into a plane. For different dimensions of the links a and
b, the shapes of the now planar toroidal surface are
illustrated in Fig. 3.15. The workspace resulting when this
surface is translated along the Z axis has no hole only when
a = b as shown in Fig. 3.15(b).
The workspace of this kind of subchain is generally the
volume between two coaxial cylinders when the translation
along the Z axis is in effect. Due to the limitation of the
rotation of the first revolute joint which is ground-
mounted, the workspace of this subchain is actually reduced
to the upper half (X 2 0) of the volume between the two
coaxial cylinders. Whenever the condition a = b exists, the
inner boundary disappears and there is no hole in the
workspace.
Case 2: s = 0 and a = +900
Since the two axes of the two revolute pairs are
perpendicular to each other and there is no offset, the
locus of point C is a torus, which is defined as the common
form of torus shown in Fig. 3.6. When the translation along
the Z axis is in effect, the workspace of the subchain can
be described as shown in Figs. 3.16 3.18, respectively.
In Fig. 3.16(a), the torus has a hole because of a > b.
When the translation d along the Z axis is in effect, we
obtain the workspace as the volume between the two coaxial
cylinders with radii of (a + b) and (a b), respectively
and height d, plus the volume of a half torus at each end.