We can now compute the inner products uL*qL and qL.qL. By incrementing the indexes in Eq. (6.10), we derive uLqL = T5d5 + a4o5 S5 o4d4a5 C5 + r5d4

 + 05 S5 (a3 C4 + a3d3 S4)

 05 C5 (-a3 4 S4 + a3d3r4 C4 + r3d3o4)

 + T5 (a3a4 S4 03d304 C4 + T3d3 3). (7.32)

Vector qL, obtained from the left hand side of Eq. (7.2), is


 qL = R3(R4 15 + 14) + 13 (7.33)


and the square of its length is given by


qL'qL = (R3(R4 15 + 14) + 13) (R3(R4 15 + 14) + 13) or

qL'qL = (15+R4-114+R4-1R3-113) (15+R4-114+R4-1R3-113), after factoring out (R3 R4) and using inner product invariance of rotations. Multiplying out the terms in parentheses and using (4.5) and (4.6) where necessary, the last equation yields


qL L = 2 [a5 C5(a4 + a3 C4 + a3d3 S4)

 + a5 S5(-a3r4S4 + a3d3T4 C4 + r3d3C4 + C4d4)

 + d5 (T4d4 +a3C4 S4 o3d3u4 C4 + d373 4)

 + a3a4 C4 + C3d3a4 S4 + 'T3d3d4]

 + a32 + d32 + a42 + d42 + a52 +d52. (7.34)

..