## 3 Experiments and Results  
### Experimental Setup  
In order to validate the tension and displacement transmission characteristics of the closed-circuit cable-pulley system, a dedicated experimental platform similar to the transmission structure of the end effector of surgical instruments was designed. As shown in Figure 8, the dedicated experimental platform includes a closed-loop cable-pulley system, a motion control system, and a displacement and force measuring system, in which a maxon motor is driven by an elmo driver to drive the driving pulley, a Gmax controller using the EtherCat protocol is used to realize the communication between the industrial computer and the elmo driver, and a digital dynamometer is used to measure the pretension of the cable-pulley system. The NDI Polaris system (Northern Digital Inc., Waterloo, ON, Canada) was used to measure the output displacement of the cable-pulley system, and the ATI six-dimensional force sensor (Analytical Technology Inc., Collegeville, PA, USA) was used to measure the output torque of the distal grasper of the cable-pulley system. The device's specific description is listed in Table 1.  
Figure 8: Experiment to investigate the transmission characteristics of a cable-pulley system. Figure 7: Control flow of the distal grasper of the cable-pulley system. Since the slip angle of the cable moving along the pulley does not exceed its wrap angle, when the pulley cannot rotate, the slip angle of the cable is equal to its wrap angle, and the tension ratio on both sides of the pulley can be calculated from Equation (13). When the pulley is rotatable, the slip angle of the cable on the pulley must be smaller than the wrap angle, and the tension ratio on both sides of the pulley can be calculated from Equation (29). The slip angle of cable \(T\) and cable \(B\) moving along the pulleys, and the tension ratios on both sides of the pulleys are, respectively, calculated and listed in Table 2.  
It can be seen from the data in Table 3 that when the pulley is fixed or rotatable, the tension ratio on both sides of the pulley changes greatly. According to Equation (30), for a given idler pulley, the shaft friction coefficient is substantially the same, and the idler pulley radius is not much different, so the tension ratio on both sides of the idler pulley is substantially close. Knowing the ratio of the tension on both sides of the pulley, it is very difficult to solve the sliding angle of the cable moving along the pulley from Equation (12), which can be solved by the classical Capstan equation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Part** & **Specification** & **Part** & **Specification** \\ \hline \multirow{2}{*}{Medical steel cable} & 7 \(\times\) 7 mm, \(\Phi\)0.54 mm, & Optical positioning & \multirow{2}{*}{NDI Polaris} \\  & Carl Stahl Corp. & and tracking system & & \\ \multirow{2}{*}{ELMO driver} & ELMO Corp. & Six-dimensional force & \multirow{2}{*}{ATI min45} \\  & ELMO Corp. & sensor & \\ \multirow{2}{*}{Gmax controller} & EtherCAT protocol, & \multirow{2}{*}{Cable-pulley system} & Single-degree-of-freedom \\  & ELMO Corp. & & \\ \multirow{2}{*}{DCX 16S motor} & Reduction ratio 44:1, & Digital dynamometer & \multirow{2}{*}{Range 200 N} \\  & maxon Corp. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The specific description of the device used in the dedicated experimental platform.