## 2 Experimental
### Center-of-Rotation Error Compensation  
Figure 1 presents the horizontal motion or rotation directions of each axis of the five-axis machine tool. Before the five-axis machine tool was used for machining, its center of rotation had to be corrected. If the center of rotation was inaccurate, errors could occur because of the cutting table's center-of-rotation positions recorded on the control system and the actual positions being different; such errors would divert the cutting track and affect machine precision of workpiece dimensions. Therefore, this study referenced the homogeneous transformation matrix proposed by [PERSON] et al. [32] to create formulas for calculating center-of-rotation errors, as well as to compensate errors in center-of-rotation positions along the \(A\) and \(C\) axes to mitigate the errors' effect on the precision of workpiece dimensions. During simultaneous machining using the five-axis machine tool, the center-of-rotation error along the \(A\) axis most directly affected the tool length. To simplify formula calculations, \(d_{y_{A}}\) and \(d_{z_{A}}\) were defined as the error values of \(Y\) and \(Z\) components of the center of rotation of the five-axis machine tool's \(A\) axis, respectively; in Figure 2, \(\theta_{A}\) is the angle of rotation of the \(A\) axis. In Figure 2, the circular dotted line with a circular center O(0,0) on the \(YZ\) plane represents a cutting track without any center-of-rotation errors along the \(A\) axis; by contrast, the solid line represents a cutting track with \(d_{y_{A}}\) and \(d_{z_{A}}\) errors along the \(A\) axis. The inferred formulas for compensating center-of-rotation errors along the \(A\) axis are presented in (2) and (3):  
Error along the \(y\)-axis:  
\[\Delta Y=-d_{y_{A}}+cos[\theta_{A}]d_{y_{A}}+sin[\theta_{A}]d_{z_{A}} \tag{2}\]  
Error along the \(Z\) axis:  
\[\Delta Z=-sin[\theta_{A}]d_{y_{A}}-d_{z_{A}}+cos[\theta_{A}]d_{z_{A}} \tag{3}\]The measured depth of cut differences \(D_{1}\) and \(D_{2}\) and two known angles of rotation, \(-45^{\circ}\) and \(-90^{\circ}\), were substituted into (3), where \(D_{1}\) and \(D_{2}\) had identical physical meanings to that of the error along the \(Z\) axis (\(\Delta\)Z), to determine the center-of-rotation error values along the \(A\) axis (\(d_{y_{A}}\) and \(d_{z_{A}}\)). This study used a \(5\,\mathrm{mm}\) ball end mill and set the table's angles of rotation along the \(A\) axis at \(0^{\circ}\), \(-45^{\circ}\), and \(-90^{\circ}\) to cut a groove on the workpiece surface. If the center of rotation of the \(A\) axis presented no error, the cut groove generated no difference in the depth of cut, as shown in Figure 3a,b. The depth of cut differences of the cut grooves were measured using 3D-Probe (Haff&Schneider, Analogue). The depth of cut at \(0^{\circ}\) was used as the base point (no error occurred at \(0^{\circ}\) along the \(A\) axis). The depth of cut values at \(-45^{\circ}\) and \(-90^{\circ}\) were measured (\(D_{1}\) and \(D_{2}\), respectively), as shown in Figure 3c. In addition, during the five-axis machine tool's cutting process, any center-of-rotation error along the \(C\) axis directly affects the eccentric state of machined workpieces. To simplify the calculation of the inferred formula, \(d_{x_{c}}\) and \(d_{y_{c}}\) were defined as center-of-rotation errors along the \(C\) axis and \(\theta_{c}\) was the angle of rotation along the \(C\) axis, as shown in Figure 4.  
In Figure 4, the circular dotted line with a circular center O(0,0) on the \(XY\) plane represents a cutting track without any center-of-rotation errors along the \(C\) axis; by contrast, the solid line represents a cutting track with center-of-rotation errors of \(d_{x_{c}}\) and \(d_{y_{c}}\) along the \(C\) axis. The inferred formulas for compensating center-of-rotation errors along the \(C\) axis are presented in (4) and (5):  
Figure 1: Direction of each axis of the five-axis machine tool (UX300). Figure 2: Center-of-rotation errors of the \(A\) axis.