Sampling-Based Adaptive Techniques for Reducing Non-Gaussian Position Errors in GNSS/INS Systems

Abstract

In this paper, we propose a novel method to reduce non-Gaussian errors in measurements using sampling-based distribution estimation. Although non-Gaussian errors are often treated as statistical deviations, they can frequently arise in practical unmanned aerial systems that depend on global navigation satellite systems (GNSS), where position measurements are degraded by multipath effects. However, nonlinear or robust filters have shown limited effectiveness in correcting such errors, particularly when they appear as persistent biases in measurements over time. In such cases, adaptive techniques have often demonstrated greater effectiveness. The proposed method estimates the distribution of observed measurements using a sampling-based approach and derives a reformed measurement from this distribution. By incorporating this reformed measurement into the filter update, the proposed approach achieves lower error levels than traditional adaptive filters. To validate the effectiveness of the method, Kalman filter simulations are conducted for drone GNSS/INS navigation. The results show that the proposed method outperforms conventional non-Gaussian filters in handling measurement bias caused by non-Gaussian errors. Furthermore, it achieves nearly twice the estimation accuracy compared to adaptive approaches. These findings confirm the robustness of the proposed technique in scenarios where measurement accuracy temporarily deteriorates before recovering.

1. Introduction

State estimation refers to the process of predicting system variables and refining those predictions using additional information, typically in the form of sensor measurements [1]. Probabilistic models are commonly used to represent uncertainty in both prediction and correction steps, allowing for systematic updates of the estimated states.

Kalman filter (KF) is a prominent probability-based filter that employs Bayesian theory to model uncertainty using Gaussian distributions for applications in various fields, including navigation, tracking, and sensor fusion [2,3,4,5,6]. However, the KF performance degrades substantially in nonlinear systems [7]. Noteworthy developments include extended Kalman filter (EKF) and unscented Kalman filter (UKF), both of which preserve the fundamental concepts of the KF. The EKF uses a linearization process via Jacobian matrices [8], while the UKF utilizes sigma points, ensuring a robust performance even in nonlinear environments [9]. In addition, beyond the KF, the particle filter (PF) presents another significant nonlinear filtering approach [10], which uses a set of particles to represent distributions, offering enhanced efficacy in modeling nonlinear dynamics.

Bayesian filtering demonstrates excellent performance. However, the inherent design of these filters, which involve correcting prediction errors using measurements, can degrade the performance when the measurements include unexpected errors. The degraded performance is noticeable in robotic navigation, wherein frequency-based sensors, such as global navigation satellite systems (GNSS), radars, and sonar, are susceptible to nonlinear errors, which are commonly attributed to multipath effects, ghost targets, or environmental factors that affect measurements.

Errors caused by characteristic non-Gaussian measurements pose challenges that traditional methods such as EKF, UKF, and PF struggle to effectively mitigate. Deriving complex nonlinear measurement equations is necessary for both EKF and UKF, which is a process that is not only challenging, but also operates accurately only under specific constraints [11,12]. Although the PF can adapt its particles to reflect the nonlinear model in state estimation, selecting an appropriate likelihood model is difficult [13]. When a Gaussian likelihood model is employed, the number of nonlinear errors present in the measurements limits the accuracy of the state estimation [14].

In this study, we introduce a novel concept called reformed measurement to address estimation errors arising from measurement errors. Our approach involves simulating the probability distribution of the observed measurements and recalculating the measurements based on this simulated distribution. Understanding the probability distribution of measurements would theoretically encompass all pertinent information, including the accuracy and true value; however, in practice, each observed measurement provides limited information and the complete distribution remains unknown. The proposed method leverages data from the predictive states of the filter, actual observed measurements, and residuals from the initially converged filter to approximate the observed probability distribution. Accurate, error-free measurements incorporating both the probabilistic characteristics of the simulations and other observed traits can be recalculated by tracking and simulating the observed distribution of measurements.

The proposed approach employs reformed measurements, which are computed to replace the directly observed ones, during the measurement update process, thereby improving the accuracy of state estimation. Consequently, the proposed approach mitigates non-Gaussian measurement errors that previous studies could not address. This enables accurate navigation in environments such as multipath areas, where such errors are typically transient and revert to normal over time.

This paper presents a reformed measurement technique, which is an innovative algorithm built upon the widely used KF framework.

• A novel method is proposed to approximate the probability distribution of observed measurements using available data such as predictions, observations, and the innovation term, enhanced by sampling techniques.
• A new approach is introduced to compute a reformed measurement from the approximated distribution, enabling robust state estimation in the presence of non-Gaussian errors, particularly biased measurements.
• The proposed reformed measurement can be modularly integrated into existing KF-based GNSS/INS systems with minimal changes to the existing KF implementation.

This paper is organized as follows. Section 2 reviews adaptive filtering techniques for handling non-Gaussian measurement errors. Section 3 provides a theoretical explanation of the proposed reformed measurement method. Section 4 presents the simulation setup and results to validate the modular implementation of the proposed algorithm. Section 5 concludes the paper and discusses directions for future research.

2. Related Works

A commonly used constraint method was employed in instances of measurement errors. This approach involves identifying stable measurement conditions or scenarios and establishing them as constraints. Although the constraint method is conceptually simple to implement, its practical application remains difficult due to the challenge of formulating universal constraints that reliably reduce measurement errors [15].

Substantial research has been conducted for adapting the Kalman filter (ADKF) framework to non-Gaussian contexts [16,17,18]. Adaptive techniques offer another method for mitigating the uncertainty in measurements. This approach adjusts the covariance, which reflects the uncertainty of either the measurement or predicted state, by utilizing the difference between the estimated and observed measurements.

Equations for predicting the state variables and measurements in the KF are [11]

$$\left\{\begin{array}{c}{\widehat{\mathbf{x}}}_k^{-}={\mathbf{F}}_{k-1}{\widehat{\mathbf{x}}}_{k-1}^{+}+{\mathit{w}}_k\\ \hspace{1em}{\mathbf{z}}_k={\mathbf{H}}_k{\widehat{\mathbf{x}}}_k^{-}+{\mathbf{v}}_k \end{array}\right.$$

(1)

where $k$, ${\widehat{\mathbf{x}}}_k^{-}$, ${\widehat{\mathbf{x}}}_{k-1}^{+}$ represent the time index, estimated value updated by the system model $\mathbf{F}$, and estimate corrected by the measurement error at the previous time. Meanwhile, the Kalman filter models the noise associated with the predictions as Gaussian, ${\mathbf{w}}_{\mathrm{k}}~N(0,\mathbf{Q})$, where $\mathbf{Q}$ denotes the process noise covariance matrix and is a symmetric positive definite matrix. The measurement ${\mathbf{z}}_k$ was calculated, where $\mathbf{H}$ represents the observation matrix that converts the predicted state variables to the same units as the observed measurements. The uncertainty of the measurement was modeled as Gaussian, ${\mathbf{v}}_{\mathrm{k}}~N(0,\mathbf{R})$, where $\mathbf{R}$ is a symmetric positive definite matrix representing the measurement noise covariance.

In addition, the covariance matrix representing the uncertainty predicted by the system model is [11,19]

$${\mathbf{P}}_{\mathrm{k}}^{-}={\mathbf{F}}_k{\mathbf{P}}_{k-1}^{+}{\mathbf{F}}_k^T+\mathbf{Q}.$$

(2)

The Kalman gain is determined by calculating the covariance of the prediction and the uncertainty of the measurement [11].

$${\mathbf{K}}_{\mathrm{k}}={\mathbf{P}}_k^{-}{\mathbf{H}}^T{\left(\mathbf{H}{\mathbf{P}}_k^{-}{\mathbf{H}}^T+\mathbf{R}\right)}^{-1},$$

(3)

where the uncertainty of the observed measurements, denoted by $\mathbf{R}$ represents the measurement noise covariance. State variables are calculated using measurements observed from an external source as [11]

$${\widehat{\mathbf{x}}}_k^{+}={\widehat{\mathbf{x}}}_k^{-}+{\mathbf{K}}_k({\mathbf{y}}_k-{\mathbf{H}}_k{\widehat{\mathbf{x}}}_k^{-}).$$

(4)

The innovation term ${\mathbf{y}}_k-{\mathbf{H}}_k{\widehat{\mathbf{x}}}_k^{-}$ reflects the discrepancy between the actual observation and the predicted output from the model. When the innovation is large, it may indicate the presence of a measurement error. To account for this, adaptive filtering techniques estimate the measurement noise covariance ${\widehat{\mathbf{R}}}_k$ at each time step $k$ based on recent innovation samples, as given by [15]:

$${\widehat{\mathbf{R}}}_k={\displaystyle \frac{1}{m}}{\mathsf{\Sigma }}_{i=1}^m\left({\mathbf{y}}_i-{\mathbf{H}}_i{\widehat{\mathbf{x}}}_i^{-}\right){\left({\mathbf{y}}_i-{\mathbf{H}}_i{\widehat{\mathbf{x}}}_i^{-}\right)}^T-{\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T.$$

(5)

Here, $m$ denotes the number of past innovations used for estimation. The summation index $i$ enumerates the stored innovation samples and is independent of the current time step $k$. In this way, the estimated covariance ${\widehat{\mathbf{R}}}_k$ represents the measurement noise covariance at the current time step, while the right-hand side of Equation (5) represents the average over the past $m$ innovation terms.

An alternative strategy adjusts the process noise covariance ${\widehat{\mathbf{Q}}}_{\mathrm{k}}$ to reduce the confidence in the prediction when measurement reliability is high.

The use of innovative techniques for managing uncertainty often yields impressive results in detecting measurement errors.

In particular, the repeated adjustment of ${\widehat{\mathbf{R}}}_k$ or ${\widehat{\mathbf{Q}}}_{\mathrm{k}}$ can cause a gradual increase in estimation uncertainty after the occurrence of measurement errors. This may lead to larger estimation errors even after the measurements return to normal [20]. Covariance adjustment methods encounter stability issues such as estimation delay in the steady state and risk of divergence. These challenges can be attributed to the inherent difficulty in accurately determining the uncertainty of measurements. Employing a variance larger than the actual measurement variance based on innovation can result in heightened overall system instability when the uncertainty of the observed measurements is not well defined.

3. Reformed Measurement via Probabilistic Distribution Simulation

3.1. Proposed Estimation Architecture

The proposed approach reconstructs the distribution of observed measurements by analyzing the discrepancies among actual observations, model predictions, and their innovation. A reformed measurement, compensated for estimated error, is then used in place of the original observation during the update step.

Figure 1 illustrates the state estimation process using measurements influenced by a non-Gaussian distribution. Although the true distribution is unknown in practice, a representative case is depicted to exemplify deviations from a Gaussian. Under standard conditions, measurements are expected to follow a distribution centered at the true state $x_k$. However, due to unexpected errors with non-Gaussian characteristics, the actual observations $y_k$ deviate from the ideal distribution.

The Kalman filter (1) uses the observed measurement values as the mean and known variance as the measurement covariance. As a result, substantial estimation errors can arise when the measurements are contaminated by error. In contrast, the adaptive technique (2) achieves more accurate final estimates by increasing the measurement variance using the innovation while employing the observed measurement value as the mean. However, residual errors persist because the exact variance is unknown and the measurement variance increases proportionally with the innovation, thereby affecting the subsequent estimates. Conversely, the proposed concept involves re-estimating the distribution from which the measurements are drawn to compute a reformed measurement ${\tilde{y}}_k$ with reduced errors for use in measurement updates. A precise simulation of the distribution in which the measurements are observed is crucial for applying this concept.

The probability distribution of the measurements cannot be computed using observed measurements alone. Therefore, uncertainty is often defined based on sensor performance assessed during the evaluation stage. In this study, the distribution of measurements was calculated by simulating samples drawn from three probability distributions representing different levels of measurement accuracy.

3.2. Sampling-Based Measurement Distribution Estimation

The key concept for calculating the reformed measurement is to simulate the measurement distribution. The first distribution used to simulate the measurement distribution is assumed to be the observed measurement distribution. This distribution was continuously updated for reflecting uncertainty in the measurements.

The measurement information is unknown, and therefore, the initial, normal distribution using the predicted value ${\mathbf{H}\widehat{\mathbf{x}}}_k^{-}\in {\mathbb{R}}^{D\times 1}$ from the system model and the known measurement covariance $\mathbf{R}\in {\mathbb{R}}^{D\times D}$ is assumed as the initial measurement distribution. $D$ represents the number of state variables.

In this work, R is assumed to be a diagonal matrix, meaning that measurement errors in different dimensions are uncorrelated. This assumption allows the multivariate measurement distribution to be factorized into independent one-dimensional distributions for each sensor channel. In the context of GNSS/INS applications, R is treated as diagonal because horizontal and vertical position errors primarily arise from independent sources, making their cross-correlation negligible [21,22]. This simplification also makes sample-based density estimation more tractable in higher dimensions.

Samples drawn from each of these probability distributions are defined as

$${{\left\{{\mathit{A}\left(j\right)}_i\right\}}_k}_{\begin{array}{c}i=\mathrm{1,2}, \dots , \mathit{m}\left(j\right)\\ j=\mathrm{1,2}, \dots , D \end{array}}~N\left({{\mathit{y}\left(j\right)-\mathbf{H}}_k\widehat{\mathbf{x}}}_k^{-}\left(j\right);\mathbf{R}(j,j)\right).$$

(6)

where ${\mathbf{y}}_{\mathrm{k}}$ represents the observed measurement and ${\mathit{A}\left(j\right)}_i$ represents the sampled value, where $j$ represents the dimension index and has a value up to dimension $D$. $i$ represents the number of samples drawn from each one-dimensional probability distribution. Therefore, the drawn sample dimensions are defined as ${\mathit{A}\left(j\right)}_i\in {\mathbb{R}}^{\mathbf{m}\left(\mathit{j}\right)\times 1}$ under the condition $j=\mathrm{1,2}, \dots ,D$, where $\mathbf{m}\left(j\right)$ represents the number of samples defined in the next section.

The second distribution is a normal distribution, which utilizes the observed measurements and known variance from the assessment stage. Samples drawn from this distribution can be represented as

$${{\left\{{\mathit{B}\left(j\right)}_s\right\}}_k}_{\begin{array}{c}s=\mathrm{1,2}, \dots , n-\mathit{m}\left(j\right)\\ j=\mathrm{1,2}, \dots , D \end{array}}~N\left({\mathit{y}}_k\left(j\right)-{\mathbf{H}}_k{\mathbf{x}}_k;\mathbf{R}(j,j)\right),$$

(7)

where $n$ denotes the total number of samples. The $\mathit{B}{\left(j\right)}_s$ represents the drawn samples and is defined in the dimensions of ${\mathit{B}\left(j\right)}_i\in {\mathbb{R}}^{\left(n-\mathit{m}\left(j\right)\right)\times 1}$, under the index of $j=\mathrm{1,2},\dots ,D$.

The third probability distribution is constructed using collected residuals until the convergence of the filter system covariance, which signifies the accuracy of the estimation.

Probability-based filters repeat predictions and corrections when in a normal state, thereby increasing the estimation accuracy. Further, accuracy indicators, such as covariance or weights, converge. Therefore, a distribution that represents the error level between the measurements and estimates can be calculated using the residuals collected until the convergence of covariance. The residual calculated at time $k$ is

$${\mathit{ϵ}}_{\mathit{k}}\left(j\right)={\left({\mathit{y}}_k\left(j\right)-{\mathbf{H}}_k{\widehat{\mathbf{x}}}_k^{-}\left(j\right)\right)}_{j=\mathrm{1,2}, \dots ,D},$$

(8)

where ${\mathit{ϵ}}_{\mathit{k}}\left(j\right)\in {\mathbb{R}}^{D\times 1}$ represents the computed residual. The number of samples collected until the covariance converged is $t$. The collected residuals are defined as a single set as

$$\mathbf{G}\left(j\right)=\{{\mathit{ϵ}}_1\left(\mathit{j}\right),\dots ,{\mathit{ϵ}}_{\mathit{t}}(j\left)\right\}.$$

(9)

The collected residuals can be considered as samples, and a sample-based probability distribution can be constructed. A representative technique to estimate probability distributions using samples is a kernel density estimation (KDE) [23]. The KDE is a technique to estimate random variables by applying a kernel function to the samples and smoothing. The KDE function was calculated using a Gaussian kernel for computational convenience. The distribution function obtained using KDE is given as

$$\mathsf{\Gamma }{\left(\mathit{G}\left(\mathit{j}\right)\right)}_{\mathit{j}=\mathrm{1,2},\dots ,\mathit{D}}={\displaystyle \frac{1}{t·\mathit{h}\left(j\right)·\sqrt{2\pi }}}{\Sigma }_{i=1}^t\mathit{exp}\left(-{\displaystyle \frac{{\mathit{ϵ}}_i\left(j\right)}{2·\mathit{h}{\left(j\right)}^2}}\right).$$

(10)

The total of $D$ probability distributions can be obtained using a one-dimensional KDE. $\mathbf{h}\left(j\right)$ represents the bandwidth corresponding to the $j$ th dimension, where $\mathbf{h}\left(j\right)=std\left(\mathit{G}\right(j\left)\right)$, and $\mathrm{s}\mathrm{t}\mathrm{d}\left(\mathit{G}\right(j\left)\right)$ represents the standard deviation of the collected sample $\mathit{G}\left(j\right)$. Samples extracted from the probability distribution calculated in Equation (10) can be defined as

$${{\left\{{\mathit{\gamma }\left(j\right)}_o\right\}}_k}_{ \begin{array}{c}o=\mathrm{1,2}, \dots , \mathit{L}\left(j\right)\\ j=\mathrm{1,2}, \dots ,D \end{array}}~\mathsf{\Gamma }(\mathit{G}(j)).$$

(11)

where $\mathbf{L}\left(j\right)$ represents the number of samples extracted from a distribution defined in a single dimension. Samples extracted from the total of three probability distributions described previously, $\left\{{\mathit{A}\left(j\right)}_{\left\{i\right\}}\right\}$, $\left\{{\mathit{B}\left(j\right)}_s\right\}$, and $\left\{{\mathit{\gamma }\left(j\right)}_o\right\}$, can be composed into a single sample set as

$${\mathit{u}}_{\mathit{k}}\left(j\right)={\left\{{\mathit{A}\left(j\right)}_i,{\mathit{B}\left(j\right)}_s, {\mathit{\gamma }\left(j\right)}_o\right\}}_{\begin{array}{c}i=\mathrm{1,2}, \dots ,\mathit{m}\left(j\right) \\ s=\mathrm{1,2}, \dots , n-\mathit{m}\left(j\right)\\ o=\mathrm{1,2} \dots ,\mathit{L}\left(j\right) \\ j=\mathrm{1,2} \dots ,D \end{array} }.$$

(12)

Here, $k$ denotes the time index of the operating filter, which corresponds to the time when the measurement is observed. The number and index based on the number of samples and size of the dimension are indicated by the subscript of the set. Therefore, the total number of the set ${\mathit{u}}_{\mathit{k}}\left(j\right)$ is $n+\mathbf{L}\left(j\right)$, and the set ${\mathbf{U}}_{\mathit{k}}\left(j\right)=\{{\left({\mathit{u}}_k\left(j\right)\right)}^{\varpi };\varpi =1,\dots ,n+\mathbf{L}(j\left)\right\}$. The new probability distribution using the samples is calculated as

$$\widehat{q}\left({\mathit{u}\left(j\right)}_k\right)={\displaystyle \frac{1}{\sqrt{2\pi }\mathit{h}\left(j\right)(n+\mathit{L}(j\left)\right)}}{\Sigma }_{\varpi =1}^{n+\mathit{L}\left(j\right)}\mathit{exp}\left(-{\displaystyle \frac{{\left({\left({\mathit{u}}_k\left(j\right)\right)}^{\varpi }\right)}^2}{2{\left(\mathit{h}\left(j\right)\right)}^2}}\right),$$

(13)

where $j=\mathrm{1,2}\dots ,D$ is the same as defined previously. The distribution of the collected samples exhibited characteristics different from a normal distribution; therefore, the optimum was unknown. Therefore, the magnitude of the calculated residual was assumed to be the standard deviation, and $\mathbf{h}\left(j\right)$ = $\left|{{\mathit{ϵ}}_k}_j\right|$ was used [24].

The newly calculated probability distribution in Equation (13) is obtained using samples drawn from three different probability distributions. These samples originate from distributions with different physical meanings such as the current observation, the prior estimate, and historical residuals, each of which influences the measurement distribution in the subsequent step.

In other words, although the sample sets $\left\{\mathit{A}\right\}, \left\{\mathit{B}\right\}$ and $\left\{\mathit{\gamma }\right\}$ are not identically and independently distributed due to their different sources, their union in Equation (12) effectively represents a sample drawn from a mixture distribution. By drawing $\mathbf{m}\left(j\right)$ samples from the first two sources and $\mathbf{L}\left(j\right)$ samples from the residual-based source, we assign implicit weights to each component. Accordingly, the kernel density estimate constructed from this combined set Equation (13) converges to an approximate mixture distribution, expressed as a weighted sum of the component densities [25].

Sample ${\mathit{A}\left(j\right)}_i$ is drawn from a distribution that assumes the probability distribution of observed measurements. Samples are obtained from a distribution based on predictions using Equation (6) when no information is available. However, after the new probability distribution in Equation (13) has been calculated, samples are drawn from the updated distribution as

$${{\left\{{\mathit{A}\left(j\right)}_i\right\}}_k}_{\begin{array}{c}i=\mathrm{1,2}, \dots , \mathit{m}\left(j\right)\\ j=\mathrm{1,2}, \dots , D \end{array}}~\widehat{q}\left({\mathit{u}\left(j\right)}_{k-1}\right).$$

(14)

Using Equation (14), ${\mathit{A}\left(j\right)}_i$ is a sample drawn from the continuously updated simulated probability distribution $\widehat{q}\left({\mathit{u}\left(j\right)}_{k-1}\right)$. Samples drawn from the three different probability distributions have a physical significance in relation to the measurements.

${\mathit{A}\left(j\right)}_i$ reflects accumulated measurement errors as a sample drawn from the repeatedly updated probability distribution. Another sample, ${\mathit{B}\left(j\right)}_s$, signifies a sample drawn from the probability distribution of observed measurements, which reflects the characteristics of the observed measurements in the simulated probability distribution regardless of errors, based on the current moment. The sample set ${\mathit{\gamma }\left(j\right)}_o$ is drawn from the probability distribution $\mathsf{\Gamma }\left(\mathit{G}\right(j\left)\right)$, which is calculated using residuals collected up to the point of filter convergence. This residual-based distribution reflects the steady-state error characteristics of the measurements and remains fixed after convergence. In other words, $\mathsf{\Gamma }\left(\mathit{G}\right(j\left)\right)$ is determined once during an initial calibration phase when the filter’s covariance has stabilized and does not change over time during subsequent filtering.

Thus, the samples ${\mathit{\gamma }\left(j\right)}_o$ drawn at any later time step are independent and identically distributed from this time-invariant distribution, serving as a consistent baseline for normal measurement error. By design, combining these ${\mathit{\gamma }\left(j\right)}_o$ samples with those from the current observation and prior estimates does not introduce time-dependent inconsistencies. Instead, it incorporates stable statistical knowledge of past error behavior into the current measurement update.

3.3. Adjuting the Number of Samples for Reformed Measurement Calculations

In this study, samples were obtained from three different probability distributions, and the KDE algorithm was used to simulate the measurement distribution. The KDE calculates the distribution based on the number of samples and the kernel function, and therefore, the measurement distribution can vary depending on the number of samples drawn from each probability distribution. For example, more samples should be drawn from both $\widehat{q}\left({\mathit{u}\left(j\right)}_k\right)$, which includes previous normal states, and $\mathsf{\Gamma }\left(\mathit{G}\right(j\left)\right)$, in cases where there are errors in the measurements to update a new measurement probability distribution. Therefore, determining the number of samples from which a distribution based on the extent of measurement errors can be drawn is crucial to accurately simulate the probability distribution of the measurements. The ratio of the measurement errors was determined using system covariance and residuals. The square of the residual element in Equation (8) can be expressed as

$${\mathsf{\gamma }}_{\mathit{k}}\left(\mathit{j}\right)={\left({\mathit{ϵ}}_{\mathit{k}}\left(\mathit{j}\right)\right)}^2.$$

(15)

Furthermore, using the system covariance ${\mathbf{P}}_{\mathrm{k}}^{-}$ in (2) and the observation matrix ${\mathbf{H}}_{\mathit{k}}$ to convert to the same physical quantity as the observed measurements, it can be represented as ${\mathbf{H}}_{\mathit{k}}{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T\in {\mathbb{R}}^{D\times D}$. At this point, if the element index of the calculated matrix is denoted as inside the parenthesis ${\mathbf{H}}_{\mathit{k}}{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T(\alpha ,\beta )$, the set of diagonal elements can be defined as

$${\mathsf{\Phi }}_{\mathit{k}}\left(\mathit{j}\right)={\left[{\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^{\mathit{T}}(j,j)\right]}_{\mathit{j}=\mathrm{1,2},\dots ,\mathit{D}}^{\mathit{T}}\in {\mathbb{R}}^{\mathit{D}\times 1}.$$

(16)

Using the accuracy of the system covariance and square of the residual, the weight can be calculated as

$${\mathbf{W}\mathbf{P}}_k\left(j\right)={\displaystyle \frac{\mathit{\Phi }\left(j\right)}{\mathit{\Phi }\left(j\right)+\mathit{{\rm Y}}\left(j\right)}}.$$

(17)

The calculated weight ${\mathbf{W}\mathbf{P}}_k\left(j\right)$ ranges from 0 to 1, indicating the normality of the current measurement. It approaches 1 when the innovation ${ϵ}_k\left(j\right)$ is small and decreases toward 0 when the residual is large. This weight is converted into a sample count as follows:

$$\mathbf{m}{\left(j\right)}_k=⌊n·\left(1-{\mathbf{W}\mathbf{P}}_k\left(j\right)\right)⌋,$$

(18)

where $n$ is the base number of samples, and the floor function is used to convert the weight into an integer, denoted by $⌊·⌋$. Intuitively, $\mathbf{m}\left(j\right)$ becomes larger when the current measurement appears inconsistent because a large residual makes ${\mathbf{W}\mathbf{P}}_k\left(j\right)$ small. Conversely, $\mathbf{m}\left(j\right)$ is smaller when the measurement closely aligns with the prediction, meaning the residual is small and ${\mathbf{W}\mathbf{P}}_k\left(j\right)\approx 1$.

In an extreme case where the residual is an outlier and ${\mathbf{W}\mathbf{P}}_k\left(j\right)$ is very small, $\mathbf{m}\left(j\right)$ will be nearly equal to $n$. This implies that almost all of the $n$ samples used in the KDE are drawn from the prior and residual-based distributions, with very few coming directly from the current observation. This adaptive allocation reduces the influence of potentially faulty measurements by up-weighting alternative sources during anomalies, while placing more trust in the current observation when it is considered reliable.

Samples ${\left\{{\mathit{\gamma }\left(j\right)}_o\right\}}_k$ extracted from probability distribution $\mathsf{\Gamma }\left(\mathit{G}\right)$ are intended to reflect error characteristics in a normal state. Significant residuals suggest an anomaly, and in such cases, the freshly estimated reformed measurement probability distribution must be minimally swayed by the distribution $\mathsf{\Gamma }\left(\mathit{G}\right)$. Therefore, the sample size should exhibit a positive association with the residual magnitude. Given that addressing a small residual should be handled with greater sensitivity than addressing a large one, we devised an exponential weighting function as

$${\mathbf{W}\mathbf{I}}_k\left(j\right)=\mathrm{e}\mathrm{x}\mathrm{p}(-{\mathsf{\gamma }}_{\mathit{k}}\left(\mathit{j}\right)).$$

(19)

Using the weighting, the number of samples $\mathbf{L}\left(j\right)$ to be drawn from the distribution $\mathsf{\Gamma }\left(\mathit{G}\right)$, multiplied by the total number of samples, is

$$\mathbf{L}\left(j\right)=⌊n·{\mathbf{W}\mathbf{I}}_k\left(j\right)⌋.$$

(20)

The measurements can be stabilized more quickly when they return to their normal status by utilizing a fixed probability distribution. The reformed measurement distribution can be determined by utilizing the sample count and Equation (13). The reformed measurement and its variance are computed by integrating the probability distribution as [24]

$${\tilde{\mathit{y}}}_{\mathit{k}}\left(j\right)={\left[{\tilde{\mathit{y}}}_1{\tilde{\mathit{y}}}_2\dots {\tilde{\mathit{y}}}_{\mathit{D}}\right]}^{\mathit{T}}={\mathsf{\Sigma }}_i^{n+L}{\mathit{u}}_{\mathit{k}}\left(j\right)·\widehat{q}\left({\left({\mathit{u}\left(j\right)}_k\right)}^i\right),$$

(21)

$${\tilde{\mathbf{R}}}_{\mathit{j},\mathit{j}}={\mathsf{\Sigma }}_i^{n+L}{\left({\mathit{u}}_{\mathit{k}}\left(j\right)\right)}^2·\widehat{q}\left({\left({\mathit{u}\left(j\right)}_k\right)}^i\right)-{\tilde{\mathit{y}}}_{\mathit{k}}\left(j\right).$$

(22)

Utilizing the reformed measurement instead of the observed measurement can effectively reduce the estimation errors over a specific timeframe, which includes nonlinear measurement errors. This was accomplished by incorporating reformed measurements in which errors were mitigated by applying a newly estimated measurement probability distribution. The calculation of the reformed measurements is presented in Algorithm 1.

Algorithm 1: Making the Reformed measurement
Input: Predicted state ${\mathbf{x}}_k^{-}$ Observed measurement ${\mathbf{y}}_k$; Observation matrix ${\mathbf{H}}_k$; Dimension $D$; Measurement covariance matrix $\mathbf{R}$; Samples $n$; Initial time $t$; Previous simulated distribution $\widehat{q}\left({\mathit{u}\left(j\right)}_{k-1}\right)$; Predicted covariance ${\mathbf{P}}_k^{-}$;
Output: Reformed measurement ${\tilde{\mathbf{y}}}_k$; Reformed covariance ${\tilde{\mathbf{R}}}_{j,j}$; Simulated distribution $\widehat{q}\left({\mathit{u}\left(j\right)}_k\right)$;
1:	Initialization; k = 1
2:	$\mathbf{i}\mathbf{f}$ $k\le t$
3:		$\mathbf{f}\mathbf{o}\mathbf{r}$ $j=1 to D$
4:			${\mathit{ϵ}}_k\left(j\right)=\left({\mathbf{y}}_k\left(j\right)-{\mathbf{H}}_k{\widehat{\mathit{x}}}_k^{-}\left(j\right)\right)$; $\mathbf{G}\left(j\right)=\{{\mathit{ϵ}}_1\left(j\right),\dots ,{\mathit{ϵ}}_k(j\left)\right\}$;
5:			${\tilde{\mathbf{y}}}_k={\mathbf{y}}_{\mathrm{k}}$; ${\tilde{\mathbf{R}}}_{j,j}=\mathbf{R}$;
6:			$\widehat{q}\left({\mathit{u}\left(j\right)}_k\right)$~$N\left({{\mathbf{H}}_k\widehat{\mathbf{x}}}_k^{-}\left(j\right);\mathbf{R}(j,j)\right)$
7:			$\mathbf{i}\mathbf{f}$ $k=t$
8:				$\mathsf{\Gamma }(\mathit{G}(j))={\displaystyle \frac{1}{t·\mathit{h}(j)·\sqrt{2\pi }}}{\mathsf{\Sigma }}_{i=1}^t\mathit{exp}\left(-{\displaystyle \frac{{\mathit{ϵ}}_i\left(j\right)}{2·\mathit{h}{\left(j\right)}^2}}\right)$
9:			$\mathbf{e}\mathbf{n}\mathbf{d} \mathbf{i}\mathbf{f}$
10:		$\mathbf{e}\mathbf{n}\mathbf{d} \mathbf{f}\mathbf{o}\mathbf{r}$
11:	$\mathbf{e}\mathbf{l}\mathbf{s}\mathbf{e}$
12:		$\mathbf{f}\mathbf{o}\mathbf{r}$ $j=1 to D$
13:			${\mathit{ϵ}}_k\left(j\right)=\left({\mathbf{y}}_k\left(j\right)-{\mathbf{H}}_k{\widehat{\mathit{x}}}_k^{-}\left(j\right)\right)$; ${\mathsf{{\rm Y}}}_k\left(j\right)={\mathit{ϵ}}_k{\left(j\right)}^2$
14:			${\mathbf{W}\mathbf{P}}_k(j)= {\displaystyle \frac{{\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T(j,j)}{{\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T(j,j)+{\mathit{ϵ}}_k\left(j\right)}}$; ${\mathit{m}}_k\left(j\right)=⌊n·\left(1-{\mathbf{W}\mathbf{P}}_k\left(j\right)\right)⌋$
15:			${\mathbf{W}\mathbf{I}}_k\left(j\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{{\rm Y}}}_k\left(j\right)\right)$; $\mathbf{L}\left(j\right)=⌊n·\mathbf{W}{\mathbf{I}}_k\left(j\right)⌋$
16:			$\mathbf{f}\mathbf{o}\mathbf{r}$ $i=1 to {\mathit{m}}_k\left(j\right)$; ${\left\{{\mathit{A}\left(j\right)}_i\right\}}_k ~ \widehat{q}\left({\mathit{u}\left(j\right)}_{k-1}\right)$ $\mathbf{f}\mathbf{o}\mathbf{r} \mathbf{e}\mathbf{n}\mathbf{d}$
17:			$\mathbf{f}\mathbf{o}\mathbf{r}$ $s=1 to( n-{\mathit{m}}_k\left(j\right))$; ${\left\{{\mathit{B}\left(j\right)}_s\right\}}_k ~ N\left({\mathbf{y}}_k\left(j\right),\mathbf{R}(j,j)\right) \mathbf{f}\mathbf{o}\mathbf{r} \mathbf{e}\mathbf{n}\mathbf{d}$
18:			$\mathbf{f}\mathbf{o}\mathbf{r}$ $o=1 to \mathbf{L}\left(j\right)$; ${\left\{{\mathit{\gamma }\left(j\right)}_o\right\}}_k ~ \mathsf{\Gamma }\left(\mathit{G}\left(j\right)\right)$ $\mathbf{f}\mathbf{o}\mathbf{r} \mathbf{e}\mathbf{n}\mathbf{d}$
19:			${\mathit{u}}_k\left(j\right)=\left\{{\mathit{A}\left(j\right)}_k,{\mathit{B}\left(j\right)}_k, {\mathit{\gamma }\left(j\right)}_k\right\}$
20:			$\widehat{q}\left({\mathit{u}\left(j\right)}_k\right)={\displaystyle \frac{1}{\sqrt{2\pi }\mathit{h}\left(j\right)(n+\mathbf{L}(j\left)\right)}}{\mathsf{\Sigma }}_{\varpi =1}^{n+\mathbf{L}\left(j\right)}\exp \left(-{\displaystyle \frac{{\left({\mathit{U}}_k\left(j\right)-{\left({\mathit{u}}_k\left(j\right)\right)}^{\varpi }\right)}^2}{2{\left(\mathit{h}\left(j\right)\right)}^2}}\right)$
21:			${\tilde{\mathbf{y}}}_k\left(j\right)={\mathsf{\Sigma }}_i^{n+L}{\mathit{u}}_k\left(j\right)·\widehat{q}\left({\left({\mathit{u}\left(j\right)}_k\right)}^i\right)$
22:			${\tilde{\mathbf{R}}}_{j,j}={\mathsf{\Sigma }}_i^{n+L}\left({\left({\mathit{u}}_k\left(j\right)\right)}^2·\widehat{q}\left({\left({\mathit{u}\left(j\right)}_k\right)}^i\right)-{\tilde{\mathbf{y}}}_k\left(j\right)\right)$
23:		$\mathbf{e}\mathbf{n}\mathbf{d} \mathbf{f}\mathbf{o}\mathbf{r}$
24:	$\mathbf{e}\mathbf{n}\mathbf{d} \mathbf{i}\mathbf{f}$
25:

4. Simulation and Analysis

4.1. Experimental Setup and Validation

The proposed method employs reformed measurements to account for and correct measurement errors, enhancing the estimation accuracy of the filter. Access to true measurement values is essential for a quantitative assessment. This study presupposes the availability of these true values via simulation data that models drone dynamics [26]. Drone navigation was estimated by integrating data from an inertial measurement unit (IMU) and GNSS. Validation includes analyzing the propagation process of the probability distribution in the observed measurements with bias error, thereby assessing the position errors of the reformed measurements. The state-estimation results calculated using an ADKF were compared with those obtained using the proposed algorithm.

Drone flights span 660 s, which encompass take-off, maneuvers for initial filter convergence, straight flight, and landing. Figure 2. shows the complete operational trajectory of the drone, thereby highlighting sections affected by errors.

Section 1 and 2 allowed errors to persist for 3 s and 5 s, respectively. The bias magnitude fluctuates between 15 and 40 m, and 6$\mathsf{\sigma }$ is selected for the GNSS error. All error scenarios were validated using Monte Carlo simulations. In addition, heavy-tailed noise models were incorporated to further broaden the error spectrum and approximate non-Gaussian disturbances frequently reported in practice [27,28]. This combination enables the simulations to capture both transient and sustained error cases, thereby strengthening the practical relevance of the evaluation framework [29,30]. Specifications of the IMU and GNSS are listed in

Table 1. Sensor specification for the simulation.

Inertial Measurement Units (IMU) [100 Hz]
	Repeatability	Noise
Accelerometer	$60 \mathrm{mg} \mathrm{r}\mathrm{m}\mathrm{s}$	$0.8 \mathrm{m}\mathrm{g}/\sqrt{\mathrm{H}\mathrm{z}}$
Gyroscope	$10°/\mathrm{s}$	$0.04°/\mathrm{s}/\sqrt{\mathrm{H}\mathrm{z}}$
Global Navigation Satellite System (GNSS) [1 Hz]
	Position error
Horizontal	$3 \mathrm{m} (1\sigma$)
Vertical	$5.5 \mathrm{m} (1\mathsf{\sigma }$)

.

An EKF was used for the state variable estimation, and the system error model of the EKF and inertial navigation algorithms utilized a well-known strap-down navigation model [31]. The EKF was utilized for state estimation, where the states are defined as position error $\mathsf{\delta }\mathbf{p}$, velocity error $\mathsf{\delta }\mathbf{v}$, attitude error $\mathsf{\delta }\mathsf{\varphi }$, accelerometer bias ${\mathbf{b}}_{\mathit{a}}$, and gyroscope bias ${\mathbf{b}}_{\mathit{g}}$, which totals 15 states. The GNSS/INS navigation was performed in the WGS84 coordinate system, and the system block diagram used for validation is shown in Figure 3.

In Figure 3, the blocks representing the EKF and the INS illustrate a typical architecture commonly employed in conventional drone navigation systems [23]. The proposed algorithm operates within the reformed measurement module, where its processing flow can be observed. In particular, the inputs to the reformed measurement module consist of GNSS and sensor specifications, or known distributions and initialization flags verified through experimentation. Therefore, a notable advantage is that the proposed method can be implemented while maintaining the existing GNSS/INS navigation system used in conventional drones. This characteristic suggests that the proposed technique can be applied in a modular form without altering the structure of the EKF framework, which in turn facilitates the implementation and testing of the algorithm.

4.2. Analysis of the Reformed Measurement Quality

We analyzed the distribution propagation process of the reformed measurements under error conditions. This process is illustrated in Figure 4. The figure presents the propagation behavior specifically along the latitude axis, which was selected for clarity. To enhance interpretability, all relevant quantities are converted into meters. Variable $k$ represents the time index, with $k$ = 0 indicating normal conditions. At $k$ = 1, an anomaly is observed in the measurements. The proposed algorithm updates the probability distribution to form one that mirrors the characteristics of the normal state even in the presence of errors. Therefore, new error-free measurements could be computed using the reconstructed measurement distribution.

A scenario at $k=5$, errors persisted in the observed measurements for five consecutive time points. The algorithm cannot depend on the continuously estimated probability distribution, and it is influenced by the observed measurements, which leading to a broader distribution that includes some errors. However, only a small portion of the distribution was affected by these errors, which suggests that it remains relatively unaffected by measurement anomalies. A slight bias in the opposite direction was detected, likely due to minor error effects on the estimates when the measurements returned to a normal state. However, subsequent measurements quickly corrected these biased errors, and normal operations resumed by $k=7$ and $k=8$. Although errors persisted across five instances, the system rapidly returned to a normal state within only two subsequent instances.

4.3. Analysis of the Navigation Results

The accuracy of the reformed measurements was evaluated through 500 Monte Carlo simulations. In these simulations, random biases ranging from 15 to 40 m were applied to Sections 1 and 2, where errors were introduced. An example case among the 500 simulations is illustrated in Figure 5.

In Section 1, errors were introduced for 3 s, and when the bias occurred, errors in the observed measurements became evident. However, the reformed measurement maintained a low error level even when errors occurred. Similarly, in Section 2, the error remained low even though errors were introduced for 5 s. Notably, after errors occurred, the proposed technique quickly returned to a normal state. The position error of the reformed measurement was RMSE 2.1243 m in Section 1 and 3.1810 m in Section 2.

In this study we compared the proposed algorithm against a normal KF and an ADKF that adjusts the measurement noise covariance in order to evaluate performance under non-Gaussian error conditions. We also incorporated two representative robust filtering approaches into our analysis, the Gaussian-mixture-based Kalman filter (GMM-KF) and the Huber-based Kalman filter (Huber-KF). The GMM-KF explicitly models measurement bias with a finite Gaussian mixture and can achieve good accuracy when the assumed mixture structure matches the true error distribution [32]. However, its performance is highly sensitive to model mismatch and parameter tuning, which limits its generalizability. The Huber-KF applies M-estimation to standardized innovations with a threshold parameter $c$, behaving like least-squares for small innovations while down-weighting large outliers [33]. This provides robustness to sporadic large errors without requiring an explicit non-Gaussian noise model.

Compared with existing robust filtering techniques, the proposed method exhibits clear methodological distinctions. The GMM-KF explicitly models measurement errors with a finite Gaussian mixture, which can provide good accuracy when the assumed mixture matches the true error distribution but is highly sensitive to model mismatch and parameter selection. The Huber-KF employs M-estimation to down-weight large innovations, offering robustness against sporadic outliers but showing limited effectiveness under persistent biases. In contrast, the proposed approach reconstructs the measurement distribution directly from sampled data without imposing a predefined noise model. This non-parametric, sampling-based strategy allows the filter to adapt to both transient and sustained non-Gaussian conditions, thereby improving its generalization capability in practical GNSS/INS applications.

The ADKF was compared when window sizes were set to 5, 10, and 20. The GMM-KF was implemented using a fixed constant-bias model, without applying the expectation-maximization algorithm, which is commonly used to learn Gaussian mixture parameters adaptively. The EM procedure requires a large number of samples and introduces additional complexity related to sample selection, which lies beyond the scope of this work. Thus, a simplified GMM-KF was adopted to assess its robustness under controlled modeling assumptions. The Huber-KF was implemented with $c$ = 1.345 a maximum of five iterations, and a convergence tolerance of ${10}^{-6}$ [34].

During this comparison, 500 Monte Carlo simulations were performed with measurement errors introduced under the same conditions using randomly assigned biases of 15 to 40 m.

Table 2. Position RMSE and Peak error with 15–40 m Non-Gaussian error.

Area Method	Total RMSE	Section 1 RMS/Peak	Section 2 RMS/Peak
Normal KF	7.4516	2.3042/10.4461	3.9481/22.5950
ADKF (5)	5.4499	0.5030/3.6556	1.1025/13.0698
ADKF (10)	5.3501	0.6438/4.2298	1.2980/7.6426
ADKF (20)	6.0195	0.9180/5.3690	1.8594/10.0596
GMM-KF	7.3581	0.8470/10.4531	1.4270/23.2291
Huber-KF	5.6856	0.5258/3.1335	0.8309/9.0361
Proposed	4.7629	0.3391/2.0432	0.5634/4.5882

summarizes the RMSE and peak error results for all filters. The proposed method achieved the lowest total RMSE and consistently outperformed the baseline and robust filters in both bias scenarios. In particular, it shows substantially reduced peak errors compared with the normal Kalman filter, indicating superior suppression of large error under non-Gaussian disturbances.

For ADKF, performance varied with the choice of window size, and the case with 10 samples provided the most favorable trade-off between stability and accuracy. The GMM-KF performed well only when the assumed 30 m bias matched the actual error, but its accuracy degraded substantially under mismatched bias settings, yielding results similar to ADKF with window sizes between 10 and 20. The Huber-KF demonstrated strong robustness, effectively suppressing outliers caused by heavy-tailed distributions and yielding superior performance in sections with randomly occurring measurement anomalies. However, its performance depends on the choice of the threshold $c$, which requires careful tuning.

The time-series plots in Figure 6 provide additional insight into the error dynamics. In both 3 s and 5 s bias scenarios, the proposed method effectively suppresses sudden error spikes and exhibits faster recovery after the bias interval. In contrast, the KF and other robust filters exhibit larger peaks and slower convergence. These results clearly demonstrate that the proposed method can maintain stable performance even in the presence of transient and persistent non-Gaussian disturbances.

In addition, bias errors were increased from 15 to 70 m in 5 m increments to evaluate stability with respect to bias size, and 100 Monte Carlo simulations were conducted for each increment. The results for each increment are shown in Figure 7.

Figure 7 demonstrates that the proposed method, which employs reformed measurements, achieves significantly more stable and accurate position estimation than the other filters under varying levels of measurement bias. The ADKF adapts measurement and process noise covariance using innovation statistics, but its performance depends heavily on the selected window size. ADKF with a 5-sample window occasionally yields low error yet shows unstable fluctuations, while the 20-sample window version produces smoother curves but suffers from increased RMSE due to inflated covariance. The 10-sample window variant offers the best compromise between stability and responsiveness among the ADKF settings.

The GMM-KF, which explicitly models measurement bias through a finite Gaussian mixture, performs competitively only when the actual bias magnitude is close to the modeled mixture mean.

Outside such matched conditions, its accuracy quickly deteriorates and becomes similar to that of the conventional KF, reflecting its sensitivity to model mismatch and parameter tuning. The Huber-KF, based on M-estimation with a threshold parameter $c$, consistently reduces the impact of large outliers and maintains lower errors than the standard KF. However, while it effectively suppresses sporadic spikes, its performance under persistent bias remains inferior to the proposed method. In contrast, the proposed method maintains fixed measurement covariance and redefines the measurement itself through distribution-based reformulation. This enables consistent performance independent of the bias magnitude, providing robustness across both transient and persistent error conditions, as observed in Sections 1 and 2.

Quantitatively, the proposed method achieved a 49.88% reduction in RMSE compared to the best-performing ADKF with window size 10 in Section 1, and a 90.64% reduction in Section 2. These results clearly support the theoretical motivation that reformed measurements can suppress non-Gaussian biases more effectively than covariance-based adaptation alone.

Figure 8 shows the bias estimation performance of the accelerometer and gyroscope under measurement error conditions. The proposed method consistently demonstrates the most stable and accurate estimation across both error sections. For accelerometer bias, the proposed method showed only a minor drift of 0.0001783 between the onset of the error and the final state, whereas the ADKF with 10-sample window exhibited a bias change of 0.011, representing a 98.38% improvement. Similarly, for gyroscope bias estimation, the proposed method reduced error by more than 98.40% relative to all baseline methods. In contrast, ADKF configurations displayed increasing estimation errors and even divergence in bias estimating under persistent error conditions.

For the GMM-KF, the method performed well when the error bias matched the modeled distribution, effectively compensating for such errors and achieving better results than the normal KF. However, under randomly assigned biases that deviated from the modeled assumptions, its performance was comparable to, or slightly worse than, that of the ADKF.

The Huber-KF, which is designed to assign reduced weights to outlier measurements, achieved bias estimation results comparable to those of the proposed method. This behavior arises because the filter effectively ignores the outlier measurements. As a result, although the bias estimation may appear accurate, the overall position estimation performance degrades when the measurements contain non-Gaussian errors.

Overall, these results confirm that the reformed measurement technique enables highly stable and accurate bias estimation even under non-Gaussian and persistent error conditions, ensuring reliable inertial navigation performance during both error periods and recovery.

5. Conclusions

We proposed a new technique that recalculates the observed probability distribution of measurements in the presence of nonlinear errors for computing the reformed measurements, thereby enhancing the estimation performance of the filter. Estimating the distribution of observed measurements directly implies the capacity to evaluate the quality of the observations.

Furthermore, the proposed sampling method allows for the revised probability distribution to incorporate fewer errors originating from measurements. This, consequently, enables the utilization of more precise measurements within a specified time frame. The proposed approach, as confirmed through simulation, has the potential to assist unmanned vehicles in navigating urban environments. This capability enables them to effectively traverse areas marked by frequent instances of multi-path scenarios. Positioning, while further exploration of the algorithm’s potential applications in diverse contexts is left for future work. This approach is thought to effectively reduce the non-Gaussian errors that could arise in frequency-based sensors like RADAR and UWB. In future research, we aim to extend this work by incorporating measurement quality monitoring with the reformed distribution and by validating the method using publicly available GNSS/INS datasets and application scenarios such as inspection drones operating near structures.