## 3 Experimental Method
Figure 4: Configuration of a working plane perpendicular to the central axis. Through the measurement and analysis process described above, the spatial coordinates \(\left(A_{x},\ A_{y},\ A_{z}\right)\) of the three-jaw chuck during measurement from the CAD model can be obtained, as well as the normal inclination angle, \(\alpha\), and offset angle, \(\gamma\), of the measurement point on each propeller blade. Here, \(\gamma\) represents the rotation angle about the z-axis of the chuck platform relative to the zeroed position. The relative coordinates of the probe \(\left(\Delta_{x},\ \Delta_{y},\ \Delta_{z}\right)\) and the advancement distance, \(T_{L}\), along the direction of the measurement axis can be calculated by combining the normal inclination angle, \(\alpha\), and the difference between the chuck coordinate, \(A_{y}\), and probe offset, \(\Delta_{y}\). Accordingly, the geometric information for multiple measurement points (four in the case of this study) can be successfully established (Figure 7). Figure 5: Construction of the tangential and normal vectors at the measurement points. Figure 6: Alignment of the measurement point with the sensor. In addition to having a graphical construction and teaching mode, the developed system supports inverse kinematics validation by using parametric formulas. The calculated parameters can be input directly to the PLC to provide precise movement instructions for each axis. When a specific point on a propeller blade, such as point A, is selected as the measurement target, its coordinates \(\left(A_{x},\ A_{y},\ A_{z}\right)\) can be obtained from the CAD model as a reference relative to the center of the propeller. The radial distance from point A to the propeller center is denoted R, and the height of the probe's rotational center is \(H_{z}\). The relative distances between the probe and point A along each axis are \(P_{x}\), \(P_{y}\), and \(P_{z}\) (distances in the x-axis, y-axis, and z-axis directions). Equations (1)-(5) are used to determine the geometric parameters of each measurement point on a propeller blade. In these equations, \(T_{0}\) denotes the initial length of the probe, and \(T_{L}\) represents the actual extension length of the probe during measurement. The parameter \(\alpha\) indicates the angle of inclination of the probe relative to the x-axis, \(\gamma\) is the angle of rotation of the blade relative to the z-axis, and \(\Delta_{y}\) is the offset between the center of the probe and the center of the propeller along the y-axis. To determine the geometric parameters of measurement point A on a blade, the rotational angle \(\gamma\) of point A with respect to the z-axis can be determined as shown in Figure 8a (top view of the propeller machining system). This angle allows for precise calcu  
Figure 7: (**a**) Measurement Point A. (**b**) Measurement Point B (**c**) Measurement Point C. (**d**) Measurement Point D.  
lation of the coordinates \(A_{x}\) and \(A_{y}\). Subsequently, through referencing of Figure 8b (side view showing the x-z geometry), Figure 8c (front view showing the \(\alpha\)-axis rotation), and Figure 8d (diagram showing the inclination angle and tangential contact point), a complete blade measurement and positioning model can be systematically established. This model effectively supports applications in automated machining and compensation control. \[Ax=R\cdot\cos(\gamma) \tag{1}\]  
\[Ay=R\cdot\sin(\gamma) \tag{2}\]  
\[\text{x-axis advancement}=P_{x}-A_{x} \tag{3}\]  
Figure 8: Composite views of the propeller machining system. (**a**) Top view of the propeller machining system (including the \(\gamma\)-axis rotation and xy plane displacement). (**b**) Side view of the propeller machining system (including x-axis and z-axis geometrical parameters). (**c**) Front view of the propeller machining system (including the \(\alpha\)-axis rotation and coordinate annotations). (**d**) Schematic of angle adjustment at the contact point between propeller and probe (\(\alpha\)-axis inclination and tangent contact).