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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 002021003 003 fts 005 20090724082633.0 006 med 007 cr mnuuuuuu 008 090724s2008 flu s 000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0002511 035 (OCoLC)427636738 040 FHM c FHM 049 FHMM 090 T56 (Online) 1 100 Chen, Shaoqiang. 0 245 Manufacturing process design and control based on error equivalence methodology h [electronic resource] / by Shaoqiang Chen. 260 [Tampa, Fla] : b University of South Florida, 2008. 500 Title from PDF of title page. Document formatted into pages; contains 70 pages. 502 Thesis (M.S.I.E.)University of South Florida, 2008. 504 Includes bibliographical references. 516 Text (Electronic thesis) in PDF format. 520 ABSTRACT: Error equivalence concerns the mechanism whereby different error sources result in identical deviation and variation patterns on part features. This could have dual effects on process variation reduction: it significantly increases the complexity of root cause diagnosis in process control, and provides an opportunity to use one error source as based error to compensate the others. There are fruitful research accomplishments on establishing error equivalence methodology, such as error equivalence modeling, and an error compensating error strategy. However, no work has been done on developing an efficient process design approach by investigating error equivalence. Furthermore, besides the process mean shift, process fault also manifests itself as variation increase. In this regard, studying variation equivalence may help to improve the root cause identification approach.This thesis presents engineering driven approaches for process design and control via embedding error equivalence mechanisms to achieve a better, insightful understanding and control of manufacturing processes. The first issue to be studied is manufacturing process design and optimization based on the error equivalence. Using the error prediction model that transforms different types of errors to the equivalent amount of one base error, the research derives a novel process tolerance stackup model allowing tolerance synthesis to be conducted. Design of computer experiments is introduced to assist the process design optimization. Secondly, diagnosis of multiple variation sources under error equivalence is conducted. This allows for exploration and study of the possible equivalent variation patterns among multiple error sources and the construction of the library of equivalent covariance matrices.Based on the equivalent variation patterns library, this thesis presents an excitationresponse path orientation approach to improve the process variation sources identification under variation equivalence. The results show that error equivalence mechanism can significantly reduce design space and release us from considerable symbol computation load, thus improve process design. Moreover, by studying the variation equivalence mechanism, we can improve the process diagnosis and root cause identification. 538 Mode of access: World Wide Web. System requirements: World Wide Web browser and PDF reader. 590 Advisor: Qiang Huang, Ph.D. 653 Variation reduction Statistical process control Engineering driven Equivalence mechanism Tolerance synthesis 690 Dissertations, Academic z USF x Industrial Engineering Masters. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.2511 PAGE 1 Manufacturing Process Design and Control Based on Error Equivalence Methodology by Shaoqiang Chen A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Ma nagement Systems Engineering College of Engineering University of South Florida Major Professor: Qiang Huang, Ph.D. Michael Weng, Ph.D. Susana LaiYuen, Ph.D. Date of Approval: May 15, 2008 Keywords: variation reduction, statistical process control, engineering driven, equivalence mechanism, tolerance synthesi s, diagnosis, root cause identification Copyright 2008, Shaoqiang Chen PAGE 2 Dedication To My Parents PAGE 3 Acknowledgements I would like to express my gratitude to my major advi sor Dr. Qiang Huang, whose expertise, understanding, and patience, added considerably to my graduate experiences. I appreciate his vast knowledge and skills in many areas. I want to thank the other members of my committee, Dr. Michael Weng, and Dr. Susana LaiYuen, for the assistance they provid ed at all levels of the research project. I wish to thank Dr. Jos ZayasCastro who provided me with the teaching assistantship. Sincerely gratitude is also given to all other faculty me mbers who have taught me many important courses in USF. Special thanks s hould be given to Ms. Gl oria Hanshaw and Ms. Jackie Stephens, who helped me with mu ch required paperwork every semester. A very sincere and warmhearted gratit ude goes to Hui Wang, who has provided all kinds of help to me during my study in USF. Without all of his help and his kind heart, I could not smoothly work out many problems in the USA. In addition, I would like to express my sincere appr eciation to Xi Zhang, Yang Tan, Qingwei Li, Diana Prieto, Patricio Rochia Athina Brintaki, a nd other IMSE friends. I also thank Don McLawhorn, who he lped me improve my English. Finally, I am forever indebted to my pa rents Xiaojun Chen and Yulian Sun, for their endless love, without which I would not have finished this thesis. PAGE 4 i Table of Contents List of Tables iii List of Figures iv Abstract vi Chapter 1 Introduction 1 1.1 Error Equivalence and Vari ation Equivalence Phenomena 3 1.2 Related Work and the State of the Arts 5 1.2.1 Literature Review for Process Modeling 6 1.2.2 Literature Review for Pr ocess Design and Optimization 8 1.2.3 Literature Review for Process Control: Root Cause Diagnosis 10 1.2.4 Summary of the Literature Review 13 1.3 Thesis Outline 14 Chapter 2 Error Equivalence Based Process Design and Optimization 16 2.1 The Error Equivalence Based Process Design and Optimization 17 2.1.1 Allocate Tolerance to Aggregated Error Sources at Each Manufacturing Stage 17 2.1.2 Error Equivalence Based Global Process Design Optimization 21 2.2 Case Study 25 2.2.1 Illustrate the Approach Using a Multistage Machining Process 25 2.2.2 Remark on Sensitivity Analysis of the Optimal Process Design 34 2.3 Chapter Summary 35 Chapter 3 Diagnose Multiple Variati on Sources under Variation Equivalence 37 3.1 Concept of Variation Equivalence and Equivalent Variation Patterns Library 38 3.1.1 Definition of Variation Equivalence 38 3.1.2 Equivalent Covariance Stru cture Analysis and Library 39 3.2 Diagnosis and Root Cause Identifi cation under Variation Equivalence 47 3.3 Case Study 51 3.3.1 Illustration of the Root Cause Diagnosis Approach Using a Machining Process 51 PAGE 5 ii 3.3.2 Remark on the ExcitationResponse Path Approach 53 3.4 Chapter Summary 54 Chapter 4 Conclusions and Future Work 56 4.1 Conclusions 56 4.2 Future Work 57 Cited References 59 Appendices 66 Appendix A: Review of EFE and Derivation of d 67 Appendix B: Prediction and Es timation of Kriging Model 69 Appendix C: Derivation of the Equi valent Covariance Structures 70 About the Author End Page PAGE 6 iii List of Tables Table 2.1 Design Variables Range under GCS (unit: mm) 32 PAGE 7 iv List of Figures Figure 1.1 Error Equivalen ce in Machining Process 4 Figure 1.2 Error Equivalence in Assembly Process 4 Figure 2.1 Procedure of Error Equi valence Based Process Design and Optimization 17 Figure 2.2 Workpiece and Locating Scheme (Wang, et al ., 2005) 26 Figure 2.3 Operation Steps 26 Figure 3.1 A 2D Machining Proce ss Example of Variation Equivalence 39 Figure 3.2 Variation Equivalence of Faulty Condition 1 42 Figure 3.3 Variation Equivalence of Faulty Condition 2 43 Figure 3.4 Variation Equivalence of Faulty Condition 3 44 Figure 3.5 Variation Equivalence of Faulty Condition 4 44 Figure 3.6 Variation Equivalence of Faulty Condition 5 45 Figure 3.7 Variation Equivalence of Faulty Condition 6 46 Figure 3.8 Variation Equivalence of Faulty Condition 7 47 Figure 3.9a Eigenvectors Gestures 51 Figure 3.9b Sample Slope and Population Slope 51 Figure 3.10 ExcitationResponse Path of Case Study Result 52 PAGE 8 v Figure 3.11 ExcitationResponse Path Usi ng Reference Vector (1 0 0)T 53 Figure 3.12 ExcitationResponse Path Usi ng Reference Vector (0 1 0)T 54 PAGE 9 vi Manufacturing Process Design and Control Based on Error Equivalence Methodology Shaoqiang Chen ABSTRACT Error equivalence concerns the mechanism whereby different error sources result in identical deviation and vari ation patterns on part features. This could have dual effects on process variation reduction: it significantly increases th e complexity of root cause diagnosis in process control, and provides an opportunity to use one error source as based error to compensate the others. There are fruitful research accomplishm ents on establishing error equivalence methodology, such as error equivalence modeling, and an error compensating error strategy. However, no work has been done on developing an effi cient process design approach by investigating erro r equivalence. Furthermore, be sides the process mean shift, process fault also manifests itse lf as variation increase. In this regard, st udying variation equivalence may help to improve the root cause identification approach. This thesis presents engineering driven approaches for process design and control via embedding error equivalence mechanisms to achieve a be tter, insightful understa nding and control of manufacturing processes. PAGE 10 vii The first issue to be studied is manuf acturing process design and optimization based on the error equivalence. Using the erro r prediction model that transforms different types of errors to the equivalent amount of one base error, the re search derives a novel process tolerance stackup mode l allowing tolerance synthesi s to be conducted. Design of computer experiments is introduced to assist the process design optimization. Secondly, diagnosis of multiple variati on sources under error equivalence is conducted. This allows for exploration and st udy of the possible equivalent variation patterns among multiple error sour ces and the construction of the library of equivalent covariance matrices. Based on the equivalent variation patterns library, this thesis presents an excitationresponse path orie ntation approach to improve the process variation sources identificati on under variation equivalence. The results show that error equivalence mechanism can significantly reduce design space and release us from considerab le symbol computation load, thus improve process design. Moreover, by studying the va riation equivalence mechanism, we can improve the process diagnosis a nd root cause identification. PAGE 11 1 Chapter 1 Introduction Variation reduction is of vital importa nce for manufacturing process and product quality improvement due to uncertainty in th e processes. It has re ceived considerable attention from the manufacturing community because of the intense global competition. There are two main categories of approaches for process variation reduction: data driven approaches such as statistical process contro l (SPC), and engineering driven approaches. SPC can detect process quality changes. Howe ver, it casts little light on engineering knowledge about the root cause, which needs efforts of engineers to figure out the sources of the quality changes. Engineering driven approaches fill this gap, and have significantly improved manufactur ing processes variation redu ction. However, there are still some process phenomena that have not been well addressed. One phenomenon named error equivalence concerns the m echanism whereby different error sources result in identical deviation and variation pa tterns on part features. This could have dual effects on process variation reduction: it signi ficantly increases the complexity of root cause diagnosis in process control, and provide s an opportunity to use one error source as based error to compensate the others. PAGE 12 2 Although there are fruitful research accomplishments on establishing error equivalence methodology, such as error e quivalence modeling, and error compensating error strategy, variation reducti on for the process design and c ontrol is still an extremely challenging issue for the following reasons: Lack of an efficient process design approach under error equivalence For early process design stage, process toleran ce strategy is crucial to control of product/process inaccuracy and imperfection. Previous tolerance synthesis has been carried out simultaneously in product design and process design. However, the traditional tolerance synthesis was conduc ted among different error sources. When the number of manufacturing stage increase, the dimension of design space will considerably increase. Thus, this fact impact s the efficiency of the early stage process design. Lack of an efficient approach for process va riation control under identical variation pattern from multiple error sources Although root cause identification draws significantly attention in recent years, there still exists a lack of consideration on the phenomenon that different error sources ma y result in the identical product feature variation pattern. Therefore, when there are multiple error sources in a manufacturing process, root cause identification of varia tion sources will be typically a challenge. Since the equivalent variation patterns could conceal the information of multiple errors and thus significantly increase th e complexity of root cause identification (diagnosis). Meanwhile, this fact may provide an opportunity to purposely study the PAGE 13 3 part feature variation patter ns of equivalent error sour ces and thus derive a more efficient variation sources identification approach. Therefore, the aforementioned issues en tail an essential analysis of error equivalence for process design and control im provement. The goal of this work is to utilize the error equivalence in manufacturing to achieve an insightful understanding of process variation for developing a better pr ocess design strategy a nd control approach. 1.1 Error Equivalence and Vari ation Equivalence Phenomena 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 tool 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 is modeled in terms of significant tool path de viations from its intended route. This thesis mainly focuses on kinematics aspect s of these three error types. PAGE 14 4 A widely observed engineering phenomenon is that different individual error sources can result in the iden tical deviation and variation pa tterns on product features in manufacturing process. For in stance, in a machining process, all aforementioned process deviations can generate the same amount of feature deviation x as shown in Fig. 1.1 (Wang, Huang, and Katz, 2005; and Wang a nd Huang, 2006). This error equivalence 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 proces s 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 proces s 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 proces s 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 proces s 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 Figure 1.1 Error Equivalence in Machining Process 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 1.2 Error Equivalence in Assembly Process PAGE 15 5 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 one error source as base erro r in the early stage of pro cess design, thus efficiently improving the tolerance strategy fo r the manufacturing process. In both cases, a fundamental understanding of this complex engineering phenomenon will assist to improve manufacturing process design and control. 1.2 Related Work and the State of the Arts Before 2005, the study on error equivalence is very limited. Most related research on process error modeling has been focused on the analysis of indi vidual error sources, e.g., the fixture errors and/or machine tool errors, how these erro rs impact the product quality, and thereby how to diagnose the er rors and conduct feedback adjustment to reduce variation. Since Wang et al ., 2005, there have been so me studies on the error equivalence for the above issues. This Sec tion reviews the related research on process modeling, process design and optimizati on, and process root cause diagnosis. PAGE 16 6 1.2.1 Literature Review for Process Modeling In this Section, we will review error equivalence modeling and process modeling. For error equivalence modeling, Wang et al. 2005, utilized the error equivalence phenomena to develop the error equivalence modeling. In the error equivalence model, different error sources (e.g., fixture error, machine tool error a nd datum error) were linearly transformed into one based error, i.e., equivale nt fixture error (EFE). After transforming the different type s of error into one based er ror, we can aggregate the equivalent errors. This will be of great benefit to the process error prediction and variation propagation modeling, since it will si gnificantly reduce the dimension of input variables, which will be introduced in de tail in Section 2.1.1 of Chapter 2. The error equivalence mechanism also helped to unders tand the process error compensating error strategy. Wang and Huang, 2006, used the equiva lent fixture error modeling in error cancellation and applied it in machining pr ocess control and deviation feedback adjustment. The process modeling in the literature is summarized as causality modeling. Models of predicting 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 the causal relationship between part features and errors, especially in a complex manufacturing system. The available model formulation includes time series model (Lawless, Mackay, and Robinson, 1999), state space models (Jin and Shi, 1999; Ding, Ceglarek, and Shi, 2000; Huang, Shi, and Yuan, 2003; Djurdj anovic and Ni, 2001; PAGE 17 7 Zhou, Huang, and Shi, 2003; and Huang and Shi, 2004), and stat e transition model (Mantripragada and Whitney, 1999). The resu lts 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 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 Based on the aforementioned research on process modeling, Wang et al., 2005, developed a multioperational machining pr ocesses variation propagation model for sequential root cause identific ation and measurement reduction by imbedding the error equivalence mechanism, which helped to better understand and model the mechanism that different error sources result in the iden tical variation pattern on part features. The derived quality predictio n model (causal model) embedded with error equivalence mechanism can reveal more physical in sights into the proc ess variation. PAGE 18 8 Summarizing the research on process e rror prediction and variation propagation modeling, we can see that causality modeling well connect the proce ss errors and product feature quality. Moreover, inte grating error equivalence into causality modeling can be a considerable benefit. Becaus e introducing error equivalence to process modeling helps to shrink the dimension of input variables. The reduction design space is of great importance to early stage process design efficiency. 1.2.2 Literature Review for Process Design and Optimization Design of a multistage machining proce ss involves tolerance allocation at each stage and design of process layou ts, in particular, the fixtur e layouts. Tolerancing strategy is therefore crucial to control of pr oduct/process inaccuracy and imperfection. Conventional tolerance synthesi s has been carried out simultaneously in product design and process design. A major goal of toleran ce synthesis at design stage is to reduce quality loss (Taguchi, 1989; Choi et al., 2000; and Pramanik et al., 2005), while tolerance synthesis for process design aims at manuf acturing cost reduction. As an example, tolerance charting (Wade, 1967; Ngoi a nd Ong, 1993, 1999; and Xue and Ji, 2005) converted the designed tolera nces of products to manuf acturing tolerances. Optimal tolerance allocation for process selection ha s also been widely studied (Nagarwala et al., 1994; Singh et al., 2004; and Wang and Liang, 2005). Recent research (Shiu et al., 2003; and Dong et al., 2005) considered deformations in manufacturing processes as well. PAGE 19 9 Simultaneous tolerance synthesis cons iders both design and manufacturing tolerances and has attrac ted more attentions in the past decade. Zhang et al., 1992, conducted optimization of design and tolera nce allocation to select processes among alternatives. An analytical model (Zha ng and Wang, 2003) was also reported to simultaneously allocate design and machin ing tolerances based on a criterion of minimum manufacturing cost. Zhang, 1997, fu rther established tolerance stackup model for assembly process. Recent research (Ye and Salustri, 200 3; and Wang and Liang, 2005) on simultaneous tolerance synthesis incorpor ated both manufacturing cost and quality loss into the optimization function. Reviews of tolerancing research are available in Bjrke, 1989, Chase and Greenwood, 1988, Jeang, 1994, Royal et al., 1991, Voelcker, 1998, Ngoi and Ong, 1998, and Hong and Chang, 2002. Simultaneous tolerance synthesis is more challenging for a multistage manufacturing process. A commonly adopted appr oach is to model the impact of process parameters on tolerance stackup (Mantripra gada and Whitney, 1999; Jin and Shi, 1999; Zhou et al., 2003; and Huang et al., 2003). Ding et al., 2005, concurrently allocated component tolerances and selected fixtures for assembly processes using a state space model. Huang and Shi, 2003, conducted a st udy on simultaneous tolerance synthesis and optimal process selection for multistage machining processes. Simultaneous tolerance synthesis, howeve r, might generate a large design space. For example, in a milling or drilling proce ss where parts are fixed under 321 locating scheme, the process variables involve toleranc es of six fixture locators or six process PAGE 20 10 variables, machine tool paths (rotation and translation with six degrees of freedom), and datum surfaces. Thus, a multistage process will incur a large design space and makes it difficult to choose optimal and unique proce ss design. One strategy is to prioritize the allocation of tolerances to different error so urces at each stage through proper selection of cost functions. Since the cost function se lection can be very s ubjective, especially when designing a new process where knowledge of cost structures is very limited, minor changes in cost functions could lead to dram atic changes in process design and tolerance allocation. 1.2.3 Literature Review for Pro cess Control: Root Cause Diagnosis Process control technology, which focuses on the detection, identification, diagnosis, and elimination of process faults can help to reduce process downtime, and hence, the operation costs. The rapid advan ces in sensing and info rmation technology that are currently being made mean that a large amount of data is readily available that requires process control methodologies to be de veloped for its interpretation. Statistical process control (SPC) (Montgomery, 2005, and the references therein) is the primary tool used in practice to improve the quality of manufacturing proce ss. Although SPC can efficiently detect a departure from normal c ondition, it is unable to pin down the process fault that caused the alarm (root cause). And it is purely statistical data driven approach that inefficiently gives the process fault physi cal explanations. Theref ore, the job of root cause identification is actually le ft to plant operators or quality engineers. In light of this PAGE 21 11 limitation of SPC, considerable research ef forts have been expended on developing the approaches for root cause identification. Sin ce process faults often manifest themselves as the shift of the mean values and the incr ease of variances, the root cause identification for process diagnosis can be cat egorized into two types: root cause identification of mean shift and of variation sources The approaches developed fo r root cause diagnosis of variation sources include va riation pattern mapping (Ceglarek and Shi, 1996, Jin and Zhou, 2006b, Li, et al., 2007), variation estimation based on physical models (Apley and Shi, 1998; Chang and Gossard, 1998; Di ng, Ceglarek, and Shi, 2002; Zhou, et al., 2003; Camelio and Hu, 2004; Carlson and Sderber g, 2003; Huang, Zhou, and Shi, 2002; Huang and Shi, 2004; and Li and Zhou, 2006) and variation pattern extraction from measurement data (Jin and Zhou, 2006a). Ceglarek, Shi, and Wu, 1994, developed 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 compone nts 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 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 PAGE 22 12 fixture faults occurring simultaneously. Thei r continuing work in 2001 presented a factor analysis (Johnson, and Wichern, 1998) appro ach to diagnose root causes of process variability by using a causality model. Di ng, Ceglarek, and Shi, 2002, derived 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 involved. The multiple error patterns are rarely orthogonal and they are difficult to disti nguish from each other. Therefore, the manufacturing process may not be diagnosable Ding, Shi, and Ceglarek, 2002, analyzed the diagnosability of multis tage manufacturing processes a nd applied the results to the evaluation of sensor distribution strategy. Variation component analysis (Rao, 1972, Rao and Kleffe, 1988) and mixed models (McCulla gh, and Nelder, 1989, Pinheiro, and Bates, 2000) are also helpful to the diagnosability and diagnosis study. By using variance component analysis, Zhou, et al., 2003, developed a more general framework for diagnosability analysis by cons idering aliasing faulty structur es for coupled errors in a partially diagnosable process. Based on state space model and linear mixed effects model, Zhou et al., 2004 developed a root cause estimation approach for manufacturing process. Further studies and research on root cause identification of multiple error sources have been achieved by Wang and Huang, 2006, util izing the error equivalence concept and error cancellation modeling. We can see that methods for diagnosis of different/equivalen t patterns of single error/variation sources and different pattern s of error/variation sources have been PAGE 23 13 developed. However, for the situation in whic h identical variation pa tterns happen, efforts are still needed for an efficient method. Becau se in a manufacturing process, an identical product feature variation pattern from mu ltiple error sources can possibly occur. 1.2.4 Summary of the Literature Review The related research work in the literature is summarized as follows: Process error prediction and variation propagation modeling. Previous research work has been done on causality modeling with an alysis of individual errors as well as equivalent errors in manufacturing pro cesses. Also, the error equivalence based process modeling has assisted in understa nding the error cance llation modeling and its application in process error root cause diagnosis and compensation. However, there is still large room for using the physical model that described the error equivalence to help understand some other i ssues, such as early stage process design and process variation so urces identification. Process design and optimization. Traditional research on process tolerance design and optimization has extensively conducted th e tolerance synthesis among different individual process errors. Th e optimization for design of process layout is also focusing on individual error sources. The design space and computation load will significantly increase as the number of ma nufacturing operation stages and process error sources increase. Therefore, previous research did not address a solution for an efficient method for the early stage process design. PAGE 24 14 Root cause diagnosis for process control. Researchers have developed many methodologies of root cause identifica tion for multistage manufacturing process diagnosis. These researches have involved in fault diagnosis for different variation patterns from single error sources, differen t variation patterns fr om different error sources. But no research work has been done on root cause identific ation for identical variation pattern from different error source, i.e., variation equivalence, while this phenomenon is an important engineer ing issue in manufacturing process. 1.3 Thesis Outline In order to achieve an insightful underst anding of manufacturi ng process variation and improve process quality, this thesis addresses the advances in: manufacturing process design and optimization strategy based on error equivalence me thodology, and error equivalence analysis for root cause dia gnosis of process variation. The following Chapters of this thesis are thus organized as follows: Chapter 2 presents the modeling of pro cess variation propagation and tolerance stackup model based on error equivalence. It ut ilizes the error equivalence mechanism to develop an efficient tolerance synthesis method for early proce ss design stage. In addition, a globally process layout optimization model is developed for searching the optimal tolerance allocation among all the po ssible process design alternatives. Chapter 3 studies the possible variation e quivalence cases in a machining process and builds the equivalent variation patterns li brary. For process diagnosis, this Chapter PAGE 25 15 develops a new approach for root cause identif ication for identical variation pattern under multiple error sources. Chapter 4 concludes the thesis. We also poi nt out prospects of future research in this Chapter. PAGE 26 16 Chapter 2 Error Equivalence Based Process Design and Optimization This Chapter aims to improve th e simultaneous process to lerance synthesis for multistage manufacturing process by incor porating an error equivalence mechanism (Wang, Huang, and Katz, 2005; Wang and Hu ang, 2006) into tolerance stackup modeling and tolerance design. We propose to redu ce design space by transf orming multiple error sources into equivalent amount of base erro rs. The reduction of design space will assist to achieve a unique solution and global optimi zation of process design. Furthermore, we also embed error equivalence with com puter experiments method to reduce the computation load for searching optimal process design. The Chapter is organized as follo ws. Section 2.1 introduces the methodology of error equivalence based tole rance synthesis and optimal process design over the allowable design region. In Section 2.2, we illustrate the methodology through a case study of multistage machining process. To eval uate the robustness of the optimal process design, Section 3 also conducts sensitivity anal ysis. Conclusions are gi ven in Section 2.3. PAGE 27 17 2.1 The Error Equivalence Based Pr ocess Design and Optimization The proposed method consists of the follo wing procedure illustrated by Fig. 2.1. First, we will generate a set of process layouts (design variables) sis through space filling design. For a given process layout si, tolerance will be allocated to aggregated error sources (process variables) at each manufactur ing stage (discussed in Section 2.1.1). The final tolerance stackup for all design variables sis will be used as responses in a Kriging model to identify the optimal fixture layout (presented in Section 2.1.2). A given process layout si Stage 1 Stage 2 Stage N Allocate tolerances to each manufacturing stage Section 2.1.1 Generate design sites sis Predict tolerance stackup through a Kriging model over all design region Identify the optimal process layoutSection 2.1.2 Final tolerance stackup Figure 2.1 Procedure of Error Equivalence Based Process Design and Optimization 2.1.1 Allocate Tolerance to Aggregated E rror Sources at Each Manufacturing Stage Tolerance synthesis requires a th orough understanding on how the process variables impact the tolerance stackup. Theref ore, the error equivalence based tolerance synthesis consists of tolerance stackup modeling and m odel based simultaneous tolerance allocation. We first present error equi valence based tolerance stackup modeling. PAGE 28 18 Tolerance stackup is mainly due to variati on sources in each stage of a manufacturing process, e.g., machine tool error, datum erro r, and fixture error in a machining process. The objective of tolerance st ackup modeling is to relate the stackup of tolerance in product features to all variation sources in a multistage manufacturing process. Existing tolerance stackup models roughly fall into three categories (Huang and Shi, 2003), namely, worst case model, root sum square model or interpolation of these two models, Monte Carlo simulation models, and physical models that study the impact of process variables on tolerance stackup. The tolera nce stackup models in the third category provide a new opportunity of simultaneously allocating product and process tolerances (Ding et al., 2005; and Huang and Shi, 2003). As discussed in Introduction, this approach unfortunately could generate a la rge design space as the number of processing stages increases. We aim to reduce the design space and im prove the tolerance stackup models in the third category. The main idea is to explore the relationship among multiple error sources, in particular, the error equivalen ce. Two types of error sources are called equivalent if they result in identical dimens ional deviation. Equivalent error sources at each manufacturing stage therefore could be a ggregated together when predicting feature deviations. In more detail, multiple types of errors xis can be transformed into a common base error thr ough transformation ix= Kixi, i=1,2,..,m (please refer to Appendix A for transformation matrices Kis). Since fixture error is easier to be controlled and monitored, we choose fixture error to be the base erro r in this paper and transform all the error PAGE 29 19 sources into equivalent amount of fixture error (EFE). The product dimensional deviation can thus be predicted thr ough the following model (Wang, Huang, and Katz, 2005): ) ( ) ( ) ( k k kj j u y (2.1) where jy(k) describes the feature deviations caused by aggregated error sources u(k) =i i *x(k) and process noise ) ( k at the kth stage. Note that in traditional error prediction model, the right hand side of E qn. (2.1) contains not just one aggregated equivalence error vector in each stage, but a high dimensional vector that consists of different individual error sources, (e.g., ) ( )] (  ) (  ) ( [ ) ( k k k k kT j j x x x y3 2 1 where[) ( k1x,) ( k2x,) ( k3x] represent machine tool, datum and fixture errors, respectively). Aggregating error enables us to focus on the process with base errors only and thereby significantly reduces process and design variable s in tolerance synthesis. The aggregated errors ) ( k u and noise term ) ( k are all assumed to follow multivariate normal distribution. It should be noted that reducing mode l dimension can also be achieved by investigating the linear depende ncy among columns in the matrix j, e.g., thorough diagnosability analysis (Zhou et al., 2003). We adopt the erro r equivalence methodology because of two reasons. First of all, the machining process involves multiple types of errors as opposed to multiple error patterns from individual error sources (e.g., multiple fault patterns of the fixture error). Secondly, it is a more engineering driven approach, i.e., direct modeling the kinematic relationships among multiple error sources. The method assists more engineering insights, e.g., e rror cancellation effect discussed in Wang and PAGE 30 20 Huang, 2006. Using Eqn. (2.1), the vari ancecovariance matrix of the feature j can be derived as Eqn. (2.2). The ) ( k uand yj are variancecovariance matrix for process variables and devi ation of feature j, respectively. I u y 2 ) () ( ) ( T j k jj. (2.2) Since diag( yj) can be directly related with tolerance (e.g., 3 as a measure of tolerance range), final tolera nce stackup can be obtained by extracting the diagonal term diag( yj) from Eqn. (2.2). With the tolerance stackup model, we can conduct equivalence error based simultaneous optimal tolerance allocation. Th e objective of optimal tolerance allocation is to allocate tolerances for process variable s that can meet the de sign specification with minimum manufacturing cost. Denote process variables T T Tk) ) ,..., ) 1 ( (( u uas and their standard deviations asT k) ,..., () ( (1)u u The variances of product features are linear combinations of 2 from the result of diag( yj), which can be denoted as 2 CT. The variance of dimensions, denoted as 2 cT can be derived from2 CT. Since larger process tolerance for dimensions will reduce the manufacturing cost, we can maximize 2 cT given design specifications for the product tolera nce and physical constraints: Max2 cT, maximize the component tolerance; s.t. 1 2b CT, constrains from design specification (2.3) 2 b 0 practical constraints of tooling Fc >0, static equilibrium constraint PAGE 31 21 where Fc is the reaction force between workpi ece and locator and is determined by clamping forces. The static equilibrium c onstraint ensures the workpiece maintaining contact with all the locators (Cai, et al., 1997). It should be noted that the tolerances are assigned to aggregated error so urces. We could further distribute tolerances to individual error sources based on the error equi valence model given in Appendix A. 2.1.2 Error Equivalence Based Gl obal Process Design Optimization The tolerance stackup is determined not only by the magnitudes of error sources (measured by standard deviations of process variables in Section 2. 1.1), but also by the process design, in particular, spatial layout of process variab les. In a machining process, the layout of fixture locators has been shown to impact tolerance stackup with a 2D example in Huang and Shi, 2003. But globa l optimization of process design was not studied therein. The main reason is that there is no unique solution for allocating tolerances to all process variables. To overcome the challenge, we transform all error sources into equivalent amount of fixture locator errors. Then the process de sign variables are just positions of fixture locators (i.e., fis). To explore the re sponse surface of tolera nce stackup under process design alternatives, we adopt the methodology of computer experiment s design. The main reason is that the tolerance s ynthesis involves heavy symbolic computational load if we explore all possible fixture layouts. The lacking of random error in the deterministic computer tolerance simulation also leads to the consideration of computer experiments PAGE 32 22 against the other traditional regression analysis. And the computer experiments design will assist to establish a surrogate predicti on model and to search the optimal process design. We search the optimal tolerance alloca tion based on a Kriging model (Matheron, 1963; Journel and Huibregts, 1978; Cressie, 1993, and Sacks, et al., 1989), which depicts the relationship between the inpu t variables (e.g. fixture layo ut) and the tolerance stackup. Kriging model has advantage over other interpol ation methods because it is more flexible and weights are not selected according to certa in arbitrary rule (Li and Rizos, 2005). The Kriging model consists of a polynomial term w f) (T and a stochastic process) ( w Z: ) ( ) ( ) (w w f wZ YT (2.4) where Y(w) is the response (in our study, it represen ts tolerance assigned to the features) at the scaled input site w = (w1, , wd), and d is the number of design variables. Note here we denote the untried site by w, while the aforementioned sis are tried sites. The stochastic process ) ( Z is assumed to be Gaussian with zeromean and a covariance between ) (1w Zand ) (2w Z at any two input sites w1 and w2, i.e., ) ( ) (2 1 2 2 1w w w wR Cov where ) (2 1w w R is a correlation func tion of the responses. A review of prediction and estimation of Kriging model is given in Appendix B. The structure of the polynomial term w f) (T and the correlation function in the stochastic process Z(w) should be determined first. According to Welch et al., 1992, more elaborate polynomial terms offer little advant age in prediction. So we set a constant for the polynomial term. For the structure of the correlation function, we choose power PAGE 33 23 exponential family correlation function that is the most popula r in the computer experiments literature. It is given by the product of st ationary one dimensional correlations as)   exp( ) (2 1 1 2 1jp j j d j jw w R w w, where2 0 jp 0 j We choose pj=2 because the correlation function with pj=2 produces smoother stochastic processes (Sacks, et al. 1989). In a twostage machining process, for instance, we have totally d =12 design variables. Then the unknown pa rameters in Kriging model include constant term the variance2of the stochastic process, and = ( 1, , 12). To construct a precise Kriging model, a good experimental design should be able to provide an overview of the respons e across the whole design region as well as precise response at certain input sites in which we are intere sted (e.g., the input site or fixture layout that yield th e optimal tolerance stackup). Searching such an experimental design invo lves a sequential procedure (Bernardo, et al ., 1992; William, et al ., 2000; and Gupta et al. 2006). The sequential procedure is consisting of initial design and design and model refinement. To make the initial design spreading ove r the whole design region, we choose the latin hypercube sampling (LHS). It is one of the most frequently used space filling design and it was introduced by McKay et al. 1979. For each component of input sites, i.e., wj, we can use a uniform distribution across each interval. For high dimensional case, only some of the LHS designs are truly space filling (Santner, et al ., 2003). Therefore, the initial model may not well predict true tolerance PAGE 34 24 responses at uncertain sites and must be refined. We first calculate the root mean square error of the predictor RMSE ( y (w)) at some tested sites, which is square root of Eqn. (B5). If the RMSE turns out to be too large, we should include these sites into the experimental design sites. The selection of untried sites can be determined by LHS design following maximin criteria. Maximin design gua rantees that no two te sted points are too close the each other, so that all the tested points are spre ad over the allowable design region. It should be noted that Gupta et al. 2006, developed a zoomin criteria to refine the Kriging model, whereby c ontour plot approach was used to show the mean square error (MSE) over the whole design region. The areas on the contour pl ot that have too large MSE will be zoomed in and added more design points. We choose to estimate the RMSE on a set of maximinLHS design sites in each refinement iteration instead of contour plot for the reasons that firstly, the input sites may have higher dimension whereas contour plot is not efficient to explore the high dimensional design space; and secondly, maximinLHS based test points se lection can effectively search the tested points spreading over the whole design region. The iterative model refinement steps are stated as: Kriging model fitting In the i th iteration, construct the Kriging model based on ni available experimental design } ,..., {1ins s S with response data} ,..., {1iny ysy, i.e.,) ( ) ( ) ( 1 1 i i i in n n n Tye y R w r ws where ineis the allone vector of length ni. i =0, 1, 2, . PAGE 35 25 Model refinement Calculate RMSE at test points generated by maximinLHS design and add the points that yield large RM SE to the experimental design. Let ii +1 and repeat these procedures. We can stop the model refinement when maximum response values do not vary significantly with iterations and the RMSE in the whole possible region is not too large (Gupta, et al. 2006). 2.2 Case Study In this Section, we will use a twostag e machining process as an example to illustrate error equivalence based tolera nce synthesis and global process design optimization. Since a multistage process consists of ope rations with and without datum changes (the latter is simpler case), the twostage example can be easily extended to a general case. 2.2.1 Illustrate the Approach Usi ng a Multistage Machining Process Figure 2.2 shows the part with features Y1~Y7, where Y1 and Y4 are two planes, and Y2, Y3, Y5, Y6, Y7 are cylindrical holes. The center of Y6 is set to be the origin of global coordinate system (GCS). Part feature can be represented as, e.g., Y1 = (0, 1, 0, 0, 131, 0)T, where the first and last three numbers represent the orientation and position of Y1, respectively. PAGE 36 26 Figure 2.2 Workpiece and Locating Scheme (Wang, et al ., 2005) Figure 2.3 Operation Steps The part goes through two operations, which are shown in Fig. 2.3. Firstly, use Y4, Y5, and Y6 as datum surfaces to mill plane Y1 and drill two holes Y2 and Y3. After that, the plane Y1 and two holes from operation one as datum surfaces to drill hole Y7. In Fig. 1, f1 ~f6 show the locating positions on datum surfaces in each operation. The coordinates of fixture locators in operation one, e.g., is f(1)1 = ( f (1)1 x, f (1)1 y, f (1)1 z)T = (7,109,0)T. Let x1( k ), x2( k ), x3( k ) denote machine tool, datum and fixt ure errors respectively. The base error in this case study is fi xture error, which can be re presented as fixture locator PAGE 37 27 deviations, e.g., ( f (1)1 z, f (1)2 z, f (1)3 z, f (1)4 y, f (1)5 y, f (1)6 x)T in operation one. Next we will illustrate the whole procedure which contains two parts. The first part is tolerance synthesis under specific fixture locating setting. For the tolerance stackup modeling, we only consider fixture and machine tool errors in operation one. However, the errors generate d from this operation may cause datum error in operation two. Denote *x1( k ) and *x2( k ) as EFE due to machine tool and datum errors in operation k respectively. Then u(2) = *x1(2) +*x2(2) + 3x(2), where *x2(2) is generated by u(1), as shown in Eqn. (A4). Th e final product feature deviation y is u u y ) 2 ( ) 1 ( (2.5) where 12 12 6 6 6 6 1 7 0 0 and y =[ yT1 yT7]T. Matrices 1 and 7 are an d ,0 0 0 18333 0 10737 0 07476 1 0 0 1 0 0 0 1 3275 0 3275 0 0 0 0 0 0 0 00833 0 00417 0 00417 0 0 0 0 0 0 0 0 0025 0 0025 0 0 0 01 .0 1025 1 1025 0 5662 1 7831 0 7831 0 0 0 0 225 1 49 0 715 0 1 27083 0 27083 0 0 26103 0 26103 0 0 0 0 0086 0 0043 0 0043 0 0 0 0 018 0 009 0 009 0 0 003 0 003 0 0 00143 0 00143 07 By Eqn. (2.2), we obtain the final pro duct features variancecovariance matrix y Denote the deviation of feature j by jy (k) = (j, j, j, xj, yj, zj )T (orientation deviation j, j, j and position deviation xj, yj, zj in three directions). By extracting its diagonal term, we have the variances of features Y1 and Y7, i.e., PAGE 38 28 ,2 2 ) 1 ( 2 ) 1 ( 2 ) 1 ( 2 2 2 2 ) 1 ( 2 ) 1 ( 2 ) 1 ( 2 2 ) 1 ( 2 ) 1 ( 2 ) 1 ( 2 2 2 ) 1 ( 2 ) 1 (3600 121 68558400 808201 68558400 79192201 160000 17161 160000 17161 14400 57600 57600 160000 160000 3 2 1 14 5 4 6 3 2 1 5 42 1 z z z y y y x z z z y yf f f f f f f f f f f fy and 4512 1245 1262626262 (1)(1)(2)(2) 626262622 (1)(1)(2)(2) 5252 (2)(2)2 7y1.1556310 1.1556310 2.0544410 2.0544410 3.2692810 4.725910 910 910 8.110 8.110 3.2yyyy zzzz yyffff ffff ff 312 3 12312 3425252 (2)(1)(1) 522 (1) 5252526262 (2)(2)(2)(1)(1) 522 (1) (1410 3.656510 1.406310 5.62510 1.84910 1.84910 7.39610 8.346710 3.210110 1.28410 yzz z yyyzz zfff f fffff f f 6645121 45 45122222222 )(2)(1)(1)(2)(2)(1) 222 (2)(2) 2222 (1)(1)(2)(2) 0.01735 0.01735 0.06813 0.06813 0.00444 0.07335 0.07335 0.01563 0.76563 0.511225 0.2401 1.50063xxyyyyz zz yyyyffffff ff ffff 31 123123 45222 (2)(1) 222222 (2)(2)(2)(1)(1)(1) 222 (2)(2) 0.0333 0.61325 0.61325 2.45298 0.08809 0.394 0.2202 0.01051 1.21551 yz yyyzzz zzff ffffff ff The cylindrical hole Y7 is critical for assembly in the subsequent operations. It is reasonable to set the tolerance for x and y positions of Y7 as the final product tolerance, which correspond to the fourth and fifth components of 27y in Eqn. (2.6), i.e.,72 x and27y. The objective is to maximize 0.527x+0.527y, (2.7a) where we assume equal importance of toleranc es along two directions and therefore equal weights are assigned. Based on the vectorial dimension and tolerancing (VD&T) scheme (2.6) PAGE 39 29 (Huang and Shi, 2003), the objective function (E qn. (2.7a)) subjects to the constraints listed below: 211b,211b,21y1byfor Y1, (2.7b) and277b,277b,277b,27x7bx,27y7by,27z7bzfor Y7. Here1b,1bxand 1bzneed not to be considered becau se the orientation component of plane Y1 is free in y direction, and the location component of the plane is free in x, z directions, respectively. As an example, we choose 0.1radian2 for1b,1b,7b,7b,7b, and assign 5mm2 to1by,7bx,7by,7bz, respectively. Set 1.73m m for all elements of 2bin Eqn. (2.3). Then the tolerances or the ma ximum allowable standard deviations for the aggregated error sources are (0.01, 0.01, 0.01, 0.01, 1.415, 1.732, 1.732, 0.01, 1.135, 0.01, 0.01, 1.327)Tmm, and 2 T c =4.99mm2. When more information is available at late stage of process design, e.g., the cost ratio between fixture and machine tool, we could further distribut e the tolerances for aggregated error sources. For example, we could allocate 80% of tolerance band for aggregated EFE to machine tool error to re duce the cost of the major equipment. In operation one (no datum error o ccurs), we allocate 80% of (1) u to ) 1 (1 *x i.e., ) 1 (1 *x =(1)8 0u where ) 1 (3x and ) 1 (1*x denote the standard deviat ion of fixture error and EFE due to machine tool error in operation one, respectively. Variancecovariance matrix for machine tool error in the first stage will be T) (1) ( (1)1 2 ) 1 ( 1 2 ) 1 (1 1K K *x x, (2.8) where ) 1 (1 *x=diag(0.822 (1) u). Appendix A gives the details of K1 and K2 matrices. PAGE 40 30 Solving Eqn. (2.8) and extracting diag() 1 (1x), we have ) 1 (1x= (1.3856mm, 0.008mm, 0.0091mm, 2.7328105radian, 8.1650105radian, 0.0028radian), where the first three numbers represent the standard deviations of machine tool translational error, and the last three are corresponding to the standard de viations of the rotational error in three directions, respectively. Since the trajectory of machine tool head may vary significantly among different product features, we set the tightest tolerance for the machine tool error in all directions, i.e., ) 1 (1x= (8 m, 8 m, 8 m, 2.7328105radian, 2.7328105radian, 2.7328105radian). For operation two, datum erro r is introduced from operation one. By Eqn. (A4), variance for EFE due to datum error is diag(TK K u (1)) = diag() 2 (2 *x) = 2 ) 2 (2 *x= (0.069mm2, 1.1294mm2, 0.5988mm2, 0.00264mm2, 0.0095mm2, 1.7929mm2). Further allocation of tolerance for datum error can be found by diag(T) (2) ( (2)1 ) 2 ( 12K Kx) = diag() 2 (2 *x). (2.9) However, the solution for datum tolerances is not unique since K1 is a 618 matrix (Eqn. (A3)). Therefore, we can not simply obtain diag() 2 (2x) by diag(1 1(2)K) 2 (2 *xT) (2) (1 1K). Due to the characteristics of K1, it is necessary to first spec ify tolerance for one element of secondary datum surface and two elemen ts of tertiary datum surface. Denote x2 = (vI, pI, vII, pII, vIII, pIII), where I, II, III represent primary, secondary and tertiary datum surfaces, respectively. The v and p represent rotational and translational error of the datum surfaces in three directions. (e.g., xvI) 2 (represents the rotati onal error of primary datum surface in x direction in operation 2). Assign 1109 radian2 to2 ) 2 (yv, 1107radian2 to 2 ) 2 (yv, and 1106mm2 to 2 ) 2 (xp. Solving Eqn. (9) leads to ) 2 (2x= (xv) 2 (, zv) 2 (, PAGE 41 31yp) 2 (, xv) 2 (, yv) 2 (, zp) 2 (, yv) 2 (, zv) 2 (, xp) 2 () = (0.0035radian, 1.3995104radian, 7.5 m, 2.473105radian, 3.1623105radian, 3.2 m, 3.1623104radian, 0.0221radian, 1 m). To distribute tolerances to fixture and machine tool errors in operation two, we have ) 2 (1*x= 0.8((2) u) 2 (2*x) =(1.1914mm, 0.00024mm, 0.4802mm, 0.0413mm, 0.0004mm, 0.4368mm). To calculate tolerance for machine tool error in operation 2, we have T) (2) ( (2)1 2 ) 2 ( 1 2 ) 2 (1 1K K *x x. (2.10) Thus, 2 ) 2 (1x= (0.4616mm2, 0.6356mm2, 4.0206mm2, 0.00023radian2, 1.8936108radian2, 0.0000158radian2). By setting equal tolerances for translational and rotational deviations in three directions, we obtain) 2 (1x= (0.6794mm, 0.6794mm, 0. 6794mm, 0.000137radian, 0.000137radian, 0.000137radian). Based on the work of first part, we can conduct global process design optimization within allowable fixture locatin g setting range. Recall that all the error sources have been transformed into EFE. Hence the input site w is related to the fixture layout and can be obtained as follows. Under 321 locating scheme, only locators 1, 2, 3 in the example can be movable over the allo wable design region since the positions of locators 4, 5, and 6 are fixed with locating hole s. Each of locators 1, 2, and 3 can move on the primary datum plane in two directio ns. Therefore, there are twelve design variables involved in total for two machining processes, i.e., = (f(1)1x, f(1)1y, f(1)2x, f(1)2y, f(1)3x, f(1)3y, f(2)1x, f(2)1z, f(2)2x, f(2)2z, f(2)3x, f(2)3z). The allowable design region PAGE 42 32 for each design variables is summarized in Table 2.1. The input variables for are usually coded into [0, 1]d. i.e., w, where, d=12, 1 0 iw, i = 1, 2, , 12. Here we can choose uniform distribution within the [0, 1]d interval. Table 2.1 Design Variables Range under GCS (unit: mm) Operation 1 f(1)1x f(1)1y f(1)2x f(1)2y f(1)3x f(1)3y Range 0~400 10~130 0~400 10~130 0~400 10~130 Operation 2 f(2)1x f(2)1z f(2)2x f(2)2z f(2)3x f(2)3z Range 0~360 0~80 0~360 0~80 0~360 0~80 Since three locators have same allowabl e ranges, they may overlap each other when we generate design sites, this can be prevented by checking deterministic locating condition, i.e., the Jacobian matrix of the fixture layout should be of full rank (Cai, et al., 1997). As mentioned in Section 2, the reactio n force should be nonnegative (we choose >0.5kN here) at the locating points so th at the locators contact the workpiece. Considering the feature dimensions of the workpiece and clamping limitations, we determine the resultant clamping force and torque at the origin as follow: for operation one, FA= (52kN, 28kN, 25kN), TA= (10136Nm, 18300Nm, 4489Nm); for operation two, FA= (45kN, 294kN, 158kN), TA= (149Nm, 302Nm, 51Nm). In addition, the static constraint can help to reduce the number of optimal fixture layouts that correspond to the maximum value ofy PAGE 43 33 Before selecting the initial design sites, we should determine the number of design sites. Number of points n0 for initial experimental design should be chosen to balance the experiment running time and fidelity of Kr iging model. It was suggested (Bernardo et al., 1992) that n0 should be chosen at most three times the number of unknown parameters in the Kriging model. For our EFE based tole rance study, we have totally 14 unknown parameters (12j s, 1 constant and 1 process variance2). Thus, we should let n0 be at least 14 and no more than 42. Here, we choose 16 points, which give rise to a 16 12 maximinLHS design. Unknown parameters in Kriging model can be estimated by maximum likelihood estimation (MLE) criterion, i.e., optimizing th e objective function Eqn. (B1) in Appendix B. There are many searching algorithms availa ble such as simplex search, pattern search methods, and Powells conjugate direction s earch method. In our study, we choose Torczon pattern search method, because To rczon, 1997, proved that pattern search methods can converge to stationary points. Fu rthermore, pattern search method can easily be extended for constrained optimization. After the initial design, we choose 52 maximinLHS design sites for model refinement iteration (at least three times number of unknown parameters in the Kriging model) We choose the extra design sites whose RMSEs are larger than 85% of the largest RMSE of the total tested sites. In our experiment, the largest RMSE of tested sites is around 0.224, the RMSE test gives rise to another 7 points (where RMSE > 0.224*0.85 = 0.19) to be added to the design. PAGE 44 34 When maximum response values do not vary significantly with iterations (Gupta, et al., 2006) and the RMSE in the whole possible region is not too large, we can stop the refinement steps, and obtain a model with = 4.6957, = (0.1, 0.85, 0.1, 0.1, 0.725, 0.1, 1.6, 0.6, 0.1, 0.1, 0.6, 0.6). The optimal solution w* that yields maximum tolerance y in the Kriging model can be obtained by simplex search, i.e.,*w= (0.4375, 0.4688, 0.1875, 0.1016, 0.7344, 0.3438, 0, 0.4219, 0.7813, 0.7344, 0.4531, 0.5469)T. The corresponding fixture layout is = (175, 55.625, 75, 4.2188, 293.75, 38.125, 0, 33.75, 281.25, 58.75, 163.125, 43.75)Tmm, with ) ( *w y = 5.029mm2, and RMSE () ( *w y) = 0.0872mm2. The yielded reaction forces from six locators are Fc = (18.2633kN, 21.3421kN, 12.7536kN, 22.0025 kN, 5.6110 kN, 25.3354 kN) for operation one, and Fc = (0.5672 kN, 0.6833 kN, 0.8252kN, 0.6773kN, 0.7525kN, 0.6631kN) for operation two. Based on these optimal design variables, we can implement the to lerance synthesis by the similar approach presented in Section 2.1. 2.2.2 Remark on Sensitivity Analysis of the Optimal Process Design The sensitivity analysis is to study the robustness of the optimal fixture layout obtained. The idea of the sensitivity analysis is to study impact of s ubtle perturbation at the optimal design point on the response, i.e., to evaluate sensitivity coefficients w wvariable desin Optimal iw y ) ( i = 1, ,12. Through analysis of these values, we can find out the sensitive direction along which small movement of locators has significant PAGE 45 35 impact on the tolerance stackup. Sensitivity directions could provide guidelines for fixture design, e.g., we should set tight tole rance for fixture locator assembly along these directions. After computation, at the optimal design point, the sensitivity coefficients are (0.005, 0.0164, 0.0119, 0.0056, 0.0272, 0.0030, 0.3202, 0.0119, 0.0043, 0.0216, 0.0081, 0.0184). We can see that locator 1 in op eration 2 has a relativ ely large sensitivity coefficient in x direction, and sensitivity coefficien t is relatively large at locator 3 in x direction in operation 2. The assembly toleran ce for these directions should be stricter than other directions. 2.3 Chapter Summary This Chapter develops a new process de sign and optimization method to seek an efficient and optimal process tolerance design for multistage machining processes. The idea is to reduce the design space in optim ization problem by developing a tolerance stackup model in which multiple error sources ar e aggregated into one base error through error equivalence transformation. The m odel based process tolerance design and optimization has a hierarchical structur e, i.e., assign the tolerances to the aggregated/bundled error sources first and then distribute them to individual error sources at each stage through cost analysis. Compared with a flat structure by which tolerances are directly assigned to indi vidual error sources, the hierar chical structure can avoid dramatic, complete change of tolerance al location and process design due to subtle change of cost functions. PAGE 46 36 In the mean time, the proposed method also searches optimal tolerance stackup as well as process design by exploring all possibl e combinations of process design variables. Computer experiments method is employed to establish the surrogate model for tolerance stackup prediction and optimal process desi gn. Space filling method (LHS along with maximin criterion) will first generate random design points and we obtain optimal tolerance stackup at each design point. A Krig ing model is then derived and refined by sequentially adding more design points into the regions with high uncertainty. One can further distribute the assigned tolerance for base errors am ong individual error sources when more process information is availa ble. We illustrate the approach through a twostage machining process where all errors were transformed to equivalent fixture errors. It has been demonstrated that consideration of error equivalence mechanism could significantly relieve the computation load of tolerance optimization problem and Kriging model fitting. The robustness of optimal tolera nce to process variati on is evaluated by a sensitivity analysis. In the twostage machin ing process, we analyze the sensitivity of tolerance stackup to the optimal layout of fixt ure locators. The sensi tivity analysis shows that the optimal design is more sensitive along some direction. The results provide a guideline to design the manufacturing process. PAGE 47 37 Chapter 3 Diagnose Multiple Variation Sources under Variation Equivalence This Chapter aims to improve the root cause diagnosis by utilizing the variation equivalence phenomenon. There are deviational error e quivalences among different individual error sources, i.e., the fixture er ror, machine tool error and datum error can generate the same deviation pattern on product feature. In the varia tion point of view, the equivalent phenomena also happen among the va riations of differen t error sources under certain conditions. This makes the process di agnosis and root cause identification of multiple variation sources more challenging. Meanwhile, based on error equivalence, we can study the equivalent prope rties among different variatio n sources by connecting the physical equivalence phenomena to mathem atical formulation. Moreover, through exploring possible equiva lent variations cases, we can construct an equivalent variation patterns library, which are useful for variatio n patterns mapping in pro cess fault diagnosis. All of these will help to improve root cause identifica tion of process fault among multiple variation sources. This Chapter is organized as follows. Section 3.1 introduces the concept of variation equivalence and cons tructs the equivalent vari ation patterns library. The diagnosis and root cause id entification under variation eq uivalence is presented in PAGE 48 38 Section 3.2. The Section 3.3 verifies the a pproach by illustrating a case study. Conclusion is given in Section 3.4. 3.1 Concept of Variation Equivalence a nd Equivalent Variation Patterns Library Previous application of error equivalence methodology on process diagnosis and root cause identification has focused on dia gnosing and distinguishing process deviation (mean shift). For instance, Wang et al ., 2006 utilized the EFE concept and the error compensating error strategy to improve the process diagnosis and root cause identification. However, besides deviation, pr ocess faults also manifest themselves as variation increases. Thus, it is also possible th at equivalence occur in terms of variation. We can call this phenomenon as variation equi valence, which concerns that different error sources may result in identical product feature variation pattern. This Section will give the definition of variation equiva lence and explore the possible variation equivalence cases in machining process, wh ich are used to cons truct the equivalent variation patterns library. 3.1.1 Definition of Variation Equivalence The definition of variation equivalence is that an identical part feature variation pattern can be generated by differ ent process variation sources. To understand the definition, we can use a simple machining process to explain. The operation is to mill the top surface of the block part, which is shown in Fig. 3.1. If PAGE 49 39 machine tool translational error in y directi on has large variation, it will cause large variation of the part top surface position. Th e large variation of the top surface position can also be caused by large variation of th e primary datum surface (the bottom surface) position. Similarly, if the two locators that are in touch with the primary datum surface are loose and have positive correlation, the part top surface would have the same variation pattern too. De note the part feature as y (in this case it is the top surface position). And denote the part feature variation caused by variation source s as Var( ys), where s = f m and d corresponding to fixture error, m achine tool error and datum error, respectively. We will have Var( y ) = Var( yf) = Var( yd) = Var( ym). Figure 3.1 A 2D Machining Process Example of Variation Equivalence 3.1.2 Equivalent Covariance St ructure Analysis and Library Based on the variation equiva lence concept, we can e xplore possible equivalent product feature variation patterns and link physic al explanations of variation equivalence Var( y ) f1 xf2 xf3 y x y o Machine tool path Var( ym) Var( yf) = Var( ym) = Var( yd) Fixture error variation Machine tool error variation Datu m error variation f1 xf2 xf3 y x y o f1 xf2 xf1 x =f2 x Positive correlation (=1) between two locators Machine tool path Var( yf) f1 xf2 xf3 y x y o p 1 yy Machine tool path Var( yd) PAGE 50 40 to mathematical formulation and analysis. Recall that the erro r equivalence based causality process model yj( k ) = ju ( k ) + ( k ) (Wang, Huang, and Katz, 2005) presents the relationship between pro duct feature deviation yj( k ) and integrated process equivalent fixture errors (EFE) u ( k ) in k th stage. And u ( k ) =i i*x ( k ). The *xi( k )s are the equivalent fixture error tran sformed from different individua l error sources, e.g., from datum error and machine tool error 1 2 1x K x* and212*x Kx (Wang, et al ., 2005), where the K1 and K2 matrices are error equivalence m odeling transformation matrices in Appendix A. Taking the covariance of this mode l, we can obtain the covariance structures of part feature and those of integrat ed process equivalent fixture error I u y2 ) () ( ) ( T j k jj. (3.1) The Eqn. (3.1) denotes the relationshi p between the covariance structure of process EFE and that of part feature. Thus we can connect the part feature variation patterns to the variation patte rns of EFE. Since our task is to develop an efficient approach for root cause identification among multiple variation sources under variation equivalence, studying() ukinstead ofy jwill more helpful to our research. Thus we will explore possible equivalent product feature variation patterns th at are connected to()uk. Taking the covariance for both sides of the e rror equivalence transformation model (Eqns. (A1)~(A3)), we can obtain the covariance stru ctures of EFE due to machine tool error and datum error as 122 xK KT and211 xK KT. Specifically, the general covariance structures for 2D case are: PAGE 51 41 M =2 11213 2 12223 2 13233 yyxyyxxxyxy yyxxyyxxyxy xyxyxyxyxxyfffff fffff fffff ( 3.2a ) and D =2 1111121 2 1121121 2 232 pyyxvxxpyyxxvxx pyyxxvxxpyyxvxx pxxyvyyfff fff f ( 3.2b ) where f1x, f2x and f3y denote the three locators coordina te under part coordinate system (PCS), with indices 1, 2 representing the tw o in touch with primary datum surface, and 3 for the one on secondary datum surface. The is the variance of machine tool rotational error, xx is the variance of mach ine tool translational er ror in x direction, and yy is the variance of machine tool transla tional error in y direction. And v1xx is the variance of primary datum normal vector error in x direction; p1yy is the variance of primary datum surface position in y direction, v2yy is the variance of secondary datum normal vector error in y direction, p2xx is the variance of secondary datum surface position in y direction. The details with regarding to th e derivation of Eqns. (3 .2a) and (3.2b) are illustrated in the Appendix C. These two equations connect the physica l meaning of vari ation patterns in machining processes to the mathematical explanations. By analyzing M and D under possible faulty/malfunction conditions, we may construct the equivalent covariance patterns library and obtain some information of how the covariance patterns change under different variation sources settings. For simplic ity, we use a block pa rt to explore all the possible variation equivalence cases between machine tool error and datum error. The PAGE 52 42 product feature here is the t op surface of the block part with the process of milling the top surface. The library pattern s are listed as follows: Faulty condition 1: product feature (top su rface) has large normal vector variation. In this case, we can see from Fig. 3.2 th at the block part top surface normal vector will have large variation. This product featur e variation pattern can be generated from large variation of machine tool angle error as well as from large variation of primary datum surface normal vector error. The covariance matrices of EFE due to machine tool error and due to datum error are given by Eqn. (3.3a) and (3.3b). Those subscripts with letter N denote the normal condition values. Figure 3.2 Variation Equivale nce of Faulty Condition 1 M =2 11213 2 12223 2 13233 yyNxyyNxxxyxy yyNxxyyNxxyxy xyxyxyxyxxNyfffff fffff fffff (3.3a) D =2 1111121 2 1121121 2 232 pyyNxvxxpyyNxxvxx pyyNxxvxxpyyNxvxx pxxNyvyyfff fff f (3.3b) x y o x y o Machine tool path v1xx PAGE 53 43 Faulty condition 2: there is larg e positional variation in x direction. The Fig 3.3 illustrates this case. There will be no negative impact on milling the top surface in this case. However, if the operation is to drill a hole in the top surface, the impact will be significant. The covariance matrices of EFE due to machine tool error and due to datum error are given by Eqn. (3.4a) and Eqn. (3.4b). Figure 3.3 Variation Equivale nce of Faulty Condition 2 M =2 11213 2 12223 2 13233 yyNxNyyNxxNxyNxyN yyNxxNyyNxNxyNxyN xyNxyNxyNxyNxxyNfffff fffff fffff (3.4a) D =2 1111121 2 1121121 2 232 pyyNxvxxNpyyNxxvxxN pyyNxxvxxNpyyNxvxxN pxxyvyyNfff fff f (3.4b) Faulty condition 3: there is larg e positional variation in y direction. This faulty condition will significantly affect the product features displacement, which is explained by Fig. 3.4. The covariance matr ices of EFE due to machine tool error and due to datum error are given by Eqn. (3.5a) and (3.5b). x y o p2xxxx x y o PAGE 54 44 Figure 3.4 Variation Equivale nce of Faulty Condition 3 M =2 11213 2 12223 2 13233 yyxNyyxxNxyNxyN yyxxNyyxNxyNxyN xyNxyNxyNxyNxxNyNfffff fffff fffff (3.5a) D =2 1111121 2 1121121 2 232 pyyxvxxNpyyxxvxxN pyyxxvxxNpyyxvxxN pxxNyvyyNfff fff f (3.5b) Faulty condition 4: part feature has la rge variations of both normal vector and position in y direction. The Fig. 3.5, Eqn. (3.6a) and (3.6b) explain this case physically and mathematically. Figure 3.5 Variation Equivale nce of Faulty Condition 4 x y o x y o Machine tool path yy p1yyv1xx x y o x y o Machine tool pathyy p1yy PAGE 55 45 M =2 11213 2 12223 2 13233 yyxyyxxxyxy yyxxyyxxyxy xyxyxyxyxxNyfffff fffff fffff (3.6a) D = 2 1111121 2 1121121 2 232 pyyxvxxpyyxxvxx pyyxxvxxpyyxvxx pxxNyvyyfff fff f (3.6b) Faulty condition 5: part feature has la rge variations of both normal vector and position in x direction. The Fig. 3.6 illustrates this case. We can see that this case is similar to Faulty condition 1. The covariance matrices of EFE due to machine tool error and due to datum error are given by Eqn. (3.7a) and (3.7b). Figure 3.6 Variation Equivale nce of Faulty Condition 5 M =2 11213 2 12223 2 13233 yyNxyyNxxxyxy yyNxxyyNxxyxy xyxyxyxyxxyfffff fffff fffff (3.7a) D =2 1111121 2 1121121 2 232 pyyNxvxxpyyxxvxx pyyNxxvxxpyyxvxx pxxyvyyfff fff f (3.7b) x y o x y o p2xxv1xx PAGE 56 46 Faulty condition 6: there ar e large variations of positi on in both x and y directions. This case has variations of displacements in two directions, which is explained by Fig. 3.7, Eqn. (3.8a) and (3.8b). Figure 3.7 Variation Equivale nce of Faulty Condition 6 M =2 11213 2 12223 2 13233 yyxNyyxxNxyxyN yyxxNyyxNxyxyN xyxyNxyxyNxxyNfffff fffff fffff (3.8a) D =2 1111121 2 1121121 2 232 pyyxvxxNpyyxxvxxN pyyxxvxxNpyyxvxxN pxxyvyyNfff fff f (3.8b) Faulty condition 7: a ll the positional and normal variations are large. This will be the most general case, in which all fault will occur in the process. The Fig. 3.8 shows this case. The covariance matrices of EFE due to machine tool error and due to datum error are given by Eqn. (3.9a) and (3.9b). x y o x y o yy p2xx xx p1yy PAGE 57 47 Figure 3.8 Variation Equivale nce of Faulty Condition 7 M =2 11213 2 12223 2 13233 yyxyyxxxyxy yyxxyyxxyxy xyxyxyxyxxyfffff fffff fffff (3.9a) D =2 1111121 2 1121121 2 232 pyyxvxxpyyxxvxx pyyxxvxxpyyxvxx pxxyvyyfff fff f (3.9b) 3.2 Diagnosis and Root Cause Identifi cation under Variation Equivalence Process root cause diagnosis usually contains two steps, with mapping the product feature variation patterns to the library patterns as first step, followed by distinguishing variation sources that cause the identified product feature variation patterns. The second step is more challenging under variation equivalence. Although some research work (e.g., Jin and Zhou, 2006b) has mentioned this case, there is still not an efficient approach developed in the literature. In this Section, we develop a so called excitationresponse path approach that is able to distingu ish multiple variation sources under variation equivalence. x y o x y o yy v1xx p2xx p1yy xx PAGE 58 48 In this thesis, we will assume th at there are only machine tool error and datum error in the machining pro cess. To identify from whic h variation sources a product feature fault pattern is generated, it is equi valently to distinguish the variation sources between M and D. In M and D, we can see that given a spec ific fixture locator layout, the covariance structure will change accord ing to changes of variation sources magnitudes. Therefore, in order to develop a variation sources identification approach, we can conduct some analysis of the covariance structures of M and D under different variation source magnitude settings. One way to represent the covariance structure is to use the eigenvectors of the covariance matrices. Denote ai( M) and ai( D) as ith eigenvectors of M and D. To analyze how ai( M) and ai( D) change as variation sources magnitudes change, we can compute the eigenvectors gestur es under different variation sources values. Denote aref as reference vector. By computing the angles between the eigenvectors and aref, we can obtain the eigenvect ors gestures information. Denote m, d as the angles set between aref and ai( M), ai( D), respectively; and m, d as the variation sources set for ai( M) and ai( D). The points set of ( m, m), ( d, d) will form two curve of eigenvectors angles VS variation sources values, for which we call excitationresponse path. Given a fixture layout, the matrix will be fixed. Furthermore, Var( yd) = Var( ym) under variation equivalence. We thus have M D. If take faulty condition 6 for example, we will haveyy +2 1 x Nf = 1 pyy +2 11 x vxxNf, yy +2 2 x Nf = 1 pyy +2 21 x vxxNf, and x x +2 3 yNf = 2pxx +2 32 yvyyNf, respectively. If N = 1 vxxN = 2vyyN we will PAGE 59 49 see that under the same value se tting of variation sources, i.e.,yy = 1 pyy and x x = 2 pxx the m, will be equal to d at each point. In this situation, the curve of ( m, m) and ( d, d) over lab each other, which ma kes the two variation sources undistinguishable. Therefore, to utilize the excitation response path for variation sources identification, we must make some assu mptions. First of all, the normal condition variation sources magnitudes are assumed to be different, e.g., yyN p1yyN. Besides, we assume that there is single product feature fault pattern for each product feature. The reason for first assumption has been aforementi oned. It is also practical in that datum variation is usually larger than machine tool variation. This assumption will enable the curves of ( m, m) and ( d, d) to be different with a proper selected aref. We do not consider multiple product feature variation pa tterns, because that our key issue for this topic is to distinguish multiple variation s ources from the same product feature variation pattern. Thus the second assumption is also necessary. Under these assumptions, we can conduc t a sequential testing procedure to distinguish the variation sources. The root cause identification procedures will be: Plot the excitationresponse pat h for possible variation sources. Here we assume that there are variations of m achine tool error and datum error, corresponding to two excitationresponse paths. If the two varia tion sources simultaneously contribute to the detected part feature variation pa ttern, this will result in a mixture excitationresponse path. For the mixture path we can assume weight coefficients for PAGE 60 50 both variation sources, i.e., mixed = MM + DD, where M + D = 1. In this case, there will be totally three excitationresponse paths in the plot. Estimate two covariance matrices of proc ess errors from two consecutive samples. The estimationui= uSi with uSi denoting the sample covariance matrices, i = 1, 2, and u= ( T )1Ty Calculate the first eigenvector angles of the two sample covariance matrices. By selecting a reference vector aref, we can obtain two eigenvectors angles i, i = 1, 2. Estimate the variation sources values. Take faulty condition 3 for instance, iyy = uSi(1, 1) 2 1 x Nf and 1pyy = uSi(1, 1) 2 11 x vxxNf, i = 1, 2. The uSi(1, 1) denote the element of the intersection between the first row and the first column in uSi. Compare the slope of the line that connects the two sample points with the slope of excitationresponse path and distinguish the variation sources. For example, if the two points (1yy 1) and (2yy 2) is close to the machin e tool excitationresponse path, and the slope of the lin e that connects the two points is similar two the slope on the excitationresponse path, the variation so urces will be identified as from machine tool error. It is vise versa for variation sources from datum error. The rationale behind these procedures is th at the eigenvectors of different samples covariance matrices will be not the same and they usually have a deviate range. However, in general, if the sample size is large enough, the sample eigenvectors gestures should be very similar to the population eigenvector (i.e the eigenvector on th e excitationresponse path). The two samples are corresponding to two points on the exci tationresponse path PAGE 61 51 graph. Thus the slope of the line segment that connects the two points thus will be similar to the slope of the tangent at the population curve point. The Fig. 3.9a and 3.9b illustrates this rationale. Population Eigenvector Sample Eigenvectors(0 0 1)(0 1 0)( 1 0 0 ) Sample Point1 Sample Point2 Population Slope Sample SlopeEigenvector AnglesVariation Sources Magnitudes Figure 3.9a Eigenvectors Gest ures Figure 3.9b Sample Slope and Population Slope 3.3 Case Study We will use a case st udy in this Section to verify this approach. In the case study, we use the block part shown in Fig. 3.1 as the example. The machining process is to mill the top surface. 3.3.1 Illustration of the Root Cause Dia gnosis Approach Using a Machining Process The fixture locaters layout are specified as f1x = 20, f2x = 400, f3y = 50. Suppose that the tolerance for translational error is 0.55mm, and the tolerance for rotational error is 0.0007radian (0.041degree). The machining pr ocess is still to m ill the top surface of a block part. There are two possible variation sour ces in the process, machine tool error and datum error. Normally, the machine tool a ngular error has smalle r variation than the datum error. Thus, we suppose that xxN = 0.2mm2, yyN = 0.2mm2, xyN = 0.000001mm2, PAGE 62 52N = 0.0000008radian2, v1xxN = 0.000001, p1yyN = 0.3mm2, v2yyN = 0.000001, p2xxN = 0.3mm2, white noise = 0.00000001mm2. We choose aref = (0 0 1)T. Suppose that product feature variation pattern of faulty condition 3 is detected, no mixture of two variation sources, and the true variation source is machine tool error with yy = 0.36mm2 (but actually we do not exactly know this valu e and which variation source occurs in the process). Here we collect two consecutive samples, with size n1 = n2 = 200. The first eigenvector angles of firs t and second samples are 1 = 89.2007degree and 1 = 88.3643degree. The estimated vari ation sources magnitudes are 1yy = 0.3731mm2 and 2yy = 0.3458mm2, respectively. The results are summarized in Fig. 3.10. 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 87.5 88 88.5 89 89.5 90 90.5 Variation Source Magnitudes, Unit: mm2Eigenvector Angles, Unit: DegreeExcitationResponse Path for Faulty Condition 3 Equivalent Datum Equivalent Machine Tool (03731, 892007) (03458, 883643) (03600, 887699) Credible Interval Figure 3.10 ExcitationResponse Path of Case Study Result In the excitationresponse path plot, it is obvious that the sample data and its slope is more close to the population curve of mach ine tool. In light of this, we can determine that the variation source is from machine tool. PAGE 63 53 3.3.2 Remark on the ExcitationResponse Path Approach In the case study, we choose (0 0 1)T as reference vector. The datum error curve is a horizontal line, which can be distinguished fr om machine tool error curve. However, if we choose (1 0 0)T and (0 1 0)T as reference vectors. The result will be different, and the root cause identification will be impossible. Because that the slopes of the two variation sources curve are the same, which makes the two curves parallel to each other. This is illustrated by Fig. 3.11 and Fig. 3.12. Therefore, for the excitationresponse path approach, a reference vector that can distingu ish the slopes of different variation sources curves is necessary. 0.31 0.32 0.33 034 0.35 0.36 0.37 0.38 0.39 0.4 49 49.5 50 50.5 51 51.5 52 52.5 Variation Sources Magnitudes, Unit: mm2Eigenvectors Angles, Unit: Degree Equivalent Datum Equivalent Machine Tool Figure 3.11 ExcitationResponse Path Using Reference Vector (1 0 0)T PAGE 64 54 0.31 0.32 0.33 034 0.35 0.36 0.37 0.38 0.39 0.4 37.5 38 38.5 39 39.5 40 40.5 41 Variation Sources Magnitudes, Unit: mm2Eigenvectors Angles, Uint: Degree Equivalent Datum Equivalent Machine Tool Figure 3.12 ExcitationResponse Path Using Reference Vector (0 1 0)T 3.4 Chapter Summary In this Chapter, the variation equi valence concept is presented and the equivalent variation patterns in machining process are explored. Based on the variation equivalence concept, we explore the possible product feature equivalent variation patterns among different variation sources, and construct th e equivalent variation pattern library. By utilizing the library covarian ce structures and conducting some excitationresponse path analysis, we find that different variation s ources can be distinguished under variation equivalence. The case study well verifies this approach. However, there is still limitation w ith regarding to this approach. For each faulty condition, a proper reference ve ctor should be carefully sele cted. Otherwise, the root cause identification may fail if there is not a reference vector that can significantly PAGE 65 55 distinguish the slopes of differ ent variation sources excitatio nresponse paths. For faulty condition 4 to faulty condition 7, each vari ation source has more than one variation elements. In this case, the excitationresponse paths will not be a curve, but a surface, or even a volume. This will makes the visualized testing procedure more challenge, especially for 3D cases. PAGE 66 56 Chapter 4 Conclusions and Future Work 4.1 Conclusions Manufacturing process design and control re lies not only on an efficient process variation modeling, but also on many other variation reduction st rategies. For early manufacturing process design stage, the effici ency of the design strategy usually relies on the dimensionality of the design space. For a good process control strategy, a method for efficiently diagnosing different variation sources is a must. The work in this thesis aims to develop efficient process design and process control stra tegies based on improving the understanding of error equivale nce and variation phenomena, th at is, different types of process errors and variation can result in the identical product feature deviation and variation. The implication of error equivalence mechanism can greatly impact the early stage design and quality cont rol in manufacturing processes. The major contributions of this thesis are summarized as follows: Process design and optimization based on error equivalence concept. Due to the fact that different error sources can generate the same product featur e deviation pattern, we can modeling the process variation propagation based on one error, i.e., the equivalent error or based error. An erro r equivalence based process tolerance stackup PAGE 67 57 model can thus be developed, and tole rance allocation can be conducted under a specified spatial layout. Meanwhile, embe dding error equivalence into computer experiment design can assist us to search global optimal tolerance allocation among all the possible process tolerance design. Introducing the error equivalence mechanism into the to the process design significantly reduces the design space and relieve us from the considerable symbolic computation load, which results in a costeffective design strategies. Process control: root cause identifica tion of variation so urces under variation equivalence. The variation equivalence phenomena expose the traditional manufacturing process diagnosis to the chal lenge that different variation sources may result in identical product feature varia tion patterns. Through exploring the possible product feature equivalent variation patte rns among multiple error sources, we can construct the equivalent covariance structure library. Meanwhile, an excitationresponse path orient ation approach is develope d to improve the variation sources root cause identification. The simula tion study results show that this approach enables multiple variation s ources to be distinguishable under variation eq uivalence. 4.2 Future Work This study aims to improve manuf acturing process design and control by using error equivalence methodology. In addition to the results obtained in process tolerance design, optimization, and process root cau se diagnosis of variation sources under PAGE 68 58 variation equivalence, we can further expand th e diagnosis approach to the processes that contain random effects. Since in practical machining processes, large variation random effects may occur due to unknown factors. The mixing of random effects with variation equivalence will lead the root cause dia gnosis to a more challenging situation. Furthermore, the limitations of the excitation response path approach drive us to improve the testing procedures for higher dimens ions of variation sources cases. PAGE 69 59 Cited References Agapiou, J. 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Zhou, S., Chen, Y., and Shi, J., 2004, Root Cause Estimation and Statistical Testing for Quality Improvement of Multistage Manufacturing Processes, IEEE Transactions on Automation Science and Engineering, 1(1), pp. 73~83. Zhou, S., Ding, Y., Chen, Y., and Shi, J., 2003, Diagnosability Study of Multistage Manufacturing Pr ocesses Based on Linear MixedEffects Models, Technomatrics, 45, pp. 312~325. Zhou, S., Huang, Q., and Shi, J., 2003, State Space Modeling of Dimensional Variation Propagation in Mu ltistage Machining Process Using Differential Motion Vectors, IEEE Transaction on Robotics and Automation, 19, pp. 296~309. PAGE 76 66 Appendices PAGE 77 67 Appendix A: Review of EFE and Derivation of d Wang et al., 2006, gave the derivation of EFE T) (III III II II I I 1 2p v p v p v K x*, (A1) and 1 2 1x K x*, (A2) where 3 2 1 1G G G K. (A3) For the specific form of K1 and K2, refer to Wang, Huang, Katz, 2006. If the coordinates are under GCS, the K1 and K2 matrices are changed accordingly in each operation. E.g., in our example, for operation 1 11 22 33 2 44 55 66001(1)(1)0 001(1)(1)0 001(1)(1)0 (1), 010(1)0(1) 010(1)0(1) 1000(1)(1)yx yx yx zx zx zyff ff ff ff ff ff K for operation 2 11 22 33 2 44 55 66010(2)0(2) 010(2)0(2) 010(2)0(2) (2), 001(2)(2)0 001(2)(2)0 1000(1)(1)zx zx zx yx yx zyff ff ff ff ff ff K and 11 44 1222366 55 33(2)0(2)010 (2)(2)0001 (2)(2)0(2)010,(2),(2)0(2)(2)100 (2)(2)0001 (2)0(2)010.xz xy xz yz xy xzff ff ffff ff ff GGG PAGE 78 68 Appendix A: (Continued) To calculate) 2 (* 2x using the feature deviation from operation 1 with the nominal fixture layout (the nominal location of six locators in ope ration 2), we can derive the relation between ) 2 (* 2x and u (1) after linearization as: ) 2 (* 2x = Ku (1), (A4) where K is the coefficient matrix. Then the EFE due to datum errors will be linearly add to operation 2 in the stackup model. The EF E due to datum errors calculated thus obtained are: .5 4 6 3 2 1 3 2 1 3 2 1 5 4 3 2 1 5 4 3 2 1 5 4) 1 ( 3275 0 ) 1 ( 3275 0 ) 1 ( ) 1 ( 183333 0 ) 1 ( 108575 0 ) 1 ( 07476 1 ) 1 ( 183333 0 ) 1 ( 108575 0 ) 1 ( 07476 1 ) 1 ( 09375 0 ) 1 ( 046875 0 ) 1 ( 046875 0 ) 1 ( 423125 0 ) 1 ( 576875 0 ) 1 ( 510417 0 ) 1 ( 255208 0 ) 1 ( 255208 0 ) 1( 798125 0 ) 1 ( 201875 0 ) 1 ( 510417 0 ) 1 ( 255208 0 ) 1 ( 255208 0 ) 1 ( 048125 0 ) 1 ( 951875 0 ) 2 ( y y x z z z z z z z z z y y z z z y y z z z y yf f f f f f f f f f f f f f f f f f f f f f f f* 2x PAGE 79 69 Appendix B: Prediction and Estimation of Kriging Model After obtaining an experimental design } ,..., {1 ns s S with corresponding responses } ,..., {1 ny ysy the unknown parameters in the correlation function have to be estimated, which is obtained by MLE crite ria, and boils down to the minimization of the function ) det ln ln ( 2 12R n (B1) where R is correlation coefficient matrix, and 11121 21222 12(,)(,)(,) (,)(,)(,) (,)(,)(,).n n nnnnRRR RRR RRR ssssss ssssss R ssssss Then, using generalized least square estimation (GLS), the unknown parameters and 2 can be estimated as sy R F F R F 1 1) ( T T, (B2) and ) ( ) ( 1 1 2 F y R F ys s Tn, (B3) where T n)] ( ),..., ( [1s f s f F is the regression design matrix. As for these parameter estimations, the best linear unbiased predictor (BLUP) is: ) ( ) ( ) ( ) ( 1 T F y R w r w f ws Ty, (B4) where T n TR R)] ( ..., ), ( [1w s w s r is a column matrix of correlation between the stochastic processes at given experimental design sites and untried input site. The mean squared error (MSE) was given by Sacks et al., 1989, as: ) ( ) ( ] ) ( ) ( [ 1 { )) ( (1 2w r w f R F F 0 w r w f wT T T T Ty MSE. (B5) PAGE 80 70 Appendix C: Derivation of the E quivalent Covariance Structures The derivation of the equivalent covariance matrices are based on the error equivalence model in Wang et al., 2005. For 3D case, we can obtain the covariance matrices of EFE due to machine tool error and datum error as 122 xK KT and211 xK KT, where 1xand 2xare the covariance matrices of individual machine tool error and datum error. The details of K1 and K2 matrices are presented in Appendix A. For 2D case, the derivation can be ba sed on 3D derivation. We can suppose there is no error for primary datum surface in 3D case, which will result in 2D case K1, i.e., x2 = [ 0 0 0 0 0 0 vIIx vIIy 0 pIIx pIIy 0 vIIIx vIIIy 0 pIIIx pIIIy 0]T. (C1) Plug Eqn. (C1) into Eqn. (2.1), we can obtain the 2D case product feature deviation yj(k), i.e., [0 0 0 pIIyvIIxf1x pIIyvIIxf2x pIIIxvIIIyf3y]T. Extracting the coefficients of individual datum error, we thus obtain K1 as K1 = 1 2 30010000 0010000 0000010x x yf f f (C2) By setting 0 to the elements that are relevant to z direction, covariate terms between normal vector and position, or primary datum surface in 3D case2x, e.g., I IzIIzp = 0, I IxIIyvp = 0, or I xxv = 0, we can obtain Eqn. (3. 2b). For equivalent covariance structure of machine tool error, the derivation is similar, and K2 = 1 2 301 01 10 x x y f f f (C3) PAGE 81 About the Author Shaoqiang Chen received a Bachelors De gree in Mechanical Engineering from Shanghai Jiao Tong University, Shanghai, Chin a in 2004. He is currently a MSIE student in the Department of Industrial and Manage ment Systems Engineering at the University of South Florida, Tampa, Florida, USA. While in the MSIE program at the Univer sity of South Florid a, Shaoqiang Chen focused on the research of manufacturing pr ocess design and process control. Currently, he is a member of IIE and a member of INFORMSUSF student chapter. 