Assembly Root Cause Analysis A Way To Reduce Dimensional Variation In Assembled Products

The International Journal of Flexible Manufacturing Systems, 15, 113–150, 2003
c
2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Assembly Root Cause Analysis: A Way to Reduce
Dimensional Variation in Assembled Products
JOHAN S. CARLSON johan.carlson@fcc.chalmers.se
Fraunhofer-Chalmers Centre for Industrial Mathematics, Chalmers Science Park, SE-412 88 Göteborg, Sweden
RIKARD SÖDERBERG riso@mvd.chalmers.se
Wingquist Laboratory, Chalmers University of Technology, Product and Production Development,
SE-412 96 Göteborg, Sweden
Abstract. The objective of root cause analysis (RCA) is to make the trouble shooting dimensional error efforts in
an assembly plant more efficient and successful by pinpointing the underlying reasons for variation. The result of
eliminating or limiting these sources of variation is a real and long term process improvement. Complex products
are manufactured in multileveled hierarchical assembly processes using positioning fixtures. A general approach
for diagnosing fixture related errors using routine measurement on products, rather than from special measurements
on fixtures, is presented. The assembly variation is effectively tracked down into variation in the fixture tooling
elements, referred to as locators. In this way, the process engineers can focus on adjusting the locators affected
by most variation. However, depending on the assembly process configuration, inspection strategy, and the type
of locator error, it can be impossible to completely sort out the variation caused by an individual locator. The
reason for this is that faults in different locators can cause identical dimensional deviation in the inspection station.
Conditions guaranteeing diagnosability are derived by considering multiple uncoupled locator faults, in contrast
to previous research focusing on single or multiple coupled locator faults. Furthermore, even if an assembly is
not diagnosable, it is still possible to gain information for diagnosis by using a novel approach to find an interval
for each locator containing the true underlying locator variation. In this way, some locators can be excluded
from further analysis, some can be picked out for adjustment, and others remain as potential reason for assembly
variation. Another way around the problem of diagnosability is to make a higher level diagnosis by calculating
the amount of variation originating from different assembly stations. Also, a design for diagnosis approach is
discussed, where assembly and inspection concepts allowing for root cause analysis are the objective.
Key Words: design for diagnosis, diagnosability, fault isolation, hierarchical assembly, multivariate statistical
analysis, root cause analysis
1. Introduction
Vehicle problems, such as wind noise, water leakage, door closing effort, gap, and flush
variation are examples of quality and functionality defects that are controlled by the dimen-
sional accuracy of the structural frame of an autobody. Today the dimensional quality in the
automotive industry is often monitored using routine measurements made in-line during
the production process.
Statistical process control is a set of powerful on-line quality tools aimed at reducing
variability around target values utilizing measurements. A review of the major tools is
found in, for example, Montgomery (1991). The most sophisticated SPC tool is the control
chart proposed by Shewhart (1931). It is based on his theory of variability in manufactured

114 CARLSON AND SÖDERBERG
products. Shewhart divides the reasons for variation in a production process into common
causes and assignable causes of variation.
The common causes are many and essentially unavoidable since each cause only has a
slight effect on the product. The cumulative effect of the common causes gives rise to a
certain avoidable level of variation inherent in the product and process concepts. In addition,
there often exist a few other assignable causes each of which has a significant effect on the
product. These assignable causes are often operator errors, defects in the raw material,
machine, and tooling errors.
The control chart is an excellent device for distinguishing between common causes and
assignable causes. A typical control chart is a graphical display of measurements or a sample
key feature, such as sample mean or sample variance, versus sample number or time. A
number of horizontal lines on the chart together with a set of rules are used to decide
whether assignable causes are present or if the process should be left alone. To change a
process when it is operating with only common causes is called tampering. Deming (1986)
gives a number of examples where tampering not only increases the product variation but
also demoralizes the working staff. If the control chart indicates that a process is out of
control then the process engineers must track down and eliminate the assignable cause.
For manufacturing processes of simple parts, patterns on the control chart may provide
enough diagnostic information to an experienced operator to perform a root cause analysis.
However, Gou, and Dooley (1992) state:
Experience shows that many SPC attempts fail to produce meaningful results because
the lack of diagnostic support for the effort.
Furthermore, in an extensive investigation of dimensional quality problems in the BIW
during the launch of a sport vehicle, Ceglarek and Shi (1995) wrote:
The complexity of the automotive body prevents measurement data from being sufficient
to localise root causes of dimensional variation. It is necessary to include knowledge
about the product and the assembly process before the root causes can be determined.
The aim of root cause analysis is to give diagnostic support for the effort to reduce dimen-
sional variation in assembled products in the automotive and other industries.
1.1. Related work
One reason for fixtures being at the center of interest is that the experience from automotive
industry suggests fixture failures as a major reason for dimensional variation in the structural
frame of an autobody. For example, in the above mentioned investigation by Ceglarek and
Shi about 70% of all root causes were fixture related.
Throughout this article, the term locators refers to the contact points between the fixture
locating elements and the part, and the locator layout is termed locating or positioning
scheme. Robust fixture design is considered by Cai, Hu, and Yuan (1997) and Wang (1999),
where the fixture locating scheme is designed to minimize the influence of fixture error on

ASSEMBLY ROOT CAUSE ANALYSIS 115
workpiece position. The philosophy of robust design is combined with the philosophy of
axiomatic design by Söderberg and Carlson (1999) to achieve an assembly concept that is
also easy to adjust and control. However, even for well-designed products and assembly
processes, the assembly tools such as positioning fixtures may become worn or broken
during production and cause dimensional variation.
Wearring and Cola (1991) proposed a procedure for troubleshooting dimensional varia-
tion issues in an assembly plant. The procedure provides a method for indirect diagnosis
through verification of assembly changes by guesswork. Fault isolation in the autobody
assembly process using inline measurements was addressed in a method developed by Hu
and Wu (1992). The method uses principal component analysis (PCA) to identify sources
of variation. A monitoring algorithm to detect and help diagnose root causes using the hier-
archical structure of the BIW was presented by Roan and Hu (1995). By only considering
fixture related problems, Ceglarek and Shi (1996) combined PCA with a fixture fault model
into an effective method for single fixture fault diagnosis. However, as pointed out by Apley
and Shi (1998) the method can erroneously point out a locator as the root cause in some
cases where the variation is actually due to a multiple fixture fault that does not include
the locator pointed out. The single fixture fault diagnosis prevails for some special types of
multiple faults, which is discussed in Ding, Ceglarek, and Shi (2000c). The multiple fixture
fault problem is solved by Apley and Shi (1998) and Carlson (1999). Both methods are
based on a linear fault model which uses the fixture layout and the geometry of the part to
relate fixture faults to the corresponding displacement at points for measurements. The fault
model used by Carlson is more general since it deals with a arbitrary deterministic locating
scheme in contrast to a 3–2–1 locating scheme for thin panels. Further, Carlson only uses
the measurement mean vector and covariance matrix while Apley and Shi use the individual
observations. While both methods are suitable for theoretical analysis only Apley and Shi
derived a test for fixture failure with a desired probability of false alarm. The power of these
methods for fixture diagnosis is determined by the locator layout and measurement location
scheme. A method for choosing optimal sensor location was developed by Khan, Ceglarek,
Shi, Ni, and Woo (1999).
In automotive industry multifixture and multistage assemblies are often used. Jin and Shi
(1999) developed a state space model describing the locator variation propagation through
the assembly process. The state space model was refined by Ding, Ceglarek, and Shi (2000b)
and used for multistage diagnosis in Ding et al. (2000c) and for process-oriented tolerance
synthesis in Ding, Ceglarek, and Shi (2000a). The challenge of multifixture assembly di-
agnosis is also considered by Carlson, Söderberg, and Lindkvist (2000). The are two main
differences between the approaches for diagnosis. First, the fault model in Ding et al. (2000a)
is an exact linearization of the locator propagation through the assembly system, in contrast
to Carlson et al. (2000) who use numerical linearization. Secondly, Carlson et al. (2000)
handle multiple locator faults, instead of single locator faults.
Lastly, methods for diagnosis have been focused on rigid parts, which can be justified
since structural parts have greater impact on the final assembly than the nonstructural ones.
However, Hu (1997) addresses nonstructural parts in a theory for predicting and diagnosing
variation in an assembly system and Rong, Ceglarek, and Shi (2000) use the single fixture
fault method (Ceglarek and Shi, 1996) to diagnose compliant beam structure assemblies.

Part
Variation
Measurement
Variation
Error
Fixture
Assembly Tool
Variation
Assembly
Design
Key Characteristics
Variation
Figure 1. Cause and effect diagram for the subassembly dimensional error.
1.2. Scope and organization of the paper
Complex products such as autobodies are assembled in a multileveled hierarchy using
assembly tools. Part variation and assembly tool related errors propagate in accordance
with the assembly concept into subassembly dimensional errors, see Figure 1. In this work,
a methodology for diagnosing dimensional errors caused by the multileveled assembly
process using measurements is presented.
A multilevel hierarchical assembly is introduced in Section 2. Two different, commonly
used, assembly positioning types used in hierarchical assembly processes are discussed
in Section 2.1. The impact of fixture failure on the subassembly dimensional accuracy
is demonstrated and captured in a fixture failure model presented in Section 3.1. The
fixture failure model includes process and product knowledge and is the basis for diag-
nosis. By a fixture failure control procedure explained in Section 3.3, the fixture fault
related error is isolated. If the fixture error has a significant impact on the assembled prod-
uct quality then a method for diagnosis outlined in Section 3.4 is used to track down
the root cause into locator variation. Locator variation is inherent or has been developed
over time and may arise from a worn, loose, bent, or broken fixture element. Unfortu-
nately, the multifixture assembly system is not often fully diagnosable since there can
exist many combinations of different locator variation that can manifest into the same as-
sembly error. However, the problem of diagnosability is partially resolved in Section 4
by calculating the minimal and maximal locator variation possible. In this way, some lo-
cators can be excluded from further analysis and others remain as potential reason for
assembly variation. Another way around the problem of diagnosability is to make a higher
level diagnosis by calculating the amount of variation originating from different assembly
stations. Section 5 presents computer simulations illustrating and verifying the proposed
method, whereas Section 6 presents a case study from an assembly plant. A good idea
is to try to find assembly and inspection concepts allowing for root cause analysis. This

design for diagnosis approach is presented in Section 7.1. Finally, conclusions follow in
Section 8.
2. Hierarchical assembly structures
From a logistic perspective, the assembly process for a complex product such as an automo-
tive body is carried out in both serial and parallel subprocesses. Parts within a subassembly
are typically assembled serially, whereas independent subassemblies on the same hierar-
chical level may be assembled in parallel. Figure 2 describes in two dimensions how two
subassemblies, A1/B1 and A2/B2 are assembled in parallel in Station 1. In Station 2, A1/B1
is assembled with C and in Station 3 A2/B2 is assembled with A1/B1/C. Finally, the whole
assembly is inspected in the inspection station. The assembly structure, generated in Sta-
tion 1, Station 2, and Station 3, may be described hierarchically as in Figure 2. Geometrical
deviation caused by fixture errors in a complex assembly, like the one in Figure 2, is often
difficult to identify. If the final position of A1/B1 and A2/B2 in relation to C is critical,
then a number of fixture errors may occur that leads to similar positioning errors in the final
assembly. We describe in the following article a general approach that allows individual sta-
tion, fixture, and locator errors to be identified. The approach is based on routine inspection
of final assemblies and requires no additional inspection of individual fixtures.
Figure 2. Hierarchical assembly structure.

Figure 3. Fixture contact positioning.
2.1. Assembly types
An assembly station for welding two parts together typically consists of two fixtures, one
for each part. The type of mating and positioning conditions in such station may, however,
differ. We will here describe two basic location types that often occur.
2.1.1. Fixture contact positioning. In this type of positioning, both parts are fully con-
strained in all 12 degrees of freedom by the two fixtures. Each fixture controls all six degrees
of freedom of each part. The three clamping points lock one translation (TZ) and two ro-
tations (RX and RY). The hole locks two translations (TX and TY) and the slot locks one
rotation (RZ). During welding the two parts are joined together by a number of welding
points in the overlap region. Figure 3 illustrates the situation.
2.1.2. Mixed contact positioning. Figure 4 illustrates another typical positioning situation
where fixture contacts and part contacts take place in perpendicular planes. In this type of
positioning, part A is fully constrained in all six degrees of freedom by its fixture. Part B
is controlled by what we will call a mixed contact. The position of part B is determined by
both its fixture and by part A. Three degrees of freedom (TX, TY, and RZ) are controlled
by the fixture whereas three degrees of freedom (TZ, RX, and RY) are controlled by part
A. In this example we also introduce the general name convention for locators that will
be used throughout the following article. Here, A1, A2, and A3 refer to the three contact
points that constitutes the plane, controlling one translation and two rotations. B1 and B2
refer to the two contact points that constitutes a line, controlling one translation and one
rotation. C refers to the last contact point, controlling the last translation. In Figure 4, B1
and B2 of part A are located close to the contact surface between part A and part B, that is,
the rotation center is close to the contact surface.
3. Multilevel hierarchical assembly RCA
The objective of root cause analysis (RCA) is to make the trouble shooting dimensional er-
ror efforts in an assembly plant more efficient and successful by pinpointing the underlying

Figure 4. Mixed contact positioning-perpendicular planes.
reasons for variation. The reasons are effectively tracked down if the amount of varia-
tion that originates from each individual locator used in an assembly system is calculated.
However, depending on the assembly process configuration, inspection strategy, and the
type of locator error it may be impossible to completely sort out the variation caused
by an individual locator. The reason for this is that faults in different locators can cause
identical dimensional deviation in the inspection station. When it comes to types of lo-
cator errors, standard methodology assumes that only one locator is the main reason for
assembly problems, see, for example, Ceglarek and Shi (1996) or Ding et al. (2000c). An-
other type, and the most general, is multiple coupled locator errors, which is treated in
Apley and Shi (1998) and Carlson et al. (2000). However, in most cases it is reasonable
to assume multiple but uncoupled locator errors, which is the basic assumption of this
article. Based on this assumption the locator variances is estimated using the method of
maximum likelihood and the uncertainty in these estimates is handled with large sample
confidence intervals. The concept of diagnosability is introduced from both a locator and
a station perspective. Even if an assembly is not diagnosable, it is still possible to gain
information for diagnosis and in this article we find an interval for each locator containing
the true underlying locator variation. In this way, some locators may be excluded from
further analysis seeing that the maximal possible locator variation is within specification.
Similarly, other locators can be pointed out for adjustment, if the minimal possible vari-
ation is above specification. Finally, some locators may be potential reason for assembly
variation, having minimal variation below and maximal variation above the specification
limits.

Figure 5. A two-fixture assembly system.
3.1. Fixture fault model
The method for diagnosis is based on a fixture fault model that explains how the inspection
points are displaced when different types of fixture errors are present. The main idea of the
diagnosis algorithm is to calculate to what extent the subassembly inspection data variation
can be explained with the fixture fault model.
We will use the subassembly A1/B1 introduced in Figure 2 to describe the fixture fault
model. This simplification is not due to limitations in the methodology, but is made to sim-
plify the discussion. The left part and right part in Figure 5 are positioned in the x − y plane
with the combination pin/hole P1 and pin/slot P2, respectively, pin/hole P3 and pin/slot
P4. The effect of fixture failures on the subassembly geometry is demonstrated by an ex-
ample. The joining geometry obtained when pin P4 is dislocated in the Y direction during
the assembly is shown in Figure 6. If the subassembly is positioned using the left most
hole and right most slot in the next process step, then the dimensional error in the assembly
Figure 6. Joining geometry if the assembly fixture pin P4 is dislocated in the Y direction.

Figure 7. Reorientation of the subassembly when positioned using the left most hole and right most slot.
fixture results in the misaligned subassembly shown in Figure 7. Next, this effect is specified
and inserted into a fixture fault model relating small locator displacement to the resulting
displacement at points for inspection. Point locations on the product are chosen for inspec-
tion because they are critical. The critical points can be divided into two classes (Ceglarek
and Shi, 1995): key product characteristics (KPC) for which dimensional variation affects
the product quality and key control characteristics which monitors process parameters that
affect the KPC. The subassembly A1/B1 is inspected at the points Mea01–Mea12, see
Figure 5. During the measurement procedure the subassembly A1/B1 is positioned using
the same hole and slot as in the assembly station, but with aid of an inspection fixture. Since
inspection fixtures are controlled and calibrated often we assume that their influence on the
measurement result is negligible.
The layout of the locators and the local surface geometry of the parts at the points for
contact between the locators and the parts introduces a set of geometric constraint equa-
tions. Solving these equations gives a nonlinear function ǫ = f (δ) that relates assembly
locator displacement δ to the resulting displacement ǫ at points for inspection. The loca-
tor displacements encountered in automotive industry are usually small and a first-order
Taylor expansion of f gives a result close to the exact relationship. This phrase is not very
precise since it is not clear what small and close mean, however, a second-order analysis
by Carlson (2001) proves when the linear analysis prevails and hence makes this vague
phrase more precise. The first-order Taylor expansion is determined by calculating the
first partial derivatives of f . For a single deterministic locating scheme the calculation of
the partial derivatives is straightforward, see Cai et al. (1997), Carlson (1999), and Wang
(1999). In the multifixture assembly setting this linearization is, however, slightly more
complicated and by considering some particular cases, Jin and Shi (1999) and Ding et al.
(2000b) managed to derive explicit formulas for the linearization. In this article numeri-
cal approximation of the derivatives is calculated using a software developed for robust
design and tolerancing (RDT). The hierarchical assembly structure is modeled in the
software by defining assembly sequences, locating schemes, and inspection points, and in
this way variation accumulation between different assembly operation is captured in the
model.
For the two-fixture assembly system, displacement in the Y direction of the pins P1,
P2, P3, and P4 is represented by δ1(1), δ1(2), δ2(1), and δ2(2) respectively. Displacement

in the X direction of the pins P1 and P4 is represented by δ1(3) and δ2(3) respectively.
Let ǫ be the displacement at the points Mea1–Mea12, evaluated in the part surface normal
direction at each point, caused by a locator displacement δ. The matrix A whose columns
are the partial derivatives of f maps each locator displacement into an approximation d of ǫ.
Thus,
d =

























0.588 −1.000 0 0 −0.411 0
0.168 −0.285 0 0 −0.117 0
0.252 −0.428 0 0 −0.176 0
0.084 −0.142 0 0 −0.058 0
−0.168 0.285 0 0 0.117 0
−0.588 1.000 0 0 0.411 0
−0.411 0 0 −1.000 0.588 0
−0.117 0 0 −0.285 0.168 0
0.058 0 1 0.142 −0.084 −1
0.176 0 1 0.428 −0.252 −1
0.117 0 0 0.285 −0.168 0
0.411 0 0 1.000 −0.588 0


























= [A1 A2]
δ1
δ2
, (1)
where the sensitivity matrix A has been blocked into [ ∂ f /∂δ1 ∂ f /∂δ2 ] to highlight which
part each locator is placed on. The ith column vector, ai , in the sensitivity matrix A is
the resulting displacement in the points Mea1–Mea12 when a unit displacement in the
ith locator is present. The column vector ai will be referred to as the ith locator fault
signature.
Generally for a hierarchical assembly involving multiple stations and fixtures, the lin-
earized displacement d can be written as
d =
I

i=1
J(i)

j=1
Ai j δi j =

A11 A12 . . . AI J(I)

A

δT
11δT
12 . . . δT
I J(I)
T

δ
,
where Ai j is the linear relationship between a locator fault, in station i and fixture j,
and the resulting displacement at the inspection station. The total number of locators is,
l = 6
I
i J(i), and the locators on each fixture are always ordered in the following way,
δi j = [ A1 A2 A3 B1 B2 C ]
T
.
In the above equation we have assumed that all measurements, d, are collected in a single
inspection station at the end of the subassembly, however, this is generalized to distributed
inspection in Section 7.2.

3.2. Inspection data model
The basis for diagnosis is that a sample of N subassemblies has been collected for routine
inspection and that the dimensional deviations from nominal at p inspection points for the
kth subassembly are recorded in a vector xk. We assume that each observation xk can be
modeled as a linear stack up of fixture error, δ, and independent unmodeled error, z, not
related to the assembly fixture system, that is,
xk = Aδ + z.
Furthermore, the fixture error and unmodeled errors are assumed normally distributed,
which means that xk is p-variate normally distributed with mean vector µx and covariance
matrix x ,
xk ∼ Np(µx , x ). (2)
These assumptions imply that the subassembly covariance matrix x is a linear stack up of
the fixture variation d and an unmodeled variation z, that is,
x = d + z, (3)
and that the fixture covariance matrix originates from locator covariances δ propagating
through the sensitivity matrix A, that is d = Aδ AT
.
3.3. Fixture failure control
In this section we estimate the amount of variation in the inspection data that originates from
fixtures in the assembly system. By a fixture failure we mean that there exists unacceptable
high locator variation in at least one locator. A failure index is introduced to determine if a
fixture failure is present. If the failure index indicates a fixture failure then we analyze the
disparity further and track down the reason for failure.
To this end the p-dimensional space of all possible observations is decomposed into two
orthogonal subspaces. One is the subspace where fixture errors are mixed with the residual
errors and the other is the subspace where there are no fixture errors. These subspaces are
defined by the sensitivity matrix A and will henceforth be called, respectively, the failure
subspace and noise subspace. To obtain this decomposition we split up each observation
x into two parts. The first part is the orthogonal projection of x onto the column space of
the sensitivity matrix A and the second part is the orthogonal projection of x onto the null
space of the transposed sensitivity matrix AT
. Mathematically,
x = UrUT
r x + Up−rUT
p−r x, (4)
where Ur is an orthonormal basis matrix for the r-dimensional column space of the sensi-
tivity matrix A and Up−r is an orthonormal basis matrix for the (p − r)-dimensional null

space of AT
. These basis matrices are calculated using the singular value decomposition
(SVD), see, for example, Golub and van Loan (1996).
Straightforward calculations using (2)–(3) give
UrUT
r x ∼ N

UrUT
r µx , d + UrUT
r zUrUT
r

Up−rUT
p−r x ∼ N

Up−rUT
p−r µx , Up−rUT
p−r zUp−rUT
p−r

.
This proves that decomposition (4) has the properties claimed above, since the covariance
matrix for UrUT
r x is built up by both fixture failure and residual error while the covariance
matrix for Up−rUT
p−r x is independent of fixture errors.
A fixture failure variation index ψ is defined as
ψ =
Trace

UrUT
r xUrUT
r

Trace(x )
, (5)
which compares the amount of variation in the failure subspace with the total variation. In
practice, we have to estimate the variation index ψ since the population parameters x and
µx are unknown. An estimate is obtained by replacing the population parameters with their
sample counter parts,
x̄ =
1
N
N

k=1
xk
Sx =
1
N
N

k=1
(xk − x̄)(xk − x̄)T
.
The uncertainty in the estimate is handled using approximative confidence intervals. In this
case a confidence interval is a random interval calculated from the sample which contains
the failure index with a specified probability (1 − α). The asymptotic confidence interval
used for approximation is derived by Carlson et al. (2000) in Appendix A. The confidence
limits for the fixture failure variation index given that α = 0.05 are given by
ψ̂ ± 1.96
√
τ̂2, where
(6)
τ̂2
= 2
(1 − ψ̂)2
Trace

UT
r SxUr
2
+ ψ̂2
Trace

UT
p−r SxUp−r
2
(N − 1)(Trace(Sx))2
,
and ψ̂ is the estimated index obtained by replacing the population covariance matrix x
with its sample counterpart Sx in equation (5).
The fixture failure index takes on values between 0 and 1. In case of a negligible fixture
error the index will be close to 0 and when a locator variation dominates the corresponding
index becomes close to 1. If the confidence interval for variation index ψ is to the right of a
threshold value ψ0 then we conclude that an unacceptable locator variation is present. The
threshold value ψ0 has to be set by the operating process engineers and should reflect when

it is worthwhile to adjust the fixture system. However, a pointer is isotropic subassembly
variation x = σ2
Ip that implies a fixture failure variation index ψ = r/p.
To sum up, the fixture failure confidence interval is compared with a threshold value and
a conclusion about the state of the assembly fixture system is drawn. If a fixture failure has
occurred, then we want to make a diagnosis by pointing out the fixture element or elements
causing the unacceptable level of variation. This is done by finding the maximum likelihood
estimator (m.l.e.) of the locator covariance matrix δ. Since locators error are assumed to
be multiple but uncoupled, the locator covariance matrix is diagonal with locator variance
elements λ1, λ2, . . . , λl.
3.4. Fault isolation: Maximum likelihood estimation of constraint
A m.l.e. is obtained by finding the parameter values that maximizes its log likelihood
function. Since the measurements {xk}N
k=1 are assumed p-variate normal distributed with
mean vector µx and covariance matrix x , the log likelihood can be simplified to
L(x; µ, ) = −
N
2
log det(2πx ) −
N
2
Trace

−1
x Sx

−
N
2
Trace

−1
x (x̄ − µx )(x̄ − µx )T

, (7)
see Mardia, Kent, and Bibby (1997, p. 97). When the fixture failure variation index is large
we assume the subassembly inspection data to have the following covariance structure:
x = Aδ AT
+ σ2
Ip. (8)
That is to say, the subassembly covariance matrix is built up by fixture variation and un-
modeled errors that are independent and with equal variation in all directions.
The amount of variation originating from each locator is estimated by finding the locator
covariance matrix δ and noise factor σ2
that maximize the log likelihood function (7).
Note that the log likelihood only depends on the measurement data through the sample
mean x̄ and covariance matrix Sx and hence the m.l.e. will only depend on x̄ and Sx . The
maximization may be done numerically by Fishers scoring method (see, e.g., Rao, 1973).
3.4.1. Large sample confidence region for the M.L.E parameters estimates. Increasing
the number of inspected subassemblies, N, makes the sample information better and the
uncertaintyinthediagnosisless.Theuncertaintyinthediagnosisishandledusingconfidence
regionsandinthissectionwederivelargesampleconfidenceregionsforthelocatorvariances
δ.
It is convenient to have the parameters defining the subassembly covariance matrix gath-
ered into a vector, hence let θ = [λ1 λ2 . . . λl σ2
]. The m.l.e. θ̂ is asymptotically normally
distributed with mean θ and covariance matrix (2/N)G−1
ˆ
θ
, where
G ˆ
θ =
∂2
L
∂θi ∂θj
= Trace

−1
x
∂x
∂θi
−1
x
∂x
∂θj

.

and the partial derivatives of the subassembly covariance matrix are
∂x
∂λi
= ai ai
T
(9)
∂x
∂σ2
= Ip
(see, e.g., Jöreskog, 1981).
A 100(1 − α)% confidence region R(X) for the parameters θ is given by the set
R(X) =

θ : (N/2)(θ − θ̂)
T
G ˆ
θ (θ − θ̂) ≤ χ2
l+1(α)

,
where χ2
l+1(α) is the α quantile of the χ2
-distribution with l + 1 degrees of freedom.
4. Diagnosability and fault isolation
As we saw in Section 3.4, the fixture system is diagnosed by finding the m.l.e. of the
locator covariance matrix. However, depending on the assembly process configuration and
inspection strategy, it may be impossible to completely sort out the variation caused by an
individual locator. The reason for this is that faults in different locators can cause identical
dimensional deviation in the inspection points. For example, if pin P1 in Figure 5 is displaced
1 unit in the X-direction during the assembly then the resulting subassembly will be identical
to the one achieved by displacing pin P3 (−1) unit in the X-direction. The following
definition clarifies what we mean with complete locator diagnosability.
Definition 1 (Complete locator diagnosability). An assembly is said to be locator i diag-
nosable if it is possible to determine the amount of variation caused by locator i irrespective
of the locator variation level present at all locators. If this is true for all locators, then the
assembly is complete locator diagnosable.
In the next section conditions on the sensitivity matrix A guaranteeing complete and locator
i diagnosability are derived. Nevertheless, even if a problem is not completely locator
diagnosable it may still be possible to gain information for diagnosis purpose and as an
illustration we show how a multiple locator fault is successfully pinpointed despite the lack
of locator diagnosability. Furthermore, Section 4.2 discusses the possibility to sort out the
amount of variation originating from different assembly stations.
4.1. Diagnosability and fault isolation on locator level
Complete locator diagnosability is mathematically checked by investigating whether the
parameters θ = [λ1 λ2 . . . λl σ2
] are uniquely determined by x , that is, if θ1 = θ2 implies
x (θ1) = x (θ2). Hence, complete locator diagnosability in the case of multiple uncoupled
locator faults means that
A1 AT
+ σ2
1 Ip = A2 AT
+ σ2
2 Ip ⇔ 1 = 2 and σ1 = σ2.

This equivalence holds if and only if there is more inspection points than locators, that is,
p l and each nonzero diagonal matrix Y makes
AY AT
= 0.
The first condition p l guarantees that A(1 −2)AT
= (σ2
2 −σ2
1 )Ip only has the trivial
solution σ1 = σ2. Since, for a non trivial solution the right side is of rank p, while the
maximum rank of the left side is l. The second condition ensures that A(1 − 2)AT
= 0
if and only if 1 = 2.
By using the Vec and Kronecker operator ⊗ together with a matrix E of size [l × l,l],
which kth colon (k = 1, 2, . . . ,l) has a one in row l(k − 1) + k and zeros elsewhere, to
rewrite the above relation (Graham, 1981),
(A ⊗ A)E

T
ET
Vec(Y)

y
= 0, (10)
it is revealed that the assembly sensitivity matrix must correspond to a full rank T matrix to
be complete locator diagnosable. An equivalent condition for diagnosibility can be found
in Ding et al. (2002).
If T is rank deficient then there exist a entire set of diagonal locator covariance matrices
that maximizes the likelihood function. Suppose that λ∗
is a particular m.l.e. then the
solutions are the set of all positive vectors of the form
λ = λ∗
+ λ0, (11)
where λ0 is any vector in the null space of T , retaining the positivity condition on λ. We
can see that locator i diagnosibility corresponds to all vectors in the null space of T having
its ith element equal to zero, λ0(i) = 0. If we use the SVD to calculate an orthonormal
basis matrix Vl−ρ for the (l − ρ)-dimensional null space of T , then each possible λ0 can
be written as a linear combination of the colons in the basis matrix. For locator i this
means,
λi = λ∗
i + Vl−ρ(i, :)γ,
where γ is a ρ-dimensional vector. Consequently, an assembly is locator i diagnosable if
the ith row of Vl−ρ is zero, that is,
Vl−ρ(i, :) = 0. (12)
However, in spite of this being ambiguous it is often possible to gain information for
diagnosis purpose by examining this set of possible locator covariance matrices.
4.1.1. Incomplete diagnosability. Even if the assembly is neither complete nor locator
i diagnosable it might still be possible to perform a successful root cause analysis by
calculating the minimal and maximal individual locator variance in the solution set (11).

In this way, a locator variation interval can be derived for each locator. These intervals can
then be compared with an allowed level of variation. An interval completely above the limit
means that the locator should be adjusted. Conversely, an interval below the limit means that
the locator should be left alone. In general, nothing can be said about locators with locator
interval containing the limit, besides being potential root causes. The minimal and maximal
locator variance are found solving two linear programming (LP) problems for each k,
max
λ
λk
V T
ρ λ = V T
ρ λ∗
, λ ≥ 0
(13)
and
min
λ
λk
V T
ρ λ = V T
ρ λ∗
, λ ≥ 0,
(14)
where Vρ is an orthonormal basis matrices for the ρ-dimensional column space of T T
. A
m.l.e. of the particular solution λ∗
, required in the LP problems above, can be found by
first using Vρ to reparametrize the l-dimensional parameter vector λ with an ρ-dimensional
vector γ in the following way:
x = A diag(Vργ)AT
+ σ2
Ip.
And then, apply Fishers score method with the revised derivatives
∂x
∂γk
= A diag(Vρ(:, k))AT
(15)
to get the m.l.e. γ̂, which is mapped into a particular locator variation m.l.e. λ∗
= Vργ̂.
Both LP problems are guaranteed to have solutions since there exist at least one feasible
solution, λ∗
, and the problems are bounded. Problem (14) is bounded by the positive
condition on the locator vector, λ ≥ 0, and Problem (13) is bounded by the following
reason. Allowing one of the locator variances to tend to infinity makes the total variation
of the measurement covariance matrix x go to infinity, which constricts that all vectors in
the solution set (11) correspond to the same measurement covariance matrix.
4.1.1.1. Numerical experiment. The incomplete diagnosable assembly in Figure 5 is used
to demonstrate a successful and unsuccessful root cause analysis. In both examples we use
a specification limit set to 0.2/3 ≈ 0.0667 which corresponds to a contact process between
the fixtures and parts with capability 1 and tolerance ±0.2. Furthermore, an infinite sample
size, N = ∞, is assumed to avoid uncertainty in the m.l.e. of λ∗
. The m.l.e. uncertainty
effect on a nondiagnosable assembly is discussed in the next section.
In the first example the variation at pin P1 is 0.032
and 0.022
in the Y- and X-direction,
respectively, and the variation at P2 is 0.122
in the Y-direction. Pin P3 is affected by variation
of size 0.172
and 0.052
in the Y- and X-direction, respectively, and in P4 the variation is

0.042
in the Y-direction. In brief, the main root cause for low assembly quality is the large
variation at pin P2 and P3 in the Y-direction. The root cause analysis is performed by first
finding a m.l.e. of the level of variation in the locators, λ∗
. Next, locator variation intervals
containing the underlying locator variation are calculated by solving the LP problems (13)
and (14). The result of this root cause analysis is presented in Figure 8, where we can see that
the variation intervals at locators 2 and 4 are above the specified allowed level, whereas the
other locator intervals are under the allowed level. This means that the root cause analysis
correctly pinpoints the large variation at pin P2 and P3.
In the second example the variation at pin P1 is 0.032
and 0.122
in the Y- and X-direction,
respectively, and the variation at pin P2 is 0.012
in the Y-direction. The level of variation
at pin P3 is set to 0.022
and 0.062
in the Y- and X-direction, respectively, and the variation
at P4 is 0.032
in the Y-direction. In other words, the main root cause is the large variation
at pin P1 in the X-direction. The same procedure as in the first example has been used to
get the results presented in Figure 9, where the root cause analysis fails since the variation
intervals corresponding to P1 and P3 in the X-direction are neither above nor below the
specification limit. Still, the other four locators can be excluded from the class of potential
Figure 8. Successful root cause analysis of an incomplete diagnosable assembly. Locators 2 and 4 are pointed
out as the reasons for assembly variation.

Figure 9. Partial successful root cause analysis of a non-diagnosable assembly. Locators 3 and 6 are potential
root causes and the other four locators are not.
faulty locators. Summing up, the relation between the levels of variation at different locators
determines if it is possible to make a successful root cause analysis when the assembly is
nondiagnosable.
4.1.2. Large sample confidence regions for incomplete diagnosability. The root cause
analysis of a nondiagnosable assembly has in practice two uncertainty factors. First, there
is the uncertainty in the m.l.e. of the particular locator variances λ∗
, which is common
to the diagnosable case and handled by using a confidence region. Secondly, there is the
uncertainty related to the solution set in equation (11), which was handled by solving the
LP problems (13) and (14). In this section these two aspects are combined to calculate lo-
cator variation confidence intervals similar to the ones achieved previously, when assuming
infinite sample size.
We begin by finding the m.l.e. γ̂ and a 100(1 − α)% confidence region,

γ : (N/2)(γ − γ̂)T
G ˆ
γ (γ − γ̂)) ≤ χ2
ρ (α)

, (16)

where (2/N)G−1
γ̂ is an estimate of the asymptotic covariance matrix of the estimators γ̂.
The matrix G ˆ
γ is calculated by replacing the derivatives ∂x /∂λ in equation (9) with the
derivatives ∂x /∂γ in equation (15). Next, we map the m.l.e. and confidence region back to
the original variables using the relation λ = Vργ. Then, the minimal and maximal locator
variation for each locator k is found by solving these two optimization problems,
max
λ
λk
(λ − λ∗
)T
V T
ρ −1
γ̂ Vρ(λ − λ∗
) = χ2
ρ (α), λ ≥ 0.
(17)
and
min
λ
λk
(λ − λ∗
)T
V T
ρ −1
γ̂ Vρ(λ − λ∗
) = χ2
ρ (α), λ ≥ 0.
(18)
The root cause procedure for incomplete diagnosable assemblies is tested on the sub-
assembly in Figure 5. To make a graphical presentation possible, only three out of the
six locators are included in the analysis. A remark is that this restricted version of the
subassembly is locator 1 diagnosable.
4.1.2.1. Numerical experiments. The variation at pin P2 is 0.122
in the Y-direction, at P1
the variation is 0.022
in the X-direction, and at P3 the variation is 0.052
in the X-direction.
The sample size used is N = 100, the noise level is set to σ2
= 0.001, and the confidence
coefficient used is 1 − α = 0.95. Figure 10, shows the m.l.e. of γ and the corresponding
95% confidence ellipse defined in equation (16). The boundary of the ellipse is mapped
back to the original parameters λ = Vργ in Figure 11. The line through the particular
locator variation m.l.e. λ∗
intersecting the λ1 − λ2 plane and λ1 − λ3 plane, illustrates
only the uncertainty due to the incomplete diagnosability. But, by solving the optimization
problem (17) and (18), also the uncertainty in the m.l.e. is included. This is illustrated by the
lines touching the boundary of the ellipse and the corresponding stars. The lines intersecting
the boundary of the ellipse and the λ1 − λ2 plane determine a 95% confidence interval for
the variation in locator 2. Similarly, the intersections with the λ1 − λ3 plane determine a
95% confidence interval for the variation in locator 3. Locator 1 is only affected by the
m.l.e. uncertainty, since the assembly is locator 1 diagnosable.
The locator variation confidence intervals in Figure 12 correctly pinpoint locator 1 as the
root cause.
4.2. Diagnosability and fault isolation on station level
If an assembly is incomplete diagnosable from a locator perspective, it may still be possible
to estimate the amount of variation caused by the set of assembly fixtures in the different
stations. Therefore, it is logical to make a root cause analysis on the station level before
the variation is tracked down to the locator level. In this way, the process engineers can

Figure 10. 95% confidence ellipse for γ∗.
concentrate on the faulty station even if the root cause analysis gives ambiguous results on
the locator level.
The total amount of variation ψi originating from station i is the sum of locator variation
in station i, that is,
ψi =
J(i)

j=1
6

k=1
λi j (k)

ψi j
.
Definition 2 (Complete station diagnosability). An assembly is said to be station i diag-
nosable if we can determine the amount of variation caused by station i irrespective of the
conditions in the other stations. Furthermore, if this is true for all stations, then a complete
diagnosability between stations is possible.
The total variation in station i is estimated by finding a particular m.l.e. λ∗
in exactly the
same way as before and then the maximal variation from station i is found by solving the

Figure 11. The interaction between incomplete diagnosability and uncertainty in the m.l.e.
LP problem
max
λ
ψi =
J(i)

j=1
6

k=1
λi j (k)
(19)
V T
ρ λ = V T
ρ λ∗
, λ ≥ 0
and the minimal variation from station i is determined by solving
min
λ
ψi =
J(i)

j
6

k=1
λi j (k)
(20)
V T
ρ λ = V T
ρ λ∗
, λ ≥ 0.
Let us introduce a vector vi,k that is built up by the row elements representing the locators
in station i of the kth column vector in the matrix Vl−ρ. A condition guaranteeing diagnos-
ability can be found by investigating a typical solution maximizing the total variation from

Figure 12. RCA of a nondiagnosable assembly using 95% locator standard deviation confidence intervals.
station i
max ψi =
J(i)

j=1

ψ∗
i j +
l−r

k=1
αi,kvi,k( j)

=
J(i)

j=1
ψ∗
i j +
l−r

k=1
αi,k
J(i)

j=1
vi,k( j) (21)
and a typical solution minimizing the total variation from station i
min ψi =
J(i)

j=1

ψ∗
i j +
l−r

k=1
βi,kvi,k( j)

=
J(i)

j=1
ψ∗
i j +
l−r

k=1
βi,k
J(i)

j=1
vi,k( j). (22)

We can see from equations (21) and (22) that if
J(i)
j=1 vi,k( j) = 0 for each k, then max ψi =
min ψi independent of the underlying fault and hence station i is diagnosable.
Obviously, it is straightforward, if desirable, to include diagnosability and fault isolation
on the fixture level using the methodology lined out in this section.
5. Numerical experiments
In the following subsections we will in detail analyze two different scenarios based on the
hierarchical assembly process described in Figure 2 in Section 2. The parts A1, B1, A2, B2,
and C are assembled together and finally inspected in an inspection station. We will in the
following subsection describe two different ways to assemble the subassemblies A1/B1 and
A2/B2 to part C. We start with describing Station 1, which is common for the two scenarios.
All figures shown are from the simulation system (RDT) used throughout the analysis.
During RC analysis, the parts are first assembled in their fixtures, stability analysis is then
performed to generate the sensitivity matrices.
5.1. Parallel assembly of subassembly in Station 1
Figure 13 describes Station 1 in the simulation model where A1 and B1 are assembled in
Fixture A1/B1 and A2 and B2 are assembled in Fixture A2/B2. The locating schemes used
here are so-called 3-point locating schemes with one point supporting three directions (A1,
B1, and C), one point supporting two directions (A2 and B2), and one point supporting one
direction (A3). Since the master location system of a part constitutes its reference coordinate
system, no variation is assumed in these points. Variation is therefore applied to the master
Figure 13. Parallel assembly of A1/B1 and A2/B2 in Station 1.

Figure 14. Parallel assembly of A1/B1 and A2/B2 with C.
locators as variation in the mating surfaces or points (the fixture). The locating directions
(variation direction) for the fixture locators are marked with arrows. The assembly type in
this station is the “fixture contact positioning,” described in Section 2.1.1.
5.2. Parallel assembly of subassemblies in Station 2
Figure 14 describes Station 2 where A1/B1 and A2/B2 are assembled with part C. The
assembly type in this station is the “mixed contact positioning—perpendicular planes,”
described in Figure 4 in Section 2.1.2. In this station the two subassemblies A1/B1 and
A2/B2 are held in position by their two fixtures. The fixtures control three degrees of
freedom whereas the remaining three degrees of freedom are controlled by the surface
contact with part C.
5.3. Serial assembly of subassemblies in Stations 2 and 3
Figure 15 describes the second scenario. Here A1/B1 and A2/B2 are assembled with part
C in two different stations. The assembly conditions are the same as for the parallel case
described in the previous subsection.

Figure 15. Serial assembly of A1/B1 in Station 2 and A2/B2 in Station 3.

Figure 16. Inspection station and inspection points.
5.4. Inspection of final assembly
Figure 16 describes the final inspection station and the inspections points. As can be seen
in the figure, only inspection points on the original parts A1, B1, A2, and B2 are used.
The assembly is positioned in the inspection fixture in the same way as part C was lo-
cated in the previous assembly station. The inspection points are named as shown in
Figure 16.

Table 1. Locator standard deviations for parallel assembly of subassemblies in Station 2.
Station 1a Station 1b Station 2
Locator Fix. A1 Fix. B1 Fix. A2 Fix. B2 Fix. A1/B1 Fix. A2/B2 Fix. C
A1 0.00 0.00 0.00 0.00 0.00 0.00 0.02
A2 0.00 0.00 0.00 0.00 0.00 0.00 0.19
A3 0.00 0.00 0.00 0.00 0.00 0.00 0.00
B1 0.03 0.02 0.03 0.02 0.03 0.04 0.00
B2 0.01 0.05 0.01 0.03 0.01 0.01 0.00
C 0.05 0.04 0.04 0.06 0.17 0.03 0.03
5.5. RCA of the parallel assembly of subassemblies
An inspection data covariance matrix x = Aδ AT
for the parallel assembly of subassem-
blies in Figure 14 is generated in accordance with Table 1 and used for RCA. Dimensional
error in some of the locators does not have any effect on the inspection points. These lo-
cators are identified in table having 0.00 variance. The result of the station level RCA is
Figure 17. Station RCA for parallel assembly of subassemblies in Station 2.

Figure 18. Locator RCA for parallel assembly of subassemblies in Station 2.
shown in Figure 17. We can see that the parallel assembly of subassemblies is completely
station diagnosable and that most of the variation is originating from Station 2. A more
detailed locator level RCA is presented in Figure 18. Note that locators without effect on
the inspection points are not presented in the figure, making locators B1 on Fixture A1 and
Fixture B1 corresponding to locators 1 and 4 in Station 1a, respectively. Problems with
Table 2. Locator standard deviations for the serial assembly of subassemblies in Stations 2 and 3.
Station 1a Station 1b Station 2 Station 3
Locator Fix. A1 Fix. B1 Fix. A2 Fix. B2 Fix. A1/B1 Fix. C1 Fix. A2/B2 Fix. C2
A1 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.07
A2 0.00 0.00 0.00 0.00 0.00 0.19 0.00 0.01
A3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
B1 0.03 0.02 0.03 0.02 0.03 0.00 0.04 0.00
B2 0.01 0.05 0.01 0.03 0.01 0.00 0.01 0.00
C 0.05 0.04 0.04 0.06 0.17 0.03 0.03 0.02

Figure 19. Station RCA for the serial assembly of subassemblies in Stations 2 and 3.
locator C on fixture A1/B1 and locator A2 on fixture C, both in station 2, are correctly
pointed out as the main reasons for assembly variation.
5.6. RCA of the serial assembly of subassemblies
An inspection data covariance matrix for the parallel assembly of subassemblies in Fig-
ure 14 is generated in accordance with Table 2 and used for RCA. Also the serial assembly
of subassemblies is completely station diagnosable and we can see in Figure 19 that most of
the variation is still originating from Station 2. A more detailed RCA on the locator level is
presented in Figure 20. From a diagnosable perspective the serial assembly is less attractive
than the parallel since only 8 out of 24 locators are diagnosable, compared to 9 out of 21 for
the parallel assembly design. For example, the problem with locator C on fixture A1/B1 can
no longer be pointed out since it is equally probable that locator C on fixture C1 is faulty.
6. A case study
A case study has been carried out to test the industrial relevance of the proposed RCA
method. Figure 21 shows an assembly where a rear bumper is joined with a vehicle floor. The

Figure 20. Locator RCA for the serial assembly of subassemblies in Stations 2 and 3.
Figure 21. A multifixture bumper assembly.

Figure22. Leftplotshowsbumperproductiondeviationfromtarget,afterassemblywiththefloor,in14inspection
points on the bumper. Right plot shows 95% confidence intervals for the locator variation.
assembly corresponds to a mixed contact positioning situation, explained in Section 2.1.2,
where three degrees of freedom are controlled by the contact with the floor and the remaining
three are controlled with a bumper fixture. The assembly process is monitored by measuring
14 points on the bumper, after assembly, using a coordinate measuring machine. The result
from 20 inspections is shown in Figure 22. The 20 inspections indicate an unacceptable
level of variation and since the bumpers were known to be close to target before assembly
this suggests an assembly quality problem.
The first step of the root cause analysis is to calculate the sensitivity matrix A, which
is done by building a simulation model of the assembly in RDT. Next, we calculate the
fixture variation index confidence interval (6) to decide if a fixture failure is present in
the assembly system. It is clear that a fixture failure is present, since the 95% confidence
interval is [0.92, 0.97], which is close to 1. The last step of the RCA is to estimate the
locator variation, following the procedure in Section 3.4, and the result is shown in the right
part of Figure 22.
Figure 23. Part B positioned using a slot with an angle relative the slot of part A.

The bar plot shows the 95% confidence interval of the variation in the contact points
(A1, A2, A3) between the floor and the bumper and the variation in a pin/hole (B1, C)
and a pin/slot (B2) contact between the fixture and the bumper. We can see that the main
reason for assembly variation is the pin/hole contact between the fixture and the bumper
controlling the translation (C). In practice, most of the assembly variation was eliminated
after the process engineers made a change in the design of the pine/hole contact.
7. Design and inspection for diagnosis
7.1. Design for diagnosis
There are assembly designs that are better suited for diagnosis than others. The parallel
assembly design is better than the serial one from a diagnosis perspective and in this
section we give another example of design for diagnosis. Also, the possibility of gaining
diagnosability by adding inspection points in the assembly station is investigated.
Figure 5 describes the normal way to locate two parts during assembly. A hole is used to
lock translation in X and in Y and a slot is used to lock rotation around Z. After assembly,
the hole in part A and the slot in part B are used as locators for the subassembly in the next
Figure 24. Successful RCA, locator 3 is pointed out as the main reason for assembly variation.

assembly step. From a fault diagnosis perspective, locating the two parts the way Figure 5
describes is not very good. If the relative position between part A and part B in X direction
is not according to specification after assembly, it is not possible to judge whether the error
results from a disturbance in the locator hole of part A or the one in part B. To allow for
diagnosis, location of the two parts could instead be designed as in Figure 23. The slot of
part B is here located with an angle relative to the slot of part A, which results in a coupled
solution making the assembly complete diagnosable. This is illustrated in Figure 24, where
the locator variation is from the second numerical example in Section 4.1.1. After assembly,
when the hole of part A and the slot of part B is used to locate the subassembly, this solution,
however, results in an increased nominal gap in the pin/slot contact. This is due to the fact
that the direction of the slot is not parallel with the line between the hole and the center of
the slot after assembly. That is, the location direction of the slot is not perpendicular to its
direction.
7.2. Inspection for diagnosis
The number and the location of inspection points on the parts affects the uncertainty in the
locator covariance estimate and hence the possibility to perform a successful RCA. In many
Figure 25. Improved diagnosability by inspection of Mea02 in the assembly station.

cases, however, the degree of diagnosability cannot be increased by adding more inspection
points in the inspection station. Therefore, we use the small subassembly in Figure 5 to
investigate the possibility to gain diagnosability by also adding some points for inspection
in the assembly station.
Let Ak
i j be the sensitivity matrix for station i, fixture j, and inspection k. Faults δ11 in
fixture A and faults δ12 in fixture B propagate into dimensional deviation in the assembly
station, d1
, and in the inspection station, d2
,
d1
= A1
11δ11 + A1
12δ12
d2
= A2
11δ11 + A2
12δ12,
or in block matrix form
d =
d1
d2
=
A1
11 A1
12
A2
11 A2
12

A
δ.
Figure 26. Complete diagnosability by inspection of Mea03 in the assembly station.

Performing a RCA on the second numerical experiment in Section 4.1.1 using the block
matrix A achieved above by adding inspection point Mea02 on part A gives the result in
Figure 25. Now, the assembly is locators 1, 2, 4, and 5 diagnosable, but both locators 3 and
6 are still potential root causes. On the other hand, if the inspection point Mea03 is instead
added then the assembly becomes completely diagnosable. Figure 26 shows the successful
RCA result.
In practice, the inspection for diagnosis approach corresponds, for example, to combine
measurements from in-line optical coordinate measuring machines or automated coordinate
checking fixtures (see Wang, 1999) with measurements from off-line coordinate measuring
machines.
8. Conclusions
Geometrical variation in the structural frame of an automotive body, body in white (BIW),
can cause quality problems in the final product such as wind noise, water leakage, door clos-
ing effort, gap, and flush variation. That is to say, any critical assembly dimension on the
BIW is a product performance characteristic and in a competitive economy the way to stay
in business is to continuously reduce the performance characteristics variation about their
target values, see Taguchi (1986). The major sources of variations in assembled products are
component variation and assembly tool related variation, which propagate in accordance
with the product/process concept used. Taguchi proposes that engineering knowledge and
designed experiments/computer simulations should be used during design to find prod-
uct/process concepts that are insensitive (“forgiving”) to variability. Robust design, the
approach proposed by Taguchi, can be used off-line as a countermeasure against variability.
However, even for very well-designed products and assembly processes the assembly tools
such as fixtures become worn and broken and therefore statistical process control (SPC)
is needed as on-line countermeasures against manufacturing variations. Unfortunately, us-
ing routine measurements on the assembled product to track down fixture related errors is
often hard because of the complexity of an autobody assembly and sometimes it is even
impossible to sort out the variation caused by a fixture tooling element.
We have developed a methodology based on knowledge of the product and assembly
process to diagnose multileveled hierarchical assemblies. If the assembly is diagnosable
then multiple fixture faults are successfully pinpointed. Even nondiagnosable assemblies
are handled, where it is often possible to help the process engineers by narrowing down
the set of potential faulty elements. Or, to single out which assembly station to put in the
trouble-shooting efforts.
In the future, it might be necessary to find product/process concept that are not only
insensitive to variation but also possible to diagnose. This important idea of design for
diagnosis has been discussed and exemplified several times in the article.
Nomenclature
d Linearized displacement vector of inspection points
δi j Locator displacement vector in assembly station i and fixture j

Ai j Linear relationship between locator displacement in station i and
fixture j, and the displacement at inspection points
I Number of subassembly stations
J(i) Number of fixtures in station i
l = 6
I
i J(i) Total number of locators in the assembly
N Number of inspected assemblies (sample size)
p Number of inspection points
x Measurement vector
µx Mean vector of x
x Covariance matrix of x
δ Locator covariance matrix
λk The kth diagonal element of δ
z Noise covariance matrix
r Rank of matrix A
Ur Orthonormal basis matrix for r dimensional column space of A
Up−r Orthonormal basis matrix for p − r dimensional null space of AT
ψ Fixture failure variation index
Ip The p times p identity matrix
σ2
Noise factor
L Simplified log likelihood function of measurement vector x
θ Parameter vector containing the locator variations λk and the noise
factor σ2
(2/N)G−1
θ̂
Estimate of the asymptotic covarianc matrix of θ̂
T Diagnosibility matrix of size [p × p,l]
ρ Rank of matrix T
λ Locator variance vector
Vl−ρ Orthonormal basis matrix for the l − ρ dimensional null space of T
Vρ Orthonormal basis matrix for the ρ dimensional column space of T T
γ Parameter vector for the column space of T T
(2/N)G−1
γ̂ Estimate of the asymptotic covariance matrix of γ̂
ψi The total locator variation in station i
ψi j The total locator variation in station i at fixture j
Ak
i j The linear relationship in station i and fixture j, at inspection k
Acknowledgments
The authors wish to acknowledge the support of the Swedish Agency for Innovations Sys-
tems (Vinnova). The authors are also grateful to the referees for their valuable suggestions
and comments.
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