# Inverse Solution Error Analysis and Correction of Beam Steering System Based on Risley Prisms
## 4 Simulation and Experiment
### Experiment  
The experimental setup we used is shown in Figure 11. The laser is an MLL-III-640 produced by Changchun New Industry Optoelectronics Technology Co., Ltd.; the wavelength is 640 \(\pm\) 5 nm, and the divergence angle is less than 1 mrad (full angle). The prism wedge angle is 17.2\({}^{\circ}\), the refractive index is 1.94595, and the prism diameter is 48 mm. Each bearing of the prism is fitted with an encoder made by Renishaw to measure the angle of the rotation of the prism. The resolution of the encoder is 0.02\({}^{\prime\prime}\). The focal length of the camera lens is 81.09 mm (measured by the image magnification method); the camera made by Basler has 1024 \(\times\) 1024 pixels, and the pixel size is 5.3 \(\upmu\)m \(\times\) 5.3 \(\upmu\)m. The resolution of the camera after pixel subdivision is 1.348\({}^{\prime\prime}\), and the deflection angle of the light is calculated from the coordinates of the spot on the camera sensor obtained by the centroid extraction algorithm. The field of view of the camera is 3.8287\({}^{\circ}\). Since the camera FOV is much smaller  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Pointing Error**} \\ \cline{2-3}  & **Maximum** & **RMS** \\ \hline
**Before correction** & 0.4122\({}^{\circ}\) & 0.2266\({}^{\circ}\) \\
**After correction** & 0.07873\({}^{\circ}\) & 0.0291\({}^{\circ}\) \\
**Descent rate** & 80.9\% & 87.16\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation results of inverse solution error correction.  
Figure 10: Simulation results of inverse solution error correction. (**a**) Inverse solution error in the whole pointing field before correction. (**b**) Inverse solution error in the whole pointing field after correction. (**c**) Maximum error in each deflection angle before correction. (**d**) Maximum error in each deflection angle after correction.  
than the pointing field, the experiments are only conducted in the angular range of the camera FOV.  
We set \(\Delta\theta=1.62^{\circ}\), \(3.24^{\circ}\), \(4.86^{\circ}\) and let the two prisms rotate for one circle in the same direction and speed, sampling at \(1^{\circ}\) intervals. A point every \(10^{\circ}\) was selected as the fitted data, for a total of 108 sets of data, and the remaining data were used as the validation set. Figure 12 shows the comparison of the pointing errors before and after error fit. Table 5 shows the maximum values, RMS values and descent rates of the forward solution error before and after error correction in the three experiments.  
Figure 11: Experimental platform and schematic diagram.  
Figure 12: Comparison of the forward solution errors before and after correction in the experiment. (**a**) Before correction. (**b**) After correction.  
We chose the deflection angle \(\Phi=0.1^{\circ}\) and the azimuth angle \(\Theta\) from \(1^{\circ}\) to \(360^{\circ}\) as the target pointing position to verify the inverse solution correction algorithm; the results are shown in Figure 13. Table 6 shows the maximum value, RMS value and descent rate of inverse solution error before and after error correction. The maximum pointing error is \(0.0997^{\circ}\) when using the two-step method directly for the inverse solution, and \(0.0059^{\circ}\) when using the correction method. The maximum error and the RMS value of errors decreased by \(94.08\%\) and \(95.18\%\), respectively. The experiment proves the effectiveness of the reverse solution correction method.