For the manipulator shown in Fig. 3.19 and with the
base arrangement in equilateral triangle form (see Fig.
3.37), Eqs. (4.3) for subchains 1, 2 and 3 have the
following forms (see Eq. (3.25) (3.27))
FEl = Xm2 + (m + h)2 (al + b1)2 = 0 (4.7)
FE2 = Xm2 + (-Ym/2 + J2Zm/2 + h)2 (a2 + b22 = 0
(4.8)
FE3 = Xm2 + (-Y/2 3Zm,/2 + h)2 (a3 + b3)2 = 0
(4.9)
We now calculate the gradient of the first equation FE1 = 0,
VFEl as
dFE1/3Xm 2Xm
VFE = FE1/Ym = 2(Ym + h) (4.10)
IFEI/aZm 0
(VFEII = / (2FEI/3XXm)2 + (FEI/oYm)2 + (FEl/oZm)2
= 2 Xm2 + (Ym + h)2
= 2(al + bl) (4.11)
Substituting expressions (4.10) and (4.11) into Eq. (4.5),
we obtain
XmL
X = Xm - (4.12)
al + b1
(Ym + h)L
Y = Y + h)L (4.13)
al + bI
Z = Zm
(4.14)