Factor graph optimization (FGO) is an estimation framework that has gained significant interest in the GNSS community since 2021, when the technique was used by the winner of the Google smartphone decimeter challenge. The technique considers all available observations over a large time window, and also considers probabilistic constraints linking the unknown variables at consecutive epochs. FGO has shown improved accuracy compared to classical algorithms such as snapshot weighted least squares or Kalman filter. However, this performance increase comes at the price of more intensive computations as the time window gets longer.

This paper proposes a characterization of the accuracy of a FGO solution depending on the time window length. We focus our study on FG combining GNSS code observations and a random walk model for the position and receiver clock bias states. Under the hypothesis of a constant geometry between the receiver and the satellites, we provide the analytical formula of the solution's accuracy.

We derive the formula of the asymptotic accuracy that the FGO solution reaches when the time window tends towards infinity. It mainly depends on the ratio between the observation and motion model uncertainties, and on the relative position of the considered epoch in the time window.

We then compute the time window length required to reach a desired fraction of the asymptotic accuracy, thus providing a justification for limiting the FG time window length, and therefore limiting the computational requirement of such technique. Tables yielding the recommended factor graph length are provided for different fractions of the asymptotic accuracy, observation to motion uncertainty ratio, and for an epoch located in the middle of the time window and for the last epoch.

The proposed FG time window limit is then tested on real data to verify the analytical formulas provided in this paper. We process 1 Hz data from an IGS station considering different factor graph time window lengths and different process noise for the random walk process. The experiments show that increasing the factor graph length beyond a certain limit does not result in an accuracy increase, thus confirming the proposed limitation of factor graph length.

We believe that the results of this paper provide useful guidance for designing a FGO-based solution, by providing a justification on the factor graph length to be considered.

This work proposes a factor graph optimization solution to solve the Precise Point Positioning (PPP) navigation problem in challenging, fault-prone environments. While kinematic PPP solutions already achieve high precision (~20 cm), their performance can degrade under multiple simultaneous faults, sensor degradation and non-line-of-sight (NLOS) signal disruptions. Such conditions can lead to solution instability, large biases, or complete outages. Here, we focus on enhancing the overall precision and availability by leveraging a robust factor graph formulation that supports both sliding-window (real-time) and full-batch operation, integrating real-time PPP products (Galileo High Accuracy Service (HAS) or SSRZ) and combining data from multiple sensors.

Our approach systematically compares three methods for state estimation: (A) recursive filtering, (B) smoothing, and (C) batch least squares. These methods differ primarily in how past and current information is incorporated to estimate the integer and real-valued parameters critical to PPP. By applying these estimation strategies within a simple, controlled simulation, we highlight their relative strengths and weaknesses in terms of accuracy, computational load, and robustness to sensor or measurement faults, all within a unified factor-graph representation.

In this study, we perform an extensive performance evaluation via Monte Carlo simulation. The metrics considered include positioning performance, solution availability, and computational complexity. The factor-graph-based solution demonstrates substantially lower RMSE than Kalman Filtering (KF), and Rauch-Tung-Striebel smoother (RTS) baselines, with reductions on the order of 40–60% in both horizontal and vertical components, indicating clear positioning benefits.

We further discuss the implications of these findings for a broader inertial and multi-sensor stack, as part of an ongoing effort to develop resilient navigation solutions for autonomous systems. In particular, we emphasize factor graph-based PPP both as a state-of-the-art ground truth estimator (via large batch windows) and as a powerful real-time processor (via fixed-lag, sliding-window inference), against which low-latency navigation solutions can be benchmarked. Preliminary conclusions suggest that hybrid solutions, where real-time filtering is complemented by periodic smoothing or batch processing, may offer the most advantageous balance of robustness and availability.

This work presents findings from a project on Factor Graph Optimization (FGO) for GNSS-based positioning, covering the design and implementation of a modular test platform and results from an extensive test campaign in representative automotive environments. The objectives are twofold: (1) to assess the practical software implementation of estimators for undifferenced and uncombined GNSS observation models in PPP and RTK using the FGO framework; and (2) to evaluate the improvements in trajectory estimation quality that FGO-based smoothing can offer over the conventional Extended Kalman Filter (EKF).

Because classic forward–backward smoother formulations imply that the latest smoothed state is equivalent to the filtered estimate, the real-time benefits of smoothing are not immediately evident and received little systematic investigation beyond hypothesis. We demonstrate how an FGO-based incremental smoother—mainly through consistent relinearization and the improved capability for error detection—can affect online performance. We identify conditions under which smoothing provides advantages over filtering with respect to trajectory estimation quality.

We report results from an extensive test campaign across diverse automotive scenarios, comparing FGO-based estimators against EKF baselines. The findings characterize when and how FGO yields measurable improvements in positioning accuracy, trajectory consistency, ambiguity handling, and continuity under degraded geometry, while quantifying computational trade-offs.

The study provides practical guidance for deploying smoothing-based estimators for PPP and RTK in real-time navigation systems.