## 2 Fundamentals of the APofRCI Method, Taking into Account Torus Milling Cutter Cutting Blade Wear
### Method to Avoid Undercutting the Machinted Surface  
In order to develop an angular positioning method for round cutting inserts of the torus milling cutter, dedicated to the multi-axis milling process, it is first necessary to determine the minimum required inclination angle \(\delta\). The minimum inclination angle required is to ensure that the torus milling cutter axis is positioned in such a way that no undercutting of the workpiece surface occurs. Three surface types were selected in terms of shape: biconcave, saddle concave-convex, and saddle convex-concave. The selected sculptured surfaces are described by a radius of curvature radius \(\rho\) parameter (the inverse of the curve curvature) in the feed direction \(\rho_{1}\) and in the direction perpendicular to the feed \(\rho_{2}\), respectively. A negative sign was assumed for the radius of convex curvature and a positive sign for the radius of concave curvature, as illustrated in Figure 3.  
Figure 3: Types of sculptured surfaces.  
Figure 2: Possible cases of the round cutting insert rotation: (**a**) overlapping of a new segment with a worn segment—wrong; (**b**) too large a gap between segments—wrong; (**c**) segments adjacent to each other—correct.  
A sphere of radius \(R\) was used to determine the position of the tool in a way that excludes the possibility of the torus undercutting the machined surface. This method was first defined by [PERSON] (2016). The common part of the sphere and the torus is the contact diameter, and the local position of the tool determines the contact between the torus and the machined surface at the \(CP_{i}\) contact point, as shown in Figure 4. The inclination angle \(\delta\) of the tool axis is defined relatively to the normal vector \(n(CP_{i})\) of the machined surface at the contact point \(CP_{i}\). In generalizing the problem, the machined surface was assumed to be a strictly tangential plane. This plane, in turn, is a simplified model of any type of sculptured surface at the contact point between the torus and the machined surface.  
The radius of sphere R can be determined using Equation (3).  
\[R=\frac{R_{T}}{\sin(\delta)}+r_{p}, \tag{3}\]  
where \(R_{T}\) is the radius of the torus milling cutter (the tool rotation axis of distance of the torus from the center of the RCI outline defining it); \(\delta\) is the inclination angle; \(r_{p}\) is the radius of the round cutting insert (the circle defining the torus).  
The undercutting of the machined surface by the torus does not occur when Equation (4) is fulfilled.  
\[R\leq\rho_{1},\rho_{2}, \tag{4}\]  
where \(\rho_{1}\) is the smallest radius of concave curvature in the feed direction; \(\rho_{1}\) is the smallest radius of concave curvature in the step direction.  
The above condition must be met for concave contours (for which the curvature has a positive sign). For convex contours, for \(\delta\geq 0^{\circ}\), undercutting does not occur.  
Hence, Equation (5) applies to calculate the minimum inclination angle \(\delta_{min}\), depending on the nominal diameter.  
\[\delta_{min}=\sin^{-1}\left(\frac{R_{T}}{\rho-r_{p}}\right), \tag{5}\]  
It is assumed that there are no large changes in curvature in the immediate vicinity of the contact point \(CP_{i}\) in the CWE that could lead to undercutting. Therefore, once calculated, the \(\delta_{min}\) parameter is used as a constant value along the entire contour of the machined surface, and the tool axis is oriented in the Cartesian space relative to the normal vector \(n(CP_{i})\) of the machined contour.  
Figure 4: Method of detecting undercutting of the machined surface by the torus of the cutter.