## 4 Experimental Results of Applying Robust-EKF
### Applying Robust-EKF
\(H_{\rho,G}\) is computed as :  
\[H_{\dot{\rho},G}=\left[\begin{array}{cccccc}\frac{\partial\rho_{c,G}^{1}}{ \partial x}&0&\frac{\partial\rho_{c,G}^{1}}{\partial y}&0&\frac{\partial\rho_ {c,G}^{1}}{\partial z}&0\\ \frac{\partial\rho_{c,G}^{2}}{\partial x}&0&\frac{\partial\rho_{c,G  
where \(\frac{\partial\rho_{c,G}^{i}}{\partial x}=\frac{-\left(x_{G}^{i}-x_{u}\right)}{ \sqrt{\left(x_{G}^{i}-x_{u}\right)^{2}+\left(y_{G}^{i}-y_{u}\right)^{2}+\left(z_ {G}^{i}-z_{u}\right)^{2}}}\), \(\frac{\partial\rho_{c,G}^{i}}{\partial y}\)\(=\)\(\frac{-\left(y_{G}^{i}-y_{u}\right)}{\sqrt{\left(x_{G}^{i}-x_{u}\right)^{2}+\left(y_{G}^{i}-y_{u} \right)^{2}+\left(z_{G}^{i}-z_{u}\right)^{2}}}\), \(\frac{\partial\rho_{c,G}^{i}}{\partial z}\)\(=\)\(\sqrt{\left(x_{G}^{i}-x_{u}\right)^{2}+\left(y_{G}^{i}-y_{u}\right)^{2}+\left(z_{G}^{i}-z_{u} \right)^{2}}\), with \(i=1,\ \ldots,\ N_{gPS}\).  
\(H_{\rho,E}\), \(H_{\rho,R}\) and \(H_{\rho,E}\), \(H_{\rho,R}\) are computed similarly to \(H_{\rho,G}\), and \(H_{\rho,G}\) respectively.  
_Step 2:__Rerforming filter_  
Following step 1 and step 2 in Section 3.1, from Equations (58) and (60) we have the equation as Equation (19). We compact it as Equation (20), the compact regression equation is as follows:  
\[\tilde{Z}_{k}=\tilde{H}_{k}X_{k}+\tilde{e}_{k} \tag{64}\]  
The covariance matrix of \(\tilde{e}_{k}\) is \(\tilde{R}_{k}=\left[ \begin{array}{cc}P_{k}^{-}&0\\ 0&R_{k}\end{array} \right]=L_{k}L_{k}^{T}\) ; where \(R_{k}=E(e_{k}e_{k}^{T}\) ) is the measurement covariance matrix and \(P_{k}^{-}=E\left[\delta_{k}\delta_{k}{}^{T}\right]\) is the covariance matrix of the predicted error, and both matrices are needed to determine \(L_{k}\). The determination of \(R_{k}\) is not straightforward and in this work the Exponential model is proposed to estimate it. The Exponential model formulation [23] depends on the elevation angle and is given by:  
\[S_{l}=x_{0}+x_{1}e^{-\frac{E(l)}{2}}\left(m\right) \tag{65}\]  
where \(l\) refers to the \(lth\) satellite; \(S_{l}\) is the elevation uncertainty \(\left(m\right)\); \(x_{0}\) and \(x_{1}\) are constant; \(El(l)\) is the elevation angle of the \(lth\) satellite ( in \(degrees\)) and \(x_{2}\) is a scaled value of the elevation error (in \(degrees\)). \(x_{0}\), \(x_{1}\) and \(x_{2}\) must be estimated empirically. After performing many tests, we chose the values \(x_{0}=0.8\), \(x_{1}=0.4\) and \(x_{2}=12\). The normalization elevation uncertainty \(\sigma_{l}^{2}\) is given by [7]:  
\[\sigma_{l}^{2}=S_{l}/\left(\prod_{l=1}^{n_{s}}S_{l}\right)^{n_{s}} \tag{66}\]  
\(n_{s}\) being the number of the tracked satellites. The covariance of the measurement noise of GPS satellite is determined as follows:  
\[R_{k,G}=\left[ \begin{array}{ccc}\sigma_{1}^{2}&0&0\\ 0&\ddots&0\\ 0&0&\sigma_{N_{gps}}^{2}\end{array} \right] \tag{67}\]  
where \(N_{gps}\) is the number of the tracked GPS satellites. The covariance of the measurement noise of Galileo (\(R_{k,E}\)) and GLONASS (\(R_{k,R}\)) satellites are determined the same way as GPS satellite (\(R_{k,G}\)). Therefore, the covariance of the measurement noise of GPS, Galileo and GLONASS satellites is written as:  
\[R_{k}=\left[ \begin{array}{cccccc}R_{k,G}&0&0&0&0&0\\ 0&\gamma R_{k,G}&0&0&0&0\\ 0&0&R_{k,E}&0&0&0\\ 0&0&0&\gamma R_{k,E}&0&0\\ 0&0&0&0&R_{k,R}&0\\ 0&0&0&0&0&\gamma R_{k,R}\end{array} \right] \tag{68}\]  
where \(\gamma R_{k,G}\), \(\gamma R_{k,E}\) and \(\gamma R_{k,R}\) are the covariance matrices of the pseudo-range rates of GPS, Galileo and GLONASS, respectively. \(\gamma\) is the covariance pseudo-range rate to the covariance pseudo-rangeratio and is empirically chosen; After performing several tests, the value \(\gamma~{}=~{}0.01\) was chosen in this paper. The covariance prediction \(P_{k}^{-}\) is calculated by:  
\[P_{k}^{-}=A_{k}P_{k-1}A_{k}^{T}+Q_{k} \tag{69}\]  
where \(Q_{k}\) is the process covariance matrix.