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Kalman Filtering and Neural Networks - Chapter VII: THE UNSCENTED KALMAN FILTER

In this book, the extended Kalman filter (EKF) has been used as the standard technique for performing recursive nonlinear estimation. The EKF algorithm, however, provides only an approximation to optimal nonlinear estimation. In this chapter, we point out the underlying assumptions and flaws in the EKF, and present an alternative filter with performance superior to that of the EKF. This algorithm, referred to as the unscented Kalman filter (UKF), was first proposed by Julier et al. [1–3], and further developed by Wan and van der Merwe [4–7] | Kalman Filtering and Neural Networks Edited by Simon Haykin Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-36998-5 Hardback 0-471-22154-6 Electronic 7 THE UNSCENTED KALMAN FILTER Eric A. Wan and Rudolph van der Merwe Department of Electrical and Computer Engineering Oregon Graduate Institute of Science and Technology Beaverton Oregon U.S.A. 7.1 INTRODUCTION In this book the extended Kalman filter EKF has been used as the standard technique for performing recursive nonlinear estimation. The EKF algorithm however provides only an approximation to optimal nonlinear estimation. In this chapter we point out the underlying assumptions and flaws in the EKF and present an alternative filter with performance superior to that of the EKF. This algorithm referred to as the unscented Kalman filter UKF was first proposed by Julier et al. 1-3 and further developed by Wan and van der Merwe 4-7 . The basic difference between the EKF and UKF stems from the manner in which Gaussian random variables GRV are represented for propagating through system dynamics. In the EKF the state distribution is 221 222 7 THE UNSCENTED KALMAN FILTER approximated by a GRV which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV which may lead to suboptimal performance and sometimes divergence of the filter. The UKF address this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV and when propagated through the true nonlinear system captures the posterior mean and covariance accurately to second order Taylor series expansion for any nonlinearity. The EKF in contrast only achieves first-order accuracy. No explicit Jacobian or Hessian calculations are .