Utilize the generating functions to enumerate the specialized mechanisms with design constraints

This study presents a mathematical method to calculate the generating function of the specialized mechanisms in type synthesis with design constraints. First, the link permutation group is derived from a candidate kinematic chain according to the combinatorial theory. Next, according to the types of design constraints, joint, path, and adjacent (or non-adjacent) pair groups are derived from the link group respectively. Then, the generating function of these groups can be calculated mathematically based on four well defined operators and basic groups. The generating function is a polynomial to express the results of specialized mechanisms subjected to a specific design constraint. The benefit of the method is that the results of the type synthesis can be calculated directly by mathematical manner without any inspection of isomorphism.