Process Design
Resume:
THREE-DIMENSIONAL ASSEMBLY SYNTHESIS FOR ROBUST DIMENSIONAL
INTEGRITY BASED ON SCREW THEORY
Byungwoo Lee
Department of Mechanical Engineering
University of Michigan
USA
********@*****.***
Kazuhiro Saitou
Department of Mechanical Engineering
University of Michigan
USA
****@*****.***
designs of a rectangular box. In contrast to design in
ABSTRACT
(a) with no in-process adjustability of the critical
This paper presents a 3D extension of our previous dimensions (length between sections 1 and 3), design
work on the synthesis of assemblies whose in (b) provides slip planes such that relative location
dimensional integrity is insensitive to the dimensi- of parts can be adjusted along the critical dimension.
onal variations of individual parts. Assuming that
1 1
assemblies can be built in the reverse sequence of (b)
(a)
decomposition, the method recursively decomposes a
4
given product geometry into two subassemblies until 2
4
2
parts become manufacturable. At each recursion,
joints are assigned to the interfaces between two 3
3
Figure 1 Two box designs (a) without and (b) with
subassemblies to ensure the two criteria for robust
adjustable height during assembly (Lee and
dimensional integrity, in-process dimensional
Saitou, 2003a).
adjustability and proper part constraints. Screw
Theory is utilized as a unified 3D representation of The dimensional integrity of an assembly is also
the two criteria. A case study on an automotive space affected by the post-assembly distortion due to the
frame is presented to demonstrate the method. internal stress induced by joining parts with
dimensional mismatches. A solution is to ensure a
KEYWORDS proper constraint of parts at each assembly step. For
Design for assembly, design optimization, computer- example, part 1 in Figure 2 (a) is not properly
aided design, assembly synthesis. constrained and therefore the post-assembly
distortion might occur, if the length of sections 2 and
1. INTRODUCTION 4 are slightly different due to manufacturing
Structural enclosures of modern mechanical variation. With two slip planes perpendicular to each
products, such as ship hulls, airplanes and other, the design in (b) can absorb manufacturing
automotive bodies, typically are made of hundreds or variations within section 1 and 2-3-4, provided that
thousands of parts due to their geometric complexity variations in angles are negligible.
and sizes. As the number of parts increases, however, In addition to the decomposition of product geometry
achieving the dimensional integrity of the final and the assignment of joint types at part interfaces,
assembly becomes more difficult due to the inherent the assembly sequence also influences the in-process
variations in manufacturing and assembly operations. 1 1
(a) (b)
A solution is to adjust critical dimensions during an
2 4 2 4
assembly when parts are located and fully
constrained in fixtures. This in-process dimensional
3 3
adjustment is typically facilitated by slip planes,
mating surfaces at joints that allow a small amount of Figure 2 Two box designs (a) without and with (b)
relative motions. For example, Figure 1 shows two proper constraints (Lee and Saitou, 2003a).
585
dimensional adjustability and proper part constraints. feasible solutions for any 2D enclosure geometry.
In an assembly sequence in Figure 3 (a), the critical Assuming that assemblies can be built in the reverse
dimension (total length) is not adjustable since there sequence of decomposition, the algorithm recursively
is no slip plane when the total length is determined decomposes a given product geometry into two
with the addition of part 1. On the other hand, the subassemblies until parts become manufacturable. At
sequence shown in (b) provides the slip plane at the each recursion, joints are assigned to the interfaces
assembly step where the critical dimension is between two subassemblies to ensure in-process
achieved, to absorb a variation in length. As another dimensional adjustability and proper part constraints.
example, the sequence in Figure 4 (b), where each 1
(b)
(a)
critical dimension is independently adjusted at each
step, is more desirable than the sequence in (a), 2
2 4
where both dimensions are adjusted by one step,
inevitably requiring a compromise between two 3
3
critical dimensions. 1
1
2 3 1 2
4
2
4
2
3
1 2 3 3
1 2 3
Figure 5 Assembly sequences (a) without and (b) with
(a) (b) proper constraints (Lee and Saitou, 2003a).
Figure 3 Assembly sequences (a) without and (b) with
This paper presents a 3D extension of the algorithm,
in-process adjustability (modified from Whitney,
where Screw Theory (Ball, 1900) is utilized as a
et al., 1999).
unified 3D representation of in-process adjustability
and proper part constraints. Dissimilar to our
2 3
2
1
previous work that assumes joints with arbitrary
mating angles, they are selected from a library of
feasible joints specific to the application domain. A
2 3
1
1 2 3
case study on an automotive space frame is presented
to demonstrate the method.
(a) (b) 2. RELATED WORKS
Previous works related to assembly synthesis in
Figure 4 Assembly sequences where two dimensions are
general are reviewed in (Lee and Saitou, 2003a). Due
adjusted (a) at one step and (b) independently at
two steps (modified from Whitney et al., 1999). to the space limit, this section focuses on the works
relevant to the 3D extension of the method.
Figure 5 illustrates an effect of the assembly
The advantages of properly constrained assemblies
sequence on proper part constraints, where the
are well known to practitioners in precision
sequence in (a) causes improper constraint of part 1
machinery design and several methods have been
at the second step, whereas all parts are properly
proposed in literatures including: Kinematic Design
constrained at all steps in the sequence in (b).
(Whitehead, 1954), Minimum Constraint Design
As illustrated so far, the in-process adjustability and (Kamm, 1990) and Exact Constraint Design (Kriegel,
proper part constraints are effective tools for 1995; Blanding, 1999). These works describe
achieving high dimensional integrity of an assembly disadvantages of over-constraint through examples
without requiring high part tolerances (Blanding, and provide good practices as well as analytical
1999). The use of these tools in complex assemblies methods to compute constraints. In these works, the
can be a very tedious task due to the coupling most commonly cited merit of properly constraint
between the product decomposition, joint design is repeatability which leads to high precision.
assignments, and assembly sequences. As a remedy, Recently, there has been a trial to analyze and
we have previously designed (Lee and Saitou, 2003a) classify key features that enables properly constraint
a correct and complete algorithm to fully enumerate design (Downey, et al., 2002).
Byungwoo Lee & Kazuhiro Saitou
586
where M, E, and A are the sets of nodes
A universal analytical method for motion and
constraint analysis dates back to Screw Theory, a representing members, edges representing
pioneering work by Ball (1900). Since then, Screw connections between two members, and edges
Theory has been applied to areas of mechanism, representing KCs, respectively (Figure 6 (b)).
robotics and machine design. Among others, A decomposition is a transition of a configuration
Waldron (1964) utilized the screw theory to build a into two or more subconfigurations by removing
general method which can determine all relative connections between two members.
degrees of freedom (DOF) between any two rigid A part is a configuration that is not decomposed
bodies making contacts to each other. Recently, further under given criteria, eg., a minimum part
Konkar and Cutkosky (1995) has proposed a size. A part may consist of one or more members.
recursive algorithm which computes motions allowed A joint library is a set of joint types available for
by mating features within mechanisms. Adams and a specific application domain (Figure 7).
Whitney (1999) have extended this method by An (synthesized) assembly is a set of parts and
providing a dual method to compute the state of joints that connect every part in the set to at least
constraint of parts and applied it to rigid body one of other parts in the set.
assemblies with mating features such as pin-slot joint. Assembly synthesis is a transformation of a
product geometry into an assembly.
While these works provide tools for analyzing
constraints in a given assembly and simple design
(a) (b) 1
guidelines, they do not address a systematic synthesis 1
4
of an assembly with desired constraint characteristics
such as in-process dimensional adjustability and kc2 4
2
kc2
proper part constraints, as discussed in this paper.
kc1
kc1
2
3. TERMINOLOGY* 3 3
Figure 6 (a) product geometry of a beam based product
Since the assembly synthesis deals with objects yet to
and (b) its configuration graph.
be decomposed into an assembly of separate parts, a
few terms need to be defined to avoid confusion with
generic meanings used in other literatures.
A product geometry is a geometric representation
of a whole product as one piece (Figure 6 (a)).
Figure 7 An example of joint library for 3-D beam based
A member is a section of a product geometry assemblies consisting of lap, butt and lap-butt.
allowed to be a separate part. A pair of members
is connected when they meet at a certain point in
4. SCREW THEORY
the product geometry.
A configuration is a group of members which are In Screw Theory, a screw is defined as a pair of a
connected to at least one member within the straight line (screw axis) in a 3D Cartesian space and
group. A product geometry is a configuration, so a scalar (pitch). It is commonly represented by screw
as a part (as defined below). coordinates, a pair of two row vectors = (s; s0) in
The Key Characteristics (KCs) are defined by Lee 3D Cartesian coordinates, where s is a unit vector
and Thornton (1996) as product features, parallel to the screw axis and
manufacturing process parameters, and assembly
s0 = r s + ps (2)
features that significantly affect a product s
performance, function and form. In this paper, KC In the equation, r is the position vector of a point on
refers to a critical dimension to be achieved in the screw axis and p is the pitch, which can be
assemblies. recovered using
A configuration graph (or simply configuration if
unambiguous in the context) is a triple
C = (M, E, A) (1)
The terminology and formalization in this section are
summarized from Ball (1900), Hunt (1978), Roth (1984)
Previously defined in Lee and Saitou (2003b). and Konkar and Cutkosky (1995).
Three-dimensional assembly synthesis for robust dimensional integrity based on screw theory 587
s s0 The intersection of screw matrices is the set of
p= (3) screws common to the screw matrices and can be
s s
computed through double reciprocals:
A screw with an infinite pitch does not follow
n n
IS = reciprocal(U reciprocal(S i ))
Equation (2), instead it is denoted by using zero (6)
vector for s and having s0 represent the unit vector
i
i =1 i =1
parallel to the screw axis.
Since a twist and a wrench are also screws, the
Two types of special screws, a twist and a wrench, definitions of reciprocal, union, and intersection hold.
are utilized in this paper. A twist is a screw
Woo and Freudenstein (1970) presents kinematic
representing a motion of a rigid body simultaneously
properties of various joint types in screw coordinates,
rotating around and translating along an axis. Using
which are borrowed to build twist matrices of beam
screw coordinates, it is denoted as = ( ; v), where
joint types.
is the angular velocity and v is the linear velocity
of a point on the body (or its extension) located at the
origin of global reference frame. A wrench is a screw (a) (b)
representing a force along and a moment around an z z
axis exerted on a rigid body. Using screw coordinates,
it is denoted as = (f; m), where f is the force and m x x
y
is the moment that a point on the body (or its y
extension) located at the origin of global reference
Figure 8 Lap (a) and lap-butt joint (b) of a beam based
frame should resist.
model and the local coordinate frames for twists.
Two screws, 1 = (s1; s01) and 2 = (s2; s02), are
reciprocal to each other, if and only if they satisfy: Figure 8 (a) shows a typical lap joints found in beam-
based structures. When it is attached to another beam,
s1 s02 + s01 s2 = 0. (4)
the tab allows planar motion parallel to x-y plane.
If a twist is a reciprocal of wrench (or vise Also, if we assume that the length of the tab is very
versa), does no work to a rigid body moving small compared to the length of the beam, it can be
according to . treated as a line contact along y-axis, thus allowing
the rotation about y-axis. Thus, a lap joint at its local
When a body can receive linear combinations of
coordinate frame can be modeled as a twist matrix:
several screws (either twist or wrench), this set of
screws are typically represented as a matrix where 0 1 0 0 0 0
each screw in the set forms a row vector of the
0 0 1 0 0 0
matrix. This matrix is called a screw matrix. As its = (7)
Tlap
row space is the screw space, the rank of a screw 0 0 0 1 0 0
matrix is equal to the dimension of the screw space. 0 0 0 0 1 0
The function reciprocal(S) returns a screw matrix, of
Similarly, a butt joint in Figure 8 (b) allows the
which row space includes all reciprocal screws to
motion parallel to y-z plane, can be modeled as:
screws contained in S. It can be obtained by
exchanging the former three columns and the letter 100000
three columns of the null space of S.
Tbutt = 0 0 0 0 1 0 (8)
The union of screw matrices represents the sum of 000001
screw spaces and it can be obtained by simply
stacking them on top of one another: In twist matrices in equation (7) and (8), each row
represents an independent motion, and each non-zero
S1
number represents rotation or translation along a
S2
n
corresponding axis x, y, z, vx, vy or vz. For
U Si = M (5)
example, in the first row in Equation (7) has 1 at the
i =1
second column, which means the lap joint allows
Sn
rotational motion about y-axis. In the third row, it has
1 at the fourth column, meaning translation along the
Byungwoo Lee & Kazuhiro Saitou
588
must be connected before and after decomposition.
x-axis. As these matrices are used only to give
The 3rd and 4th conditions specify two
information on which DOFs are constrained for a
subconfigurations do not share any members.
joint type, amplitude of each twist (row) of these
twist matrices, in this paper, does not have significant
A joint is assigned to each connection broken by a
meaning.
binary decomposition, which can be represented as a
mapping d : CS d a JL, where JL is a joint library.
Once the twist matrix is obtained for a joint type, the
reciprocal wrench matrix can be computed as With the joint assignment, a (binary) decomposition
described above, and the wrench matrices d can be uniquely specified as d = (Ma, d, (Mb, Mc)).
corresponding to twist matrices in (7) and (8) are: See Figure 10 for an example. Note that a feasible
joint type may depend on the local geometry at the
001000
Wlap = reciprocal(Tlap ) = (9) joint location. For example, feasible joint types
000100 between two perpendicular beams would be different
from that for two coaxial beams.
100000
1
Wbutt = reciprocal(Tbutt ) = 0 0 0 0 1 0 1
(10) cut-set
4
000001
kc2
kc 4
2
Each non-zero number now represents force or
kc1
kc1
2
moment along a corresponding axis fx, fy, fz, mx, my
3
or mz - that the joint can constrain. For example, in 3
the first row in Equation (9) has 1 at the third column,
which means the lap joint can support a force along 1 1
z-axis. 4
kc2
5. 3D ASSEMBLY SYNTHESIS 4
2
kc1
2
5.1. Binary decomposition 3
3
The assembly synthesis algorithm in (Lee and Saitou,
Figure 9 A binary decomposition in product geometry
2003a) adopted in this paper assumes every assembly (left) and configuration graph (right).
step combines two subassemblies. Conversely, the
algorithm decomposes a configuration into two Z
(sub)configurations, by removing some connections, 1
1
j1
which is equivalent of finding a cut-set (Foulds,
L
4
1991) of the configuration graph. In the following, X
j2
Y kc2
CSd and KCd denote the cut-set and the set of KCs 4
2
kc1
broken by a decomposition d, respectively. For the
L
2
decomposition shown in Figure 9, CSd = {(1, 2), (3, 3
3
4)} and KCd = {kc, kc2}.
Any configuration Ca = (Ma, Ea, Aa) decomposed to Figure 10 Joint types assigned to the subconfiguratins in
Figure 9. The L represents a lap joint from a
two subconfigurations Cb = (Mb, Eb, Ab) and Cc = (Mc,
lower-index node to a higher-index node.
Ec, Ac), must satisfy the following conditions:
Mb and Mc .
5.2. The 1st decomposition rule for in-
(Ma, Ea), (Mb, Eb) and (Mc, Ec) are connected.
process dimensional adjustability
Ma = Mb Mc.
Mb Mc = . (9) Let us consider how to assign appropriate joint types
for those decompositions which have at least one
st
The 1 condition states subconfigurations should be
broken KC. Recall Figure 3, which has a slip plane
nonempty. The 2nd condition states the configurations
between parts 2 and 3 such that the KC can be
delivered. The assembly sequence in Figure 3 (b)
In a configuration graph, edges representing KCs are not shows that it is desirable that a slip plane is provided
counted to a cut-set.
Three-dimensional assembly synthesis for robust dimensional integrity based on screw theory 589
rank(( W )( Wa ))
at the very assembly operation where KC is realized,
(e )
no matter how subassemblies are assembled before. d
e CSd a KC d
(12)
This can be stated in the reverse course as follows: = rank( W ) + rank( Wa )
(e )
no matter how a subconfiguration is decomposed d
e CSd a KC d
further, when KCs are broken by a decomposition,
Furthermore, as the proper constraint design in its
joints assigned to the cut-set, in combination, should
rigorous definition avoids under-constraint as well as
allow motions compatible with the KCs. This
over-constraint, the combined constraints from joints
statement has been refered to as the 1st
and KCs should cover six DOFs, such that no DOF
decomposition rule for in-process dimensional
could be left unconstrained when two parts are being
adjustability (Lee and Saitou, 2003a).
assembled. In other words, the dimension of the
A KC, in this paper, is assumed to be a critical combined wrench space, i.e., the rank of the union of
dimension between parts only achieved by joint and KC wrench matrices, should be equal to six.
adjustment during assembly of the parts. Thus the Combined with equation (12), we can now conclude
dimension noted as a KC will be constrained by a the 1st rule of decomposition for in-process
fixture, according to which parts being assembled dimensional adjustability with:
will be adjusted. In this context, KC constrains
rank(( W )( Wa ))
relative DOFs between two parts; hence it is natural (e )
d
e CSd a KC d
to model a KC as a wrench matrix. The approach (13)
= rank( W ) + rank( Wa ) = 6.
taking tolerance relations as constraints can be found (e )
d
in the area of computer-based tolerance modeling, e CSd a KC d
and a recent study by Wu et. al. (2003) shows the
Consider the product geometry decomposed in
number and the type of DOFs constrained for each
Figure 9 and joint assignment shown in Figure 10,
tolerance relation in standard tolerance classes. In
which has two lap joints, j and j2 for edges cut by
this paper, we consider only distance and angularity
the decomposition. Suppose the location of j and j2
between lines (beams axes). The distance between
in global reference frame X-Y-Z are (3, 0, 0) and (0,
lines constrain only one translational DOF between
4, 0). Then, based on the local coordinate frame of
two points where the KC is anchored, thus it is
lap joint shown in Figure 8 and orientation of j and
modeled as a wrench whose axis passes these points.
j2, Wlap (Equation (9)) can be transformed to j and
The angularity between lines constrains only one
j2 in global reference frame. Then the union of joint
rotational DOF between two lines and it is modeled
wrench matrices can be computed:
as a wrench with infinite pitch whose axis is the
vector product of the two lines direction vectors. 001000
U = W j U W j2 ~ 0 0 0 1 0 0
st
(14)
W
The 1 decomposition rule for in-process (e)
d
e CSd
dimensional adjustability, in other words, states that 000010
the DOFs constrained by KCs should not be
constrained by the joints, thus avoiding conflicts. The wrench matrix in (14) has 1 at the third, fourth
Once wrench matrices are associated to joints and and fifth column, meaning that it supports force
KCs broken by a decomposition, this rule can be along Z-axis, moments about X and Y axis,
stated in the screw theory s terminology: for a respectively. On the other hand, the decomposition in
decomposition, the wrenches representing joints and Figure 9 has broken two KCs, kc and kc2. The
KCs should not constrain the same DOF, thus union of these KCs is:
satisfying:
0 1 0 0 0 1.5
U Wa = Wkc1 U Wkc 2 = . (15)
( W )( Wa ) = O (11) 00000 1
(e ) a KCd
d
e CSd a KC d
Note that Wkc (upper row) represents the distance
Since the rank of the intersection of the joint and KC
KC between member 1 and 3 (translation along Y-
matrices is zero as shown in equation (11), by the
axis) and Wkc2 (lower row) represents the angularity
theorem from linear algebra, it is obvious that the
KC between member 1 and 2 (rotation about Z-axis).
rank in equation (12) is merely summation of ranks
of joint and KC matrices:
The result has been reduced to the Row Reduced Echelon
Form for easy interpretation.
Byungwoo Lee & Kazuhiro Saitou
590
The union of the joint twist matrix (Equation (14)), which is also equivalent to:
and KC twist matrix (Equation (15)) is:
rank( Wa ) = rank(Wa ) . (18)
0 1 0 0 0 0 a KC d a KC d
0 0 1 0 0 0 The two KCs shown in Figure 9, each with single
( U W U
)U( wrench make the KC matrix of rank 2 as shown in
Wa ) ~ 0 0 . (16)
0 0 1 0
(e)
d
Equation (15), thus satisfying Equation (18).
e CSd a KCd
0 0 0 0 1 0
0 0 0 0 0 1 5.4. The decomposition rule for in-
process proper constraint
It shows that the parts are constrained in X-axis
neither by joints nor by KCs. Although it does not In Figure 5, it has been shown that joints should be
satisfy Equation (13), it does satisfy Equation (12), perpendicular to each other to have subassemblies
which implies at least that there is no conflict being assembled properly constrained. Similarly to
between joints and KCs. As this decomposition does drawing the first and second decomposition rule for
not satisfy Equation (13), the assembly synthesis in-process dimensional adjustment, this assembly
process will discard it. rule has been inversed to the decomposition rule for
non-forced fit in our previous work (Lee, B. and
5.3. The 2nd decomposition rule for in- Saitou, K., 2003a), which allows only mutually
process dimensional adjustability perpendicular joints to be broken by a decomposition.
This rule is simplified and limited to two-
As discussed in Figure 4, when multiple KCs in the
dimensional space assuming over-constraints in
same direction are realized at an assembly step, the
rotation are minimal.
adjustment of one KC will affect the dimension of
The idea of this rule is that there should be no over-
the other KCs. Viewing KCs as constraints, this
constraint at each assembly step, hence the
happens when two or more KCs constrain the same
decomposition rule (renamed as the decomposition
DOF of a subassembly at an assembly step. However,
rule for in-process proper constraint) should not
for complex assemblies, detecting over-constrained
allow any combination of joints yielding over-
tolerance relationship is not straightforward from the
constraint of parts. In other words, joints placed for
engineering drawings because tolerances are
connections broken by a decomposition, i.e., the
specified on parts, not subassemblies, which are
joints corresponding to CSd, should not constrain the
defined by assembly sequences. Therefore, a clumsy
same DOF more than once. Except that joints serve
assembly planning might cause a subassembly s
as constraints, instead of KCs, this rule is identical to
DOF to be constrained by several KCs. In order to
the 2nd rule of in-process dimensional adjustability,
avoid this situation, one should plan assembly steps
thus satisfying:
such that, in every assembly step, subassemblies
being assembled are free of over-constraining KCs.
CSd, ( W )( W )=O
C (19)
(e ) (e )
Accordingly, the 2nd decomposition rule for in- d d
e CSd \C
eC
process dimensional adjustability in (Lee and Saitou,, which is also equivalent to:
2003a) states a decomposition can break only KCs
independent to each other ** . In other words, KCs rank( W )= rank(W ). (20)
(e ) (e )
broken by a decomposition, i.e., the KCs in KCd, d d
e CSd e CSd
should not constrain the same DOF more than once.
For the decomposition depicted in Figure 10, each of
In such cases, the intersection of the wrench matrix
the two joints j and j2 has rank 2 (Equation (9)).
corresponding to any subset of KCd and the wrench
However, the union of corresponding wrench
matrix of its complement set must result in the zero
matrices has rank 3, which does not satisfy Equation
matrix:
(20). In order to check what DOFs are over-
KC d, ( Wa ) ( Wa ) = O
K (17) constrained, we can intersect the wrench matrices:
a KC d \K
aK
W j I W j 2 = recip(recip( W j ) U recip( W j2 ))
= recip(Tj U Tj2 ) (21)
In 2D cases in the previous works, only KCs
= [0 0 1 4 3 0] O
perpendicular to each other were allowed to be stricter.
Three-dimensional assembly synthesis for robust dimensional integrity based on screw theory 591
The results states that the joints over-constrain the Ma has a closed loop (cannot extrude such parts).
Ma has a connection point where three or more
translational DOF along Z-axis, which yields locked
members meet (cannot extrude such parts).
moment about X-axis with unit of 4 and moment
Ma has members lie on more than one plane (difficult
about Y-axis with unit of -3. It occurs because j
to handle/fixture).
itself constrains parts both in translation along Z-axis
and the moment about X-axis at the same time j2 The product geometry shown in Figure 6 has two
combined with j constrain the moment about X-axis KCs and a closed loop thus stop_de returns false,
again. And j2 and j cooperate in the same way to subject to further decomposition.
result in the locked moment about Y-axis.
M1 M2
5.5. Unified decomposition rule for in-
process proper constraint
1 2 3
According to Equation (18), the set of KCs related to
a decomposition should be linearly independent.
decompose
Similarly, the set of joints assigned for broken
assemble
M3 M5 M7
connections should be linearly independent
according to Equation (20). Further, as these sets
should be linearly independent to each other by M M6
Equation (13), these three equations in combination
requires the independency of constraints, regardless
of KC or joint, and full rank when unionized. Thus,
Figure 11 A partial AND/OR graph of the 2-D rectangular
combining Equation (13), (18) and (20), we can unify
box in Figure 1.
three decomposition rules into:
rank(( W )( Wa ))
5.7. AND/OR graph of assembly
(e )
d
e CSd a KC d
(22) synthesis
= rank(W )+ rank(Wa ) = 6.
(e )
Figure 11 shows a partial AND/OR graph of
d
e CSd a KC d
assembly synthesis (Lee and Saitou, 2003a) for the
Finally, a predicate of a decomposition d = (Ma, d, 2D rectangular box shown in Figure 1. Each node in
(Mb, Mc)) for complying the all three rules is given white background contains a subset of members (Ma
as de: 2M0 (2E0 a JL) (2M0 2M0 ) a {true, false}, M0) and each node in black background contains
where de(Ma, d, (Mb, Mc)) is true if and only if joint assignment i : CS i a JL . A set of three lines
Equation (22) is satisfied. However, it is often the which connects a configuration Ma, joint assignment
case that a under-constraints are unavoidable during i, and two subconfigurations (Mb, Mc) is a hyper-
assembly synthesis due to the limited choice of joints,
edge, represented as (Ma, i, (Mb, Mc)) which is also
Equation (22) may be relaxed to abandon the full
the representation of a decomposition defined earlier.
rank.
The AND/OR graph of assembly synthesis is then
represented as a triple:
5.6. Part manufacturability
AO = (S, J, F) (23)
The decomposition stops when the resulting
where S is a set of nodes representing configurations,
subconfigurations become manufacturable by a
J is a set of nodes representing joint assignments, and
chosen manufacturing process. In the following case
F is a set of hyper-edges (Ma, i, (Mb, Mc)) satisfying
study on frame structures, components are assumed
the following necessary conditions.
to be extruded and bent. Therefore, a predicate of a
configuration Ma for stopping decomposition is given
1. stop_de(Ma) = false.
as stop_de: 2M0 a {true, false}, where stop_de(Ma) is 2. de(Ma, i, (Mb, Mc)) = true. (24)
false (i.e.. decomposition continues) if and only if
Then AO = (S, J, F) is recursively defined as:
any of the following conditions are satisfied:
1. If stop_de(M0) = false, M0 S.
Ma has a KC (KCs can not be achieved by the
tolerances of extrusion and bending).
Byungwoo Lee & Kazuhiro Saitou
592
2. For Ma S, if i, Mb, Mc such that f = (Ma, i, with 73 feasible subassemblies. Due to the space
limit, we have extracted a part of the AND/OR graph
(Mb, Mc)) satisfies necessary conditions (19), then
i J, Mb, Mc S and f F. containing the assembly designs with minimum
number of parts and under-constrains. The extracted
3. No element is in S, J and F, unless it can be
graph has 26 feasible decompositions with 28
obtained by using rules 1 and 2. (25)
feasible subassemblies, a half of which is depicted in
Figure 14 as the product geometry is symmetrical
The recursive definition in Equation (25) can be
about its XZ-plane. In Figure 14, white nodes are
easily transformed to an algorithm build_AO that
parts and grey nodes are subassemblies. Joint
generates AO from initial configuration C0 = (M0, E0,
assignments are represented as black nodes with
A0) and joint library JL by recursively decomposing a
numbers, which represent the number of under-
configuration into two subconfigurations (Lee and
constraints related to the assembly step. The number
Saitou, 2003a), whose details are omitted due to
of parts and the number of under-constraints do not
space limitation. Using stop_de and de as defined
exhibit any trade-off in this example, since in general
earlier, one can run build_AO with any 3D
assemblies with fewer parts have less chance to be
configurations to enumerate all possible assemblies
under-constrained. All assemblies shown in Figure
(decompositions and joint assignments) and
14 have 7 parts and 2 under-constraints. There are no
accompanying assembly sequence that satisfy the
assemblies with fewer parts or under-constraints.
rules for in-process dimensional adjustability and
proper part constraint.
0
6. CASE STUDY 0
A frame structure in Figure 12 is decomposed using A
the joint types in Figure 13. Since the initial attempt 0
yields no assembly synthesis without under-
constraints, the Equation (22) is relaxed to allow B
under-constraints.
0
kc7
kc2
kc1 C
kc8 0
0
kc3 kc6
Z
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