# An Improved Adaptive Iterative Extended Kalman Filter Based on Variational Bayesian
## 3 Illustrative Example
### Two-Dimensional Target Tracking
When the iteration count reaches \(N=10\), the ARMSE values tend to stabilize. This iterative improvement is significant, despite the anticipated increase in computational demands. In Figure 2, a detailed examination is conducted to measure the impact of the adjustment factor \(\tau\) on the performance of the proposed VBAIEKF algorithm. The adjustment factor \(\tau\) systematically varies within the range of \(\tau=1,2,3,5\ldots,10\). The results demonstrate that throughout the entire random process, the algorithm is uniformly influenced within a predetermined stable range. Notably, the operation of the algorithm is subject to a slight but not significant influence by the adjustment factor and minimization and precision enhancement. The VBAIEKF exhibits low sensitivity to parameters, displaying minimal influence from varying parameters. Therefore, \(\tau=6\) was selected. This influence consistently remains within a specific stable range throughout the random process. At \(\tau=6\), the algorithm achieves an optimal balance between error algorithm comparison experiments. In Figure 3, we conducted a comprehensive analysis to evaluate the impact of the forgetting factor \(\rho\) on the performance of the proposed VBAIEKF algorithm. Systematic variations of the variable \(\rho\) within a certain range can identify various factors affecting the algorithm's performance. Affecting the \(\rho\) algorithm's performance, \(\rho=1-\exp(-4)\), the impact of the forgetting factor significantly diminishes, thereby accelerating the convergence speed, reducing the RMSE, and making the algorithm output trajectory more perfect. The current configuration appears to have achieved the optimal balance between convergence speed, accuracy, and overall stability. Therefore, we chose \(\rho=1-\exp(-4)\) for the subsequent comparative analysis. Table 1 provides a comprehensive summary of the relative performance of the proposed VBAIEKF, GA-VB [32], and RKF-ML [33] at various levels of noise. Table 1 lists several key metrics: average root mean square error (ARMSE) for position and velocity, and average execution time. The results show that VBAIEKF has better convergence properties and adaptability. Compared to GA-VB and RKF-ML, the VBAIEKF shows a decrease in both the average position ARMSE and average velocity ARMSE, indicating improved accuracy in position and velocity estimation. Furthermore, this algorithm reduces the average execution time, highlighting its computational efficiency. Under various noise levels, using example noise \(\bar{R}=\text{diag}\big{(}(5\;\;\text{m})^{2},(0.05\;\text{rad})^{2}\big{)}\) as a reference, the convergence ranges under noises \(\bar{R}=\text{diag}\big{(}(1\;\text{m})^{2},(0.01\;\text{rad})^{2}\big{)}\) and \(\bar{R}=\text{diag}\big{(}(10\;\;\text{m})^{2},(0.1\;\text{rad})^{2}\big{)}\) are computed. From Table 1, it is evident that the VBAIEKF algorithm under different levels of noise exhibits a position convergence range of \(+0.058\%\)\(\sim\)\(+0.58\%\) and a velocity convergence range of \(-0.17\%\)\(\sim\)\(+0.21\%\), significantly smaller than RKF-ML and GA-VB. This indicates that the algorithm possesses excellent convergence and adaptability. Comparative analysis  
Figure 1: \(ARMSE_{pos}\) and \(ARMSE_{vol}\) for different iterations \(N\). Figure 4 presents a thorough comparative analysis, wherein the proposed VBAIEKF algorithm is juxtaposed with existing optimization algorithms GA-VB and RKF-ML in terms of the noise \(\bar{R}=\text{diag}\big{(}(5\,\text{m})^{2},(0.05\,\text{rad})^{2}\big{)}\), a specific number of iterations (\(N=10\)). The comparative analysis is based on the root mean square absolute error (ARMSE) of velocity and position estimates. The results underscore the efficacy of the VBAIEKF algorithm; compared to the existing optimization algorithms RKF-ML and GA-VB, the VBAIEKF exhibited significant improvements in \(RMSE_{pos}\) and \(RMSE_{vel}\), with increases of 21.81% and 22.11%, respectively.