manipulators. Although this technique can equally be developed using Eqs. (7.4) or (7.6), it will be based on Eq. (7.2) for convenience.



 Two-Dimensional Iterative Technique


 Since we only need to know 2 of the joint variables to solve for the whole solution set, the inverse kinematics problem of six-DOF manipulators can be reduced to finding the values of the first two joint variables only, and getting closed-form values for the remaining variables. A numerical technique aimed at finding the values of 81 and E2 can be implemented by defining a system of two nonlinear equations in 81 and e2,


 f(e1,82) = 0 (7.8)

 g(ElE2) = 0, (7.9)


that can be solved using an iterative method such as a twodimensional Newton-Raphson.

 From the left hand side of Eq. (7.2), two vectors uL and qL, corresponding to vectors u and q, (vectors u and q relate to pose Q as shown in Eq. (6.4)), are given by


 UL = R3 R4 R5 z (7.10)

and


(7.11):


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

..