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Nonlinear Estimation

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

2.5 Generation of deterministic sampling points and weights

Unscented Kalman filter (GHF)

$\bullet$ 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.

$\bullet$ A matrix $\Psi$ is generated using $2n+1$ weighted sigma points.

$\bullet$ A scaling parameter $k$ is used such that $\sqrt{(n+k)P_{x,h}}_{i}$ gives the $ith$ row or column of matrix square root of $(n+k)P_{x,h}$. This can be done using cholesky decomposition.

$\bullet$ Unscented transformation can be represented as

1) Generation of sigma points \begin{equation*} \psi_{0,h}=x_{h} \end{equation*} for $i$=$1$ to $n$ \begin{equation*} \psi_{i,h}=x_{h}+(\sqrt{(n+k)P_{x,h}})_{i} \end{equation*} for $i$=$n+1$ to $2n$ \begin{equation*} \psi_{i,h}=x_{h}-(\sqrt{(n+k)P_{x,h}})_{i} \end{equation*}

2) Generation of weights \begin{equation*} W_{0}=k/(n+k) \end{equation*} for $i$=$1$ to $2n$ \begin{equation*} W_{i}=\dfrac{1}{2(n+k)} \end{equation*}