Classical Extended Kalman Filter Research Articles

3D Indoor Position Estimation Based on a UDU Factorization Extended Kalman Filter Structure Using Beacon Distance and Inertial Measurement Unit Data.

The development of the GPS (Global Positioning System) and related advances have made it possible to conceive of an outdoor positioning system with great accuracy; however, for indoor positioning, more efficient, reliable, and cost-effective technology is required. There are a variety of techniques utilized for indoor positioning, such as those that are Wi-Fi, Bluetooth, infrared, ultrasound, magnetic, and visual-marker-based. This work aims to design an accurate position estimation algorithm by combining raw distance data from ultrasonic sensors (Marvelmind Beacon) and acceleration data from an inertial measurement unit (IMU), utilizing the extended Kalman filter (EKF) with UDU factorization (expressed as the product of a triangular, a diagonal, and the transpose of the triangular matrix) approach. Initially, a position estimate is calculated through the use of a recursive least squares (RLS) method with a trilateration algorithm, utilizing raw distance data. This solution is then combined with acceleration data collected from the Marvelmind sensor, resulting in a position solution akin to that of the GPS. The data were initially collected via the ROS (Robot Operating System) platform and then via the Pixhawk development card, with tests conducted using a combination of four fixed and one moving Marvelmind sensors, as well as three fixed and one moving sensors. The designed algorithm is found to produce accurate results for position estimation, and is subsequently implemented on an embedded development card (Pixhawk). The tests showed that the designed algorithm gives accurate results with centimeter precision. Furthermore, test results have shown that the UDU-EKF structure integrated into the embedded system is faster than the classical EKF.

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.

Real-Time Leak Diagnosis in Water Distribution Systems Based on a Bank of Observers and a Genetic Algorithm

The main contribution of this paper is to present a novel solution for the leak diagnosis problem in branched pipeline systems considering the availability of pressure head and flow rate sensors on the upstream (unobstructed) side and the downstream (constricted) side. This approach is based on a bank of Kalman filters as state observers designed on the basis of the classical water hammer equations and a related genetic algorithm (GA) which includes a fitness function based on an integral error that helps obtaining a good estimation despite the presence of noise. For solving the leak diagnosis problem, three stages are considered: (a) the leak detection is performed through a mass balance; (b) the region where the leak is occurring is identified by implementing a reduced bank of Kalman filters which localize the leak by sweeping all regions of the branching pipeline through a GA that reduces the computational effort; (c) the leak position is computed through an algebraic equation derived from the water hammer equations in steady-state. To assess this methodology, experimental results are presented by using a test bed built at the Tuxtla Gutiérrez Institute of Technology, Tecnológico Nacional de México (TecNM). The obtained results are then compared with those obtained using a classic extended Kalman filter which is widely used in solving leak diagnosis problems and it is highlighted that the GA approach outperforms the EKF in two cases whereas the EKF is better in one case.

New approach for multidimensional prognostic of stochastic time series based on sparse-based fractional Levy quaternion extended Kalman filter

Multidimensional data are frequently encountered in numerous industrial systems, standard techniques regarding statistical multichannel prognostic methods naturally cannot fully exploit and capture the coupled nature and spatiotemporal relationship of the available information among channels. To alleviate a bottleneck problem, in this paper, a new approach named sparse-based fractional Levy quaternion extended Kalman filter (SFLQEKF) is formulated for the prognostic of stochastic time series. Initially, the stochastic time series is decomposed into the static component and the dynamic component resorting to the piecewise representation through a new improved sparse low-rank matrix (SLRM) approximation algorithm namely singular value penalty resonance sparse signal decomposition (SVP-RSSD). Afterward, both static component and dynamic component are predicted individually through the proposed fractional Levy quaternion extended Kalman filter (FLQEKF) method, the final time series of interest can be obtained by integrating the predicted static- and dynamic components correspondingly, in which a new predictive derivation of the proposed FLQEKF model is formulated for the first time by considering fractional-order characteristic and Levy flight process damping term, hence the issue of oscillating indeterminacy induced from classical extended Kalman filter (EFK) can be addressed. Eventually, the prognostic behavior of the proposed approach is investigated via the stochastic time series conducted from two experimental cases, the predicted results illustrate the availability and applicability of the proposed approach compared with three state-of-the-art benchmarks.

Sequential method for fast neural population activity reconstruction in the cortex from incomplete noisy measurements

During recent years there has been a growing interest in stochastic dynamic neural fields employed for modeling and predictions in biomedical and technical systems. In this paper, given some incomplete noisy data available from sensors, we propose and explore a state estimation method for fast restorations of membrane potential in the cortex based on such measurements and the Amari equation used for simulations of neural population activity in a stochastic setting. Our novel technique relies upon a Galerkin-type spectral approximation utilized within the conventional state-space approach. Translating a stochastic system into its state-space form creates a straightforward and fruitful way to the data-driven parameter estimation, filtering, prediction and smoothing. The present study is particularly focused on establishing a nonlinear stochastic Galerkin-spectral-approximation-induced system of large size, which is further estimated by the traditional extended Kalman filter (EKF). The efficiency of calculations is the main purpose of our research. That is why the fast filtering solution devised is based on processing the incoming data incrementally, that is, by processing measurements one at a time, rather than handling them as a unified high-dimensional vector. Such sequential filters suit well for dealing with large data sets as well as with real-time on-line computations. Also, their derivation and substantiation is of great interest in the context of neural network training because of large stochastic systems arisen there. In comparison to the batch filtering, our novel algorithm reduces the computational cost of membrane potential reconstructions in terms of the amount of grid nodes N accepted in the underlying spacial discretization, significantly. Apart from its computation efficiency, this sequential method is more robust to round-off errors committed within a computer-based finite precision arithmetics than the classical EKF because of the (N × N)-matrix inversion elimination from such membrane potential calculations. The superior performance of our technique is examined and confirmed in comparison to the batch one on two known scenarios in the dynamic neural field modeling.

Autonomous orbit determination and timekeeping in lunar distant retrograde orbits by observing X‐ray pulsars

<h3>Abstract</h3> A server satellite that is parked in a stable distant retrograde orbit (DRO) in the vicinity of the Moon can provide navigation service for satellites in the Earth– Moon system. In this study, X-ray pulsar navigation (XPNAV) is investigated to determine the server’s orbital position and velocity, and maintain its onboard timing autonomously to ensure the autonomy of the server. First, the feasibility and potential advantages of XPNAV in DRO are analyzed qualitatively based on the unique properties of DRO: long-term orbital stability and short-term, nearly uniform-velocity rectilinear motion. Second, high-fidelity X-ray photon timestamps are simulated to serve as the original measurements for XPNAV. A classical extended Kalman filter is used to estimate the orbital states (i.e., position and velocity) and the clock timing of the satellite in DRO, wherein the pulse phase and Doppler frequency are estimated as intermediate measurements. Finally, Monte Carlo simulations are conducted to assess the variation in the XPNAV performance with the DRO orbital period, clock noise, and detector effective area. The simulation results show that XPNAV in DRO can achieve considerably improved position accuracy than that in low Earth orbit using the same hardware while maintaining the long-term stability of the satellite onboard clock, thus demonstrating the capability of XPNAV to be used in an operational spaceflight mission around or beyond the Moon.

Intelligent Sensor Fusion in High Precision Satellite Attitude Estimation Utilizing an Adaptive-Network-Based Fuzzy Inference System

In this study, Adaptive Network-Based Fuzzy Inference System (ANFIS) is presented with sensor data fusion approach to estimate satellite attitude. The active sensors are sun and earth sensors. Satellite attitude dynamic, including attitude quaternion and angular velocities are estimated simultaneously utilizing the measured values by the sensors. The Extended Kalman Filter (EKF) is employed to verify and evaluate the efficiency of the presented method. Additionally, the neural networks with Radial Basis Function (RBF) and Multi-Layer Perceptron (MLP) are also designed to prove the superiority of the proposed ANFIS network among the smart methods of sensor data fusion for satellite attitude estimation. Root Mean Square Error (RMSE) as a numerical criterion and graphical analysis of residues are utilized to evaluate the simulation results. The simulations confirm that the obtained estimations from ANFIS network have more accuracy in modeling of nonlinear complex systems compared to EKF, MLP and RBF networks. In general, using intelligent data fusion, especially ANFIS, reduces attitude estimation error and time in comparison to the classical EKF method.

A Real-Time BLE/PDR Integrated System by Using an Improved Robust Filter for Indoor Position

Indoor position technologies have attracted the attention of many researchers. To provide a real-time indoor position system with high precision and stability is necessary under many circumstances. In a real-time position scenario, gross errors of the Bluetooth low energy (BLE) fingerprint method are more easily occurring and the heading angle of the pedestrian will drift without acceleration and magnetic field compensation. A real-time BLE/pedestrian dead-reckoning (PDR) integrated system by using an improved robust filter has been proposed. In the PDR method, the improved Mahony complementary filter based on the pedestrian motion states is adopted to estimate the heading angle reducing the drift error. Then, an improved robust filter is utilized to detect and restrain the gross error of the BLE fingerprint method. The robust filter detected the gross error at different granularity by constructing a robust vector changing the observation covariance matrix of the extended Kalman filter (EKF) adaptively when the application is running. Several experiments are conducted in the true position scenario. The mean position accuracy obtained by the proposed method in the experiment is 0.844 m and RMSE is 0.74 m. Compared with the classic EKF, these two values are increased by 38% and 18%, respectively. The results show that the improved filter can avoid the gross error in the BLE method and provide high precision and scalability in indoor position service.

An Improved Extended Kalman Filter for Radar Tracking of Satellite Trajectories

Nonlinear state estimation problem is an important and complex topic, especially for real-time applications with a highly nonlinear environment. This scenario concerns most aerospace applications, including satellite trajectories, whose high standards demand methods with matching performances. A very well-known framework to deal with state estimation is the Kalman Filters algorithms, whose success in engineering applications is mostly due to the Extended Kalman Filter (EKF). Despite its popularity, the EKF presents several limitations, such as exhibiting poor convergence, erratic behaviors or even inadequate linearization when applied to highly nonlinear systems. To address those limitations, this paper suggests an improved Extended Kalman Filter (iEKF), where a new Jacobian matrix expansion point is recommended and a Frobenius norm of the cross-covariance matrix is suggested as a correction factor for the a priori estimates. The core idea is to maintain the EKF structure and simplicity but improve its accuracy. In this paper, two case studies are presented to endorse the proposed iEKF. In both case studies, the classic EKF and iEKF are implemented, and the obtained results are compared to show the performance improvement of the state estimation by the iEKF.

Leak diagnosis in pipelines based on a Kalman filter for Linear Parameter Varying systems

This paper proposes a new approach for the leak diagnosis problem in pipelines based on the use of a Kalman filter for Linear Parameter Varying (LPV) systems. Such a filter considers the availability of flow and pressure measurements at each end of the pipeline. The proposed methodology relies on an LPV model derived from the nonlinear description of the pipeline. For the Kalman filter design purposes, the LPV model is transformed into a polytopic representation. Then, using such a representation, the LPV Kalman filter is designed by solving a set of Linear Matrix Inequalities (LMIs) offline. In the online implementation, the observer gain is calculated as an interpolation of those gains previously computed at the vertices of the polytopic model. The main advantages of this approach are: a) the embedding of the nonlinearities in the varying parameters allows the quasi-LPV system to be obtained which is equivalent to the original nonlinear one, and; b) the use of the well-known LMIs to compute the Kalman gain allows the extension to the LPV case. Those aspects are the main advantages with respect to the classic design of the Extended Kalman Filter (EKF) that requires a linearization procedure and the solution of the Ricatti equation at each iteration. To illustrate the potential of this method, a test bed plant built at Cinvestav-Guadalajara is used. Additionally, the results presented are compared with those results obtained through the classical EKF showing that LPV Kalman observer outperforms the classical EKF.

Machine Learning Gyro Calibration Method Based on Attitude Estimation

This paper proposes a novel machine learning (ML) gyro calibration method that can achieve higher-accuracy gyro calibration and attitude estimation than the classical extended Kalman filter (EKF) approach when high-accuracy measurements are unavailable. The gyro calibration process is modeled as a time-series problem based on the standard EKF output. Then, a designed ML model is trained by the collected time-series data so that it can conduct gyro calibration by generating an ML correction to the gyro measurement. The proposed ML calibration method does not make assumptions in the form of gyro measurement errors, but directly learns it from data when high-accuracy information is available. Therefore, it is possible to outperform the EKF bias calibration when there is only low-accuracy information. To validate the method, a torque-free CubeSat is simulated using sun sensors and magnetometers to generate higher- and lower-accuracy attitude measurements, respectively. The simulation results show that the ML gyro calibration achieves smaller residual errors compared with the standard EKF. Meanwhile, the EKF attitude estimation accuracy is also improved, as the attitude integration is more accurate using the ML-calibrated gyro measurement. Four ML models based on different principles are examined, including multilayer perceptron, convolutional neural network, recurrent neural network, and Gaussian processes. It is found that, usually, a more sophisticated ML model can capture more gyro error information, but all four models can achieve similar performance with well-tuned parameters.

Jet transport-based nonlinear state and parameter estimation for geostationary spacecraft

Based on the jet transport technique, this paper proposes a novel nonlinear Kalman filter for simultaneously estimating the spacecraft state vector and uncertain parameters, either physically related with the spacecraft or with the measurement procedure. Two different coordinate representations, including Cartesian and hybrid geostationary orbital elements, are exploited in the new nonlinear estimators. The performance and sensitivity analyses of the proposed jet transport-based nonlinear estimators are assessed by numerical simulations and compared with the classical extended Kalman filter.

An AEKF-SLAM Algorithm with Recursive Noise Statistic Based on MLE and EM

Extended Kalman Filter (EKF) has been popularly utilized for solving Simultaneous Localization and Mapping (SLAM) problem. Essentially, it requires the accurate system model and known noise statistic. Nevertheless, this condition can be satisfied in simulation case. Hence, EKF has to be enhanced when it is applied in the real-application. Mainly, this improvement is known as adaptive-based approach. In many different cases, it is indicated by some manners of estimating for either part or full noise statistic. This paper present a proposed method based on the adaptive-based solution used for improving classical EKF namely An Adaptive Extended Kalman Filter. Initially, the classical EKF was improved based on Maximum Likelihood Estimation (MLE) and Expectation-Maximization (EM) Creation. It aims to equips the conventional EKF with ability of approximating noise statistic and its covariance matrices recursively. Moreover, EKF was modified and improved to tune the estimated values given by MLE and EM creation. Besides that, the recursive noise statistic estimators were also estimated based on the unbiased estimation. Although it results high quality solution but it is followed with some risks of non-positive definite matrices of the process and measurement noise statistic covariances. Thus, an addition of Innovation Covariance Estimation (ICE) was also utilized to depress this possibilities. The proposed method is applied for solving SLAM problem of autonomous wheeled mobile robot. Henceforth, it is termed as AEKF-SLAM Algorithm. In order to validate the effectiveness of proposed method, some different SLAM-Based algorithm were compared and analyzed. The different simulation has been showing that the proposed method has better stability and accuracy compared to the conventional filter in term of Root Mean Square Error (RMSE) of Estimated Map Coordinate (EMC) and Estimated Path Coordinate (EPC).

IDENTIFICATION OF STRUCTURAL PARAMETERS AND UNKNOWN EXCITATIONS BASED ON THE EXTENDED KALMAN FILTER

The classical extended Kalman filter (EKF) method is capable of accurately identifying structural parameters with known external excitations. However, in some practical situations, the excitations are difficult or impossible to measure. A time-domain approach based on EKF is proposed in this paper for the simultaneous identification of structural parameters and unknown inputs. A projection matrix is introduced in the observation equation, based on which the structural parameters are identified. The unknown inputs are determined by means of least squares estimation using the estimated parameters. The effectiveness and robustness of the proposed approach is verified through three numerical examples including a four-story benchmark model, a piecewise linear structure and a Duffing hysteretic structure. The numerical results show that the proposed approach can not only accurately identify the parameters of linear and nonlinear structures, but also satisfactorily estimate the unknown external excitations.

An EKF-Based Fixed-Point Iterative Filter for Nonlinear Systems.

In this paper, a fixed-point iterative filter developed from the classical extended Kalman filter (EKF) was proposed for general nonlinear systems. As a nonlinear filter developed from EKF, the state estimate was obtained by applying the Kalman filter to the linearized system by discarding the higher-order Taylor series items of the original nonlinear system. In order to reduce the influence of the discarded higher-order Taylor series items and improve the filtering accuracy of the obtained state estimate of the steady-state EKF, a fixed-point function was solved though a nested iterative method, which resulted in a fixed-point iterative filter. The convergence of the fixed-point function is also discussed, which provided the existing conditions of the fixed-point iterative filter. Then, Steffensen’s iterative method is presented to accelerate the solution of the fixed-point function. The final simulation is provided to illustrate the feasibility and the effectiveness of the proposed nonlinear filtering method.

Iterative learning based model identification and state of charge estimation of lithium‐ion battery

This work focuses on the accurate identification of lithium-ion battery's non-linear parameters by using an iterative learning method. First, the second-order resistance-capacitance model and its regression form of the battery are introduced. Then, when the battery repeatedly implements a discharge trial from the state of charge (SOC) 100 to 0%, an iterative learning based recursive least square (IL-RLS) algorithm is presented to accurately identify the non-linear parameters of the regression model. The essential idea of the IL-RLS algorithm is to improve the current parameter estimates by learning the predictive errors of the previous trials. After that, the parameters are identified as the functions of SOC by using the IL-RLS, which are verified by comparing with the results of the classic identification method for current pulses. As a result, an application-oriented SOC estimation scheme is proposed, where the IL-RLS calibrates the battery parameters offline and the classic extended Kalman filter (EKF) estimates the SOC in real-time. Finally, based on the EKF as well as the parameters identified by the IL-RLS, one static and three dynamic operating conditions are given to show the efficiency of the IL-RLS, where all the SOC estimation errors are <2%.

Extended Kalman Filter with Input Detection and Estimation for Tracking Manoeuvring Satellites

The technique of tracking a non-cooperative manoeuvring satellite is important for Space Situation Awareness (SSA). However, the classical extended Kalman filter cannot work successfully in this situation. Motivated by this problem, a novel Extended Kalman Filter with Input Detection and Estimation (EKF/IDE) method is proposed in this paper for tracking a non-cooperative satellite with impulsive manoeuvres. The impulsive manoeuvre is modelled as an unknown acceleration without any prior information. An unbiased minimum-variance input and state estimation method is introduced to estimate the manoeuvre acceleration. An approach based on the Mahalanobis distance of the manoeuvre estimate error is proposed for manoeuvre detection. With the impulsive manoeuvre being detected and estimated accurately, an adaptive extended Kalman filter is proposed to estimate the state of the target. An approach of covariance inflation is proposed to deal with the manoeuvre during the unobserved period. Simulations and Monte Carlo experiments are implemented to demonstrate the feasibility and validity of the proposed method. The results of simulations show that the proposed method can accurately detect and estimate unknown impulsive manoeuvres of a non-cooperative satellite. Through the compensation of the estimated manoeuvre, the estimation of position and velocity after the manoeuvre maintains the accuracy of the pre-manoeuvre period, demonstrating the robustness of the proposed method against unknown manoeuvres.

Augmented unbiased minimum-variance input and state estimation for tracking a maneuvering satellite

The technique of tracking a maneuvering satellite is significantly important for Space Situation Awareness (SSA). Traditional Kalman filters (KF) cannot robustly track unknown maneuvers. Motivated by the problem of tracking a non-cooperative maneuvering satellite, an augmented unbiased minimum-variance input and state estimation (AUMVISE) method is developed for estimating the state and the maneuver acceleration in this study. The maneuver acceleration estimate of the proposed method is proven to be more accurate than the original unbiased minimum-variance input and state estimation (UMVISE) method. Approaches based on the measurement residuals and the acceleration estimates are developed for maneuver start and end detection. The filter is switched between the AUMVISE method and the classical extended Kalman filter (EKF) to obtain an accurate tracking result during both the maneuver period and the non-maneuver period. Simulation results show that the proposed method has a suitable maneuver detection delay and outperforms the UMVISE method in estimating the state and maneuver acceleration.

Design and implementation of a model predictive observer for AHRS

A GPS-aided Inertial Navigation System (GAINS) is used to determine the orientation? position and velocity of ground and aerial vehicles. The data measured by Inertial Navigation System (INS) and GPS are commonly integrated through an Extended Kalman Filter (EKF). Since the EKF requires linearized models and complete knowledge of predefined stochastic noises? the estimation performance of this filter is attenuated by unmodeled nonlinearity and bias uncertainties of MEMS inertial sensors. The Attitude Heading Reference System (AHRS) is applied based on the quaternion and Euler angles methods. A moving horizon-based estimator such as Model Predictive Observer (MPO) enables us to approximate and estimate linear systems affected by unknown uncertainties. The main objective of this research is to present a new MPO method based on the duality principle between controller and observer of dynamic systems and its implementation in AHRS mode of a low-cost INS aided by a GPS. Asymptotic stability of the proposed MPO is proven by applying Lyapunov's direct method. The field test of a GAINS is performed by a ground vehicle to assess the long-time performance of the MPO method compared with the EKF. Both the EKF and MPO estimators are applied in AHRS mode of the MEMS GAINS for the purpose of real-time performance comparison. Furthermore? we use flight test data of the GAINS for evaluation of the estimation filters. The proposed MPO based on both the Euler angles and quaternion methods yields better estimation performances compared to the classic EKF.

$H_{\infty }$-Based Nonlinear Observer Design for State of Charge Estimation of Lithium-Ion Battery With Polynomial Parameters

This paper focuses on the state-of-charge (SOC) estimation of the Lithium-ion battery in electric vehicles based on an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -based nonlinear observer. First, the second-order RC equivalent circuit model is introduced by utilizing the physical behavior of the battery. Then, the parameters in the second-order RC model are identified as polynomial functions with respect to SOC. Meanwhile, the battery model with constant parameters is also introduced for comparison. Due to that the battery model is undetectable, an one-sided Lipschitz condition is proposed to ensure that the nonlinear function in output equation can play a positive role in the observer design. After that, a nonlinear observer design criterion is presented based on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> method, which is formulated as linear matrix inequalities. Compared with existing nonlinear observer-based SOC estimation methods, the proposed observer design criterion does not depend on any estimates of the unknown variables. Consequently, the convergence of the proposed nonlinear observer is guaranteed for various operating conditions. Finally, one static and three dynamic operation conditions are given to show the efficiency of the proposed nonlinear observer by comparing with the classic extended Kalman filter and the nonlinear observer for constant parameters.