or after computing the z-components of the terms in parentheses,


uL.qL = T4d4 + a3a4 S4 a3d304 C4 + r4d3

 + U4 S4 (a2 C3 + a2d2 S3)

 04 C4 (-a2r3 S3 + C2d2T3 C3 + r2d203)

 + T4 (a2a3 S3 o2d2a3 C3 + T2d2'2). (6.10)


This last equation shows that 93 and 94 must be known before we can compute uL.qL. With el known, 93 and 84 can be obtained by solving Eq. (6.2) as described in Chapter 5. The coordinates of vectors u and q and the inner-products u.q, and q.q are necessary for the 4-DOF inverse kinematic method of Chapter 5. Equation (6.5) yields


 t C1 + ty S1

 u = R- t = -71 tx S1 + Tity C1 + ltz (6.11)

 t1 x S1 lty C1 + Tltz and from Eq. (6.6),


 Px C1 + Py Sl al

 q = -rlpx S1 + Py C1 + lPz (6.12)

 alPx S1 alPy C1 + rlPz


where we have assumed (R1-1 11) = [al, 0, 0]T since dl=0 by proper positioning of frame Fo.

..