# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 1 Introduction
The motivation of this paper is to propose a universal method to identify instant centers of planar linkages, especially for single-loop multi-DOF planar linkages. Based on coupled loop chain characteristic and definition of instant center, a single-loop multi-DOF planar linkage is changed into the two-loop linkage with the added virtual links using the proposed criteria, and the unchanged instant centers are identified according to Aronhold-Kennedy theorem, then all the instant centers of the original single-loop multi-DOF planar linkage can be obtained by the instant center identification process graph. The essence of the proposed method is to cover the shortage of the lacking link relationships of the single-loop linkage using virtual link operation, then obtain the indeterminate instant centers. The virtual link operation is an auxiliary mean which does not affect the motion of the original single-loop linkage. Moreover, the instant center identification of the single-loop multi-DOF planar linkage can be the basic of the instant center identification for multi-loop multi-DOF planar linkage which is discussed in our further research. That is, the proposed method may solve the instant center problem of the planar linkages no matter how many number of the links the linkages contain. The first contribution of this paper is to provide a virtual link method to identify instant centers of the single-loop multi-DOF planar linkage, which is a graphical method and sticks to the definition of instant center. Compared to the previous literature [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22], the proposed method has three advantages: simplification, concision, and validity. Firstly, to build virtual link operation, a few steps are carried out and normally, only one virtual link added operation is required. There is no need to choose the specific link and to perform repetitive complex steps [9; 10; 12; 14; 15; 16]. Secondly, the location operation is completely decided by the drawing graph, which also can be programmed and automated in AutoCAD. The closure equations are unnecessary [11; 20]. Thirdly, the proposed method is only based on instantaneous configuration of the linkage and the definition of instant center, and to ensure universality, the parameters of the virtual links are arbitrary. As a result, the application of this method is universal no matter what the degrees of freedom and the components of the joints of the linkage are [17; 18; 19]. The second contribution is to redefine three types of instant centers. The new classification is more accurate and detailed compared to the existing classification, which brings convenience for further research about instant centers. The third contribution is that the instant center problem of the single-loop five-bar, six-bar planar linkages with only rotation joints and several prismatic joints is solved. Note that the instant centers of single-loop five-bar, six-bar planar linkages with several prismatic joints are the first time to be located, to our best knowledge. The proposedmethod provides a geometry loop insight to reveal the relationship between the formation of the instant center and the motion of the mechanism, and a new research idea for the study of instant centers. This paper is organized as follows. In Section 2, the mathematics definition is presented and three types of instant centers are redefined. In Section 3, based on coupled loop chain characteristic and definition of instant center, two criteria to determine instant centers are provided, and the instant center identification process graph is introduced. Instant centers of the Stephenson six-bar linkage are identified to explain how the proposed method works. Then, the steps of instant center identification of single-loop multi-DOF N-bar (N \(\geq\) 5) planar linkages are summarized. Subsequently, the mathematical proof is shown in Section 4.