# Two-Dimensional Profile Simulation Model for Five-Axis CNC Machining with Bull-Nose Cutter
## 1 Introduction  
Milling is an important machining process widely used in manufacturing free-form surfaces. Bull-nose cutters have superior characteristics, such as curvature adaptiveness, so they are increasingly used in CNC machining. Applying bull-nose cutters for machining complex convex surfaces can reduce machining time and improve machining quality compared to ball-end cutters [1]. Surface topography is defined as the surface geometry of the machined workpiece after cutting, which includes the surface profile and surface quality [2] and is closely related to the fatigue strength and mating accuracy of the workpiece. The better the surface topography, the higher the workpiece's fatigue performance, corrosion resistance, stability, and reliability [3]. Further investigations have revealed that surface topography is affected by tool type, tool path [4; 5; 6], machining parameters [4; 7; 8; 9], and tool orientation [3; 8; 10; 11; 12; 13]. Because surface topography has a large impact on the performance of a workpiece, a large number of scholars have conducted research on surface topography prediction methods. Currently, surface topography prediction methods can be mainly categorized into experimental methods and analytical models. In this regard, extensive cutting experiments have been carried out, and empirical models have been established through data analysis. [PERSON] et al. [14] conducted cutting experiments on five-axis milling with bull-nose cutters, analyzed the effect of tool orientation and workpiece radius of the curvature on surface roughness and topography, and developed an empirical model. [PERSON] et al. [15] developed a neural milling process model based on experimental data to predict surface roughness during milling. [PERSON] et al. [16] analyzed experimental data and developed a surface roughness prediction model using neural networks and genetic algorithms. [PERSON] et al. [17] proposed a GAN algorithm to predict surface roughness. [PERSON] et al. [18] investigated the effects of different cutting parameters on surface roughness by experiments and proposed a mathematical model to predict the signal-to-noise (S/N) ratio. [PERSON] et al. [19] developed a mathematical relationship between the surface roughness, cutting speed, feed rate, and cutting depth based on experimental data. [PERSON] et al. [20] carried out experiments and studied the variation of surface roughness of Inconel 718 under different machining conditions and developed second-order surface roughness models to predict the surface roughness of a workpiece. It should be indicated that although these proposed models obtained by cutting experiments for topography or roughness of the surface are accurate, they are only suitable for the materials studied. Furthermore, the formation process of milling surface topography cannot be theoretically analyzed. Reviewing the literature also indicates that numerous analytical models have been proposed for surface topography. [PERSON] et al. [21] established a trajectory model for the cutting edge of a ball-end cutter, then calculated the intersection between the trajectory and the cutting plane in the vertical feed direction, and finally established a surface topography analysis and prediction model for five-axis CNC milling. [PERSON] et al. [22] established a surface roughness prediction model to study the flat-end cutter milling process, considering the runout and minimum chip thickness in the calculations. [PERSON] et al. [23] established a motion trajectory equation of the cutting edge to obtain a surface topography prediction model of flat-end and ball-end cutters. [PERSON] et al. [24] proposed an iterative method to simulate the surface topography of milling with a ball-end cutter. [PERSON] et al. [25] analyzed the motion of the cutting edge of a ball-end cutter and established a surface topography model based on the Z-Map model. [PERSON] et al. [26] established a cutting-edge trajectory model to simulate the surface topography of a micro-ball-end cutter.