# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 2. Mathematic Definition and Classification of Instant Center  
Instant center is defined as a point where the relative velocity of the two rigid bodies is zero in a planar motion, but the absolute velocity may be not. For a given planar linkages, the positions of instant centers are varying yielding to the input condition when the planar linkage continuously moves. In fact, the continuous motion of the linkage is formed by lots of instantaneous configurations. Each instantaneous configuration corresponds to one specified input at this instant. That is, an instantaneous configuration coming from the continuous motion linkage is a momentary configuration in which the corresponding momentary input is specified. Moreover, different instantaneous configurations correspond to different specified ratios of the inputs. If the same linkage moves from one instantaneous configuration to the other instantaneous configuration, the specified ratios of the inputs should be accordingly changed. In this paper, instant centers of the instantaneous configuration are discussed, i.e., the linkages researched in our paper are all discussed on the condition that the linkages are all the instantaneous configurations, only one specified ratio of the inputs corresponds to the corresponding instantaneous configuration. Since the momentary input condition is specified in the instantaneous configuration, the passive angles of the linkage can be decided at this instant. In another word, the link relationships (i.e., instant centers) of the instantaneous configuration can be all obtained at the exact points at this instant based on mathematic definition above. In Figure 1, the location of the instant center I\({}_{13}\) can be expressed as followed. \[\mathrm{V}_{13}=\mathrm{W}_{1}\times\mathrm{r}_{1}=\mathrm{W}_{3}\times \mathrm{r}_{3} \tag{1}\]  
where V\({}_{13}\) is absolute velocity of the links 1 and 3, and W\({}_{i}\) and r\({}_{i}\) (i = 1 or 3) denote the angular velocity of corresponding link and distance between the binary link from the point, respectively. Equation (1) represents the fact that the absolute velocity V\({}_{13}\) of the instant center I\({}_{13}\) only relatives to the angular velocity W\({}_{i}\) and the corresponding distance r\({}_{i}\) in a instantaneous configuration where the momentary input condition is specified and the link parameters of the planar linkage are determinate. It means that the identification of instant center in the instantaneous configuration is only related to the positions and motions (i.e., distances and angular velocities discussed in above) of the participant links (links 1 and 3 in Figure 1) and is irrelevant to the other link of the instantaneous configuration. In another word, when the positions and motions of participant links are the same regardless of whether the remaining links of the linkage changed or not, the corresponding instant center is the same. This property, evolved from the definition, is the theoretical basis to research the single-loop multi-DOF planar linkage in the following sections of this paper. In addition, it is used and proved in Ref. [32] for invariant link rotatability of N-bar kinematic chains. It is worth noting that the property above is effective in instantaneous configurations, but it fails when the linkage is in a continuous motion. The reason is that the passive angular velocity of the participant links (for example links 1 and 3 in Figure 1) is timely drove by the input link (link 4) and the limits coming from sizes of the links. Although the distancesand the angular velocities of other links of the moving planar linkage do not occur in Equation (1), they still influence the positions and motions of the participant links. All the locations of the instant centers are continuously varying. However, for instantaneous configurations, each instantaneous configuration corresponds to one specified input at this instant. The ratio of the inputs of an instantaneous configuration is specified, the momentary input is known and invariable.