Journal of
Journal
Manufacturing
of Manufacturing
Systems Systems
Vol. 25/No.
Vol.225/No. 2
2006
2006
Identification of Dimensional Variation Patterns
on Compliant Assemblies
Jaime A. Camelio, Dept. of Mechanical Engineering and Engineering Mechanics, Michigan Technological
University, Houghton, Michigan, USA
Hyunjune Yim, Dept. of Mechanical Engineering, Hongik University, Seoul, South Korea
Abstract
Dimensional imperfection of the final body assembly may be traced back to any combination of the
three types of faults: non-nominal shape or size of
parts, non-nominal location or deteriorated condition of fixtures, and welding distortion faults.
Because of its importance, the dimensional quality of automotive body has been pursued in the assembly plants of leading manufacturers, mainly by
applying statistical process control (SPC) methods
to the assembly measurement data. In general, assembly products are measured at the end of the assembly line because it is not economically viable to
measure parts at each station and it is difficult to
measure tooling misalignment. When the SPC gives
a precursor for dimensional quality deterioration, the
root cause of the situation needs to be identified and,
if possible, eliminated. Most advancement in the root
cause identification technology has been oriented
toward fixture faults (Ceglarek and Shi 1995, 1996;
Hu and Wu 1992; Liu and Hu 2005; Apley and Shi
1998; Chang and Gossard 1998; Camelio, Hu, and
Yim 2005; Wang and Nagarkar 1999; Ding, Ceglarek,
and Shi 2002). This is mainly because fixture faults
have been found to be the major cause for defective
automotive body assembly (Ceglarek and Shi 1995).
Significant research has been conducted on fixture fault diagnosis, including the use of principal
component analysis (PCA) and designated component analysis (DCA). PCA is a systematic methodology to diagnose fixture faults by extracting the
principal components from the measured data on the
subassembly (Hu and Wu 1992). Because of its entire dependence on the measurement data, the root
cause of principal components extracted may not
This paper presents a methodology to diagnose sources
of dimensional variation for compliant parts from the measurement data of final assemblies. The method is developed
for a single-station assembly process. The proposed diagnosis tool is based on applying a predictive variation propagation model to determine the part-to-part and tooling interaction
in the assembly system. The variation propagation model
allows identifying the impact of different faulty component
patterns in the final assembly product. Using the predictive
assembly fault patterns and the designated component analysis, the contribution of each fault in the total system variation
may be identified. The methodology incorporates an optimal
sensor placement algorithm to determine the key measurement points in the assembly. Two case studies were conducted to illustrate that the methodology is capable of identifying
part variation patterns from assembly measurement data,
even under significant levels of noise. Although the methodology is presented for a single assembly station, it can be
extended to a multiple-station assembly scenario using a
multistation variation propagation model.
Keywords: Assembly System Diagnosis, Compliant Assembly, Variation Reduction
1. Introduction
Quality improvement in automotive body assembly is one of the major tasks undertaken by the leading automobile manufacturers. Among various
quality measures, the dimensional quality of the final body assembly, also called body in white, has
been regarded as the most important characteristic
because its deterioration affects many quality aspects
of the vehicle—for example, noise and vibration
characteristics such as wind noise, convenience of
use such as the amount of effort required to open
and close the door, and the aesthetic appearance.
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
lems on the assembly. The objective of the present
work is to develop a method to detect part variation
patterns in an assembly station from the measurements on the station subassembly output and not
from the incoming components. The significance of
this work lies in that an assembly may exhibit dimensional variations in the absence of faults in the
fixture and welding process if the incoming parts
have dimensional variation or non-nominal shapes,
which may not be detected or may be misinterpreted
by the methods of fixture fault diagnosis previously
developed. It is understood that decoupling the effects of parts variation from those of fixture faults or
those of welding faults is not trivial and could be a
challenging research topic. In the present study, however, all fixtures and the welding processes are assumed to be perfect, leaving the decoupling problem
as a future research topic.
As in the case of fixture faults, detection of part
dimensional variation may be considered as an inverse problem of the forward problem—the generation of assembly dimensional variation caused by
the component variation fed into the assembly process. There exist a great number of models and
analysis methods for this forward problem, in the
categories called ‘variation simulation’ and ‘tolerance analysis.’ Most of the methods, available in the
form of commercial software, assume rigid behavior of parts. In such methods, the assembly variation
is determined by “adding” the variation of parts according to its geometric and kinematic relations
(Greenwood and Chase 1988), regardless of the
“adding” method used–worst-case analysis, root sum
squares analysis, or Monte Carlo simulation.
Noting the importance of the compliant behavior of the assembly components, Liu and Hu (1997)
developed a mechanistic variation simulation
method by integrating engineering structural models with statistical analysis. In this model, the deformation of parts during the clamping process and
the subsequent springback of the assembly are analyzed by FEA. As a result, the assembly variation is
expressed as linear function of part variation
through the use of sensitivity matrices. The present
work may be considered as an attempt to solve the
inverse problem of the one modeled in Liu and Hu
(1997). The solution technique for this inverse problem will be the least-squares method, facilitated by
the optimal placement of sensors (Camelio, Hu, and
Yim 2005).
always have a physical meaning. To address this
problem, the method of designated component analysis (DCA) was developed by Liu and Hu (2005).
This approach first defines mutually orthogonal dimensional variation patterns (called diagnostic vectors) for a part, each of which is associated with
one or more fixture faults. Then, actual measurement data are decomposed into the components of
the diagnostic vectors, disclosing the major components and thus the associated fixture faults. DCA
is a powerful method for simple problems, but it is
not always obvious how to define physically meaningful, mutually orthogonal patterns for complex
part geometries.
More physics-based diagnostic vectors, which are
only constrained to be linearly independent with
each other, were derived by Ceglarek and Shi
(1996). The diagnostic vectors are defined by
simple geometric analyses of rigid body movement
for each fixture fault. The same diagnostic vectors
were used for diagnosis of multiple fixture faults
using the least squares method by Apley and Shi
(1998). This diagnosis approach was adopted by
Camelio, Hu, and Yim (2005) for the fixture fault
diagnosis in the assembly of complaint parts, where
the diagnostic vectors corresponding to fixture
faults were obtained by finite element analyses
(FEA). Furthermore, Camelio, Hu, and Yim (2005)
also presented a methodology to determine the optimal locations of sensors on the parts by applying
the effective independence (EfI) method so that
different diagnostic vectors can readily be distinguished from each other.
In addition, Ding, Ceglarek, and Shi (2002) developed a method to identify fixture faults in multistage manufacturing processes by applying a
mapping procedure that combines PCA and the pattern recognition approach to a state-space multistage
model. In Ding, Ceglarek, and Shi’s approach, the
parts were all assumed to be rigid and perfect, and
fixture faults were the only cause of non-nominal
product dimensions. Camelio, Hu, and Ceglarek
(2003) incorporated the effects of part variation and
weld gun variation into the state-space model of a
multistation assembly system, yet they did not address the diagnosis problem.
As mentioned earlier, fixture faults are not the only
cause leading to the deteriorated dimensional quality of a product. Variations in the parts and in the
welding process can also result in dimensional prob66
Journal of Manufacturing Systems
Vol. 25/No. 2
2006
2. Methodology
vector, often assumed to have a known covariance
matrix Σ w = σ2w I , where I is the identity matrix.
The problem studied in this paper is how to find
the part variation vector, u, from the measurement
vector, y. The following sections provide detailed
descriptions of the steps taken by the diagnosis methodology.
2.1 Problem Statement
An automotive body is an assembly consisting of
hundreds of sheet metal parts. The product key dimensions, or key product characteristics, are typically measured at several intermediate assembly
stations as well as at the end of the body assembly
line. Each measurement system consists of a set of
sensors that measures displacements at some points
of the assembly or subassembly. The deviations of
these dimensions with respect to their nominal values are caused by various sources of variation: part
geometry errors, faults on fixtures, and/or errors in
the joining process.
As indicated in the introduction, the objective of
this work is to develop a methodology to identify
the variation patterns of the parts based on the data
measured on the assemblies. It is assumed that the
fixtures and the assembly process are both perfect,
thus leaving the parts variation as the only source of
variation. In an initial effort to investigate this issue,
the present study considers a single assembly station where two or more compliant parts are assembled, as shown in Figure 1. The proposed
methodology will ultimately be extended to a series
of multiple assembly stations.
In the single assembly station shown in Figure 1,
the parts variation, denoted by u, yields the assembly variation, denoted by v. These variation vectors,
u and v, represent deviations of specific points on
the parts/components or assembly from their nominal positions in a determined direction. The assembly variation, v, is in general measured using a
coordinate measurement machine (CMM) or an optical CMM. Noise from the measurement process or
from the unmodeled characteristics is also considered in the model. The noise vector, w, is a random
2.2 Modeling of the Problem
Based on the concept of geometric covariance
(Camelio, Hu, and Marin 2004), it can be assumed
that the deformation vector of the assembly components can be decomposed in the contribution of different deformation patterns. Therefore, the part
variation vector u can be modeled as a linear combination of q patterns, b1, b2, …, bq, each of which
may represent a potential variation pattern of the parts,
as will be explained in section 2.3. Then, the deformation vector for the parts can be written as
u = x1 ⋅ b1 + x2 ⋅ b2 + … + xq ⋅ bq = B ⋅ x
where xi (i = 1, 2, …, q) are the coefficients of the
linear combination and represent the weight of the
each pattern in u. The dimensions of the vectors, bi
and u, are both assumed to be p × 1; x vector dimensions are q × 1; and B dimensions are p × q. p corresponds to the number of input sources or key control
characteristics (KCC).
As a result of the part variation given by Eq. (1),
the assembly dimensions will vary as shown in Figure 1. Several variation analysis models have been
presented to describe the relationship between the
parts variation and assembly variation. For the spotwelding process of compliant parts, as in the automotive body assembly, it is possible to predict the
assembly variation from the components/parts variation (Liu and Hu 1997). This prediction is made us-
Noise, w
Components
Variation
u
(1)
Assembly
Variation
Assembly Process
v
Measurements
Measurement
System
Figure 1
Propagation of Variations in an Assembly Process
67
y
Journal of Manufacturing Systems
Vol. 25/No. 2
2006
ing the sensitivity matrix defined by Liu and Hu’s
model. The sensitivity matrix can be obtained for
each assembly process by conducting finite element
analyses and using the method of influence coefficients. These finite element analyses require information on the nominal geometry of the parts, the
fixturing scheme of each part, and the locations of
the welding spots. This information must be obtained
from the existing assembly line or from the design
drawings. The method of Liu and Hu (1997) may be
expressed mathematically as
⎧ v1 ⎫ ⎡ s11
⎪v ⎪ ⎢s
⎪ ⎪ ⎢ 21
v=⎨ 2⎬=⎢
⎪#⎪ ⎢ #
⎪⎩vm ⎪⎭ ⎢ s
⎣ m1
s12
s22
#
sm 2
" s1 p ⎤ ⎧ u1 ⎫
⎥
" s2 p ⎥ ⎪⎪u2 ⎪⎪
⎨ ⎬ = S⋅u
% # ⎥⎪ # ⎪
⎥
" smp ⎥⎦ ⎪⎩u p ⎪⎭
(4) that the measurement vector, y, has the same dimension, m, as the assembly variation vector, v. These
m points of measurement must be carefully selected,
as will be discussed in detail in section 2.4, and thus
the sensitivity matrix, S, must be defined such that S ⭈
u yields v at those m selected points.
The columns of the matrix D, often called the diagnostic vectors, may be interpreted as the effect of
deformation pattern bi represented by each diagnostic vector, di = S ⭈ bi , on the assembly variation measurement. The components of the vector x (Eq. 3)
may accordingly be interpreted as the weights of those
components existing in the measurement y. From
Eqs. (3) and (4), the problem of the present study
may be interpreted as finding the weights of diagnostic vectors x from the assembly measurement data
y given the diagnostic matrix D. In other words, the
diagnostic problem is to estimate the amount of variation in the measured data that can be explained by
each designated pattern or fault. A fault is defined as
a source of variation instead of a shift in the mean.
Therefore, if the variance of a pattern (xi) is significant, a fault will be identified.
The covariance matrix of the designated components, xi, can be calculated as follows:
(2)
where the element of the sensitivity matrix, sij, measures the sensitivity of the assembly variation at the
i-th point to the incoming part variation at the j-th
point. This model is based on several assumptions:
(1) deformation on parts in the assembly process is
within the linear elastic range; (2) the material is
homogeneous and isotropic; (3) fixtures and welding guns are rigid; (4) there is no thermal deformation due to welding; and (5) the stiffness matrix for
non-nominal part shapes remains the same as that
for nominal shapes. The validity of these assumptions is presented in Liu and Hu (1997).
Note that the assembly variation vector, v, in Eq.
(2) has dimensions of m × 1, generally different from
those of the parts variation vector, u. The value of m
here corresponds to the number of measurement points
on the assembly, which will be discussed shortly.
The measurement vector, y, in Figure 1 is simply
modeled as the sum of the assembly variation and
noise. That is, using Eqs. (1) and (2),
y = v + w = S⋅u + w = S⋅B⋅x + w
Σ Y = D ⋅ Σ X ⋅ DT + Σ w
From Eq. (5), the covariance matrix of the measurements points, ⌺Y, can be seen as a linear stackup of the variation effect of the designated patterns,
D ⭈ ⌺X ⭈ DT, and the nonmodeled effects variance, ⌺w.
2.3 Definition of Part Variation Patterns
The part variation vectors, b1, b2, …, bq, in Eq.
(1) are defined such that they correspond to patterns typically observed in the specific component
fabrication process. For example, a deformation
pattern for an assembly part/component can be a
bending effect from the stamping process or a twisting effect from the material handling equipment.
These patterns can be determined by experts or
measured from available past data from suppliers.
Figure 2 shows some examples of deformation patterns for a rectangular plate part that has been
clamped along its left-hand edge. For complex assembly geometries, such patterns will be constructed using finite element simulations or obtained
from previous components measurement data
(Camelio, Hu, and Marin 2004).
(3)
Given that the matrices S and B are constant and
known, Eq. (3) can be simplified as follows:
D = S⋅B
y = D⋅x + w
(5)
(4)
where D, often called the diagnostic matrix, is defined as an m × q matrix. The matrix D represents the
effect of the different deformation patterns of the components on the assembly. It should be noted from Eq.
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
(a) Bending about x-axis, b1
(b) Twisting about y-axis, b2
z
y
x
(c) Bending about y-axis, b3
Figure 2
Examples of Part Variation Patterns (Camelio, Hu, and Marin 2004)
the locations of the measurement points are very important. To facilitate the fault identification, m measurement points must be selected such that the
diagnostic vectors, that is, the column vectors of D in
Eq. (4), may easily be differentiated from each other.
Camelio, Hu, and Yim (2005) proposed a methodology for optimal placement of sensors. This
method is essentially an iterative numerical method
that finds the most effective sensor locations by starting with a great number of candidate locations. The
candidate locations are typically at the nodes of the
finite element mesh of the assembly components.
The method eliminates the least-effective sensor in
each iteration step, based on the effective independence (EfI) criterion. This iteration proceeds until
the desired number of sensors are left. During this
iterative process, the number of rows of the sensitivity matrix, S, in Eq. (2) and thus that of the diagnostic matrix, D, in Eq. (3), are reduced from the number
of all finite element nodes to the desired number of
measurement points. The effectiveness of the sensor locations obtained by the procedure assures that
they provide the greatest linear independency among
the diagnostic vectors, each associated with a single
fault of the parts.
In this paper, the optimal sensor placement problem is solved using a genetic algorithm (GA) (Goldberg
1989). The objective function in this method is to
maximize the determinant of the Fisher information
Considering the patterns shown in Figure 2 and
defining the sources of variation as three points along
the edge in the x-direction (points along the right-hand
edge of the plate, one at the midpoint and two at the
ends), the matrix B will have the following form:
B = [ b1 b2
⎡1 1 1 ⎤
b3 ] = ⎢⎢1 0 0 ⎥⎥
⎢⎣1 −1 1 ⎥⎦
(6)
Each pattern vector, bi, is defined as the deviations from nominal at the sources of variation, which
are often located at the welding points. Thus, the iT
th pattern vector is written as bi = {bi1, bi2, …, bip} ,
where p is the total number of welding points over
the parts, and bij is the deviation from the nominal at
the j-th welding point for the i-th pattern. As will be
discussed in section 3.1, the pattern vectors, bi, must
be linearly independent of each other. It must be
noted that the vectors bi may be also defined for deviations occurring in the in-plane directions, however, for simplification, in this example they are only
defined as deviation normal to the surface plane on
a reduced set of key control characteristic points.
2.4 Placement of Measurement Points
Because the problem is to identify the part fault
patterns from the measurement data on the assembly,
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Journal of Manufacturing Systems
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T
matrix, Q = D D. Among the subsets consisting of m sensor locations, constructed
from all candidate sensor locations (in this
case, corresponding to the finite element
nodes of the assembly), the one associated
with the maximum value of determinant of
Q is sought using the GA, and it is selected
as the optimal set of sensor locations. The
GA method has been found to be significantly faster than the iterative elimination
method in Camelio, Hu, and Yim (2005).
Off-Line Preparation
Define Parts Variation
Patterns, H
Locate Sensors and Calculate
Diagnostic Matrix, D
On-Line Diagnosis
2.5 Diagnosis Methodology
The entire diagnosis methodology for the
problem defined and modeled above consists of two steps, as shown in Figure 3, the
off-line preparation procedure, and the
online diagnosis procedure. The off-line
procedure mostly has been explained in the
previous sections. That is, the part variation pattern matrix, B, is first established by
assuming typical patterns of parts (see section 2.3). Then, the sensitivity matrix, S, is
obtained by conducting finite element
analyses and using the method of influence
coefficients. In this step, the number of rows of S is
the total number of the finite element nodes of the
assembly. The final step in the off-line procedure is
to construct the diagnostic matrix, D, as S ⭈ B and to
reduce its number of rows on D to the desired number of measurements, m, by the optimal sensor placement method described in section 2.4.
The diagnosis procedure is conducted online as
shown in Figure 3. The measurement data are obtained online from the m measurement sensors,
which provide y(j), j = 1, 2, …, N, for the j-th assembly. For each measured part, a measurement
(j)
vector, y , is obtained. The corresponding weight
(j)
vector, x , is estimated using the least-squares
method, as
(
xˆ ( j ) = DT D
)
−1
Construct Sensitivity
Matrix, S
DT y ( j )
Calculate Assembly Patterns
Significance (LSM)
Measurement
Data, y
Calculate Patterns Correlation
Identification of
Multiple Part Faults
Figure 3
Part Variation Diagnosis Methodology
After the weight vector, x, is estimated using Eq.
(7) for N assemblies, the variance of the contribution of each diagnostic vector (corresponding to
each of the q part/component variation patterns) is
calculated as
(
1 N ( j)
∑ xˆi − xˆ i
N − 1 j =1
i = 1,2," , q
σi2 =
),
2
(8)
where xˆ i = ∑ j =1 xˆ i( j ) N . The matrix form of the covariance can be expressed as
N
(
∑ xˆ = DT D
)
−1
(
DT ∑ y D DDT
)
−1
It must be noted that the covariance of the measurement data, ⌺y, includes the effect of the deformation patterns variation, ⌺x, and the effect of the
noise variation, ⌺w. Therefore, to estimate the contribution of each deformation pattern, the noise effect should be removed from the estimation ∑ xˆ . The
effect of the noise in the estimation of x is ∑ xˆ ( w) and
corresponds to the effect on the measurement data if
only noise is present.
(7)
(j)
where xˆ ( j ) is the least-squares estimator of x . The
estimator xˆ ( j ) differs from the real contribution because is has the noise effect included. This approach
T
requires the matrix D D to be invertible. In other
T
words, D D must be of full rank. This restriction will
be further discussed in section 3.1.
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
(
∑ xˆ ( ) = DT D
w
)
−1
(
DT ∑ w D DDT
)
3. Special Considerations
−1
3.1 Diagnosability
Diagnosability is defined here as the capability of
the proposed diagnostic procedure to detect and isolate a specific pattern or multiple patterns of part
variation. The diagnosability analysis investigates
under which conditions the proposed methodology
will be able to identify the faults.
From Eq. (7), the estimate x̂ of the part deformation pattern contributions to the assembly variation
T
T
cannot be obtained if D D is not invertible. D D will
be invertible if the columns of D are linearly independent or, equivalently, if D is full rank (Strang
1988). Recalling that D = S ⭈ B [Eq. (4)] and given
that the part patterns (columns of the matrix, B) are
forced to be linearly independent as discussed in
T
section 2.3, D D will be invertible if the columns of
S are linearly independent. Then, the diagnosability
condition is given by
The percent contribution of each pattern, Ci, to
the total variation can be calculated individually as
a proportion of the measurement data variation explained by each pattern. The total variation of the
system is defined as the sum of the variance of all
patterns. The effect of each pattern is calculated as
the effect of the estimated variance of each significant pattern in the total variation of the significant
patterns. Therefore, the contribution of each pattern
is obtained as follows:
(
C =
trace ( Σ
) ×100%,
−Σ ( ))
diag Σ xˆ − Σ xˆ (
i
w)
w
xˆ
xˆ
(9)
i = 1,2," , q
where Σ xˆ − Σ xˆ ( w ) denotes a matrix that includes only
the significant patterns. The significant patterns may
be identified using a hypothesis testing presented
in section 3.2. The function diag is defined as the
diagonal of a matrix. The trace function is defined
as the sum of the variances of all significant components. If the i-th diagnostic vector (or equivalently the i-th assembly pattern) is found to be
significant, the i-th part variation pattern, bi, is regarded as a significant source for the assembly
variation.
In addition, because multiple faults may result in
significant correlations among the assembly variation patterns, the correlations between patterns may
be calculated in Eq. (8).
rank ( S ) = p
where p is the number of columns in S and corresponds to the number of sources of variation. The
rank function counts the number of independent columns in a matrix. It should be clear that this independence requirement means the linear independence of
vectors, not their statistical independence. In fact,
designated components are often correlated with each
other as measurements are made. The correlation
matrix is used in the diagnosis methodology.
Furthermore, if D is not of full rank, the sensor
placement procedure explained in section 2.4 also
fails because the determinant of the Fisher information matrix becomes zero.
Even in the cases where D is not of full rank, partial diagnosis can be done. A full-rank submatrix,
Dind, may be constructed by selecting a linearly independent set of columns of D. The other columns
of D constitute the complement submatrix, Dcind , then
D = ⎡⎣ D ind Dcind ⎤⎦ . By definition, the columns (or
assembly patterns) in the submatrix, Dcind , are linear
combinations of the patterns in matrix Dind. The leastsquares estimation can then be performed only for
the linearly independent patterns in Dind as
N
ηij =
r
r
∑ xˆ i( ) xˆ (j )
r =1
N
( ) ( )
r
∑ xˆ i( )
r =1
2 N
r
∑ xˆ (j )
,
2
(11)
(10)
r =1
i, j = 1,2," , q
The correlation among the patterns can be useful to identify the root cause in the presence of
multiple faults occurring simultaneously. It should
be noted that it is only meaningful to analyze the
correlation among the significant patterns. However, statistical independence among the patterns
is usually expected.
(
x̂ ind = DTind ⋅ D ind
)
−1
DTind y
(12)
where x̂ ind is a subvector of x consisting only of
those components of x corresponding to Dind. The
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significance of these independent assembly patterns
is evaluated by Eqs. (8) and (9).
In the latter case of limited diagnosability, the dependency among assembly patterns means that the
diagnosis method cannot identify and isolate the difference among the dependent patterns. Patterns dependent on each other can be grouped into ⍀j, j = 1,
2, …. Patterns in a group, ⍀j, are dependent on each
other but independent of patterns in other subgroups.
If one or more of the patterns in a group is identified
as significant, it means that any pattern in the group
can be the source of variation.
pothesis testing for the variance of a normal population, the test in Eq. (15) may be conducted to identify if the contribution of a designated pattern is
statistically significant. The interpretation of the test
is that for each deformation pattern i the variance of
the pattern must be statistically greater that the variance of the noise w to diagnose a pattern.
(
(
)
−1
= σ2w DT D
)
(
DT ⎤ Σ y ⎡ DT D
⎦⎥ ⎣⎢
−1
)
−1
DT ⎤
⎦⎥
w)
w
xi
2
w
xi
xi
2
2
i = 1,… , q
(15)
To evaluate the hypothesis, we will use the test statistic:
χ20 =
( m − 1) Sx2
(σ )
(w)
i
2
(16)
xˆi
2
where Sx is the estimate of the variance of the deformation pattern i, and m is the number of observations in the data y. The null hypothesis H0 will be
rejected if
i
χ20 > χ2α ,m −1
(17)
If the null hypothesis is rejected for a specific deformation pattern j, then the pattern is a significant
contributor of the system variation.
4. Case Studies
The proposed methodology is illustrated with two
examples. The first example considers the assembly
of two rectangular sheet metal parts. This simple
example is used to clarify each step of the methodology without including difficulties coming from
complex geometries. A second example considers a
realistic automotive example, that is, the assembly
of a fender and its reinforcement.
T
(13)
4.1 Assembly of Two Metal Sheets
The proposed diagnosis methodology is first exemplified with the assembly of two sheet metal parts,
as shown in Figure 4. Each part has dimensions of
400 mm x 400 mm x 1 mm. The material is mild steel
with Young’s modulus E = 207 GPa and Poisson’s
ratio, ␯ = 0.3. Both parts are constrained along one
edge parallel to the y-axis. The parts are joined together at three weld points (designated as W1, W2,
and W3 in Figure 4) located along the mating edge.
where Σ y = Σ w = σ2w I has been used.
From Eq. (4), the covariance matrix for x̂ can be
calculated as
Σ xˆ = Σ x + Σ(xˆ
2
xi
H1 :
3.2 Statistical Significance Under Noise
Using Eq. (9), the contribution of each part variation pattern to the total variation of the assembly can
be determined. However, it is necessary to examine
the statistical significance of each fault identified.
For example, even for the in-control case (i.e., when
no faults are present), the noise in the measured data
may generate large values for Ci. Therefore, a large
value of Ci does not necessarily imply that the i-th
pattern is a significant source of variation. The statistical significance of each fault may be determined
using statistical hypothesis testing.
The proposed approach assumes that the covariance matrix of w, ⌺w, is known and xi, i = 1, 2, …, q,
(deformation patterns deviation) exhibit a normal
distribution. Based on these assumptions, the covariance matrix for x, ∑ xˆ , can be estimated for in-control conditions (⌺Y = ⌺w). The noise contribution of
the measuring system and the locating fixture is included in the term ⌺w. The covariance matrix of x̂ ,
w
∑(x̂ ) , solely due to noise can be estimated from Eq.
(7) as follows:
∑ xˆ ( w ) = ⎡ DT D
⎣⎢
( σ ) = ( σ( ) )
( σ ) > ( σ( ) )
H0 :
(14)
Therefore, if there is a fault present the estimation
of the contribution of the faults will be larger than
the estimation of x solely due to noise. Using hy-
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
part variation patterns. The matrix, D, originally has dimen400 mm
sions of 242 × 6. Using the
W1 (12)
400 mm
optimal sensor placement apW2 (17)
proach described in section 2.4,
W3 (2)
Part 2
three optimal measurement
points were selected as the
Part 1
Z
welding points. This means that
X
at these welding points the asY
sembly variation patterns may
be most easily differentiated
Figure 4
Assembly Example of Two Rectangular Plates
from each other. The measurement points selected are shown
The procedure in Figure 3 is followed. Six part
as Nodes 12, 17, and 2 in Figure 4. The resulting
variation patterns, bi, are considered in this example.
diagnostic matrix, D, of reduced dimensions 3 × 6 is
Each part is assumed to exhibit three patterns (simiD = [ d1 d 2 d3 d 4 d 5 d6 ]
lar to the patterns shown in Figure 2): bending along
the y-axis, twisting along the x-axis, and bending
0.281 0.265 0.112
0.281 0.265 ⎤
⎡ 0.112
⎢
along the x-axis. Therefore, the part variation pat0
0
−0.019 ⎥⎥
= ⎢ 0.103
−0.019 0.103
terns for the two parts may be written as the out-of⎢⎣ 0.112 −0.281 0.265 0.112 −0.281 0.265 ⎥⎦
plane deviations of the parts from the nominal at the
welding points. A similar approach can be considFrom matrix D, it can be seen that d1 = d4, d2 = d5,
ered for in-plane rigid motion. The resulting part
and d3 = d6. Recall that there is one-to-one correvariation pattern matrix, B, with dimensions of 6 × 6
spondence between the assembly patterns, di’s, and
is found to be
the part variation patterns, bi’s. Then, the dependence
among di’s observed above means that the diagnosis methodology cannot distinguish between the
B = [ b1 b2 b3 b 4 b5 b6 ]
same type of part variation patterns of the two parts.
⎡0.574 0.707 0.707
⎤
The reason for this is, of course, the symmetry of
⎢0.574
⎥
∅3×3
0
0
⎢
⎥
the problem, that is, identical part geometries, and
⎢0.574 −0.707 0.707
⎥
identical part variation patterns for the two parts. In
=⎢
⎥
0.574
0.707
0.707
general, if there were lack in complete symmetry,
⎢
⎥
⎢
⎥
there would have been no diagnosability problem.
∅3×3
0.574
0
0
⎢
⎥
Following the method in section 3.1, the follow0.574 −0.707 0.707 ⎦
⎣
ing groups may be defined for the dependent patterns: ⍀1 = {d1, d4}; ⍀2 = {d2, d5}; ⍀3 = {d3, d6}.
where b1, b2, b3 represent the variation patterns for
Selecting the first members of the three groups to
one part, b4, b5, b6 for the other part, and ∅3×3 deform a submatrix of the diagnostic matrix gives
notes a zero matrix having dimensions of n × m.
Then, using the finite element method and the
D ind = [ d1 d 2 d3 ]
method of influence coefficients, the sensitivity ma⎡0.112 0.281 0.265 ⎤
trix, S, is obtained for this problem. The finite ele0
= ⎢⎢ 0.103
−0.019 ⎥⎥
ment mesh of each part has 121 nodes, and thus the
sensitivity matrix has dimensions of 242 × 6. As
⎢⎣0.112 −0.281 0.265 ⎥⎦
mentioned previously, sensitivity matrix represents
Allowing only one part to have variation, a simulathe response of the assembly to a unit deviation of
tion study was performed. Cases of the three part variaparts at each welding point.
tion patterns, b1, b2, b3, are exclusively simulated. For
Next, using Eq. (4), the diagnostic matrix, D, is
each case, the number of assemblies simulated was N
computed. Recall that the columns of D represent
= 100, and the faulty pattern was assumed to have a
the assembly variation patterns corresponding to the
400 mm
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
M5
Fault
Faults
d1
d2
d3
M1
M2
M3
M4
No Fault
0%
20%
40%
60%
80%
Noise to Signal Ratio
Figure 5
Simulation Results for Rectangular Plate Assembly
variance of 1 mm2. In addition, different levels of
noise have been simulated, with a variance of a given
percentage of the pattern variance.
Figure 5 shows the results of the study. The presence of a part fault will be identified as 1, and a zero
value means that the fault is not significant based in
the hypothesis testing. Figure 5 shows the capability
of the methodology to detect a part fault with respect to the level of noise. It can be seen that under
the noise level over 7 percent, Pattern 1, d1, will have
a missed alarm. The probability of a missed alarm is
lower for Pattern 2, d2; that is, Pattern 2 is diagnosable for noise level up to 49 percent. This seems to
imply that the second pattern produces stronger signal than the others, thus making it more discernible
even under high level of noise. Similar results show
that the probability of false alarm is low for same
level of noise as the simulated in Figure 5.
Figure 6
Finite Element Mesh of Fender-Reinforcement Assembly
the y-direction) at Nodes 2769 (L3), 4598 (L4), and
472 (L5). For the reinforcement component, the locating scheme consists of a hole-pin (constraining
displacements in the x-, y- and z-directions) at Node
10815 (L1), a slot-pin (constraining displacement in
the y- and z-direction) at Node 11988 (L2), and a
locating pad (constraining displacements in the ydirection) at Node 10805 (L3). The parts are joined
together at four welding points that constrain Nodes
1001/11103 (W 1), 4530/12163(W 2), 1395/10178
(W3), and 2699/11628 (W4). Fixtures and welding
locations are shown in Figure 7 for the fender and in
Figure 8 for the reinforcement.
In this study, only two component deformation
patterns of the reinforcement are considered: bending and twisting. The two reinforcement variation
patterns are defined, in terms of the out-of-plane (ydirection) deviations from the nominal at the welding locations, as
4.2 Assembly of Automotive Fender
A more realistic case study was conducted by
simulating a fender-reinforcement assembly. Figure
6 shows the finite element mesh for the assembly.
The assembly consists of two parts: (1) a fender component, and (2) a reinforcement that is assembled to
the top of the fender along its longitudinal direction.
Each part was assumed to be located following
the 3-2-1 locating scheme (Cai, Hu, and Yuan 1996).
For the fender, the locating scheme consists of a hole
(constraining displacements in the x- and z-directions) at Node 3098 (L1), a slot (constraining displacements in the z-direction) at Node 855 (L2), and
three locating pads (constraining displacements in
⎡
⎢
⎢
B = [ b1 b2 ] = ⎢
⎢
⎢
⎢⎣
∅ 4×2
0.5
0.5
0.5
0.5
⎤
0 ⎥⎥
0.707 ⎥
⎥
−0.707 ⎥
0 ⎥⎦
The matrix B includes the key control characteristics (sources of variation) for both parts, the fender
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
L1
W2
4530
W3
1395
3098
W4
2699
W1
1001
W1
1001
L5
L2
L1
472
855
10815
L4
L2
L3
4598
855
2769
W2
12163
W3
1395
W4
11628
L3
Figure 8
Fixture and Welding Locations for Reinforcement
2769
points reduces the dimensions of D to 5 × 2. The
reduced matrix, D, is
⎡
⎢
⎢
D = [ d1 d 2 ] = ⎢
⎢
⎢
⎢⎣
Figure 7
Fixture and Welding Locations for Fender
and the reinforcement. However, this case study only
includes failure modes on the reinforcement. Given
that the reinforcement is more rigid than the fender
itself, the effect of the fender can be neglected. Therefore, it should be noted that the patterns have zeros
for the welding points in the fender component.
Using the method of influence coefficients, the
sensitivity matrix, S, is obtained. The input variables
are the sources of variation defined at the welding
points, Wi, i = 1, …, 4. The output variables are at
the 7317 finite element nodes in the assembly. The
sensitivity matrix has dimensions of 7313 × 8.
Using Eq. (4), the diagnostic matrix, D, may be
computed, which has dimensions of 7317 × 2. Among
the potential 7317 locations for measurements, only
five measurement points were selected using the
optimal sensor placement methodology based on
genetic algorithms described in section 2.4. The
measurement points selected are Nodes 1612 (M1),
3013 (M2), 4962 (M3), 4966 (M4), and 5707 (M5), as
shown in Figure 6. It must be noted that based on
the genetic algorithm, the five selected sensors may
not be an optimal solution and only be a good set of
sensors. In addition, to identify two faults only two
sensors are needed. Therefore, some of the measurement points are close together in areas were the deformation or the signal or the sensor will be larger.
Accordingly, this selection of five measurement
0.5558
0.5633
0.8187
0.8147
0.3026
−0.2544 ⎤
−0.2602 ⎥⎥
0.0764 ⎥
⎥
0.0799 ⎥
0.3887 ⎥⎦
This reduced matrix, D, is of full rank. This means
that the two assembly variation patterns, resulting
from the two part variation patterns, are independent of each other. Both patterns are thus fully diagnosable with the five measurement points. The
diagnosis methodology was evaluated similarly to
the previous example. The methodology was capable
of successfully diagnosing the two part variation
patterns—bending and twisting with noise levels
under 30% of the system variation.
5. Conclusions
A methodology has been developed for the diagnosis of the variation contribution of assembly components using measurement data from the final
assembled products. The method assumes that typically occurring part variation patterns are known from
the characteristics of the specific manufacturing process of the parts. Using the previously developed
method of influence coefficients for compliant parts
assembly, the sensitivity of components geometrical variation on the assembly product is determined.
Finally, by using the sensitivity matrix and selecting
a set of optimal sensor locations, the contribution of
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Journal of Manufacturing Systems
Vol. 25/No. 2
2006
assembly variation patterns (which constitute a diagnostic matrix) as measured by the sensors is computed. Because these assembly variation patterns
correspond to the part variation patterns, it is possible
to diagnose particular part variation patterns by identifying the significant assembly variation patterns.
Diagnosability of part variation patterns using the
proposed methodology has also been studied in relation to the full-rankness of the diagnostic matrix.
Furthermore, the effects of the measurement noise,
which may also represent the unmodeled behavior
on the diagnosis, were discussed. To illustrate the
capability of the proposed methodology, two case
studies were performed: an assembly of two identical rectangular plates, and a fender-reinforcement
assembly. In both cases, the methodology developed
in this study was capable of identifying the part variation patterns even under some levels of noise. In the
first simple case, some diagnosability issues occurred
because the diagnostic matrix turned out to be less
than full rank due to the complete symmetry of the
problem. Given that, in general, real problems do
not possess such complete symmetry, as in the second case of fender-reinforcement assembly, such a
problem is not expected to occur in real cases.
Having established a methodology for the diagnosis of part variation in a single-station assembly,
the next natural problem to study will be the extension of the method to multiple-station assembly process. No major obstacle is expected in doing so,
except the increased complexity of calculations.
Camelio, J.; Hu, S.J.; and Marin, S. (2004). “Compliant assembly
variation analysis using components geometric covariance.” ASME
Journal of Mfg. Science and Engg. (v126, n2), pp355-360.
Camelio, J.; Hu, S.J.; and Yim, H. (2005). “Sensor placement for
effective diagnosis of multiple faults in fixturing of compliant
parts.” ASME Journal of Mfg. Science and Engg. (v127, n1),
pp68-74.
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automotive body assembly.” Mfg. Review (v8, n2), pp139-154.
Ceglarek, D. and Shi, J. (1996). “Fixture failure diagnosis for
autobody assembly using pattern recognition.” ASME Journal of
Engg. for Industry (v118), pp55-66.
Chang, M. and Gossard, D.C. (1998). “Computational method for
diagnosis of variation-related assembly problems.” Int’l Journal
of Production Research (v36, n11), pp2985-2995.
Ding, Y.; Ceglarek, D.; and Shi, J. (2002). “Fault diagnosis of multistage manufacturing processes by using state space approach.”
ASME Journal of Mfg. Science and Engg. (v124), pp313-322.
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(v110), pp232-235.
Hu, S.J. and Wu, S.M. (1992). “Identifying root causes of variation in
automobile body assembly using Principal Component Analysis.”
Transactions of NAMRI/SME (v20), pp311-316.
Liu, S.C. and Hu, S.J. (1997). “Variation simulation for deformable
sheet metal assemblies using finite element methods.” ASME Journal of Mfg. Science and Engg. (v119), pp368-374.
Liu, Y.G. and Hu, S.J. (2005). “Assembly fixture fault diagnosis
using Designated Component Analysis.” ASME Journal of Mfg.
Science and Engg. (v127, n2), pp358-368.
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Authors’ Biographies
Dr. Jaime Camelio is currently an assistant professor in the Mechanical Engineering-Engineering Mechanics Dept. at Michigan Technological University. Previously he was a consultant in the Automotive/
Operations Practice at A.T. Kearney Inc. Dr. Camelio obtained his BS
and MS in mechanical engineering from the Catholic University of
Chile in 1994 and 1995, respectively. In 2002, he received his PhD
from the University of Michigan. After graduation, he held a
postdoctoral research position and later a research scientist position in
the Dept. of Mechanical Engineering at the University of Michigan,
Ann Arbor. His research interests are in assembly systems modeling,
uncertainty management, systems diagnosis/prognosis, and
remanufacturing. Dr. Camelio received the 2007 SME Kuo K. Wang
Outstanding Young Manufacturing Engineer award.
Acknowledgments
The authors acknowledge the case study data provided by Mr. Eric Chen from General Motors Corporation.
References
Apley, D. and Shi, J. (1998). “Diagnosis of multiple fixture faults in
panel assembly.” Journal of Mfg. Science and Engg. (v120),
pp793-801.
Cai, W.; Hu, S.J.; and Yuan, J.X. (1996). “Deformable sheet metal
fixturing: principles, algorithms, and simulations.” Journal of Mfg.
Science and Engg. (v118), pp318-324.
Camelio, J.; Hu, S.J.; and Ceglarek, D. (2003). “Modeling variation
propagation of multi-station assembly systems with compliant parts.”
ASME Journal of Mechanical Design (v125, n4), pp673-681.
Hyunjune Yim is director of the PACE Center and a professor in
the Mechanical and System Design Engineering Dept. at Hongik
University (Seoul, Korea). He received his bachelor’s (1984) and
master’s (1986) degrees in mechanical engineering from Seoul National University and PhD (1993) in mechanical engineering from
the Massachusetts Institute of Technology. Dr. Yim’s current research
interests include tolerance analysis and design, digital manufacturing, and digital product development.
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