Geometry-Aware Set-Membership Multilateration: Directional Bounds and Anchor Selection

Abstract

In this paper, we study anchor selection for range-based localization under unknown-but-bounded measurement errors. We start from the convex localization set $\X=\Xd\cap\Hset$ recently introduced in \cite{CalafioreSIAM}, where $\Xd$ is a polyhedron obtained from pairwise differences of squared-range equations between the unknown location $x$ and the anchors, and $\Hset$ is the intersection of upper-range hyperspheres. Our first goal is \emph{offline} design: we derive geometry-only E- and D-type scores from the centered scatter matrix $S(A)=AQ_mA\tran$, where $A$ collects the anchor coordinates and $Q_m=I_m-\frac{1}{m}\one\one\tran$ is the centering projector, showing that $λ_{\min}(S(A))$ controls worst-direction and diameter surrogates for the polyhedral certificate $\Xd$, while $\det S(A)$ controls principal-axis volume surrogates. Our second goal is \emph{online} uncertainty assessment for a selected subset of anchors: exploiting the special structure $\X=\Xd\cap\Hset$, we derive a simplex-aggregated enclosing ball for $\Hset$ and an exact support-function formula for $\Hset$, which lead to finite hybrid bounds for the actual localization set $\X$, even when the polyhedral certificate deteriorates. Numerical experiments are performed in two dimensions, showing that geometry-based subset selection is close to an oracle combinatorial search, that the D-score slightly dominates the E-score for the area-oriented metric considered here, and that the new $\Hset$-aware certificates track the realized size of the selected localization set closely.

Figures (5)

• Figure 1: Localization with three anchors in $\mathbb{R}^2$.
• Figure 2: Illustration of the localization construction on a three-anchor example. For visual clarity, this figure uses larger interval widths than the numerical experiments of Section V, so that the annuli and the nested sets are easier to distinguish. In the left panel, the blue solid circles and red dotted circles are the upper and lower range boundaries, the stars are the anchors, and the orange cross is the true target. The black dashed polygon is $\mathcal{X}_d$, the purple region is $\mathcal{X}=\mathcal{X}_d\cap\mathcal{H}$, and the green region is the true feasible set $\mathcal{X}_{\mathrm{true}}$. The right panel enlarges the neighborhood of $\mathcal{X}_{\mathrm{true}}$.
• Figure 3: Hybrid coordinate-box bound $B_{\mathrm{hyb}}$ versus the exact coordinate-box area of the actual localization set $\mathcal{X}$ over all subsets in the Monte Carlo experiment. Both axes are on logarithmic scale.
• Figure 4: Empirical CDF of the box-area ratio to the oracle over the $60$ Monte Carlo trials. The D-score slightly dominates the E-score on this area-based metric and is therefore the practical default, while both remain close to the oracle.
• Figure 5: Geometry sweep for the actual localization set $\mathcal{X}$. Left: exact vertical width, the $\mathcal{X}_d$-based bound, the $\mathcal{H}$-support bound, and their hybrid minimum. Right: pure polyhedral diameter bound and ball-induced diameter bound.

Theorems & Definitions (13)

• Remark 1: Noise Model Equivalence
• Proposition 1
• proof
• Corollary 1
• proof
• Proposition 2
• proof
• Remark 2
• Proposition 3
• proof