## 3 Federated Kalman Filter Design
### Kalman Filter Algorithm for Inertial/Odometer Integrated Navigation
#### 3.6.3. Information Fusion Process  
The federated filter global state estimation \(\hat{\mathrm{X}}(\mathrm{k})\) and covariance matrix \(\mathrm{P}(\mathrm{k})\) reset the state estimation \(\hat{\mathrm{X}}_{\mathrm{i}}(\mathrm{k})\) and covariance matrix \(\mathrm{P}_{\mathrm{i}}(\mathrm{k})\) of the subfilter so that the subfilter can update recursively according to the following relationship:  
Figure 4: Federated filter architecture.  
\[\widehat{X}_{i}(k)=\widehat{X}(k) \tag{28}\]  
\[P_{i}(k)=\beta_{i}^{-1}P(k) \tag{29}\]  
According to the principle of information conservation, the information distribution coefficient should satisfy the following condition:  
\[\sum_{i=1}^{2}\beta_{i}=1 \tag{30}\]  
Different \(\beta_{i}\) can obtain different federated filter structures and characteristics (fault tolerance, accuracy, and calculation amount). The better the performance of the sub-navigation system, the larger the corresponding information distribution coefficient.  
6.4 Information Distribution Method Based on Orthogonality of Innovation and Piecewise Smooth Activation Function  
The working scene of the track detector often has a large number of unfavorable factors such as water area, mountains, tunnels, high-voltage cables, etc. It is therefore easy for the signal of a satellite positioning system to lose its lock through a multipath effect or electromagnetic interference, which can lead to noise in the output. The reliability of the measurement of the satellite positioning system is proportional to the dilution of precision (DOP) value of the satellite signal, where a smaller DOP indicates higher accuracy [34; 35]. In this article, according to the nature of innovation orthogonality, the DOP value was used to judge whether each component of the quantity measurement was an outlier, and this information was then used to determine the information distribution coefficient.  
The noise variance measurement and the DOP value have the following relationship:  
\[R_{i}=N_{i}DOP_{i}+R_{i0} \tag{31}\]  
where \(R_{i}\) is credibility of the \(i_{th}\) observed quantity, \(N_{i}\) is the scaling factor of the satellite positioning system signal precision and the signal DOP value, and \(R_{i0}\) is adjustment amount for noise measurement.  
The information allocation coefficient was determined by the piecewise smooth activation function shown in the following equation:  
\[f_{i}(r_{i})=\left\{\begin{array}{c}1\;r_{i}\ll d_{i}\\ d_{i}/r_{i}\;r_{i}>d_{i}\end{array}\right.\;\;i=1,2,\cdots,m \tag{32}\]  
where \(r_{i}\) and \(d_{i}\) are obtained from \(R_{i}\). The specific derivation and calculation process are detailed in [28].  
The state vector fusion shown in Equation (32) occurs when the measurement is not an outlier, the gain of activation function is 1, and the information distribution coefficient \(\beta_{1}=1\). When the measurement is an outlier (\(r_{i}>d_{i}\) and the activation function \(<1\), according to the dilution of precision of current measurement), the weighted distribution of \(\beta_{1}\) and \(\beta_{2}\) in different precision ranges is realized.