## 1 Introduction  
To machine free-form parts used widely in aerospace, die and mould industries, cutter of the machine tool is required to follow the programmed toolpath under computer numerical control (CNC) commands [1; 2]. The curvilinear toolpath generated by computer-aided manufacturing (CAM) software is usually approximated by successive linear segments (linear-segment toolpath or G01 blocks) [3]. However, the transition corners between adjacent linear segments lead to tangential and curvature discontinuities, which cause fluctuation in the feedrate, acceleration, and jerk commands in the feedrate scheduling process that generates machine tool vibration and poor surface finish [4]. To avoid this behavior, the continuity of the linear-segment toolpath needs to be improved. Several approaches have been proposed in the existing literature for three-axis linear-segment toolpathsmoothing, which can be categorized as local corner-smoothing method and global curve-fitting method. The local corner-smoothing method utilizes parametric curves to locally replace the sharp corners, such as B-spline [5; 6; 7], Bezier [8; 9; 10], and Pythagorean-hodograph (PH) curves [11; 12]. [PERSON] et al. [7] propose G4-continuous toolpath via a quintic B-spline curve. Optimized by curvature variation energy (CVE), the chord error is seriously constrained and curvature extrema can be analytically calculated. Then the jerk-smooth feedrate scheduling scheme is developed based on the bidirectional scanning algorithm. [PERSON] et al. [10] generated a G4 interpolative trajectory model using a symmetric nine-degree Bezier curve with confined chord error and analytical curvature extrema for trajectory smoothing, and employs jerk-smooth feedrate mode to perform time-optimal feedrate scheduling. [PERSON] et al. [12] adopt a curvature-continuous PH curve as a transition to blend sharp corners. The blending algorithm can guarantee the approximation error exactly. The control points, the arc length and the curvature of the transition curve also have analytical expressions. The high-order geometric continuity leads to high-smoothness toolpath and feedrate profile. But the rising complexity may prevent these methods from promoting to five-axis processing. On the other hand, the global curve-fitting method utilizes a parametric curve to smooth the linear-segment toolpath globally by interpolating or approximating G01 data points. However, this method suffers from below-standard fitting error. [PERSON] et al. [13] propose a general, fast and robust B-spline fitting scheme with PIA method to generate G2 tool path under confined chord error for high speed interpolation of micro-line tool path. [PERSON] et al. [14] propose a progressive and iterative approximation for least squares incorporating energy term (ELSPIA) to deal with chord error constraint. [PERSON] et al. [15] derive the explicit expression for the Hausdorff distance between a line segment and a curve segment, then a time-parameterized curve-fitting algorithm is presented to combine path smoothing and trajectory planning. Five-axis CNC machine tools provide better accessibility with the help of additional rotary axes [16; 17]. The smoothing of five-axis linear-segment toolpath brings about extra difficulties compared to three-axis. First, the tool orientation must be smoothed to ensure smooth rotary motion in addition to the tool-tip position smoothing. Second, given that the tool-tip position and tool orientation are smoothed independently, the two smoothed trajectories must be synchronized [18]. The local corner-smoothing method and the global curve-fitting method mentioned above are extended and used to smooth tool-tip position and tool orientation respectively. The local corner-smoothing method applied to five-axis linear-segment toolpath can guarantee high-order continuity while respect error tolerance and synchronization of the tool-tip position and tool orientation is achieved by adjusting the joint points of the linear and curvilinear segments [19; 20; 21].