## 3 Results and Discussion  
### Tool Path Generation for Sinusoidal Microstructure Fabrication on Curved Surface  
The tool path generation for sinusoidal microstructure fabrication on a planar surface is carried out first. The design of the planar sinusoidal grid is completed in the Cartesian coordinate system, and the \(Z\) coordinate of each point can be represented by the corresponding \(X\) and \(Y\) coordinates, the independent sine waves in which are superimposed to form a sine grid. Therefore, the \(Z\) coordinate of each point can be expressed by Equation (1):  
\[Z=f(\rho,\theta)=\mathrm{A}_{x}\sin\left(\frac{2\pi\rho\cos\theta}{\lambda_{x} }\right)+\mathrm{A}_{y}\sin\left(\frac{2\pi\rho\cos\theta}{\lambda_{y}}\right) \tag{1}\]  
where A is the distance from the projection of any point of the tool path on the \(X\)-\(Y\) plane to the coordinate origin (\(\mu\)m) and \(\lambda\) is the angle of rotation relative to the initial point (\({}^{\circ}\)). When \(\rho\) and \(\theta\) take different values according to a certain rule, a tool path is formed. The curved, sinusoidal grid is formed by superimposing independent sine waves in the \(X\) and \(Y\) directions on the base surface of the curved surface. For the curved surface microstructure fabrication using the FTS system, the movement in the \(Z\) direction can be replaced by the fast tool auxiliary axis. During the machining process, the stroke of the diamond tool fixed on the FTS post is superimposed on the ball trajectory, while the FTS device achieves a high-frequency response in the \(Z\) direction. The generated sine mesh surface and tool path are shown in Figure 2.  
Furthermore, to avoid the over-cutting phenomenon, the compensation amount of the tool nose trajectory is calculated, which is needed for obtaining the movement trajectory of the arc center of the tool nose as the coordinate of the tool location point. In order to simplify the calculation algorithm, save the time of generating data points and improve the processing efficiency, a new compensation method based on the tool tip arc segmentation is established. The generation method of tool path for the fabrication of sinusoidal surface microstructure is illustrated in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Symbol** & **Values** \\ \hline Rake angle & \(\gamma\) & 0\({}^{\circ}\) \\ \hline Relief angle & \(\alpha\) & 7\({}^{\circ}\) \\ \hline Tip angle & \(\epsilon\) & \textless{}80\({}^{\circ}\) \\ \hline Arc radius & \(\mathrm{R_{a}}\) & \textless{}121.67 \(\mu\)m \\ \hline Cutting edge radius & \(\mathrm{R_{\epsilon}}\) & 50 nm \\ \hline \end{tabular}
\end{table}
Table 2: Geometric parameters of the utilized diamond tool.  
Figure 2: Sinusoidal mesh surface and tool path. (**a**) Curved sine mesh surface; (**b**) Curved sine grid tool path.  
In the programming coordinate system for a spherical surface, the highest point of the spherical surface is \(-\mathrm{R_{t}}\), and the position of the center point of the tool tip arc is zero. In particular, the starting position of any point in the machining process of sinusoidal microstructures is the zero position in the program coordinate, as shown by the dotted line zero position in Figure 3a. According to the analysis of the actual processing situation, only the arc edge of the lower half can realize the turning operation, so only the lower half-circle area needs to be analyzed. In order to save calculation time of the tool path generation, according to the radius of the tool tip arc used, the discretization accuracy is set as \(\mathrm{e^{-5}}\), which can ensure the calculation of the corresponding \(Z\)-direction distance at least every 7 nm in the process of generating data points. Therefore, the interval is set between \(-\mathrm{R_{t}}\):\(\mathrm{R_{t}}\) according to the discretization accuracy of \(\mathrm{e^{-5}}\), and the semi-circular region is discretized by dispersing the points to obtain the position point parameter matrix relative to the zero point, as shown in Figure 3b.