# A Novel Anti-Jamming Technique for INS/GNSS Integration Based on Black Box Variational Inference
## 4 Experimental Results and Analyses
### Experiments with Simulation Data  
The validity of the proposed algorithm was verified by simulations. The simulation conditions are set as follows. The initial position is latitude 45.783898 degrees north and longitude 126.69458 degrees east. The Simulation lasts 60 min. The specifications of the INS and GNSS are listed in Table 2.  
Interference conditions are set as follows. The ranges of interference: (20 min-25 min) and (40 min-45 min). Velocity error of GNSS: 3 m/s. Position error of GNSS: 15 m.  
According to the above interference settings, the variation in GNSS velocities' noise and position noise caused by interference are shown in Figures 7 and 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**INS** & & \multicolumn{2}{c}{**GNSS**} \\ \hline Gyro bias & 0.02\({}^{\circ}\)/h & Velocity & 1 m/s \\ Accelerometer bias & 5 mg & Position & 8 m \\ Update rate & 100 HZ & Update rate & 1 HZ \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of the INS and GNSS. Figure 7: Variation in GNSS velocities’ noise. To evaluate the accuracy of the methods, the RMSE of velocity and position were used as performance metrics. It was defined as  
\[\text{RMSE}\triangleq\frac{1}{M}\sum_{i=1}^{M}\left(x_{k}^{l}-\hat{x}_{k}^{l}\right) \tag{37}\]  
where \(x_{k}^{l}\) denotes the true navigation parameters, and \(\hat{x}_{k}^{l}\) is the estimated navigation parameters. \(M=500\) represents the total number of Mont Carlo runs. A previous study proved the influence of different sample numbers on the BBVI method [21], and concluded that the algorithm has the fastest convergence speed when \(S=50\). Thus, \(S=50\) was selected as the sample number in this paper. In addition, the parameters of the proposed method were set as  
\[\eta=10,\ \gamma=0.5,\ \boldsymbol{g}=0,\ \boldsymbol{G}=0,\ \varepsilon=1 \times 10^{-10}.\]  
Before the navigation parameters of each algorithm were shown, we used Inv-Gamma, Inv-Wishart and Gaussian prior models to estimate the MNCM values. We take the north velocity for example, and the estimated MNCM values and the true MNCM values of the north velocity are shown in the Figure 9.  
For a clearer description of the estimated MNCM values, a larger version of Figure 10 is shown in Figure 9.  
Figure 8: Variation in GNSS position’s noise. Figure 9: MNCM values of the north velocity. Rvn is the MNCM value of the north velocity. It can be seen from Figures 9 and 10 that all three models can estimate the MNCM values of north velocity. It shows that all three methods can play an anti-jamming role. However, the MNCM values estimated by Inv-Gamma and Inv-Wishart prior models were both smaller than the true MNCM values, and the MNCM estimated by the Gaussian prior model is more accurate than the other two models. This would lead to inaccurate estimation of navigation parameters by the Inv-Gamma or Inv-Wishart models. Therefore, the anti-jamming effect is not as good as the proposed algorithm. We evaluated the anti-jamming effect of each algorithm from the estimated navigation parameters. The RMSEs of the velocity and position from the existing VBAKF and the proposed algorithm are shown in Figures 11 and 12, respectively. Table 3 shows the RMSEs of the velocity and position from existing VBAKF and the proposed algorithm, respectively. Figures 11 and 12 and Table 3 show that the proposed algorithm with the prior Gaussian distribution has smaller RMSEs than the existing VBAKF algorithm with the prior distribution of Inv-Gamma or Inv-Wishart. The velocity and position accuracy of the proposed method is 3-4 times and 1-2 times higher than the existing VBAKF method, respectively. It also can be seen that the RMSEs of the velocity and position for the VBAKF based on the prior distribution of Inv-Gamma are similar to those estimated by the VBAKF based on the prior distribution of Inv-Wishart. The latter has a slightly higher accuracy compared with the former. The simulation results and analyses proved that the prior distribution of MNCM is closer to the Gaussian distribution.