ai, bi perpendicular distance between successive
joint axes zl, z2 and z3, respectively.
sbi, abi offset along axis zbi and twist angle between
zai and Zbi, respectively.
HCi approaching zero for infinitesimal platform.
As shown in Figs. 3.20 and 3.21 when sbi 7 0 or sbi = 0
respectively, if the revolute joint Bi makes a complete
rotation, the locus of the point Ci is a circle with respect
to the moving coordinate system BixbiYbizbi. It can also be
expressed with respect to the coordinate system AiXaiYaizai
from Eq. (2.5) as follows:
Xai = biCSbi + ai (3.5)
Yai = biSSbiCabi SbiSabi (3.6)
zai = biS9biSabi + sbiCabi (3.7)
The locus, generated by the point Ci turning around the
revolute joint Bi and the cylindric joint Ai, without
translating along the axis of the cylindric joint, can be
obtained by turning the circle, now represented by Eqs.
(3.5) and (3.6), about the zai axis. The shape of this
locus is the surface of a general form of torus as described
in section 3.3.1. The subworkspace of this subchain is the
volume swept by this toroidal surface as it translates along
the axis of the cylindric joint.