PaperlayerView exact revision on arXiv

Source-linked AI summary

NLOS-Aware TDOA-Based UAV Localization: Cram\'er-Rao Bounds, Measurement Modeling, and Experimental Validation

Cole Dickerson, Saad Masrur, Özgür Özdemir, Ismail Güvenç

arXiv:2610.30053v1eess.SP

TL;DR

Non-cooperative UAVs may not report their positions, and NLOS biases can leave practical TDOA localizers biased. The paper develops NLOS-aware bounds and a calibrated measurement model, finding that LOS sensor count strongly tracks median error and that bias can leave measured errors above the bounds.

  • Problem

    Non-cooperative UAVs may not report their positions, while localizers that ignore NLOS biases are biased and have an error floor.

    Source: §I · pp. 1–2

  • Method

    The paper develops random-bias, deterministic, and hybrid NLOS-aware CRLBs and an empirically calibrated TDOA measurement model evaluated with five AERPAW flights.

    Source: §I · p. 1, §IX · p. 13

  • Results

    Median horizontal error rose from 33 m with three LOS towers to 510 m with one, while measured error remained tens of meters above the bound and barely decreased from 2.5 MHz to 5 MHz.

    Source: A. Effect of Altitude, Bandwidth, and… · p. 9, §IX · p. 13

  • Takeaways & Limitations

    Outlier rejection lowered pooled 90th-percentile error from 850 m to 99 m, whereas Kalman and particle filtering did not lower it further because residual error was dominated by bias.

    Source: §I · p. 2

    The bandwidth error-floor prediction assumes fixed biases; wider bandwidth may separately reduce biases by resolving multipath.

    Source: D. Bias-Induced Error Floor · p. 5

Abstract

from arXiv · show

Passive time-difference-of-arrival (TDOA) localization is well suited for tracking non-cooperative unmanned aerial vehicles (UAVs) that do not report their positions. However, its accuracy degrades under the mixed line-of-sight (LOS) and non-line-of-sight (NLOS) propagation that is typical of low-altitude flight. In this paper, we develop NLOS-aware Cram\'er--Rao lower bounds (CRLBs) and an empirically calibrated measurement model for TDOA-based UAV localization, and we evaluate both with measurements from five UAV flights on the Aerial Experimentation and Research Platform for Advanced Wireless (AERPAW). The analysis yields an identifiability condition that relates the number of sensors to the number of NLOS links, together with a bias-induced error floor that a wider bandwidth cannot remove. The model parameters are estimated by matching simulated and measured error distributions. Experimental results show that the error depends most strongly on the number of sensors in LOS, while a higher altitude and a wider bandwidth lower the median error. Relocating a sensor that is always in NLOS into the open flight area can lower the median horizontal bound by two orders of magnitude if the relocated sensor is in LOS. The fitted model reproduces the median errors of the 2.5 MHz and 5 MHz flights but underestimates the heaviest error tails. Finally, rejecting outlier fixes lowers the error far more than Kalman or particle filtering, because the remaining error is dominated by bias.

I. INTRODUCTION

Passive TDOA can locate non-cooperative UAVs without their reported positions, but NLOS links add excess-delay bias that degrades accuracy. This paper extends prior LOS-focused analysis with NLOS-aware bounds and a measurement-validated error model for aerial TDOA localization.

I. INTRODUCTION

  • NLOS-aware CRLB analysis evaluated against aerial TDOA measurements, together with a model reproducing measured errors, had not been reported.The earlier AERPAW evaluation did not model NLOS bias or the measured error distribution, and did not process position estimates.Source: §I · p. 1
  • The paper develops random-bias, deterministic-bias, and hybrid NLOS-aware CRLBs for 3D TDOA localization.The hybrid bound equals the random-bias bound when its prior matches bias statistics and tends toward the deterministic bound as the prior becomes uninformative.Source: §I · p. 1
  • The deterministic bound is finite only when sensors exceed NLOS links by at least four, or by three when altitude is known; knowing altitude is equivalent to one additional LOS sensor.For the AERPAW deployment, position and biases cannot be separated at any point in 3D.Source: §I · pp. 1–2
  • A practical localizer that ignores NLOS biases is biased, with an error floor that wider bandwidth lowers neither.The position offset depends to first order on the biases and sensor geometry, not bandwidth.Source: §I · p. 2
  • The fitted measurement model reproduces median errors of four 2.5 MHz and 5 MHz flights within 25%, while deviations in error tails and at 1.25 MHz indicate uncaptured effects.The model combines correlated shadowing, attitude and antenna-pattern fluctuations, heavy-tailed NLOS excess delay, mobility noise, and detection outages.Source: §I · p. 2

A. Localization Bounds and LOS TDOA Methods

C. UAV Localization Experiments

Prior UAV localization experiments did not compare measured error with a localization bound or model NLOS bias. This paper's scope combines those elements with real measurements and an error model for a deployed ground TDOA network localizing UAVs in 3D.

C. UAV Localization Experiments

  • Prior UAV localization experiments neither compared measured error with a localization bound nor modeled NLOS bias.This paper combines NLOS-aware CRLB analysis with a measurement-validated error model for a deployed ground TDOA network localizing UAVs in 3D.Source: C. UAV Localization Experiments · p. 2

A. TOA and TDOA Measurements

  • TDOA range differences cancel the unknown transmit time, using arrival-time differences between synchronized sensors.The measurement model includes range bias on NLOS links and ranging noise.Source: A. TOA and TDOA Measurements · p. 3
  • The TDOA measurements share reference-sensor noise, which correlates all measurements through a common term.The model forms range differences relative to a reference sensor.Source: A. TOA and TDOA Measurements · p. 3

B. LOS Cram´er–Rao Lower Bound

The LOS CRLB provides a baseline for analyzing TDOA localization bounds. Extending it to NLOS links requires accounting for both attenuation-driven noise and positive excess-delay bias, with the bound depending on what is known about those biases.

B. LOS Cram´er–Rao Lower Bound

  • The LOS CRLB bounds the error covariance of any unbiased position estimator using the inverse Fisher information matrix.The analysis neglects information carried by the position dependence of the measurement covariance.Source: B. LOS Cram´er–Rao Lower Bound · p. 3
  • With known altitude, the horizontal bound estimates only horizontal position using the first two Jacobian columns.In the stated setup, poor vertical geometry can make the horizontal bound with jointly estimated altitude much larger than the known-altitude bound.Source: B. LOS Cram´er–Rao Lower Bound · p. 4

IV. NLOS-AWARE CRLBS AND THE BIAS-INDUCED ERROR FLOOR

  • An NLOS link both attenuates received power, increasing ranging noise, and adds a positive excess-delay bias.The bounds assume each link's LOS state is known; random, deterministic-unknown, and hybrid bias models represent different levels of bias knowledge.Source: §IV · p. 4

A. Model A: Random-Bias CRLB

The random-bias bound incorporates NLOS bias uncertainty and attenuated SNR in the TDOA covariance. The deterministic model treats each NLOS bias as an unknown parameter, yielding an identifiability condition based on the number of LOS sensors.

A. Model A: Random-Bias CRLB

  • The random-bias bound combines NLOS bias uncertainty and attenuated-link SNR in the TDOA covariance, and is finite for non-degenerate geometry.For non-Gaussian biases with the same variances, it bounds the exact CRLB from above.Source: A. Model A: Random-Bias CRLB · p. 4

B. Model B: Deterministic Unknown Biases

  • Each unknown NLOS bias adds a parameter, so the deterministic bound is finite only when enough LOS sensors constrain position.The condition is sufficient for non-degenerate geometry, including when any three LOS-sensor Jacobian rows are linearly independent.Source: B. Model B: Deterministic Unknown Biases · p. 4
  • 3D localization requires at least four LOS sensors; when this holds, the deterministic bound equals the LOS-only bound.Excluding NLOS sensors loses no information but cannot restore identifiability.Source: B. Model B: Deterministic Unknown Biases · pp. 4–5
  • With known altitude, the identifiability condition becomes N ≥ 3 + K_NLOS, making known altitude equivalent to one additional LOS sensor.Source: B. Model B: Deterministic Unknown Biases · p. 5

C. Hybrid Bound

A Gaussian prior on unknown NLOS biases gives a hybrid bound connecting random- and deterministic-bias models. The paper also shows that fixed biases create an error floor that wider bandwidth does not remove.

C. Hybrid Bound

  • The hybrid bound uses a Gaussian prior on each unknown NLOS bias and connects the random- and deterministic-bias bounds.When the prior matches the bias statistics it equals the random-bias bound; as the prior becomes uninformative it tends to the deterministic bound.Source: C. Hybrid Bound · p. 5

D. Bias-Induced Error Floor

  • For a localizer that ignores NLOS biases, the MSE separates into a variance term and a bias term determined by biases, geometry, and relative link weights.The variance term is the CRLB evaluated with attenuated NLOS SNRs.Source: D. Bias-Induced Error Floor · p. 5
  • With fixed biases, the bias-induced position offset does not depend on bandwidth, so increasing bandwidth cannot remove the error floor.The result applies to the ML-weighted estimator under the stated bandwidth and fixed-bias assumptions.Source: D. Bias-Induced Error Floor · p. 5
  • Doubling bandwidth reduces the variance contribution by a factor of 2, while leaving the bias term unchanged under the stated per-sample SNR model.Source: D. Bias-Induced Error Floor · p. 5
  • If the receiver integrates over fixed time, the variance contribution instead halves per bandwidth doubling; resolving multipath may separately reduce the biases.Source: D. Bias-Induced Error Floor · p. 5

V. MEASUREMENT MODEL AND POSITION ESTIMATION

The measurement model generates synthetic range measurements along measured UAV trajectories for parameter estimation against measured errors. It represents link conditions, time-varying propagation and attitude effects, NLOS biases, mobility noise, and detection outages.

V. MEASUREMENT MODEL AND POSITION ESTIMATION

  • The per-measurement model generates synthetic range measurements along measured trajectories for comparison with measured errors.The position estimator operates on those measurements, and model parameters are estimated from measured errors in the following section.Source: §V · p. 5

A. Per-Measurement Signal and Range Model

  • Received SNR includes free-space link budget and receiver noise figure, with correlated shadowing and attitude-dependent antenna-pattern fluctuations.Shadowing correlation is higher during hover and smaller in transit; attitude-related variation is also smaller in hover.Source: A. Per-Measurement Signal and Range… · p. 6
  • NLOS links have log-normal ranging bias, while quasi-static multipath bias and transit mobility noise also contribute to range error.The two biases are frozen during hover and redrawn independently at each transit index.Source: A. Per-Measurement Signal and Range… · p. 6
  • A sensor is detected only above the SNR threshold, and no position estimate is produced when fewer than three sensors are detected.The model therefore produces nearly constant error offset in hover and fluctuating error in transit.Source: A. Per-Measurement Signal and Range… · p. 6

B. TDOA Position Estimation

The position estimator solves for horizontal position using detected sensors, known altitude, and TDOA measurements. It discards solutions whose normalized residual exceeds a threshold.

B. TDOA Position Estimation

  • The estimator operates in 2D with known altitude and uses the highest-SNR detected sensor as the TDOA reference.It solves for horizontal position with the Levenberg–Marquardt algorithm.Source: B. TDOA Position Estimation · p. 6
  • Table III lists fitted parameters for raw and gated measured errors alongside fixed parameters; hover and transit values are separated by a slash.Source: Table III · p. 6
  • A solution is discarded when its residual norm divided by the number of TDOAs exceeds the threshold.This gate models the geolocation software's consistency check.Source: B. TDOA Position Estimation · p. 6

VI. MODEL PARAMETER ESTIMATION AND POST-PROCESSING

The section estimates measurement-model parameters from measured errors and describes post-processing of position fixes. For parameter fitting, simulated and measured error distributions are matched across the flights.

VI. MODEL PARAMETER ESTIMATION AND POST-PROCESSING

  • Post-processing first removes gross outliers and then smooths the retained position fixes.The two stages address fixes hundreds of meters from truth and smaller fix-to-fix scatter, respectively.Source: §VI · p. 6

A. Measurement-Model Parameter Estimation by Error-Distribution Matching

  • The fitted seven parameters include site- and hardware-specific quantities that cannot be measured directly from the released position-estimate dataset.Other parameters are fixed at typical values to keep the number of unknowns small; the detection threshold enters only through its sum with the estimated NF.Source: A. Measurement-Model Parameter… · p. 6
  • A simulation-based minimum-distance estimator fits the model because the likelihood is intractable, matching simulated and measured error distributions rather than a few moments.The error is heavy-tailed, so its moments can be dominated by a few extreme errors.Source: A. Measurement-Model Parameter… · pp. 6–7
  • For each flight, the fit simulates the model and estimator on the actual geometry, trajectory, and LOS states, then uses Nelder–Mead updates to minimize the cost.The simulations use two replicates per flight with fixed seeds, making the cost deterministic in the parameter vector.Source: Figure 2 · p. 7, A. Measurement-Model Parameter… · p. 7
  • The cost combines the two-sample KS distance between measured and modeled error CDFs with a valid-estimate fraction term, weighted by η = 0.6.The KS distance is evaluated over 220 logarithmically spaced error values from 1 m to 3.2 km; the valid-fraction term constrains NF and τres.Source: A. Measurement-Model Parameter… · p. 7

B. Post-Processing: Outlier Rejection and Filtering

The section describes position-fix rejection and filtering, the AERPAW flight dataset, and the stated settings for evaluating localization bounds. The supplied results material identifies per-flight error metrics but does not include their values.

B. Post-Processing: Outlier Rejection and Filtering

  • Outlier rejection uses a CEP threshold and a speed gate; retained fixes are then processed with constant-velocity Kalman, random-walk Kalman, or random-walk particle filtering.The speed gate rejects a fix when its distance from the last accepted fix divided by elapsed time exceeds vmax.Source: B. Post-Processing: Outlier Rejection… · pp. 7–8

VII. EXPERIMENTAL SETUP AND DATASET

  • The experiments used four GPS-synchronized Keysight RF sensors on towers LW2–LW5 and a UAV-mounted USRP B205mini transmitting a 3.32 GHz channel-sounding waveform.The setup used a known waveform and known altitude, which the paper characterizes as favorable for passive localization of non-cooperative UAVs.Source: §VII · p. 8
  • The five flights, each flown once, cover altitudes of 40 m, 70 m, and 100 m and bandwidths of 1.25 MHz, 2.5 MHz, and 5 MHz.The common waypoint path includes hovers at B, D, E, and G; the system produces a fix about every 4 s.Source: §VII · p. 8

VIII. EXPERIMENTAL AND NUMERICAL RESULTS

  • The table reports measured horizontal localization error per flight for valid fixes, including the 90th percentile.The supplied caption gives the metrics and scope but no table values.Source: Table IV · p. 8

VII. EXPERIMENTAL SETUP AND DATASET

  • Tower LW5 was in NLOS throughout every flight, while LW1 had no sensor and was used in the sensor-placement study.The link states were derived by ray tracing modeled buildings and trees along the trajectories.Source: §VII · p. 8

VIII. EXPERIMENTAL AND NUMERICAL RESULTS

  • Bounds are evaluated at 5 MHz using measured trajectories, actual sensor positions, per-link LOS states, and fitted NLOS parameters; the random-bias bound uses σb,i = 54 m.The paper states this bias choice can only lower the bound because the fitted log-normal bias has a larger spread.Source: §VIII · p. 8

A. Effect of Altitude, Bandwidth, and LOS Conditions

Measured localization error is most strongly associated with the number of LOS towers; altitude and bandwidth also affect the median. The bounds explain how LOS availability, sensor placement, and NLOS bias shape identifiability and residual error.

B. NLOS CRLBs, Identifiability, and Sensor Placement

  • 0.2 m with three LOS towers versus 53–70 m with fewer: the altitude-known random-bias bound is bimodal because uncertain NLOS links affect position when fewer than three towers are in LOS.The median vertical bound is 0.8–2.6 km when altitude is unknown, raising the median horizontal bound to 123–146 m.Source: B. NLOS CRLBs, Identifiability, and… · p. 9, Table V · p. 9, Figure 7 · p. 10
  • 45%: with altitude known, the deterministic-bias bound is finite at only 45% of trajectory points; it is unbounded everywhere for the deployed 3D network.One and two additional LOS sensors raise the 3D finite-bound fraction from zero to 45% and 91%, respectively; known altitude has the effect of one additional sensor.Source: B. NLOS CRLBs, Identifiability, and… · pp. 9–10, Figure 8 · p. 11

A. Effect of Altitude, Bandwidth, and LOS Conditions

B. NLOS CRLBs, Identifiability, and Sensor Placement

  • 38 m to 0.24 m: replacing always-blocked LW5 with assumed-LOS LW1 lowers the 40 m median horizontal bound by over two orders of magnitude.The 90th percentile falls from 83 m to 1.3 m; the gain is optimistic because LW1's field-wide LOS state was not ray traced.Source: B. NLOS CRLBs, Identifiability, and… · p. 9, Figure 7 · p. 10

C. Per-Measurement Error and the Bias Floor

  • Tens of meters: measured errors remain far above the sub-1 m bound at points with three LOS towers, consistent with a bias-induced floor.With equal TDOA weights, a 54 m LW5 bias predicts a 19–27 m median offset, comparable to the measured 28–40 m at 2.5 MHz and 5 MHz.Source: C. Per-Measurement Error and the Bias… · p. 10, Figure 9 · p. 11
  • 59 m to 52 m: the 40 m median error falls only modestly from 2.5 MHz to 5 MHz, while mean and 90th-percentile errors do not decrease with bandwidth.The bias floor can be reduced by excluding or de-weighting NLOS links or improving sensor geometry.Source: C. Per-Measurement Error and the Bias… · pp. 10–11
  • 32–34 m: at waypoint E, hover errors persist across three 5 MHz flights despite only about a 6 m predicted offset from the LW5 bias.A static 40–55 m bias on a LOS link could produce the observed error, but is improbable under the fitted model, pointing to an unmodeled site-specific effect.Source: C. Per-Measurement Error and the Bias… · p. 11

D. Measurement-Model Fit and Parameter Sensitivity

Outlier rejection substantially reduces pooled errors, while position filtering adds little because residual error is bias-dominated. The fitted measurement model matches typical errors in the 2.5 MHz and 5 MHz flights but misses the heaviest tails and narrowband degradation.

D. Measurement-Model Fit and Parameter Sensitivity

  • Within 25%: modeled median errors match measurements for the four 2.5 MHz and 5 MHz flights, but the model underestimates the 90th percentile at 40 m and 70 m by factors of 2.2–4.4.At 1.25 MHz, modeled median error is 77 m versus 117 m measured; the model lacks gross-error mechanisms and narrowband degradation.Source: D. Measurement-Model Fit and Parameter… · p. 11, Figure 10 · p. 12
  • 14%: refitting after outlier rejection shifts median NLOS bias from 54 m to 62 m, while σlog, NF, and τres each change by about 40%.The latter parameters trade off with one another and with the missing CEP gate.Source: D. Measurement-Model Fit and Parameter… · p. 11

E. Outlier Rejection and Filtering

D. Measurement-Model Fit and Parameter Sensitivity

  • 18 m to 121 m: raising modeled NLOS bias from 10 m to 120 m increases the modeled median error; excess loss and mobility noise change the median by less than 15%.A fixed bias of µb lowers the modeled 90th percentile from 258 m to 200 m, still below the measured 1132 m.Source: D. Measurement-Model Fit and Parameter… · p. 12, Figure 10 · p. 12

E. Outlier Rejection and Filtering

  • 30 m: weak RW Kalman smoothing lowers the post-rejection median from 33 m but raises the 90th percentile from 99 m to 102 m; stronger smoothing raises both errors.The particle filter raises the 90th percentile to 771 m; remaining error is dominated by bias, which position-domain filtering cannot remove.Source: E. Outlier Rejection and Filtering · p. 13

IX. CONCLUSION

  • The model reproduces typical errors but underestimates the heaviest tails; the study also identifies one flight per configuration and unavailable TDOAs as limitations.The conclusion calls for validation on more flights and sites and modeling gross errors and narrowband degradation.Source: §IX · p. 13

APPENDIX A PROOF OF PROPOSITION 1

The appendix proves the deterministic-bound identifiability condition by reducing the augmented Jacobian's rank to that of the LOS measurement rows. It also derives the bias-induced WLS error floor and its independence from bandwidth.

APPENDIX A PROOF OF PROPOSITION 1

  • Full column rank of the augmented Jacobian requires at least three LOS rows in non-degenerate geometry when the reference sensor is LOS.Eliminating the NLOS bias columns leaves the LOS position rows; their rank must be three.Source: Appendix A · p. 13
  • With an NLOS reference, full rank requires N − KNLOS ≥ 4; the LOS rows must determine three coordinates and the reference bias offset.The proof eliminates other NLOS rows using their own free bias columns.Source: Appendix A · p. 13
  • When the rank condition fails, a joint position-and-bias perturbation leaves every TDOA mean unchanged, so no unbiased estimator has finite variance.With known altitude, the rank requirement drops to two and the condition becomes N ≥ 3 + KNLOS.Source: Appendix A · p. 13

APPENDIX B PROOFS OF PROPOSITION 2 AND COROLLARY 1

  • The WLS error separates into a deterministic bias offset and zero-mean noise; its mean-squared error is at least the squared bias offset.For ML weights, increasing bandwidth lowers the covariance term but does not change the bias offset.Source: Appendix B · pp. 13–14
Loading 2610.30053v1…