However, these direct measurement methods and instruments still cannot directly obtain the 21 geometric errors of the three linear axes of machine tools. On the other hand, a sensitivity analysis has been employed to identify the key geometric error factors that have a significant impact on machine tool accuracy. Once these key factors are determined, they can be measured and compensated, thereby reducing the number of geometric errors that require measurement and improving error compensation efficiency. This approach is known as key error identification, which focuses on studying the influence of error elements on machine tool accuracy and serves as the basis for machine tool design and manufacturing. [PERSON] [16] utilized measurement error data and error modeling to identify key errors in three-axis CNC machine tools. [PERSON] [17] proposed a sensitivity coefficient model for 3-RPS symmetrical parallel robot leg based on a normalized description. [PERSON] [18] developed a rapid modeling and compensation method for volumetric errors using the global sensitivity analysis method. Although these methods ultimately reduce the number of errors that need to be measured, it is still necessary to obtain the 21 geometric errors before identifying the key geometric error elements. Therefore, the rapid and high-precision acquisition of the 21 geometric errors of the three linear axes plays a crucial role in conducting critical error identification and subsequent error compensation. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Geometric Errors** & **X-Axis** & **Y-Axis** & **Z-Axis** \\ \hline Position error & \(\delta_{x1(X)}\) & \(\delta_{y(Y)}\) & \(\delta_{x1(Z)}\) \\ Horizontal straightness & \(\delta_{y(X)}\) & \(\delta_{x1(Y)}\) & \(\delta_{y(Z)}\) \\ Vertical straightness & \(\delta_{x1(X)}\) & \(\delta_{x1(Y)}\) & \(\delta_{x1(Z)}\) \\ Yaw & \(\varepsilon_{x(X)}\) & \(\varepsilon_{x(Y)}\) & \(\varepsilon_{x1(Z)}\) \\ Pitch & \(\varepsilon_{y(X)}\) & \(\varepsilon_{x(Y)}\) & \(\varepsilon_{y(Z)}\) \\ Roll & \(\varepsilon_{x(X)}\) & \(\varepsilon_{y(Y)}\) & \(\varepsilon_{x(Z)}\) \\ Squareness error & \(S_{XY}\), \(S_{ZX}\), \(S_{YZ}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: 21 geometric errors for the three linear axes of machine tools. We have proposed direct methods for simultaneously measuring six-degree-of-freedom geometric errors of linear guides [19; 20]. Based on these, we first proposed a 21-geometric-error measurement method and system for the three linear axes with one-step installation and developed a measurement system, which enabled the direct, rapid, and high-precision acquisition of all 21 geometric errors [21]. Thus, to efficiently identify the key error factors that have a significant impact on machine tool accuracy, this paper first utilizes the 21 geometric errors obtained from the developed measurement system to determine the volumetric errors of the machine tool. Then, a sensitivity analysis model for these 21 geometric errors is built based on the global sensitivity analysis method. Finally, experimental measurements of the 21 geometric errors for the three linear axes of the vertical machining tool TH5656 were conducted, achieving the rapid and accurate identification of the key error factors.