where CS and SG are shorthand for cos(8) and sin(8),
respectively. Similarly, Sc denotes the relative rotation
angle of joint C, db and dc denote the translations of
joints B and C along the moving axes z2 and z3,
respectively, and c denotes the length of link c. It is
noted that db, d, and c are zero for this subchain, but all
these notations are used throughout the following sections.
Premultiplying both sides of Eq. (2.4) by A2-1A1-1
yields
bCOb = CxC8, + CyS8a a (2.7)
bS8b = -CxSSaCab + CyCSaCab + CzSab daSab (2.8)
sb = CxS9SaSb CyCGaSab + CzCab daCab (2.9)
Squaring and adding Eqs. (2.7), (2.8) and (2.9) yields
b2 Sb2 = Cx2 + Cy2 + C 2 + da2 2Czda -
2aCxC8a 2aCySa + a2 (2.10)
From Eq. (2.9), we have
1
da = -- (CxSeSab CyCeaSab + CzCab sb)
Cab
Substituting da into Eq. (2.10) yields
E1X4 + E2X3 + E3X2 + E4X + E5 = 0 (2.11)
where
X = tan(9a/2)
D1 = CzCab sb
D2 = C2 y C 2 + C 2 + a2 b2 Sb2