SLAM as a Stochastic Control Problem with Partial Information: Optimal Solutions and Rigorous Approximations

Visual Place Recognition (VPR) in lifelong deployment requires continuous adaptation to new environments without catastrophic forgetting. In this paper, we propose SLAM, a Structured and Localized Analytic Manifold adaptation framework. Our framework elegantly unifies uncertainty-aware smoothing via Unscented transformation, topological space partitioning through a Gaussian Mixture Model (GMM), and $H_\infty$ robust bound optimization into a singular, unified closed-form analytical recursion. Exhaustive ablation studies demonstrate that while the synergistic combination of uncertainty smoothing and localized mapping (U+G configuration) achieves the state-of-the-art nominal accuracy of 27.5%, the full deployment of the $H_\infty$ bound does not require an architectural split; rather, it introduces a mathematically guaranteed minimax robust bound. This formulation enables the system to seamlessly modulate the intrinsic trade-off between nominal placement precision and worst-case disturbance attenuation through a single regularization parameter.

Traditional Simultaneous Localization and Mapping (SLAM) algorithms rely heavily on the static environment assumption, which severely limits their applicability in real-world spaces populated by moving entities, such as pedestrians. In this work, we propose DynoSLAM, a tightly-coupled Dynamic GraphSLAM architecture that integrates socially-aware Graph Neural Networks (GNNs) directly into the factor graph optimization. Unlike conventional approaches that use rigid constant-velocity heuristics or deterministic single-agent neural priors, our framework formulates pedestrian motion forecasting as a stochastic World Model. By utilizing Monte Carlo rollouts from a trained GNN, we capture the multimodal epistemic uncertainty of human interactions and embed it into the SLAM graph via a dynamic Mahalanobis distance factor. We demonstrate through extensive simulated experiments that this stochastic formulation not only maintains highly accurate retrospective tracking but also prevents the optimization failures caused by the deterministic "argmax problem". Ultimately, extracting the empirical mean and covariance matrices of future pedestrian states provides a mathematically rigorous, probabilistic safety envelope for downstream local planners, enabling anticipatory and collision-free robot navigation in densely crowded environments.

In safety-critical robotics applications, guaranteed and practical uncertainty quantification (UQ) in perception is vital. Many existing works either offer no formal containment guarantee, rely on restrictive modeling assumptions, or focus only on pose estimation rather than a complete SLAM pipeline. This paper presents provably guaranteed UQ algorithms for 3D-3D landmark-based SLAM. The algorithms consist of three basic UQ modules: forward UQ for mapping, backward UQ for pose tracking, and pose compound. Each module produces a certified uncertainty set; when the input uncertainty bounds are deterministic, the output sets inherit deterministic guarantees, i.e., they provably contain the true poses and landmarks. Specifically, we use polytopes to represent uncertainty sets, enabling tractable computations and a unified treatment of pose uncertainty. To enhance algorithms' practical usability, we incorporate conformal prediction to calibrate measurement uncertainty from data with prescribed probability. Simulations and experiments demonstrate that the proposed algorithms provide both strong theoretical guarantees and practical usability. The code is open-sourced at https://github.com/LIAS-CUHKSZ/Polytopic-SLAM-Uncertainty-Quantification.