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Int J Adv Manuf Technol (2013) 64:6383DOI 10.1007/s00170-012-4016-4ORIGINAL ARTICLEManufacturing variation modelsin multi-station machining systemsJos Vicente Abelln-Nebot Fernando Romero Subirn Julio Serrano MiraReceived: 28 July 2011 / Accepted: 19 February 2012 / Published online: 25 March 2012 Springer-Verlag London Limited 2012Abstract In product design and quality improvementfields, the development of reliable 3D machining vari-ation models for multi-station machining processes isa key issue to estimate the resulting geometrical anddimensional quality of manufactured parts, generaterobust process plans, eliminate downstream manufac-turing problems, and reduce ramp-up times. In the lit-erature, two main 3D machining variation models havebeen studied: the stream of variation model, oriented toproduct quality improvement (fault diagnosis, processplanning evaluation and selection, etc.), and the modelof the manufactured part, oriented to product and man-ufacturing design activities (manufacturing and producttolerance analysis and synthesis). This paper reviewsthe fundamentals of each model and describes step bystep how to derive them using a simple case study. Thepaper analyzes both models and compares their maincharacteristics and applications. A discussion about thedrawbacks and limitations of each model and somepotential research lines in this field are also presented.Keywords Multi-station machining processes Part quality SoV MoMP Manufacturingvariation modelJ. V. Abelln-Nebot (B) F. Romero Subirn J. Serrano MiraDepartment of Industrial Systems Engineering and Design,Universitat Jaume I, Castelln de la Plana, 12071, Spaine-mail: abellanesid.uji.esF. Romero Subirne-mail: fromeroesid.uji.esJ. Serrano Mirae-mail: jserranoesid.uji.esNomenclatureCSCoordinate systemDDesign CSRReference CSFkFixture CS at station kHk,jCS of jth part-holder surface at station kMkMachine tool CS at station kMk,oCS of the oth machining operation atstation kMk,oiCS of the cutting tool tip when machiningsurface SiGGauge CSGpCS of the pth positioning gauge surfaceSlCS of the lth locating datum surfaceSiCS of the ith machined surfaceSmCS of the mth measurement datum surfaceSqCS of the qth inspected part surface or toler-anced surfacedRFSmall translational deviations of F with re-spect to (w.r.t.) R,?dRFx,dRFy,dRFz?TRFSmall orientational deviations of F w.r.t. R,?RFx,RFy,RFz?TxRFDifferential motion vector of F w.r.t. R,?(dRF)T,(RF)T?TTR,FGSmall displacement torsor (SDT) of F atR expressed in the G coordinate system,TR,FG=?R,FDR,F?R,FOrientation deviations of the SDT of F at R,?,?TDR,FTranslational deviations of the SDT of F atR,?u,v,w?TtRFPosition vector of F w.r.t. RRFOrientation vector of F w.r.t. R64Int J Adv Manuf Technol (2013) 64:63831 IntroductionTraditionally, product design has been separated frommanufacturing process design along the product de-velopment cycle increasing ramp-up times, productchanges costs, and variability of product quality. Thisproduct-oriented approach, named as over-the-walldesign due to the sequential nature of the design activ-ities, prevents the integration of design and manufac-turing activities to improve product development 1.In order to overcome this limitation, manufacturershave begun to investigate the ways to simultaneouslyevaluate product designs and manufacturing processesin an attempt to eliminate downstream manufacturingproblems and reduce ramp-up times. For this purpose,product design requires the application of process-oriented approaches through 3D manufacturing vari-ation models to integrate product and manufacturingprocess information 2. However, the application of3D manufacturing variation models is nowadays lim-ited, specially in multi-station machining processes(MMPs) where a large number of machining operationsare conducted in different stations with different fixturedevices. In fact, in product design and quality improve-ment fields, the development of reliable 3D variationmodels of MMPs is a key issue to estimate the resultinggeometrical and dimensional quality of manufacturedparts.In the literature, one can distinguish two importantgroup of researchers dealing with the development of3D variation models for MMPs: (a) a first group ofresearchers more focused on the quality improvementfield, mostly universities from USA, and (b) a sec-ond group of researchers focused on product design,mostly universities from France and Canada. The re-search conducted by the first group that we nameas Industrial Operations school (“IO school”) is fo-cused on modeling the dimensional and geometricalvariation of manufactured parts by the state-spacemodel which is commonly applied in control theory 3.Through this model, a large number of quality im-provement activities can be conducted such as partquality control, optimal placement of inspection sta-tions, robust process planning, process-oriented tol-erancing, etc. The research conducted by the secondgroup that we name as Product Design school (“PDschool”) is focused on modeling dimensional and geo-metrical part quality variations considering the systemworkpiece/fixture/machine tool as a mechanical assem-bly. By this consideration, well-known approaches forthe analysis of mechanical assemblies can be applied.Specifically, the PD school applies the concept of smalldisplacement torsors (SDTs) to describe and propagatethe surface deviations along the different machiningstations. The application of the PD school investiga-tions are mainly focused on tolerance analysis andsynthesis.Both schools have been very active during lastdecade, and their 3D manufacturing variation modelshave been used in a large number of applications. Sur-prisingly enough, these schools have been working inparallel, without using potential synergies or adaptingthe advances from one school to the other. Moreover,despite the success of both schools and the large num-berofpublicationsrelatedto them,thereis no reviewinthe literature that analyzes and compares both 3D man-ufacturing variation models. This paper contributes tofill this literature gap, providing a critical comparisonof both models, paying special attention to (a) fixturesand processes supported, (b) limitations of virtual in-spection, (c) GD&T conformance, (d) applicability,(e) modeling accuracy, and (f) modeling complexity.Furthermore, the review also exposes future lines ofresearch that should be addressed by both schools. Thepaper is organized as follows: Section 2 describes thefundamentals of the 3D manufacturing variation modelapplied by the IO school and its main applications.Similarly, Section 3 describes the 3D manufacturingvariation model applied by the PD school together withits main applications. Section 4 presents two modeling2D examples (extensible to any 3D case study) to show,step by step, the implementation of both models, ana-lyzingthemsymbolicallyandnumerically.Section5dis-cusses both 3D variation models, comparing their maindrawbacks and advantages. Finally, Section 6 concludesthe paper and outlines some potential research lines inthe field.2 Variation modeling and propagation by the IOschool: the stream of variation model2.1 FundamentalsManufacturing variability in MMPs and its impact onpart quality can be modeled by capturing the math-ematical relationship between the sources of varia-tion of manufacturing process variables that are crit-ical to part quality (named key control characteris-tics (KCCs) of the process) and the deviations of thefunctional features of the product (named key productcharacteristics (KPCs). This relationship is modeledthrough a non-linear function y1= f1(u1,u2,.,un),where y1is a KPC and u1,u2,.,unare the KCCs inthe MMP. By the assumption of small variations, thenon-linear model can be linearized through a TaylorInt J Adv Manuf Technol (2013) 64:638365Fig. 1 Manufacturing variation propagation in a MMPseries expansion, and the part quality variation can bedefined as?y1=f1(u)u1?u= u(u1 u1) + +f1(u)un?u= u(un un) + 1,(1)where u = u1,u2,.,unT, 1contains the high-ordernon-linear residuals of the linearization, and the lin-earizationpointisdefinedby u = u1, u2,., unT.Thislinear approximation can be considered good enoughfor many MMPs 4. Considering that there are MKPCs in the part which are stacked in the vector Y =?y1,., ?yMT, Eq. 1 can be re-written in a matrixform asY = ? U + ,(2)where U = ?u1,.,?unTand ?uj= uj ujfor j =1,.,n defines the small variations of the KCCs ina MMP; ? is the matrix?f1(u)u1?u= u,.,f1(u)un?u= u?;.;?fM(u)u1?u= u,.,fM(u)un?u= u?; and is the stackedvector of the high-order non-linear residuals.In MMPs, the derivation of Eq. 2 is a challeng-ing task. Researchers from USA universities haveproposed the adoption of the well-known state-spacemodel from control theory 3 to represent mathemat-ically the relationship between the sources of variationof a MMP and the deviation of the machined surfacesat each station, including how the deviation of previousmachined surfaces influences at current station whenthesesurfacesareusedaslocatingdatums.Inthisrepre-sentation, dimensional deviations of part surfaces fromnominal values are defined by 61 vectors named statevectors, in the form of xk,i= (dii)T,(ii)TT, wheredii= diix,diiy,diizTis the small translational deviationand ii= iix,iiy,iizTis the small orientation devia-tion of the local CS of the ith part surface. The notationi refers to the nominal CS and i to the current CS. Thedeviation of all part surfaces at the station k are stackedin the state vector xk= xTk,1,.,xTk,i,.T. Note thateach feature deviation, xk,i, is expressed in its own CS.In a machining station, three main sources of vari-ation can be distinguished: datum-induced deviations,fixture-induced deviations, and machining-induced de-viations. The state-space model defines analyticallyhow these three main sources of error influence onthe final part quality deviation. To illustrate how thesethree main sources of variation influence on part qual-ity, we consider an N-station machining process shownin Fig. 1 and the kth machining station with the work-piece and the fixture device shown in Fig. 2. At this kthstation, the sources of variation below exist.Fig. 2 Sources of variationand state-space modelformulation for station k66Int J Adv Manuf Technol (2013) 64:6383First, the deviations of the datum surfaces used forlocating the workpiece deviate the workpiece locationfrom its nominal value. This term can be estimated asxdk= Akxk1, where xk1is the vector of part surfacedeviations from upstream machining stations and Aklinearly relates the datum deviations with the machinedsurface deviation due to the locating deviation of theworkpiece.Secondly, the fixture-induced deviations deviate theworkpiece location on the machine tool table and pro-duce a machined surface deviation. This term can beestimated as xfk= Bfkufk, where ufkis the vector of lo-cator deviations and Bfkis a matrix that linearly relateslocator deviations with the machined surface deviation.Thirdly, the operation or machining deviations suchas those due to geometrical and kinematic errors, tool-wear errors, etc. deviate the cutting tool tip duringmachining, and thus, the machined surface is deviatedfrom its nominal value. This term is modeled as xmk=Bmkumk, where umkis the vector that defines the KCCsrelated to operation or machining deviations and Bmkis a matrix that linearly relates these KCCs with themachined surface deviations.Therefore, for an N-station machining process, thederivation of the state-space model can be defined in ageneric form asxk= Akxk1+ Bkuk+ wk,k = 1,., N,(3)where Bkukrepresents the deviations introducedwithin station k due to the KCCs (related to fixturingand machining) and it is defined as BfkBmk (ufk)T,(umk)TTand wkis the unmodeled system noise andlinearization errors.The derivation of the state-space model in MMPs,named the stream of variation (SoV) model, was firstlypresented by Huang et al. 5. Djurjanovic et al. 6expanded Huangs work in order to explicitly de-rive the linear equations that model the relationshipsbetween fixtures, locating datum, and measurementdatum features. In their research work, a complexmathematical derivation was required, and the method-ology proposed was not straightforward to be applied.The SoV model derivation was improved by 4 whoapplied the differential motion vector (DMV) conceptfrom robotics to represent the geometric deviation ofeach machined feature. In their work, a step-by-stepmethodology was proposed in order to derive the ma-trices Akand Bkat each station using product andprocess information (i.e., part geometry and fixture lay-outs). Previous works were limited to 321 orthogonalfixture layouts based on locators and generic cuttingtoolpathdeviationswithoutexplicitlyincludingspecificmachining-induced errors. Loose et al. 7 extendedthe state-space model formulation by including gen-eral non-orthogonal fixture layouts based on locators.More recently, Abellan-Nebot et al. 8 expanded theformulation of matrix Bkin order to include commonmachining sources of variation such as those due to toolwear, thermal expansions, cutting tool deflections, andgeometrickinematic machine tool errors.2.2 Virtual part quality inspection and verificationSomewhere along the MMP, an inspection station canbe placed in order to inspect the KPCs and verifywhether the workpiece/part is within specifications.Following the state-space model formulation from con-trol theory 3, a virtual inspection after the kth machin-ing station can be conducted using the expression:yk= Ckxk+ vk,(4)where ykrepresents the deviations of the inspectedKPCs, Ckxkare the deviations of the KPCs thatare defined as a linear combination of the devia-tions of workpiece features at the kth station, and vkis the measurement noise of the inspection process.In a similar way to xk, vector ykis defined asyTk,1,.,yTk,q,.,yTk,MT, where yk,qis the inspecteddeviation of the qth KPC (denoted as Sq) defined bythe vector yk,q= (dSmSq)T,(SmSq)TT, where Smis themeasurement datum surface and M is the number ofKPCs inspected. In Eq. 4, matrix Ckdepends on whatKPCs they are and which measurement datums areused to locate the part in the inspection station. Howto derive matrix Ckis explained in detail in 4.In order to express the part quality measurements byan explicit linear function of the KCCs presented alongthe MMP and considering that the inspection station isplaced at the end of the MMP (after machining stationN), Eqs. 3 and 4 can be combined and rewritten in theinputoutput form as:Y = ? U + ,(5)where the vectors Y and U are the stacking quality vec-tors after inspection and the vectors of sources of error,respectively, from the stations k = 1,2,., N. In Eq. 5,vector Y is defined as Y = yTN,1,yTN,2,.,yTN,MT, vec-tor U is defined as U = uT1,uT2,.,uTNT, and matrices? and are defined as:? = MN,1,.,MN,N,(6) = MN,1,.,MN,N w1,.,wNT+ vN,(7)Int J Adv Manuf Technol (2013) 64:638367whereMN,j= CN?N,jBj,j N,MN,j= CN?N,j,j N,(8)?N,j=?AN1AN2Aj, if j 0,(39)2H4,S2=?2H3 1M5+ 1H1 2H4? 0.(40)If Eqs. 39 and 40 hold, the deviation of the jointworkpiece/fixture at the secondary locating datum in Xdirection (expressed in the fixture CS) at each station isdefined as:v1H2,S4= pF1B?1H1 1H2?,(41)v2H4,S2= pF2I?2H3 1M5+ 1H1 2H4?.(42)For these workpiece/fixture configurations, the re-sulting final part variability is defined as follows:KPC1: The deviation of KPC1is defined asKPC1= max?TTZ1,P6A?y?,?TTZ1,P6B?y?,(43)Int J Adv Manuf Technol (2013) 64:638379where:?TTZ1,P6A?y= v1H1+ v1M5+ v2H3 v2M6 52M6+ 27.5 ?1M5 2H3 1H1?,(44)?TTZ1,P6B?y= v1H1+ v1M5+ v2H3 v2M6+ 52M6+ 37.5 ?1M5 2H3 1H1?,(45)Note that deviations of part-holder surfaces H2andH4do not influence on this KPC.KPC2: The deviation of KPC2is defined asKPC2= max?TTZ2,P7A?y?,?TTZ2,P7B?y?,(46)where?TTZ2,P7A?y= v2H4 v2M7 102H4 2.52M7 25 ?1M5 2H3 1H1?,(47)?TTZ2,P7B?y= v2H4 v2M7 102H4+ 2.52M7 20 ?1M5 2H3 1H1?.(48)In this case, note that deviations of the part-holdersurface H2do not influence on this KPC. Further-more, the machining deviations of surface S6do noteither influence.KPC3: the deviation of KPC3is defined asKPC3=?TTZ1,P6A?y?TTZ1,P6B?y?,(49)where?TTZ1,P6A?yand?TTZ1,P6B?yare defined byEqs. 44 and 45, respectively. By substituting, thedeviation is defined as |101M5+ 102M6 102H3101H1|. Similar to the example shown previously,as this KPC is a parallelism relationship, trans-lational machining deviations at any station donot influence. Furthermore, deviations of part-holder surfaces H1and H3in X or Y direction donot influence, and only their orientation deviationinfluences on the parallelism relationship.NotethatifEqs.39and40donotapply,itmeansthatthe workpiecefixture assembly is different than theconfigurations analyzed and the resulting positioningdeviation torsor varies. To analyze those cases, Eqs. 41and 42 should be re-evaluated considering the points Aand E as contact points between workpiece and part-holder at station 1 and 2, respectively.4.2.2 Numerical resolutionSimilar to the SoV case study, the case study ofthe MoMP is numerically solved analyzing the worstcase and the statistical approach. The variability rangefor each manufacturing process variable is shown inTable 9. Unlike the SoV case study, where the locatorsvariability is independent to each other, in the MoMPthe variability of a surface depends on different torsorparameters and those parameters are not independentto each other since they are subjected to the geometri-cal/dimensional tolerance of the surface. For this casestudy, part-holder surfaces are planar and a positiontolerance zone applies for each one.Table 9 Precision of part-holder surfaces and machined surfaces for the MoMP case studyStation 1Precision of part-holder surface 1:t = 0.1 mmPart-holder torsor constraints:0.12 v1H1+ 37.5 1H10.120.12 v1H10.120.175 1H10.175Precision of part-holder surface 2:t = 0.02 mmPart-holder torsor constraints:0.022 v1H2+ 10 1H20.0220.022 v1H20.0220.0220 1H20.0220Machining precision:0.01 u1M5 0.010.01 v1M5 0.010.001 1M5 0.001Station 2Precision of part-holder surface 3:t = 0.1 mmPart-holder torsor constraints:0.12 v2H3+ 37.5 2H30.120.12 v2H30.120.175 2H30.175Precision of part-holder surface 4:t = 0.02 mmPart-holder torsor constraints:0.022 v2H4+ 10 2H40.0220.022 v2H40.0220.0220 2H40.0220Machining precision:0.01 u2M6 0.010.01 v2M6 0.010.001 2M6 0.001Machining precision:0.01 u2M7 0.010.01 v2M7 0.010.001 2M7 0.001Dimensional deviations in millimeters and angular deviations in radian80Int J Adv Manuf Technol (2013) 64:6383Table 10 Numerical resolution according to the WC and the STanalysis for the MoMP case studyKPC1KPC2KPC3WC (Pro/E)0.1630.1140.047WC0.1620.1140.047ST (6 interval)0.0910.0370.018Dimensions in millimetersWC worst case, ST statisticalThe numerical resolution for worst-case analysis re-quiresasearchalgorithm.Inthiscasestudy,itisappliedtwo algorithms sequentially. Firstly, a genetic algorithm(GA) in order to find a region close to the optimal solu-tion. Secondly, the solution provided by the GA is usedas the initial point in a mesh adaptative direct searchalgorithm for tuning the optimal result. The resolutionwas repeated ten times to ensure the optimization con-vergence, so the global minimum reached provides theworst-case solution. The numerical resolution for thestatistical analysis requires the evaluation of thousandsof samples through Monte Carlo simulations. For thiscase study, 5,000 Monte Carlo simulations were runassuming that all torsor parameters are normally dis-tributed with the 6 ranges shown in Table 9 subjectedto their tolerance constraints. In order to validate themodel,thesameworst-caseanalysisusingPro/EngineerWildfire 5.0 as explained in Section 4.1.2 was con-ducted. Table 10 shows the KPC variations accordingto the type of analysis conducted.5 Comparison and discussion5.1 Fixture and processes supportedThe SoV model is based on a linear system of equationsdescribed in a matricial form in a similar way as thewell-know state-space model from control theory. Forthis reason, the SoV model has been used for modelingisostatic fixtures based on punctual locators accordingto the 321 layout. Hyperstatic fixtures based on lo-cators or other workholding devices such as vises orfour-jaw chucks have not been considered. The MoMPovercomes this limitation since the formulation letsconsidernon-punctualcontactsformodelingthefixtureassembly. Therefore, hyperstatic fixtures with a definedhierarchy of contacts and industrial fixtures such asvises or chucks can be derived.In terms of machining systems, the SoV model dealsonly with milling operations since in this process, alldegreesoffreedomareconstrained,andtheapplicationof the SoV model in its matricial form for diagnosis andprocess improvement is straightforward. The MoMPhasbeenderivedforbothmillingandturningprocesses,making it more applicable in real MMPs.5.2 Limitations of virtual inspectionAt the virtual inspection station, the SoV model con-ducts a point-based inspection similar to that conductedby a coordinate measuring machine (CMM) includingin the model the measurement error in the inspectionprocess. The virtual inspection and verification canbe conducted straightforward for the worst-case andstatistical analysis by evaluating one single expression(without conducting a large number of simulations orusing an optimization algorithm). In the case of theMoMP, the deviation of the inspected surfaces is an-alyzed through a virtual gauge, so the inspection is asurface-based process where the precision of the gaugesystem can be considered. According to the toleranceaccumulation, the MoMP can conduct both worst-caseand statistical analysis. However, the worst-case analy-sis requires the resolution of a complex optimizationproblem where the global solution (and thus, the realworst-case solution) may not be reached. On the otherhand, the statistical analysis requires time-consumingMonte Carlo simulations to infer, from thousands ofsimulated samples, the statistical results of the virtualinspection. Thus, the resolution of both worst-case andstatistical analysis are quite complex in comparisonwith the same analysis conducted under the SoV model.This is because the SoV model includes the individualsource of errors in the manufacturing process (e.g., in-dividual locator errors in isostatic 3-2-1 fixture devices)which are independent to each other, whereas in theMoMP models, the sources of error are described bythe parameters of the SDTs of surfaces which may benot independent. In other words, the SoV model isapplied only in simple isostatic fixture configurationsbased on locators that greatly facilitates both worst-case and statistical analysis. For more complex fixturesbased on surfaces, the SoV cannot be applied and itsresolution should be conducted applying the MoMPsolving a complex optimization problem for both worst-case and statistical cases.5.3 GD&T conformanceThe SoV model virtually inspects the parts as theywere inspected in a CMM. Therefore, the SoV modelis oriented to vectorial dimensioning and tolerancing(VD&T) specifications, and common product designspecifications based on GD&T should be translatedin order to be applied correctly. The mathematicalInt J Adv Manuf Technol (2013) 64:638381translation of some GD&T specifications such as par-allelism, angularity, and perpendicularity to be used bytheSoVwasinvestigatedin53.Kongetal.54alsoin-cluded into the SoV model other GD&T specificationssuch as material condition modifiers.Unlike SoV model, the MoMP describes surfacedeviations using SDTs and uses gauges for inspec-tion. These two characteristics make the MoMP morefriendlytodealwithGD&TspecificationsthantheSoVmodel. For instance, the MoMP has been successfullyappliedformanufacturingtolerancingaccordingtoISOstandards and GD&T specifications 49. The MoMPhas been also applied to deal with material conditionmodifiers and incomplete datum frames 39, 42.5.4 ApplicabilityThe state-space model formulation in the SoV modelmakes it to be preferred to the MoMP model whenthe applicability of the model is mainly focused onpart quality improvement activities such as fault diag-nosis,processdiagnosabilityanalysis,dimensionalqual-ity control, and fixture maintenance planning. How-ever, this applicability can be only conducted in millingprocesses with isostatic fixtures. In this field, the for-mulation of the MoMP makes it more difficult to beapplied since there is no single matrix equation to bestraightforward analyzed as there is in the SoV model.Unlike the SoV model, the MoMP is oriented toproduct design activities. This is mainly because of theabilityofSDTstomodelsurfacetolerancesandbecauseof the contact between surfaces is not only based onpunctual contacts but also on surface contacts.5.5 Model accuracyConsidering the modeling mechanism, both SoV andMoMP models are based on kinematic chains andhomogeneous transformation matrices. Both assumesolid rigid parts and they cannot deal with form errors(intrinsic part tolerances), so they are focused on posi-tion and orientation errors (extrinsic part tolerances).In manufacturing context, angular deviations due tomanufacturing error remain very small, so the gen-eral matrix approach can be simplified using SDT orDMV 55. Simulations conducted with Pro/Engineerare shown in Tables 6 and 10, and they reveal that, forthe worst-case analysis in the case study, the estimationerror of KPCs is below 1 m for both approaches.It should be noted that both models require acomprehensive information of the product and themanufacturing process such as geometry of nomi-nal part surfaces, manufacturing and fixture capabili-ties, geometrical information of fixture configurations,placement of the inspection stations, capability of theinspection systems, etc. The large quantity of a prioriengineering knowledge required and its level of accu-racy has an important impact on the final accuracy ofthe model, which might prevent their application inlarge-scale MMPs 56. However, for low and medium-scale systems, the application of SoV and MoMP mod-els is highly recommended to eliminate downstreammanufacturing problems and reduce ramp-up times.5.6 Modeling complexityThe derivation of the SoV model is, in authors opinion,more oriented to MMPs than the MoMP due to theadoption of the state-space model of control theory,and it is more easily to be automated due to its matrixformulation. The multi-station representation by theSoV model is clearer than that from the MoMP model,and the inclusion of the inspection stations is alsostraightforward. Furthermore, the resolution of worst-caseandstatisticalanalysisisalsoeasierandfastersinceit requires only the evaluation of a system of equa-tions in matrix form. However, it should be noted thatthere are more limitations applied in the SoV model,so complex MMPs based on turningmilling processesand non-punctual isostatic fixtures should require theapplication of the MoMP.6 Conclusion and future workThe development of 3D manufacturing variation mod-els for integrating product and manufacturing processdesign is essential to improve product development,eliminate downstream manufacturing problems, andreduce ramp-up times. In the literature, two main 3Dmanufacturing variation models in MMPs have beenstudied: the SoV model and the MoMP. This paperhas presented both SoV and MoMP models, analyzingand comparing their main characteristics and applica-tions. Furthermore, the methodology to derive bothmodels has been described step by step by a simple2D case study (extensible for any 3D part). As futurework, some potential improvements on manufacturingvariation propagation models in MMPs should be con-ducted. As some ideas, the authors suggest:Both SoV and MoMP models are restricted to solidrigid components. Industrial machining operationsalso require the analysis of manufacturing variabil-ity on complaint parts due to deformations inducedby clamp forces or cutting tool forces.82Int J Adv Manuf Technol (2013) 64:6383Both SoV and MoMP models do not include formerrors. Although form errors are considered to benot as critical as dimension or orientation errorsfor 3D manufacturing variation, research effortsshould be conducted to include in somehow thesegeometrical variations.SoV model deals with fixtures composed of loca-tors,buttheirextensiontocovercommonindustrialfixturessuchasvisesorchucksshouldbeaddressed.SoV model deals with non-orthogonal 321 fixturelayouts, but common hyperstatic fixture schemesin industry such as N21 layouts are still unad-dressed.The derivation of the MoMP includes generic ma-chine tools capabilities by defining different torsorcomponents. However, machine tool capabilitiescan be expressed by their sources of variation them-selves such as cutting tool deflections, tooling wear,etc. The inclusion of additional chain of torsorsto explicitly indicate the effect of these machiningvariations could be interesting for future applica-tions in manufacturing tolerancing.The application of the MoMP for process toleranceallocation is still unaddressed. This process toler-ance allocation should be analyzed by integratingthe different cost functions of the manufacturingprocess such as fixture maintenance operations orcutting tool replacement policies among others.AcknowledgementsThis work has been partially supported byFundaci Caixa-Castell Bancaixa (Research Promotion 2007,2009, and 2011).References1. Ceglarek D, Huang W, Zhou S, Ding Y, Kumar R, Zhou Y(2004) Time-based competition in multistage manufacturing:stream-of-variation analysis (SOVA) methodologyreview.Int J Flex Manuf Syst 16:11442. 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