# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 3 Results and Discussions
### Surface Roughness  
Arithmetic average roughness measurement is a well-established entity to evaluate the surface quality and it has been a crucial part of many theoretical and experimental studies. It is the average absolute deviation of the profile height \(y(x)\) from the mean line within the sample line. It is easy to measure and gives a general view of the height profile, however, it is not sensitive to small height variation and it does not give any information about wavelength. Conversely, the root mean square roughness (\(R_{q}\)) is the square root of the mean of the squared of roughness. It indicates the deviation of the distribution of the surface heights. Although \(R_{a}\) is the most commonly used entity, the \(R_{q}\) is more capable of large deviations from the mean line. Figure 16 demonstrates the arithmetic average roughness height profile at a point on the part surface 1 of the eyeglass frame before and after polishing under 1 mm evaluation and 0.25 mm sample length. Besides the \(R_{a}\) and \(R_{q}\), skewness (\(R_{sk}\)) and kurtosis (\(R_{ku}\)) are also mentioned to understand the peaks and valleys formation during polishing steps. Before polishing, the surface has plenty of high peaks and deep valleys. After the 1 st polishing cycle, this profile became negatively skewed (\(R_{sk}=-1.726\)) because more high peaks are removed compared to the valleys. Moreover, fine grains of the polishing wax created shallow valleys that make the profile leptokurtic (\(R_{ku}=7.854\)). The second polishing cycle pushed it towards the normally distributed profile (\(R_{ku}=2.946\)), however still have fewer peaks (\(R_{sk}=-0.695\)). The initial experimental analyses substantiate that the used of LCW and polishing wax can provide an average minimum \(R_{a}\) of 0.017 \(\upmu\)m. Further visualization of the surface topography of the eyeglass frame at the same point under the highly zoomed microscope is present in Figure 17 that shows the magnified photos taken at different polishing stages. Figure 17a shows the rough surface image before polishing with deep scratches. However, an effective polishing smoothed the surface as shown in Figure 17b with few shallow scratches left. The first polishing cycle improved 84% in the surface roughness approximately. The second polishing cycle further improved the surface as shown in Figure 17c. It gives gloss to the surface without any visible scratch. By acknowledging the fact that the eyeglass frame is made of plastic (synthetic resin) and can easily deform its shape particularly at the edges if exposed to the tool for a longer time. Therefore, the higher _SFPM_ and contact force are avoided. The minimum polishing force of 1 N for the \(F_{N}\) is used throughout the experiments. Table 1 shows the complete experimental runs conducted at the part surface one (\(S_{1}\)) to find the optimized input polishing parameters for all the part surfaces. \(R_{a}\) as an output of the polishing process was measured for different combinations of three input parameters. Every experiment was started with the average initial surface roughness of 0.214 \(\upmu\)m. The polishing cycles were planned to achieve the target \(R_{a}\) of 0.02 \(\upmu\)m in a way that the first cycle was the workpiece motion against wheel rotation and in the later cycles, it would be in the direction of wheel rotation. Figure 15: Equipment used in experiments: (**a**) surface roughness tester (Mitutoyo, SJ-210); (**b**) gloss meter (NHG60M); (**c**) tachometer; and (**d**) microscope (ZEISS Axio Scope A1). Figure 18 shows the \(R_{a}\) curve against the feed rate for two different _SFPM_ and three polishing cycles for the part surface one. A little disagreement in the graph from the trend line can be due to impurities in the polishing material. Increment in the feed rate increased the resultant roughness; however, with proper _SFPM_ this rate could slowdown effectively. Suitable feed rate and _SFPM_ could increase the polishing efficiency and achieve the desired surface quality in a short time.