Since the manipulator's workspace with infinitesimal
platform is the common intersection of the three
subworkspaces of the three dyads that support the platform,
in general the ends of these subworkspaces need not be
considered. Therefore, only the projection of the
subworkspace onto the xy plane is of interest. For this
reason, henceforth r will designate only the xy projection
of the vector from the center Ai to the cylindric boundary
of the subworkspace. Accordingly:
r2 = xai + Yai2 = xi2 + i2
= (biCebi + ai)2 + (biS8biCabi sbiSabi)2 (3.8)
Subworkspace with offset sh = 04 (infinitesimal
platform). For the sake of simplicity and ease of
visualization, we will discuss the subworkspace with
infinitesimal platform and with offset sbi = 0 first. Eq.
(3.8) becomes
r2 = (biCebi + ai)2 + bi2S2ebiC2abi (3.9)
The projections on the oixiYi plane of the circles
described by Ci, which are ellipses, can be represented by
Eqs. (3.8) and (3.9) in terms of the parameter ebi. Figs.
3.20 and 3.21 show these ellipses when ai > bi. When
4See page 102 for subworkspace with offset sbi # 0.