1 0 0 74 0 04
0 0


0

-04 74

0


 a4

, k4 = 0 is the position vector

 d4


of B4, and R and p are the usual rotation matrix and position vector of end-effector pose P. Explicitly, we get



 nx4 S5 + bxU4 C5 + txT 4
 u = n y4 S5 + b y4 C5 + tyT4 (6.34)

 nzU4 S5 + bza 4 C5 + tzT4


and


(-nxa4d4 + bxa4) S5

 (nxa4 + b a4d4) (-n ya4d4 + b ya4) S5

 (n ya4 + by c4d4) (-nza4d4 + bza4) S5

 (nza4 + bz"4d4)


C5 tx 4d4 + Px C5 tyT4d4 + Py C5 tz74d4 + Pz


 Expressions of inner-products u.q and q.q in terms of S5 and C5 can be obtained from Eqs. (6.32) and (6.33),


 u.q =[R R51G4 z] R (R5 (-G4 k4) + p]-


and


B4 =


(6.35)

..