Y. Zhong et al. / Computer-Aided Design 45 (2013) 1253–1275 1257
feature surfaces from each part, to construct an ABox A
AL
by gather-
ing all OWL class assertions which represent instances of assembly
feature surfaces and OWL property assertions which represent as-
sembly constraint relations between assembly feature surfaces of
parts, and to lay foundation for the construction of spatial relation
layer.
Each part of an assembly can be regarded as a closed geometry
which is constituted by a number of feature surfaces. In all of the
feature surfaces of a part, the feature surfaces, which have assem-
bly constraint relations with the feature surfaces of other parts, are
called as the assembly feature surfaces of this part. Each part of an
assembly has at least one assembly feature surface, therefore the
assembly constraint relations between parts can be decomposed
into the assembly constraint relations between assembly feature
surfaces of each part.
Let RealFeatureSurface be an OWL class which denotes a real
assembly feature surface. Let has-AssemblyRelation be an OWL
object property which denotes an assembly constraint relation
between assembly feature surfaces of parts. The assembly feature
surfaces and assembly constraint relations between assembly
feature surfaces of parts in assembly feature surface layer can be
formally defined as follows.
Definition 2. Let p
i
= {s1(p
i
), s2(p
i
), . . . , s
m
(p
i
)} be the i-th part
in an assembly, where s
1
(p
i
), s
2
(p
i
), . . . , s
m
(p
i
) are the m assem-
bly feature surfaces of part p
i
. Let p
j
= {s
1
(p
j
), s
2
(p
j
), . . . , s
n
(p
j
)}
be the j-th part in this assembly, where s
1
(p
j
), s
2
(p
j
), . . . , s
n
(p
j
)
are the n assembly feature surfaces of part p
j
. If there is an as-
sembly constraint relation between the assembly feature surfaces
s
u
(p
i
) (u = 1, 2, . . . , m) and s
v
(p
j
) (v = 1, 2, . . . , n), then we can
obtain the OWL class assertions RealFeatureSurface(s
1
(p
i
)),
RealFeatureSurface(s
2
(p
i
)), . . . , RealFeatureSurface(s
m
(p
i
)),
RealFeatureSurface(s
1
(p
j
)), RealFeatureSurface(s
2
(p
j
)), . . . ,
RealFeatureSurface(s
n
(p
j
)), and the OWL property assertions
has-AssemblyRelation(s
u
(p
i
), s
v
(p
j
)) and
has-AssemblyRelation(s
v
(p
j
), s
u
(p
i
)). The ABox A
AL
is constructed
by gathering all these assertions.
3.3. Spatial relation layer
Spatial relation layer is the last layer of the extended model.
Its main functions are to extract the spatial relations between ge-
ometrical features of each pair of assembly feature surfaces, to
construct an ABox A
SL
by gathering all OWL class assertions which
represent instances of geometrical features and OWL property as-
sertions which represent spatial relations between geometrical
features of each pair of assembly feature surfaces, and to lay foun-
dation for assembly tolerance representations.
From the language of geometrical product specifications [60],
there are seven types of feature surfaces: spherical surface,
cylindrical surface, planar surface, helical surface, revolute surface,
prismatic surface, and complex surface. Zhang et al. [45] increased
the number of types of feature surfaces from seven to eleven. They
are inner (outer) spherical surface, inner (outer) cylindrical surface,
planar surface, inner (outer) helical surface, inner (outer) revolute
surface, and inner (outer) prismatic surface. According to the
classifications of lower and higher pairs [60], we add inner (outer)
complex surface into the eleven types of feature surfaces above.
So the number of types of feature surfaces is thirteen. The possible
constraint relations and corresponding associated derived features
of the thirteen types of feature surfaces are shown in Table 1.
From Table 1, we know that the associated derived feature of
each type of feature surface is a single point, a single straight line,
a single plane, or their combination. Therefore the spatial relations
among associated derived features are essentially the spatial
relations among points, straight lines, and planes. In practice,
Table 1
Possible constraint relations and corresponding associated derived features of the
thirteen types of feature surfaces. ISS (OSS) stands for inner (outer) spherical
surface, ICS (OCS) stands for inner (outer) cylindrical surface, PLS stands for planar
surface, IHS (OHS) stands for inner (outer) helical surface, IRS (ORS) stands for inner
(outer) revolute surface, IPS (OPS) standards for inner (outer) prismatic surface, IXS
(OXS) stands for inner (outer) complex surface.
Type Feature surfaces Associated derived features
C01 (OSS, OSS) (Point, Point)
C02 (OSS, ISS) (Point, Point)
C03 (OSS, PLS) (Point, Plane)
C04 (OSS, OCS) (Point, Line)
C05 (OSS, ICS) (Point, Line)
C06 (OSS, OHS) (Point, (Point, Line))
C07 (OSS, IHS) (Point, (Point, Line))
C08 (PLS, PLS) (Plane, Plane)
C09 (PLS, OCS) (Plane, Line)
C10 (PLS, OHS) (Plane, (Point, Line))
C11 (PLS, ORS) (Plane, (Point, Line))
C12 (PLS, OPS) (Plane, (Line, Plane))
C13 (OCS, OCS) (Line, Line)
C14 (OCS, ICS) (Line, Line)
C15 (OCS, OHS) (Line, (Point, Line))
C16 (OCS, ORS) (Line, (Point, Line))
C17 (OCS, OPS) (Line, (Line, Plane))
C18 (OHS, OHS) ((Point, Line), (Point, Line))
C19 (OHS, IHS) ((Point, Line), (Point, Line))
C20 (ORS, ORS) ((Point, Line), (Point, Line))
C21 (ORS, IRS) ((Point, Line), (Point, Line))
C22 (OPS, OPS) ((Line, Plane), (Line, Plane))
C23 (OPS, IPS) ((Line, Plane), (Line, Plane))
C24 (OXS, IXS) ((Point, Line, Plane), (Point, Line, Plane))
the spatial relations among points, straight lines, and planes are
coincidence, disjoint, inclusion, parallel, perpendicular, intersection,
and nonuniplanar. The details are shown in Table 2.
Furthermore, each type of real feature surface has constraint
relation with its ideal feature surface. The details are shown in
Table 3.
With the geometrical features and spatial relations in Tables 2
and 3, the geometrical features and spatial relations between geo-
metrical features of each pair of assembly feature surfaces can be
described by OWL class and property assertions, respectively. Let
RealPlanar be an OWL class which denotes a real planar surface.
Let adf (x) be the associated derived features of real feature sur-
face x. Let ConstraintPlane and ConstrainedPlane be OWL classes
which denote a constraint plane and constrained plane, respec-
tively. Let has-CoincidenceRelation and has-ConstraintRelation be
OWL object properties which denote coincidence and constraint
relations, respectively. Let ideal-planar denote an ideal planar sur-
face. The spatial relations among geometrical features of each pair
of mutual constraint assembly feature surfaces in spatial relation
layer can be formally defined as follows.
Definition 3. Let p
i
= {s
1
(p
i
), s
2
(p
i
), . . . , s
m
(p
i
)} be the i-th part
in an assembly, where s
1
(p
i
), s
2
(p
i
), . . . , s
m
(p
i
) are the m assem-
bly feature surfaces of part p
i
. Let p
j
= {s
1
(p
j
), s
2
(p
j
), . . . , s
n
(p
j
)}
be the j-th part in this assembly, where s
1
(p
j
), s
2
(p
j
), . . . , s
n
(p
j
)
are the n assembly feature surfaces of part p
j
. Let OWL ob-
ject properties has-CoincidenceRelation, has-DisjointRelation, has-
InclusionRelation, has-ParallelRelation, has-PerpendicularRelation,
has-Intersection Relation, has-NonuniplanarRelation, and
has-ConstraintRelation be coincidence relation, disjoint relation,
inclusion relation, parallel relation, perpendicular relation, inter-
section relation, nonuniplanar relation, and constraint relation,
respectively. Let OWL object property has-SpatialRelation be the
elements of the set which comprises of the above eight OWL
object properties. Let adf (s
u
(p
i
)) and adf (s
v
(p
j
)) be the asso-
ciated derived features of the real feature surfaces s
u
(p
i
) and
s
v
(p
j
), respectively. Let ifs(s
u
(p
i
)) and ifs(s
v
(p
j
)) be the instances
of the ideal feature surfaces of s
u
(p
i
) and s
v
(p
j
), respectively.