Loosely Coupled Factor Graph Optimization for Pseudolite-Augmented Navigation

Chih-Chun Chen; Heike Vallery; Lipeng Tan; Shiyu Bai

arxiv: 2605.24980 · v1 · pith:CFOPGKK5new · submitted 2026-05-24 · 💻 cs.RO

Loosely Coupled Factor Graph Optimization for Pseudolite-Augmented Navigation

Chih-Chun Chen , Lipeng Tan , Shiyu Bai , Heike Vallery This is my paper

Pith reviewed 2026-06-30 00:46 UTC · model grok-4.3

classification 💻 cs.RO

keywords factor graph optimizationpseudolitesGNSSIMU fusionpositioning accuracynavigationleast-squares methods

0 comments

The pith

Loosely coupled factor graph optimization fuses GNSS and pseudolite least-squares solutions with IMU data to cut mean 3D positioning error 22.8 to 41.3 percent versus standard least-squares methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a loosely coupled factor graph optimization framework that combines least-squares position solutions from four high-elevation GNSS satellites and up to two pseudolite transmitters with inertial measurement unit data. It evaluates performance in low-visibility scenarios over 80-second windows. A sympathetic reader would care because the approach demonstrates measurable error reductions and additional gains from pseudolite geometry without requiring tight sensor integration. The work shows the framework outperforms both plain least-squares and a GNSS-IMU baseline under the tested conditions.

Core claim

The paper claims that its loosely coupled factor graph optimization, which treats GNSS and pseudolite least-squares solutions as factors alongside IMU measurements, achieves a 22.8% to 41.3% reduction in mean 3D error compared to standard least-squares methods. It further claims that adding pseudolite transmitters improves accuracy over a GNSS-IMU baseline, with the degree of improvement depending on transmitter geometry.

What carries the argument

Loosely coupled factor graph optimization that incorporates least-squares position estimates as factors with IMU data.

If this is right

Positioning error drops in GNSS-degraded settings with limited satellite visibility.
Pseudolite addition yields further accuracy gains beyond GNSS-IMU fusion alone.
The size of the improvement varies with the geometric arrangement of the pseudolites.
The method maintains its advantage across an 80-second time window.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same loose coupling structure could be tested with additional pseudolites or other ranging sources.
Pseudolite placement planning based on geometry might be used to maximize the observed gains.
The framework may apply to urban or indoor navigation where satellite signals are obstructed.

Load-bearing premise

The least-squares solutions computed from four GNSS satellites plus up to two pseudolites remain sufficiently accurate and unbiased inputs for the factor graph to correct over the 80 s window.

What would settle it

A dataset from four GNSS satellites and two pseudolites over 80 seconds where the factor graph optimization produces no mean 3D error reduction or reductions below 22.8 percent compared to the input least-squares solutions.

Figures

Figures reproduced from arXiv: 2605.24980 by Chih-Chun Chen, Heike Vallery, Lipeng Tan, Shiyu Bai.

**Figure 1.** Figure 1: Overview of the factor graph. II. METHODS A. State Variables and Factor Formulations The state variables at time ti are defined as: xi = R e b,i, p e b,i, v e b,i, ba,i, bg,i , (1) where R e b ∈ SO(3) denotes the rotation matrix from the body frame (b) to the Earth-centered, Earth-fixed frame (ECEF, e), p e b , v e b ∈ R 3 are the position and velocity expressed in ECEF frame, ba, bg ∈ R 3 are the accele… view at source ↗

**Figure 2.** Figure 2: Partial trajectory results (50 s-80 s) for the GPSonly and GPS+2PL cases using LS and FGO, compared with ground truth. For readability, only the final 30 s of the experiment are shown. is 22.8%-41.3%. The largest gain appears in the GPS-only setting. Compared to LS, FGO produces smoother trajectories and is less sensitive to measurement noise. This demonstrates the robustness of FGO-based sensor fusion un… view at source ↗

read the original abstract

In Global Navigation Satellite System (GNSS)-degraded environments, pseudolites (PLs) provide additional signal sources to enhance positioning performance, but their integration in optimization-based frameworks remains limited. This paper presents a loosely coupled factor graph optimization (FGO) framework that fuses the GNSS/PL least-squares (LS) solutions with inertial measurement unit (IMU) data. The evaluation considers low GNSS visibility scenarios with four high-elevation GNSS satellites and up to two PL transmitters over an 80~s window. FGO achieves a 22.8\% to 41.3\% reduction in mean 3D error compared to standard LS methods. Compared to a GNSS-IMU baseline, incorporating PL transmitters further improves positioning accuracy, with performance depending on geometry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows a 23-41% mean 3D error drop from loosely coupling LS fixes (4 GNSS + ≤2 PLs) into an IMU factor graph over 80 s, but the gains rest on those LS inputs being accurate enough that the optimizer does not need to correct their biases.

read the letter

The main thing to know is that this loosely coupled FGO pipeline fuses per-epoch LS position solutions from limited GNSS plus pseudolites with IMU data and reports 22.8-41.3% lower mean 3D error than plain LS, plus further gains over a GNSS-IMU baseline when the PLs are included. The benefit varies with geometry.

What is new is the concrete application of standard FGO to pseudolite augmentation in low-visibility robotics navigation. The work is incremental rather than foundational, but it fills a narrow gap by showing how to plug existing LS outputs into an existing optimization framework without rewriting the measurement model.

It does well by sticking to a controlled test case (four high-elevation satellites, up to two PLs, 80 s windows) and by noting that performance depends on geometry. That keeps the claim grounded.

The soft spot is exactly the stress-test concern: because the coupling is loose, the graph treats the LS fixes as direct observations. Any residual bias or variance from the sparse geometry gets baked in; there is no joint estimation of clock, troposphere, or geometry errors that a tighter raw-pseudorange factor could provide. The abstract supplies no error bars, no factor-construction details, and no trajectory plots, so it is hard to judge whether the reported percentages survive realistic input noise. If the full paper includes those elements and shows the LS inputs are already low-bias, the result holds; otherwise the improvement is partly an artifact of the test conditions.

This is for navigation researchers who already run FGO pipelines and need a quick way to add PLs in degraded GNSS settings. A reader working on sensor fusion for ground robots or UAVs in urban or indoor environments would get practical value from the setup and numbers.

It deserves peer review. The empirical comparison is straightforward to check once the methods and statistics are visible, and the problem is real even if the method is not revolutionary.

Referee Report

2 major / 1 minor

Summary. The paper proposes a loosely coupled factor graph optimization (FGO) framework fusing GNSS/pseudolite (PL) least-squares (LS) position solutions with IMU data to improve navigation in GNSS-degraded environments. Evaluation focuses on low-visibility cases using four high-elevation GNSS satellites plus up to two PL transmitters over 80 s windows; the central claim is a 22.8%–41.3% reduction in mean 3D error versus plain LS, with further gains over a GNSS-IMU baseline that depend on geometry.

Significance. If the empirical gains prove robust, the work supplies a practical, loosely coupled route for adding PL signals to optimization-based navigation pipelines without requiring raw-pseudorange factors. The geometry-dependent results highlight a concrete operational regime where PL augmentation can help, which is useful for field robotics and urban navigation applications.

major comments (2)

[Abstract] Abstract: the headline 22.8%–41.3% mean 3D error reduction is stated without error bars, number of trials, statistical tests, or trajectory plots, so it is impossible to judge whether the reported percentages are statistically distinguishable from the LS baseline.
[Abstract] Abstract (and implied methods): the loosely coupled architecture treats per-epoch LS fixes (computed from only four GNSS + ≤2 PLs) as direct measurements; any residual clock, tropospheric or geometry bias in those fixes is therefore propagated unchanged into the FGO. The manuscript supplies no analysis showing that these LS solutions remain sufficiently unbiased over the 80 s window for the claimed improvement to be attributable to the factor-graph formulation rather than to the quality of the input fixes.

minor comments (1)

The abstract would be clearer if it briefly indicated how the LS position solutions are converted into factors and what IMU pre-integration or bias-state model is employed inside the graph.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract and the implications of the loosely coupled architecture. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the headline 22.8%–41.3% mean 3D error reduction is stated without error bars, number of trials, statistical tests, or trajectory plots, so it is impossible to judge whether the reported percentages are statistically distinguishable from the LS baseline.

Authors: We agree that the abstract would benefit from greater transparency on the experimental statistics. The reported reductions are obtained from repeated evaluations across multiple 80 s windows under the specified low-visibility geometry. We will revise the abstract to state the number of trials performed and reference the error bars, statistical comparisons, and trajectory plots already present in the results section, enabling readers to assess distinguishability from the LS baseline. revision: yes
Referee: [Abstract] Abstract (and implied methods): the loosely coupled architecture treats per-epoch LS fixes (computed from only four GNSS + ≤2 PLs) as direct measurements; any residual clock, tropospheric or geometry bias in those fixes is therefore propagated unchanged into the FGO. The manuscript supplies no analysis showing that these LS solutions remain sufficiently unbiased over the 80 s window for the claimed improvement to be attributable to the factor-graph formulation rather than to the quality of the input fixes.

Authors: The loosely coupled formulation is chosen precisely to integrate existing LS position solutions with IMU data without requiring raw pseudorange factors. The FGO enforces consistency across epochs via IMU preintegration, which can reduce the influence of per-epoch LS errors. We acknowledge that the current manuscript does not contain an explicit bias-propagation study. We will add a dedicated paragraph in the methods section that quantifies typical LS biases under the four-GNSS-plus-PL geometry and demonstrates, via comparison against a non-optimized fusion baseline, that the observed gains exceed those attributable to input quality alone. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical comparison of estimators on shared data

full rationale

The paper reports an empirical evaluation of loosely-coupled FGO versus plain LS on the same GNSS/PL+IMU dataset in low-visibility conditions. The central performance numbers (22.8–41.3 % mean 3-D error reduction) are measured outcomes of running two estimators on identical inputs; no equation, parameter fit, or self-citation is shown to define the reported metric by construction. The method description fuses per-epoch LS fixes as measurements inside the graph, but the claimed improvement is an observed difference, not a tautology. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the evaluation geometry and LS-to-FGO interface assumptions are implicit but not enumerated.

pith-pipeline@v0.9.1-grok · 5662 in / 1122 out tokens · 40105 ms · 2026-06-30T00:46:57.197157+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

[1]

Gnss position in- tegrity in urban environments: A review of literature,

N. Zhu, J. Marais, D. Betaille, and M. Berbineau, “Gnss position in- tegrity in urban environments: A review of literature,”IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 9, pp. 2762–2778, 2018

2018
[2]

Pseudolites to support location services in smart cities: Review and prospects,

T. Liu, J. Liu, J. Wang, H. Zhang, B. Zhang, Y . Ma, M. Sun, Z. Lv, and G. Xu, “Pseudolites to support location services in smart cities: Review and prospects,”Smart Cities, vol. 6, no. 4, pp. 2081–2105, 2023

2081
[3]

Estimation techniques for low-cost inertial navigation,

E.-H. Shin, “Estimation techniques for low-cost inertial navigation,” Ph.D. dissertation, University of Calgary, Calgary, Canada, 2005

2005
[4]

Indoor position- ing with smartphone by using doppler observations from asynchronous pseudolite system,

X. Lu, L. Chen, N. Shen, Y . Dai, T. Zhou, and R. Chen, “Indoor position- ing with smartphone by using doppler observations from asynchronous pseudolite system,”IEEE Internet of Things Journal, 2024

2024
[5]

High- precision indoor positioning technology based on multi-channel array pseudolite,

L. Huang, B. Yu, Y . Li, H. Zhang, J. Cheng, X. Liang, and S. Liu, “High- precision indoor positioning technology based on multi-channel array pseudolite,”IEEE Transactions on Instrumentation and Measurement, vol. 74, 2025

2025
[6]

Dellaert and M

F. Dellaert and M. Kaess,Factor Graphs for Robot Perception. Foundations and Trends in Robotics, V ol. 6, 2017. [Online]. Available: http://www.cs.cmu.edu/∼kaess/pub/Dellaert17fnt.pdf

2017
[7]

Gnss/multisen- sor fusion using continuous-time factor graph optimization for robust localization,

H. Zhang, C.-C. Chen, H. Vallery, and T. D. Barfoot, “Gnss/multisen- sor fusion using continuous-time factor graph optimization for robust localization,”IEEE Transactions on Robotics, vol. 40, pp. 4003–4023, 2024

2024
[8]

Factor graph optimization for gnss/ins integration: A comparison with the extended kalman filter,

W. Wen, T. Pfeifer, X. Bai, and L.-T. Hsu, “Factor graph optimization for gnss/ins integration: A comparison with the extended kalman filter,” NA VIGATION: Journal of the Institute of Navigation, vol. 68, no. 2, pp. 315–331, 2021

2021
[9]

Imu preinte- gration on manifold for efficient visual-inertial maximum-a-posteriori estimation,

C. Forster, L. Carlone, F. Dellaert, and D. Scaramuzza, “Imu preinte- gration on manifold for efficient visual-inertial maximum-a-posteriori estimation,” inRobotics: Science and Systems XI, 2015

2015
[10]

Low-cost pseudolites: a loop-back time synchronization scheme for pseudolites using real-time clock offset calibration for reliable PNT,

M. S. Hameed, M. Philips-Blum, T. Pany, D. Sanroma, and S. Lakshma- iah, “Low-cost pseudolites: a loop-back time synchronization scheme for pseudolites using real-time clock offset calibration for reliable PNT,” in Proceedings of the 37th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2024), 2024, pp. 1444–1460

2024
[11]

Development and Demonstration of Compact Galileo E1/E6 Pseudolite with online Synchronization to GNSS time,

D. Sanroma, S. Lakshmaiah, M. S. Hameed, M. Philips-Blum, and T. Pany, “Development and Demonstration of Compact Galileo E1/E6 Pseudolite with online Synchronization to GNSS time,”European Jour- nal of Navigation, vol. 24, no. 3, pp. 20–30, 2024

2024
[12]

Validation of Pseudolites with a loop-back time synchronization scheme using a GNSS signal simulator,

M. S. Hameed, M. Philips-Blum, T. Pany, D. Sanroma, and S. Laksh- maiah, “Validation of Pseudolites with a loop-back time synchronization scheme using a GNSS signal simulator,” in2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), 2025

2025
[13]

borglab/gtsam,

F. Dellaert and G. Contributors, “borglab/gtsam,” May 2022. [Online]. Available: https://github.com/borglab/gtsam)

2022

[1] [1]

Gnss position in- tegrity in urban environments: A review of literature,

N. Zhu, J. Marais, D. Betaille, and M. Berbineau, “Gnss position in- tegrity in urban environments: A review of literature,”IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 9, pp. 2762–2778, 2018

2018

[2] [2]

Pseudolites to support location services in smart cities: Review and prospects,

T. Liu, J. Liu, J. Wang, H. Zhang, B. Zhang, Y . Ma, M. Sun, Z. Lv, and G. Xu, “Pseudolites to support location services in smart cities: Review and prospects,”Smart Cities, vol. 6, no. 4, pp. 2081–2105, 2023

2081

[3] [3]

Estimation techniques for low-cost inertial navigation,

E.-H. Shin, “Estimation techniques for low-cost inertial navigation,” Ph.D. dissertation, University of Calgary, Calgary, Canada, 2005

2005

[4] [4]

Indoor position- ing with smartphone by using doppler observations from asynchronous pseudolite system,

X. Lu, L. Chen, N. Shen, Y . Dai, T. Zhou, and R. Chen, “Indoor position- ing with smartphone by using doppler observations from asynchronous pseudolite system,”IEEE Internet of Things Journal, 2024

2024

[5] [5]

High- precision indoor positioning technology based on multi-channel array pseudolite,

L. Huang, B. Yu, Y . Li, H. Zhang, J. Cheng, X. Liang, and S. Liu, “High- precision indoor positioning technology based on multi-channel array pseudolite,”IEEE Transactions on Instrumentation and Measurement, vol. 74, 2025

2025

[6] [6]

Dellaert and M

F. Dellaert and M. Kaess,Factor Graphs for Robot Perception. Foundations and Trends in Robotics, V ol. 6, 2017. [Online]. Available: http://www.cs.cmu.edu/∼kaess/pub/Dellaert17fnt.pdf

2017

[7] [7]

Gnss/multisen- sor fusion using continuous-time factor graph optimization for robust localization,

H. Zhang, C.-C. Chen, H. Vallery, and T. D. Barfoot, “Gnss/multisen- sor fusion using continuous-time factor graph optimization for robust localization,”IEEE Transactions on Robotics, vol. 40, pp. 4003–4023, 2024

2024

[8] [8]

Factor graph optimization for gnss/ins integration: A comparison with the extended kalman filter,

W. Wen, T. Pfeifer, X. Bai, and L.-T. Hsu, “Factor graph optimization for gnss/ins integration: A comparison with the extended kalman filter,” NA VIGATION: Journal of the Institute of Navigation, vol. 68, no. 2, pp. 315–331, 2021

2021

[9] [9]

Imu preinte- gration on manifold for efficient visual-inertial maximum-a-posteriori estimation,

C. Forster, L. Carlone, F. Dellaert, and D. Scaramuzza, “Imu preinte- gration on manifold for efficient visual-inertial maximum-a-posteriori estimation,” inRobotics: Science and Systems XI, 2015

2015

[10] [10]

Low-cost pseudolites: a loop-back time synchronization scheme for pseudolites using real-time clock offset calibration for reliable PNT,

M. S. Hameed, M. Philips-Blum, T. Pany, D. Sanroma, and S. Lakshma- iah, “Low-cost pseudolites: a loop-back time synchronization scheme for pseudolites using real-time clock offset calibration for reliable PNT,” in Proceedings of the 37th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2024), 2024, pp. 1444–1460

2024

[11] [11]

Development and Demonstration of Compact Galileo E1/E6 Pseudolite with online Synchronization to GNSS time,

D. Sanroma, S. Lakshmaiah, M. S. Hameed, M. Philips-Blum, and T. Pany, “Development and Demonstration of Compact Galileo E1/E6 Pseudolite with online Synchronization to GNSS time,”European Jour- nal of Navigation, vol. 24, no. 3, pp. 20–30, 2024

2024

[12] [12]

Validation of Pseudolites with a loop-back time synchronization scheme using a GNSS signal simulator,

M. S. Hameed, M. Philips-Blum, T. Pany, D. Sanroma, and S. Laksh- maiah, “Validation of Pseudolites with a loop-back time synchronization scheme using a GNSS signal simulator,” in2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), 2025

2025

[13] [13]

borglab/gtsam,

F. Dellaert and G. Contributors, “borglab/gtsam,” May 2022. [Online]. Available: https://github.com/borglab/gtsam)

2022