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    2 Literature review on quadrature based nonlinear estimation (Cont'd...)

    2.2 Unscented Kalman filter (cont'd...)

    Let $P_{x,h}$ denote the covariance of an $n$-dimensional random variable $x_{h}$, $P_{y,h}$ be the covariance of measurement vector $y_{h}$ and $P_{xy,h}$ be the covariance of state and measurement vectors. A matrix $\Psi$ is generated using $2n+1$ weighted sigma points. A scaling parameter $k$ is used such that $\sqrt{(n+k)P_{x,h}}_{i}$ gives the $i^{th}$ row or column of matrix square root of $(n+k)P_{x,h}$. This can be done using cholesky decomposition. Unscented transformation can be represented as:

    1) Generation of sigma points

    \begin{equation*} \psi_{0,h}=x_{h}, \end{equation*} \begin{equation*} \psi_{i,h}=x_{h}+(\sqrt{(n+k)P_{x,h}})_{i}, for i=1,2,...,n \end{equation*} \begin{equation*} \psi_{i,h}=x_{h}-(\sqrt{(n+k)P_{x,h}})_{i}, for i=n+1, n+2,...,2n \end{equation*}

    The UKF was first proposed by Julier and Uhalmar [Julier 1996] and after that many applications of this filter have been made in the field of target tracking, navigation etc. Here review of the literature related to theoretical development of unscented Kalman filter, or sigma point filter and their variants is discussed.

    In the introductory paper [Julier 1996] , the authors proposed a new algorithm for nonlinear filtering where a general method for predicting mean and covariance has been proposed by a nonlinear transformation. Using this transformation a new filter has been proposed. The transformation is known as unscented transformation and the filter is known as unscented Kalman filter (UKF).

    After that a series of paper have been published by the same authors, which established the new filtering theory in a solid foundation. In a similar paper [Julier 1997], Julier and Uhlmann have described the unscented transformation and algorithm of the newly proposed filter named as UKF. An application of this filter to a real life system has been presented. In [Julier 2000] , the algorithm of UKF developed previously has been presented and discussed. The performance of EKF and UKF has been compared for a test problem. In general for an nth order system the number of sigma points required for computation are 2n+1. In [Julier 2002a] it has been shown that the number of sigma point can be reduced to (n+1). The major advantage of this reduction is the reduction of computational load of the filter.

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