## 3 XNAV for Spacecraft in Formation  
### Measurement Model  
The diagram of XNAV for spacecraft in formation is shown in Figure 1. Assume that spacecraft A and spacecraft B observe the same pulsar; thus, according to (1), the pulse phases are given by  
\[\phi_{i}^{A}=\phi_{A}(t_{i})=F_{0}[g^{A}(t_{i})-T_{0}]+\frac{F_{1}}{2}[g^{A}( t_{i})-T_{0}]^{2}, \tag{20}\]  
\[\phi_{i}^{B}=\phi_{B}(t_{i})=F_{0}[g^{B}(t_{i})-T_{0}]+\frac{F_{1}}{2}[g^{B}( t_{i})-T_{0}]^{2}, \tag{21}\]  
where  
\[g^{A}(t_{i})=t_{i}+\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}+\mathbf{r}_{E,i})}{c}+\sum_{k= 1}^{K}\frac{2\mu_{k}}{c^{3}}ln|\mathbf{n}\cdot\mathbf{r}_{k,i}^{A}+\Big{\|}\mathbf{r}_{k,i} ^{A}\Big{\|}, \tag{22}\]  
\[g^{B}(t_{i})=t_{i}+\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{B}+\mathbf{r}_{E,i})}{c}+\sum_{k= 1}^{K}\frac{2\mu_{k}}{c^{3}}ln|\mathbf{n}\cdot\mathbf{r}_{k,i}^{B}+\Big{\|}\mathbf{r}_{k,i} ^{B}\Big{\|}, \tag{23}\]  
where \(\mathbf{r}_{i}^{A}\) is the position of spacecraft A relative to the Earth at \(t_{i}\) and \(\mathbf{r}_{i}^{B}\) is the position of spacecraft B relative to the Earth at \(t_{i}\). Given that the distance between the two spacecraft is far less than the distances between the celestial bodies and the spacecraft, we assume that \(\mathbf{r}_{k,i}^{A}\approx\mathbf{r}_{k,i}^{B}\), and we denote it as \(\mathbf{r}_{k,i}\).  
We use pulse phase delay between spacecraft A and spacecraft B as a measurement:  
\[\Delta\phi_{i}^{A,B}=\phi_{i}^{A}-\phi_{i}^{B} \tag{24}\]  
Substituting (20) and (21) into (24), we have  
\[\Delta\phi_{i}^{A,B}=\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}-\mathbf{r}_{i}^{B})}{c}\cdot[ F_{0}+F_{1}\cdot(t_{i}+\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}+\mathbf{r}_{E,i})}{c}+\sum_{k=1}^{ K}\frac{2\mu_{k}}{c^{3}}\ln|\mathbf{n}\cdot\mathbf{r}_{k,i}+\|\mathbf{r}_{k,i}\|-T_{0})- \frac{F_{1}}{2}\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}-\mathbf{r}_{i}^{B})}{c})]. \tag{25}\]  
Since \(c\) is about \(3\times 10^{8}\) m/s, \(F_{1}\) is smaller than \(10^{-10}\) HZ/s for most pulsars. Assume that the distance between spacecraft A and B is shorter than 100 km. Therefore, \(\frac{F_{1}}{2}\frac{\mathbf{n}\cdot(\mathbf{r}_{1}^{A}-\mathbf{r}_{i}^{B})}{c}\) is close to 0, and we have  
\[\Delta\phi_{i}^{A,B}=\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}-\mathbf{r}_{i}^{B})}{c}\cdot [F_{0}+F_{1}\cdot(t_{i}+\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}+\mathbf{r}_{E,i})}{c}+ \sum_{k=1}^{K}\frac{2\mu_{k}}{c^{3}}\ln|\mathbf{n}\cdot\mathbf{r}_{k,i}+\|\mathbf{r}_{k,i} \|-T_{0})]. \tag{26}\]  
We also define \(F_{i}^{A,B}\) as follows:  
\[F_{i}^{A,B}=F_{0}+F_{1}\cdot(t_{i}+\frac{\mathbf{n}\cdot(\mathbf{r}_{i}^{A}+\mathbf{r}_{E, i})}{c}+\sum_{k=1}^{K}\frac{2\mu_{k}}{c^{3}}\ln|\mathbf{n}\cdot\mathbf{r}_{k,i}+\|\mathbf{r}_{k,i} \|\|-T_{0}). \tag{27}\]  
Substituting (27) into (25), we have  
\[\Delta\phi_{i}^{A,B}=F_{i}^{A,B}\frac{\mathbf{n}\cdot\left(\mathbf{r}_{i}^{A}-\mathbf{r}_ {i}^{B}\right)}{c}. \tag{28}\]  
Based on Equation (28), the difference in pulse phases of spacecraft A and spacecraft B reflects their distance in the direction of the pulsar. Assuming that \(\Delta x_{i}^{A,B}=\mathbf{x}_{A}(t_{i})-\mathbf{x}_{B}(t_{i})\), \(y_{i,AB}=\Delta\phi_{i}^{A,B}\) and \(h_{A,B}(\Delta x_{i}^{A,B})=F_{i}^{A,B}\frac{\mathbf{n}\cdot\left(\mathbf{r}_{i}^{A}- \mathbf{r}_{i}^{B}\right)}{c}\), then Equation (28) can be rewritten as a standard measurement model of  
\[y_{i,AB}=h_{A,B}\Big{(}\Delta\mathbf{x}_{i}^{A,B}\Big{)}. \tag{29}\]