# A Feedrate Planning Method for the NURBS Curve in CNC
## 3 Simulation and Experimental Results
### Simulation  
The curvature of the N-shaped NURBS curve shown in Figure 9 alternates between small and large serval times along the arc length, which is typical and complex enough to be used as a simulation test object to verify the effectiveness of the CCCP method. In order to improve the \"fitting effect\" of the CCC in the CCCP method, the NURBS curve shown in Figure 9 is first divided into sub-curves according to the monotonicity of the curvature and whether the feedrate-constraint value is greater than the command value (this dividing method will be used in the CCCP method, if no other dividing method specified). Then, the CCC constructing and feedrate planning are performed on these sub-curves. The final planning result is shown as \(C_{2}(s)\) in Figure 10a, and its corresponding chord error, normal acceleration and tangential acceleration are shown as subplots b, c and d in Figure 10, respectively, which do not exceed their upper limits, indicating the effectiveness of this method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(T\) & Interpolation period & 800 μs \\ \(\varepsilon\) & Chord error & 1 μm \\ \(a_{\text{n}}\) & Normal acceleration & 100 mm/s\({}^{2}\) \\ \(a_{\text{T}}\) & Tangential acceleration & 360 mm/s\({}^{2}\) \\ \(F_{\text{cmd}}\) & Command feedrate & 2000 mm/min \\ \(F_{\text{ppm}}\) & Speed of the electric spindle & \(12,000\) r/min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters in the simulation and experiment. Figure 8: The architecture of CNC machining experimental platform. The above are the results of feedrate planning of the NURBS curve in Figure 9 by using the curvature-monotonicity and command-feedrate-threshold dividing + CCCP method, while if the previously mentioned planning methods of command-feedrate-threshold dividing + minimum-constraint (CFTM) in the literature [19] and curvature-threshold dividing + minimum-constraint (CTM) in the literature [20; 21] are used, the corresponding feedrate-planning results are shown in Figure 11a,b, respectively. Figure 10: Feedrate planning results of the N-shaped NURBS curve based on CCC: (**a**) Feedrate; (**b**) chord error; (**c**) normal acceleration; and (**d**) tangential acceleration. Figure 9: N-shaped NURBS curve and its feedrate-constraint value. In Figure 11a, \(C_{2}(s)\) and \(C_{1}(s)\) do not fit closely enough, leading the poor machining efficiency, while in Figure 11b, although \(C_{2}(s)\) and \(C_{1}(s)\) fit more closely and the machining efficiency is improved, a large number of new sub-curves are generated, which increase the amount of calculation. In addition, feedrate in Figure 11b changes discontinuously, and the problem of \"stepping\" will have a negative impact on the machining quality. The relevant data such as the number of sub-curves and machining time corresponding to the planning results shown in Figure 11a,b above are shown in Table 2. If \(C_{2}(s)\) is directly valued as \(C_{1}(s)\) without considering the Acc/Dec process, the corresponding machining time is the ideal value of the actual machining time. The difference between the machining time of other planning methods and the ideal value is the time of the Acc/Dec process. In Table 2, the Acc/Dec time is reduced by 76% from 0.90 s to 0.22 s, when the minimum-constraint principle in the CFTM method is replaced by the CCCP method. While the Acc/Dec time can be reduced by 44% when the minimum-constraint principle in the CTM method is replaced by the CCCP method, and its corresponding feedrate-planning result is shown in Figure 12. While reducing the Acc/Dec time, the problem of \"stepping\" in Figure 11b can be alleviated, and the smoothness of the machining process can also be improved.