## 3 Navigation Framework
### Observation Model
### Simulation Results and Discussions

#### 5.2.1 Convergence and Error Analysis of Tracking

We first study the convergence and error of our tracking scheme by executing a 3-second trajectory once with the experimental method and parameters in Section 5.1, which contains 100 discrete samplings of users' positions. Other simulations conducted within the range of our parameter settings follow the same path. The number of operations is hard to generalize as it depends on the choice of GD updating step size and other parameters. However, the convergence characteristics of the algorithm are worthy of study. As illustrated in Figure 4, this cluster of curves represents the convergence process of the optimization algorithm in 100 discrete sampling time slots, where the ordinate denotes the value of the objective function, and the abscissa denotes the number of iterations. The phase shift matrix of all RISs in each sampling time slot is randomly generated. We set the value of the objective function MSE at the end of the first tracking time slot as the metric for subsequent time slots, where the optimization procedure will repeat until it satisfies the stopping criteria. It can be found that the optimization of the phase shift design converges well, which validates the effectiveness of Algorithm 2. Further, our algorithm exploits temporal correlation and continuity during the tracking process, requiring fewer adjustments in subsequent time slots. Thus, except for the first optimization of RISs' phase shift at the first tracking time slot, the optimization process takes much fewer operations in the subsequent time slots. The accuracy of our proposed tracking scheme is evaluated by the root mean square error (RMSE) of tracking errors. As indicated in Figure 5, the abscissa denotes 100 consecutive time slots and the ordinate denotes the RMSE of the tracking errors. The RMSE of the tracking errors with random phase shift is divergent, as the LoS paths between users and the BS are presumed to be blocked, which is a relatively harsh tracking scenario. However, our tracking scheme with optimized phase shift can achieve significant accuracy and can approximate the theoretical BCRB, which validates the effectiveness of Algorithm 1. Further, in the optimized case, it can be found that at some certain sampling slots, the RMSE of tracking errors is larger than other slots. This indicates that even if our tracking scheme is temporarily underperforming, it will not cause major interference with the subsequent tracking process, since the E



#### 5.2.2 Effect of the Number of BS Antennas

We first investigate the effect of the number of BS antennas on the RMSE of tracking with varying transmit power. In this and all the subsequent simulations, the perfor

\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]1 The convergence of Algorithm 2. \(P=10\,\mathrm{dBm}\); other parameters follow \[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]2

\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]3 The performance illustration of Algorithm 1. \(P=10\,\mathrm{dBm}\); other parameters follow \[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]4

mance of our tracking scheme is evaluated by the RMSE of tracking errors averaging from 30 conducted 3-second trajectories, each containing 100 discrete samplings of users' positions. The tracking error with random phase shift on RISs, the averaging of which has no practical significance, will not be further studied because it is divergent. With other variable parameters fixed at \(K=3\), \(N_{R}=60\), \(\{\sigma_{x_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\text{m}^{2}/\text{s}^{2}\), \(\{\sigma_{y_{m}}^{2}\}_{m\in\mathcal{M}}=0.5\,\text{m}^{2}/\text{s}^{2}\), \(\{\sigma_{z_{m}}^{2}\}_{m\in\mathcal{M}}=0\,\text{m}^{2}/\text{s}^{2}\), the transmit power \(P\) varying from 0 dBm to 15 dBm and the number of BS antennas \(N_{B}\) is set by \(8\times 3\), \(8\times 6\), and \(8\times 9\) as a comparison, respectively.