Extended Kalman Filter Method 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.

The accuracy of vehicle position and heading angle improvement based on dual-antenna GNSS/INS/Barometer integration using extended Kalman filter

Abstract In the field of autonomous driving progresses, obtaining the precise position and attitude of vehicles through measurement technology is vital. An integrated GNSS/INS/Barometer (Global Navigation Satellite System/Inertial Navigation System/Barometer) fusion method is proposed to improve vehicle positioning and heading angle accuracy. An extended Kalman filter (EKF) based on horizontal dilution of precision (HDOP) is designed to fuse the INS error and the GNSS measurement. Furthermore, GNSS dual antennas are utilized to increase a state variable in EKF based on HDOP in addition to position and velocity. The kinematic constraint of zero velocity in the lateral and vertical directions of vehicle movement is used to measure the mounting angle to reduce error. The position and heading angle accuracy of the proposed method are 0.483m and 1.358°, which represent reductions of 37.03% and 43.86% compared to the conventional EKF method, respectively. The experimental results demonstrate that the proposed method can measure accurate position and heading angle in both open and occluded environments.

A switched adaptive strategy for state estimation in MEMS-based inertial navigation systems with application for a flight test

Abstract This paper proposes a switched model to improve the estimation of Euler angles and decrease the inertial navigation system (INS) error, when the centrifugal acceleration occurs. Depending on the situation, one of the subsystems of the proposed switched model is activated for the estimation procedure. During global positioning system (GPS) outages, an extended Kalman filter (EKF) operates in the prediction mode and corrects the INS information, based on the system error model. Compared with previous works, the main advantages of the proposed switched-based adaptive EKF (SAEKF) method are (i) elimination of INS error, during the centrifugal acceleration, and (ii) high accuracy in estimating the attitude and positioning, particularly during GPS outages. To validate the efficiency of the proposed method in various trajectories, an experimental flight test is performed and discussed, involving a microelectromechanical (MEMS)-based INS. The comparative study shows that the proposed method considerably improves the accuracy in various scenarios.

Application of the adaptive method to determine the process noise in the extended Kalman filter to estimate the parameters of the two dimensional inverse heat transfer problem

To deal with inverse nonlinear heat transfer problems in a two-dimensional heat conduction problem, this paper presents a hybrid approach combining fuzzy logic and the extended Kalman filter (EKF). The proposed algorithm has been applied to the real-time reconstruction of the temperature field, time-varying convective heat transfer coefficient, and time-varying boundary heat flux using temperature data from a set of sensors. The effects of the covariance of the initial estimation error, the time step of data acquisition by the sensors, and the number of installed sensors on the precision and stability of the estimation results has been investigated by numerical experiments. To compare and validate the results, all these parameters were investigated using the EKF method. The results show that the adaptive fuzzy extended Kalman filter (FEKF) method for estimating temperature, convection coefficient, and heat flux at the boundaries is precise and stable. The comparison shows that the performance of the FEKF method in parameter estimation is better than the EKF method, and the highest increase in precision and stability of the results is related to the convective heat transfer coefficient.

Test mass charge estimation for the space inertial sensor with extended Kalman filter

Charge Management System (CMS), aimed at mitigating charge-induced noise on an isolated free-falling test mass (TM), is a crucial component of space inertial sensors in various spaceborne gravitational missions. The estimation of TM charge is one of the tasks of CMS, directly impacting CMS performance. However, current methods for TM charge estimation suffer from slow response and significant estimation errors. This paper presents a new charge estimation method that combines the force modulation principle with an extended Kalman filter (EKF). We analyzed the relationship between the TM surface potential and TM motion, then established a mathematical model. The estimation of TM charge is accomplished through the EKF algorithm. A charge estimation simulation model was developed in Simulink, and experiments were conducted to evaluate the performance of the charge estimation methods under varying charge conditions. The experimental results demonstrate that the EKF method obtains more accurate estimation results and faster convergence rates, compared with Kalman Filter (KF) and Quadrature demodulation (QD) methods. The KF and QD methods show larger errors when the estimated charge is greater than 106e order, meanwhile the QD method exhibits slower convergence rates. The presented method is expected to be beneficial for the CMS of on-orbit inertial sensors.

Research on Excitation Estimation for Ocean Wave Energy Generators Based on Extended Kalman Filtering

Wave energy generation methods have significant energy costs. The implementation of sophisticated control techniques in wave energy generators can lower the cost of power generation by optimizing the energy recovered from wave energy converters (WECs). To determine control inputs, most control systems rely on knowledge of the wave excitation force, including information on past, present, and future excitation forces. For the excitation of WEC devices, wave excitation force can only be inferred and predicted because it is an unmeasurable quantity. One of the more widely used observers in wave excitation estimates at the moment is the Kalman filter, but its use is primarily restricted to linear Kalman filtering. The mooring system is an integral component of floating wave energy producers. The mooring force of the device is actually nonlinear; however, the majority of current studies on excitation estimates for wave energy producers based on Kalman filter methods employ an ideal motion model based on the linearization of the mooring force. This paper, in an attempt to make things more realistic, creates a WEC system with highly nonlinear mooring forces, suggests a way to build a wave excitation force estimator for a nonlinear WEC system using the extended Kalman filtering method, and assesses the impact of various factors, such as measurement noise, random phase, and the number of equal-energy methods dividing the frequency, on the accuracy of the wave excitation force estimate.

A New Fault Prognosis of MFS System Using Integrated Extended Kalman Filter and Bayesian Method

This paper presents a new fault prognosis approach for a multifunctional spoiler (MFS) system which employs an extended Kalman filter (EKF) and Bayesian theorem method for prognosis. The MFS is an important part of an aircraft spoiler control system (SCS), and thus, prognosis and health management (PHM) of this system improves the safety of the aircraft. To monitor the system, residual estimation based on the EKF method is utilized to observe the progress of the failure in the system. Then, a new measure is introduced by using a transformation to estimate degradation path (DP) of the failure in the system. Furthermore, a new recursive Bayesian method is invoked to predict the RUL of the system using the estimated DP data. Finally, for performance assessment, relative accuracy (RA) is utilized to evaluate the accuracy of the proposed method.

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.

Degradation prediction of proton exchange membrane fuel cells with model uncertainty quantification

This study presents a new approach for state of health (SOH) estimation and degradation trend prediction of proton exchange membrane fuel cells (PEMFCs) considering model uncertainty quantification. The model uncertainty is considered as completely unknown in this paper, and the Bayesian framework-based model uncertainty and state simultaneous estimation (Bayes-MUSE) method is proposed to quantify the model uncertainty from the perspective of the form of its effect on the system state transfer, and this method of quantifying model uncertainty is first applied to PEMFC long-term aging prediction and RUL estimation. The proposed Bayes-MUSE method is compared with extended Kalman filter (EKF), Unscented Kalman filter (UKF) and Unscented Particle filter (UPF) methods in terms of SOH estimation, noise immunity, aging prediction performance and RUL estimation using two aging test datasets, and the method is verified to have excellent noise immunity and the highest long-term aging prediction and RUL estimation accuracy. Compared with EKF, UKF and UPF methods, the proposed Bayes-MUSE method has a maximum improvement of 93.95% and 52.2% in RUL estimation accuracy of the two data sets under quasi-dynamic and static conditions, respectively.

ROEKF-MPC Estimator for Satellite Attitude and Gyroscope Bias Estimation

Micro-electro-mechanical system (MEMS) based gyroscopes are commonly used for satellite's attitude determination and control system in recent years. They are packaged in a small form factor and have lower power consumption. They provide a low-cost solution to the emerging NewSpace industry. MEMS gyroscopes exhibit time-zero null bias with variation over temperature. To overcome this problem, this paper proposes a technique for self-calibration of gyroscope bias based on reduced order Extended Kalman Filter (EKF) working alongside an estimator based on the model predictive control (MPC) approach. The proposed technique is referred as the ROEKF-MPC Estimator. Both simulation results and experimental verification using an in-house developed spacecraft simulator are presented. Unlike the EKF method, the proposed method can estimate the gyroscope bias instantly and it is robust against changes in temperature. In addition, the proposed method employing a reduced order EKF is 28.4% computationally more efficient than the typical higher order EKF method. Results show that the pointing performance is better than 0.35deg making the proposed method very attractive for most NewSpace applications. While the proposed method allows for self-calibration of gyroscope bias, its performance is affected by the gyroscope noise and requires the satellite's attitude to be nearly in steady state. This limitation will be addressed in detail.

Real time dynamic inclination identification method based on extended Kalman filtering and vibration monitoring

Structural vibration includes both translational and rotational degrees of freedom, but the latter are often ignored. The paper establishes a dynamic inclination measurement model based on the existing accelerometer, and constructs state and measurement equations that comprehensively represent translational and rotational degrees of freedom. Then, the extended Kalman filtering method is used to linearize and discretize the state equation and measurement equation, and the filtering process of dynamic Inclination Identification is derived. The feasibility of the method was verified through an example of a simply supported beam structure.

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.

Improved Terahertz Time-Domain Spectroscopy via the Extended Kalman Filter

To implement terahertz time-domain spectroscopy (THz-TDS) systems that have short integration time and maintain performance, the extended Kalman filter (EKF) has been used. Performance of EKF with THz-TDS on vapor samples and solid samples is untested. Previous EKF methods were not adaptive for use on multiple THz-TDS systems. We provide an adaptive EKF method that can work on any THz setup, to improve maximum measurable absorption coefficient. Our adaptive EKF method responds to THz pulse properties. The adaptive EKF is exhaustively demonstrated for six different time-constant data sets spanning 1 ms to 300 ms for THz-TDS on vapor with electro-optic THz detection. The acquisition time saved is quantified and shown to significantly reduce acquisition time from 2.58 ms to 1 ms. This improvement of 2.58 times will have a significant effect to move spectral imaging from taking hours to taking minutes. Vapor absorption lines are maintained. We show the adaptive EKF maintain correct spectral information when compared to the high-resolution transmission molecular absorption (HITRAN) database. The adaptive EKF is also demonstrated for THz-TDS on vapor with photoconductive THz detection, thus demonstrating adaptability, and is applied for THz-TDS of a solid sample of polyethylene terephthalate glycol.

A key-term separation based least square method for Hammerstein SOC estimation model

In this paper, a Hammerstein state of charge (SOC) estimation model for lithium-ion batteries with two inputs and single output is constructed to map the relation between battery SOC and its discharge current/voltage, and a key-term separation based least square (KT-LS) algorithm is studied to identify the model parameters so as to identify the battery SOC. The thought is, by adopting the key-term separation principle, to decompose the key-term from the linear element and to substitute the expression of the nonlinear element only into the decomposed key-term, thus to express the model output as a linear auto-regression form about parameters. Then the least square algorithm is performed to optimize parameters in the reformed auto-regression form of the Hammerstein SOC system. The advantages of the explored the KT-LS algorithm contains the least parameters required, thus the calculation burden of the KT-LS algorithm is small compared with the over-parameterized model based algorithm. In the simulation experiment part, the performance of the KT-LS algorithm under different conditions are analyzed under UDDS and LA92 conditions.; the performance comparison between the KT-LS algorithm and the key-term separation based stochastic gradient (KT-SG) algorithm is carried out; the performance of the KT-LS algorithm based Hammerstein model method is compared with that of the back propagation (BP) neural network, the Long Short Term Memory (LSTM) neural network and the Extended Kalman filter (EKF) method. The simulation results show that the KT-LS algorithm based Hammerstein SOC model can effectively estimate the SOC of lithium-ion batteries.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

Cost-Efficient Real-Time Adaptive Location Tracking With Interacting Multiple Transition Model for Implantable Medical Device

One of the main tasks connected with implantable medical devices, such as wireless capsule endoscope (WCE) is location tracking of a capsule moving inside a human body. To improve the localization accuracy in a real situation, a transition model to describe the capsule movement inside the body should be appropriately considered, where the trajectory segments are composed of different movement models. In this paper, in addition to the conventional random way point model, we constructed an interacting multiple model (IMM), which is a combination of two different movements to represent suitable WCE movement. Then, we proposed a combined extended Kalman and particle filter method with respect to the balance between the accuracy and computational cost for realizing precise real-time WCE location tracking. Compared with conventional tracking methods, the proposed tracking location method with the IMM transition model achieved 30% better estimation accuracy and 98% lower computational cost.

Implementation Fuzzy and Extended Kalman Filter for Estimation of High and Low Stock Price Travel Company

Competition in the business world is getting tougher from year to year both within a country and abroad. There are a large number of companies competing with one another, especially entering the free market share in Asia, namely the Asean Economic Community (AEC). In the current development of modern economy, Indonesia is making efforts to increase its economic growth. For this, developments in any fields are made. Among others is the service industry such as accommodation, travel, and transportation services. Considering that Indonesia is a country comprised of many islands with a variety of natural beauty, it has the very potential for tourist resort attraction. This kind of thing leads to the growth of the Travel, tourism and hotel industry to support development of tourism. With such rapid service industry development, supported by promising business opportunities, investors for such sector are encouraged. The right way to reduce risk for investors interested is to develop a system for estimating the stock prices. Therefore, in this study, the high and low stock price estimation method applied for travel companies adopted developed Kalman Filter, a comparison of two Kalman Filter development methods, namely Extended Kalman Filter (EKF) and Fuzzy Kalman Filter (FKF) as a chart for investors to take into consideration in their investment decision making. The simulation results showed that the EKF method had higher accuracy than the FKF method with an error by the EKF of 3.5% and that by the FKF of 8.9%.

An Enhanced Fusion Algorithm With Empirical Thermoelectric Models for Sensorless Temperature Estimation of Li-ion Battery Cells

An enhanced dual extended Kalman filter method is presented in this article for estimating and tracking the state-of-temperature of lithium-ion battery cells. A simple but effective dynamic and measurement empirical fit models are proposed and utilized to estimate the state-of-temperature concurrently with the state-of-charge. The proposed dual estimator improves the estimation accuracy of the temperature state by accounting for the variations in the state-of-charge. To test the performance of the proposed estimation method, two independent lithium-ion battery cell datasets were used to derive the empirical models and run the estimation algorithm. The obtained results show a promising performance of the estimation method in terms of the high estimation accuracy even in the case when the measurement contains high-magnitude noise or when the estimation algorithm is inaccurately initialized. The proposed models and the estimation algorithm are derived and experimentally tested in this article.