## 3 Navigation Framework
### Observation Model
\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]5 Performance illustration with different numbers of RISs \(K\) set by 2 and 3 and transmit power \(P\) varying from 0 dBm to 15 dBm. Other variable parameters are fixed at \(N_{R}=60\), \(N_{B}=72\), \(\{\sigma_{x_{n}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{y_{n}}^{2}\}_{m\in\mathcal{M}}=0.5\,\mathrm{m}^{2}/\mathrm{s}^{2}\), \(\{\sigma_{z_{n}}^{2}\}_{m\in\mathcal{M}}=0\,\mathrm{m}^{2}/\mathrm{s}^{2}\). ## 6 Conclusions

In summary, concerning the mobility of users and the potential obstruction of LoS paths, a multi-user tracking system based on EKF is designed by directly utilizing the non-linear relationship between the signals received at the BS antennas and the coordinates of the moving targets, where the temporal correlations between each user's coordinate changes are fully exploited. An optimization scheme of passive phase shift design at the RISs is devised by minimizing the derived BCRB and solving the problem by leveraging the GD method. Numerical results indicate that the accuracy of our tracking scheme is affected by a series of factors. With the increment of transmit power, the number of BS antennas, users' mobility, and the number of RISs or RIS elements, the proposed method attains higher tracking accuracy and approaches closer to the BCRB in the conducted simulations. High-precision multi-user tracking via a single BS can be realized by abundant RISs deployed and optimized, even in harsh localization environments. Future works will entail the tracking of multiple users under more variable scenarios, i.e., monitoring the existence of the LoS path between the RISs and the users and switching the adopted tracking model under varied circumstances. Conceptualization, Y.M. and R.W. ; methodology, Y.M. and R.W. ; software, Y.M. and R.W. ; validation, Y.M. and R.W. ; formal analysis, Y.M. and R.W. ; investigation, Y.M. ; resources, Y.M. and R.W. ; data curation, Y.M. and R.W. ; writing--original draft preparation, Y.M. ; writing--review and editing, Y.M., R.W., E.L. and I.S. ; visualization, Y.M. ; supervision, R.W., E.L. and I.S. ; project administration, R.W., E.L. and I.S. ; funding acquisition, R.W., E.L. and I.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China under Grant 62271352, the Shanghai Science and Technology Innovation Action Plan Project No. 21220713100 and the Natural Science Foundation of Shanghai under Grant 22 ZR1465100. Not applicable. The data presented in this study are available in this article. The authors declare no conflicts of interest. ## Appendix A

\[\hat{L}(\mathbf{\Omega}|\mathbf{\Omega}^{r}) \geq L(\mathbf{\Omega}),\]6





## References

* [PERSON] et al. (2021) [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; et al. Integration of communication and sensing in 6G: A joint industrial and academic perspective. In Proceedings of the 2021 IEEE 32 nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Helsinki, Finland, 13-16 September 2021; pp. 1-7. * [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] An overview of signal processing techniques for joint communication and radar sensing. _IEEE J. Sel. Top. Signal Process._**2021**, _15_, 1295-1315. [CrossRef]
* [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON] Integrated sensing and communications: Toward dual-functional wireless networks for 6G and beyond. _IEEE J. Sel. Areas Commun._**2022**, _40_, 1728-1767. [CrossRef]
* [PERSON] et al. (2020) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; et al. 6G white paper on localization and sensing. _arXiv_**2020**, arXiv:2006.01779. * [PERSON] et al.