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locus of all points HiC constitutes the external boundary of
the so called complete rotatability subworkspace (CRSW)
determined by the subchain i. The CRW (Complete
Rotatability Workspace) of the manipulator is the common
region of the three CRSWs. Moreover, let us consider the
other point HiN on the line of the gradient VFEi, and with
the same distance outside of the boundary as HiC on the
inside. The sense of CiHiN is the same as that of the
gradient. One can easily verify that the locus of these
HiN points constitutes the so-called nonrotatability
subsurface (NRSS). The region between the CRSW and NRSS is
called the partial rotatability subworkspace (PRSW). After
the three PRSWs of each of the three subchains have been
obtained, one can find the manipulator's PRW (Partial
Rotatability Workspace) as the common regions of the three
PRSWs.
It should be noticed that for the complete rotatability
subworkspace to exist, the short principal radius of
curvature of the surface of the external boundary should be
longer than L, the length of CiH of the platform. Otherwise
the arc drawn by point Ci centered at HiC is on the outside
of the SWIP as shown in Fig. 4.3, and therefore, point Ci
would prevent the platform from rotating about H. For the
manipulator with the R-L actuators with offset sbi = 0, the
external boundary of the SWIP is the cylinder of radius
(ai + bi) (see Fig. 3.21). Since the short principal radius
of curvature of a cylinder is the radius of the cylinder