# Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link
## 4 Whole-Constellation Centralized Extended Kalman Filter  
The whole-constellation centralized extended Kalman filter (WCCEKF) is one of the centralized data processing methods. According to this method, a main satellite and a back-up satellite are assigned to complete the task of data processing. The other satellites in the constellation need to send their measurement data, state vectors, and corresponding covariance matrices to the main satellite for orbit determinations. For all the satellites in the constellation, the states and corresponding improving values from Equations (1) and (2) are collected and stored in a state vector \(\mathbf{X}_{k}\) and an improving values vector \(\delta\mathbf{x}_{k}\)[6; 28], as in  
\[\mathbf{X}_{k}=\left[ \begin{array}{cc}\left(\mathbf{X}^{1}\right)^{T}&\left(\mathbf{X}^ {2}\right)^{T}&\cdots&\left(\mathbf{X}^{i}\right)^{T}&\cdots&\left(\mathbf{X}^{n} \right)^{T}\end{array} \right]^{\text{T}} \tag{26}\]  
\[\delta\mathbf{x}_{k}=\left[ \begin{array}{cc}\left(\delta\mathbf{x}_{k}^{1}\right)^{ T}&\left(\delta\mathbf{x}_{k}^{2}\right)^{T}&\cdots&\left(\delta\mathbf{x}_{k}^{i} \right)^{T}&\cdots&\left(\delta\mathbf{x}_{k}^{n}\right)^{T}\end{array} \right]^{ \text{T}} \tag{27}\]  
where \(n\) is the number of satellites in the system. In this way, the state equation for all the satellites can be obtained through Equation (24). \[\delta\mathbf{x}_{k}=\mathbf{\Phi}_{k-1}\delta\mathbf{x}_{k-1}+\mathbf{G}_{k}\mathbf{w}_{k-1} \tag{28}\]  
The state transition matrix \(\mathbf{\Phi}_{k}\) and integral coefficient matrix \(\mathbf{G}_{k}\) are diagonal matrices and can be expressed as:  
\[\mathbf{\Phi}_{k-1}=\text{\emph{diagonal}}\Big{[} \begin{array}{cc}\left(\mathbf{ \Phi}_{k-1}^{1}\right)&\left(\mathbf{\Phi}_{k-1}^{2}\right)&\cdots&\left(\mathbf{\Phi} _{k-1}^{i}\right)&\cdots&\left(\mathbf{\Phi}_{k-1}^{n}\right)\end{array} \Big{]} \tag{29}\]  
\[\mathbf{G}_{k}=\text{\emph{diagonal}}\Big{[} \begin{array}{cc}\left(\mathbf{G}_{k}^ {1}\right)&\left(\mathbf{G}_{k}^{2}\right)&\cdots&\left(\mathbf{G}_{k}^{i}\right)& \cdots&\left(\mathbf{G}_{k}^{n}\right)^{T}\end{array} \Big{]} \tag{30}\]  
The state noise vector, \(\mathbf{w}_{k-1}\), stores the noise for all the satellites in the constellation,  
\[\mathbf{w}_{k-1}=\Big{[} \begin{array}{cc}\left(\mathbf{w}_{k-1}^{1}\right)&\left( \mathbf{w}_{k-1}^{2}\right)&\cdots&\left(\mathbf{w}_{k-1}^{i}\right)&\cdots&\left(\mathbf{ w}_{k-1}^{n}\right)\end{array} \Big{]}^{\text{T}}, \tag{31}\]  
and it has the statistical characteristics as follows:  
\[E\big{[}\mathbf{w}_{k-1}\big{]}=0,\quad E\bigg{[}\mathbf{w}_{k-1}^{i}\left(\mathbf{w}_{k- 1}^{j}\right)^{T}\bigg{]}= \begin{cases}\mathbf{Q}_{k-1}^{i}&i=j\\ 0&i\
eq j\end{cases}  \tag{32}\]  
\[E\Big{[}\mathbf{w}_{k-1}\mathbf{w}_{k-1}^{T}\Big{]}=\mathbf{Q}_{k-1}=\text{\emph{diagonal}} \Big{[} \begin{array}{cc}\mathbf{Q}_{k-1}^{1}&\mathbf{Q}_{k-1}^{2}&\cdots&\mathbf{Q}_{k-1 }^{i}&\cdots&\mathbf{Q}_{k-1}^{n}\end{array} \Big{]} \tag{33}\]  
The measurement equation, which is a combination of Equations (8) and (12), is then derived as:  
\[z_{k}=\mathbf{H}_{k}\delta\mathbf{x}+\varepsilon_{k} \tag{34}\]  
where \(\mathbf{H}_{k}=\left[0 \begin{array}{cc}\cdots&\left(\mathbf{H}^{i}\right)^{\text{T} }\cdots&0\end{array} \cdots\left(\mathbf{H}^{i}\right)^{\text{T}}\cdots&0\right]^{ \text{T}}\), \(z_{k}=z_{k}^{ij}(t_{k})\), and \(\varepsilon_{k}=\varepsilon^{ij}(t_{k})\) for the _ith_ satellite and _jth_ satellite with ISL; \(\mathbf{H}_{k}=\left[0 \begin{array}{cc}\cdots&\left(\mathbf{H}^{i}\right)^{\text{T} }\cdots&0\end{array} \right]^{\text{T}}\), \(z_{k}=z_{k}^{ig}(t_{k})\), and \(\varepsilon_{k}=\varepsilon^{ig}(t_{k})\) for GAS-to-_ith_ satellite measurements. Next, the measurement covariance matrix \(R_{k}=E(\varepsilon_{k}\varepsilon_{k}^{T})\), the initial state vector \(\overline{\mathbf{X}}_{0}=E(\mathbf{X}_{0}^{*})\), and the initial state vector covariance \(\overline{\mathbf{P}}_{0}=E[\mathbf{X}_{0}^{*}\mathbf{X}_{0}^{*T}]\) are defined.