Adaptive Minimum Error Entropy Cubature Kalman Filter in UAV-Integrated Navigation Systems

Highlights

What are the main findings?

• The MEECKF is derived, which simultaneously tackles nonlinear errors and non-Gaussian measurement noise in UAV navigation systems.
• The AMEECKF is developed by designing a kernel bandwidth adjustment factor that realizes real-time kernel bandwidth correction based on innovations, solving the accuracy limitation of fixed kernel bandwidth.

What is the implication of the main finding?

• The AMEECKF breaks traditional Kalman filters’ limitations in adapting to non-Gaussian noise and fixed kernel bandwidth, offering a more robust navigation filtering framework.
• For UAV-integrated navigation, the AMEECKF mitigates navigation accuracy loss caused by multipath effects and dynamic environments, significantly improving UAV flight safety and ensuring stable navigation tasks in complex scenarios.

Abstract

Small unmanned aerial vehicles are now commonly equipped with integrated navigation systems to obtain high-precision navigation parameters. However, affected by the dual impacts of multipath effects and dynamic environmental changes, their state estimation process is vulnerable to interference from measurement outliers, which in turn leads to the degradation of navigation accuracy and poses a threat to flight safety. To address this issue, this research presents an adaptive minimum error entropy cubature Kalman filter. Firstly, the cubature Kalman filter is introduced to solve the problem of model nonlinear errors; secondly, the cubature Kalman filter based on minimum error entropy is derived to effectively curb the interference that measurement outliers impose on filtering results; finally, a kernel bandwidth adjustment factor is designed, and the kernel bandwidth is estimated adaptively to further improve navigation accuracy. Through numerical simulation experiments, the robustness of the proposed method with respect to measurement outliers is validated; further flight experiment results show that compared with existing related filters, this proposed filter can achieve more accurate navigation and positioning.

1. Introduction

Nowadays, small unmanned aerial vehicles (UAVs) have been widely outfitted with integrated navigation systems based on GNSS/INS (Global Navigation Satellite System/Inertial Navigation System) to acquire high-precision navigation parameters [1,2,3]. In the information fusion module of such systems, the Kalman filter and its various derivatives remain the mainstream technical solutions [4,5,6]. However, the GNSS/INS-integrated navigation model itself possesses significant nonlinear characteristics, whereas the classical Kalman filter can only process simplified linear models, resulting in navigation accuracy that fails to meet the requirements of complex scenarios. Simultaneously, small UAV navigation systems commonly utilize low-cost sensors and single-point GNSS devices, which are susceptible to interference from two main factors in practical applications: signal distortion caused by multipath effects and measurement deviations induced by dynamic changes in complex environments. These interferences can introduce numerous outliers into the measurement, further exacerbating the degradation of navigation accuracy. The combination of these problems has become a key bottleneck restricting the safety of small UAVs performing tasks such as reconnaissance, inspection, and material delivery.

Addressing the nonlinearity of the GNSS/INS-integrated navigation model, developers have proposed various high-precision nonlinear filtering methods [7,8,9]. Under the premise that noise satisfies a Gaussian distribution, academia has proposed several nonlinear Kalman filtering schemes based on different numerical integration techniques, typical representatives encompass the Extended Kalman Filter (EKF) [10], the Unscented Kalman Filter (UKF) [11], as well as the Cubature Kalman Filter (CKF) [12]. Of these, the CKF addresses the Gaussian weighted integral via the third-degree spherical-radial cubature rule. This design not only fundamentally eliminates errors introduced by the linearization process but also effectively mitigates the accuracy degradation problem encountered by the UKF in high-dimensional systems [13]. Reference [14] uses the CKF to address the nonlinear problems of the UAV roll torque model and updates the thermal state parameters of the UAV during flight via this algorithm. Based on this, one of the focuses of this paper is the use of nonlinear filtering methods in UAV-integrated navigation.

Two core premises serve as the basis for the derivation of the Kalman filter: firstly, the optimization goal is the Minimum Mean Square Error (MMSE), and secondly, it is assumed that both the system noise and the measurement noise satisfy a Gaussian distribution [15,16,17]. However, in the practical application contexts of GNSS/INS systems, the noise distribution often deviates from this ideal Gaussian characteristic, preventing the Kalman filter’s state estimation from achieving optimal performance [18,19]. Specifically, under the dual influence of multipath effects and dynamic changes in complex environments, GNSS signals are vulnerable to impulsive noise interference, directly leading to a significantly increased probability of abnormal values in GNSS measurement data. The measurement noise contaminated by such impulsive noise gradually exhibits significant non-Gaussian distribution characteristics—this kind of noise, with statistical properties that depart from the Gaussian distribution due to external interference and has higher tail probability density, is usually defined as heavy-tailed non-Gaussian noise [20,21]. Under such non-ideal noise conditions, the accuracy of the GNSS/INS system can drastically decline, making it difficult to meet the reliability requirements of practical applications. To suppress the interference of heavy-tailed measurement noise’s effect on filtering results and the improvement of the integrated navigation system’s stability, researchers have proposed various robust filters, providing more suitable solutions for navigation state estimation in non-Gaussian scenarios. The H∞ filter is one effective solution [22], the core idea of this algorithm is to treat noise as a random signal with energy properties, eliminating the need to rely on presuppositions or estimations of the noise’s statistical characteristics, thus exhibiting a certain degree of adaptability in non-Gaussian scenarios. However, this filter still has obvious limitations in practical application: its filtering performance relies on the internal parameter settings, but there remains a lack of universal and efficient methods for determining the optimal values of these parameters. This key issue greatly restricts the promotion and application of the H∞ filter in navigation systems. Literature [23] proposed a Huber-based Kalman Filter (HKF) by combining the minimum L1 and L2 norms. The theoretical basis of the HKF is the generalized maximum likelihood estimation, which is commonly referred to as M-estimation. It uses the Huber cost function to ensure the filtering residuals are bounded, reducing the effect of heavy-tailed noise. However, when the current measurement information is completely useless, the HKF still assigns it a certain weight, which can reduce filtering accuracy. Over recent years, drawing on information theory, a set of high-performance robust filters have been developed—among which those based on the Maximum Correntropy Criterion (MCC) and the Minimum Error Entropy (MEE) criterion stand as typical representatives. Correntropy, serving as a local similarity measurement function, exhibits insensitivity to outliers and enables effective resolution of the outlier issue in noise. It has been widely used in the design of robust filters. Literature [24] proposed a Kalman Filter based on the MCC (MCCKF) for linear systems, this filter constructs a cost function by maximizing the correntropy associated with the prediction residual and the estimation residual. However, when facing significant outliers and more complex non-Gaussian noises, the MCCKF ability to suppress the weight of outliers weakens, leading to poorer filtering accuracy. The MEE stands as another key criterion in information theory. The MEE-based Kalman Filter (MEEKF) uses propagation equations to generate a prior estimate, and then employs the fixed-point iteration method to update the posterior estimate. Since the MEEKF directly minimizes the error entropy and assigns lower weight to larger outliers, it can well handle non-Gaussian noise interference containing outliers in the system [25]. It remains crucial to keep in mind that within the MEEKF, the selection of the kernel bandwidth significantly affects its performance. In practical integrated navigation systems, it is impossible to pre-select an appropriate kernel bandwidth value. Therefore, the optimal kernel bandwidth is not a “fixed value”. When the noise distribution follows a Gaussian function, the kernel bandwidth should be as large as possible, as a large kernel bandwidth implies almost the same filtering accuracy with lower computational cost. When outliers appear in the noise, the kernel bandwidth should be reduced in a timely manner, ensuring filtering stability while reducing the kernel bandwidth to a certain value to improve filtering accuracy. Therefore, how to construct a robust filter that can effectively inhibit the interference induced by heavy-tailed non-Gaussian noise is another research focus of this paper.

To further enhance the operating performance of the Kalman filter in GNSS/INS systems, and to mitigate the interference from nonlinear errors and measurement uncertainty on the filtering results, this paper proposes an Adaptive Minimum Error Entropy Cubature Kalman Filter (AMEECKF), this method utilizes the CKF to handle the nonlinearity of the model, introduces the MEE criterion to deal with measurement outlier interference, and designs a kernel bandwidth adaptive estimation approach for real-time adjustment of the kernel bandwidth value, fully leveraging the advantages of MEE criterion-based filtering to improve state estimation’s adaptability and robustness. Finally, the AMEECKF is employed in an integrated SINS/GNSS navigation system, and comparative experiments are designed using real sensor data to verify the performance of the proposed filter.

In this paper, the core contributions are presented as follows:

• Aiming at the adverse effects of nonlinear errors and non-Gaussian measurement noise on navigation systems, the Minimum Error Entropy Cubature Kalman Filter is derived, which significantly improves the robustness of navigation systems against nonlinear errors and measurement outliers.
• To address the issue that fixed kernel bandwidth has a significant impact on filtering accuracy, a kernel bandwidth adjustment factor is designed, the kernel bandwidth is corrected in real time based on innovations, thereby developing the AMEECKF.
• The superior performance of AMEECKF within UAV-integrated navigation systems is verified through extensive experiments, which provides a new and effective technical paradigm for the development of inertial sensor-based multi-sensor fusion navigation systems.

The remaining sections are arranged as follows: Section 2 introduces the CKF and the MEE criterion; Section 3 derives the MEECKF; Section 4 constructs the kernel bandwidth adaptive method, deriving the AMEECKF; Section 5 verifies the effectiveness exhibited by the proposed method through numerical simulations and flight experiments; finally, Section 6 presents the discussion of this paper.

2. Related Work

This section contains the CKF and the MEE criterion, laying the foundation for the subsequent algorithm derivation.

2.1. Cubature Kalman Filter

The CKF approximates the prior and posterior probability distributions of the state vector by propagating a set of deterministically selected sample points (referred to as volume points) through the nonlinear system. It generates 2n volume points, which have equal weights and are symmetrically distributed around the current state estimate. Subsequently, these volume points are directly propagated through the nonlinear functions; based on the propagated volume points, the mean and covariance of the transformed state are then calculated. This method avoids any linearization process and can achieve third-order or higher estimation accuracy under the Gaussian assumption. The specific implementation steps of this filter are as follows:

Take into account the nonlinear dynamic system given below [12]:

$$\left\{\begin{array}{cc}{\mathit{x}}_k\hfill & =f\left({\mathit{x}}_{k-1}\right)+{\mathit{v}}_{k-1}\hfill \\ {\mathit{z}}_k\hfill & =h\left({\mathit{x}}_k\right)+{\mathit{w}}_k\hfill \end{array}\right.$$

(1)

Initialization:

$${\widehat{\mathit{x}}}_0=E\left({\mathit{x}}_0\right)$$

(2)

$${\mathit{P}}_0=\mathrm{E}\left[({\mathit{x}}_0-{\widehat{\mathit{x}}}_0)({\mathit{x}}_0-{\widehat{\mathit{x}}}_0{)}^{\mathrm{T}}\right]$$

(3)

Time Update:

• ${\mathit{P}}_{k-1}$ and ${\mathit{x}}_{i,k-1}$ is given below:

  $${\mathit{P}}_{k-1}={\mathit{S}}_{k-1}{\mathit{S}}_{\mathit{k}\mathbf{-}\mathbf{1}}^{\mathit{T}}$$

  (4)

  $${\mathit{x}}_{i,k-1}={\mathit{S}}_{k-1}{\mathit{\xi }}_i+{\widehat{\mathit{x}}}_{k-1}$$

  (5)

  where ${\mathit{\xi }}_i=\sqrt{n}[1{]}_i$, $i=1,2,\dots ,2n$.
• Compute ${\widehat{\mathit{x}}}_{k/k-1}$ and ${\mathit{P}}_{k/k-1}$:

$${\mathit{x}}_{i,k/k-1}^{*}=f\left({\mathit{x}}_{i,k-1}\right)$$

(6)

$${\widehat{\mathit{x}}}_{k/k-1}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}{x}_{i,k/k-1}^{*}$$

(7)

$${\mathit{P}}_{k/k-1}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}{\mathit{x}}_{i,k/k-1}^{*}({\mathit{x}}_{i,k/k-1}^{*}{)}^{\mathrm{T}}-{\widehat{\mathit{x}}}_{k/k-1}({\widehat{\mathit{x}}}_{k/k-1}{)}^{\mathrm{T}}+{\mathit{Q}}_{k-1}$$

(8)

Measurement Update:

• Calculate ${\mathit{P}}_{k/k-1}$ and ${\mathit{x}}_{i,k/k-1}$:

  $${\mathit{P}}_{k/k-1}={\mathit{S}}_{k/k-1}{\mathit{S}}_{k/k-1}^{\mathrm{T}}$$

  (9)

  $${\mathit{x}}_{i,k/k-1}={\mathit{S}}_{k/k-1}{\mathit{\xi }}_i+{\widehat{\mathit{x}}}_{k/k-1}$$

  (10)
• Estimate the predicted measurement:

  $${\mathit{z}}_{i,k/k-1}=h\left({\mathit{x}}_{i,k/k-1}\right)$$

  (11)

  $${\widehat{\mathit{z}}}_{k/k-1}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}{\mathit{z}}_{i,k/k-1}$$

  (12)
• Calculate the updated state estimate along with the associated error covariance matrix:

  $${\mathit{P}}_{xz,k}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}\left[{\mathit{x}}_{i,k|k-1}-{\widehat{\mathit{x}}}_{k|k-1}\right]{\left[{\mathit{z}}_{i,k|k-1}-{\widehat{\mathit{z}}}_{k|k-1}\right]}^{\mathrm{T}}$$

  (13)

  $${\mathit{P}}_{zz,k}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}\left[{\mathit{z}}_{i,k|k-1}-{\widehat{\mathit{z}}}_{k|k-1}\right]{\left[{\mathit{z}}_{i,k|k-1}-{\widehat{\mathit{z}}}_{k|k-1}\right]}^{\mathrm{T}}+{\mathit{R}}_k$$

  (14)

  $${\mathit{K}}_k={\mathit{P}}_{xz,k}/{\mathit{P}}_{zz,k}$$

  (15)

  $${\widehat{\mathit{x}}}_k={\widehat{\mathit{x}}}_{k/k-1}+{\mathit{K}}_k\left({\mathit{z}}_k-{\widehat{\mathit{z}}}_{k/k-1}\right)$$

  (16)

  $${\mathit{P}}_k={\mathit{P}}_{k/k-1}-{\mathit{K}}_k{\mathit{P}}_{z,k}{\mathit{K}}_k^T$$

  (17)

2.2. Minimum Error Entropy Criterion

Suppose $X$ and $Y$ are random variables, where the error is defined as $e=Y-X$. Under the framework of the MEE criterion’s concept [25], the information carried by the error $e$ is measurable using Renyi entropy:

$$H_{\beta }\left(e\right)={\displaystyle \frac{1}{1-\beta }}\mathit{log}\left(V_{\beta }\left(e\right)\right)$$

(18)

where $V_{\beta }\left(e\right)$ denotes the information potential, $\beta \left(\begin{array}{c}\beta \ne 1,\beta >0\end{array}\right)$ is the Renyi entropy’s specific order. The second-order Renyi entropy $(\beta =2)$ has been successfully utilized within information-theoretic learning while also being defined as:

$$V_2\left(\begin{array}{c}e\end{array}\right)=\int P^2\left(x\right)dx=E\left[\begin{array}{c}P\left(e\right)\end{array}\right]$$

(19)

where P(⋅) is the corresponding probability density function (PDF) belonging to the error variable $e$. The Parzen window method is commonly used to obtain an estimate of the PDF:

$$\widehat{P}\left(x\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\mathrm{G}}_{\sigma }\left(e-e_i\right)$$

(20)

where ${\mathrm{G}}_{\sigma }$ denotes the kernel function, $\sigma$ is the corresponding kernel bandwidth, and ${\left\{e_i\right\}}_{i=1}^N$ represents the error samples. Due to its universality, the Gaussian kernel function ${\mathrm{G}}_{\sigma }\left(x\right)=\exp \left(-x^2/\left(2{\sigma }^2\right)\right)$ is typically chosen. Since the logarithmic function is monotonic, minimizing the MEE can be considered equivalent to maximizing the information potential. Combining (18)–(20), the specific second-order Renyi entropy may be redefined empirically as:

$${\widehat{H}}_2\left(e\right)=-\mathit{log}{\displaystyle \frac{1}{N}}\sum _{i=1}^N\widehat{P}\left(e_i\right)=-\mathit{log}{\displaystyle \frac{1}{N^2}}\sum _{i=1}^N\sum _{j=1}^N{G}_{\sigma }\left(e_i-e_j\right)$$

(21)

From the definition of the error entropy criterion, a smaller second-order Renyi entropy ${\widehat{H}}_2(e)$ directly indicates reduced uncertainty in the error variable and increased stability of the system error, thereby signifying the convergence of the filtering algorithm. Therefore, the essence of the MEE criterion is to minimize the error’s quadratic entropy. Since the negative logarithm is monotonically decreasing, minimizing ${\widehat{H}}_2(e)$ implies maximizing ${\widehat{V}}_2(e)$, i.e.,:

$${\widehat{V}}_2\left(\begin{array}{c}e\end{array}\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N\widehat{P}\left(e_i\right)={\displaystyle \frac{1}{N^2}}\sum _{i=1}^N\sum _{j=1}^N{\mathrm{G}}_{\sigma }\left(e_i-e_j\right)$$

(22)

3. Minimum Error Entropy Cubature Kalman Filter

This section builds upon the CKF by introducing and integrating the MEE criterion, thereby deriving and constructing the CKF based on this criterion. First, the statistical linearization method is utilized for approximating the specific nonlinear equation [26]:

$${\mathit{z}}_k=h\left({\widehat{\mathit{x}}}_{k|k-1}\right)+{\mathit{H}}_k{\mathit{\epsilon }}_{k/k-1}+{\mathit{\nu }}_k$$

(23)

where ${\mathit{H}}_k={\left({\mathit{P}}_{k|k-1}^{-1}{\mathit{P}}_{xz,k|k-1}\right)}^T$ represents the observation matrix, and ${\mathit{\epsilon }}_{k/k-1}={\mathit{x}}_k-{\widehat{\mathit{x}}}_{k|k-1}$. Combining ${\mathit{\epsilon }}_{k/k-1}$ with the measurement yields the following augmented model:

$$\left[\begin{array}{c}{\widehat{\mathit{x}}}_{k/k-1}\\ {\mathit{z}}_k-h\left({\widehat{\mathit{x}}}_{k|k-1}\right)+{\mathit{H}}_k{\widehat{\mathit{x}}}_{k/k-1}\end{array}\right]=\left[\begin{array}{c}{\mathit{I}}_n\\ {\mathit{H}}_k\end{array}\right]{\mathit{X}}_k+{\mathit{\mu }}_k$$

(24)

where ${\mathit{I}}_n$ signifies the $n\times n$ dimensional identity matrix. In addition, ${\mathit{\mu }}_k$ denotes the augmented noise, written as:

$${\mathit{\mu }}_k=\left[\begin{array}{c}-{\mathit{\epsilon }}_{k/k-1}\\ {\mathit{\nu }}_k\end{array}\right]$$

(25)

Suppose that the specific covariance matrix for the augmented noise is positive definite, it can be obtained as:

$$E\left[{\mathit{\mu }}_k{\mathit{\mu }}_k^T\right]={\mathsf{\Theta }}_k{\mathsf{\Theta }}_k^T=\left[\begin{array}{cc}{\mathsf{\Theta }}_{p,k/k-1}{\mathsf{\Theta }}_{p,k/k-1}^T& \mathbf{0}\\ \mathbf{0}& {\mathsf{\Theta }}_{r,k}{\mathsf{\Theta }}_{r,k}^T\end{array}\right]$$

(26)

where ${\mathsf{\Theta }}_k,{\mathsf{\Theta }}_{p,k/k-1}$, and ${\mathsf{\Theta }}_{r,k}$ are obtained from the Cholesky decomposition of $E\left[{\mathit{\mu }}_k{\mathit{\mu }}_k^T\right]$, ${\mathit{P}}_{k/k-1}$, and ${\mathit{R}}_k$, respectively. Multiplying both sides of (8) by ${\mathsf{\Theta }}_k^{-1}$ yields:

$${\mathit{D}}_k={\mathit{W}}_k{\mathit{x}}_k+{\mathit{e}}_k$$

(27)

where

$${\mathit{D}}_k={\mathsf{\Theta }}_k^{-1}\left[\begin{array}{c}{\widehat{\mathit{x}}}_{k/k-1}\\ {\mathit{z}}_k-h\left({\widehat{\mathit{x}}}_{k/k-1}\right)+{\mathit{H}}_k{\widehat{\mathit{x}}}_{k/k-1}\end{array}\right]$$

(28)

$${\mathit{W}}_k={\mathsf{\Theta }}_k^{-1}\left[\begin{array}{c}{\mathit{I}}_n\\ {\mathit{H}}_k\end{array}\right]$$

(29)

$${\mathit{e}}_k={\mathsf{\Theta }}_k^{-1}\left[\begin{array}{c}-{\mathit{\epsilon }}_{k/k-1}\\ {\mathit{\nu }}_k\end{array}\right]$$

(30)

The specific elements constituting each variable in (28), (29), and (30) are represented as ${\mathit{D}}_k={\left[D_{1,k},D_{2,k},\dots ,D_{L,k}\right]}^T$, ${\mathit{W}}_k={\left[\begin{array}{c}{W}_{1,k},W_{2,k},\dots ,W_{L,k}\end{array}\right]}^T$, ${\mathit{e}}_k={\left[\begin{array}{c}{e}_{1,k},e_{2,k},\dots ,e_{L,k}\end{array}\right]}^T$. Here, $L=m+n$, $n$ stands for the dimensionality of the state variable, while $m$ denotes the dimensionality of the measurement vector

According to Equation (22), the cost function is:

$$J_L\left({\mathit{x}}_k\right)={\displaystyle \frac{1}{L^2}}\sum _{i=1}^L\sum _{j=1}^L{\mathrm{G}}_{{\sigma }_2}\left(e_{i,k}-e_{j,k}\right)$$

(31)

The ${\widehat{\mathit{x}}}_k$ is derived through the maximization of this cost function:

$${\widehat{\mathit{x}}}_k=\mathit{arg}\underset{x_k}{max}{J}_L\left({\mathit{x}}_k\right)=\mathit{arg}\underset{x_k}{max}{\displaystyle \frac{1}{L^2}}\sum _{i=1}^L\sum _{j=1}^L{\mathrm{G}}_{{\sigma }_2}\left(e_{i,k}-e_{j,k}\right)$$

(32)

Computing the partial derivative of the cost function:

$${\displaystyle \frac{\partial }{\partial X_k}}{J}_L({\mathit{x}}_k)=={\displaystyle \frac{2}{L^2{\sigma }^2}}{\mathit{W}}_k^T{\mathsf{\Psi }}_k{\mathit{e}}_k-{\mathit{W}}_k^T{\mathsf{\Phi }}_k{\mathit{e}}_k=0$$

(33)

where

$${\left[\begin{array}{c}{\mathsf{\Phi }}_k\end{array}\right]}_{ij}={\mathrm{G}}_{{\sigma }_2}\left(e_{j,k}-e_{i,k}\right)$$

(34)

$${\mathsf{\Psi }}_k=diag\left\{\sum _{i=1}^L{\mathrm{G}}_{{\sigma }_2}\left(e_{1,k}-e_{i,k}\right),\dots ,\sum _{i=1}^L{\mathrm{G}}_{{\sigma }_2}\left(e_{L,k}-e_{i,k}\right)\right\}$$

(35)

For solving Equation (33), the fixed-point iteration method can be employed:

$${\widehat{\mathit{x}}}_k=g\left(\widehat{\mathit{x}}\right)=\left({\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{W}}_k{)}^{-1}\right({\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{D}}_k)$$

(36)

where

$${\mathsf{\Lambda }}_k={\mathsf{\Psi }}_k-{\mathsf{\Phi }}_k=\left[\begin{array}{cc}{\mathsf{\Lambda }}_{xx,k}& {\mathsf{\Lambda }}_{zx,k}\\ {\mathsf{\Lambda }}_{xz,k}& {\mathsf{\Lambda }}_{zz,k}\end{array}\right]$$

(37)

In the equation, ${\mathsf{\Lambda }}_k\in {\mathrm{R}}^{L\times L}$, the expressions for its sub-elements ${\mathsf{\Lambda }}_{xx,k}\in {\mathrm{R}}^{n\times n}$, ${\mathsf{\Lambda }}_{x,k}\in {\mathrm{R}}^{m\times n}$, ${\mathsf{\Lambda }}_{zx,k}\in {\mathrm{R}}^{n\times m}$, and ${\mathsf{\Lambda }}_{z,k}\in {\mathrm{R}}^{m\times m}$ are given as follows:

$${\mathsf{\Lambda }}_{xx,k}=({\mathsf{\Lambda }}_{ij,k}{)}_{n\times n}=({\mathsf{\Phi }}_{ij,k}{)}_{n\times n}-({\mathsf{\Psi }}_{ij,k}{)}_{n\times n}$$

(38)

$${\mathsf{\Lambda }}_{xz,k}={\left({\mathsf{\Lambda }}_{ij,k}\right)}_{m\times n}={\left({\mathsf{\Phi }}_{ij,k}\right)}_{m\times n}-{\left({\mathsf{\Psi }}_{ij,k}\right)}_{m\times n}$$

(39)

$${\mathsf{\Lambda }}_{zx,k}=({\mathsf{\Lambda }}_{ij,k}{)}_{n\times m}=({\mathsf{\Phi }}_{ij,k}{)}_{n\times m}-({\mathsf{\Psi }}_{ij,k}{)}_{n\times m}$$

(40)

$${\mathsf{\Lambda }}_{zz,k}=({\mathsf{\Lambda }}_{ij,k}{)}_{m\times m}=({\mathsf{\Phi }}_{ij,k}{)}_{m\times m}-({\mathsf{\Psi }}_{ij,k}{)}_{m\times m}$$

(41)

From Equations (28), (29), and (37), the specific expressions for ${\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{W}}_k$ and ${\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{D}}_k$ in (36) are:

$$\begin{array}{c}{\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{W}}_k=\left[{\left({\mathsf{\Theta }}_{p,k/k-1}^{-1}\right)}^T{\mathsf{\Lambda }}_{xx,k}+{\mathit{H}}_k^T{\left({\mathsf{\Theta }}_{r,k}^{-1}\right)}^T{\mathsf{\Lambda }}_{xz,k}\right]{\mathsf{\Theta }}_{p,k/k-1}^{-1}\hfill \\ +\left[{\left({\mathsf{\Theta }}_{p,k/k-1}^{-1}\right)}^T{\mathsf{\Lambda }}_{zx,k}+{\mathit{H}}_k^T{\left({\mathsf{\Theta }}_{r,k}^{-1}\right)}^T{\mathsf{\Lambda }}_{zz,k}\right]{\mathsf{\Theta }}_{r,k}^{-1}{\mathit{H}}_k^T\hfill \end{array}$$

(42)

$$\begin{array}{c}{\mathit{W}}_k^T{\mathsf{\Lambda }}_k{\mathit{D}}_k=\left[({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{xx,k}+{\mathit{H}}_{\dot{k}}^T({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{xz,k}\right]{\mathsf{\Theta }}_{p,k/k-1}^{-1}{\widehat{\mathit{x}}}_{k/k-1}\hfill \\ +\left[({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{zx,k}+{\mathit{H}}_k^T({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{zz,k}\right]{\mathsf{\Theta }}_{r,k}^{-1}{\mathit{z}}_k^T\hfill \end{array}$$

(43)

Substituting (42) and (43) into (36) yields:

$$\begin{array}{c}{\widehat{\mathit{x}}}_k=({\mathsf{\Omega }}_1+{\mathsf{\Omega }}_2{\mathsf{\Omega }}_3{)}^{-1}({\mathsf{\Omega }}_1{\widehat{\mathit{x}}}_{k/k-1}+{\mathsf{\Omega }}_2{\mathit{z}}_k)\hfill \\ =({\mathsf{\Omega }}_1+{\mathsf{\Omega }}_2{\mathsf{\Omega }}_3{)}^{-1}{\widehat{\mathit{x}}}_{k/k-1}+({\mathsf{\Omega }}_1+{\mathsf{\Omega }}_2{\mathsf{\Omega }}_3{)}^{-1}{\mathsf{\Omega }}_2{\mathit{z}}_k\hfill \end{array}$$

(44)

where

$${\mathsf{\Omega }}_1=\left[\begin{array}{c}({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{xx,k}+{\mathit{H}}_k^T({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{xz,k}\end{array}\right]{\mathsf{\Theta }}_{p,k/k-1}^{-1}$$

(45)

$${\mathsf{\Omega }}_2=\left[\begin{array}{c}({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{zx,k}+{\mathit{H}}_k^T({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{zz,k}\end{array}\right]{\mathsf{\Theta }}_{r,k}^{-1}$$

(46)

$${\mathsf{\Omega }}_3={\mathit{H}}_{\mathit{k}}$$

(47)

Employing $(A+BCD{)}^{-1}=A^{-1}-A^{-1}B(C^{-1}+D{A}^{-1}B)D{A}^{-1}$, and substituting the following symbols: ${\mathsf{\Omega }}_2\to A$, ${\mathsf{\Omega }}_2\to B$, $I_L\to \mathrm{C}$, and ${\mathsf{\Omega }}_3\to D$, Equation (44) is rearranged as follows:

$${\widehat{\mathit{x}}}_k={\widehat{\mathit{x}}}_{k/k-1}+{\overline{\mathit{K}}}_k\left({\mathit{z}}_k-{\mathit{H}}_k{\widehat{\mathit{x}}}_{k/k-1}\right)$$

(48)

where

$${\overline{\mathit{K}}}_k=\left[{\overline{\mathit{P}}}_{k/k-1}+{\mathit{H}}_k^T{\overline{\mathit{P}}}_{k/k-1}+\left({\overline{\mathit{P}}}_{zx,k/k-1}+{\mathit{H}}_k^T{\overline{\mathit{R}}}_k\right){\mathit{H}}_k\right]\left({\overline{\mathit{P}}}_{zx,k/k-1}+{\mathit{H}}_k^T{\overline{\mathit{R}}}_k\right)$$

(49)

$${\overline{\mathit{P}}}_{k/k-1}=({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{xx,k}{\mathsf{\Theta }}_{p,k/k-1}^{-1}$$

(50)

$${\overline{\mathit{P}}}_{xz,k/k-1}=({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{xz,k}{\mathsf{\Theta }}_{p,k/k-1}^{-1}$$

(51)

$${\overline{\mathit{P}}}_{zx,k/k-1}=({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^T{\mathsf{\Lambda }}_{zx,k}{\mathsf{\Theta }}_{r,k}^{-1}$$

(52)

$${\overline{\mathit{R}}}_k=({\mathsf{\Theta }}_{r,k}^{-1}{)}^T{\mathsf{\Lambda }}_{zz,k}{\mathsf{\Theta }}_{r,k}^{-1}$$

(53)

Finally, the estimated error covariance is updated:

$${\mathit{P}}_k=(\mathit{I}-{\overline{\mathit{K}}}_k{\mathit{H}}_k){\mathit{P}}_{k/k-1}(\mathit{I}-{\overline{\mathit{K}}}_k{\mathit{H}}_k{)}^T+{\overline{\mathit{K}}}_k{\mathit{R}}_k{\overline{\mathit{K}}}_k^T$$

(54)

4. Adaptive MEE Cubature Kalman Filter

This section designs a kernel bandwidth adjustment factor to derive and construct AMEECKF. Simultaneously, an analysis and discussion regarding the algorithm’s iteration count are presented.

4.1. Kernel Bandwidth Adjustment Factor

The innovation is incorporated into the iterative process of the MEECKF to tune the specific kernel bandwidth in real-time. The so-called innovation refers to the measurement innovation, and incorporates the one-step predicted state, thus including information from the system matrix. The innovation vector ${\mathit{r}}_k$ is defined as:

$${\mathit{r}}_k={\mathit{z}}_k-{\widehat{\mathit{z}}}_{k/k-1}$$

(55)

The kernel bandwidth adjustment factor, ${\lambda }_k$, is constructed utilizing the innovation and the predicted error covariance matrix:

$${\mathit{P}}_{zz,k/k-1}={\displaystyle \frac{1}{2n}}\sum _{i=1}^{2n}\left[{\mathit{z}}_{i,k|k-1}-{\widehat{\mathit{z}}}_{k|k-1}\right]{\left[{\mathit{z}}_{i,k|k-1}-{\widehat{\mathit{z}}}_{k|k-1}\right]}^{\mathrm{T}}+{\overline{\mathit{R}}}_k$$

(56)

$${\lambda }_k=\left\{\begin{array}{cc}trace\left({\mathit{P}}_{zz,k/k-1}\right)/trace\left({\mathit{r}}_k{\mathit{r}}_{\mathit{k}}^{\mathit{T}}\right)\hspace{1em}\hfill & trace\left({\mathit{r}}_k{\mathit{r}}_{\mathit{k}}^{\mathit{T}}\right)>trace\left({\mathit{P}}_{zz,k/k-1}\right)\hfill \\ 1\hfill & trace\left({\mathit{r}}_k{\mathit{r}}_{\mathit{k}}^{\mathit{T}}\right)\le trace\left({\mathit{P}}_{zz,k/k-1}\right)\hfill \end{array}\right.$$

(57)

where $trace\left(\cdot \right)$ denotes the trace of a matrix. When $trace\left({\mathit{r}}_k{\mathit{r}}_k^T\right)\le trace\left({\mathit{P}}_{zz,k/k-1}\right)$, it indicates relatively accurate filtering posterior estimates, and thus adjusting the kernel bandwidth is unnecessary. When $trace\left({\mathit{r}}_k{\mathit{r}}_k^T\right)>trace\left({\mathit{P}}_{zz,k/k-1}\right)$, it indicates a large innovation value, which significantly impacts the posterior calculation; in this case, the kernel bandwidth needs to be reduced to capture outlier noise. After obtaining the value of the kernel bandwidth adjustment factor, ${\mu }_k$, the kernel bandwidth is adjusted as follows:

$${\sigma }^{k+1}={\lambda }_k{\sigma }^k$$

(58)

4.2. AMEECKF Algorithm Flow

By introducing the kernel bandwidth adjustment strategy based on the innovation method into the minimum error entropy with a reference point, the AMEECKF with parameter adaptive capability is obtained.

In summary, the AMEECKF is described in Algorithm 1.

Algorithm 1: Adaptive Minimum Error Entropy Cubature Kalman Filter (AMEECKF).

Step 1: Initialization Given ${\mathit{x}}_0,$${\mathit{P}}_0,$$\sigma ,$$\epsilon$. Step 2: Time Update Perform the time update, obtaining ${\widehat{\mathit{x}}}_{k/k-1}$ and ${\mathit{P}}_{k/k-1}$, respectively.    Compute ${\mathit{D}}_k$ and ${\mathit{W}}_k$ using Equations (28) and (29), respectively. Step 3: Fixed-Point Iteration    Set the iteration counter $t=1$ and ${\widehat{\mathit{x}}}_k^0={\widehat{\mathit{x}}}_{k/k-1}$.     ${\tilde{\mathit{e}}}_k={\mathit{D}}_k-{\mathit{W}}_k{\widehat{\mathit{x}}}_k^{t-1}$     ${\tilde{\mathsf{\Lambda }}}_{\mathrm{k}}=\left[\begin{array}{cc}{\tilde{\mathsf{\Lambda }}}_{\mathrm{x}\mathrm{x},\mathrm{k}}& {\tilde{\mathsf{\Lambda }}}_{\mathrm{z}\mathrm{x},\mathrm{k}}\\ {\tilde{\mathsf{\Lambda }}}_{\mathrm{x}\mathrm{z},\mathrm{k}}& {\tilde{\mathsf{\Lambda }}}_{\mathrm{z}\mathrm{z},\mathrm{k}}\end{array}\right]$     ${\tilde{\mathit{P}}}_{k/k-1}=({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^{\mathrm{T}}{\tilde{\mathsf{\Lambda }}}_{xx,k}{\mathsf{\Theta }}_{p,k/k-1}^{-1}$     ${\tilde{\mathit{P}}}_{xz,k/k-1}=({\mathsf{\Theta }}_{r,k}^{-1}{)}^{\mathrm{T}}{\tilde{\mathsf{\Lambda }}}_{xz,k}{\mathsf{\Theta }}_{p,k/k-1}^{-1}$     ${\tilde{\mathit{P}}}_{zx,k/k-1}=({\mathsf{\Theta }}_{p,k/k-1}^{-1}{)}^{\mathrm{T}}{\tilde{\mathsf{\Lambda }}}_{zx,k}{\mathsf{\Theta }}_{r,k}^{-1}$     ${\tilde{\mathit{R}}}_k=({\mathsf{\Theta }}_{r,k}^{-1}{)}^{\mathrm{T}}{\tilde{\mathsf{\Lambda }}}_{zz,k}{\mathsf{\Theta }}_{r,k}^{-1}$     ${\tilde{\mathit{K}}}_k=\left[{\tilde{\mathit{P}}}_{k/k-1}+{\mathit{H}}_k{\tilde{\mathit{P}}}_{k/k-1}+\left({\tilde{\mathit{P}}}_{zx,k/k-1}+{\mathit{H}}_k^T{\mathit{R}}_k\right){\mathit{H}}_k\right]\left({\tilde{\mathit{P}}}_{zx,k/k-1}+{\mathit{H}}_k^T{\mathit{R}}_k\right)$     ${\widehat{\mathit{x}}}_k^t={\widehat{\mathit{x}}}_{k/k-1}+{\tilde{\mathit{K}}}_k\left({\mathit{z}}_k-{\mathit{H}}_k{\widehat{\mathit{x}}}_{k/k-1}\right)$     Employ Equation (56) to compute the ${\tilde{\mathit{P}}}_{zz,k/k-1}$     Employ Equation (57) to compute the tuning factor ${\lambda }_k$     Employ Equation (58) to compute the kernel bandwidth value ${\sigma }^{k+1}$     If $∥{\widehat{\mathit{x}}}_k^t-{\widehat{\mathit{x}}}_k^{t-1}∥/{\widehat{\mathit{x}}}_k^t\le \epsilon$       set ${\widehat{\mathit{x}}}_k={\widehat{\mathit{x}}}_k^t$, exit the iteration, and proceed to Step 4.     else       set $t+1\to t$ and continue the for loop.     End Step 4: Covariance Update and Next Step    Update $k+1\to k$.    Calculate ${\mathit{P}}_k$.    Return to Step 2.

The iteration termination conditions of the AMEECKF are shown in Algorithm 1. In practical applications, a fixed number of iterations is usually set to simplify the algorithm flow, and the appropriate number of iterations needs to balance the algorithm’s computational efficiency and estimation accuracy. For linear systems, a single iteration can meet the required threshold conditions. However, for nonlinear systems, a single iteration often fails to meet the accuracy requirements. Therefore, in the experimental verification section, an in-depth investigation into the reasonable value of the number of iterations will be conducted through comparative experiments.

5. Experiments and Analysis

The AMECKF performance will be validated through two examples and compared against the CKF, the Huber-based CKF (HCKF) [26], and the Maximum Correntropy Criterion Cubature Kalman Filter (MCCCKF) [27].

5.1. Numerical Simulations and Analysis

To evaluate the performance of AMEECKF, this study utilizes the Univariate Nonstationary Growth Model (UNGM) to conduct numerical simulations [28]. The discrete state-space equations for this model are as follows:

$$\left\{\begin{array}{cc}{x}_k\hfill & =0.5x_{k-1}+25{\displaystyle \frac{x_{k-1}}{1+x_{k-1}^2}}+8\cos (1.2(k-1))+w_{k-1}\hfill \\ y_k\hfill & =x_k^2/20+{\nu }_k\hfill \end{array}\right.$$

(59)

where $w_{k-1}$ is zero-mean Gaussian process noise with variance $Q_{k-1}$, ${\nu }_k$ is zero-mean measurement noise with variance $R_k$, and $k$ denotes the simulation time step. Root Mean Square Error (RMSE) and Average RMSE (ARMSE) are used to measure the deviation between estimates and the true state. They are defined as:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}\left(k\right)=\sqrt{{\displaystyle \frac{1}{M}}\sum _{k=1}^M{\left({\widehat{x}}_k-x_k\right)}^2}$$

(60)

$$\mathrm{A}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{K}}\sum _{k=1}^K\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}\left(k\right)$$

(61)

where $M$ refers to the number of Monte Carlo experiments. RMSE at each time step is calculated from a set of samples obtained from $M$ Monte Carlo experiments at that same sampling time. $K$ represents the total number of time steps. Two different measurement noise scenarios are considered. In Scenario 1, all noise satisfies:

$${\nu }_k\sim N\left(\mathrm{0,1}\right)$$

(62)

Under Scenario 2, the measurement noise is configured as heavy-tailed noise, used to describe Gaussian noise contaminated by outliers. The measurement noise satisfies:

$${\nu }_k\sim 0.9N\left(0,1\right)+0.1N\left(0,400\right)$$

(63)

To further study the impact of the number of iterations on performance, experiments are designed in Scenario 2 to compare the average number of iterations and ARMSE corresponding to different $\epsilon$ values. $M=100$, and the total number of time steps per experiment is $K=100$. The experimental results are presented in

Table 1. Average number of iterations and ARMSE corresponding to different thresholds.

Filter	Average Number of Iterations	ARMSE
CKF	—	24.9063
$\mathrm{AMCCCKF}(\epsilon ={10}^{-1}$)	1.3029	15.3124
$\mathrm{AMCCCKF}(\epsilon ={10}^{-2}$)	3.3156	14.9213
$\mathrm{AMCCCKF}(\epsilon ={10}^{-4}$)	4.2132	14.9201
$\mathrm{AMCCCKF}(\epsilon ={10}^{-6}$)	6.0157	14.9186

.

As can be seen from Table 1, a smaller threshold $\epsilon$ leads to a higher average number of iterations, increasing the algorithm’s time complexity. Simultaneously, a smaller $\epsilon$ corresponds to a smaller ARMSE value, indicating higher algorithm accuracy. It should also be noted that when $\epsilon <{10}^{-2}$, further decreasing the threshold yields very limited improvement in accuracy while significantly increasing computation time. Therefore, the threshold $\epsilon$ is set to ${10}^{-2}$. For nonlinear systems, to simplify the algorithm flow, the number of iterations can be configured to a constant value. Setting the iteration count to $N_m=3$ in nonlinear systems can satisfy the requirement for the threshold $\epsilon ={10}^{-2}$.

Next, the comparative experiments include the four filters mentioned earlier. The four filters are set with the same initial parameters: ${\widehat{x}}_0=0$, $P_0=1$, $M=50$, $K=60$. The process noise is Gaussian with its covariance set to $Q_k=1$. For the MCCCKF, the kernel bandwidth is set to ${\sigma }_w=2.0$, and the number of iterations is $N_m=3$. The RMSE curves for the two experimental scenarios are presented in Figure 1 and Figure 2, respectively.

Table 2. ARMSE for Four Filters.

Filter	Scenario 1	Scenario 2
CKF	4.4190	15.8242
HKF	7.7609	10.9411
MCCCKF	5.7684	8.4375
AMEECKF	4.6424	6.5628

presents the ARMSE values for the four filters.

Figure 1 shows that in Scenario 1, the CKF performs significantly better than the other filters, this is because the CKF is founded on the assumption that the measurement noise distribution is Gaussian, allowing it to accurately match the actual noise characteristics and thus achieve optimal estimation performance during filtering.

In contrast, the other three filters assume the measurement noise is non-Gaussian, which contradicts the actual noise distribution in this scenario. This leads to an inability to fully utilize the true statistical properties of the noise during state estimation, consequently affecting filtering accuracy.

Furthermore, the results in Table 2 clearly indicate that, under Gaussian noise, the CKF achieves the smallest ARMSE among the four filters. This further quantitatively verifies the excellent performance of the CKF when the noise conforms to a Gaussian distribution, demonstrating that, by virtue of its accurate noise assumption, the CKF exhibits significant advantages in estimation accuracy and stability.

From Figure 2, it is evident that when the noise exhibits heavy-tailed characteristics, the performance of the filters varies significantly. The CKF, using the MMSE criterion as its optimal estimation criterion, lacks robustness in heavy-tailed noise. Due to the presence of significant outliers, the CKF cannot effectively handle these disturbances, leading to severely degraded filtering performance, with RMSE rising sharply and fluctuating drastically.

HCKF’s performance is enhanced in comparison with the CKF because it considers the non-Gaussian noise to some extent and optimizes the processing of measurement information. However, the HCKF still has shortcomings; it still assigns some weight to measurement information that is useless or even interfering, meaning the filtering process remains affected by measurement outliers. Therefore, its accuracy is still inferior to that of the MCCCKF.

The MCCCKF performs better than both CKF and HCKF because it achieves insensitivity to measurement outliers through specific mechanisms, maintaining good filtering stability and accuracy in heavy-tailed non-Gaussian noise environments, thus exhibiting strong robustness against such noise.

Notably, when faced with larger measurement outliers, the AMEECKF demonstrates a stronger ability to suppress error interference. It can more effectively identify and mitigate the adverse effects of measurement outliers on the filtering process, keeping its RMSE consistently at a relatively low level. It exhibits the best performance among the four algorithms, fully highlighting its advantages in environments with numerous outliers.

The ARMSE results in Table 2 show that among the four filters, the AMEECKF has the smallest ARMSE, this indicates that the proposed algorithm exhibits stronger robustness in heavy-tailed noise scenarios compared to the other algorithms, enabling more stable and accurate state estimation.

5.2. Flight Experiment

Due to the effects of GNSS multipath and dynamic environmental changes, the measurements of an actual GNSS/INS system can be corrupted by measurement outliers. The contaminated measurement noise no longer follows a Gaussian distribution, forming a typical heavy-tailed non-Gaussian noise, this degrades the positioning accuracy, in severe cases, may even trigger filter divergence, failing to meet the positioning reliability requirements for scenarios such as UAV flight. To address the insufficient robustness of traditional algorithms in GNSS/INS systems under non-Gaussian noise environments, the AMEECKF is applied to this system. Through comparative experiments with algorithms like CKF, HCKF and MCCCKF, this paper systematically verifies the robustness of AMEECKF.

The coordinate systems for navigation: the inertial frame (i-frame) is set as the reference frame; the North-East-Down (NED) geographic frame denoted as n-frame; the Front-Right-Down (FRD) frame denoted as b-frame; simultaneously, the Earth-Centered Earth-Fixed (ECEF) frame is set as another reference frame [29,30].

The INS error equation can be expressed as follows [31]:

$$\dot{\mathit{\varphi }}={\mathit{C}}_{\omega }^{-1}\left[\left(\mathit{I}-{\mathit{C}}_n^{n^{\prime }}\right){\widehat{\mathit{\omega }}}_{in}^n+\delta {\mathit{\omega }}_{in}^n-{\mathit{C}}_b^{n^{\prime }}{\mathit{\epsilon }}^b\right]+{\mathit{C}}_{\omega }^{-1}{\mathit{C}}_b^{n^{\prime }}{\mathit{w}}_g^b$$

(64)

$$\begin{array}{c}\delta {\dot{\mathit{\nu }}}^n=\delta \mathit{g}+\left[\mathit{I}-{\left({\mathit{C}}_n^{n^{\prime }}\right)}^T\right]{\mathit{C}}_b^{n^{\prime }}{\widehat{\mathit{f}}}^b-\left(2{\widehat{\mathit{\omega }}}_{ie}^n+{\widehat{\mathit{\omega }}}_{en}^n\right)\times \delta {\mathit{\nu }}^n-\left(2\delta {\mathit{\omega }}_{ie}^n+\delta {\mathit{\omega }}_{en}^n\right)\times {\mathit{\nu }}^n\hfill \\ +{\left({\mathit{C}}_n^{n^{\prime }}\right)}^T{\mathit{C}}_b^{n^{\prime }}{\mathsf{\nabla }}^b+{\left({\mathit{C}}_n^{n^{\prime }}\right)}^T{\mathit{C}}_b^{n^{\prime }}{\mathit{w}}_a^b\hfill \end{array}$$

(65)

$$\delta \dot{L}={\displaystyle \frac{\delta {\nu }_x}{R_m+h}}-\delta h{\displaystyle \frac{{\nu }_x}{{\left(R_m+h\right)}^2}}$$

(66)

$$\delta \dot{\lambda }={\displaystyle \frac{\delta {\nu }_y\sec L}{R_n+h}}+\delta L{\displaystyle \frac{{\nu }_y\tan L\sec L}{R_n+h}}-\delta h{\displaystyle \frac{{\nu }_y\sec L}{{\left(R_n+h\right)}^2}}$$

(67)

$$\delta \dot{h}=-\delta {\nu }_z$$

(68)

where

$${\mathit{\omega }}_{ie}^n={\left[\begin{array}{ccc}{\omega }_{ie}\cos L& 0& -{\omega }_{ie}\sin L\end{array}\right]}^T$$

(69)

$$\delta {\mathit{\omega }}_{ie}^n={\left[\begin{array}{ccc}-\delta L{\omega }_{ie}\sin L& 0& -\delta L{\omega }_{ie}\cos L\end{array}\right]}^T$$

(70)

$${\mathit{\omega }}_{en}^n={\left[\begin{array}{ccc}{\displaystyle \frac{{\nu }_y}{R_n+h}}& -{\displaystyle \frac{{\nu }_x}{R_m+h}}& -{\displaystyle \frac{{\nu }_y\tan L}{R_n+h}}\end{array}\right]}^T$$

(71)

$$\begin{array}{c}\delta {\mathit{\omega }}_{en}^n=\left[\begin{array}{c}{\displaystyle \frac{\delta {\nu }_y}{R_n+h}}-\delta h{\displaystyle \frac{{\nu }_y}{{\left(R_n+h\right)}^2}}\\ -{\displaystyle \frac{\delta {\nu }_x}{R_m+h}}+\delta h{\displaystyle \frac{{\nu }_x}{{\left(R_m+h\right)}^2}}\\ -{\displaystyle \frac{\delta {\nu }_y\tan L}{R_n+h}}-\delta L{\displaystyle \frac{{\nu }_y{\sec }^{2}L}{R_n+h}}+\delta h{\displaystyle \frac{{\nu }_y\tan L}{{\left(R_n+h\right)}^2}}\end{array}\right]\end{array}$$

(72)

$${\mathit{C}}_{\omega }^{-1}={\displaystyle \frac{1}{\cos {\varphi }_y}}\left[\begin{array}{ccc}\cos {\varphi }_z& -\sin {\varphi }_z& 0\\ \sin {\varphi }_z\cos {\varphi }_y& \cos {\varphi }_z\cos {\varphi }_y& 0\\ -\cos {\varphi }_z\sin {\varphi }_y& \sin {\varphi }_z\sin {\varphi }_y& \cos {\varphi }_y\end{array}\right]$$

(73)

The ${\mathit{\epsilon }}^b$ and ${\mathsf{\nabla }}^b$ are represented as follows:

$${\dot{\mathit{\epsilon }}}_i^b=\mathbf{0}$$

(74)

$${\dot{\mathsf{\nabla }}}_i^b=\mathbf{0}$$

(75)

Definition of the state vector:

$$\mathit{x}={\left[\begin{array}{ccccc}\mathit{\varphi }& \mathit{\delta }\mathit{s}& \mathit{\delta }\mathit{p}& {\mathit{\epsilon }}^{\mathit{b}}& {\mathsf{\nabla }}^{\mathit{b}}\end{array}\right]}^T$$

(76)

Here, $\mathit{\varphi }=\left[\begin{array}{ccc}{\varphi }_x& {\varphi }_y& {\varphi }_z\end{array}\right]$ denotes the attitude error, $\mathsf{\delta }\mathit{s}=\left[\begin{array}{ccc}\mathsf{\delta }{s}_{\mathrm{x}}& \mathsf{\delta }{s}_{\mathrm{y}}& \mathsf{\delta }{s}_{\mathrm{z}}\end{array}\right]$ stands for the velocity error, and $\mathsf{\delta }\mathit{p}=\left[\begin{array}{ccc}\mathsf{\delta }L& \mathsf{\delta }\lambda & \mathsf{\delta }h\end{array}\right]$ represents the position error. The gyro’s constant drift is ${\mathit{\epsilon }}^{\mathrm{b}}=\left[\begin{array}{ccc}{\mathit{\epsilon }}_{\mathrm{x}}^{\mathrm{b}}& {\mathit{\epsilon }}_{\mathrm{y}}^{\mathrm{b}}& {\mathit{\epsilon }}_{\mathrm{z}}^{\mathrm{b}}\end{array}\right]$, while the accelerometer’s zero-bias is ${\mathsf{\nabla }}^{\mathrm{b}}=\left[\begin{array}{ccc}{\nabla }_{\mathrm{x}}^{\mathrm{b}}& {\nabla }_{\mathrm{y}}^{\mathrm{b}}& {\nabla }_{\mathrm{z}}^{\mathrm{b}}\end{array}\right]$.

The system state equation is:

$$\dot{\mathit{x}}\left(t\right)=f\left(\mathit{x}\left(t\right)\right)+\mathit{w}\left(t\right)$$

(77)

In this context, $f\left(\cdot \right)$ constitutes a nonlinear function made up of Equations (64)–(68), (74) and (75), while the process noise vector is given by $\mathit{w}\left(\mathrm{t}\right)=[\begin{array}{ccc}({\mathit{C}}_{\mathsf{\omega }}^{-1}{\mathit{C}}_{\mathrm{b}}^{{\mathrm{n}}^{\prime }}{w}_{\mathrm{g}}^{\mathrm{b}}{)}^{\mathrm{T}}& ({\mathit{C}}_{\mathrm{b}}^{\mathrm{n}}{w}_{\mathrm{a}}^{\mathrm{b}}{)}^{\mathrm{T}}& {\mathbf{0}}_{1\times 9}\end{array}{]}^{\mathrm{T}}$.

Applying the improved Euler discretization approach to Equation (77) for discretization, the discrete state equation is:

$${\dot{\mathit{x}}}_k=f\left({\mathit{x}}_{k-1}\right)+{\mathit{w}}_k$$

(78)

The measurement equation is thus presented as follows:

$${\mathit{z}}_k=\left[\begin{array}{c}{\nu }_{k,x}-{\nu }_{k,x}^g\\ {\nu }_{k,y}-{\nu }_{k,y}^g\\ {\nu }_{k,z}-{\nu }_{k,z}^g\\ L-L^g\\ \lambda -{\lambda }^g\\ h-h^g\end{array}\right]=\left[\begin{array}{c}\delta {\nu }_x\\ \delta {\nu }_y\\ \delta {\nu }_z\\ \delta L\\ \delta \lambda \\ \delta h\end{array}\right]+{\mathit{\nu }}_k=h({\mathit{x}}_k)+{\mathit{\nu }}_k$$

(79)

where ${\mathit{\nu }}_k$ refers to the measurement noise.

To validate the practical performance of the AMEECKF, experimental data were collected using a fixed-wing UAV platform. This platform comprises three core modules: the Skywalker X8 UAV airframe (Wuhan Skywalker Technology Co., Ltd., Wuhan, China), a sensor data acquisition system, and an onboard navigation system, ensuring the reliability of the measurements and reference data.

The sensor data acquisition system is responsible for obtaining raw GNSS/INS measurement data. It consists of a French SBG Ellipse2-N inertial navigation system and a single-point GNSS antenna. The Ellipse2-N integrates a three-axis gyroscope, accelerometer, magnetometer, and a single-point GNSS receiver. It can synchronously collect raw data such as acceleration, angular rate, and GNSS information, providing the foundation for subsequent state estimation. The airborne navigation system is used to record reference values of navigation parameters for evaluating the accuracy of filters. Its core components are an RTK-GNSS device and an Ellipse2-E inertial navigation system. Assisted by RTK-GNSS, the Ellipse2-E can provide high-precision reference values for attitude, velocity, and position under dynamic conditions, offering accurate benchmarks for quantitative analysis of navigation errors.

Table 3. Sensor Parameters.

Sensor	Parameters	Numerical Value
Gyroscope	Constant deviation	$0.2°/\mathrm{s}$
	Random walk	$0.15°/\sqrt{\mathrm{h}}$
	Sampling frequency	50 Hz
Accelerometer	Zero bias	$5\times {10}^{-3} \mathrm{g}$
	Random walk	$5\times {10}^5 \mathrm{g}\cdot \sqrt{\mathrm{s}}$
	Sampling frequency	50 Hz
Magnetometer	Zero bias	$0.1\mathsf{\mu }\mathrm{T}$
	Measurement range	$\pm 5\times {10}^3\mathsf{\mu }\mathrm{T}$
	Sampling frequency	50 Hz
Single-point GNSS receiver	Speed error range	0.1 m/s
	Position error range	2 m
	Sampling frequency	10 Hz

shows the performance indicators of the gyroscope, accelerometer, and single-point GNSS receiver within the sensor data acquisition system.

Table 4. Performance Parameters of Airborne Navigation Systems.

Parameters	Performance
Roll/Pitch Angle Uncertainty (1σ)	0.05°
Yaw Angle Uncertainty (1σ)	0.2°
Position Uncertainty (1σ)	0.01 m

presents the performance indicators of the airborne navigation system used in the experiment—consisting of an Ellipse2-E Inertial Navigation System and an RTK device.

In the GNSS/INS-integrated navigation system, the filter takes the biases of the gyroscope and accelerometer, along with the biases of position, velocity, and attitude, as core states for real-time online estimation. When GNSS data (with an update frequency of 10 Hz) arrive, the system uses the position and velocity information it provides to compare with the results of pure inertial calculation. The improved CKF is then applied to estimate the biases of the gyroscope and accelerometer, as well as the biases of position, velocity, and attitude. This more accurate bias estimation is subsequently used in the high-frequency (50 Hz) inertial calculation within the next GNSS update interval, thereby continuously compensating for the source of drift.

The uncertainty of all parameters refers to one standard deviation (1σ), which corresponds to a confidence interval of approximately 68%.

The duration of sensor data acquisition was 200 s. The initial position latitude and longitude were (34.0277615, 108.6926932), with an altitude of 599.87 m. This initial position was set as the origin. The fixed-wing UAV platform is displayed in Figure 3, and the UAV’s flight trajectory during this period is displayed in Figure 4.

The four filters are set with the same initial parameters, the initial covariance matrices ${\mathit{P}}_0,{\mathit{Q}}_k\mathrm{a}\mathrm{n}\mathrm{d}{\mathit{R}}_k$ were set, respectively, as:

$${\mathit{P}}_0=\mathrm{diag}[(0.1{)}^2\cdot {\mathit{I}}_{3\times 3},(0.1{)}^2\cdot {\mathit{I}}_{3\times 3},(2{)}^2\cdot {\mathit{I}}_{3\times 3},{\left(0.2\right)}^2\cdot {\mathit{I}}_{3\times 3},(5\times {10}^{-3}{)}^2\cdot {\mathit{I}}_{3\times 3}]$$

(80)

$${\mathit{Q}}_k=\mathrm{diag}[(0.15{)}^2\cdot {\mathit{I}}_{3\times 3},(5\times {10}^{-5}{)}^2\cdot {\mathit{I}}_{3\times 3},{\mathbf{0}}_{9\times 9}]$$

(81)

$${\mathit{R}}_k=\mathrm{diag}\left[{\left(0.1\right)}^2\cdot {\mathit{I}}_{3\times 3},{\left(2\right)}^2\cdot {\mathit{I}}_{3\times 3}\right]$$

(82)

RMSE was still used to evaluate algorithm performance. The attitude angle error comparison graphs for the four filters are displayed in Figure 5, Figure 6 and Figure 7, and the corresponding RMSE values for the attitude angles are shown in

Table 5. RMSE for Attitude Angles (°).

Filter	Roll Angle	Pitch Angle	Yaw Angle
CKF	0.3675	0.3749	1.0660
HCKF	0.3564	0.2995	0.8486
MCCCKF	0.2904	0.1282	0.3124
AMEECKF	0.1575	0.0782	0.2037

.

From Figure 5, Figure 6 and Figure 7, the CKF yields the worst accuracy in estimated attitude angles, this is because during UAV maneuvers, GNSS experiences multipath effects, coupled with constantly changing flight conditions introducing measurement outliers. The HCKF and MCCCKF show better attitude estimation performance than CKF due to their inherent robustness to measurement outliers. Regarding the AMEECKF, it results in the smallest attitude angle errors, indicating that among these filters, AMEECKF possesses the strongest robustness against measurement noise containing outliers. Furthermore, the angle error for the AMEECKF is controlled within 0.7°, achieving high attitude estimation accuracy.

From Table 5, CKF exhibits the maximum RMSE values. Although the attitude estimation performance of HCKF and MCCCKF is superior to CKF, their RMSE values are still higher than those of the AMEECKF. The AMEECKF exhibits the smallest RMSE values. This demonstrates that compared to CKF, HCKF, and MCCCKF, the proposed algorithm enhances the attitude estimation accuracy.

Figure 8, Figure 9 and Figure 10 shows the position error comparison for the algorithms, and

Table 6. RMSE for the corresponding location (m).

Filter	X-Axis Position	Y-Axis Position	Z-Axis Position
CKF	0.8261	0.6882	0.5649
HKF	0.5026	0.4986	0.4355
MCCCKF	0.4485	0.3264	0.2083
AMEECKF	0.3224	0.2515	0.1397

displays the corresponding position RMSE values.

From Figure 8, Figure 9 and Figure 10, similar to the attitude estimation results, the CKF corresponds to the largest position errors. Although the position errors of the HCKF and MCCCKF are smaller than those of CKF, their error values still exceed 1.0 m, failing to meet the accuracy requirements for position estimation in UAV navigation systems. Moreover, compared to the MCCCKF, the AMEECKF demonstrates better suppression of larger outliers, controlling the position error within 1.0 m and achieving higher position estimation accuracy.

From Table 6, it can be seen that the position RMSE values corresponding to the AMEECKF are the smallest compared to the results from the CKF, HCKF, and MCCCKF. Since the RMSE directly reflects the magnitude of position estimation error, smaller values indicate higher position estimation accuracy.

6. Discussion

Aiming at the problem that GNSS/INS systems mounted on low-cost UAVs are susceptible to interference from heavy-tailed non-Gaussian measurement noise, leading to degraded navigation accuracy, this paper proposes an AMEECKF. This algorithm leverages the efficient processing capability of the CKF for system nonlinear models, establishing a foundation of basic accuracy for state estimation in nonlinear scenarios. Utilizing the insensitivity of correntropy to outliers, correntropy theory is introduced to mitigate the effects of heavy-tailed measurement noise, replacing the traditional MMSE criterion with the MEE criterion. Simultaneously, a kernel bandwidth adaptive adjustment factor is constructed to dynamically tune the kernel bandwidth according to the real-time characteristics of the measurement data, further reducing the interference of measurement outliers on filtering accuracy and enhancing the filter’s robustness in complex noise environments.

For demonstrating the filter’s effectiveness, this paper designed numerical simulations and UAV flight experiments, comparing AMEECKF against CKF, HCKF, and MCCCKF. Specific experimental results show that in the dynamic flight environment of UAVs, the AMEECKF can effectively suppress the interference of the nonlinearity of the system model on navigation accuracy and ensure the stable operation of the integrated navigation system. Meanwhile, when confronted with significant measurement outliers, the algorithm still demonstrates excellent robustness and has notable advantages in the estimation accuracy of navigation parameters, thereby comprehensively enhancing the overall accuracy of the GNSS/INS-integrated navigation system.

In summary, the AMEECKF effectively addresses the interference challenge posed by heavy-tailed non-Gaussian measurement noise on low-cost UAV GNSS/INS-integrated navigation systems. It provides reliable technical support for the high-precision application, possessing considerable practical engineering value. Meanwhile, it should be noted that the GNSS/INS system in this paper employs a loose coupling model. While this model possesses significant advantages of low computational complexity and a linear measurement model, and these features help reduce the difficulty of engineering implementation, it has lower overall accuracy compared with the tight coupling model that can deeply fuse GNSS raw observation data and still has certain limitations in high-precision navigation scenarios. Future research may attempt to implement the model proposed in this paper within the GNSS/INS tight coupling framework. It may also try to extend the algorithm in this paper to multi-sensor fusion navigation scenarios involving inertial sensors combined with visual cameras, LiDAR, and airspeed sensors—so as to further explore its broader applicability in multi-sensor-integrated navigation and meet complex and ever-changing navigation requirements.