Ci (i = 1, 2 and 3) should be as small as possible, which is
consistent with controllability of end-effector orientation.
When the lengths are infinitesimal or, in the limit, zero,
the largest possible workspace with complete, but
uncontrollable rotatability of the platform results. In
this chapter, the equations of the workspace of the
manipulator with infinitesimal dimensions of the platform
are derived, i.e., with joints Cl, C2 and C3 infinitesimally
close to each other. In chapter 4, the workspace of a
similar parallel manipulator with a finite size platform
will be determined. Of course, controllability of the
orientation of the platform is reduced sharply as the
spherical joints approach one another. Therefore it must be
realized that there must be a practical trade-off between
the distances of the spherical joints from one another and
the controllability of platform orientation.
3.3 The Subworkspace Analysis of the Manipulator
The toroidal surface (torus) is the locus of a point
attached to a body that is jointed back to the reference
system through a dyad of two serially connected revolute
pairs. A general R-R dyad with a point C which traces the
surface of a general form of torus is shown in Fig. 3.3. It
is similar to the subchain (R-L)-R-S when the platform is
assumed to be infinitesimal and without the consideration of
translation along the axis of the cylindric (R-L) joint.
The shapes of the torus are illustrated first, then the