## 3 Experiments and Results
### Measurement of the Output Displacement and Torque of the Distal Grasper of the Cable-Pulley System
By comparing the model output force and the actual measurement output force, the maximum error of the output force and position of the cable-pulley system still reach 4.84 degrees and 0.081 N\(\cdot\)m, but the maximum error of the output force and position occurs during the release of a distal grasper of the cable-pulley system; that is, the moment when the direction of motion of the driving pulley changes. During the grasping process of the distal grasper of the cable-pulley system, the maximum error of the output force does not exceed 0.53 N, so the output force of the model can be used as the force saturation limit threshold. The accuracy of the displacement and output torque estimation of the cable-pulley system is not high; it is mainly caused by the identification errors of the model's parameters, the measurement errors, and the modeling errors introduced by neglecting the friction between the driving pulley and its support rolling bearing. Figure 11: Comparison of the output displacement and torque estimation of the cable-pulley system with measured data: (**a**) The estimated output displacement of the distal grasper of the cable-pulley system; (**b**) The estimated output torque of the distal grasper of the cable-pulley system. Position and Grasping Force Control of the Distal Graspers of the Surgical Instrument with Transmission Backlash Compensation  
The surgical instrument grasping experiment is shown in Figure 13. During the experiment, the initial pretension of the cable that is driving the distal two graspers of the surgical instrument is adjusted to 25 N, and three markers of the NDI Optotrak system (with a measurement accuracy of 0.1 mm) are pasted on the wrist of the surgical instrument and the distal two graspers, respectively. The movement state of the graspers is obtained by measuring the angle between the two graspers. Figure 14a shows the comparison curve between the measured values and the expected values of the output position of the distal graspers of the surgical instrument with transmission backlash compensation. By compensating for transmission backlash, the measured output positions of the graspers are very close to the desired output values, and the trend change of the measured values is consistent with the desired values. The maximum position error between the measured values and the expected values is 3.49 degrees and the average position error is 1.03 degrees, which indicates that the proposed transmission backlash compensation algorithm can significantly improve the position control accuracy of surgical instruments. Figure 12: Output displacement and torque curves of the distal grasper with position compensation: (**a**) Output displacement of the distal grasper with position compensation; (**b**) Output torque of the distal grasper with position compensation. Figure 13: Validation of the transmission backlash compensation for the surgical instrument. Although the deviations between the measured values and the estimated values of the distal graspers cannot be evaluated by installing a force sensor on the distal graspers of the surgical instrument, the deviations between the desired values and the estimated values of the output torque of the distal graspers can be analyzed. Figure 14b shows the comparison curve between the desired values and estimated values of the output torque of the distal graspers of the surgical instrument with transmission backlash compensation. The estimated values of the grasping torque are very close to the desired output values, and the trend change of the two is consistent. The maximum error between the estimated values and the desired values of the grasping torque shall not exceed 0.025 N.m. That is, for the distal grasper, with a length of 15 mm, the maximum grasping force error between the estimated values and the desired values shall not exceed 1.47 N. Therefore, the estimated grasping torque can be used as a torque saturator to limit the excessive grasping force exerted by the graspers, so as to avoid soft tissue damage.