Process Yield Improvement
Through Optimum Design of
Fixture Layouts in 3D Multistation
Assembly Systems
T. Phoomboplab
D. Ceglarek
Warwick Digital Laboratory, WMG
University of Warwick,
Coventry CV4 7AL, UK;
Department of Industrial and Systems
Engineering,
University of Wisconsin–Madison,
Madison, WI 53706
Fixtures control the positions and orientations of parts in an assembly process. Inaccuracies of fixture locators or nonoptimal fixture layouts can result in the deviation of a
workpiece from its design nominal and lead to overall product dimensional variability
and low process yield. Major challenges involving the design of a set of fixture layouts for
multistation assembly system can be enumerated into three categories: (1) highdimensional design space since a large number of locators are involved in the multistation system, (2) large and complex design space for each locator since the design space
represents the area of a particular part or subassembly surfaces on which a locator is
placed, (here, the design space varies with a particular part design and is further expanded when parts are assembled into subassemblies), and (3) the nonlinear relations
between locator nominal positions and key product characteristics. This paper presents a
new approach to improve process yield by determining an optimum set of fixture layouts
for a given multistation assembly system, which can satisfy (1) the part and subassembly
locating stability in each fixture layout and (2) the fixture system robustness against
environmental noises in order to minimize product dimensional variability. The proposed
methodology is based on a two-step optimization which involves the integration of genetic algorithm and Hammersley sequence sampling. First, genetic algorithm is used for
design space reduction by estimating the areas of optimal fixture locations in initial
design spaces. Then, Hammersley sequence sampling uniformly samples the candidate
sets of fixture layouts from those predetermined areas for the optimum. The process yield
and part instability index are design objectives in evaluating candidate sets of fixture
layouts. An industrial case study illustrates and validates the proposed methodology.
关DOI: 10.1115/1.2977826兴
Keywords: fixture layout, multistation assembly, process yield, instability index, genetic
algorithm, Hammersley sequence sampling
1
Introduction
Fixture design is one of the most important design tasks during
process design for a new product development since it involves
defining the locations and orientations of parts during assembly
processes as well as providing physical support, which can greatly
affect product dimensional variations and process yield. Generally, fixture design process can be divided into three stages which
are 共1兲 fixture planning, 共2兲 fixture configuration, and 共3兲 fixture
construction 关1兴. In the fixture planning stage, issues related to the
number of fixtures needed, the type of fixtures, the orientation of
fixture corresponding to orientation, and the joining or machining
operations, which fixtures have to handle are identified. The fixture configuration stage determines the layout of a set of locators
and clamps on a workpiece surface such that the workpiece is
completely restrained. Finally, the fixture construction stage involves constructing fixture components and then installing them to
support the workpiece. Specifically for complex assemblies such
as an automotive body, a ship hull, and an aircraft fuselage, fixture
layout design, which falls under the domain of the fixture planning and fixture configuration stages, is a primary concern and it
Contributed by the Manufacturing Engineering Division of ASME for publication
in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received
August 17, 2007; final manuscript received July 18, 2008; published online October
10, 2008. Review conducted by Shivakumar Raman. Paper presented at the 2007
International Conference on Manufacturing Science and Engineering 共MSEC2007兲,
Atlanta, GA, October 15–17, 2007.
involves adjusting the design nominalof locator positions in order
to eliminate mean shifts and reduce variations of all key product
characteristics 共KPCs兲.
However, research on fixture layout design in a multistation
assembly process is limited because of the lack of a methodology
to predict product dimensional variations and a process yield during the product/process design phase 关2–4兴. The absence of such a
methodology poses special challenges in assessing a performance
of a fixture layout design and its impact on a process yield. Currently, researchers have developed a variation propagation model
for multistation assembly processes using a state-space representation 关5–7兴, which allows prediction of KPC variations under
known statistical characterizations of key control characteristics
共KCCs兲. The extension of this work to three-dimensional 共3D兲
rigid body assembly processes has led the development of the
so-called stream-of-variation 共SOVA兲 methodology 关8,9兴.
Nevertheless, the fixture layout design for multistage assembly
processes continues to pose various challenges. For example, multistage assembly processes usually involve a large number of locators since these processes consist of a large number of parts
assembled in many assembly stations. For instance, a typical automotive body assembly consists of 200–250 sheet metal parts
assembled in 60–100 assembly stations with 1700 to 2100 locators 关4,10兴. In addition, the locating positions used in one station
may be reused in the different stations. Thus, a fixture layout
design methodology optimizing a fixture layout independently for
each assembly station may not necessarily lead to a good design
Journal of Manufacturing Science and Engineering
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solution because of the interdependencies between the fixture layouts in assembly systems. The challenges in designing a set of
fixture layouts for multistation assembly system can be summarized into three categories as follows.
共1兲 High-dimensional design space. Multistation automotive
assembly processes have a very large number of locators
since the number of locators increases proportionally to the
number of parts and assembly stations.
共2兲 Large and complex design space for each locator. The design space of a locator can be defined as any allowed position of the locator on a corresponding workpiece surface.
Thus, the design space of each locator varies with the size
and surface shape of the workpiece. In addition, the design
space is further expanded since fixtures need to be designed
not only for individual parts but also for all intermediate
subassemblies.
共3兲 Nonlinear relations between locator nominal position and
KPCs. Deviations of locators from their nominal positions
have nonlinear effects on KPC variations.
This paper addresses the aforementioned challenges by proposing a methodology for optimizing fixture layouts in all assembly
stations simultaneously. This is to ensure that the variation propagation throughout the whole assembly line is considered to
achieve the dimensional quality threshold of the final product. The
SOVA model as presented in Refs. 关8,9兴 is applied to assess the
performance of fixture layout design on product dimensional
variation. The proposed methodology is based on the integration
of the genetic algorithm 共GA兲 with Hammersley sequence sampling 共HSS兲. The instability index is also incorporated into the
proposed methodology to ensure that a fixture layout design meets
the locating stability requirement. The methodology is conducted
in two steps. First, the genetic algorithm is used for design space
reduction by estimating the areas of optimal fixture locations in
initial design spaces. Then, Hammersley sequence sampling uniformly samples the candidate sets of fixture layouts from those
predetermined areas for the optimum. GA is selected in the first
step since it can handle the nonlinearity between the KPC and
locator positions. The shortcoming of GA as a time-consuming
heuristic optimization technique is alleviated by incorporating the
instability index and discretizing the continuous design spaces of
all locators to improve convergence to an optimal solution. The
instability index helps to expedite the search capability by eliminating those fixture layouts that do not meet the locating stability
requirement. Discretization of continuous design spaces reduces
the initial candidate design space of locator positions, which GA
can select. HSS is conducted in the second step to compensate for
any potentially missed optimum solutions by uniformly sampling
the candidate locator positions in the areas defined by conducting
GA search in the first step. The uniform sampling of HSS evenly
selects the representatives of fixture layouts in all assembly stations, which increase the probability to select near-optimal locator
positions. The rest of this paper is organized as follows. The stateof-the-art in fixture layout design is reviewed in Sec. 2. The problem formulation, as well as the descriptions of a process yield
calculation based on the SOVA model and instability index, is
described in Sec. 3. The proposed multifixture layout design methodology is presented in Sec. 4. A case study illustrating the application of the proposed methodology on an automotive underbody
assembly and a comparison of optimization algorithms are presented in Sec. 5. Finally, conclusions are drawn in Sec. 6.
2
Literature Review
In general, the fixture layout design has to satisfy four functional requirements, which are 共i兲 locating stability; 共ii兲 deterministic workpiece location; 共iii兲 clamping stability; and 共iv兲 total
restraint 关1,11–13兴. Locating stability is related to the design of a
fixture layout that can provide static equilibrium of a workpiece
when it is placed on fixtures. Second, the fixture should provide
061005-2 / Vol. 130, DECEMBER 2008
the deterministic location for the workpiece to ensure position
accuracy during operation. Third, clamping stability involves determining the sequence of clamping and its layout that does not
disturb the stability and position accuracy of a workpiece established by locators in the previous two functional requirements.
Last, clamps should completely restrain the workpiece to withstand any forces and couples to maintain the workpiece in an
accurate position. Past research in fixture layout design has proposed approaches to address these functional requirements
关14–24兴.
Locating stability is one of the most important requirements in
fixture design since a workpiece has to satisfy this requirement
before achieving other functional requirements. Locating stability
is mainly concerned with static equilibrium under the given fixturing condition in the presence of manufacturing forces. In addition, the fixture layout design has to ensure that all locators maintain contact with the workpiece throughout the manufacturing
operation. Issues involving locating stability begin when the
workpiece is placed on locators as these locators provide a support
against gravity forces until the workpiece is processed. Thus, locating stability also involves fixture force and kinematic analysis
to estimate the necessary clamping forces to maintain a workpiece
in equilibrium. Roy and Liao 关14兴 presented a quantitative evaluation of the part location stability based on screw theory. The
method proposed by Roy and Liao 关14兴 can help in designing a
fixture configuration in automated fixture design environment.
Deterministic workpiece location involves designing the locator
positions or a fixture layout to provide a unique and accurate
position and orientation of a workpiece with respect to its fixture
reference frame 关15兴. Common challenges involving the fixture
layout design that will meet this functional requirement include
the positioning accuracy, which is subject to a random manufacturing error of fixture elements, geometric variability of the workpiece, and workpiece positioning errors induced by fixture position. In general, the position variability of the workpiece can be
predicted from the statistical characterization of the dimensioning
and tolerancing scheme assigned to the fixtures and their contact
points on the workpiece. Thus, determining the fixture layout,
which is not sensitive to these variation sources, can minimize the
workpiece positional variability. Researchers have responded to
this challenge by proposing various methods and sensitivity indices in order to determine the optimal locator positions in a fixture
layout. For example, Cai et al. 关16兴 proposed a variational method
to design a robust fixture layout using the Euclidean norm of the
fixture sensitivity index. Wang 关15兴 and Wang and Pelinescu 关17兴
determined the optimal fixture layout by maximizing the determinant of the information matrix 共D-optimality兲. Carlson 关18兴 assessed the fixture locating scheme in terms of workpiece position
errors by using the quadratic sensitivity equation. However, all
aforementioned methodologies are limited to fixture layout design
involving a single workpiece.
Clamping stability and total restraint are functional requirements that are related to determining the clamping positions and
forces, which do not affect the part locating stability and position
accuracy of workpiece provided by the locators. Clamps apply
forces on the workpiece against any external force to ensure total
restraint 关11兴. The challenge in designing clamping locations is to
minimize the workpiece deformations under clamping and external forces. Many studies have addressed these challenges by
adopting the finite element method 共FEM兲 to design locator and
clamp layouts. For instance, Menassa and DeVries 关19兴 proposed
the integration of Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and FEM simulation to determine the fixture layout
that can minimize deflection of the workpiece. Cai et al. 关20兴
proposed the fixture layout optimization for deformable sheet
metal parts based on nonlinear programming and FEM analysis.
In a similar vein, Krishnakumar and Melkote 关21兴 employed GA
to optimize a fixture layout that can minimize the deformation of
the machined surface due to clamping and machining forces. In
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Table 1 State-of-the-art in fixture layout design
Fixture design functional requirements
Fixture
layout
design
complexity
Single
workpiece
Multistation
assembly
processes
3D
2D
3D
Locating
stability
Deterministic
workpiece location
Roy and
Liao 关14兴
Wang 关15兴
Cai et al. 关16兴
Wang and Pelinescu
关17兴
Carlson 关18兴
Clamping
stability
Total restraint
Menassa and DeVries 关19兴
Cai et al. 关20兴
Krishnakumar and Melkote 关21兴
Li and Melkote 关22兴
DeMeter 关23兴
Marin and Ferreira 关24兴
Camelio et al. 关25兴
Kim and Ding 关26兴
Izquierdo et al. 关27兴
Proposed in this paper
addition, there are also several approaches proposed to determine
the clamping design. For instance, Li and Melkote 关22兴 proposed
an approach to minimize the maximum positional errors by considering the workpiece dynamics during machining. DeMeter 关23兴
proposed a technique to determine the optimal locator layout
based on min-max loading criteria. Marin and Ferreira 关24兴 presented the method to optimize the 3-2-1 fixture layout based on
screw theory. Nevertheless, these studies mainly focus on design
of fixture layout for a single workpiece. Currently, most of the
fixture layout design methodologies are limited to a single workpiece fixture layout design. Table 1 summarizes the current stateof-the-art research in the area of fixture layout design and presented in the context of fixture functional requirements.
Current research related to fixture layout design for multistation
assembly processes is limited because of the challenges in developing a variation propagation model and computational complexity. Recently, Camelio et al. 关25兴 presented the fixture layout design for compliant part assembly by considering the effects of part
variation, tooling variation, and assembly springback. Kim and
Ding 关26兴 proposed a methodology to design multifixture layouts
in multistation assembly based on a station-indexed state-space
model 关5–7,28兴. The extension of Kim and Ding 关26兴 in designing
fixture layouts for a product family can be found in the work of
Izquierdo et al. 关27兴. Kim and Ding 关26兴 involved determining a
set of fixture layouts for all assembly stations that are insensitive
to the variations of random manufacturing errors of fixture elements, geometric variability of the workpiece, and workpiece positioning errors induced by fixturing position. The methodology
developed by Kim and Ding 关26兴 uses E-optimality to minimize
the eigenvalue of the information matrix and exchange algorithm
共EA兲, first proposed by Cook and Nachtsheim 关29兴, to determine
the optimal set of fixture layouts. The design objective of the
methodology proposed by Kim and Ding 关26兴 is similar to the
design objective of this paper in designing a set of multiple fixture
layouts, which is robust to environmental noise. However, there
are fundamental differences between both methodologies in terms
of two-dimensional 共2D兲 versus 3D problem formulation as well
as in specifics of the developed methodologies as elaborated below.
The design problem addressed by Kim and Ding 关26兴 is limited
to 2D assembly processes while this paper focuses on 3D assembly processes. Designing fixture layout in 2D leads to three simplifications, as follows: 共i兲 the locating stability functional requirement is not taken into consideration, 共ii兲 the variations caused by
mating joints between the two parts are not included in the 2D
model, and 共iii兲 the design space dimensionality, as well as nonlinearity between the KCC locator positions and KPCs in 2D fixture layout design is significantly less than those in the 3D fixture
layout design problems since no out-of-plane variation is included
in the 2D model. These three simplifications significantly limit the
industrial application of 2D fixture layout design methodology
Journal of Manufacturing Science and Engineering
since 共i兲 the locating stability consideration is mandatory and must
be achieved before meeting other functional requirements and 共ii兲
there is a variety of part-to-part joints in the assembly processes
such as lap, butt, and T-joints, which have different impacts on
product dimensional variations. This paper considers the fixture
layout design for 3D assembly processes, which addresses the
aforementioned challenges in 2D problems.
Furthermore, the methodology proposed in this paper also differs from Kim and Ding’s 关26兴 approach with respect to the following specifics.
共1兲 An evaluation index to assess fixture layout design. Kim
and Ding 关26兴 minimized the sensitivity index of information matrix in their approach while the approach proposed
in this paper integrates the following two objectives: 共i兲
maximize the percentage of KPCs conforming to specifications and 共ii兲 minimize part locating instability. Evaluation
indices used in both methodologies have different advantages and disadvantages. The sensitivity index used by Kim
and Ding 关26兴 is less computationally intensive than the
calculation ratio of KPCs conforming to design specifications by using simulation techniques such as the Monte
Carlo approach. However, sensitivity index is difficult to
interpret and to provide explicit relations between the index
and product quality while the percentage of KPCs conforming to design specifications can explicitly indicate product
dimensional quality. Moreover, part locating stability is
used in this paper to ensure that fixture layout design satisfies necessary fixturing functional requirement, which is not
considered by Kim and Ding 关26兴.
共2兲 Searching algorithm to select optimal locator positions.
Kim and Ding 关26兴 used EA with enhanced computational
capabilities done by increasing the exchange rate in each
iteration and reducing the candidate fixture locations by
using an experience-based approach. Since 2D fixture layout design optimizes only the layout of two locators per
part, four-way and two-way locating pins, the reduction of
design space based solely on experience is feasible. On the
other hand, the methodology proposed in this paper is analytically based and conducts two-step optimization: 共i兲 initial reduction of design space by using GA approach and
共ii兲 uniform sampling for the optimum multiple fixture layouts by using HSS approach. Instead of relying on the designers’ experience, in the first step, GA is employed to
reduce the size of all locator design spaces, especially important for the design spaces of NC blocks, which have
highly nonlinear relations with KPCs. The first step integrates GA with the part instability index, which further enhances search performance by eliminating the candidate
fixture layouts that do not meet the location stability requirement. Overall, GA performs an initial search to deterDECEMBER 2008, Vol. 130 / 061005-3
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Table 2 Comparative analysis of Kim and Ding †26‡ and the proposed methodology
Kim and Ding 关26兴
Design
problem
Proposed in this paper
2D fixture layout design
• Functional requirement:
Deterministic workpiece location
• Fixture elements: only four-way and twoway locating pins
• Mating joint: only 2D lap-joint
design between parts
• Design space: 2D design space
with moderate nonlinearity between
KCCs and KPCs
Multifixture
layout design
methodologies
Exchange algorithm
• Evaluation index: Minimizing
sensitivity index based on Eoptimality approach
• Enhanced capability in searching
optimal locator positions: Increase
number of exchanges per iteration
and reduce the candidate space by
experience-based approach
mine areas that have a higher probability of containing optimal positions for each locator. Then, HSS conducts
uniform sampling in the areas around the locator position
identified by GA for optimal locator positions. Two-step
optimization, which integrates GA and HSS, is very beneficial in a large design space problem with nonlinear relations between locator nominal positions and KPCs, especially when determining the locations of NC blocks in 3D
fixture layout design. A comparative analysis of the methodologies proposed in this paper with the approach proposed by Kim and Ding 关26兴 is summarized in Table 2.
3D fixture layout design
• Functional requirement: Locating
stability and deterministic workpiece
location
• Fixture elements: four-way locating
pins, two-way locating pins, and NC
blocks
• Mating joint: lap joints, butt joints, Tjoints, and mixed joints in 3D assembly
• Design space: 3D design space and
high nonlinearity between KCCs and
KPCs
Two-step optimization which
integrates GA and HSS
• Evaluation indices: 共i兲 Maximizing
percentages of KPCs conforming to
specifications and 共ii兲 minimizing the
instability index
• Enhanced capability in searching
optimal locator positions: Using GA
and instability index as strategic design
space reduction and using HSS to
conduct local search
The above relations of locator positions, fixture layouts, and a
set of fixture layouts are illustrated in Fig. 1.
In a single assembly station, all locator positions are required to
provide the locating stability for workpieces. Additionally, the set
of fixture layouts in a multistage assembly process has to be robust to environment noise, which results in minimum dimensional
variations of the final product. In this paper, a process yield and an
instability index are used as criteria for determining an optimal
fixture system. The design parameters are the locator positions,rk,
in a set of fixture layout ⌳. Therefore, the optimization scheme is
expressed as
maximize⌳
3
Problem Description
subject to
Let us denote that there are N locators in a given multistation
assembly process. A locator position in the assembly system is
described as
rk = 兵x,y,z其k,
k = 1,2, . . . ,N
共1兲
where 兵x , y , z其k represents the Cartesian coordinates of a locator rk
In this paper, a binary assembly process is taken into consideration where two parts are assembled at each station. A fixture
layout L in an assembly station includes information about locator
positions, which are used for locating two parts: a root part 共S兲
and a mating part 共M兲. Thus, a fixture layout L is represented as
a collection of two distinct sets.
L = 兵S,M其
S = 兵ri其,
M = 兵r j其,
共2兲
i = 1, . . . ,n
共3兲
j = 1, . . . ,n − m
共4兲
where n = 6, if the corresponding part is a rigid body, n ⬎ 6, if a
corresponding part is a compliant part, and m is a number of
degree-of-freedom 共DOF兲, which the part-to-part joint constrains
a mating part.
Let ⌳ represents a set of fixture layouts for a given multistation
assembly process with p assembly stations. A set of fixture layouts, ⌳, is expressed as
⌳ = 兵L1,L2, . . . ,L p其
061005-4 / Vol. 130, DECEMBER 2008
共5兲
yield共⌳兲
␼共⌳兲 ⬍ 0
共6兲
where yield 共·兲 is a function for calculating process yield, and ␼共·兲
is a function for assessing the instability index for a set of fixture
layouts in an assembly system.
The remaining part of Sec. 3 is organized as follows. The review of design evaluation indices and method to calculate process
yield based on SOVA model are presented in Sec. 3.1. To assess
and compare the location stability between the two sets of fixture
layouts, this paper adopts the instability index based on screw
theory proposed by Roy and Liao 关14兴, which is presented in Sec.
3.2.
3.1 Design Evaluation Indices. A fundamental aspect of fixture layout performance is its robustness against environmental
Fig. 1 Fixture layout representation for a multistage assembly
process
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noise to minimize product dimensional variability, which is characterized by KPC variations. In assembly processes, the variations
of KPCs are caused by two main factors, which are 共1兲 variability
of locator positions and 共2兲 part-to-part interactions 关9兴. The position variability of a locator depends on the dimensioning and
tolerancing scheme assigned to locators and geometrical shape
errors of a workpiece. Product dimensional variability caused by
part-to-part interactions stem from errors related to part-to-part
mating features, which characterize variation propagation between
parts 关10兴.
The SOVA model is used in this paper as an assembly response
function to assess the robustness of a designed set of fixture layouts against the aforementioned variation sources. The SOVA
model allows evaluating KPC variations since it defines relations
between KPC and KCC vectors, as shown in Eq. 共7兲. The details
in formulating the SOVA model are discussed in Refs. 关8,9兴.
冤 冥 冤
KPC1
KPC2
=
]
KPCm
m⫻1
SOVA matrix
a11
a12 . . . a1,n
a21
a22 . . . a2,n
]
]
am,1 am,2
]
. . . am,n
冥冤 冥
KCC1
KCC2
]
m⫻n
KCCn
n⫻1
or KPC = A共⌳兲 · KCC
共7兲
where A共⌳兲 is the SOVA matrix, which is formulated based on
the dimensional relationships among fixture layout design, partto-part mating features, and the assembly sequence. In practical,
the relations between KCC variations, including locator position
variations, and KPC variations are non-linear. Huang et al. 关8,9兴
proposed the technique to approximate the relations into linear
model, as shown in Eq. 共7兲. Each element in the SOVA matrix,
ai,j, represents as sensitivity of KCC j on KPCi, which consists
information regarding to nominal positions of locators and partto-part joint. Locator position adjustment while conducting fixture
layout optimization results in different structure of SOVA matrix,
which then can be used to compare the performance of each set of
fixture layouts. If the variations of KPCs illustrated by variance
analysis or simulation are reduced after a fixture layout is adjusted, it indicates that the fixture layout increases in its robustness. We use the SOVA model as the assembly response
function,A, to evaluate the impact of KCC variations, u, on the
KPC variations, y. The relationship in Eq. 共7兲 can be expressed as
y = Au
共8兲
In general, KPC variations can be reduced by 共i兲 tightening the
tolerances of KCCs, u or 共ii兲 optimizing the assembly response
function represented as a SOVA matrix, A, to be robust to KCC
variations. Tightening KCC tolerances is the most straightforward
approach to reducing KPC variations. However, its trade-off involves increasing tooling cost in order to produce tooling at higher
precision. Optimizing the assembly process is more appealing in
practice since KCC tolerance ranges can be increased and the
assembly process is still able to achieve the same KPC variation
levels. Relaxing KCC tolerances usually leads to lower production
cost. Fixture layout design is one approach that can increase assembly process robustness.
In a fixture layout design, we aim to determine the locator
nominal positions, which can minimize the KPC variations while
the tolerances that control variations of KCCs, u, are constant.
Different locator positions contribute to the alteration of elements
in the SOVA matrix, A. The robustness of the SOVA matrix resulting from altering the locator positions can be assessed by two
approaches: 共i兲 a loss function based on sensitivity indices and 共ii兲
an estimated percentage of nonconforming items.
In the sensitivity index approach, the product dimensional quality is measured by the variations of yTy = uTATAu. To minimize
the variations of yTy, the robustness of the SOVA matrix, A, has
to be improved in order to be insensitive to the KCC variation
Journal of Manufacturing Science and Engineering
inputs, u. The sensitivity index can be defined as the variations of
output signals to input noise 关26兴, which can be expressed as
S=
yTy uTATAu
=
u Tu
u Tu
共9兲
The sensitivity index, S, has to be minimized such that the
significant variations of uTu contribute to minor variations of yTy.
If the KCC variations of vector u are constant, the KPC variations
depend on the assembly response function A. The challenge is to
select the design index to assess ATA. Several measures are proposed based on optimality criteria in experimental design. Kim
and Ding 关26兴 provided the analysis of the three optimality criteria
in fixture layout design which are 共i兲 D-optimality 共min
det共ATA兲兲, 共ii兲 A-optimality 共min tr共ATA兲兲, and 共iii兲 E-optimality
共min ␭max共ATA兲; ␭max is the extreme eigenvalue兲. The advantages
and disadvantages of these three optimality criteria for fixture layout design are discussed below:
D-optimality is to minimize the determinant of a matrix ATA,
共min det共ATA兲兲. The advantage of D-optimality in fixture layout
design is that it minimizes both the variances and the covarinces
of matrix ATA. It is equivalent to minimizing the overall process
m
variations; min det 共ATA兲 = min 兿i=1
␭i, where ␭i is an eigenvalue.
D-optimality is very effective to evaluating the design problems,
which inherent highly nonlinear relationships such as fixture layout design. However, the singularity of matrix ATA is a major
obstacle to the use of D-optimality in multistage fixture layout
design.
A-optimality is to minimize the trace of matrix ATA, min
tr共ATA兲, which is the summation of sensitivities of all KCC-KPC
pairs in the assembly processes. Nevertheless, A-optimality does
not consider the dimensional variation impact from covariances
within matrix ATA. Thus, A-optimality does not imply that the
percentage of nonconforming items will be reduced since the covariances among the locator nominal positions on KPC variations
are high.
E-optimality is to minimize the extreme eigenvalue of matrix
ATA, min ␭max共ATA兲. E-optimality is similar to D-optimality,
which considers both variances and covariances of all pairs of
KCC-KPC, but E-optimality considers only ␭max共ATA兲. Thus,
E-optimality can avoid the singularity of matrix ATA during computation, and it is aligned with the Pareto principle in quality
engineering 关26兴. However, minimizing only the maximum eigenm
value, ␭max共ATA兲 cannot guarantee that overall variations, 兿i=1
␭ i,
of the new set of fixture layouts design are decreased. It leaves the
possibility that several principle components dominate the overall
variations of matrix ATA, and the summations of these eigenvalues can contribute to larger variations even though its extreme
eigenvalue is lower than the previous fixture layout design. Therefore, it is difficult to decide that process increases its robustness
by assessing only the extreme eigenvalue.
On the other hand, the percentage of nonconformance items can
be used to evaluate the performance among fixture layout designs
by maintaining constant u. In general, process performance is
measured by process capability indices, C p or C pk, where C p can
be defined as 共USL− LSL兲 / 6␴; USL and LSL are the upper and
lower specification limits, respectively, and ␴ is the standard
variation of a single KPC variable. In multivariate cases, the KPC
tolerance/specification region in multivariate m space is the volume of the hyper-rectangular cube 关30兴, which can be defined as
m
兿 共USL − LSL 兲
i
i
共10兲
i=1
The KPC variations of a multivariate process can be assessed
by using chi-square distance defined as
␹20 = 共y − ␮兲⬘
兺
−1
共y − ␮兲
共11兲
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Fig. 2 KPC variations compared with KPC tolerance region: „a… before optimizing fixture layouts „b… after optimizing fixture layouts
However, C p in evaluating multivariate normality KPC variations cannot be obtained directly by dividing the volume of the
KPC hyper-rectangular cube specification shown in Eq. 共10兲 with
the actual process chi-square distance expressed in Eq. 共11兲 because KPC tolerances/specifications are a hypercube while the
chi-square distance has elliptical probability region. Thus, to determine C p, it is necessary to estimate the KPC tolerance region
into an ellipsoid shape. As a result, when the process is centered at
the target and C p = 1, this implies that 99.73% of the process variations are inside the estimated KPC tolerance ellipsoid. Taam et al.
关30兴 proposed an approach to calculate C p by approximating the
KPC tolerance hypercube with the largest ellipsoid that can lie
inside the KPC tolerance hypercube. However, to estimate the
largest ellipsoid volume is difficult in the case where m ⬎ 3.
The process yield proposed in this paper is similar to C p in a
multivariate process. However, instead of focusing on estimating
the KPC tolerance ellipsoid shape, the process yield defines the
probability that the KPC variation vector, y, lies in KPC tolerance
hypercube, as illustrated in Fig. 2共b兲. The process yield provides
an understandable design criterion for design engineers to evaluate their process design. Moreover, the process yield does not
depend on the multivariate normality distribution assumption. To
benchmark the robustness of any two processes in the case that
100% probability of KPC variation vectors, y, lies in the KPC
tolerance hypercube can also be performed by integrating concept
of sensitivity indices to process yield. A-optimality and
E-optimality can be used to analyze the variances or principle
components of interest.
In this paper, the process yield is used as the quality index in
assessing the performance of a set of fixture layouts. Yield is
defined as a function of KPCs, which represents the probability of
all KPCs simultaneously being within their respective specification ranges as shown
再
m
yield 共⌳兲 = Pr 艚 LSLi 艋 KPCi 艋 USLi
i=1
冎
共12兲
where LSLi and USLi are the lower and upper specification limits
for KPCi, respectively.
Yield can be estimated by using Monte Carlo technique by
simulating k KCC vectors, KCC1, KCC2 , . . . , KCCk, where
KCCi = 关KCC1 . . . KCCn兴Ti . A variation of each KCC expressed in
Cartesian coordinate, 共␦x , ␦y , ␦z兲, is randomly generated based on
its statistical characterizations. Then, the KCCi ; i = 1 , . . . , k, is
substituted into Eq. 共7兲 to obtain a vector of, KPC variations,
KPC1 , . . . , KPCk, where KPC = 关KPC1 . . . KPCm兴T.
⌽共KPCi兲 is a function to provide a response whether all KPCs
are in-specification windows. If all KPC variations are within
specification windows; LSL 艋 KPCi 艋 USL, then ⌽共KPCi兲 = 1;
otherwise ⌽共KPCi兲 = 0. Thus, yield can be expressed as:
3.2 Instability Index. The instability index, ␼, is adopted
from Roy and Liao 关14兴 to compare the locating stability between
two fixture layouts. In this paper, a binary assembly process is
taken into consideration where at each station a root part is located on the fixture layout S, and then a mating part located by
fixture layout M, and a part-to-part joint is assembled to the root
part. Thus, the root part has to be in static equilibrium and is fully
constrained by fixture layout S before being assembled to a mating part. The static equilibrium of a root part located by fixture
layout S, as shown in Fig. 3, can be expressed in matrix form as
冤
0
0
0
−1
−1
0
0
0
0
0
0
−1
1
1
1
0
0
0
r1y
r2y
r3y
0
0
r6z
− r1x − r2x − r3x − r4z − r5z
0
0
0
r4y
r5y
0
− r6x
冥冤 冥 冤 冥
0
F1
F2
0
F3
− Wg
F4
+
− Wgrgy
F5
Wgrgx
F6
0
=0
共14兲
where Fi, i = 1 , 2 , . . . , 6 represent supporting and locating forces;
r1x , r1y , . . . , r6z represent the x , y , z coordinates of six locators in
the fixture layout; rgx , rgy , rgz represent the x , y , z coordinates of
the center of gravity of the workpiece; Wg represents the weight of
the workpiece.
If there are only supporting forces from three NC blocks, F1,
F2, and F3, against the weight of a root part, Wg, in order to
maintain static equilibrium, Eq. 共14兲 can be simplified to
冤
1
1
1
r1y
r2y
r3y
− r1x − r2x − r3x
冥冤 冥 冤 冥
F1
− Wg
F2 + − Wgrgy = 0
F3
Wgrgx
共15兲
To calculate the instability index, the following information is
required: 共1兲 a new wrench, wd, of external forces and couples to
rebalance the static equilibrium in an adjusted fixture layout and
共2兲 twist caused by root part weight, tmg. The wrench and twist
can be obtained as described below.
共1兲 A new wrench, wd, of external forces and couples. When
k
兺 ⌽共KPC 兲
i
yield 共⌳兲 =
i=1
k
⫻ 100%
061005-6 / Vol. 130, DECEMBER 2008
共13兲
Fig. 3 3-2-1 fixture layout for prismatic workpiece
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Step 1: Fixture Planning
1.1 Define the number and types of locators required in the
assembly process
Output: Λ = L1, L 2 , K, L p
{
}
1.2 Define the design spaces for N locators
Output: ℜ system = {ℜ1 ℜ 2 K ℜ N }
1.3 Discretize design space, ℜi , into t nodes
Outputs: ℜi → Oi = {O1 O2 K Ot }i
{
O system = O 1 , O 2 , K , O N
Fig. 4 External wrench, wd, and twist caused by gravity force,
tmg
Step 2: Design space reduction by Genetic Algorithm (GA)
2.1 Design space reduction by GA
maximizeΛ Yield(Λ)
the position of any NC block used to locate a root part is
adjusted while conducting fixture layout optimization, the
previous static equilibrium condition, as shown in Eq. 共15兲,
is altered and equilibrium needs to be determined again.
For example, if the position of NC block No.1 is changed
from 共x1 , y 1兲 to 共x⬘1 , y 1兲, the new equilibrium equation with
a new external force and moment vector 共we兲 to balance the
root part locating stability can be expressed as in Eq. 共16兲
where the force vector, 关F1 F2 F3兴T, acting at the NC
block is unchanged.
冤
subject to
1
1
ry1
ry2
ry3
− r⬘x1 − rx2 − rx3
fz
{
}
2.1 Define design spaces of interest, A i , around GA candidate
nodes
Output: ℜnew = {A1 A 2 K A N }
Step 3: Local search by Hammersley Sequence Sampling
(HSS)
3.1 Projection of design space of interest, A i , into 2D plane
冥冤 冥 冤 冥 冤 冥 冤 冥
− Wg
F1
ϖ ( Λ) < 0
1
2
N
, O GA
, K , O GA
Output: O GA = O GA
we
1
}
D
= {B1 B 2 K B N }
Output: ℜ 2new
3.2 Uniform sampling sets of fixture layouts by HSS
Output: Λ1, Λ 2 ,K, Λ n
0
F2 + − Wgrgy + M x = 0
0
F3
Wgrgx
My
共16兲
Step 4: Evaluation of sampled sets of fixture layouts
4.1 Evaluation of sets of fixture layout generated by HSS
against design objective
maximizeΛ Yield(Λ )
Both the force and moments of we can be obtained from
Eq. 共16兲, and they can be presented in a wrench form as
wd = 关0
0
fz
Mx
0兴T
My
共17兲
共2兲 Twist caused by root part weight, tmg: To obtain the twist
tmg of a root part caused by its weight, let us assume that
the workpiece undergoes an infinitesimal movement caused
by gravity force Wg, as shown in Fig. 4. This movement
can be expressed as a twist about the origin of coordinates:
tmg = ␻x
␻y
␻z
vx
vy
v z
subject to ϖ ( Λ ) < 0
Output: an optimum set of fixture layouts Λ optimum
Fig. 5 The
methodology
1/rgx
0
0
0
− 1兴
共20兲
The instability index defined by using virtual work can be interpreted as follows.
共1兲 ␼ ⬎ 0 represents the positive work done by wrench wd in
accomplishing twist tmg. This positive virtual work implies
that the adjustment of a locator position in a new fixture
layout reduces the workpiece stability. Therefore, in the
proposed methodology in this paper, it is concluded that the
adjusted fixture layout is worse than the pre-adjusted one.
共2兲 ␼ = 0 represents that no work is done by wrench wd in
accomplishing twist tmg. The virtual work ␼ = 0 can be interpreted that there is no improvement in locating stability
condition after the adjustment of a locator position.
共3兲 ␼ ⬍ 0 represents the negative work done by wrench wd in
Journal of Manufacturing Science and Engineering
of
the
proposed
accomplishing twist tmg. The negative virtual work implies
that the adjustment of locator position into a new fixture
layout increases the workpiece stability. Therefore, in the
proposed methodology in this paper, it is concluded that the
adjusted fixture layout is better than the pre-adjusted one.
共19兲
The instability index represents the virtual work done by the
wrench wd against the twist tmg. The instability index, ␼, can be
obtained by
␼ = f xv x + f y v y + f zv z + M x␻ x + M y ␻ y + M z␻ z
procedure
共18兲
where ␻x , ␻y , ␻z are the components of part angular displacement, and vx , vy , vz are components the of part translational displacement. For example, the twist of gravity
force, Wg, about the origin of coordinates, as shown in Fig.
4, is
tmg = 关− 1/rgy
optimization
4
Methodology
In keeping with the previous discussion, multiple fixture layouts of all assembly stations are required to be designed simultaneously results in a high-dimensional optimization problem.
Moreover, the design space of each locator is large, and the locator nominal positions have nonlinear relationships with KPCs. In
order to address these challenges, this paper proposes a two-step
optimization approach based on genetic algorithm for design
space reduction and Hammerley sequence sampling for direct
search optimization scheme in a design space predetermined by
GA. Yield and instability index are incorporated to the proposed
design approaches as design objectives. The procedure of the proposed methodology in designing multiple fixture layouts is shown
in Fig. 5. The proposed methodology is based on the following
assumptions.
1. All parts are rigid body.
2. The Locator-part constraint is characterized by frictionless
point contact.
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Fig. 6 Mating part degrees of freedom allocation between part-to-part joint
and locators
3. The part-to-part joints always maintain full contact to each
other.
4. Locators are not only to determine the positions and orientations of the part but also function as clamps in constraining
parts.
5. Only fixture and part-to-part joint errors are considered,
which are considerably small compared with part and assembly dimensions.
Step 1. Fixture planning.
Step 1.1. Define the number and types of locators required in
the assembly process.
The number and types of locators required to constrain root, S,
and mating, M, parts in each assembly station 共L = 兵S , M其兲 are
defined in this step. It also can be observed that the number of
locators in each assembly station depends on the part-to-part joint
design and assembly sequence. To illustrate this, let us assume an
assembly process of two rigid parts, each with 6 DOFs. The first
part positioned in the assembly station called a root part is fully
constrained by fixture locators. The second part positioned in the
assembly station is called a mating part and has its 6 DOFs constrained by part-to-part joint and fixture locators. The potential
allocation of mating part 6 DOFs to be constrained by fixture
locators 共M兲 and by part-to-part joint is shown in Fig. 6.
Furthermore, the assembly sequence in selecting of root and
mating part in each station directly affects the fixture layout design. An example of fixture layouts for S and M parts is illustrated
in Fig. 7. The root part in Fig. 7共a兲 is located by a typical 3-2-1
fixture layout. The locators for the root part S consist of two
locating pins, four-way pin 共P4way兲 and two-way pin 共P2way兲, and
three net contact blocks 共NC1–3兲. Two locating pins constrain 3
DOFs in the X-Y plane, where P4way controls the part translation
in the X and Y axes, and P2way controls the rotation of the root part
about Z axis. Three NC blocks constrain the remaining 3 DOFs,
which are translation in Z axis and rotations about the X and Y
axes. Thus, the locators required for the root part in 3-2-1 fixture
locating scheme can be defined as
r
r
, P2way
,NCr1,NCr2,NCr3其
S = 兵P4way
共21兲
Then, the mating part is located by part-to-part joint and the
remaining of DOFs are constrained by fixture locators, M. The
DOFs of a part-to-part joint, which constrain the mating part,
affect the fixture planning, as shown in Fig. 6. The part-to-part
joint shown in Fig. 7共b兲 constrains the 3 DOFs of the mating part:
translation in Z axis and rotations about X and Y axes. Therefore,
the remaining DOFs of the mating part are constrained by P4way
and P2way fixture locators.
Fig. 7 3-2-1 fixture layout for a single station assembly
061005-8 / Vol. 130, DECEMBER 2008
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Rsystem = 兵R1
R2
...
R N其
共27兲
Step 1.3 Discretize design space, Ri, into nodes.
To reduce the computational effort of GA, the continuous design space, Ri, for each locator is discretized into nodes. These
nodes represent the candidate design space to be considered in the
next step. The design space of locator ri discretized into t nodes,
as shown in Fig. 9, can be expressed as
Oi = 兵O1
O2
...
O t其 i
where O j = 共x j,y j,z j兲 苸 Ri
共28兲
and j = 1, . . . ,t
Fig. 8 Example of part-to-part joints used in automotive body
assembly
In an automotive body assembly, the following three types of
part-to-part joints are widely used: 共1兲 lap joint, 共2兲 butt joint, and
共3兲 T-joint 关4,10兴, as shown in Fig. 8. In this paper, a part-to-part
joint is assumed to constrain 3 DOFs of a mating part. A set of
fixture locators required for the mating part M where part-to-parts
are
lap joint, butt joint, and T-joint can be defined as follows,
respectively:
m
m
, P2way
其
Mlap = 兵P4way
共22兲
m
m
,NCm
Mbutt = 兵P2way
1 ,NC2 其
共23兲
m
m
,NCm
MT = 兵P2way
1 ,NC2 其
共24兲
Thus, the fixture layout for the single assembly station with a
predetermined part-to-part joint 共Mjoint兲 can be defined as
r
r
, P2way
,NCr1,NCr2,NCr3,Mjoint其
L = 兵S,M其 = 兵P4way
共25兲
Step 1.2. Define the design space for each locator.
The design space, Ri, of each locator, ri, is defined as an area
on a workpiece that the locator can be placed. Design engineers
justify the locator design space by considering other design constraints in the subsequent fixture construction stage such as allowable maximum deformation of workpiece or the potential difficulty in locator installation and calibration. The continuous design
space of a locator ri can be defined as
Ri = 兵共x,y,z兲其,
x 苸 关lx,ux兴,
y 苸 ly,uy
z = f i共x,y兲 共26兲
where 共x , y , z兲 represents the Cartesian coordinate of a locator ri;
lx,y and ux,y are the lower and upper boundaries in the x and y
axes, respectively; and z = f i共x , y兲 is the workpiece surface shape
function.
For example, in Fig. 9 the design space Ri covers the whole
part. Let us assume that a set of fixture layouts ⌳ consists of N
locators in a given assembly system. Then, the design space of the
assembly process can be expressed as
The continuous design spaces 共R1 , . . . , RN兲 of N locators in a
given assembly system after the discretization into nodes can be
expressed as
Osystem = 兵O1,O2, . . . ,ON其
共29兲
Step 2. Design space reduction by genetic algorithm (GA).
Step 2.1: Design space reduction by GA.
Since the design space for each locator can be relatively large
and its position has the nonlinear relations with multiple KPCs, it
is necessary to reduce the design space to the area that potentially
contains the optimal locator position. The design space reduction
is conducted by using the GA approach. In each iteration, GA
selects one node from each candidate space Oi; i = 1 , . . . , N, to
formulate a candidate set of fixture layouts, ⌳ = 兵O1j , . . . , ONj 其. The
GA optimization scheme is expressed as determining a set of fixture layout ⌳, which maximizes the process yield subjected to a
constraint of instability index,␼共⌳兲, as shown below:
maximize⌳
subject to
yield共⌳兲
␼共⌳兲 ⬍ 0
共30兲
In general, the genetic algorithm adopted in this paper involves
four major steps, as illustrated in Fig. 10. In the first step, the
Cartesian coordinates of locator positions, which are aimed to be
optimized, are modeled into a chromosome vector. In the second
step, the initial population size in each generation is defined. Initial populations in this paper are selected randomly with uniform
distribution function. Then, the process yield of individual population is evaluated subjected to locating stability requirement, as
shown in Eq. 共30兲. In the last step, the chromosome of population
is improved by selecting the best individual to reproduce in the
next generations. The reproduction process to improve the chromosome involves two functions: 共i兲 crossover function and 共ii兲
mutation function. Crossover function is to combine two individuals, or parent, to produce a new individual. In this paper, crossover
function randomly selects chromosomes from parents by generating binary vector, which have a length equal to a number of chromosome in a population. If an element of a binary vector is 1, a
chromosome is selected from the first parent. On the other hand, a
chromosome is selected from the second parent if an element of
Fig. 9 Illustration of the proposed methodology Steps 1.2–3.1
Journal of Manufacturing Science and Engineering
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Fig. 11 Sets of fixture layouts generation by HSS
2D
Rnew
= 兵B1
Fig. 10 Step 2 design space reduction by genetic algorithm
binary vector is 0. Mutation function is to create small random
changes in individual population, which help to prevent GA
trapped in local optima. GA is terminated when there is no improvement in a process yield or the number of iterations reaches
the maximum number of generations designated in the GA procedure. Denote the ⌬␼ as the improvement in the instability index,
i
, which is
i.e., ⌬␼ = ␼共⌳i兲 − ␼共⌳i−1兲. Figure 9 shows a node OGA
the optimal position of locator ri selected by GA. The optimal
positions of all locators in the system selected by GA can be
expressed as
1
2
N
OGA = 兵OGA
,OGA
, . . . ,OGA
其
Rnew = 兵A1
Ai = 兵共x,y,z兲其,
A2
x 苸 关l⬘x ,u⬘x 兴,
A N其
共32兲
y 苸 关l⬘y ,u⬘y 兴,
z = f i共x,y兲 共33兲
...
where l⬘x,y and u⬘x,y are the lower and upper boundaries of x and y
coordinates in design space of interest Ai, respectively; z
= f i共x , y兲 is the workpiece surface shape function.
Step 3. Local search by Hammersley sequence sampling (HSS).
Step 3.1. Projection of design space of interest, Ai, into 2D
plane.
The direct sampling of potential locator positions on part surface is very complex and time consuming since geometrical information of part surface has to be included in the HSS algorithm.
Therefore, the sampling procedure can be simplified by selecting
the locator positions in 2D plane. The 2D plane is obtained by
projecting design space of interest Ai into a given plane. Design
space of interest in 3D space, Ai, is projected into 2D space, Bi, as
shown in Fig. 9. The 2D space, Bi, can be expressed as follows:
Bi = 兵共x,y兲其,
x 苸 关l⬘x ,u⬘x 兴
and
y 苸 关l⬘y ,u⬘y 兴
共34兲
where Bi is the projection area in 2D plane of Ai, which consists
of a set of points within the lower and upper boundaries of x and
y coordinates.
Design spaces of interest for all locators in 2D plane can be
expressed as
061005-10 / Vol. 130, DECEMBER 2008
...
B N其
Bi:共x j,y j兲 → Ai:共x j,y j,z j = f共x j,y j兲兲 = ri,
⌳ = 兵r1, . . . ,rN其
共31兲
Step 2.2. Define design space of interest around GA candidate
nodes.
The area around each node predetermined by GA in Eq. 共31兲 is
defined as the new design space so-called design space of interest,
Ai. The design space of interest covers the area that GA did not
consider, which might contain the optimal locator position. The
size of design space of interest is related to the grid size defined in
Step 1. It can be seen in Fig. 9 that design space of interest of
locator ri is significantly smaller than an initial design space, Ri.
The new design space of the assembly system can be expressed as
B2
共35兲
Step 3.2. Uniform sampling sets of fixture layouts by HSS.
The optimal positions of all locators are searched by using
HSS. To sample the locator positions, first the transformed 2D
design spaces 兵B1 B2 . . . BN其 are formulated into
N-dimensional hypercube design space. The example hypercube
with three 2D plane design spaces 共B1–3兲 is shown in Fig. 11共a兲.
The samples are selected uniformly in the hypercube shown in
Fig. 11共a兲. Then, the sample points are projected onto each facet
of the hypercube, which is a 2D plane design space, as shown in
Fig. 11共b兲. Finally, the sampling locator positions in the 3D design
space 兵A1 A2 . . . AN其 can be obtained by substituting the
coordinates of samples in the 2D design spaces into corresponding
part surface functions; z = f i共x , y兲 ; i = 1 , . . . , N, as shown in Fig.
11共c兲.
The transformation of locator position sampled in 2D plane, Bi,
into 3D design space, Ai, is mathematically expressed as
i = 1, . . . ,N 共36兲
共37兲
Step 4. Evaluation of sampled sets of fixture layouts.
The n sets of fixture layouts; ⌳1 , ⌳2 , . . . , ⌳n, generated by using HSS are evaluated by 共i兲 formulating SOVA models, A共⌳兲, for
each set of fixture layouts; and 共ii兲 conducting Monte Carlo simulation to obtain a process yield. The optimization problem is formulated as follows:
maximize⌳
subject to
5
yield共⌳兲
␼共⌳兲 ⬍ 0
共38兲
Case Study: Floor Pan Assembly
The developed methodology is illustrated and validated by applying it to automotive underbody assembly process. Floor Pan
subassembly consists of four parts; floor pan left 共FPL兲 and right
共FPR兲, and bracket left 共BrktL兲 and right 共BrktR兲, assembled in
three stations, as shown in Fig. 12. The dimensional quality of the
floor pan assembly is evaluated by 12 KPCs and reported as process yield. The manufacturer aims to design the fixture layouts in
three assembly stations to improve the robustness of the floor pan
assembly process by increasing the process yield from a current
level of 85% without tightening any KCC tolerances, i.e., without
increasing tooling cost. The process yield is assessed by Monte
Carlo simulation in which the variations of fixture locators and
part-to-part joints are set according to their tolerances. Tolerances
of locators are assumed to be ⫾0.3 mm. while part-to-part joint
tolerances are assumed to be ⫾0.5 mm. for linear variation and
⫾0.25 degree for angular variation. The proposed multifixture
layout optimization methodology is applied to this case study. The
comparative study in terms of optimization performances among
the proposed methodology and other fixture layout optimization
algorithms are also shown in this section.
Step 1: Fixture planning.
Step 1.1. Define the number and types of fixtures required in
the assembly process.
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⌳ = 兵L1,L2,L3其
共40兲
Step 1.2. Define the design space for each locator.
The design spaces of all locators in all assembly stations are
defined in Tables 4–6. The design space for each locator is shown
in Figs. 13–18. The coordinates of center of gravity and weight of
each root part in each assembly station are shown in Table 7.
Step 1.3. Discretize design space, Ri, into nodes.
The design spaces of all four-way and two-way locating pins
are discretized into grid of 50⫻ 50 mm2. Similarly, the design
spaces of NC blocks are discretized into grid of 100⫻ 110 mm2.
Figure 19 shows one example of dicretization of design space R3
representing NC block NC1s
1 into 60 nodes. All 21 discretized
design spaces in the system can be expressed as
Osystem = 兵O1,O2, . . . ,O21其
Fig. 12 Floor pan assembly
The assembly sequence, part-to-part joints, and required locators in each assembly station are shown in Table 3. The fixture
layouts for three assembly stations and a set of fixture layouts for
the system can be formulated as follows
L1 = 兵S1,M1其,
L2 = 兵S2,M2其
and
L3 = 兵S3,M3其
共39兲
共41兲
Step 2. Design space reduction by GA.
Step 2.1. Design space reduction by GA.
In each iteration, GA selects one node from each of 21 candidate design spaces in Eq. 共41兲 to formulate a set of fixture layouts
⌳, which is then evaluated by using a process yield and an instability index, ␼. The optimization scheme is expressed, as shown
in Eq. 共30兲. The genetic algorithm configurations are selected as
follows: 共1兲 chromosomes are formulated by Cartesian coordinates of 21 locators; 共2兲 population size has 50 individuals in each
generation; 共3兲 selection of parents for reproduction is based on
ranking for the most fitness and stochastic uniform selection with
Table 3 Required locators in each assembly station
Station No.
Parts
Joints
Required locators
FPL 共root part兲
FPR 共mating part兲
Lap joint
1s
1s
1s
1s
, P2way
, NC1s
S1 = 兵P4way
1 , NC2 , NC3 其
1m
1m
M1 = 兵P4way , P2way其
Station 1
Station 2
Subassembly 共FPL+ FPR兲
Bracket left 共mating part兲
Lap joint
2s
2s
2s
2s
S2 = 兵P4way
, P2way
, NC2s
1 , NC2 , NC3 其
2m
2m
M2 = 兵P4way
, P2way
其
Station 3
Subassembly
共FPL+ FPR+ BrktL兲
Bracket right 共mating
part兲
Lap joint
3s
3s
3s
3s
S3 = 兵P4way
, P2way
, NC3s
1 , NC2 , NC3 其
3m
3m
, P2way
其
M3 = 兵P4way
Table 4 Design spaces of locators in Station 1
Root part
Station 1
Locators
Design
spaces
Mating Part
1s
P4way
1s
P2way
NC1s
1
NC1s
2
NC1s
3
1m
P4way
1m
P2way
R1
R2
R3
R4
R5
R6
R7
Table 5 Design spaces of locators in Station 2
Root part
Station 2
Locators
Design
spaces
Mating Part
2s
P4way
2s
P2way
NC2s
1
NC2s
2
NC2s
3
2m
P4way
2m
P2way
R8
R9
R10
R11
R12
R13
R14
Table 6 Design spaces of locators in Station 3
Root part
Station 3
Locators
Design
spaces
Mating part
3s
P4way
3s
P2way
NC3s
1
NC3s
2
NC3s
3
3m
P4way
3m
P2way
R15
R16
R17
R18
R19
R20
R21
Journal of Manufacturing Science and Engineering
DECEMBER 2008, Vol. 130 / 061005-11
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Fig. 16 Locator design spaces of bracket left „BrktL, mating
part… in Station 2
Fig. 13 Locator design spaces on floor pan left „FPL, root
part… in Station 1
a crossover fraction of 0.8; 共4兲 mutation function is based on the
Gaussian distribution; 共5兲 crossover function is based on using a
binary vector as criterion; and 共6兲 the GA is terminated at 5000
generations or there is no improvement in a process yield within
50 consecutive generations. The locator positions selected by GA
are shown in Tables 8–10 and Figs. 20–22.
Step 2.2. Define design space of interest around GA candidate
nodes.
The area around each node predetermined by GA is defined to
be the design space of interest, Rnew = 兵A1 A2 . . . A21 其. However, in this case study, some locators are required to be defined
with additional constraints. First, both bracket left and right have
limited areas to place four-way and two-way locating pins. Thus,
2m
the positions of four-way and two-way pins for both parts 共P4way
,
2m
3m
3m
P2way, P4way, and P2way兲 are assigned to be the same as the locations selected by GA. Second, to minimize the need for locating
holes on the parts, some locators are reused and have the same
position in different assembly stations. These locators are 共1兲
1s
2s
3s
1s
2s
3s
P4way
, P4way
, and P4way
; 共2兲 P2way
, P2way
, and P2way
; 共3兲 NC1s
3 ,
3s
1s
2s
3s
2s
NC2s
,
and
NC
;
共4兲
NC
,
NC
,
and
NC
;
and
共5兲
NC
and
1
1
1
2
2
3
NC3s
3 . Therefore, the design spaces of interest in each group of
these locators are equivalent to the integration of areas around
each node within the group. The design spaces of interest for all
Fig. 17 Locator design spaces of the root part in Station 3 „the
coordinates of the boundaries are the same, as shown in Fig.
15…
Fig. 18 Locator design spaces of bracket right „BrktR, mating
part… in Station 3
Table 7 Center of gravity coordinates of a root part in three
assembly stations
Fig. 14 Locator design spaces on floor pan right „FPR, mating
part… in Station 1
Fig. 15 Locator design spaces of the root part „FPL+ FPR… in
Station 2
061005-12 / Vol. 130, DECEMBER 2008
Center of gravity
coordinates
Station 1
共FPL兲
Station 2
Subassembly
共FPL+ FPR兲
Station 3
Subassembly
共FPL+ FPR+ BrktL兲
X
Y
Z
Weight 共kg兲
1250
−290
0
60
1250
0
0
120
1340
−110
0
150
Fig. 19 An example of design space discretization
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Table 8 Locator positions selected by GA in Station 1
Root part
Mating part
Station 1
Coordinates
1s
P4way
1s
P2way
NC1s
1
NC1s
2
NC1s
3
1m
P4way
1m
P2way
X
Y
Z
900.00
−250.00
0.00
1700.00
−250.00
0.00
1650.00
−180.00
0.55
750.00
−400.00
0.00
1250.00
−510.00
8.94
1700.00
231.35
0.00
800.00
231.35
0.00
Table 9 Locator positions selected by GA in Station 2
Root part
Mating part
Station 2
Coordinates
2s
P4way
2s
P2way
NC2s
1
NC2s
2
NC2s
3
2m
P4way
2m
P2way
X
Y
Z
750.00
−300.00
0.00
1700.00
−300.00
0.00
1350.00
−620.00
25.34
1550.00
−290.00
0.00
1050.00
191.35
1.39
1556.00
−154.00
90.00
1556.00
−551.00
90.00
Table 10 Locator positions selected by GA in Station 3
Root part
Mating part
Station 3
Coordinates
3s
P4way
3s
P2way
NC3s
1
NC3s
2
NC3s
3
3m
P4way
3m
P2way
X
Y
Z
750.00
−300.00
0.00
1700.00
−300.00
0.00
1450.00
−620.00
25.34
1550.00
−290.00
0.00
1150.00
301.35
1.39
1556.00
135.00
90.00
1556.00
532.00
90.00
locators are shown in Fig. 23. This case study uses 2 ⫻ 2 grid size
around the nodes to define design space of interest.
Step 3. Local search by HSS.
Step 3.1. Projection of design space of interest, Ai, into 2D
plane.
The boundaries of the 2D planes, B, are shown in Tables 11–13.
Step 3.2. Uniform sampling sets of fixture layouts by HSS.
One thousand sets of fixture layouts, ⌳i ; i = 1 , . . . , 1000, are
sampled from the hypercube of 21 2D plane design spaces, B1–21
by using HSS. The sampled locator positions in 2D plane design
spaces, B1–21, are projected onto part surfaces to obtain the actual
positions in 3D design space, A1–21 by using CAD software to
acquire the coordinates on workpiece.
Step 4. Evaluation of sampled sets of fixture layouts.
One thousand sets of fixture layouts are evaluated for their process yield and stability. The set of fixture layouts that has the
highest process yield and satisfy stability requirement is considered to be the optimum set of fixture layouts. The locator positions
of the optimum set of fixture layouts are shown in Tables 14–16
and Figs. 24 and 25. The process yield of the optimum set of
fixture layouts is 96.16% while the process yield of the initial
industrial design is around 85%.
The performance of the proposed optimization algorithm is
Fig. 20 Locator positions selected by GA in Station 1
Fig. 22 Locator positions selected by GA in Station 3
Fig. 21 Locator positions selected by GA in Station 2
Fig. 23 Locator design spaces of interest
Journal of Manufacturing Science and Engineering
DECEMBER 2008, Vol. 130 / 061005-13
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Table 11 The upper and lower boundary coordinates of the projected 2D plane design space in Station 1
Locator
1s
P4way
1s
P2way
NC1s
1
NC1s
2
NC1s
3
1m
P4way
1m
P2way
Design space of
interest
Projected 2D plane
design space
Upper boundary of
2D plane 共x,y兲
Lower boundary of
2D plane 共x,y兲
A1
A2
A3
A4
A5
A6
A7
B1
B2
B3
B4
B5
B6
B7
共950, −250兲
共1750, −250兲
共1750, −180兲
共850, −300兲
共1550, −400兲
共1750,281兲
共850,281兲
共750, −350兲
共1650, −350兲
共1450, −400兲
共750, −500兲
共1150, −620兲
共1650,231兲
共750,231兲
benchmarked with other optimization algorithms in terms of 共i兲
methodology ffectiveness measured by closeness of its solution to
the global optimum and 共ii兲 computational efficiency measured by
time to converge to optimum solutions. However, it is considerably difficult to determine the global optimum in nonlinear optimization problems by all available optimization algorithms without an exhaustive search 关26兴. Instead of searching for the global
optimum, this optimization algorithm proposed in this paper provides an optimum set of fixture layouts, which contributes to a
significant improvement in the process yield and workpiece sta-
bility with a minimal computational effort. The methodology effectiveness and computational efficiency of the proposed optimization algorithm are compared with those of sequential quadratic
programming 共SQP兲, simplex search, genetic algorithm, and EA
used by Kim and Ding 关26兴. The performances of optimization
algorithms are summarized in Table 17.
A gradient-based search such as sequential quadratic programming is used in determining an optimal fixture layout design
关16,19兴. The disadvantage of a gradient-based search is that the
design solutions are easily entrapped in a local optimum since its
Table 12 The upper and lower boundary coordinates of the projected 2D plane design space in Station 2
Locator
2s
P4way
2s
P2way
NC2s
1
NC2s
2
NC2s
3
2m
P4way
2m
P2way
Design space of interest
Projected 2D plane
design space
Upper boundary of 2D
plane 共x,y兲
Lower boundary of 2D
plane 共x,y兲
A8共=A1兲
B8
A9共=A2兲
B9
A10共=A5兲
B10
A11共=A3兲
B11
A12
B12
A13
B13
A14
B14
共950, −250兲
共1750, −250兲
共1550, −400兲
共1750, −180兲
共1250,191兲
共1556, −154兲
共1556, −551兲
共750, −350兲
共1650, −350兲
共1150, −620兲
共1450, −400兲
共950,401兲
共1556, −154兲
共1556, −551兲
Table 13 The upper and lower boundary coordinates of the projected 2D plane design space in Station 3
Locator
3s
P4way
3s
P2way
NC3s
1
NC3s
2
NC3s
3
3m
P4way
3m
P2way
Design space of
interest
Projected 2D
plane design
space
Upper boundary
of 2D plane 共x,y兲
Lower boundary
of 2D plane 共x,y兲
A15共=A1兲
A16共=A2兲
A17共=A5兲
A18共=A3兲
A19共=A12兲
A20
A21
B15
B16
B17
B18
B19
B20
B21
共950, −250兲
共1750, −250兲
共1550, −400兲
共1750, −180兲
共1250,191兲
共1556,135兲
共1556,532兲
共750, −350兲
共1650, −350兲
共1150, −620兲
共1450, −400兲
共950,401兲
共1556,135兲
共1556,532兲
Table 14 The optimal locator positions in Station 1
Root part
Station 1
Coordinates
X
Y
Z
Mating part
1s
P4way
1s
P2way
NC1s
1
NC1s
2
NC1s
3
1m
P4way
1m
P2way
800.00
−270.00
0.00
1700.00
−270.00
0.00
1743.00
−205.00
1.42
777.00
−275.00
0.00
1308.00
−611.00
29.62
1700.00
240.00
0.00
800.00
240.00
0.00
Table 15 The optimal locator positions in Station 2
Root part
Station 2
Coordinates
X
Y
Z
Mating part
2s
P4way
2s
P2way
NC2s
1
NC2s
2
NC2s
3
2m
P4way
2m
P2way
800.00
−270.00
0.00
1700.00
−270.00
0.00
1308.00
−611.00
29.62
1743.00
−205.00
1.42
1099.00
299.00
0.00
1556.00
−154.00
90.00
1556.00
−551.00
90.00
061005-14 / Vol. 130, DECEMBER 2008
Transactions of the ASME
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Table 16 The optimal locator positions in Station 3
Root part
Station 3
Coordinates
X
Y
Z
3s
P4way
3s
P2way
NC3s
1
NC3s
2
NC3s
3
3m
P4way
3m
P2way
800.00
−270.00
0.00
1700.00
−270.00
0.00
1308.00
−611.00
29.62
1743.00
−205.00
1.42
1099.00
299.00
0.00
1556.00
135.00
90.00
1556.00
532.00
90.00
searching algorithm is based on the steepest ascent/descent direction. Moreover, it is difficult for the gradient-based search to obtain the derivative information where a process yield and an instability index are the quality measures. To illustrate this, the
process yield is multivariate probability density function of random KPC variables, which depend on locator position variations.
Therefore, to formulate the explicit model showing relationships
between the process yield and nominal locator positions, which
allow obtaining derivative information, is infeasible in the highdimensional problems. In addition, the gradient-based search does
not consider the workpiece stability because of the difficulty in
formulating the model that can represent instability index and
yield measure simultaneously.
A simplex search can be used to determine the optimal fixture
layouts. Although a simplex search is a direct search method,
which does not require gradient derivative information, the design
solutions obtained from a simplex search also easily converges to
local optimum. A simplex search was applied on the floor pan
subassembly case study by using fminsearch function available in
MATLAB. The design spaces are defined as continuous and the
Fig. 24 The optimal locator positions selected by HSS
Fig. 25 The optimal locator positions selected by HSS on „a…
bracket left and „b… bracket right
Table 17 Comparison of optimization methods
Methodology
effectiveness
Optimization
methodologies
Gradient-based search
Simplex search
Genetic algorithm
Exchange algorithm
Proposed methodology
Mating part
Computational
efficiency
Process yield
Workpiece
stability
consideration
共Yes/No兲
Computational
time 共s兲
—
88.24%
95.40%
—
96.16%
No
Yes
Yes
No
Yes
—
3600
8103
—
2062
Journal of Manufacturing Science and Engineering
searching operation is terminated in 1 h. The sets of fixture layouts sampled by the simplex search are validated for workpiece
stability first, and then the process yields are calculated. The result
shows that the process yield of the optimal set of fixture layouts
increases only 2.1% from an initial design and 7.9% lower than
process yield of the optimum fixture layouts obtained from the
proposed method in this paper.
The performance of genetic algorithm in determining the optimal set of fixture layouts was also conducted. Although genetic
algorithm can avoid the design solutions converging to local optima, it takes considerably computational time in continuous design spaces. For example, genetic algorithm was applied on the
floor pan subassembly case study to determine the optimum fixture layouts. It takes 8103 s in computational time to obtain a set
of fixture layouts that can meet 95% yield and satisfy workpiece
stability requirement, while the proposed methodology can
achieve the same result within 2062 s.
The performances of the exchange algorithm proposed by Kim
and Ding 关26兴 is also studied and compared with that of the proposed methodology. In terms of algorithm effectiveness, both exchange algorithm and the proposed methodology can provide optimal sets of fixture layouts, which meet the dimensional quality
requirement. However, the optimal result obtained from EA is
sometimes practically infeasible since the workpiece stability is
not incorporated in its design criteria, sensitivity index. Therefore,
comparison between the proposed methodology and EA in terms
of the computational efficiency is difficult because both methods
use the different assessment indices.
In addition, the proposed multi-fixture layout optimization
methodology can be used to eliminate the limitation of KCC tolerance optimization in improving the assembly process robustness. In some design problems, a tolerance optimization methodology might not be able to identify a set of KCC tolerances which
can achieve the process yield requirement 关33兴. The framework
which can integrate multiple design synthesis tasks 共e.g., multifixture layout optimization and KCC tolerance optimization兲 in
optimizing assembly process design can be found in Phoomboplab
and Ceglarek 关34兴. Additionally, the proposed methodology can be
extended into fixture workspace synthesis for reconfigurable assembly 关35兴.
6
Conclusions
This paper presents a methodology to improve a process yield
by optimizing the locator positions in a multistation assembly system. The performance of fixture layouts is assessed by a process
yield, which represents the robustness of fixturing system in terms
of a final product dimensional quality. In addition, fixture locating
stability is taken into consideration to ensure that the design of
fixture layouts is feasible in practical. The variation sources in real
industrial assembly processes, which are locator positions and
part-to-part joint variations are also taken into consideration. The
proposed methodology is based on two-step optimization, which
integrates heuristic algorithm 共GA兲 with a low-discrepancy sampling technique 共HSS兲. The application of the proposed methodology is illustrated through a case study using an automotive underbody assembly where process yield greatly increases from 85%
to 96% after optimizing the locator positions with no increase of
tooling cost.
DECEMBER 2008, Vol. 130 / 061005-15
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Acknowledgment
n ⬅ nm · nm−1 . . . n2 · n1 · n0 = n0 + n1R + n2R2 + ¯ + nmRm
The authors gratefully acknowledge the financial support of the
UK EPSRC Star Award EP/E044506/1 and US NSF-CAREER
Award DMII-0239244. The authors also appreciate the fruitful
discussions with Professor W. Huang, University of Massachusetts, Professor James Kong, Oklahoma State University and Dr.
Ying Zhou from Dimensional Control Systems, Inc.
where R1 , R2 , . . . , Rk−1 are the first k − 1 prime numbers and m
= 关logR n兴 = 关ln n / ln R兴 共the square brackets denote the integer
part兲. A unique fraction between 0 and 1 called the inverse radix
number can be constructed by reversing the order of the digits of
p around the decimal point as follows:
Nomenclature
The Hammersley points on a k-dimensional cube are given by
the following sequence:
KPC
KCC
ri
Li
Si
Mi
⌳i
⫽
⫽
⫽
⫽
⫽
⫽
⫽
s
P4way
⫽
s
P2way
⫽
NCsi ⫽
m
P4way
⫽
m
P2way
⫽
NCm
i ⫽
Fi
Rx,y,z
M x,y,z
Wg
wd
f
m
l⬘x,y and u⬘x,y
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
tmg
␻
v
␼
关A共⌳兲兴
LSL, USL
Ri
Oi
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
Oi
OGA
z = f i共x , y兲
Ai
Bi
⫽
⫽
⫽
⫽
⫽
key product characteristic
key control characteristic
vector defining the location of a locator
fixture layout for a single station
fixture layout for a root part
fixture layout for a mating part
a set of fixture layouts for a particular assembly system
four-way pin in a fixture layout for a root part
共Si兲
two-way pin in a fixture layout for a root part
共Si兲
NC block in a fixture layout for a root part
共Si兲
four-way pin in a fixture layout for a mating
part 共Mi兲
2-way pin in a fixture layout for a mating part
共Mi兲
NC block in a fixture layout for a mating part
共Mi兲
supporting and locating force of a locator
the resultant force
moment of resultant force
weight of a part
external wrench to balance locating stability
force in external wrench wd
moment in external wrench wd
lower and upper boundary of x and y coordinated in design space of interest Ai
twist of a workpiece due to weight
angular displacement in twist tmg
translation displacement in twist tmg
instability index
SOVA matrix
lower and upper specification limits
design space for a locator
a set of nodes discretized from design space
Ri
a Cartesian coordinate of a discretized node
a node selected by GA
surface function of a workpiece
design space of interest
2D projection of design space of interest 共Ai兲
Appendix: Hammersley Sequence Sampling
HSS is used in Step 3 of the proposed methodology. The challenge in the local search step is high-dimensional design space,
which usually requires a large number of iterations or samples.
HSS is a sampling technique that selects samples uniformly in a
hypercube design space, which requires fewer samples to converge to the solution within desired variance compared with other
sampling technique or space filling technique such as Number–
Theoretical Net 关31兴 or Latin hypercube 关32兴. Kalagnanam and
Diwekar 关32兴 provided a procedure for selecting N Hammersley
points in k-dimensional hypercube. Any integer n共n
苸 兵1 , 2 , . . . , N其兲 can be written in radix-R notation 共R is a prime
number兲 as follows:
061005-16 / Vol. 130, DECEMBER 2008
␾R共n兲 = n0n1n2 ¯ nm = n0R−1 + n1R−2 + ¯ + nqR−m−1
zk共n兲 =
冉
冊
n
, ␾R1共n兲, ␾R2共n兲, . . . , ␾Rk−1共n兲 ,
N
n = 1,2, . . . ,N
The Hammersley points generated in a unit hypercube are
xk共n兲 = 1 − zk共n兲
References
关1兴 Chou, Y.-C., Chandru, V., and Barash, M. M., 1989, “A Mathematical Approach to Automatic Configuration of Machining Fixtures: Analysis and Synthesis,” ASME J. Eng. Ind., 111共2兲, pp. 299–306.
关2兴 Shalon, D., Gossard, D., Ulrich, K., and Fitzpatrick, D., 1992, “Representing
Geometric Variations in Complex Structural Assemblies on CAD Systems,”
ASME Paper No. DE-44/2, pp. 121–132.
关3兴 Ceglarek, D., Huang, W., Zhou, S., Ding, Y., Kumar, R., Zhou, Y., 2004,
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DECEMBER 2008, Vol. 130 / 061005-17
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