# Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link
## 3 Fundamental Equations for Measurement and Motion
### Equations for Motion  
Satellites can be affected by a variety of factors when operated in orbit. For navigation satellites, only the gravitational forces, solar radiation pressure, and relativistic effects are considered [27]. The gravitational forces include the attractions from the earth, the moon, and other planets in the solar system. The dynamic equation for the \(ith\) satellite can be written as:  
\[\dot{\mathbf{X}}^{i}(t)=f^{c}(\mathbf{X}^{i},\mathbf{w}^{i}) \tag{13}\]  
where \(f^{c}\) is a continuous function, \(\mathbf{w}^{i}\) is the system disturbances that have the following properties:\[E\big{[}\mathbf{w}^{i}(t)\big{]}=0,\quad E\Big{[}\mathbf{w}^{j}(t)(\mathbf{w}^{i}(\tau))^{T} \Big{]}= \begin{cases}\mathbf{Q}^{i}(t)&t=\tau\\ 0&t\
eq\tau\end{cases}  \tag{14}\]  
\(E[\cdot]\) denotes the expected value, and \(\mathbf{Q}^{i}(t)\) is a covariance matrix which is symmetric, non-negative, and definite. Here, the system disturbances are simulated by Gaussian white noise. Because the continuous function \(f^{c}\) and the system disturbance \(\mathbf{w}^{i}\) are not coupled with each other, Equation (13) can be written as:  
\[\dot{\mathbf{X}}^{i}(t)=\mathcal{F}^{c}(\mathbf{X}^{i}(t))+\mathbf{G}\mathbf{w}^{i} \tag{15}\]  
In which \(\mathbf{G}=\Big{[} \begin{array}{ccc}\mathbf{0}&\mathbf{I}&\mathbf{0}\end{array} \Big{]}^{ \mathrm{T}}\) is the coefficient matrix, \(\mathbf{I}\) is the identity matrix, and \(\mathbf{0}\) is the zero matrix. Equation (15) is then linearized by Taylor expansion at the reference state vector \(\mathbf{X}^{i*}\):  
\[\dot{\mathbf{X}}^{i}(t)=\mathcal{F}^{c}(\mathbf{X}^{i*}(t))+\left.\frac{\partial \mathcal{F}^{c}(\mathbf{X}^{i})}{\partial\mathbf{X}^{i}}\right|_{\mathbf{X}^{i*}}(\mathbf{X}^ {i}(t)-\mathbf{X}^{i*}(t))+\cdots+\mathbf{G}\mathbf{w}^{i} \tag{16}\]  
From the equation above, the state increments are then derived as:  
\[\delta\dot{\mathbf{x}}^{i}(t)=\mathbf{F}(t)\delta\mathbf{x}^{i}(t)+\mathbf{G}\mathbf{w}^{i} \tag{17}\]  
where \(\delta\dot{\mathbf{x}}=\dot{\mathbf{X}}-\mathcal{F}^{c}(\mathbf{X}^{i*}(t))\), and \(\mathbf{F}(t)\) is the dynamic partial derivative matrix [6]:  
\[\mathbf{F}(t)=\left.\frac{\partial\mathcal{F}^{c}(\mathbf{X})}{\partial\mathbf{X}}\right| _{\mathbf{X}^{*}}=\left[ \begin{array}{ccc}0&\mathbf{I}&0\\ \frac{\partial\mathcal{F}^{c}}{\partial\tau}&\frac{\partial\mathcal{F}^{c}}{ \partial\tau}&\frac{\partial\mathcal{F}^{c}}{\partial\tau_{D}}\\ 0&\mathbf{0}&0\end{array} \right]_{|\mathbf{X}^{*}} \tag{18}\]  
Equation (17) is the state equation of the stochastic linear continuous systems and its general solution is:  
\[\delta\dot{\mathbf{x}}^{i}(t)=\mathbf{\Phi}^{i}(t,t_{0})\delta\dot{\mathbf{x}}^{i}_{0}(t) +\mathbf{G}\!\int_{t_{0}}^{t}\mathbf{\[PERSON]}^{i}(t,\tau)\mathbf{w}^{i}(\tau)\mathrm{d}\tau \tag{19}\]  
in which \(\mathbf{\[PERSON]}^{i}(t,t_{0})\) is the system state transition matrix and the solution of the following equations:  
\[\mathbf{\[PERSON]}^{i}(t,t_{0})=\mathbf{F}(t)\mathbf{\[PERSON]}^{i}(t,t_{0}),\quad\mathbf{\[PERSON]}^{i}(t_{ 0},t_{0})=\mathbf{I} \tag{20}\]  
where \(\mathbf{I}\) is the identity matrix with the same dimensions as dynamics matrix, \(\mathbf{F}(t)\). The state transition matrix \(\mathbf{\[PERSON]}^{i}(t,t_{0})\) has the following features:  
\[\mathbf{\[PERSON]}^{i}(t,\tau)\mathbf{\[PERSON]}^{i}(\tau,t_{0})=\mathbf{\[PERSON]}^{i}(t,t_{0}),\quad \Big{[}\mathbf{\[PERSON]}^{i}(t,\tau)\Big{]}^{-1}=\mathbf{\[PERSON]}^{i}(\tau,t) \tag{21}\]  
In the actual computation process, discretization needs to be implemented for Equation (19). \[\delta\dot{\mathbf{x}}^{i}_{k}=\mathbf{\[PERSON]}^{i}(t_{k},t_{k-1})\delta\dot{\mathbf{x}}^{i}_ {k-1}+\mathbf{G}\!\int_{t_{k-1}}^{t_{k}}\mathbf{\Phi}^{i}(t_{k},\tau)\mathbf{w}^{i}_{k-1}( \tau)\mathrm{d}\tau \tag{22}\]  
In a sampling interval from \(t_{k-1}\) to \(t_{k}\), the white noise \(\mathbf{w}^{i}_{k-1}(\tau)\) can be considered as a constant.