Measurement Noise Covariance Matrix Research Articles

A Combined UWB/IMU Localization Method with Improved CKF.

Aiming at the problem that ultra-wide band (UWB) cannot be accurately localized in environments with large noise variations and unknown statistical properties, a combinatorial localization method based on improved cubature (CKF) is proposed. First, in order to overcome the problem of inaccurate local approximation or even the inability to converge due to the initial value not being set near the optimal solution in the process of solving the UWB position by the least-squares method, the Levenberg-Marquardt algorithm (L-M) is adopted to optimally solve the UWB position. Secondly, because UWB and IMU information are centrally fused, an adaptive factor is introduced to update the measurement noise covariance matrix in real time to update the observation noise, and the fading factor is added to suppress the filtering divergence to achieve an improvement for the traditional CKF algorithm. Finally, the performance of the proposed combined localization method is verified by field experiments in line-of-sight (LOS) and non-line-of-sight (NLOS) scenarios, respectively. The results show that the proposed method can maintain high localization accuracy in both LOS and NLOS scenarios. Compared with the Extended Kalman filter (EKF), unbiased Kalman filter (UKF), and CKF algorithms, the localization accuracies of the proposed method in NLOS scenarios are improved by 25.2%, 18.3%, and 11.3%, respectively.

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.

Adaptive IMM Smoothing Algorithms for Jumping Markov System With Mismatched Measurement Noise Covariance Matrix

Adaptive IMM Smoothing Algorithms for Jumping Markov System With Mismatched Measurement Noise Covariance Matrix

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.

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.

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.

Real-time water demand pattern estimation using an optimized extended Kalman filter

This study presents a hybrid approach for the estimation of real-time water demand multipliers using the Kalman filter (KF) and extended Kalman filter (EKF). Multiple Linear Regression (MLR) and Nonlinear Regression (NLR) models were applied to predict water demand multipliers at each time step with historical flow data. The estimation performance of EKF is highly affected by the state noise covariance matrix (Q) and the measurement noise covariance matrix (R). An inappropriate value ofQandRsignificantly degrades the EKF’s performance and makes the filter diverge. So, the particle swarm optimization algorithm (PSO) was used to tune the noise covariance matricesQandRat each time step of EKF. Then the optimal values of noise covariance matrices are inserted in the real-time water demand multiplier estimation process. To find the optimal values ofQandR, the mean absolute percentage error (MAPE) between measured and simulated pressure was minimized. The proposed method was evaluated in Net1 and Net3 benchmark networks. The root means square error (RMSE) of EKF-PSO estimated water demand multiplier for Net1 and Net3 were 0.063 and 0.198, respectively. The simulation results indicated that the EKF-PSO algorithm was more accurate than the conventional EKF algorithm. Moreover, the KF-PSO performed poorly when dealing with nonlinear hydraulic systems.

Variational Bayesian and generalized maximum-likelihood based adaptive robust nonlinear filtering framework

An adaptive robust nonlinear filtering framework is proposed based on the variational Bayesian (VB) method and generalized maximum likelihood estimation (GM estimation) to simultaneously handle the outliers and uncertain measurement noise covariance matrix (MNCM) for state estimation. This framework utilizes the cubature criterion to solve nonlinear integration problems. By embedding GM estimation in the VB method, the uncertain MNCM is modified while robust estimation is performed. The modified MNCM also provides more accurate model parameters for GM estimation, achieving the unity of adaptability and robustness. The VBCEECKF algorithm was obtained by applying the strong robust centered error entropy criterion to the adaptive robust nonlinear filtering framework. The simulation results of target tracking under different noise conditions verify the superiority of the adaptive robust filter compared to existing methods.

Application of adaptive extended Kalman algorithm based on strong tracking fading factor in Stat-of-Charge estimation of lithium-ion battery

Accurately estimating the state of charge (SOC) of a lithium-ion battery is the key to a battery management system (BMS). This paper proposes an adaptive extended Kalman algorithm (ASTEKF) based on strong tracking fade factor to address the issues that the extended Kalman filter algorithm cannot track the system state in real time and cannot accurately estimate the measurement noise covariance matrix when estimating the state of charge of lithium-ion batteries. The extended Kalman algorithm incorporates the fade factor of the strong tracking filter to improve tracking performance. The adaptive algorithm uses the relationship between prior residual and posterior residual to re-determine the value of the fading factor. On this basis, the window-opening method is introduced to ensure the stability of the fading factor while making it self-adaptive. The measurement noise covariance matrix is updated according to the predicted and estimated values, and the measurement noise covariance matrix is estimated and corrected in real time. Finally, a pulse discharge experiment is performed, and the estimated and experimental results are compared. According to the findings, the maximum error of the ASTEKF algorithm in SOC estimation is reduced by 24.6%, the average error is reduced by 25.5%, and the root mean square error is reduced by 64.7% when compared to the conventional EKF algorithm.

A Robust Adaptive Extended Kalman Filter Based on an Improved Measurement Noise Covariance Matrix for the Monitoring and Isolation of Abnormal Disturbances in GNSS/INS Vehicle Navigation

Global Navigation Satellite Systems (GNSS) integrated with Inertial Navigation Systems (INS) have been widely applied in many Intelligent Transport Systems. However, due to the influence of various factors, such as complex urban environments, etc., accurately describing the measurement noise statistics of GNSS receivers and inertial sensors is difficult. An inaccurate definition of the measurement noise covariance matrix will lead to the rapid divergence of the position error of the integrated navigation system. To overcome this problem, this paper proposed a Robust Adaptive Extended Kalman Filter (RAKF) method based on an improved measurement noise covariance matrix. By analyzing and considering the position accuracy factors, measurement factor, and position standard deviation in GNSS measurement results, this paper constructed the optimal measurement noise covariance matrix. Based on the Huber model, this paper constructed a two-stage robust adaptive factor expression and obtained the robust adaptive factors with and without abnormal disturbances. And robust adaptive filtering was carried out. To assess the performance of this method, the author conducted experiments on land vehicles by using a self-developed POS system (GNSS/INS combined navigation system). The classic Extended Kalman Filter algorithm (EKF), Adaptive Kalman Filter (AKF) algorithm, Robust Kalman Filter (RKF) algorithm, and the proposed method were compared through data processing. Experimental results show that compared with the classical EKF, AKF, and RKF, the positioning accuracies of the proposed method were improved by 72.43%, 2.54%, and 47.82%, respectively, in the vehicle land experiment. In order to further evaluate the performance of this method, the vehicle data were subjected to different times and degrees of disturbance experiments. Experimental results show that compared with EKF, AKF, and RKF, the heading angle accuracy had obvious advantages, and its accuracy was improved by 34.65%, 31.53%, and 18.36%, respectively. Therefore, this method can effectively monitor and isolate disturbance and improve the robustness, reliability, accuracy, and stability of GNSS/INS integrated navigation systems in complex urban environments.

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 Improved Multiple-Outlier Robust Filter Based on Maximum Correntropy Criterion for Integrated Navigation

In the integrated navigation of global navigation satellite system (GNSS) and Inertial navigation system (INS), the Kalman filter is frequently employed for state estimation. GNSS signals are subject to interference from the outside environment, and it is difficult for the classical Kalman filter to handle observations with outliers. Numerous robust filters have been proposed based on the maximum correntropy criterion (MCC), which is insensitive to non-Gaussian noise, to address the issue of state estimation in the presence of outliers. However, a fixed prior covariance will restrict the use of a maximum correntropy criterion Kalman filter (MCKF) in integrated navigation systems because GNSS signals can be affected by both time-varying noise and outliers. Existing filters consider the prior measurement noise covariance matrix (MNCM) as a deterministic matrix and can only amplify signals unidirectionally during iteration cycles. The weight of effective observations is reduced despite these filters having strong robustness. To overcome this shortcoming, this paper proposes a τ -based maximum correntropy criterion Kalman filter (τ -MCKF). A strategy based on the Chisquare test is designed to optimize the covariance of GNSS observations. By introducing τ to bi-directionally adjust the initial MNCM, τ -MCKF makes full use of effective observations. Then, the τ -corrected MNCM is used to weigh different elements of the innovation vector, and a robust estimation is calculated based on the multiple-outlier robust Kalman filter (MORKF) framework. The GNSS/INS integrated navigation system is taken as an example to verify the effectiveness of the filter through simulation and vehicular experiments.

Variational adaptive Kalman filter for unknown measurement loss and inaccurate noise statistics

Considering a common situation that the measurements are obtained from independent sensors and the accurate noise statistics are not available, we propose a novel variational adaptive Kalman filter (KF), which can selectively treat measurement loss and adaptively estimate inaccurate state and measurement noise covariance matrices. Firstly, a multiple inverse-Wishart mixture (MIWM) distribution is utilized to modeled state transition probability density function (PDF), which reduces the dependence on the pre-selected nominal state noise covariance matrix (SNCM). Then, a modified measurement model is constructed and a new measurement likelihood PDF is provided, where the measurement losses from different sensors are considered independently and the measurement noise covariance matrix (MNCM) is modeled as an inverse Gamma distribution. Finally, based on the modified state transition and measurement likelihood PDFs, a novel variational adaptive KF is derived by variational Bayesian method, and the feasibility and superiority of the filter are demonstrated by the numerical simulation.

A Fast Variational Bayesian Adaptive Extended Kalman Filter for Robust Underwater Direction-of-Arrival Tracking

The underwater direction-of-arrival (DOA) tracking using hydrophone array is an important research subject in passive sonar signal processing. To deal with the unknown measurement noise results from underwater environmental noise, the existing variational Bayesian adaptive extended Kalman filter (VB-AEKF) technique for robust underwater DOA tracking jointly estimates the measurement noise covariance matrix and the bearing angle of a target. The iteration process and the nonlinear measurement model result in significant computational complexity of the VB-AEKF. By deriving computational process which is equivalent to VB-AEKF but more computationally efficient, a fast variational Bayesian adaptive extended Kalman filter (FVB-AEKF) for robust DOA tracking is proposed. The proposed method is tested via Monte Carlo simulations of an underwater DOA tracking scenario and an experiment in the South China Sea in July 2021. The results of the simulations and the experiment verified the accuracy and the computational efficiency of the FVB-AEKF.

An Improved UWB/IMU Tightly Coupled Positioning Algorithm Study.

The combination of ultra-wide band (UWB) and inertial measurement unit (IMU) positioning is subject to random errors and non-line-of-sight errors, and in this paper, an improved positioning strategy is proposed to address this problem. The Kalman filter (KF) is used to pre-process the original UWB measurements, suppressing the effect of range mutation values of UWB on combined positioning, and the extended Kalman filter (EKF) is used to fuse the UWB measurements with the IMU measurements, with the difference between the two measurements used as the measurement information. The non-line-of-sight (NLOS) measurement information is also used. The optimal estimate is obtained by adjusting the system measurement noise covariance matrix in real time, according to the judgment result, and suppressing the interference of non-line-of-sight factors. The optimal estimate of the current state is fed back to the UWB range value in the next state, and the range value is dynamically adjusted after one-dimensional filtering pre-processing. Compared with conventional tightly coupled positioning, the positioning accuracy of the method in this paper is improved by 46.15% in the field experimental positioning results.