## 4 Simulations
### Simulations and Discussions of Aligning with Dynamic Assembly Objects  
The mean square error (MSE) is chosen as an index to compare the performances of different strategies for aligning with dynamic assembly objects. It represents the average image error between the current and desired image features in the servo process, and is defined as  
\[MSE=\frac{1}{N}{\sum_{i=1}^{N}{\|\Delta I_{i}\|}}\]  
where \(\Delta I_{i}\) is the image error at the \(i\)th iteration and \(N\) is the number of iterations. The same target is adopted to move in planar and 3D trajectories. Both 2D motion in a double leaf rose path and 3D motion in a triple leaf rose path with different noise levels are simulated. These different paths of the target center are denoted as follows:  
\[P_{rose2}=\left[{ \begin{array}{*{20}{c}}{450+300\cos(3\omega N\Delta T)\sin( \omega N\Delta T)}\\ {800+300\cos(\omega N\Delta T)\sin(3\omega N\Delta T)}\\ {820}\end{array} }\right]\]  
\[P_{rose3}=\left[{ \begin{array}{*{20}{c}}{450+250\sin(3\omega N\Delta T)\sin( \omega N\Delta T)}\\ {800+300\cos(\omega N\Delta T)\sin(3\omega N\Delta T)}\\ {820+vN\Delta T}\end{array} }\right]\]  
Figure 5: Detailed result of the improved PHUVS with the KF-SVSF method. (**a**) Trajectories of partial feature points in the image plane. (**b**) Norm of image error. (**c**) Norm of position error. (**d**) Norm of attitude error. (**e**) Norm of error vector. (**f**) Joint velocities of the manipulator. where \(\omega=\pi/30\) rad\(\cdot\)s\({}^{-1}\) is the angular velocity in the \(X-Y\) plane, \(v=2.5\) mm\(\cdot\)s\({}^{-1}\) is the translation velocity along the \(Z\) axis, \(\Delta T=0.02\)\(s\) is the sampling period and \(N\) is the number of iterations. The threshold of the KF-SVSF boundary layer was defined as  
\[\psi_{S}=diag\Big{[} \begin{array}{cccccccc}128&42&1360&42&128&1360&16&16\end{array}  \Big{]}\]  
Table 3 gives the MSEs of all the above tests. All of these MSEs are the average of 100 experiments. Figures 6 and 7 show the image errors of the different strategies for the double leaf rose and triple leaf rose paths with a noise mean of \(|V|=0.5\), and Figures 8 and 9 show the camera trajectories of the different strategies. The IBUVS is the least robust method to image noise, and its tracking precision is unacceptable. The improved PHUVS, which directly controls manipulator joints, has better performance than the original PHUVS, as shown in Figure 2a. Meanwhile, the improved PHUVS with the KF-SVSF method shows better tracking precision than the improved PHUVS with the KF. Figure 8: Camera trajectories of different strategies in a double leaf rose path. (**a**) Camera trajectory of IBUVS. (**b**) Camera trajectory of the original PHUVS. (**c**) Camera trajectory of the improved PHUVS with the KF method. (**d**) Camera trajectory of the improved PHUVS with the KF-SVSF method. Figure 7: Image error of different strategies in a triple leaf rose path. (**a**) Image error of IBUVS. (**b**) Image error of the original PHUVS. (**c**) Image error of the improved PHUVS with the KF method. (**d**) Image error of the improved PHUVS with the KF-SVSF method. The abilities of the strategies to avoid the joint limit and handle an unstable conditional number are examined as well. The joint limit was set as \(\pm 90^{\circ}\), and the minimum acceptable singular value of the total Jacobian was set as \(\sigma_{0}=10^{-4}\). Figure 10 shows the unlimited and limited joint angles for these two trajectories. Figure 11 shows the logarithm of the minimum singular value \(\sigma_{\min}\) of the estimated total Jacobian \(\left(J_{t}\right)_{K}\) and processed Jacobian \(\left(J_{S}\right)_{K}\) in each period. In each tracking trajectory, all joint angles of the super-redundant manipulator are limited to the acceptable range. When unstable  
Figure 9: Camera trajectories of different strategies in a triple leaf rose path. (**a**) Camera trajectory of IBUVS. (**b**) Camera trajectory of the original PHUVS. (**c**) Camera trajectory of the improved PHUVS with the KF method.