## 3 XNAV for Spacecraft in Formation
### Procedure of Pulsar-Based Navigation Using Pulse Phase Delay  
In this section, we discuss the procedure of pulsar-based navigation using pulse phase delay between two spacecraft. We assume that the true position and velocity of spacecraft A is known, and the position and velocity of spacecraft B can be estimated by measuring pulse phase delay between spacecraft A and B. By measuring \(\Delta\phi_{i}^{A,B}\), we can obtain the distance of the direction of the pulsar between spacecraft A and spacecraft B. Using one measurement, we can determine a ring of coordinate triples for spacecraft B. Moreover, given that the dynamic model of spacecraft B can be modeled in advance, we use an unscented Kalman filter (UKF) to estimate the state of spacecraft B using the measurement and dynamic model. The procedure of pulsar-based navigation using pulse phase delay is shown in Algorithm 1. At first, we apply (30) and (31) to \(\Big{\{}\tau_{j}^{A}\}_{j=1}^{N_{A}}\) and \(\Big{\{}\tau_{j}^{B}\}_{j=1}^{N_{B}}\) and obtain \(\Big{\{}\phi\Big{(}\tau_{j,SSB}^{A}\Big{)}\Big{\}}_{j=1}^{N}\) and \(\Big{\{}\phi\Big{(}\tau_{j,SSB}^{B}\Big{)}\Big{\}}_{j=1}^{N}\), respectively. Then, we estimate \(\hat{F}_{i}^{orbit,A},\hat{F}_{i}^{orbit,B}\) by (17) and estimate \(\Delta\phi_{i}^{A,B}\) by (33). The state of spacecraft B is then estimated using UKF. ``` Input:\(\mathbf{x}_{A}(t_{i}),\Big{\{}\tau_{j}^{A}\}_{j=1}^{N_{A}},\Big{\{}\tau_{j}^{B} \}_{j=1}^{N_{B}}\) = 1,2,..



## 4 Experiments

In this section, we used the simulation data of Crab pulsar and real data of Crab pulsar derived from NICER and Insight-HXMT to verify the proposed pulsar-based navigation method using pulse phase delay. ### Experiments with Simulation Data

Here, the Crab pulsar with simulation parameters shown in Table 1 was taken as the observed pulsar. Figure 2 shows the template of Crab pulsar. The initial orbit elements of spacecraft A and spacecraft B are shown in Table 2. Both spacecraft were assumed to observe Crab pulsar with observation \(t_{obs}=100\) s. In this section, we assumed that the true position and velocity of spacecraft A was known, and the observation data of Crab pulsar were used to estimate state of spacecraft B with Algorithm 1. The initial state errors of spacecraft B were set as [5 km, 5 km, 5 km] and [5 m/s, 5 m/s, 5 m/s]. The initial state of spacecraft A and B was used to recover empirical profiles with \(\left\{\phi\left(\tau_{j,SSB}^{HXMT}\right)\right\}\) and \(\left\{\phi\left(\tau_{j,SSB}^{NICER}\right)\right\}\) via (13). The recovered empirical profiles are shown in Figure 3. Due to the distance between spacecraft A and B, pulse phase of Spacecraft A and B in the same exposure were different. Using (33) to estimate \(\Delta\hat{\phi}_{i}^{A,B}\), the value of estimated pulse phase delay was 0.2650, and \(\Delta\hat{\phi}_{i}^{A,B}\), calculated by (29), was 0.2652. It can be seen that \(\Delta\hat{\phi}_{i}^{A,B}\) was close to \(\Delta\hat{\phi}_{i}^{A,B}\), computed via (29). Therefore, the pulse phase delay can reflect the distance between spacecraft A and B. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Orbital Elements** & **Spacecraft A** & **Spacecraft B** \\ \hline Inclination/\({}^{\circ}\) & 51.6995 & 63.2 \\ Semi-major axis/km & 6781.5888 & 6922.0854 \\ Eccentricity & 8.63 \(\times\) 10\({}^{-4}\) & 4.76 \(\times\) 10\({}^{-4}\) \\ Right ascension of the & 137.2927 & 252.4538 \\ Ascending node/\({}^{\circ}\) & 124.2185 & 121.7897 \\ Argument of perigee/\({}^{\circ}\) & 183.5224 & 279.4325 \\ True anomaly/\({}^{\circ}\) & \multicolumn{1}{c}{} & \\ \end{tabular}
\end{table} 
Table 2: Orbital elements of spacecraft A and B.

Figure 2: Template of the Crab pulsar. To verify the proposed pulsar-based navigation method using pulse phase delay, we chose the real data when Insight-HXMT and NICER simultaneously observe the Crab pulsar. The exposures on Crab pulsar from NICER and Insight-HXMT are shown in Figure 6. As shown in the figure, 13 March 2018 (MJD:58190) Insight-HXMT and NICER did not observe Crab pulsar simultaneously.