## 1 Introduction  
With the development of manufacturing technology, the requirement of machining efficiency and quality is becoming more and more important in the field of molds, impellers, aerospace and other complex parts [1]. Some commercial computer-aided design (CAD)/ computer-aided manufacturing (CAM) software is also widely used to generate the tool path of complex parts, which is usually represented by straight-line segments [2, 3]. However, due to the discontinuity of the tangent and curvature of the corner point between the linear tool paths, the speed, acceleration and jerk are discontinuous and the phenomena of acceleration and deceleration occur frequently. These factors will reduce the machining efficiency and produce violent vibration, which will also affect the machining quality and accuracy [4]. Therefore, it is of great significance to improve the smooth transition performance of the linear tool path. In recent years, the corner smoothing method has been an effective method for achieving a high-speed continuous motion along a linear tool path, which has been widely studied. [PERSON] et al. [5] and [PERSON] et al. [6] inserted an arc in the middle of the double linear tool path to make the tool path reach G\({}^{1}\) continuity (i.e., speed-continuous). Subsequently, higher-order curves, such as Bezier, B-spline, PH or other curves were used to smooth corners and achieve G\({}^{2}\) continuity (i.e., curvature-continuous), which can obtain good acceleration characteristics and low machining vibration. [PERSON] et al. [7] developed a curvature smoothing algorithm based on a cubic Bezier curve to smooth the linear tool path of high-speed machining. This method realized G\({}^{2}\) continuity and improved the feed speed. A similar curve with eight control points lying in the straight segment and inside the tool path was constructed to smooth the linear tool path in [PERSON] et al. [8] and they further optimize the curvature variation energy (CVE) to obtain good motion performance. The similar distribution of control points also appears in the following literature. [PERSON] et al. [9] utilized triple B-spline to connect adjacent liner tool paths and achieve G\({}^{2}\) continuity. The accuracy of the tool path and the smoothness of the feed speed were improved. [PERSON] et al. [10], [PERSON] et al. [11] and [PERSON] et al. [12] also used the same B-spline for realizing smaller curvature extrema to increase the transition speed. [PERSON] et al. [13] used quintic B-spline to smooth the path corners to ensure G\({}^{2}\) continuity and optimal curvature. [PERSON] [14] developed a corner smoothing algorithm with G\({}^{2}\) continuity based on the quintic PH curve. [PERSON] et al. [15] also used the quintic PH curve to smooth the corners of a three-axis tool path, which can accurately limit the approximation error. [PERSON] et al. [16] applied clothoid splines to develop a novel corner smoothing method with G\({}^{2}\) continuity. In addition, some scholars [17; 18; 19; 20; 21] found that there are potentials of the G\({}^{3}\) (i.e., curvature-smooth) continuous tool path in suppressing machining vibration in experiments. [PERSON] et al. [17] used two quartic Bezier curves to obtain a G\({}^{3}\) continuous tool path where the curvature variation energy was optimized and the G\({}^{3}\) tool path could generate smaller jerk than a G\({}^{2}\) tool path. Similarly, in our previous work, we introduced a method of generating a G\({}^{3}\) continuous tool path from a symmetric Bezier curve [18]. [PERSON] et al. [19] and [PERSON] et al. [20] applied quintic B-spline curves to obtain tool paths with G\({}^{3}\) continuity. An analytical NURBS curve was inserted into the corners between linear segments for improving the smoothness of the linear toolpath and the machining efficiency [21]. In addition, others also extend the smoothing methods with G\({}^{3}\) continuity to four-axis and five-axis tool paths [22; 23; 24]. [PERSON] et al.