## 2 System Models of Integrated Navigation
### Integrated Navigation Measurement Model  
#### 2.2.1 Navigation Measurement Model Based on X-Ray Pulsar  
The X-ray pulsar navigation measurement uses the pulse TOA as the navigation measurement information, which is suitable for the probe on the SSBE mission that is an in-orbit operation for a long period of time [47]. The X-ray pulsars provide a unique pulse signal that can be detected by the X-ray sensor mounted on the probe. Figure 1 presents the fundamental principle of the X-ray pulsar measurement of the pulse TOA. The X-ray pulsar measurement is the difference between the pulse TOA observed by the X-ray sensor on the probe and the pulse TOA predicted by the pulse arrival time model on the SSB [48]. \(t_{ps}^{k}\) is the pulse TOA of the \(k\)th X-ray pulsar measured from the probe, and \(t_{SSB}^{k}\) is the time it takes the pulse of the \(k\)th X-ray pulsar to reach the SSB. \(\hat{n}^{k}\) is the line of sight direction vector of the \(k\)th pulsar along the SSB, it can be considered constant throughout the solar system [49]. \(\mathbf{r}_{pb}\) is the position vector of the probe with respect to the SSB, \(\mathbf{r}_{pb}=\mathbf{r}_{ps}-\mathbf{r}_{SSB}\). Here, \(\mathbf{r}_{SSB}\) is the position vector of the SSB relative to the center of Sun, which could be provided by the JPL DE441 ephemeris tables. \(c\) is the speed at which light travels. The offset projection reflected on \(t_{SSB}^{k}-t_{ps}^{k}\) could be used to estimate the position of the probe in the following expression [27]. \[t_{SSB}^{k}-t_{ps}^{k}=\frac{\hat{n}^{k}\cdot\mathbf{r}_{pb}}{c} \tag{4}\]  
The TDOA between the pulse of the X-ray pulsar and the SSB could be used as the observation value. Considering the influences of the general relativistic effect, the time conversion model from the probe to the SSB is obtained as follows [49]. \[ \begin{split} t_{SSB}^{k}-t_{ps}^{k}=&\frac{1}{c} \hat{n}^{k}\cdot\mathbf{r}_{pb}+\frac{1}{2 cD_{0}^{k}}\left[\left(\hat{n}^{k}\cdot \mathbf{r}_{pb}\right)^{2}+2\left(\hat{n}^{k}\cdot b\right)\left(\hat{n}^{k}\cdot \mathbf{r}_{pb}\right)\right.\\ &\left.-2\left(\mathbf{b}\cdot\mathbf{r}_{pb}\right)-\mathbf{r}_{pb}^{2} \right]+\frac{2\mu_{S}}{c^{3}}\ln\left|\frac{\hat{n}^{k}\cdot\mathbf{r}_{pb}+\left| \mathbf{r}_{pb}\right|}{\hat{n}^{k}\cdot\mathbf{b}+\left|\mathbf{b}\right|}+1\right|\\ \end{split}  \tag{5}\]  
Here, \(D_{0}^{k}\) is the range from the \(k\)th X-ray pulsar to the SSB. \(\mathbf{b}\) is the position vector from the SSB to the heliocentric, \(\mathbf{b}=\mathbf{r}_{SSB}\). \(k\) is the number of pulsars applied for navigation estimates, \(k=1,2,\ldots,m\). When three X-ray pulsars are observed simultaneously, the high-precision three-dimensional position estimation information could be obtained. Figure 1: The fundamental principle of the X-ray pulsar measurement pulse TOA. Assume the XNAV measurement value is \(\mathbf{Z}^{X}(t)\), which can be expressed as follows:  
\[\mathbf{Z}^{X}(t)=\left[c\left(t_{SSB}^{1}-t_{PS}^{1}\right),c\left(t_{SSB}^{2}-t_{ ps}^{2}\right),\cdots,c\left(t_{SSB}^{k}-t_{PS}^{k}\right)\right]^{T} \tag{6}\]  
The observed quantity of the X-ray pulsar measurement can be expressed as follows:  
\[\mathbf{Z}^{X}(t)=\mathbf{h}^{X}[\mathbf{X}(t),t]+\mathbf{v}(t) \tag{7}\]  
where \(\mathbf{v}(t)\) is measurement noise. \(\mathbf{h}^{X}[\mathbf{X}(t),t]\) is the XNAV measurement model, and the measurement model for multi-pulsar navigation [36] can be expressed as follows:  
\[\mathbf{h}^{X}[\mathbf{X}(t),t]=\left[\mathbf{h}_{1}[\mathbf{X}(t),t],\cdots,\mathbf{h}_{i}[\mathbf{X} (t),t],\cdots,\mathbf{h}_{k}[\mathbf{X}(t),t]\right]^{T} \tag{8}\]  
where \(\mathbf{h}_{k}[\mathbf{X}(t),t]\) is the XNAV measurement equation for the \(k\)-th X-ray pulsar. \[