## 4 Error Simulation and Analysis
### Datum Transfer Error  
The accuracy of the datum transfer can be affected by several errors. The four single variables are added term by term to the equivalent model in Section 4.2. The effect of a single error source on the transfer accuracy is analyzed by the simulation to obtain an error sensitivity ranking, which in turn allows the control of each error source. Figure 8a-d show the error of the \(X\), \(Y\), and \(Z\) coordinate axis vectors when theodolite collimation error, laser tracker collimation error, laser tracker reflective target sphere manufacturing eccentricity error, and cubic prism coordinate system establishment error are applied to the datum transfer, respectively. The horizontal coordinates are the range of error values, and the vertical coordinates are the axis transfer errors. As a result of the analysis, when the measurement error of the theodolite varies within 5\", the \(X\)-axis turning angle transfer error varies within 0.5\", and the \(Y\)-axis and \(Z\)-axis transfer error gradually increases within 8.5\" and 12.7\". As the laser tracker aiming error varies from within 0.05 mm, the \(XYZ\) axis angular transfer error gradually increases in the 4.1\", 6.5\", and 2.8\" ranges. As the cubic prism coordinate system establishment error varies within 5\", the \(XYZ\) axis angular transfer error gradually increases within 2.8\", 3.5\", and 11.2\". In summary, every single error has a different degree of influence on the accuracy of the coordinate axis transfer. The most significant effect on the \(Z\)-axis rotation is caused by errors in the measurement of the theodolite and errors in the establishment of the cubic prism coordinate system. Laser tracker collimation error and reflective target ball manufacturing eccentricity error have the most significant effect on the \(Y\)-axis rotation angle. As the role of each error has a certain distribution interval, it is necessary to ensure that the total error accumulated after the benchmark transfer is within the design allowable range, and the simulation process is as follows:  
Figure 7: Simulation of angular errors in the establishment of a multi-source data datum alignment coordinate system. (**a**) probability distribution curve of the angular error in the \(X\)-axis. (**b**) probability distribution curve of the angular error in the \(Y\)-axis. (**c**) probability distribution curve of the angular error in the \(Z\)-axis. Set the simulation model parameters as the theoretical values for the simulation scenario, including mechanical feature points, the azimuth of the M1&Instruments holder to the M2 holder, and the mounting error of each component. 2. Calculation of the theoretical values of the attitude angles of the M1&Instruments holder and the M2 holder according to the theoretical parameters of the simulation set in step (1), in combination with the computer model. 3. Transfer simulation based on the effect of each measurement error to obtain the angular values containing random errors between the datum platform, the M1&Instruments holder, and the M2 holder. 4. The results of the calculation in step (3) are compared with the theoretical true value in step (2) to produce a simulation error value for the datum transfer process. The theoretical values of the simulation were obtained from the computer theoretical model as shown in Table 3.  
The mathematical model of the transformation matrix is established by computer simulation and brought into the above simulation parameters. Simulation of the datum transfer relationship in (1), (2), and (3) gives the theoretical angular values of the datum platform, the M1&Instruments holder, and the M2 holder. The angular values in the above transfer process are all constants. The Monte Carlo mathematical model is used for benchmark transfer simulation. The errors listed in Table 4 were selected and brought into the transfer chain with the appropriate expectation and standard deviation, and the root-mean-square synthesis law was met between the errors.