## 3 Four-Axis Machining Strategy of Five-Axis Machine Tool
### Interpolation Detection  
To confirm the viability of the interpolation program, the impeller shunted blade's machining trajectory is produced by the UG12.0 software. The interpolation algorithm is applied to realize the process section and interpolates the tool position data to verify whether the five-axis machine tool code has five-axis linkage coordinates. Partial machine tool coordinate data for the blade track processing are displayed in Table 2, while Table 3 displays the corresponding machine tool coordinate data after transformation. The conversion from five-axis linkage coordinates to four-axis linkage machine coordinates is achieved via interpolation, as can be observed from the data in the tables. The nonlinear error is a unique type of error associated with five-axis CNC machine tools and is a significant factor contributing to the machine's geometric errors. Both five-axis and four-axis linkage setups exhibit nonlinear errors due to the rotational shaft's influence. The nonlinear errors of the trajectories are carefully analyzed and compared both before and after interpolation to determine their impact on the machining process. Figure 8 delineates the interpolation intention tool of the five-axis CNC machine tool, illustrating the intention behind the interpolation process. It is noteworthy that nonlinear error typically manifests as a three-dimensional error that can be challenging to express and resolve. However, in this study, the nonlinear error is represented as a two-dimensional graph, making it more convenient to understand and analyze. The tool position data of the adjacent tool position points M and N are respectively \((x_{0},y_{0},z_{0},i_{0},i_{0},j_{0},k_{0})\) and \((x_{1},y_{1},z_{1},i_{1},j_{1},k_{1})\), and its corresponding machine tool coordinates are \((X_{0},Y_{0},Z_{0},B_{0},A_{0})\) and \((X_{1},Y_{1},Z_{1},B_{1},A_{1})\). L(t) represents the ideal interpolation trajectory, while Q(t) represents the actual track of the machine tool, which is subject to the effects of the rotation axis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**X** & **Y** & **Z** & **B** & **A** \\ \hline
150.891 & \(-\)1.827 & \(-\)38.483 & 40.823 & 336.463 \\
149.828 & \(-\)1.885 & \(-\)37.6 & 40.5 & 336.789 \\
148.746 & \(-\)1.942 & \(-\)36.712 & 40.173 & 337.115 \\
147.649 & \(-\)1.998 & \(-\)35.824 & 39.844 & 337.438 \\
146.552 & \(-\)2.048 & \(-\)34.946 & 39.516 & 337.756 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Machine tool coordinates before interpolation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**X** & **Y** & **Z** & **B** & **A** \\ \hline
150.891 & \(-\)1.827 & \(-\)38.483 & 40.823 & 336.463 \\
150.915 & \(-\)1.871 & \(-\)38.515 & 40.823 & 336.626 \\
149.805 & \(-\)1.841 & \(-\)37.568 & 40.5 & 336.626 \\
149.828 & \(-\)1.885 & \(-\)37.6 & 40.5 & 336.789 \\
149.852 & \(-\)1.929 & \(-\)37.633 & 40.5 & 336.952 \\
148.722 & \(-\)1.899 & \(-\)36.688 & 40.173 & 336.952 \\
148.746 & \(-\)1.942 & \(-\)36.712 & 40.173 & 337.115 \\
148.769 & \(-\)1.985 & \(-\)36.745 & 40.173 & 337.276 \\
147.626 & \(-\)1.955 & \(-\)35.792 & 39.844 & 337.276 \\
147.649 & \(-\)1.998 & \(-\)35.824 & 39.844 & 337.438 \\
147.672 & \(-\)2.038 & \(-\)35.856 & 39.844 & 337.597 \\
146.529 & \(-\)2.007 & \(-\)34.914 & 39.516 & 337.597 \\
146.552 & \(-\)2.048 & \(-\)34.946 & 39.516 & 33 Therefore, Equation (19) establishes the equation of the interpolation trajectory using time to as a parameter, accounting for the influence of the rotational axis on tool motion. \[ \begin{cases}X(t)=X_{0}+(X_{1}-X_{0})\times t\\ Y(t)=Y_{0}+(Y_{1}-Y_{0})\times t\\ Z(t)=Z_{0}+(Z_{1}-Z_{0})\times t\\ B(t)=B_{0}+(B_{1}-B_{0})\times t\\ A(t)=A_{0}+(A_{1}-A_{0})\times t\end{cases}  \tag{19}\]  
The maximum deviation amount \(\varepsilon_{max}\) of Q(t) from the ideal tool point trajectory L(t) between adjacent tool points can be estimated as the nonlinear error. The actual tool tip trajectory Q(t) is determined using the machine tool's kinematic transformation model, following the process outlined below.