can be solved for 83. Another way to obtain 83 is by using the equations
(T3u4C4+a3 T4) S3 + 64S4 C3 = C2ux + S2uy (6.25) and
04S4 S3 (r364C4+63T4) C3 =
-T 2S2ux + T2C2Uy + U2Uz, (6.26)
derived from Eqs. (5.25) and (5.26) by incrementing the indexes. With 81, 82, and 83 known, uL-qL can be computed as in Eq. (6.10) and f(81) is then given by Eq. (6.8).
The ability to compute f(81) when e is given is
sufficient to implement a practical Newton-Raphson algorithm for finding the zeros of function f. The algorithm can be programmed according to the following steps:
Step 1. Obtain an initial estimate for 01. As for all iterative methods, the closer the initial estimate of e is to a true solution, the faster the convergence will be. If the end-effector of the robot is tracking a trajectory given as a finite set of end-effector poses, a good estimate for finding the solution set for a pose along the trajectory is the value of 81 corresponding to the preceding pose on the trajectory.
Step 2. Compute 83 and 84 and then f(81) as described earlier.
..