ComputerAided_Design_Engineering_amp_Manufactur.pdf

direction) the face, respectively. The only acceptable solutions for the condition of disjointness between
the two objects are � i, j �, � �. It must be emphasized that the value for clearance, �, is selected based
on the accuracy of the robot and the application.
As an example, consider two polyhedral objects A and B (Figure 3.20). If nAi denotes the “outward”
normal for faces i � 1, …, 6, on object A, and vBj denotes the vertices j � 1,…, 8 for the object B. The
above formulation simply stems from the substitution of components of a vertex of object B into general
equation of the face on object A. This formulation determines upon which side of the face the vertex in
question resides. It can be seen that for vertex (1) belonging to object B to be inside the region bounded
by the faces (1–6) of object A, the solution to the set of equations resulting from the formulation must
all be less than zero. In other words, the following relationships must hold:
© 2001 by CRC Press LLC
Faces on
object A
General
constraints
on object
Constrains
ci for face i
on object
Vertex vj (j � 1 :
60, 20, 0)
Resultant
�i, i
A
A on object B
1 �y�c1�0 10 �20�10 �10
2 x�c2�0 �50 60�50 10� terminate
3 y�c 3�0 �30 20�30 �10
4 �x�c 4�0 �10 �60�10 �50
5 z�c 5�0 �30 0�30 �30
6 �z�c 6�0 00 0�0 0
FIGURE 3.20 Illustration of two convex polyhedral models. (Reprinted from Reference 37, with permission from
Elsevier Science.)
�1, 1
�2, 1
.
.
�6, 1
�
�
�
nA1x vB1x nA2x vB1x nA6x vB1x � � � � 0
nA1y vB1y nA2y vB1y nA1z vB1z nA2z vB1z (3.36)
If the results of the above equations are all less than zero, then the vertex of object B is within the
region bounded by the faces of object A. In other words, there will be interference between the two
C A1
� � � � 0
nA6y vB1y nA6z vB1z C A2
� � � � 0
C A6