a sufficient condition for closed-form solutions is that three consecutive axes be intersecting. The inverse kinematics problem then reduces to finding the zeros of a quartic polynomial. In the literature, It seems to be common knowledge that three consecutive joint axes that are parallel is another sufficient geometric condition for
closed form solutions.
The analysis of Chapter 5 and Appendix A showed that under certain conditions, the reduced system of equations (7.20)-(7.23) included constraint equations of the form
ri = 0. (7.50)
The quantities ri, i=l, ,4, are functions of 81 and 92, as we have seen earlier. By looking for conditions under which a joint variable value can be directly obtained from an equation having the form of Eq. (7.50), we find two more sufficient six-DOF robot structure conditions for closed-form solutions (excluding the already known conditions of three parallel or three intersecting axes).
When the first two joint axes of a manipulator are parallel so that ai=0, then ci=0, ri=l, and the z-components of vectors u and q, given by Eqs. (7.16) and (7.17), become
uz= 2 (-ty C12 + tx S12) + r2tz (7.51)
and
(7-.52)-
Sqz = -a2' (Px S12 Py C12) + T2 (Pz-d2)':
..