# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 4 Simulation Results  
The 3-DOF displacement measurement of the proposed interferometer is simulated to verify the high accuracy. To verify the advantage that there is no measurement error caused by installation errors, the deviation between the nominal value and the true value of the interferometers' installation positions and orientations is set as shown in Table 1.  
The simulation scheme is shown in Figure 8. A total of 10 groups of 3-DOF displacement in the range of the interferometer are randomly selected as the true values. By substituting the true values of displacements and the true values of installation parameters into the interferometer model (4), the readings of the six interferometers are simulated. Then, substituting the readings into Equation (4) and taking the displacements and the installation parameters as unknowns, the displacement computation is simulated by using the proposed method in Section 3. In solving Equation (4) with Algorithm 2, the nominal value of the installation parameters is taken as the initial iteration, and the initial iteration of the displacements is the computation result in the last time step. As the simulation results, the iteration error curves of the 48 unknowns are shown in Figures 9-12, where Figures 9 and 10 are the iteration error curves of the installation position \((x_{d},y_{d})\) and orientation \(\alpha\) of the 6 interferometers; Figures 11 and 12 are the iteration error \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Laser Number** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(x_{d}(\times 10^{-3}\) mm) & \(-0.96\) & \(0.61\) & \(-1.40\) & \(0.67\) & \(1.10\) & \(-0.74\) \\ \(y_{d}(\times 10^{-3}\) mm) & \(-0.21\) & \(0.23\) & \(-1.10\) & \(2.60\) & \(-0.48\) & \(0.38\) \\ \(\alpha(\times 10^{-3}\) mrad) & \(0.51\) & \(-1.20\) & \(0.09\) & \(-1.70\) & \(-1.00\) & \(0.91\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Installation errors of lasers. Figure 8: Scheme of the simulation process. Figure 7: (**a**) Eigenvalues of covariance matrix. (**b**) Independent factors. The symbol \({}^{\star}\) in the figure represents the data point. Further, the error reduction ratio (ERR) is defined as follows to verify the conclusion in Section 3.3 that only the unknowns with high IDF can be solved accurately: for the general nonlinear equation system \(\mathbf{F}(\mathbf{X})=\mathbf{0}\) where \(\mathbf{X}=\left[x_{1},\ldots,x_{N}\right]^{T}\), the ratio of the variable error after iteration to the variable error before iteration is defined as ERR  
\[\text{ERR}_{n}=-20\log\left|\frac{\hat{x}_{n}-x_{n}^{*}}{x_{n}^{(0)}-x_{n}^{*}}\right| \tag{31}\]  
where \(\text{ERR}_{n}\) is the ERR of \(x_{n}\), \(x_{n}^{*}\) is the true value, \(x_{n}^{(0)}\) is the initial iteration, and \(\hat{x}_{n}\) is the convergence solution. For 3-DOF displacement computation, namely the solution of Equation (4), there are a total of 48 unknowns including displacements and installation parameters. Arrange the 48 unknowns as Equation (30), the ERR of each unknown is shown in Figure 13. It can be seen from Figure 13 that the magnitude of ERRs corresponds to the IDFs shown in Figure 7b, which verifies that only the iteration of unknowns with a high IDF can converge to the true value. To test the numerical accuracy of the proposed algorithm, 200 displacement measurements are simulated for the case that there is noise in the readings of interferometers, and the root mean square (RMS) of noise is set as 0.1 nm to simulate the practical engineering. The measurement error of displacement is shown in Figure 14, from which the mean and the standard deviation (STD) of displacement \(x\) error are 0.155 nm and 1.884 nm, and the mean and STD of displacement \(\theta\) error are 3.549 \(\times\) 10\({}^{-8}\) mrad and 5.871 \(\times\) 10\({}^{-7}\) mrad. Figure 12: (**a**) Detail of iteration error of \(x\) and \(y\). (**b**) Detail of iteration error of \(\theta\). Figure 13: ERRs of the 48 unknowns.