LEO- and RIS-Empowered User Tracking: A Riemannian Manifold Approach

Low Earth orbit (LEO) satellites and reconfigurable intelligent surfaces (RISs) have recently drawn significant attention as two transformative technologies, and the synergy between them emerges as a promising paradigm for providing cross-environment communication and positioning services. This paper investigates an integrated terrestrial and non-terrestrial wireless network that leverages LEO satellites and RISs to achieve simultaneous tracking of the three-dimensional (3D) position, 3D velocity, and 3D orientation of user equipment (UE). To address inherent challenges including nonlinear observation function, constrained UE state, and unknown observation statistics, we develop a Riemannian manifold-based unscented Kalman filter (UKF) method. This method propagates statistics over nonlinear functions using generated sigma points and maintains state constraints through projection onto the defined manifold space. Additionally, by employing Fisher information matrices (FIMs) of the sigma points, a belief assignment principle is proposed to approximate the unknown observation covariance matrix, thereby ensuring accurate measurement updates in the UKF procedure. Numerical results demonstrate a substantial enhancement in tracking accuracy facilitated by RIS integration, despite urban signal reception challenges from LEO satellites. In addition, extensive simulations underscore the superior performance of the proposed tracking method and FIM-based belief assignment over the adopted benchmarks. Furthermore, the robustness of the proposed UKF is verified across various uncertainty levels.