## 3 XNAV for Spacecraft in Formation
### Procedure of Pulsar-Based Navigation Using Pulse Phase Delay
On 14 March 2018 (MJD:58191), there are nine exposures that meet the criteria. We select these nine exposures and align the start time of each exposure. The start times of the selected exposures are shown in Table 4.

Given that there are only nine exposures in which Insight-HXMT and NICER simultaneously observe Crab pulsar, we verified the proposed pulsar-based navigation method by comparing pulse phase delay calculated by (28) and that estimated by (33). With the pulse phases of events received by Insight-HXMT and NICER, \(\left\{\phi\left(\tau_{l,SSB}^{HXMT}\right)\right\}\) and \(\left\{\phi\left(\tau_{l,SSB}^{NICER}\right)\right\}\) were calculated by (20) and (21) using the real position of NICER and Insight-HXMT. The empirical profiles of NICER and Insight-HXMT in each exposure were then recovered using \(\left\{\phi\left(\tau_{l,SSB}^{HXMT}\right)\right\}\) and \(\left\{\phi\left(\tau_{l,SSB}^{NICER}\right)\right\}\) via (13). The empirical profiles in each exposure are shown in Figure 7. Given the distance between NICER and Insight-HXMT in the direction of pulsar, the pulse phases of NICER and Insight-HXMT in the same exposure were different. In accordance with (24), the pulse phase delay \(\Delta\delta_{l}^{HXMT,NICER}\) reflects the distant between NICER and Insight-HXMT in the direction of the pulsar. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number** & **Start Time [UTC]** \\ \hline
1 & 14 March 2018 02:06:27 \\
2 & 14 March 2018 03:40:51 \\
3 & 14 March 2018 05:14:53 \\
4 & 14 March 2018 06:43:16 \\
5 & 14 March 2018 08:16:06 \\
6 & 14 March 2018 09:50:46 \\
7 & 14 March 2018 11:22:42 \\
8 & 14 March 2018 12:58:12 \\
9 & 14 March 2018 14:34:19 \\ \hline \end{tabular}
\end{table} 
Table 4: The start times of the selected nine exposures. Figure 6: Exposure on Crab pulsar from Insight-HXMT and NICER. Next, we used the estimation error of \(\Delta\hat{\psi}_{i}^{HXMT,NICER}\) to reflect the measurement error of the distance between Insight-HXMT and NICER. The pulse phase delays \(\Delta\hat{\psi}_{i}^{HXMT,NICER}\), \(i=1,\ldots,9\) were estimated by (33) using the empirical profiles of NICER and Insight-HXMT shown in Figure 6. The real pulse phase delays were calculated by (28) using the real positions of NICER and Insight-HXMT. Furthermore, the estimation error of pulse phase delay was defined as the difference between estimated pulse phase delay and real pulse phase delay. Table 5 shows the estimation errors of the pulse phase delay of nine exposures. The results revealed that the estimated pulse phase delay can accurately reflect the distance between Insight-HXMT and NICER, thus verifying the feasibility of the proposed pulsar-based navigation using the pulse phase delay. Figure 7: _Cont._

Finally, we compared the performance of the separate estimation method for pulse phase delay used in (19) and the proposed direct estimation method. In the proposed direct estimation method, pulse phase delays \(\Delta\hat{\phi}_{i}^{HXMT,NICER},\ i=1,\ldots,9\) were directly estimated by (33) using the empirical profiles of Insight-HXMT and \(NICER\) in each exposure. In the respective estimation method, \(\Delta\hat{\phi}_{i}^{HXMT}\) and \(\Delta\hat{\phi}_{i}^{NICER}\) were estimated, respectively, by (19) using empirical profiles and templates, after which \(\Delta\hat{\phi}_{i}^{HXMT,NICER},\ i=1,\ldots,9\) were then calculated by (24). The evaluation criterion of the estimation performance is defined as RMS (root mean square) error of the pulse phase delay. We defined CPU time cost as evaluation criterion of computational cost. The computation environment contained Intel Core [EMAIL_ADDRESS] GHZ (Intel, Santa Clara, CA, USA) and python 3.8. Table 6 shows the RMS error and mean CPU time cost of the proposed direct estimation method and the separate estimation method for pulse phase delay. The results showed that the estimation