Sigma point Kalman filter for bearing only tracking

Performance comparison of Kalman−based filters for nonlinear structural finite element model updating

The Extended Kalman Filter (EKF) has been the state of the art in integrated navigation systems and especially in Pedestrian Dead-Reckoning PDR for foot-mounted Inertial Measurements Units. However in most related work with PDR, indirect filtering approach is used based on linear error Kalman Filter (KF). In this work, it is proposed to outperform this approach by the use of Direct filtering approach, which involves the non-linearity in the propagation of the orientation, velocity and in some models, in position coordinates, where the EKF can not achieve optimal estimation. We propose then, the usage of the most recent algorithms developed in the last decade; Sigma Point Kalman Filters (SPKF), especially based on Cubature rule, called Cubature Kalman Filter (CKF) as the integration algorithm for the inertial measurements. The CKF improves the mean and covariance propagation consequently comparing with EKF and previous SPKF (UKF, CDKF). Although the CKF provides a better estimate of the orientation, velocity and position with Zero velocity UPdaTes (ZUPT) measurements and Zero Angular Rate UpdaTes (ZARUT), additional sensors are necessary to measure other states such as yaw angle and to estimate properly the gyroscope bias. We studied then the possibility to integrate electronic compass as additional measure and also Map street data base or Map building data base depending on the type of navigation "Indoor", "Outdoor". In order to get much better estimation based on the Cubature rule, it is proposed to synthesize CKF in order to get robust estimation against nonlinearity of the process and the multiple measurement sensors.

T HIS Note describes a robust modification of the divided difference filtering technique. The robust technique relies on Huber’s generalized maximum likelihood approach to estimation [1]. Specifically,Huber’smethod is a combinedminimum ‘1and ‘2norm estimation technique, which exhibits robustnesswith respect to deviations from the commonly assumed Gaussian error probability density functions, for which the least–squares or minimum ‘2-norm technique exhibits a severe degradation in estimation accuracy [2]. TheHuber-based estimates are robust in the sense that theyminimize the maximum asymptotic estimation variance when applied to contaminated Gaussian densities. The Huber technique was originally developed as a generalization of maximum likelihood estimation, applied first to estimating the center of a probability distribution in [1] and generalized tomultiple linear regression in [2– 4]. The Kalman filter is a recursive minimum ‘2-norm technique and therefore exhibits sensitivity to deviations in the true underlying error probability distributions [5]. For this reason, theHuber technique has been further extended to dynamic estimation problems. Boncelet and Dickinson [6] first proposed to solve the robust filtering problem by means of the Huber technique at each measurement, by expressing the discrete-time Kalman filter as a sequence of linear regression problems. The authors do not provide any simulation results to validate the proposed technique. Kovacevic et al. [7] follow thework of [6] and develop a robust Kalman filter using the Huber technique applied to a linear regression problem at each measurement update. References [8–10] express the dynamic filtering problem as a sequential linear regression to be solved by the Huber technique and apply the filter to underwater vehicle tracking, power system state estimation, and spacecraft rendezvous navigation, respectively. The increase in computation due to the use of the Huber technique was found in [10] to be small. It should be noted that [6–10] apply the Huber methodology to linearized filters. The divided difference filter is one of several new estimation techniques that are collectively known as sigma-point Kalman filters (SPKF). The first-order (DD1) and second-order (DD2) divided difference filters [11,12] are generalizations of the filter introduced by Schei [13] and are two examples of SPKF-class estimators; other examples can be found in [14–16]. Like the basic Kalman filter, the SPKFs seek to determine a state estimate that minimizes the ‘2 norm of the residuals. The SPKF technique differs from the standard Kalmanfilter in the sense that the SPKFs do not linearize the dynamic system for the propagation, but instead propagate a cluster of points centered around the current estimate to form improved approximations of the conditional mean and covariance. Specifically, the divided difference filters make use of multidimensional interpolation formulas to approximate the nonlinear transformations. As a result of this approach, the filter does not require knowledge or existence of the partial derivatives of the system dynamics and measurement equations. SPKFs have the additional advantage over the basic Kalman filter in that they can easily be extended to determine second-order solutions to the minimum ‘2-norm filtering problem, which increases the estimation accuracywhen the system andmeasurement equations are nonlinear. It is important to note that the SPKFs use a minimum ‘2-norm measurement update and are therefore subject to the same sensitivity to non-Gaussian measurement errors as the Kalman filter. Therefore, the purpose of this Note is tomodify the DD1 andDD2measurement update equations by making use of the Huber technique to provide robustness against deviations from Gaussianity without a large increase in computation. This Note first provides a short review of the DD1 and DD2 filters and then shows how the measurement update can be expressed in terms of a standard regression problem, which can be solved using the robust Huber technique. The filtering techniques are then applied to a benchmark problem that involves estimating the trajectory of an entry body from discrete-time range data measured by a radar tracking station. The simulation is conducted using Monte Carlo techniques for both Gaussian and non-Gaussian cases. The computational cost associated with each filter is provided.

The iterative extended Kalman filter (IEKF) algorithm is dependent on the previous data and estimation accuracy of the noise. The noise will result in filtering precision decrease and divergence. This paper puts forward an adaptive algorithm based on Levenberg-Marquardt method and two-factor for IEKF, which optimizes the iterative process by using Levenberg-Marquardt method, adjusts the prediction covariance by suboptimal fading factor. A simplified calculation method of suboptimal fading factor is also presented. The previous process noise is suppressed by exponential fading factor, and the observation noise is corrected by the residual and innovation, so that filter gain becomes larger, which improves the accuracy and robustness of the filter algorithm. The experiment results indicate the improved algorithm can effectively suppress the filter divergence. Compared with the classical extended Kalman filter (EKF), IEKF, and the modified IEKF, the filter accuracy and robustness of the improved algorithm are sufficient, and the algorithm can better meet the requirements of engineering applications.

A particle filter guided by iterated extended Kalman filter (IEGPF) is proposed to solve the problem of particle impoverishment. Firstly, particles are divided into two groups adaptively according to the maximum likelihood ratio, which is defined to measure how well the particles, drawn from transition prior density, match the likelihood model. Particles in one group are drawn from transition prior density, while those in the other group are drawn from the Gaussian approximate posterior density, obtained by running an iterated extended Kalman filter (IEKF). The results of numerical simulations, conducted at different observation noise level, show that IEGPF can produce more accurate estimate than extended Kalman particle filter (EPF) and unscented Kalman particle filter (UPF), whereas the computational cost is comparable to that of SIR.

In this paper, vision-based relative navigation of two spacecraft is addressed using the sparse Gauss-Hermite quadrature filter. The relative navigation provides the estimation of the relative attitude and relative orbit between spacecraft, which is a challenging filtering problem since it is a highly nonlinear and high dimensional estimation problem. Many filters have been used to solve this problem, such as, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). However, these filters are not accurate enough when the initial uncertainty is large or the nonlinearity is significant. Moreover, although the conventional Gauss-Hermite quadrature filter is more accurate than the UKF, it cannot be used since a huge number of points is required. It is shown in this paper that the sparse Gauss-Hermite quadrature filter can achieve better estimation accuracy than the EKF, the UKF, and the cubature Kalman filter without excessive computation load.

Fast non-intrusive estimation of liquid-phase interface and boundary heat flux in participating media by the Kalman filtering methods

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.

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

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.

Two estimation techniques, the extended Kalman filter (EKF) and the iterative extended Kalman filter (IEKF), have been applied to a nonlinear time-varying system that has nonmeasurable state variables. An iterative solution to a fed-batch fermentation process is reported using the EKF based on measurements of the oxygen and carbon dioxide concentrations. The results demonstrate that this estimation technique can be successfully applied to complex biological processes. If the nonlinearities of these systems are sufficiently important or if a long delay in the estimation cannot be permitted for a particular process, then the IEKF can be selected. >

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

The stochastic nature of noise signals affects the vehicle’s internal states and the outputs, resulting in the difficulty in estimation. The unknown or time-varying nature of noise signals if not taken into account for estimation, the results will diverge and be highly deteriorated. In this paper, a maximum likelihood principle (MLP) based adaptive robust extended kalman filter for estimating the states of the adopted non-linear vehicle model is proposed. An observability test is done for the purpose of estimation. A covariance matching (CM) based robust adaptive high forgetting factor based fault tolerant technique is also employed on the robust adaptive extended and unscented kalman filters for comparison purpose. The Robustness of the filter is analyzed by varying the parameter of the vehicle through a local sensitivity analysis. The results show that the MLP based approach to the extended kalman filter performs well in three simulations for sinusoidal steering, Double Lance Change, J-Turn, Fishhook, Slalom maneuver in comparison to robust adaptive unscented kalman filter. Friction coefficient of 0.8 (dry road) and 0.4 (wet road) is chosen for the simulation. The sideslip angle RMSE value for MLP based estimation is obtained as 2.62e-05, 4.545e-06 for Sine and DLC maneuver.

This paper presents the efficient Unscented Kalman filter (UKF) based estimation and control of a 3-DOF Omni bundle robot. The angular trajectory tracking becomes more difficult under highly non-linear measurement noise and process noise. A rapidly converging Unscented Kalman Filter (UKF) is presented in the paper. The extended Kalman filter (EKF) provides less accurate results in the highly non-linear system and sometime may become unstable because of convergence related issues, whereas in UKF the results in the non-linear system are better than EKF. UKF is more accurate in state and noise estimation. The computed torque technique is taken as a model to apply UKF. After, estimation the noise is removed. The experiment is conducted on Omni bundle with process and measurement noise. The results prove the superiority of presented UKF over conventional UKF and EKF.

PurposeAutonomous orbit determination using geomagnetic measurements is an important backup technique for safe spacecraft navigation with a mere magnetometer. The geomagnetic model is used for the state estimation of orbit elements, but this model is highly nonlinear. Therefore, many efforts have been paid to developing nonlinear filters based on extended Kalman filter (EKF) and unscented Kalman filter (UKF). This paper aims to analyze whether to use UKF or EKF in solving the geomagnetic orbit determination problem and try to give a general conclusion.Design/methodology/approachThis paper revisits the problem and from both the theoretical and engineering results, the authors show that the EKF and UKF show identical estimation performances in the presence of nonlinearity in the geomagnetic model.FindingsWhile EKF consumes less computational time, the UKF is computationally inefficient but owns better accuracy for most nonlinear models. It is also noted that some other navigation techniques are also very similar with the geomagnetic orbit determination.Practical implicationsThe intrinsic reason of such equivalence is because of the orthogonality of the spherical harmonics which has not been discovered in previous studies. Thus, the applicability of the presented findings are not limited only to the major problem in this paper but can be extended to all those schemes with spherical harmonic models.Originality/valueThe results of this paper provide a fact that there is no need to choose UKF as a preferred candidate in orbit determination. As UKF achieves almost the same accuracy as that of EKF, its loss in computational efficiency will be a significant obstacle in real-time implementation.