# A Feedrate Planning Method for the NURBS Curve in CNC
## 4 Conclusions  
The feedrate-constraint value of the NURBS curve varies along the arc length, which makes the feedrate planning in the CNC machining of the NURBS curve difficult. For this reason, this paper analyzes the problems and corresponding causes of feedrate-planning methods in literatures, and proposes the CCCP method.  
Figure 14: Machining result of the butterfly-shaped NURBS curve.  
Figure 15: Machining result of apple-shaped NURBS surface.  
In this method, a composite feedrate-constraint curve CCC with concise expressions is constructed, in which both the geometric and kinematic constraints of the feedrate are considered. Then, the feedrate planning of the NURBS curve is carried out based on CCC and the Acc/Dec model. The CCCP method avoids the iterative calculation of the point-by-point planning method, and also does not have the conservative problem caused by the minimum-constraint principle in the segment-by-segment planning method. Therefore, the CCCP method can balance the computational and machining efficiency.  
The simulations of N/butterfly-shaped NURBS curves show that, without increasing the number of sub-curves, the Acc/Dec time can be respectively reduced by 44% and 42% for replacing the minimum-constraint principle in the CTM method with the CCCP method. The experimental results verify the real-time performance and stability of the CCCP method. Moreover, the machining quality can also be guaranteed for both free-form curves and surfaces. Hence, the CCCP method can balance the computational and machining efficiency, while ensuring the machining quality. For the parametric interpolation of NURBS curves, the CCCP method proposed in this paper offers an effective strategy to plan the feedrate, which will promote the application of NURBS curves in CNC machining.  
In view of the limited research time, the CCCP method presented in this paper is currently implemented in single NURBS curve CNC machining in a 3-axis machine tool, and has not been applied to double NURBS curve paths in 5-axis CNC machining, which will be the focus of further research.  
Conceptualization, P.G., Y.W. and G.Y.; methodology, P.G. and G.Y.; software, P.G.; validation, P.G., Z.S., H.Z. and P.Z.; writing--original draft preparation, P.G. and G.Y.; writing--review and editing, P.G., Y.W. and F.L.; visualization, H.L.; project administration, Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (No. 51275462) and Science and Technology Innovation 2025 Major Project of Ningbo (No. 2018B10069).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.  
Non-uniform rational B-splines  
AccAcceleration  
DecDeceleration  
CCCA critical constraint curve of the feedrate  
CCPThe CCC-based feedrate-planning method  
\(C_{1}(s)\)A curve with the arc length as the abscissa value and the feedrate-constraint  
value as the ordinate value  
\(C_{2}(s)\)A curve with the arc length as the abscissa value and the feedrate value as  
the ordinate value  
CFTMA feedrate-planning method base on command-feedrate-threshold dividing  
+ minimum-constraint principle  
CTMA feedrate-planning method base on curvature-threshold dividing +  
minimum-constraint principle  
\(T\)Interpolation period, us  
\(\rho\)Radius of curvature, mm  
\(\epsilon\)Chord error, \(\mu\)m  
\(a_{n}\)Normal acceleration, mm/s\({}^{2}\)  
\(a_{\rm T}\)Tangential acceleration, mm/s\({}^{2}\)