Linear Kalman Filter Research Articles

A Gyroless Attitude Estimation for Ground Vehicle Under Severe Dynamic Conditions

In this paper, we propose an inertial measurement unit (IMU) based accurate attitude estimation algorithm for land vehicles using only accelerometer and magnetometer data along with a set of nonholonomic constraints on vehicle motion. Currently, most of the IMU based attitude estimation methods deploy gyroscopes along with accelerometer measurements which work efficiently only either for static or quasi-static conditions. However, during prolonged dynamic phases of vehicle motion, the accelerometer measurements are corrupted due to external acceleration which deteriorates the attitude estimation accuracy. In this paper, we illustrate that vehicle and sensor constraints allow the removal of external acceleration from accelerometer measurement in a magnetically clean environment. This, in turn, enables to design and use a linear Kalman filter (KF) using only accelerometer and magnetometer data for accurate attitude estimation. The proposed algorithm is tested and compared with three different state-of-the-art algorithms in simulation and real-world experiments under various dynamic conditions. The results illustrate that vehicle attitude information can be obtained accurately and reliably during long term fast dynamic conditions, using the designed KF.

Simplified radius-directed linear Kalman filter algorithm for lightning-induced state of polarization compensation in optical ground wire

Lightning strokes can cause an ultrafast rotation of the state of polarization (RSOP) in an optical ground wire cable, which leads to communication interruptions. To address this, a simplified radius-directed linear Kalman filter (S-RD-LKF) for blind polarization demultiplexing was developed. The ultrafast RSOP under thunderstorm weather conditions was simulated based on the Heidler model, and the simulation results indicated that the proposed S-RD-LKF algorithm effectively compensated for the RSOP impairment caused by the first and subsequent lightning strokes. Moreover, the RD-LKF algorithm and the proposed S-RD-LKF algorithm exhibited a similar performance; however, the computation complexity of the proposed algorithm was lower.

State Estimation of Gas-Lifted Oil Well Using Nonlinear Filters.

The focus of this work is the extension of nonlinear state estimation methods to gas-lifted systems. The extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) were used to estimate the nonlinear states. Brief descriptions of the filters were first presented starting from the linear Kalman filter. Hypothesis tests on the expectation of the residuals were performed to show how close to optimal the estimation methods are and it showed the UKF estimates to be slightly better than EKF while PF performs the worst. The PF has poor accuracy using residual visualisation, hypothesis test and the root mean squared error (RMSE) values of the residuals. The gas-lifted system exhibits casing heading instability where the states show oscillatory behaviour depending on the value of the input but the results here do not change in a known way for each filter as the input is changed from the non-oscillatory region to the oscillatory region. Therefore, for this noise distribution and model assumption, either the EKF or UKF can be used for nonlinear state estimation with UKF better preferred if computational cost is not considered when control solutions are used in gas-lifted system.

Angular Velocity Estimation and Error-Covariance Analysis from Accelerometers Using Least Squares

In this work angular velocity solutions are presented from accelerometers. Two main formulations are presented. The first is based on using multiple vector sets of accelerometers, and the second is based on using multiple single-axis accelerometers. Linear least-squares approaches are derived for both, along with error-covariance expressions for the angular velocity estimates. Nonlinear least-squares approaches are also derived for both, which also provide the Cramér–Rao lower bound as a byproduct. All approaches have a sign ambiguity in the angular velocity solutions, which is resolved using a simple linear Kalman filter. Simulation results are shown to compare the formulations, along with the linear and nonlinear least-squares solutions.

A State of the Art in Simultaneous Localization and Mapping (SLAM) for Unmanned Ariel Vehicle (UAV): A Review

Abstract For the past decade, the main problem that has attracted researchers’ attention in aerial robotics is the position estimation or Simultaneous Localization and Mapping (SLAM) of Unmanned Aerial Vehicles (UAVs) where the GPS signal is poor or denied. This article reviews the strengths and weaknesses of existing methods in the field of aerial robotics. There are many different techniques and algorithms that are used to overcome the localization and mapping problem of these UAVs. These techniques and algorithms use different sensors, such as Red Green Blue-Depth (RGB_D), Light Detecting and Ranging (LIDAR), and Ultra-wideband (UWB). The most common technique is used, i.e., probability-based SLAM, which uses two algorithms: Linear Kalman Filter (LKF) and Extended Kalman Filter (EKF). LKF consists of five phases and this algorithm is just used for linear system problems. However, the EKF algorithm is used for non-linear systems. Aerial robots are used to perform many tasks, such as rescue, transportation, search, control, monitoring, and different military operations because of their vast top view. These properties are increasing their demand as compared to human service. In this paper, different techniques for the localization of aerial vehicles are discussed in terms of advantages and disadvantages, practicality and efficiency. This paper enables future researchers to find the suitable SLAM solution based on their problems; either the researcher is dealing with a linear problem or a non-linear problem.

Universal prediction algorithm in maneuvering target tracking

The paper considers the methods to increase the accuracy of trajectory processing of the results of radar measurements in automated air traffic management systems (AATMS). The recursive filters traditionally used in the secondary processing path of radar information of modern AATMS systems are briefly considered: alpha-beta filter, linear and extended Kalman filters, multi-model filter. Along with multi-model Kalman filters it is proposed to use a universal filter based on the universal prediction theorem, the accuracy of which is asymptotically no worse than any other filter in the class of continuous functions. The algorithm for the operation of a universal alpha-beta filter was developed. As a result of modeling, it was obtained that the mean square of the tracking error of the trajectory of a maneuvering target for the universal filter during the change of the motion model turned out to be lower than for traditional filters.

A practical filter error method for aerodynamic parameter estimation of aircraft in turbulence

It is common for aircraft to encounter atmospheric turbulence in flight tests. Turbulence is usually modeled as stochastic process noise in the flight dynamics equations. In this paper, parameter estimation of nonlinear dynamic system with both process and measurement noise was studied, and a practical filter error method was proposed. The linearized Kalman filter of first-order approximation was used for state estimation, in which the filter gain, along with the system parameters and the initial states, constituted the parameter vector to be estimated. The unknown parameters and measurement noise covariance were estimated alternately by a relaxation iteration method, and the sensitivities of observations to unknown parameters were calculated by finite difference approximation. Some practical aspects of the method application were discussed. The proposed filter error method was validated by the flight simulation data of a research aircraft. Then, the method was applied to the flight tests of a subscale aircraft, and the aerodynamic stability and control derivatives were estimated. All the estimation results were compared with the results of the output error method to demonstrate the effectiveness of the approach. It is shown that the filter error method is superior to the output error method for flight tests in atmospheric turbulence.

Application of LSTM algorithm combined with Kalman filter and SOGI in phase-locked technology of aviation variable frequency power supply.

With the development of computer technology, the concept of computer automatic control has gradually penetrated the research field of aircraft power control, and intelligent power control systems have become mainstream research. The present work aims to improve the performance of the broadband phase-locked loop (PLL) based on the linear Kalman filter. Specifically, this paper first introduces linear Kalman filter and second-order generalized integrator (SOGI). Then, SOGI is added to PLL based on the linear Kalman filter. The purpose is to use the infinite gain effect of SOGI at the central angular frequency to eliminate the time-varying angular frequency component in the error when the system inputs SOGI to achieve a better filtering effect. Then, the system's stability analysis and parameter settings are carried out to establish an intelligent phase-locked method of aviation variable frequency power supply. Finally, simulation experiments are performed. The experimental results demonstrate that PLL via the linear Kalman filter with SOGI can solve the problem that the output phase angle contains high-frequency components when the power supply voltage distortion rate is 10%. This scheme has a strong anti-interference ability under power grid voltage imbalance. The accuracy of the Long and Short-term Memory network used here is about 80%, which can well realize the intelligent aviation power frequency conversion control method. The research reported here provides a reference for establishing smart phase-locked technology of aviation variable frequency power supply.

Social Distance Evaluation in Transportation Systems and Other Public Spaces using Deep Learning

Abstract This research put forward an efficacious real-time deep learning-based technique to automate the process of monitoring the social distancing in transportation systems (e.g., bus stops, railway stations, airport terminals, etc.) and other public spaces with the purpose to mitigate the impact of coronavirus pandemic. The proposed technique makes use of the YOLOv3 model to segregate humans from the background of each image of a surveillance video and the linear Kalman filter for tracking the humans’ motion even in case in which another object or person overlaps the trajectory of the person under analysis. The performance of the model in human detection is extremely high as demonstrated by the accuracy of the model that reaches values higher than 95%. The detection algorithm can be applied for alerting people to keep a safe distance from each other when they are in crowded places or in groups.

Real-Time Updating High-Order Extended Kalman Filtering Method Based on Fixed-Step Life Prediction for Vehicle Lithium-Ion Batteries.

Lithium-ion batteries have become an important power source in low-carbon transportation energy, and the safe operation and remaining useful life prediction are of great significance. Aiming at the shortcomings of existing methods, such as low prediction accuracy and a short prediction period, this paper proposes a real-time update high-order extended Kalman filter method based on fixed-step life prediction for vehicle lithium batteries based on the principle of combining models and data. First, the state model describing the parameters in the dynamic energy attenuation model is established, and the energy attenuation model is regarded as the observation model of the system to meet the requirements of establishing the Kalman filter. Secondly, the multi-step prediction equation of the state model is established by iterative recursion. At the same time, the multi-step prediction equation between the existing energy output value and the future output value is established based on the multi-dimensional Taylor network (MTN). The multiplicative noise term introduced in the dynamic modeling process is regarded as the hidden variable of the system to meet the requirements of establishing the multi-step linear predictive Kalman filter. Finally, the effectiveness of the new method is verified by digital simulation examples.

Adaptive model predictive control for actuation dynamics compensation in real-time hybrid simulation

Hybrid simulation is used to obtain the dynamic response of a system whose components consist of physical and numerical substructures. The coupling of these substructures is achieved by actuation systems, which are commanded in closed-loop control setting. To ensure high fidelity of such hybrid simulations, performing them in real-time is necessary. However, real-time hybrid simulation poses challenges since the inherent dynamics of the actuation system introduce time delays, thus modifying the dynamic response of the investigated system. Therefore, a tracking controller is required to adequately compensate for such time delays. In this study, a novel tracking controller is proposed for dynamics compensation in real-time hybrid simulations. It is based on adaptive model predictive control, a linear time-varying Kalman filter, and a real-time model identification algorithm. Within the latter, auto-regressive exogenous polynomial models are identified in real-time to estimate the changing plant dynamics. A parametric virtual case study, encompassing a virtual motorcycle, is used to validate the performance and robustness of the proposed controller. Results demonstrate the effectiveness of the proposed controller for real-time hybrid simulations.

State of charge estimation of lithium-ion phosphate battery based on weighted multi-innovation cubature Kalman filter

State of charge (SoC) is one of the most important parameters of battery manage system (BMS). To obtain a better result of SoC estimation, an accurate battery model and effective SoC estimation algorithm are indispensable. In this work, a Thevenin model is established and linear Kalman filter (LKF) is used for online parameters identification. Results show that convergence of parameters identification of LKF is better than that of the recursive least squares (RLS) and recursive least squares with forgetting factor (FFRLS). In addition, the equilibrium potential equation (EPE) is used to fit the relationship between open circuit voltage (OCV) and SoC instead of 9th order polynomial. Then, by the weighted calculation of the innovation vector based on error distribution and time distribution, the weighted multi-innovation cubature Kalman filter (WMICKF) is proposed for the SoC estimation. The experimental data are obtained based on a prismatic LiFePO4 battery, which is tested under urban dynamometer driving schedule (UDDS) test and new European driving cycle (NEDC) test at room temperature. The results show that the WMICKF outperforms the classical cubature Kalman filter (CKF) and the multi-innovation CKF (MICKF). The SoC estimation error of WMICKF can be limited within 1% (0.91%), which is better than 1.30% of CKF and 2.71% of MICKF. In addition, the mean absolute error (MAE), root mean square error (RMSE) and coefficient of determination (R-square) are calculated for comprehensive assessment of the proposed method. Furthermore, the robustness of WMICKF is verified under different initial SoC error and different types of noise disturbance. Results show that under these uncertain factors, the proposed WMICKF is still reliable for accurate SoC estimation.

Comparative Analysis of Three Structures of Second-Order Generalized Integrator and Its Application to Phase-Locked Loop of Linear Kalman Filter

The present work explores the power quality problems of a microgrid in the aviation field, such as frequency offset or waveform distortion, caused by voltage imbalance and nonlinear load, to ensure the efficient and stable flight of aircraft, unmanned aerial vehicles, and other aircraft. There are many problems such as excessive harmonics, voltage imbalance, and direct current (DC) component in aviation variable frequency power supply voltage in microgrid under different conditions. Therefore, three kinds of second-order generalized integrators (SOGIs) with different structures are combined with linear Kalman filter phase-locked loops (PLLs). Besides, intelligent sensors are utilized for signal processing. Finally, simulation experiments are conducted to compare three SOGIs. The results show that the system is stable when the parameters k1 = 738.9553, k2 = 1092108.98405, k3 = 1477.91, and Ts = 0.000125 s and k ranges in [0.5, 3]. The angular frequency of PLL output is very low under the problems of many harmonics, three-phase voltage imbalance, and DC component ω g . The angular frequency ω g output by the PLL finally changes linearly with a change rate of 400 πrad/s2, so that the output phase angle θ g reaches a stable state. Thus, the proposed steady-state linear Kalman filter phase locking can accurately phase-lock the aviation variable frequency power supply. It provides an important reference for the power supply module in the microgrid to select the appropriate second-order generalized integrator to realize the accurate phase locking of the phase-locked loop under different conditions.

A Variable Parameter Method Based on Linear Extended State Observer for Position Tracking

This paper presents a control strategy with a linear extended state observer (LESO) and Kalman filter to achieve a high performance of the motion control system. The moment of inertia of the system, which is variable with the robotic joint motion, is estimated in the established model. A LESO with variable gain is designed, which could estimate the states and the total disturbance of the plant without a precision mathematical model. The disturbance caused by variable load and unknown dynamics can be compensated based on the LESO, while the moment of inertia is variable. In order to restrain the process noise and measure the noise of the system, the Kalman filter was applied. Tracking differentiator was utilized to avoid the overshoot of the system for the step signal. The designed control strategy with the LESO and the Kalman filter could improve the tracking performance for the servo system with parametric uncertainties, unknown dynamics, and disturbances. The effectiveness of the proposed method is implemented and validated in the experiment of the robotic joint, for which desired servo tracking performance is achieved with the conditions of load variation and sudden disturbance.

Modeling and observer design for aluminum manufacturing

This paper addresses modeling, identification, and observer design of the Hall-Héroult process. This process is the main core of aluminum manufacturing. More specifically, by building upon a blend of physical-chemical constitutive relations and system identification tools, this paper first provides a model describing the interplay of Anode-Cathode Distance (ACD), alumina concentration, and pot pseudo-resistance. The proposed model is then used to tune a linear Kalman filter generating online estimates of the plant state variables. The proposed approach is validated via data taken from a real industrial plant.

Phase current estimation based shunt active power compensation

Nonlinear and unbalanced loads introduce distortions to the power network in the form of harmonics and asymmetric load conditions. Through the rising penetration of residential renewable plants, energy is fed back into the grid. By shaping the output of those power converters the distortions can be mitigated. The technique of shunt active power filtering allows the instantaneous compensation of a dedicated nonlinear load, by measuring its distorted consumption. The goal of this article is to present an extension to this method which allows to compensate for a given area utilizing the advanced metering infrastructure in a smart grid. Given a compensation unit, instantaneous voltage measurements are collected from neighboring consumers. By utilizing the model of the distribution line, a linear Kalman filter allows the estimation of the phase currents. These values replace the dedicated nonlinear load measurements, and by compensating for line currents, the power quality of a given transformer area is improved. Analyzing the model based approach against real world like scenarios and elaborating the robustness shows the potential for real world applicability.

Effect of Weighted PDOP on Performance of Linear Kalman Filter for RTK Drone Data

The study established a theoretical relation between weighted position dilution of precision (WPDOP) and standard deviation (STD) of positioning error ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon /\sigma _{R})$ </tex-math></inline-formula> derived by linear Kalman filter (KF). The obtained results exposed a rectangular hyperbolic relationship of WPDOP and STD of error. The RTK Drone data were collected around 50 min of interval (07:37 to 08:28 A.M.) on May 12, 2021, with respect to the base station located at 13.841°N; 100.288°E over Thailand region, and a numerical test has been carried out to demonstrate the proposed algorithm. The simulation results verified the theoretical hyperbolic relationship with the distinct flatness of the curve at the base and drone receiver. This different flatness from the experiments was anticipated because of the position errors for the highly nonlinear dynamics motion of the drone receiver while the base receiver is fixed at one location.

3-D Refuse-Derived Fuel Particle Tracking-by-Detection Using a Plenoptic Camera System

Multiple particle tracking-by-detection is a widely investigated issue in image processing. The paper presents approaches to detecting and tracking various refuse-derived fuel particles in a industrial environment using a plenoptic camera system, which is able to yield 2D gray value information and 3D point clouds with noticeable fluctuations. The presented approaches, including an innovative combined detection method and a post-processing framework for multiple particle tracking, aim at making the most of the acquired 2D and 3D information to deal with the fluctuations of the measuring system. The proposed novel detection method fuses the captured 2D gray value information and 3D point clouds, which is superior to applying single information. Subsequently, the particles are tracked by the linear Kalman filter and 2.5D global nearest neighbor (GNN) and joint probabilistic data association (JPDA) approach, respectively. As a result of several inaccurate detection results caused by the measuring system, the initial tracking results contain faulty and incomplete tracklets that entail a post-processing process. The developed post-processing approach based merely on particle motion similarity benefits a precise tracking performance by eliminating faulty tracklets, deleting outliers, connecting tracklets, and fusing trajectories. The proposed approaches are quantitatively assessed with manuelly labeled ground truth datasets to prove their availability and adequacy as well. The presented combined detection method provides the highest F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -score, and the proposed post-processing framework enhances the tracking performance significantly with regard to several recommended evaluation indices.

A Review of Dynamic Phasor Estimation by Non-Linear Kalman Filters

Phasor estimation under dynamic conditions has been under study recently by relaxing the amplitude and phase of the static phasor. This paper will review some methods to estimate dynamic phasor by nonlinear Kalman filters. Nonlinear Kalman filters have found an application in the tracking of time-varying amplitude and phase. Five nonlinear Kalman filtering methods for dynamic phasor estimation are examined in this paper. These methods are: EKF1 stands for first-order Extended Kalman filter, EKF2 stands for second-order Extended Kalman, UKF stands for Unscented Kalman filter, GHKF stands for Gauss Hermite Kalman filter, and finally CKF stands for Cubature Kalman filter. This paper describes the theoretical processes of these methods and demonstrates their effectiveness in dynamic phasor estimation by some test signal simulations in MATLAB. The simulation section shows that nonlinear Kalman filters give more accuracy than linear Kalman filters when the phasor is relaxed by modulated amplitude and phase. Moreover, comparative assessments among the performance of five nonlinear Kalman filters are done for dynamic phasor estimation, and also their performances are compared with the other methods which have already been published. According to the simulation results, EKF1 gives the highest accuracy during steady-state ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 10^{-13}$ </tex-math></inline-formula> ) because the signal model is more similar to the estimation model of EK1 during the steady-state condition. However, the other non-linear Kalman filters show better performances in dynamic conditions. When the phasor is a time-varying amplitude and phase, filters give the same accuracy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.5 \%$ </tex-math></inline-formula> ). A step-change in amplitude and phase creates different overshoot and response times, but EKF2 shows the least overshoot (3.2%) and the longest response time (7.6 ms). Computation burden and noise indices can discriminate the methods from other viewpoints. The computation burden of GHKF is drastically increased when the number of states gives rise. CKF shows an appropriate performance when the number of samples and the number of the state increased in the input signal. EKF1 is not a good solution for noise infiltration when SNR is less than 40 dB, but CKF gives the highest accuracy in high noise levels. Compared to the six already pblished methods, CK shows the best performance with a reasonable estimation error ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.1101 \%$ </tex-math></inline-formula> ) and simulation time (0.6146 ms).

Extended Target Tracking With a Lidar Sensor Using Random Matrices and a Virtual Measurement Model

Random matrices are widely used to estimate the extent of an elliptically contoured object. Usually, it is assumed that the measurements follow a normal distribution, with its standard deviation being proportional to the object’s extent. However, the random matrix approach can filter the center of gravity and the covariance matrix of measurements independently of the measurement model. This work considers the whole chain from data acquisition to the linear Kalman Filter with extension estimation as a reference plant. The input is the (unknown) ground truth (position and extent). The output is the filtered center of gravity and the filtered covariance matrix of the measurement distribution. A virtual measurement model emulates the behavior of the reference plant. The input of the virtual measurement model is adapted using the proposed algorithm until the output parameters of the virtual measurement model match the result of the reference plant. After the adaptation, the input to the virtual measurement model is considered an estimation for position and extent. The main contribution of this paper is the reference model concept and an adaptation algorithm to optimize the input of the virtual measurement model.