## 3 Navigation Framework
### Observation Model
The corresponding update process of \(\{\psi_{k,p}\}_{k\in\mathcal{K},p\in\mathcal{P}}\) is given by

\[\psi_{k,p}^{r+1}=\psi_{k,p}^{r}-\frac{1}{\beta_{r}}\
abla_{\psi_{k,p}}L(\psi_{ k,p}^{r}). \tag{35}\]

This procedure will repeat until it satisfies the stopping criteria. The whole procedure entailing the RISs' phase shift design is sketched in Algorithm 2. ```
Initialization\(\mathbf{\Omega}^{t}\). Set iteration index \(r=0\). repeat  Compute \(\mathbf{G}_{t+1}\) and \(L(\mathbf{s}_{t+1})\) according to (19) and (28), respectively. For each element in \(\mathbf{\Omega}_{k}\), calculate \(\
abla_{\psi_{k,p}}L(\psi_{k,p}^{r})\) by (34). For each element in \(\mathbf{\Omega}_{k}\), update \(\psi_{k,p}^{r+1}\) by (35). Set \(r=r+1\). until the fractional decrement of the target value is below a certain threshold. Output:\(\mathbf{\Omega}^{t+1}\)
```

**Algorithm 2** Optimization for RISs' phase shift

## 5 Results

A series of simulations are conducted in this section to assess the validity of the proposed tracking algorithm. With different parameter settings, the performance of our tracking algorithm against the BCRB is compared. ### Simulation Scenario and Parameter Settings

Unless otherwise indicated, we focus on the tracking scenario as sketched in Figure 3.

For the channel settings, we consider a simple scenario where each single-antenna user transmits a single data symbol \(p=1\) at a unique frequency. We set carrier frequency \(f_{c}=28\,\mathrm{GHz}\), subcarrier spacing \(\Delta f=120\,\mathrm{kHz}\), the frequency of the \(m\)-th user \(f_{m}=f_{c}+m\Delta f-B/2\), and the signal bandwidth \(B=M\Delta f\). The gain of the RIS unit element \(G_{p}=\pi\), and the gain of the antenna of users and BS are \(G_{m}=1\) and \(G_{b}=1\), respectively. In our tracking scheme, the LoS paths between BS and users are presumed to be blocked. The interference power caused by large-scale fading and reflection loss is \(20\,\mathrm{dB}\) weaker than the LoS path in general [51]. Thus, at the receiver, we fix the noise variance per subcarrier at \(\sigma^{2}=-125\,\mathrm{dBm}\), and at the transmitter, we consider that transmitted power per subcarrier \(P\) varies from \(0\,\mathrm{dBm}\) to \(15\,\mathrm{dBm}\). For system geometry, the only BS is located at \(\mathbf{p}_{B}=\left[16,0,2\right]\mathrm{m}\), and \(K=3\) RISs are deployed at \(\{\mathbf{p}_{k}\}_{k\in K}\), including \(\left[0,0,3\right]\mathrm{m}\), \(\left[8,-8,3\right]\mathrm{m}\) and \(\left[8,8,3\right]\mathrm{m}\). The BS consists of a uniform planar array of \(8\times 3\) elements lying on the \(YZ\) plane and the RISs are equipped with a uniform linear array of \(60\times 1\) elements on the \(YZ\), \(XZ\), and \(XZ\) planes, respectively. According to \(f_{c}\), the antenna spaces on the BS array and the element spaces on the RISs' array are \(d_{space}=0.54\,\mathrm{cm}\). Notably, the elements in RISs are mainly deployed horizontally as long stripes, by which the radiating near-field scope is extended over about \(18\,\mathrm{m}\). For the kinematic model of users, \(M=3\) single-antenna users move in a uniform linear pattern with \(\sigma_{x}^{2}=0.5\,\mathrm{m}^{2}/s^{2}\), \(\sigma_{y}^{2}=0.5\,\mathrm{m}^{2}/s^{2}\) and \(\sigma_{z}^{2}=0\). The initial positions of users \(\{\mathbf{p}_{m}\}_{m\in\mathcal{M}}^{0}\) are \(\left[10,0,1\right]\mathrm{m}\), \(\left[4,-4,1\right]\mathrm{m}\), and \(\left[4,4,1\right]\mathrm{m}\), the first of which has an initial velocity of \(-2\,\mathrm{m}/\mathrm{s}\) in the \(x\) direction, and the other two users both have initial velocities of \(-1\,\mathrm{m}/\mathrm{s}\) in the \(x\) direction and \(1\,\mathrm{m}/\mathrm{s}\) and \(-1\,\mathrm{m}/\mathrm{s}\) in the \(y\) direction, respectively. The initial state is assumed to be relatively accurately measured with minor errors generated based on \(\mathbf{Q}\) randomly. For clearer declarations, the parameter settings of simulations are given in Table 3.

\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]0 The simulation scenario.