# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 1 Introduction
[PERSON]'s study [29] explored the potential of using Recurrence Quantification Analysis (RQA) indicators to assess the stability of the milling process for an aluminum alloy. In these experiments, cutting tests were conducted with a gradually increasing depth of cut. The study revealed that five recursive measures--DET/RR, LAM/DET, LENT, DIV, and VENT--are correlated with changes in the stability of the milling process and are suitable for process monitoring. Meanwhile, [PERSON] et al. investigated the relationship between RQA indicators and the dynamics of the milling process [31]. Their research was carried out during a milling process of a composite material, with the spindle speed being the variable process parameter. Changes in the spindle speed induced vibrations and the resulting change in the DET indicator exhibited a nonmonotonic trend. The study concluded that the DET indicator is effective for evaluating the dynamics of the milling process, particularly in mitigating vibrations at higher harmonics. Furthermore, cross-recurrence plot analysis (CRQA) was employed to assess the stability of the turning process [28]. Six CRQA indicators were tested, and the results demonstrated that the parameter values change rapidly upon the onset of vibrations. This analysis, based on spindle power signals, used vibration data solely for identifying process stability. From these findings, it can be inferred that the type of material being machined does not significantly affect the relationship between the RQA indicators and the stability of the cutting process. Furthermore, a study by the authors [32] established that six fundamental RQA indicators are related to the stability of milling processes for the Inconel 718 alloy. In particular, with the occurrence of vibrations, the parameter values exhibited significant increases or decreases. Recursive analysis can be used not only to monitor the stability of the cutting process but also to detect defects during milling. [PERSON], [PERSON] et al. investigated the application of recurrence plots and RQA indicators for defect detection in the milling process. Their study demonstrated that indicators such as DET, TT, VMAX, RPDE, and recursive times T1 and T2 are effective in identifying material defects [25]. In contrast, another study [26] indicated that a larger set of indicators exhibit a relationship with the occurrence of defects. These indicators include LAM, DET, TRANS, L, ENTR, RPDE, CC, TT, T1, and T2. In particular, these indicators show significant changes in their values at the location of the defect, with variations of up to 60%. Correlation studies between milling technological indicators and RQA measures were carried out by [PERSON] [33]. He demonstrated that the recursive indicators L and LAM were related to the feed per revolution and cutting speed. Furthermore, the LAM indicator showed a correlation with surface roughness. The study was performed during the milling of thin-walled composite parts, although no statistical analysis was conducted on the results. In a subsequent study, [PERSON] explored the use of RQA indices to detect the type of tool and material composition [27]. He found that the DET, L, and ENTR indices were sensitive to variations in both the type of tool and the grade of composite material. The face milling process has also been analyzed using recursive analysis. [PERSON] et al. examined the potential to identify transient and steady states by cutting through the RQA indicators derived from the vibration signal [34]. Their study proved that the RR, DET, TT, and LAM indicators exhibited value changes corresponding to shifts in the state of the cutting process. These changes were primarily attributed to variations in the cutting speed and depth of cut. However, the correlation between RQA indicators and cutting parameters was not investigated. Recursive analysis has also been employed to assess surface roughness after turning [35]. In this study, the signal used was acoustic emission.