Exploiting Double-Bounce Paths in Snapshot Radio SLAM: Bounds, Algorithms and Experiments

TL;DR

The numerical and experimental results demonstrate that the double-bounce NLoS paths which share at least one incidence point (IP) with the single-bounce NLoS paths improve the estimation accuracy of the UE state and existing IPs of single-bounce NLoS paths.

Abstract

Radio-based simultaneous localization and mapping (SLAM) has the potential to provide precise user equipment (UE) localization and environmental sensing capabilities by exploiting radio signals. Most existing approaches leverage line-of-sight (LoS) and single-bounce non-line-of-sight (NLoS) paths solely, while higher-order NLoS paths are treated as disturbance. In this paper, we investigate the benefits of leveraging double-bounce NLoS paths for solving the bistatic snapshot radio SLAM problem. We derive the Cramer-Rao bound (CRB) for joint estimation of the UE state and landmark positions when double-bounce NLoS paths are present. In addition, we propose an algorithm to identify double-bounce NLoS paths and leverage them into joint UE and landmarks estimation. The derived bounds are validated through simulated data, and the proposed algorithms are evaluated using experimental millimeter wave (mmWave) measurements harnessing beamformed 5G cellular reference signals. The numerical and experimental results demonstrate that the double-bounce NLoS paths which share at least one incidence point (IP) with the single-bounce NLoS paths improve the estimation accuracy of the UE state and existing IPs of single-bounce NLoS paths. Importantly, exploiting double-bounce NLoS paths enhances environmental mapping capabilities by revealing landmarks that are unobservable with single-bounce NLoS paths alone.

Figures (10)

• Figure 1: Illustration of the bistatic radio SLAM problem including LOS, single-bounce, and double-bounce NLOS paths. Double-bounce NLOS paths enable improved environmental sensing capabilities since they can be used to illuminate and detect landmarks and other objects that cannot be observed with single-bounce paths alone.
• Figure 2: Visualization of ${\mathbf{J}}_i({\mathbf{x}})$ in \ref{['eq:fim-per-path']} for different path types. LOS path visualized on the left ($i \in \mathcal{I}_\textrm{LoS}, \; \mathcal{M}_i = \emptyset$), single-bounce path in the middle ($i \in \mathcal{I}_\textrm{SB}, \; \mathcal{M}_i = \{{\mathbf{m}}_2\}$), and double-bounce path shown on the right ($i \in \mathcal{I}_\textrm{DB}, \; \mathcal{M}_i = \{{\mathbf{m}}_2, {\mathbf{m}}_4\}$). In the example, there are four IPs overall ($N=4$) and the non-zero elements of ${\mathbf{J}}_i({\mathbf{x}})$ are illustrated using blue markers.
• Figure 3: Examples of rank-deficient single-bounce SLAM geometries. In the examples, the LOS is assumed to exist and one single-bounce IP is considered such that $\dim({\mathbf{x}}) = 6$. In scenario 1, as indicated by the number inside the marker, the rank of the FIM is $\textrm{rank}({\mathbf{J}}({\mathbf{x}})) = 5$ and null space of the system is $\textrm{null}({\mathbf{H}}({\mathbf{x}})) = \left(-100100\right)^\top$ meaning that we can change the $x$ coordinate of the UE and the clock bias (in equal but opposite values) without changing the measurements. In scenario 3, $\textrm{rank}({\mathbf{J}}({\mathbf{x}})) = 5$ and $\textrm{null}({\mathbf{H}}({\mathbf{x}})) = \left(100-110\right)^\top$, whereas in scenario 2, $\textrm{rank}({\mathbf{J}}({\mathbf{x}})) = 4$ and $\textrm{null}({\mathbf{H}}({\mathbf{x}})) = \left(-100100000010\right)^\top$.
• Figure 4: An example of a rank-deficient double-bounce SLAM geometry. The example considers the LOS, one single-bounce NLOS and one double-bounce NLOS that shares one of the IP with the single-bounce path. The dimension of the joint state is $\dim({\mathbf{x}}) = 8$, the rank of the FIM is $\textrm{rank}({\mathbf{J}}({\mathbf{x}})) = 7$ and null space of the system is $\textrm{null}({\mathbf{H}}({\mathbf{x}})) = \left(00000010\right)^\top$.
• Figure 5: On top, the performance bounds with LoS and one single-bounce NLoS path. On bottom, the performance bounds with LoS, one single-bounce NLoS and one double-bounce NLoS path, which shares the first interaction point with the single-bounce NLoS path (case 2). In the figures, the IP of the single-bounce NLoS path is varied $x, \, y \in [-10\,\text{m},\, 10\,\text{m}]$, whereas the second IP of the double-bounce NLoS path is fixed to $x = 0\,\text{m},\, y = 5\,\text{m}$. The BS location illustrated using (), the UE location with (), and the second IP of the double-bounce NLoS path using ().