## 3 Geometric Error Weight
Analyses show that \(\delta_{xY}\), \(\delta_{xZ}\), \(\delta_{xA}\), \(\delta_{xC}\) are the key error components of \(E_{x}\). While the X- and Z-axes' movement increases error weight owing to error accumulation, it does not constitute a high-weight error. When the C-axis rotates, the weight of \(\delta_{xY}\), \(\delta_{xZ}\), \(\delta_{xA}\), \(\delta_{xC}\), \(\delta_{xA}\), \(\delta_{xC}\), \(\epsilon_{yA}\), \(\epsilon_{yC}\), \(S_{XC}\), and \(S_{ZA}\) decreases or increases, and the weight of \(\delta_{yY}\), \(\delta_{yZ}\), \(\delta_{yA}\), \(\delta_{yC}\), \(\epsilon_{xA}\), \(\epsilon_{xC}\), \(S_{YZ}\), and \(S_{YC}\) increases or decreases accordingly. This is because the error model established  
Figure 5: Error weight analysis for \(E_{y}\). Figure 6: Error weight analysis for \(E_{z}\). Figure 7: Error weight analysis for \(E_{V}\). by this method changes the position of the coordinate system after the C-axis is rotated in the reference coordinate system. For example, when the workpiece coordinate system of the C-axis rotates 90\({}^{\circ}\) around the Z-axis, the X-axis coincides with the Y-axis of the reference coordinate system while the X-axis coincides with the negative direction of the Y-axis. A-axis rotation will also increase the weight of low-weight errors \(\varepsilon_{xA}\), \(\varepsilon_{zC}\), and \(S_{YA}\); however, they are still not high-weight error components. Similarly, \(\delta_{yY}\), \(\delta_{yz}\), \(\delta_{yA}\), and \(\delta_{yC}\) are the key error components of \(E_{y}\), while \(\delta_{zY}\), \(\delta_{zZ}\), \(\delta_{zA}\), \(\delta_{zC}\) are the key error components of \(E_{z}\). The rotation of the A-axis affects the weight of the error, but does not affect the dominant position of the high-weight error on the measurement accuracy. \(\delta_{XY}\), \(\delta_{xZ}\), \(\delta_{xA}\), \(\delta_{xC}\), \(\delta_{yY}\), \(\delta_{yzZ}\), \(\delta_{yA}\), \(\delta_{yC}\), \(\delta_{zY}\), \(\delta_{zZ}\), \(\delta_{zA}\), and \(\delta_{zC}\) are the high-weight errors of \(E_{V}\). The geometric error weight is determined using the differential geometric error weight calculation method. The results show that the positioning and straightness errors of each axis significantly influence the measurement accuracy. During the measurement, the motion of the rotation axis increases the weights of some low-weight error components. Although these errors do not become high-weight error components, the weights of the key error components exhibit a smaller reduction. Additionally, the motion of the rotation axis also changes the position of the coordinate system such that some error components change periodically in the one-way error weight analysis; however, it does not affect the comprehensive error model. Owing to the structure of the measuring machine, the probe is always perpendicular to the measured surface, and its tangential position measurement error produces only a negligible second-order error [31]. Therefore, for the comprehensive error, the main high-weight errors are \(\delta_{yY}\), \(\delta_{yZ}\), \(\delta_{yA}\), \(\delta_{yC}\), \(\delta_{zY}\), \(\delta_{zZ}\), \(\delta_{zA}\), and \(\delta_{zC}\). In the precise design and manufacturing of a measuring machine, attention should be given to the influence of high-weight error components. The geometric error weight analysis results were used for the design and manufacture of the subsequent measuring machines.