## 1 Introduction
Uncertainty analysis and calibration were implemented to predict the system. As evaluated by the authors of this work, the proposed measurement method can also solve the problems of coaxiality between the measuring devices and the rotary table. The system developed in this work achieves an angular error of 0.00121\({}^{\circ}\) for real workpieces, which is smaller than the error achieved by the commercial system, which is approximately 0.0022\({}^{\circ}\). In turn, in [30], the angular positioning error of the rotary axis caused by the tilt movement error and the spindle radial movement error was analyzed and experimentally verified. In addition, the offset value introduced to the encoder was analyzed and experimentally verified. Angular positioning is also an important parameter that determines the proper operation of force systems. Hence, in [31], a non-contact optical sensing method was developed to simultaneously detect the linear and angular position of the rotor of a prototype 2 DOF-SRM reluctance motor. In [32], the authors proposed a method for calibrating the angular positioning deviation of a precision rotary table of a coordinate measuring machine based on a measurement system with a laser marker. The Levenberg-Marquardt algorithm and the singular value distribution transformation were used to calculate the coordinates of the laser marker station. The fixed-interval angular deviation was calibrated using a geometric relationship model between the coordinates of the laser marker station and the rotation angle of the turntable. The work in [33] presents an angular position measurement system based on a sensor consisting of a single transmitter and two receiving coils. The mutual inductance between the transmitter coils was spatially modulated using conductive lenses. The measurement results form the basis for estimating the absolute angle. The advantage of this solution is resistance to many harmful factors, such as moisture, poor lighting, or oil contamination. Therefore, this system could be successfully used for angular positioning, e.g., indexable cutter blades. Angular positioning is also important from a military point of view. In [34], a system was developed based on an algorithm to calculate the angular position of the rocket in real time when it is ready to launch from the rocket launcher. For this purpose, video streams recorded from three different cameras were used. Finally, the algorithm was implemented on a programmable SoPC chip using FPGA programmable array gates. In addition to systems based on sensors or optics, flexible mechanical systems are also used for precise angular positioning [35]. These are usually flexible micro-angular position measurement systems. However, in order to measure along with determining the positioning of the circular cutting edge insert, it is necessary to know after what time this insert needs to be turned in the cutter body due to wear. For this, there is a need to develop tool wear and life models. In terms of multi-axis milling, [PERSON] et al. [11] proposed such a model, but it is a linear model. The linear model does not predict the variable nature of wear progression during the operational wear progression phase. Hence, in this thesis, a non-linear estimation is proposed that predicts this variable nature. The purpose of this study is to develop the basis for an angular positioning method for the circular cutting insert of a torus cutter that takes into account its wear and life. To achieve this, the parameters for the orientation of the tool axis with respect to the normal vector of the machined surface and the technique for avoiding undercutting the machined surface in multi-axis milling were first defined. A relationship was then determined for the working angle of the cutting blade, which is also a parameter controlling the orientation of the tool axis, and in its range, the wear progress of the active cutting edge segment is observed.