# Using Fuzzy Control for Feed Rate Scheduling of Computer Numerical Control Machine Tools
## 4 Discussion  
The \(\infty\) shape experimental data indicate that although the method by [PERSON] et al. (2011) has the largest jerk, the cutting speed is mostly as low as 20 mm/min; therefore, it shows the best tracking error while sacrificing the processing time. The proposed method has the same processing time as the method by [PERSON] et al. (2019). The maximum tracking error is better than 0.041 mm, minimum tracking error is better than 0.006 mm, and average tracking error is better than 0.039 mm. Compared with the method by [PERSON] and [PERSON] (2010), the maximum tracking error is less than 0.004 mm, minimum tracking error is better than 0.001 mm, and average tracking error is less than 0.005 mm; however, our method requires only half the processing time. In experiment 1, the proposed system increases the cutting accuracy by 41% under the same cutting time; moreover, it decreases the cutting time by 50% under approximative same cutting accuracy (0.023 and 0.028).  
The trident shape experimental data prove that the method by [PERSON] et al. (2011) has the best path error performance, although it somewhat sacrifices the processing time.  
Figure 28: Cutting butterfly shape using our method with maximum feed rate of 185 mm/min: (**a**) feed rate, (**b**) acceleration, and (**c**) jerk.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  &
\begin{tabular}{c} **Max Speed** \\ **(mm/min)** \\ \end{tabular} & \multicolumn{3}{c}{**Tracking Error (mm)**} & \multicolumn{1}{c}{**Time (s)**} \\ \cline{2-6}  & **MAX** & **MIN** & **Average** & **Standard Deviation** & \\ \hline [PERSON] et al. (2019) & 300 & 0.09 & \(\sim\)0 & 0.041 & 0.0194 & 73 \\ \hline [PERSON] and [PERSON] (2010) & 185 & 0.061 & \(\sim\)0 & 0.029 & 0.0130 & 120 \\ \hline [PERSON] et al. (2011) & 58 & 0.028 & \(\sim\)0 & 0.017 & 0.0048 & 729 \\ \hline \multirow{3}{*}{Our method} & 300 & 0.061 & \(\sim\)0 & 0.032 & 0.0124 & 67 \\ \cline{2-6}  & 185 & 0.042 & \(\sim\)0 & 0.026 & 0.0102 & 109 \\ \cline{2-6}  & 58 & 0.024 & \(\sim\)0 & 0.015 & 0.0045 & 360 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Tracking error and processing time for butterfly shape.  
Figure 29: Cutting butterfly shapes using our method with maximum feed rate of 300 mm/min: (**a**) feed rate, (**b**) acceleration, and (**c**) jerk.  
The proposed method uses a maximum cutting speed of 300 mm/min, in which case its processing time is less than 1 min as with the method by [PERSON] et al. [9], and its maximum tracking error is better than 0.029 mm and average tracking error is better than 0.008 mm. Compared with the method by [PERSON] and [PERSON] [10], the average tracking error is poorer than 0.004 mm; yet the processing time is better than 82 s. In experiment 2, the proposed system increases the cutting accuracy by 70% under the approximative same cutting time (46 and 45). Even though our method shows higher average tracking errors in comparison with [10] and [11], the machining time can be saved around 2.8 and 6.5 times, respectively.  
The butterfly shape experimental data illustrate that the maximum tracking error of 0.004 mm is better than that of the method by [PERSON] et al. [11], average tracking error is better than 0.002 mm, and processing time is half for the same maximum speed. Under the same maximum speed as in the method by [PERSON] and [PERSON] [10], the maximum tracking error is better than 0.019 mm, average tracking error is better than 0.003 mm, and processing time is less than 11 s. Our method uses the same maximum speed as the method by [PERSON] et al. [9], and the maximum tracking error is better than 0.029 mm, average tracking error is better than 0.009 mm, and processing time is less than 6 s. In experiment 3, the average tracking error and machining time are all decreasing using the proposed method overall.