## 3 Navigation Framework
### Observation Model
 \tag{25}\]

Given initialized \(\mathbf{J}(\mathbf{s}_{0})\), the Fisher information matrix at time \(t+1\) can be recursively calculated, and the BCRB of users' state estimation error can be attained by

\[ \begin{split}\mathbf{C}(\mathbf{s}_{t+1})&=\mathbf{ J}^{-1}(\mathbf{s}_{t+1})\\ &=\bigg{[}\Big{[}\mathbf{Q}_{t}+\mathbf{F}_{t+1}\mathbf{J}^{-1}( \mathbf{s}_{t})\mathbf{F}_{t+1}^{\mathsf{T}}\Big{]}^{-1}+\mathbf{G}_{t+1}^{ \mathsf{T}}\mathbf{R}_{t+1}^{-1}\mathbf{G}_{t+1}\bigg{]}^{-1}.\end{split}  \tag{26}\]

Note that \(\mathbf{J}(\mathbf{s}_{t})\) should be parameterized by \(\mathbf{s}(t)\) theoretically but may not always be available. Instead, we rely on the predictions made during the previous time interval. Since the diagonal elements of the BCRB include the lower bound of the MSE of both users' position and velocity estimates, (26) cannot be used directly as the metric of tracking accuracy. Here, we denote

\[\mathbf{T}_{m}= \begin{bmatrix}\mathbf{I}_{3\times 3}&\mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}\end{bmatrix} , \tag{27}\]

and \(\mathbf{T}=\text{diag}(\mathbf{T}_{1},\dots,\mathbf{T}_{m},\dots,\mathbf{T}_{ M})\). Thus, the metric of tracking accuracy can be prescribed as

\[ \begin{split} L(\mathbf{s}_{t})&=tr(\mathbf{C}(\mathbf{p}_{1}, \dots,\mathbf{p}_{m},\dots,\mathbf{p}_{M}))\\ &=tr(\mathbf{TC}).\end{split}  \tag{28}\]

The derived BCRB serves as a benchmark for the proposed tracking scheme, based on which an optimization scheme for the phase shift of RISs is proposed in the subsequent section. ### GD-Based RIS Phase Shift Optimization

To achieve high-precision multi-user tracking, we design the phase shift induced at the RISs to minimize the BCRB mentioned above, i.e.,

\[ \begin{split}\min_{\Omega}& L(\Omega)\\ s.t.&\Big{|}c_{k,p}\Big{|}=1,\forall\,k\in\mathcal{K},p\in \mathcal{P}.\end{split}  \tag{29}\]

where \(\Omega\) denotes the set of RISs' reflection coefficient matrices \(\{\mathbf{\Omega}_{k}\}_{k\in\mathcal{K}}\). To tackle the optimization problem, we utilize the classical GD-based method. By applying the second-order Taylor expansion at the given point \(\Omega^{\prime}\), the approximate function is given by [46]

\[\hat{L}(\Omega|\Omega^{\prime})=L(\Omega^{\prime})+\
abla_{\Omega}L(\Omega^{ \prime})^{\mathsf{T}}(\Omega-\Omega^{\prime})+\frac{\alpha_{r}}{2}\|\Omega- \Omega^{\prime}\|^{2}. \tag{30}\]As long as the GD updating step \(\alpha_{r}\) is chosen properly, which can be determined by the Armijo rule [47], the approximation function \(\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r})\) satisfies the following three properties[48; 49]

\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\] \[\hat{L}(\mathbf{\Omega}^{r}|\mathbf{\Omega}^{r}) =L(\mathbf{\Omega}^{r}), \tag{31}\] \[\
abla_{\mathbf{\Omega}}\hat{L}(\mathbf{\Omega}^{r}|\mathbf{ \Omega}^{r}) =\
abla_{\mathbf{\Omega}}L(\mathbf{\Omega}^{r}).\]

Then, the optimal solution of \(\mathbf{\Omega}\) at the \((r+1)\)-th iteration is given by

\[\mathbf{\Omega}^{r+1}=\arg\min_{\mathbf{\Omega}}\hat{L}(\mathbf{\Omega}| \mathbf{\Omega}^{r}). \tag{32}\]

It is proved in [46] that the optimal \(\mathbf{\Omega}^{r+1}\) follows the update rule of the GD method, which is given by

\[\mathbf{\Omega}^{r+1}=\mathbf{\Omega}^{r}-\frac{1}{\alpha_{r}}\
abla_{\mathbf{ \Omega}}L(\mathbf{\Omega}^{r}). \tag{33}\]

Notably, the calculation of \(\
abla_{\mathbf{\Omega}}L(\mathbf{\Omega}^{r})\) can be converted into the gradient calculation of the phase shift \(\{\psi_{k,p}\}_{k\in\mathcal{K},p\in\mathcal{P}}\) of each element, which is given by [50]

\[\
abla_{\psi_{k,p}}L(\psi_{k,p}^{r})=-\frac{1}{\sigma^{2}}tr\left\{\mathbf{J}^ {-1}(\mathbf{s}_{t+1})\mathbf{T}\mathbf{J}^{-1}(\mathbf{s}_{t+1})\left\{\left[ \frac{\partial\mathbf{G}_{t+1}}{\partial\psi_{k,p}}\right]^{\mathsf{T}} \mathbf{G}_{t+1}+\mathbf{G}_{t+1}^{\mathsf{T}}\frac{\partial\mathbf{G}_{t+1}}{ \partial\psi_{k,p}}\right\}\right\}, \tag{34}\]

the detailed derivation of which is shown in Appendix B.