## 4 Kinematic Analysis of the Rope Drive Mechanism
### Inverse Kinematics Analysis of the Rope Drive Mechanism  
The inverse kinematics solution based on the theory of spinors is built on several basic subproblems, commonly referred to as the Paden-Kahan subproblems. The rope drive designed in this paper has four revolving shafts. The revolving shaft 1 and revolving shaft 2 are mutual orthogonal to the point \(q_{1}\), the revolving shaft 3 and revolving shaft 4 are mutual orthogonal to the point \(q_{2}\). In addition, the plane where the axis 1 and axis 2 are located is parallel to the plane where the revolving shaft 3 and revolving shaft 4 are located. If a point lies on the axis of rotation, its coordinates remain unchanged, no matter how it rotates around the axis of rotation. Hence, for the intersection point \(q_{2}\):  
\[ \begin{split} q_{2}=\begin{bmatrix}L_{1}+L_{2}&0&0\end{bmatrix}^ {T}\\ e^{\theta_{3}\widehat{\xi}_{3}}e^{\theta_{4}\widehat{\xi}_{4}}q_{2}=q_{2} \end{split}  \tag{10}\]  
Set  
\[ \begin{split}\widehat{\xi}_{T}g(\theta)\widehat{\xi}_{T}g(0)^{-1} =e^{\theta_{1}\widehat{\xi}_{1}}e^{\theta_{2}\widehat{\xi}_{2}}e^{\theta_{3} \widehat{\xi}_{3}}e^{\theta_{4}\widehat{\xi}_{4}}=g_{1}\end{split}  \tag{11}\]  
Then  
\[q=e^{\theta_{1}\widehat{\xi}_{1}}e^{\theta_{2}\widehat{\xi}_{2}}e^{\theta_{3} \widehat{\xi}_{3}}e^{\theta_{4}\widehat{\xi}_{4}}q_{2}=e^{\theta_{1}\widehat{ \xi}_{1}}e^{\theta_{2}\widehat{\xi}_{2}}q_{2}=g_{1}q_{2}= \begin{bmatrix}q_{x }&q_{y}&q_{z}\end{bmatrix} ^{T} \tag{12}\]  
Figure 8: The operating space. (**a**) Three-dimensional diagram; (**b**) _XOY_ planar projection; (**c**) _YOZ_ planar projection; (**d**) _XOZ_ planar projection. Therefore, \(\theta_{2}\) can be calculated through the Paden-Kahan subproblem 2. As shown in Figure 9, point \(p=q_{2}=\left(L_{1}+L_{2}\quad 0\quad 0\right)\) rotates by the angle \(\theta_{2}\) around the axis \(\xi_{2}\) to point \(p_{1}\); then, point \(p_{1}\) rotates by the angle \(\theta_{1}\) around the axis \(\xi_{1}\) to point \(p\). Take a point \(t_{1}\) located on the axis \(\xi_{1}\) and a point \(t_{2}\) located on the axis \(\xi_{2}\). \[t_{1}= \begin{bmatrix}L_{1}&0&1\end{bmatrix} t_{2}= \begin{bmatrix}L_{1}&1&0 \end{bmatrix}  \tag{13}\]  
Based on the principle of keeping distance constant:  
\[\left\|e^{\theta_{2}\cdot\widehat{\xi}_{2}}p-t_{1}\right\|=\left\|q-t_{1}\right\| \tag{14}\]  
Then, define  
\[ \begin{split}\delta=\left\|q-t_{1}\right\|,u=p-t_{2},v=t_{1}-t_{ 2}\\ u^{\prime}=u-\omega_{2}\omega_{2}{}^{T}u,v^{\prime}=v-\omega_{2}\omega_{2}{} ^{T}v,\delta^{\prime 2}=\delta^{2}-\left\|\omega_{2}{}^{T}(p-t_{1})\right\|\end{split}  \tag{15}\]  
where \(u^{\prime},\ v^{\prime}\), and \(\delta^{\prime}\) are the projections of \(u,\ v\) and \(\delta\) on the plane perpendicular to the axis \(\xi_{2}\)  
\[ \begin{split}\theta_{0}&=a\tan 2(\omega_{2}^{T}(u^{ \prime}\times v^{\prime}),u^{\prime T}v^{\prime})\\ \theta_{2}&=\theta_{0}\pm ar\cos(\frac{\left\|u^{ \prime}\right\|^{2}+\left\|v^{\prime}\right\|^{2}-\delta^{2}}{2\left\|u^{ \prime}\right\|\left\|v^{\prime}\right\|})\end{split}  \tag{16}\]  
where \(\theta_{0}\) is the angle between vectors \(u^{\prime}\) and \(v^{\prime}\)  
\[ \begin{split} u&=\begin{bmatrix}q_{x}-L_{1}&q_{y}&q_{z}-1\\ \end{bmatrix}^{T}\\ v&=\begin{bmatrix}0&-1&1\\ \end{bmatrix}^{T}\\ \omega_{2}&=\begin{bmatrix}0&1&0\end{bmatrix}^{T}\end{split}  \tag{17}\]  
Having determined the angle \(\theta_{2}\), \(\theta_{1}\) can be calculated through the Paden-Kahan subproblem 1, as shown in Figure 10.  
\[ \begin{split} p_{1}=e^{\theta_{2}\cdot\widehat{\xi}_{2}}p= \begin{bmatrix}p_{x}&p_{y}&p_{z}\end{bmatrix}\\ e^{\theta_{1}\cdot\widehat{\xi}_{1}}q_{2}=p_{1}\end{split}  \tag{18}\]  
Then  
\[ \begin{split} u&=p_{1}-r=\begin{bmatrix}p_{x}-L_{1}&p_{y}&p_{z} \end{bmatrix}\\ v&=q-r=\begin{bmatrix}L_{2}&0&0\end{bmatrix}^{T}\\ \omega_{1}&=\begin{bmatrix}0&0&1\end{bmatrix}^{T}\end{split}  \tag{19}\]  
Figure 9: (**a**) The spiral motion of joints \(1\) and \(2\); (**b**) projection of the spiral motion. where  
\[