# 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
#### 5.1.2 Single-Loop Five-Bar Planar Linkage with Prismatic Joints  
The instant center of a prismatic joint lies on the line which is perpendicular to the motion pair, for example, L45 in Figure 11. There are three cases of single-loop five-bar planar linkages with the different number of prismatic joints. Taking the linkages in Figure 11a for example, the instant centers are shown in Table 4. Contrasted Table 2 with Table 4, the difference of the instant centers between the single-loop five-bar linkage with one prismatic joint and the single-loop five-bar linkage with only rotation joints is that one first instant center (I45 in Table 2) is changed into an A type third instant center (I45 in Table 4). Using the proposed method, the instant center problem of this kind linkages can be solved as follows. The virtual links can be added in two ways: (1) after added operation, the five-bar linkage is changed into a virtual four-bar loop with only rotation joints and a five-bar loop with one prismatic joint; (2) the five-bar linkage is changed into a virtual five-bar loop with only rotation joints and a four-bar loop with one prismatic joint, shown in the Figures 12 and 13. For Figure 12, the linkage with the prismatic joint is changed into a virtual Stephenson linkage with one prismatic joint containing loop1-2-3'-6' with rotation joints and loop1'-6'-3'-4-5 with the prismatic joint. However, some A type third instant centers, such as L14, L24, L46, and L45, in this case, cannot be obtained even under exceptional conditions that the angle between the link 4 and the sliding block is 90\({}^{\circ}\). As for Figure 13a, using the corresponding virtual link operation, the linkage is transformed into a virtual Stephenson linkage including loop 1'-2-3-4'-6' with rotation joints and loop 1'-6'-4'-5 with the prismatic joint. Although the instant centers L14, L56, and L45 in the loop1'-6'-4'-5 cannot be located, the instant centers of loop 1'-2-3-4'-6' with rotation joints are solvable seeing from the case in Figure 9. The corresponding virtual link operations are shown in Figure 13b. The instant center I14 is the common instant center of the loop 1'-2-3-4'-6' and loop 1'-6'-4'-5 with the prismatic joint. In another word, the instant center I14, obtained from the loop 1'-2-3-4'-6', can be used to identify the rest instant centers L56, and L45 in loop 1'-6'-4'-5 with the prismatic joint. The solution for single-loop five-bar planar linkage with prismatic joint sounds like an iterative process. The instant centers of the five-bar loop with rotation joints are solved, and then used in the identification of the four-bar loop with the prismatic joint. The other cases are shown in Figure 13c,d in which the five-bar planar linkage has two prismatic joints and three prismatic joints, respectively. The five-bar planar linkage with two prismatic joints (Figure 13c) can be changed into a four-bar loop with one prismatic joint and a five-bar loop with another prismatic joint discussed in Figure 11a. The five-bar planar linkage with three prismatic joints (Figure 13d) can be changed into a four-bar loop with one prismatic joint and a five-bar loop with another two prismatic joints discussed in Figure 11b. For simplicity, this repeated discussion is omitted. \begin{table}
\begin{tabular}{c c c} \hline \hline First instant center & I12 123 134 I15 \\ \hline Secondary instant center & none \\ \hline Third instant center & A type & I13 114 I24 I25 I35 I45 \\ B type & none \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Instant centers of five-bar planar linkage with one prismatic joint. I\({}_{36}\), and I\({}_{46}\) are shown in Table 5, then the instant center identification process graph is established in Figure 14. Similar to the instant center identification of single-loop five-bar planar linkage, the instant center identification is shown as follows.