Challenges/shortages of existing research: Both a topology graph TG and a contracted graph CG are the important tool for the type synthesis of mechanisms and have been studied. Let (b, t, q, p, h) be (binary, ternary, quaternary, pentagonal, hexagonal) link, respectively. Existing TG may include (b, t, q, p, h, …), and existing CG may include (t, q, p, h, …) and exclude b. Therefore, their derivation and isomorphism identification are quite complicated.

The significance of this research: A novel high-contracted topology graph Gj/i including n (q, p, h, …) is proposed in this paper. Since Gj/i excludes (b, t), Gj/i must be more simple than complex TG and CG, and the derivation and isomorphism identification of various Gj/i must be more easily than that of the complex TG and CG.

This paper focuses on the derivation and isomorphism identification of various Gj/i for deriving TG and type synthesis of the complex closed robotic mechanisms with more mechanical advantages and useful functions. First, the conceptions for deriving Gj/i and the relations between Gj/i and the basic links are explained, the relative rules are determined for representing Gj/i and reducing isomorphism of Gj/i using the character strings and the digit groups, and many different Gj/i are represented based on determined rules. Second, many different Gj/i (j = 0, …, 17) are constructed using a circle, several composite curves, and vertex points on the circle based on the character strings and the digit groups. Third, a condition for identifying isomorphism Gj/i is discovered and proved using the composite curves and their connections in Gj/i. Some isomorphism Gj/i are identified from constructed different Gj/i. Finally, different Gj/i (G7/7, G11/11) are illustrated to derive TGs, several complex closed robotic mechanisms are synthesized using derived TGs for creating robotic gripper, a long-stroke robotic pumping unit, and a tunnel robotic excavator.