## 5 Conclusions  
In this paper, a comprehensive study on the systematic error in the CoM method considering both the detector sampling frequency limitation and the sampling window size limitation was presented. The BA-ELM was applied to predict the corresponding systematic error of the actual star centroid position and then improve the star centroiding accuracy of the CoM method. The input weights matrix and hidden layer biases were optimized with BA in an off-line training process, which promoted the prediction accuracy and stability of the basic ELM model. Several simulations were implemented to test the BA-ELM compensation method, and the results indicate that the proposed method can effectively eliminate the systematic error under different Gaussian widths of starlight energy distribution, and it achieves higher accuracy in estimating the star centroid location after compensation than do other methods. Future work will focus on optimizing the search ability of BA to obtain better parameters in ELM and further validate the compensation method with a real star sensor. In summary:  
1. Through a frequency field and numerical simulations approach, the characteristics and causes of two types of systematic error in the CoM method, namely, discrete approximation error and truncation error, were illustrated, and the relationships between the systematic error and actual star centroid position under different Gaussian widths were obtained. This helped us to design the compensation method.
2. BA was adopted to improve the prediction accuracy and stability of the ELM. The random parameters in the BA-ELM model were optimized during the off-line training process. Through the verification of the results, the proposed BA-ELM compensation method was shown to effectively eliminate the systematic error and enhance the accuracy of the CoM method in estimating star centroid locations.
3. Comparing the simulation results of the BP neural network method, analysis compensation method, LSSVR compensation method, and basic ELM showed that the BA-ELM compensation method performed better in terms of prediction accuracy than did the other methods during the systematic error compensation process.  
Funding acquisition, D.W., Z.S.; Investigation, W.Z., G.L., and Z.L.; Methodology, X.W.; Supervision, D.W. and Z.S.; Validation, X.W.; Writing--original draft, X.W.; Writing--review and editing, J.X.  
This research was funded by National High Technology Research and Development Program of China 863 Program (No.Y512171800) and Youth Innovation Promotion Association (No.1188000111).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Training Time** & **RMSE** & **Prediction Error** \\ \hline BP & 549.8 ms & \(3.6230\times 10^{-4}\) & \(\leq 2.0\times 10^{-3}\) \\ AC & – & \(1.8460\times 10^{-4}\) & \(\leq 5.0\times 10^{-4}\) \\ ELM & 11.5 ms & \(5.5251\times 10^{-6}\) & \(\leq 4.0\times 10^{-5}\) \\ LSSVR & 204.6 ms & \(3.0387\times 10^{-5}\) & \(\leq 6.0\times 10^{-5}\) \\ BA-ELM & 50,400 ms & \(2.4532\times 10^{-8}\) & \(\leq 3.0\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of the BA-ELM compensation method with other compensation methods.