## 3 Experiments and Results
### Measurement of the Output Displacement and Torque of the Distal Grasper of the Cable-Pulley System
For the distal grasper, with a length of 15 mm, the average error and maximum error of the output force of the cable-pulley was 2.7 N and 13.4 N, respectively, which indicates that the displacement and force transmission accuracy of the cable-pulley system was very low; and without visual servo control by the operator, the operation cannot be completed accurately. In order to validate the displacement and torque transmission characteristics of the proposed model, a distal grasper driven by a pulley was designed for grasping operation experiments. During the experiment, the tensile modulus of the cable was 1684 N.m, and the pretension of the two cable was 25 N. The driving pulley driven by the motor moves along the sinusoid as \(\theta(t)=1.74\ sin(2\pi\theta t),f=12.5\) Hz. The input torque of the cable-pulley system is the product of the motor current and the moment constant. The actual output displacement and force of the distal grasper of the cable-pulley system are measured by an NDI locator and force sensor, respectively. Due to the limited position of the distal grasper of the cable-pulley system, the NDI locator and force sensor can only measure the output displacement and output torque of the distal grasper in every half cycle time. Figure 11a is an estimated curve of the output displacement of the distal grasper of the cable-pulley system. The estimated output displacement is very close to the actual measured output displacement and has the same trend. The maximum position estimation error is 3.17 degrees. Since there are obvious errors between the desired output displacements and the measured outputs, and the maximum displacement error is 13.96 degrees, the cable-pulley system needs position compensation to eliminate transmission backlash. Figure 11b \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Pretension** & **MaxPosErr 1** & **AvgPosErr 1** & **MaxTorqueErr 1** & **AvgTorqueErr 1** \\
**(N-m)** & **(Rad)** & **(Rad)** & **(N-m)** & **(N-m)** \\ \hline
12.5 & 0.342 & 0.1436 & 0.196 & 0.041 \\
50 & 0.323 & 0.1368 & 0.182 & 0.052 \\
150 & 0.354 & 0.1538 & 0.201 & 0.073 \\ \hline \hline \end{tabular} 1 MaxPosErr (maximum displacement transmission error); AvgPosErr (average displacement transmission error); MaxTorqueErr (maximum torque transmission error); AvgTorqueErr (average torque transmission error).  
\end{table} 
Table 4: Transmission error of position and torque of cable pulley with the pretension of 50N. Figure 10: The displacement and torque transmission characteristics of cable-pulley systems exerting different pretensions: **(a)** Transmission backlash of the cable-pulley system exerting different pretensions; **(b)** Output torque of the cable-pulley system exerting different pretensions. is an estimated curve of the output torque of the distal grasper of the cable-pulley system. The estimated output torque is very close to the actual measured output torque and has the same trend of change. The maximum output torque estimation error is 0.109 N\(\cdot\)m. There is an obvious error between the desired output torques and the measured outputs. The maximum output torque error is 0.215 N\(\cdot\)m. For the distal grasper, with a length of 15 mm, the maximum output force error of the distal grasper can reach 14.3 N, so it is impossible to achieve precise force control operation of the distal grasper. Figure 12 shows the output displacement and output torque curves of the distal grasper of the cable-pulley system with position compensation. Figure 12a shows that, after position compensation, the output displacement of the cable-pulley system is very close to the desired output, and the transmission backlash of the cable-pulley system is significantly reduced. Figure 12b shows that, after the transmission backlash compensation, the output force of the cable-pulley system is very close to the desired output with the same change trend, and there is basically no delay between them.