Some Relations Between Extended and Unscented Kalman Filters (original) (raw)
2012, IEEE Transactions on Signal Processing
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. Index Terms-extended Kalman filter, unscented Kalman filter, transformations Gustaf Hendeby received his M.Sc. degree in Electrical Engineering and Applied Physics in 2002 and his Ph.D. in Automatic Control in 2008, both from Linköping University, Sweden. He remained at Linköping University as assistant professor until 2009 when he joined the German Research Center for Artificial Intelligence (DFKI). Since 2011 he works in the Competence Unit Informatics, Division of Information Systems at the Swedish Defense Research Agency (FOI) in Linköping, Sweden. Dr. Hendeby's main research interests are stochastic signal processing, sensor fusion, and change detection, especially for nonlinear and non-Gaussian systems. He has experience in both theoretical analysis as well as practical implementation aspects.
Related papers
The unscented Kalman filter for nonlinear estimation
2000
The Extended Kalman Filter (EKF) has become a standard technique used in a number of nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system, estimating parameters for nonlinear system identification (e.g., learning the weights of a neural network), and dual estimation (e.g., the Expectation Maximization (EM) algorithm) where both states and parameters are estimated simultaneously.
—The purpose of this paper is to point out a confusing phenomenon in the teaching of Kalman filtering. Students are often confused by noting that the a posteriori error covariance of the discrete Kalman filter (DKF) is smaller than the error covariance of the continuous Kalman filter (CKF), which would mean that the DKF is better than CKF since it gives a smaller error covariance. However, simulation results show that CKF gives estimates much closer to the true states. We will provide a simple qualitative argument to explain this phenomenon.
2006
The Kalman Filter developed in the early sixties by R.E. Kalman is a recursive state estimator for partially observed non-stationary stochastic processes. It gives an optimal estimate in the least squares sense of the actual value of a state vector from noisy observations.
Criteria for When the Extended Kalman Filter Works and Issues with Sigma Point Kalman Filters
The extended Kalman filter (EKF) is frequently tried for solving nonlinear estimation problems, but often fails. There have been no published objective criteria that can be used to determine whether the EKF will work. The EKF is the most basic nonlinear Gaussian filter approximation (NGFA), where the estimate is correct to first order and the covariance correct to second order. For a filter observation with a nonlinear measurement function an NGFA uses a Taylor series expansion about the mean to compute the mean and covariance of the measurement. We show that a requirement for the EKF to work is that the part of the fourth order covariance term of the measurement covariance that involves the products of the second order derivatives of the measurement function must be negligible in comparison to the covariance of the observation error. We also show that when this condition is not met we can implement an NGFA using the standard Kalman filter update equations where instead of the observation error covariance we use the sum of the observation error covariance and the above fourth order covariance term. We also discuss the limitations of the various sigma point filters because they do not implement this fourth order term correctly.
International Journal of Engineering Sciences & Research Technology, 2013
This paper answers several questions of centralized Kalman-Filters in multi-sensor fusion, fault detection and isolation in sensors, optimal control in linear-quadratic Gaussian problem, an algorithm in fuzzy based approach to adaptive Kalman-Filtering additionally in multi-state multi-sensor fusion. Generally, Kalman-Filters comprise a number of types and topologies depending on use and computing complexity of applied processors. State estimation provided by a Kalman-Filter is crucial in this thesis. Kalman-Filter performs optimal estimation of an unknown system state through filters behavior. This thesis supposes some models of promising linear Kalman-Filter simulated beyond MATLAB and Simulink program especially utilized in the fields of steering-controls or navigations, etc.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.