curves in the diametral section become banana-shaped, as
shown in Fig. 3.11(c), when a < b.
General Form. The diametral section of the torus is
shown in Fig. 3.4(d). For different twist angles (30, 45
and 750) and dimensions of the links (a > b, a = b, and a <
b) with each specified twist angle, the shapes of the torus
are shown in Figs. 3.12 3.14, respectively. The closed
curves in the diametral section, while still more or less
banana-shaped, but now they are tilted over.
Equations (3.1) (3.4) are all of the fourth degree
and all forms of torus are thus quartic surfaces. The curve
of intersection between a torus and a general plane is a
quartic; also, in general, a straight line cuts any torus in
four points (real, imaginary, or coincident).
The volume and shape of the workspace are very
important for applications since they determine capabilities
of the robot. In order to obtain the optimum workspace, the
volume of the subworkspace of the corresponding subchain
generally should be as large as possible. Since most of
today's available industry robots have 0 or 900 twist
angles, we will discuss the following two cases with the
conditions of s = 0 and twist angle a = 0 or +900 applied,
respectively.
Case 1: s = 0 and a = 0 (or n)
Since the axes of the two revolute pairs are parallel
and there is no offset, the toroidal surface degenerates