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3D Protraction Technique for Anchor-Free, Range-Free Wireless Localization in Industrial Warehouses via Multi-Level Binary Reachability

Peyman Moeini, Mark Coates

arXiv:2609.09330v1eess.SP

TL;DR

Industrial warehouses constrain fixed-node placement, while metric localization can require calibration, synchronization, specialized hardware, or surveyed coordinates. 3D-PT turns binary detections across ordered transmit-power levels into reliability-weighted ordinal relations for relative 3D localization. In simulated warehouse networks, four levels outperform connectivity baselines, with accuracy gains flattening beyond four.

Abstract

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Three-dimensional (3D) localization is difficult in industrial facilities because fixed-node placement is constrained by hardware, power, networking, installation, and maintenance costs. This paper presents the 3D Protraction Technique (3D-PT), an anchor-free, range-free method that converts reciprocal packet detections at ordered transmit-power levels into reliability-weighted ordinal distance relations. It estimates a relative 3D configuration without Received Signal Strength Indicator (RSSI) magnitudes, estimated ranges, surveyed anchors, or prior coordinates. In a simulated multi-height warehouse, four transmit-power levels reduce mean network-diameter-normalized RMSE by 16.0\% relative to MDS-MAP(P) in both small- and medium-network regimes, and by 23.9\% and 34.0\% relative to CATL-reproduction, respectively. Accuracy largely saturates beyond four power levels. The results show that multi-level binary reachability provides useful geometric information beyond a single maximum-power connectivity graph.

I. INTRODUCTION

Industrial 3D localization faces constrained fixed-node placement and burdensome metric-measurement requirements. 3D-PT instead derives relative geometry from reliability-weighted ordinal relations across transmit-power levels.

I. INTRODUCTION

  • Industrial warehouses constrain fixed-node placement, while RSSI-, time-, or angle-based localization may require calibration, synchronization, specialized hardware, or surveyed coordinates.Metal racks, machinery, stored material, and walls also create difficult propagation conditions.Source: §I · p. 1
  • Multiple transmit-power levels add ordinal proximity information to AF–RF localization: pairs detected at lower power are likely closer than pairs detected only at higher power.These binary observations can be used without exposing received-power magnitudes to the localization algorithm.Source: §I · p. 1
  • 3D-PT converts reciprocal detection signatures into ordinal proximity and reliability scores, selects local and global relations, and optimizes a relative 3D configuration.The method uses robust, scale-normalized ordinal penalties.Source: §I · p. 1

II. RELATED WORK

  • MDS-MAP(P) merges overlapping local maps,, but local embedding errors can propagate; CATL’s hop-based approach assumes dense, roughly uniform nodes,.Source: §II · pp. 1–2
  • Unlike MDS-MAP(P) and CATL, 3D-PT retains observations across ordered transmit-power levels rather than reducing them to one maximum-power graph.It extends 2D-PT to 3D and adds global pair-order relations to reference-centered local comparisons.Source: §II · p. 2

III. SYSTEM MODEL AND OBJECTIVE

  • The model uses binary packet detections across ordered power levels and requires a connected maximum-power graph; without anchors or metric ranges, the estimate is relative up to translation, rotation or reflection, and scale.The localization algorithm receives no surveyed anchors, ranges, RSSI magnitudes, timing, angular measurements, floor labels, or obstacle geometry.Source: §III · p. 2

IV. 3D PROTRACTION TECHNIQUE

3D-PT represents a trial embedding as one shared set of candidate 3D node coordinates, evaluates pairwise distances induced by that configuration, and optimizes agreement with selected ordinal relations.

IV. 3D PROTRACTION TECHNIQUE

  • A trial embedding is a temporary 3D node configuration considered during optimization, with each node assigned a candidate coordinate.Source: §IV · p. 2
  • The candidate coordinates form a common matrix Y, and each node-pair distance is the Euclidean distance induced by that shared configuration.Pairs are represented through the common configuration rather than assigned separate coordinates.Source: §IV · p. 2
  • The objective penalizes violations of selected local and global distance orderings together with a scale-normalization term; lower L(Y) indicates greater agreement.The optimized relative configuration is reported as X̂.Source: §IV · p. 2

A. Binary Protraction Scores

Binary Protraction Scores correct reciprocal detection signatures for power-order reversals, then summarize projected reachability across levels as ordinal proximity.

A. Binary Protraction Scores

  • Because wireless variability can produce power-order reversals, each reciprocal signature is projected onto the set of nondecreasing sequences.A reversal is a 1-to-0 transition across increasing transmit-power levels.Source: A. Binary Protraction Scores · p. 2
  • The projected sequence is converted back to binary form, and its average defines the ordinal proximity score.Source: A. Binary Protraction Scores · p. 2
  • s_ij is the fraction of projected power levels with reciprocal reachability; lower-power reachability earns a larger score, without treating the score as a range estimate.Equal weighting avoids assuming that power-level index is linearly related to physical power.Source: A. Binary Protraction Scores · p. 2

B. Pair Reliability

Pair reliability captures observation-quality effects that the ordinal proximity score alone omits, including directional disagreement and power-order reversals.

B. Pair Reliability

  • The proximity score does not preserve directional disagreement or power-order reversals, so it omits two observation-quality effects.Source: B. Pair Reliability · p. 2
  • The exponent β_r controls how strongly reciprocity and monotonicity inconsistencies reduce pair reliability.Its value is specified in Table I.Source: B. Pair Reliability · p. 3
  • Pair reliability incorporates directional agreement and monotonic consistency, with κ_ij representing the resulting reliability.Source: B. Pair Reliability · p. 3

C. Local Candidate Relations and Penalties

3D-PT builds local ordinal relations by ranking nodes around each reference node, selecting reliable comparisons, and weighting their penalties by score separation and link reliability.

C. Local Candidate Relations and Penalties

  • Local candidates propose that one node is closer to a reference than another, using close–close, close–far, or stratified rank comparisons.The rules select which triplets to consider; their exact rank counts are specified in Section V.Source: C. Local Candidate Relations and… · p. 3
  • The selected local set retains candidates with positive score gaps and sufficiently reliable constituent links, using observations before coordinate optimization.Selection does not use true coordinates or the current trial embedding.Source: C. Local Candidate Relations and… · p. 3
  • Each local triplet supplies noisy evidence for an ordinal distance preference, penalized with a bounded-influence robust softplus.The robustification parameter controls the penalty, while τ controls smoothness and μ_ijk sets the requested ordinal separation.Source: C. Local Candidate Relations and… · p. 3
  • Local relation weights increase with score gaps and constituent-link reliability, combine the two link reliabilities geometrically, and are normalized to unit mean over T.The normalized weights multiply the local penalties in the overall objective.Source: C. Local Candidate Relations and… · p. 3

D. Global Relations and Ordinal Embedding

3D-PT adds global ordinal comparisons between reliable unordered node pairs, then combines them with local penalties and scale control to estimate a relative configuration.

D. Global Relations and Ordinal Embedding

  • Global candidates compare unordered node pairs ranked by decreasing reliable score, extending the information beyond local triplets.The selected set requires a minimum global score gap and minimum reliability for each participating pair.Source: D. Global Relations and Ordinal… · p. 3
  • A selected global quadruple proposes an ordinal distance comparison and receives a margin and weight that emphasize larger score gaps.Global and local score gaps are ordinal evidence, not metric distances.Source: D. Global Relations and Ordinal… · pp. 3–4
  • The objective combines local and global ordinal penalties with a scale-normalization penalty; optimization uses spectral initializations and coarse-to-fine Adam refinement.The finite rank-three solution with the lowest objective is centered and reported as the estimate.Source: D. Global Relations and Ordinal… · p. 4

V. EXPERIMENTAL METHODOLOGY

The evaluation uses simulated packet detections in a multi-height warehouse and compares 3D-PT configurations with connectivity-based baselines under specified trial, budget, and scoring rules.

V. EXPERIMENTAL METHODOLOGY

  • Evaluation uses a 140 × 90 × 14 m warehouse with four attenuating rack structures and nodes at five nominal heights, perturbed by mounting-height jitter.The channel model includes path loss, shadowing, Rician fading, receiver variation, non-reciprocity, and transmit-power error; localization methods receive no RSSI, true ranges, or environment geometry.Source: §V · p. 4
  • Small-network trials use n = 4–15 and medium-network trials n ∈ {10, 20, 30, 40, 50}; both compare K = 1 and K = 4, with a separate K = 1–10 sensitivity experiment.The K = 1 and K = 4 configurations use 40 packets at +8 dBm and 40 packets at each of {−10, −4, 2, 8} dBm, respectively.Source: §V · p. 4
  • MDS-MAP(P) and CATL-reproduction receive the same total packet budget as 3D-PT but reduce observations to one reciprocal graph at maximum power.MDS-MAP(P) uses fixed two-hop local patches, while CATL-reproduction is a centralized reproduction with fixed parameters.Source: §V · p. 4
  • Algorithm parameters were selected using separate development simulations and fixed before publication trials; the reported numerical settings were not changed after inspecting those results.The development and publication runs used the same warehouse and channel models but independent geometry and channel realizations.Source: §V · pp. 4–5
  • Only trials with a connected reciprocal maximum-power graph are eligible; NRMSE is 3D RMSE normalized by true network diameter, and Table II is scoped to small-network mean NRMSE (%).Estimates are similarity-aligned to the true configuration; mean curve estimates use successful trials and bootstrap confidence intervals.Source: §V · p. 5, Table II · p. 5

A. Small-Network Performance

In the small-network regime, PT K = 4 has lower mean NRMSE than MDS-MAP(P), CATL-reproduction, and PT K = 1 across n = 4–15. Its success-rate and accuracy comparisons at n = 4 require separate interpretation.

A. Small-Network Performance

  • 19.78% mean NRMSE for PT K = 4 across n = 4–15, versus 23.54% for MDS-MAP(P), 25.99% for CATL-reproduction, and 24.17% for PT K = 1.The corresponding descriptive reductions are 16.0%, 23.9%, and 18.2%; for n = 5–15, paired reductions versus both baselines were statistically significant.Source: A. Small-Network Performance · p. 5
  • PT K = 4 succeeds in 99.2% of n = 4 trials versus 47.2% for MDS-MAP(P), but its advantage reverses among the 227 trials where both succeed.Across n = 5–15, PT K = 4 succeeds in every connected trial; MDS-MAP(P) reaches 100% from n = 6, while CATL-reproduction is less reliable in the smallest networks.Source: A. Small-Network Performance · p. 5

B. Effect of the Number of Power Levels

Four transmit-power levels capture most of the ordinal accuracy benefit in the n = 10 sensitivity experiment. At medium network sizes, PT K = 4 has the lowest mean NRMSE, with increased computation as network size grows.

B. Effect of the Number of Power Levels

  • At n = 10, mean PT NRMSE falls from 23.64% at K = 1 to 20.07% at K = 4, a 15.1% reduction, then remains within 19.90–20.33% for K = 4–10.Runtime rises from 1.37 s at K = 1 to 2.21 s at K = 4 and 4.95 s at K = 10; packet counts are 40, 160, and 400, respectively.Source: B. Effect of the Number of Power Levels · p. 5, Figure 1 · p. 6

C. Medium-Network Performance

  • 17.18% mean NRMSE for PT K = 4 averaged across medium-network sizes, 16.0% below MDS-MAP(P), 34.0% below CATL-reproduction, and 18.1% below PT K = 1.PT K = 4 has the lowest mean NRMSE at every tested size; PT K = 4 and MDS-MAP(P) succeed in all 1,250 connected trials, while CATL-reproduction succeeds in 97.84%.Source: C. Medium-Network Performance · p. 5
  • Mean PT runtime grows from 2.31 s at n = 10 to 43.3 s at n = 50, versus 0.0049–0.249 s for MDS-MAP(P) and 0.055–0.401 s for CATL-reproduction.The stated implementation direction is reducing candidate-relation count and multi-restart optimization cost.Source: C. Medium-Network Performance · pp. 5–6

D. Interpretation and Limitations

The interpretation attributes multi-level PT’s gains to preserved transition information and global comparisons, while accuracy gains level off near four power levels. Conclusions remain bounded by the simulation setting and the method’s relative, connected-graph output.

D. Interpretation and Limitations

  • Preserving each pair’s transition along the power ladder separates pairs merged in the maximum-power graph, while global comparisons discourage larger-scale distortion.The gain is largest against CATL-reproduction, whose hop-based construction is more sensitive to sparse or nonuniform connectivity.Source: D. Interpretation and Limitations · p. 6
  • Beyond K = 4, accuracy changes little despite rising packet and runtime costs; K = 4 is not universal because radio and propagation conditions can shift saturation.Power spacing and receiver thresholds may also shift the saturation point.Source: D. Interpretation and Limitations · p. 6
  • PT K = 1 versus K = 4 compares complete per-level configurations, not fixed-total-packet ablations, so it measures practical configuration benefit rather than level-count-only gain.Source: D. Interpretation and Limitations · p. 6
  • Conclusions are simulation-based and depend on the stated warehouse and channel model; 3D-PT returns relative geometry, requires a connected maximum-power graph, and uses a slower non-convex multi-restart solver.Source: D. Interpretation and Limitations · p. 6

VII. CONCLUSION

  • Four power levels improve mean NRMSE over single-level PT and both connectivity baselines, with gains persisting from very small to 50-node networks.Physical validation and faster optimization remain necessary next steps.Source: §VII · pp. 6–7
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