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CAD system.
3.2.2 Relationships Between Parts
In an assembly, the position and orientation of a part is usually
associated with another part. Figure 3(b) shows a plate (PP1)
and a screw (PP2). The relative placement of the screw is
constrained by the counter-bore hole on the plate. The relationships
between the screw and the plate are defined as follows:
As we can see in Eqs. (8) and (9), it is essential to calculate
the matrix Mp and Mr to determine the position and orientation
of the screw with reference to the plate. Mp and Mr can be
derived from the spatial constraints (SC). This derivation
requires the task of inferring the configuration of a part in an
assembly, which will be discussed in the next section.
We have presented a representation of the injection mould
assembly in a computer. At this stage, it might be worthwhile
to summarise the benefits of this representation. Assemblies
are represented as a collection of subassemblies that in turn
may consist of subassemblies and/or component parts, and a
component part can further be considered as the assembly of
form features. Such hierarchical relationships imply an ordering
on the assembly sequence and a parent–child link. The featurebased
representation not only allows designers to work at a
high-level of abstraction while designing inpidual parts, but
also extends the feature paradigm to assembly modelling,
because this representation allows a component to be changed
parametrically with the other components consequently having
their positions changed accordingly. The object-oriented representation
can combine both the data structure and operation
in an object. The encapsulated operational functions in an
assembly object can help to automate the routine processes
such as the pocketing and interference check.
4. Inferring Part Configuration in the
Assembly
As we can see from Eqs (8) and (9), the positions and
orientations of parts in an assembly are eventually represented
by the transformation matrices. For the sake of convenience,
the spatial relationships are usually specified by the high-level
mating conditions such as “mate”, “align” and “parallel”. Thus,
it is essential to derive automatically the explicit transformation
matrices between parts from implicit constraint relationships.
Three techniques to infer the configurations of parts in an
assembly have been discussed in Section 2. Because the symbolic
geometric approach can locate all solutions to constraint
equations with polynomial time complexity, we use this
approach to determine the positions and orientations of parts
in an assembly. To implement this approach in assembly
modelling software, a large amount of programming is required.
Therefore, a simplified geometric approach is proposed to
determine the positions and orientations of parts in an assembly.
In the symbolic geometric approach, determining positions
and orientations of parts is performed symbolically by generating
a sequence of actions to satisfy each constraint incrementally.
The information required to satisfy each constraint
incrementally is stored in a table of “plan fragments”. Each
plan fragment is a procedure that specifies a sequence of
measurements and actions that move parts in such a way as
to satisfy the corresponding constraint. The plan fragment also
records the object’s new degrees of freedom (DOFs) and
associated geometric invariants. Conceptually, Kramer’s plan
fragment table is a 3D dispatch table. We use TDOF to
represent translational degrees of freedom and RDOF to represent