## 1 Introduction
[PERSON] and [PERSON] [10] provided a method for modeling such interactions, taking into account cutting edge topography, material deformation, and cutting edge trajectory errors. Using this method, a reliable prediction of the surface topography generation in ball end milling was achieved. [PERSON] and [PERSON] et al. [12] investigated the quality of the Inconel 718 surface during the end milling process. The surface roughness measured in the feed direction was found to be lower than that in the pick direction. [PERSON] and [PERSON] et al. [13] developed a computer-supported milling simulation algorithm referred to as 'ball mill'. This was used to predict the workpiece roughness in multi-axis finishing milling using ball end tools. Based on this algorithm, considering the individual movements of the cutting tool and workpiece due to the milling kinematics, the undeformed chip geometry, cutting force components, tool deflections, and final surface topography expected can be determined. The research results of the above scholars are relatively complete and representative, which have great reference value. During the machining process, due to the close contact of tool and workpiece material, the materials abrasive, adhere to, and diffuse each other, resulting in wear. Abrasive wear occurs during the entire machining process, and the hard particles shed from the tool body form a three-body wear pattern between the tool and the workpiece surface, which has a great negative impact on processing quality. Related research shows that adjusting cutting parameters is an effective improvement method [14]. In addition, improving the lubrication conditions can significantly reduce the milling force and milling temperature [15; 16], which is the root cause of wear generation and expansion. Similarly, adding a coating to the tool also has a significant effect of reducing wear [17]. The relationship between wear and machining quality is also complicated, which can be divided into directly and indirectly affects. For example, the tool path affects the tool wear distribution characteristics and vibration characteristics, thereby affecting the machining quality [18; 19]. For difficult-to-machine materials, the above factors have a more significant impact on wear. These factors significantly affect the tool wear rate and machining quality [20]. However, the machining of the workpiece primarily depends on the geometric creation [21; 22]. The surface roughness \(R_{d}\) perpendicular to the feed direction is particularly significant [12]. These factors are closely related to other geometric factors, including the tool wear. This paper focuses on establishing the relationship between the tool wear and workpiece surface roughness with time-varying. As current relevant research results do not fully include time, tool design, and cutting parameters, these models cannot be directly used for the control prediction and tool design in the service performance. The realization of machining requirements depends on tool performance, and the tool lifetime should be further linked to the corresponding machining requirements and quality. With the progress of time, the overall research trend of surface roughness in academia has gradually shifted from the establishment of analytical models, to the application of simulation software and algorithms. [PERSON] and [PERSON] [23; 24] developed an empirical model for the prediction of surface roughness in finish turning. The workpiece material hardness, feed, cutting tool point angle, depth of cut, spindle speed, as well as cutting time were all considered in this model. A nonlinear regression analysis, along with logarithmic data transformation, was applied in the development of the empirical model. [PERSON] and [PERSON] [25] developed a mathematical model for the surface roughness in terms of the cutting speed, feed rate, and axial depth of cut in end milling (EN32M).