6 for five-degree-of-freedom arms. Assuming Eq. (7.45) is to be solved, we define a function f of 9i by
f(91) = uL-qL u.q (7.49)
where vectors UL, qL, u, and q are defined as earlier. Given a value of 81, vectors u and q are computed from Eq. (7.46), the values of the remaining joint variables are computed in closed form from Eq. (7.45) as indicated in Chapter 6 and Appendix B, the inner product uL.qL can then be obtained as in Eq. (6.10) with the proper index adjustments, and the value of f is then given by Eq. (7.49). As we have seen before, the ability to compute the function f for a given value of E1 allows the implementation of a practical one-dimensional Newton-Raphson algorithm. Therefore, we can conclude that a six-degree-of-freedom manipulator with two consecutive pairs of intersecting or parallel joint axes or three consecutive joint axes that are parallel and/or intersecting two at a time can be solved by use of a one-dimensional iterative technique instead of the two-dimensional method required for an arm of arbitrary architecture.
Closed-Form Solution
Some six-degree-of-freedom manipulators with simple geometries do not require any iterative method since- they can be solved in closed-form. Pieper (1968) has shown that
..