## 2 BDS-3 Satellites POD Method
### Satellite Equations of Motion and Observation Models  
Following [PERSON]'s theorem of motion, satellites are subjected to a combination of multiple perturbing forces. Therefore, the equations of motion need to be established in an inertial coordinate system. Combining the various perturbing forces, the equations of motion of a navigation satellite can be written as [28]:  
\[\overset{\cdot}{r}(t)=-\frac{GM_{E}}{r^{3}}r+f_{1}(r,\overset{\cdot}{r},p,t)=f _{0}(r,t)+f_{1}(r,\overset{\cdot}{r},p,t) \tag{1}\]  
where \(t\) is for a specific moment of observation, \(r\), \(\overset{\cdot}{r}\), \(\overset{\cdot}{r}\) are the position, velocity, and acceleration of the satellite's center of mass at the moment \(t\), respectively. \(p\) is the vector of model parameters to be estimated with the POD filter. \(f_{0}\) represents the gravitational force of the Earth. \(f_{1}\) indicate other perturbations (spherical harmonic expansion of the Earth's gravity field, non-gravitational perturbations, third-body perturbations, etc). The second-order differential equation can be written as a system of first-order differential equations. The form is as follows:  
\[\left\{ \begin{array}{l}\dot{\mathbf{r}}=v\\ \dot{v}=\ddot{r}\\ \dot{p}=0\end{array} \right. \tag{2}\]  
The initial state of the satellite can be recorded as:  
\[\left\{ \begin{array}{l}r(t_{0})=v\\ v(t_{0})=\dot{r}\\ p(t_{0})=p_{0}\end{array} \right. \tag{3}\]  
The first-order differential equation can be written as:  
\[\left\{ \begin{array}{l}\dot{X}=F(X,t)\\ X(t_{0})=X_{0}\end{array} \right. \tag{4}\]  
By solving the satellite variational equation, the state transfer matrix can be derived so that the satellite's precise orbital parameters can be obtained from the initial orbital parameters. Assuming that \(X^{*}\) is the satellite reference orbit and linearizing the differential Equation (4) (neglect the higher order terms), we obtain:  
\[\dot{X}=F(X^{*},t)+\left.\frac{\partial F(X,t)}{\partial X}\right|_{*}(X-X^{*} )=\dot{X^{*}}+\left.\frac{\partial F(X,t)}{\partial X}\right|_{*}(X-X^{*}) \tag{5}\]  
of which,  
\[\frac{\partial F(X,t)}{\partial X}=\left[ \begin{array}{ccc}\frac{\partial \dot{r}}{\partial r}&\frac{\partial\dot{r}}{\partial\dot{r}}&\frac{\partial \dot{r}}{\partial p}\\ \frac{\partial\ddot{r}}{\partial\dot{r}}&\frac{\partial\ddot{r}}{\partial\dot {r}}&\frac{\partial\ddot{r}}{\partial p}\\ \frac{\partial\dot{p}}{\partial r}&\frac{\partial\dot{p}}{\partial\dot{r}}& \frac{\partial\dot{r}}{\partial\dot{p}}\end{array} \right]=\left[ \begin{array}[] {ccc}0&I&0\\ \frac{\partial\ddot{r}}{\partial\dot{r}}&\frac{\partial\ddot{r}}{\partial\dot {r}}&\frac{\partial\ddot{r}}{\partial p}\\ 0&0&0\end{array} \right] \tag{6}\]  
\(x(t)=X-X^{*}\), \(A(t)=\frac{\partial F(X,t)}{\partial X}\), Equation (5) can be expressed as:  
\[\dot{x}(t)=A(t)\cdot x(t) \tag{7}\]  
This can be recorded as:  
\[x(t)=\psi(t,t_{0})\cdot x_{0} \tag{8}\]  
\[\psi(t,t_{0})=\left[ \begin{array}{ccc}\frac{\partial r}{\partial r_{0}}& \frac{\partial\dot{r}}{\partial\dot{r}_{0}}&\frac{\partial\tau}{\partial p} \\ \frac{\partial\dot{r}}{\partial r_{0}}&\frac{\partial\dot{r}}{\partial\dot{r} _{0}}&\frac{\partial\ddot{r}}{\partial p}\\ \frac{\partial p}{\partial r_{0}}&\frac{\partial\dot{p}}{\partial\dot{r}_{0}} &\frac{\partial p}{\partial p}\end{array} \right] \tag{9}\]  
Further decomposition of \(\psi(t,t_{0})\):  
\[\phi(t,t_{0})=\frac{\partial x}{\partial x_{0}}=\left[ \begin{array}{ccc} \frac{\partial r}{\partial\dot{r}_{0}}&\frac{\partial\tau}{\partial\dot{r}_ {0}}\\ \frac{\partial\dot{r}}{\partial\dot{r}_{0}}&\frac{\partial\dot{r}}{\partial\dot {r}_{0}}\end{array} \right]_{6\times 6} \tag{10}\]\[S(t)=\frac{\partial x}{\partial p}= \begin{bmatrix}\frac{\partial\tau}{\partial p} \\ \frac{\partial\dot{r}}{\partial p}\end{bmatrix}  \tag{11}\]  
Additionally, there are:  
\[ \begin{split}\phi(t_{0},t_{0})=l_{6*6}\\ S(t_{0})=0_{6*n_{d}}\end{split}  \tag{12}\]  
where \(n_{d}\) denotes the vector dimension corresponding to the relevant perturbing force and \(I_{6*6}\) denotes the unit matrix.