and
rmin2 = (ai bi)2 = r2i2, (3.14)
which are intuitively correct.
The relationship between r and ebi can be expressed in
a Cartesian coordinate system as shown in Fig. 3.22. It is
obvious that in the subworkspace where r2i < r (=
Sx2 + y2) < rji, for a given position of point Ci, which
implies a given value of r, there are two corresponding
solutions of 8bi. The subworkspace is a two-root region or
two-way accessible region. In other words, there are two
sets of Sai and ebi, or two kinematic branches of the i-th
dyad for reaching a given position of Ci. On the boundaries
where x + y2 = rli or r2i, there is only one solution
for 8bi. Thus the boundary surfaces are one-root regions.
(Recall that this is a planar R-R case).
In another case, when Sabi 0 (abi 0 or n) and ai
biS2abi, two additional roots of 9bi are found from Eq.
(3.12). Therefore in addition to rli and r2i there are
another two values of r, say r3i and r4i, which are also
limiting values of Eq. (3.9).
Substituting Eq. (3.12) into Eq. (3.9) yields
r3i2 = r4i2 = cot2abi(bi2S2obi ai2)
rmax2 (al + bi )2 = rli 2
(3.13)
(3.15)