## 5 Results and Discussion  
Based on the layout and size chain relationship of the drafting mechanism, the position coordinates of the end of the hook needle in the coordinate system of the rope drive could be obtained as \(\left[1.6\text{ mm }\quad 1.2\text{ mm }\quad-1.6\text{ mm}\right]\). Then, the inverse kinematics solution algorithm provided was used to obtain the rotation angles of each joint of the rope drive at this position. Four sets of solutions, shown in Table 1, were obtained. According to the rotation angle calculated earlier, only one set of solutions satisfied the requirements. We substituted this set of solutions into the forward kinematics solution equation to obtain the end pose of the rope drive. \[ \begin{split}\theta_{1}=\quad 29.978812\quad\theta_{2}=29.998971\quad \theta_{3}=30.013216\quad\theta_{4}=29.987864\\ \widehat{\xi}g(\theta)=e^{\theta_{1}\widehat{\xi}}_{1}e^{\theta_{2 }\widehat{\xi}}_{2}e^{\theta_{3}\widehat{\xi}}_{3}e^{\theta_{4}\widehat{\xi}} _{4}\widehat{\xi}_{3}\widehat{\xi}(0)=\left[\begin{array}{ccc}0.129487&-0.8 08012&0.574759&1.625538\\ 0.574759&0.533493&0.620512&1.209326\\ -0.808012&0.250000&0.533493&-1.616612\\ 0.00000&0.00000&0.000000&1.000000\end{array}\right]\end{split}  \tag{21}\]  
By comparison, the calculated data and the end coordinates were basically consistent. Therefore, it can be concluded that the forward and inverse kinematics analysis algorithm based on screw theory proposed in this paper has good numerical stability and accuracy. At the same time, the results also verified that the weaving mechanism designed in this study could complete the basic steps of weaving. For the dynamic simulation analysis of the weaving mechanism, three-dimensional models of the rope drive and the hook were established and then imported into the software Adams. The specific simulation steps were as follows: first, the constrains of each part were \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\theta_{1}\) & **(Degree)** & \(\theta_{2}\) & \(\theta_{3}\) & \(\theta_{4}\) \\ \hline  & 29.978812 & 29.998971 & 30.013216 & 29.987864 \\ \hline  & 209.988781 & \(-209.987698\) & 30.013216 & 29.987864 \\ \hline  & 29.978812 & 29.998971 & \(-149.889769\) & \(-209.99878\) \\ \hline  & 209.988781 & \(-209.987698\) & \(-149.889769\) & \(-209.99878\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The calculated rotation angles of the joints. Figure 10: (**a**) The spiral motion of joint 1; (**b**) projection of the spiral motion. defined based on the designed weaving mechanism; second, the drive for each joint was set, and the direction of movement was adjusted properly; third, the calculated motion parameters for the weaving mechanism derived from forward and inverse kinematics were imported to the editor of Adams; fourth, we set the points at the end of the lead system and the hook as the target, then started the drive to obtain the trajectory coordinates of the target points. Last, the trajectory coordinates were processed through software Matlab2019 to draw a three-dimensional map of the trajectory, as shown in Figure 11. The red line and blue line represent the end trajectories of the lead system and the hook, respectively. The lead system runs from the initial position A to position B, and the hook runs from the initial position C to position B. The trajectories converge at point B to complete the hook work, which verified the accuracy of the forward and inverse solutions calculated through screw theory. In addition, the dynamic simulation also indicated the rationality and feasibility of the designed weaving mechanism in small, enhanced grids. Through trajectory simulation analysis of the model, the coordinates of the intersection point B located in the trajectory of the weaving mechanism could be obtained as \(\left[1.60928\text{ mm }\quad 1.21097\text{ mm }\quad-1.59492\text{ mm}\right]\) and were basically consistent with the calculated results of forward and inverse kinematics.