## 2 BDS-3 Satellites POD Method
### Satellite Equations of Motion and Observation Models
For the POD of navigation satellites, the equations of motion and the equations of observation correspond to the dynamical and geometrical observation information of the satellite, respectively, and the solution of the orbit is to use these two aspects of information to solve for the parameters related to the initial state of the satellite. Suppose that, at the moment \(t_{i}\) the observation noise is \(V_{i}=\left[ \begin{array}{c}v_{PC}(t_{i})\\ v_{LC}(t_{i})\end{array} \right]\), the difference between the observed quantity and its valuation is \(L_{i}=\left[ \begin{array}{c}l_{PC}(t_{i})\\ l_{LC}(t_{i})\end{array} \right]\); the vector of orbital parameters to be estimated is \(x(t_{i})=dX^{s}\). Make \(H_{i}=\left[ \begin{array}{c}-u_{r}^{s}\\ -u_{r}^{s}\end{array} \right]\), the other parameter vectors to be estimated are \(y_{i}=\left[dX_{r}\quad dRm\quad t_{r}\quad t^{s}\quad N\right]^{T}\), and make \(B_{i}=\left[ \begin{array}{cccc}u_{r}^{s}(t_{i})&1&1&-1&0\\ u_{r}^{s}(t_{i})&1&1&-1&0\end{array} \right]\), where \(H_{i}\) and \(B_{i}\) are the design matrices for the parameter vector \(x(t_{i})\) and \(y_{i}\), respectively. Substituting into Equation (17) gives:  
\[v_{i}=H_{i}x(t_{i})+B_{i}y_{i}-l_{i} \tag{17}\]  
Combining Equations (8) and (17) yields:  
\[v_{i}=H_{i}x(t_{i})+B_{i}y_{i}-l_{i}=H_{i}\psi(t_{i},t_{0})x_{0}+B_{i}y_{i}-l_ {i} \tag{18}\]  
Make \(A_{i}=\left[H_{i}\psi(t_{i},t_{0})\quad B_{i}\right];x_{i}=\left[ \begin{array} []{c}x_{0}\\ y_{i}\end{array} \right]\). Substituting into Equation (18) gives:  
\[v_{i}=A_{i}x_{i}-l_{i} \tag{19}\]  
The observations of all satellites from \(t_{1}\) to \(t_{n}\) within the observation arc can be expressed in Equation (19), after which a least squares processing algorithm is used to perform repeated iterations of the solution until the result of the solution reaches the set limits. Using the orbits and clock offset of LEO and BDS satellites as unknown parameters, the use of LEO satellite on-board data can effectively increase the number of observable arc segments and enhance the geometrical structure of station satellites, thus improving the orbit determination accuracy of BDS satellites. \[L_{sta}=G(x_{bds},x_{sta},x_{0},t)+v_{sta} \tag{20}\]  
\[L_{leo}=G(x_{bds},x_{sta},x_{0},t)+v_{leo} \tag{21}\]  
where \(L_{sta}\) and \(L_{leo}\) denotes BDS observations obtained by ground-based and satellite-based receivers, respectively, \(x_{bds}\) and \(x_{leo}\) denote the BDS orbital parameters and LEO satellite orbital parameters, respectively, \(x_{sta}\) denotes the station-related parameter, \(x_{0}\) denotes the parameters related to the observed quantity, such as the carrier phase ambiguity, clock offset parameters, etc., and \(v_{sta}\) and \(v_{leo}\) correspond to the station and LEO satellite measurement noise, respectively. The observation equation is linearized and written in matrix form, denoted as:  
\[l=\left[l_{sta}\quad l_{leo}\right]^{T} \tag{22}\]\[A= \begin{bmatrix}\dfrac{\partial G}{\partial x_{bds}}&0&\dfrac{\partial G}{ \partial x_{sta}}&\dfrac{\partial G}{\partial x_{0}}\\ \dfrac{\partial F}{\partial x_{bds}}&\dfrac{\partial F}{\partial x_{leo}}&0& \dfrac{\partial F}{\partial x_{0}}\\ \end{bmatrix}  \tag{23}\]  
\[\delta x= \begin{bmatrix}\delta x_{bds}&\delta x_{leo}&\delta x_{sta}&\delta x_{ 0}\end{bmatrix} ^{T} \tag{24}\]  
The corresponding least squares solution can be expressed as:  
\[\delta\dot{x}=\left(A^{T}PA\right)^{-1}A^{T}Pl \tag{25}\]  
\[\hat{Q}_{x}=\left(A^{T}PA\right)^{-1}=\left(A^{T}_{sta}P_{sta}A_{sta}+A^{T}_{ leo}P_{leo}A_{leo}\right)^{-1} \tag{26}\]  
where \(P= \begin{bmatrix}P_{sta}\\ &P_{leo}\end{bmatrix} \), \(P_{sta}\), and \(P_{leo}\) represent the weight matrix, ground-based weight matrix, and satellite-based receivers weight matrix. \(\hat{Q}_{x}\) is a covariance matrix.