## 4 Simulations
### Simulations and Discussions of Aligning with Static Assembly Objects  
To align with static objects, the various errors after the system reaches the steady state are the most reasonable index for comparing the performances of different uncalibrated visual servo strategies. Several tests were performed with different noise levels to check the robustness of the IBUVS, original PHUVS, improved PHUVS with KF method and improved PHUVS with KF-SVSF method. A square target was chosen with a side length of 200 mm and 49 feature points. Different types of initial pose errors, including a pure translation error \([\Delta\chi,\Delta\theta]=[350\text{ mm},350\text{ mm},100\text{ mm},0^{\circ},0^{\circ},0^{\circ}]\), a pure rotation error \([\Delta\chi,\Delta\theta]=[0\text{ mm},0\text{ mm},0\text{ mm},-10^{\circ},-10^{\circ},45^{\circ}]\) and a general error \([\Delta\chi,\Delta\theta]=[350\text{ mm},350\text{ mm},100\text{ mm},-10^{\circ},\,-10^{\circ},45^{\circ}]\), were set according to the expected camera motion. The threshold of the boundary layer in the KF-SVSF model is defined as  
\[\psi_{S}=diag\Big{[}\begin{array}{cccccc}64&36&420&36&64&420&10&10\end{array} \Big{]}\]  
Table 2 shows the errors of all the above tests. The improved PHUVS with the KF-SVSF method performs better than the IBUVS, original PHUVS and improved PHUVS with the KF. Sometimes, the IBUVS does not converge. Figure 5 shows the detailed test results of the improved PHUVS with the KF-SVSF method for general motion with a noise mean of \(|V|=0.5\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Task**} & \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**Noise Mean**} \\ \cline{3-10}  & & \multicolumn{2}{c}{\(|V|\)=0.2} & \multicolumn{8}{c}{\(|V|\)=\(0.5\)} & \multicolumn{8}{c}{\(|V|\)=1} \\ \cline{3-10}  & & \(\Delta\chi\) & \(\Delta\theta\) & \(\Delta I\) & \(\Delta\chi\) & \(\Delta\theta\) & \(\Delta I\) & \(\Delta\chi\) & \(\Delta\theta\) & \(\Delta I\) \\ \hline \multirow{4}{*}{\begin{tabular}{c} Pure \\ Translation \\ \end{tabular} } & IBUVS & 0.425 & 0.014 & 1.121 & 0.823 & 0.032 & 4.541 & 1.225 & 0.067 & 6.346 \\  & OPHUVS & 0.183 & 0.004 & 0.991 & 0.302 & 0.017 & 2.453 & 0.395 & 0.031 & 3.637 \\  & KF & 0.181 & 0.004 & 0.987 & 0.289 & 0.015 & 2.278 & 0.379 & 0.029 & 3.308 \\  & KF-SVSF & **0.174** & **0.004** & **0.886** & **0.273** & **0.015** & **2.052** & **0.364** & **0.029** & **3.081** \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Pure \\ Rotation \\ \end{tabular} } & IBUVS & 0.341 & 0.023 & 1.427 & 0.537 & 0.045 & 5.253 & 0.873 & 0.069 & 7.641 \\  & OPHUVS & 0.126 & 0.004 & 0.932 & 0.268 & 0.016 & 2.466 & 0.385 & 0.034 & 3.826 \\  & KF & 0.124 & 0.004 & 0.929 & 0.251 & 0.015 & 2.349 & 0.374 & 0.031 & 3.527 \\  & KF-SVSF & **0.117** & **0.003** & **0.906** & **0.242** & **0.014** & **2.235** & **0.304** & **0.029** & **3.174** \\ \hline \multirow{4}{*}{
\begin{tabular}{c} General \\ Motion \\ \end{tabular} } & IBUVS & 0.374 & 0.029 & 1.435 & 0.911 & 0.049 & 6.213 & 1.352 & 0.072 & 8.214 \\  & OPHUVS & 0.187 & 0.006 & 0.813 & 0.261 & 0.018 & 2.716 & 0.406 & 0.033 & 3.826 \\ \cline{1-1}  & KF & 0.185 & 0.006 & 0.811 & 0.243 & 0.017 & 2.529 & 0.394 & 0.032 & 3.672 \\ \cline{1-1}  & KF-SVSF & **0.169** & **0.006** & **0.798** & **0.228** & **0.015** & **2.482** & **0.363** & **0.031** & **3.396** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Steady-state error of different strategies in the static positioning test. \(\Delta\chi\) is in millimeters, \(\Delta\theta\) is in degrees and \(\Delta I\) is in pixels. IBUVS: image-based uncalibrated visual servo. OPHUVS: original PHUVS.