# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 1 Introduction  
Milling is one of the most common machining methods in industrial practice. One type of this process is face milling, which is carried out using face milling cutters [1]. These cutters are characterized by an entering angle of less than 90\({}^{\circ}\), which means that they do not have cutting edges on the cylindrical surface. The toroidal milling cutter is a specific type of face milling tool, which is equipped with round cutting inserts. Its distinguishing feature is the fact that this cutter does not have a fixed entering angle. Moreover, the thickness of the cutting layer depends on the feed or the depth of the cut. As the depth of the cut increases, so does the thickness of the cutting layer [1]. Furthermore, changes in the depth of the cut also result in variations in the entering angle (Figure 1). As the entering angle and the cross-sectional area of the cutting layer change, the cutting conditions also vary [2,3]. This variation, in turn, affects the dynamics of the milling process, particularly the relative proportions of the cutting force components. Consequently,the load on the tool changes in both the axial and radial directions. Since the axial and radial tool stiffnesses are different, the direction of the resultant cutting force has a significant effect on tool stability. Therefore, selecting appropriate milling parameters--especially the feed rate and depth of cut--is crucial, as these parameters play a decisive role in the efficiency and stability of the milling process. The dynamics of the milling process have a crucial impact on the surface quality of the machined material [4; 5; 6]. In toroidal milling, the key cutting parameters are the depth of cut and the feed per tooth, both of which influence the cross-sectional area of the cutting layer. And these factors in turn determine the magnitude of the cutting force and the ratios of its components. The cutting force is a fundamental parameter of milling processes [7] and is frequently used to study process performance [8]. However, the analysis of cutting force components in milling can be a challenging task due to the interrupted action of the cutting blades, which causes signal fluctuations [9]. Consequently, extensive research has been dedicated to analyzing force signals, and various signal measures and statistical parameters have been proposed [10]. One of these methods is recursive analysis, which is capable of studying time-varying signals. Recurrence analysis and Recurrence Plot (RP) indicators are widely used in the literature to analyze problems across various scientific fields (Figure 2). These methods, originally popular in medicine, have also been adopted in areas such as biology, mathematics, physics, engineering, and chemistry [11; 12; 13; 14; 15; 16; 17]. Figure 1: The influence of the depth of the cut on the cutting layer and entering angle: (**a**) \(a_{p}\) equal 15% of insert radius, (**b**) \(a_{p}\) equal 75% of insert radius. Figure 2: Number of journal articles with recurrence plot keywords divided into scientific fields published from 1980 to 2024 \"[[https://www.lens.org/](https://www.lens.org/)]([https://www.lens.org/](https://www.lens.org/)) (accessed on 12 November 2024)\". In the context of cutting processes, Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA) can be applied to classical machining operations such as turning, milling, and drilling [18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. These statistical methods are used to analyze signals from the cutting process, typically including components of the cutting force, vibration, and cutting power [20; 28]. However, they have been used most frequently to investigate process stability [4; 29]. The identification of vibrations in cutting processes using recursive analysis has been the focus of numerous recent studies. [PERSON] et al. demonstrated that the recurrence plot analysis of cutting force components can effectively detect vibrations in the machining process [30].