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Gaussian Variational Inference with Non-Gaussian Factors for State Estimation: A UWB Localization Case Study

Andrew Stirling, Mykola Lukashchuk, Dmitry Bagaev, Wouter Kouw, James R. Forbes

TL;DR

The paper addresses robust state estimation for robotic localization under non-Gaussian noise by extending ESGVI to matrix Lie group states and introducing non-Gaussian (Skew-Laplace) measurement factors. It demonstrates that an ESGVI approach operating on SE(2) with skewed noise yields competitive translation and orientation accuracy while providing principled uncertainty estimates, validated on both simulated and real UWB data. The work compares against robust MAP baselines, showing comparable RMSE and promising uncertainty calibration, and offers an open-source Python implementation for broader adoption. Overall, the contributions enable principled, uncertainty-aware localization in environments with NLOS and multipath effects, at the cost of higher computation which is a target for future acceleration.

Abstract

This letter extends the exactly sparse Gaussian variational inference (ESGVI) algorithm for state estimation in two complementary directions. First, ESGVI is generalized to operate on matrix Lie groups, enabling the estimation of states with orientation components while respecting the underlying group structure. Second, factors are introduced to accommodate heavy-tailed and skewed noise distributions, as commonly encountered in ultra-wideband (UWB) localization due to non-line-of-sight (NLOS) and multipath effects. Both extensions are shown to integrate naturally within the ESGVI framework while preserving its sparse and derivative-free structure. The proposed approach is validated in a UWB localization experiment with NLOS-rich measurements, demonstrating improved accuracy and comparable consistency. Finally, a Python implementation within a factor-graph-based estimation framework is made open-source (https://github.com/decargroup/gvi_ws) to support broader research use.

Gaussian Variational Inference with Non-Gaussian Factors for State Estimation: A UWB Localization Case Study

TL;DR

The paper addresses robust state estimation for robotic localization under non-Gaussian noise by extending ESGVI to matrix Lie group states and introducing non-Gaussian (Skew-Laplace) measurement factors. It demonstrates that an ESGVI approach operating on SE(2) with skewed noise yields competitive translation and orientation accuracy while providing principled uncertainty estimates, validated on both simulated and real UWB data. The work compares against robust MAP baselines, showing comparable RMSE and promising uncertainty calibration, and offers an open-source Python implementation for broader adoption. Overall, the contributions enable principled, uncertainty-aware localization in environments with NLOS and multipath effects, at the cost of higher computation which is a target for future acceleration.

Abstract

This letter extends the exactly sparse Gaussian variational inference (ESGVI) algorithm for state estimation in two complementary directions. First, ESGVI is generalized to operate on matrix Lie groups, enabling the estimation of states with orientation components while respecting the underlying group structure. Second, factors are introduced to accommodate heavy-tailed and skewed noise distributions, as commonly encountered in ultra-wideband (UWB) localization due to non-line-of-sight (NLOS) and multipath effects. Both extensions are shown to integrate naturally within the ESGVI framework while preserving its sparse and derivative-free structure. The proposed approach is validated in a UWB localization experiment with NLOS-rich measurements, demonstrating improved accuracy and comparable consistency. Finally, a Python implementation within a factor-graph-based estimation framework is made open-source (https://github.com/decargroup/gvi_ws) to support broader research use.
Paper Structure (16 sections, 32 equations, 5 figures, 1 table)

This paper contains 16 sections, 32 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Background and Related Work
  4. Probabilistic Model Specification
  5. State of the Art
  6. Gaussian Variational Inference
  7. ESGVI with Matrix Lie Group States
  8. Simulation and Experimental Results
  9. Simulation Results
  10. Experimental Setup
  11. Experimental Results
  12. Limitations
  13. Conclusion
  14. Appendix
  15. Matrix Lie Groups
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Depiction of obstacle-induced signal propagation effects that lead to errors in range-based localization.
  • Figure 2: Experimental setup showing the Clearpath Husky platform equipped with four UWB tags operating in a cluttered environment with four fixed UWB anchors. UWB modules are circled in blue, and obstacles are outlined in red.
  • Figure 3: Example trajectory (Trajectory 6) driven by the Husky through a cluttered environment, comparing estimator outputs against ground-truth data recorded by a motion capture system.
  • Figure 4: Calibrated UWB range error histogram across all trajectories with fitted noise models.
  • Figure 5: Distribution of orientation RMSE, translation RMSE, and aNEES across 10 experimental trajectories for each estimator. The consistency bounds for the aNEES are computed using the median trajectory length.