## 2 The Model for GNSS/INS/LEO/5G Integrated Navigation
### The Structure of Federated Filtering Algorithm
#### 2.1.5 Federated Main Filter  
The inertial navigation system, as a reference system, is separately combined with GNSS, LEO, and 5G systems to construct a set of local error state Kalman filters, which can be regarded as the first stage of federated filter integrated navigation. In the second stage of federated filter integrated navigation, the output of each sub-filter is combined to form the integrated navigation result. In this paper, the federated main filter is used for information fusion.  
The federated master filter is used for information fusion, and the state vector is:  
\[\hat{X}_{fk}=\left\{\delta\psi,\delta v,\delta r,b_{a},b_{g},\rho,\dot{\rho}\right\} \tag{20}\]  
The relationship between the state vector of the main filter and the sub-filter is:  
\[\hat{X}_{fk}=P_{fk}\sum\limits_{j=1}^{N}P_{jk}^{-1}\hat{X}_{jk} \tag{21}\]  
where \(P_{jk}^{-1}\) is the covariance matrix of sub-filter \(j\). \(\hat{X}_{jk}\) is the state estimate of sub-filter \(j\). \(N\) is the number of sub-filter. \(P_{fk}\) is the federated filtering covariance matrix, that is:  
\[P_{fk}=\left(\sum\limits_{j=1}^{N}P_{jk}^{-1}\right) \tag{22}\]  
The covariance matrix of the federated main filter \(P_{fk}\) is obtained by summing the inverse covariance matrices of each sub-filter, and then inverting. Then, find the sum of the product of the state values of each sub-filter \(\hat{X}_{jk}\) and the covariance inverse matrix \(P_{jk}^{-1}\). Finally, the final state solution after federated filtering \(\hat{X}_{fk}\) is obtained by multiplying the \(P_{fk}\) matrix with the sum \(\sum\
olimits_{j=1}^{N}P_{jk}^{-1}\hat{X}_{jk}\). By combining the state values and covariance of each sub-filter, the federated filter value is finally formed according to Formula (21), which is the mathematical significance of federated filtering.  
According to the different applications of federated filtering to state vector and covariance matrix, this paper mainly compares the following federated filtering combinations:  
1. Federation no reset combination, FNR:  
In FNR mode, Formulas (21) and (22) are used as the combination of single point fusion algorithms [29]. When the fusion algorithm is updated, it can be directly used to update the federated integrated navigation output without feedback on the state estimation and covariance of the local filter.
2. Federation fusion reset combination, FFR:  
In the FFR mode, the state estimation and error covariance matrix obtained from the information fusion of the federated main filter is fed back to the sub-filter. The sub-filter replaces the corresponding state vector, covariance matrix, and the feedback covariance matrix is multiplied by the coefficient \(\beta_{j}\)[28]. \(\beta_{j}\) is the information distribution factor of sub-filter j, which satisfies the principle of information fusion; that is, for N sub-filters, there are \(\sum\
olimits_{j=1}^{N}\beta_{j}=1\).
3. Federation Zero reset combination, FZR:  
In FZR mode, after the state estimation result of the sub-filter is input to the main filter, the state values of all the sub-filters are set to zero, and the corresponding covariance matrix element values are set to the initial value [29].