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)

..