## 2 Mathematical Model Descriptions
### The Lagrange Expression
\right] \tag{6}\]  
Differentiating \(R_{e1}\) and \(R_{e2}\) with respect to time yields:  
\[\dot{R}_{e1}=\dot{\theta}_{1}\big{[}A_{\theta_{1}}(\theta_{1})\big{]}[A(\psi_ {1})]\mathrm{e}_{1}+\dot{\psi}_{1}\big{[}A(\theta_{1})\big{]}\big{[}A_{\psi_{ 1}}(\psi_{1})\big{]}\mathrm{e}_{1} \tag{7}\]  
\[\dot{R}_{e2}=\dot{\theta}_{2}\big{[}A_{\theta_{2}}(\theta_{2})\big{]}[A(\psi_{ 2})]\mathrm{e}_{2}+\dot{\psi}_{2}\big{[}A(\theta_{2})\big{]}\big{[}A_{\psi_{ 2}}(\psi_{2})\big{]}\mathrm{e}_{2} \tag{8}\]  
Let the net displacement of \(J_{D2}\) be expressed as:  
\[\theta_{2}=(\theta_{1}+\psi_{1})-\mu(\theta_{1}+\psi_{1})\text{ and }\psi_{2}= \psi_{1}-\mu\psi_{1} \tag{9}\]  
where \(\mu\) is a perturbation function of the rigid and elastic body motion (\(\theta_{1}\) and \(\psi_{1}\)), characteristic of a disturbance by a Hooke's joint. For a finite range of \(\mu\), the product \(\mu\times(\theta_{1}+\psi_{1})\) will be finite and periodically convergent. \(\mu\) used in this paper has been partially developed in [24] and is extended here in Appendix A.1:  
\[\mu=\frac{a\tan(\theta_{1}+\psi_{1})}{(\theta_{1}+\psi_{1})\big{\{}\gamma\tan ^{2}(\theta_{1}+\psi_{1})+1\big{\}}} \tag{10}\]  
where \(a=1-\gamma\) and \(\gamma=\cos\beta\). \(\theta_{2}\) was determined by eliminating \(\psi_{1}\) in Equation (9), after which:  
\[\theta_{2}=\theta_{1}-\frac{an}{(\gamma\eta^{2}+1)} \tag{11}\]  
where \(n=\tan\theta_{1}\). The first derivative of Equation (11) with respect to time gives the relationship of angular velocity between input and output shaft of the Cardan joint as follows:  
\[\dot{\theta}_{2}=\dot{\theta}_{1}(1-g)\text{ where }g=\frac{an}{\gamma\eta^{2 }+1}\Bigg{(}\frac{n^{2}+1}{n}-\frac{2\gamma\eta\big{(}n^{2}+1\big{)}}{\gamma \eta^{2}+1}\Bigg{)} \tag{12}\]  
Differentiating \(R_{e1}\) and \(R_{e2}\) with respect to time gives:  
\[\dot{R}_{e1}=\dot{\theta}_{1}\big{[}A_{\theta_{1}}(\theta_{1})\big{]}[A(\psi_ {1})]\mathrm{e}_{1}+\dot{\psi}_{1}\big{[}A(\theta_{1})\big{]}\big{[}A_{\psi_{ 1}}(\psi_{1})\big{]}\mathrm{e}_{1} \tag{13}\]  
\[\dot{R}_{e2}=\dot{\theta}_{2}\big{[}A_{\theta_{2}}(\theta_{2})\big{]}[A(\psi_ {2})]\mathrm{e}_{2}+\dot{\psi}_{2}\big{[}A(\theta_{2})\big{]}\big{[}A_{\psi_{ 2}}(\psi_{2})\big{]}\mathrm{e}_{2} \tag{14}\]  
where \(\big{[}A_{\theta_{1}}(\theta_{1})\big{]}=d[A(\theta_{1})]/d\theta_{1}\), \(\big{[}A_{\phi_{1}}(\psi_{1})\big{]}=d[A(\psi_{1})]/d\psi_{1}\), \(\big{[}A_{\psi_{2}}(\psi_{2})\big{]}=d[A(\psi_{2})]/d\psi_{2}\). Upon substituting for \([A_{\theta_{1}}(\theta_{1})]\) and \([A_{\psi_{1}}(\psi_{1})]\) in Equations (14) through (15), \(\dot{R}_{e1}\) and \(\dot{R}_{e2}\) can be presented by:  
\[\dot{R}_{e1}=\left\{ \begin{array}{c}\dot{X}_{1}-\dot{\theta}_{1}\epsilon_{x1 }\alpha_{1}-\dot{\theta}_{1}\epsilon_{y1}\dot{\alpha}_{1}-\dot{\psi}_{1}\eta_ {1}\\ \dot{Y}_{1}+\dot{\theta}_{1}\epsilon_{x1}\dot{\delta}_{1}-\dot{\theta}_{1} \epsilon_{y1}\dot{\alpha}_{1}+\dot{\psi}_{1}\mu_{1}\end{array} \right\}\text{ and }\dot{R}_{e2}=\left\{ \begin{array}{c}\dot{X}_{2}-\dot{\theta}_{e2}\alpha_{2}- \dot{\theta}_{2}\epsilon_{y2}\dot{\delta}_{2}-\dot{\psi}_{2}\eta_{2}\\ \dot{Y}_{2}+\dot{\theta}_{2}\epsilon_{x2}\dot{\delta}_{2}-\dot{\theta}_{2} \epsilon_{y2}\alpha_{2}+\dot{\psi}_{2}\mu_{2}\end{array} \right\} \tag{15}\]  
where  
\[ \begin{array}{c}\alpha_{1}=\sin\theta_{1}+\psi_{1}\cos\theta_{1};\text{ }\delta_{1}=\cos\theta_{1}-\psi_{1}\sin\theta_{1}\\ \eta_{1}=\epsilon_{x1}\sin\theta_{1}+\epsilon_{y1}\cos\theta_{1};\text{ }\eta_{11}=\epsilon_{x1}\cos\theta_{1}-\epsilon_{y1}\sin\theta_{1}\end{array}  \tag{16}\]  
and  
\[ \begin{array}{c}\alpha_{2}=\sin\theta_{2}+\psi_{2}\cos\theta_{2};\text{ }\delta_{2}=\cos\theta_{2}-\psi_{2}\sin\theta_{2}\\ \eta_{2}=\epsilon_{x2}\sin\theta_{2}+\epsilon_{y2}\cos\theta_{2};\text{ }\mu_{2}=\epsilon_{x2}\cos\theta_{2}-\epsilon_{y2}\sin\theta_{2}\end{array}  \tag{17}\]The pairs \(e_{x1}\), \(e_{y1}\) and \(e_{x2}\), \(e_{y2}\) are components of e\({}_{1}\) and e\({}_{2}\) in \(x_{1}\), \(y_{1}\) and \(x_{2}\), \(y_{2}\) coordinate.