# Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link
## 4 Mathematical Proof  
Loop equation is a common mathematical tool to analyze the kinematics of planar linkages. It is employed here to verify the validity of virtual link operation in the instantaneous configuration. Taking the single-loop five-bar planar linkage (red part in Figure 7a, i.e., loop1-2-3-4-5), for example, the known variables are the input angles \(\theta_{2}\), \(\theta_{5}\) and the link parameters a\({}_{1}\), a\({}_{2}\), a\({}_{3}\), a\({}_{4}\), and a\({}_{5}\), i.e., the passive angles \(\theta_{3}\), and \(\theta_{4}\) are unknown. Choosing the angle \(\theta_{4}\) as the output, the loop equation of the single-loop five-bar planar linkage can be expressed as  
Mathematical proof for single-loop five-bar planar linkage. Loop ABCDE:  
\[\text{a}_{1}+\text{a}_{2}\text{e}^{\text{i}\theta_{2}}+\text{a}_{3}\text{e}^{ \text{i}\theta_{3}}-\text{a}_{4}\text{e}^{\text{i}\theta_{4}}-\text{a}\text{e} ^{\text{i}\theta_{5}}=0 \tag{2}\]  
Using [PERSON] formula, Equation (2) can be written as the following two equations:  
\[\text{a}_{1}+\text{a}_{2}\text{cos}\,\theta_{2}+\text{a}_{3}\text{cos}\, \theta_{3}-\text{a}_{4}\text{cos}\,\theta_{4}-\text{a}_{5}\text{cos}\,\theta_{ 5}=0 \tag{3}\]  
\[\text{a}_{2}\text{sin}\,\theta_{2}+\text{a}_{3}\text{sin}\,\theta_{3}-\text{a} _{4}\text{sin}\,\theta_{4}-\text{a}_{5}\text{sin}\,\theta_{5}=0 \tag{4}\]  
Figure 7: (**a**) Single-loop two-DOF five-bar planar linkage, (**b**) Two-loop virtual Stephenson planar linkage. Eliminating the unknown angle \(\theta_{3}\) in Equations (3) and (4) according to the trigonometric function \((\sin\theta_{3})^{2}+(\cos\theta_{3})^{2}=1\),  
\[\theta_{4}=\mathrm{M}_{1}(\theta_{2},\,\theta_{5}) \tag{5}\]  
Differentiating Equation (5) with respect to time, yields,  
\[W_{4}=\overset{\bullet}{\theta}_{4}=M_{2}(\overset{\bullet}{\theta}_{2}, \overset{\bullet}{\theta}_{5}) \tag{6}\]  
where \(\mathrm{M}_{i}\) (i = 1, 2) are the coefficients in the light of \(\theta_{2}\) and \(\theta_{5}\), which can be obtained with mathematical software Maple. Note that \(\mathrm{M}_{i}\) is constant. Equation (6) shows that the angular velocity \(W_{4}\) of the single-loop five-bar planar linkage is only related to the input angle \(\theta_{2}\), \(\theta_{5}\). After adding the virtual links, the single-loop five-bar linkage (Figure 7a) is changed into a virtual Stephenson linkage (Figure 7b). According to discussion above, the sizes and positions of the virtual links 1', 3', and 6' are arbitrary, hence, there are no known parameters added. Note that the links 1 and 3 (Figure 7a) and the links 1'and 3' in Figure 7b are different. Seeing from Figure 7a, the link 3 has only one angular velocity, the derivative of angle \(\theta_{3}\) in loop ABCDE. However, the link 3' (Figure 7b) has three angular velocities: the derivative of angle \(\theta_{3}\) in loop ABCDE, the derivative of angle \(\theta_{31}\) in loop GFCDE, the derivative of angle (\(\theta_{31}+2\pi-\beta-\eta\)) in loop ABFG. As for the links 1 and 1', the similar result can be concluded. Therefore, the instant center \(\mathrm{I}_{17^{3}}\) in the virtual Stephenson linkage is not same to the instant center \(\mathrm{I}_{13}\) in the single-loop five-bar planar linkage. In fact, the virtual link operation is an assumption. The virtual links are unreal. The virtual Stephenson linkage is actually still the single-loop five-bar linkage. The inputs and the motions of the links (not including the virtual links) in the virtual Stephenson linkage are the same compared to the original single-loop five-bar linkage. The instant centers which are formed by the virtual links 1', 3', and 6', such as \(\mathrm{I}_{172}\), \(\mathrm{I}_{1^{3}}\), \(\mathrm{I}_{1^{4}}\), \(\mathrm{I}_{1^{5}}\), \(\mathrm{I}_{1^{6}}\), \(\mathrm{I}_{23^{7}}\), \(\mathrm{I}_{26}\), \(\mathrm{I}_{3^{4}}\), \(\mathrm{I}_{35}\), \(\mathrm{I}_{36^{7}}\), \(\mathrm{I}_{46^{7}}\), and \(\mathrm{I}_{56^{6}}\) in Figure 7b, do not change the motion of the original five-bar linkage.