# Path Planning under Force Control in Robotic Polishing of the Complex Curved Surfaces
## 3 Results and Discussions
### Glossiness  
Glossiness or specular reflection is another quality parameter for the polished surface that measures the visual appearance of the product. From Equation (11), glossiness (\(G\)) can be measured as the ratio of the specular reflectance of the target surface (\(R_{T}\)) to the specular reflectance of the polished glass (\(R_{o}\)) with a standard refractive index of 1.567 [35; 36]. \[G= \frac{R_{T}}{R_{o}}\times 100\% \tag{11}\]  
From the Fresnel equation, the specular reflectance (\(R\)) can be calculated as in Equation (12) [35]. It substantiates that incident angle \(i\) and refractive index \(n\) affect the spectral reflectance. Therefore, measurements at different angles were adopted in various standards. \(20^{\circ}\), \(60^{\circ}\), and \(80^{\circ}\) are mostly used \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Polishing** & **Selected** & **Polishing** & **SFPM (3750)** & **Feed Direction** & **Contact Force** & **Feed Rate** \\
**Cycle** & **Tool** & **Material** & **Wheel** & **Wheel Speed** & **F\({}_{\text{N}}\) (N)** & **(mm/s)** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{LCW} & Dry finish & \multirow{2}{*}{152} & \multirow{2}{*}{2387} & Against wheel rotation & \multirow{2}{*}{1} & \multirow{2}{*}{45} \\  & & & Paste (P126) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Economical polishing parameters for each part surface of the eyeglass frame. Figure 20: Arithmetic and root mean roughness of all the six selected points at different polishing stages. angles in many measuring standards like ASTM D523 and DIN 67530. However, the 60\({}^{\circ}\) is the most adopted angle in many gloss meters as it is used in majority standards. \[R= \frac{1}{R_{S}+R_{P}}=\frac{1}{2}\Bigg{[}\Bigg{(}\frac{cos\;i-\sqrt{n^{2}- sin^{2}\;i}}{cos\;i+\sqrt{n^{2}-sin^{2}\;i}}\Bigg{)}^{2}+\Bigg{(}\frac{n^{2} cos\;i-\sqrt{n^{2}-sin^{2}\;i}}{n^{2}cos\;i+\sqrt{n^{2}-sin^{2}\;i}}\Bigg{)}^{2} \Bigg{]} \tag{12}\]  
Similarly, many studies have been established the relation between specular reflectance and surface roughness [35; 37]. For example, [PERSON] and [PERSON] derived the following equation [38]:  
\[R_{S}=R_{o}\;exp\Bigg{[}-\frac{(4\pi\sigma)^{2}}{\lambda^{2}}\Bigg{]}+R_{o}\; \frac{2^{5}\pi^{4}}{m^{2}}\left(\frac{\sigma}{\lambda}\right)^{4}(\Delta \theta)^{2} \tag{13}\]  
Here, \(\sigma\) is the root mean square roughness, \(\Delta\theta\) is acceptance angle, \(m\) is the root mean square slope of the surface profile, and \(\lambda\) is a wavelength, which is assumed to be greater than the root mean square roughness. Surfaces with high roughness provide less gloss compared to the less rough surface. Table 3 shows the glossiness comprehensive measurements at selected point P3 using gloss meter for best mean value with tolerance 10 G. A similar procedure was used throughout all sample surfaces. Five readings on the same points on every sample part surface were taken for the measurement. Figure 21 presents the glossiness of all the selected six points before and after polishing. It also presents the improvement in the gloss after polishing. It shows the significant improvement in the glossiness of the part surfaces at different points between 62.5 and 68.7 G. It provided an average 88% improvement in glossiness in a short time just with two polishing cycles. In manufacturing industries, machining economics is crucial to optimize the machine process. This optimization to get the target can be achieved by controlling the machining influencing variable parameters. Figure 22 shows the comparison in achieving the target roughness of 0.02 \(\upmu\)m by three different no. of polishing cycles at their maximum efficiencies as in the _PE_ vs. \(R_{a}\) plot. This comparison