Quality Stability of Multi-Station Assembly Process Based on Variation Stream

Transactions of Tianjin UniversityISSN1006-4982pp409-415Vol.13 No.6Dec.2007Quality Stability of Multi- Station Assembly ProcessBased on Variation StreamWANG Lei(王磊)', GUO Wei(郭伟)' , ZHANG Conghui(张聪慧)”, ZHAO Jiali(赵家黎)'(1. School of Mechanical Engineering, Tianjin University, Tianjin 300072, Chin;2. School of Management, Tanjin University, Tanjin 300072, China)Abstract: To analyze the physical structure of assembly process and assure product quality, thequality stability of multi-station assembly process was investigated. First, the assembly process wasmodeled as a one-dimensional discrete variant system by state space equation based on variationstream. Then, the criterion to judge whether the process is stable or not and the index, stability de-gree, to show the level of stability were proposed by analyzing the bounded-input bounded-output(BIBO) stability of system. Finally, a simulated example of a sheet metal assembly process withthree stations, was provided to verifty the effectiveness of the proposed method.Keywords: multi-station assembly process; variation stream; quality stabilit;tabilit degreeIn recent years, product quality has ben one of the constructed the catastrophe model of machining vanaton bymostimportant topics for the improvement of enterprises'the catastrophe theory. Agrawal and Lawless4- 6J proposedIndustrial statistics shows that about 60%一70% of product an autoregressive model to analyze the variation transmissionquality problems come from the manufacturing process.and forecast problems on the basis of the relations amongTherefore the object of quality control should be transformed mulivariate quality characteristics, measuring eror and los-from product to process. It means that all factors afectinging data etc. Mantriparagada et al7.8J thought the matrixthe product quality in process are monitored and optimized transform which expresses the relationship between part .char-by identifying, analyzing, and contolling the process to as- acters could express the variation propagation, and forecastsure them in prescriptive value.the success possibility of assembly process. According to theIn manufacturing process, the relations between pro-concept of datum flow chain, Mantriparagada et al7,8J prop-cess parameters, such as fixture layout and tool parameter, osed that each station of assembly process be treated as a se-etc, and quality characteristics are complex. For example,ries of discrete events and the discrete dynamic system be ex-a quality characteristic variation comes posibly from some pressed by adopting state transform equation. They also ap-process parameter variations directly, or from other quality plied the control theory to analyzing the assembly process andcharacteristic variations, that is, impacted by other process improving mating feature. Based on the research result ofparameter variations indirectly, and possibly rooted in this Mantriparagada et al, Huang et al9 -B3] adopted the statestation or before. It is known that, instead of being absolute space equation to depict the variation propagation in a multi-and static, the variation of quality characteristic in manu-station body assembly process, and then determined the rootfacturing process transmits, changes, and impacts each oth- causes of quality-related problems. The variation stream the-er in every station under the efect of process parameters. ory is thus established owing to the above pioneer work,This is called variation stream.which has an active ffct on improving quality control meth-Many scholars researched variation stream at diferent ods and enhancing product quality.aspects. Luo et al2,3] introduced the iterative model toBased on the aforementioned work, we tory to investi-show the dynamic characteristics of variation stream and gate the quality stability of multi-station assembly processAccepted date:2007-07-17.WANG Lei, bom in 1975, male, Dr, lecturer.中国煤化工、and Tianjin Science and* Supored by the National High-Tech Research and Development Plan (NatiTechnology Key Poject (No.05YFCDGX08700).YHCNMHG'Correspnodence to GUO Wei, E mail: wguo@ tju. edu. cn.Transactions of Tanjin UniversityVol.13 No.6 2007by adopting state space equation. New quality stability cri- time-varying system and described by the state spaceterion and stability index, stability degree, are first pro- equation. And the station sequence is considered as timeposed based on the bounded-input bounded- output (BIBO)series. The ideal state is expressed asconcept, which can help evaluate process physical strucure[X; =Ai-1Xi-1i=1,.,*,Nand improve the relative process parameters. As a result,lY;= CXi∈{1,2,.,N}the variation of final product quality is reduced. The simu- where X and Y:represent the perfect value of product fea-lated example of a sheet metal assembly process with threeture dimension and quality characteristics after station i re-stations is provided to verify the effectiveness of the pro-spectively; A;-1 is a system matrix that shows the productposed method.transformnation from station i- 1 to station i. In particular,1 Model of variation stream for multi-Ao denotes the unit matrix. C;, as an observation matrix,station assembly processlabels the number of sensors and the location of measuredFig.1 shows the structure of an N-station assemblypoints, which is the zero matrix when the station is notprocess, which can be treated as a one-dimensional discretemeasured.-切xXuXSuation 1StationiStationN .Fg.1 Diagram of皿asembly procss with N stationsDue to the existence of stochastic variation and system alyze the stability of multi-station assembly process is to re-variation in real assembly process, the system model can be alize the impact of each station physical structure on productexpressed asvariation, and ensure the product quality characteristics tofX =Ai-1X{-1+ B;μ:+ξ(2)have a leas variation bound under the same input error. Sta-lY{=CX{ +η:bility includes state stability and BIBO stability. State sta-InEq. (2), x and Y{ reresent the actual value of bility means if vey small itial variation can only arouseproduet fature dimension and quality charctersties afer very small vaiation of the gytem departure from belancestation i, respectively. μ; is the variation input of stationstate, then the system is of state stability. BIBO stabilityi. In this peper, the frture faultl is cnided tobe the means for ech bodiput, if the cupup is bounded,variation source because fixture layout is the main processthen the system is of BIBO stability. Given that the partsparaneter impacting product quality characteristics in theintroduced in the assembly process have no fault (i.e. theassembly process-14). B; is an input matrix which reflectssystem initial state is zero) , obviously , the quality stabilitythe efect of fixture structure on product. ξiand η; are theissue of multi- station assembly process can be regarded asthe BIBO stability problem.noise terms having no effect.At present, there is no full theory to judge the BIBOThen, subracting Eq.(1) from Eq.(2), we can getstability of discrete time varying system. According to Ref.the following variation stream model for multi-station assem-[ 15 ] and the characteristics of variation stream, we proposebly process:the fllwing theorem to identify the quality stability.[x:=X -X=A:-1xi-1+Bjμ:+弘(3)Theorem 1 The process is BIBO stable if and only ifly:= Y- Y;=Cx;+ η: .the zero- input response of Ea. (3) is uniformn asymptotically2Stability analysis for multi-station stable,中国煤化工T,j is convergent;B; anc; defined asassembly processHCNMHGFrom the viewpoint of variation stream, the aim to an-一410WANG Lei et al: Quality Stability of Multi Sation Assembly Process Based on Variation Stream[I,j= A.-1A2***A,i≥j(4)「 y1lT,s=IyzProof First, assume that there is a positive constant:M such that II μ4ll≤M for every i, where“|●|" is- yN-the 2-norm.C1 B0..0 ][μ1]The zero-input response of Eq. (3) is uniform asymp-C2Tz,BC2 B242|+totically stable, which means that there exist positive con-stants C1 and C2 such that I1 T,j II≤c1e~e2(-), whenCwTw,1B\ CvTv,2B2 .. CvByJL μNJξiand ηi are zero for i≥j.[ C1T1.0][81 1HiηlH.From Eq.(3), we obtain lyll≤llC:I.|x;|| +CzT2,ox+| :(9)Because B;and C; are bounded and 气，η; are also. cNTv,oJ.LεNJbounded in assembly process, there exist positive constantswhereE;= ZT.5+ n;,i=1,2,..,N. Then, de-M;,M2,M3 and M4 such that II B;|≤M, ICll≤.M2, ξill≤Mzand |I η:|≤M4 for all i.fine H asThe general solution of Eq. (3) can be obtained as「CqBiCfollows:H=C2T2,Bx:= TI,0%o+n,(BH+引)(5)L CwTw,Bi CwTw,2B2 . CvByJThen(10)|xl≤cllxol! + 2 C;e-(i-).In Eq. (10), matrix H is composed of system matrixA;, input matrix B; and observation matrix Ci, which(MM + Mz)(6) completely includes the process parameter information andSorelects the process physical structure. xo shows the part er-xll≤cllxol +cq(MM|+ M;)/c2 (7ror in the process, i.e. the initial state of system. Accord-. lyll≤M2[crl|xoll +c(MM; +ing to the aforementioned assumption, the initial state is ve-M3)/c2]+ M4.(8) ro. & is the summation of all modeling uncertainties andThus, the BIBO stability of process is proved com- noise temms and is supposed to be a constant in conimonpletely.cases. From above all, the property of H determines theBased on Theorem 1, the criterion to analyze the qual- variation magitude of quality characteristics. So stable de-ity stability of multi- station assembly process is obtained as gree P can be expressed asfollows (supposing the system variation ξ and η: are very(11)φ=THTrsmall):(1) Judge the unifom asympotical stability of processwhere I.llp is the F-norm.under zero input, i.e. the convergence of Ti,j;3 Case study(2) Find whether B; andC: are bounded or not.The assembly process of sheet metal is taken as an ex-In addition, for a stable process, a quantitative indexample to validate stability criterion and the concept of stableis required to show the stability level. Hence, we proposedegre. Suppose fixture fault is the main variation source inthe concept of ' 'stable degree" that is defined as the degreethe assembly process, and the fixture layout is shown inof output quality characteristics close to the ideal value, i. .Fig.2, which locates the part using two pins correspondinge. the larger the stable degree is, the less the variation ofto one locating hole and one locating slot in the part. Fix-quality characteristic is. Its expression can be obtained byture fault is expressed as(OL(x),OL(y),OL'(y)). The .the fllowing steps.First, convert Eq. (3) to the direct input-ouputposition deviation of a random point on product representsform:the pro中国煤化工(oM(x),0M(y),a).Ety characteristics is(ON(YHC N M H Gon deviation of mea-sured points .一411一Transactins of Tianjin UniversityVol.I3 No.6 2007Put3 N2●MaD， 200 350L'Ls (M3)图Pat2 N1Fg.2 Diagram of fixture layoutPart1L1 (M1)C5L3 (M2)53.1 Validation of stability criterionFirst, two cases are judged by the proposed stability(b) Station2criterion.Case one is shown in Fig.3, which has four parts 8-Pant380sembling in three stations. In accordance with Ref. [ 16],the detailed expression of A;, B; and C; can be obtained asshown in Eqs.(12)- -(18).Riu6 (M4) L95 (M3)FPart2vn°|↑Par250_ .100|z3 (M2750H(MI)_ 12 g(M2)_ 4o 450(e) Station3(a) Station1Fig.3 Ilustratio of case one (unit: mm)0000 0.8e-3 1 0-0.8e-3 - 346.2e-306x6A1=-101(12)0-0.32000.32- 3060.8e-3 0 0 0.8e-30.64O%x6I6x6)12x12000000.9e-3-0.9e-3 00 1000-23.5e-2 0 0 1 00 -76.5e-2 0Agx3A2=0.9e-00010(13)-1 - 40.5e-240.5e-2 0-5.5e-25.5e-20 0 000-0.9e-3 103x9中国煤化工YHCNMHG一412一.WANG Lei et al: Quality Stability of Multi Station Asembly Process Based on Variation Streamn010BI=0 -1.5e-3 1.5e-3_ 0 -2.2e-3 2.2e-3(14)0yx612x610 -0.8e-3 0.8e-B2=) 346.2e-3 653.8e-3 0 0 0(15)0 -0.8e-30.8e-30-∞03x6)12x6(1-0.9e-30.9e-3- 190.8e-3809.5e-30x30.9e" 3Bz=1428.6e-3- 428.6e-3(16)0 -0.9e-303x30 -3.3e-3 3.3e-3/12x602x310-50002x3 .350Cz=(17)0-35001650)4x12 .I 0 -2101C=[0*x10 i 48012x3(18)Then, the process to judge quality stablity is as fol- ed into case two as shown in Fig.4, i.e. adding one locat-lows:ing pin L' for part 3 on station 2 and changing the locating(1) Judge the convergenceOf Tr,jpoint Ls into L' in station 3. In this case, the process toBy calcultion, IIT2,lI= IAill =306.486 1,judge quality stability is as follows ( the mathematic expres-| T3,II = |A2A1 | =307.018 2sionof A;, B; and Ci in this section is omitted due toDue toll T2,Il < II T3,l，T,j is not convegent. space limitaion);(2) Judge B; and C are bounded or not(1) Judge the convergence of Tr,jBy calculation, |BII =1.4203, I B2lI =∞，By calculation, II T2,I = |AiII =306.486 1,|lB;I| =45.6542， IC2I =738.241 8,|Ts,Il= II A2A II =304.1607| C;II =523.928 4r H T.j isconver-中国煤化工Because Ti,; is not convergent and B2 has no bound, gent;from Theorem 1, the first case is unstable.TYHCNMHGedornotTherefore station 2 and station 3 in Fig. 3 are convert-By calculation, I|BIlI =1.4203,一413-Transactions of Tianjin UniversityVol.13 No.6 2007F Pant3N2.0 rLs'.200元350; 0.6- Meineto047Covariance 1.017 2Covariance: 0.981 7L5 (M3)8↑0.4-Part2 NiPat1●4(M1) c 542| 0u3 (M2) I4Poeition error (e) /mmFig.5 Varation distribution chart of Nzin case one(a) Stucion2Part580us'3401300 N3abo.2-[6(M4) L7.0年directionLS (M3)Mean: -0.069 1y directionC 0.8个 Covariance: 0.381 4_ Mean: 0.032 7Purt2Covariance 0.327 40.4L3 (M2)o'-25-2.0-15-1.0-05 0 0.5 1.01.5 2.0 25(b) Stutico 3Position error (e) /mmFIg.4 Station2 and statio3 in case twoFg.6 Variation distribution chart of Nzin case two|B2I =36.8356,1 B3I =45.6542,commonly the prescriptive tolerance due to the restrictionIC2I =738.241 8, IIC3ll =523.928 4insufficiency of part 3, but product quality characteristics inBecause T.j is convergent and B and C are bound-the second case are preserved in the prescriptive toleranceby adding one locating pin Ls in part3. So the first case ised, obviously, the second case is stable.The center limit theorem of probability points out thatnot stable; the second case is stable. Therefore, the stabil-the distribution of the sum of several random variables,ity criterions proposed in this paper are valid.which are independent of one another and have identical.2 Validation of stability degreedistribution, tends to be nommal distribution. So, two casesSability degree is adopted to scale the stable level ofare simulated to prove whether these judgments are right or the stable assembly process with diferent process parame-not by adopting normal distribution.ters.Suppose the tolerance of the position of locating pinsTake the second case for instance, the process param-in the fixture and quality characteristics is士1 mmn. For the eter in this assembly process is the distance from loeatingfirst cae, 50-group data, ineluding 15 random mumbers hole to locaing slot, i.e. LL2,LsLs,LsLs, and L6L.from -1 to +1 in each group, are generated by MATLABFirst, as it is shown in Fig.3 and Fig.5, LLz=8s a sarmple to represent the position variation of all locating 650,LzL4=450, LsL =400, and L6Ln= 300, by cal-points in every station in Fig.3, which are substituted intoculation,Eq. (3) with matrix described in Eqs.(12)- -(18). As aresult, the position variation of measured points N;can beφ1=πzr,=0.0069obtained after calculating. Adding three numbers from - 1Then, enlarge the distance from locating hole to locat-to +1 in each goup data, the position variation of mea- ing slot, Ll =750, L3L4 =650, LsLj = 500,L6L=sured points N; for the second case can be obained in the 500, by calculation,8same way. Here, only the variation distribution charts of中国煤化工N2 in two cases are given in Fig.5 and Fig.6.Fig.5 and Fig. 6 show that under the same fxture.MHCN M H Gegve incrwses wih .fault, product quality characteristics in the first case exceed the enlangement of the distance from locating hole to locat-414WANG Lei et al: Quality Stability of Multi Station Assembly Process Based on Variation Streaming slot. Using the same way and fixture fault data in theProdutirity Research, 2004, 135(3): 129- -130 (in Chi-case of enlarging the distance from locating hole to locatingnese).slot, the variation distribution chart of N2 is created as[2]Luo Zhenbi, Wang Jinsong. Machining eror flow model inmanufacturing processes[J]. Chinese Jourmal of Mechani-shown in Fig.7.cal Engineering, 1994,30 (1): 112- -118 (in Chinese)..4-[3] Luo Zhenbi, Wang Jinsong. 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Modeling and controlingtion extension of product quality characteristics is reducedvariation in mechanical assemblies using state transitionwith the enlargement of the distance from locating hole tomodels[C]. In: IEEE International Conference on Roboticslocating slot, i.e. the process has better stability. The efi--and Automation. Luven, Belgium, 1998.ciency of stability degree is also proved.[8] Mantriparagada R, Whitney D E. Modeling and cntollingvariation propagation in mechanical assemblies using state4 Conclusionstansition models[J]. IEEE Transactions on Robotics andIn this paper, the quality stability issue of multi-sta-Automation, 1999, 15(1): 124- -140.tion asembly process has been investigated in order to ana- [9] Huang Qiang, Shi Jianjn. Stream of variation modeliglyze the physical structure of assembly process and assureand analysis of serial-parallel multistage manufacturing sy8-the product quality. The main conclusions are listed as fol-tems[J]. 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Annals of the CIRP, 1997, 46(1): 1- 6.reduce the variation of product quality characteristics.[13] Camelio J, Hu SJ, Ceglarek D. Modeling variation propa-A sheet metal assembly process including three sta-gation of multi-station assembly systems with compliantparts[J]. Joumal of Mechanical Design, 2003, 125(4):tions, is provided to vernify the efectiveness of the proposed673- -681.method. Whereas, there have been some assumptions and[14] Ding Yu, Jin Jionghua, Ceglarek D et al . Proces orientedsimplifications for fixture type and variation source of thetolerancing for multi-station assembly systems[J]. IIEassembly process, the further development of the research,Transations, 2005, 37(6): 493- -508.more relative process parameters and variation sources will[15] Yang Xiao. The Sability Analysis of Multi Dimensions Sys-be included, and the results in this paper can be applied totem [ M ]. 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