown in the left panel of Fig. 4.2. At the same time, inline probing system provides the thermal errors in z direction of machine tool spindl e (right panel of Fig. 4.2). The nonstationary nature of the data is obvious. Figure 4.1 Thermal Sensor Locations on a Machine Tool For engineering processes, the common trea tment on nonstationarity is to take the first or second order difference on original data and check the first two moments for adequacy test (Box and Jenkins, 197 0). Define a difference operator Dcc( q ) = Diag [(1q1)d1, (1q1)d2, , (1q1)dj, ], where dj = 1 or 2 ( j =1,2, , c or p ) and q1 is the back shift operator, i.e., (1q1) z(t)= z(t)z(t1). Therefore, to obtain stationary time series, the temperature and thermal error data are transformed as S #4 Column Spindle Motor Spindle Driving Box S #1 S #8 S #7 Backside View S #6 S#11 S #9 S #5 S #3 S #10 S #2 11 Thermal Sensors X Z Y PAGE 79 69D( q )(SNc) = [D( q )s(1) D( q )s(2) D( q )s(N)]T, (4.1) D( q )(Np) = [D( q )(1) D( q )(2) D( q )(N)]T. (4.2) For simplicity, denote [D( q )s(i) D( q )s(i+1) D( q )s(j)]T as S( i j ) and [D( q )(i) D( q )(i+1) D( q )(j)]T as ( i j ) with i < j 0 50 100 150 20 22 24 26 28 30 32 34 36 38 40 Machine Tool Temperature TimeTemperature (Celsius Degree) 0 50 100 150 5 0 5 10 15 20 25 30 35 40 45 Thermal Error in z Direction TimeDeformation(m) Figure 4.2 Machine Tool Temperature and Thermal Error Data The order dj in D( q ) is determined by the nature of data. For the example data in Fig. 4.2, we can take the first and second or der difference of the temperature and thermal error data. The first and second order differe nces of the temperature and thermal error data are shown in Fig. 4.3. We can also compute the mean and variance of these data differences. Among the total 120 observations (Fig. 4.2), two segments are randomly selected. Segment 1 contains observation No. 5 to 45 and Segment 2 contains the observation No. 50 to 90. The mean and varian ce of differences of these two segments are shown Tables A.1 and A.2 in Appendi x D. We can see that the second order differences in two segments are very small a nd it is not necessary to consider the second order difference in the model (Box and Je nkins, 1970). Therefore, the first order difference is sufficient for the temperature and thermal error data in this example. PAGE 80 70 0 50 100 150 1 0 1 2 The first order diff. of temperature 0 50 100 150 10 5 0 5 10 0 50 100 150 1 0.5 0 0.5 1 0 50 100 150 10 5 0 5 10 Time Index The first order diff. of thermal error The second order diff. of temperature The second order diff. of thermal error Time Index Figure 4.3 Stationarity Treatment Rather than in a unique format pr esented in (Shi and MacGregor, 2000), construction of descriptor data X and response Y for the modeling study is depending upon whether in situ measurement of thermal deformation is suffici ently available during the process. There are commonly two situations. The first one is that thermal sensing information is sufficient and accurate mode l prediction can be achieved for real time thermal error compensation. Another situation is that thermal sensing information may be inadequate for model based compensation whereas inline monitoring of machine thermal status and prediction of pro cess degradation is important to continuous maintenance of product quality. If the in situ measurements of thermal error are available, for example, using process intermittent probing, the lagged vari ables both in temperature and thermal error histories can be included in screening procedure of lagged input variable, and PAGE 81 71 determining the model structure. Suppose the speculated maximum time lag is l where l can be chosen as a number large initially, the input vector X and output vector Y can be represented as X = [S(1, N l +1) S(2, N l +2) S( l N ) (1, N l +1) (2, N l +2) ( l 1, N 1)], (4.3) Y = ( l N ). (4.4) Here, block matrix S( i N l + i ) is regarded as the data collection for variable difference t(i)t (i1), where t is the variable vector of temperature. Matrix ( i l + i ) contains data collection of the variable difference m q(i)m q(i1). Eq. (4.3) enables the screening procedure to consider the lagged variables bo th in temperature history and thermal error history. If the in situ measurement of thermal error is unavailable, the X can only be formed with time sequences of temperature measurements: X = [S(1, N l +1), S(2, N l +2), , S( l N )]. (4.5) When using data matrices (4.4) and (4.5), we can avoid stopping normal production and monitor thermal error in situ. 4.2.2 Latent Variable Modeling of Machine Tool Dynamic Error Latent variable modeling is a method for constructing predictive models when the factors are many and highly collinear (Bur nham, Viveros, and MacGregor, 1999). The general model structure is X = TPT + E, (4.6) Y = TQT + F, (4.7) PAGE 82 72 where P is X loadings and Q is Y loadings. X and Y are assumed to have the common underlying latent variables LV with LV= [ LV1, , LVA]T ( A k + m ). LV reduces X and Y spaces into a low dimensional subspace spanned by LV. The subspace is expected to grasp the most relevant info rmation and structures from X and Y spaces. LV has the nice property that its elements are orthogonal to each other. E ach realization of LV forms the corresponding row of score matrix T, which can be direc tly computed from X as: T = X W(PTW)1, (4.8) where W is the weights of X. The unmodeled noise terms are E and F. Y can be expressed in a regression form as Y = XG + F (4.9) with G = W(PTW)1QT. Among different model fitting approach es, such as principal component regression (PCR), partial least squares or proj ection to latent struct ures (PLS), canonical correlation analysis (CCA), and reduced rank analysis (RRA), PLS chooses LV by maximizing the covariance between historical information in X and Y (Shi and MacGregor, 2000) and it has been widely a pplied in chemical processes for process calibration and process monito ring (Nomikos and MacGrego r, 1995). The PLS algorithm (Westerhuis, Kourti, and MacGregor, 1998) is adopted in this research. A variable screening procedure is then implemented to choose the number of sensors and maximum time lag for the model. Training data matrices X and Y need to be mean centered and scaled to unit variance prior to fitting the model, while the ne w data matrices are still denoted as X and Y in this chapter. We add operator ~ on the top of the not ation to represent the scaled variable or PAGE 83 73 data matrix (scale for each column). Hence, the input vector X and output vector can be represented as X=[S(1, n l +1) S(2, n l +2) S( l n ) (1, n l +1) (2, n l +2) ( l 1, n 1)], (4.10) =( l n ). X is an n l +1 by rl +6( l 1) matrix consisting of the da ta collection of temperatures and thermal errors. Here, the block matrix S( i n l + i ) contains n l +1 scaled temperature data vectors for {s(i)~s(nl+i)}i=1,2, l and can be regarded as the data collection of the variable difference t (i)t (i1), i =1,2 l over a period from i to n l + I Similarly, matrix ( i n l + i ) includes n l +1 scaled thermal error vectors for {(i)~ (nl+i)} i=1,2, l and is an ( n l +1)period (from i to n l + i ) data collection of the variable difference m q(i)m q(i1). Temperatures will be used for input and therma l errors will be for autoregressive terms in the model. includes the data collection of thermal errors m q(n). By LVM fitting procedure, we fit the regression coefficient G in Eq. (4.9) to the data in Eq. (4.10). Hence, the first order differences of errors at time period n can be represented as the function of error sources in the previous periods: 12()(1)()()(1)()()(1) 11 [][]pp nnlnlnllnlnl mmmm ll qqAqq tt (4.11) where () lA is a 66 square coefficient matrix and its nonzero entr ies come from the entries in G corresponding to auto regressive terms. () lB is a 611 coefficient matrix and its nonzero entries come from the entries in G corresponding to the temperature variables (see the example of the coe fficient matrices in the case study). p1 and p2 represent the maximum time lag for temperature and thermal error in the model. Time lag p1 is for A and p2 for B, n n0. n0 is the starting period when the adjustment applies. PAGE 84 74 Scaling the data back with the mean and va riance from the training dataset, we have 12()(1)()()(1)()()(1)()(1) 00 11 [][]pp nnlnlnllnlnlnn mmmm ll qqAqq ttDD, (4.12) where D0( n ) is the intercept term that is the linear combination of the means of the original data. A( l ) and B( l ) are the coefficient matrices after scaling back the data. Considering Eq. (2.9), we get 1 2()(1)()1()(1) 1 ()()(1)()(1) 00 1[] [][].p nnlnlnl l p lnlnlnn l mmKAKmm KBttKDD (4.13) Denote q1 as the backward operator, e.g., q1m(n) represents m(n1). Canceling (1q1) on both hand sides of Eq. (4.13) leads to 12()()1()()()() 0 11 pp nlnllnln ll mKAKmKBtK, (4.14) where () 0 nK is a matrix that is rela ted to the initial condition t(n0), m(n0) and intercept term () 0 nD. Eq. (4.14) is the fitted model for the quasistatic EFE thermal error. It will predict the thermal error at the next period based on all the previous information such as the temperatures and thermal errors collected. It can also be applied to the equivalent error compensation (or automatic process adjustment) that has been discussed in Chapter 3. Design of process adjustment algorithm w ill be discussed in the next chapter. Since the thermal errors in Fig. 4.2 are along z direction only, the equivalent amount of fixture error (EFE) is the same as thermal errors (see Fig. 4.4). We make the notations of EFE and thermal errors interchangeable in Section 4.3. PAGE 85 75 Figure 4.4 Equivalent Fixture Error of Fig. 4.2 As to the situation one mentioned in Sec tion 4.2.1, we fit the model to data for variable screening. Given th e 120 observations in Fig. 4.2, suppose the maximum time lag is 6 (or a larger number) for (t) 11 and s(t) 111 (note: is one dimensional for this case). Using the first 94 observations as training set and construct X and Y by Eqs. (4.3) and (4.4), we can see that after first order difference, l =61=5. So, according to Eq. (4.3), total lc + l 1=(61)11+(61)1=59 candidate variables in X9059 need to be screened to fit the corresponding thermal error data Y901. The number of latent variables is determined by the percentage of variance they can explain. To start, just assume there are 30 latent variables and fit the model e xpressed by Eqs. (4.6) and (4 .7). Based on the index of variable importance for projecti on (VIP: a variable with VIP greater than 0.8 is assumed to be significant) (Wold, 1994) and regression coefficients G, 29 input variables in X are screened out (Table A.3 in the Appendix E). Table A.3 suggests that sensor 9 seems to be insignificant in the thermal error model. However, in the experiment, sensor 9 is mounted onto the places near spindle motor and spindle bearing (Fig. 4.1), which a ppear to be major influencing heat sources. By further investigating the significant factors, (1q1) (t2) and (1q1) (t1), the lagged deformation information, which are related with temperature information from sensor 9, are found in the model. f1 f2 f3 Nominal tool path Deviated tool path f3 z z Nominal tool path f1f2 PAGE 86 76 The maximum time lag for temperature data and the thermal error is 5. The model could also be refitted with several larger ti me lags to be certain no significant lagged variables were missed. Sensors 1, 3, 4, 5, 7, 8, and 10 all appear at least 3 times with different lags in the model, which indicat es that those 3 sensors might have more complicated thermal dynamic behaviors than the rest. Since inline probing is usually not easily accessible during production, LV modeling with only lagged temperature variables is a main focus in this chapter. For the rest of this section, a latent variable model will be built for inline monitoring of machine tool thermal errors, in the case no sufficient information is available for error compensation. We still use the first 94 observations for model fitting and assume 30 latent variables without autoregressive terms. The re st of data will be us ed for model testing. With the data matrices X9055 (Eq. (4.5)) and Y901 (Eq. (4.4)), the model fitting procedure includes tw o steps, i.e., Screen sensors and find A latent variables; Refit model with significant sensors and A latent variables. The screening procedure remains the same and the result is given by Table A.4. Based on the percentage of va riance explained by the latent variables (Table A.5 in Appendix E), choose 9 latent variables out of 30, i.e., A = 9, which explain 86.954% of variation in X and 96.6966% of variation in Y (see the ninth latent variable in Table A.5). Therefore, 24 input variables (denoted as x241 ) in Table A.4 are used to refit the model to get LV91= xTW(PTW)1, W, T, P, Q and G. PAGE 87 77 To test model accuracy, first, we can us e the temperature observations No. 95 to No. 119 as input for predicting thermal errors. After taking the first order difference, the new 25 observations are mean centered and sc aled with the mean and variance obtained from training dataset. Denot e the preprocessed data as Xnew. By Eq. (4.8), the new score Tnew is obtained as Tnew = XnewW(PTW)1. By Eq. (4.9), the predicted Y is Y = XnewW(PTW)1QT and the residuals are F=YY The first 60 observations in another experiment are used to predict the thermal errors in a similar manner. Y needs to be added with mean and s caled back with the variance from the training dataset. Given the first new initial thermal error, we integrate the postprocessed Y and compare it with the observed thermal errors (Fig. 4.5). As can be seen, the residuals are small and this result is satisfactory. If sensing information is sufficient, th e prediction power of the model developed by SI theory is normally higher than the one obtained from LVM method. If not, LVM method is expected to have better perform ance. The reason is that LVM method is to model the underlying structure in X and Y, rather than to model the impact of X on Y. Therefore, reduced sensing information does not limit LVMs capability to find out some basic structures from the data, i.e., getting T = XW(PTW)1 from X and Y. The heat sources in a machine tool are abundant and mo re sensors are needed to well describe the temperature field. Thus LVM based inline monitoring method is more appropriate for the given situation. PAGE 88 78 93 98 103 108 113 118 123 20 10 0 10 20 30 40 50 Thermal Error ( m)Time Index 0 10 20 30 40 50 60 20 10 0 10 20 30 40 Thermal Error ( m)Time Index Prediction error for subsequent data in the same experiment Prediction error for another experiment Predicted Thermal Error Predicted Thermal Error Observed Thermal Error Observed Thermal Error Residual Residual Figure 4.5 Model Prediction and Residuals 4.3 InLine Monitoring of Dynamic Equi valent Errors of Machine Tool The first step of inline monitoring is to obtain historical temperature and thermal error data collected from a machine tool under normal working conditions. Then these data are used to fit a model using the same technique as introduced in Section 4.2. Its basic idea is to fit a latent variable model (Eqs. (4.6) and (4.7)) and monitor the process based on latent structures captured by utilizi ng the process information (e.g. temperature) and historical product information (e.g. thermal error) (Kourti and MacGregor, 1996). This section demonstrates the method to build control charts and to monitor machine tool thermal error. PAGE 89 79 We still use the data in Fig. 4.2 as an example. Suppose the first 94 observations on temperature and thermal error data are available for the training stage. During production, only temperature measurements are available. We need to find out whether the thermal behavior of the machine tool is changed to increase thermal errors. With the model fitted in Section 4.2. 2 (the second situation), we have T, P, Q and LV91. If the 24 variables are denoted as vector x241, then LV91 can be expressed as LVT = xTW(PTW)1, (4.15) with W(PTW)1 given in Table A.7 in Appendix E. Use ta to denote the on e realization of LV (or one row in matrix T) and ta = [ t1, t2, , tA]T with A = 9 in the example. Define a Hostellings T2 statistic in terms of latent variables as 2 2 2 1 A a a at T s (4.16) where 2 as stands for the variance of ta. It can be estimated from eigenvalues of the sample covariance matrix of T, i.e., SLV. Since t1, t2, , tA are orthogonal to each other, SLV is a diagonal matrix and the estimator of 2 as is given by 2[]aaasdiagLVS. (4.17) The control limit can be set by the F distribution (Johnson and Wichern, 1998), 2 2 ,(1) () ()AnAAn TF nnA (4.18) where FA, nA( ) is the upper 100(1)% critical point of F distribution with degree of freedom of ( A nA ). PAGE 90 80 In the example, Diag [SLV] = [7.6719, 3.1620, 2.5583, 1.0428, 1.0312, 1.2627, 0.3845, 0.2626, 0.5375], i.e., 2 1s = 7.6719 and 2 9s = 0.5375. 10 5 0 5 10 10 5 0 5 10 10 33 56t Score 1t S core 2 Ellipse Format Chart Figure 4.6 Ellipse Format Chart We can see from Table A.5 that latent va riable 1 accounts for the majority of the variance for the dependent variab les. If only first two latent variables are considered, the elliptic control chart can be employed to m onitor the stability of the machine tool. To build phase I control chart, the 2 T statistic is approximated by 2 distribution with 2 degrees of freedom (Johnson and Wichern, 1998), i.e., 22 22 12 2 22 12() tt T ss (4.19) where 2 2() is the upper 100(1)% critical point of 2 distribution with 2 degrees of freedom. For level = 0.05 (2 2(0.05)5.9915), the ellipse format chart for the 90 observations is shown in Fig. 4.5 (after the fi rst order difference, 90 data points are left for charting). In Fig. 4.6, points 10, 33, and 56 are beyond the limit. The scores of those 3 PAGE 91 81 points need to be eliminated from T (not from X) before building phase II control chart. After the elimination, Diag [SLV] = [7.7625, 2.4835, 2.6186, 1.0622, 1.0390, 1.1998, 0.3514, 0.2604, 0.5361]. This will be used for constructing phase II control limits. 8 6 4 2 0 2 4 6 8 6 4 2 0 2 4 6 t Score 1t Score 2Ellipse Format Chart for Future Observation6 3 5 2 96 3 5 2 96 3 5 2 96 3 5 2 9 Figure 4.7 Control Ellipse for Future Observations By Eq. (4.18), the phase II c ontrol chart is designed as 22 2 new,1new,2 2 2,2 22 12tt 2(1) () (2)nn TF ssnn (4.20) where n is the number of observations fo r constructing the control limits ( n = 903 = 87 in the example). 2 1s and 2 2s are computed from phase I, i.e., 2 1s = 7.2636 and 2 2s = 2.3244. 2 new,1t and 2 new,2t come from the future observations a nd they are the first two entries of tnew. For a new observation (xnew)241 (tnew)T= (xnew)TW(PTW)1. (4.21) Suppose observations No. 95 to No. 119 are new measurements. The control ellipse based on Eq. (4.18) is shown in Fig. 4.7 ( = 0.05, and F2, 85(0.05)=3.1038), which suggests that the machine tool thermal condition is stable. PAGE 92 82 4.4 Isolation of Lagged Variables and Sensor s Responsible for the OutofControl Signal Although the control charts ca n identify outofcontrol signa ls, they are unable to determine the root causes. Contribution plots ha ve been suggested to isolate the variables responsible for the outofcontrol signals (Kourti and MacGregor, 1996). The idea is to check the standardized scores (i.e., ta/ sa) with high values and to further investigate the variables that have the large contributions to those scores. In Section 4.3, points 10, 33, and 56 are id entified to be outofcontrol. The scores of these 3 points (the rows in T corresponding to these points) are listed in Table A.8. For each realization of LV (or each point), plot ta/ sa with a = 1,2, ,9 on the same graph (Fig. 4.8). With 95% confidence of type I error, the Bonferroni limit for the graph in Fig. 4.8 is 2.7 (Alt, 1985). We can see that the s econd score component (i.e., the second latent variable) of those 3 points is the main cont ributing factors to the outofcontrol signal. PAGE 93 83 1 2 3 4 5 6 7 8 9 1 0 1 2 3 Standardized Score in Point 10 ComponentsStandardized Score in Poin t 1 2 3 4 5 6 7 8 9 2 1 0 1 2 3 Standardized Score in Point 33 ComponentsStandardized Score in Point 1 2 3 4 5 6 7 8 9 4 2 0 2 4 Standardized Score in Point 56 ComponentsStandardized Score in Point Figure 4.8 Standardized Scores in Points 10, 33, and 56 Since tT= xTW(PTW)1 (refer to Table A.7 for W(PTW)1), we can further investigate the contributions of lagged variable in x and contributing sensors responsible for the signals. For each realization of LV, the contribution of variable xj (the j th component in x241, j =1,2,..,24 and t =10, 33, and 56 for this exam ple) to the score of the a th score component is defined to be (Kourti and MacGregor, 1996) Contributiona,j= xjeaj, (4.22) where eaj is the j th component in the a th column (corresponding to a th score component) of matrix W(PTW)1. To the second latent variable in points 10, 33, and 56, the contributions of 24 variables (see Table A.4 for the variables) are shown in Fig. 4.9. Clearly, variable No.21, i.e. (1q1) s7 (t), makes the largest contri butions to the signals. PAGE 94 84 0 5 10 15 20 25 0.5 0 0.5 1 1.5 Point 10 Variable No.Contribution to Score Component 2 0 5 10 15 20 25 0.5 0 0.5 1 Point 33 Variable No.Contribution to Score Component 2 0 5 10 15 20 25 1.5 1 0.5 0 0.5 Point 56 Variable No.Contribution to Score Component 2 Figure 4.9 Lagged Variable Contri butions to Score Component 2 Since the 24 lagged variables are from 10 sensors (sensor 9 is excluded by the screening procedure), study on the aggregated contributions from each sensor might provide valuable information for root cause de termination, i.e., finding out the main heat sources that lead to thermal errors. Th e aggregated contribution can be found by summing up the contribution of lagged variables corresponding to each sensor (see the correspondence in Table A.6). Fig. 4.10 show s the contributions of 10 sensors to the second latent variable of points 10, 33, and 56. Although this latent va riable is the main contributing factor to the outo fcontrol signal in these 3 points, the patterns in terms of sensor contributions are quite different. In these points the he at sources at sensors 3, 4, and 7 are the main factors causing the changes of machine tool thermal condition. Further investigation should be taken to find out the physical reas ons, such as spindle bearing PAGE 95 85 overheating or coolant not func tioning. All the sensor contributions for point 56 for score component 2 take negative values because th e data was sampled at cooling down cycle. 1 2 3 4 5 6 7 8 9 10 0.5 0 0.5 1 1.5 Sensor No.Sensor ContributionPoint 10 1 2 3 4 5 6 7 8 9 10 0.5 0 0.5 1 Sensor No.Sensor ContributionPoint 33 1 2 3 4 5 6 7 8 9 10 1.5 1 0.5 0 Sensor No.Sensor ContributionPoint 56 Figure 4.10 Sensor Contributi ons to Score Component 2 If more latent variables are beyond limits in Fig. 4.8, we can study the overall average lagged variable c ontribution and sensor contri bution. The procedure of computing the overall average variable contri bution is similar to that in Kourti and MacGregor (1996). 4.5 Summary This chapter models the error equivale nce for the dynamic process errors and develops the inline monitoring of the equi valent dynamic process errors and process degradation caused by thermal errors. Machine tool thermal errors are selected as an example to demonstrate the dynamic error equivalence modeling and process monitoring PAGE 96 86 in machining. The thermal sensors and ma ximum time lag are chosen according to a screening procedure applied to results of LVM. The in line monitoring of dynamic equivalent errors is achieved by theories of statistical qu ality control: first T2 control chart is built to detect out of limit signal; then bar plots of normalized scores and contribution are created to identify the major contributi ng latent variables, the contribution of each lagged variable and sensor to the thermal errors. These procedures show that LVM method provides interesting results in vari able screening, mo del prediction, and especially in inline monitori ng and root cause identificati on. LVM method is especially appropriate for multivariate measurements and illconditioned data, and it could also provide a benchmark to judge whether the se nsing information is sufficient to perform dynamic error compensation. The success of applying LVM method in mon itoring is due to the property of LV model, i.e., finding out the latent variables that maximize the covariance between process variable (e.g., temperature) and product va riable (e.g., thermal errors). LVM method captures the thermal patterns from the hist orical data collected from an incontrol machine tool. Future observation is assumed to be outofcontrol signal if the pattern changes are detected. Once an outofcontrol si gnal is detected, the study shows that the lagged variable and sensor contribution plot s are very helpful to determine the root causes. PAGE 97 87 Chapter 5 Error Compensation Based on Dynamic Error Equivalence for Reducing Dimensional Variation in Discrete Machining Processes* Traditional SPC technique has been wide ly employed for the process monitoring in discrete manufacturing. Ho wever, SPC does not consider any adjustment that prevents the process drifting from the target. Furthe rmore, many inline adjustment approaches, such as thermal error compensation and avoidance, are designed only for machine tool error reduction. This chapter intends to fully utilize the engineering process information and propose an alternative compensation stra tegy that could automatically reduce the overall process variations. Based on the model of dynamic equivalent errors developed in Chapter 4, a SPC integrated errorcancelingerror APC methodology is derived to compensate for both time invariant and dynamic errors by adjusting the base error. The performance of the adjustment algorithm su ch as stability and sensitivity is then evaluated. A self updating scheme for the ad justment algorithm has been proposed to track the latest process information as well. This process adjustment has been simulated using the data collected from a real machining process. The results show that this algorithm can improve the machining accuracy and reduce the process variations. The work in this chapter ha s appeared in Wang, H. and Huang, Q., 2007, U sing Error Equivalence Concept to Automatically Adjust Discrete Manufacturing Processe s for Dimensional Variation Reduction, ASME Transactions, Journal of Manufacturing Science and Engineering, 129, 644652. PAGE 98 88 In Section 5.1, an error e quivalence adjustment algorith m is derived to counteract the machining process variation. Its integrati on with SPC is discussed in Section 5.2. The adjustment algorithm is implemented via a case study in Section 5.3. Section 5.4 evaluates the performance of the APC methodology such as stab ility and sensitivity when a change in the dynamics of process occurs Conclusions are given in Section 5.5. 5.1 Automatic Process Adjustment Ba sed on Error Equiva lence Mechanism For a manufacturing process with causal relationship x = f (u1, u2, ,up) + the traditional error compensation strategy is to minimize individual process errors uis so as to reduce output deviation u. As pointed out in Chapter 3, since error equivalence also implies the cancellation among process erro rs, this allows us to develop a new compensation strategy, i.e., treating all proce ss error sources as a system and using one error to compensate for the others. For instan ce, with the development of flexible fixture whose locator length is adjust able, it is feasible to compensate for the overall process errors in the machining process by changing locator length. In this new strategy, the outputs of the adjustment algorithm and pro cess will be monitored using SPC methods. The main purpose is to monitor unexpected ev ents such as adjustment device failure. It should be noted that compen sation cost is a critical fa ctor to be considered in real applications. It is not discussed in this study because cost issue is often case dependent. Using the observed feature deviation x(n) at time period n as input, the proposed error equivalence based algorithm 1uG generates adjustment c(n) to counteract *(1) 2 pn ii u for the ( n +1)th time period. Let c(n) be the cumulative amount of adjustment. x(n+1) is PAGE 99 89x(n+1) = *c(n) + **(1) 2 pn ii u+(n+1). (5.1) The adjustment c(n) should be able to cancel E(x) and reduce the process variation. The adjustment algorithm can be designed to re duce the mean squared deviation of product feature, i.e., min E(1)2[]n x. As proposed by Capilla, et al. (1999), we can treat a simpler problem of minimizing an instan taneous performance index, min (1)2 []n x. Taking the first derivative of (1)2 []n x and equaling it to zero, the adjustment rule can be summarized as follows. When the process errors *ius are all static, the adju stment to reduce the mean shift of the process output is ()***1*() 1 ().n nTTk j jSk cu x (5.2) where S is the set for the static errors. Eq. (5 .2) is in fact the same as Eq. (3.9). Considering static and dynamic pro cess errors, the process adjustment c(n) using error equivalence turns out to be ()**(1)**() {} ({}),nnk jijikkn jSiDjSiDg cuuuu *()(1)**1*()* ()nnTTn ij iDjS uc xu, (5.3) where D is the set for dynamic errors. * j jS u and *(1)n i iD u are the process static and dynamic equivalent errors based on the least square estimation, respectively. Dynamic errors can be represented by *(1)*() ({})nk iik iDiDg uu, where g(.) is the fitted dynamic model of process errors. Since the pr ocess will compensate for the same amount of error 2p j j u at each time period, the proposed sequential root cause identification procedure can be applied to identify the error sources. PAGE 100 90 In the machining processes, the dynamic m achine tool error can be represented by an ARX model with the temperatures of machine tool as input (see Eq. (4.13)). Substituting process model (4.13) into Eq. (5.1 ), the prediction for feature deviation at period n+1 is 12(1)*()*()1()*()()**() 330 10 []pp nnlnllnln j lljS x c KAKm KBt uK. (5.4) The adjustment rule for machining process is then c(n)12()1()()()*() 330 10pp lnllnln j lljS KAKmKBtuK, c(n0)=0, and ()(1)**1*()* ()nlnlTTnl j jS mc xu. (5.5) Static error is obtained by di rect measurement or by equation ***1** ()TT ji jSiD u xu, (5.6) where x and i iDu are the measurements of feature deviation and dynamic errors when fitting the error model g(.). Applicable conditions of compensation strategy. The base error u1 is not random because of the adjustment. Although the adjustment c is expected to compensate for the remaining process errors 2 =i p ix, it becomes a new random error source because of the variability in the actuator. Therefore, the adjusted total process error *au has 2 E()E()E()p aiiiucKu and 2 Cov()Cov()Cov()pT aiiiiucKuK. (5.7) 1xG normally aims to keep the process output x on the target and with the minimum variation. The commonly used adju stment algorithm is to let E(c) = 2 E()p iiiKu or * E()u= 0. However, the generalized variance of error au or Det(* Cov()au) is not necessary to be smaller than the one without adjustment, where Det(.) represents the PAGE 101 91 determinant of the matrix in the parentheses. Cleary, if Det(Cov() c) Det(1 Cov()u), the new compensation strategy will uniformly reduce process variation. If Det(Cov()c) > Det(1 Cov()u) but the increase of tota l process variation (Det(* var()au)Det(* Cov()u)) /Det(* Cov()u) is insignificant, the compensati on might be acceptable as well. For instance, the precision of fixture is usually much higher than the workpiece and machine tool. An adjustable fixtur e equipped could have lowe r precision or larger Det(Cov() c). The minor percentage of fixtur e variation in the tool pro cess errors might justify the application of error compensation because it brings the process on the target. Compensation is normally not effective if Det(Cov()c)>Det(1 Cov()u) and (Det(* Cov()au)Det(* Cov()u))/Det(* Cov()u) is appreciable. The conventional compensation strategy aims to offset E()iu and reduce Cov()iu individually. It will be effective if ther e are only a few process errors dominating in E()u and Cov()u. Otherwise, a large number of adjustme nts are needed to compensate for all error sources in order to keep the process output x on target. Under this condition, these two compensation strategies can be applied complementarily. The error sources with the largest variations can be counteracted using conven tional methods to reduce Cov()u, whereas the new compensation strategy is to achieve * E()au= 0. 5.2 SPC Integrated Process Adjust ment Based on Error Equivalence In real application, process adjustment as shown in Eq. (5.5) has to consider the following practical problems: PAGE 102 92Over Adjustment. Over adjustment may increase the production cost and process variation. However, the adjustment does not ne ed to be implemented in the periods when: The process errors are not significant comp ared to the assigned tolerance of base errors (denoted by 1 u). We can predict the process erro rs in the next period and test if the predicted errors are within tolerance. The adjustment is beyond the accuracy limit of the device. Therefore, in the early stage of adjustment system design, we s hould choose the device whose accuracy limit matches the assigned tolera nce of base errors. Fast varying errors. The adjustment in Eq. (5.5) only compensates for the slow varying dynamic errors (quasistatic errors), which ar e relatively constant between the adjacent periods. Large process variation within one peri od can lead to large adjustment errors in x. In order to identify such process change, the samples of outputs {x(n)} of the manufacturing process within one period can be monitored by quali ty control charts. Unexpected process errors. On some occasions, unexpected process errors (e.g., variation of adjustable fixture locator, hot chips during machining) ha ve not been considered in {ui} and thus the adjusted process could show a large variation. Integration of SPC and APC is an economic way to reduce the variation of adjusted process though it has been rarely applied in a discrete manufacturing proce ss. Monitoring the estimated noise, i.e., ()n=x(n)**() 2 pn ii u*c(n1) can help to detect if unexpect ed errors impact the process output. When the unexpected errors take plac e, we can also update the process error model to track the latest information about er rors and make a closer prediction. With the updating scheme, the coefficients in function g(.) also change with period n. In the machining process, suppose we measure the temperature and thermal error every period, PAGE 103 93 and the measurement data ar e available at the period 1~n0. The updating adjustment procedure can be proposed as follows: At the beginning of period n0+k, data, including part f eatures (measured by CMM) {x(n0+kl)}, thermal errors (measured by inline probes) { qm(n0+kl)}, and temperatures (measured by thermal sensors) {t(n0+kl)}, are collected to compute the locator adjustment c(n0+k1)c(n0+k2). k=1, 2, 3, (Eq. (5.5)). Then cut the parts after the adjustment. With the updating scheme, the fitted coefficient matrices {A(l)} and {B(l)} in Eq. (5.5) also change with period n (or equivalently, updating iteration). So, it is reasonable to denote them as {An(l)} and {Bn(l)}. At the end of period n0+k, measure the parts and take the average of measurement results to estimate x(n0+k). Increase k and repeat the above procedures. Since SPC is an effective tool to enhan ce the process robustness, we can develop a SPC strategy for the adjusted process by co llecting the information of process output x(n) and adjustment output c(n) for each sample product. Then we can do the following: Monitor samples of feature deviation x within one period to determine whether the period length is appropriate for quasistatic error assump tion. Shorter period duration might be necessary when quality control chart signals an alarm. Monitor the part features x(n) by multivariate control charts, c(n) by multivariate EWMA chart and the noise estimation ()n for all the samples to identify whether unexpected errors or process change occurs. Update the adjustment algorithm when the control chart indicates outofcontrol of x(n), c(n), or systematic trend of ()n. PAGE 104 94 Within the device accuracy limit, the incremental adjustment c(n)c(n1) should be applied (to compensate for the quasistatic errors) only when cumulative adjustment c(n) exceeds control limits determined by the tolerance of the base error 1u and meanwhile incremental adjustment c(n)c(n1) exceeds device accuracy limits. Both error tolerance and device accuracy limits define a dead band for the adjustment. The SPC integrated adjustment based on error equivalence can be shown by Fig. 5.1. To simplify the representation, the fi gure only shows the adjustment scheme for compensating static errors. Manufacturing processes( *) 2 p ii u 1uG c x u* a()*()*() {} *()()**1*()* ({}), ()nnk jikkn jSiD knTTk ij iDjSg cuu uc xuError equivalence adjustment Statistical quality control EWMA chart monitoring c Monitoring x Dead band Manufacturing processes( *) 2 p ii u 1uG c x u* a()*()*() {} *()()**1*()* ({}), ()nnk jikkn jSiD knTTk ij iDjSg cuu uc xuError equivalence adjustment Statistical quality control EWMA chart monitoring c Monitoring x Dead band Figure 5.1 Adjustment Based on Error Equivalence 5.3 Simulation of Error Equivalence Process Adjustment We use the same single stage milling proce ss to implement the process adjustment as in Chapter 2. The process performs cutting on two planes X1 and X2 as shown in Fig. 2.9 in Chapter 2. Thickness along the z direction lz and y direction ly are the part features to be controlled (the nominal th ickness of the finished part is lz =15.240.1mm and ly=96.50.1mm). PAGE 105 95 In this simulation, we use the data (Fig 5.2(a)) from the experiment. There are 11 thermal sensors mounted on the CNC milling machine to collect data (r=11). The thermal deviation is measured along two di rections: the a ngular deviation around x axis and translational deformation along z direction of the tool head (see Fig. 5.2(b)). The left panel of Fig. 5.2(a) shows the readings fr om 11 thermal sensors. The middle and right panels show the measurement of thermal errors The data are collected in each adjustment period. 0 50 100 150 20 22 24 26 28 30 32 34 36 38 40 Time PeriodTemperature (Celcius degree) 0 50 100 150 0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Deformation along z direction (mm) 0 50 100 150 0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Deformation around x directionTime Period Time Period Figure 5.2 (a) Machine Tool Te mperature and Error Data Figure 5.2 (b) Thermal Error Measurements z Tool path Tool hea d PAGE 106 96 We have derived in Chapter 3 to be: 00.02630.0263000 0.01580.00790.0079000 000000 00.13790.13791.33681.33681 0.08280.04140.04141.50.50 1.30330.84831.1517000 0000.02630.02630 000000 0.01580.00790.0079000 0026320.26321.20261.2026 1 0.1580.0790.0791.5050 0.22121.61060.3894000 (5.8) Suppose the maximum time lag in the model is 5, and N=95, the fitted coefficient matrix G is ( 4 )( 4 )( 0 )( 0 )( 4 )( 1 ) 11 111 111 ( 4 )( 4 )( 0 )( 0 )( 4 )( 1 ) 11 111 12200 00Tvvvvaa wwwwaa G where vi(l) and ai(l) are fitted coefficients. Then coefficient matrices )( ~ lA and )( ~ lB are () 1 () () 2000000 000000 00000 00000 000000 000000l l la a A and ()()()() 12311 () ()()()() 12311 6110000 0000 0000 0000llll l llllvvvv wwww B (5.9) The static kinematic errors, after being tran sformed to equivalent fixture error, are assumed to be *j jSu=[0.4 0 0.35 0 0 0]T mm. The measurement noise (n) is assumed to follow N(0, (0.002mm)2) for displacement and N(0, (0.001rad)2) for orientation. For each adjustment period, 5 parts go through th e cutting operation. We use average of 5 PAGE 107 97 measurements to estimate the real feature deviation for each period. Thermal error and temperature for 95 periods (So, n0=95) are available before the adjustment is applied. The measurements of temperature from i~95+i periods and thermal error from i~94+i are used to estimate the adjustment of locator pins for the (95+i)th period, i=1,2,,20. The adjustment algorithm is updated afte r measuring the parts at the (95+i)th period. The accuracy of the locator movement is assumed to be f=0.003mm and the criterion for stopping the compensation is 0.01 c(n)c(n1)0.01mm. The values of adjustments for 6 locators are given by the so lid line in the Fig. 5.3. The dash dot line represents the value of 3 f. The adjustments for locators 4, 5, and 6 are zero since the EFEs of errors introduced on these lo cators are zero in this example. 94 99 104 109 114 06 05 04 03 02 01 0 01 Adjustment Period or PartsAmount of Adjustment(mm) 94 99 104 109 114 012 01 008 006 004 002 0 002 94 99 104 109 114 05 04 03 02 01 0 01 02 Amount of Adjustment(mm) Amount of Adjustment(mm)Adjustment Period or Parts Adjustment Period or Parts Locator 1 Locator 2 Locator 3 Figure 5.3 EFE Adjustment The effect of the automatic process ad justment can be evaluated by monitoring the thickness of the part ly and lz. The mean of such distance (in each period) is estimated by the average of 4 edge lengths along y and z directions at that period. The variance in each period is estimated by the variance of the 4 edge lengths. PAGE 108 98 94 104 114 149 15 151 152 153 154 Adjustment PeriodAverage thickness (mm) 94 104 114 0 01 02 03 04 05 06 07 Adjustment PeriodStandard deviation of thickness (mm) 94 104 114 9646 9647 9648 9649 965 9651 9652 9653 Adjustment Period 94 104 11 4 0 001 002 003 004 005 006 Adjustment PeriodAverage thickness (mm) Standard deviation of thickness (mm) Figure 5.4 Monitoring Thickness and St andard Deviation of Edge Length Fig. 5.4 shows the mean and standard devi ation of the thickness for 20 adjustment periods (periods 95~114). There is no adjustme nt applied in period 95. We can see that, after the process adjustment, the mean of the thickness is within specification limit (0.01mm) and variance is grea tly reduced. We conclude th at the proposed adjustment algorithm can significantly increas e the product quality. It sh ould be noticed that the thickness ly has less mean shift than that lz. This is because plane X2 tilts around x axis and the distances between edges ly are smaller along z direction. Such edge layout leads to edge lengths with less variance and mean shift. 5.4 Adjustment Algorithm Evaluation Since the adjustment algorithm may have unstable modes, it is necessary to estimate the performance such as stability a nd sensitivity. The stability of the adjustment algorithm means that an error in the output can be cancelle d by an adjustment sequence that converges to zero. One can obtain the stability of the algorithm by inspecting the poles of the transfer function of Eq. (5.5). Sensitivity refers to how the quality could be PAGE 109 99 affected whenever moderate changes occur in the algorithm parameters. This can be analyzed by differentiating Eq (5.5) with respect to coefficients in function g(.). Introducing backward operator q1, Eq. (5.5) can be represented as 1 1 21()1() 33 1 ()1**1*()()()()1**() 3330 101[] ()p llln n l ppp lTTlnllnln nnnjj llljSjSq qq IKAKc KAK xKBtKAKuuK (5.10) The stability of the algorithm is gove rned by the entries in 66 matrix 1()111 33 1[]p ll n lq IKAK. If the roots of denominator of each entry contain the poles inside the unit circle in q plane, the algorithm is stable. It clear that the adjustment algorithm is always stable if the thermal error model does not contain autoregressive term, i.e., An(l)=0. When autoregressive terms are included in the model, the algorithm may be unstable though the prediction accuracy may increase. The designed algorithm at certain periods may contain unstable poles (poles outside unit circle). This may cause the adjust ment exhibit fluctuati on and large output if the parameters An(l) and Bn(l) in the algorithm had been unchanged as n increase. The solution for unstable output can be to use th e model without autoregressive term since such algorithm is always stable. Another so lution is to introduce the updating scheme which makes the adjustment output capture the latest process information. In this case, Eq. (5.10) is not strictly prope r to evaluate the stability for only one adjustment period because model for m(nl1) is different from m(nl). In practice, the proposed algorithm can achieve satisfactory results. This ha s been validated by the results from the simulation study in Section 5.3. PAGE 110 100 Another important issue is the sensitivity of the algorithm to the modeling errors that can feasibly occur. If there are moderate changes of modeling parameters (entries in matrices A() l n) and B() l n), we are more interested in ho w the quality of the product could be affected. Such change may be due to seve ral reasons, including se nsor reading errors and change of lubrication condition. To study sensitivity, expand Eq. (5.5) as 1 2 123 ()()()() 12 11 11 ()()()() 10 01 311 ()()()()()() 20 1101 () 6[], [],1,2,3, ,4,5, 0,p nllnl jjyii li p llnln iiyiii li pp nlnllnln jjziiijziii lili ncaafhm vwftukj cafhmwftukj c (5.11) where hi is the function of fixture coordinates f1, , f6. Differentiating both hand sides of Eq. (5.11) leads to 11 22 1233 ()()()()() 12 1111 1111 ()()()() 1 0101 311 ()()()()() 2 1101,1,2,3, ,pp nnllnll jiiiijy lili pp nllnll iiyii lili pp nnllnll jjziijzii lilichmahmfa tvftwj cfhmaftwj () 64,5, 0.nc (5.12) m(nl) is only related to the previously fitted mo del and is not affected by the fitting error of An(l) and Bn(l). It can be considered as a consta nt when we conduct the sensitivity analysis. For the example in Section 5.3, s ubstituting the values of coordinates yields 1 1 22 1 1 2() 112321231 11 1111 0101 () 12321231 11 11 01(2516.322.1)(1.30.81.2) 19.2, (107.57095)(1.30.81.2)pp n ll pp iiii lili pp n j ll p ii licmmmammma tvtw cmmmammma tv 2 1 211 01 11 () 1232 101 () 682.5,2,3, (13.38.411.5)10,4,5, 0.p ii li pp n j ii lli ntwj cmmmatwj c (5.13) PAGE 111 101 To simplify the representation, time indices (nl) and l are dropped in this equation. We can conclude the following about the adjustment algorithm at time period n, There is no adjustment on the locator 6. Deviation of coefficients a1 (nl) and vi(nl) does not affect the adjustment c4 (n) and c5 (n); and a1 (nl) has the same effect on the adjustment of c1 (n), c2 (n), and c3 (n). The adjustment for locators 2 and 3 are more likely to be affected by the fitting errors. Locators 4 and 5 are less sensitive to the f itting error. This is because the thermal error occurs is only around z and along x directions. The EFEs on locators 1, 2, and 3 have more impact on the feat ure deviation than on locator s 4 and 5. Locator 6 never affects feature deviation al ong these two directions. 94 99 104 109 11 4 14.6 14.8 15 15.2 15.4 15.6 15.8 16 Ad j ustment PeriodAverage thickness along z direction (mm ) 94 99 104 109 11 4 96.48 96.49 96.5 96.51 96.52 96.53 96.54 96.55 96.56 A d j ustment PeriodAverage thickness along y direction(mm) Figure 5.5 Effect of Parameters Cha nge in Process Adjustment Algorithm PAGE 112 102 The updating scheme can effectively enha nce the sensitivity robustness of the adjustment algorithm. We have simulated the feature deviation when there are changes of 50%, 200%, 350% and 500% in the coefficients v6 (0) and w6 (0) in matrix B105 (0). Fig. 5.5 shows an example when there are changes up to 500% in the coefficients. We can notice a large variation of feature lz at period 104 and 105. Feature ly is not too much affected. After period 105, the feature lz falls within the specification limit since the adverse effect of the fitting error has been counteracted by the updated model. 5.5 Summary APC and its integration with traditional SPC have not been sufficiently addressed in discrete machining processe s. Regarding the error compensation, the conventional method in machining processes is to compensate for the multiple errors individually. Based on the dynamic error equiva lence model developed in Chapter 4, this chapter derives a novel SPC integrated errorcancelingerror APC methodology to compensate for joint impact of errors in the machining process. As an alternative strategy, an APC methodology by using one type of erro r to compensate for others has been proposed. The method shows an advantage that it compensates for the overall process variation without interrupting production in the machining processes. The applicable condition of this new compensati on strategy is also discussed. This chapter first develops an error e quivalence adjustment method based on the engineering process causal m odel and statistical model of dynamic equivalent errors. It uses prediction from the statistical process erro r model to compensate for the errors in the future periods. Second, SPC is applied to th e adjusted process to identify the unexpected PAGE 113 103 process errors. When SPC signals an alert, the fitted model is updated to obtain the latest information of the dynamic process. The adjustment algorithm is implemented using the data collected from a milling process. It has been shown that the error equivalence adjustment can effectively improve the mach ining accuracy and reduce the variation. In addition, a discussion on the applicable c ondition of compensation strategy shows that the variation of adjustment to the base error must be relatively small compared with that of the base error itself. Finally, the perf ormance of designed adjustment algorithm is analyzed. It has been demonstrat ed that the proposed updating scheme is effective to tune the parameters and stabilize its output. The se nsitivity of adjustment output to the change of model parameters is also studied. It help s to find out the para meters that contribute most to the deviations in the adjustment outputs. PAGE 114 104 Chapter 6 Conclusions and Future Work 6.1 Conclusions Process quality improvement usually relies on the modeling of process variations. Models that can reve al the physics of fundamental e ngineering phenomena could provide better insights into the process and significan tly enhance the quality. The work in this dissertation aims to improve the understanding of error equi valence phenomenon, that is, different types of process errors can result in the same feature de viation on parts. The implication of error equivalence mechanism can greatly impact the prediction and quality control in manufacturing processes. The majo r contributions of th is dissertation are summarized as follows Error equivalence modeling. A rigorous mathematical de finition of error equivalence is introduced. An error transformation is proposed to establish the mathematical formulation of error equivalence phenomenon. By the kinematic analysis, equivalent errors are transformed into one base error. In machining processes, the base error is chosen to be fixture deviation and other t ypes of errors, including datum and machine tool errors, are transformed to the fixture er ror. A process causal model is derived to depict how the base errors affect the features of part s. The error equivalence is investigated for both static and dynamic process errors. The model serves as the base for quality prediction and control. PAGE 115 105 Sequential root caus e diagnosis strategy. Due to the error equivalence mechanism, errors may cancel each other on the part features and may conceal the process information for process diagnosis. The proposed sequential diagnostic methodology based on error equivalence overcomes the difficulty by conducti ng diagnosability analysis, identifying the existence of pro cess variations, and distinguishing the multiple error sources. Errorcancelingerror compensation strategy integrated with SPC. The error cancellation is further expl ored and a novel errorcancel ingerror APC strategy is proposed, i.e., treating all erro r sources as one system and using the base error to automatically compensate or adjust the ot hers for process variation reduction. An error equivalence adjustment al gorithm is designed to compensate both time invariant and dynamic errors. By monitoring outputs from the manufacturing pr ocess as well as adjustment algorithm, SPC could enhance th e robustness of the controlled process. In this dissertation, the studies and an alyses are based on a machining process. However, error equivalence methodology for pro cess control is generic and can be easily extended to other discrete manufacturing processes. PAGE 116 106 6.2 Future Work This study aims to establish error equivale nce theory and obtain insights into this fundamental phenomenon for improved proce ss variation control. In addition to the results obtained in the modeling, diagnosis and error compensation, we can further expand the impact of erro r equivalence on the life cy cle of product design and manufacturing. The error equi valence can facilitate tole rance synthesis and optimal tolerance allocation in a comp lex manufacturing process. For example, process tolerance can be allocated only to the to tal amount of equivalent error at the initial design stage. This would lead to reducing the dimension of design space. Then the tolerance would be further distributed for individual error sources at late stages of pr ocess design when more process information becomes available. 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PAGE 125 115 Appendices PAGE 126 116 Appendix A: Infinitesimal Analysis of Wo rkpiece Deviation Due to Fixture Errors If there are small deviations on these 6 locators as (f1z f2z f3z f4y f5y f6x)T, the change of orientation and position of rigid wo rkpiece in the 3D space can be analyzed by (Cai, et al., 1997). 1f qJ Ef, (A.1) where for prismatic workpiece, Jacobian Matrix J is J v Ixv Iyv Iz2f 1zv Iyf 1yv Iz2f 1zv Ixf 1xv Iz2f 1yv Ixf 1xv Iyv Ixv Iyv Iz2f 2zv Iyf 2yv Iz2f 2zv Ixf 2xv Iz2f 2yv Ixf 2xv Iyv Ixv Iyv Iz2f 3zv Iyf 3yv Iz2f 3zv Ixf 3xv Iz2f 3yv Ixf 3xv Iyv IIxv IIyv IIz2f 4zv IIyf 4yv IIz2f 4zv IIxf 4xv IIz2f 4yv IIxf 4xv IIyv IIxv IIyv IIz2f 5zv IIyf 5yv IIz2f 5zv IIxf 5xv IIz2f 5yv IIxf 5xv IIyv IIIxv IIIyv IIIz2f 6zv IIIyf 6yv IIIz2f 6zv IIIxf 6xv IIIz2f 6yv IIIxf 6xv IIIy where vj=(vjx vjy vjz)T is the orientation vector of datum surface j and the index k is dropped in the equations in Appe ndix A. The Jacobian matrix J is definitely full rank because the workpiece is deterministically lo cated. The inverse of Jacobian therefore exists. Matrix is v Ix v Iy v Iz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 vIxvIyvIz000000000000 000000 vIxvIyvIz000000000 000000000 vIIxvIIyvIIz000000 000000000000 vIIxvIIyvIIz000 000000000000000 vIIIxvIIIyvIIIz When it is clear in the text, index k is dropped in the above equation. E is an 186 matrix, that is, 1 1 1 2 2 3 186 E00000 0E0000 00E000 000E00 0000E0 00000E, where E1=(0 0 1)T, E2=(0 1 0)T, and E3=(1 0 0)T. (A.2) (A.3) PAGE 127 117 Appendix B: Proof for Proposition in Chapter 2 Proof. If the variables u1, u2, ..., um can be grouped to Eq. (2 .25), we can expand Eqs. (2.24) and (2.25) and make them equal. Then we get kjpi=gij. Substituting it into yields 11211 12222 12... ... = ... ...m m nnmnkpkpkp kpkpkp kpkpkp whose rank is not larger th an 1. On the other hand, if rank(H) is less than 1, there exists at most one row that is linearly independent. The conclusion is obvious. PAGE 128 118 Appendix C: Proof for Corollary in Chapter 2 Proof. This can be proved by substituting Eq. (2.4) into the expression 1111011TT TFFFFT jPdfMmMPjX=HHHHHHHX and conducting a lengthy computation. It can be found that equa lities among the coefficient matrices are determined by the symmetry of matrix 1111 FF dfMmM HHHHH. Since Hd, Hf, and Hm are skewsymmetric, 1111 FF dfMmM HHHHH is also skewsymmetric if FHM=I88. Nonidentity matrix FHM can affect the symmetry of 1111 FF dfMmM HHHHH, which yields different coefficient matrices for d, f, and m. Therefore, the MCS and the FCS must coincide with each other for the proposed grouping method. PAGE 129 119 Appendix D: Determine Difference Order for D(q) Table A.1 First Order Difference Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5 Sensor 6 Mean Segment 1 0.046 0.257 0.111 0.107 0.089 0.003 Segment 2 0.042 0.078 0.072 0.004 0.053 0.018 Variance Segment 1 0.007 0.126 0.080 0.035 0.045 0.007 Segment 2 0.007 0.137 0.056 0.030 0.038 0.006 Table A.1 First Order Difference (Continued) Sensor 7 Sensor 8 Sensor 9 Se nsor 10 Sensor 11 Thermal Error Mean Segment 1 0.194 0.059 0.030 0.039 0.000 0.481 Segment 2 0.049 0.025 0.027 0.023 0.019 0.378 Variance Segment 1 0.140 0.001 0.003 0.009 0.005 9.053 Segment 2 0.106 0.001 0.003 0.015 0.004 9.191 Table A.2 Second Order Difference Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5 Sensor 6 Mean Segment 1 0.0041 0.0093 0.0003 0.0079 0.0025 0.0069 Segment 2 0.0003 0.0003 0.0054 0.0042 0.008 0.0013 Variance Segment 1 0.0019 0.04 0.0764 0.043 0.0306 0.005 Segment 2 0.0031 0.0353 0.0425 0.0331 0.0161 0.0051 Table A.2 Second Order Difference (Continued) Sensor 7 Sensor 8 Sensor 9 Se nsor 10 Sensor 11 Thermal Error Mean Segment 1 0.0317 0.0002 0.0024 0.0019 0.0066 0.1921 Segment 2 0.0107 0.0008 0.0007 0.0018 0.001 0.0716 Variance Segment 1 0.1275 0.0006 0.0003 0.0056 0.0018 7.6983 Segment 2 0.0763 0.001 0.0004 0.0045 0.0018 5.9466 PAGE 130 120 Appendix E: Screened Variables Table A.3 Screened Variables With Autoregressive Terms No. Predictor B VIP No. Predictor B VIP 1 (1q1)s1 (t4) 0.013050.86065 16 (1q1)s8 (t1) 0.099531.16198 2 (1q1)s4 (t4) 0.032410.81609 17 (1q1)s10 (t1) 0.028450.92829 3 (1q1)s5 (t4) 0.006340.83828 18 (1q1)s1 (t) 0.082471.59173 4 (1q1)s8 (t4) 0.091790.86482 19 (1q1)s2 (t) 0.020800.83598 5 (1q1)s10 (t4) 0.023640.80570 20 (1q1)s3 (t) 0.068072.47004 6 (1q1)s3 (t3) 0.348480.80145 21 (1q1)s4 (t) 0.051542.25059 7 (1q1)s3 (t2) 0.077690.95171 22 (1q1)s5 (t) 0.134031.91569 8 (1q1)s4 (t2) 0.270980.90617 23 (1q1)s6 (t) 0.104941.22256 9 (1q1)s7 (t2) 0.565761.18210 24 (1q1)s7 (t) 0.741622.85025 10 (1q1)s1 (t1) 0.075590.87789 25 (1q1)s8 (t) 0.099771.86957 11 (1q1)s3 (t1) 0.379201.63392 26 (1q1)s10 (t) 0.057741.22009 12 (1q1)s4 (t1) 0.214411.51635 27 (1q1)s11 (t) 0.050221.06204 13 (1q1)s5 (t1) 0.049981.05582 28 (1q1)(t2) 0.204951.07594 14 (1q1)s6 (t1) 0.490690.93271 29 (1q1)(t1) 0.660461.80827 15 (1q1)s7 (t1) 0.385061.79123 Table A.4 Screened Variables Without Autoregressive Terms No. Predictor B VIP No.Predictor B VIP 1 (1q1)s1 (t4) 0.075460.87825 13 (1q1)s8 (t1) 0.01998 1.16039 2 (1q1)s5 (t4) 0.142130.84481 14 (1q1)s10 (t1) 0.06269 0.91400 3 (1q1)s8 (t4) 0.064490.86229 15 (1q1)s1 (t) 0.05278 1.61400 4 (1q1)s3 (t2) 0.138630.94842 16 (1q1)s2 (t) 0.07197 0.83951 5 (1q1)s4 (t2) 0.203170.88582 17 (1q1)s3 (t) 0.13796 2.45840 6 (1q1)s7 (t2) 0.371191.21801 18 (1q1)s4 (t) 0.02998 2.24474 7 (1q1)s1 (t1) 0.094050.85471 19 (1q1)s5 (t) 0.04362 1.94720 8 (1q1)s3 (t1) 0.287251.64462 20 (1q1)s6 (t) 0.24511 1.22009 9 (1q1)s4 (t1) 0.227531.54242 21 (1q1)s7 (t) 0.75227 2.84803 10 (1q1)s5 (t1) 0.001471.06443 22 (1q1)s8 (t) 0.13120 1.88072 11 (1q1)s6 (t1) 0.595380.92221 23 (1q1)s10 (t) 0.24731 1.23607 12 (1q1)s7 (t1) 0.031431.80960 24 (1q1)s11 (t) 0.03777 1.06831 PAGE 131 121 Appendix F: Results of Partial Least Square Estimation Table A.5 Percentage of Varian ce Explained by Latent Variables Number of Latent Variables Model Effects (%) Depe ndent Variables (%) Current Total Current Total 1 15.591715.591776.182176.1821 2 20.107235.69897.707783.8898 3 9.368445.06726.959190.8489 4 12.61657.68331.436692.2855 5 13.086270.76951.316193.6016 6 7.624578.3940.850994.4525 7 2.518980.9131.383495.836 8 5.093486.00640.265696.1016 9 0.947686.9540.59596.6966 10 0.994887.94880.39597.0916 Table A.6 Regression Coefficient B No. Predictor B No. Predictor B 1 (1q1)s1 (t4) 0.00295 13 (1q1)s8 (t1) 0.03124 2 (1q1)s5 (t4) 0.07369 14 (1q1)s10 (t1) 0.05272 3 (1q1)s8 (t4) 0.04627 15 (1q1)s1 (t) 0.09544 4 (1q1)s3 (t2) 0.02239 16 (1q1)s2 (t) 0.03551 5 (1q1)s4 (t2) 0.14135 17 (1q1)s3 (t) 0.25085 6 (1q1)s7 (t2) 0.19833 18 (1q1)s4 (t) 0.07654 7 (1q1)s1 (t1) 0.06708 19 (1q1)s5 (t) 0.14277 8 (1q1)s3 (t1) 0.00332 20 (1q1)s6 (t) 0.03152 9 (1q1)s4 (t1) 0.13779 21 (1q1)s7 (t) 0.63287 10 (1q1)s5 (t1) 0.00365 22 (1q1)s8 (t) 0.15118 11 (1q1)s6 (t1) 0.05853 23 (1q1)s10 (t) 0.02636 12 (1q1)s7 (t1) 0.13353 24 (1q1)s11 (t) 0.00588 PAGE 132 122 Appendix F: Results of Partial Le ast Square Estimation (Continued) Table A.7 Matrix W(PTW)1 0.0981 0.1184 0.2937 0.0671 0.1614 0.0115 0.1211 0.1543 0.2777 0.0853 0.1030 0.3320 0.1636 0.0416 0.2867 0.2415 0.0031 0.1945 0.0915 0.1248 0.3311 0.1683 0.0010 0.1158 0.1513 0.4230 0.1306 0.0976 0.2586 0.1109 0.2005 0.1183 0.1166 0.0912 0.0588 0.2703 0.0797 0.2724 0.1579 0.0732 0.2737 0.7027 0.0143 0.1231 0.1023 0.1446 0.1463 0.0200 0.4585 0.4373 0.2628 0.3876 0.0295 0.0456 0.0982 0.2468 0.1376 0.0260 0.0408 0.0275 0.0757 0.3548 0.7202 0.2310 0.1041 0.1130 0.2678 0.0693 0.0939 0.1550 0.0113 0.2258 0.2093 0.1019 0.3140 0.4380 0.1832 0.2984 0.2134 0.1396 0.1070 0.1242 0.2688 0.0965 0.1842 0.1653 0.0548 0.0829 0.2448 0.3003 0.0906 0.1443 0.3645 0.0643 0.1818 0.0501 0.0926 0.1017 0.2684 0.2656 0.0230 0.1141 0.1265 0.4077 0.4207 0.0419 0.3707 0.1089 0.1548 0.2160 0.0510 0.1519 0.1224 0.2794 0.0759 0.1968 0.1371 0.0765 0.2734 0.3562 0.2578 0.2401 0.2263 0.4808 0.1649 0.0098 0.2389 0.0403 0.1355 0.1708 0.3586 0.5769 0.3515 0.1920 0.0056 0.1083 0.0747 0.1138 0.0946 0.0687 0.2293 0.3266 0.3766 0.5756 0.3585 0.3941 0.3279 0.0025 0.0572 0.0339 0.1391 0.1223 0.0245 0.3242 0.3285 0.0280 0.2948 0.5396 0.7230 0.1123 0.5779 0.4749 0.2946 0.0201 0.1390 0.0745 0.2116 0.2126 0.1252 0.0170 0.1537 0.1731 0.1173 0.0282 0.2887 0.1699 0.2162 0.4279 0.0117 0.0748 0.3942 0.5384 0.4333 0.6301 0.6588 0.7654 0.4969 0.3907 0.4897 0.2819 0.1268 0.1118 0.1892 0.1004 0.0881 0.1786 0.3599 0.0845 0.1728 0.0233 0.1074 0.0166 0.2916 0.0056 0.0537 0.1426 0.0017 0.1526 0.0162 0.1654 0.1552 0.2400 0.1773 0.2017 0.0706 0.2082 Table A.8 Scores for Points 10, 33, and 56 t10 T 1.4300 4.4685 0.59610.64650.13290.70591.2548 0.1687 0.3506 t33 T 1.3451 4.3501 0.75060.13930.68870.97640.0422 0.1284 1.2530 t56 T 3.3721 5.3679 1.24821.01361.38812.78001.5272 0.9628 0.1017 PAGE 133 About the Author Hui Wang received a Bachelors Degree in Mechanical Engineering from Shanghai Jiao Tong University, Shanghai, China in 2001, and an M.S.E. in Mechanical Engineering at the University of Michigan, Ann Arbor in 2003. He is currently a Ph.D. student in the department of Industrial a nd Management Systems Engineering at the University of South Florida, Tampa. While in the Ph.D. program at the Univer sity of South Florida, Hui Wang focuses on the research of modeling and quality c ontrol for manufacturing processes including traditional discrete manufacturing and micro/na no manufacturing. Curre ntly, he has four publications in ASME Transactions, Journal of Manuf acturing Science and Engineering IIE Transactions on Quality and Reliability and Journal of Manufacturing Systems He also made paper presentations at annual m eetings of ASME, IERC, and INFORMS. He is a member of INFORMS and IIE. 