Stream-of-Variation (SOVA) Modeling—Part II: A Generic 3D Variation Model for Rigid Body Assembly...

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Wenzhen HuangDepartment of Mechanical Engineering,

University of Massachusetts,Dartmouth, MA 02747

e-mail: [email protected]

Jijun LinEngineering Systems Division,

School of Engineering,Massachusetts Institute of Technology,

Cambridge, MA 02139-4307e-mail: [email protected]

Zhenyu KongSchool of Industrial Engineering and

Management,Oklahoma State University,

Stillwater, OK 74078e-mail: [email protected]

Dariusz CeglarekDept. of Industrial and Systems Engineering,

University of Wisconsin-Madison,Madison, WI 53706

e-mail: [email protected]

Stream-of-Variation „SOVA…Modeling—Part II: A Generic 3DVariation Model for Rigid BodyAssembly in MultistationAssembly ProcessesA 3D rigid assembly modeling technique is developed for stream of variation analysis(SOVA) in multi-station processes. An assembly process is modeled as a spatial indexedstate transition dynamic system. The model takes into account product and process fac-tors such as: part-to-fixture, part-to-part, and inter-station interactions, which representthe influences coming from both tooling errors and part errors. The incorporation of thevirtual fixture concept (Huang et al., Proc. of 2006 ASME MSEC) and inter-stationinteraction leads to the generic, unified SOVA model formulation. An automatic modelgeneration technique is also developed for surmounting difficulties in modeling based onfirst principles. It enhances the applicability in modeling complex assemblies. The devel-oped SOVA methodology outperforms the current simulation based techniques in compu-tation efficiency, not only in forward analysis of complex assembly systems (toleranceanalysis, sensitivity analysis), but it is also more powerful in backward analysis (toler-ance synthesis and dimensional fault diagnosis). The model is validated using industrialcase studies and series of simulations conducted using standardized industrial software(3DCS Analyst). �DOI: 10.1115/1.2738953�

Introduction

1.1 Problem Description. Multi-station assembly processesenerally refer to the processes involving multiple work-stationso manufacture a complex assembly. For example, an automotiveody assembly process, consisting of up to 70 stations and severalundreds of sheet panels, is a typical multi-station assembly pro-ess to fabricate the body-in-white of an automobile.

Dimensional quality, represented by product dimension vari-bility, is one of the most critical challenges in industries such asutomotive, aerospace, shipbuilding, appliance, etc. The scale andomplexity of the multistage manufacturing systems �MMS� inhese industries greatly increase the difficulties in modeling,nalysis, and control of dimensional variation in both design andanufacturing phases. Lack of modeling tools has pushed the

roblems downstream �e.g., ramp-up and launch�, which can incurhuge amount of cost. For example, two-thirds of engineering

hanges in automotive and aerospace industries are caused byimensional faults �1�.

The overall importance of dimensional variation in manufactur-ng processes is also reflected by extensive research focused onimensional variation modeling, faults diagnosis, and toleranceesign �2–15,28�. Statistical analysis of variation propagation inulti-station assembly processes is important to quality and pro-

uctivity improvement. The complexity of a manufacturing pro-ess puts high demands on process modeling, design optimization,nd fault diagnosis to ensure the dimensional integrity ofroducts.

There are two main error sources in an assembly system: fixtureocating error and part error. In general, the variation propagationn rigid body assemblies is driven by three fundamental mecha-

Contributed by the Manufacturing Engineering Division of ASME for publicationn the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedugust 14, 2006; final manuscript received March 27, 2007. Review conducted by
hivakumar Raman.
32 / Vol. 129, AUGUST 2007 Copyright © 2

ded 15 Aug 2007 to 137.205.181.120. Redistribution subject to ASM

nisms: �1� part-to-part interaction, �2� part-to-fixture interaction,and �3� inter-station interactions, which is in fact a specific type of�2� between stations. In Type I assembly, there is only part-to-partinteraction, whereas for Type II assembly, both �1� and �2� inter-actions exist. They have been identified by industry as the twomost frequent causes of engineering changes �16�. Inter-stationinteraction has been identified as one of the contributors in multi-station assembly processes �17�.

Efforts have been made in SOVA modeling in multi-stationassembly processes. A state space model was initially introducedfor variation propagation in �18,19� and was recently developed in�4,6,28� for 2D assembly systems. More recently, an effort hasbeen made to take into account both tooling and part errors in the3D single station SOVA model �14�. Therefore, further efforts aredesirable to provide a complete 3D SOVA model for multi-stationassembly systems.

1.2 Related Work. The assembly modeling and variationcontrol of multi-station manufacturing processes have become anemerging area on the boundaries of engineering and statistics re-search and have grown rapidly since its inception in the early1990s.

Earlier research primarily preoccupied with statistical descrip-tions of variation patterns from in-line measurement data �2� orrule-based fault isolation method �20�. These diagnostic methodsare primarily heuristic knowledge based. Engineering model-based diagnostics of fixture fault in the manufacturing processwas initiated in �1� on the assumptions of single fault, single fix-ture, and rigid part. From then on, subsequent efforts have beenmade on more general assumptions, such as multiple faults, com-pliant components, and multiple fixtures �3,8,17,21,22,27�.

Data driven techniques were developed in parallel for diagno-sis. Lawless and co-workers �23–25� used an autoregressive �AR�model to investigate dimensional variation in assembly processesas well as in machining processes. They identify interrelations
between their statistical model and the physical process by esti-
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ating the parameters of the AR model from real process mea-urements. The amount of variation attributable to different stagesf a process is then studied to identify opportunities for variationeduction. Huang and Ceglarek �9,10� developed statistical modalnalysis techniques for form error pattern identification and diag-osis in stamped parts.

The aforementioned research activities have significantly ad-anced process monitoring and diagnosis for dimensional controlf assembly processes. However, lack of models describing theverall assembly process has imposed a significant constraint onhe capability of variation prediction and root cause diagnosis inesign and manufacturing.

As such, efforts have already been initiated in the last decade toharacterize variation propagation in multi-station assembly pro-esses. Jin and Shi �18� introduced a state space model �SSM� forariation modeling; a spatial indexed model and observation equa-ion were established. In a similar vein, �19� employed a stateransition model to describe the propagation of variation in assem-ly processes. The quality state was defined as kinematical devia-ions of parts in an assembly, which were produced from variationactors at each station and transmitted through the system. Theseodels took advantage of the well-established techniques in mod-

rn control theory for design analysis, monitoring, and diagnosis.ing et al. �4� developed a 2D SSM for modeling multistage sheetetal assembly processes. The associated assumptions include:

1� rigid part, �2� 3-2-1 fixture locating scheme, �3� in-plane de-iation �2D�, �4� fixture locator errors, and �5� lap joint. Thisodel adopted the concepts and methods from control theory and

sed rigid 2D kinematics to establish linear models with explicitesign parameters. The linear explicit SSM model has the advan-ages of highly efficient computation, model transparency �clearhysical explanation�, and incorporation of sensing informationor diagnosis. These developments provided a unified frameworkor both forward analysis �e.g., sensitivity and tolerance analysis�nd backward analysis �e.g., synthesis and diagnosis�. A compre-ensive review of 2D SSM modeling techniques was given in �6�nd summarized in Table 1.

In spite of these developments, the numerical simulation basedethod for variation analysis of rigid body assembly has already

een developed and used in commercialized variation analysisoftware such as 3DCS. The focus is on forward design simulationnd tolerance analysis. Numerical algorithms and Monte CarloMC� simulation are the major tools for the calculation. Majoroncerns of the MC based techniques are the tedious computationnd tremendous processing time required, in particular, in com-lex high-dimensional design synthesis and diagnosis problems.fforts have also been made for improvement. �11,26� introducedumber-theoretical net quasi-MC which can improve the compu-ation efficiency of MC based techniques in tolerance designroblems.

To improve the model generality and applicability of currentSM, it is desirable to develop models on more general assump-

ions. More recently, Huang and co-workers �10,14� developed aingle station SOVA model for 3D rigid part assembly. Fixture/arts errors and different types of joints are taken into account inhe model. A concept of generalized virtual fixture was created toormulate variation caused by part mating errors and fixture errorsn a unified framework. This work provided an infrastructure forurther expansion of single station SOVA model to a generic 3D

Table 1 Related work on state spa

Mantripragada andWhitney �19�

Fixture error NoPart manufacturing error Yes

2D/3D N/A

OVA model in multi-station systems.

ournal of Manufacturing Science and Engineering


1.3 Objective and Organization. This paper aims at devel-oping a variation propagation model for 3D rigid part assembly.

This 3D SOVA model provides a linear analytical tool for sys-tem evaluation and synthesis, thus going beyond the current 2DSOVA model and numerical simulation techniques. To achievemodel generality and applicability, the model should be able to:�1� include part and fixture error sources, �2� take into accountdifferent types of joints, �3� formulate 3D variation propagation inmulti-stations, and �4� be implemented easily. Since the model isdirectly derived from first principles, it tends to be too compli-cated and thus impractical for complex assemblies in application.Another focus of this paper is, therefore, on an efficient modelgeneration method.

This paper is organized into six sections. Section 2 starts withassumptions used in the paper, and then identifies and formulatesvariation factors in a multi-station assembly process. In Sec. 3.1,we initially introduce three theorems for modeling error accumu-lation, whereby the 3D SOVA model is developed in Secs.3.2–3.4. An automatic SOVA model generation method is devel-oped in Sec. 3.5. In Sec. 4, a simulation example, as well as afloor-pan and a truck cab assembly as examples from the automo-tive industry are presented for model validation. The conclusion isgiven in Sec. 5.

2 Assumptions and Variation FactorsThe underlying assumptions are initially introduced; the varia-

tion factors within a station and between stations are then formu-lated. Three main contributors of product dimensional variationare formulated in this section.

2.1 Assumptions. The assumptions are introduced below:

• Rigid assembly, i.e., all parts are taken as a rigid body in theassembling process; therefore, part deviation can be repre-sented by any specified reference point �RP�;

• Rigid mating interface, i.e., the mating plane error is mod-eled as a plane undergoing a rigid body movement from itsnominal position;

• Fixture errors and part errors;• 3-2-1 fixture locating scheme, the generalized fixture con-

cept in �14� is used when a part or a subassembly is posi-tioned by both fixture and part mating interface;

• Four typical joint types: lap joint, butt joint, mixed joint, andT joint;

• Small errors �compared with part and assemblydimensions�.

The first four assumptions have been widely accepted in theliterature and in the industrial variation control practices�1–4,18–20�. The rigid assumption is valid for the open assembly,which does not produce a structural closure with lock-in stresses�12�. A complete assembly model involves both rigid and compli-ant components �i.e., hybrid model� and requires more compre-hensive assumptions such as the n-2-1 locating scheme and partcompliancy, which is beyond the scope of this paper. Althoughpart mating features are confined to the typical joint types in sheetmetal panel assembly, the more general types of joints in me-chanical assemblies can actually be modeled by using the tech-

modeling of multi-station assembly

Jin and Shi�18�

Ding, Ceglarek,and Shi �4�

Proposed3D SOVA

Yes Yes YesNo No Yes2D 2D 2D and 3D

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niques introduced in �10,14�. The small error assumption is intro-

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uced for model linearization. The fixture and part dimensionalrrors are usually two to three orders smaller than the partimensions.

2.2 Station-Level Variation Factors. At station level therere two major variation sources: fixture imperfection and partanufacturing imperfection. Fixture imperfection leads to part po-

itioning error or deviation in an assembly. Manufacturing imper-ection refers to dimensional or geometrical errors, generated onart mating features in part fabricating processes. To model thetream of variation in a single station, a method has been devel-ped in �14� that integrated these error sources in a unified formatith a generalized (virtual) fixture concept. The deviation of aart in an assembly is represented in the global coordinate systemGCS�, whereas fixture and part errors are represented in the localoordinate system �LCS�. In the modeling process, the effects ofxture errors and part errors are formulated and transformationsf parts deviations between GCS and LCS are introduced.

2.2.1 Fixture Error. We consider a generic fixture locatingcheme in Fig. 1 that represents a 3-2-1 locating scheme in a GCS.n this section, we will provide the relationship between the fix-ure error in LCS and RP deviation in the GCS. The variation flowhrough part-part interfaces and fixtures can be unitedly formu-

Fig. 1 Generic 3-2-1 fixture layout in GCS

Therefore, using Eqs. �6�–�9� we have

34 / Vol. 129, AUGUST 2007


lated with a “generalized fixture” or “virtual fixture” as shown in�14�; therefore, the fixture model provides a fundamental basis inthe multi-station SOVA modeling.

Denote the errors on locators P1– P5 in the LCS as

�f = ��x1,�y1,�x2,�y2,�z3,�z4,�z5�� 1�

The xyz LCS centers at the four-way pin P1. Let us define aprimary plane orthogonal to the four-way pin P1’s orientation asdirection z. The normal vector of this primary plane is denoted asZ�nx ,ny ,nz�, which is expressed in XYZ GCS. Three in-plane de-grees of freedom �displacements in x and y directions and rotationaround the z axis� are fully controlled by P1 and P2. The otherthree degrees of freedom �translation in z direction, rotation aboutx, and y axes� are controlled by locators P3– P5.

Given the normal vector of primary plane: z�nx ,ny ,nz� in XYZGCS. Local x and y axes in the primary plane can be determinedby

z = nxi + nyj + nzk �2�

x = Z � z = − nyi − nxj �3�

y = z � x = − nxnzi − nynzj + �ny2 − nx

2�k �4�

For any LCS with direction cosines of x, y, and z as lt, mt, andnt, t� �x ,y ,z� in a GCS, the coordinate transformation matrix � isgiven in �10,14�. Using the results in Eqs. �2�–�4�

� = �lx mx nx

ly my ny

lz mz nz � = � − ny − nx 0

− nxnz − nynz �ny2 − nx

2�nx ny nx

� �5�

Denote a deviation on a part by �Pr on any specified RP, wehave �14�

�Prglobal = � 03�3

03�3 �−1

�Prlocal = �−1�Pr

local �6�

�P caused by fixture error in LCS can be derived in �10�
r
�Prlocal = ��Pp

�Po� = ��x,�y,�z,��,��,��r = Fs�f �7�

=�1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 0 0 D35 D36 D37

0 0 0 0 D45 D46 D47

0 0 0 0 D55 D56 D57

sin �

L−

cos �

L−

sin �

L

cos �

L0 0 0

� �x1

�y1

�x2

�y2

�z3

�z4

�z5

�local

�8�

here, L is the length of projection vector P1P2 in primary plane.is the angle between x axis and P1P2 in primary plane. Dij can

e derived with P1– P5 coordinates �10�. In the Fs matrix there iso element associated with z components of P1– P5. Thus, we canrst project P1– P5 onto the primary plane, and then construct thes matrix based on x and y coordinates of P1– P5:

Fs = � 03�3

03�3 �−1

Fs �9�

�Prglobal = Fs�f local �10�

Obviously Fs, and � are fixture specific. If a part is fully con-strained by a fixture, Eq. �10� provides the relationship of thefixture error and part deviation. It can be extended to the partbeing constrained by a fixture and mating features �10,14�.

2.2.2 Part Manufacturing Imperfection. If part k−1 is as-sembled with the deviation �Pr�k−1� in GCS. When part k is join-ing in through part k−1, the deviation �Prk will be caused by
upstream deviation �Pr�k-1�, mating feature errors between k-1
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nd k, and fixture errors on part k.In �14�, the mating features, which partially constrain a part in

n assembly, have been formulated in a generalized fixture. Theocally defined mating feature errors have a similar effect on partsositioning, as presented in Sec. 2.2.1. To accommodate arbitraryosition and orientation of mating feature planes with respect toCS, we define a LCS on each mating feature plane. The trans-

ormations for �Pr�k−1� and �Prk between GCS and LCS are firstntroduced �Fig. 2�. The superscript “m” on part deviations repre-ents it is in mating feature LCS. Then the error stack up from part−1 to part k is formulated.There are three sequential steps for coordinate transformations

nd error stack formulation as illustrated below:

�1� �Pr�k−1� in GCS → �Pr�k−1�m in LCS

�2� �Pr�k−1�m in LCS → �Prk

m in LCS

�3� �Prkm in LCS → �Prk in GCS

Steps 1 and 3 are GCS and LCS coordinates transformations.ased on Eq. �7�, the deviation of RP expressed in two coordinate

ystems is

�Prm = ��Pr

global �11�

Applying Eq. �7� to part k−1 and part k, respectively, it yields

�Pr�k−1�m = �k−1�Pr�k−1� �Prk = �k−1

−1 �Prkm �12�

Step 2 formulates the error stack-up process. According to theheory in �14�, the deviation of a RP on part k is determined by:

�1� The deviation of the RP on part k−1, i.e., �Pr�k−1�,�2� Local mating transformation: ��k−1�,�3� The errors of mating feature plane k−1 �between part k and

part k−1�, i.e., ��b�k−1� and ��ak,�4� Fixture error on part k, i.e., �fk.

Considering all these four factors, the error propagation throughating feature can be expressed as

�Prk = Jk−1��Pr�k−1�� ,��b�k−1�� ,��bk� ,�fk��T �13�

here

Ji = �i−1Ji�i 0

0 I

25�25

�14�

Ji = Bi The ith joint is a butt joint

Li The ith joint is a lap joint

Mi The ith joint is a mixed joint

Ti The ith joint is a T joint� �15�

here Bi, Li, Mi, and Ti are joint-specific matrices defined in10,14�.

2.3 Reorientation-Induced Variation. Shiu et al. �17� re-

Fig. 2 Coordinate transformation between mating planes

ealed the reorientation phenomenon in the multi-station process.



The reorientation is defined as repositioning due to transferringsubassemblies between stations that consist of different dataschemes. For example, after part 1 and part 2 are joined togetherat station i, the subassembly is transferred to station i+1 �Fig. 3�.If only pin-holes on part 2 are used to position the subassembly“1+2” at station i+1, part 1 deviates from its nominal positioneven if fixtures are perfect at station i+1. This station-station tran-sition induced deviation is defined as the reorientation error. Thedashed line represents the nominal positions of parts, while thesolid line indicates the actual positions.

We consider the deviation accumulation of part J in the assem-bly process at the ith station. In a similar vein as in 2D assemblysystems �4,18,28�, the state vector �deviation� XJ�i� of an assem-bly at station i comes from XJ�i−1�, fixture and mating featureerror-induced deviation E1

J, and reorientation-induced deviationE2

J; that is,

XJ�i� = XJ�i − 1� + E1J�i� + E2

J�i� �16�

Equation �16� is illustrated in Fig. 4. Two deviation terms, E1J

and E2J in Eq. �16�, are given by the following Lemmas.

Case 1. If part J is first merged into the assembly stream as asingle part in station i, we can use the single station assemblymodel to calculate E1

J�i�. The reorientation error E2J�i� is zero.

Case 2. Suppose part J is already in subassembly s�i−1� and itis joined into the main assembly in station i along with a rootsubassembly, which is fully constrained by a fixture. The accumu-lated deviation can thus be expressed as

�PrJ�i� = �Pr

J�i − 1� + E1I �i� + E2

J�i� �17�

where E1J is caused by fixture error of Ps1�i�– Ps5�i�; E2

J is due tosubassembly error of Ps1�i��i−1�– Ps5�i��i−1�. All terms in Eq.�17� are expressed in GCS. E1

J and E2J are given by the following

Lemmas 1 and 2.Lemma 1. Suppose part J transits with subassembly s from

Fig. 3 Deviation from reorientation

Fig. 4 Deviation accumulation on part J at station i

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tation i−1 to i, the fixture error-induced deviation E1J�i� is

E1J�i� = Q�Ps1�i�,Pr

J�E1s�i� �18�

here E1s�i� is the RP deviation on subassembly s�i�. Let ushoose Ps1 as RP. From Eq. (10), it yields

E1s�i� = Fs�i��fs�i� �19�

nd

E1s�i� = ��xrs�i�,�yrs�i�,�zrs�i�,��rs�i�,��rs�i�,��rs�i�� 20�

�fs�i� = ��xPs1,�yPs1,�xPs2,�yPs2,�zPs3,�zPs4,�zPs5�� 21�

fs�i� is the fixture error expressed in the local coordinate system

f subassembly s�i�, and Fs�i� is defined in Eq. (9). The transfor-ation between the deviations of any two points in the same part

an be expressed as [10,14]


J�E1s�i� �22�

Q�Ps1�i�,PrJ� = I3�3 Q�Ps1�i�,Pr

J�0 I3�3

�23�

Q�Ps1�i�,PrJ� = �0 z − y

− z− 0− x

y − x 0� �24�

In the matrix Q�Ps1�i� , PrJ�, we express Pr

J in the local coordi-ate system which centers at Ps1�i� and has parallel axes to globaloordinate axes. If there are ki−1 parts in subassembly s�i�, theeviation of all RPs due to subassembly s�i� fixture error can bexpressed as

E1s�i��i� = �E1

1�i�,E12�i�, . . . ,E1

ki−1�i�,�T �25�

r E1s�i��i�=M�Ps1�i� , Pr�E1s�i�.

Substituting Eq. �22� into Eq. �25� yields

E1s�i��i� = M�Ps1�i�,Pr�Fs�i��fs�i� �26�

M�Ps1�i�,Pr� = �Q�Ps1�i�,Pr

1�

Q�Ps1�i�,Pr2�

]

Q�Ps1�i�,Prki−1�

� �27�

The Lemma 2 below characterizes the motion of transferringarts between stations. Suppose a subassembly s�i−1� be posi-ioned by Ps1�i�– Ps5�i� at station i, if there is no fixture deviationt the current station i but there is deviation accumulated ins1�i�– Ps5�i� at the previous station i−1, then the subassemblyndergoes rotation and translation when moving from station i1 to station i.Lemma 2. E2

J�i� at station i is caused by the deviation of locat-ng points accumulated up to station i−1, therefore,


J�E2s�i� �28�

The translation and rotation of subassembly s�i−1� when mov-ng from station i−1 to station i can be expressed as a linearombination of deviation accumulated in its locating pointss1�i�– Ps5�i� at the previous station i−1. Thus, E2s�i� can be ex-
ressed as
36 / Vol. 129, AUGUST 2007


E2s�i� = − Fs�i��x1,�y1

,�x2,�y2

,�z3,�z4

,�z5�T = − Fs�i��L

�29�

Here, k= Psk�i��i−1�, �XPs1�i��i−1�–�ZPs5�i��i−1� are deviations of

the locating points Ps1�i�– Ps5�i� at the previous station i−1, andthey could be in different parts. As in Lemma 1 the same trans-formations can be used for calculating the accumulated deviationof �XPs1�i��i−1�–�ZPs5�i��i−1�. If there are ki−1 parts in subassembly

s�i�, the deviation of all RPs due to reorientation of subassemblys�i−1� is

E2s�i��i� = �E2

1�i�,E22�i�, . . . ,E2

ki−1�i�,�T �30�

which can be written as

E2s�i��i� = M�Ps1�i�,Pr�E2s�i� �31�

where M�Ps1�i� , Pr� is defined in Eq. �27�.

3 State Space Model for 3D Assembly in MMSTo improve the generality and applicability, in this section, the

2D state space model �4,19,28� is generalized and extended for 3Dassembly in multi-station processes. The same framework isadopted with modification in model matrices for incorporation offixture/part errors and 3D geometric factors.

The stream of variation in an MMS is illustrated in Fig. 5 for anN-station process; all the vectors are defined below. x�i��Rnt�1 isthe accumulated deviation; U�i��Rm�i�x1 is the fixture/part devia-tion contributed from station i; Y�i��Rq�i�x1 is the measurementobtained on station i; superscripts nt, m�i�, q�i� are dimensions ofthe three vectors, respectively; W�i�, V�i� are mutually indepen-dent noise.

The stream of variation in this MMP is characterized by thefollowing equations:

X�i� = A�i − 1�X�i − 1� + B�i�P�i� + W�i� i = 1,2, . . . ,N

�32�

Y�i� = C�i�X�i� + V�i� �i� � �1,2, . . . ,N� �33�where the first equation, the state equation, implies that part de-viation on station i is influenced by two sources: the accumulatedvariation up to station i−1 and the variation contribution on sta-tion i; the second equation is the observation equation. Detailedexpressions of system matrices A�i�, B�i�, and C�i� will be derivedin this section.

3.1 Theorems. Suppose there are ki−1 parts assembled to-gether from upstream processes. They form a subassembly s�i� instation i−1, and we add ki−ki−1 parts sequentially to subassemblys�i� at station i. At the end of station i, there are ki parts in thesubassembly/product. Therefore, the state vector X�i� is updatedstation by station, which is shown in Eq. �32�

X�i� = �X1�i�, . . . ,Xki−1�i��Xki−1+1�i�, . . . ,Xki�i��Xki+1�i�, . . . ,Xnt�i��T

�34�

where nt is the total number of parts in final product.In the following discussion, we will introduce two theorems

and then derivate the system matrices A�i−1�, B�i�, and C�i� in

Fig. 5 An assembly process with N stations

Eq. �32�.



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THEOREM 1. Suppose in the station i there are totally ki−1 partsn root subassembly s�i�, the reorientation-induced error E2

s�i��i�or the subassembly s�i� can be expressed as

E2s�i��i� = − M�Ps1�i�,Pr�Fs�i�G�i��

X1�i − 1�X2�i − 1�

]

Xki−1�i − 1�� 35�

here

G�i� = F2Q�i�S�i� �36�

F2 = �f ij�7�15 with f11 = f22 = f34 = f45 = f59 = f6,12 = f7,15 = 1,

otherwise, f ij = 0 �37�

Q�i� = diag�Qi� i = 1 – 5 �38�

Qi = �I3�3 Qi� i = 1 – 5 �39�

Qi = � 0 z − y

− z 0 x

y − x 0� �40�

x ,y ,z�i, i=1–5 is the P1– P5 coordinates in the root subassem-ly’s local coordinate system.

S�i� is a selecting matrix, which selects parts that have locatorsn root subassembly at station i:

S�i� = �ijI6�6�30�6ki−1i = 1, . . . ,5, j = 1, . . . ,ki−1 �41�

uppose the root subassembly’s locators P1– P5 belong to parts,b ,c ,d ,e� �1,ki−1�, respectively:

1m = �1 if m = a

0 if m � a� 2m = �1 if m = b

0 if m � b�

3m = �1 if m = c

0 if m � c� 4m = �1 if m = d

0 if m � d�

5m = �1 if m = e

0 if m � e�

Proof. Comparing Lemma 2, Eqs. �29� and �31� with Eq. �36�,e only need to prove

�L = G�i��X1�i − 1�,X2�i − 1�, . . . ,Xki−1�i − 1��T �42�

The proof is very straightforward. From Eq. �36�, G�i�F2Q�i�S�i�. The S�i� matrix selects RP deviation of the parts,hich have five locating points P1– P5. Q�i� further translates theeviation of this RP error to locating points’ error accumulated upo station s�i�. Finally, the F2 matrix selects the necessary compo-ent of the deviation.

THEOREM 2. Suppose in the station i there are ki−1 parts in rootubassembly s�i�, and after applying reorientation- and fixturerror-induced deviation, the new state vector of part 1–ki−1 be-
omes


� X1�i�]

Xki−1�i�� = Ms�i��i��

X1�i − 1�X1�i − 1�

]

Xki−1�i − 1�Xki−1+1�i − 1�

]

Xnt�i − 1�

� + M�Ps1�i�,Pr�Fs�i��fs�i�

�43�

where

Ms�i��i� = �I − M�Ps1�i��,Pr�Fs�i�G�i�F1�i� �44�

F1�i� = �I6ki−1�6ki−106ki−1�6�nt−ki−1��6ki−1�6nt

�45�

Proof. We can express Eq. �16� as a system vector equation forsubassembly s�i�:

� X1�i�]

Xki−1�i�� = � X1�i − 1�

]

Xki−1�i − 1�� + E2

s�i��i� + E1s�i��i� �46�

Substituting Eqs. �26� and �35� into Eq. �46� yields

� X1�i�]

Xki−1�i�� = �I − T�i�G�i��F1�i�� X1�i − 1�

]

Xki−1�i − 1�� + T�i��fs�i�

�47�

where T�i�=M�Ps1�i� , Pr�Fs�i�. With the definitions in Eqs. �44�and �45�, Eq. �43� can be obtained. In Eq. �43�, the first term is theaccumulated error of subassembly s�i� after reorientation but be-fore applying any fixture error to subassembly in station i, whichis determined by state vector of station i−1.

THEOREM 3. Suppose in the station i there are totally ki−1 partsin root subassembly s�i�, and we add ki–ki−1 parts sequential tothe subassembly s�i� in the station i, the propagation ofreorientation-induced error to downstream parts ki−1+1�ki canbe expressed as

E2s�i��i� = �

RXki−1+1�i�RXki−1+2�i�

]

RXki�i�� = R�i − 1��

X1�i − 1�X2�i − 1�

]

Xki−1�i − 1�Xki−1+1�i − 1�

]

Xnt�i − 1�

� �48�

where the superscript “R” means the reorientation-induced devia-tion component for state vector:

R�i − 1� = H�i − 1�F3�i�Ms�i��i� �49�

F3�i� = �06�6�ki−1−1� I6�6�6�6ki−1�50�

H�i − 1� = �Jki−1

Jki−1+1Jki−1

]

Jki−1 ¯ Jki−1

�6�ki−ki−1��6

�51�

Jj = JjF4 �52�

where Ji is defined in Eq. �14�. F4 is defined as

AUGUST 2007, Vol. 129 / 837


R

f−

w

w

a

8

Downloa

F4 = �I6�6,06�6,06�6,06�7�T �53�

Proof. Selecting matrix F3�i� selects deviation of the part ki−1’s
P after reorientation. H�i−1� calculates how the part ki−1’s
19��7+2�19� 19�19 19�m�i�

38 / Vol. 129, AUGUST 2007


deviation affects downstream assembling through part-to-partjoints. According to Eq. �13�, ��b�k−1�, ��ak, and �fk are all set

to zero, because the E2s�i��i� term only considers how part ki−1’s RP

deviation propagates through joints.

3.2 State Equation

3.2.1 A�i−1�

A�i − 1� = 06nt�6nt

if i = 1

Ms�i��i�

06�nt−ki−1��6nt

6nt�6nt

+ �06ki−1�6nt

R�i − 1�06�nt−ki��6nt

�6nt�6nt

if i � 1� �54�

A�i−1� depends on locating schemes in processes. At the first station, the part deviation error due to the previous station is zero. Thus,or i=1, A�0�=06nt�6nt

. If i�1, A�i−1� represents the error accumulation of RPs and changing of locating scheme. Ms�i��i� and A�i1� are defined in Theorem 1 and Theorem 3.

3.2.2 B�i� and U�i�. Define the input error vector as

�55�

B�i� = ��i�6nt�m�i� if i = 1

M�Ps1�i�,Pr�W0

��i�

6nt�m�i�if i � 1 � �56�

here

��i� =��W0,J1W1, . . . Jki−1Wki−1�06�nt−ki−1��m�i�� if i = 1

�Jki−1W1,Jki−1+1W2, . . . ,Jki−1Wki−ki−1

�06�nt−ki−1��m�i�� if i � 1� �57�

here Ji is given in Eq. �14� as Ji.

m�i� = �19ki − 6 if i = 1

19�ki − ki−1 + 1� − 6 if i � 1� �58�

nd

W0 = �Fs�i�6�7�0 ¯ 0�6�m�i�

W0 = F3�i�M�Ps�i�,Pr�W0

W1 = W0

U0

25�m�i�

U0 = �019�7�I19�19�0 ¯ 0�19�m�i�

W2 = Jki−1W1

U1

25�m�i�

U1 = �019��7+19��I19�19�0 ¯ 0�19�m�i�

W3 = Jki−1+1 W2

U2

25�m�i�

U2

= �0 �I �0 ¯ 0�

Wj = Jki−1+j−2 Wj−1

Uj−1

25�m�i�

Uj−1 = �019��7+�j−1��19��I19�19�0 ¯ 0�19�m�i� �59�

where W0 selects the deviation of parts Ki−1’s RP point, which isthe starting point for error accumulation in station i.

3.3 Observation Equation. If sensors are put along the as-sembly line, the observation equation could be obtained. If thereare rJ measurement points on part J at station i, then the deviationof each point on part J is

y J�i� = Q�Pr

J,MLP J�XJ�i� = 1,2, . . . ,rJ �60�

where

Q�PrJ,MLP

J� = I3�3 Q�PrJ,MLP

J�0 I3�3

�61�

Q�PrJ ,MLP

J� is defined in Eq. �24� wherein �x ,y ,z� is in thelocal coordinate system, which centers at Pr

J and has parallel axeswith respect to global coordinate axes. Thus, the measurement
vector of part J is


w

T

w

d

awpesr

w

Xfu

c

rt

w�

a

a

o

J

Downloa

YJ�i� = �y1J�i�,y2

J�i�, . . . ,yrJJ �i��6rJ�6

T �62�

hich can be written as YJ�i�=CJ�i�XJ�i�, where

CJ�i� = �Q��PrJ,MLP1

J�, . . . ,Q��PrJ,MLPrJ

J ��6rJ�6T �63�

hen, the deviation vector of all the measurement points is

Y�i� = �Y1�i�T,Y2�i�T, . . . ,Ynt�i�T��6�J=1nt rJ��1

T= C�i�X�i� �64�

here C�i� is a diagonal matrix of submatrices Ck�i�, i.e.

C�i� = diag�Ck�i��, k = 1, . . . ,nt �65�

If we consider noises W�i� and V�i�, the complete model foreviation propagation is expressed in Eqs. �32� and �33�.

3.4 Variation Propagation Model. The state space model ofmultistage assembly process is represented in Eqs. �32� and �33�,hich take the same form as in 2D SSM. Therefore, a similarrocedure as in 2D SSM �4� can be adopted. Suppose there is onlynd-of-line observation; that is, i=N in Eq. �33�. Adopting thetate transition matrix ��· , · � from control theory, the input-outputelationship can be represented as

Y = �i=1

N

C��N,i�B�i�U�i� + C��N,0�X�0� + � �66�

here the state transition matrix is

��N,i� = A�N − 1�A�N − 2� ¯ A�i� and ��i,i� = I �67�

�0� is the initial condition, resulted from manufacturing imper-ection of stamped parts, and � is the summation of all modelingncertainty and sensor noise terms

� = �i=1

N

C��N,i�W�i� + V �68�

Define: ��i�=C��N , i�B�i� and ��0�=C��N ,0�; then Eq. �68�an be simplified as

Y = �i=1

N

��i�U�i� + ��0�X�0� + � �69�

Considering that U�i� is a sequence of uncorrelated Gaussianandom vectors with zero mean, the input-output variance rela-ionship can be obtained from Eq. �66� as

�Y = �i=1

N

��i��p�i��T�i� + ��0��0�T�0� + �e �70�

here � represents the covariance matrix of a random vectorr.v.�, that is

�x = E��x − E�x��x − E�x��T� ∀ r.v. x �71�

nd �0 is given as initial conditions.The uncertainty term � can be estimated from process toler-

nces, sensor specifications, and/or through historic data. So can

r



the covariance matrix �0 of initial condition, which can be ob-tained based on incoming inspection. The system can then beapproximated with a pure linear relationship as

�Y = �i=1

N

��i��p�i��T�i� �72�

which indicates the variation of final product is contributed fromvariation of stamped part �input from precedent process� andvariations of fixture errors at all stations.

3.5 Automatic SOVA Model Generation. The SSM SOVAmodel is appealing in both forward and backward applications forvariation control in multi-station manufacturing processes. How-ever, direct derivation of the SOVA model based on first principlesfor a multi-station assembly system, as presented above in Secs. 2and 3.1–3.4, is lengthy, tedious, and prone to error. Relying onexisting simulation-based software, an automatic SOVA modelgeneration approach is developed below to surmount thisdifficulty.

The coefficients of the first-order Taylor series expansion of anassembly model can be generated by any numerical simulationbased algorithm in current standard industrial software for varia-tion analysis such as 3DCS. We take 3DCS as an example to illus-trate the method for SOVA model generation. The GeoFactor in3DCS represents the linear relationship between input and output.To automatically generate the SOVA model, we first build the3DCS model according to assembly station sequence. At eachstation, we define input tolerances and RPs �as measurementpoints� for all parts. We then run GeoFactor station by station andrecord the GeoFactors of each station in matrices �ij�j=1– i�.Equation �33� can be expressed as

�X�1�X�2�]

]

X�N�� = �

�11 0 0 0 0

�21 �22 0 0 0

] ] � 0 0

] ] ] � 0

�N1 �N2 ¯ ¯ �NN

��U�1�U�2�]

]

U�N�� 11

= �ij

06�nt−ki�ui

�73�

where nt is the total number of parts in final product after stationN, Ki is the total number of parts assembled from station 1 to i,and uj is the total number of inputs rom station 1 up to station i.

In Eq. �73�, matrices �ij �j=1– i� are constructed directly fromGeoFactors in station i. Adapting state transition matrix ��· , · �from linear control theory, the state vector-input relationship canbe represented as

X�i� = �j=1

i

��i, j�B�j�U�j� + ��N,0�X�0� �74�

Equation �74� then becomes

�X�1�X�2�]

]

X�N�� = �

B�1� 0 0 0 0

A�1�B�1� B�2� 0 0 0

A�2�A�1�B�1� A�2�B�2� B�3� 0 0

] ] ] � 0

A�N − 1� ¯ A�1�B�1� A�N − 1� ¯ A�2�B�2� ¯ ¯ B�N��

U�1�U�2�]

]

U�N��

AUGUST 2007, Vol. 129 / 839


w

mttgqt

iba

masttmSga“

4

wtbamw

pia

8

Downloa

X = �U �75�

here, X= �X��1� , . . . ,X��N��, U= �U��1� , . . . ,U��M�� and

� = �aij� aij = �k=1

i−1

A�k� � B�j� i � j aij = 0 otherwise

A method is developed to automatically generate the SOVAodel based on a simulation based method. A standardized indus-

rial software platform �e.g., 3DCS�, which is a MC based simula-ion tool for 3D rigid assemblies, can be used for SOVA modeleneration. Once the SOVA model is created, the entire subse-uent forward and backward analyses can be conducted based onhis linear model efficiently.

Comparing Eqs. �73� and �75�, we get

B�1� = �11

A�1� = �21�11−1 B�2� = �22

A�2� = �31�21−1 B�3� = �33

] ]

A�N� = �N1��N−1�1−1 B�3� = �NN

�76�

The linear model of observed variables key product character-stics �KPCs� and variation sources in Eqs. �66� and �69� can alsoe obtained in the same way. This model allows for backwardpplications in design synthesis and diagnosis.

The process of automatic model generation is to use the nu-erical simulation to transform a nonlinear assembly model into

n explicit linear SOVA model. It requires only a few runs ofimulation. Once the model is created, all the subsequent applica-ions, which are usually very computation intensive with simula-ion techniques, are conducted by using the generated SOVA

odel. Therefore, it can greatly enhance the applicability ofOVA model for complex assemblies. It allows for seamless inte-ration of current numerical simulation algorithms such as 3DCS

nd SOVA modeling techniques. The latter can be developed as anadd on” module and embedded in existing software.

Model ValidationThree cases were designed for model validation. The first case

as to compare the developed 3D SOVA model with MC simula-ion on the model without linearization, which was taken as aenchmark. The second and third cases used a floor-pan assemblynd a real automotive body assembly. The comparisons wereade between the SOVA models and standardized industrial soft-are �3DCS�.

Case I. A three-part assembly is designed for validation pur-oses. The assembly is shown in Fig. 6 with three parts assembledn three stations. The assembly sequence and datum shift schemere shown in Fig. 7. The error sources are fixture errors and mat-

Fig. 6 Case study for validation

Table 2 Coordinat

P11 P12 P13 P14 P15 P2

�5,5� �25,5� �10,15� �15,25� �30,10� �45,

40 / Vol. 129, AUGUST 2007


ing feature planes’ error. The fixture errors are assumed followingnormal distribution N�0,0.5�. The translation error components ofmating features are also assumed following N�0,0.5�. Rotationalerror components of these features are assumed followingN�0, �5� /180� /6�; i.e., the tolerance range for rotations will belimited within ±2.5 deg. For the sake of simplicity, all data pointsare in one plane Z=0. There are two mating feature planes. Thefirst one is lap joint with 30 deg rotation with respect to the Z axis,and the second one is butt joint with −15 deg rotation with respectto the Z axis. Since fixture deviations at the measurement stationare substantially smaller, they could be regarded as zero. The statespace model is built up to represent error propagation in multi-station assembly. The design nominal dimension for fixture loca-tor part locating points �PLPs� and measurement locating points�MLPs� are given in Tables 2 and 3, respectively.

MC simulation is based on the complete model without anylinear approximation and the run number is set as 10,000. All thecoordinates of locators and RPs on features are given in Tables2–4, respectively. The standard deviations of measurement points�MLPk, k=1–4� on all parts are shown in Table 2. The relativeerror is defined as 100% � ��MC−�SOVA� /�MC. All relative errors�1%, this accuracy is satisfactory. The total time spent on MC inthis case study is 132.06 s, most of the time is spent on randomnumber generation and the associated calculations, while theSOVA model took only 0.11 s, i.e., 0.0833% of the time used byMC.

Case II. Case II is a floor-pan assembly with four parts as-sembled in three stations, as shown in Fig. 8. This subassembly isused on an underbody of an automotive body assembly. Thepoints, straight lines, and triangles in Fig. 8, which represent ei-ther true fixture locating points or virtual fixture locating points�14�, are used for locating parts. On each part, a RP is selected asa KPC point. Using the approach presented in Sec. 3.5, the SOVAmodel is automatically generated through 3DCS. Defining similarerror input, the SOVA model and a 3DCS model have been estab-lished and analyzed. The same relative errors were defined forcomparison. The results are presented in Fig. 9. The maximumrelative error is about 5–6%.

Case III. A truck cab assembly produced by one of the majordomestic automobile manufacturing companies in the U.S. wasused for model validation. The assembly system has 42 stations,111 assembly operations, 1169 KCCs �tolerances�, and 390 KPCs�measurement points�. The SOVA model was constructed by theautomatic model generation method developed in Sec. 3.5. In themodeling processes, the identical KPC and KCC points were as-signed, and the 3DCS Analyst software was used to generate SOVAmodel matrices in Eqs. �73� and �76�.

Comparison was made between all the calculated KPCs out of3DCS Analyst and the SOVA model with the same input KCCs.

Fig. 7 PLP layout and shift schemes

of PLPs „Units:cm…

P22 P23 P31 P33 P34

�55,15� �55,40� �70,20� �90,10� �90,40�

es

1

10�



TfsrK

swiTnnt

5

sTssTicS

J

Downloa

he inputs were composed of all the KCC tolerances obtainedrom the product design. Normal distributions of all the dimen-ions were assumed in the simulation. The maximum relative er-or of all the variances of the KPCs was 5.7% and about 99% ofPCs had the relative errors less than 3%, as shown in Fig. 10.Once the SOVA model was generated, as a contrast to MC

imulation based techniques, very high computation efficiencyas achieved in tolerance analysis, since the covariance analysis

n the SOVA model involved only simple matrix manipulations.his makes it particularly useful when multiple tolerance designseed to be evaluated or in tolerance synthesis in which a largeumber of tolerance analyses is required in optimization itera-ions.

Conclusions and Future WorkA 3D rigid assembly modeling technique is developed for

tream of variation analysis �SOVA� in multi-station processes.he assembly process is modeled as a spatially indexed statepace model which takes into account product and process factorsuch as: part-to-fixture, part-to-part, and inter-station interactions.hese interactions represent the influences coming from both tool-

ng errors and part errors. The incorporation of generalized fixtureoncept and inter-station interaction leads to the generic, unifiedOVA model.

Table 3 Coordinates of MLPs „Units:cm…

MLP MLP1 MLP2 MLP3 MLP4

�X ,Y ,Z� �20,20� �40,20� �50,30� �80,10�

Table 4 Coordinates of mating feature RPs. „Units:cm…

M M1 M2 M3

�X ,Y ,Z� �40,0� �60,10� �100,0�

Fig. 8 Floor-pan assembly in three assembly stations

Fig. 9 Relative error comparison SOVA versus 3DCS



An automatic model generation technique is developed for sur-mounting difficulties in direct model creation based on first prin-ciples. It has been validated by case studies in this paper andindustrial cases. It greatly enhances the applicability in modelingcomplex assemblies.

The proposed model can be used for variation control in com-plex assemblies such as the products in the automotive, aerospace,and shipbuilding industries. This model allows for forward andbackward application which includes variation prediction, designevaluation, tolerance analysis and synthesis, system optimization,and fault diagnosis. For forward application, product variation canbe predicted and the sensitivity indices at different levels such ascomponent, station, and system levels can be defined and evalu-ated for design evaluation and improvement. For backward appli-cation, the model provides a linear causality relationship betweenvariation sources and observations, which can enhance the capa-bility for diagnosis and design synthesis. Monte Carlo based simu-lation, in contrast, requires a large number of runs and thus canmake the backward analysis computationally intractable. In com-parison, the developed model is very computationally efficient, inparticular for backward applications, thus makes it possible forcomplex system tolerance synthesis, design optimization, and di-agnosis. Some of the latest works in these applications can befound in �8,15�.

AcknowledgmentThe authors gratefully acknowledge the financial support of the

Advanced Technology Program �ATP� award provided by the Na-tional Institute of Standards and Technology �NIST�. The authorswould also like to thank Dr. Z. Ying, Mr. K. Ramesh, and Mr. G.Suren of DCS Inc. for their enthusiastic support and insightfuldiscussions.

Nomenclature�Pri � the deviation of reference point on part i

�xi ,�yi ,�zi � deviations of locator i�� ,�� ,�� angular deviations of a part

Fs � inverse of locator matrix

Ji � homogeneous transformation matrix forpart i

� � aggregated homogeneous transformationmatrix

� � error source input matrix�� ji � feature j deviation on part i

�f � fixture locator deviation� � local-global coordinate transformation

matrixXJ�i� � state �deviations� vector at station iE1

J�i� � part/fixture induced variation vector at

Fig. 10 Relative errors in ND33 truck cab assembly analysis„3DCS versus SOVA…

station i

AUGUST 2007, Vol. 129 / 841


R

8

Downloa

E2J�i� � reorientation induced variation vector at

station i

Q�Ps1�i� , PrJ� � point-to-point homogeneous

transformationA�i� � transformation matrix for reorientation

induced variationsB�i� � transformation matrixC�i� � observation matrixP�i� � local variation vectorY�i� � observation vector at station iU�i� � deviation from local variation source at

station iV�i� � measurement noiseW�i� � unmodeled noiseS�i� � selecting matrix

Wi�i=0,1 , . . . � � transformation matrix for part-to-partvariation propagation

��N , i� � state transition matrix �from station i tostation N�

�x ,�Y � covariance matrices of x and Y

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Stream-of-Variation (SOVA) Modeling—Part II: A Generic 3D Variation Model for Rigid Body Assembly...

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