# Research on Distributed Autonomous Timekeeping Algorithm for Low-Earth-Orbit Constellation
## 4 Experiment and Results Analysis
### Frequency Stability Evaluation  
Frequency stability is a critical metric for evaluating time reference performance. All satellite clocks involved in the calculation are divided into clock ensemble 1 and clock ensemble 2, with _TA_1 and _TA_2 established using the D-KPW algorithm, respectively. In Figure 8, the Allan deviations of _TA_1 and _TA_2 are visualized.  
The results indicate that _TA_1, using short-term stability as the weight index, reduces the impact of frequency white noise but exhibits poor long-term stability. Conversely, _TA_2, using long-term stability as the weight index, mitigates random walk noise but suffers from poor short-term stability.  
Figure 9 demonstrates that the long-term drift of _TA_1 relative to _TA_2 can be predicted and corrected using a Kalman filter, thereby establishing a time reference with improved short-term and long-term stability. To further validate the algorithm's performance, its stability is compared against the equal weight averaging algorithm, AT1 algorithm, ALGOS algorithm, Kalman filter algorithm, and D-KPW algorithm. The experimental parameters are configured as presented in Table 2.  
Figure 8: Allan deviation of _TA_1 and _TA_2. **(a)** Allan deviation of each clock in clock ensemble 1 and _TA_1; **(b)** Allan deviation of each clock in clock ensemble 2 and _TA_2.  
Figure 7: Allan deviation changes with time at different \(L\) values. **(a)** Allan deviation over time with a sampling time of 1000 s; **(b)** Allan deviation over time with a sampling time of 100,000 s.  
Figure 10 presents the Allan deviation of time references generated by several algorithms, while Table 3 displays the Allan deviation values at different sampling intervals.  
The experimental results indicate that the frequency stability of the D-KPW (Control) algorithm reaches \(7.40\times 10^{-15}\) and \(7.86\times 10^{-15}\) at sampling times of 1000 s and 10,000 s, respectively, outperforming all comparative algorithms in terms of short-term stability. At sampling intervals of 100,000 s and 1,000,000 s, the frequency stability reaches \(7.83\times 10^{-15}\) and \(2.30\times 10^{-15}\), slightly lower than ALGOS's \(6.02\times 10^{-15}\) at 100,000 s. Overall, the long-term stability surpasses that of all the comparative algorithms. The proposed D-KPW algorithm integrates the strengths of the weighted average algorithm and the Kalman filter. Compared to the Kalman filter, it offers reduced computational complexity and improved performance. In comparison with the weighted average algorithm, it effectively utilizes  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Weight Calculation Index** &
\begin{tabular}{c} **Requirement for Prior** \\ **Information** \\ \end{tabular} \\ \hline Equal Weight & Same weight & No \\ ALGOS & \(\sigma^{2}(10^{4})\) & No \\ AT1 & Predicted residual error & No \\ Kalman & Estimate the correction of the & Yes \\ D-KPW & \(\sigma^{2}(10^{5})\) & Yes \\ D-KPW (Control) & Calculate \(TA1\) weight with \(\sigma^{2}(10^{5})\), & Yes \\  & calculate \(TA2\) weight with \(\sigma^{2}(10^{5})\) & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental parameters of different timescale algorithms.  
Figure 10: Allan deviation of time references generated by several algorithms.  
Figure 9: Allan deviation of \(TA\) calculated by D-KPW (Control).  
prior information. D-KPW (Control) further exploits the clock ensemble's characteristics, effectively suppressing various noise sources, thereby enhancing both the short-term and long-term stability.