## 3 Experiments
### Verification of Kinematic Model of Surgical Robot
#### 3.2.3 Experiments of Repetitive Positioning Accuracy of Continuum Robots  
Repetitive positioning accuracy of the robot's end-effector refers to the precision with which the robot can replicate its trajectory when performing the same task. This metric is a critical indicator of the robot's precision and stability. To validate the repetitive positioning accuracy (RP) of the continuum robot, an experimental platform was established using a high-precision optical localization system, as shown in Figure 12A. We employed the AP-STD-200 optical localization system, developed by Guangzhou Aimooe Technology Co. Ltd. (Guangzhou, China), with an error margin of less than 0.12 mm.  
Figure 11: Verification of parameter identification algorithm via numerical simulation. (**a**) Training curve of the fitness value for the Particle Swarm Optimization (PSO) algorithm. (**b**) Identified errors of parameters via PSO via numerical simulation.  
Figure 12: Measurement scheme for repetitive positioning accuracy of continuum robots. (**A**) Measuring platform. (**B**) Testing targets.  
The experiments on the repetitive positioning accuracy of the continuum robot were conducted in accordance with the GB/T 1.1--2009 standard. Four sets of measurement target points within the workspace of the slave manipulator arm were selected to cover the full range of motion: up, down, left, and right. These poses were sequentially labeled as Pose 1, Pose 2, Pose 3, and Pose 4, as shown in Figure 12B.  
The master robot computer controls the slave manipulator arm to sequentially reach the four sets of target points in the following order: P1\(\rightarrow\)P2\(\rightarrow\)P3\(\rightarrow\)P4. This procedure is repeated five times, and in each repetition, upon reaching each target point, the position parameters output by the optical navigation system's upper-level computer application are recorded. The repetitive positioning accuracy (RP) of the end-effector of the slave manipulator arm is then calculated as follows:  
\[RP=\bar{I}+3S_{i} \tag{22}\]  
\[\text{where }\ S_{i}=\sqrt{\frac{\sum\limits_{j=1}^{n}\left\langle l_{j}-\bar{l} \right\rangle^{2}}{n-1}},\ \bar{I}=\frac{1}{n}\sum\limits_{j=1}^{n}l_{j},\ \ l_{j}=\sqrt{\left(x_{j}-\overline{x}\right)^{2}+\left(y_{j}-\overline{y} \right)^{2}+\left(z_{j}-\overline{z}\right)^{2}},\ \ \overline{x}=\frac{1}{n}\sum\limits_{j=1}^{n}x_{j},\]  
\(\overline{y}=\frac{1}{n}\sum\limits_{j=1}^{n}y_{j},\ \overline{z}=\frac{1}{n}\sum \limits_{j}^{n}z_{j}\). \(x_{j},y_{j},z_{j}\) denotes the end-effector position of the robot at the \(j\)-th instance.  
The repetitive positioning accuracy of the manipulator arm, both with and without parameter identification, is illustrated in Figure 13. Before applying parameter identification, the repetitive positioning accuracy of the end-effector for the four sets of measurement target points was 9.62 mm, 3.03 mm, 3.93 mm, and 1.32 mm, respectively. After applying parameter identification to correct the robot's kinematic parameters, the repetitive positioning accuracy of the end-effector for the same four target points improved to 4.53 mm, 2.14 mm, 2.45 mm, and 0.91 mm, respectively. These results demonstrate a significant improvement in repetitive positioning accuracy, with the overall control error reduced to within 5 mm. The repetitive positioning accuracy improved on average by approximately 1.97 mm, with an average percentage improvement in repetitive positioning accuracy of approximately 37.74%.