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Logistic-aided Huber M-estimator for robust GNSS positioning

Zhengdao Li, Penggao Yan, Li-Ta Hsu

Abstract

This paper develops a logistic-aided Huber (LAH) M-estimator for robust GNSS positioning under long-tailed, multipath-affected measurement errors. The key idea is to leverage a logistic measurement error assumption and establish a one-to-one approximation between the logistic-based loglikelihood (i.e., quasi-log-cosh) and the Huber kernel by matching their score functions. This yields closed-form tuning rules for the scale and threshold parameters in the Huber estimator, grounded on logistic error statistical properties. We further show that the proposed LAH estimator preserves comparable efficiency and robustness to the connected logistic-based least quasi-log-cosh (LQLC) estimator. Both Monte Carlo simulations with long-tailed measurement errors and a one-hour urban GNSS dataset confirm that the proposed logistic-statistics-based tuning improves positioning accuracy and precision while suppressing large error spikes. Specifically, LAH reduces the 2D RMSE/STD by 28.03%/38.83% versus conventional 95%-efficiency-based Huber tuning in simulation, and reduces the overall 3D RMSE/STD by 4.85%/16.68% in real-world experiments while suppressing large positioning error spikes by up to 51%.

Logistic-aided Huber M-estimator for robust GNSS positioning

Abstract

This paper develops a logistic-aided Huber (LAH) M-estimator for robust GNSS positioning under long-tailed, multipath-affected measurement errors. The key idea is to leverage a logistic measurement error assumption and establish a one-to-one approximation between the logistic-based loglikelihood (i.e., quasi-log-cosh) and the Huber kernel by matching their score functions. This yields closed-form tuning rules for the scale and threshold parameters in the Huber estimator, grounded on logistic error statistical properties. We further show that the proposed LAH estimator preserves comparable efficiency and robustness to the connected logistic-based least quasi-log-cosh (LQLC) estimator. Both Monte Carlo simulations with long-tailed measurement errors and a one-hour urban GNSS dataset confirm that the proposed logistic-statistics-based tuning improves positioning accuracy and precision while suppressing large error spikes. Specifically, LAH reduces the 2D RMSE/STD by 28.03%/38.83% versus conventional 95%-efficiency-based Huber tuning in simulation, and reduces the overall 3D RMSE/STD by 4.85%/16.68% in real-world experiments while suppressing large positioning error spikes by up to 51%.
Paper Structure (11 sections, 32 equations, 5 figures, 2 tables)

This paper contains 11 sections, 32 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Logistic-aided Huber (LAH) M-estimator
  3. Logistic error model and LQLC estimator
  4. Approximate relationship between Huber and QLC loss function
  5. Theoretical validation on the approximation
  6. Efficiency similarity
  7. Robustness similarity
  8. Experimental results
  9. Monte Carlo simulation
  10. Real-world experiment
  11. Conclusion and future work

Figures (5)

  • Figure 1: QLC (with unit scale parameter) and the approximated LAH kernel in view of (a) loss function and (b) score function.
  • Figure 2: Efficiency analysis for LQLC and LAH estimators: (a) the change in efficiency of both estimators when the logistic scale parameter $s$ increases from 0.0 to 3.0, and (b) ARE of LQLC estimator compared to LAH estimator.
  • Figure 3: Robustness analysis for LQLC and LAH estimators: (a) the change in residual GES ($\gamma_{res}$) of both estimators when the logistic scale parameter $s$ increases from 0.0 to 6.0, and (b) relative GES of LQLC estimator compared to LAH estimator.
  • Figure 4: Monte Carlo simulation results: (a) true and maximum-likelihood-estimated scale parameters for measurements from the 8 anchors; (b) 2D positioning in X-Y coordinate; (c) 2D positioning error throughout the $1\times 10^5$ experiments; (d) empirical CDF of 2D positioning errors.
  • Figure 5: Real-world experiment results: (a) elevation-dependent scale models based on Gaussian and logistic distributions; (b) 2D positioning in longitude-latitude coordinate; (c) 3D positioning error throughout the $1\times 10^5$ experiments; (d) empirical CDF of 3D positioning errors.