# LEO Precise Orbit Determination with Inter-satellite Links
## 3 Simulation and Method
### POD Methods  
For any ISL observation in the constellation, it can form a formula as follow:  
\[v_{ij}=-\frac{\Delta x_{ij}}{\widehat{S}_{ij}}\hat{x}_{i}-\frac{ \Delta y_{ij}}{\widehat{S}_{ij}}\hat{y}_{i}-\frac{\Delta z_{ij}}{\widehat{S}_ {ij}}\hat{z}_{i}+\frac{\Delta x_{ij}}{\widehat{S}_{ij}}\hat{x}_{j}+\frac{ \Delta y_{ij}}{\widehat{S}_{ij}}\hat{y}_{j}+\frac{\Delta z_{ij}}{\widehat{S}_ {ij}}\hat{z}_{j}-\overline{l_{ij}}+\varepsilon_{ij} \tag{12}\]  
where \(v_{ij}\) is the residual of ISL measurement; \((\hat{x}_{i},\hat{y}_{i},\hat{z}_{i})\) and \(\left(\hat{x}_{j},\hat{y}_{j},\hat{z}_{j}\right)\) are the corrections of satellite coordinates; \(\Delta x_{ij}\), \(\Delta y_{ij}\), and \(\Delta z_{ij}\) are coordinate differences calculated by rough starting orbital data; \(\overline{l_{ij}}\) denotes the difference between link observations and \(S_{ij}\); \(\varepsilon_{ij}\) is the measurement noise.  
As long as the numbers of link observations in the constellation are greater than the numbers of satellite coordinates to be sought, the orbits of the entire constellation can be precisely determined under high-precision inter-satellite geometric constraints. According to the principle of the whole network adjustment, precise orbits are calculated as follow:  
\[\hat{x}=\left(B^{T}PB\right)^{-1}\!BPI \tag{13}\]  
where \(\hat{x}\) is corrections of coordinates of satellites in ECI; \(B\), \(P\), and \(l\) (which is formed by \(\overline{l_{ij}}\) in each link) denote the coefficient matrix, weight matrix, and residual matrix, respectively.  
In the absence of ground stations or anchor stations to provide a spatial reference, relying solely on inter-satellite orbit determination will result in overall constellation rotation [25]. In essence, the above coordinate solving process is similar to the rank-deficient free network adjustment. Without considering the scale information of the free network, an n-dimensional ranging network needs to provide \(n\) position references and \(\frac{n(n-1)}{2}\) direction references. Since the inter-satellite ranging network is a 3D mesh shape, the rank deficit is 6. The spatial reference of the entire constellation can be determined by simply giving the coordinates of the two LEO satellites. In our study, we simulate the coordinates of the two LEO satellites determined by the navigation satellite data to solve the rank deficit problem in the network adjustment.  
In addition to the abovementioned method, it is also possible to use the center of gravity reference to perform ISL POD without relying on satellite coordinate data. Assuming that the sum of the corrections of all satellite coordinates in the entire constellation (i.e., the sum of vector \(\hat{x}\)) is zero, the auxiliary matrix \(S\) is introduced, and it satisfies the following formula:  
\[S^{T}\hat{x}=0 \tag{14}\]  
In an ISL network, the specific form of the \(S\) matrix is detailed in [PERSON] et al. [42]. Then, precise orbits determined using center of gravity reference can be calculated as:  
\[\hat{x}=\left(B^{T}PB+SS^{T}\right)^{-1}\!BPI \tag{15}\]