## 3 XNAV for Spacecraft in Formation
### Procedure of Pulsar-Based Navigation Using Pulse Phase Delay
\begin{table}
\begin{tabular}{c c} \hline \hline
**Exposure Number** & **Estimation Error [Cycle]** \\ \hline
1 & \(2.63\times 10^{-3}\) \\
2 & \(-2.82\times 10^{-3}\) \\
3 & \(2.80\times 10^{-3}\) \\
4 & \(3.89\times 10^{-4}\) \\
5 & \(-9.16\times 10^{-4}\) \\
6 & \(3.39\times 10^{-5}\) \\
7 & \(-2.51\times 10^{-3}\) \\
8 & \(-2.44\times 10^{-4}\) \\
9 & \(-1.42\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Estimation errors of the pulse phase delay. Figure 7: Empirical profiles of each exposure. performance of the proposed direct estimation method was close to that of respective estimation method. Furthermore, the CPU time cost of the proposed direct estimation method was about 36.99% lower than that of respective estimation method. ## 5 Conclusions

This study derived a pulsar-based navigation method using pulse phase delay between spacecraft. A direct estimation method for pulse phase delay is proposed to eliminate reliance on templates of pulsars for pulse phase delay estimation. The proposed method is verified with the simulation data of the Crab pulsar and the real data of the Crab pulsar form Insight-HXMT and NICER. Real data experiment results show that the pulse phase delay estimated by the real data of the Crab pulsar can accurately reflect the distance between Insight-HXMT and NICER. Moreover, compared with the estimation method that estimates the pulse phase of two spacecraft, the proposed direct estimation method can estimate pulse phase delay between two spacecraft by running the pulse phase estimation algorithm just once. The experimental results show that the estimation performance of the proposed direct estimation method is close to that of the respective estimation method and that the CPU time cost of the proposed method is about 36.99% lower. Notably, the transform model shown as (2) only considers the [PERSON] delay and Shapiro delay. An accurate transform model that considers the Einstein delay and high-order terms of relativistic effects should be considered in further studies to provide an accurate navigation service for spacecraft near gravitational sources. Conceptualization, [PERSON] and [PERSON]. ([PERSON]); methodology, [PERSON] and [PERSON]; software, [PERSON] and [PERSON]; validation, [PERSON], [PERSON] and H.Y. ([PERSON]); formal analysis, [PERSON]; investigation, [PERSON]; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON]; visualization, [PERSON]; supervision, [PERSON] ([PERSON]); project administration, [PERSON] and H.Y. ([PERSON]); funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript. This research received no external funding. The data presented in this study are available at [[[http://archive.hxmt.cn/iapi/opendownload/10014825/1L/z14](http://archive.hxmt.cn/iapi/opendownload/10014825/1L/z14) YsMOXkM3 EO4 jUpFM3 ugAQ]([http://archive.hxmt.cn/iapi/opendownload/10014825/1L/z14](http://archive.hxmt.cn/iapi/opendownload/10014825/1L/z14) YsMOXkM3 EO4 jUpFM3 ugAQ)] [19 May 2024] [P0111605054] [NICER] [[[https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/1013010125/](https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/1013010125/)]([https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/1013010125/](https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/1013010125/))] (19 May 2024) [1013010125] [NICER] [[[https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/101](https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/101)]([https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/101](https://heasarc.gsfc.nasa.gov/FTP/nicer/data/obs/2018_03/101)) 3010126/] (19 May 2024) [1013010126]. The authors declare no conflicts of interest. ## References

* [PERSON] and [PERSON] (2021) [PERSON]; [PERSON] Adaptive extended Kalman filtering strategies for spacecraft formation relative navigation. _Acta Astronaut._**2021**, _178_, 700-721. [CrossRef]
* [PERSON] et al.