Since rotations are orthogonal transformations, they leave inner products invariant, hence
Ru Rv = u v (4.5)
for any rotation matrix R and any vectors u and v. A
special case of (4.5) that is very useful is
Ru v = u R-1v. (4.6)
These properties are extremely efficient in eliminating algebraic terms and unnecessary joint variables when applied to Eqs. (4.3) and (4.4) if it is further recognized that
Ri-lli = [ai,diai,dii]T (4.7)
and
Ri-l = [0,oi,ri], (4.8)
where z = [ 0, 0, 1]T, are always independent of 8ei. Also, due to the frame assignments discussed above,
R6 z = R6-1 z =
in all cases since frame F6 is chosen to force a6 =0.
By repeated use of Eqs. (4.5) and (4.6), we obtain four reduced equations from Eqs. (4.3) and (4.4).
tz equation.
tz = t z = (R z) z
t = (R1 R2 R3 R4 R5 R6 z) z
tz = (R1 R2 R3 R4 R5 z) z
t = z (R5-1 R4-1 R3-1 R2- R1-1 z) (4.9)
..