## 2 Analytical Two-Link Model and Equations of Motion
### Equations of Motion of the Model  
From the model, we can obtain the following governing equations of motion (a detailed derivation is given in Appendix A). \[ \begin{split}&\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t}\bigg{[} \frac{-(m_{0}+m_{1})l_{1}\sin\theta}{\cos\theta_{0}}+m_{1}l_{g1}\sin(\theta+ \theta_{0})+m_{2}l_{1}\tan\theta_{0}\cos(\theta+\theta_{0})\bigg{]}\\ &+\frac{\mathrm{d}^{2}\varphi}{\mathrm{d}t^{2}}\bigg{[}\frac{-(m _{0}+m_{1}+m_{2})l_{2}\sin\varphi}{\cos\theta_{0}}+m_{2}l_{g2}\sin(\varphi+ \theta_{0})\bigg{]}\\ =&\bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)} ^{2}\bigg{[}\frac{(m_{0}+m_{1})l_{1}\cos\theta}{\cos\theta_{0}}-m_{1}l_{g1} \cos(\theta+\theta_{0})+m_{2}l_{1}\tan\theta_{0}\sin(\theta+\theta_{0})\bigg{]} \\ &+\bigg{(}\frac{\mathrm{d}\varphi}{\mathrm{d}t}\bigg{)}^{2} \bigg{[}\frac{(m_{0}+m_{1}+m_{2})l_{2}\cos\varphi}{\cos\theta_{0}}-m_{2}l_{g2} \cos(\varphi+\theta_{0})\bigg{]}\\ &+k_{0}\bigg{[}l_{1}\big{(}1-\frac{\cos\theta}{\cos\theta_{0}} \big{)}+l_{2}\bigg{(}1-\frac{\cos\varphi}{\cos\theta_{0}}\bigg{)}-h_{d}\bigg{]} +c_{0}\bigg{[}\bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)}\frac{l_{1} \sin\theta}{\cos\theta_{0}}+\bigg{(}\frac{\mathrm{d}\varphi}{\mathrm{d}t} \bigg{)}\frac{l_{2}\sin\varphi}{\cos\theta_{0}}\bigg{]}\\ &+N\cos\theta_{0}+f\sin\theta_{0}-(m_{0}+m_{1}+m_{2})a\omega^{2} \sin\theta_{0}\cos(\omega t),\end{split}  \tag{2}\]  
Figure 3: Restoring moment of the first link. \(k_{11}\) and \(k_{12}\) are the rotation stiffness of the shoulder with or without head contact, respectively. \(\theta_{c}\) is the angle at which the shoulder begins to contact the head. \[\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\bigg{[}\frac{-m_{1}l_{1}l_{g1 }\sin\theta}{\cos\theta_{0}}\sin(\theta+\theta_{0})+\frac{m_{2}l_{1}^{2}\cos \theta}{\cos\theta_{0}}\cos(\theta+\theta_{0})+m_{1}l_{g1}^{2}+I_{1}\bigg{]}\] \[+\frac{\mathrm{d}^{2}\varphi}{\mathrm{d}t^{2}}\bigg{[}m_{2}l_{1} l_{g1}\cos(\varphi-\theta)-\frac{(m_{1}l_{g1}+m_{2}l_{1})l_{2}\sin\varphi\sin( \theta+\theta_{0})}{\cos\theta_{0}}\bigg{]}\] \[= \bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)}^{2}\bigg{[} \frac{(m_{1}l_{g1}+m_{2}l_{1})l_{1}\cos\theta\sin(\theta+\theta_{0})}{\cos \theta_{0}}\bigg{]} \tag{3}\] \[+M_{k1}-c_{1}\bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)} +k_{2}(\varphi-\theta)+c_{2}\bigg{[}\bigg{(}\frac{\mathrm{d}\varphi}{\mathrm{ d}t}\bigg{)}-\bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)}\bigg{]}\] \[+Nl_{1}\sin\theta+fl_{1}\cos\theta-\big{(}m_{1}l_{g1}+m_{2}l_{1} \big{)}a\omega^{2}\cos\theta\cos(\omega t),\]  
\[\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\bigg{[}\frac{m_{2}l_{1} l_{g2}\cos\varphi}{\cos\theta_{0}}\cos(\theta+\theta_{0})\bigg{]}+\frac{ \mathrm{d}^{2}\varphi}{\mathrm{d}t^{2}}\bigg{[}\frac{-m_{2}l_{2}l_{g2}\sin \varphi}{\cos\theta_{0}}\sin(\varphi+\theta_{0})+m_{2}l_{g2}^{2}+I_{2}\bigg{]}\] \[= \bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t}\bigg{)}^{2}\bigg{[} \frac{m_{2}l_{1}l_{g2}\cos\varphi\sin(\theta+\theta_{0})}{\cos\theta_{0}}\bigg{]} +\bigg{(}\frac{\mathrm{d}\varphi}{\mathrm{d}t}\bigg{)}^{2}\bigg{[}\frac{m_{2}l _{g2}\cos\varphi\sin(\varphi+\theta_{0})}{\cos\theta_{0}}\bigg{]} \tag{4}\] \[-k_{2}(\varphi-\theta)-c_{2}\bigg{[}\bigg{(}\frac{\mathrm{d} \varphi}{\mathrm{d}t}\bigg{)}-\bigg{(}\frac{\mathrm{d}\theta}{\mathrm{d}t} \bigg{)}\bigg{]}+Nl_{2}\sin\varphi+fl_{2}\cos\varphi-m_{2}l_{g2}a\omega^{2} \cos\varphi\cos(\omega t).\]  
There are four unknown quantities in these three equations, namely the angles of the two links \(\theta\) and \(\varphi\), the normal force \(N\), and the friction force \(f\). One additional equation is required to obtain the solution. This is obtained next by considering the different states of motion. Unlike the case where the attack angle is not considered (\(\theta_{0}=0\)), it is impossible to obtain an equation for the normal force independently.