Robust Cubature Kalman Filter Research Articles

Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements.

Handling the challenge of missing measurements in nonlinear systems is a difficult problem in various scientific and engineering fields. Missing measurements, which can arise from technical faults during observation, diffusion channel shrinking, or the loss of specific metrics, can bring many challenges when estimating the state of nonlinear systems. To tackle this issue, this paper proposes a technique that utilizes a robust cubature Kalman filter (RCKF) by integrating Huber's M-estimation theory with the standard conventional cubature Kalman filter (CKF). Although a CKF is often used for solving nonlinear filtering problems, its effectiveness might be limited due to a lack of knowledge regarding the nonlinear model of the state and noise-related statistical information. In contrast, the RCKF demonstrates an ability to mitigate performance degradation and discretization issues related to track curves by leveraging covariance matrix predictions for state estimation and output control amidst dynamic disruption errors-even when noise statistics deviate from prior assumptions. The performance of extended Kalman filters (EKFs), unscented Kalman filters (UKFs), CKFs, and RCKFs was compared and evaluated using two numerical examples involving the Univariate Non-stationary Growth Model (UNGM) and bearing-only tracking (BOT). The numerical experiments demonstrated that the RCKF outperformed the EKF, EnKF, and CKF in effectively handling anomaly errors. Specifically, in the UNGM example, the RCKF achieved a significantly lower ARMSE (4.83) and ANCI (3.27)-similar outcomes were observed in the BOT example.

Enhancing the reliability of shipborne INS/GNSS integrated navigation system during abnormal sampling periods using Bi-LSTM and robust CKF

Developing a reliable navigation system is a crucial challenge for intelligent ships. Integrating Inertial Navigation Systems (INS) with shipborne Global Navigation Satellite Systems (GNSS) provides a viable solution. However, the system's reliability heavily relies on GNSS availability. This paper proposes a method to enhance the reliability of shipborne INS/GNSS integrated navigation systems during GNSS abnormal sampling periods. The proposed method consists of two parts. Firstly, using specially designed input and output parameters, a pseudo-GNSS prediction model based on a Bidirectional Long Short-Term Memory Recurrent Neural Network (Bi-LSTM) is constructed to generate predicted pseudo-GNSS measurements during periods of GNSS loss, effectively filling in the gaps left by the absence of actual GNSS data. Secondly, Huber's M-estimation-based Robust Cubature Kalman filter (RCKF) is integrated into the data fusion algorithm, and a switching mechanism between RCKF and the Cubature Kalman filter (CKF) is established based on the source of GNSS data utilized for measurement update, to mitigate the impact of non-Gaussian distribution prediction noise introduced by pseudo-GNSS prediction model. Experimental validation using real ship navigation data in a natural marine environment confirms the proposed method's capability to maintain navigation accuracy during GNSS loss for shipborne INS/GNSS integrated navigation systems. The algorithm's superiority becomes more evident as the duration of GNSS loss increases, underscoring its effectiveness in enhancing shipborne integrated navigation system reliability.

A Novel Adaptive Robust Cubature Kalman Filter for Maneuvering Target Tracking with Model Uncertainty and Abnormal Measurement Noises.

The features of measurement and process noise are directly related to the optimal performance of the cubature Kalman filter. The maneuvering target model's high level of uncertainty and non-Gaussian mean noise are typical issues that the radar tracking system must deal with, making it impossible to obtain the appropriate estimation. How to strike a compromise between high robustness and estimation accuracy while designing filters has always been challenging. The H-infinity filter is a widely used robust algorithm. Based on the H-infinity cubature Kalman filter (HCKF), a novel adaptive robust cubature Kalman filter (ARCKF) is suggested in this paper. There are two adaptable components in the algorithm. First, an adaptive fading factor addresses the model uncertainty issue brought on by the target's maneuvering turn. Second, an improved Sage-Husa estimation based on the Mahalanobis distance (MD) is suggested to estimate the measurement noise covariance matrix adaptively. The new approach significantly increases the robustness and estimation precision of the HCKF. According to the simulation results, the suggested algorithm is more effective than the conventional HCKF at handling system model errors and abnormal observations.

An Adaptive Robust Cubature Kalman Filter Based on Sage-Husa Estimator for Improving Ship Heave Measurement Accuracy

High-pass and Kalman filtering are two digital filtering methods for ship heave acceleration signals. However, the traditional Kalman method has dynamic model errors due to the dependence on the Fast Fourier Transform (FFT) identification model. The presence of factors such as unknown statistical properties of the noise and anomalous measurement information at the same time can cause degradation in the accuracy of the heave solution or even lead to Kalman filter divergence. This study proposes an adaptive robust cubature Kalman filter based on the Sage-Husa noise estimator (SARCKF) for ship heave motion estimation to obtain higher heave measurement accuracy. Unlike the traditional method, which adjusts the process covariance matrix by prior knowledge, the new way estimates the statistical characteristics of noise online by Sage estimator, improving the real-time filtering performance. The Huber robust method is used to correct the measurement covariance matrix to suppress the outliers in the measurement information. Then, based on the strong tracking theory, an adaptive factor is derived and combined with the Huber method to form a new cost function, effectively compensating for the model error. The simulation results show that SARCKF can still make the estimation accuracy of heave displacement reach 5% of the maximum amplitude when the statistical characteristics of noise are unknown and the model has errors. In addition, the proposed method has stronger robustness than the traditional strong tracking filter.

A robust Cubature Kalman filter for nonlinear systems subject to randomly occurring measurement anomalies without a priori statistic

In this work, we investigate the problem of state estimation for a class of nonlinear systems subjected to randomly occurring measurement anomalies (ROMAs) without a priori statistic. To address the problem, first, a novel measurement model is constructed, in which the anomalous measurements and anomaly probability are modeled as Gaussian mixture distribution (GMD) and Beta distribution, respectively. Different from the existing researches assuming that the statistical information of anomalous measurements is known in advance, the model does not require a priori statistical knowledge of anomalous measurements. Moreover, by adaptive learning of the anomaly probability, the measurement model is identical with the classical cubature Kalman filter (CKF) in the absence of measurement anomalies. Then, the variational Bayesian inference (VBI) is employed to approximately calculate the joint posterior distribution of the system state and unknown parameters, and a robust filter is derived. Finally, the effectiveness of our filter is demonstrated by the numerical simulation.

A Gaussian-multivariate Laplacian mixture distribution based robust cubature Kalman filter

In this paper, the state estimation problems of nonlinear systems with outlier-corrupted measurements are investigated. First, to model the non-Gaussian noises caused by the randomly occurring measurement outliers, we propose a new Gaussian-multivariate Laplacian mixture (GMLM) distribution and analyze its distribution characteristics. Then, the measurement likelihood probability density function (PDF) is formulated as the GMLM distribution and further expressed as a hierarchical Gaussian expression. Furthermore, by variational Bayesian (VB) method, a robust cubature Kalman filter is derived (GMLMRCKF). Finally, simulation test and real date are utilized to evaluate the effectiveness of our GMLMRCKF, the results illustrate that the proposed filter has better estimation accuracy and consistence in the case of non-stationary heavy-tailed noises than the existing robust filters, i.e., it has almost the same performance as the standard CKF in the absence of outliers and better robust performance in the presence of unknown outliers.

Environmentally Adaptive Event-Driven Robust Cubature Kalman Filter for RSS-Based Targets Tracking in Mobile Wireless Sensor Network

This article presents an environmentally adaptive event-driven robust square root cubature Kalman filter (EAERCKF) for received-signal-strength-based (RSS-based) moving targets tracking in resource-constrained mobile wireless sensor network (MWSN) under the motion modeling and RSS quantization uncertainties. First, an environmentally adaptive event-driven mechanism, scheduling dynamically and approximately the desired amount of anchors regardless of the anchors distribution density near the moving target, is designed to reduce energy consumption, communication rates, and computation complexity of the MWSN system. Second, a robust square root cubature Kalman filter is developed to estimate the motion state of the moving target, where the covariances of the process noise and the measurement noise assumed to be time-varying uniform distributions are generated randomly. Third, the bounds of the uniformly distributed covariances describing the motion modeling and RSS quantization uncertainties are given to improve the stability, accuracy, and consistency of the estimator. Finally, the results of numerical simulations demonstrate that our proposed EAERCKF estimation method has superiority in simplifying the determination of the random noise covariances with satisfactory target tracking performance and approximating the desired number of triggered anchors.

A Stochastic Event-Triggered Robust Cubature Kalman Filtering Approach to Power System Dynamic State Estimation With Non-Gaussian Measurement Noises

In power system communication and control, the wide-area measurement system (WAMS) is usually adversely affected by noisy measurements and data congestion, posing great challenges to the stability and functionality of modern power grids. This study proposes a stochastic event-triggered robust dynamic state estimation (DSE) method for non-Gaussian measurement noises, using the cubature Kalman filter (CKF) technique. To reduce the computational burden and data transmission congestion resulting from centrally processing the measurement data, the proposed event-triggered robust CKF (ET-RCKF) is deployed at a local level with appropriate system formulation. The proposition of the novel robust DSE strategy is detailed in this brief, with its stability mathematically analyzed and proven, and simulation study on the IEEE 39-bus benchmark test system verifies the effectiveness of the proposed ET-RCKF approach. This novel DSE method is able to cope with non-Gaussian measurement noises and produce highly satisfactory estimation results, leading to wide applicability in real-world power system applications.

Robust Cubature Kalman Filter With Gaussian-Multivariate Laplacian Mixture Distribution and Partial Variational Bayesian Method

This paper explores the problem of nonlinear state estimation in the presence of outlier-contaminated measurements. First, to deal with the non-stationary non-Gaussian noises caused by randomly occurring measurement outliers, we propose a new Gaussian-multivariate Laplacian mixture (GMLM) distribution and construct it as a hierarchical Gaussian expression. Next, utilizing the GMLM distribution and existing variational Bayesian (VB) method, a robust cubature Kalman filter is derived (VB-GMLMRCKF). Then, considering the high computational complexity of the existing VB inference process, a new partial VB (PVB) method is developed, which can separately estimate state vector and mismatched measurement noise covariance matrix. Building upon the VB-GMLMRCKF and PVB approach, a novel robust cubature Kalman filter is derived (PVB-GMLMRCKF). Finally, a target tracking model is utilized to evaluate the PVB-GMLMRCKF in terms of estimation accuracy, estimation consistency and computational efficiency.

A Computationally Efficient Robust Cubature Kalman Filter With Multivariate Laplace Distribution

This paper investigates the nonlinear state estimation under heavy-tailed process and measurement noises (PAMNs), the noises may be induced by sensor failures, measurement loss, modeling errors, environmental changes, or malicious cyberattacks. Considering that the multivariate Laplacian (ML) distribution has obvious heavy-tailed characteristic, we employ the distribution to describe heavy-tailed PAMNs. Furthermore, to improve the computational efficiency of the existing variational Bayesian (VB) iteration process, we design an improved VB iteration method, which can separately calculate the posterior distributions of the state vector and unknown noise parameters. Employing the ML distribution and improved VB inference process, a computationally efficient robust cubature Kalman filter (CEMLRCKF) is derived. Simulation and vehicle experiment results illustrate the superiority of the proposed filter.

Minimum Mixture Error Entropy-Based Robust Cubature Kalman Filter for Outlier-Contaminated Measurements

This letter investigates the robust state estimation of the nonlinear systems with outlier-contaminated measurements. Due to the advantage of mixture error entropy with two kernel bandwidths in handling non-Gaussian noise caused by the outliers, a novel minimum mixture error entropy (MMEE) criterion-based robust cubature Kalman filter is proposed, in which the cost function is constructed by MMEE criterion, and the nonlinear measurement model is linearized by the statistical linear regression method. By a benchmark target tracking scenario with non-Gaussian measurement noise and INS/GNSS loose combination vehicle tracking experiment, the effectiveness of the proposed filter is demonstrated.

Research on Mono-Pulse Beam Angle Tracking Algorithm of Phased Array Antenna Based on CKF

Phased array antenna beam tracking and communication system based on mono-pulse system is an alternative satellite-to-earth communication technology solution for future planetary spacecraft, which meets the mission requirements of agile scanning, multimode, wide field of view, and small size. This paper firstly outlines the technical scheme of mono-pulse angle tracking based on the planar phased array antenna. A simplified mathematical model of angle measurement is summarized. Secondly, the beam pointing of the spacecraft communication antenna is frequently affected by the attitude and presents a nonlinear dynamic change. A robust two-dimensional angular tracking filtering algorithm based on cubature Kalman filter (CKF) is proposed, in order to solve this complex high-order nonlinear dynamic estimation problem. Finally, the comprehensive performance of the proposed algorithm is simulated and analyzed under two conditions of stable flight and high dynamic flight of the spacecraft. When the angular velocity is within 4°/s, the tracking accuracy can be better than 0.65°, and the tracking accuracy will be better than 1° when the angular velocity is within 8°/s. Compared with the basic CKF algorithm, the robust CKF tracking accuracy is significantly improved, while the resource consumption is slightly reduced. The research results can provide reference for the engineering implementation of related fields in the future.

Improved Robust High-Degree Cubature Kalman Filter Based on Novel Cubature Formula and Maximum Correntropy Criterion with Application to Surface Target Tracking

Robust nonlinear filtering is an important method for tracking maneuvering targets in non-Gaussian noise environments. Although there are many robust filters for nonlinear systems, few of them have ideal performance for mixed Gaussian noise and non-Gaussian noise (such as scattering noise) in practical applications. Therefore, a novel cubature formula and maximum correntropy criterion (MCC)-based robust cubature Kalman filter is proposed. First, the fully symmetric cubature criterion and high-order divided difference are used to construct a new fifth-degree cubature formula using fewer symmetric cubature points. Then, a new cost function is obtained by combining the weighted least-squares method and the MCC loss criterion to deal with the abnormal values of non-Gaussian noise, which enhances the robustness; and statistical linearization methods are used to calculate the approximate result of the measurement process. Thus, the final fifth-degree divided difference–maximum correntropy cubature Kalman filter (DD-MCCKF) framework is constructed. A typical surface-maneuvering target-tracking simulation example is used to verify the tracking accuracy and robustness of the proposed filter. Experimental results indicate that the proposed filter has a higher tracking accuracy and better numerical stability than other common nonlinear filters in non-Gaussian noise environments with fewer cubature points used.

Polarized light compass-aided inertial navigation under discontinuous observations environment.

In recent years, integrated polarized light/inertial heading measurement systems have been widely used to obtain autonomous heading measurements of small unmanned combat platforms in the case of satellite navigation rejection. However, existing polarized light/inertial heading measurement systems have certain limitations. For example, they can only measure the heading angle in environments where continuous observations can be obtained. When encountering a complex environment with trees and/or tall buildings, the measured heading angle will contain sharp noise which greatly affects its accuracy. In particular, when encountering an underpass, it will lead to the complete loss of lock of the polarized light compass signal. Therefore, for the problem of sharp noise arising from a complex environment, a robust Cubature Kalman filter (CKF) data-fusion algorithm is proposed and verified by experiments. It is proved that the robust CKF algorithm has a certain ability to filter out the effects of poor measurements. After application of the robust CKF algorithm, the Root Mean Square Error (RMSE) of the heading angle reaches 0.3612°. This method solves the problem of low precision and poor stability of the polarized light/inertial system when high buildings and/or trees are contained in a complex environment. Next, in view of the problem that the polarized light compass signal is completely lost due to passing through an underground passage, a random forest regression (RFR) neural network model is established and introduced into the combined system. Simulated and outdoor experiments are carried out to verify the designed model using data obtained with a vehicle. The RMSE of the heading angle obtained in the experiment is 1.1894°, which solves the problem that the polarized light/inertial system cannot utilize discontinuous observations when attempting to detect the carrier heading angle.

Minimum Cauchy Kernel Loss Based Robust Cubature Kalman Filter and Its Low Complexity Cost Version With Application on INS/OD Integrated Navigation System

This article focuses on addressing the non-linear state estimations problem of measurement noise with non-Gaussian distribution, which is critical for the performance of the inertial navigation system (INS)/odometer (OD) integrated navigation system. Although many existing robust filters have been proposed to address the non-Gaussian issue, their performance is not quite ideal for non-Gaussian noise in real applications, such as impulsive and noises from multimodal distributions. Meanwhile, the computational complexity burden impedes the application of robust non-linear algorithms in INS/OD integrated navigation systems. This paper proposes a minimal Cauchy kernel Cubature Kalman filter (MCK-CKF) to address the non-Gaussian problem in general non-linear systems. In addition, a simplified MCK-CKF (SMCK-CKF) is proposed to reduce the computational cost in INS/OD integrated navigation system, meanwhile ensuring outstanding performance similar to MCK-CKF. First, we derive the MCK-CKF based on the new cost function, which is obtained by a combination of weighted least square (WLS) and MCK. Then, for the INS/OD integrated navigation non-linear system, which has a linear measurement equation, the SMCK-CKF is established by introducing the approximate method to calculate the matrix inversion in the combination of Kalman filter measurement updated and MCK-CKF framework. The computational complexity of the CKF/MCK-CKF/SMCK-CKF is analyzed and compared. The powerful robustness for tackling various forms of non-Gaussian interference of MCK-CKF is validated by simulation on a typical non-linearity system. Furthermore, the vehicle’s INS/OD integrated navigation experiment demonstrates that the MCK-CKF/SMCK-CKF have remarkable robustness, and the SMCK-CKF has outstanding time efficiency.

Robust cubature Kalman filter based on maximum correntropy and resampling-free sigma-point update framework

The problem of nonlinear filtering for GNSS/INS under non-Gaussian noise is studied in this paper, where an improved maximum correntropy cubature Kalman filter (MCCKF) is proposed by utilizing resampling-free sigma-point update framework (SUF). The MCCKF has better robustness than CKF in case non-Gaussian noise appears. However, only the Gaussian moments are circulated in MCCKF, which is less efficient when the nonlinearity of dynamic function is strong. The proposed algorithm named RMCCKF, imports the re-scaled measurement and state prediction covariance that produced by maximum correntropy criterion into the direct construction of posterior sigma points. By constraining the posterior covariance of MCCKF and modifying the sigma points directly, the resampling-free SUF spreads extra information of dynamic model over the posterior sigma points update. The proposed algorithm is verified by using numerical simulation and field test on GNSS/INS. Experiment results indicate that, RMCCKF achieves better performance than CKF and MCCKF, especially when the filtering problem is of high dimension. On the GNSS/INS of land vehicle navigation, RMCCKF not only improves the position but also reduces the heading error from 1.77 degree to 0.26 degree, compared with MCCKF.

Variational Bayesian-Based Robust Cubature Kalman Filter With Application on SINS/GPS Integrated Navigation System

In this article, we focus on addressing the nonlinear filtering problem with unknown measurement noise covariance and measurement outliers, which may be encountered in the application in strapdown inertial navigation system/global positioning system integrated navigation system. Although the existing methods, such as the adaptive Kalman filter, are widely used in the integrated navigation system, their estimation accuracy is poor, this paper proposes a variational Bayesian-based robust cubature Kalman filter to address this problem, which not only retains the adaptivity when the measurement noise is unknown but also exhibits robustness in the presence of measurement outliers, First, the variational Bayesian method is applied to the estimation of measurement noise, then the maximum correntropy criterion is introduced to replace the minimum mean square error criterion, which improves the robust performance of the filter. The numerical simulation demonstrates that the proposed filter outperforms the existing filters both in estimation accuracy and robustness, and the effectiveness of the proposed filter is verified on the integrated navigation system.

A variational Bayesian based robust cubature Kalman filter under dynamic model mismatch and outliers interference

This paper proposes a variational Bayesian-based cubature Kalman filter to solve the state estimation problem of nonlinear discrete-time systems under dynamic model mismatch and outliers interference, which occurs in the maneuvering target tracking system. In the proposed filter, the measurement noise is divided into the normal measurement noise and the outlier noise, and two inverse Wishart distributions with different degree of freedom parameters are used to model the unknown normal measurement noise covariance and the unknown outlier noise covariance, respectively. To overcome outliers interference, the judgment factor which follows the Bernoulli-beta distribution is designed to identify the type of measurement noise. Meanwhile, the input vector which follows the Gaussian distribution is introduced to realize the dynamic model revision. The state, the judgment factor, the normal measurement noise covariance, the outlier noise covariance, and the input vector are jointly estimated under the variational Bayesian framework. The numerical simulation is illustrated to verify the performance of the proposed filter and the corresponding results show that the proposed filter has good state estimation performance and robustness.

Singular value decomposition‐based iterative robust cubature Kalman filtering and its application for integrated global positioning system/strapdown inertial navigation system navigation

The issue of non-linear robust state estimation in the integration of a strapdown inertial navigation system and global positioning system is addressed in this study. Based on the cubature Kalman filtering frame, a non-linear robust filter called a robust cubature Kalman filter (RCKF) was introduced to address the outliers and the inaccurate model. It has been found that the determination of an optimal restriction parameter is crucial for maintaining the robustness and accuracy of the non-linear robust filter. Unfortunately, the value of this restriction parameter is always determined by experience. In this study, an iterative strategy was proposed to adaptively attain the optimal restriction parameter without much previous experience. To improve the computational stability of the iterative non-linear robust filter, a singular value decomposition strategy was adopted simultaneously. Two case studies indicate that the iterative RCKF can achieve greater robustness and accuracy using the methodology discussed in this study.

Cubature Kalman Filter With Both Adaptability and Robustness for Tightly-Coupled GNSS/INS Integration

Tightly-coupled GNSS/INS (Global Navigation Satellite System/Inertial Navigation System) integration is of importance to vehicle positioning. However, this integration technology has difficulty in achieving optimal positioning solutions for the dynamic systems involving strong nonlinearity and systematic modelling error. This paper proposes a new methodology to address the problem of tightly-coupled GNSS/INS integration. This methodology rigorously derives a novel adaptive CKF (Cubature Kalman Filter) with fading memory for kinematic modelling error and a new robust CKF with emerging memory for observation modelling error, using the concept of Mahalanobis distance without involving artificial empiricism. Based on this, a new CKF with both adaptability and robustness is further developed by fusing the results of the standard CKF, adaptive CKF and robust CKF via the principle of interacting multiple model (IMM). Simulation and experiment results together with comparison analysis prove that the proposed methodology can curb the interferences of both kinematic and observation modelling errors on state estimation, leading to improved positioning accuracy for vehicle positioning via tightly-coupled GNSS/INS integration.