## 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
\begin{table}
\begin{tabular}{c c} \hline \hline
**Case** & \\
**Number** & **Graphs of Articulated Manipulators** & Fixed Base, \(\bullet\): End-Effector \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._of the connection positions. Therefore, the equation for the number of non-isomorphic manipulator graphs that can be obtained in Case 1-2 is as follows:  
\[N_{\text{Case1-2}}=\left|SP_{X}\right|\ \times\ \left|AP_{Y}\right|\ \times 1\times 2 \times 2. \tag{5}\]  
Graphs \(X\) and \(Y\) of Figure 3b,d serve as examples to show that 16 vertex pair combinations are available since \(\left|SP_{X}\right|=4\) and \(\left|AP_{Y}\right|=4\). When \(SP_{X1}\) = (1-1, 1-2) and \(AP_{Y1}\) = (1-1, 2-1) pairs are used, two 2-DOF intermediate fractionated graphs can be created depending on which vertex is used as a cut-vertex from the two vertices in \(AP_{Y1}\), as shown in Figure 7a. After that, a fixed base and an end-effector can be connected to each intermediate graph in two ways, as shown in Figure 7b. Therefore, the number of non-isomorphic manipulator graphs generated in this example is calculated as follows, and all corresponding graphs are presented in Table 2b. \[N=\left|SP_{X}\right|\ \times\ \left|AP_{Y}\right|\ \times 1\times 2\times 2 =4\times 4\times 1\times 2\times 2=64. \tag{6}\]  
Case 1-3: An Asymmetrical Vertex Pair of Graph \(X\) and a Symmetrical Vertex Pair of Graph \(Y\) (\(AP_{X}\) and \(SP_{Y}\))  
In contrast to Case 1-2, an asymmetrical vertex pair is selected from graph \(X\), and a symmetrical vertex pair is selected from graph \(Y\). However, the principle of generating non-isomorphic manipulator graphs is the same as in Case 1-2. Therefore, the equation for the number of non-isomorphic manipulator graphs that can be obtained in Case 1-3 is as follows:  
\[N_{\text{Case1-3}}=\left|AP_{X}\right|\ \times\ \left|SP_{Y}\right|\ \times 2 \times 1\times 2. \tag{7}\]  
Graphs \(X\) and \(Y\) in Figure 3b,d serve as examples to show that the number of non-isomorphic manipulator graphs generated for Case 1-3 is calculated as follows, and all corresponding graphs are presented in Table 2c. \[N=\left|AP_{X}\right|\ \times\ \left|SP_{Y}\right|\ \times 2\times 1\times 2=2\times 3 \times 2\times 1\times 2=24. \tag{8}\]  
Case 1-4: An Asymmetrical Vertex Pair of Graph \(X\) and an Asymmetrical Vertex Pair of Graph \(Y\) (\(AP_{X}\) and \(AP_{Y}\))  
When two asymmetrical vertex pairs one each from graphs \(X\) and \(Y\) are used, the number of possible pair combinations is \(\left|AP_{X}\right|\ \times\ \left|AP_{Y}\right|\), the number of 2-DOF intermediate fractionated graphs generated for a specific pair combination is \(2\times 2=4\), and there are two options for connecting a fixed base and an end-effector to a specific intermediate  
Figure 7: Method of generating non-isomorphic graphs in Case 1-2: (**a**) two intermediate graphs generated by connecting two vertex pairs; (**b**) two options for connecting an end-effector and a fixed base to each intermediate graph. graph. Therefore, the equation for the number of non-isomorphic manipulator graphs that can be obtained in Case 1-4 is as follows:  
\[N_{\text{Case1-4}}=\left\lvert AP_{X}\right\rvert\ \times\ \left\lvert AP_{Y} \right\rvert\ \times 2\times 2\times 2. \tag{9}\]  
Graphs \(X\) and \(Y\) in Figure 3b,d serve as examples to show that a total of 8 vertex pair combinations are available since \(\left\lvert AP_{X}\right\rvert=\) 2 and \(\left\lvert AP_{Y}\right\rvert=\) 4. In one case, as shown in Figure 8a, four 2-DOF intermediate fractionated graphs can be generated by using a combination of \(AP_{X1}\) = (1-1, 2-1) and \(AP_{Y1}\) = (1-1, 2-1) pairs. After that, there are two ways to determine the connection point of a fixed base and an end-effector to each intermediate graph, as shown in Figure 8b. Therefore, the number of non-isomorphic manipulator graphs generated in this example is calculated as follows, and all corresponding graphs are presented in Table 2d.