e5 can be directly computed by use of Eqs. (6.42) or (6.46) respectively.
In the special geometries described in Chapter 5, cases 5, 7, and 10, we did not find a constraint of the form ri=0, yet a five-DOF arm having one of these particular geometries can still be solved in closed form. We now study these
special cases as they apply to five-degree-of-freedom robots.
Case 7: Three joint axes are such that they either intersect or they are parallel two at a time. This type of structure is studied in cases 5, 7, and 10 of Chapter 5. Assuming this geometry concerns axes 3, 4, and 5 of the five-DOF arm, Eq. (6.2) should be used. From Chapter 5,
case 5 and case 10, we see that the last two equations of the reduced system, Eqs. (6.17) and (6.18) have the form
o2(qx S2 qy C2) = r3 a2(qy S2 + qx C2) =r4
where qx, qy, r3, and r4 are all linear expressions in S( and C1. A quartic polynomial equation in t1 = tan(91/2) is readily obtained by squaring and adding the last two equations,
qx2 + qy2 = (r3/o2)2 + (r4/a2)2
and substituting S1 = 2t1/(l+tl2) and C1 = (1-t12)/(l+t12). This polynomial can be solved for 81 and the solution set
..