Measurement Covariance Matrix Research Articles

Adaptive Multi-Sensor Joint Tracking Algorithm with Unknown Noise Characteristics.

In this study, to solve the low accuracy of multi-space-based sensor joint tracking in the presence of unknown noise characteristics, an adaptive multi-sensor joint tracking algorithm (AMSJTA) is proposed. First, the coordinate transformation from the target object to the optical sensors is considered, and the observation vector-based measurement model is established. Then, the measurement noise characteristics are assumed to be white Gaussian noise, and the measurement covariance matrix is set as a constant. On this premise, the traditional iterative extended Kalman filter is applied to solve this problem. However, in most actual engineering applications, the measurement noise characteristics are unknown. Thus, a forgetting factor is introduced to adaptively estimate the unknown measurement noise characteristics, and the AMSJTA is designed to improve the tracking accuracy. Furthermore, the lower bound of the proposed algorithm is theoretically proved. Finally, numerical simulations are executed to verify the effectiveness and superiority of the proposed AMSJTA.

Adaptive multi-object tracking based on sensors fusion with confidence updating

Multi-object tracking (MOT) systems typically rely on object detection results for tracking, so the accuracy of the MOT system is significantly affected by the error of the detector. Changes in error usually lead to unstable tracking. Regarding this problem, we proposed an adaptive MOT method based on detection confidence. At first, we use a simple data fusion method to combine the detection results of LiDAR and camera to reduce the large number of false detections. And then we used a factor based on confidence to adjust the estimating covariance matrix and measurement covariance matrix adaptively. The algorithm can judge which is more reliable between prediction and detection, and choose which is more important in the update step. Meanwhile, we set a factor based on confidence to control the search range in the data association module. Our method reduces the impact of detector error while ensuring accuracy and speed, and improves the robustness of the MOT algorithm. Through experiments conducted on the KITTI multi-object tracking dataset, our method has demonstrated significant advantages over state-of-the-art (SOTA) methods in terms of both accuracy and processing speed. The results of MOTA for 90.02% and FPS for 262.

Adaptive dynamic kalman filter for high-performance tilt-angles estimation

Estimation of attitude is a potential task for autonomous vehicles as it directly affects the velocity and position estimates as well as the overall system autonomy. The estimation of attitude angles utilizing gyro measurements always suffers from increasing drift with time. Although accelerometer measurements can provide an absolute estimate of pitch and roll angles for static and low dynamics conditions, they have two drawbacks the effect of additional external acceleration and the inability to capture the high dynamics motion. In this paper, an adaptive Kalman filter (AKF) is utilized for integrating the gyros and accelerometer measurements for enhancing the roll and pitch angles estimates by compensating for the effect of linear acceleration as well as adaptively adapting the Kalman filter measurement covariance matrix. The proposed algorithm is evaluated using real offline measurements for a flying vehicle. Further, a comparative analysis is carried out with the original Kalman filter (KF) algorithm and with the imufilter from Sensor Fusion and Tracking Toolbox from Matlab. The results presented satisfactory performance and enhancement for roll and pitch angles estimate.

Enhanced square root CKF with mixture correntropy loss for robust state of charge estimation of lithium-ion battery

The measurement data of lithium-ion battery collected in complex operating conditions may be contaminated by non-Gaussian noises (or outliers), a novel robust state of charge (SOC) estimation approach that enhances the robustness of square-root cubature Kalman filter (SRCKF) by incorporating a mixture correntropy loss (MCL) is proposed to overcome the issue of non-Gaussian measurement noise interference. Although the SRCKF can perform non-destructive state estimation for nonlinear system, it is sensitive to the non-Gaussian noises (or outliers) due to the use of mean square error (MSE) loss. To overcome this limitation, the MCL is used as a cost to replace the MSE in CKF framework, which is constructed by mixture of two Gaussian kernel functions with different kernel widths and has outstanding robustness. Moreover, we establish an effective nonlinear regression model that incorporates noise and state error information into the MCL function, and then a fixed-point iteration method is utilized to update the posterior state estimation. Additionally, to further suppress the influence of abnormal noise on the posterior state estimation, we introduce the exponential function of the innovation vector to adjust the measurement covariance matrix in real-time. Accordingly, an enhanced MCL-SRCKF is developed for SOC estimation considering the measurement noises with non-Gaussian distribution. Results from various simulation environments demonstrate that the proposed method achieves high estimation accuracy in different cases.

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.

An integrated adaptive Kalman filter for improving the reliability of navigation systems

Abstract Integrated GPS/INS using Kalman filter is the best technique for improving navigation accuracy. Assuming that the covariance matrices are known and constant, a conventional Kalman filter (CKF) is usually used, however, when they are unknown and time-varying, several adaptive estimation approaches have to be developed to estimate the statistical information of the measurement (R), process (Q), and state (P) covariance matrices. In many situations, blunders/faults in the measurement model and/or sudden changes in the dynamic model may occur during the navigation period. Therefore, the CKF, as well as the adaptive Kalman filter (AKF) will exhibit abnormal behavior and may lead the filter to be suboptimal or even diverge. In this study, the Sage-Husa adaptive Kalman filter (SHAKF) and innovation-based adaptive Kalman filter (IAKF) approaches are employed for adapting the measurement covariance matrix(R). In the case of abrupt changes in the dynamic model, the state covariance matrix (P) is adapted using the strong tracking filter (STF). The performance of these adaptive approaches is evaluated before and after simulating a fault of different sizes in the measurement and dynamic models. The results show that with a large window width, the SHAKF outperforms the CKF and IAKF. However, when the system encounters any fault either in the measurement or dynamic model, the SHAKF loses its optimality and diverges. The sensitivity of the SHAKF to the fault is because the R matrix accumulates with the propagation of the recursive noise estimator. On the other hand, the IAKF and STF provide better performance than both the CKF and SHAKF because the gain matrix is adaptively adjusted to mitigate the influence of the fault, and therefore, they behave normally when a fault of any size occurs in the measurement and/or dynamic model.

Statistically optimal estimation of surface mass anomalies by directly using GRACE level-2 spherical harmonic coefficients as measurements

SUMMARYPoint-mass inversion is widely employed in GRACE level-2 data processing. Conventionally, the spherical harmonic (SH) coefficients are used indirectly: a set of pseudo measurements is generated first using the SH coefficients through SH synthesis; then the point-mass inversion is done with these pseudo measurements. To be statistically optimal, the covariance matrix of pseudo measurements should be calculated and used to appropriately weigh the parameter estimation. In this work, we propose a statistically optimal point-mass inversion scheme by directly using the SH coefficients as measurements. We prove the equivalence between this direct approach and the conventional indirect approaches. We also demonstrated their comparable performance through both simulation and real GRACE data processing. Choosing and calculating pseudo measurements, propagating covariance matrix and potentially dealing with the singularity of the covariance matrix involved in the conventional indirect approaches are avoided in the proposed direct approach. This statistically optimal direct approach can readily be employed in mascon inversion of GRACE data and other radial basis functions-based approaches in regional gravity modeling.

A novel sampling methodology exploiting the least correlated-columns for angle of arrival estimation

Estimating Angle of Arrival (AOA) parameter from the measured data has gained much attention, particularly in the 5G-and-beyond wireless communication networks. With these communication networks, it is highly expected that the size and number of the data sets are huge and much larger than the rival traditional networks. To this end, the way of sampling and dealing with collected data in the array signal processing stage will have a significant impact on the signal estimation parameters. Thus, this manuscript proposes a novel sampling methodology to extract the collected information optimally by exploiting the least correlated and dependent columns within the measured Covariance Matrix (CM). The suggested method, Least Correlated Column Sampling (LCCS), is then adopted to implement a robust AOA estimation algorithm called Least Correlated Column Projection Matrix (LCCPM). A theoretical analysis is derived and presented to justify that the proposed technique minimizes correlations between adjacent columns and then compared with current sampling methodologies in terms of the position selection and corresponding correlation. The analytical results show that the LCCS methodology selects the least dependent columns among the rival methods. This, in turn, improves the AOA algorithm performance and enables it to localize closely-separated sources with high accuracy. To verify the theoretical claims and reveal the advantages of the suggested approach, a numerical example is given and then intensive Monte Carlo simulations are conducted over a wide range of scenarios where the proposed method performance is systemically compared with existing techniques. The obtained results show that the proposed algorithm is superior and outperforms all the preceding methodologies in terms of higher estimation resolution, lower Root Mean Square Error (RMSE), and higher Probability of Successful Detections (PSDs). The results also illustrate that the proposed method achieves higher noise immunity, less sensitivity to correlated signals, and detects a higher number of angles in comparison with conventional and recently-developed methods.

Variational Bayesian implicit unscented Kalman filter for celestial navigation using time delay measurement

Sunlight arrival time delay is an innovative measurement for celestial navigation, which provides the distance information between the spacecraft and the nearby celestial body. For deep space detectors, the time delay measurement noise is time-varying, and its statistics cannot be accurately determined. The too-large deviation between the measured noise covariance matrix (R) and the noise statistics seriously reduces navigation accuracy. A variational Bayesian implicit unscented Kalman filter (VBIUKF) method is proposed and applied to celestial navigation using time delay measurement in this paper. VBIUKF adaptively estimates R using the innovation sequence and the predicted measurement covariance matrix to make R closest to the measurement noise statistics. Simulation results indicate that VBIUKF can obtain high accuracy navigation results under time-varying measurement noise.

Nonnegative block-sparse Bayesian learning algorithm for EEG brain source localization

Localizing electrical sources on the cortex surface from scalp recorded electroencephalogram (EEG) is challenging, due to ill-posed problem, noises/artifacts contamination, etc. Applying sparse Bayesian learning (SBL) in this field can automatically give sparse solution to ill-posed problem, while most of the SBL-based algorithms require precise or estimated version of noise statistics information. As EEG signals are more likely to stem from locally synchronized neural masses, modeling source block-sparsity on the cortex surface would bring benefits. In this paper, we develop an EEG brain source localization algorithm in SBL framework, with innovative modeling at sensor level as well as source level. For sensor-level modeling, the distribution of sample covariance matrix of multi-electrode measurements is considered, to circumvent the requirement of noise covariance matrix information. The innovation of source-level modeling is that, with block-sparsity prior used, the block-sparse signal reconstruction problem is transformed to an atom-sparse one, in which variance parameters of brain regions are to be estimated. As these parameters are nonnegative, their priors are modeled by nonnegative Gaussian, which was neglected by previous studies, and ultimately a nonnegative block-SBL (NNBSBL) algorithm is proposed in expectation–maximization (EM) approach. Simulations demonstrate that the proposed NNBSBL algorithm has excellent performance in variations of source number, source locations, homoscedastic/heteroscedastic noises, signal-to-noise ratio (SNR), and number of samples, compared to benchmark and state-of-the-art algorithms. The performance of the proposed algorithm is also evaluated through real P300 EEG data, which is proved consistent with the conclusions of P300 source locations in literature.

Multifrequency array calibration in presence of radio frequency interferences

Radio interferometers are phased arrays producing high-resolution images from the covariance matrix of measurements. Calibration of such instruments is necessary and is a critical task. This is how the estimation of instrumental errors is usually done thanks to the knowledge of referenced celestial sources. However, the use of high sensitive antennas in modern radio interferometers (LOFAR, SKA) brings a new challenge in radio astronomy because they are more sensitive to Radio Frequency Interferences (RFI). The presence of RFI during the calibration process generally induces biases in state-of-the-art solutions. The purpose of this paper is to propose an alternative to alleviate the effects of RFI. For that, we first propose a model to take into account the presence of RFI in the data across multiple frequency channels thanks to a low-rank structured noise. We then achieve maximum likelihood estimation of the calibration parameters with a Space Alternating Generalized Expectation-Maximization (SAGE) algorithm for which we derive originally two sets of complete data allowing closed-form expressions for the updates. Numerical simulations show a significant gain in performance for RFI corrupted data in comparison with some more classical methods.

Unified Detection of Attacks Involving Injection of False Control Commands and Measurements in Transmission Systems of Smart Grids

Transmission system operation and monitoring, in the context of smart grids, are heavily dependent on the correctness or aptness of supervisory control commands. These command channels are a part of the SCADA system, which is vulnerable to cyber-attacks. Maliciously injected commands can cause a wide range of issues including blackouts. However, the literature addressing this problem is limited. In order to fill this gap, a generalized framework is proposed to achieve simultaneous detection of attacks against various types of control devices/equipments used in transmission systems, even if they are carried out stealthily. First, generalized mathematical models of such stealthy attacks are presented. Then, based on these attack models, changes in the measurement covariance matrix that occur during an attack are exploited to formulate the proposed approach. We first mathematically characterize these changes using the eigenvectors and trace of the covariance matrix and then use them to develop a detection metric. It is observed that when the model of a false data injection (FDI) attack is considered, the developed detection metric is also capable of detecting such attacks. The algorithm that results from the detection metric is a generalized framework for detection of attacks against all types of transmission system controls and measurements. Moreover, the algorithm is non-iterative, computationally inexpensive, and independent of both the type of state estimation and communication technology used. The proposed algorithm is found to be effective, when tested on the IEEE 118-bus system.

Extended Target Tracking With a Lidar Sensor Using Random Matrices and a Virtual Measurement Model

Random matrices are widely used to estimate the extent of an elliptically contoured object. Usually, it is assumed that the measurements follow a normal distribution, with its standard deviation being proportional to the object’s extent. However, the random matrix approach can filter the center of gravity and the covariance matrix of measurements independently of the measurement model. This work considers the whole chain from data acquisition to the linear Kalman Filter with extension estimation as a reference plant. The input is the (unknown) ground truth (position and extent). The output is the filtered center of gravity and the filtered covariance matrix of the measurement distribution. A virtual measurement model emulates the behavior of the reference plant. The input of the virtual measurement model is adapted using the proposed algorithm until the output parameters of the virtual measurement model match the result of the reference plant. After the adaptation, the input to the virtual measurement model is considered an estimation for position and extent. The main contribution of this paper is the reference model concept and an adaptation algorithm to optimize the input of the virtual measurement model.

Anomaly Detection in Wind Tunnel Experiments by Principal Component Analysis

Principal component analysis (PCA) was used to detect anomalies in wind tunnel measurements. Data were compiled from previous measurements for three two-dimensional airfoils in the Virginia Tech Stability Wind Tunnel with similar experimental arrangements. Measurements included the surface pressure distribution about the airfoil, total pressure distribution in the airfoil wake, static pressure in the wake, angle of attack, freestream velocity, flow temperature, and ambient pressure. These data were used to train the PCA scheme through an eigendecomposition of the measurement covariance matrix. A low-order reconstruction of this covariance matrix was then used to assess whether new measurements not included in the training set were anomalous by comparing predictions of the expected result with measured values. Results show that the method is very good at detecting anomalies in data from airfoils included in the training set as well as an additional airfoil outside of this set. Predictions of a particular measurement can be further improved by biasing the data used to construct the covariance matrix with observations that are more similar to the measurement of interest. However, this biasing reduces the ability of the method to predict results and therefore detect anomalies for experimental conditions outside of the range of the input data.

ZUPT-Aided INS Bypassing Stance Phase Detection by Using Foot-Instability-Based Adaptive Covariance

In this paper, we propose a Foot-Instability-Based Adaptive (FIBA) covariance to dynamically adjust the covariance matrix for the pseudo-zero-velocity measurements in the Zero velocity UPdaTe (ZUPT)-aided Inertial Navigation Systems (INS). The proposed ZUPT-aided INS using the FIBA covariance is implemented in an Extended Kalman Filter (EKF) framework, which utilizes a time-varying measurement covariance matrix that is updated in each iteration according to the FIBA covariance. The FIBA covariance is designed to have a very high value during the swing phases in a gait cycle, and the value significantly decreases during the stance phases. As a result, the proposed method eliminates a need to use a binary stance phase detector in implementation of the ZUPT-aided INS. Properties of EKF innovation sequences in the algorithm were studied, and two series of indoor pedestrian navigation experiments were conducted to demonstrate the navigation performance of the proposed system. In the first series of experiments, which included cases of walking and running, localization solutions produced by the system using the FIBA covariance demonstrated 36% and 64% improvements in navigation accuracy along the horizontal and vertical directions, respectively. In the second series of experiments, which included a pedestrian walking on different indoor terrains, such as flat planes, stairs, and ramps, the navigation accuracy of the system using the FIBA covariance reduced horizontal and vertical position errors by 12% and 45%, respectively, as compared to the conventional ZUPT-aided INS.

On the general conditions of existence for linear MMSE filters: Wiener and Kalman

State estimation is a fundamental task in a plethora of applications. Two important classes of linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman filters (KFs), the latter being the recursive form of the former for linear discrete state-space (LDSS) dynamic models. The minimum set of uncorrelation conditions regarding states and measurements which allows to formulate a general KF form has been recently introduced in the literature, but for both WFs and KFs a standard key assumption is that the measurement noise covariance is invertible. In this contribution we provide i) an alternative derivation of the WF in the case of a non-invertible measurement covariance matrix, and ii) the proof that under the minimum set of uncorrelation conditions lately released, the KF can always be formulated irrespective of the measurement covariance matrix rank. Notice that these results also apply for Kalman predictors and smoothers, linearly constrained WFs and minimum variance distortionless response (MVDR) filters, therefore being a remarkable result of broad interest.

A novel adaptive Kalman filter for Euler-angle-based MEMS IMU/magnetometer attitude estimation

This paper introduces a novel Euler-angle-based adaptive attitude filter for micro-electro-mechanical system inertial measurement unit (IMU)/magnetometer fusion. An intelligent coordinate switch algorithm is proposed to overcome the singularity problem that exists in the Euler-angle-based attitude calculation approach. This method rotates the physical IMU body frame and forms a quasi-computational coordinate to perform the attitude and filter update. In addition, a novel noise estimation theorem based on redundant measurement systems is introduced, proofed and employed to adaptively tune the measurement covariance matrix R in the filter. This algorithm uses gyroscope short-term accuracy to evaluate the attitude performance derived from different sensors and optimally assigns weight in the filter update. Simulated and practical experiments are carried out to test the validation of the proposed adaptive attitude filter. The results achieved demonstrate that this approach can provide a promising orientation solution.

Gridless DOA Estimation and Root-MUSIC for Non-Uniform Linear Arrays

The problem of gridless direction of arrival (DOA) estimation is addressed in the non-uniform array (NUA) case. Traditionally, gridless DOA estimation and root-MUSIC are only applicable for measurements from a uniform linear array (ULA). This is because the sample covariance matrix of ULA measurements has Toeplitz structure, and both algorithms are based on the Vandermonde decomposition of a Toeplitz matrix. The Vandermonde decomposition breaks a Toeplitz matrix into its harmonic components, from which the DOAs are estimated. First, we present the `irregular' Toeplitz matrix and irregular Vandermonde decomposition (IVD), which generalizes the Vandermonde decomposition to apply to a more general set of matrices. It is shown that the IVD is related to the MUSIC and root-MUSIC algorithms. Next, gridless DOA is generalized to the NUA case using IVD. The resulting non-convex optimization problem is solved using alternating projections (AP). A numerical analysis is performed on the AP based solution which shows that the generalization to NUAs has similar performance to traditional gridless DOA.

Analysis of changes in the coordinates of the “Tavaksay” geodynamic polygon

This article presents the results of GNSS and laser measurements at the “Tavaksay” geodynamic polygon. Describes the classical methods for determining the change in coordinates based on triangulation and leveling. Ways to improve the accuracy of local microplate tectonics using navigation systems are outlined here.The coordinates of the benchmarks were calculated in the rectangular GaussKruger coordinate system and the spatial system B, L, H for CK42 and WGS84.The trilateration method for determining the distances between points of the geodynamic network is analyzed. An assessment of the accuracy of the coordinates of points and the distances between them, depending on the number of measurement cycles, has been made.The accuracy of the trilateration and GNSS method for this network is graphically presented. The diagonal elements of the covariance matrix of GNSS measurements are investigated. It is proposed to make homogeneous GNSS and laser measurements at the tops of the hills at a certain time interval to study local deformation processes.

An Improved Tobit Kalman Filter with Adaptive Censoring Limits

This paper deals with the Tobit Kalman filtering (TKF) process when the measurements are correlated and censored. The case of interval censoring, i.e., the case of measurements which belong to some interval with given censoring limits, is considered. Two improvements of the standard TKF process are proposed, in order to estimate the hidden state vectors. Firstly, the exact covariance matrix of the censored measurements is calculated by taking into account the censoring limits. Secondly, the probability of a latent (normally distributed) measurement to belong in or out of the uncensored region is calculated by taking into account the Kalman filter residual. The designed algorithm is tested using both synthetic and real data sets. The real data set includes human skeleton joints’ coordinates captured by the Microsoft Kinect II sensor. In order to cope with certain real-life situations that cause problems in human skeleton tracking, such as (self)-occlusions, closely interacting persons, etc., adaptive censoring limits are used in the proposed TKF process. Experiments show that the proposed method outperforms other filtering processes in minimizing the overall root-mean-square error for synthetic and real data sets.