A Dynamic Grid-based Q-learning for Noise Covariance Adaptation in EKF and its Application in Navigation

This paper presents an extended Kalman filter for pose estimation using noise covariance matrices based on sensor output. Compact and lightweight nine-axis motion sensors are used for motion analysis in widely various fields such as medical welfare and sports. A nine-axis motion sensor includes a three-axis gyroscope, a three-axis accelerometer, and a three-axis magnetometer. Information obtained from the three sensors is useful for estimating joint angles using the Kalman filter. The extended Kalman filter is used widely for state estimation because it can estimate the status with a small computational load. However, determining the process and observation noise covariance matrices in the extended Kalman filter is complicated. The noise covariance matrices in the extended Kalman filter were found for this study based on the sensor output. Postural change appears in the gyroscope output because the rotational motion of the joints produces human movement. Therefore, the process noise covariance matrix was determined based on the gyroscope output. An observation noise covariance matrix was determined based on the accelerometer and magnetometer output because the two sensors’ outputs were used as observation values. During a laboratory experiment, the lower limb joint angles of three participants were measured using an optical 3D motion analysis system and nine-axis motion sensors while participants were walking. The lower limb joint angles estimated using the extended Kalman filter with noise covariance matrices based on sensor output were generally consistent with results obtained from the optical 3D motion analysis system. Furthermore, the lower limb joint angles were measured using nine-axis motion sensors while participants were running in place for about 100 s. The experiment results demonstrated the effectiveness of the proposed method for human pose estimation.

<p>Inspection of large space structures is imperative for long term mission success. One solution is to utilize a second free flying spacecraft capable of performing inspection in orbit. An Extended Kalman Filter (EKF) is used to perform estimations on the relative position, velocity, angular velocity, and attitude through the use of navigation markers. Simulation takes place using MATLAB to compare the true values with the estimated values using the EKF. The initial covariance (P), process noise covariance (Q), and measurement noise covariance (R) matrices were tuned for a space structure with three navigation points. The largest recorded errors over 100 iterations occurred during the initial estimation yielding 26.78 centimeters in relative position, 1.80 centimeters per second in relative velocity, 0.0444 radians per second in relative angular velocity, and a difference of 0.0172 in the unit vector of relative attitude. After allowing 20 seconds of settling time the maximum errors were reduced to 5.0 centimeters in relative position, 0.40 centimeters per second in relative velocity, 0.0046 radians per second in relative angular velocity, and a difference of 0.0020 in the unit vector of relative attitude. The paper also discusses the application of training algorithms to tune the EKF parameters for future consideration.</p>

The electrification of the transport sector increases the demand for a battery management system that allows the Li-ion battery to work in a sophisticated way to reach its maximum. The main focus of the battery management system is the estimation of the battery’s state of charge (SOC) which is an indicator to determine the driving range of an electric vehicle. However, battery modeling has a direct influence on SOC estimation. In this article, the equivalent circuit model (ECM) and extended Kalman filter (EKF) algorithm are combined for SOC estimation of the Li-ion battery. Since the performance of Kalman filter algorithms strictly depends on noise covariance matrices, incorrect values reduce the accuracy and make divergence in state estimation. Therefore, it is motivated to analyze the impact of process noise and measurement noise covariance matrix values on SOC estimation using Kalman filter algorithms. Ten different combinations of noise covariance matrices are employed to analyze the effect of these matrices on SOC estimation. As a result, the tuning of the filter is critical for improvement in state estimation regardless of the type of the battery model and filtering technique. The matrix values are chosen to take into account the convergence rate and estimation error. To validate the effectiveness of the chosen noise covariance matrices, the simulation results of the ECM based EKF estimation method is superimposed with the discharge curves available in the data sheet. The simulation results show minimal estimation error and a good convergence rate for tuned EKF.

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

Real-time state of charge estimation for electric vehicle power batteries using optimized filter

Robust adaptive smooth variable structure Kalman filter for spacecraft attitude estimation

State of Charge (SOC) estimation plays a crucial role in managing the performance and longevity of electric vehicle (EV) batteries. Various estimation techniques, including various Kalman Filters (KF), have been employed for this purpose. The mathematical process and limits of different KF family algorithms are analysed. In this study, an impact of nonlinear KF parameters on SOC estimation accuracy for EV batteries is analyzed. Specifically, the performance of Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF) is examined. EKF is most preferred for SOC estimation whereas introducing sigma points in UKF is the basic difference between EKF and UKF. In comparison to EKF and UKF, PF is more flexible since it applies to nonlinear systems with non-Gaussian distributions. Utilising a sequence of noisy measurements, KF is a recursive method that approximates the state of a dynamic system. It requires accurate parameters such as process noise covariance (Q) and measurement noise covariance (R) to work effectively. In this work, the effect of these parameters on state of charge estimation using nonlinear KFs has been analysed under different settings of noise covariance matrices. Tuning these parameters is essential for optimizing the performance of the KF and ensuring accurate and reliable state estimation in real-world applications. To find parameter values that provide the best balance between accuracy, robustness, and computational efficiency, the tuning process involves experimentation, simulation, and validation using real-world data. According to the results observed, the filter parameters such as three diagonal elements of Q are selected as 2e-8, 0.005, 0.003 and R as 2e-6. By evaluating the state estimation accuracy under different driving scenarios, robustness and sensitivity of filtering technique in estimating SOC for EV batteries has been studied. This analysis contributes to the optimization of SOC estimation algorithms, enhancing the reliability and efficiency of EV battery management systems.

ABSTRACTIn complex indoor environments, the ultrawide band (UWB) positioning systems enhanced by Kalman filters (KFs) often struggle with large variations in accuracy and reliability due to outlier observations in the state system. This paper presents a cumulative adaptive extended KF (CA‐EKF) to mitigate this issue by optimizing state estimation accuracy. The proposed technique adaptively adjusts process and measurement noise covariance matrices based on residual and innovation vectors, effectively suppressing filtering divergence and significantly enhancing positioning precision. Performance evaluation demonstrates the superiority of CA‐EKF over traditional extended KF (EKF), innovation adaptive KF (IAKF), and Sage–Husa adaptive KF (SHAKF). Monte Carlo simulations under multiple outlier scenarios and nonlinear motion trajectories confirm CA‐EKF's robustness, achieving lower positioning errors, smoother error evolution, and improved resilience against multiple simultaneous outliers. Quantitative results highlight that CA‐EKF consistently outperforms other filters in terms of running‐average root mean square error (RA‐RMSE), mean squared error (MSE), and standard deviation (STD), reducing error accumulation over time and enhancing tracking accuracy. Notably, cumulative distribution function (CDF) analysis confirms that 99% of positioning errors with CA‐EKF remain below 7.65 cm significantly outperforming competing methods. These results establish CA‐EKF as a highly effective solution for high‐precision UWB tracking applications in dynamic environments.

The paper deals with the estimation of the means and covariance matrices of the noises of a linear time-varying system described by the state-space model. The proposed measurement difference method is based on a statistical analysis of differenced, linearly transformed, and shifted augmented measurement vector resulting in a system of linear matrix equations for consistent estimation of both; noise means and noise covariance matrices. The theoretical results are discussed and illustrated in a numerical example.

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

Nowadays, magnetic and inertial measurement units (MIMUs) have become useful tools for determining the attitude of moving objects. An attitude calculation algorithm (ACA) is programed into an MIMU, which determines the attitude by fusing three built-in sensors, namely, gyroscope, accelerometer, and magnetometer. Attitude is often represented by Euler angles, quaternion, or rotation matrix. Despite that great development has been achieved, how to eliminate the negative effect of magnetic disturbance on the pitch and roll estimation is still difficult for an ACA. This article aims to propose an Extended Kalman Filter (EKF)-based approach for problem solving. Inspired by the fact that the Euler angles are computed by using some elements of the rotation matrix, a diagonalizable regulation matrix (DRM) is multiplied to the right half part of the gain matrix (which is multiplied to the residual of the magnetometer measurement in the update stage of EKF). The determination of DRM is based on the principle that these elements can be appropriately regulated by the DRM so as to make the magnetometer measurement only participate in the computation of azimuth. No further revision to EKF is made except the Kalman gain matrix, hence the proposed EKF behaves similar as traditional EKF, except its ability of eliminating the negative effect of magnetic disturbance on the pitch and roll estimation. The superiority of the proposed EKF is validated by comparing it with four other competing algorithms through simulation and experimental experiments.

Tracking a ballistic target in its reentry mode by considering the radar measurements is a highly complex problem in nonlinear filtering. Kalman Filter (KF) is used to estimate the position of target when the measurements are corrupted with noise. If the measurements are nonlinear (radar measurements) then Extended kalman filter (EKF) is used. For obtaining reliable estimate of the target state, filter has to be tuned before the operation which is offline. Tuning an EKF is the process of estimating the process noise covariance matrix (Q) and measurement noise covariance matrix (R). This paper presents a new method of tuning the EKF using different evolutionary algorithms.

In this paper, normalised state vectors are used to demonstrate the state equations of Extended Kalman Filter (EKF) based sensorless Permanent Magnet Synchronous Motor (PMSM) drive. Based on the normalised EKF equations, Simple Genetic Algorithm (SGA) is employed to optimize the noise covariance and weight matrices of EKF parameters, which thereby reduces the parameter adjusting time and ensures stability of filters in estimations of position and speed. The simulations for SGA training are carried out by MATLAB/Simulink. The experimental sensorless drive system employing Field Oriented Control (FOC) method and SGA is implemented on STM32F103. The simulating and experimental results indicate the effectiveness of the proposed method.

In the case of a non-linear system, the dynamic state of the targets (position, velocity, and acceleration) is estimated by an extended Kalman filter (EKF). The theory of EKF is established on the assumption that measurements follow Gaussian distribution. However, in practice, this assumption falls short and limits the application of EKF. In literature, to deal with the non-Gaussianity, the maximum correntropy criterion (MCC)-based EKF (EKF-MCC) has been studied well. The MCC, an information-theoretic criterion, claims to effectively deal with the system's non-Gaussianity. Nevertheless, like EKF, EKF-MCC also approximates the known system non-linearity with a Jacobian. The Jacobian provides the first-order approximation of the non-linearity and hinders the estimation accuracy achieved by EKF-MCC, particularly for complex target motion models. Therefore, in this work, firstly, we propose to use EKF-MCC for estimating the dynamic state of the target from non-Gaussian measurement. After that, utilizing MCC, we propose reproducing kernel Hilbert space (RKHS) based non-linear estimation of system non-linearity and using it with EKF-MCC. Amid non-linear estimation utilizing MCC, the proposed filter is named EKF-MCC-RKHS. The simulation performed to estimate the dynamic states of the complex constant acceleration (CA) target motion model validates the superiority of EKF-MCC-RKHS over recently introduced EKF-MCC and traditional EKF.

Traffic state estimation problems must deal with imperfect information. Imperfect information may occur as a result of misalignments between reality and the assumptions or simplifications made when developing the models as well as incomplete or corrupted data. One of the well-known methods of handling imperfect information is the Kalman filter algorithm and its variations, such as the extended Kalman filter and the unscented Kalman filter. While the standard Kalman filter and the extended Kalman filter have been widely applied to the state estimation of linear systems, the unscented Kalman filter has been reported the better choice in the multiclass traffic state estimation. In many applications of Kalman filters to traffic estimation problems, the model and measurement noise covariance matrices are normally estimated. When there is a mismatch between the true and the assumed noise distribution, however, the filter often suffers from performance degradation and even divergence in certain situations. To this end, this paper presents a more efficient and accurate algorithm embedded in the unscented Kalman filter to simultaneously estimate the traffic state and the model noise distribution statistics. The proposed method is facilitated through a simultaneous update of the model noise covariance matrix in the predicted covariance equations of the unscented Kalman filter algorithm. It is found through simulation that the proposed algorithm may improve the model performance over the standard algorithm.