## 2 Sensitivity Analysis for 21 Geometric Errors of Three Linear Axes of Machine Tools  
The 21 geometric errors of the three linear axes of machine tools can be obtained through a one-step installation and three-step automated measurement process using our previously developed measurement system [21]. Once these 21 geometric errors are determined, it becomes possible to establish the error compensation model using a homogeneous coordinate transformation matrix. By synthesizing error elements, the volumetric errors of the machine tool can be determined, which refers to the maximum deviation between the actual and ideal positions of a cutting tool in the \(X\)-, \(Y\)-, and \(Z\)-directions of a machining workspace. Take a typical vertical machine tool as an example. Its model for volumetric errors (\(\Delta x\), \(\Delta y\), \(\Delta z\)) in the \(X\)-, \(Y\)-, and \(Z\)-directions is given by [22]:  
\[\left\{ \begin{array}{l}\Delta x=-\delta_{x(X)}-\delta_{x(Y)}+\delta_{x(Z)}- y\varepsilon_{z(X)}-2\varepsilon_{y(X)}-2\varepsilon_{y(Y)}+yS_{XY}+zS_{XZ}\\ \Delta y=-\delta_{y(X)}-\delta_{y(Y)}+\delta_{y(Z)}+2\varepsilon_{x(X)}+2 \varepsilon_{x(Y)}-zS_{yZ}\\ \Delta z=-\delta_{z(X)}-\delta_{z(Y)}+\delta_{z(Z)}+y\varepsilon_{x(X)}\\ \end{array} \right. \tag{1}\]  
To identify the key geometric error elements that have a significant impact on machine accuracy, a sensitivity analysis of the 21 geometric error elements is conducted based on the global sensitivity analysis method [18]. This involves calculating the partial derivatives of the volumetric errors (\(\Delta x\), \(\Delta y\), \(\Delta z\)) with respect to each of the 21 geometric error elements, resulting in the derivation of a sensitivity coefficient matrix \(K\) for volumetric errors  
\[K= \begin{bmatrix}\frac{\partial\Delta x}{\partial x_{(X)}}&\frac{\partial \Delta x}{\partial\delta_{y(X)}}&\cdots&\frac{\partial\Delta x}{\partial S_{XY} }&\frac{\partial\Delta x}{\partial S_{XZ}}&\frac{\partial\Delta x}{\partial S _{YZ}}\\ \frac{\partial\Delta y}{\partial x_{(X)}}&\frac{\partial\Delta y}{\partial y(X) }&\cdots&\frac{\partial\Delta y}{\partial S_{XY}}&\frac{\partial\Delta y}{ \partial S_{XZ}}&\frac{\partial\Delta y}{\partial S_{YZ}}\\ \frac{\partial\Delta z}{\partial s_{x(X)}}&\frac{\partial\Delta z}{\partial s _{y(X)}}&\cdots&\frac{\partial\Delta z}{\partial S_{XY}}&\frac{\partial\Delta z }{\partial S_{YZ}}&\frac{\partial\Delta z}{\partial S_{YZ}}\\ \end{bmatrix}  \tag{2}\]  
wherein the six-degree-of-freedom geometric errors for the \(X\)-, \(Y\)-, and \(Z\)-axes, as well as the three squareness errors from left to right are represented. The order of the six errors is position error, horizontal straightness, vertical straightness, pitch, yaw, and roll. Next, substituting Equation (1) into Equation (2) yields a sensitivity coefficient matrix:  
\[K= \begin{bmatrix}-1&0&0&-y&-z&0&0&-1&0&0&0&-z&0&0&1&0&0&0&y&-z&0\\ 0&-1&0&0&0&z&-1&0&0&0&z&0&0&1&0&0&0&0&0&-z\\ 0&0&-1&0&0&y&0&0&-1&0&0&0&1&0&0&0&0&0&0&0\end{bmatrix}  \tag{3}\]  
The matrix \(K\) signifies the contribution of each geometric error on the volumetric errors of the machine tools. A higher sensitivity coefficient for a specific error element indicates that the volumetric error is more responsive to that particular error element, thus suggesting its significance as a key geometric error element impacting machining accuracy. To assess the relative impact of each geometric error element on the volumetric errors, the sensitivity coefficients are normalized in \(X\)-, \(Y\)-, and \(Z\)-directions, respectively:\[k_{xi}=\frac{|K_{xi}|}{\sum|K_{xi}|},\ k_{yi}=\frac{\left|K_{yi}\right|}{\sum|K_{yi }|},\ k_{zi}=\frac{\left|K_{zi}\right|}{\sum|K_{zi}|} \tag{4}\]  
The magnitude of volumetric errors caused by the geometric error elements is not solely determined by their sensitivity coefficients, but also by their magnitudes. Geometric error elements with higher sensitivity coefficients, controlled at lower levels during machine tool design and assembly, can lead to reduced volumetric errors.