## 5 Experiments
### Evaluation of System Performance
#### 5.2.2 Dynamic Tracking Experiment  
In the dynamic tracking experiment, the target was fixed to the end-effector of a universal robot and taken to move along different paths. To ensure reliability, both complex planar and 3-D paths were tested. The planar and 3D paths of the target were set as  
\[P_{rose2}=\left[ \begin{array}{c}840+200\cos(3\omega N\Delta T)\sin(\omega N \Delta T)\\ 1690\\ 200+250\cos(\omega N\Delta T)\sin(3\omega N\Delta T)\end{array} \right]\]  
\[P_{rose3}=\left[ \begin{array}{c}840+250\sin(3\omega K\Delta T)\sin(\omega K \Delta T)\\ 1690+vK\Delta T\\ 200+250\cos(\omega K\Delta T)\sin(3\omega K\Delta T)\end{array} \right]\]  
where \(\omega=\pi/6\) rad\(\cdot\)s\({}^{-1}\) is the angular velocity in the \(X-Y\) plane, \(v=1.25\) mm\(\cdot\)s\({}^{-1}\) is the translation velocity in the \(Z\) direction, \(\Delta T=0.04\)\(s\) is the sampling period and \(K\) 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 6 gives the mean square error (MSE) of different strategies tracking two different paths, and these results are the average of five repeated experiments. Figures 16-19 show the image errors of the different strategies, and the joint angle and joint angular velocity of the proposed improved PUHVS with the KF-SVSF method, in the double leaf rose path and the triple leaf rose paths respectively. Obviously, the improved PUHVS with the KF-SVSF method shows better accuracy and stability than other strategies. During the tracking process, the tracking error is always kept in an acceptable range. It is worth mentioning that, due to the adoption of the SVF method, the system smoothly completes the servo process without severe disturbance. It can also be seen that, because the redundancy resulting from the improved homography-based task function is used for the design of the controller to limit the joint angle, there was no case in which the joint angle exceeded the limit, and the entire experiment was successfully completed. Figure 15: Joint velocities of the improved PHUVS with the KF-SVSF method in the static positioning experiment. It is worth noting that, in our experiment, we calculated the corresponding joint increment command \(\Delta q=\dot{q}\cdot\Delta T\) through the sampling period \(\Delta T\) instead of continuous joint velocities commands \(\dot{q}\) to send to the robot. During the execution of each sampling period, joint velocities were automatically planned when moving the increment \(\Delta q\). As shown in Figures 17a and 19a, the joint angle was still relatively smooth. However, in Figures 5f, 15, 17b and 19b, high-frequency oscillations were present in joint velocities caused by various noises. If the manipulator were controlled in a speed mode in actual tasks, this might cause vibrations and it might be necessary to add a filter to cut high frequencies. This would add a delay and possibly make this method unsuitable for real-time operations in space. Researchers have noticed this problem and found that PHUVS experiences a much smaller variation amplitude in joint velocities than IBUVS [23]. However, this level of improvement is not sufficient to completely solve the problem, and this is still a shortcoming that needs to be overcome in future applications. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method Path** & **Double Leaf Rose** & **Triple Leaf Rose** \\ \hline IBUVS & 9.43 & 10.19 \\ \hline OPHUVS & 7.32 & 7.74 \\ \hline IPHUVS with KF & 6.28 & 6.61 \\ \hline IPHUVS with KF-SVSF & 5.61 & 5.89 \\ \hline \end{tabular} MSE is in pixels.  
\end{table} 
Table 6: MSE of different strategies in the dynamic tracking experiment. Figure 16: Image error of different strategies in the experiment of the double 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.