If the value of e1 used to compute pose matrix Q in Eq. (6.3) does correspond to a solution set, then Eq. (6.2) will hold, vectors uL and qL will be exactly equal to u and q, respectively, and function f will equal zero. In other

words solution sets of Eq. (6.1) correspond to zeros of function f defined in Eq. (6.9). Hence, the inverse

kinematics problem of 5-DOF robot manipulators reduces to solving the one-dimensional equation


 f(e1) = 0.


 The zeros of f can be found by use of any suitable onedimensional technique such as Newton-Raphson or the secant method. Once 91 is known, the solution set can be completed by solving Eq. (6.2) in closed form as we showed in Chapter 5. The solution set can then be checked for consistency with Eq. (6.1) to determine whether the one found is extraneous or not because the zeros of f are not always part of a solution set of the manipulator.

 Computinq f(el_. Using Eqs. (6.7) and (6.8), the inner product uL-qL is given by


 UL qL = (R2 R3 R4 z) (R2(R3 14 + 13) + 12).


 If we apply properties (4.5) and (4.6) repeatedly, this last equation becomes


 uL-qL = z.(R4-114) + z.(R4-1R3-113) + z.(R4-1R3-1R2-112)

..