Traditional Extended Kalman Filter Research Articles

LEO-Enhanced GNSS/INS Tightly Coupled Integration Based on Factor Graph Optimization in the Urban Environment

Precision point positioning (PPP) utilizing the Global Navigation Satellite System (GNSS) is a traditional and widely employed technology. Its performance is susceptible to observation discontinuities and unfavorable geometric configurations. Consequently, the integration of the Inertial Navigation System (INS) and GNSS makes full use of their respective advantages and effectively mitigates the limitations of GNSS positioning. However, the GNSS/INS integration faces significant challenges in complex and harsh urban environments. In recent years, the geometry between the user and the satellite has been effectively improved with the advent of lower-orbits and faster-speed Low Earth Orbit (LEO) satellites. This enhancement provides more observation data, opening up new possibilities and opportunities for high-precision positioning. Meanwhile, in contrast to the traditional extended Kalman filter (EKF) approach, the performance of the LEO-enhanced GNSS/INS tightly coupled integration (TCI) can be significantly improved by employing the factor graph optimization (FGO) method with multiple iterations to achieve stable estimation. In this study, LEO data and the FGO method were employed to enhance the GNSS/INS TCI. To validate the effectiveness of the method, vehicle data and simulated LEO observations were subjected to thorough analysis. The results suggest that the integration of LEO data significantly enhances the positioning accuracy and convergence speed of the GNSS/INS TCI. In contrast to the FGO GNSS/INS TCI without LEO enhancement, the average enhancement effect of the LEO is 22.16%, 7.58%, and 10.13% in the north, east, and vertical directions, respectively. Furthermore, the average root mean square error (RMSE) of the LEO-enhanced FGO GNSS/INS TCI is 0.63 m, 1.21 m, and 0.85 m in the north, east, and vertical directions, respectively, representing an average improvement of 41.91%, 13.66%, and 2.52% over the traditional EKF method. Meanwhile, the simulation results demonstrate that LEO data and the FGO method effectively enhance the positioning and convergence performance of GNSS/INS TCI in GNSS-challenged environments (tall buildings, viaducts, underground tunnels, and wooded areas).

High-Order Extended Kalman Filter for State Estimation of Nonlinear Systems

In general, the extended Kalman filter (EKF) has a wide range of applications, aiming to minimize symmetric loss function (mean square error) and improve the accuracy and efficiency of state estimation. As the nonlinear model complexity increases, rounding errors gradually amplify, leading to performance degradation. After multiple iterations, divergence may occur. The traditional extended Kalman filter cannot accurately estimate the nonlinear model, and these errors still have an impact on the accuracy. To improve the filtering performance of the extended Kalman filter (EKF), this paper proposes a new extended Kalman filter (REKF) method that utilizes the statistical properties of the rounding error to enhance the estimation accuracy. After establishing the state model and measurement model, the residual term is used to replace the higher-order term in the Taylor expansion, and the least squares method is applied to identify the residual term step by step. Then, the iterative process of updating the extended Kalman filter is carried out. Within the Kalman filter framework, a higher-order rounding error-based extended Kalman filter (REKF) is designed for the joint estimation of rounding error and random variables, and the solution method for the rounding error is considered for the multilevel approximation of the original function. Through numerical simulations on a general nonlinear model, the higher-order rounding error-based extended Kalman filter (REKF) achieves better estimation results than the extended Kalman filter (EKF) and improves the filtering accuracy by utilizing the higher-order rounding error information, which also proves the effectiveness of the proposed method.

Performance enhancement of diffuse fluorescence tomography based on an extended Kalman filtering-long short term memory neural network correction model.

To alleviate the ill-posedness of diffuse fluorescence tomography (DFT) reconstruction and improve imaging quality and speed, a model-derived deep-learning method is proposed by combining extended Kalman filtering (EKF) with a long short term memory (LSTM) neural network, where the iterative process parameters acquired by implementing semi-iteration EKF (SEKF) served as inputs to the LSTM neural network correction model for predicting the optimal fluorescence distributions. To verify the effectiveness of the SEKF-LSTM algorithm, a series of numerical simulations, phantom and in vivo experiments are conducted, and the experimental results are quantitatively evaluated and compared with the traditional EKF algorithm. The simulation experimental results show that the proposed new algorithm can effectively improve the reconstructed image quality and reconstruction speed. Importantly, the LSTM correction model trained by the simulation data also obtains satisfactory results in the experimental data, suggesting that the SEKF-LSTM algorithm possesses strong generalization ability and great potential for practical applications.

Target Tracking Algorithm Based on Adaptive Strong Tracking Extended Kalman Filter

Kalman filtering is a common filtering method for millimeter-wave traffic radars. The proposal is for an Adaptive Strong Tracking Extended Kalman Filter (EKF) algorithm that aims to address the issues of classic EKF’s low accuracy and lengthy convergence time. This method, which incorporates time-varying fading effects into the covariance matrix of the traditional EKF, is based on the ST algorithm. It allows the recalibration of the covariance matrix and precise filtering and state estimation of the target vehicle. By altering the fading and attenuating factors of the ST algorithm and using orthogonality principles, many fine-tuned fading factors produced from least-squares optimization are introduced together with regionally optimum attenuation factors. The results of Monte Carlo experiments indicate that the average velocity inaccuracy is reduced by at least 38% in comparison to existing counterparts. The results validate the efficacy of this methodology in observing vehicular movements in metropolitan regions, satisfying the prerequisites of millimeter-wave radar technology for traffic monitoring.

An Extended Kalman Filter Design for State-of-Charge Estimation Based on Variational Approach

State of charge (SOC) is a very important variable for using batteries safely and reliably. To improve the accuracy of SOC estimation, a novel variational extended Kalman filter (EKF) technique based on least square error method is herein provided by establishing a second-order equivalent circuit model for the battery. It was found that when SOC decreased, resistance polarization occurred in the electrochemical model, and the parameters in the equivalent RC model varied. To decrease the modeling error in the equivalent circuit model, the system parameters were identified online depending on the SOC’s estimated result. Through the SOC-estimation process, the variation theorem was introduced, which enabled the system parameters to track the real situations based on the output measured. The experiment results reveal the comparison of the SOC-estimation results of the variational EKF algorithm, the traditional EKF algorithm, the recursive least square (RLS) EKF algorithm, and the forgotten factor recursive least square (FFRLS) EKF algorithm based on different indices, including the mean square error (MSE) and the mean absolute error (MAE). The variational EKF algorithm provided in this paper has higher estimation accuracy and robustness than the traditional EKF, which verifies the superiority and effectiveness of the proposed method.

Robust adaptive smooth variable structure Kalman filter for spacecraft attitude estimation

This paper proposes a novel robust adaptive filter that combines smooth variable structure filter and extended Kalman filter for the spacecraft attitude estimation with model uncertainty originating from gyro drift, parameter error, and installation error. The proposed filter employs the extended Kalman filter based on quaternion and fuses an adaptive factor and smooth variable structure filter to improve robustness to the model uncertainty and noise. To improve the robustness of the extended Kalman filter, an adaptive factor that tunes the predicted covariance matrix is utilized to design the attitude estimation strategy. Moreover, to further decrease the impact of the suddenly rapid maneuvering and improve its robust effect of model uncertainty induced by dynamic error, gyro drift, and so on, a smooth variable structure estimation strategy founded on sliding mode theory is derived and utilized for the satellite attitude estimation. Thereby, a novel filtering algorithm is proposed by incorporating an adaptive factor and a smooth variable structure estimation strategy. Simulation results indicate that our proposed algorithm delivers exceptional robustness and achieves high precision and accuracy in satellite attitude estimation compared with the traditional extended Kalman filter.

Cooperative localization for master–salve multi-AUVs based on range measurements

Underwater localization has consistently remained a prominent technical challenge for autonomous underwater vehicles (AUVs). The advent of cooperative localization techniques has emerged as a novel avenue for enhancing localization accuracy. The master–slave cooperative localization mode has gained widespread adoption due to its cost-effectiveness in implementation. In view of the complexity of underwater noise characteristics, in the multi-AUVs cooperative localization system, this paper addresses scenarios involving distance-dependent noise in a master–slave-based multi-AUVs cooperative localization system. To tackle the negative impact of distance-dependent noise and the non-linearity of the distance function, a two-step algorithm is proposed that combines maximum likelihood estimation and the Gaussian belief propagation algorithm (ML-GBP) to estimate the positions of AUVs. The maximum likelihood estimation is employed to cope with the interference caused by distance-dependent noise, and subsequently, the Gaussian belief propagation algorithm, based on range observations and reference information, is used to achieve accurate estimation of AUV positions and implement position correction. Simulation results demonstrate that the proposed ML-GBP algorithm outperforms traditional extended Kalman filter (EKF) and nonparametric belief propagation (NBP) methods by enhancing the localization accuracy of the system while exhibiting superior performance in terms of computational complexity and system communication overhead.

A Comparative Study of Factor Graph Optimization-Based and Extended Kalman Filter-Based PPP-B2b/INS Integrated Navigation

Recently, factor graph optimization (FGO)-based GNSS/INS integrated navigation has garnered widespread attention for its ability to provide more robust positioning performance in challenging environments like urban canyons, compared to traditional extended Kalman filter (EKF)-based methods. In existing GNSS/INS integrated navigation methods based on FGO, the primary approach involves combining single point positioning (SPP) or real-time kinematic (RTK) with INS by constructing factors between consecutive epochs to resist outliers and achieve robust positioning. However, the potential of a high-precision positioning system based on the FGO algorithm, combining INS and PPP-B2b and that does not rely on reference stations and network connections, has not been fully explored. In this study, we developed a loosely coupled PPP-B2b/INS model based on the EKF and FGO algorithms. Experiments in different urban road and overpass scenarios were conducted to investigate the positioning performance of the two different integration navigation algorithms using different degrades of inertial measurement units (IMUs). The results indicate that the FGO algorithm outperforms the EKF algorithm in terms of positioning with the combination of GNSS and different degrades of IMUs under various conditions. Compared to the EKF method, the application of the FGO algorithm leads to improvements in the positioning accuracy of approximately 15.8%~45.9% and 19%~41.3% in horizontal and vertical directions, respectively, for different experimental conditions. In scenarios with long and frequent signal obstructions, the advantages of the FGO algorithm become more evident, especially in the horizontal direction. An obvious improvement in positioning results is observed when the tactical-grade IMU is used instead of the microelectron-mechanical system (MEMS) IMU in the GNSS/INS combination, which is more evident for the FGO algorithm than for the EKF algorithm.

SOC estimation of lithium-ion batteries based on the improved Kalman filter

Nowadays, the environmental crisis and energy crisis have been known by people around the world as issues that have to be taken seriously. It is recognized that electric vehicles are more environmentally friendly than traditional fuel cars, but there are still many immature aspects of electric vehicle technology. Battery technology is one of them, and lithium battery is a common power storage tool for electric vehicles. Therefore, it is very important to monitor the status of the battery pack in real-time. Improving the practical capacity, safety, and service life of lithium batteries all have an important impact on the development of electric vehicles. In battery management systems (BMS), the accurate prediction of the battery state of charge (SOC) is particularly important. In this paper, the 18650 lithium battery was selected for the research experiment, and the circuit model of second-order RC was selected. Modeling by MATLAB, the SOC-OCV curve of the battery was determined through the data obtained from the charging and discharging experiment (HPPC), the original least squares method was improved, the forgetting factor was introduced, and the model was identified by the parameters based on the experimental data. The traditional SOC prediction method added noise adaptive rules and iterative theory in the extended Kalman filter(EKF), formed the adaptive Kalman filter (AIEKF) to estimate the SOC of lithium battery, and estimated the SOC using the American urban road cycle (UDDS) data. The results show that the AIEKF algorithm has fewer errors and advantages than the traditional EKF algorithm.

State of Charge Estimation for Power Battery Using Improved Extended Kalman Filter Method Based on Neural Network

In order to enhance the accuracy of the traditional extended Kalman filter (EKF) algorithm in the estimation of the state of charge (SoC) of power batteries, we first derived the state space equation and measurement equation of lithium power batteries based on the Thevenin battery model and the modified Ampere-Hour integralalgorithm. Then, the basic principles of EKF, backpropagation neural networks (BPNNs), and a biogeography-based optimization (BBO) algorithm were analyzed, and the arc curve mobility model was used to improve the global search ability of the BBO algorithm. By combining these three algorithms, this paper proposes a BP neural network method based on the BBO algorithm. This method uses the BBO algorithm to optimize the incipient weight and threshold of the BP neural network and uses this improved neural network to modify the estimated value of the extended Kalman filter algorithm (BBOBP-EKF). Finally, the BBOBP-EKF algorithm, the extended Kalman filter algorithm based on the BP neural network (BP-EKF), and the EKF algorithm are used to estimate the error value of the SOC of a power battery, and according to the experimental data, it was confirmed that the proposed BBOBP-EKF algorithm has been improved compared to other algorithms with respect to each error index term, in which the maximum error is 1% less than that of the BP-EKF algorithm and 2.4% less than that of the EKF algorithm, the minimum error is also the smallest, and the estimation accuracy is improved compared to the traditional algorithms.

Split-KalmanNet: A Robust Model-Based Deep Learning Approach for State Estimation

Estimation of the state of a discrete-time state-space model from noisy measurements is a crucial aspect of signal processing. The extended Kalman filter (EKF) is widely used as a low-complexity solution based on a state evolution and measurement model of the state-space model. However, obtaining precise information about these models can be difficult in practice, and model mismatch greatly reduces state estimation accuracy. In this paper, we introduce Split-KalmanNet, a robust EKF algorithm that utilizes the power of deep learning for state estimation. Split-KalmanNet is inspired by the recent KalmanNet and calculates the Kalman gain using the Jacobian matrix of a measurement function and two recurrent neural networks (RNNs) with a split structure. The RNNs independently learn the covariance matrices of the prior state estimate and the innovation from data. The proposed split structure in calculating the Kalman gain allows for compensation of both state and measurement model mismatch effects. Numerical simulations show that Split-KalmanNet outperforms traditional EKF and the state-of-the-art KalmanNet algorithm in various scenarios of model mismatch.

Passive Radar Tracking in Clutter Using Range and Range-Rate Measurements

Passive bistatic radar research is essential for accurate 3D target tracking, especially in the presence of missing or low-quality bearing information. Traditional extended Kalman filter (EKF) methods often introduce bias in such scenarios. To overcome this limitation, we propose employing the unscented Kalman filter (UKF) for handling the nonlinearities in 3D tracking, utilizing range and range-rate measurements. Additionally, we incorporate the probabilistic data association (PDA) algorithm with the UKF to handle cluttered environments. Through extensive simulations, we demonstrate a successful implementation of the UKF-PDA framework, showing that the proposed method effectively reduces bias and significantly advances tracking capabilities in passive bistatic radars.

High-precision State of Charge Estimation of Lithium-ion Batteries Based on Joint Compression Factor Particle Swarm Optimization-Forgetting Factor Recursive Least Square-Adaptive Extended Kalman Filtering

Accurate state of charge (SOC) estimation is an important basis for battery energy management and the applications of lithium-ion batteries. In this paper, an improved compression factor particle swarm optimization-forgetting factor recursive least square (CFPSO -FFRLS) algorithm is proposed, in which the forgetting factor is optimized to identify more accurate parameters for high-precision SOC estimation of lithium-ion battery. In order to improve the SOC estimation accuracy, a dual noise update link is introduced to the traditional extended Kalman filter (EKF), which enhances the algorithm’s ability to adapt to noise by updating the process and measurement noises in real time. The experimental results of parameter identification and SOC estimation show that the CFPSO-FFRLS algorithm proposed significantly improves the accuracy of parameter identification, and the joint CFPSO-FFRLS-AEKF algorithm can accurately estimate the SOC of lithium-ium battery under different working conditions. Under HPPC, BBDST and DST working conditions, the mean absolute errors of SOC estimation are 1.14%, 0.78% and 1.1%, which are improved by 42.71%, 65.79% and 39.56% compared with FFRLS-EKF algorithm, and the root mean square errors are 1.18%, 0.99% and 1.11%, improved by 44.86%, 65.98% and 51.74%, respectively.

Robust Estimation of Vehicle Dynamic State Using a Novel Second-Order Fault-Tolerant Extended Kalman Filter

<div>The vehicle dynamic state is essential for stability control and decision-making of intelligent vehicles. However, these states cannot usually be measured directly and need to be obtained indirectly using additional estimation algorithms. Unfortunately, most of the existing estimation methods ignore the effect of data loss on estimation accuracy. Furthermore, high-order filters have been proven that can significantly improve estimation performance. Therefore, a second-order fault-tolerant extended Kalman filter (SOFTEKF) is designed to predict the vehicle state in the case of data loss. The loss of sensor data is described by a random discrete distribution. Then, an estimator of minimum estimation error covariance is derived based on the extended Kalman filter (EKF) framework. Finally, experimental tests demonstrate that the SOFTEKF can reduce the effect of data loss and improve estimation accuracy by at least 10.6% compared to the traditional EKF and fault-tolerant EKF.</div>

Intelligent Deep Learning Estimators of a Lithium-Ion Battery State of Charge Design and MATLAB Implementation—A Case Study

The main objective of this research paper was to develop two intelligent state estimators using shallow neural network (SNN) and NARX architectures from a large class of deep learning models. This research developed a new modelling design approach, namely, an improved hybrid adaptive neural fuzzy inference system (ANFIS) battery model, which is simple, accurate, practical, and well suited for real-time implementations in HEV/EV applications, with this being one of the main contributions of this research. On the basis of this model, we built four state of charge (SOC) estimators of high accuracy, assessed by a percentage error of less than 0.5% in a steady state compared to the 2% reported in the literature in the field. Moreover, these estimators excelled by their robustness to changes in the model parameters values and the initial “guess value” of SOC from 80–90% to 30–40%, performing in the harsh and aggressive realistic conditions of the real world, simulated by three famous driving cycle procedure tests, namely, two European standards, WLTP and NEDC, and an EPA American standard, FTP-75. Furthermore, a mean square error (MSE) of 7.97 × 10−11 for the SOC estimation of the NARX SNN SOC estimator and 5.43 × 10−6 for voltage prediction outperformed the traditional SOC estimators. Their effectiveness was proven by the performance comparison with a traditional extended Kalman filter (EKF) and adaptive nonlinear observer (ANOE) state estimators through extensive MATLAB simulations that reveal a slight superiority of the supervised learning algorithms by accuracy, online real-time implementation capability, in order to solve an extensive palette of HEV/EV applications.

Double Extended Kalman Filter Algorithm Based on Weighted Multi-Innovation and Weighted Maximum Correlation Entropy Criterion for Co-Estimation of Battery SOC and Capacity.

Most of the traditional extended Kalman filter algorithms for the co-estimation of SOC and capacity of lithium-ion batteries are designed based on the minimum mean square error (MMSE) criterion, which may show superior performance in Gaussian noise scenes. However, due to the complexity of the battery operating environment, it is likely to face non-Gaussian noise (especially outlier noise), at which time the performance of the traditional extended Kalman filter algorithms will be seriously weakened. To solve the above problems, this paper first proposes a double extended Kalman filter algorithm based on weighted multi-innovation and weighted maximum correlation entropy (WMI-WMCC-DEKF) for the co-estimation of battery SOC and capacity. In this paper, the performance of the target algorithm is verified and compared by generating different types of noise from three noise models: weak Gaussian mixture noise, strong Gaussian mixture noise, and outlier noise. The maximum absolute error value (MAE) and root mean square error value (RMSE) of the WMI-WMCC-DEKF algorithm can achieve the highest performance improvement of 69.3 and 84.2% (SOC), 61.3, and 94.2% (capacity), respectively. The experimental results fully prove that the target algorithm has excellent performance against three kinds of noises.

Soft Short-circuit Fault Diagnosis of the Lithium-ion Battery Pack Based on an Improved EKF Algorithm

Background: Lithium-ion batteries are widely used in new energy vehicles and energy storage systems due to their superior performance. However, lithium batteries are prone to safety problems in the use process, so the fault diagnosis technology of lithium batteries has attracted more attention. Objective: This study aimed to ensure the safety of lithium batteries and accurately and timely diagnose the soft short circuit (soft SC) fault of lithium battery. Methods: Aiming at the energy storage lithium battery pack, this study proposed a soft short-circuit fault diagnosis method for the lithium-ion battery pack based on the improved Extended Kalman Filter (EKF) algorithm. First, the 1st-order RC equivalent circuit model of normal battery and soft SC fault battery was established, and model parameters were identified using Recursive Least Squares with Forgetting Factor (FFRLS). Then, using the improved EKF, the state of charge (SOC) of a single cell was estimated, and the difference between the calculated SOC and the estimated SOC by the coulomb counting method was used to detect soft SC faults and compared them with the reference data. Finally, the SC resistance value indicated the severity of the fault. Results: The proposed method could accurately diagnose the soft short circuit fault, and the error was found to be lower than the traditional EKF algorithm. The estimation error was about 0.4% for the battery with slight failure and about 1.5% for the battery with serious failure. Conclusion: The experimental results showed that the improved EKF algorithm could estimate the SOC difference more accurately, and the effect of soft SC fault diagnosis was better. At the same time, it could quantitatively identify the size of the short circuit resistance, which is very helpful for the subsequent management of the battery system.

Research on Indoor AGV Fusion Localization Based on Adaptive Weight EKF Using Multi-sensor

To solve the problem of large accumulative errors of wheel odometer and inertial measurement unit (IMU), which make pose calculation mistakes due to friction coefficient of the ground or bumps and collisions, furthermore, affect path planning and navigation of automated guided vehicle (AGV), an extensible fusion localization method based on multiple sensors is proposed. Based on the traditional extended Kalman filter (EKF), the internal positioning data is used for prediction, while the external positioning data is used for correction. According to the states of different sensors, the adaptive weight method is applied to eliminate the influence of system accumulative error and provide continuous and accurate positioning data. The experimental results show that compared to single-sensor, the fusion localization method has an increase in quantity, which can reach an accuracy within 5~8cm. It can deal with non-line-of-sight (NLOS) landmark occlusion situations, and can navigate in dim light or dark environments. The system ensures the accuracy of fusion data through visual positioning and ensures the stability of the system through the odometer and IMU. New positioning sensors are also possible to be added into the system for data fusion.

A Fast Continuous Self-Calibration Method for FOG Rotational Inertial Navigation System Based on Invariant Extended Kalman Filter

For a given device, the rapidity and accuracy of device error coefficient calibration and attitude alignment of the rotational inertial navigation system mainly depend on the convergence rate of the nonlinear filtering algorithm. The traditional extended Kalman filter method has no theoretical convergence guarantee for nonlinear systems and requires a long time in the actual calibration process. Since the attitude is subject to the 3-D rotation transformation group, the error transfer equation can be better linearized by defining the error on the lie group to use the new observation information fully. This article proposes a primary method of using the invariant extended Kalman filter to carry out an inertial navigation system. In the continuous rotation of the framework, the accelerometer, gyroscope, and framework angle are used to calibrate the error coefficient of the accelerometer and gyroscope and estimate the base alignment error angle. Simulation and experiment results show that the proposed method can accomplish alignment and calibration in 10 min under a large initial attitude error.

An EKF-Based Attitude Estimator for Eliminating the Effect of Magnetometer Measurements on Pitch and Roll Angles

Nowadays, magnetic and inertial measurement units (MIMUs) have become useful tools for determining the attitude of moving objects. An attitude calculation algorithm (ACA) is programed into an MIMU, which determines the attitude by fusing three built-in sensors, namely, gyroscope, accelerometer, and magnetometer. Attitude is often represented by Euler angles, quaternion, or rotation matrix. Despite that great development has been achieved, how to eliminate the negative effect of magnetic disturbance on the pitch and roll estimation is still difficult for an ACA. This article aims to propose an Extended Kalman Filter (EKF)-based approach for problem solving. Inspired by the fact that the Euler angles are computed by using some elements of the rotation matrix, a diagonalizable regulation matrix (DRM) is multiplied to the right half part of the gain matrix (which is multiplied to the residual of the magnetometer measurement in the update stage of EKF). The determination of DRM is based on the principle that these elements can be appropriately regulated by the DRM so as to make the magnetometer measurement only participate in the computation of azimuth. No further revision to EKF is made except the Kalman gain matrix, hence the proposed EKF behaves similar as traditional EKF, except its ability of eliminating the negative effect of magnetic disturbance on the pitch and roll estimation. The superiority of the proposed EKF is validated by comparing it with four other competing algorithms through simulation and experimental experiments.