# Recurrence Quantification Analysis (RQA) of Toroidal End Tool Milling Process
## 2 Methodology of the Experiment
### Method of Analysis  
To analyze cutting forces, the classical statistical parameters were used, i.e., mean value (MEAN), standard deviation (STD), skewness (SKEW), and kurtosis (KURT). Moreover, the recurrence plot (RP) technique was engaged. This technique offers a visual method for interpreting hidden patterns in dynamical systems by reconstructing their phase space using time-delay embedding. Originally introduced by [PERSON] et al. [39], an RP is represented as a square matrix, where each point corresponds to instances where the system revisits a previous state. Specifically, the RP highlights moments (with both axes \(i\) and \(j\) representing time) when the phase space trajectory returns to nearly the same region. The parameters \(m\), \(d\), and \(\varepsilon\) denote the embedding dimension, time delay, and threshold, respectively. Detailed explanations for selecting these parameters can be found in [40; 41]. The Recurrence Plot (RP) method has evolved to incorporate quantitative measures derived from both point and line patterns within the plot, a technique known as Recurrence Quantification Analysis (RQA). RQA is a nonlinear analysis approach introduced by [PERSON] and [PERSON] [42], which quantifies recurrence patterns in the system's phase space. It includes various complexity measures based on the density of recurrence points and the structures formed by diagonal and vertical lines within the plot. More detailed information on RP and RQA techniques can be found in [41]. In this study, we evaluated all RQA measures and selected six that most effectively indicated stability in milling processes. Among these, Recurrence Rate (RR) calculates the ratio of recurrence points to all possible points in the RP, while Determinism (DET) measures the fraction of recurrence points forming diagonal lines. Additional measures include Average Diagonal Length (L), Maximum Diagonal Length (LMAX), Entropy (ENT), Trapping Time (TT), and Length of Longest Vertical Line (VMAX). The embedding parameters \(m\) and \(d\) were estimated using the average mutual information function and the false nearest neighbors method. With these parameters, RPs were created to illustrate the time instances when the system's phase space trajectory revisits a similar region. Mathematically, an RP is defined by the following matrix:  
\[M_{ij}=\theta\big{(}\varepsilon-|s_{i}-s_{j}|\big{)}, \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**C** & **Mn** & **Si** & **P** & **S** & **Cr** & **Mo** \\ \hline
0.38–0.45 & 0.6–0.9 & max 0.4 & max 0.025 & max 0.035 & 0.9–1.2 & 0.15–0.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Chemical composition of 42 CrMo4 steel (%). \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Unit** & **Value** \\ \hline Density & g/cm\({}^{3}\) & 7.83 \\ Specific heat capacity & kJ/(kg\(\cdot\)K) & 0.473 \\ Linear expansion coefficient & K\({}^{-1}\) & 11.2 \\ Thermal conductivity & kW/(cm\(\cdot\)K) & 0.42 \\ Young’s modulus & GPa & 217 \\ Hardness & HB & 220 \\ Poisson’s ratio & - & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of 42 CrMo4 steel. where \(\theta\) represents the Heaviside step function, \(\varepsilon\) is a parameter that sets the threshold, and \(s_{i}\) and \(s_{i}\) and \(s_{j}\) are delay vectors. If a recurrence is detected, then \(M_{ij}\) is assigned the value 1, otherwise it is equal to 0. These outcomes are visualized in recurrence plots (RPs), where black dots represent \(M_{ij}=1\) and white dots indicate \(M_{ij}=0\). However, interpreting patterns in RPs is not a sufficient objective, so Recurrence Quantification Analysis (RQA) is often used to provide statistical descriptions of RPs.