Vector t is given by

 t = R z = R1 R2 R3 R4 z


where z is the third canonical unit vector z = [ 0, 0, 1]T Since twist angle a4 is equal to 0,


 R4 z = [04 S4, -U4 C4, T4]T = [0, 0, 1]T = z, and the expression for t simplifies to


 t = R1 R2 R3 z. (5.5)


Multiplying by R1-1 yields


 R1 1 t = R2 R3 z

and the inner product of each side of this equality with vector z provides


 z (RI-1 t) = z (R2 R3 z).


Eq. (4.5) applied to both sides of this last equation gives


 R1 z t =R2-1 z R3 z or

 (R1 z).t (R2-1 z).(R3 z) = 0. (5.6)

 Since R2-1 = [0, 02, 2] T does not depend on 829 this last equation is independent of joint variables 2 and 4.

 Subtracting vector 1- from both sides of Eq. (5.4) and multiplying by R-Q yields

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