## 7 Simulation
### Simulation and Analysis of Trajectory Planning  
To validate the kinematic model and trajectory planning method of the continuum space robot, trajectory planning is conducted based on explicit time theory and singularity avoidance methods. Table 2 presents some initial parameters of the continuum space robot system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Quantity** \\ \hline \(m_{0}\) & \( \begin{bmatrix}104.97&0&0\\ 0&34.97&0\\ 0&0&103.34\end{bmatrix} \)kg \(\cdot\) m\({}^{2}\) & The moment of inertia of the base \\ \(b\) & \( \begin{bmatrix}0.35&0&0.5\end{bmatrix} ^{\mathrm{T}}\) & The position of the continuum \\ \(\rho\) & \( \begin{bmatrix}0.35&0&0.5\end{bmatrix}   
Move the end effector from the initial pose to the final pose, using trapezoidal velocity profiles for both linear and angular velocities [25]. If one directly attempts to invert the positive kinematic model (33), and given that the momentum of the system is already specified to be zero, the equation for the inverse solution becomes  
\[\dot{\mathbf{q}}=(J_{S})^{-1}\mathbf{V}_{\alpha} \tag{73}\]  
Due to the occurrence of singular points during the robot's motion, where the determinant of is close to zero, the planned joint angular velocities are excessively large, which is unrealistic in practical situations. An improved approach involves directly avoiding singularities by incorporating a damping term to reduce joint velocities at singular points [25]. Therefore, the joint angular velocities planned based on DLS method are  
\[\dot{\mathbf{q}}=\sum_{i=1}^{6}\frac{\delta_{i}}{\delta_{i}^{2}+\lambda_{i}^{2}}v_ {i}u_{i}^{\rm T}\mathbf{V}_{\alpha} \tag{74}\]  
Its meaning is consistent with Equation (61). However, the introduction of singularity avoidance methods can result in pose errors in the robot system, and correction is not achievable. During the operation of the robot system, errors will continue to accumulate. Therefore, this paper proposes a singularity avoidance trajectory planning method based on pose feedback and improves the precision of the system's trajectory planning based on explicit time stability theory. The relevant parameter settings for the proposed trajectory planning method (61) are \(T_{p}=5\), \(m=0.25\), \(\epsilon_{c}=3\). To validate the proposed trajectory planning method, comparative experiments were conducted. In addition to the singularity avoidance direct inversion used in (74), finite-time algorithms and the commonly used proportional feedback in space robot trajectory planning were also employed. The joint angular velocities planned based on the finite time [26] are  
\[\dot{\mathbf{q}}=\sum_{i=1}^{6}\frac{\delta_{i}}{\delta_{i}^{2}+\lambda_{i}^{2}}v_ {i}u_{i}^{\rm T}\left(-\frac{c}{2^{\alpha}}\frac{\mathbf{e}^{2\alpha}}{\mathbf{e}}- \mathbf{\xi}\right) \tag{75}\]  
where \(\alpha=0.7\) and \(c=0.2\). The joint angular velocities planned based on the proportional feedback method are  
\[\dot{\mathbf{q}}=\sum_{i=1}^{6}\frac{\delta_{i}}{\delta_{i}^{2}+\lambda_{i}^{2}}v_ {i}u_{i}^{\rm T}(-\mathbf{K}\mathbf{e}-\mathbf{\xi}) \tag{76}\]  
where \(\mathbf{K}=diag(k_{1}\ \ldots\ k_{n})\) is the proportional coefficient matrix; here, \(k_{i}=3\). Figures 8 and 9 illustrate the end effector pose errors of the continuum space robot during the trajectory planning process. Line a represents the proposed explicit-time trajectory planning method, line b is based on the finite-time trajectory planning method, line c is based on proportional feedback for trajectory planning, and line d is based solely on the DLS method without pose feedback. From line d, it can be observed that using only singularity avoidance introduces pose errors, and these errors persist after the completion of the trajectory planning task. On the contrary, trajectory planning methods based on pose error feedback, represented by lines a, b, and c, can converge the errors to zero.