# Research on Kalman Filter Fusion Navigation Algorithm Assisted by CNN-LSTM Neural Network
## 2 Kalman Filtering GNSS/INS Fusion Navigation Basic Principles
### Mathematical Model of GNSS/INS Fusion Navigation
#### 2.2.1 System Equations  
In the tightly coupled mode, the state variables consist of two parts [25]: One part is the error state of the INS, including attitude, velocity, position, and sensor biases, totaling 15 dimensions. Its state equations are as follows:  
\[X_{1}=F_{1}(t)X_{1}(t)+\Gamma(t)W(t) \tag{3}\]  
In the equation provided, this equation  
\[X_{1}=\left[\phi_{E},\phi_{N},\phi_{UL},\delta v_{E},\delta v_{N},\delta v_{U}, \delta L,\delta\lambda,\delta h,\varepsilon_{x},\varepsilon_{y},\varepsilon_ {z},\
abla_{x},\
abla_{y},\
abla_{z}\right]^{T}\]  
is established based on the strapdown INS error equation. The state transition matrix is given by  
\[F_{1}=\begin{bmatrix}(F_{N})_{9\times 9}&(F_{S})_{9\times 9}\\ 0&(F_{M})_{6\times 6}\end{bmatrix}_{15\times 15} \tag{4}\]  
In the equation provided, \(F_{N}\) is the \(9\times 9\) error matrix of the INS, corresponding to the basic error equation of the INS. Within the transition matrix, \(F_{S}\) is represented as:  
\[F_{S}=\begin{pmatrix}C_{b}^{n}&O_{3\times 3}\\ O_{3\times 3}&C_{b}^{n}\\ O_{3\times 3}&O_{3\times 3}\end{pmatrix} \tag{5}\]  
\[F_{M}=Diag\bigg{[}-\frac{1}{T_{rx}},-\frac{1}{T_{ry}},-\frac{1}{T_{rz}},-\frac {1}{T_{ax}},-\frac{1}{T_{ay}},-\frac{1}{T_{az}}\bigg{]} \tag{6}\]  
The other part of the error state comprises errors from the GNSS. In tightly coupled integration, two time-dependent errors are usually removed: one is the distance error \(\delta t_{ru}\) equivalent to the error in Equation (1), and the other is the distance error \(\delta t_{u}\) equivalent to the clock frequency error, \(\delta t_{u}\) typically modeled as a first-order Markov process. As this system involves two satellite navigation systems, the error states of the GNSS are represented by the distance errors equivalent to the clock errors of the GPS and BeiDou systems, denoted as \(\delta t_{u1}\) and \(\delta t_{u2}\), respectively, and the distance rate errors equivalent to the clock frequency errors, denoted as \(\delta t_{ru1}\) and \(\delta t_{ru2}\), respectively.  
Therefore, the error state of GNSS is given by  
\[X_{G}(t)=[\delta t_{u1},\delta t_{u2},\delta t_{ru1},\delta t_{ru2}]^{T} \tag{7}\]  
The corresponding differential equation is  
\[\begin{split}&\delta t_{u}=\delta t_{ru}+\omega_{t}\\ &\delta t_{ru}=-\beta\delta t_{ru}+\omega_{ru}\end{split} \tag{8}\]  
where \(\beta\) is the correlation time. Expressing the equation in matrix form yields:  
\[X_{G}(t)=F_{G}(t)X_{G}(t)+\Gamma_{G}(t)W_{G}(t) \tag{9}\]  
Hence,  
\[\begin{bmatrix}\delta t_{u1}\\ \delta t_{u2}\\ \delta t_{ru1}\\ \delta t_{ru2}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-\beta_{t_{ru2}}\end{bmatrix}\begin{bmatrix}\delta t_{u1}\\ \delta t_{u2}\\ \delta t_{ru1}\\ \delta t_{ru2}\end{bmatrix}+\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\omega_{u1}\\ \omega_{u2}\\ \omega_{ru1}\\ \omega_{ru2}\end{bmatrix} \tag{10}\]Combining the error state equations of the INS and the GPS, the state equation of the pseudorange and pseudorange rate combination system is obtained as follows:  
\[\begin{bmatrix}X_{1}(t)\\ X_{G}(t)\end{bmatrix}=\begin{bmatrix}F_{1}(t)&0\\ 0&F_{G}(t)\end{bmatrix}\begin{bmatrix}X_{1}(t)\\ X_{G}(t)\end{bmatrix}+\begin{bmatrix}\Gamma_{1}(t)&0\\ 0&\Gamma_{G}(t)\end{bmatrix}\begin{bmatrix}W_{1}(t)\\ W_{G}(t)\end{bmatrix} \tag{11}\]