# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 1 Introduction  
The concept of instant center is proposed by [PERSON] [1], which refers to the zero velocity point for two rigid bodies in a planar motion. It is not only used to analyze kinematics of linkages such as absolute velocity and angular velocity, but also to research singular configurations [2; 3; 4; 5], configuration synthesis [6] and dynamics modeling [7]. Generally, instant centers are obtained by Aronhold-Kennedy theorem [8], but if the linkages contain more links or link loops, the traditional method fails. The instant center which cannot be directly located by Aronhold-Kennedy theorem, is called \"indeterminate instant center\" [9] or indeterminate secondary instant center in Refs. [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Moreover, it exists in almost all linkages besides single-loop four-bar linkage and Stephenson linkage. Therefore, the location of indeterminate secondary instant center is a difficulty of kinematics analysis of planar linkages, which attracts many researchers' interests. [PERSON] [10] proposed a graphical linkage reduction method to find coordinated centers of curvature by changing the pentagonal loop into four-bar loop. The disadvantage of this method is that the joint-joining operation may have to be carried out twice and each time in a different way. Based on singular coefficient matrix of the derived velocity equations, [PERSON] and [PERSON] [11]presented an analytical method to locate instant centers of single or multiple degrees of freedom linkages. This method has tedious calculation. [PERSON] [12] raised a graphic method of trial-and-error type to identify the right position of the center. The defect is that the process is complex and inefficient. [PERSON] and [PERSON] [9; 14; 15; 16] presented both analytical method and graphical technique to solve instant center problem of single-DOF, two-DOF planar indeterminate linkage and three-DOF planar six-bar linkage. The essence of this method is first to determine one indeterminate secondary instant center by combing two arbitrary possible positions, then to reverse all the indeterminate instant centers, but its implementation method is roundabout. Based on instant centers of the cam pair residing on the extension line of a primary adjacent link, [PERSON] [17; 18; 19] introduced a virtual cam method to help locate key instant centers of the linkages up to ten-bar. This method is a graphical one and is accurate in AutoCAD, but the applicable rate is not high. In addition, in some cases, it needs to work with [PERSON]'s method [9; 14; 15; 16], then the linkages can be solved. [PERSON] [20] presented an algorithm to calculate the positions of indeterminate secondary instant centers of the indeterminate linkages. Although the analytical method uses only pieces of information of the linkage configuration, the difficulty is how to write the equations together with the closure equations of the mechanism to allow the computation of the instant centers' positions as a function of the generalized coordinate chosen to identify the mechanism configuration. Obviously, it is not easy. [PERSON] and [PERSON] [21] proposed the concept of instant center walk and instant center circuit and established the recursive formula to compute the coordinates of the instant centers. However, the drawback is that the application is only for single-DOF indeterminate linkages at non-singular configurations. According to the fact that an indeterminate secondary instant pole of two-DOF spherical linkage lies somewhere on the unique great circle for a specific configuration of the linkage, [PERSON] [22] provided two techniques to convert a single-DOF spherical mechanism to a two-DOF one, and then locate all the indeterminate secondary instant centers. The main issue is how to use the techniques, and it is not universal for different linkages. [PERSON] [23; 24] presented a screw theory approach for the computation of instant rotation axes of the spherical linkages requiring the solution and comparison of two quadratic equations.