pz equation
p = R1 R2 R3 R4 q
with
q = 15 + R4 1(14 + R3-1(13 + R2-1(12 + RI-I1I))), so that
Pz = p z = q (R4- 1 R3-1 R2-1 RI-I) z. (4.10)
p.t equation.
p t = R5-1q z (4.11)
p.p equation.
p.p = p2 = q.q = q2. (4.12)
Since Rl-llI and Rl-lz are independent of ei (Eqs. (4.7) and (4.8)), vector q and Eqs. (4.9)-(4.12) are easily seen to be independent of the first and last joint variables and therefore form a system of 4 equations in 4 unknowns. Figure 4.1 illustrates this discussion. With 16=0, vector t, which coincides with z5, and the position vector p of the origin of frame F6 are invariant in the rotation R6
(rotation about z5 which can only affect the end-effector orientation). Rotation about z has no effect on the zcomponent of any vector expressed in frame F0. Hence, pz and t are independent of I as well. Finally, since rotation about z0 moves all the robotic structure as a bloc, it does not affect the length of vector p or the inner product of t and p. ,
..