Noise Covariance Matrix Research Articles

VB-Based Gaussian Sum Cubature Kalman Filter for Adaptive Estimation of Unknown Delay and Loss Probability

The traditional Kalman filter assumes that all measurements can be obtained in real time, which is invalid in practical engineering. Therefore, a variational Bayesian- (VB-) based Gaussian sum cubature Kalman filter is proposed to solve the nonlinear tracking problem of multistep random measurement delay and loss (MRMDL) with unknown probability. First, the measurement model with MRMDL is modified by Bernoulli random variables. Then, the expression of the likelihood function is reformulated as a mixture of multiple Gaussian distributions, and the cubature rule is used to improve the estimation accuracy under the framework of Gaussian sum filter in the process of time update. Finally, by constructing a hierarchical Gaussian model, the unknown and time-varying measurement delay and loss probability are estimated in real time with the state jointly using the VB method in the measurement update stage. The algorithm does not need to calculate the equivalent noise covariance matrix so as to avoid the possible division by zero operation, which improves the stability of the algorithm. Simulation results for a target tracking problem show that the proposed algorithm has a better performance in the presence of MRMDL and can estimate the unknown measurement delay and loss probability accurately.

Beam-Space Post-Doppler Reduced-Dimension STAP Based on Sparse Bayesian Learning

The space–time adaptive processing (STAP) technique can effectively suppress the ground clutter faced by the airborne radar during its downward-looking operation and thus can significantly improve the detection performance of moving targets. However, the optimal STAP requires a large number of independent identically distributed (i.i.d) samples to accurately estimate the clutter plus noise covariance matrix (CNCM), which limits its application in practice. In this paper, we fully consider the heterogeneity of clutter in real-world environments and propose a sparse Bayesian learning-based reduced-dimension STAP method that achieves suboptimal clutter suppression performance using only a single sample. First, the sparse Bayesian learning (SBL) algorithm is used to estimate the CNCM using a single training sample. Second, a novel angular Doppler channel selection algorithm is proposed with the criterion of maximizing the output signal-to-clutter-noise ratio (SCNR). Finally, the reduced-dimension STAP filter is constructed using the selected channels. Simulation results show that the proposed algorithm can achieve suboptimal clutter suppression performance in extremely heterogeneous clutter environments where only one training sample can be used.

An accurate and robust visual-inertial positioning method

The human–machine integrated coordinate measurement is a promising coordinate measurement method with high flexibility and efficiency for the complex working environments. The cameras installed on the head-mounted measurement device achieves accurate global positioning by observing the uncoded LED landmarks, and then combines with the local measuring to obtain 3D coordinates. However, limited by the frame rate of the camera, the fast movements of the operator’s head may cause landmark misidentification and visual positioning failure. In order to improve the robustness, a visual-inertial positioning method is proposed in this paper. An inertial measurement unit (IMU) is added to compensate for the deficiency of the visual positioning and enhance the dynamic performance. An adaptive extended Kalman filter (EKF), which adjusts the measurement noise covariance matrix based on the visual positioning uncertainty, is established to obtain the optimal state estimation. And an efficient initialization procedure is presented to implement the initial registration of uncoded landmarks based on the normal distribution transform algorithm and to determine the initial state of the IMU. Furthermore, the residual chi-square test is employed to detect false pose estimate in real time and to avoid positioning failure. The experiments demonstrate that the proposed method has high static positioning accuracy (0.681 mm) and high dynamic positioning robustness. The adaptive EKF realizes reliable landmark identification under fast movements and provides a higher accuracy than the common EKF.

A MCKF-based cascade vector tracking method designed for ship navigation

In ship navigation, the most widely used technology is global navigation satellite systems (GNSS), which provide the ship’s position and velocity continuously over a period of time. However, when ships are blocked by port buildings or bridges, the quality of signals received from shipborne GNSS receivers may be reduced, resulting in inaccurate ship positioning that poses a risk to navigational safety. In an occluded environment, the measurement process during the signal processing of shipborne GNSS receivers is nonstationary and prone to measurement anomalies, which can contaminate measurement noise with outliers. To address this problem, a cascaded non-coherent vector tracking loop is designed, with the maximum correntropy Kalman filter serving as a cascaded carrier/code pre-filter for shipborne GNSS receivers. The measurement noise covariance matrix of the pre-filter is adaptively calculated and corrected using the carrier-to-noise ratio and the maximum correntropy criterion, respectively. The algorithm proposed is more sensitive to outliers than the traditional tracking methods and can effectively solve the state estimation problem under the condition of measurement anomalies. Specifically, the algorithm offers ships with more precise position and velocity estimations and lower signal tracking errors than traditional tracking methods under both static and dynamic conditions, as demonstrated by shipboard experiments. The horizontal positioning is increased by 88.4% and the horizontal velocity error is reduced by 62.1% in the occluded environment under dynamic conditions.

A comparison of indirect and direct filter designs from data for LTI systems: the effect of unknown noise covariance matrices

Existing literature on model-based filter design for stochastic LTI systems assumes complete correspondence between the system and its model. When the system is not completely known, the standard indirect model-based (two-steps) filtering solution consists of: (i) identify a model of the system from measured input/output data; (ii) design a Kalman filter based on the estimated model. The performance of this indirect approach are limited by the model and noise covariance matrices accuracy. To overcome such limitations, this paper investigates a direct (one-step) solution to the filtering problem for SISO LTI systems in the Prediction Error Method (PEM) identification framework. Simulation results indicate the effectiveness of the direct filtering approach, especially when the noise covariance matrices are misspecified.

Research of adaptive algorithms of recurrent filtration at processing data from GNSS when radio signals lose from navigation satellites

Algorithms for linear and adaptive Kalman filtering of navigation data received from satellite navigation systems are considered. A comparison of the Kalman linear filter algorithms, the Yazvinsky adaptive filter, the adaptive filter with feedback on the updated sequence and the adaptive filter without a priori information about the noise covariance matrices is made. The algorithms are tested in a practical experiment — navigation information from GNSS is obtained by using a GNSS receiver installed on a car moving in an urban environment. The comparative analysis results of the algorithms usage for adaptive modifications of the Kalman filter are presented. The results showed that the Yazvinsky filter works worse when radio signals from navigation satellites disappear, and the adaptive filter algorithm without a priori information about the noise covariance matrices does not actually filter the data. Keywords GNSS, navigation information, Kalman filter, Yazvinsky filter, adaptive Kalman filter

A Novel Adaptive Noise Covariance Matrix Estimation and Filtering Method: Application to Multiobject Tracking

This paper presents a novel online adaptive method for estimating the process and measurement noise covariance matrices in Kalman filters (KFs) to address the challenge of varying noise characteristics in practical applications. Specifically, the proposed method decomposes the noise covariance matrix into an element distribution matrix and a noise intensity and employs an improved Sage filter to estimate the element distribution matrix. Additionally, a calibration and correction method is introduced to accurately determine and adaptively correct the online bias of the noise intensity. The unbiasedness and convergence of the proposed method are mathematically proven under the condition that the system is detectable. Moreover, this method is applied to multiobject tracking (MOT) based on KFs and light detection and ranging (LiDAR), and it is evaluated on the KITTI dataset and the official KITTI server. The experimental results demonstrate that the proposed method achieves significantly improved MOT performance based on KFs and outperforms other LiDAR-based methods on the KITTI leaderboard. This method provides a new approach for enhancing the performance of KFs and assisting with MOT, and it has practical feasibility for real-world applications.

Structural Dynamic Response Reconstruction Based on Recurrent Neural Network–Aided Kalman Filter

In structural health monitoring (SHM), an important issue is the limited availability of measurement data due to the spatial sparsity of sensors installed on the structure. These measurements are insufficient to accurately depict the actual dynamic behavior and response of the structure. Therefore, full‐field (i.e., every degree of freedom) structural response reconstruction based on sparse measured data has drawn a lot of attention in recent years. Kalman filter (KF) is an effective technology for response reconstruction (also known as state estimation), providing an optimal solution for systems that can be well‐represented by a fully known Gaussian linear state‐space model. This implies that both the process noise and measurement noise follow known zero‐mean Gaussian distribution, which is impractical in many civil engineering applications considering the unavoidable modeling errors and variations of environmental conditions. To address this challenge, a data‐physics hybrid‐driven method, i.e., KalmanNet, is proposed in this study for response reconstruction of partially known systems. By integrating a recurrent neural network (RNN) module into the KF framework, KalmanNet can efficiently learn and compute the Kalman gain using available monitoring data, without any Gaussian assumptions or explicit noise covariance specifications (e.g., covariance matrices of process and measurement noise). Both numerical and experimental investigations are conducted to validate this method. The results demonstrate that under the influence of non‐Gaussian noise and modeling errors, KalmanNet can effectively and accurately reconstruct the structural response from sparse measurements in real‐time and has higher accuracy and robustness compared to traditional KF even with optimal parameter settings.

Q-Learning-Based Noise Covariance Matrices Adaptation in Kalman Filter for Inertial Navigation

Q-Learning-Based Noise Covariance Matrices Adaptation in Kalman Filter for Inertial Navigation

Noise Covariances Identification by MDM: Weighting, Recursion, and Implementation

The problem of noise covariance matrix identification of stochastic linear time-varying state-space models is addressed. The measurement difference method (MDM) is generalized to time-varying dimensions of the measurement and control. Three MDM identification techniques that differ in weighting used in the underlying least squares method are proposed. The techniques differ in estimate quality and computational complexity. In addition, recursive forms are designed for two techniques. The performance of the proposed techniques is analyzed using two numerical examples. The implementation of techniques is enclosed with the paper.

A robust GNSS sensors in presence of signal blockage for USV application

Unmanned surface vehicle (USV) can navigate autonomously via the global navigation satellite system (GNSS). However, the traditional GNSS scalar tracking loop easily loses lock in low carrier-to-noise ratio (CNR) situations, such as signal occlusion and weak signals. Meanwhile, an increase in the carrier/code phase error leads to an increase in the measurement error of the navigation filter, which decreases the accuracy of the position estimation. To solve this problem, this paper proposes a carrier and code tracking structure based on a forward and backward Kalman filter to dynamically adjust the gain of the vector tracking loop. The carrier and code phase errors calculated by the loop discriminators were linearly transformed into pseudo-range rate and pseudo-range errors after filtering and smoothing, which were used as the measurements of the navigation filter. The signal CNR was used to adaptively adjust the measurement noise covariance matrix of the loop filters. The field tests used a commercial receiver’s navigation solution as the reference. In the stationary test, the proposed structure reduced the localization error by 44.3% compared with the traditional methods. In kinematic experiments, the proposed structure reduced the carrier and code phase errors in a harsh signal environment and improved the positioning accuracy at the source. The test results demonstrate that the proposed GNSS tracking method can provide a possible solution for the development of navigation systems for USV.

Optimal Pose Estimation and Covariance Analysis with Simultaneous Localization and Mapping Applications

This work provides a theoretical analysis for optimally solving the pose estimation problem using total-least-squares for vector observations from landmark features, which is central to applications involving simultaneous localization and mapping. First, the optimization process is formulated with observation vectors extracted from point-cloud features. Then, error-covariance expressions are derived. The attitude and position estimates obtained via the derived optimization process are proven to reach the bounds defined by the Cramér–Rao lower bound under the small-angle approximation of attitude errors. A fully populated observation noise-covariance matrix is assumed as the weight in the cost function to cover the most general case of the sensor uncertainty. This includes more generic correlations in the errors than previous cases involving an isotropic noise assumption. The proposed solution is verified using Monte Carlo simulations, a Gazebo simulation in a robotics operating system, and an experiment with an actual light detection and ranging sensor to validate the error-covariance analysis.

An invariant error-based SINS/DVL/USBL tightly coupled integrated navigation algorithm and its robust state estimator

The existing strapdown inertial navigation system (SINS)/Doppler velocity log (DVL)/ultra-short baseline system (USBL) integrated navigation error models directly apply algebraic differences to define the vector error, which is not strictly accurate. Considering that the invariant error definitions derived from Lie group theory have been used in the navigation field to obtain more accurate linear error models, in this paper, a new SINS/DVL/USBL integrated navigation system is investigated based on the invariant error definition and tightly coupled structure. In order to suppress the effect of measurement outliers on navigation accuracy, a new statistical similarity measure (SSM)-based multiple adaptive outlier-robust state estimator (MAORSE) is proposed. It decomposes the noise covariance matrix into multiple sub-matrices. Through the estimation and adjustment of sub-matrices, the influence of abnormal observations on the filtering results is weakened while realizing the full utilization of normal observation information. The test results demonstrate the effectiveness and superiority of the proposed invariant error-based SINS/DVL/USBL tightly coupled integrated navigation algorithm and SSM-based MAORSE.

Equivalent circuit modeling and state-of-charge estimation of lithium titanate battery under low ambient pressure

The low ambient pressure during the flight of aircraft has a significant impact on the performance and safety of the onboard power battery. In order to ensure the safe operation of the battery system, it is very important to accurately estimate and manage the state-of-charge (SOC) of the battery. In this work, the equivalent circuit modeling (ECM) of lithium titanate battery (LTB) is studied in detail, and the influence of analog circuit model parameters in low ambient pressures is discussed for the first time. The forgetting factor recursive least square algorithm is introduced to accurately identify the ECM parameters of the LTB under different pressures online. The particle swarm optimization algorithm is innovatively proposed to optimize the covariance matrix of the Kalman filter algorithm. The verification shows that the root mean square error of the ECM of LTB under different ambient pressures is less than 0.025. In the SOC estimation process, the noise covariance matrixes of the extended Kalman filter and the unscented Kalman filter are optimized by the particle swarm algorithm. The optimized SOC estimation absolute error is less than 3 %, especially at 96 kPa and 30 kPa, where the absolute error is less than 2 %.

Bayesian Optimization for Fine-Tuning EKF Parameters in UAV Attitude and Heading Reference System Estimation

In various applications, the extended Kalman filter (EKF) has been vital in estimating a vehicle’s translational and angular motion in 3-dimensional (3D) space. It is also essential for the fusion of data from multiple sensors. However, for the EKF to perform effectively, the optimal process noise covariance matrix (Q) and measurement noise covariance matrix (R) must be chosen correctly. The use of EKF has been challenging due to the need for an easy mechanism to select Q and R values. As a result, this research focused on developing an algorithm that can be easily applied to determine Q and R, allowing us to harness the full potential of EKF. Accordingly, an EKF innovation consistency statistics-driven Bayesian optimization algorithm was employed to achieve this goal. Q and R values were tuned until the expected result met the performance requirement for minimum error through improved measurement innovation consistency. The comprehensive results demonstrate that when the optimum Q and R, as tuned by the suggested technique, were used, the performance of the EKF significantly improved.

Parameter Identification of Permanent Magnet Synchronous Motor with Dynamic Forgetting Factor Based on H∞ Filtering Algorithm

To address system parameter changes during permanent magnet synchronous motor (PMSM) operation, an H∞ filtering algorithm with a dynamic forgetting factor is proposed for online identification of motor resistance and inductance. First, a standard linear discrete PMSM parameter identification model is established; then, the discrete H∞ filtering algorithm is derived using game theory reducing state and measurement noise influence. A cost function is defined, solving extremes values of different terms. A dynamic forgetting factor is introduced to the weighted combination of initial and current measurement noise covariance matrices, eliminating identification issues from different initial values. On this basis, a dynamic forgetting factor is added to weigh the combination of the initial measurement noise covariance matrix and the current measurement noise covariance matrix, which eliminates the influence of the discrimination error caused by the different initial values. Finally, the identification model is built in MATLAB/Simulink for simulation analysis to verify the feasibility of the proposed algorithm. The simulation results show the proposed H∞ filtering algorithm rapidly and accurately identifies resistance and inductance values with significantly improved robustness. The forgetting factor enables quick stable recognition even with poor initial values, enhancing PMSM control performance.

Direction of Arrival Estimation With Gain-Phase Uncertainties in Unknown Nonuniform Noise

This paper considers the problem of direction-of-arrival (DOA) estimation for narrowband source signals under the coexistence of gain-phase uncertainties and nonuniform noise. Firstly, by exploiting the low-rank property of the covariance of signal components, and the diagonal property of noise covariance, an optimization problem with low-rank constraints is formulated to estimate the nonuniform noise covariance matrix, which is then solved efficiently by the alternating direction method of multipliers (ADMM). Secondly, a linear transformation operator is employed to construct a rank reduction (RARE) estimator, and subsequently an initial DOA and gain-phase uncertainties are obtained via one-dimensional (1-D) spectral search. With initial estimates, an augmented DOA estimation is finally achieved by constructing a weighted and structured sparse total least-squares (WSS-TLS) optimization problem, which is solved using the block coordinate descent algorithm. Simulation results show the superior performance of the proposed method, and it achieves better tradeoff between computational complexity and performance, compared to existing methods.

Soc Estimation of Li-ion Battery Based on Adaptive CKF Algorithm

A second-or der RC equivalent circuit model was established to improve the estimation accuracy of state of charge (SOC) of power Li-ion batteries, and the model parameters were identified by the recursive least square method with forgetting factor (FFRLS). On this basis, an adaptive cubature kalman filter (ACKF) algorithm was proposed to adaptively modify the process noise covariance matrix and the measurement noise covariance matrix to improve the SOC estimation accuracy. Finally, the SOC estimation algorithm was verified by MATLAB simulations. The results show that compared with UKF and CKF algorithms, the proposed algorithm has higher estimation accuracy and robustness, and can meet the application requirements.

Efficiently building receive arrays with electromagnetic simulations and additive manufacturing: A two-layer, 32-channel prototype for 7T brain MRI.

We propose a comprehensive workflow to design and build fully customized dense receive arrays for MRI, providing prediction of SNR and g-factor. Combined with additive manufacturing, this method allows an efficient implementation for any arbitrary loop configuration. To demonstrate the methodology, an innovative two-layer, 32-channel receive array is proposed. The design workflow is based on numerical simulations using a commercial 3D electromagnetic software associated with circuit model co-simulations to provide the most accurate results in an efficient time. A model to compute the noise covariance matrix from circuit model scattering parameters is proposed. A 32-channel receive array at 7 T is simulated and fabricated with a two-layer design made of non-geometrically decoupled loops. Decoupling between loops is achieved using home-built direct high-impedance preamplifiers. The loops are 3D-printed with a new additive manufacturing technique to speed up integration while preserving the detailed geometry as simulated. The SNR and parallel-imaging performances of the proposed design are compared with a commercial coil, and in vivo images are acquired. The comparison of SNR and g-factors showed a good agreement between simulations and measurements. Experimental values are comparable with the ones measured on the commercial coil. Preliminary in vivo images also ensured the absence of any unexpected artifacts. A new design and performance analysis workflow is proposed and tested with a non-conventional 32-channel prototype at 7 T. Additive manufacturing of dense arrays of loops for brain imaging at ultrahigh field is validated for clinical use.

Underwater Doppler-bearing maneuvering target motion analysis based on joint estimated adaptive unscented Kalman filter

Noncooperative maneuvering target motion analysis is one of the challenging tasks in the field of underwater target localization and tracking for passive sonar. Underwater noncooperative targets often perform various maneuvers, and the targets are commonly modeled as a combination of constant-velocity models and coordinate-turn models with unknown turning rates. Traditional algorithms for Doppler-bearing target motion analysis are incapable of processing noncooperative maneuvering targets because the algorithms rely on a priori information of the turning rate and the center frequency. To address these shortcomings, this paper proposes the joint estimated adaptive unscented Kalman filter (JE-AUKF) algorithm. The JE-AUKF places the center frequency and turning rate into the state vector and constructs a time-varying state model that self-adapts to a maneuvering target. The JE-AUKF also introduces a time-varying fading factor into the process noise covariance matrix to improve the tracking performance. Simulations and sea trials are conducted to compare the performance of the JE-AUKF with the iterative unscented Kalman filter, the interacting multiple model-unscented Kalman filter, the interacting multiple model-iterative unscented Kalman filter, and the interacting multiple model-joint estimated unscented Kalman filter. The result shows that the JE-AUKF achieves better tracking performance for noncooperative maneuvering targets.