This paper presents the synthesis of Stephenson III six-bar linkages for function generation. The approach is similar to that of the Stephenson II linkage except additional reductions allow a multihomogeneous degree of 55,050,240 to be computed. A multihomogeneous homotopy was computed for a numerically general system to obtain 834,441 nonsingular solutions which are used to construct efficient parameter homotopies that can be solved for specific function generator task requirements. The linkage solutions found by parameter homotopy are sorted into cognate pairs and analyzed to verify performance. An example is presented of a function generator that creates a specified torque profile to cancel the effects of spasticity in the wrists of stroke survivors.