# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 1 Introduction
Although this method is simpler than the previous literature, the calculation is relatively complex. [PERSON] [25] utilized the location of the instant centers to perform acceleration analysis of a four-bar mechanism. On the basis of an adequate literature analysis, [PERSON] and [PERSON] [6] presented a synthesis method based on optimization in which the identification of instant center is included in the objective function for rigid-body guidance synthesis Moreover, this proved approach is robust, accurate and efficient. Depended on the fact that the positions of instant centers can determine the velocity coefficients and the virtual work of the external forces of the mechanism, [PERSON] [7] provided a dynamic model and an algorithm to solve the dynamic problems of single-DOF planar linkage. It is simple to use and numerically effective. In addition, since the relationships between the positions of the instant centers and the absolute velocities, instant centers are possibly used in some other applications such as dynamic model of the spherical mechanisms [26; 27], wrench capability analysis of the redundant mechanisms [28], dynamic balancing analysis [29], and even commercial packages. For dynamic model of the spherical parallel mechanism, based on the principle of virtual work, the dynamic model can be built with the Jacobian matrices including angular velocities. As we all know, in a certain mechanism, once the instant centers are located, the corresponding angular velocities are decided. That is, the instant centers are able to address the dynamic model. For wrench capability analysis of the redundant planar parallel manipulator, joint torques, which are decided by the forces and moments acting on the end-effector, are the kinematic condition to sustain the wrench. They can be deduced by the derivatives of absolute velocities obtained from the instant centers. For dynamic balancing analysis of a given mechanism [29], the positions of instant centers can directly be used to calculate the angular momentum which effects whether the sum of all forces and moments acting on the based are zero. For commercial packages, the identification of instant centers may be available as a plug-in of the AutoCAD tool since the instant centers have a widespread application and the proposed method is suited to be programmed. As for real devices where imperfections are unavoidable in production processes, another application of instant center is found inspired by Profs. [PERSON] and [PERSON]'s paper [30]. In their research, the nonlinear dynamical circuits are built to help investigate the performance of a novel control strategy for imperfect systems, and the instant center may be used to calculate the dynamical equations which regulate the behavior of the circuits. Although instant center, which is a basic kinematic property, is a useful tool to analyze kinematics of the linkages [2; 3; 4; 5; 31], rigid-body synthesis [6], dynamics modeling [7], and dynamic balancing analysis [29] of planar linkages, and it can be used in spatial mechanisms [26; 27], redundant mechanisms [28], imperfect systems of real devices [30], and is even available in commercial packages, the identification of instant center is difficult since the lack of relationship of the interval links in a N-bar loop (N \(\geq\) 5). As discussed above, the existing methods can be classified three types: (1) graphical method [10; 12; 17; 18; 19; 22], (2) analytical method [11; 20; 23; 24], and (3) both analytical method and graphical method [9; 14; 15; 16]. The graphical method has the advantage of visualization, but the defects are complex process and low applicable rate. The analytical method has the merit of high accuracy, but the calculation is normally complicated. Both analytical method and graphical method has both advantages, but the implementation method is roundabout. The generality and simplicity of instant center identification method is still a challenge.