## 2 Analytical Two-Link Model and Equations of Motion
### Friction Models in the Slip and Stick States  
There are two states in which the motion of the wiper blade is governed by different kinds of friction. The slip state corresponds to the case where the relative velocity between the tip of the wiper blade and the windshield is nonzero. The friction acting on the tip of the wiper blade is dynamic in this case. The wiper blade is in the stick state when the relative velocity is continuously zero, in which case static friction acts on the blade. The dynamic friction is uniquely determined by the relative velocity. As long as the stick state is maintained, the absolute value of the static friction can be taken arbitrarily in the range from zero to the maximum static friction. Hence, different models are required for the slip and stick states. The dynamic friction in the slip state is modeled as a function of the relative velocity. The displacement along the \(y\)-axis of the tip of the wiper blade, which corresponds to \(v_{3}\) in Figure 2, can be expressed as follows:  
\[v_{3}=a[1-\cos(\omega t)]-(C-l_{1}\cos\theta-l_{2}\cos\varphi)\tan\theta_{0}+ l_{1}\sin\theta+l_{2}\sin\varphi, \tag{6}\]  
where \(C\) is a constant representing the distance from the origin to the wiping surface. The relative velocity can be expressed as follows:  
\[\dot{v_{3}}=a\omega\sin(\omega t)+\frac{\mathrm{d}\theta}{\mathrm{d}t}\left[ \frac{l_{1}\cos(\theta+\theta_{0})}{\cos\theta_{0}}\right]+\frac{\mathrm{d} \varphi}{\mathrm{d}t}\left[\frac{l_{2}\cos(\varphi+\theta_{0})}{\cos\theta_{ 0}}\right]. \tag{7}\]  
In the motion of the rubber in contact with the glass, the change in the coefficient of dynamic friction under wet conditions due to the thin film of water in between the rubber blade and glass is particularly critical. A thin film of water can be formed between the wiper blade and glass because the pressure of the fluid gradually increases as the sliding velocity increases [14]. The lubrication effect of the thin film of water leads to a reduction in the coefficient of dynamic friction [13]. Thus, the relationship between the relative velocity of the tip of the second link \(\dot{v_{3}}\) and the coefficient of dynamic friction \(\mu_{d}\) is assumed to be as follows in the case \(\dot{v_{3}}\
eq 0\):  
\[\mu_{d}=\mathrm{sign}(\dot{v_{3}})\{A\exp(-B|\dot{v_{3}}|)+E\}, \tag{8}\]  
where the coefficient of dynamic friction is assumed to be equal to the coefficient of the maximum static friction when \(\dot{v_{3}}\) is zero and has a negative correlation with \(\dot{v_{3}}\). The parameters \(A\) and \(E\) define the coefficient of the maximum static friction \(\mu_{max}\) as follows:  
\[\mu_{max}=A+E. \tag{9}\]  
The parameter \(B\) reflects the magnitude of the negative correlation. \(A\), \(B\), and \(E\) are all positive constants, and \(\mathrm{sign}\)\((\dot{v_{3}})\) is the signum function, expressed as  
\[\mathrm{sign}(\dot{v_{3}})= \begin{cases}1,&\dot{v_{3}}>0\\ -1,&\dot{v_{3}}<0\end{cases} . \tag{10}\]  
The dynamic friction \(f_{d}\) is then expressed as  
\[f_{d}=-\mu_{d}N. \tag{11}\]  
The normal force \(N\) is always positive, except when the jump phenomenon occurs. Therefore, the dynamic friction \(f_{d}\) always acts in the direction opposite the relative velocity \(\dot{v_{3}}\). In the stick state, the relative velocity remains at zero continuously and the equilibrium should be maintained. The absolute value of the friction force should be less than the maximum static friction and balance the tangential force of the wiper blade.