# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 3 Virtual Link Method for Instant Center Identification  
### Criteria for Instant Center Identification  
In order to explain the formation of the two proposed criteria, the instant center identification of the Stephenson linkage containing two-loop, in Figure 3, is taken as an example as follows.  
Figure 3: Instant centers of the Stephenson linkage.  
Figure 2: Two types of instant centers in (**a**) single-loop five-bar planar linkage; (**b**) single-loop six-bar planar linkage.  
The Stephenson linkage has C6\({}^{2}\) = 15 instant centers (Figure 3). The seven instant centers I12, I23, I34, I14, I35, I56, and I16 are the first instant centers, and the secondary instant centers are the centers I13, and I24. The rest of the instant centers, such as I15, I36, I46, I25, I26, and I45, are the A type third instant centers shown in Table 1. Since no B type third instant center exists, instant centers of the Stephenson linkage are easy to be located with Aronhold-Kennedy theorem. In order to efficiently use the existing first or secondary instant centers to locate the third instant centers, the instant center identification process graph is introduced, which makes the location operation visible and efficient. The essence of the instant center identification process graph is the visualization of third instant center identification using Aronhold-Kennedy theorem. For example, the third instant center identification process graph of Stephenson linkage is shown in Figure 4. The instant center identification process graph is built by the third instant centers and the lines formed by other instant centers except for themself. The symbols \"\(\surd\)\", \"\(\times\)\", solid \"\(\downarrow\)\" and hollow \"\(\downarrow\)\" of the graph denote determinate instant center, indeterminate instant center, determinate line and indeterminate line, respectively. The steps of the identification process are shown as follows. Firstly, the third instant centers are all listed on the top. Secondly, using Aronhold-Kennedy theorem, the third instant center can be divided into four instant centers on two possible Aronhold-Kennedy lines. As for instant center I15, the four centers are the instant centers I16, I56, I13, and I35. Thirdly, if the four instant centers are all determinate instant centers, i.e., four \(\surd\), then the two lines can be decided (i.e., two solid \(\downarrow\)). For instant center I15, the two lines are line I16I56 and line I13I35. Finally, the third instant center can be identified in which the two determinate lines cross. For Figure 4, all the instant centers of the Stephenson linkage can be directly obtained with Aronhold-Kennedy theorem, but for single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage with B type third instant centers, the situation becomes complex. In this paper, the concept of coupled loop chain characteristic and virtual link operation are proposed to solve the instant center identification of single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage.  
\begin{table}
\begin{tabular}{c c c} \hline First instant Center & I12 I23 I34 I14 I35 I56 I16 \\ \hline Secondary Instant Center & I13 I24 \\ \hline Third instant center & A type & I15 I36 I46 I25 I26 I45 \\ B type & none \\ \hline \end{tabular}
\end{table}
Table 1: Instant centers of the Stephenson linkage.  
Figure 4: The instant center identification process graph of the Stephenson linkage.