## 3 Graph Generation Methods of Articulated Manipulators
### Graph Generation Methods for Manipulators with 2-DOF Fractionated Intermediate Kinematic Chains
#### 3.2.1 Case 1: Two Different Non-Fractionated Graphs  
When enumerating articulated manipulator graphs using two different 1-DOF non-fractionated graphs \(X\) and \(Y\), it is necessary to consider whether the vertex pairs are symmetrical or asymmetrical in order to enumerate only non-isomorphic graphs. Case 1-1: A Symmetrical Vertex Pair of Graph X and a Symmetrical Vertex Pair of Graph Y (SP\({}_{\rm X}\) and SP\({}_{\rm Y}\))  
Each symmetrical vertex pair from graphs \(X\) and \(Y\) is chosen, whose number of possible pair combinations is \(\mid SP_{X}\mid\times\mid SP_{Y}\mid\), and a vertex is chosen from each pair and connected as a cut-vertex to generate a 2-DOF intermediate fractionated graph. Since each of the vertex pairs is 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 in Case 1-1 is 1 \(\times\) 1 = 1. Finally, the fixed base and the end-effector are connected to the two vertices that are not used as cut-vertices among vertices of two pairs. Since the two vertices are not topologically identical, new manipulator graphs can be obtained by exchanging the connection positions of the fixed base and the end-effector. Based on the generation process, the equation for the number of non-isomorphic manipulator graphs that can be obtained in Case 1-1 is derived as follows:  
\[N_{\rm Case1-1}=\mid SP_{X}\mid\times\mid SP_{Y}\mid\times 1\times 1\times 2. \tag{3}\]  
Graphs \(X\) and \(Y\) in Figure 3b,d serve as examples to show that a total of 12 vertex pair combinations are available since \(\mid SP_{X}\mid\) = 4 and \(\mid SP_{Y}\mid\) = 3. Among them, when \(SP_{X1}\) = (1-1, 1-2) and \(SP_{Y2}\) = (2-1, 2-2) pair combinations are used, one 2-DOF intermediate fractionated graph is obtained as shown in Figure 6a. After that, a fixed base and an end-effector can be connected to the intermediate graph in two ways, as shown in Figure 6b. Therefore, the number of generated non-isomorphic manipulator graphs for Case 1-1 in this example is calculated as follows, and all corresponding graphs are presented in Table 2a. \[N=\mid SP_{X}\mid\times\mid SP_{Y}\mid\times 1\times 1\times 2=4\times 3\times 1 \times 1\times 2=24. \tag{4}\]  
\begin{table}
\begin{tabular}{c c Figure 6: Method of generating non-isomorphic graphs in Case 1-1: (**a**) an intermediate graph generated by connecting two vertex pairs; (**b**) two options for connecting an end-effector and a fixed base to the intermediate graph. \begin{table}
\begin{tabular}{c} \hline \hline
**Case** \\ **Number** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Graphs of articulated manipulators generated for Case 1: (**a**) Case 1-1; (**b**) Case 1-2; (**c**) Case 1-3; (**d**) Case 1-4. Case 1-2: A symmetrical Vertex Pair of Graph \(X\) and an Asymmetrical Vertex Pair of Graph \(Y\) (\(SP_{X}\) and \(AP_{Y}\))  
A symmetrical vertex pair from graph \(X\) and an asymmetrical vertex pair from graph \(Y\) are selected, whose number of possible pair combinations is \(\mid SP_{X}\mid\times\mid SP_{Y}\mid\), and a vertex is chosen from each pair and connected as a cut-vertex to generate a 2-DOF intermediate fractionated graph. Since the vertex pair of graph \(X\) is a symmetrical vertex pair, the fractionated graph generated is the same regardless of which of the two vertices of the pair is selected as a cut-vertex. Conversely, since graph \(Y\) is asymmetrical, new fractionated graphs are created depending on which of the two vertices is used as a cut-vertex. Therefore, the number of 2-DOF intermediate fractionated graphs generated for a specific vertex pair combination is \(1\times 2=2\). Finally, a fixed base and an end-effector are connected to two vertices that are not used as cut-vertices among two pairs of vertices. As a result, two different manipulator graphs are obtained according to the exchange