###### Abstract
This method's ability to lower the production cost of high-end parts is of paramount importance for the low-end machine tool industry as it unlocks the full cost-effective potential of five-axis CNC machines. Moreover, this method provides an alternative, feasible, and low-cost solution for small and medium-sized enterprises to manufacture parts with complex surface geometries. At present, several research institutions and manufacturing enterprises employ NC programming software like CATIA, UG, Master CAM, and CIMATRON to undertake programming and processing of impellor parts [10]. [PERSON] et al. [11] investigated the five-axis NC machining technology of integral impellors and leveraged the numerical control module of standard software to execute control over the impellor and attain its NC machining tool path. Generally, the research on the five-axis machining trajectory primarily centers around refining the interpolation of the five-axis trajectory while evading potential interference and collisions. In the literature, many algorithms have been developed for tool path generation and tool-gauging avoidance to achieve highly efficient tool path planning without gauges [2; 12]. [PERSON] et al. [13] established a theoretical foundation and proposed an implementation approach for attaining a multi-axis CNC system with fewer axes, along with presenting an algorithm for a multi-axis CNC system with additional axes. However, the actualization process requires the transformation of the machine tool, and the tool path planning algorithm is intricate, leading to low processing efficiency in machining complex surfaces. The two simplest approaches for tool path generation comprise using curves of constant parameters or intersection curves of the parametric surface and a series of vertical planes. However, in both the iso-parameter and iso-plane methodologies, the machining paths do not align with a constant scallop height and, as a result, lead to inadequate surface precision [14; 15; 16]. The iso-scallop machining method was proposed to address this issue; however, the algorithms are complicated. Literature studies have also investigated errors associated with five-axis machining processes [17; 18]. For example, [PERSON] et al. [19] conducted an experimental study to test the motion error of the machine axis in the five-axis machining process; the systematic errors were identified and could be effectively compensated. High-precision axial error measurement using a frequency-modulated interferometer was developed to be a suitable solution for noncontact and high-precision spindle error measurements in the machining process [20]. [PERSON] et al. [21] introduced a series of groundbreaking techniques, employing deep learning and reinforcement learning technologies in five-axis machining to predict contour errors and provide compensation. These methods accounted for non-linear issues such as clearance and friction involving the feed shaft in an adaptive manner. [PERSON] et al. [22] explored how surface properties could be utilized to screen for tool interference with the component surface in sculptured surface machining. The approach involves translating the tool contact points into tool position points; however, the computation is time consuming. [PERSON] et al. [23; 24] proposed algorithms for rapidly detecting and correcting collision between a manually pre-defined tool and an arbitrary workpiece. The workpiece is represented as a point cloud, and the tool is modeled through implicit equations. Although prior research predominantly gravitates towards optimizing the trajectory of a five-axis system, a few studies involve the machining trajectory transformation algorithm from a greater-axis system to a fewer-axis system. Aiming at the particular circumstance of the tool position data track in a five-axis system, [PERSON] et al. [25] presented a local corner transition algorithm with a global motion planning strategy that considers a more rational axial acceleration limit. [PERSON] et al.