Extendable Factor Graph Optimization

• Extendable Factor Graph Optimization is a modular, probabilistic framework that unifies variable estimation and constraint integration in complex systems.
• It employs scalable inference via message-passing and nonlinear optimization, enabling efficient sensor fusion and multi-agent coordination.
• Its design supports seamless integration of new sensors, constraints, and domain-specific innovations to enhance performance in robotics, navigation, and control.

An extendable Factor Graph Optimization (FGO) framework is a unifying probabilistic modeling and optimization paradigm in which variables representing system states (e.g., positions, velocities, sensor biases) and measurements (e.g., sensor observations, prior knowledge, or constraints) are jointly represented as nodes and factors in a bipartite graph structure. Unlike classic methods limited to either directed graphical models (Bayesian networks) or undirected models (Markov random fields), the factor graph formalism inherently allows explicit, modular decomposition of complex objectives, supports combinatorial or continuous variables, and permits straightforward addition of heterogeneous constraints. In this context, “extendable” refers to both the theoretical generality—ability to model arbitrary and hybrid systems—and algorithmic modularity, enabling seamless incorporation of new sensors, physical constraints, or application-specific priors. Factor graph optimization underlies a wide spectrum of modern inference, estimation, and control problems, notably in robotics, navigation, communication, and multi-agent systems.

1. Unified Representation: Variables, Factors, and Explicit Factorization

An FGO framework employs a bipartite graph G = (X, F, E) with variable nodes X (system states) and factor nodes F (data constraints, priors, physical laws). The joint probability of the system is factorized as

$P(x_1, \ldots, x_N) = \prod_{k=1}^K g_k(x_{C_k}),$

where each factor $g_k$ encodes the local dependency among a subset $x_{C_k} \subseteq X$ (Frey, 2012). This structure explicitly exposes the underlying factorization—contrasting with hidden or implicit decompositions in chain graphs—enabling arbitrary combinations of directed and undirected influences and facilitating model extension. In practical frameworks, each measurement, physical constraint (kinematics, dynamics), or environmental prior is encoded as a separate factor node connected only to its relevant variables. For example, in GNSS/inertial fusion systems, position, velocity, and clock bias states are linked to pseudorange, Doppler, and motion model factors, supporting easy extension as new data types or constraints become available (Blanco-Claraco et al., 2021, Wen et al., 2020).

2. Inference via Scalable Message Passing and Nonlinear Optimization

Factor graphs allow for unified inference via message-passing algorithms (e.g., sum-product, max-product), in both directed and undirected models. For example, the marginal probability for a variable $x$ is computed as

$P(x) \propto \prod_{f \in n(x)} m_{f \to x}(x),$

with messages $m_{f \to x}$ from adjacent factors, formulated as

$$m_{f \to x}(x) = \sum_{x' \setminus x} f(x', x) \prod_{y \in n(f) \setminus \{x\}} m_{y \to f}(y)$$

(Frey, 2012). Real-world applications—where factors are highly nonlinear and/or continuous—employ iterative nonlinear optimization schemes (e.g., Gauss–Newton, Levenberg–Marquardt) over an aggregate cost function that sums squares of all factor residuals. The graphical structure exposes the problem sparsity, enabling efficient exploitation by solvers such as GTSAM or Ceres (Blanco-Claraco et al., 2021, Xie et al., 2020). This unifies inference across a range of problem classes, including batch, incremental, and sliding-window approaches; in large-scale systems, marginalization is used to eliminate old states while preserving historical information.

3. Modularity and Extendability: Incorporation of Heterogeneous Data and Constraints

A haLLMark of the extendable FGO framework is its modularity in incorporating new sensors, constraints, or application couplings:

• Equality constraints: Incorporated as explicit error factors (e.g., ZUPTs in pedestrian inertial navigation (Hu et al., 13 May 2025); holonomic position/velocity/dynamics constraints in multi-body systems (Blanco-Claraco et al., 2021)).
• Inequality constraints: Modeled via differentiable penalties or hinge-loss terms (e.g., joint limits in robot arms (Xie et al., 2020), softmax-based inter-foot distance bounds (Hu et al., 13 May 2025), friction cones for contacts).
• Complex model fusion: Integration of sensor and control objectives (e.g., unified MPC and pose estimation in UAVs using JPCM (Yang et al., 23 Apr 2024, Yang et al., 2023)), multi-sensor plug-and-play architectures supporting asynchronous or partially missing measurements (e.g., GNSS/IMU/DVL/pressure in underwater vehicles (Song et al., 2023), GNSS-UWB synchronization via time-invariant offsets (Zhao et al., 9 Jul 2024)), and simultaneous robot and infrastructure calibration (e.g., UWB anchor/robot localization (Liu et al., 28 Mar 2025)).
• Problem-specific innovations: Custom priors (e.g., plane constraints for cooperative vehicle localization (Zhuang et al., 2023)), robust error models (Huber/Cauchy loss functions for outlier-resistant GNSS (Suzuki, 12 Feb 2025)), and continuous clustering/neural parameterizations for factor node adaptation in communication inference (Rapp et al., 2022).
• Multi-agent and high-dimensional extension: Centralized or decentralized architectures for cooperative task-solving (Zhuang et al., 2023), scalable batch/incremental approaches for large time or agent populations (Zhang et al., 2022).

Modularity ensures that each new information source or constraint is encoded as a self-contained factor, supporting straightforward maintenance and extension.

4. Constraints, Augmented Lagrangians, and Constrained Optimization

Standard FGO models natively support equality constraints as factors. For more complex constraints, particularly inequality or nonlinear constraints in control and planning, extensions such as embedding augmented Lagrangian (AL) methods within the factor graph have been developed (Bazzana et al., 2023). Here, the cost is augmented as

$$\mathcal{L}(x, \lambda, \rho) = F(x) + \sum_i \lambda_i^\top f_i(x) + \|f_i(x)\|^2_{P_i} + \sum_j \mu_j^\top(g_j(x) + q_j) + \|g_j(x) + q_j\|^2_{P_j},$$

where equality ($f_i$) and inequality ($g_j$) constraints are handled via corresponding dual variables ($\lambda, \mu$) and penalty matrices, and $q_j$ are slack variables for inequality constraints. This approach enables explicit, iterative treatment of constraints in the standard Gauss-Newton linearization, with dual variable updates embedded directly in the graph’s optimization cycle (Bazzana et al., 2023, Xie et al., 2020, Hu et al., 13 May 2025).

5. Application Domains and Empirical Performance

Extendable FGO frameworks have demonstrated strong empirical performance across diverse domains:

• GNSS/INS integration: Robustness in urban canyon environments, with significantly lower mean and standard deviation positioning errors compared to EKF, and resilience to non-Gaussian outlier distributions (Wen et al., 2020, Suzuki, 12 Feb 2025, Song et al., 1 Oct 2025).
• Multi-sensor fusion: Real-time, online, continuous-time multi-sensor localization for vehicles in urban contexts (Zhang et al., 2022, Zhang et al., 2023).
• Robotics and motion planning: Whole-body robot dynamics and kinodynamic motion planning, facilitating constraint enforcement and high-dimensional scalability for multi-DoF systems (Xie et al., 2020, Blanco-Claraco et al., 2021).
• Pedestrian and networked navigation: Constrained FGO for drift mitigation and robust outlier-handling through ZUPTs and biomechanical penalties, outperforming EKF in position error distributions (Hu et al., 13 May 2025).
• Cooperative and infrastructure calibration: Simultaneous UWB anchor calibration and robot localization with rapid convergence and negligible manual intervention, enabling fast and reliable emergency deployments (Liu et al., 28 Mar 2025).
• Control and estimation integration: Tightly joined positioning and control for UAVs using explicit joint factorization, enhancing trajectory tracking in GNSS-degraded and disturbance-rich settings (Yang et al., 23 Apr 2024, Yang et al., 2023).
• Integrity monitoring and safety: Position error bounding for safety-critical GNSS/INS/Vision integration by factor graph–based residual analysis, with 100% availability after timely fault exclusion (Tian et al., 30 Oct 2024).
• Communication and inference: Learnable structure and neural message passing on factor graphs for symbol detection and inference in cyclic, high-complexity communication networks (Rapp et al., 2022).

These domains leverage the same foundational mechanics—explicit factorization, modular factor addition, scalable optimization, and tractable enforcement of physical or domain constraints.

6. Future Directions and Framework Adaptability

Ongoing research seeks to further generalize FGO by:

• Expanding real-time, adaptive, and scalable inference: Development of sliding-window, fixed-lag, and incremental algorithms to support large-scale and real-time processing, mitigating computational bottlenecks in high-frame-rate or long-duration deployments (Blanco-Claraco et al., 2021, Song et al., 2023).
• Enhanced constraint representations: Improved handling of nonconvex inequalities, physical boundary conditions, and temporal logic constraints, including active-set/soft-penalty and AL formulations (Bazzana et al., 2023).
• Greater integration of learning: Structural optimization of factor graphs via machine learning and data-driven clustering or neural message-passing, bridging model-based and data-driven paradigms (Rapp et al., 2022).
• Autonomous infrastructure management: Simultaneous online calibration of networked infrastructure (e.g., UWB anchors, environmental sensors) during robot operation (Liu et al., 28 Mar 2025).
• Safety, explainability, and certification: Incorporation of integrity monitoring metrics and verifiable error bounds, targeting deployment in safety-critical, certified navigation and control systems (Tian et al., 30 Oct 2024).

A plausible implication is that, given their flexibility and extensibility, FGO frameworks will remain the backbone for complex, multi-modal, and multi-agent estimation, inference, and control tasks demanding both accuracy and tractable modular extension.

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This comprehensive overview draws on implementation structures, mathematical formulations, and experimental outcomes across several domains, highlighting the versatility and adaptability of the extendable factor graph optimization paradigm.