# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 3 Results and Discussions
### Surface Roughness
To find the highest \(PE\) by optimizing the polishing parameters, a graph between the \(R_{a}\) and \(PE\) for all three polishing cycles with _SFPM_ of 2950 and 3750 was plotted in Figure 19. It shows that the target roughness was achieved with all three polishing cycles at different feed rates. The maximum feed rates  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \begin{tabular}{c} **Exp.** \\ **No.** \\ \end{tabular}  & \begin{tabular}{c} **Feed Rate** \\ **(mm/s)** \\ \end{tabular}  & \begin{tabular}{c} **No. of** \\ **Cycle** \\ \end{tabular}  & _SFPM_ & \(R_{a}\) & \begin{tabular}{c} **PE = \(\frac{\text{AR}_{a}}{T}\)** \\ \end{tabular}  & \begin{tabular}{c} **Exp.** \\ **No.** \\ \end{tabular}  & \ \begin{tabular}{c} **Feed Rate** \\ **(mm/s)** \\ \end{tabular}  & \begin{tabular}{c} **No. of** \\ **Cycle** \\ \end{tabular}  & _SFPM_ & \(R_{a}\) & \(PE\) = \(\frac{\text{AR}_{a}}{T}\) \\ \hline
1 & 15 & 1 & 2950 & 0.032  
with a 10% safety margin to get the desired roughness in one, two, and three polishing cycles were 13 mm/s, 45 mm/s, and 54 mm/s respectively. However, the two polishing cycles with 3750 _SFPM_ shows the maximum efficiency to get the target. Therefore, optimized polishing parameters (feed rate = 45 mm/s, no. of cycles = 2, and _SFPM_ = 3750) from the graph were selected for polishing the all remaining 17 part surfaces of the eyeglass frame. Table 2 summarizes the optimized polishing parameters taken form the initial experimentation of the eyeglass frame polishing on surfaces \(S_{1}\). It is supposed that all the part surfaces had approximately similar initial \(R_{a}\) as \(S_{1}\). It suggested two polishing cycles with _SFPM_ of 3750 by LCW utilizing dry finish paste (P126 by Menzerna). Different combinations of wheel diameter and rotation speed were possible, which generated _SFPM_ of 3750. A wheel with a 152 mm diameter and rotation speed 2387 rpm was adopted to polish. Figure 19: Plot of polishing efficiency against roughness for different _SFPM_ and polishing cycles. Figure 18: Plot of roughness against feed rate for different surface feet per minute (_SFPM_) and polishing cycles. Figure 20 demonstrates the surface roughness evolution for rough to mirror-like polishing in terms of \(R_{a}\) and \(R_{q}\) at all six selected points in different part surfaces. To minimize the chance of error, every measurement was repeated three times at the same point. It can be seen that the roughness dropped severely in the first polishing step, while it slowed down in the second step because of the feed direction in both steps. However, the second polishing cycle provided gloss to the surface. Before polishing, \(R_{a}\) was ranging from 0.204 to 0.215 \(\upmu\)m for six selected points that shows all part surfaces had nearly the same initial roughness. After the first polishing cycle, roughness reduced approximately 83% on average at the selected points. Average \(R_{a}\) and \(R_{q}\) were 0.0185 and 0.0233 \(\upmu\)m respectively with the approximately 90% improvement in the surface height profile after two polishing steps. The major accomplishment of this robotic polishing was the resultant even surface quality in the freeform surface with economical parameters. \(R_{a}\) values after polishing reflected this evenness with a very small variation from 0.018 to 0.021 \(\upmu\)m.