## 3 Graph Generation Methods of Articulated Manipulators
### Graph Generation Methods for Manipulators with 2-DOF Fractionated Intermediate Kinematic Chains
#### 3.2.2 Case 2: Two Identical Non-Fractionated Graphs  
When two identical 1-DOF non-fractionated graphs \(X\) and \(X\) are used to generate articulated manipulator graphs, slightly different methods need to be applied to enumerate only non-isomorphic graphs depending on the symmetry of their vertex pairs and whether the two pairs are identical or different. Case 2-1: Two Identical Symmetrical Vertex Pairs (\(SP_{Xi}\) and \(SP_{Xj}\), Where \(i=j\))  
The same symmetrical vertex pair is selected from graphs \(X\) and \(X\), and the number of possible pair combinations is \(\mid SP_{X}\mid\). Then two vertices, one from each pair, are selected and interconnected as cut-vertices to create a 2-DOF intermediate fractionated graph. As in Case 1-1, since both vertex pairs are symmetrical, the generated fractionated graphs will be the same regardless of which vertex is selected as the cut-vertex. Therefore, the number of 2-DOF intermediate fractionated graphs generated for a specific combination of vertex pairs is one. Finally, the two vertex pairs that are not used as cut-vertices have a fixed base and an end-effector connected to them. Unlike Case 1-1, the two vertices in the fractionated graph are topologically identical. This means that exchanging the connection positions of the fixed base and the end-effector will result in duplicate manipulator graphs. Based on the above generation process, the expression for the number of non-isomorphic manipulator graphs that can be obtained in Case 2-1 is derived as follows:  
\[N_{\text{Case2-1}}=\mid SP_{X}\mid\times 1\times 1. \tag{12}\]  
Graphs \(X\) and \(X\) in Figure 3a as an example show that a total of 4 vertex pair combinations are available since \(\mid SP_{X}\mid\) = 4. In one case, when \(SP_{X1}\) = (1-1, 1-2) and \(SP_{X1}\) = (1-1, 1-2) pair combination is used, one 2-DOF intermediate fractionated graph can be obtained as shown in Figure 9a. After that, a fixed base and an end-effector can be connected to the intermediate graph in one way, as shown in Figure 9b. Therefore, the number of non-isomorphic manipulator graphs generated in this example can be calculated as follows, and the four graphs are presented in Table 3a. \[N=\mid SP_{X}\mid\times 1\times 1=4\times 1\times 1=4. \tag{13}\]  
Figure 9: Method of generating non-isomorphic graphs in Case 2-1: (**a**) an intermediate graph generated by connecting two vertex pairs; (**b**) one option for connecting an end-effector and a fixed base to the intermediate graph. Case 2-2: Two Different Symmetrical Vertex Pairs (\(SP_{Xi}\) and \(SP_{Xi}\), Where \(i\
eq j\))  
Two different symmetrical vertex pairs are chosen in graphs \(X\) and \(X\), and the number of possible pair combinations is \({}_{|SPX|}\) C\({}_{2}\), where \({}_{n}\)C\({}_{2}\) = \(n\times(n-1)/2\). After that, even if any of the two vertices of each pair is selected as a cut-vertex, the generated fractionated graph is the same. Hence, the number of 2-DOF intermediate fractionated graphs generated for a specific vertex pair combination is one. Finally, by exchanging the connection positions of the fixed base and the end-effector, two non-isomorphic manipulator graphs can be obtained for each 2-DOF intermediate fractionated graph. Therefore, the equation for the number of non-isomorphic manipulator graphs that can be obtained in Case 2-2 is as follows:  
\[N_{\text{Case2-2}}={}_{|SPX|}\text{C}_{2}\times 1\times 2. \tag{14}\]  
Graphs \(X\) and \(X\) in Figure 3b,d serve as examples to show that a total of \({}_{|SPX|}\) C\({}_{2}\) = 6 vertex pair combinations are available. In one case, when the \(SP_{X1}\) = (1-1, 1-2) and \(SP_{X3}\) = (2-1, 2-3) pair combination is used, one 2-DOF intermediate fractionated graph is obtained as shown in Figure 10a. After that, a fixed base and an end-effector can be connected to the intermediate graph in two ways as shown in Figure 10b.