Student’s t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter for GPS Navigation

Abstract

Global Navigation Satellite System (GNSS) receivers may produce measurement outliers in real-world applications owing to various circumstances, including poor signal quality, multipath effects, data loss, satellite signal loss, or electromagnetic interference. This can lead to a noise distribution that is non-Gaussian heavy-tailed, affecting the effectiveness of satellite navigation filters. This paper presents a robust Extended Kalman Filter (EKF) based on the Maximum Correntropy Criterion with a Student’s t kernel (STMCCEKF) for GPS navigation under non-Gaussian noise. Unlike traditional EKF and Gaussian-kernel MCCEKF, the proposed method enhances robustness by leveraging the heavy-tailed Student’s t kernel, which effectively suppresses outliers and dynamic observation noise. A fixed-point iterative algorithm is used for state update, and a new posterior error covariance expression is derived. The simulation results demonstrate that STMCCEKF outperforms conventional filters in positioning accuracy and robustness, particularly in environments with impulsive noise and multipath interference. The Student’s t-distribution kernel efficiently mitigates heavy-tailed non-Gaussian noise, while it adaptively adjusts process and measurement noise covariances, leading to improved estimation performance. A detailed explanation of several key concepts along with practical examples are discussed to aid in understanding and applying the Global Positioning System (GPS) navigation filter. By integrating cutting-edge reinforcement learning with robust statistical approaches, this work advances adaptive signal processing and estimation, offering a significant contribution to the field.

1. Introduction

The Global Positioning System (GPS) [1,2] offers several advantages in modern satellite navigation systems, including global coverage, resistance to weather-related disruptions, and high positioning accuracy. However, real-world environments such as tunnels, dense urban areas with tall buildings, and forests present diverse topographies and atmospheric conditions that can obstruct GPS signals or cause satellite failures. These disruptions may degrade system performance and lead to positioning errors. In such challenging conditions, especially under non-Gaussian noise, ensuring algorithmic robustness becomes essential. When the noise is assumed to follow a Gaussian distribution, estimators based on Mean Square Error (MSE) are commonly employed, such as the Extended Kalman Filter (EKF).

To address this, it is logical to design cost functions that directly influence the learning process, transferring the information within data samples into system parameters. Information Theoretic Learning (ITL) provides a framework for such learning without assuming specific data distributions or relying on particular learning architectures, only requiring data availability. This perspective blurs the traditional boundaries between supervised and unsupervised learning and offers significant utility for engineering applications [3,4,5].

Recent advances have introduced nonlinear filtering methods based on entropy concepts as alternatives for GPS signal processing. However, ITL methods utilizing entropy measures, such as Minimum Error Entropy (MEE) [6,7] and Maximum Correntropy Criterion (MCC) [8,9,10,11], have shown strong performance in non-Gaussian settings, particularly by suppressing the effects of impulsive noise via kernel-based approaches. By applying entropy-based criteria like MEE, optimal estimation becomes more feasible. Specifically, the Mean Error Entropy Extended Kalman Filter (MEEEKF) leverages the full randomness of residuals to improve estimation accuracy. This technique has demonstrated effectiveness in multipath parameter estimation and loop tracking in GPS code tracking systems.

Correntropy is a statistical measure that captures both second and higher-order statistics, making it particularly effective in non-Gaussian noise environments. Chen et al. [12] proposed the Maximum Correntropy Kalman Filter (MCCKF) as a robust alternative to conventional Kalman filtering. Unlike traditional approaches based on the Minimum Mean Square Error (MMSE) criterion, the MCCKF uses the MCC for optimization, enhancing resilience against impulsive or non-Gaussian noise. The filter estimates state and covariance using standard propagation equations and updates posterior estimates through a fixed-point algorithm, improving robustness where traditional Kalman Filters may fail. It assesses relationships between variables using the probability density function (PDF). Within this framework, MCC serves as a robust cost function in filtering applications, especially in vehicle navigation systems. Subsequently, a robust statistical linearization approach based on the Maximum Correntropy Criterion (MCC) is applied to compute the posterior state and covariance estimates. The MCC, developed through Information Theoretic Learning, has been widely adopted in filter designs involving non-Gaussian noise environments. To further enhance numerical stability, a square root formulation of the Kalman filter has been introduced.

The studies in [13,14] introduce a new filter known as the maximum correntropy unscented filter (MCUF), developed to enhance the robustness of the unscented Kalman filter (UKF) in the presence of impulsive noise. In the MCUF framework, the unscented transformation (UT) is employed to generate prior estimates of both the state and covariance matrices. The robustness and adaptability for GNSS integration have been verified by using the cubature Kalman filter (CKF) [15]. Additionally, cubature rules are utilized to compute the second-order statistics of Gaussian random variables undergoing nonlinear transformations. Building on this foundation, the current research investigates an adaptive and robust CKF. The adaptive capability is achieved using the variational Bayesian (VB) method to dynamically estimate the measurement noise covariance [16]. Meanwhile, robustness is established by leveraging MCC to effectively suppress outliers. This method addresses several key challenges in real-world filtering: handling non-Gaussian noise, adapting to variable and unpredictable noise covariances, and ensuring efficient performance under dynamic conditions.

A more recent advancement is the robust Student’s t-based stochastic cubature filter (RSTSCF), designed for nonlinear state-space models, where the relationship between state variables and observations is complex. By applying cubature integration rules, the RSTSCF efficiently estimates system states. It is particularly well suited for scenarios involving heavy-tailed process and measurement noise. The benefits of kernel adaptive filters, including linearity and convexity in reproducing kernel Hilbert space (RKHS) and the ability to approximate nonlinear functions using universal kernels have attracted significant attention. In the fields of machine learning and nonlinear signal processing, especially under non-Gaussian conditions, correntropy has emerged as a compelling alternative to traditional Mean Square Error (MSE)-based criteria. This trend has motivated the development of a new kernel adaptive filtering method, termed the kernel maximum correntropy (KMC) method [17,18,19,20,21].

The RSTSCF also introduces a powerful framework for addressing heavy-tailed noise through the use of the Student’s t-distribution, known for its robustness against outliers and impulsive disturbances, which is common in navigation and tracking applications [22]. The degrees of freedom in the distribution allow control over tail heaviness, offering flexibility in noise modeling. In this work, a generalized Gaussian model incorporating a Student’s t kernel is proposed, as it more effectively captures the characteristics of heavy-tailed non-Gaussian noise compared to traditional Gaussian assumptions. This paper introduces a novel filtering approach, the Student’s t kernel Maximum Correntropy Criterion Extended Kalman Filter (STMCCEKF), designed to improve the robustness of the EKF in the presence of impulsive noise. To address the limitations of EKF and Gaussian MCCEKF under heavy-tailed GPS noise, we propose a robust STMCCEKF that employs a Student’s t kernel for enhanced outlier suppression. Unlike prior methods, it adapts better to impulsive, non-Gaussian noise through a tailored fixed-point iteration and revised covariance update, offering improved accuracy and robustness in challenging navigation scenarios. Although this work focuses on GPS-based navigation, the proposed STMCC-EKF framework applies to general GNSS systems, as the underlying measurement and error models are common across GPS.

The remainder of this paper is organized as follows. A brief review of the Maximum Correntropy Criterion EKF is provided in Section 2. In Section 3, the basic description of the STMCCEKF is presented. The STMCCEKF with adaptive kernel bandwidth (STMCCEKF-AKB) is introduced. The proposed STMCCEKF-AKB’s performance compared to the EKF, MCCEKF, and STMCCEKF techniques is assessed using illustrative examples based on simulation experiments in Section 4. Finally, conclusions are given in Section 5.

2. Review on Extended Kalman Filter with Maximum Correntropy Criterion

The probability density function of the Student’s t-distribution [23] can be written as follows:

$$S(\mu ,\Sigma ,\upsilon )={\displaystyle \frac{\Gamma \left(\frac{\upsilon +d}{2}\right)}{\Gamma (\frac{\upsilon }{2}){\upsilon }^{\frac{d}{2}}{\pi }^{\frac{d}{2}}{\left|\Sigma \right|}^{\frac{1}{2}}}}{(1+{\displaystyle \frac{1}{\upsilon }}{(y-\mu )}^T{\Sigma }^{-1}(y-\mu ))}^{-{\displaystyle \frac{\upsilon +d}{2}}}$$

(1)

This formula describes the probability density of the random variable t given the degrees of freedom $\upsilon$. As $\upsilon$ approaches infinity, the usual normal distribution is approached by the Student’s t-distribution.

$$S_{\upsilon ,\sigma }(e)={\kappa }_{\upsilon ,\sigma }(X,Y)={(1+{\displaystyle \frac{\left|\right|X-Y|{|}^2}{\upsilon {\sigma }^2}})}^{-({\displaystyle \frac{\upsilon +2}{2}})}(e=X-Y)$$

(2)

In practical scenarios, it is challenging to obtain the joint probability density function of random variables. Thus, as indicated by the equation, the sample mean estimate is utilized below [24]:

$${\widehat{V}}_{v,\sigma }(X,Y)={\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N{\mathrm{S}}_{v,\sigma }(e_i)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\kappa }_{v,\sigma }\left(x_i-y_i\right)}$$

(3)

Substituting the Gaussian kernel function in the MCC with the Student’s t kernel function provides stronger robustness against outliers, enhancing stability and performance.

2.1. MCCEKF Algorithm 1: MCCEKF-1

The Maximum Correntropy Criterion (MCC) forms the foundation for deriving a nonlinear extension of the linear Kalman filter, as detailed in [25]. Alternatively, a new objective function can be constructed to enhance robustness by adopting the Weighted Least Squares (WLS) approach. The goal, therefore, is to improve the objective function [26,27]

$$J_{\mathrm{MCC}-1}=G_{\sigma }\left(\right||z_k-h({\widehat{x}}_{k|k-1})|{|}_{R_k^{-1}})+G_{\sigma }(\left|\right|{\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1})\left|{|}_{P_{k|k-1}^{-1}}\right)$$

(4)

Several variants of the MCCEKF have been developed. For the first type, the estimator is derived by taking the derivative of the objective function $J_{\mathrm{MCC}-1}$ with respect to ${\widehat{x}}_{k|k}$ and setting its value to zero:

$${\displaystyle \frac{\partial J_{\mathrm{MCC}-1}}{\partial {\widehat{x}}_{k|k}}}=0$$

(5)

and therefore the following equation can be obtained:

$${\widehat{x}}_{k|k}=f({\widehat{x}}_{k-1|k-1})+{\displaystyle \frac{G_{\sigma }(||z_k-h({\widehat{x}}_{k|k-1})|{|}_{R_k^{-1}})}{G_{\sigma }(||{\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1})|{|}_{P_{k|k-1}^{-1}})}}{H}_k^{\mathrm{T}}(z_k-h({\widehat{x}}_{k|k-1}))$$

(6)

From Equation (5), we have

$$-{\displaystyle \frac{1}{{\sigma }^2}}{G}_{\sigma }(||z_k-h({\widehat{x}}_{k|k-1})|{|}_{R_k^{-1}})H_k^T{R}_k^{-1}(z_k-h({\widehat{x}}_{k|k-1}))+{\displaystyle \frac{1}{{\sigma }^2}}{G}_{\sigma }(||{\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1})|{|}_{P_{k|k-1}^{-1}})P_{k|k-1}^{-1}({\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1}))=0$$

(7)

which can be represented as

$$P_{k|k-1}^{-1}{\widehat{x}}_{k|k}-P_{k|k-1}^{-1}f({\widehat{x}}_{k-1|k-1})=L_k{H}_k^T{R}_k^{-1}(z_k-h({\widehat{x}}_{k|k-1}))$$

where

$$L_k={\displaystyle \frac{G_{\sigma }(||z_k-h({\widehat{x}}_{k|k-1})|{|}_{R_k^{-1}})}{G_{\sigma }(||{\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1})|{|}_{P_{k|k-1}^{-1}})}}$$

(8)

Since $G_{\sigma }\left(\right||{\widehat{x}}_{k|k}-f({\widehat{x}}_{k-1|k-1})|{|}_{P_{k|k-1}^{-1}})\approx G_{\sigma }(\left|\right|{\widehat{x}}_{k|k-1}-f({\widehat{x}}_{k-1|k-1})\left|{|}_{P_{k|k-1}^{-1}}\right)=1$, we obtain

$$L_k=G_{\sigma }\left(\right||z_k-h({\widehat{x}}_{k|k-1}){\left|\right|}_{R_k^{-1}})$$

(9)

The resulting estimator is the first version of Maximum Correntropy Criterion-based Extended Kalman Filter (MCCEKF-1), featuring an updated gain defined by

$$K_k={(P_{k|k-1}^{-1}+L_k{H}_k^T{R}_k^{-1}{H}_k)}^{-1}{L}_k{H}_k^T{R}_k^{-1}=L_k{P}_{k|k-1}{H}_k^{\mathrm{T}}{[L_k{H}_k{P}_{k|k-1}{H}_k^{\mathrm{T}}+R_k]}^{-1}$$

(10)

The following summarizes the computation procedures:

(1)

Initialize the state vector and the state covariance matrix: ${\widehat{x}}_{0|0}$ and $P_{0|0}$.

(2)

Predict the state vector ${\widehat{x}}_{k|k-1}$ and the state covariance matrix $P_{k|k-1}$.

(3)

Compute the measurement innovation using the Maximum Correntropy Criterion and derive the corresponding scaling factor $L_k$.

(4)

Calculate the modified Kalman gain matrix $K_k$ incorporating the robustification factor.

(5)

Update the state vector ${\widehat{x}}_{k|k}$ and the error covariance matrix $P_{k|k}$.

(6)

Repeat Steps (2)–(5) for each subsequent time step in the estimation process.

2.2. MCCEKF Algorithm 2: MCCEKF-2

The second type of MCCEKF is reviewed in this section. Based on the correntropy properties proposed by Liu et al. [28], the MCCEKF can be derived involving the regression approach. By constructing the system in a nonlinear recursive form, the formulated equations for the state and measurement [28] are as follows:

$$\left[\begin{array}{c}{\widehat{x}}_{k|k-1}\\ z_k\end{array}\right]=\left[\begin{array}{c}{x}_k\\ h\left(x_k\right)\end{array}\right]+{\mathsf{\delta }}_k$$

(11)

where ${\mathsf{\delta }}_k={\left[\begin{array}{cc}({\widehat{x}}_{k|k-1}-x_k)& v_k\end{array}\right]}^T$

$$\begin{array}{c}{\rm E}\left[{\mathsf{\delta }}_k{\mathsf{\delta }}_k^{\mathrm{T}}\right]=\left[\begin{array}{cc}\mathrm{E}\left[(x_k-{\widehat{x}}_{k|k-1}\left)\right(x_k-{\widehat{x}}_{k|k-1}{)}^{\mathrm{T}}\right]& 0\\ 0& \mathrm{E}\left[v_k{v}_k^{\mathrm{T}}\right]\end{array}\right]\\ =\left[\begin{array}{cc}{P}_{k|k-1}& 0\\ 0& R_k\end{array}\right]=\left[\begin{array}{cc}{B}_P{B}_P^{\mathrm{T}}& 0\\ 0& B_R{B}_R^{\mathrm{T}}\end{array}\right]=B_k{B}_k^{\mathrm{T}}\end{array}$$

(12)

where $B_k$ is obtained through Cholesky decomposition, combining $P_{k|k-1}$ and $R_k$, which are the prediction error and measurement noise covariance matrices, respectively. In this context, $B_{k,P}$ and $B_{k,R}$ can be obtained from Cholesky decomposition of $P_{k|k-1}$ and $R_k$, respectively. Multiplying $B_k^{-1}$ by both sides of Equation (11) yields the following model:

$$B_k^{-1}\left[\begin{array}{c}{\widehat{x}}_{k|k-1}\\ z_k\end{array}\right]=B_k^{-1}\left[\begin{array}{c}{x}_k\\ h\left(x_k\right)\end{array}\right]+B_k^{-1}{\mathsf{\delta }}_k$$

(13)

$$D_k=B_k^{-1}\left[\begin{array}{c}{\widehat{x}}_{k|k-1}\\ z_k\end{array}\right], g_k=B_k^{-1}\left[\begin{array}{c}{x}_k\\ h\left(x_k\right)\end{array}\right], e_k=B_k^{-1}{\mathsf{\delta }}_k$$

(14)

$$D_k=g_k+e_k$$

(15)

and $e_{k,i}=d_{k,i}- g_{k,i}$. According to the definition of correntropy, the cost function can be written as

$$J_{\mathrm{MCC}-2}(x_k)={\displaystyle \frac{1}{L}}{\displaystyle \sum _{i=1}^L{G}_{\sigma }(d_{k,i}-g_{k,i})}$$

(16)

where $L=n+m$.

Taking the derivative of $J_{\text{MCC-2}}$ with respect to $x_k$ to find the optimality condition, we obtain ${\displaystyle \frac{\partial J_{\mathrm{MCC}-2}}{\partial x_k}}=0$

$${\left({\displaystyle \frac{\partial g_k}{\partial x_k}}\right)}^{\mathrm{T}}{\Lambda }_k(D_k-g_k)=0$$

(17)

$${\Lambda }_k=\left[\begin{array}{cc}{\Lambda }_x& 0\\ 0& {\Lambda }_z\end{array}\right]$$

(18)

where

$${\Lambda }_x=\mathrm{diag}(G_{\sigma }(e_{k,1}),\dots ,G_{\sigma }(e_{k,n}\left)\right)=\mathrm{diag}(\exp (-{\displaystyle \frac{e_{k,1}^2}{2{\sigma }^2}}),\dots ,\exp (-{\displaystyle \frac{e_{k,n}^2}{2{\sigma }^2}}\left)\right)$$

(19)

$${\Lambda }_z=\mathrm{diag}(G_{\sigma }(e_{k,n+1}),\dots ,G_{\sigma }(e_{k,n+m}\left)\right)=\mathrm{diag}(\exp (-{\displaystyle \frac{e_{k,n+1}^2}{2{\sigma }^2}}),\dots ,\exp (-{\displaystyle \frac{e_{k,n+m}^2}{2{\sigma }^2}}\left)\right)$$

(20)

After derivation, we have

$$B_P^{-\mathrm{T}}{\Lambda }_x{B}_P^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{B}_R^{-\mathrm{T}}{\Lambda }_z{B}_R^{-1}(z_k-h(x_k\left)\right)=0$$

(21)

Express ${\tilde{P}}_{k|k-1}$ and ${\tilde{R}}_k$ as

$${\tilde{P}}_{k|k-1}=B_P{\Lambda }_x^{-1}{B}_P^T; {\tilde{R}}_k=B_R{\Lambda }_z^{-1}{B}_R^T$$

(22)

and thus,

$${\tilde{P}}_{k|k-1}^{-1}=B_P^{-\mathrm{T}}{\Lambda }_x{B}_P^{-1}; {\tilde{R}}_k^{-1}=B_R^{-\mathrm{T}}{\Lambda }_z{B}_R^{-1}$$

(23)

and we have

$${\tilde{P}}_{k|k-1}^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h(x_k\left)\right)=0$$

(24)

Since the measurement model is given by

$$z_k=h\left(x_k\right)+v_k=h\left({\widehat{x}}_{k|\mathrm{k}-1}\right)+H_k(x_k-{\widehat{x}}_{k|\mathrm{k}-1})+v_k$$

we have

$${\tilde{P}}_{k|k-1}^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h({\widehat{x}}_{k|\mathrm{k}-1})-H_k(x_k-{\widehat{x}}_{k|\mathrm{k}-1}))=0$$

and thus,

$$({\tilde{P}}_{k|k-1}^{-1}+H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}{H}_k)x_k=({\tilde{P}}_{k|k-1}^{-1}+H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}{H}_k){\widehat{x}}_{k|k-1}+H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h({\widehat{x}}_{k|\mathrm{k}-1}\left)\right)$$

(25)

Finally, we obtain

$$x_k={\widehat{x}}_{k|k-1}+{({\tilde{P}}_{k|k-1}^{-1}+H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}{H}_k)}^{-1}{H}_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h({\widehat{x}}_{k|\mathrm{k}-1}\left)\right)$$

(26)

Let the Kalman gain

$$K_k={({\tilde{P}}_{k|k-1}^{-1}+H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}{H}_k)}^{-1}{H}_k^{\mathrm{T}}{\tilde{R}}_k^{-1}={\tilde{P}}_{k|k-1}{H}_k^{\mathrm{T}}{(H_k{\tilde{P}}_{k|k-1}{H}_k^{\mathrm{T}}+{\tilde{R}}_k)}^{-1}$$

(27)

we obtain

$$x_k={\widehat{x}}_{k|k-1}+K_k(z_k-h({\widehat{x}}_{k|k-1}\left)\right)$$

According to the fixed-point iteration algorithm, the filter estimates state using standard propagation equations and updates posterior estimates, which is given by

$${\widehat{x}}_k^{(t)}={\widehat{x}}_{k|k-1}+K_k^{(t-1)}(z_k-h({\widehat{x}}_{k|k-1}^{(t-1)}\left)\right)$$

(28)

where

$$K_k^{(t-1)}={\tilde{P}}_{k|k-1}^{(t-1)}{H}_k^T{(H_k{\tilde{P}}_{k|k-1}^{(t-1)}{H}_k^{\mathrm{T}}+{\tilde{R}}_k^{(t-1)})}^{-1}$$

(29)

Finally, the state covariance matrix in its updated form can be written as

$${\tilde{P}}_k^{\left(t\right)}=(I-K_k^{(t-1)}{H}_k){\tilde{P}}_{k|k-1}^{(t-1)}{(I-K_k^{(t-1)}{H}_k)}^T+K_k^{(t-1)}{\tilde{R}}_k{(K_k^{(t-1)})}^T$$

(30)

3. Student’s t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter

According to Huang et al. [24], one can derive the STMCCEKF. From Equation (14), we obtain

$$B_P^{-1}{\widehat{x}}_{k|k-1}=B_P^{-1}{x}_k+B_P^{-1}{\mathsf{\delta }}_{k,1}, B_R^{-1}{z}_k=B_R^{-1}h\left(x_k\right)+B_R^{-1}{\mathsf{\delta }}_{k,2}$$

(31)

where

$${\mathsf{\delta }}_{k,1}={({\widehat{x}}_{k|k-1}-x_k)}^{\mathrm{T}}, {\mathsf{\delta }}_{k,2}=v_k^{\mathrm{T}}$$

Furthermore, defining $e_x$ and $e_z$ as

$$e_x=B_P^{-1}{\mathsf{\delta }}_{k,1}=B_P^{-1}{\widehat{x}}_{k|k-1}-B_P^{-1}{x}_k; e_z=B_{k,R}^{-1}{\mathsf{\delta }}_{k,2}=B_{k,R}^{-1}{z}_k-B_{k,R}^{-1}h\left(x_k\right)$$

The above equations can be rewritten as

$$e_x=B_P^{-1}(x_k-{\widehat{x}}_{k|k-1}); e_z=B_R^{-1}(z_k-h(x_k\left)\right)$$

(32)

By considering Student’s t, the cost function is given as

$$J_{\mathrm{MCC}-\mathrm{ST}}={\displaystyle \frac{1}{L}}\left[{\displaystyle \sum _{i=1}^n{S}_{\upsilon ,\sigma }(e_{x,i})}+{\displaystyle \sum _{j=1}^m{S}_{\upsilon ,\sigma }(e_{z,j})}\right]$$

(33)

where $L=n+m$. Considering the Student’s t kernel-based MCC, in order to obtain the optimal state estimation, we can solve the following equation:

$${\displaystyle \frac{1}{L}}\left[{\displaystyle \frac{\partial }{\partial x_k}}{\displaystyle \sum _{i=1}^n{S}_{\upsilon ,\sigma }(e_{x,i})}+{\displaystyle \frac{\partial }{\partial x_k}}{\displaystyle \sum _{j=1}^m{S}_{\upsilon ,\sigma }(e_{z,j})}\right]=0$$

(34)

After derivation, we obtain

$${\displaystyle \sum _{i=1}^n(\frac{\upsilon {\sigma }^2}{\upsilon {\sigma }^2+e_{x,i}^2})}{S}_{\upsilon ,\sigma }(e_{x,i})B_{P,i}^{-\mathrm{T}}{B}_{P,i}^{-1}(x_k-{\widehat{x}}_{k|k-1})-{\displaystyle \sum _{j=1}^m(\frac{\upsilon {\sigma }^2}{\upsilon {\sigma }^2+e_{z,j}^2})}{S}_{\upsilon ,\sigma }(e_{z,j})H_k^{\mathrm{T}}{B}_{R,j}^{-\mathrm{T}}{B}_{R,j}^{-1}(z_k-h(x_k\left)\right)=0$$

(35)

Replacing the summation form in Equation with its matrix form, we obtain

$$\begin{array}{l}{\Lambda }_x^{(t-1)}=diag(\begin{array}{ccc}({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{x,1})}{\upsilon {\sigma }^2+e_{x,1}^2}}),& \dots & ,({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{x,n})}{\upsilon {\sigma }^2+e_{x,n}^2}})\end{array})\\ {\Lambda }_z^{(t-1)}=diag(\begin{array}{ccc}({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{z,1})}{\upsilon {\sigma }^2+e_{z,1}^2}}),& \dots & ,({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{z,m})}{\upsilon {\sigma }^2+e_{z,m}^2}})\end{array})\end{array}$$

(36)

We can obtain the following equation:

$$B_P^{-\mathrm{T}}{\Lambda }_x{B}_P^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{B}_R^{-\mathrm{T}}{\Lambda }_z{B}_R^{-1}(z_k-h(x_k\left)\right)=0$$

(37)

By using ${\tilde{P}}_{k|k-1}=B_P{\Lambda }_x^{-1}{B}_P^T$ and ${\tilde{R}}_k=B_R{\Lambda }_z^{-1}{B}_R^T$ we have

$${\tilde{P}}_{k|k-1}^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h(x_k\left)\right)=0$$

(38)

$${\tilde{P}}_{k|k-1}^{-1}(x_k-{\widehat{x}}_{k|k-1})-H_k^{\mathrm{T}}{\tilde{R}}_k^{-1}(z_k-h({\widehat{x}}_{k|\mathrm{k}-1})-H_k(x_k-{\widehat{x}}_{k|\mathrm{k}-1}))=0$$

3.1. Algorithm for the Student’s t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter

The state estimate equation is given by

$$x_k={\widehat{x}}_{k|k-1}+{\tilde{K}}_k(z_k-h({\widehat{x}}_{k|k-1}\left)\right)$$

and the obtained Kalman gain is

$${\tilde{K}}_k={\tilde{P}}_{k|k-1}{H}_k^T{(H_k{\tilde{P}}_{k|k-1}{H}_k^T+{\tilde{R}}_k)}^{-1}$$

The covariance matrix for updated state can be

$${\tilde{P}}_k=(I-{\tilde{K}}_k{H}_k){\tilde{P}}_{k|k-1}{(I-{\tilde{K}}_k{H}_k)}^{\mathrm{T}}+{\tilde{K}}_k{\tilde{R}}_k{\tilde{K}}_k^{\mathrm{T}}$$

The computational steps for the Student’s t kernel for MCCEKF (STMCCEKF) are as follows:

(1)

$k=1$, select the error tolerance $\epsilon$ and the kernel bandwidth $\sigma$;

(2)

For the prior estimation steps,

a.

Calculate the prior state estimate and the prior error covariance matrix

$$\begin{array}{c}{\widehat{x}}_{k|k-1}=f\left({\widehat{x}}_{k-1}\right)\\ P_{k|k-1}={\Phi }_{k-1}{P}_{k-1}{\Phi }_{k-1}^{\mathrm{T}}+Q_{k-1}\end{array}$$

b.

Use Cholesky decomposition for $B_{k,P}$ and $B_{k,R}$.

c.

The prior estimation process, the predicted measurement, and its associated covariance matrix are as follows:

$$B_k^{-1}\left[\begin{array}{c}{\widehat{x}}_{k|k-1}\\ z_k\end{array}\right]=B_k^{-1}\left[\begin{array}{c}{x}_k\\ h\left(x_k\right)\end{array}\right]+B_k^{-1}{\mathsf{\delta }}_k$$

(3)

For posterior estimation,

a.

$t=1$, ${\widehat{x}}_k^0={\widehat{x}}_{k|k-1}$.

b.

Perform iteration:

(i)

$e_x=B_P^{-1}(x_k-{\widehat{x}}_{k|k-1})$; $e_z=B_R^{-1}(z_k-h(x_k\left)\right)$

(ii)

$\begin{array}{l}{\Lambda }_x^{(t-1)}=diag(\begin{array}{ccc}({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{x,1})}{\upsilon {\sigma }^2+e_{x,1}^2}}),& \dots & ,({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{x,n})}{\upsilon {\sigma }^2+e_{x,n}^2}})\end{array})\\ {\Lambda }_z^{(t-1)}=diag(\begin{array}{ccc}({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{z,1})}{\upsilon {\sigma }^2+e_{z,1}^2}}),& \dots & ,({\displaystyle \frac{\upsilon {\sigma }^2{S}_{\upsilon ,\sigma }(e_{z,m})}{\upsilon {\sigma }^2+e_{z,m}^2}})\end{array})\end{array}$

(iii)

${\tilde{P}}_{k|k-1}^{(t-1)}=B_{k,P}{\Lambda }_{k,x}^{(t-1)}{B}_{k,P}^{\mathrm{T}}$; ${\tilde{R}}_{k|k-1}^{(t-1)}=B_{k,R}{\Lambda }_{k,z}^{(t-1)}{B}_{k,R}^{\mathrm{T}}$

(iv)

${\tilde{K}}_k^{(t-1)}={\tilde{P}}_{k|k-1}^{t-1}{H}_k^T{(H_k{\tilde{P}}_{k|k-1}^{(t-1)}{H}_k^{\mathrm{T}}+{\tilde{R}}_k^{(t-1)})}^{-1}$

(v)

${\widehat{x}}_{k|k-1}^{(t)}={\widehat{x}}_{k|k-1}^{(t-1)}+{\tilde{K}}_k^{(t-1)}(z_k-h({\widehat{x}}_{k|k-1}))$

(vi)

${\tilde{P}}_k^{(t)}=(I-K_k^{(t-1)}{H}_k){\tilde{P}}_{k|k-1}^{(t-1)}{(I-K_k^{(t-1)}{H}_k)}^T+K_k^{(t-1)}{\tilde{R}}_k{(K_k^{(t-1)})}^T$

(vii)

Compare the estimation for the current steps with the previous steps for convergence check:

$${\displaystyle \frac{‖{\widehat{x}}_k^{\left(t\right)}-{\widehat{x}}_k^{\left(t-1\right)}‖}{{\widehat{x}}_k^{(t-1)}}}\le \epsilon$$

(viii)

If the above condition holds, set ${\widehat{x}}_k={\widehat{x}}_k^{(t)}$, $P_k={\tilde{P}}_k^{\left(t\right)}$, and $t=t+1$.

(4)

Repeat Steps (2)–(3) for each subsequent time step in the estimation process.

3.2. Algorithm for Kernel Bandwidth Adaptation

The innovation sequence shows the difference between the predicted measurement ${\widehat{z}}_{k|k-1}$ and the actual measurement $z_k$,

$${\upsilon }_k=z_k-{\widehat{z}}_{k|k-1}=z_k-h\left({\widehat{x}}_{k|k-1}\right),$$

(39)

while the weighted innovation, $K_k(z_k-{\widehat{z}}_{k|k-1})$, acts as a correction to the predicted estimation ${\widehat{x}}_{k|k-1}$ with the estimated ${\widehat{x}}_k$. The covariance matrix of the innovated sequence can be expressed as

$$C_{{\upsilon }_k}=E[{\upsilon }_k{\upsilon }_k^{\mathrm{T}}]=H_k{P}_{k|k-1}{H}_k^{\mathrm{T}}+R_k$$

(40)

Given that the length of the window is one and the degree of divergence $\xi$ is used as the record of the innovation covariance matrix,

$$\xi ={\upsilon }_k^{\mathrm{T}}{\upsilon }_k=tr({\upsilon }_k{\upsilon }_k^{\mathrm{T}})$$

(41)

This variable can be adjusted for the adaptive filtering or utilized to identify the divergence or outliers.

For the STMCCEKF based on adaptive kernel bandwidth mechanism, the kernel size is modified in the following way in the measurement-specific processing for outliers:

$${\sigma }_{j,k}={\lambda }_{j,k}\cdot {\sigma }_{\max }, \mathrm{for} j=1,2,\dots ,m$$

(42)

where ${\lambda }_{j,k}$ denotes the adaptive factor for the $j$th measurement element at time step $k$, $z_{j,k}$, with the maximum kernel size ${\sigma }_{\max }$ = 12.

Further, ${\lambda }_{j,k}$ must be suitably adjusted for varying noise intensities. The correlation between the innovation factor and its covariance matrix $C_{{\upsilon }_k}$ is the first point to consider in identifying the presence of measurement outliers, and ${\alpha }_{j,k}$ can be defined as

$${\alpha }_{j,k}={\displaystyle \frac{C_{{\upsilon }_{j,k}}}{{\upsilon }_{j,k}^2}}={\displaystyle \frac{H_k{P}_{jj,k|k-1}{H}_k^{\mathrm{T}}+R_k}{{\upsilon }_{j,k}^2}}$$

(43)

where $C_{{\upsilon }_{j,k}}$ is the jth diagonal element of the innovation covariance matrix, with ${\upsilon }_{j,k}$ being the innovation term ${\upsilon }_k$ for the jth element, and ${\upsilon }_{j,k}=z_{j,k}-{\widehat{z}}_{j,k|k-1}$. The ${\lambda }_{j,k}$ is obtained by using the relation

$${\lambda }_{j,k}=1-\exp (-{\alpha }_{j,k})$$

(44)

See reference [13] for further details.

4. Results and Discussion

To evaluate the performance of the proposed algorithm, two different environments are designed for testing. Environment 1 focuses on the filters’ ability to mitigate noise induced by multipath effects, where satellite signals are reflected by surrounding surfaces and cause interference. In this case, outliers with varying magnitudes are inserted at specific time segments, allowing examination of the filters’ sensitivity to changes in outlier strength and timing. Environment 2 is constructed to evaluate the filters’ robustness under time-varying observations with heavy-tailed noise and randomly occurring outliers.

This setting is intended to assess how well each filter handles non-Gaussian noise characteristics and abrupt disturbances, providing insight into their ability to maintain estimation accuracy under dynamic and unpredictable noise conditions. To facilitate comparison, the filters are grouped according to their underlying design. One set of comparisons involves the EKF, MCCEKF-1, MCCEKF-2, and STMCCEKF (with fixed kernel bandwidth), aiming to evaluate the influence of different kernel functions and robust modeling strategies under various noise scenarios. Another comparison includes the EKF, STMCCEKF, and STMCCEKF-AKB to highlight improvements introduced by integrating the Student’s t-distribution, with fixed and adaptive kernel bandwidths. These structured tests provide a comprehensive assessment of each filter’s effectiveness in handling distinct types of noise and interference.

The testing environment used Matlab 2022a, with the Satellite Navigation (Satnav) Toolbox 3.0 [29,30] to generate the necessary satellite data for GPS positioning, including satellite positions, velocities, and pseudoranges. The simulation will involve navigating a vehicle in motion. Figure 1 represents the motion trajectory of the simulated vehicle. Figure 2 illustrates a skyplot showing the spatial distribution of visible satellites during the simulation period, with each satellite labeled by its PRN (Pseudorandom Noise number) identifier to provide a clear understanding of satellite geometry and visibility conditions. The distribution of visible satellites is illustrated in the satellite skyplot shown in Figure 2, which visualizes the azimuth and elevation of the observed satellites. The plot consists of concentric circles, each representing a 10° elevation increment, ranging from 0° at the outermost ring (horizon) to 90° at the center (zenith). The multipath effect, along with other stochastic errors, is captured in the noise term as ${\mathit{ϵ}}_{\mathit{i}}$ is commonly used in GNSS observation models when exact modeling is impractical or environment-dependent. After introducing various types of noise interference, the equation representing the pseudorange between a satellite and a receiver in a satellite-based navigation system, such as GPS, can be expressed as [24]

$${\rho }_i=c\left(t_u-t_{si}\right)=\sqrt{{(x_i-x_u)}^2+{(y_i-y_u)}^2+{(z_i-z_u)}^2}+c{t}_b+{\nu }_{\rho i}+d_{iono,i}+d_{tropo,i}+{ϵ}_i$$

(45)

where the pseudorange error is represented by $v_{{\rho }_i}$, and the receiver thermal noise, multipath, ionospheric delay, and tropospheric delay are the error sources which affect the accuracy of GPS measurements. These delays must be estimated and corrected to improve the positioning accuracy. The majority of inaccuracies are subsequently addressed by using the differential GPS (DGPS) mode, although the effects of multipath and the receiver’s thermal noise cannot be completely avoided in the current investigation, and thus, the remaining error becomes ${\delta }_{mp}+{\delta }_{noise}$, where ${\delta }_{noise}$ and ${\delta }_{mp}$ represent the additional multipath error, which can be in the form of outliers. Atmospheric delays are mitigated using the Klobuchar and Saastamoinen models for ionospheric and tropospheric corrections, respectively. However, due to model limitations and environmental variability, residual errors, including multipath and unmodeled biases, persist. The STMCCEKF addresses these uncertainties through its robust Student’s t kernel, which down-weights the influence of such non-Gaussian errors during measurement updates, enhancing estimation accuracy in challenging conditions.

Figure 3a illustrates the probability density function of the Student’s t-distribution with degrees of freedom $\upsilon =3$, exhibiting a symmetric, bell-shaped curve similar to the Gaussian distribution but with heavier tails. These heavier tails indicate a higher probability of extreme observations, a characteristic arising from the distribution’s ability to model variance uncertainty, making it more robust against the outliers. Figure 3b compares the Student’s t-distribution (blue) with the Gaussian distribution (red), showing that while both distributions peak at the mean, the Gaussian distribution has a sharper peak and thinner tails, implying a stronger assumption of normally distributed errors. This property is particularly advantageous in filtering and estimation problems where real-world measurement noise deviates from Gaussian nature, such as in MCCEKF with a Student’s t kernel, enhancing resilience to sporadic large deviations and ensuring more stable state estimation.

4.1. Environment 1: Effect of Outliers on Data with Changing Standard Deviation over Time

In this section, the performance of various filters in suppressing outliers in a GPS multipath environment will be validated. The setup for this environment is the multipath effect noise ${\delta }_{mp}$, which follows a Student’s t-distribution with 3 degrees of freedom $(\upsilon =3)$, as illustrated in Figure 3. The thermal noise ${\delta }_{noise}$ is modeled as Gaussian noise with mean and standard deviations. To simulate realistic environments, the additional disturbances induced by outliers are injected into the pseudorange measurements at predetermined time steps. The outliers are incorporated into the multipath error by scaling according to predefined values. These configurations will allow for a comprehensive assessment of each filter, which handles and mitigates the impact of outliers and noise in the given environment.

Figure 4 illustrates the resulting error sequence over the full simulation period, where 30 outliers are introduced based on varying standard deviations. These outliers are embedded in the Student’s t-distributed multipath noise, producing heavy-tailed behavior and occasional sharp deviations. The induced pseudorange errors span a wide range, from tens up to more than a hundred meters, emulating the abrupt multipath disruptions commonly encountered in urban GPS environments. This setup enables a robust evaluation of each filter’s ability to handle diverse noise characteristics and mitigate severe measurement faults.

Figure 5 analyzes the performance differences among MCCEKF-1, MCCEKF-2, and STMCCEKF with fixed kernel bandwidth. In the experiments, both MCCEKF variants adopt a fixed kernel bandwidth of $\sigma =50$ but differ in their kernel function or implementation settings, which leads to slight variations in robustness to outliers. Additionally, the STMCCEKF with a fixed kernel bandwidth is ${\sigma }_{\mathrm{ST}}=30$. These kernel bandwidths play a vital role in determining the performance of each filter by affecting the sensitivity to noise and outliers. Figure 6 compares the position errors of the EKF and STMCCEKF filters, with fixed and adaptive kernel bandwidths, under multipath interference. During periods without outliers, both filters exhibit similar performance. However, when significant anomalies are introduced, STMCCEKF demonstrates improved robustness by effectively suppressing most of the induced noise.

From Figure 5 and Figure 6, it can be observed that, during periods without outliers, MCCEKF and EKF show similar performance. However, when noises are introduced, STMCCEKF suppresses most of the noise caused by these disturbances. Nonetheless, there are occasional instances where the kernel bandwidth is insufficient to handle the noises effectively. After using STMCCEKF, the performance improves in terms of noise suppression. It is important to point out that the MCCEKF exhibits greater sensitivity to measurement noise, whereas the MCCEKF variant incorporating the Student’s t kernel effectively suppresses noise, demonstrating its resilience against outliers. This contrast further validates the STMCCEKF’s capability in attenuating measurement uncertainties, thereby improving state estimation accuracy in non-Gaussian noise environments. The STMCCEKF-AKB is based on the Student’s t-distribution and incorporates an adaptive kernel bandwidth that adjusts according to the noise characteristics.

Figure 7 shows the variation in kernel bandwidth over time for Environment 1. Smaller kernel bandwidths correspond to the filter’s response to outliers, highlighting its ability to adjust dynamically for robust performance under diverse noise conditions. This improvement can be attributed to the use of the Student’s t-distribution with a fixed degree of freedom ($\upsilon =3$) and an adaptive kernel bandwidth, which dynamically adjusts to changing noise conditions. The adaptive mechanism allows STMCCEKF to respond flexibly to varying levels of measurement uncertainty, thereby enhancing its resilience to both noise and outliers. This adaptive strategy significantly improves the filter’s ability to suppress measurement anomalies under non-Gaussian conditions. These configurations show that the estimation performance is highly influenced by the choice of kernel mechanism and the ability to adapt to dynamic noise environments. While STMCCEKF filters with fixed kernel bandwidth may suffer from parameter rigidity, STMCCEKF-AKB achieves better overall robustness by combining heavy-tailed modeling with adaptive kernel tuning.

4.2. Environment 2: Impact of Outliers in a Time-Varying Noise Environment

In this section, to evaluate the characteristics of various filters, an environment was designed that includes both time-varying observation noise and outliers. This setup aims to verify whether the proposed filters perform better in handling time-varying observation noise and to observe if they can also effectively suppress outliers within this context. The goal is to achieve simultaneous handling of time-varying observation noise and outliers within a single filtering framework, thereby improving overall filtering accuracy.

In this experiment, the characteristics of pseudorange error illustrated in Figure 8 represent a time-varying, non-Gaussian environment constructed by injecting thermal noise of different intensities into two specific time intervals. In addition, ten outliers with varying magnitudes were randomly inserted throughout the signal to simulate abrupt disturbances. This setup reflects a more complex and realistic noise condition compared to static Gaussian assumptions. Traditional Gaussian-based filters like the EKF may struggle under such conditions due to their reliance on fixed noise statistics. In contrast, robust filtering approaches, such as those incorporating the Student’s t-distribution or adaptive noise covariance estimation, are more suitable for handling such non-stationary noise environments. These methods can better accommodate dynamic fluctuations and suppress outlier effects, thereby enhancing overall estimation robustness.

Figure 9 shows the adaptive kernel bandwidth variation over time in Environment 2. The kernel bandwidth decreases sharply at several points, reflecting the filter’s response to abrupt noise changes of outliers. This behavior highlights the filter’s ability to adjust dynamically for enhanced robustness under non-Gaussian conditions.

Based on Figure 10, it can be observed that the performance of MCCEKF and STMCCEKF is comparable to that of EKF during periods without anomalies. However, when anomalies occur, both MCCEKF and STMCCEKF demonstrate superior robustness, with STMCCEKF exhibiting a more effective suppression of large deviations in position errors. As illustrated in the test environment, time-varying noise combined with randomly injected outliers creates a challenging environment for state estimation. Although MCCEKF adopts a wider kernel bandwidth $\sigma$, its ability to mitigate the effects of outliers is limited. In contrast, STMCCEKF employs a Student’s t-distribution kernel with adaptively adjusted bandwidth $\sigma$ and degrees of freedom $\upsilon$, where $\sigma$ is capped at 12 to maintain numerical stability.

This configuration enhances its adaptability to heavy-tailed noise and confirms the critical role of $\upsilon$ in improving robustness against non-Gaussian disturbances. In STMCCEKF, both the process and measurement noise are modeled using the Student’s t-distribution, which inherently reduces sensitivity to extreme values. As a result, the presence of outliers has minimal impact on the overall estimation accuracy. Additionally, STMCCEKF incorporates a generalized stochastic integration rule (SIR), enabling it to effectively combine the statistical properties of both Student’s t and Gaussian distributions for enhanced performance under complex noise conditions. Although the proposed STMCCEKF introduces additional computations due to the use of the adaptive Student’s t kernel and the fixed-point iteration, its computational complexity remains feasible for real-time implementation. The iteration typically converges within 3–7 steps per time update, and the associated operations primarily involve matrix/vector multiplications and scalar evaluations, which are computationally efficient and parallelizable. With modern embedded hardware and optimization techniques, the STMCCEKF can be implemented in real-time navigation systems without significant performance bottlenecks.

5. Conclusions

This study presents a hybrid Student’s t-based maximum entropy criterion algorithm for robust GPS positioning in non-Gaussian noise environments. By replacing the conventional Gaussian kernel with a Student’s t-distribution, the proposed filter achieves improved estimation accuracy, particularly under heavy-tailed noise and in the presence of outliers. Experiments were conducted in two simulated environments: one focusing on multipath-induced noise, and the other involving time-varying heavy-tailed disturbances. In both cases, the Student’s t-based filter consistently outperformed the extended Kalman filter and its maximum correntropy-based variants.

Furthermore, both robust filters outperformed the extended Kalman filter when significant anomalies were present. Although the experiments were based on simulated data, the Student’s t-distribution, derived from Bayesian inference with unknown variance, effectively captures impulsive noise characteristics and abrupt state transitions. The simulation results confirm that STMCCEKF outperforms standard MCCEKF in terms of robustness and estimation precision. While the STMCCEKF filter with fixed kernel bandwidth may suffer from parameter rigidity, the proposed STMCCEKF-AKB demonstrated superior overall robustness and estimation accuracy by combining heavy-tailed modeling with adaptive kernel tuning. This highlights the advantage of incorporating the degrees of freedom parameter, which plays a key role in controlling the influence of outliers.

Future work should involve collecting real-world GNSS data or conducting field tests to validate whether the filter retains its effectiveness in practical environments. In addition, future work will also focus on validating the STMCCEKF using real GPS data to assess its performance under practical conditions with multipath and dynamic noise. The STMCCEKF offers robust and accurate state estimation under non-Gaussian noise, making it suitable for GPS-based applications such as autonomous vehicles, UAVs, and mobile robots operating in multipath-affected or dynamic environments. The integration of a generalized stochastic integration rule allows the filter to combine the statistical properties of both Student’s t and Gaussian distributions, offering improved performance under complex noise conditions. This hybrid entropy-based framework has the potential to be extended to other filtering architectures, such as the unscented Kalman filter or the cubature Kalman filter. Its flexibility makes it a promising approach for robust state estimation in challenging GNSS environments.