## 2 Principle
After generating an angular error of \(\varepsilon_{xY}\), the laser interferometer signal in the Z-direction will generate an error of \(\Delta L\), where \(\Delta L=L_{actual}-L_{ideal}=L_{actual}(1-\cos\varepsilon_{xY})\). Through the precision manufacturing of the straight axis, the Angle error does not exceed \(1^{\prime}\). Assuming that the Angle error is \(1^{\prime}\), the maximum stroke of the Z-axis is 80 mm, and in the Z-direction, the displacement due to the Angle error is 3.38 nm. The resulting error is negligible during the full travel measurement. Therefore, the changes in the laser interferometer signal caused by the geometric error of the Y-axis guideway are mainly due to the positioning error \(\delta_{yY}\) and the Z-direction straightness error \(\delta_{zY}\). Similarly, the Z-axis guideway produces six geometric errors \(\delta_{xZ}\), \(\delta_{yZ}\), \(\delta_{zZ}\), \(\varepsilon_{zZ}\), \(\varepsilon_{yZ}\), and \(\varepsilon_{zZ}\) in the process of movement, which have an impact on the measurement of the main positioning error \(\delta_{zZ}\) and straightness error \(\delta_{yZ}\). Because of the structural factors of the measuring machine, the positional errors generated in the X-direction are corrected using the measuring system at a later stage. Figure 1: Structure model of measuring machine. In practice, the probe position may deviate from the ideal position, as shown in Figure 2b. Assuming the angle between the actual probe position and the ideal position is \(\theta\), the probe measuring point relative to the intersection of the laser interferometer distance \(\Delta L_{s}\), in the Y-direction, has a deviation value of \(\Delta L_{sy}\), where \(\Delta L_{sy}=\Delta L_{s}\)sin\(\theta\). In the Z-direction the deviation value is \(\Delta L_{sz}\), where \(\Delta L_{sz}=\Delta L_{s}(1-\cos\theta)\). When the A turntable rotates, due to the existence of positioning error, the actual staying position will deviate from the ideal position; the deviation angle is \(\theta\), the positioning error is controlled within 1 arc sec, the size of \(\Delta L_{s}\) is 100 mm, which is calculated in the formula, and the resulting error is very small, which is negligible relative to the full stroke. The verticality error between the Y-axis and the Z-axis arises from production and installation errors of the guide rail and the Y and Z standard flat crystals. Consequently, the laser lines from the Y and Z laser interferometers are not strictly vertical, leading to verticality and squareness errors of the angle \(S_{YZ}\) between them. When the mobile platform moves along the Z-direction, a position change in the size \(L_{S_{YZ}}\) occurs in the Y-direction. \[L_{S_{YZ}}\approx Z\cdot S_{YZ} \tag{1}\]  
Similar to the linear axis, the rotating axis has six position-dependent geometric errors (PDGEs). The instantaneous error of the rotating axis generates six degrees of freedom, resulting in three position errors (\(\delta_{xC}\), \(\delta_{yC}\), and \(\delta_{zC}\)) and three angle errors (\(\varepsilon_{xC}\), \(\varepsilon_{yC}\), and \(\varepsilon_{zC}\)), as shown in Figure 3a. Three angular errors cause the rotation axis to rotate around the X-, Y-, and Z-axes, and three position errors cause the rotation axis to shift slightly in the X-, Y-, and Z-directions. The absolute representation method was used to define the position-independent geometric errors (PIGEs) of the rotating axis [29] based on the base coordinate system of the platform. For example, considering the C-axis of the rotating axis, each rotating axis contains four PIGEs. The four errors of the rotating shaft from the assembly are two squareness errors (\(S_{XC}\) and \(S_{YC}\)) and two position errors (\(O_{XC}\) and \(O_{YC}\)), as shown in Figure 3b. The value of the two position errors is equal to the position error term when the C-axis rotation angle is \(0^{\circ}\) and thus can be incorporated into the C-axis error, eliminating the need for separate error matrix calculations.