## 2 Preliminaries of State Estimation Methods Using Kalman Filter
### Unscented Kalman Filter (UKF)  
With methods similar to the EKF, the UKF generally has better performance and reduced complexity due to the sampling strategy and avoids the divergence of errors found in the case of EKF. It approximates the weighted density distribution of the nonlinear function and handles non-additive noise without requiring the calculation of Jacobian matrices [42], thus providing greater accuracy and faster convergence for a nonlinear problem.  
[PERSON] and [PERSON] devised the UKF and applied a strategy close to nonlinear distribution [45]. The sampling points in the UKF are called sigma points and are generally smaller in number. The \(2n+1\) symmetric-sigma sampling strategy is the most widely accepted approach [46], where n is the state vector dimension. These sigma sampling points then have a non-linear transform applied on them to obtain a non-linearly transformed set of points with a mean and covariance of \(x\) and \(P(x)\), respectively [47]. The UKF consists of the following processing steps: (i) Initialisation of state error covariance matrix as well as the state vector; (ii) selecting sigma sampling points at moment \(k-1\) based on the state vector and error covariance matrix and calculation of weighted values; (iii) updating time through the chosen sampling points and propagation of the covariance and mean through the state equation; (iv) updating measurements by the selected sampling points through an observation equation that is non-linear; and (v) KF coefficients being updated. The state equations for this process are shown in Equations (11) and (12). The selection of sigma points is defined by (18)-(21) as  
\[x_{0}=\hat{x}_{k},\hskip 14.226378 ptx_{i}=\hat{x}_{k}+\left(\sqrt{(n+\Delta)P} \right)_{k},\hskip 14.226378 ptx_{i+n}=\hat{x}_{k}-\left(\sqrt{(n+\Delta)P} \right)_{k},\hskip 14.226378 pti=1,\ldots,n, \tag{18}\]  
\[W_{0}^{m}=\Delta/(\Delta+n),\hskip 14.226378 ptW_{i}^{m}=1/2(\Delta+n),\hskip 14.226378 pti= 1,\ldots,2n, \tag{19}\]  
\[W_{0}^{m}=W_{0}^{c}+(1-\alpha^{2}+\beta),\hskip 14.226378 ptW_{i}^{c}=1/2( \Delta+n),\hskip 14.226378 pti=1,\ldots,2n, \tag{20}\]  
\[\Delta=\alpha^{2}(k+n)-n, \tag{21}\]  
where the scaling constant is represented by \(\Delta\), the spread of the selected sigma points is given by \(\alpha\), k is the secondary scaling constant--mostly considered as zero (21)--\(\beta\) incorporates previous information about the state variables distribution, and for the posterior sigma points, \(W_{i}^{m}\) is the the \(i\)th weighted sample mean and \(W_{i}^{c}\) is the \(i\)th weighted covariance (20). The steps for updating the time of the UKF are  
\[\epsilon_{i}=f(x_{i}), \tag{22}\]  
\[\hat{x}_{k+1/k}=\sum W_{i}^{m}\epsilon_{i}, \tag{23}\]  
\[P_{k+1/k}=\sum W_{i}^{c}(\epsilon_{i}-\hat{x}_{k+1/k})(\epsilon_{i}-\hat{x}_{k +1/k})^{T}. \tag{24}\]  
where \(\epsilon\) is the function representing the non-linear system (22) and \(\hat{x}_{k+1/k}\) is the prior state estimation (23), (24). Similarly, measurement updates of the UKF can be found according to (25)-(28):  
\[Z_{i}=h(\epsilon_{i}), \tag{25}\]  
\[\hat{z}_{k+1/k}=\sum W_{i}^{m}Z_{i}, \tag{26}\]  
\[P_{zz}=\sum W_{i}^{c}(Z_{i}-\hat{Z}_{k+1/k})(Z_{i}-\hat{Z}_{k+1/k})^{T}, \tag{27}\]  
\[P_{xz}=\sum W_{i}^{c}(\epsilon_{i}-\hat{x}_{k+1/k})(\epsilon_{i}-\hat{z}_{k+1 /k})^{T}. \tag{28}\]  
Finally, the filter update of the UKF proceeds as follows:  
\[K_{k+1}=P_{xz}P_{zz}^{-1}, \tag{29}\]  
\[\hat{x}_{k+1}=\hat{x}_{k+1/k}+K_{k}(y_{k+1}-\hat{z}_{k+1/k}), \tag{30}\]  
\[P_{k+1/k+1}=P_{k+1/k}-K_{k+1}P_{zz}K_{k+1}^{T}. \tag{31}\]  
where \(K\) is the Kalman gain (29) and \(\hat{x}_{k+1}\) is posterior state estimation (30), (31). Since the UKF deals with \(2n+1\) sigma points, as well as the sampling and re-sampling of data, it is slower than the EKF.