## 3 Methodology
### MDDA Augmented GNSS/INS Integration
#### 3.6.5 LC Integration Model  
As shown in Figure 3, after choosing the best set of the satellite by the MDDA, the receiver can determine the position and the velocity LC integration fuses the final six-element results of the GNSS receiver position and velocity with the INS navigation solution.  
The state vector is defined as  
\[\delta x_{15\times 1}^{I}=\left[\delta p_{3\times 1}^{I}\ \delta vel_{3\times 1 }^{I}\ \delta e_{3\times 1}^{I}\ \delta\omega_{3\times 1}\ \delta ax_{3\times 1}\right]^{T} \tag{35}\]  
where \(\delta p_{3\times 1}^{I}\) is the error vector of position, \(\delta vel_{3\times 1}^{I}\) is the error vector of velocity, \(\varepsilon_{3\times 1}^{I}\) is the attitude error vector, \(\delta\omega_{3\times 1}\) is the gyroscope drift error vector, and \(\delta ax_{3\times 1}\) is the accelerometer bias error vector. The measurement vector of the LC model contains the difference between the INS prediction and GPS reading as follows:  
\[\begin{bmatrix}p_{INS}^{I}-p_{GPS}^{I}\\ vel_{INS}^{I}-vel_{GPS}^{I}\end{bmatrix} \tag{36}\]  
The \(G\), \(F\), and \(H\) matrices in (31) and (32) can be expressed as follows:  
\[G_{1\times 15}=\left[\sigma_{p,1\times 3}\ \sigma_{vel,1\times 3}\ \sigma_{e,1\times 3 }\ \sigma_{\omega,1\times 3}\ \sigma_{ax,1\times 3}\right] \tag{37}\]  
where \(G\) is the vector of noise distribution and includes the standard deviation of the state vector \(x^{l}\).  
\[F_{15\times 15}=\begin{bmatrix}0_{3\times 3}&F_{p}&0_{3\times 3}&0_{3\times 3 }&0_{3\times 3}\\ 0_{3\times 3}&0_{3\times 3}&F_{vel}&0_{3\times 3}&C_{b}^{I}\\ 0_{3\times 3}&F_{e}&0_{3\times 3}&C_{b}^{I}&0_{3\times 3}\\ 0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&F_{av}&0_{3\times 3}\\ 0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&0_{3\times 3}&F_{ax}\end{bmatrix} \tag{38}\]  
\[F_{p}=\begin{bmatrix}\frac{1}{K_{m}+\Delta t}&0&0\\ 0&\frac{1}{\cos\left(R_{n}+\Delta t\right)}&0\\ 0&0&-1\end{bmatrix} \tag{39}\]\[F_{pel}=\begin{bmatrix}0&f_{d}&-f_{e}\\ -f_{d}&0&f_{n}\\ f_{e}&-f_{n}&0\end{bmatrix} \tag{40}\]  
where \(f_{n}\), \(f_{e}\), and \(f_{d}\) are accelerations of the body along the NED directions; we can transform from ENU to NED and vice versa with the following transformation matrix [54]:  
\[C_{NED}^{ENU}=C_{ENU}^{NED}=\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&-1\end{bmatrix} \tag{41}\]  
\[F_{\varepsilon}=\begin{bmatrix}0&\frac{-1}{R_{n}+all}&0\\ \frac{1}{(R_{n}+all)}&0&0\\ 0&\frac{\tan\theta}{R_{n}+all}&0\end{bmatrix} \tag{42}\]  
\[F_{ax}=\begin{bmatrix}-\beta_{ax_{x}}&0&0\\ 0&-\beta_{axy}&0\\ 0&0&-\beta_{ax_{z}}\end{bmatrix} \tag{43}\]  
\[F_{\omega}=\begin{bmatrix}-\beta_{\omega_{x}}&0&0\\ 0&-\beta_{\omega_{y}}&0\\ 0&0&-\beta_{\omega_{z}}\end{bmatrix} \tag{44}\]  
The \(H\) matrix is the design matrix of measurements and, in LC integration, the measurements exactly fit the states of the position and velocity errors; hence,  
\[H_{6\times 15}=[I_{6\times 6}\,0_{6\times 9}] \tag{45}\]  
Sequentially, after substitution in Equations (31) and (32), we can obtain the system and measurement LC model, respectively. Finally, the covariance matrix \(R\), which has the measured states' variance in its diagonal, can be expressed as  
\[R=diag[\sigma_{\phi}^{2}\;\sigma_{\lambda}^{2}\;\sigma_{alt}^{2}\;\sigma_{vel _{a}}^{2}\;\sigma_{vel_{d}}^{2}\;\sigma_{vel_{d}}^{2}] \tag{46}\]  
Also, the prediction covariance matrix \(P_{15\times 15}\)  
\[P_{15\times 15}=diag[\sigma_{p,3\times 3}^{2}\;\sigma_{vel,3\times 3}^{2}\; \sigma_{\epsilon,3\times 3}^{2}\;\sigma_{\omega,3\times 3}^{2}\;\sigma_{\alpha,3\times 3}^{2}] \tag{47}\]