# Research on Distributed Autonomous Timekeeping Algorithm for Low-Earth-Orbit Constellation
## 5 Conclusions  
The rapid development of low-Earth-orbit (LEO) constellations will make them an integral part of future satellite navigation systems. To enhance the autonomous timekeeping capabilities of these systems, this paper proposes an autonomous timekeeping algorithm designed specifically for LEO constellations, which optimizes the clock resource utilization in the presence of numerous satellite nodes and complex noise conditions. The effectiveness of the proposed method is evaluated in terms of stability and predictability, comparing the D-KPW (Control) algorithm with classic methods such as equal-weight averaging, AT1, ALGOS, Kalman filtering, and the D-KPW algorithm introduced in this paper. The D-KPW (Control) algorithm synthesizes clock error data from 48 satellites to establish a time reference. Electing sampling periods of 1000 s, 10,000 s, 100,000 s, and 1,000,000 s, the frequency stability of the time reference is \(7.40\times 10^{-15}\), \(7.86\times 10^{-15}\), \(7.83\times 10^{-15}\), and \(2.30\times 10^{-15}\), respectively. The D-KPW (Control) algorithm generates a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**RMSE** & **3 Days** & **5 Days** & **10 Days** & **20 Days** \\ \hline Equal weight & \(1.27\times 10^{-9}\) & \(1.77\times 10^{-9}\) & \(4.65\times 10^{-9}\) & \(7.30\times 10^{-9}\) \\ ALGOS & \(2.02\times 10^{-9}\) & \(1.78\times 10^{-9}\) & \(4.23\times 10^{-9}\) & \(4.44\times 10^{-9}\) \\ AT1 & \(4.61\times 10^{-9}\) & \(3.85\times 10^{-9}\) & \(5.38\times 10^{-9}\) & \(4.43\times 10^{-9}\) \\ Kalman & \(2.31\times 10^{-9}\) & \(2.50\times 10^{-9}\) & \(2.44\times 10^{-9}\) & \(4.52\times 10^{-9}\) \\ D-KPW & \(1.64\times 10^{-9}\) & \(2.18\times 10^{-9}\) & \(3.23\times 10^{-9}\) & \(4.76\times 10^{-9}\) \\ D-KPW(Control) & \(7.82\times 10^{-10}\) & \(8.01\times 10^{-10}\) & \(1.03\times 10^{-9}\) & \(1.55\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMSE of different algorithms’ prediction errors (s). Figure 14: Prediction residual of time reference established by different algorithms. (**a**) RMSE of 3-day prediction error; (**b**) RMSE of 5-day prediction error; (**c**) RMSE of 10-day prediction error; (**d**) RMSE of 20-day prediction error. time reference with frequency stability that is superior to all other algorithms. The RMSE of the prediction error for the time reference established by the D-KPW (Control) algorithm is \(7.82\times 10^{-10}\) s, \(8.01\times 10^{-10}\) s, \(1.03\times 10^{-9}\) s, and \(1.55\times 10^{-9}\) s for 3, 5, 10, and 20 days, respectively, demonstrating superior performance compared to other algorithms. Due to the lack of publicly available data for LEO satellites, this paper primarily uses GNSS satellite clock error data to simulate LEO spaceborne clocks, only expanding the number of timekeeping nodes. In the future, with the launch of LEO constellations and the deployment of high-performance onboard clocks, sufficient LEO satellite clock data will become available to further validate the algorithm. For LEO satellites, further clock data errors aside from random noise will be introduced, which could affect the accuracy of the LEO constellation time reference. Owing to cost considerations, the large LEO constellation might not be fully equipped with the same high-quality clocks as those in today's MEOs. In addition, during the transmission of satellite signals, there is a risk of spoofing, so the security of the time information must be taken into account when establishing the time reference. Since time comparison is achieved through the satellite's bidirectional link, it is possible to incorporate additional authentication information into the link. Addressing these issues will be a key focus in the next phase of research. All authors contributed to the study's conception and design. Conceptualization, S.Y. ; Methodology, S.Y., Z.L., and S.N. ; Software, S.Y. ; Validation, J.P.; Writing--original draft, S.Y. ; Writing--review and editing, J.P., M.M., H.G., Z.L., and S.N. ; Visualization, M.M.