## 1 Introduction  
The continuous small line segments are widely used in CNC machining [1; 2; 3], which are universal and easy to calculate. However, the direction of the machining movement will change abruptly at corners of adjacent line segments, which brings many difficulties to the realization of precise and efficient CNC machining [4; 5; 6]. One significant difficulty is to constrain the feedrate reasonably at corners [7; 8]. Too low a feedrate at the corner will increase the machining time, while too high will cause the machine tool to vibrate and damage the machining quality. A reasonable constraint on the feedrate at the corner of continuous small line segments is very important for achieving high-speed and high-precision machining. Two commonly used constraint methods are the angle-constraint method and the curvature-constraint method. The angle-constraint method first calculates the angle of the corner of adjacent line segments and then substitutes it into a mathematical model to obtain the corresponding feedrate-constraint value. As in the literature [9; 10], scholars assume that the magnitude of the velocity (a vector) of the machine tool remains constant during machining a corner and that the change of the velocity direction is completed within one interpolation cycle, which generates a steering acceleration during the process. Then, according to the relationship among the steering acceleration, the angle, and the velocity magnitude, the feedrate-constraint value can be achieved from the angle, under the constraint that the steering acceleration does not exceed the preset upper limit. The literature [11] assumes that there exists a virtual transition arc at the corner. Firstly, the radius of the arc is obtained according to the corner's angle and the maximum allowable value of transition error. Then, the feedrate-constraint value of the transition arc is calculated according to the radius and the preset upper limit of the normal acceleration, which will be used as the feedrate-constraint value at the corner. In the literature [12], considering that the length of each line segment is usually not exactly the sum of integer interpolation cycles, the interpolation across line segments at the corner will cause the trajectory error in machining. According to the relationship among the trajectory error, the corner's angle, and the feedrate, the feedrate-constraint value can be obtained from the angle under the constraint that the trajectory error does not exceed the preset upper limit. The angle-constraint method above has good performance for the area with long line segments in the machining path (hereafter referred to as \"Line Area\"), but for the continuous small line segments discretized from the curve (hereafter referred to as \"Curve Area\"), the calculated angle will be small and unstable due to the short line segments, which will eventually cause the feedrate-constraint value to be overlarge and unstable. The curvature-constraint method first calculates the radius of curvature of the continuous small line segments and then obtains the feedrate-constraint value based on the radius of curvature. For example, the Syntec CNC [13] uses a virtual arc to approach the path of two adjacent line segments and calculates the chord error. If the chord error is small enough, the line segments will be recognized as Curve Area and its radius of curvature will be calculated. Similarly, the literature [14] estimates the radius of curvature corresponding to the current corner based on the geometric information of several adjacent small line segments around this corner, while the literature [15] first groups small line segments by the length of the line segment and the angle of the corner and then calculates the radius of curvature corresponding to each group. After that, the feedrate-constraint value can be obtained from the radius of curvature and the preset upper limit of normal acceleration.