# An Improved Adaptive Iterative Extended Kalman Filter Based on Variational Bayesian
## 1 Introduction  
Given the extensive use of nonlinear systems [1; 2] in various domains, including radar [3; 4], communication [5; 6], navigation [7; 8], control [9; 10], finance [11], and statistics [12; 13], the nonlinear dynamics of these systems require sophisticated filtering techniques for robust and precise estimation [14; 15]. However, the effectiveness of these filtering techniques significantly diminishes in the presence of heavy-tailed noise and errors. According to previous research [16; 17], this leads to a marked decline in estimation accuracy. Numerous filters have been proposed to address the issue of heavy-tailed noise impacting system tracking. Notable among these are the robust variational Bayesian filter [18; 19; 20], Student's t-based filters (RSTFs) [21; 22], [PERSON]'s M-estimation-based filters [23; 24], and the maximum correlation entropy-based filters [25]. While these filters demonstrate enhanced tracking performance and effectively mitigate the negative impact of heavy-tailed noise, they face challenges in precise parameter selection and computational complexity. The need for accurate parameter calibration, a potentially time-consuming and obstructive task, combined with the computational demands of these methods, may limit their practical applicability in real-time scenarios. Consequently, there is an urgent need to improve and streamline the parameter selection process, thereby reducing computational complexity. As a key technique in nonlinear recursive filtering, the Extended Kalman Filter (EKF) is widely recognized and utilized [16; 26; 27]. Its popularity stems from several advantages, such as obviating the need for calculating nominal trajectories, its straightforward methodology, and ease of implementation. However, enhancing the functionality and effectivenessof state estimation filters in real-world applications remains crucial. The Kalman filter algorithm based on Huber [28; 29], for example, shows resilience to outliers and superior performance in non-Gaussian settings. Yet, it faces increased computational complexity, requires fine-tuning of parameters, and may be less efficient in Gaussian environments. The Interactive Multiple Model (IMM)-based Kalman filter [30; 31] offers improved estimation accuracy and robustness amidst model uncertainties, but this benefit comes with increased computational and implementation complexities, as well as reliance on precise model selection and meticulous parameter tuning. A robust Gaussian approximate filter, integrating unknown scale matrices and degrees of freedom parameters, was introduced in the literature (GA-VB) [32]. To enhance this algorithm, the variational Bayesian (VB) method was employed to modify the state and posterior probability density functions (PDFs) for each parameter individually, aiming to reduce the overall complexity while inadvertently increasing sensitivity to parameters. Concurrently, literature (RKF-ML) [33] presented a robust state estimation algorithm using the maximum likelihood (ML) distribution to model heavy-tailed noise. This method notably avoids the need for manually selecting the degrees of freedom (DOF) parameters, thus being less sensitive to free parameters. However, it requires a VB approach to adaptively estimate ML distribution parameters in a closed recursive state, adding to the overall complexity. These algorithms underscore the trade-offs between complexity and sensitivity in algorithm design, highlighting the necessity for a nuanced approach that carefully balances these factors to achieve optimal performance in state estimation algorithms under challenging conditions. Addressing parameter sensitivity and algorithmic complexity, this paper introduces a new approach: an improved variational Bayesian (VB)-based adaptive iterative extended Kalman filter (VBAIEKF). This method aims to enhance the accuracy of filters handling imprecise heavy-tailed noise while simultaneously addressing parameter sensitivity and reducing algorithmic complexity.