# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 2. Mathematic Definition and Classification of Instant Center
In literatures above [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24], instant centers are classified into two types: primary instant center and secondary instant center. For a given linkage, primary instant center is constantly the coincident point for a pair of rigid bodies of a planar linkage. The other instant centers are all secondary instant centers no matter whether they can be located with Aronhold-Kennedy theorem. However, in this paper, instant centers are classified into three types in view of the access: first instant center, secondary instant center and third instant center. First instant center is same to primary instant center of the existing literatures. It is the easiest to be obtained and is also the base to locate the secondary and third instant centers, such as I\({}_{12}\), I\({}_{23}\), I\({}_{14}\), and I\({}_{34}\) in Figure 1. The instant centers, which can be directly received according to the established first instant centers using Aronhold-Kennedy theorem, are called as secondary instant center. It is different to the concept of secondary instant center of the existing literatures. In the new classification, all the secondary instant centers can be located with Aronhold-Kennedy theorem. They have exact positions. For Figure 1, the secondary instant centers are I\({}_{13}\), I\({}_{24}\). Apart from first and secondary instant centers, the remaining instant centers of the linkages are third instant center. Since its indeterminacy, the identification of third instant center is a difficulty for single-loop multi-DOF planar linkages. Compared to secondary instant center, third instant center can also be determined by Aronhold-Kennedy theorem, but the difference is that secondary instant center can be acquired straightforwardly using the existing first instant centers but third instant center is not. The third instant center I\({}_{13}\), formed by the link 1 and link 3 in Figure 2a, is this case. The instant center I\({}_{13}\) is a point lying on the extension line of link 5, but the exact location is unknown. That is, the location of third instant center needs other addition information. According to whether third instant center exists on a known line (at least two instant centers on the line are known, normally, the extension line of the link), two types third instant center can be concluded. When a third instant center lies on a known line, the third instant center is an A type third instant center. If not, it is a B type. Taking third instant centers in Figure 2a for example, the instant center I\({}_{13}\), I\({}_{14}\), I\({}_{24}\), I\({}_{25}\), and I\({}_{35}\) (also represented as L\({}_{13}\), L\({}_{14}\), L\({}_{24}\), L\({}_{25}\), and L\({}_{35}\), L means that the instant centers are on the lines, but their exact locations are unknown) are all the A type third  
Figure 1: Planar four-bar linkage with instant centers. instant centers. Obviously, there is no B type third instant center. For Figure 2b, the instant center \(\rm I_{14}\), \(\rm I_{25}\) and \(\rm I_{36}\) are all the B type third instant centers. What is noteworthy is that the shortest topological distance of the two links coming from the instant center can be used to distinguish which type the third instant center belongs to. The discrimination process is shown as follows. If the shortest topological distance is less than two units, which means that there is only one link among the two links, the corresponding instant center is an A type third instant center. For Figure 2a, the link 2 is the only link for the A type instant center \(\rm I_{13}\) formed by the links 1 and 3. The B type third instant center \(\rm I_{14}\) has two links (i.e., links 2 and 3) between the formed links 1 and 4 in Figure 2b. Compared to the listed literatures, the new classification is more detailed and accurate.