## 3 Geometric Error Weight  
Twenty-three error components were considered in the instrument's conceptual design, as shown in Table 1.  
Through analysis of the structural and geometric errors, a spatial volume error model of 23 geometric errors was established by excluding the linear axis angle error, which had minimal influence on the measurement results. The weights of the 23 geometric errors were calculated using the differential solution method. The weight coefficient matrix of the spatial geometric error related to the motion position of the measurement instrument is as follows:  
\[W= \begin{bmatrix}\frac{\partial E_{x}}{\partial\delta_{xY}}&\frac{\partial E_{ x}}{\partial\delta_{yX}}&\cdots&\frac{\partial E_{x}}{\partial\delta_{zA}}&\frac{ \partial E_{x}}{\partial\delta_{XC}}&\frac{\partial E_{x}}{\partial\delta_{YC }}\\ \frac{\partial E_{y}}{\partial\delta_{xY}}&\frac{\partial E_{y}}{\partial\delta_ {xY}}&\cdots&\frac{\partial E_{y}}{\partial\delta_{xZ}}&\frac{\partial E_{y}}{ \partial\delta_{yC}}\\ \frac{\partial E_{z}}{\partial\delta_{xY}}&\frac{\partial E_{z}}{\partial\delta _{zY}}&\cdots&\frac{\partial E_{z}}{\delta\delta_{ZA}}&\frac{\partial E_{z}}{ \partial\delta_{XC}}&\frac{\partial E_{z}}{\partial\delta_{YC}}\end{bmatrix}  \tag{8}\]  
From left to right, Equation (8) includes Y-axis positioning errors, two Y-axis straightness errors, Z-axis positioning errors, two Z-axis straightness errors, six A-axis PDGEs, six C-axis PDGEs and five squareness errors. The described volume error model was used to obtain the corresponding error weight coefficient matrix. The weight coefficient of each error element in each direction position is defined as follows:  
\[W_{V_{i}}=\frac{\partial E_{V}}{\partial\varepsilon_{i}},W_{Ex_{i}}=\frac{ \partial E_{x}}{\partial\varepsilon_{i}},W_{Ey_{i}}=\frac{\partial E_{y}}{ \partial\varepsilon_{i}},W_{Ez_{i}}=\frac{\partial E_{z}}{\partial\varepsilon_ {i}} \tag{9}\]  
where \(W_{V_{i}}\), \(W_{Ex_{i}}\), \(W_{Ey_{i}}\), \(W_{Ez_{i}}\) (\(i=\delta_{xY},\delta_{yY},\delta_{zY},\varepsilon_{XY}\)...) represent the weight coefficients of the error component \(\varepsilon_{i}\) to \(E_{V}\), \(E_{x}\), \(E_{y}\), and \(E_{z}\). \(E_{x}\), \(E_{y}\), and \(E_{z}\) are the volume errors of the measurement points in three directions of the spatial coordinate system. \(E_{V}\) denotes the entire measurement space of the measuring machine. This is expressed as follows:  
\[E_{V}=\sqrt{(E_{x})^{2}+(E_{y})^{2}+(E_{z})^{2}} \tag{10}\]  
The physical meaning of the error weight coefficient is the change in the spatial position error caused by the unit variation in the 23 geometric errors in the measurement machine. The greater the weight coefficient \(W\) of the error element, the greater the influence of the error element on the spatial positioning accuracy of the machine. \begin{table}
\begin{tabular}{c c} \hline \hline
**Type** & **Element** \\ \hline Position error & \(\delta_{xY}\), \(\delta_{yY}\), \(\delta_{zY}\), \(\delta_{zY}\), \(\delta_{zZ}\), \(\delta_{yZ}\), \(\delta_{zZ}\), \(\delta_{xC}\), \(\delta_{yC}\), \(\delta_{zC}\), \(\delta_{XA}\), \(\delta_{yA}\), \(\delta_{ZA}\) \\ Angle error & \(\varepsilon_{xC}\), \(\varepsilon_{zC}\), \(\varepsilon_{xA}\), \(\varepsilon_{yA}\), \(\varepsilon_{zA}\) \\ Squareness error & \(S_{XC}\), \(S_{YC}\), \(S_{YZ}\), \(S_{YA}\), \(S_{ZA}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The 23 different errors in the model. Figures 4-7 present the error weight analyses of the five groups of positions. The diagram revealed that the position error has a significantly greater impact on volume error than the angle error, and the high-weight error components are the positioning and straightness errors. For the angle error, the influence of the squareness error on the measurement accuracy was significantly higher than that of the angular motion error. The error component with a weight coefficient greater than 0.05 is considered the high-weight error component.