ComputerAided_Design_Engineering_amp_Manufactur.pdf

FIGURE 3.19 Topological representation for the convex polyhedral models. (Reprinted from Reference 37, with
permission from Elsevier Science.)
• Clusters of objects in the fixture workstation
• Convex polyhedral models describing each cluster
• Faces on each model
• Vertices bounding each face
• Dimensional information for each vertex.
The necessary information required to perform the interference detection analysis can be derived from
the above topology. This information includes the following:
• A list of faces on a cluster
• A list of vertices on a cluster
• A table of face normals for each cluster.
The approach for this phase of the interference detection consists of checking the vertices of the convex
polyhedral models of one fixture module against the face normals of the convex polyhedral models for
the second fixture module cluster. The reverse operation is also carried out, where the vertices of the
second fixture module cluster are checked against the face equations of the first fixture module cluster.
Let A and B be objects represented by convex sets �q R , q � 1, … , g. For two-dimensional space,
u � 2, and for three-dimensional space, u � 3. If we define the set of unit outward normals for the faces
bounding the model as the set of constants satisfying the equations for the
faces as C, and the set of vertices attached to each face as then the general formulation
may be written as follows:
u
�
N n : n R u � { � , n � 1},
V v : v R u
� { � },
© 2001 by CRC Press LLC
�i, j
�
3
�
nAik vBjk k�1
� � �
(3.35)
where i represents the identification number of the face on object A (i.e., i � 1, …, m), j represents the
identification number of vertex attached to the object B (i.e., j � 1, …, p), k signifies the coordinates
x, y, and z, and � is the specified clearance. In the above formulation, the solutions � i, j �, �, and � 0
signify that the vertex is outside (i.e., positive direction) the face, on the face, and inside (i.e., negative
C Ai