## 2 Methods  
### Decoupling Model for Motion of Multi-Joints  
The minimally invasive thyroid surgery robot (Figure 3) features a six-degree-of-freedom arm, including rotary, shoulder, elbow, and wrist joints, driven by a combination of gear transmission and a closed-loop wire system [21]. This design mimics the structure and mobility of the human arm. The robot's kinematic model utilizes three-space mapping theory for cable-driven technology, integrating a  
Figure 3: Joint and drive design and kinematic model [21]. (**a**). Motion model for continuum manipulator. (**b**). Constant curvature assumption. (**c**). Single joint closed-loop cable-driven model. Figure 2: Decoupling model for motion of multi-joints. (**a**). Coordinate system schematic. (**b**). Rigidly hinged joint. (**c**). The torques exerted on the remaining joints when shoulder joint 1 rotate. (**d**) The coupling effect between multiple joints. constant curvature model for the wrist joints and a closed-loop wire drive model for the shoulder and elbow joints (Figure 3c). This approach ensures precise control over the position and orientation of the end effector. For the hand-driven arm utilizing wire drive technology, the driving wire will pass through multiple joints, leading to the issue of motion coupling between the joints (Figure 2d). Table 1 illustrates the relationship of motion coupling between joints. Rotation of the proximal joint affects distal joints, causing their rotation, but rotation of a joint by gear transmission does not influence other joints. Table 1 summarizes the effects of motion coupling between the joints. According to the closed-loop wire model, if the rotation angle of the transmission wheel is \(\theta_{d}\) and the rotation radius is \(R\), the following relationships hold:  
\[\Delta L=\frac{\theta_{d}\pi R}{180} \tag{1}\]  
When the driving motor rotates clockwise, the joint also rotates clockwise. Let the clockwise rotation of the joint be defined as positive and use the rotation function \(\text{sgn}(\theta)\) to represent the direction of joint rotation. The following relationship holds:  
\[\text{sgn}(\theta)=\left\{ \begin{array}{l}1,\theta>0\\ 0,\theta=0\\ -1,\theta<0\end{array} \right. \tag{2}\]  
As shown in Figure 2c, when shoulder 1 (joint 2) rotates by an angle \(\theta_{2}\), the remaining distal joints exhibit coupled motion. The variation in the drive wire length for the \(i\)-th joint, as well as the coupled drive wire length variation from the \(i\)-th joint to the \(j\)-th joint, can be expressed as follows:  
\[\Delta L_{\text{d}ii}=2h_{i}\cdot\sin(\theta_{i}/2)+\text{sgn}(\theta_{i})(2 r-2r\cdot\cos(\theta_{i}/2)) \tag{3}\]  
\[\Delta L_{2j}=2h_{ij}\cdot\sin(\theta_{i}/2)+\text{sgn}(\theta_{i})(2r-2r \cdot\cos(\theta_{i}/2)),(j=3,4,5,6) \tag{4}\]  
Let the joint angle vector \(\theta=[\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5},\theta_{6}]^{T}\) represent the angular change of each joint and the driving angle vector \(\theta_{d}=[\theta_{d1},\theta_{d2},\theta_{d3},\theta_{d4},\theta_{5}, \theta_{6}]^{T}\) represent the angular change of each reducator motor. The vector of steel wire length variation, \(\Delta L=[0,\Delta L_{d2},\Delta L_{d3},\Delta L_{d4},\Delta L_{d5},\Delta L_{ d6}]^{T}\), corresponds to the change in steel wire length for