This shows that r2 and r3, as given in (7.25) and (7.26), become linear functions of S12 and C12. When joint axes 3 and 4 are parallel and joint axes 5 and 6 are parallel, the reduced system oe equations (7.20)(7.23) becomes similar to that of case 2 of Chapter 5, a3uy S3 + a3ux C3 = rI (7.53) 0 = r2 (7.54) c4a5 S5 = r3 (7.55) a3qy S3 + a3qx C3 + a4a5d4 S5 + a4a5 C5 = r4. (7.56) Equation (7.54) yields two possible values for G12, each of which will provide two possible values of 05 from Eq. (7.55) when substituted in the expression of r3. The remaining joint values can then be computed in closed form. A similar development occurs when axes 3 and 4 are parallel and axes 5 and 6 intersect. To summarize, a six-DOF manipulator has a closed-form solution if one of the following conditions is satisfied: 1. Three consecutive joint axes are parallel. 2. Three consecutive joint axes intersect at one point. 3. The arm is formed of three sets of two parallel axes. This structure is illustrated in Figure 7.1(a). 4. The arm has two sets of two parallel joint axes followed or preceded by two intersecting axes. These structures are illustrated in Figure 7.1(b) and 7.1(c). ..