# Tilt-Induced Error Compensation with Vision-Based Method for Polarization Navigation
## 4 Discussion
### Verification of Polarization Orientation Error Compensation Method  
In this experiment, the UAV is equipped with a polarized light sensor to obtain polarized light information at various tilt degrees, aiming to verify the feasibility and accuracy of using a horizontal attitude angle to compensate for the polarized light heading under inclined conditions. The heading data before and after compensation were compared with the real heading data obtained by the high-precision benchmark, and the experimental results are shown in Figures 12 and 13.  
Figure 12 illustrates the comparison results between the heading angle of the polarized light and the reference prior to and following compensation. The figure demonstrates how the raw polarized light data exhibit substantial fluctuations and deviation from the reference due to the tilt factor affecting the measurements. This deviation is particularly noticeable at  
Figure 12: Polarized light heading angle output results before and after compensation.  
Figure 13: Comparison of polarized light heading angle error before and after compensation.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Performance** & **SIFT** & **Optical Flow** &
\begin{tabular}{c} **Traditional** \\ **Horizon** \\ \end{tabular} & **Proposed** \\ \hline RMSE (\({}^{\circ}\)) & 0.7499 & 0.4390 & 0.2995 & 0.2383 \\ STD (\({}^{\circ}\)) & 0.6739 & 0.3872 & 0.2956 & 0.2366 \\ \hline \end{tabular}
\end{table}
Table 5: Roll angle error comparison table of different attitude estimation methods.  
certain sample points, where the angle reaches significant extremes. After compensation, the heading angle of the polarized light aligns more closely with the reference, showing a marked reduction in error. The results clearly highlight that the compensation method substantially reduces the misalignment, confirming the effectiveness of the horizontal attitude angle in correcting heading errors. Figure 13 illustrates the comparison of the heading error in polarized light before and after compensation. The pre-compensation error, represented by the gray dashed line, shows high variability and spikes, which suggests significant inaccuracies in the heading calculation. This is primarily due to the effects of tilt and environmental factors. After compensation (red line), the error is significantly reduced, with a marked decrease in error amplitude and frequency. The post-compensation curve shows a much more stable and smaller range of error, further confirming that the proposed compensation method effectively minimizes heading errors, improving overall heading accuracy.  
Table 6 provides a quantitative comparison of the heading error before and after the application of the compensation method. It lists the root-mean-square error (RMSE), standard deviation (STD), and mean heading error for both the pre-compensation and post-compensation cases. The results show a substantial improvement in all metrics post-compensation: The RMSE decreases from 5.0869\({}^{\circ}\) to 1.1485\({}^{\circ}\), a reduction of approximately 77.4%, indicating a significant reduction in the overall heading error. The standard deviation (STD) also decreases, from 5.0633\({}^{\circ}\) to 1.1441\({}^{\circ}\), showing that the compensation method not only reduces the average error but also stabilizes the data, making the measurements more consistent. The mean heading error shifts from a positive value of 0.5005\({}^{\circ}\) (indicating a slight overestimation) to a negative value of -0.1039\({}^{\circ}\), indicating a slight underestimation post-compensation. This shift suggests that the compensation method helps to bring the heading angle closer to the true value.