physics-aware ekf tracking

This work addresses state estimation for discrete-time nonlinear stochastic systems with finite-dimensional states and infinite-dimensional measurements—such as image fields—arising in visual localization and tracking. Conventional methods struggle to rigorously model the relationship between image gradients and the Jacobian of the observation function with respect to the state. To overcome this, we propose an extended Kalman filter (EKF) grounded in infinite-dimensional random field modeling. Crucially, we establish, for the first time from a systems-theoretic perspective, that image gradients correspond precisely to the Fréchet derivative of the observation functional with respect to the state—thereby providing a rigorous mathematical foundation for gradient-based visual state estimation. Evaluated on monocular visual-inertial navigation for unmanned aerial vehicles, our approach reduces root-mean-square error by an order of magnitude compared to VINS-MONO, demonstrating substantial improvements in both estimation accuracy and robustness.

This work proposes a data-driven state estimation algorithm that integrates the extended Kalman filter (EKF) with Koopman operator theory to address the challenge of modeling complex or poorly calibrated sensors. By lifting nonlinear observations into a linearly observable Koopman space, the method enables closed-form learning of a linear Gaussian observation model directly from ground-truth data—without requiring an explicit sensor model or iterative optimization. Crucially, Jacobian matrices are computed online to preserve the recursive structure and real-time performance of the EKF. Evaluated on a real-world quadrotor localization task, the approach substantially outperforms conventional EKF implementations reliant on imperfect geometric models and data-driven calibration baselines, achieving significant improvements in estimation accuracy, consistency, and computational efficiency.

This paper addresses state estimation under Stiefel manifold geometric constraints. We propose an extended Kalman filter (EKF) framework rigorously defined on the Stiefel manifold, departing from conventional Euclidean-space EKFs. Our method performs linearization in the tangent space, and employs exponential and logarithmic maps for observation updates—thereby preserving orthogonality constraints exactly. Theoretically, we derive the recursive filtering equations and covariance propagation rules intrinsic to the Stiefel manifold. Algorithmically, the framework supports arbitrary St(n,p) manifolds and accommodates common subcases including the unit sphere S² and the 4×2 orthogonal matrix manifold. Simulation results demonstrate that, compared to standard EKF applied naively in the ambient Euclidean embedding space, our approach achieves significantly higher estimation accuracy and superior constraint satisfaction—validating its effectiveness and robustness for non-Euclidean state estimation.

Traditional Kalman filtering (KF) suffers from limited state estimation accuracy due to oversimplified state-space models. To address this, we propose an AI-enhanced filtering framework that deeply integrates model-driven and data-driven paradigms. We systematically introduce two novel AI-KF fusion paradigms—task-oriented and state-space-model-oriented—and embed deep neural networks directly into the KF architecture, enabling adaptive modeling of unknown dynamics while preserving physical interpretability. Our method supports partial state-space modeling and end-to-end joint training. Experiments across diverse nonlinear and time-varying systems demonstrate significant improvements in tracking accuracy and robustness over conventional approaches. Furthermore, we fully open-source the implementation, establishing the first reproducible benchmark and design paradigm for AI-augmented filtering.

Robot state estimation faces growing challenges from platform diversity and task complexity, while traditional discrete-time filtering and smoothing methods suffer from sampling-rate limitations and temporal misalignment. This paper proposes a unified formal framework for continuous-time state estimation, systematically integrating major modeling paradigms—including spline interpolation, Gaussian process regression, Bayesian smoothing, and continuous-time optimization—for the first time. We present the most comprehensive survey and taxonomy to date, clarifying methodological evolution, state representation strategies, and application-specific advancements. Furthermore, we identify and formally characterize key open problems, highlighting emerging research directions: differentiable modeling, asynchronous multi-sensor fusion, and real-time computation. Our framework significantly improves estimation accuracy, temporal resolution flexibility, and downstream planning and control performance. By bridging theoretical rigor with practical applicability, this work advances both the foundations and deployment of continuous-time estimation in robotics.

This work addresses the issue of overconfident filtering in nonlinear state-space models caused by misspecification in either the dynamics or observation model. To mitigate this, the authors propose a Prediction-oriented (PrO) online filtering approach that does not strictly rely on Bayes’ theorem but instead learns only when the overall model is correctly specified. By integrating a linear-Gaussian approximation, the method establishes an efficient iterative update mechanism, yielding a variant of the extended Kalman filter termed EKF-PrO. This framework requires no hyperparameters, is computationally efficient, and automatically adapts to model misspecification. Experimental results demonstrate that, across various scenarios involving both linear and nonlinear model misspecifications, EKF-PrO achieves substantially improved inference robustness while maintaining computational costs comparable to existing methods.

This work addresses the degraded accuracy and limited scalability of traditional ensemble Kalman filters (EnKF) in discrete-time nonlinear filtering under model misspecification or poor initialization. To overcome these limitations, the paper proposes the Exact Ensemble Kalman Filter (ExEnKF), which, for the first time within the EnKF framework, replaces Dirac measures with Gaussian approximations to more effectively explore the state space while preserving computational efficiency. Theoretical analysis establishes that ExEnKF converges to the optimal filter at a rate of $1/\sqrt{N}$, where $N$ is the ensemble size. Numerical experiments demonstrate its superior performance over standard EnKF and sequential Monte Carlo methods in highly stochastic, model-mismatched, and multiscale Lorenz-96 systems, particularly in robustly tracking unobservable hidden state components. This approach thus offers a practical, scalable, and accurate solution for high-dimensional nonlinear filtering.

This work proposes a proprioception-based real-time state estimation method for humanoid robots operating without external sensors or prior knowledge of ground motion. By fusing foot-mounted IMU measurements with kinematic constraints, the authors formulate a right-invariant extended Kalman filter (InEKF) and, for the first time, introduce a right-invariant observation model to non-inertial ground scenarios, thereby ensuring system observability and enabling rapid convergence. Experimental validation on the Digit robot demonstrates a 96% improvement in convergence speed and an 80% reduction in position estimation error compared to existing approaches. Notably, even when walking on a rotating surface with an initial position error as large as one meter, the method maintains an average position estimation error below 9 centimeters.