## 4 Experimental Results of Applying Robust-EKF
### Applying Robust-EKF  
From Equation (10), we can write the compact user state of the three systems GPS, Galileo and GLONASS as follows:  
\[X=\left[ \begin{array}{cccccccccccc}x_{u}&v_{x}&y_{u}&v_{y}&z_{u}&v_{z}&b_{G}& d_{G}&b_{E}&d_{E}&b_{R}&d_{R}\end{array} \right]^{T} \tag{55}\]  
where \([x_{u},y_{u},z_{u}]\) is the user position; \([v_{x},v_{y},v_{z}]\) is the user velocity; \(b_{G}\), \(b_{E}\) and \(b_{R}\) are the errors in range due to the user's clock bias with GPS, Galileo and GLONASS time, respectively; \(d_{G}\), \(d_{E}\) and \(d_{R}\) are the user's clock drifts with GPS, Galileo and GLONASS time respectively. We then proceed to the steps of the robust Extended Kalman Filter:  
_Step 1_: _Linearizing equations_  
By linearizing the function \(f(X_{k-1})\), the state transition matrix \(A_{k}\) is obtained as a 6 \(\times\) 6 block diagonal matrix of the form:  
\[A_{k}=\left.\frac{\partial f}{X}\right|_{X=X_{k-1}}=\left[ \begin{array}{ ccccc}A_{x}&0&0&0&0&0\\ 0&A_{y}&0&0&0&0\\ 0&0&A_{z}&0&0&0\\ 0&0&0&A_{b,G}&0&0\\ 0&0&0&0&A_{b,E}&0\\ 0&0&0&0&0&A_{b,R}\end{array} \right] \tag{56}\]  
where \(A_{x}=A_{y}=A_{z}=A_{b,G}=A_{b,E}=A_{b,R}=\left[ \begin{array}{cc}1&\Delta T \\ 0&1\end{array} \right]\), and \(\Delta T\) is the time interval between two epochs. The predicted state is then determined by:  
\[X_{k}^{-}=A_{k}X_{k-1} \tag{57}\]  
The relation between the true state (\(X_{k}\)) and its prediction \((X_{k}^{-})\) can be written as:  
\[X_{k}^{-}=X_{k}-\delta_{k} \tag{58}\]The observation equation is defined as: \(Z_{k}=h\left(X_{k}\right)+e_{k}\), where  
\[Z_{k}= \begin{bmatrix}Z_{\rho,G}\\ Z_{\rho,G}\\ Z_{\rho,E}\\ Z_{\rho,E}\\ Z_{\rho,R}\\ \end{bmatrix}  \tag{59}\]  
where \(Z_{\rho,G}\) and \(Z_{\rho,G}\) are the pseudo range (\(\rho_{G}\)) and the pseudo range rate (\(\rho_{G}\)) matrices of \(N_{gps}\) GPS satellites tracked (dimension (\(N_{gps}\times 1\)) ); \(Z_{\rho,E}\) and \(Z_{\rho,E}\) are the pseudo range (\(\rho_{E}\)) and the pseudo range rate ( \(\rho_{E}\)) matrices of \(N_{gal}\) Galileo satellites tracked (dimension (\(N_{gal}\times 1\))); \(Z_{\rho,R}\) and \(Z_{\rho,R}\) are the pseudo range (\(\rho_{R}\) ) and the pseudo range rate ( \(\dot{\rho}_{R}\)) matrices of \(N_{glo}\) GLONASS satellites tracked (dimension (\(N_{glo}\times 1\)) ). The \(\rho_{G}\), \(\dot{\rho}_{G}\), \(\rho_{E}\), \(\dot{\rho}_{E}\), \(\rho_{R}\), and \(\dot{\rho}_{R}\) are calculated in Equation (9). Following step 1 in Section 3.1, the linearized \"Observation equation\" using the first order Taylor series expansion with the predicted state vector \(X_{k}^{-}\) is written as:  
\[Z_{k}-h\left(X_{k}^{-}\right)+H_{k}X_{k}^{-}=H_{k}X_{k}+e_{k} \tag{60}\]  
where the matrix \(H_{k}\) is determined as the derivative of the function \(h(X_{k}^{-}):\ H_{k}=\left.\frac{\partial h}{\partial X}\right|_{X=X_{k}^{-}}\)  
\[H_{k}=\left[ \begin{array}{cccccc}H_{\rho,G}&1&0&0&0&0&0\\ H_{\dot{\rho},G}&0&1&0&0&0&0\\ H_{\rho,E}&0&0&1&0&0&0\\ H_{\dot{\rho},E}&0&0&0&1&0&0\\ H_{\rho,R}&0&0&0&0&1&0\\ H_{\dot{\rho},R}&0&0&0&0&1\end{array} \right] \tag{61}\]  
\(H_{\rho,G}\), \(H_{\rho,E}\) and \(H_{\rho,R}\) are the Jacobians of the pseudo-range measurement model for GPS, Galileo and GLONASS data with the dimensions (\(N_{gps}\times 6\)), (\(N_{gal}\times 6\)) and (\(N_{glo}\times 6\)), respectively; \(H_{\dot{\rho},G}\), \(H_{\dot{\rho},E}\) and \(H_{\dot{\rho},R}\) are the Jacobians of the pseudo-range rate measurement model for GPS, Galileo and GLONASS data with the dimensions (\(N_{gps}\times 6\)), (\(N_{gal}\times 6\)) and (\(N_{glo}\times 6\)), respectively.