## 7 Simulation
### Simulation and Analysis of Trajectory Planning
Line b shows that the finite-time trajectory planning method, although having a smaller overall error, exhibits noticeable singularity issues at 1.7 s, and the errors cannot be quickly eliminated after the trajectory planning task. Line c, representing the widely used proportional  
feedback method in space robot trajectory planning, shows no apparent singularity or oscillation issues and can effectively eliminate errors after the trajectory planning task. However, the planning accuracy and error convergence speed are lower than line a. Line a demonstrates that the proposed method can keep the trajectory planning error of the robot consistently at a low level and converge to zero at an earlier stage. As shown in Figure 10, line a represents the joint velocity planned by the proposed method and line b represents the joint velocity planned by the predefined time method proposed in reference [24]. It can be seen that the joint angular velocity obtained by the proposed trajectory planning method is bounded and continuous, and effectively reduces joint velocity. The joint trajectory obtained using the proposed method is shown in Figure 11.  
Figure 8: Position error in trajectory planning of end effector (no initial error). Figure 9: Attitude error in trajectory planning of end effector (no initial error). To visually demonstrate the differences in trajectory planning methods, a three-dimensional plot of the end effector's position trajectory is presented in Figure 12. Considering a scenario where there is a deviation between the initial position of trajectory planning and the expected initial position, with the expected initial joint angles and end effector pose consistent with Table 2, the actual initial position is \(\mathbf{r}_{\epsilon 0}^{\prime}=[1.01~{}0.72~{}0.83]^{\mathrm{T}}\) (m), the initial orientation is \(\mathbf{q}_{\epsilon 0}=[0.32~{}-0.70~{}0.64~{}0.00]^{\mathrm{T}}\), and the initial joint angles are \(\mathbf{q}=[0.83~{}0.83~{}0.83~{}0.83~{}0.83~{}0.83]^{\mathrm{T}}\) (rad). Each line in the figure illustrates the conver  
Figure 11: Joint angles planned by the proposed method. Figure 10: Joint angular velocities planned by the proposed method. (a) explicit-time trajectory planning method. (b) finite-time trajectory planning method. gence of errors. Line d, which does not use pose feedback, fails to eliminate the existing error. Among the trajectory planning methods based on pose feedback, line a, representing the proposed trajectory planning method, demonstrates the fastest convergence and highest accuracy. Figures 13 and 14 illustrate the end-effector orientation error of the continuum space robot during the trajectory planning process. From curve d, it is evident that relying solely on a singularity avoidance strategy leads to orientation errors, which persist even after the trajectory planning task is completed. In contrast, trajectory planning methods incorporating orientation error feedback (curves a, b, and c) successfully drive the error to zero. Although curve b (finite-time trajectory planning) exhibits relatively low overall error, it struggles to rapidly eliminate residual errors after completing the trajectory planning task. Curve c (the proportional feedback method, widely used in space robot trajectory planning) does not exhibit significant singularity or oscillation issues and effectively eliminates post-task errors. However, its planning accuracy and error convergence rate are lower than those of curve a. Curve a demonstrates that the proposed method consistently maintains a low trajectory planning error and achieves error convergence to zero at an earlier stage. \[\dot{\mathbf{q}}=\sum_{i=1}^{6}\frac{\delta_{i}}{\delta_{i}^{2}+\lambda_{i}^{2}}v_ {i}u_{i}^{\rm T}\mathbf{V}_{\alpha} \tag{74}\]0 Three-dimensional plot of position trajectory errors. \[\dot{\mathbf{q}}=\sum_{i=1}^{6}\frac{\delta_{i}}{\delta_{i}^{2}+\lambda_{i}^{2}}v_ {i}u_{i}^{\rm T}\mathbf{V}_{\alpha} \tag{74}\]1 Position error in trajectory planning of end effector (initial error exists).