Measurement Noise Covariance Research Articles

Robust adaptive Kalman filter for structural performance assessment

SummaryStructural performance assessment is a critical stage in the health monitoring for the maintenance and risk management of engineering structures, in which the interstory drift is a key indicator. In this paper, a robust adaptive Kalman filter is proposed for an interstory drift estimation problem to show the health condition of steel structures in the case that the statistics or internal dynamics describing the signals and measurements are not known precisely. More precisely, we build an adaptive current Jerk model (ACJM) where the model parameters are updated in each time step to presuppose the statistics characterization of the steel dynamic, while the unknown measurement noise covariance is adapted based on a fixed‐lag innovation with respect to measurements. Moreover, a robust adaptive Kalman filter is designed for the modelling uncertainties in each time increment by solving a minimax game: one player tries to select an “actual” model far from the proposed ACJM with an exponential decay tolerance, while its “hostile” player, namely, the optimum filter, is designed by minimizing the estimation error according to the selected “actual” model. Finally, some simulation and experimental results show the effectiveness of the proposed algorithm.

An Iteratively Extended Target Tracking by Using Decorrelated Unbiased Conversion of Nonlinear Measurements.

Extended target tracking (ETT) based on random matrices typically assumes that the measurement model is linear. However, nonlinear measurements (such as range and azimuth) depending on locations of a series of unknown scattering centers always exist in many practical tracking applications. To address this issue, this paper proposes an iteratively extended target tracking based on random matrices by using decorrelated unbiased conversion of nonlinear measurements (ETT-IDUCM). First, we utilize a decorrelated unbiased converted measurement (DUCM) method to convert nonlinear measurements depending on unknown scatters of target extent in polar coordinates into the ones in Cartesian coordinates with equivalent measurement noise covariances. Subsequently, a novel method, combining iteratively extended Kalman filter (IEKF) updates with variational Bayesian (VB) cycles is developed for precise estimation of the target's kinematic state and extension. This method leverages the synergy between external IEKF iterations, which use the estimated state as a new prediction and input for DUCM, and internal VB iterations, which realize a closed-form approximation of the joint posterior probability. This approach progressively enhances estimation accuracy. Simulation results demonstrate the ETT-IDUCM algorithm's superior precision in estimating the target's kinematic state and extension compared to existing methods.

Seamless Micro-Electro-Mechanical System-Inertial Navigation System/Polarization Compass Navigation Method with Data and Model Dual-Driven Approach.

The integration of micro-electro-mechanical system-inertial navigation systems (MEMS-INSs) with other autonomous navigation sensors, such as polarization compasses (PCs) and geomagnetic compasses, has been widely used to improve the navigation accuracy and reliability of vehicles in Internet of Things (IoT) applications. However, a MEMS-INS/PC integrated navigation system suffers from cumulative errors and time-varying measurement noise covariance in unknown, complex occlusion, and dynamic environments. To overcome these problems and improve the integrated navigation system's performance, a dual data- and model-driven MEMS-INS/PC seamless navigation method is proposed. This system uses a nonlinear autoregressive neural network (NARX) based on the Gauss-Newton Bayesian regularization training algorithm to model the relationship between the MEMS-INS outputs composed of the specific force and angular velocity data and the PC heading's angular increment, and to fit the integrated navigation system's dynamic characteristics, thus realizing data-driven operation. In the model-driven part, a nonlinear MEMS-INS/PC loosely coupled navigation model is established, the variational Bayesian method is used to estimate the time-varying measurement noise covariance, and the cubature Kalman filter method is then used to solve the nonlinear problem in the model. The robustness and effectiveness of the proposed method are verified experimentally. The experimental results show that the proposed method can provide high-precision heading information stably in complex, occluded, and dynamic environments.

An Adaptive Unscented Kalman Filter for the Estimation of the Vehicle Velocity Components, Slip Angles, and Slip Ratios in Extreme Driving Manoeuvres.

This paper presents a novel unscented Kalman filter (UKF) implementation with adaptive covariance matrices (ACMs), to accurately estimate the longitudinal and lateral components of vehicle velocity, and thus the sideslip angle, tire slip angles, and tire slip ratios, also in extreme driving conditions, including tyre-road friction variations. The adaptation strategies are implemented on both the process noise and measurement noise covariances. The resulting UKF ACM is compared against a well-tuned baseline UKF with fixed covariances. Experimental test results in high tyre-road friction conditions show the good performance of both filters, with only a very marginal benefit of the ACM version. However, the simulated extreme tests in variable and low-friction conditions highlight the superior performance and robustness provided by the adaptation mechanism.

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 meticulous covariance adaptive Kalman filter for satellite attitude estimation

Aiming at the problems of model errors, non-Gaussian noise and measurement anomaly in the spacecraft attitude estimation system, this article proposes an improved adaptive filtering method based on covariance matching, which solves the problems of simultaneous dynamics model error and measurement model error in the attitude estimation system, and at the same time, effectively reduces the effects of non-Gaussian noise and large outlier situations occurring in the vector measurement sensor. Firstly, an adaptive filtering algorithm based on the innovation sequence estimation covariance is investigated under the framework of multiplicative extended Kalman filter (MEKF), which is used to correct process noise covariance, then the Sage–Husa adaptive Kalman filtering (SHAKF) method is combined to correct the measurement noise covariance, and finally the meticulous covariance adaptive multiplicative extended Kalman filter is designed. The proposed algorithm uses both innovation and SHAKF methods to correct the two covariance matrices simultaneously. Several attitude estimation simulation scenarios are set up to simulate the proposed algorithm in the presence of model errors, non-Gaussian noise, and large outlier. The simulation results demonstrate that the proposed algorithm outperforms the conventional algorithms in terms of estimation accuracy and robustness.

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.

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

Velocity estimation of a rigid body using measurements from low-cost inertial and magnetic sensors plays an important role in various applications. The extended Kalman filter (EKF) is widely used for this purpose. However, EKF's estimation performance relies on the knowledge of process and measurement noise covariance matrices, and this information is generally unavailable. In this work, we introduce a solution that combines two techniques: the generation of velocity pseudo-measurements using a Bidirectional Long Short-Term Memory (BiLSTM) network, and the Q-learning method for online adaptation of noise covariance matrices. The performance of the proposed solution is validated using real experimental datasets, demonstrating that Q-learning can select appropriate noise covariance matrices to enhance velocity estimation.

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.

Fault-Tolerant Cubature Kalman Filter for Engineering Estimation Control Systems.

The cubature Kalman filter (CKF) overcomes the limitations of the Kalman filter in strong nonlinear systems, which has been widely used in many fields. However, in practical engineering, the abnormal measurement information obtained by the sensor causes the measurement noise covariance to change, which may deteriorate the filtering performance and even cause the filter failure. The fault-tolerant filter can deal with the state estimation problem for the systems with abnormal measurements. The key of the fault-tolerant filter is to forcefully correct filter innovation by using a fading factor. The fault-tolerant filter technology has been extensively applied in many practical systems, but it is still lack of reasonable theoretical analysis. To this end, the measurement noise model is established and the magnitude of the noise deviation is analyzed. The filtering performance under abnormal measurement is analyzed by three mean squared errors (MSEs), which are the ideal MSE, the filter calculated MSE and the true MSE. In order to solve the influence of sampling approximation deviation of CKF on fault detection, an improved fault detection algorithm is proposed. The performance of fault-tolerant CKF is analyzed from two views. The first view is about comparing the filter calculated MSEs of CKF and of fault-tolerant CKF, the second view is about comparing the relative closeness of the filter calculated MSE to the true MSE for the two algorithms. Numerical examples further verify these conclusions.

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.

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.

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.

SOC estimation for lithium-ion battery based on AGA-optimized AUKF

The window size for covariance matching (CM) of the adaptive unscented Kalman filter (AUKF) affects the state of charge (SOC) estimation performance due to changes with time in the distribution of error innovation sequence (EIS). A new adaptive genetic algorithm (AGA) to address this problem is proposed in this paper. The proposed AUKF (AGA-AUKF1) obtains its best window size determined by the AGA. A novel AUKF (AGA-AUKF2) is proposed to prevent the uncertainty caused by time-varying EIS with the combination of AGA and CM methods. Firstly, the influence of different temperatures on the prediction performance of the algorithm is investigated by FUDS data. The influence of various parameters on the algorithm is further analyzed by FUDS. For different temperatures, initial SOC values, and initial measurement noise covariance, the SOC estimation results show that the accuracy of AGA-AUKFs is better than AUKF. The population size and termination algebra simulation results indicate that the proposed AGA performs well in parameter optimization. Subsequently, the SOC estimation capability of two proposed methods in different working conditions is analyzed by BJDST and US06. The results show that AGA-AUKF2 has better accuracy and robustness than AGA-AUKF1.

Extended target tracking under multitarget tracking framework for convex polytope shapes

This paper discusses the problem of extended target tracking for a single 2D extended target with a convex polytopic shape and known dynamics. Extended targets are those that produce multiple measurements for a single frame. One of the major challenges in extended target tracking is the joint uncertainty in the shape and the kinematics of the target. Another challenge is the lack of visibility due to self-occlusion in targets with a finite extent (as opposed to zero extent for point targets). To address these challenges, we develop a framework for tracking single (or widely separated) extended targets. This framework is based on the existing point multitarget tracking framework by modeling different parts of an extended target as separate targets. An algorithm is developed using the proposed framework for tracking convex polytope-shaped targets. The proposed shape function consists only of the boundary of the target since the center may not be observable. The algorithm is capable of dynamically changing the number of parameters used to describe the shape as more parts of the target become visible over time. The performance of the algorithm is evaluated for various scenarios using root mean square error (RMSE) of velocity, center, and intersection over union (IoU) metrics. It is seen that the algorithm is able to handle the self-occlusion problem and estimate the whole target shape even when different parts of the target are visible at different frames, for various shapes, and for various conditions of measurement noise covariance and number of measurements. New faces are added to the shape estimate as more parts of the target become visible. The algorithm is able to conserve the parts of the target that were visible in the previous frames but are no longer visible.

Synchronous identification of nonlinear structural parameters and unknown external excitation based on improved UKF

The traditional Unscented Kalman Filter (UKF) approach for nonlinear structural identification usually requires knowledge of external excitation. However, in real application, obtaining the input excitation of nonlinear structures can be challenging due to measurement error or difficulty in acquiring the accurate information. Consequently, the real-time application of the UKF method will be constrained. To address this issue, this paper develops an improved UKF-based synchronous identification method of nonlinear structural parameters and unknown external excitation. The proposed method starts with a preliminary estimate of the unknown excitation using the current predicted values of structural dynamic responses and parameters. Subsequently, the external excitation is further identified based on the updated state vector. In addition, to reduce the effect of measurement noise on the nonlinear structural identification and ensure the accuracy of the identified results, a Kalman Filter (KF) process is embedded in the UKF to optimize measurement noise covariance matrix in real time. The practicability and accuracy of the improved UKF method are confirmed by conducting numerical simulations on a single degree-of-freedom (S-DOF) and a six degree-of-freedom (six-DOF) nonlinear structural systems due to unknown seismic excitation. Moreover, a shake table test of a five-storey steel frame subjected to seismic excitation is employed to attest the feasibility of the proposed method. Both numerical and experimental results demonstrate that the improved UKF method can effectively achieve synchronous identification for nonlinear structural parameters and unknown excitation.

An adaptive outlier-robust Kalman filter based on sliding window and Pearson type VII distribution modeling

Due to inaccurate noise modeling and only using current measurement, existing robust filtering algorithms cannot obtain good state estimation results when measurement is interfered by outliers. In this article, a Pearson type VII (PTV) distribution-based adaptive sliding window outlier-robust Kalman filter (PTVSWAKF) is proposed. According to the characteristics of the parameters, the PTV and inverse Wishart distributions are respectively modeled as prior distributions for heavy-tailed measurement noises (HMN) and covariance. Two different Gamma distributions are modeled as priori distributions for the two degree of freedom (DOF) parameters to achieve adaptive updating of them. By the smooth posterior distribution of the sliding window state values, the state, measurement noise covariance, auxiliary random variable and DOF parameters are jointly iterated and updated to solve the outlier interference problem through variational Bayesian. Finally, the simulation experiments verify the proposed PTVSWAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

Determination of the dynamics of the wheel of a mobile machine

A sensor, which consists of a three-axis accelerometer, a gyroscope, and a magnetometer, has been developed to determine the dynamics of the wheel of a mobile machine. The sensor is connected to a microcontroller, which transmits the received data via a 2.4 GHz channel. It is mounted coaxial to the center of the wheel. At the first processing stage, the accelerometer, gyroscope, and magnetometer data values are adjusted. The corrected accelerations and angular velocities signal are processed using a Butterworth filter. Filter Madgwick determines the angles of orientation of the sensor in space. In the next step, we remove the centrifugal component from the accelerations. Next, the gravitational component is subtracted from the accelerations, and its real value is obtained. The speed of the wheel is obtained by integrating accelerations. The Kalman filter processes the angular velocities, acceleration, and velocities. The measurement noise covariance matrix and the evaluation process matrix are calculated based on sensor errors. The results of experimental studies of the proposed method and sensor for determining the wheel dynamics of mobile machines on tractors with a 4x4 wheel arrangement were carried out. The sensors were installed on all wheels of the tractor. The developed wheel dynamics sensor allows us to determine the slippage of the engines of a mobile machine. This method of determining slipping is entirely accurate and does not require intervention in the design of the machine. The determination of the speed of rotation of the wheels of a mobile machine is based on the determination of the angular speed of rotation, which is measured with a gyroscope and accelerometer located in the center of the wheel of a mobile machine. Keywords: sensor, dynamics, wheel, mobile car, filter.

Attitude Determination System for a Cubesat Experiencing Eclipse.

In the context of Kalman filters, the predicted error covariance matrix Pk+1 and measurement noise covariance matrix R are used to represent the uncertainty of state variables and measurement noise, respectively. However, in real-world situations, these matrices may vary with time due to measurement faults. To address this issue in CubeSat attitude estimation, an adaptive extended Kalman filter has been proposed that can dynamically estimate the predicted error covariance matrix and measurement noise covariance matrix using an expectation-maximization approach. Simulation experiments have shown that this algorithm outperforms existing methods in terms of attitude estimation accuracy, particularly in sunlit and shadowed phases of the orbit, with the same filtering parameters and initial conditions.