## 3 Navigation Framework
### Observation Model  
Approaches starting from intermediate parameters demand proper and accurate characterization of the measurements. We consider estimating the users' positions in a direct way, where the non-linear relationships between the signals received at the BS antennas and the coordinates of the moving users are directly utilized. Thus, the measurement vector is defined as  
\[\mathbf{z}_{t}\triangleq g(\mathbf{s}_{t})+\mathbf{v}_{t}= \begin{bmatrix} \widehat{\mathbf{z}}_{1}^{\mathsf{T}},\dots,\widehat{\mathbf{z}}_{m}^{\mathsf{ T}},\dots,\widehat{\mathbf{z}}_{M}^{\mathsf{T}}\end{bmatrix} ^{\mathsf{T}},\forall\,m \in\mathcal{M}, \tag{14}\]  
where is the measurement vector of the \(m\)-th user, and  
\[\widehat{\mathbf{y}}_{m}= \begin{bmatrix}\widehat{y}_{m\geqslant 0},\dots, \widehat{y}_{m\geqslant b},\dots,\widehat{y}_{m\geqslant N_{B}-1}\end{bmatrix} ^{ \mathsf{T}},\forall\,b\in\mathcal{B}, \tag{15}\]  
and \(\mathbf{v}_{t}\) denotes a zero-mean white noise sequence unavoidably caused by measurement, the covariance matrix of which is defined as  
\[\mathbf{R}=\text{diag}(\sigma^{2},\dots,\sigma^{2})_{2N_{B}M\times 2N_{B}M}. \tag{16}\]  
The detailed Jacobian calculation is provided in Appendix A, and the EKF procedure is sketched in Algorithm 1. ```
Initialization Define initial distribution \(\mathbf{s}_{0}\thicksim\mathcal{N}(\mathbf{\hat{s}}_{0\mid 0},\mathbf{P}_{0\mid 0})\). Prediction Step Given \(\mathbf{s}_{t+1}=\mathbf{Fs}_{t}+\mathbf{w}_{t}\), where \(\mathbf{w}_{t}\thicksim\mathcal{N}(0,\mathbf{Q}_{t})\). \[\mathbf{\hat{s}}_{t+1\mid t}=\mathbf{Fs}_{t\mid t}.\] (17)

\[\mathbf{P}_{t+1\mid t}=\mathbf{FP}_{t\mid t}\mathbf{F}^{\mathsf{T}}+\mathbf{Q} _{t}.\] (18) Update Step Given \(\mathbf{z}_{t+1}=g(\mathbf{\hat{s}}_{t+1\mid t})+\mathbf{v}_{t+1}\), where \(\mathbf{v}_{t+1}\thicksim\mathcal{N}(0,\mathbf{R}_{t+1})\). \[\mathbf{G}_{t+1}=\frac{\partial g}{\partial\mathbf{s}}(\mathbf{\hat{s}}_{t+1 \mid t}). \tag{19}\]

\[\mathbf{K}_{t+1}=\mathbf{P}_{t+1\mid t}\mathbf{G}_{t+1}^{\mathsf{T}}(\mathbf{ G}_{t+1}\mathbf{P}_{t+1\mid t}\mathbf{G}_{t+1}^{\mathsf{T}}+\mathbf{R}_{t+1})^{-1}. \tag{20}\]

\[\mathbf{\hat{s}}_{t+1\mid t+1}=\mathbf{\hat{s}}_{t+1\mid t}+\mathbf{K}_{t+1}( \mathbf{z}_{t+1}-g(\mathbf{\hat{s}}_{t+1\mid t})). \tag{21}\]

\[\mathbf{P}_{t+1\mid t+1}=\mathbf{P}_{t+1\mid t}-\mathbf{K}_{t+1}\mathbf{G}_{t+ 1}\mathbf{P}_{t+1\mid t}. \tag{22}\]

## 4 RIS-Aided Tracking Error Bound and Optimization

### Derivation of BCRB

The inaccuracy in modelling users' dynamic patterns and the limitations of the adopted measurement method may bring uncertainty to the users' state estimates. Denote the users' estimated state as \(\mathbf{\hat{s}}_{t+1}\); the MSE of the users' state estimation satisfies

\[ \begin{split}\mathbb{E}_{\mathbf{s}_{t+1}\mathbf{z}_{t+1}}& \{(\mathbf{\hat{s}}_{t+1}(\mathbf{z}_{t+1})-\mathbf{s}_{t+1})( \mathbf{\hat{s}}_{t+1}(\mathbf{z}_{t+1})-\mathbf{s}_{t+1})^{\mathsf{T}}\}\\ &\geq\mathbf{J}^{-1}(\mathbf{s}_{t+1}),\end{split}  \tag{23}\]

where \(\mathbb{E}_{\mathbf{s}_{t},\mathbf{z}_{t}}\{\cdot\}\) denotes mathematical expectations for users' states and observations, \(\mathbf{J}(\mathbf{s}_{t+1})\) is the Bayesian Fisher information matrix (BFIM) of users' states \(\mathbf{s}_{t}\), which can be expressed as [43]

\[\mathbf{J}(\mathbf{s}_{t+1})=\mathbf{J}_{\mathsf{P}}(\mathbf{s}_{t+1})+ \mathbf{J}_{\mathsf{D}}(\mathbf{s}_{t+1}), \tag{24}\]

where \(\mathbf{J}_{\mathsf{P}}(\mathbf{s}_{t+1})\) and \(\mathbf{J}_{\mathsf{D}}(\mathbf{s}_{t+1})\) are the BFIM of the users' a priori information [44] and the data of observations[45], respectively. Specific expressions of them are

\[ \begin{split}\mathbf{J}_{\mathsf{P}}(\mathbf{s}_{t+1})& =[\mathbf{Q}_{t}+\mathbf{F}_{t+1}\mathbf{J}^{-1}(\mathbf{s}_{t}) \mathbf{F}_{t+1}^{\mathsf{T}}]^{-1}\\ \mathbf{J}_{\mathsf{D}}(\mathbf{s}_{t+1})&=\mathbf{G }_{t+1}^{\mathsf{T}}\mathbf{R}_{t+1}^{-1}\mathbf{G}_{t+1}.\end{split}