Since ([A1](1,4))2 + ([A1](2,4))2 = ([HOA3-1A2-1](1,4 )2 +
([HOA3-1A2-1](2,4))2 is true, we can obtain the following
equation:
a2 = [-nx(c + bCSc) + bSxSec axSc + px]2 +
[-ny(c + bCSc) + bSyS8c aySc + py]2 (2.22)
Equation for ea is obtained since [A1](2,4) / [All(1,4)
[HOA3-1A2-1](2,4) / [HOA3-1A2-11(1,4) holds, and can be
expressed as
a= tan-l( -ny(c + bCec) + sybS9c aySc + py (2.23)
8, = tan- --- ) (2.23)
-nx(c + bcec) + sxbSSc axsc + Px
It is observed that [Al](3,4) = [HOA3-1A2-1](3,4) directly
implies the translation of joint A as
da = -nz(c + bCec) + szbSec azsc + Pz (2.24)
Let tan(e)/2) = X; then substituting C8c = (1 X2)/(1 + X2)
and SEc = 2X/(1 + X2) into Eqs. (2.22) yields
E1X4 + E2X3 + E3X2 + E4X + E5 = 0 (2.25)
where
D1 = (nx2 + ny2)b2
D2 = (sx2 + Sy2)b2
D3 = 2b(nx2c + ny2c + nxaxsc + nyaysc Pxnx Pyny)
D4 = -2b(nxSxC + nysyc + sxaxsc + yaysc sxpx -sypy)