# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 4 Conclusions  
The milling of flat surfaces using a toroidal cutter is a multifaceted cutting process, where variations in technological parameters lead to dynamic changes in cutting conditions. Specifically, an increase in the feed per tooth modifies the specific cutting force, while alterations in the depth of cut affect the entering angle. Therefore, the dynamics of toroidal milling are not influenced solely by the cross-sectional area of the cutting layer but by a combination of additional factors. To achieve a comprehensive assessment of the dynamics of toroidal milling, an in-depth analysis was conducted based on signals from the three cutting force components F\({}_{\text{f}}\), F\({}_{\text{RN}}\), and F\({}_{\text{a}}\), while maintaining stable cutting conditions throughout the study. In order to analyze the dynamics of the process, the influence of other factors on the machining process, such as tool wear and vibrations, was eliminated. An analysis of statistical parameters of cutting force components indicated that only the mean value (MEAN) and standard deviation (STD) are influenced by the milling parameters. Both the MEAN and STD increase proportionally with increases in the feed per tooth \(\left\langle f_{z}\right\rangle\) and depth of cut \(\left(a_{p}\right)\), suggesting that they are directly related to the cross-sectional size of the cutting layer, independent of other factors affecting milling dynamics. Consequently, MEAN and STD are reliable indicators for determining the cross-sectional area of the cutting layer and can be calculated from any cutting force component. In contrast, other statistical parameters, such as kurtosis (KUR) and skewness (SKEW), show random variation and are not related to the dynamics of the milling. However, these parameters may serve as indicators for assessing the stability of the cutting process. Recurrence analysis demonstrated that distinct forms of recurrence plots were obtained for each cutting force component. However, the dynamics of the toroidal milling process could not be reliably assessed using only RP analysis, as changes in milling parameters did not significantly alter the shapes and intensity of areas of the plots. Consequently, an analysis of Recurrence Quantification Analysis (RQA) measures was conducted. Although the variation in the RQA indicators with respect to the feed per tooth \(f_{z}\) and depth of cut \(a_{p}\) was insufficient, the response surface methodology proved useful in modelling the RQA indicators. In particular, the axial force component (F\({}_{\text{a}}\)) exhibited markedly different analysis results compared to the feed force component (F\({}_{\text{f}}\)) and the normal to the feed force component (F\({}_{\text{f}}\)). For F\({}_{\text{f}}\) and F\({}_{\text{f}}\)N, three RQA indicators, DET, ENT, and VMAX, were associated with the milling parameters, although they varied in model forms and influential factors. The VMAX indicator and its \(a_{p}\) factors for each cutting force component were characterized by an increased \(p\)-value. This means that their usefulness is questionable and the real results may differ from the model ones. In contrast, the RQA indicators of the axial force component were exclusively related to the depth of cut, with DET\({}_{\text{Fa}}\), ENT\({}_{\text{Fa}}\), VMAX\({}_{\text{Fa}}\), RRFa, and LMAX\({}_{\text{Fa}}\) proving to be effective in assessing the milling dynamics. Notably, DET\({}_{\text{Fa}}\) and LMAX\({}_{\text{Fa}}\) appeared primarily dependent on the cross-sectional size of the cutting layer and their models are linear, like the cross-sectional area of the cut layer. Models of ENT\({}_{\text{Fa}}\), VMAXFa, and RRFa have quadratic factors, and this suggests their relationship with the variable entering angle. The models obtained, apart from the VMAX indicator, are characterized by low values of the \(p\)-value parameter, which indicates a high probability of obtaining results that coincide with the models.