Sliding window factor graph optimization (SW-FGO) is widely recognized for its robustness, yet its theoretical relationship with the extended Kalman filter (EKF) remains a subject of debate. This paper establishes the sufficient conditions to bridge SW-FGO with the iterated extended Kalman filter (IEKF). We introduce recursive FGO (Re-FGO), a conceptual perspective that employs a two-stage marginalization pipeline to mathematically degenerate the factor graph optimization to the IEKF recursive update. By enforcing the Markov assumption and a single-state window, we prove the theoretical equivalence between the IEKF and Re-FGO. This degeneration is validated through simulations and real-world urban GNSS and INS tightly coupled fusion experiments. The results confirm that Re-FGO exactly reproduces IEKF estimation behavior, demonstrating that the two-stage marginalization pipeline is foundational to enforce structural consistency, thereby successfully uniting graph-based smoothing and filtering paradigms under unified optimization principles.

The receiver autonomous integrity monitoring (RAIM) and Global Navigation Satellite System (GNSS) fault detection literature uses a group of statistical detectors that are often introduced with different names, coordinate systems, and derivation styles. This makes it difficult for new researchers to determine whether two methods use different information or only express the same inconsistency through different statistics. This paper provides a detector-centered tutorial review of statistical snapshot fault detection with a unified whitened linearized model. The chi-squared detector, parity-space detector, Baarda w-test, range comparison detector, jackknife detector, solution separation detector, and generalized likelihood-ratio test are derived with consistent notation. For each detector, the statistic construction, null and alternative distributions, threshold rule, and minimum detectable bias (MDB) are presented. A relationship map was developed to distinguish exact equivalence, projection relations and linear transformations among these detectors. An illustrative validation was then conducted with a real satellite geometry collected in an urban environment and synthetic Gaussian faults. The results verify the relationship checks, detection-probability behavior, and MDB calculations in a reproducible setting.

To provide backup and augmentation to global navigation satellite system (GNSS), Doppler shift from Low Earth Orbit (LEO) satellites can be employed as signals of opportunity (SOP) for position, navigation and timing (PNT). Since the Doppler positioning problem is non-convex, local searching methods may produce two types of estimates: a global optimum without notice or a local optimum given an inexact initial estimate. As exact initialization is unavailable in some unknown environments, a guaranteed global optimization method in no need of initialization becomes necessary. To achieve this goal, we propose a certifiably optimal LEO Doppler positioning method by utilizing convex optimization. In this paper, the certifiable positioning method is implemented through a graduated weight approximation (GWA) algorithm and semidefinite programming (SDP) relaxation. To guarantee the optimality, we derive the necessary conditions for optimality in ideal noiseless cases and sufficient noise bounds conditions in noisy cases. Simulation and real tests are conducted to evaluate the effectiveness and robustness of the proposed method. Specially, the real test using Iridium-NEXT satellites shows that the proposed method estimates an certifiably optimal solution with an 3D positioning error of 140 m without initial estimates while Gauss-Newton and Dog-Leg are trapped in local optima when the initial point is equal or larger than 1000 km away from the ground truth. Moreover, the certifiable estimation can also be used as initialization in local searching methods to lower down the 3D positioning error to 130 m.

Sliding window-factor graph optimization (SW-FGO) has gained more and more attention in navigation research due to its robust approximation to non-Gaussian noises and nonlinearity of measuring models. There are lots of works focusing on its application performance compared to extended Kalman filter (EKF) but there is still a myth at the theoretical relationship between the SW-FGO and EKF. In this paper, we find the necessarily fair condition to connect SW-FGO and Kalman filter variants (KFV) (e.g., EKF, iterative EKF (IEKF), robust EKF (REKF) and robust iterative EKF (RIEKF)). Based on the conditions, we propose a recursive FGO (Re-FGO) framework to represent KFV under SW-FGO formulation. Under explicit conditions (Markov assumption, Gaussian noise with L2 loss, and a one-state window), Re-FGO regenerates exactly to EKF/IEKF/REKF/RIEKF, while SW-FGO shows measurable benefits in nonlinear, non-Gaussian regimes at a predictable compute cost. Finally, after clarifying the connection between them, we highlight the unique advantages of SW-FGO in practical phases, especially on numerical estimation and deep learning integration. The code and data used in this work is open sourced at https://github.com/Baoshan-Song/KFV-FGO-Comparison.