## 3 Simulation and Experimental Results
### 2D Machining Simulation  
The 2D machining path shown in Figure 6 contains corners, arcs, and reversals, which are typical and complex. Therefore, it can be used for the simulation to compare the effect of the angle-constraint method and FNA method. Figure 5: The architecture of the experimental platform. Figure 6: The 2D machining path of the continuous small line segments. The radius of curvature \(\rho\) at Are1 the in above figure is 3 mm, and the curve here is discretized into small continuous line segments with the chord error \(\varepsilon=3\) um. Therefore, the angle of the corner of adjacent segments is  
\[\theta=2a\sin\frac{\sqrt{2\rho\varepsilon-\varepsilon^{2}}}{\rho}=5.13^{\circ} \tag{56}\]  
Therefore, the feedrate-constraint value obtained by using the virtual transition-arc constraint method [11] is  
\[f_{1}(\theta)=\sqrt{\sigma a_{\mathrm{n}}\cos\frac{\theta}{2}/(1-\cos\frac{ \theta}{2})}=2823\;\mathrm{mm}/\mathrm{min} \tag{57}\]  
while, according to the constraint of the radius of curvature and the normal acceleration, the feedrate-constraint value should be  
\[f_{2}(\rho)=\sqrt{a_{\mathrm{n}}\rho}=1548\;\mathrm{mm}/\mathrm{min} \tag{58}\]  
Similarly, the feedrate-constraint values at Area2, Area3, and Area4 can also be achieved, as shown in Table 2.  
The angle-constraint method and the FNA method were used to calculate the feedrate-constraint value of the machining path in Figure 6, respectively. Then, the feedrate planning and interpolation were performed, and the results are shown in Figure 7.  
As can be seen from Figure 7, for Line Area in Figure 6, both the angle-constraint method and the FNA method effectively constrained the feedrate, while for Curve Area, the feedrate-constraint values obtained from these two methods were significantly different. The machining paths in Area1 and Area2 (or Area3 and Area4) had the same radius of curvature, but the angles were different. Therefore, if the angle-constraint method is used to calculate the feedrate-constraint value, the machining paths with the same radius of curvature will be machined by different feedrates, as shown in Figure 7. Especially in Area1 and Area4, the feedrate exceeded the theoretical value obtained by Equation (58). While in Area1-Area4 of Figure 7, the feedrates obtained based on the FNA method were all nearly same as the theoretical values in Table 2.  
In addition, the small line segments in Area2 and Area3 were relatively longer; so, the velocity direction changed more at the corner. As a result, in Figure 7, the feedrate-constraint value obtained by the FNA 1 method was obviously lower than the theoretical value, while the feedrate-constraint value obtained by the FNA method was almost equal to the theoretical value, which verified the effectiveness of the machining trajectory prediction in Section 2.2. Therefore, the machining trajectory prediction can suppress the accuracy problem of continuous small line segments and make the calculating results of the proposed method more reasonable and stable. Similarly, the feedrate-constraint values obtained by the FNA 2 method in Area1 and Area2 (or Area3 and Area4) had a difference over 50 mm/min, while the difference of the FNA method was less than 10 mm/min, which verified the effectiveness of the acceleration filtering in Section 2.3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Area** & **Area1** & **Area2** & **Area3** & **Area4** \\ \hline Radius of the curvature \(\rho\) (mm) & 3 & 3 & 5 & 5 \\ Chord error \(\varepsilon\) (μm) & 3 & 10 & 10 & 3 \\ Length of the line segment (μm) & 268 & 490 & 632 & 346 \\ Angle of adjacent line segments \(\theta\) (\({}^{\circ}\)) & 5.13 & 9.36 & 7.25 & 3.97 \\ Feedrate-constraint value by the angle (mm/min) & 2823 & 1546 & 1997 & 3000 \\ Feedrate-constraint value by the curvature (mm/min) & 1548 & 1548 & 2000 & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric information of the machining path and its feedrate-constraint value.