## 3 Navigation Framework
### Observation Model
As illustrated in Figure 6, from three groups of simulations with different numbers of BS antennas, it is easy to find that the accuracy of our tracking algorithm can approximate the theoretical BCRB. With the increment of BS antennas, our tracking algorithm performs significantly better and approaches closer to the BCRB. This is reasonable as the increment of BS antennas extends the dimension of measurements, which contributes to the accuracy of the estimation procedure. Moreover, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power. #### 5.2.3 Effect of Users' Mobility

To study the effect of the mobility of users, we have a fixed sampling frequency, which means that the localization and optimization procedure is conducted at the same time slot length. Thus, users' mobility can be characterized by both covariance matrix \(\mathbf{Q}\). With other variable parameters fixed at \(K=3\), \(N_{R}=60\), \(N_{B}=24\), \(\{\sigma_{z_{m}}^{2}\}_{m\in\mathcal{M}}=0\,\text{m}^{2}/\text{s}^{2}\), the transmit power \(P\) varies from 0 dBm to 15 dBm. Controlled simulations are conducted where the velocities of users are fixed, \(\{\sigma_{z_{m}}^{2}\}_{m\in\mathcal{M}}\) and \(\{\sigma_{y_{m}}^{2}\}_{m\in\mathcal{M}}\) in \(\mathbf{Q}\) vary within the range of 0.1 m\({}^{2}/\text{s}^{2}\), 0.5 m\({}^{2}/\text{s}^{2}\), and 1 m\({}^{2}/\text{s}^{2}\). It is obvious in Figure 7 that the tracking of trajectories with minor covariance attains higher accuracy and can reach a relatively good accuracy even under a high-covariance circumstance. Moreover, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power. #### 5.2.4 Effect of the Number of RISs and RIS Elements

We exploit the influence of both the number of RISs and RIS elements. To assess the influence of the number of RISs, with other variable parameters fixed at \(N_{R}=60\), \(N_{B}=72\), \(\{\sigma_{s_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{y_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{z_{m}}^{2}\}_{m\in\mathcal{M}}=0\,\mathrm{m}^{2}/\mathrm{s}^{2}\), the transmit power \(P\) varying from \(0\,\mathrm{dBm}\) to \(15\,\mathrm{dBm}\) and the RIS deployed at \([0,0,3]\,\mathrm{m}\) is removed in the \(K=2\) case. As indicated in Figure 8, the performance of tracking in the \(K=3\) case is significantly better than in the \(K=2\) case. With the increment in the number of RISs, our tracking algorithm approaches much closer to the BCRB as well. To exploit the effect of the number of RIS elements, we consider other variable parameters fixed at \(K=3\), \(N_{B}=24\), \(\{\sigma_{s_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{y_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{z_{m}}^{2}\}_{m\in\mathcal{M}}=0\,\mathrm{m}^{2}/\mathrm{s}^{2}\), the transmit power \(P\) varying from \(0\,\mathrm{dBm}\) to \(15\,\mathrm{dBm}\), and the numbers of RISs' elements \(N_{R}\) varying in \(60\times 1\), \(60\times 2\), and \(60\times 3\). As illustrated in Figure 9, with the increment in the number of RIS elements, the performance of our tracking algorithm improves. Further, we can find that with the increment in the number of elements, the tracking accuracy of the proposed scheme is closer to the BCRB. Moreover, as indicated by the gaps between three groups of lines with different \(N_{R}\), the improvement of accuracy with the increment of \(N_{R}\) becomes less significant when the same number of elements is added. From the above two sets of experiments, it can be concluded that with abundant RISs deployed and optimization of their phase shift, high accuracy can be achieved in harsh localization environments. In all cases, the performance of our tracking scheme improves and approximates BCRB more closely with the increment of transmit power.