## 1 Introduction
### Theoretical Form Error Caused by Overcut of Tool Edge in HFUVAS  
In diamond sculpturing of a microlens array, there exists a theoretical form error between the designed microlens array and the machined microlens array induced by overcutting of the tool edge. Take sculpturing of a single lenslet with an aperture of AC\({}_{1}\)EC\({}_{2}\) as shown in Figure 3. When the diamond tool moves along the target line of ABCDE, the curvature radius of the lenslet contour perpendicular to the cutting direction increases to the largest at point C and then decreases. Since the tool radius keeps constant which is equal to the largest curvature radius at point C, the tool edge will overcut the material when the tool moves along line ABCDE except at the point of C. To be specific, at the position of point C, the lenslet contour perpendicular to the cutting direction is line C\({}_{1}\)CC\({}_{2}\), whose curvature radius is equal to the tool radius. While at point D, the curvature radius of the lenslet contour, namely the radius of line D\({}_{1}\)DD\({}_{2}\) as depicted on the blue circle is smaller than the tool radius, resulting in the tool edge overcutting the line D\({}_{1}\)DD\({}_{2}\). As a result, overcut of the tool edge in HFUVAS leads to a deviation between the designed structure and the machined structure. It is essential to generate the machined structured surface and identity the theoretical form error between the designed microlens array and the microlens array generated by tool edge sculpturing in order to evaluate the machining accuracy for this machining technique. In this process, two coordinate systems, namely the local tool coordinate system (\(O_{t}x_{t}y_{t}z_{t}\)) and the workpiece coordinate system (\(O_{w}x_{w}y_{w}z_{w}\)) are defined as depicted in Figure 4a. Considering the tool rake angle is \(0^{\circ}\), tool edge in the local tool coordinate system is determined by Equation (7) [21]:  
\[\left\{ \begin{array}{l}x_{t}=Rcos\theta_{t}\\ y_{t}=0\\ z_{t}=-Rsin\theta_{t}\end{array} \right.,\theta_{t}\in[\theta_{min},\theta_{max}] \tag{7}\]  
where \(R\) is the tool nose radius. \(\theta_{t}\) represents the angular position of the point on the cutting edge, \(\theta_{min}\) and \(\theta_{max}\) are the minimum and maximum angle of the effective tool edge. To generate the machined structure by tool sculpturing, the coordinates (\(x_{t},\ y_{t},z_{t}\)) of all points with a certain sampling on the tool edge should be transformed as (\(x_{tw},\ y_{tw},\ z_{tw}\)) in the workpiece coordinate system. The tool edge center is expected to be transformed to the original point \(O_{w}\) of the workpiece coordinate system. The relationship between the workpiece coordinate and the local coordinate for the tool edge is given by Equation (8):  
\[\left[ \begin{array}{l}x_{tw}\\ y_{tw}\\ z_{tw}\\ 1\end{array} \right]=\left[ \begin{array}{l}1\;0\;0\;0\\ 0\;1\;0\;0\\ 0\;0\;1\;R\\ 0\;0\;0\;0\end{array} \right]\left[ \begin{array}{l}x_{t}\\ y_{t}\\ z_{t}\\ 1\end{array} \right] \tag{8}\]  
Figure 3: A schematic diagram of diamond sculpturing of microlens array. Coordinates of all the points on the tool edge in the workpiece coordinate system can be calculated by taking into account of the tool path, which is given by Equation (9):  
\[\left\{ \begin{matrix}x_{w}=x_{tw}+p_{1}(k_{n}-1/2)\\ y_{w}=y_{tw}+v_{c}t\\ z_{w}=z_{tw}+z(p_{1}(k_{n}-1/2),v_{c}t)\end{matrix} \right. \tag{9}\]  
where \(p_{1}\) is the pitch of lenslet in the feed distance, or the distance between two target lines, \(k_{n}\) is the number of the target line (\(k_{1}=1,\ k_{2}=2,\ k_{3}=3\) as shown in Figure 4b). \(v_{c}\) and \(t\) represent the tool's normal cutting velocity and moving time at each target line. \(z(p_{1}(k_{n}-1/2),v_{c}t)\) is the \(z\) coordinate value of the cutting point (\(p_{1}(k_{n}-1/2),v_{c}t\)) in the workpiece coordinate system. By determining all the tool points with respect to target lines, surface generation of the microlens array machined by HFUVAS can be simulated and analyzed.