## 2 Materials and Methods
### Closed-Circuit Cable-Pulley System Transmission Backlash and Grasping Force Estimation
#### 2.3.1. Cable Elongation and Grasping Force of the Cable-Pulley System during the Working Phase
\begin{array}{l}\frac{M_{in}\kappa_{1}\kappa_{2}}{ EAR_{0}\mathbb{R}_{5}(\kappa_{11}+\kappa_{12})}+\frac{(\mathbb{r}_{par}^{b}L_{ end}^{b}\kappa_{11}-\mathbb{r}_{par}^{f}L_{total}^{f}\kappa_{12})}{ EARS_{0}(\kappa_{11}+\kappa_{22})},\ if\text{sgn}(\dot{\theta}_{in})=1\\ \frac{M_{in}\kappa_{2}\kappa_{22}}{EAR_{0}\mathbb{R}_{5}(\kappa_{21}+\kappa_{22 })}+\frac{(\mathbb{r}_{par}^{b}L_{total}^{b}\kappa_{21}-\mathbb{r}_{par}^{f}L_{ total}^{f}\kappa_{22})}{EARS_{0}(\kappa_{21}+\kappa_{22})},\ other\end{array} \right., \tag{45}\]  
Substituting Equation (45) into Equation (38), the output displacement of the cable-pulley system can be expressed as:  
\[\left\{ \begin{array}{l}\theta_{out}=\frac{\theta_{in}R_{0}}{\mathbb{R}_{5}}- \left(\frac{M_{in}\kappa_{1}\kappa_{12}}{EAR_{0}\mathbb{R}_{5}(\kappa_{11}+ \kappa_{22})}+\frac{(\mathbb{r}_{par}^{b}L_{total}^{b}\kappa_{11}-\mathbb{r}_{ par}^{b}L_{total}^{f}\kappa_{12})}{EARS_{0}(\kappa_{11}+\kappa_{12})} \right),\ if\text{sgn}(\dot{\theta}_{in})=1\\ \theta_{out}=\frac{\theta_{in}R_{0}}{\mathbb{R}_{5}}-\left(\frac{M_{in}\kappa_{ 21}\kappa_{22}}{EAR_{0}\mathbb{R}_{5}(\kappa_{21}+\kappa_{22})}+\frac{( \mathbb{r}_{par}^{b}L_{total}^{b}\kappa_{21}-\mathbb{r}_{par}^{b}L_{total}^{b} \kappa_{22})}{EARS_{0}(\kappa_{21}+\kappa_{22})}\right),\ other\end{array} \right., \tag{46}\]  
3.2 Cable Elongation and Grasping Force of the Cable-Pulley System during the Delay and Transition Phases  
To derive the formulas of the elongation of the cable-pulley system during the delay and transition phases according to Equations (21) and (36)-(38), it is necessary to solve the 4 th nonlinear equations, which are very complicated and difficult to solve. In order to simplify the analysis process, the cable elongation and grasping force of the cable-pulley system during the delay phase and the transition phase are taken into consideration in a unified way. When at time \(t_{0}\), the driving pulley transitions from the counterclockwise working phase to the delay and transition phases, if and only if the direction of the torque \(M_{in}(t_{1})\) driving the driving pulley changes, and the direction of the speed \(\dot{\theta}_{in}(t_{1})\) of the driving pulley does not change, and the tension on both sides of the driven pulley satisfies \(T_{4,5}^{t}(t_{0}-1)>T_{4,5}^{b}(t_{0}-1)\), then the cable-pulley system is in the delayed and transitional phases. At this time, the output displacement and output torque of the driven pulley remain unchanged, that is, \(\theta_{out}(t_{0})\) and \(M_{out}(t_{0})\). Using a similar analysis, a similar conclusion can be inferred when the driving pulley changes from a clockwise working phase to the delay and transition phases. According to the above analysis, the calculation formulas of the output displacement and torque of the cable-pulley system during the entire transmission phase can be obtained as:  
\[\left\{ \begin{array}{l}M_{out}(t_{1})=(1-flag)M_{out}(t_{1})+M_{out}(t_{0}) flag\\ \theta_{out}(t_{1})=(1-flag)\theta_{out}(t_{1})+\theta_{out}(t_{0})flag\\ flag=\Big{(}1-\text{ssgn}\Big{(}\dot{\theta}_{in}(t_{1})M_{in}(t_{1})\Big{)} \Big{)}\text{ssgn}\Big{(}\dot{\theta}_{in}(t_{1})\Big{(}T_{4,5}^{t}(t_{1})-T_ {4,5}^{b}(t_{1})\Big{)}\Big{)}M_{out}(t_{0})\end{array} \right., \tag{47}\]  
where \(M_{out}(t_{0})\) and \(M_{out}(t_{1})\) are the output torque of the cable-pulley system at time \(t_{0}\) and \(t_{1}\), respectively; \(\theta_{out}(t_{0})\) and \(\theta_{out}(t_{1})\) are the output displacements of the cable-pulley system at time \(t_{0}\) and \(t_{1}\), respectively; and the angular velocity of the driving pulley, \(\text{ssgn}(. )=0.5+0.5\) sgn(.) is a switch function. Combining Equations (25) and (47), the output grasping torque \(M_{grasp}\) can be calculated as:  
\[M_{grasp}=M_{out}-M_{fric}, \tag{48}\]