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Probabilistic Positioning Via Ray Tracing With Noisy Angle of Arrival Measurements

Vincent Corlay, Viet-Hoa Nguyen, Nicolas Gresset

TL;DR

This work tackles indoor NLoS positioning using uplink AoA measurements and a digital twin. It combines reverse ray tracing with Monte Carlo AoD sampling to generate per-BS maps and fits Gaussian mixture models to obtain $p(x|y_i)$, which are fused via $p(x|\mathbf{y}) \propto \prod_i p(x|y_i)$ to estimate UE position. The approach yields two implementation modes: an online mode with ray launching and pdf fitting, and an offline mode where pdf parameters are precomputed and stored for fast online retrieval; this substantially reduces online computation and allows robust performance with few launched rays. Simulation results demonstrate that the GMM-based method matches or surpasses high-ray baselines under realistic AoA noise while maintaining a low failure rate and offering clear complexity and storage advantages. Overall, the method provides a scalable, offline-capable solution for NLoS indoor positioning in industrial environments.

Abstract

We consider the positioning problem in non line-of-sight (NLoS) situations, where several base stations (BS) try to locate a user equipment (UE) based on uplink angle of arrival (AoA) measurements and a digital twin of the environment. Ray launching in a Monte Carlo manner according to the AoA statistics enables to produce a map of points for each BS. These points represent the intersections of the rays with a xy plane at a given user equipment (UE) elevation. We propose to fit a parametric probability density function (pdf), such as a Gaussian mixture model (GMM), to each map of points. Multiplying the obtained pdfs for each BS enables to compute the position probability of the UE. This approach yields an algorithm robust to a reduced number of launched rays. Moreover, these parametric pdfs may be fitted and stored in an offline phase such that ray tracing can be avoided in the online phase. This significantly reduces the computational complexity of the positioning method.

Probabilistic Positioning Via Ray Tracing With Noisy Angle of Arrival Measurements

TL;DR

This work tackles indoor NLoS positioning using uplink AoA measurements and a digital twin. It combines reverse ray tracing with Monte Carlo AoD sampling to generate per-BS maps and fits Gaussian mixture models to obtain , which are fused via to estimate UE position. The approach yields two implementation modes: an online mode with ray launching and pdf fitting, and an offline mode where pdf parameters are precomputed and stored for fast online retrieval; this substantially reduces online computation and allows robust performance with few launched rays. Simulation results demonstrate that the GMM-based method matches or surpasses high-ray baselines under realistic AoA noise while maintaining a low failure rate and offering clear complexity and storage advantages. Overall, the method provides a scalable, offline-capable solution for NLoS indoor positioning in industrial environments.

Abstract

We consider the positioning problem in non line-of-sight (NLoS) situations, where several base stations (BS) try to locate a user equipment (UE) based on uplink angle of arrival (AoA) measurements and a digital twin of the environment. Ray launching in a Monte Carlo manner according to the AoA statistics enables to produce a map of points for each BS. These points represent the intersections of the rays with a xy plane at a given user equipment (UE) elevation. We propose to fit a parametric probability density function (pdf), such as a Gaussian mixture model (GMM), to each map of points. Multiplying the obtained pdfs for each BS enables to compute the position probability of the UE. This approach yields an algorithm robust to a reduced number of launched rays. Moreover, these parametric pdfs may be fitted and stored in an offline phase such that ray tracing can be avoided in the online phase. This significantly reduces the computational complexity of the positioning method.
Paper Structure (21 sections, 3 equations, 5 figures)

This paper contains 21 sections, 3 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Technical background
  3. Reverse ray-tracing positioning
  4. Monte-Carlo ray launching (proposed in Nguyen2023)
  5. Processing the map of points in Nguyen2023
  6. Statistical modeling and notations
  7. Probability density function fitting
  8. Formulation of the problem
  9. Choice of the continuous parametric distribution
  10. Exploiting the fitted probability density functions for positioning
  11. First implementation mode
  12. Second implementation mode
  13. Complexity analysis
  14. Complexity of ray tracing
  15. Complexity of the square method of Nguyen2023 and GMM fitting
  16. ...and 6 more sections

Figures (5)

  • Figure 1: Left: Standard triangulation approach via LoS-based AoA at the BS. Right: Situation where some LoS AoA are not available for several BS. In the example, two of the three LoS paths are blocked by clutters. In this case, reverse ray tracing using a digital twin can be considered.
  • Figure 2: Map of points corresponding to the positions where the rays launched by each BS cross the xy plane (having the dimensions provided in Section \ref{['sec_sim_env']}).
  • Figure 3: Green: Map of points obtained when launching the rays from one BS with a given AoA and a given error statistics. Blue: Contour plot of the clusters of the fitted GMM.
  • Figure 4: Considered scene for the simulations.
  • Figure 5: Cumulative density function of the positioning error.