## 1 Introduction  
In the conceptual design phase of a new machine, it is important to find the optimal kinematic chain (KC) under given conditions because its structure has a great influence on the kinematic and dynamic characteristics, and the KC cannot be changed in later stages such as dimensional synthesis or structure design. For example, parallel manipulators have relatively low inertia, high stiffness, and a large payload capacity versus serial manipulators because of their structural arrangements [1]. In the case of a variable compression ratio engine, since it has a two-degree-of-freedom (DOF) kinematic structure, the movement of its piston can be adjusted according to driving situations, which result in high performance and fuel efficiency [2]. In addition, an engine's shaking force, which causes vibration and wears, can be reduced by selecting an appropriate kinematic structure [3]. For this reason, many studies have been conducted to propose methods for generating the KCs or to present a database. In the early times, structural synthesis to generate KCs relied on the designer's intuition or experience [4, 5]. However, it was difficult to enumerate all the KCs without any omission and isomorphism by simply relying on intuition and experience. Since then, several studies have been conducted to enumerate the KCs systematically. [PERSON] et al. [6] generated two DOF fractionated KCs (FKCs) with up to 15 links and non-fractionated KCs (non-FKCs) with up to 13 links. [PERSON] et al. [7] proposed an automatic structural synthesis method of 3-DOF KCs. Furthermore, to enumerate the KCs systematically, graphs that can easily represent the KCs have been widely used. [PERSON] [8] presented a method to synthesize 4-, 6-, and 8-link non-FKCs based on graph theory. [PERSON] and [PERSON] [9, 10] proposed a method for the structural synthesis of planar and single-loop spatial mechanisms using graph theory and vertex-edge incidence matrices. In addition, one of the essential contents when enumerating KCs is to avoid generating duplicate isomorphic KCs. One of the effective ways to solve the duplication problem is utilizing the concept of symmetry group. The concept enables researchers to generate only non-isomorphic KCs and skip an isomorphism test [1], which is complicated and time-consuming. Therefore, many studies have used this concept to enumerate non-isomorphic KCs. [PERSON] and [PERSON] [11] and [PERSON] et al. [12; 13] utilized the concept of symmetry group theory and proposed an enumeration method of isomorphism-free non-FKCs. Based on the concept of symmetry group theory, [PERSON] and [PERSON] [14] generated 1- to 4-DOF non-FKCs with two to six independent loops, and [PERSON] [15] developed a program for enumerating 1- to 3-DOF non-FKCs with two to six independent loops. Recently, with the development of the automation industry, the demand for multi-DOF manipulators is rapidly increasing. As a result, many studies have been conducted on parallel manipulators. [PERSON] [16] proposed a method to determine the structure of parallel manipulators with given DOFs (from three to six) using graph theory, group theory, and screw theory approaches. [PERSON] and [PERSON] [17] developed a systematic method to create 4-DOF and 5-DOF overconstrained parallel manipulators based on the screw theory and illustrated several examples of parallel manipulators. [PERSON] [18] used an evolutionary morphology approach and the conditions derived from the theory of linear transformation to synthesize parallel robot structures according to the joint combination for each DOF. [PERSON] [19] used adjacency matrices and presented a method for generating planar or spatial parallel mechanisms. Conversely, unlike parallel manipulator research, most studies on designing articulated manipulators take an approach of dynamics [20], structural topology [20; 21], or control [22; 23], and there have only been a few structural synthesis studies on articulated manipulators.