# Inverse Solution Error Analysis and Correction of Beam Steering System Based on Risley Prisms
## 3 Causes and Correction of Inverse Solution Error
### Causes of Inverse Solution Error
The blue curve is the ideal TLS and the black curve is the TLS with error impacts. (**a**) Beam-pointing field deviation caused by the incident light tilt. (**b**) Beam-pointing field radius increase caused by prism tilt. (**c,d**) Beam-pointing field displacement and TLS distortion caused by the prism rotation axis tilt error; the prism rotation axis tilt is constant in (**c**), and this error is a function of the prism rotation angle in (**d**). (**e**) The actual TLS obtained in experiment. The pointing field deviation and the TLS distortion also cause multiple solutions in the first step. As shown in Figure 6b, there are two ways of rotation in the first step, corresponding to the two prisms' included angles. The errors generated by these two ways in the second step are different. In fact, each TLS in the actual pointing field intersects with the blue circle corresponding to a solution. When analyzing the inverse solution error in the entire pointing field, only the maximum error is taken; the curve of the inverse solution error versus \(\Delta\Theta\) is shown in Figure 7. We separately analyze the variation trend of the pointing errors with \(\Delta\Theta\) caused by \(H_{I}\), \(V_{I}\), \(H_{R1}\), \(V_{R1}\), \(H_{R2}\), \(V_{R2}\). The above errors are independent of each other, and the values of these errors are set to 0.1\({}^{\circ}\). For the same error value, the inverse solution error caused by the incident light tilt is the largest. The error caused by the rotation axis tilt of prism 2 is much larger than that caused by the prism 1 rotation axis tilt, which is due to the fact that prism 2 will cause a greater pointing-field deviation than prism 1 under the same error. Table 2 summarizes the magnitude and causes of the inverse solution error generated by different parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Magnitude of Error** & **Causes of Error** \\ \hline \(H_{I}=0.1^{\circ}\) & 0.2002\({}^{\circ}\)-0.3233\({}^{\circ}\) & Pointing field deviation \\ \(V_{I}=0.1^{\circ}\) & 0.2002\({}^{\circ}\)-0.3233\({}^{\circ}\) & Pointing field deviation \\ \(H_{R1}=0.1^{\circ}\) & 0.0084\({}^{\circ}\)-0.0100\({}^{\circ}\) & Point field deviation \& TLS deformation \\ \(V_{R1}=0.1^{\circ}\) & 0.0042\({}^{\circ}\)-0.0051\({}^{\circ}\) & Point field deviation \& TLS deformation \\ \(H_{R2}=0.1^{\circ}\) & 0.0059\({}^{\circ}\)-0.1241\({}^{\circ}\) & Point field deviation \& TLS deformation \\ \(V_{R2}=0.1^{\circ}\) & 0.0063\({}^{\circ}\)-0.1255\({}^{\circ}\) & Point field deviation \& TLS deformation \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The magnitude and causes of the inverse solution error generated by different parameters. Figure 6: (**a**) Generation principle of the inverse solution error. The green circle represents the circle corresponding to the deflection angle of the initial position in the ideal pointing field, and the blue circle represents the circle corresponding to the deflection angle of the target position. The red circle represents the circle corresponding to a solution in the actual pointing field. (**b**) The principle of generating multiple solutions in inverse solution. The meanings of the green and blue circles are the same as in Figure 6a. The red and yellow circles represent the circles corresponding to the two solutions in the actual pointing field, respectively.