## 2 Materials and Methods
### Single-Cable-Pulley Transmission
\begin{array}{l}T(\theta)=T_{1}\lambda(\rho,\mu,\theta)\\ =T_{1}\left(\begin{array}{l}\frac{-1-\rho+\sqrt{(1+\rho)^{2}+4\mu^{2}\rho}} {2\sqrt{(1+\rho)^{2}+4\mu^{2}\rho}}\exp\left(\frac{-1-\rho-\sqrt{(1+\rho)^{2}+ 4\mu^{2}\rho}}{2\mu}\theta\right)\\ +\frac{1+\rho+\sqrt{(1+\rho)^{2}+4\mu^{2}\rho}}{2\sqrt{(1+\rho)^{2}+4\mu^{2} \rho}}\exp\left(\frac{-1-\rho+\sqrt{(1+\rho)^{2}+4\mu^{2}\rho}}{2\mu}\theta \right)\end{array}\right)\end{array} , \tag{12}\]  
Since the tension ratio on both sides of the pulley is related to the direction of the cable moving on the pulley, it is defined that:  
\[T_{2}=\left\{ \begin{array}{l}T_{1}/\lambda(\rho,\mu,\theta_{slip}),\;if\; \text{sgn}(\dot{\theta}_{in})=1\\ T_{1}\lambda(\rho,\mu,\theta_{slip})\;,\;\text{other}\end{array} \right., \tag{13}\]  
As the cable subjected to tensile force is elastically elongated, according to [PERSON]'s law, the strain generated in the infinitesimal cable segment can be expressed as:  
\[\varepsilon(\theta)=\frac{d\delta}{Rd\theta}=\frac{T(\theta)}{EA}\;\Rightarrow \;d\delta=\frac{T(\theta)R}{EA}d\theta, \tag{14}\]  
where \(T(\theta),\varepsilon(\theta)\), and \(d\delta\) are the tension, stress and strain of the infinitesimal cable segment at position \(\theta\), respectively. As the cable is preloaded by the tension \(T_{pre}\), the cable elongation caused by the change of the outcoming force \(T_{pre}\) can be formulated as:  
\[\Delta L=L_{T_{2}}-L_{T_{pre}}, \tag{15}\]  
Figure 3a is the tension distribution diagram of the cable in the process of loading. Combining Equations (12)-(14), the following can be formulated as:  
\[\begin{array}{l}\Delta L=\int_{0}^{\theta_{slip}}\frac{T(\theta)R}{EA}d \theta-\frac{T_{pre}R}{EA}\theta_{slip}=\left\{\begin{array}{l}\frac{T_{1} R\varphi(\theta_{I},\theta_{slip})}{EA(\rho,\mu,\theta_{slip})}-\frac{T_{pre}R}{EA} \theta_{slip},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\  
Figure 3b is the tension distribution diagram of the cable in the process of unloading. The outcoming tension is firstly increased from \(T_{1}\) to \(T_{pk}\), and then gradually reduced to \(T_{2}\). At this time, the cable only partially moves on the pulley.  
For the cable between the intervals \(\theta_{slip}-\alpha\leq\theta\leq\theta_{slip}\), the following is obtained from the Equation (12) as:  
\[T_{b}=T_{pk}/\lambda(\rho,\mu,\alpha)=T_{2}\lambda(\rho,\mu,\alpha)\ \Rightarrow T_{b}=\sqrt{T_{pk}T_{2}}, \tag{17}\]  
\[\lambda(\rho,\mu,\alpha)=\sqrt{T_{pk}/T_{2}}, \tag{18}\]  
From Equation (15), the cable elongation in the sliding interval \(\theta_{slip}-\alpha\leq\theta\leq\theta_{slip}\) can be expressed as:  
\[\Delta L_{\alpha}=\left\{ \begin{array}{ll}\frac{T_{pk}\rho(\rho,\mu, \alpha)}{E\Lambda(\rho,\mu,\alpha)}-\frac{T_{pr}R}{E\Lambda}\alpha,\ if\ \text{sgn}(\dot{\theta}_{in})=1\\ \frac{T_{pk}R}{E\Lambda(\rho,\mu,\alpha)}-\frac{T_{pr}R}{E\Lambda}\alpha,\ other\end{array} \right., \tag{19}\]  
where \(\text{sgn}(\dot{\theta}_{in})\) denotes the direction of the cable moving on the pulley; if the cable moves counterclockwise on the pulley, then \(\text{sgn}(\dot{\theta}_{in})=1\), otherwise \(\text{sgn}(\dot{\theta}_{in})=0\). From Equation (15), the cable elongation in the sliding interval \(0<\theta\leq\theta_{slip}-\alpha\) is:  
\[\Delta L_{(\theta_{slip}-\alpha)}=\left\{ \begin{array}{ll}\frac{T_{pk}R \varphi(\rho,\mu,\left(\theta_{slip}-\alpha\right))}{E\Lambda(\rho,\mu, \left(\theta_{slip}-\alpha\right))}-\frac{T_{pr}R}{E\Lambda}\alpha(\theta_{ slip}-\alpha),&if\ \text{sgn}(\dot{\theta}_{in})=1\\ \frac{T_{pk}R}{E\Lambda}\sigma\left(\rho,\mu,\left(\theta_{slip}-\alpha \right)\right)-\frac{T_{pr}R}{E\Lambda}(\theta_{slip}-\alpha),&\ other\end{array} \right., \tag{20}\]  
From Equations (19) and (20), the total elongation of the cable can be calculated as:  
\[\Delta L=\Delta L_{a}+\Delta L_{(\theta_{slip}-\alpha)}\] \[=\left\{