Conventional Extended Kalman Filter Research Articles

Research on the fusion positioning system of UWB/LiDAR based on the algorithm of SPF and KF

In order to overcome the disadvantages of single sensor localization in complex indoor environments, this paper proposes a design scheme for the ultra-wide band (UWB) and LiDAR co-location fusion system. To attain the ideal positioning effect of this system, a segmented point fusion approach based on Euclid's theorem is put forward to optimize the conventional extended Kalman filter and unscented Kalman filter algorithms. It can effectively reduce the positioning error of UWB in non-line-of-sight environments and under the multipath effect, and also assist LiDAR in correcting trajectory drift in sparsely textured scenes. Through both simulations and experiments, the feasibility of the proposed algorithm in the UWB/LiDAR positioning system is verified. The results reveal that the positioning error of the enhanced multi-sensor fusion algorithm is reduced by 22% compared with the original algorithm and 25.7% compared with the single sensor positioning method.

UWB/IMU integrated indoor positioning algorithm based on robust extended Kalman filter

Abstract As a fundamental method for integrated indoor positioning, Ultra wideband (UWB)/ Inertial Measurement Unit (IMU) offers more stable performance compared to separate UWB sensor and IMU sensor positioning and uses the least squares (LS) and extended Kalman filter (EKF) for positioning processing, however, the performance of conventional LS and EKF will be seriously affected when the contamination rate of measurement noise is high. To address this problem, this paper proposes a robust extended Kalman filter (REKF) based algorithm for robot-integrated indoor positioning, which combines UWB and IMU positioning methods to obtain a more robust system performance and optimal positioning accuracy. In this paper, a vehicle equipped with a UWB/IMU sensor and four anchor points is set up to perform linear, circular, and trajectory motions in an experimental environment with visible obstacles. The experimental results show that compared with EKF, REKF improves the localization precision and accuracy in the three motions by 20.98%, 35.51%, and 47.06%, respectively. REKF effectively improves the accuracy and robustness of the robot’s indoor positioning, and provides an improved and practical method for integrated indoor positioning algorithms to make indoor positioning more accurate and reliable.

An Integrated Navigation Method Aided by Position Correction Model and Velocity Model for AUVs.

When autonomous underwater vehicles (AUVs) perform underwater tasks, the absence of GPS position assistance can lead to a decrease in the accuracy of traditional navigation systems, such as the extended Kalman filter (EKF), due to the accumulation of errors. To enhance the navigation accuracy of AUVs in the absence of position assistance, this paper proposes an innovative navigation method that integrates a position correction model and a velocity model. Specifically, a velocity model is developed using a dynamic model and the Optimal Pruning Extreme Learning Machine (OP-ELM) method. This velocity model is trained online to provide velocity outputs during the intervals when the Doppler Velocity Log (DVL) is not updating, ensuring more consistent and reliable velocity estimation. Additionally, a position correction model (PCM) is constructed, based on a hybrid gated recurrent neural network (HGRNN). This model is specifically designed to correct the AUV's navigation position when GPS data are unavailable underwater. The HGRNN utilizes historical navigation data and patterns learned during training to predict and adjust the AUV's estimated position, thereby reducing the drift caused by the lack of real-time position updates. Experimental results demonstrate that the proposed VM-PCM-EKF algorithm can significantly improve the positioning accuracy of the navigation system, with a maximum accuracy improvement of 87.2% compared to conventional EKF algorithms. This method not only improves the reliability and accuracy of AUV missions but also opens up new possibilities for more complex and extended underwater operations.

Observation vector reconstruction-based nonparametric nonlinear restoring force identification for granules-structures coupled vibrating system

The nonlinear restoring force (NRF) generated by the collision and friction between particles and structures is the leading cause of the complex dynamic response of the granules-structures coupled vibrating system (GSCVS). Identification of NRF can provide critical information for post-event damage diagnosis and structural design of immersed structures. However, the spatial distribution and dynamic response of the particles near the structures are diverse and complex, making it difficult to describe the NRF with an accurate mathematical model. This paper proposed a data-based nonparametric method to estimate the NRF in the GSCVS. A nonparametric model of NRF that considered the additional effects of particles on both sides of the structures and consisted of system response and undetermined coefficients was developed. The observation vector of the conventional Extended Kalman Filter (EKF) was reconstructed by the sparse measurement of the strain response. The reconstructed observation vector contains three response components: translational displacement, translational acceleration, and rotational acceleration, in which the rotational acceleration response is difficult to measure in engineering applications. The proposed EKF based on observation vector reconstruction (EKF-OVR) can identify the undetermined coefficients in the nonparametric model, and then the NRF can be calculated. Numerical studies showed that EKF-OVR achieved higher accuracy and noise robustness than the conventional EKF and the data fusion based EKF. A dynamic experimental study on granules-beam coupled vibrating system (GBCVS) was conducted, and the proposed algorithm was employed to identify the NRF of the GBCVS. The effects of excitation amplitude, particle size, and immersed depth on NRF are analyzed, and it is found that higher harmonic components in the NRF led to period doubling and chaos of the beam response.

Tuning the EKF using estimated measurements (EKF-EM) for balanced performances

In this paper, the conventional Extended Kalman Filter (EKF) algorithm has been systematically extended to the estimated measurement (EKF-EM) paradigm. The derivation of the EKF-EM state and state error estimates establishes the relation of the robustness and sensitivity metrics to the measurement projected Kalman gain. Recently, these metrics are being used ad-hoc for obtaining desired EKF performances in various signal processing applications. The EKF-EM is used to develop a process noise adjustment algorithm to tune the EKF for balanced performances. This proposed method is applied and validated in 3 diverse applications:(a) ensure convergent comparable filter performances for an originally divergent EKF in the 3D Lorenz attractor problem, (b) obtain spectral and audio improvement of a practical noise corrupted speech signal and (c) determine a suitable process model out of several options for balanced filter performances in a realistic 2D ballistic target tracking scenario.

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 0.483 m 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.

Air-ground multi-agent system cooperative navigation based on factor graph optimization SLAM

In this study, a cooperative navigation algorithm centered on factor graph optimization—simultaneous localization and mapping (FGO-SLAM) is presented for an air-ground multi-agent system. The algorithm prioritizes the control of error statuses during the position and attitude estimation procedure throughout the entire back-end optimization process. In the conventional extended kalman filtering (EKF) algorithm, periodic cumulative errors may arise, introducing uncertainty to the estimation process. The application of the FGO algorithm not only mitigates deviation but also stabilizes errors, thereby eliminating the accumulation of periodic errors. In comparison to the practical EKF-SLAM, FGO-SLAM serves as a semi-offline optimization system that leverages key frames to minimize computational load. During multi-agent simulations, when two or more agents have overlapping field views, landmark data is merged, enhancing the optimization effectiveness. Through simulation experiments, the proposed algorithm demonstrates a 40% reduction in position error and a 41% reduction in attitude error, affirming the efficacy of FGO-SLAM for cooperative navigation.

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

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

A novel parallel constrained extended Kalman filter for improving navigation algorithm – case study: gas pipeline

In many real navigation problems, moving objects may have some state or measurement constraints along their way. Using these constraints in conventional Extended Kalman Filter (EKF) equations results large matrices which are computationally time-consuming. In this paper, a new Constrained Navigation Filter (CNF) is proposed as a parallel to reduce the computational burden of the conventional EKF algorithm while increasing the positioning accuracy. So a methodology has been developed for Strap-down Inertial Navigation System (SINS) based on MEMS IMU applied on Pipeline Inspection Gauge (PIG) to sense data at constant sampling rate of 108 km of the pipeline. The results verified that using such a hybrid approach has improved positional accuracy 8.97% in comparison with that of the latest methods like EKF/ Pipe Line Junctions (PLJ). Also, the proposed method is 2.277 times better than EKF/PLJ in the algorithm runtime.

Nonlinear state estimator design and implementation for speed control of a BLDC motor drive using square‐root extended H∞$\text{H}_{\infty}$ filter

AbstractSensorless speed control of a brushless DC (BLDC) motor, basically a combined control‐observer design, plays a crucial role in several industries, including electrical and aerospace applications. The key idea is to use only electrical (current) sensors and the remaining mechanical states (speed and rotor position) to be estimated using square‐root extended filter (SREF) for a BLDC motor. These estimated states are then fed‐back to the controller. In most of the designs, it is assumed that the BLDC motor plant parameters are constant; however, in reality, their parameters vary due to various intrinsic and extrinsic conditions. Compared to the conventional extended Kalman filter or its square‐root version, the proposed SREF for the BLDC motor has an inherent tendency to deal with parameter uncertainties and hence improves the robustness of the overall system. The robustness of the proposed scheme is tested at various reference speeds for a closed‐loop BLDC drive; further parametric uncertainties in stator resistance, detuning of inductance, and change in load are considered. The proposed control‐observer strategy using SREF is compared with its counterpart square‐root extended Kalman filter (SREKF), and the experimental results validate the efficacy of SREF for speed control of BLDC motor drive during the nominal conditions and in the presence of parametric uncertainties.

Neural extended Kalman filters for learning and predicting dynamics of structural systems

Accurate structural response prediction forms a main driver for structural health monitoring and control applications. This often requires the proposed model to adequately capture the underlying dynamics of complex structural systems. In this work, we utilize a learnable extended Kalman filter (EKF), named the neural extended Kalman filter (neural EKF) throughout this article, for learning the latent evolution dynamics of complex physical systems. The neural EKF is a generalized version of the conventional EKF, where the modeling of process dynamics and sensory observations can be parameterized by neural networks, therefore learned by end-to-end training. The method is implemented under the variational inference framework with the EKF conducting inference from sensing measurements. Typically, conventional variational inference models are parameterized by neural networks independent of the latent dynamics models. This characteristic makes the inference and reconstruction accuracy weakly based on the dynamics models and renders the associated training inadequate. In this work, we show that the structure imposed by the Neural EKF is beneficial to the learning process. We demonstrate the efficacy of the framework on both simulated and real-world structural monitoring datasets, with the results indicating significant predictive capabilities of the proposed scheme.

Robust state and protection-level estimation within tightly coupled GNSS/INS navigation system

In autonomous applications for mobility and transport, a high-rate and highly accurate vehicle-state estimation is achieved by fusing measurements of global navigation satellite systems (GNSS) and inertial sensors. The state estimation and its protection-level generation often suffer from satellite-signal disturbances in urban environments and subsequent poor parametrization of the satellite observables. Thus, we propose an innovative scheme involving an extended H∞ filter (EHF) for robust state estimation and zonotope for the protection-level generation. This scheme is shown as part of a tightly coupled navigation system based on an inertial navigation system and aided by the GPS/Galileo dual-constellation satellite navigation system. Specifically, GNSS pseudorange and deltarange observables are utilized. The experimental results of post-processing a real-world dataset show significant advantages of EHF against a conventional extended Kalman filter regarding the navigation accuracy and robustness under various GNSS measurement parametrizations and environmental circumstances. The zonotope-based protection-level calculation is proven valid, computationally affordable, and feasible for real-time implementations.

An enhanced dynamic soft sensor–based online estimation of missing data for water distribution system with inherent disturbances

Interruption of water supply is an unfavourable event in water distribution system (WDS). The interruptions are mainly due to water hammer (WH) transients that are caused by sudden pump/valve failures. In this paper, the effects of hammer are mathematically realized considering the effect of valve coefficient integrated with the system. Due to economic and technical limitations, the system is generally equipped with some obsolete instruments for continuous monitoring, which creates a missing data problem. The problem of missing data estimation could be addressed using a novel customized Kalman filter (CKF), a customized version of conventional Kalman filter (KF) based on process model. Initially, to avoid the intricacies of obtaining the optimal estimates of missing instants, the vacancies in the data are filled assigned by the average values of available adjacent data. The measured quantities are considered distorted with state-, input-dependent and independent uncertainties. The effect of WH on a prototypical WDS is studied for different modes of operation, and the missing data points are reconstructed for regular and irregular sampled data set using the proposed CKF. The performance of the proposed algorithm is also compared with the conventional extended Kalman filter (EKF). The study indicates that the proposed missing data estimation algorithm is highly reliable to identify the missing instances in the states of distribution system.

Partial Discharge Localization in Power Transformers Using Invariant Extended Kalman Filter

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.

An Invariant Method for Electric Vehicle Battery State-of-Charge Estimation Under Dynamic Drive Cycles

This paper proposes a novel invariant extended Kalman filter (IEKF), a modified version of the extended Kalman filter (EKF), for state-of-charge (SOC) estimation of lithium-ion (Li-ion) battery cells. Unlike conventional EKF methods where the correction term used to update the state is linearly proportional to the output error, this paper employs the IEKF where the correction term is independent of the output error, resulting in a significant reduction in the estimation error and improving the estimation accuracy. In contrast to classic method like the EKF and more contemporary ones like the square root variant of the Cubature Kalman Filter (SCKF), the IEKF can successfully mimic the nonlinear dynamics and mitigate measurement noise stochasticity. Moreover, even if the measurement model fails to fully capture the cell’s dynamics, the IEKF will still sustain a reasonable performance. Hence, IEKF outperforms the conventional EKF, and even the SCKF, which can diverge if a mismatch between the SOC measurement model and the true SOC measurement occurs. The derivation of the proposed method followed by experimental verification using commercial Li-ion battery cells are presented.

Model Predictive Controlled IM Drive based on IT2FNN Controller

Abstract In this paper, the predictive torque control (PTC) based induction motor (IM) drive using an interval type-2 fuzzy neural network (IT2FNN) controller in the speed control loop is designed and tested in simulations. The states required for the proposed motor drive are estimated by extended complex Kalman filter (ECKF). The ECKF performs online estimations of stator currents, rotor fluxes, rotor mechanical speed, and rotor resistance. Compared to conventional extended Kalman filter (EKF), which estimates the same states/parameters, the designed ECKF has less computational burden because it does not contain matrix inverse and the matrix dimensions have been reduced. In addition, the rotor resistance estimated by ECKF is updated online to the PTC system. Thus, the performance of the PTC-based IM drive is improved against variations in the rotor resistance, whose value changes with operating conditions such as frequency and temperature. In order to force both the ECKF observer and the proposed IM drive, a challenging scenario containing the wide speed range operation of the IM is designed. Simulation results confirm the performance of the proposed speed-sensorless PTC-based drive that uses an IT2FNN controller in the speed control loop and the estimation performance of the ECKF observer.

Attitude estimation of connected drones based on extended Kalman filter under real outdoor environments

In this study, we design connected multiple drones through electromagnets, and it requires an attitude estimation method that does not use magnetic sensors. In order to solve this problem, we propose an extended Kalman filter-based attitude estimation method based on the multiple global positioning system (GPS) to substitute the magnetic sensor. Additionally, we demonstrate that combining velocity vectors allows for highly accurate attitude estimation. We compare the estimation accuracy and stability of attitude estimation in simulation and real environments using two types of experiments. It is confirmed that the accuracy of yaw angle estimation of the proposed method is improved 36.1% in simulation and 29.7% in the real environments compared to conventional extended Kalman filter.

H∞ filter-based dynamic joint carrier frequency offset and linear phase noise tracking for CO-OFDM systems.

In this paper, a robust, dynamic and joint carrier frequency offset (CFO) and linear phase noise (LPN) tracking algorithm is proposed by utilizing the H∞ filter for CO-OFDM systems. The dynamic tracking is implemented by Bayesian filtering which time-recursively updates the posterior state estimates based on the prior state information and calculated gain. To improve the robustness of the dynamic tracking process against the uncertainty of the noise terms, we adopt the H∞ filter, which designs a min-max optimization problem with a performance bound-defined constraint. The H∞ filter obtains the state estimates by limiting the worst-case estimation error, which guarantees its robustness to the system uncertain noise terms and external disturbances. After the joint estimation of CFO and LPN, we then propose a decision-feedback algorithm to achieve accurate signal detection. The accuracy and robustness of the proposed algorithm are verified in an 88.9 Gb/s 16-QAM CO-OFDM system, by comparing with the conventional extended Kalman filter (EKF) and Gaussian particle filter (GPF). Simulation results show that the MSEs of CFO and LPN by H∞ filter can reach the Cramér-Rao Lower Bounds (CRLBs) as well as the Bayesian CRLB (BCRLB), and possesses excellent noise and chromatic dispersion (CD) tolerance.

ANFIS-EKF-Based Single-Beacon Localization Algorithm for AUV

Singe-beacon localization technology can help Autonomous Underwater Vehicles (AUVs) to obtain precise positions by deploying only one beacon. It is considered as a promising way, benefiting from saving much time and labor compared with traditional Long-Baseline Localization (LBL). A typical single-beacon localization scheme contains two essential questions: the initial observability problem and long-endurance trajectory tracking problem. Aiming at these core problems, a comprehensive solution for single-beacon localization is described in this paper. An multi-hypothesis initial position discriminant method is proposed firstly, which helps to achieve accurate initial location based on observability analysis. Then, an Adaptive Network Fuzzy Inference System (ANFIS)-improved Extended Kalman Filter (EKF) method is proposed, in which single-beacon measuring information is fused with off-the-shelf sensors, including DVL, Compass, etc. ANFIS-EKF can help to improve trajectory tracking precisions by restraining the heavy loss of linearization in conventional EKF. Both simulation and field tests are conducted to verify the performance of the proposed algorithms.

Modified extended Kalman filtering algorithm for precise voltage and state-of-charge estimations of rechargeable batteries

With the popularization of electric vehicles (EVs), the voltage and state-of-charge (SOC) estimations of rechargeable batteries are of great significance. The SOC parameter has been used as an indicator for delivering the electrical energy of rechargeable lithium-ion batteries (LIBs), while the voltage has been a critical parameter needed to monitor to prevent the cause of battery damage especially during the charging and discharging process. Thus, the research focus is to estimate the SOC and voltage accurately using algorithms. With the capability of avoiding major estimation errors, conventional extended Kalman Filtering (EKF) has been employed to estimate the optimal value of SOC using the parameters obtained by indirect measurements, such as voltage and current. However, the algorithm suffers from limited precision in SOC and voltage estimations, and there is still no in-depth investigation of the error reduction in voltage prediction. Although the SOC accuracy can be improved by a joint algorithm such as double Kalman filtering, optimization of EKF itself is still needed due to the superposition of nonlinear errors. This study reveals that the conventional EKF algorithm induces estimation errors especially when the current abruptly changes. In this study, the research on the modified extended Kalman filtering (MEKF) algorithm was conducted for estimating the voltage and SOC of the LIBs with great improvement in estimation accuracy. The YUASA LEV50 cell was subjected to a standard discharging rate of 0.2C at 298 K to acquire offline parameters, followed with the newly proposed dynamic estimation mathematical battery model (DBOFT) for the optimization of estimations. This is the first time to propose a method of combining gain matrix and noise to reduce the error of voltage estimation at the current turning points, which greatly improves the accuracy of voltage estimation. Specifically, the MEKF algorithm is capable of adjusting the parameters in real-time and reducing the SOC estimation error. The SOC error estimated by the MEKF algorithm was reduced to 0.0052 % under the current rate of 0.2C. Finally, the maximum estimated voltage error was reduced to 0.972 % while the SOC estimation error was reduced to 0.01016 % under the same working current. The precise voltage and SOC estimations obtained by the MEKF with higher accuracy than traditional methods were validated under experimental verification, which is crucial for the life extension of LIBs and the safety of the battery management system (BMS).