# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 3 Virtual Link Method for Instant Center Identification
### Process for Instant Center Identification  
For single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage, as shown in Figure 5a, there are \(\text{C}_{\text{N}}\)2 instant centers including no secondary instant center. According to two criteria above, the single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage can be transformed into the two-loop virtual planar linkage which is combined by a N-bar virtual loop and a (N \(-\) 1)-bar virtual loop when N is odd number or two (N \(-\) 1)-bar virtual loops when N is even number by adding the middle virtual links N + 1' in Figure 5b (the number m of the middle virtual links depends on what odd number or even number N (N \(\geq\) 5) belongs to. For example, 5 is the first odd number for N, the number m is one for single-loop five-bar planar linkage. 7 is the second odd number for N, therefore, for seven-bar linkage, the number m is two. The number m is three in nine-bar linkage. Similarly, 6 is the first even number for N, so the number m is one for single-loop six-bar linkage, and the number m is two for eight-bar linkage. It is three in 10-bar linkage. The rule of adding virtual links is to ensure that the number of the two formed virtual loops are same or the difference is 1). If the common instant centers of the two formed virtual loops are located, all the instant centers of the two-loop virtual planar linkage can be identified, and then the instant center problem of the original N-bar planar linkage can be solved with Aronhold-Kennedytheorem. Six steps are summarized for the instant center identification of single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage as follows. The corresponding flow chart is shown in Figure 6.  
Figure 5: (**a**) Single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage, (**b**) Two-loop virtual planar linkage with virtual links.  
Figure 6: The flow chart of instant center identification for single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkage.  
Step 1: Calculate the number of instant centers and classify the instant centers according to the proposed classification, then form the instant center Table.  
Step 2: Form the instant center identification process graph and choose the right links to build the common instant centers. The common instant centers can be formed by the unchanged links after virtual link operation or directly come from the virtual links. The common instant centers and virtual link operation are interdependent. In another word, the rule of adding virtual links discussed above is the rule to choose the common instant centers, to some extent.  
Step 3: Check if the number N (N \(\geq\) 5) belongs to odd number or not.  
Step 4: Based on the results from Steps 2 and 3, add the corresponding virtual links to form two-loop virtual planar linkage.  
Step 5: Use Criterions 1 and 2 to locate the common instant centers of the two-loop virtual planar linkage. If not, back to Step 3 to add new virtual links in the existing virtual loop until the common instant centers can be obtained. Step 3-Step 4-Step 5 is a iteration loop. The proposed method is suitable to be programmed.  
Step 6: Determine the remaining instant centers of the original N-bar planar linkage which fails to be located in Step 5 using Aronhold-Kennedy theorem.