# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 5 Instant Center Identification of the Single-Loop Planar Linkages  
### Single-Loop Five-Bar Planar Linkage  
#### 5.1.1 Single-Loop Five-Bar Planar Linkage with Only Rotation Joints  
The five-bar planar linkage (Figure 2a) is the simplest single-loop planar linkage beside the four-bar planar linkage. It has ten instant centers which are shown in Table 2. The instant centers \(\mathrm{I}_{12}\), \(\mathrm{I}_{23}\), \(\mathrm{I}_{34}\), \(\mathrm{I}_{45}\), and \(\mathrm{I}_{15}\) are the first instant centers, and the A type third instant centers include the instant centers \(\mathrm{I}_{13}\), \(\mathrm{I}_{14}\), \(\mathrm{I}_{24}\), \(\mathrm{I}_{25}\), and \(\mathrm{I}_{35}\). Lacking the secondary instant centers, the third instant centers cannot be determined as shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c} \hline First Instant Center & \(\mathrm{I}_{12}\)\(\mathrm{I}_{23}\)\(\mathrm{I}_{34}\)\(\mathrm{I}_{45}\)\(\mathrm{I}_{15}\) \\ \hline Secondary Instant Center & None \\ \hline Third instant center & A type & \(\mathrm{I}_{13}\)\(\mathrm{I}_{14}\)\(\mathrm{I}_{24}\)\(\mathrm{I}_{25}\)\(\mathrm{I}_{35}\) \\ B type & none \\ \hline \end{tabular}
\end{table}
Table 2: Instant centers of five-bar planar linkage.  
Using Criterions 1 and 2, the instant centers of five-bar planar linkage (Figure 2a) can be solved. After adding the virtual links, the virtual loops are formed and the secondary instant centers occur. The created secondary instant center can be used to get the A type third instant centers. Since the parameters and positions of inserted virtual links are arbitrary, the transformation of the configuration is not out of generality. In Figure 9, the original five-bar planar linkage (Figure 2a) is converted into the virtual Stephenson linkage with the virtual links 2', 5', and 6'. Note that the virtual links 2' and 5' are different to the links 2 and 5 (Figure 2a). The instant centers of the virtual Stephenson linkage are shown in Table 3. With Aronhold-Kennedy theorem, the secondary instant centers I\({}_{16}\)' and I\({}_{25}\)' (which is also the common instant center of the loop 1-2'-6'-5' and loop 2'-3-4-5'-6') and the A type third instant centers I\({}_{13}\), I\({}_{14}\), I\({}_{24}\), I\({}_{35}\), I\({}_{36}\), and I\({}_{45}\)' are easy to be located in Figure 10. Compared the links 1, 3, and 4 (red segment in Figure 9) in the five-bar planar linkage and the virtual Stephenson linkage, the locations and motions of links are invariant, that is, the unchanged instant centers I\({}_{13}\), and I\({}_{14}\) of two planar linkage are equivalent. Seeing from Figure 8, as a result, the remaining A type third instant centers (i.e., I\({}_{24}\), I\({}_{25}\), and I\({}_{35}\)) of the single-loop five-bar planar linkage are solvable using the instant center identification process graph. Note that the instant center I\({}_{25}\) obtained in loop 1-2'-6'-5' is different to the instant center I\({}_{25}\) in loop 1-2-3-4-5 since the derivative of angle of the instant center I\({}_{25}\)' coming from loop 1-2'-6'-5' and the derivative of angle of the instant center I\({}_{25}\)' coming from loop 1-2-3-4-5 are different based on the discussion in Section 4.  
\begin{table}
\begin{tabular}{c c} \hline First instant center & I\({}_{12}\)’ I\({}_{23}\) I\({}_{34}\) I\({}_{15}\) I\({}_{45}\)’ I\({}_{26^{\prime}}\) I\({}_{56^{\prime}}\) \\ \hline Secondary instant center & I\({}_{16^{\prime}}\) I\({}_{25^{\prime}}\) \\ \hline Third instant center & A type & I\({}_{13}\) I\({}_{14}\) I\({}_{24}\) I\({}_{35^{\prime}}\) I\({}_{36^{\prime}}\) I\({}_{46^{\prime}}\) \\ B type & none \\ \hline \end{tabular}
\end{table}
Table 3: Instant centers of Stephenson linkage with the virtual links.  
Figure 8: The instant center identification process graph of five-bar planar linkage without virtual links.  
Figure 10: The instant center identification process graph of five-bar planar linkage with virtual links.  
Figure 9: Single-loop five-bar planar linkage with virtual links.