completed as described earlier. Similarly, from case 7 of Chapter 5, we get the equations
a2(uy S2 + ux C2) = r, C2(ux S2 uy C2) = r2
corresponding to Eqs.(6.15) and (6.16) of the reduced system of equations. Here again a quartic polynomial equation in tI is obtained by eliminating S2 and C2.
When the three axes with the special geometry are located elsewhere along the five-DOF structure, a similar result can be obtained by using equation (6.30) or by
exchanging the roles of base and end-effector frames.
To summarize the above cases, we find that a 5-DOF robot manipulator will allow closed-form solutions if any of the following conditions is satisfied:
1. Three consecutive joint axes are parallel.
2. Three consecutive joint axes intersect.
3. There are two consecutive sets of two joint axes that are either parallel or intersecting.
4. Three consecutive joint axes are such that two intersect and two are parallel.
5. Three consecutive joint axes are such that the first two intersect and the last two intersect.
Note that these conditions are not exclusive of one another.- For example, arms that satisfy condition 5 include those that satisfy condition 2.
..