## 4 Discussion  
Minimally invasive robotic surgery systems, such as the da Vinci Surgical System, have led to significant advancements across a range of surgical procedures. However, thyroid surgery presents unique challenges due to its confined workspace and the high precision required. These spatial constraints demand robotic systems that can navigate tight spaces with exceptional flexibility while maintaining the rigidity necessary for precise instrument control--a requirement often unmet by traditional continuum robots. To address these challenges, this study proposes introducing a novel robotic system designed specifically for thyroid surgery, combining a modular structure with multiple degrees of freedom (DOF) and a closed-loop cable-driven mechanism. This hybrid system offers both flexibility and rigidity, but also introduces significant kinematic analysis challenges, particularly in handling model inaccuracies caused by manufacturing and assembly errors. Studies have shown that parameter identification can effectively solve the precision issues caused by manufacturing errors in robots [32; 33; 34]. However, most of these studies focus on serial robots, and research on flexible robots is relatively limited. This is mainly due to the difficulty of directly measuring manufacturing errors, as well as the fact that most parameter identification methods are model-dependent, which adds further complexity. Figure 14: Trajectory of the slave manipulator arm’s end-effector. The color lines represent the coordinate system established by the ArUco Marker, with red for the X-axis, green for the Y-axis, and blue for the Z-axis. The red dashed line, red ellipse, and red triangle depict the trajectory of the end-effector. Figure 15: Master–slave grasping experiment. (**A**–**D**) illustrate the four distinct phases of interaction between the end-effector of the slave manipulator arm and a ring during a ring grasping experiment. (**A**) The end-effector of the manipulator arm approaches the ring, initiating the grasping motion. (**B**) The end-effector makes contact with the ring, preparing to secure a grip. (**C**) The end-effector has successfully grasped the ring, demonstrating the effectiveness of the grasping action. (**D**) With the ring securely grasped, the end-effector begins to lift it from the cylinder, ready to move the ring to the designated location. Additionally, the coupling effects between the joints of continuum robots (as illustrated in Figure 2d) make parameter identification even more challenging. The robot discussed in this paper combines continuum and rigid robotic elements, allowing us to apply some analysis methods typically used for rigid robots. Given this advantage, we were able to investigate the joint coupling problem and proposed a decoupling method. The experiment in Section 3.1.2 demonstrates that the proposed decoupling method is effective. Based on this, we applied a parameter identification algorithm to correct the 19 parameters of the robot's kinematic model outlined in Section 2.1. Research in [32] indicates that the impact of different identification methods (such as genetic algorithms and Particle Swarm Optimization) on robot errors is within 2%. For continuum robots, factors like cable friction and hysteresis have a more significant impact on accuracy. Thus, we adopted the Particle Swarm Optimization (PSO) algorithm, which offers relatively better performance, for parameter identification. Following the parameter identification scheme outlined in Section 2.3, the input data for parameter identification include the robot's actual joint angles and motor rotation angles. The output is the robot's model parameters. The actual motor rotation angles can be directly measured, while the predicted motor angles are calculated using Equation (6), which depends on the 19 parameters being optimized and the joint angles of the robot. The initial values of the 19 parameters were obtained using Solidworks software.