## 2 Analytical Two-Link Model and Equations of Motion
### Friction Models in the Slip and Stick States
Considering the above characteristics, the static friction is defined as follows:  
\[f_{s}=\text{Sign}(\dot{v}_{3})\mu_{max}N, \tag{12}\]  
where \(\text{Sign}(\dot{v}_{3})\) is a set-valued function expressed as  
\[\text{Sign}(\dot{v}_{3})=[-1,1],\quad\dot{v}_{3}=0. \tag{13}\]  
By combining the two models for dynamic and static friction, the relationship between the coefficient of friction and the relative velocity is as shown in Figure 4. Given the relationship between the normal force \(N\) and the friction force \(f\) for the different states of motion, the behavior of the two links can be analyzed. For the four unknown quantities, in addition to Equations (2)-(4), the other equations in the slip and stick states are Equations (11) and (12), respectively. By introducing the representative time \(T=\sqrt{\frac{I_{2}+m_{2}I_{2}^{2}}{k_{2}}}\) and the representative length \(L=a\), the dimensionless forms of Equations (2)-(4) are expressed as follows:  
Figure 4: Relationship between coefficient of friction and relative velocity of wiper blade and wiping surface. \[ \begin{split}&\delta\bigg{[}-\frac{-m_{d}^{*}l_{1}^{*}\sin\theta}{\cos \theta_{0}}+m_{a}^{*}l_{g1}^{*}\sin(\theta+\theta_{0})+m_{b}^{*}l_{1}^{*}\tan \theta_{0}\cos(\theta+\theta_{0})\bigg{]}\\ &+\ddot{\varphi}\bigg{[}\frac{-l_{2}^{*}\sin\varphi}{\cos\theta_ {0}}+m_{b}^{*}l_{g2}^{*}\sin(\varphi+\theta_{0})\bigg{]}\\ =&\theta^{2}\bigg{[}\frac{m_{d}^{*}l_{1}^{*}\cos \theta}{\cos\theta_{0}}-m_{a}^{*}l_{g1}^{*}\cos(\theta+\theta_{0})+m_{b}^{*}l_ {1}^{*}\tan\theta_{0}\sin(\theta+\theta_{0})\bigg{]}\\ &+\dot{\varphi}^{2}\bigg{[}\frac{l_{2}^{*}\cos\varphi}{\cos\theta _{0}}-m_{b}^{*}l_{g2}^{*}\cos(\varphi+\theta_{0})\bigg{]}+k_{0}^{*}\bigg{[}l_ {1}^{*}\bigg{(}1-\frac{\cos\theta}{\cos\theta_{0}}\bigg{)}+l_{2}^{*}\bigg{(}1 -\frac{\cos\varphi}{\cos\theta_{0}}\bigg{)}-h_{d}^{*}\bigg{]}\\ &+c_{0}^{*}\bigg{(}\frac{\dot{\theta}l_{1}^{*}\sin\theta}{\cos \theta_{0}}+\frac{\dot{\varphi}l_{2}^{*}\sin\varphi}{\cos\theta_{0}}\bigg{)}+ N^{*}\cos\theta_{0}+f^{*}\sin\theta_{0}-{\omega^{*}}^{2}\sin\theta_{0}\cos( \omega^{*}t^{*}),\end{split}  \tag{14}\]  
\[ \begin{split}&\ddot{\theta}\bigg{[}\frac{-m_{a}^{*}l_{1}^{*}l_{g1} ^{*}\sin\theta}{\cos\theta_{0}}\sin(\theta+\theta_{0})+\frac{m_{b}^{*}l_{1}^{* 2}\cos\theta}{\cos\theta_{0}}\cos(\theta+\theta_{0})+m_{a}^{*}l_{g1}^{*2}+I_{1} ^{*}\bigg{]}\\ &+\ddot{\varphi}\bigg{[}m_{b}^{*}l_{1}^{*}l_{g1}^{*}\cos(\varphi -\theta)-\frac{m_{e}^{*}l_{2}^{*}\sin\varphi\sin(\theta+\theta_{0})}{\cos \theta_{0}}\bigg{]}\\ =&\theta^{2}\bigg{[}\frac{m_{e}^{*}l_{1}^{*}\cos \theta\sin(\theta+\theta_{0})}{\cos\theta_{0}}\bigg{]}+\dot{\varphi}^{2}\bigg{[} \frac{m_{e}^{*}l_{2}^{*}\cos\varphi\sin(\theta+\theta_{0})}{\cos\theta_{0}}+m_ {b}^{*}l_{1}^{*}l_{g2}^{*}\sin(\varphi-\theta)\bigg{]}\\ &+M_{k1}^{*}-c_{1}^{*}\dot{\theta}+k_{2}^{*}(\varphi-\theta)+c_{ 2}^{*}\big{(}\dot{\varphi}-\dot{\theta}\big{)}\\ &+N^{*}l_{1}^{*}\sin\theta+f^{*}l_{1}^{*}\cos\theta-m_{e}^{*} \omega^{*2}\cos\theta\cos(\omega^{*}t^{*}),\end{split}  \tag{15}\]  
\[ \begin{split}&\ddot{\theta}\bigg{[}\frac{m_{b}^{*}l_{1}^{*}l_{g2} ^{*}\cos\varphi}{\cos\theta_{0}}\cos(\theta+\theta_{0})\bigg{]}+\dot{\varphi} \bigg{[}\frac{-m_{b}^{*}l_{2}^{*}l_{g2}^{*}\sin\varphi}{\cos\theta_{0}}\sin( \varphi+\theta_{0})+m_{b}^{*}l_{g2}^{*2}+I_{2}^{*}\bigg{]}\\ =&\dot{\theta}^{2}\bigg{[}\frac{m_{b}^{*}l_{1}^{*}l_{ g2}^{*}\cos\varphi\sin(\theta+\theta_{0})}{\cos\theta_{0}}\bigg{]}+\dot{\varphi}^{2} \bigg{[}\frac{m_{b}^{*}l_{2}^{*}l_{g2}^{*}\cos\varphi\sin(\varphi+\theta_{0})}{ \cos\theta_{0}}\bigg{]}\\ &-k_{2}^{*}(\varphi-\theta)-c_{2}^{*}\big{(}\dot{\varphi}-\dot{ \theta}\big{)}+N^{*}l_{2}^{*}\sin\varphi+f^{*}l_{2}^{*}\cos\varphi-m_{b}^{*}l_ {g2}^{*}\omega^{*2}\cos\varphi\cos(\omega^{*}t^{*}),\end{split}  \tag{16}\]  
where the dots denote derivatives with respect to the dimensionless time \(t^{*}\).