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Localization-Uncertainty-Aware UAV Swarm Search for GNSS-Denied Maritime Surveillance

Siyuan Yang, Junning Lyu, Xiaoyu He, Lakmal Seneviratne, Shaoming He

arXiv:2610.34873v1cs.RO

TL;DR

GNSS-denied maritime swarm search must coordinate coverage and safety while localization uncertainty propagates along multi-hop ranging references. The paper combines a Fisher-information uncertainty model, rate-aware robust CBFs, and hierarchical Voronoi targeting. In 20 Monte Carlo runs, all searches completed with feasible local QPs and localization-error containment in at least 98.97% of UAV-frames.

  • Problem

    GNSS-denied maritime search relies on cooperative localization whose uncertainty accumulates along multi-hop references and affects connectivity, collision avoidance, and search control.

    Source: §I · p. 1

  • Method

    The framework approximates cascading localization covariance with Fisher information and uses its uncertainty envelope and rate in robust CBFs alongside hierarchical Voronoi target selection.

    Source: 1) Cascading uncertainty modeling for… · p. 2, A. Target Selection and Task CBFs · p. 7

  • Results

    20 Monte Carlo runs all reached 100% coverage, with all 9,800 local QP commands per run feasible and localization-error containment in 98.97–100% of UAV-frames per run.

    Source: D. Statistical Validation and… · p. 11

  • Takeaways & Limitations

    The framework demonstrates complete coverage across estimator-in-the-loop simulation, Monte Carlo deployments, polygon examples, and six MBZIRC runs.

    Source: §VIII · p. 13

    The invariance statement is conditional on envelope and rate conditions; its covariance model is approximate and its closed-loop validity is assessed empirically.

    Source: 2) Collision avoidance · p. 7

Abstract

from arXiv · show

This paper studies UAV swarm search in GNSS-denied maritime environments, where inter-UAV ranging drives cooperative localization and uncertainty propagates along the multi-hop reference chain, affecting connectivity, collision avoidance, and search effectiveness. We propose an uncertainty-aware control barrier function (CBF) framework approximating cascading localization covariance through Fisher information matrices. First, the cascading covariance model captures how ranging noise and neighbor uncertainty accumulate along the localization chain. Second, uncertainty-tightened CBFs incorporate the covariance model and its rate into margins for connectivity maintenance and collision avoidance, with conditional invariance and graceful-degradation bounds. Third, a hierarchical Voronoi-target strategy with formation keeping supports efficient forward-looking coverage. The distributed controller is validated with a standard unbiased extended Kalman filter in the loop. The framework is demonstrated on several polygon shapes and swarm sizes from 6 to 30 sUAVs. An estimator-in-the-loop run completes the search in 302.5 s, 20 Monte Carlo deployments confirm repeatable completion with all local QPs feasible and containment in at least 98.97% of UAV-frames, and six MBZIRC runs provide estimator-in-the-loop evidence. The source code is publicly available.

I. INTRODUCTION

The paper frames GNSS-denied maritime search as a joint problem of coverage, cooperative localization, and safety under cascading position uncertainty. It proposes Fisher-information-based uncertainty modeling, uncertainty-tightened control barrier functions, and hierarchical search control, then validates the framework with an estimator in the loop.

I. INTRODUCTION

  • GNSS-denied maritime search couples multi-hop localization uncertainty with connectivity, collision avoidance, and forward-looking coverage control.References propagate from coastal bUAVs into the search area because direct ranging cannot support long-range offshore search.Source: §I · p. 1

B. CBF-Based UAV Swarm Control

  • Unlike the authors’ prior work under perfect localization, this paper incorporates cascading uncertainty into control design.The paper also distinguishes maritime forward-looking sensing, which couples translation with yaw alignment, from common downward-looking circular-footprint formulations.Source: §I · p. 1, B. CBF-Based UAV Swarm Control · pp. 2–3

1) Cascading uncertainty modeling for cooperative localization:

  • The framework combines a Fisher-information approximation of cascading localization uncertainty with rate-aware, uncertainty-tightened CBFs for connectivity and collision avoidance.The CBF design includes conditional invariance and a graceful-degradation bound.Source: 1) Cascading uncertainty modeling for… · p. 2

3) Hierarchical search strategy with forward-looking sensing:

  • Squad leaders select targets through Voronoi partitioning while followers maintain a triangular formation, with distance and yaw objectives decoupled for forward-looking sensors.Only squad leaders participate in the Voronoi partition.Source: 3) Hierarchical search strategy with… · p. 2

4) Estimator-in-the-loop validation:

  • A standard unbiased EKF in the control loop completes the representative search in 302.5 s, with all QP rows feasible and no true-distance violations across 20 Monte Carlo deployments.Six high-fidelity MBZIRC runs also complete coverage with no constraint violations.Source: 4) Estimator-in-the-loop validation · p. 2

A. GNSS-denied Cooperative Localization and Maritime Search

  • The paper identifies a gap in which cooperative-localization methods typically do not integrate uncertainty into safety-critical control design.Related maritime search approaches also generally assume reliable positioning and do not address the coupling among search, localization connectivity, and collision avoidance under uncertain cooperative localization.Source: A. GNSS-denied Cooperative… · p. 2
  • Prior coverage methods use independent uncertainty bounds or omit reference-chain error accumulation, while existing CBF approaches do not combine cascading uncertainty with forward-looking sensing constraints.The paper positions its contribution as deriving uncertainty margins online from the cooperative-localization model and its rate.Source: A. GNSS-denied Cooperative… · p. 2, B. CBF-Based UAV Swarm Control · p. 3

III. PRELIMINARIES

The preliminaries model a planar GNSS-denied swarm with forward-sector sensing and range-based cooperative localization. Searching UAVs estimate position from inter-UAV ranges, while uncertainty accumulates through a multi-hop reference chain.

A. Modeling

  • The swarm model assumes fixed altitude and full communication, making search and localization planar; each UAV has planar position and yaw state with velocity and yaw-rate commands.The velocity-command model abstracts underlying attitude and translational dynamics.Source: A. Modeling · p. 3
  • Forward-sector sensors search points within both a detection radius and a horizontal field-of-view angle, coupling sensing to yaw orientation.The model describes maritime forward-looking cameras or radar rather than downward circular sensing.Source: A. Modeling · p. 3

B. Cooperative Localization

  • In GNSS-denied conditions, sUAV position estimates come from inter-UAV ranging, with yaw obtained from the IMU and treated as accurately measured.The numerical study and Gazebo implementation use the same unbiased EKF with distance-dependent ranging noise and sequential scalar range updates.Source: B. Cooperative Localization · p. 3
  • Each sUAV maintains at least two references in a multi-hop cascade, inheriting uncertainty from ranging noise and neighbor estimates; the triangular ladder references bUAVs or smaller-index sUAVs.The scalar uncertainty radius encloses the Mahalanobis-radius-3 ellipse, but is not itself an upper bound on estimator error.Source: B. Cooperative Localization · p. 4

C. uGdMS Problem Formulation

The formulation seeks decreasing uncovered area while maintaining localization connectivity and safety constraints. It introduces CBF and robust-CBF conditions for estimated states, with uncertainty-dependent tightening supporting forward invariance.

C. uGdMS Problem Formulation

  • The search objective is to drive the integral of the unsearched-point indicator over the region toward zero.The indicator is one where a point remains unsearched and zero once covered by at least one sUAV footprint.Source: C. uGdMS Problem Formulation · p. 4
  • The problem specifies N searching UAVs, at least two beacon UAVs, a search region, sensing and localization ranges, safe separation, and per-UAV uncertainty radii.Control inputs should decrease uncovered area while maintaining the stated conditions over time.Source: C. uGdMS Problem Formulation · p. 4
  • The formulation includes localization-connectivity maintenance using two assigned reference UAVs for each sUAV.The supplied passage identifies the references as α_i and β_i.Source: C. uGdMS Problem Formulation · p. 4

D. CBF and Robust CBF

  • A control barrier function defines a safe set as states where a continuously differentiable function is nonnegative and requires a class-K condition.The paper notes that CBFs have also been extended to high-order safety constraints through differentiable optimization.Source: D. CBF and Robust CBF · p. 4
  • For estimated rather than exact states, the robust-CBF construction tightens the barrier by a Lipschitz-scaled error bound and yields forward invariance when the standard CBF condition holds.The stated guarantee applies to initial estimated states within the tightened safe set.Source: D. CBF and Robust CBF · p. 4
  • For time-varying uncertainty, the barrier derivative includes the uncertainty-radius rate, and robust CBFs enforce connectivity and collision-avoidance constraints through uncertainty-dependent margins.The result depends on the radius and rate conditions holding.Source: D. CBF and Robust CBF · p. 4

IV. POSITION UNCERTAINTY PROPAGATION AND ROBUST CONSTRAINTS

This section approximates cascading position covariance from range measurements using Fisher information, then derives an uncertainty radius and its rate for robust constraint design.

A. Uncertainty Propagation

  • The range-only Fisher information matrix combines line-of-sight geometry with effective measurement variances that include neighbor-position error and sensor noise.The diagonal measurement covariance omits cross-correlations, including those from references sharing upstream localization ancestors.Source: A. Uncertainty Propagation · p. 5
  • The inverse FIM is a covariance approximation based on a Cramér–Rao lower bound, not an upper bound on estimator error or a guaranteed mission-wide envelope.The radius is a calibrated design choice; whether it remains valid across UAVs and time is assessed through closed-loop containment statistics.Source: A. Uncertainty Propagation · p. 5
  • Each UAV’s covariance depends recursively on its references’ information, so downstream sUAVs inherit upstream uncertainty as well as ranging noise.sUAVs referencing bUAVs depend only on sensor noise and geometry; farther sUAVs inherit upstream uncertainty.Source: A. Uncertainty Propagation · p. 5
  • The uncertainty radius varies with reference geometry, inherited neighbor covariances, and ranging-noise variance; its rate has a finite design bound under admitted-geometry and bounded-quantity conditions.The derivative bounds apply piecewise while the active reference set is unchanged; adding or removing a reference may cause jumps.Source: A. Uncertainty Propagation · p. 5

B. Cascading Localization Graph

The section specifies a directed localization graph and a triangular-ladder reference assignment that propagates localization from beacon UAVs into the search area.

B. Cascading Localization Graph

  • Every active localization graph must give each sUAV at least two incoming references and a directed path from a bUAV; those conditions alone do not ensure acyclicity.The adopted ordering supplies a DAG rooted at the bUAVs, including as visible references enter or leave.Source: B. Cascading Localization Graph · p. 6
  • The triangular ladder uses fixed CBF references to form overlapping triangular units, while optional range-visible references add FIM information without adding CBF edges.The figure shows an example with 3 bUAVs and 6 sUAVs.Source: B. Cascading Localization Graph · p. 6, Figure 4 · p. 6
  • Squads use ordered bUAV pairs to initialize their localization chains, and neighboring squads may share a bUAV.M = s + 1 bUAVs suffice for s squads, or M = s with cyclic sharing.Source: B. Cascading Localization Graph · p. 6
  • In the triangular ladder, the first sUAV references both bUAVs, the second references the first sUAV and one bUAV, and later sUAVs reference the two preceding UAVs.Each sUAV has two assigned CBF reference neighbors.Source: B. Cascading Localization Graph · p. 6
  • Table II illustrates assigned CBF references for two squads with seven sUAVs per squad, using global indices.Source: Table II · p. 6

C. Robust CBF Formulation

The section identifies localization connectivity and collision avoidance as the two safety-critical requirements addressed by uncertainty-aware CBF design.

C. Robust CBF Formulation

1) Localization-connectivity maintenance:

Uncertainty-tightened localization and collision CBFs account for position-error radii and their possible growth. Their safety guarantees are conditional on the envelope, rate, feasibility, and initial-barrier conditions.

1) Localization-connectivity maintenance:

  • The rate-aware localization row accounts for growth in both uncertainty radii; for a hovering beacon reference, its velocity is zero.The implementation uses γ as the class-K coefficient αc(h) = 0.1h.Source: 1) Localization-connectivity maintenance · p. 6

2) Collision avoidance:

  • Collision CBFs tighten the minimum-separation constraint by both UAVs’ uncertainty radii and their rates.The robust two-body barrier uses the relative-position error bound ϵi + ϵj.Source: 2) Collision avoidance · p. 7
  • True localization and collision distances remain within their design bounds when uncertainty envelopes and rates satisfy the stated conditions, local QPs are feasible, and tightened barriers start nonnegative.The conditional invariance statement applies to the continuous-time planar velocity-command model.Source: 2) Collision avoidance · p. 7
  • With a bounded envelope breach, localization distance may exceed dloc and collision separation may fall below dsafe by at most ¯ρ(ϵi + ϵj).Across the 20 deployments, the observed maximum breach ratio was ¯ρ ≈ 0.277; the reported per-pair localization degradation was at most about 11.9 m.Source: 2) Collision avoidance · p. 7

V. TASK-ORIENTED SEARCH STRATEGY

The search strategy assigns targets hierarchically: squad leaders partition the region, while followers form a triangular ladder toward the leader target. Distance and yaw objectives are separated for target approach and heading alignment.

A. Target Selection and Task CBFs

  • Squad leaders select targets within their Voronoi subregions, while followers form a triangular ladder toward the leader target.Only leaders participate in the Voronoi partition; each squad operates with its own partition.Source: A. Target Selection and Task CBFs · pp. 7–8
  • The alternating triangular ladder supports localization connectivity and forward coverage.Rotating the section vector creates alternating triangles, with adjacent squads using mirrored orientations.Source: A. Target Selection and Task CBFs · p. 8
  • Fig. 5 depicts leaders 7 and 14 partitioning targets for two squads, with followers arranged in the triangular ladder.The example contains 14 sUAVs and 3 bUAVs.Source: Figure 5 · p. 8, A. Target Selection and Task CBFs · p. 8

V. TASK-ORIENTED SEARCH STRATEGY

  • Distance and yaw task CBFs separate target approach from heading alignment because combining the objectives caused orbiting.The strategy is presented alongside hierarchical target selection and formation keeping.Source: §V · p. 7, A. Target Selection and Task CBFs · p. 8

1) Distance task CBF:

Distance and yaw task CBFs guide target approach and heading alignment, while a distributed QP integrates task objectives with localization and collision constraints. Constraint rows are hard; task objectives can be softened with slack.

1) Distance task CBF:

  • The distance task CBF encourages movement toward the assigned target and increases toward zero as the UAV approaches it.The distance task gain kp is positive.Source: 1) Distance task CBF · p. 8

2) Yaw task CBF:

  • The yaw task CBF uses the target azimuth as the desired heading and reaches zero when the UAV heading aligns with it.The yaw barrier is negative under heading misalignment.Source: 2) Yaw task CBF · p. 8

B. CBF Integration

  • The distributed controller gives each UAV an independent local QP based on exchanged estimates, scaling with its local neighborhood rather than a centralized stacked system.Selected localization and collision rows remain hard, while task CBFs are softened; infeasible local problems may be re-solved jointly with up to three most-violated partners.Source: B. CBF Integration · pp. 8–9

VI. NUMERICAL SIMULATIONS

The numerical study uses an unbiased EKF, sequential range updates, and a calibrated covariance contribution to drive tightened CBF rows. The calibration choices are not deterministic error bounds, and the filter can coast when no measurement is accepted.

A. Estimator Implementation

  • The distance-dependent ranging-noise model uses σr,0 = 0.5 m and d0 = 850 m, while κ calibrates the anchor-covariance contribution for omitted shared-ancestor correlation.The filter applies available range references as sequential scalar updates.Source: A. Estimator Implementation · p. 9
  • The EKF initializes from known deployment positions, processes references in increasing local-index order, and runs in about 0.5 ms per frame.It uses 3σ innovation gating and coasts on the prediction when no measurement is accepted.Source: A. Estimator Implementation · p. 9
  • The coefficient 3 in ϵi = 3√λmax(Σi) and κ = 3 are three-sigma calibration choices, not deterministic error bounds.On the same trajectories, pooled containment was 97.9% for κ = 2, 99.95% for κ = 3, and 99.999% for κ = 4.Source: A. Estimator Implementation · p. 9

B. Simulation Setup

The simulations evaluate search and safety control under stated setup constraints, then compare the estimator-in-the-loop run with a perfect-localization baseline and ablated variants. The full method completes coverage while maintaining feasible safety and communication constraints in the representative run.

B. Simulation Setup

  • The comparison and scalability runs use static margins without explicit U_i and provide coverage and completion evidence only.The numerical configuration uses a 3000 × 3000 m region and solves the QP with Gurobi.Source: B. Simulation Setup · p. 9

C. Performance Analysis

  • 302.5 s completes the estimator-in-the-loop search, 1.0 s later than the perfect-localization configuration, with every safety and communication row feasible at every frame.Coverage reaches 25%, 50%, 75%, and 100% at 50.5, 128.0, 173.5, and 302.5 s.Source: C. Performance Analysis · pp. 9–10, Figure 7 · p. 10
  • Localization-error containment holds in every representative-run frame, with RMSE 1.42 m and median, p95, and maximum errors of 0.9, 2.9, and 5.9 m.The FIM approximation is conservative in 67.6% of UAV-frames and optimistic in the rest; it maintains safe true-distance margins even in optimistic frames.Source: Figure 9 · p. 10, C. Performance Analysis · p. 10
  • The covariance radius grows along the localization chain, with UAV 7 having the largest radius in Squad 1; the run completes without triggering a constraint-violation check.Source: C. Performance Analysis · p. 10
  • The full uGdMS variant reaches 100% coverage; Pure CVT fails at 20.0%, conventional CBF violates localization range at 47.3%, and CVT+Formation reaches 42.4%.Pure CVT fails at 24 s and conventional CBF at 118 s; the reported violation is a true localization-link distance exceeding dloc.Source: Table V · p. 11, C. Performance Analysis · p. 11

D. Statistical Validation and Scalability

Twenty estimator-in-the-loop deployments complete coverage with feasible local QPs, while polygon cases illustrate coverage behavior across region shapes and swarm sizes. Six MBZIRC simulation runs also complete coverage under quadrotor flight dynamics; physical-platform experiments and 3D search under disturbances remain future work.

D. Statistical Validation and Scalability

  • Across the Monte Carlo runs, no true-distance safety or localization violation occurs: the maximum assigned-link distance is 846.8 m and the minimum inter-UAV distance is 12.25 m.The 103 containment-breach UAV-frames retain at least 165.9 m of link margin below dloc and 116.1 m above dsafe.Source: D. Statistical Validation and… · p. 11
  • All 20 Monte Carlo runs reach 100% coverage in 301.0–330.0 s, with all 9,800 local QP commands per run feasible.Containment holds in 98.97–100% of UAV-frames per run, with a median of 100%.Source: D. Statistical Validation and… · p. 11, Table VI · p. 12
  • Polygon examples cover triangular, trapezoidal, and hexagonal regions with 6–30 sUAVs and 3 bUAVs, but provide coverage evidence only; the 6-sUAV case cannot reach the farthest region within communication range.Source: D. Statistical Validation and… · p. 12, Figure 14 · p. 12, Figure 15 · p. 12

VII. GAZEBO IMPLEMENTATION EVIDENCE

  • Six MBZIRC runs complete coverage in 466 ± 1 s, with all local QPs optimal and no safety violation; the worst maximum localization link is 849.1 m.The representative run maintains localization-error containment in all 13,580 UAV-frames.Source: Table VII · p. 12, §VII · p. 12

VIII. CONCLUSION

  • The paper reports that Propositions 1–2 are supported by Monte Carlo containment and true-distance statistics; physical-platform experiments and 3D search under disturbances are future work.Source: §VIII · pp. 13–14

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