## 3 Experiments and Results
### Measurement of the Output Displacement and Torque of the Distal Grasper of the Cable-Pulley System  
Because there is no position encoder installed on the distal grasper of the cable-pulley system, its angle displacement needs to be measured by an NDI optical stereo positioner. The coordinate system of the mark points of the NDI optical stereo positioner is shown in the lower right corner of Figure 9. The positions of the two markers located on the distal grasper are represented by \(A\) and \(B\), respectively. The initial position of the grasper at time \(t_{0}\) is represented by vector \(\overset{\rightarrow}{d}=<A_{0},B_{0}>\), \(A_{0}=(x_{0}^{0},y_{0}^{0},z_{0}^{0})\) and \(B_{0}=(x_{0}^{1},y_{1}^{1},z_{1}^{0})\). The position of the distal grasper at time \(t\) is represented by vectors \(\overset{\rightarrow}{b}=<A_{t},B_{t}>\), \(A_{t}=(x_{0}^{t},y_{0}^{t},z_{0}^{t})\) and \(B_{t}=(x_{1}^{t},y_{1}^{t},z_{1}^{t})\). The expressions for calculating the angle displacement of the distal grasper using vector algebra are as follows:  
\[\alpha=\cos^{-1}\left(\frac{\overset{\rightarrow}{a}\overset{ \rightarrow}{b}}{\left|\overset{\rightarrow}{a}\right|\left|\overset{ \rightarrow}{b}\right|}\right)=\cos^{-1}\left(\frac{a_{x}b_{x}+a_{y}b_{y}+a_{ z}b_{z}}{\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}\sqrt{b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}} \right), \tag{50}\]  
\[a_{x}=x_{1}^{0}-x_{0}^{0},\ a_{y}=y_{1}^{0}-y_{0}^{0},\ a_{z}=z_{1}^{0}-z_{0}^{0 }\approx 0,\ b_{x}=x_{1}^{t}-x_{0}^{t},\ b_{y}=y_{1}^{t}-y_{0}^{t},\ b_{z}=z_{1}^{t}-z_{0}^{t}\approx 0\]  
During the experiment, the distal grasper is driven to perform a periodic motion such as a sinusoidal curve, which is a free motion in half cycle time (the exerted resistance torque is zero), and is subjected to the working surface of the force sensor used to measure the output force of the distal grasper in the other half cycle time. In order to verify the relationship between the output displacement and the torque of the cable-pulley system and its exerted pretension, the driving pulley driven by the motor moves along the sinusoid as \(\theta(t)=1.74\ sin(2\pi\acute{t}),f=12.5\) Hz. Figure 10 shows the transmission hysteresis and torque output curves of the cable-pulley system when the pretension exerted on the cable is 12.5 N, 50 N and 150 N, respectively. In addition, Table 4 shows the displacement and torque transmission errors of the cable-pulley system when the pretension is 50 N. In theory, the bigger the pretension exerted upon the cables, the smaller the transmission backlash of the cable, and the closer the output torque is to the desired output torque of the cable-pulley system; however, the actual effect is not obviously consistent with the theoretical analysis results, because the greater the pretension, the greater the friction of the transmission circuit, and the more obvious the nonlinearity transmission. Therefore, the selection of the pretension needs to weigh the influence of the pretension on the friction force. Figure 10a shows that with the three pretensions exerted upon the cables, the transmission backlash of the cable-pulley system is very large, which is close to the reference output displacement value. Figure 10b shows that, compared with the expected output value, the maximum error and the average error of the output torque with  
Figure 9: Measurement of the angle displacement of the end effector using an NDI locator. a pretension of 150 N are the largest, so the friction force of the cable-pulley system must be correspondingly compensating according to the pretensions. According to Table 4, the average error of the position transmission of the cable-pulley system applied the pretension of 50 N and reached 0.1368 radian (equal to 7.84 degrees), and the maximum transmission error of the cable-pulley system even reached 0.323 radian (equal to 57 degrees) when the direction of the driving wheel drive torque changes.