# A Novel Anti-Jamming Technique for INS/GNSS Integration Based on Black Box Variational Inference
## 2 Problem Description
### INS/GNSS Integration System Model  
The loosely integrated INS/GNSS navigation system can output information on velocity, position, and attitude with the measurement information for velocity and position. The system models as follows. As the application background in this paper is vehicle navigation, the navigation parameters of the horizontal are required to be higher. So, for the velocity and position parameters, we only take their horizontal parameters to establish the state equations. We take the errors of horizontal velocity, horizontal position, platform angles, accelerometer bias, and gyroscope bias as the state vectors [22], which can be written as  
\[\mathbf{X}=\left( \begin{array}{cccccccccccc}\delta v_{\rm E}&\delta v_{\rm N}& \delta\varphi&\delta\lambda&\phi_{\rm E}&\phi_{\rm N}&\phi_{\rm U}&\
abla_{\rm x }&\
abla_{\rm y}&\
abla_{\rm z}&\epsilon_{\rm x}&\epsilon_{\rm y}&\epsilon_{ \rm z}\end{array} \right)^{\rm T}\]  
where \(\mathbf{X}\) is the state vector, \(\delta v_{\rm E}\) and \(\delta v_{\rm N}\) are the horizontal velocity errors, \(\delta\varphi\) and \(\delta\lambda\) are the horizontal position errors, \(\phi_{\rm E}\), \(\phi_{\rm N}\), and \(\phi_{\rm U}\) are the misalignment angles, \(\
abla_{\rm x}\), \(\
abla_{\rm y}\), and \(\
abla_{\rm z}\) are the accelerometer bias, \(\epsilon_{\rm x}\), \(\epsilon_{\rm y}\), and \(\epsilon_{\rm z}\) are the gyroscope bias. The measurement vectors derived from the difference between the velocity and position from the INS and GNSS can be given by  
\[\mathbf{Z}=\left( \begin{array}{cc}\mathbf{V}_{\rm INS}-\mathbf{V}_{\rm GNSS}&\mathbf{P}_{ \rm INS}-\mathbf{P}_{\rm GNSS}\end{array} \right)^{\rm T}\]  
where \(\mathbf{Z}\) is the measurement vector, \(\mathbf{V}_{\rm INS}\) and \(\mathbf{V}_{\rm GNSS}\) are the horizontal velocity of INS and GNSS, respectively, and \(\mathbf{P}_{\rm INS}\) and \(\mathbf{P}_{\rm GNSS}\) are the horizontal position of INS and GNSS, respectively. The state equations and measurement equations of the system are established based on the navigation coordinate system. The system models can be described as  
\[\left\{ \begin{array}{ll}\dot{\mathbf{X}}=\mathbf{\Phi}\mathbf{X}+\mathbf{W}\\ \mathbf{Z}=\mathbf{H}\mathbf{X}+\mathbf{V}\end{array} \right. \tag{1}\]  
where \(\mathbf{\Phi}\) is the one-step state transition matrix, \(\mathbf{W}\) is system noise, \(\mathbf{H}=\left[ \begin{array}{cc}\mathbf{I}_{4\times 4}&\mathbf{0}_{4\times 9}\end{array} \right]\) is the measurement matrix. \(\mathbf{V}=\left[ \begin{array}{cc}n_{V_{E}}&n_{V_{N}}&n_{\varphi}&n_{\lambda} \end{array} \right]^{\rm T}\) is the measurement noise. The state equations of Equation (1) are described in detail as follows. \[\left\{ \begin{array}{ll}\delta\dot{V}_{E}=&\delta V_{E}\frac{V_{N}\tan\varphi }{R_{N}}+\delta V_{N}(2\omega_{E}\sin\varphi+\frac{V_{E}\tan\varphi}{R_{N}}) +\\ &\delta\varphi(2\omega_{E}v_{N}\cos\varphi+\frac{V_{E}V_{N}}{R_{N}}\sec^{2} \varphi)+\phi_{U}f_{N}-\phi_{N}f_{U}+\
abla_{E}^{n}\\ \delta\dot{V}_{N}=&-\delta V_{E}(2\omega_{E}\sin\varphi+\frac{2V_{E}\tan \varphi}{R_{N}})-\\ &\delta\varphi(2\omega_{E}v_{E}\cos\varphi+\frac{V_{E}}{R_{N}}\sec^{2} \varphi)-\phi_{U}f_{E}+\phi_{E}f_{U}+\
abla_{N}^{n}\\ \delta\dot{\varphi}=&\frac{\delta V_{N}}{R_{N}}\\ \delta\dot{\lambda}=&\frac{\delta V_{N}}{R_{N}}\sec\varphi+\delta\varphi\frac{ V_{E}}{R_{N}}\tan\varphi\sec\varphi\\ \dot{\varphi}_{E}=&-\frac{\delta V_{E}}{R_{N}}\delta V_{E}-\phi\omega_{E}\sin \varphi-\phi_{U}(\omega_{E}\cos\varphi+\frac{V_{E}}{R_{N}})+\varepsilon_{E}^{ n}\\ \dot{\varphi}_{N}=&-\frac{1}{R_{N}}\delta V_{E}-\delta\varphi\omega_{E}\sin \varphi-\phi_{E}(\omega_{E}\sin\varphi+\frac{V_{E}\tan\varphi}{R_{N}})-\phi_{ U}\frac{V_{E}}{R_{N}}+\varepsilon_{N}^{n}\\ \dot{\varphi}_{U}=&\frac{\tan\varphi}{R_{N}}\delta V_{E}+\delta\varphi(\omega_ {ie}\cos\varphi+\frac{V_{E}}{R_{N}}\sec^{2}\varphi)+\phi_{E}(\omega_{ie}\cos \varphi+\frac{V_{E}}{R_{N}})+\phi_{N}\frac{V_{E}}{R_{N}}+\varepsilon_{U}^{n} \\ \dot{\
abla}=&0\\ \dot{\epsilon}=&0\end{array} \right. \tag{2}\]  
where \(\left[