## 3 Simulation and Experimental Results
### 3D Machining Simulation  
The effectiveness of the FNA method was preliminarily verified by the 2D simulation in the previous section. In this section, the FNA method will be further simulated and analyzed by using the continuous small line segments of a 3D free-form surface.  
For the first 3000 lines of the ISO6983 code [39] corresponding to the machining path shown in Figure 8b, the angle-constraint method, the curvature-constraint method, the acceleration-constraint method [17], and the proposed FNA method were used to calculate the feedrate-constraint value, respectively. Then, the feedrate planning and interpolation were performed, and the final feedrate results are shown in Figure 9.  
Figure 8: The workpiece and its machining path: (**a**) the workpiece; (**b**) the machining path (simplified).  
Figure 7: The feedrate planning results of the 2D machining path: (**a**) the feedrate-constraint value obtained from different methods (FNA 1: The FNA method without the trajectory prediction. FNA 2: The FNA method without the acceleration filtering); (**b**) the feedrate planning result of the angle-constraint method; (**c**) the feedrate planning result of the FNA method.  
For area 1 with large curvature in Figure 9a, the distribution of the feedrate in the X-Y plane obtained by the angle-constraint method is shown in Figure 9b, and the feedrate was distributed from 500 mm/min to 3000 mm/min, which is rather chaotic. In a contrast, the feedrate obtained from the FNA method is shown in Figure 9e, which was maintained around 1500 mm/min, and the stability of the feedrate was obviously better than the one of the angle-constraint method and the other two constraint methods.  
In order to quantify the stability of the feedrate to make a better comparison between different methods, the feedrate values for each row at X = 43 mm are collected in Figure 9b-e, respectively, and their distribution is shown in Figure 9f. The average, range, relative range (range/average), and standard deviation were calculated (the first three points in the circles of Figure 9b-f are in the reverse region and were not taken into the statistics), and the results are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Feedrate-Constraint Method** & **Average (mm/min)** & **Range (mm/min)** & **Relative Range** & **Standard** **Deviation (mm/min)** \\ \hline Angle & 2239 & 2032 & 90.8\% & 602 \\ \hline Curvature & 1258 & 1156 & 91.9\% & 212 \\ \hline Acceleration & 652 & 445 & 68.2\% & 81 \\ \hline FNA & 1489 & 64 & 4.3\% & 17 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The statistics results of the feedrate of each row when X = 43 mm.  
Figure 9: The feedrate results of the 3D machining path: (**a**) the line segments of the free-form surface; the feedrate results of the (**b**) angle-constraint, (**c**) curvature-constraint, (**d**) acceleration-constraint and (**e**) FNA method; (**f**) the feedrate of each row when X = 43 mm.  
From the data in this table, it can be seen that the proposed FNA method worked much better than other three methods. Especially compared with the angle-constraint method, the FNA method reduced the relative range of the feedrate from 90.8% to 4.3%, and the standard deviation also decreased from 602 to 17. Therefore, the feedrate obtained by the FNA method had better stability, which is quite important for improving the CNC machining quality of the free-form surface.