# 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  
Using the two-step method for inverse solution in the error model faces two problems. First, the introduction of errors complicates the function relationship between outgoing beam and prism rotation angle. Moreover, we set the prism rotation axis tilt as a function of the rotation angle, which makes the analytical solution unobtainable. This problem can be solved by a fast numerical solution through dichotomy [17]. Second, the incident light tilt and the prism rotation axis tilt will cause the pointing field to deviate from the theoretical pointing field and the TLS distortion. They will cause pointing errors in the second step. The detailed causes of the inverse-solution error are discussed next. The simulated and experimentally measured TLS is shown in Figure 5. The blue curve is the ideal trajectory and the black curve is the trajectory with error impacts. We simulate with the actual experimental setup and system parameters. The outgoing beam is received by a camera and the TLS is the trajectory of the beam on the camera sensor. The system parameters used in the simulation are shown in Table 1.  
Figure 5a depicts the effect of the incident light tilt on the TLS, where the incident light tilt deflects the pointing field. Figure 5b depicts the TLS when the prism is tilted; it can be seen that the prism tilt changes the size of the pointing field. Figure 5c,d depict the impacts on the TLS when the prism axis tilt error is a constant and a function of the prism rotation angle, respectively. The prism axis tilt can both deviate the beam-pointing field and deform the TLS. Figure 5e shows the actual TLS obtained by experiment, which contains the effects of all errors. Figure 6 illustrates the generation principle of the inverse solution error. The ideal pointing field is a circular region that consists of an infinite number of concentric circles centered on \(O\). Each concentric circle corresponds to an included angle of prisms \(\Delta\theta\) and a deflection angle \(\Phi\) of the incident light. The tilt of the incident light deviates the pointing \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Refractive Index & 1.94595 \\ Wedge Angle & 17.2\({}^{\circ}\) \\ Lens Focal Length & 81.09 mm \\ Pixel Size & 5.3 μm \\ \hline \end{tabular}
\end{table} 
Table 1: System parameters used in simulation. field into a circular region centered on \(O^{\prime}\). Assuming that point A is the initial position of the light and point C is the target pointing position, the circles where they are located correspond to the beam deflection angles \(\Phi_{1}\) and \(\Phi_{2}\). According to the two-step method solution process, \(\Delta\theta\) is first obtained, and then one prism rotates \(|\Delta\theta-\Delta\theta^{\prime}|\) to achieve the desired beam deflection angle, at which time the beam motion trajectory is from point A to point B. The second step is to calculate the azimuth difference \(\Delta\Theta\) between point B and point C. Then the two prisms rotate \(\Delta\Theta\) in the same direction. Ideally, the beam will point from point B to point C. However, the actual trajectory of the light shifts from the blue circle to the red circle, so the actual trajectory of the light is from point B to point D. The deviation between the actual pointing position of beam D and the target position C is the inverse solution error. The prism rotation axis tilt will not only cause the beam-pointing field deviation, but also makes the TLS no longer a circle (as shown in Figure 5c,d). In a word, the inverse solution error comes from the deflection and distortion of the TLS caused by incident light tilt and prism rotation axis tilt. They lead to the beam azimuth and deflection angle changing at the same time in the second step; thus, the decoupling of azimuth and deflection angle cannot be realized. Figure 5: Impact of errors on the outgoing beam.