## 2 The Model for GNSS/INS/LEO/5G Integrated Navigation
### The Structure of Federated Filtering Algorithm
#### 2.1.2 GNSS/INS Sub-Filter  
The GNSS/INS sub-filter uses tight coupling, and its flow chart is shown in Figure 2 below. In this paper, closed-loop correction is used as the final output of integrated navigation to reduce the error of linearization. Figure 2: Tightly coupled flow chart of GNSS/INS sub-filter. In this paper, the state vector of GNSS/INS sub-filter is based on the error state model, that is:  
\[x=\left\{\delta\psi,\delta v,\delta r,b_{a},b_{g},\delta\rho_{c}^{a},\delta\rho_{ c}^{a}\right\} \tag{7}\]  
where \(\delta\psi\) is the attitude error. \(\delta v\) is the velocity error. \(\delta r\) is the position error. \(b_{a}\) is the accelerometer offset. \(b_{g}\) is the gyroscope offset. \(\delta\rho_{c}^{a}\) is the receiver clock error. \(\delta\rho_{c}^{a}\) is the receiver clock drift. The transfer matrix of the GNSS/INS sub-filter system is the first-order term:  
\[\Phi_{GNSS,\ INS}^{e}\approx\left[ \begin{array}{cccccc}I_{3}-\Omega_{ie}^{e} \tau_{s}&0_{3}&0_{3}&0_{3}&\hat{C}_{b}^{e}\tau_{s}&0&0\\ F_{21}^{e}\tau_{s}&I_{3}-2\Omega_{ie}^{e}\tau_{s}&F_{23}^{e}\tau_{s}&\hat{C}_{b }^{e}\tau_{s}&0_{3}&0&0\\ 0_{3}&I_{3}\tau_{s}&I_{3}&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&I_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&I_{3}&0&0\\ 0&0&0&0&0&1&\tau_{s}\\ 0&0&0&0&0&0&1\end{array} \right] \tag{8}\]  
where \(F_{21}^{e}\) and \(F_{23}^{e}\) are:  
\[F_{21}^{e}=\left[-\left(\hat{C}_{b}^{e}\hat{I}_{ib}^{b}\right)\wedge\right], \tag{9}\]  
\[F_{23}^{e}=-\frac{2\hat{\tau}_{ib}^{e}}{\hat{\tau}_{cS}^{e}\left(L_{b}\right) \mid\frac{\rho_{cb}^{e}}{\hat{\tau}_{cb}^{e}}\mid} \tag{10}\]  
where, the \(\wedge\) in \(\hat{C}_{b}^{e}\) means that this is an estimate of the rotation matrix. \(\hat{\gamma}_{ib}^{e}\) is the gravitational acceleration at the estimated position. \(r_{eS}^{e}\) is coordinates of the satellite in ECEF system. The system covariance matrix is:  
\[P = \left[ \begin{array}{cccccc}SD_{att3}^{2}&0_{3}&0_{3}&0_{3}&0_{3}&0 &0\\ 0_{3}&SD_{vel3}^{2}&0_{3}&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&SD_{pos3}^{2}&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&SD_{buf3}^{2}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&0_{3}&SD_{bg3}^{2}&0&0\\ 0&0&0&0&0&SD_{cd}^{a}{}^{2}&0\\ 0&0&0&0&0&0&SD_{cd}^{a}{}^{2}\end{array} \right] \tag{11}\]  
where, \(SD_{i}\) is the standard deviation of attitude angle, velocity, position, accelerometer deviation, gyroscope deviation, receiver clock error, and receiver clock drift, which is calculated in the form of constant in the simulation. The innovation matrix is:  
\[\delta z_{G,k}^{-}\ =\ \left( \begin{array}{c}\delta z_{\rho,k}^{-}\\ \delta z_{r,k}^{-}\end{array} \right),\ \delta z_{r,k}^{-}\ =\ \left(\begin{array}{c}\tilde{\rho}_{a,C}^{1}-\tilde{\rho}_{a,C}^{1-} \tilde{\rho}_{a,C}^{2}-\tilde{\rho}_{a,C}^{2-},\cdots\tilde{\rho  
The observation matrix is:  
\[H_{G,k}^{\gamma}\approx\left( \begin{array}{cccccc}0_{1,3}&0_{1,3}&u_{a1\tau}^{ \gamma T}&0_{1,3}&0_{1,3}&1&0\\ 0_{1,3}&0_{1,3}&u_{a2}^{\gamma T}&0_{1,3}&0_{1,3}&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0_{1,3}&0_{1,3}&u_{am}^{\gamma T}&0_{1,3}&0_{1,3}&1&0\\ 0_{1,3}&u_{a1\tau}^{\gamma T}&0_{1,3}&0_{1,3}&0_{1,3}&0&1\\ 0_{1,3}&u_{a2}^{\gamma T}&0_{1,3}&0_{1,3}&0_{1,3}&0&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0_{1,3}&u_{am}^{\gamma T}&0_{1,3}&0_{1,3}&0_{1,3}&0&1\end{array} \right)_{x=x_ {k}^{-}} \tag{13}\]  
where \(u_{a1}^{\gamma T}\) is the line-of-sight unit vector from the user antenna, a, to satellite, 1. The observation noise matrix is:  
\[R_{G,k}^{\gamma}\approx\left( \begin{array}{cc}{SD_{\rho}}^{2}*I_{m,m}&0_{m,m} \\ 0_{m,m}&{S\dot{D}_{\rho}}^{2}*I_{m,m}\end{array}