A Factorized Extended Kalman Filter

Comparisons of six heating source localisation measurement models are conducted where temperature or ultrasonic time of flight readings provide the measurement update to the extended Kalman filter for estimating the location of a high heat flux spot source on a flat plate. For a particular measurement model, one of two processes are used: (1) directly using the measurements as the measurement vector in the extended Kalman filter or (2) indirectly obtaining the distance from the sensor to the heating source based on the measurement and then using the obtained distance as the measurement vector in the extended Kalman filter. For the direct models, the Jacobian required by the extended Kalman filter is obtained numerically using finite differences from the finite element forward conduction solution. For the indirect models, the derivatives of the distances with respect to the state variables are obtained in closed form. Heating source localisation results and convergence behaviour are compared for the six measurement models investigated. The ellipse from ultrasonic pulse one-way time of flight measurement model produces the best results when considering accuracy of converged solution, ability to converge to the correct solution given different initial guesses, and smoothness of convergence behaviour. Additionally, extended Kalman filter, extended information filter, and least squares inverse methods are compared for a parameter identification to quantify the heat flux and convection coefficient on the plate. All three inversion methods produce similar results which is significant as future work will consider only the extended Kalman filter.

In this study, an extended Kalman filter (EKF) for estimation of spacecraft's position and velocity by single station antenna tracking data in the presence of singularities is proposed. The designed EKF estimates the position and velocity of the spacecraft on direct nonlinear range-azimuth-elevation measurements. The proposed EKF is examined through the geostationary satellite orbital motion simulations. The corruption in the results by EKF emerges from singularities caused by the highly complex elements of the Jacobian measurement matrix. The recommendations to prevent the mentioned problem are given.

This paper proposes a new move toward to assess the performance of sensorless control of interior permanent magnet synchronous motor (IPMSM) drive along with electromagnetic noise effect by using EKF. Normally in rotary condition, rotor position and speed estimation of IPMSM drive are drawn through an Extended Kalman Filter (EKF) algorithm by measuring its voltages and currents of the stator. The main drawback in developing EKF is it may not proficient to consider the effect of electromagnetic noise which is mainly produced during the time of different speed ranges. Owing to this reason this may cause to vary the motor flux linkages which are significant to find the rotor position and speed by EKF method will give approximate results. To carry on this process, we present the simulation results for sensorless speed control of IPMSM drive by using EKF algorithm with the incorporation of a noise signal which is corresponding to the frequency of electromagnetic noise signal using MATLAB/Simulink software. The armature current, rotor position, and speed estimation are analyzed under this noise signal effect and the effectiveness of the EKF for sensorless control of IPMSM drive is observed.

An estimation fusion algorithm based on a group of hybrid coordinate (HC) filters is presented to the position and velocity estimation using angle-only measurements extracted from multiple maneuverable aircrafts with onboard passive sensor. The algorithm is a hierarchical architecture which consists of several local processors and a global processor. In each local processor, an extended Kalman (EK) filter utilizes the reference Cartesian coordinate (RCC) system for state and state covariance extrapolation and utilizes the modified spherical coordinate (MSC) system for state and state covariance updating. In the global processor, a recursive least squares (RLS) estimator sequentially computes a global estimate in the local inertial Cartesian coordinate (LICC) system. The estimator is developed by utilizing RCC updated state covariance to compute each filter weight for combining the outputs of local processors. The typical case of target motion analysis is investigated through computer simulation. In simulation study the proposed algorithm is compared to each HC EK filter. The EK filter encounters slow convergence problem under both circular arc and waving flight scenarios. By using a RLS estimator, the convergence is greatly accelerated. The algorithm markedly improves the tracking accuracy as well.

As one of the most essential technologies, wireless sensor networks (WSNs) integrate sensor technology, embedded computing technology, and modern network and communication technology, which have become research hotspots in recent years. The localization technique, one of the key techniques for WSN research, determines the application prospects of WSNs to a great extent. The positioning errors of wireless sensor networks are mainly caused by the non-line of sight (NLOS) propagation, occurring in complicated channel environments such as the indoor conditions. Traditional techniques such as the extended Kalman filter (EKF) perform unsatisfactorily in the case of NLOS. In contrast, the robust extended Kalman filter (REKF) acquires accurate position estimates by applying the robust techniques to the EKF in NLOS environments while losing efficiency in LOS. Therefore it is very hard to achieve high performance with a single filter in both LOS and NLOS environments. In this paper, a localization method using a robust extended Kalman filter and track-quality-based (REKF-TQ) fusion algorithm is proposed to mitigate the effect of NLOS errors. Firstly, the EKF and REKF are used in parallel to obtain the location estimates of mobile nodes. After that, we regard the position estimates as observation vectors, which can be implemented to calculate the residuals in the Kalman filter (KF) process. Then two KFs with a new observation vector and equation are used to further filter the estimates, respectively. At last, the acquired position estimates are combined by the fusion algorithm based on the track quality to get the final position vector of mobile node, which will serve as the state vector of both KFs at the next time step. Simulation results illustrate that the TQ-REKF algorithm yields better positioning accuracy than the EKF and REKF in the NLOS environment. Moreover, the proposed algorithm achieves higher accuracy than interacting multiple model algorithm (IMM) with EKF and REKF.

This work considers the problem of position and position-uncertainty estimation for atonomous vehicles during power black-out, where it cannot be assumed that any position data is accessible. To tackle this problem, the position estimation will instead be performed using power separated and independent measurement devices, including one inertial 6 Degrees of Freedom (DOF) measurement unit, four angular wheel speed sensors and one pinion angle sensor. The measurement unit's sensors are initially characterized in order to understand conceptual limitations of the inertial navigation and also to be used in a filtering process. Measurement models are then fused together with vehicle dynamics process models using the architecture of an Extended Kalman Filter (EKF). Two different EKF filter concepts are developed to estimate the vehicle position during a safe stop; one simpler filter for smooth manoeuvres and a complex filter for aggressive manoeuvres. Both filter designs are tested and evaluated with data gathered from an experimental vehicle for selected manoeuvres of developed safe-stop scenarios. The experimental results from a set of use-case manoeuvres show a trend where the size of the position estimation errors significantly grows above an initial vehicle speed of 70 km/h. This paper contributes to develop vehicle dynamics models for the purpose of a blind safe stop.

A smart city’s main purpose is to provide intelligent responses to different problems of the rapid urban population growth. For instance, integrating fleet management solutions into intelligent transportation systems (ITS) can efficiently resolve transportation problems relying on each vehicle information. Usually, the position estimate is ensured by the integration of the Global Positioning System (GPS) and Inertial Navigation Systems (INS). For multisensor data fusion, the Extended Kalman Filter (EKF) is generally applied using the sensor’s measures and the GPS position as a helper. However, the INS are expensive and require more complex computing which induces restrictions on their implementation. Furthermore, the EKF performance depends on the vehicle dynamic variations and may quickly diverge because of environmental changes. In this paper, we present a robust low cost approach using EKF and Support Vector Machines (SVM) to reliably estimate the vehicle position by limiting the EKF drawbacks. The sensors used are a GPS augmented by a low cost wireless sensor network. When GPS signals are available, we train SVM on different dynamics and outage times to learn the position errors so we can correct the future EKF predictions during GPS signal outages. We obtain empirically an improvement of up to 94% over the simple EKF predictions in case of GPS failures.

This paper presents the integration of an Ultra Short Base Line (USBL) acoustic modem and positioning device in a two-parallel Extended Kalman Filter (EKF) multisensor navigation schema for an Autonomous Underwater Vehicle (AUV). The system consists of a first odometric EKF position estimator fed with the measurements provided by an Inertial Measurement Unit (IMU), a Doppler Velocity Log (DVL), a visual tracker and a pressure sensor, and a second corrective EKF that fuses the previous sensors with the Global Positioning System (GPS) and the corrected delayed-state Ultra Short Base Line (USBL) position data. The first filter is aimed to compute a highly reliable vehicle position estimation from the sensor suit that produces high frequency relative navigation data. This estimation is essential to correct the delay of the USBL position measurements. The objective of the second filter is to give the vehicle a global pose, eliminating the drift inherent to dead-reckoning sensors by integrating absolute positions (GPS and USBL) provided at low frequencies. The effects of the USBL integration in the global localization module are evaluated in a simulated environment, and compared to a ground truth trajectory with the pose estimates given by the first EKF, the USBL raw data and the output of the second EKF, showing an effective reduction of the trajectory error.

In weak GNSS signal environments, extending integration time is paramount to improving the GNSS receiver's sensitivity. Furthermore, sufficient coherent integration can help to separate the line-of-sight (LOS) and non LOS (NLOS) signals — primarily in the Doppler domain — for multipath mitigation. However, extending integration time is limited by the presence of the navigation message data bits. The Maximum-Likelihood (ML) estimation method has been shown as the most effective way to estimate the navigation bit boundary locations (i.e., bit synchronization) and subsequently estimate the data bit values (i.e., bit decoding) in the presence of noise alone. This paper further analyzes the performance of ML estimation method as a function of various other parameters by using the successful synchronization rate (SSR) and successful decoding rate (SDR) as the criteria. The parameters considered include the number of data bits required (i.e., integration time) and the Doppler frequency error, and both are evaluated under different signal strength scenarios. The requirements for bit synchronization are analyzed under three SSRs, which are 85%, 90% and 95%. Results indicate that, under the SSR of 90% and the SDR of 90%, bit synchronization and bit decoding are valid for signal strengths of 15 dB-Hz and 20 dB-Hz respectively, and the Doppler frequency errors should not be larger than 24 Hz and 11 Hz respectively.

The Kalman filter (KF) is the optimal estimator that minimizes the mean square error when the state and measurement dynamics are linear in nature, provided the process and measurement noise processes are modeled as white Gaussian. However, in the real world, one encounters a large number of scenarios where either the process or measurement model (or both) are nonlinear. In such cases a class of suboptimal Kalman filter implementations called extended Kalman filters (EKF) are used. EKFs operate by linearizing the nonlinear model around the current reference trajectory and then designing the Kalman filter gain for the linearized model. Recently, an alternative approach has emerged for a certain class of problems where the error in the states is estimated using a Kalman filter, rather than the state itself. This error state KF (ErKF) approach, by deriving the error state dynamics, via the perturbation of the nonlinear plant, lends itself to optimal updates in the error states and optimal prediction and updates in the error state covariance. This is because the error state dynamics are linear, thereby satisfying a condition for optimal Kalman filtering. This paper offers a comparison between the EKF and ErKF via simulations and shows that the ErKF performance is robust to a variety of aircraft maneuvers performed. Furthermore, this paper shows that the ErKF, unlike the EKF, need not be repeatedly tuned with respect to the noise covariances in order to obtain acceptable estimation performance.

Navigation includes the integration of methodologies and systems for estimating time-varying position, velocity and attitude of moving objects. Navigation incorporating the integrated inertial navigation system (INS) and global positioning system (GPS) generally requires extensive evaluations of nonlinear equations involving double integration. Currently, integrated navigation systems are commonly implemented using the extended Kalman filter (EKF). The EKF assumes a linearized process, measurement models and Gaussian noise distributions. These assumptions are unrealistic for highly nonlinear systems like land vehicle navigation and may cause filter divergence. A particle filter (PF) is developed to enhance integrated INS/GPS system performance as it can easily deal with nonlinearity and non-Gaussian noises. In this paper, a hybrid extended particle filter (HEPF) is developed as an alternative to the well-known EKF to achieve better navigation data accuracy for low-cost microelectromechanical system sensors. The results show that the HEPF performs better than the EKF during GPS outages, especially when simulated outages are located in periods with high vehicle dynamics.

The conventional extended Kalman filter (EKF) is based on Taylor's series approximation of nonlinear motion and measurement models. The EKF cannot be directly generalized to multitarget situations since the multitarget state variable X = {x₁,...,x_n} exhibits discontinuous jumps in target number: X = 0, X = {x₁}, X = {x₁, x₂}, etc. However, it is possible to extend X to a continuous multitarget state X = {x₁,..., x_n} via the concept of a point target-cluster x = (a,x) where x is interpreted as multiple targets co-located at target-state x, the expected number of which is a. We generalize the EKF to multitarget problems. We illustrate the approach by deriving predictor and corrector equations for a single-sensor, single-target EKF that integrates the functions of detection and tracking in the presence of missed detections and false alarms.

This paper proposes a novel invariant extended Kalman filter (IEKF), a recently modified version of the extended Kalman filter (EKF), to estimate partial discharge (PD) location in a transformer insulation system model. An acoustic signal measurement is utilized to localize the PD location. Unlike conventional EKF methods, where the correction term used to update the state is linearly proportional to the output error, the correction term of the proposed algorithm is independent of the output error, resulting in a fast response with a significant reduction in the estimation error. In contrast to the EKF, the proposed method can successfully mimic the nonlinear dynamics and mitigate measurement noise stochasticity. Moreover, even if the measurement model fails to fully capture the PD's dynamics, the IEKF will still sustain a reasonable performance. In contrast, conventional EKFs can easily diverge if a mismatch between the measurement model and the true measurement occurs. Experimental results are shown to verify the proposed method's performance compared to a recently published variant of the EKF.

BeiDou Navigation Satellite System (BDS) is composed of satellites in geostationary Earth orbit (GEO), medium Earth orbit (MEO) and inclined geosynchronous orbit (IGSO). However, the orbit determination of geostationary Earth orbits and of geosynchronous orbits (GSO) with small inclination angle and small eccentricity is a challenging task that is addressed in this paper using Extended Kalman Filter (EKF). The satellite positions were predicted in Earth-centred inertial (ECI) reference frame when propagated through Keplerian model and perturbation force model for different values of right ascension of ascending node (RAAN). Root mean square (RMS) errors of 9.61 cm, 6.73 cm and 11.46 cm were observed in ECI X, Y and Z satellite position coordinates of GSO respectively, whereas, the RMS errors for GEO satellite were 8.89 cm, 7.92 cm, and 0.93 cm respectively in ECI X, Y and Z coordinates; for perturbation force model with maximum value of RAAN when compared with dynamic orbit determination model. Kolmogorov-Smirnov test for EKF reported a p-value > 0.05, indicating a good fit of perturbation force model for orbit propagation. Orbit determination using EKF with perturbation force model were compared with that using EKF with Kepler's model. Wilcoxon Rank Sum test was used to compare the residuals from EKF algorithm through Kepler's model and perturbation force model. EKF with Perturbation force model showed improvement in predicting the satellite positions as compared to Kepler's model. EKF with Perturbation force model was further applied to International GNSS Service (IGS) station data and kilometre level accuracy was achieved. RMS errors of 0.75 km, 2.53 km and 1.91 km were observed in ECI X, Y and Z satellite position coordinates of GSO, respectively, whereas, the RMS errors for GEO satellite were 3.89 km, 4.20 km and 6.66 km respectively in ECI X, Y and Z coordinates for perturbation force model.

A required assumption of a Kalman filter, the most-widely-used state estimator in avionic systems, is the Gaussian and whiteness of process and measurement noise. If the assumption fails, the performance of the Kalman filter degrades, and its estimation results are no longer optimal. In fact, many avionic applications produce colored noise, and the parameters of colored noise models are typically unknown beforehand without additional information about the noise statistics. In addition, the functions of each underlying model-nonlinear dynamic and measurement models - are sometimes improper or partially unknown. To estimate the states of systems with unknown correlations of each instance of noise and uncertain modeling errors of parametric models, we propose a novel approach that incorporates the kernel embedding of distributions into the extended Kalman filter. In our approach, kernel embedding maps process and measurement residuals, defined by differences between outputs of approximate system models and collected training data, into a reproducing kernel Hilbert space to generate nonparametric models in the functional space. Results from Monte Carlo simulations demonstrate that the proposed method, compared to existing methods (e.g., extended Kalman filter and Gaussian process-based filter), improves the accuracy of state estimation under colored noise conditions.