  In the context of single-base station (BS) non-line-of-sight (NLoS)
single-epoch localization with the aid of a reflective reconfigurable
intelligent surface (RIS), this paper introduces a novel three-step algorithm
that jointly estimates the position and velocity of a mobile user equipment
(UE), while compensating for the Doppler effects observed in near-field (NF) at
the RIS elements over the short transmission duration of a sequence of downlink
(DL) pilot symbols. First, a low-complexity initialization procedure is
proposed, relying in part on far-field (FF) approximation and a static user
assumption. Then, an alternating optimization procedure is designed to
iteratively refine the velocity and position estimates, as well as the channel
gain. The refinement routines leverage small angle approximations and the
linearization of the RIS response, accounting for both NF and mobility effects.
We evaluate the performance of the proposed algorithm through extensive
simulations under diverse operating conditions with regard to signal-to-noise
ratio (SNR), UE mobility, uncontrolled multipath and RIS-UE distance. Our
results reveal remarkable performance improvements over the state-of-the-art
(SoTA) mobility-agnostic benchmark algorithm, while indicating convergence of
the proposed algorithm to respective theoretical bounds on position and
velocity estimation.
