# A Feedrate Planning Method for the NURBS Curve in CNC
###### Abstract  
The curvature of the NURBS curve varies along its trajectory, therefore, the commonly used feedrate-planning method, which based on the acceleration/deceleration (Acc/Dec) model, is difficult to be directly applied in CNC machining of a NURBS curve. To address this problem, a feedrate-planning method based on the critical constraint curve of the feedrate (CCC) is proposed. Firstly, the problems of existing feedrate-planning methods and their causes are analyzed. Secondly, by considering both the curvature constraint and the kinematic constraint during the Acc/Dec process, the concept of CCC which represents the relationship between the critical feedrate-constraint value and the arc length is proposed. Then the CCC of a NURBS curve is constructed, and it has a concise expression conforming to the Acc/Dec model. Finally, a feedrate-planning method of a NURBS curve based on CCC and the Acc/Dec model is established. In the simulation, a comparison between the proposed method and the conventional feedrate-planning method is performed, and the results show that, the proposed method can reduce the Acc/Dec time by over 40%, while little computational burden being added. The machining experimental results validate the real-time performance and stability of the proposed method, and also the machining quality is verified. The proposed method offers an effective feedrate-planning strategy for a NURBS curve in CNC machining. NURBS; curvature; feedrate-constraint value; feedrate planning; acceleration/deceleration  
8 May 2021  
26 May 2021  
27 May 2021  
[table][PERSON] et al. [13] planned the feedrate and calculated interpolation points along the curve in each interpolation period with the local optimization of machining efficiency as the goal, under the constraints of chord error, normal acceleration and others [14; 15]. However, when a constraint-satisfied feedrate cannot be planned at the current interpolation point, the strategy of backing off for a certain distance and replanning was adopted, which resulted in a large amount of calculations and it was difficult to be used in real-time CNC machining. [PERSON] et al. [16] divided a NURBS curve into multiple windows, and tried using a multi-core CPU to perform parallel calculations on these windows to reduce the total calculation time. However, the point-by-point planning method was still used to plan the feedrate in a single window. [PERSON] et al. [17] first generated an initial feedrate curve, and then adjusted the feedrate for the sample points on that curve that did not satisfy the constraint by using an iterative approach until the feedrate of all sample points met the constraint. The above point-by-point planning method suffers from the problem of requiring multiple adjustments and replanning, so the real-time performance of the algorithm cannot be guaranteed. The segment-by-segment planning method divides a NURBS curve into several sub-curves, which are taken as calculation units, and the Acc/Dec model is used to plan the feedrate in each unit. [PERSON] et al. [18] divided a NURBS curve into sub-curves at the points with the local maximum curvature, and calculated the feedrate-constraint value at junctions between different sub-curves, then the feedrate was planned accordingly. However, there is a risk that the feedrate will exceed the limit in the inner region of each sub-curve, since this algorithm only constrained the feedrate at the junctions of sub-curves. [PERSON] et al. [19] divided the regions of the NURBS curve of which the feedrate-constraint value was less than the command value into independent sub-curves, calculated the feedrate-constraint value of each sub-curve according to its minimum radius of curvature (hereinafter referred to as \"minimum-constraint principle\"), and then finally planned the feedrate. Although this method ensures that the feedrate does not exceed the limit, the minimum-constraint principle reduces machining efficiency due to its conservation. [PERSON] et al. [20] and [PERSON] et al.