## 4 Results and Discussion
### Tool Wear and Tool Life Models  
The development of a concept for an angular positioning method for the round cutting insert of a torus milling cutter begins the wear modeling stage. When modeling tool wear, all three phases of the wear curve were taken into account. This is evident from the resulting classic three-phase wear curve that was obtained for the adopted machining conditions, as shown in Figure 12a. To clarify, it should be mentioned that this drawing shows the first of the three machining tests carried out. The higher number of tests performed is due to the generally accepted principles of statistical analysis and to increase the accuracy of the final model.  
Figure 11: General procedure for the angular positioning of the RCI.  
As mentioned, wear modeling was carried out using an advanced non-linear ANOVA estimation, as the data obtained cannot be transformed into a linear model. Non-linear estimation involves calculating the relationship between a set of independent variables (i.e., cutting time \(t_{c}\)) and the dependent variable (i.e., tool wear \(VB_{Bmax}\)). It was assumed that the wear was evenly distributed over the individual cutting blades, i.e., only one cutting blade is active in the individual feed motion sequences, as also observed by [PERSON] [41].  
Analyzing the resulting data set, it was found that it could be approximated by an exponential function model, as described by Equation (10). This is an assumed estimated function. The Least-Squares Method (LSM) was adopted as the loss function, while the Levenberg-Marquardt algorithm was used as the estimation method. The method solves a system of linear equations in each iteration to calculate the gradient. A maximum number of iterations of 6000 was adopted, with a convergence criterion of six decimal places. Based on the above, a model relationship was obtained, which is described by Equation (15).  
\[VB_{Bmax}=0.0685797\cdot 1.06903^{t_{c}}, \tag{15}\]  
Table 5 shows the estimation statistics and results of the ANOVA analysis of variance.  
Figure 12: Results of non-linear estimation: (**a**) wear curve; (**b**) fitted model function and observed values; (**c**) scatter plot; (**d**) error normality chart.  
However, Figure 12b-d show the fitted model function and the observed values, a scatter plot of the observed values against the predicted values, and a probability normality plot, respectively.  
Based on the results obtained and the analysis of variance, it was concluded that, within the assumed range of variables under study, the developed model is adequate (\(F=344.5687\)) and approximates the predictor variable, i.e., tool wear during cutting, with an accuracy of 90%. This is evidenced by a determination coefficient of \(R=0.90\). Furthermore, in the scatter plot (Figure 12c), the points lay along a straight line, indicating that the model is appropriate and that the residuals (i.e., errors) are subject to a normal distribution, which, in turn, is illustrated in Figure 12d.  
Subsequently, a tool life model was developed for HSM machining, that is, in the range adopted in this work according to Table 2. As a result of the machining tests, the so-called calibration, and after performing the necessary calculations, the tool life is described by Equation (16).  
\[T_{c,0.00075}=4.2\cdot 10^{6.2}\left(\frac{2\pi n\left(R_{T}+\sqrt{2r_{p}g-g^{2} }\right)}{1000}\right)^{3.561149}f^{1.931468}, \tag{16}\]  
Again, an adequate model was obtained, since for an assumed threshold of \(VB_{Blim}=0.2\) mm, the lifetime \(T_{c}\) for the \(CP_{i}\) point (\(g=0.00075\) mm) is 16.353177 min.