80
However, this is true only when the translation along the Z
axis, d, is greater than or equal to 2b. Otherwise, there
are additional voids that can be found as shaded areas
shown in Fig. 3.16(b). When the translation along the Z
axis is less than 2b, there is a void inside the workspace,
shown with a lentil-shaped cross section in Fig. 3.16(b).
When a = b as shown in Fig. 3.6(b), there is no hole as the
translation along the Z axis is in effect and d 2b. The
workspace is the volume of the cylinder with radius of
(a i b), height d, plus the volume of this kind of half
torus at each end, as shown in Fig. 3.17(a). If the
translation along the Z axis is less than 2b, voids can be
found even if the torus has no hole at all. The lentil-
shaped cross sections of the void can be shown as shaded
areas in Fig. 3.17(b). Finally, when the torus intersect
itself as shown in Fig. 3.6(c), there is a void inside in
this torus. As the translation d along the Z axis is in
effect and is greater than or equal to 2b, we obtain the
workspace as the volume of the cylinder with radius of (a +
b) and height d, plus the volume of this kind of half torus
at each end as shown in Fig. 3.18(a). Similarly, voids can
be found if the translation along the Z axis is less than
2b, which is shown in Fig. 3.18(b).
3.3.2 Boundaries of the subworkspace and root regions
in the subworkspace (infinitesimal platform)
In order to calculate the volume of the subworkspace,
we must find the boundaries (external and internal) of the