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∩\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 offline design: we derive geometry-only E- and D-type scores from the centered scatter matrix S(A)=AQmA\tran, where A collects the anchor coordinates and Qm=Im−m1\one\one\tran is the centering projector, showing that λmin(S(A)) controls worst-direction and diameter surrogates for the polyhedral certificate \Xd, while detS(A) controls principal-axis volume surrogates. Our second goal is online uncertainty assessment for a selected subset of anchors: exploiting the special structure \X=\Xd∩\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 & tables
Fig. 1: Localization with three anchors in R2 .
Fig. 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 Xd , the purple region is X=Xd∩H , and the green region is the true feasible set Xtrue . The right panel enlarges the neighborhood of Xtrue .
Method
Mean box area of X
Rel. to oracle
Oracle subset
1.42×10−3
1.00
D-score maxdetS
1.47×10−3
1.04
E-score maxλmin(S)
1.50×10−3
1.06
Average subset
5.85×10−3
4.13
TABLE I: Subset selection over 60 Monte Carlo trials ( N=8 , k=4 ).
Fig. 3: Hybrid coordinate-box bound Bhyb versus the exact coordinate-box area of the actual localization set X over all subsets in the Monte Carlo experiment. Both axes are on logarithmic scale.
Fig. 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.
Fig. 5: Geometry sweep for the actual localization set X . Left: exact vertical width, the Xd -based bound, the H -support bound, and their hybrid minimum. Right: pure polyhedral diameter bound and ball-induced diameter bound.
We study hidden-target seeking in an anchor-free regime where neither the vehicle, the relay, nor the target has an accessible global pose. The vehicle never senses the target directly; it receives only range-bearing observations through a single relay of unknown position and orientation, with motion known only via integrated body-frame odometry. Absolute localization is fundamentally impossible: the joint configuration retains an exact three-dimensional SE(2) gauge no estimator can resolve. Yet the quantities needed for control remain fully recoverable: motion collapses calibration to a task-relevant quotient, the relay-to-odometry yaw and target displacement, identifiable in closed form from two distinct vehicle views. We introduce an O(K) multi-view self-calibration estimator with an exact first-order yaw-uncertainty certificate that propagates the cross-view correlations integrated odometry induces: the full certificate attains 95.0% pooled coverage at the nominal 95% level, versus 82.1% when correlated poses are treated as independent. The certificate drives a hybrid policy that excites until calibration is trustworthy, refuses uncertified estimates, seeks using continuously re-measured geometry, and detects relay-frame changes via persistent certified inconsistency, avoiding unbounded dead-reckoning drift. Across 200 randomized closed-loop trials, the method attains 0.064 m median station error versus 0.065 m for an oracle given the true relay yaw, despite 27 m median dead-reckoning drift over long horizons; 90 physics-based ROS 2/Gazebo trials retain 30/30 success under nominal operation, communication degradation, and relay-frame disturbances. These results show global localization is unnecessary for reliable hidden-target seeking under this single-relay model, even when both the sensing infrastructure and the vehicle's own reference frame are uncalibrated.
Indoor localization from wireless measurements remains challenging in large-scale deployments due to substantial variation in building geometry, the set of detectable access points (APs), and the heterogeneity of received signals. Existing learning-based methods often perform well only in limited settings and degrade under environmental shifts, making robust anchor-free localization across diverse indoor environments notoriously difficult. In this paper, we present OmniLoc, an environment-interactive foundation model for anchor-free user equipment localization across diverse indoor environments. To the best of our knowledge, OmniLoc is the first foundation-model-based approach built directly on wireless measurements for this task. OmniLoc is built on three key designs. First, a unified input tokenization module converts heterogeneous wireless measurements into a common representation that is more amenable to learning. Second, a geometry-aware Transformer performs AP-aware feature extraction by emphasizing dominant APs while aggregating complementary evidence from supporting APs. Third, a geometry-aware location estimation module conditions regression on geometric embeddings to produce geometrically consistent location predictions. We evaluate OmniLoc on both a large-scale in-house dataset and a public benchmark dataset. Results show that OmniLoc significantly outperforms existing methods, consistently improves existing backbones when its design components are integrated, and demonstrates strong generalization in cross-environment evaluations.
This paper studies hidden-target localization from range-bearing packets reported by a relay beacon whose global position and yaw are unknown. The vehicle knows its own trajectory but never directly senses the target; the relay packet contains only local-frame range and bearing to the vehicle and to the hidden target. Unlike bearing-only network localization, relative-frame localization, and target-enclosing control, the target is neither directly observed in the vehicle frame nor treated as a node in a relative-sensing graph. The main result characterizes the minimal motion that removes the resulting calibration ambiguity: one vehicle pose leaves a continuous yaw/translation/target gauge, whereas two distinct vehicle-relative observations from one unknown-pose relay constructively determine relay yaw (modulo 2 pi), relay position, and the anchored target in the noiseless case. A local rank corollary, a shared-target multi-beacon extension, and a trajectory-spread conditioning lemma connect relay self-calibration to finite-window excitation and native range-bearing estimation. In Monte Carlo evaluation the estimator recovers the hidden target with 5.5 mm RMSE, five times below the 30 mm per-packet range noise and thirteen times more accurate than a naive EKF baseline; it converges to the same accuracy from 2 m target offsets and 2.4 rad yaw errors, and Huber weighting preserves millimeter accuracy under 10% outlier corruption that drops the unprotected estimator to a 0.10 success rate. Trajectory spread predicts estimator quality: the two weakly excited trajectories carry condition numbers above 100 with success rates of 0.82 and 0.70, while every well-excited trajectory attains full success.