💻 computer science

Loosely Coupled Factor Graph Optimization for Pseudolite-Augmented Navigation

This paper proposes a loosely coupled factor graph optimization framework that fuses GNSS, pseudolite, and IMU data to significantly improve positioning accuracy in GNSS-degraded environments, achieving a 22.8% to 41.3% reduction in mean 3D error compared to standard least-squares methods.

Original authors: Chih-Chun Chen, Lipeng Tan, Shiyu Bai, Heike Vallery

Published 2026-05-26

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Chih-Chun Chen, Lipeng Tan, Shiyu Bai, Heike Vallery

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find your way through a dense city of tall skyscrapers (or a deep tunnel). You have a GPS device, but the tall buildings block the view of the sky, so your GPS signal is weak, bounces off walls, or disappears entirely. It's like trying to navigate by looking at the stars when the clouds are too thick to see them.

To fix this, the researchers in this paper added "fake satellites" called Pseudolites. Think of these as friendly streetlamps that stand on the ground and shout, "I am here!" to your GPS receiver. They act like extra stars in a sky that is otherwise dark.

However, just adding these extra lights isn't enough if your GPS is still glitchy. That's where the paper's main idea comes in: Factor Graph Optimization (FGO).

The Problem with the Old Way (Least Squares)

Imagine you are walking blindfolded, and every few seconds, a friend shouts your location to you.

The Old Method (Least Squares): Every time your friend shouts a location, you stop, look at that single shout, and guess where you are. If your friend shouts a wrong number because of an echo (a "multipath" error), you instantly believe it and take a wrong step. You don't remember where you were a second ago.
The New Method (FGO): This method is like having a memory. Instead of just listening to the current shout, you remember where you were, how fast you were walking, and which way you were turning (using a device called an IMU, which is like a high-tech inner ear that feels motion). You look at the whole story of your walk from the last 80 seconds, not just the last second. You ask: "Does this new shout make sense with where I was, how I was moving, and what my inner ear felt?"

How They Did It

The researchers set up a test in a parking lot at a university in Germany.

The Setup: They used a car with a special antenna on top. They turned off most real GPS satellites, leaving only four high ones (to simulate a bad signal).
The Helpers: They turned on two "Pseudolite" transmitters (the ground-based fake satellites).
The Test: They drove around for 80 seconds and compared two things:
- The old way (just listening to the shouts).
- The new way (listening to the shouts and remembering the motion).

What They Found

The results were like finding a clearer path through the fog:

Smarter Navigation: The new "memory-based" method (FGO) was significantly better. It reduced the average 3D error (how far off you are in all directions) by 22.8% to 41.3% compared to the old method.
Smoother Rides: The path drawn by the new method was much smoother. The old method jumped around wildly when the signal got noisy, but the new method ignored the "jitters" because it knew, "I was moving smoothly a second ago, so this sudden jump must be a lie."
Geometry Matters: Adding the fake satellites (Pseudolites) helped, but where they were placed mattered. Having two fake satellites in the right spots made the 3D position much more accurate, but having them in different spots helped the side-to-side (horizontal) accuracy more than the up-and-down (vertical) accuracy.

The Bottom Line

This paper proves that when you are in a place where GPS is weak (like a city canyon), you can find your way much better by:

Adding ground-based "fake satellites" (Pseudolites).
Using a smart system that remembers your past movements and checks if new location data makes sense with your history, rather than just trusting the newest piece of data blindly.

It's the difference between taking a single, shaky photo to guess where you are, versus watching a video of your entire journey to figure out exactly where you are right now.

Technical Summary: Loosely Coupled Factor Graph Optimization for Pseudolite-Augmented Navigation

Problem Statement
Accurate localization is critical for autonomous vehicles, yet Global Navigation Satellite System (GNSS) performance frequently degrades in environments such as tunnels and urban canyons due to signal blockage, multipath, and outages. While fusing GNSS with Inertial Measurement Units (IMUs) is a standard approach to mitigate these issues, IMU data suffers from drift without frequent correction. Pseudolites (PLs), which transmit GNSS-like signals, offer a solution to enhance signal availability and geometry in degraded environments. However, existing integration methods for GNSS and PLs predominantly rely on Least-Squares (LS) estimation and often fail to incorporate IMU data. Furthermore, to the authors' knowledge, no prior work has integrated PLs with other sensors using Factor Graph Optimization (FGO), a framework known for its ability to leverage historical measurements and perform iterative optimization.

Methodology
The paper proposes a loosely coupled (LC) Factor Graph Optimization (FGO) framework designed to fuse GNSS, PL, and IMU measurements.

State Definition: The system state at time $t_i$ includes the rotation matrix ( $R^e_b$ ), position ( $p^e_b$ ), velocity ( $v^e_b$ ), and biases for the accelerometer ( $b_a$ ) and gyroscope ( $b_g$ ).
Factor Formulation:
- Preintegrated IMU Factor: Models motion between consecutive states using IMU preintegration, accounting for rotation, velocity, and position residuals.
- IMU Bias Factor: Models the temporal evolution of accelerometer and gyroscope biases.
- GNSS/PL-Position Factor: Utilizes a standard Least-Squares solution derived from combined GNSS and PL pseudorange measurements to estimate the antenna position. This position is then related to the IMU state via a lever arm transformation.
Graph Structure: The factor graph appends state nodes and measurement factors as new GNSS/PL data arrives. High-rate IMU measurements between GNSS/PL epochs are preintegrated into single factors connecting the states. A prior factor constrains the initial state.
Optimization: The system solves for the maximum a posteriori (MAP) estimate by minimizing the sum of squared errors from the prior, position factors, and IMU factors.

Experimental Setup
The framework was validated using real-world data collected in a parking lot at the University of the Bundeswehr Munich.

Hardware: A rover equipped with a multi-band GNSS/PL antenna and a MEMS IMU. Two PL transmitters broadcast Galileo E1BC-like signals.
Scenario: To emulate GNSS-degraded conditions, the experiment utilized only the four highest-elevation GPS satellites over an 80-second window.
Configurations: The study compared performance across four signal configurations: GPS-only, GPS + 1 PL (PL01), GPS + 1 PL (PL02), and GPS + 2 PLs.
Baselines: The proposed FGO approach was compared against a standard LS method (without IMU) and a GNSS-IMU baseline.

Key Results
The evaluation demonstrates that the FGO framework consistently outperforms standard LS methods across all signal configurations:

Error Reduction: FGO achieved a 22.8% to 41.3% reduction in mean 3D error compared to LS. In 2D error, improvements ranged from 11.9% to 21.1%.
Trajectory Quality: FGO produced smoother trajectories and showed reduced sensitivity to measurement noise compared to LS, particularly in the GPS-only scenario where observability is limited.
Impact of Pseudolites: Incorporating PL transmitters further improved positioning accuracy compared to a GNSS-IMU baseline. However, the magnitude of improvement depended on the geometric arrangement of the transmitters.
- The GPS+2PL configuration yielded the best 3D accuracy (lowest PDOP).
- The GPS+PL02 configuration achieved the lowest 2D error, despite having worse Vertical DOP (VDOP) than the single-PL01 case.
DOP Correlation: While Dilution of Precision (DOP) trends generally aligned with positioning performance, the authors note that DOP alone does not fully determine accuracy, as factors like multipath and clock bias also influence the final error.

Significance and Claims
The paper claims to present the first integration of Pseudolites with other sensors using Factor Graph Optimization. The primary significance of this work lies in validating the effectiveness of FGO for PL-assisted navigation in environments with limited satellite visibility. The results confirm that a loosely coupled FGO framework can robustly fuse GNSS, PL, and IMU data to significantly enhance positioning accuracy over traditional LS methods. The authors conclude that while the geometric configuration of PLs influences the specific gains, the approach is a viable solution for improving navigation robustness in degraded GNSS environments. Future work is identified as investigating tightly coupled integration to further improve robustness and accuracy.

The Problem with the Old Way (Least Squares)

How They Did It

What They Found

The Bottom Line

Technical Summary: Loosely Coupled Factor Graph Optimization for Pseudolite-Augmented Navigation

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