## 5 Experiments  
Both simulation and experiments were carried out to validate the effectiveness of the presented method on the prototype of the hybrid polishing robot. As shown in Figure 2, it is mainly composed of a 6-DOF (degrees of freedom) hybrid robot, a polishing effector, a magnetic worktable, a column, and a CNC system. The hybrid robot is composed of a 3-DOF (3 UPS and UP) parallel mechanism and a 3-DOF wrist. The UP limb and the wrist form a UPS or UPRRR limb. Here, R, U, S, and P represent, respectively, revolute, universal, spherical, and prismatic joints, and the underlined P, S, and R denote the actuated prismatic, spherical, and revolute joints, respectively. The CNC system is built upon an IPC+PMAC open architecture, consisting of a host control computer responsible for reconstruction of the parametric curves, trajectory interpolation, and NC command generation, and a PMAC motion controller for servo-control of the actuated joints. Without loss of generality, a segment of NC codes was taken from the polishing trajectory to validate the effectiveness of the proposed interpolation method. According to the SOR iterative scheme mentioned in Section 2, Figure 3 shows the two fitted trajectories. Then, the optimisation algorithm described in Section 3 is used to smooth the curvature of the trajectory. The change threshold of curvature for judging outlier points is given as 0.01. As shown in Table 1, the fairness of the two trajectories is both significantly improved. The maximum curvatures of the two trajectories are reduced by as much as 79.7% and 63.3% and the maximum absolute values of shear jerks are decreased by 91.2% and 90.2%, correspondingly. Considering the optimisation and interpolation methods for the two trajectories are same, experimental results of the end-point trajectory are just shown in the following discussion for the sake of simplicity. Based on the optimisation trajectory shown in Figure 4, the discrete NC command sequences were generated and sent to the PMAC motion controller, which were mapped into the servo-command of actuated joints through an inverse kinematic model. The PMAC motion controller synchronously gathered the positions and velocities fed back from the servo-motors. The interpolation and sampling periods are 10 and 20 ms, respectively. To validate the effectiveness of the method, a comparison experiment was carried out utilizing the linear interpolation method. Figure 5 shows the experimental result, which is computed by the feedback positions of all the actuated joints. It can be seen from the partial enlarged view that the NC commands generated by the proposed method are closer to the NC codes than those found using linear interpolation, which means that the removal position of polishing process can be reached more accurately. Figures 6 and 7 show the runtime  
Figure 4: The end-point trajectory before, and after, fairing optimization: (**a**) the end-point trajectory; and (**b**) the partial enlarged view  
Figure 5: The end-point trajectory interpolated by different methods: (**a**) the end-point trajectory; and (**b**) the partial enlarged view. Figure 6: Runtime of the trajectory interpolated by the proposed method: (**a**) runtime of the trajectory; and (**b**) the partial enlarged view. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Trajectory**} & \multicolumn{2}{c}{**Before**} & \multicolumn{2}{c}{**After**} \\ \cline{2-5}  & **Maximum Curvature** & **Maximum Shear Jerk** & **Maximum Curvature** & **Maximum Shear Jerk** \\ \hline End-point & 1.249 & 0.285 & 0.253 & 0.025 \\ Reference-point & 1.380 & 0.325 & 0.506 & 0.032 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Changes in the two trajectories before, and after, fairing optimisation.