## 3 Establishment and Simulation of the Error Model
### Model Simulation Analysis  
MATLAB matrix calculation software was used to simulate the proposed error model for verification. By introducing certain installation errors and manufacturing errors, the model compensation calculation results of the six DOF motion errors of a rotary axis are compared with the preset values.  
Setting of the simulation parameters: the six DOF motion errors of the target rotary axis are set to zero at the initial timepoint. Thirteen measurement points are set from 0\({}^{\circ}\) to 360\({}^{\circ}\) with intervals of 30\({}^{\circ}\), corresponding to the twelve faces of the polyhedral prism. Except for the 0\({}^{\circ}\) and 360\({}^{\circ}\) points, the \(\varepsilon_{Z}\), \(\varepsilon_{Y}\), and \(\varepsilon_{X}\) values of other points are set to 100\({}^{\circ}\); the \(\delta_{X}\) and \(\delta_{Y}\) values are set to 100 \(\upmu\)m, and the \(\delta_{Z}\) value is set to 10 \(\upmu\)m.  
The diameter of the polyhedral prism is 100 mm, its thickness is 17 mm, the distance from the reflection surface to the center is 48.3 mm, and the width of each surface is 12.94 mm. Assuming that the distance from the light point position of the autocollimators and the laser displacement sensors on the reflection surface of the polyhedral prism to the axis in X and Y directions is 100 \(\upmu\)m, the installation errors of the polyhedral prism around the X and Y axes are 50\({}^{\circ}\), the radial installation errors along the X direction and Y direction are 10 \(\upmu\)m, and the axial installation error along the Z direction is 10 \(\upmu\)m. Because the manufacturing errors of the working angle of the 0-level polyhedral prism in practical application are within 1\({}^{\circ}\), the manufacturing errors of the working angle of each surface are set to a random number within 1\({}^{\circ}\). The perpendicularity error between the working surface and the datum plane of the polyhedral prism is \(\leq\)5\({}^{\circ}\); subsequently, the perpendicularity errors \(\varepsilon_{px}\) and \(\varepsilon_{py}\) between the working surface and the datum plane of the polyhedral prism are set to be 5\({}^{\circ}\). The parallel error between the top surface and the reference surface is less than 2 \(\upmu\)m. Because the diameter of the polyhedral prism is 100 mm, the angle between  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\varepsilon_{x}\) & \(\varepsilon_{y}\) & \(\varepsilon_{z}\) & \(\delta_{x}\) & \(\delta_{y}\) & \(\delta_{z}\) \\ \hline \(\varepsilon_{px}\) & \(\surd\) & \(\times\) & \(\times\) & \(  
the top and bottom surfaces around the X axis and the Y axis is less than 4.13\"; therefore, the angle between the two surfaces around the X and Y directions is 4.13\". The maximum distance between the origin coordinates of the measurement unit coordinate systems 4, 5, 6, and 7 and the origin coordinates of the world coordinate system is 100 mm. The angle installation errors of the autocollimators and laser displacement sensors are set to 100\". The positions of the emitted light of the autocollimators and the laser displacement sensors are located at 100 \(\upmu\)m deviation from the X axis and Y axis of their respective coordinate centers. The specific simulation results are shown in Figure 3.  
According to the simulation results, the proposed error model is able to compensate the installation errors and manufacturing errors for measuring all six DOF geometric motion errors of a rotary axis. The differences between the compensated measurement results and the errors' set values are 0.0509\", 0.0003\", 0.0003\", 0.084 \(\upmu\)m, 0.036 \(\upmu\)m, and 0.028 \(\upmu\)m, respectively. The compensation effect is significant, which verifies the accuracy and effectiveness of the compensation model.