## Appendix A Sigma-point transformation  
In this Appendix we summarize the fundamentals of sigma-point transformation, with focus on its performance in representing the probability distribution of a random variable vector undergoing a nonlinear transformation [  
Through the sigma-point transformation, the probability distribution of \(\mathbf{x}\) at time \(t_{i}\) is sampled via a set of sigma-points \(\mathbf{\hat{\chi}}_{i,\mathrm{j}}\) scattered around \(\mathbf{\hat{x}}_{i}\) according to (see Table 1):  
\[ \begin{cases}\mathbf{\Delta}\mathbf{\chi}_{i,0}&=\mathbf{0}\\ \mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{k}}&=+\psi\sqrt{\mathbf{P}_{i}}\, \mathbf{1}_{\mathrm{k}}\\ \mathbf{\Delta}\mathbf{\chi}_{i,\frac{N_{\chi}}{2}+\mathrm{k}}&=-\psi\sqrt{ \mathbf{P}_{i}}\,\mathbf{1}_{\mathrm{k}}\end{cases} \qquad\quad\mathsf{k}=1,...,\tfrac{N_{\chi}}{2} \tag{28}\]  
where: \(\psi\) is a scaling factor to be determined; \(\sqrt{\mathbf{P}_{i}}\) is the square root of the (positive definite) covariance matrix \(\mathbf{P}_{i}\); \(\mathbf{1}_{\mathrm{k}}\) is a unit vector pointing towards the \(\mathsf{k}-\)th component of the state vector in the relevant space, whose role in (28) is to sample the \(\mathsf{k}-\)th column of matrix \(\sqrt{\mathbf{P}_{i}}\); \(N_{\chi}=2N_{\mathrm{z}}\). Information conveyed by the evolved sigma-points are merged at time \(t_{i+1}\) via a weighted averaging scheme, according to:  
\[ \begin{split}\mathbf{\hat{x}}_{i+1,\mathrm{SPT}}^{-}=& \sum_{\mathrm{j}=0}^{N_{\chi}}\omega_{\mathrm{j}}\,\mathbf{\hat{ \chi}}_{i+1,\mathrm{j}}^{-}\\ =&\left(\sum_{\mathrm{j}=0}^{N_{\chi}}\omega_{ \mathrm{j}}\right)\mathbf{f}_{\mathrm{i}}(\mathbf{\hat{x}}_{\mathrm{i}})+\sum _{n=1}^{\infty}\frac{1}{n!}\left(\sum_{\mathrm{j}=0}^{N_{\chi}}\omega_{ \mathrm{j}}\,\mathbf{D}_{\mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{j}}}^{n,i} \right)\end{split}  \tag{29}\]  
and:  
\[ \begin{split}\mathbf{P}_{i+1,\mathrm{SPT}}^{-}=&\sum _{\mathrm{j}=0}^{N_{\chi}}\omega_{\mathrm{j}}^{*}\left(\mathbf{\hat{\chi}}_{i +1,\mathrm{j}}^{-}-\mathbf{\hat{x}}_{i+1,\mathrm{SPT}}^{-}\right)\left( \mathbf{\hat{\chi}}_{i+1,\mathrm{j}}^{-}-\mathbf{\hat{x}}_{i+1,\mathrm{SPT}}^{ -}\right)^{\mathrm{T}}\\ =&\sum_{\mathrm{j}=0}^{N_{\chi}}\omega_{\mathrm{j}}^{ *}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{n!}\frac{1}{m!}\left(\mathbf{D }_{\mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{j}}}^{n,i}-\sum_{\mathrm{r}=0}^{N_{ \chi}}\omega_{\mathrm{r}}\,\mathbf{D}_{\mathbf{\Delta}\mathbf{\chi}_{i, \mathrm{r}}}^{n,i}\right)\left(\mathbf{D}_{\mathbf{\Delta}\mathbf{\chi}_{i, \mathrm{j}}}^{m,i}-\sum_{\mathrm{s}=0}^{N_{\chi}}\omega_{\mathrm{s}}\,\mathbf{ D}_{\mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{s}}}^{m,i}\right)^{\mathrm{T}}\end{split}  \tag{30}\]  
where:  
\[\mathbf{D}_{\mathbf{\Delta}\mathbf{\chi}_{i,\mathrm{j}}}^{n,i}\equiv\left( \sum_{\ell=1}^{N_{\mathrm{z}}}\frac{\partial\mathbf{f}_{\mathrm{i}}}{\partial x _{\ell}}\bigg{|}_{\mathbf{x}=\mathbf{\hat{x}}_{\mathrm{i}}}\Delta\chi_{i, \mathrm{j}\ell}\right)^{n} \tag{31}\]  
Weights \(\omega_{\mathrm{j}}\) in (29) can be determined by matching the series expansions (25) and (29) up to third order; by assuming \(\omega_{\mathrm{j}}=\omega\) for \(\mathrm{j}=1,...,N_{\chi}\) this requirement results in:  
\[ \begin{cases}\omega_{0}+N_{\chi}\omega=1\\ 2\psi^{2}\omega=1\end{cases}  \tag{32}\]  
To also set the value of the scaling factor \(\psi\), a further condition needs to be established. Usually, it is suggested to match the diagonal entries of the fourth order terms in (25) and (29): this leads to \(\psi=\sqrt{3}\). In [27], partially exploiting the features of the so-called scaled sigma-point transformation [46], we proposed to adaptively set \(\psi\) so as to always fulfill the physical constraints:  
\[\boldsymbol{\vartheta}^{m}\leq\boldsymbol{\vartheta}\leq\boldsymbol{\vartheta}^ {M} \tag{33}\]where \(\boldsymbol{\vartheta}^{m}\) and \(\boldsymbol{\vartheta}^{M}\) are, respectively, the minimum and maximum (if applicable) allowed values of model parameters.