# An Improved Adaptive Iterative Extended Kalman Filter Based on Variational Bayesian
## 3 Illustrative Example
### Nonlinear Numerical UNGM  
This paper employs a nonlinear numerical UNGM [40] to demonstrate the effectiveness of VBRAIEKF. The equation representing the state space of the UNGM model is as follows:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Algorithms and Noise**} & \multicolumn{3}{c}{**Values**} \\ \hline
**Algorithms** & **Noise** & \(\boldsymbol{ARMSE_{pos}(\text{m})}\) & \(\boldsymbol{ARMSE_{vel}(\text{m/s})}\) & \(\boldsymbol{Time(\text{s})}\) \\ \hline \multirow{3}{*}{RKF-ML} & \(\bar{R}=\text{diag}\big{(}(1\,\text{m})^{2},(0.01\,\text{rad})^{2}\big{)}\) & 6.3241 (+4.25\%) & 7.5472 (\(-3.54\%\)) & 0.31657 \\  & \(\bar{R}=\text{diag}\big{(}(5\,\text{m})^{2},(0.05\,\text{rad})^{2}\big{)}\) & 6.0665 (0\%) & 7.8242 (0\%) & 0.32264 \\  & \(\bar{R}=\text{diag}\big{(}(10\,\text{m})^{2},(0.1\,\text{rad})^{2}\big{)}\) & 6.3854 (+5.26\%) & 7.6821 (\(-1.82\%\)) & 0.33187 \\ \hline \multirow{3}{*}{GA-VB} & \(\bar{R}=\text{diag}\big{(}(1\,\text{m})^{2},(0.01\,\text{rad})^{2}\big{)}\) & 4.8532 (\(-1.65\%\)) & 10.3365 (\(-1.03\%\)) & 0.11242 \\  & \(\bar{R}=\text{diag}\big{(}(5\,\text{m})^{2},(0.05\,\text{rad})^{2}\big{)}\) & 4.9348 (0\%) & 10.4445 (0\%) & 0.11754 \\  & \(\bar{R}=\text{diag}\big{(}(10\,\text{m})^{2},(0.1\,\text{rad})^{2}\big{)}\) & 4.9938 (+1.196\%) & 10.5564 (+1.07\%) & 0.12432 \\ \hline \multirow{3}{*}{The VBAIEKF} & \(\bar{R}=\text{diag}\big{(}(1\,\text{m})^{2},(0.01\,\text{rad})^{2}\big{)}\) & 4.3034 (+0.058\%) & 7.1024 (\(-0.17\%\)) & 0.10128 \\  & \(\bar{R}=\text{diag}\big{(}(5\,\text{m})^{2},(0.05\,\text{rad})^{2}\big{)}\) & 4.3009 (0\%) & 7.1147 (0\%) & 0.11215 \\ \cline{1-1}  & \(\bar{R}=\text{diag}\big{(}(10\,\text{m})^{2},(0.1\,\text{rad})^{2}\big{)}\) & 4.3259 (+0.58\%) & 7.2658 (+0.21\%) & 0.11393 \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(\boldsymbol{ARMSE_{pos}}\), \(\boldsymbol{ARMSE_{vel}}\) and the average execution time for the RKF-ML, GA-VB, and VBAIEKF under different noise.  
Figure 4: \(RMSE_{pos}\) and \(RMSE_{vel}\) for different algorithms.  
\[\left\{\begin{array}{l}x_{k}=0.5x_{k-1}+25\frac{x_{k-1}}{1+x_{k-1}^{2}}+8\cos(1.2 (k-1))+v_{k-1}\\ y_{k}=x_{k}^{2}/20+r_{k}\end{array}\right. \tag{51}\]  
where \(Q_{k-1}\) and \(r_{k}\) represent measurement noise with a mean of zero and covariance, and \(v_{k-1}\) represents non-Gaussian process noise \(R_{k}\).  
The measurement noise is selected to be the typical heavy-tailed noise [25].  
\[v_{i}\sim 0.9\mathcal{N}(0,\bar{R})+0.1\mathcal{N}(0,100\bar{R}) \tag{52}\]  
The symbol \(\bar{R}\) represents the measurement noise covariance matrix, which is defined as \(\text{diag}\big{(}(1\,\text{m})^{2},(0.01\,\text{rad})^{2}\big{)}\). By adjusting the parameters to optimize the algorithm's performance, setting the forgetting factor \(\rho=1-\exp(-4)\) significantly reduces its impact, resulting in a significant decrease in influence. By selecting \(\varepsilon=10^{-6}\), the algorithm's convergence speed is greatly enhanced. By setting the tuning factor \(\tau=6\), the algorithm attains an optimal equilibrium in error algorithm comparative experiments. Performing 10 iterations decreases the average root mean square error (ARMSE), resulting in a more precise trajectory for the algorithm's output. During the simulation experiment, GA-VB, RKF-ML, and VBAIEKF are employed to evaluate the filtering efficacy.  
Figure 5 examines the comparability of \(RMSE_{pos}\) and \(RMSE_{vel}\) for the GA-VB, RKF-ML, and the VBAIEKF algorithms. The results indicate that the estimation state of VBAIEKF demonstrates higher precision, with RMSE values lower than those of GA-VB and RKF-ML. It can more accurately estimate the expected state covariance and the actual state covariance. The improvement in estimation accuracy for both position and velocity signifies its reliability.