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  • Nonlinear Estimation

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

    2 Generation of deterministic sampling points and weights

    2.5 Cubature quadrature Kalman filter (CQKF)

    $\bullet$ Find the cubature points $[u_i]_{(i=1,2,..,2n)}$, located at the intersection of the unit hyper-sphere and it's axes as described in section $2.3$.

    $\bullet$ Solve the $n'$ order Chebyshev-Laguerre polynomial for $\alpha=(n/2 -1)$ to obtain the quadrature points ($\lambda_{i'}$).

    \begin{equation*} \begin{split} L_{n'}^{\alpha}(\lambda)=\lambda^{n'}-\dfrac{n'}{1!}(n'+ \alpha)\lambda^{n'-1}+\dfrac{n'(n'-1)}{2!}(n'+\alpha) \\ \times (n'+\alpha-1)\lambda^(n'-2)... =0 \end{split} \end{equation*}

    $\bullet$ Find the CQ points as $\xi_j=\sqrt{2\lambda_{i'}}[u_i]$ and their corresponding weights as \begin{equation*} w_j=\dfrac{1}{2n\Gamma(n/2)}\dfrac{n'!\Gamma(\alpha+n'+1)}{\lambda_{i'}[L'^{\alpha}_{n'}(\lambda_{i'})]^2} \end{equation*} for $i=1,2,..,2n$, $i'=1,2,..,n'$ and $j=1,2,..,2nn'$.

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