## 3 Experiments
### Verification of Kinematic Model of Surgical Robot
#### 3.2.2 Verification of Parameter Identification Algorithm  
The nominal values of the thread-driven model parameters were initially obtained using SolidWorks. Artificial errors were then introduced into these parameters. Subsequently, 100 sets of joint angles were randomly selected within the joint limits of the robotic arm, serving as theoretical values. The corresponding theoretical motor angles were calculated based on these joint angles. The actual motor angles were computed using the nominal values, along with the introduced parameter errors. To identify the parameter errors, the Particle Swarm Optimization (PSO) algorithm was employed. The key parameters for the PSO algorithm were set as outlined in Table 4. The convergence curve of the fitness value over 160 iterations is presented, with convergence beginning around iteration 40. The final best fitness value achieved by the PSO algorithm was 0.074. The identified parameter errors are listed in Table 5 and Figure 11b.  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Parameter Name** & **Inertia Weight** & **Individual Learning Factor** & **Social Learning Factor** & **Random Number** & **Random Number** & **Number of Particles** & **Search Space Dimension** & **Number of Iterations** \\  & \(w\) & \(c_{1}\) & \(c_{2}\) & \(r_{1}\) & \(r_{2}\) & \(N\) & **D** & \(n\) \\ \hline Parameter Value & 0.5 & 2 & 2 & 0.8 & 0.6 & 100 & 19 & 160 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter settings for the Particle Swarm Optimization (PSO) algorithm.  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Parameter Name** &
\begin{tabular}{} \end{tabular} & **Individual Learning Factor** & **Social Learning Factor** & **Random Number Factor** & **Number Factor** & **Number of Particles** & **Search Space Dimension** & **Number of Iterations** \\  & \(w\) & \(c_{1}\) & \(c_{2}\) & \(r_{1}\) & \(r_{2}\) & \(N\) & **D** & \(n\) \\ \hline Parameter Value & 0.5 & 2 & 2 & 0.8 & 0.6 & 100 & 19 & 160 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter settings for the Particle Swarm Optimization (PSO) algorithm.  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Parameter Name** &
\begin{tabular}{} \end{tabular} & **Theoretical Value Value** & **Introduced Error** & **Errors of Identification** & **Theoretical Value** & **Errors of Identification** & **Actual Value** \\ \hline \hline \(d\) & 14 & 0.1 & 0.0903 & 14 & 0.006 & 13.994 \\ \hline \hline \(D\) & 28 & 0.1 & 0.0938 & 28 & 0.005 & 27.995 \\ \hline \(R\) & 3 & 0.1 & 0.0749 & 3 & 0.005 & 2.995 \\ \hline \(h_{2}\) & 2.18 & 0.1  
Building upon the visual angle measurement method developed in Section 3.2.1, parameter errors in the cable-driven model were identified through the PSO algorithm in Section 2.3. In this study, joint angle data were collected within the range of \([-40^{\circ},40^{\circ}]\), with a sampling interval of \(5^{\circ}\). Specifically, 17 sets of angle data were collected for each of the following joints: rotary joint, shoulder joint 1, shoulder joint 2, elbow joint, wrist joint 1, and wrist joint 2. In total, 102 sets of data were gathered. Meanwhile, the actual rotational angles of each drive motor were recorded in real-time by the TWinCAT3 software (3.1 Build 4024) during the angle collection process. The collected joint angle data were input into the parameter identification method outlined in Section 3.2 for parameter identification analysis. Results in Table 5 showed that the error identification values were consistent with the simulation results. Then, the actual values of the kinematic model (Equation (6)) are applied in Section 3.2.3 and the application of the robot is in Section 3.3.