Extended Kalman Filter (EKF) Overview

• Extended Kalman Filter (EKF) is a nonlinear state estimator that linearizes process and measurement models to compute optimal, minimum-variance state predictions under Gaussian noise assumptions.
• Modern EKF variants extend to infinite-dimensional systems and manifold settings, incorporating techniques like geometric filtering and robust outlier rejection for improved accuracy.
• Adaptive and data-driven enhancements, including SINDy-EKF and neural network-based noise tuning, significantly boost real-time sensor fusion and navigation performance.

The extended Kalman filter (EKF) is a nonlinear generalization of the classical Kalman filter, widely employed for recursive state estimation in nonlinear systems where both process and measurement models exhibit nonlinearity. Originating as a solution for finite-dimensional, discrete-time nonlinear systems with additive Gaussian noise, the EKF proceeds by linearizing the transition and measurement maps about the current estimate, enabling a prediction–update cycle that integrates model dynamics and noisy observations to produce optimal (minimum-variance) estimates under locally Gaussian assumptions. Modern EKF variants now address infinite-dimensional systems, geometric state spaces (manifolds), outlier-robust filtering, consistent estimation under symmetry-induced unobservabilities, and hybrid model–data approaches leveraging sparse regression or deep learning.

1. System Models and EKF Formulation

The canonical EKF operates on a discrete-time, nonlinear stochastic system with state $\chi_k\in\mathbb{R}^n$, governed by

$\chi_{k+1} = f_k(\chi_k) + w_k$

where $f_k:\mathbb{R}^n\to\mathbb{R}^n$ is $C^1$, and $w_k\sim\mathcal{N}(0,Q)$ is process noise. Measurements take the general form

$\zeta_k = g_k(\chi_k) + v_k$

with $g_k:\mathbb{R}^n\to\mathbb{R}^m$, $v_k\sim\mathcal{N}(0,R)$ measurement noise.

In standard applications, the EKF proceeds via:

• Prediction:
  • $\hat x_{k|k-1} = f_{k-1}(\hat x_{k-1})$
  • $P_{k|k-1} = F_{k-1} P_{k-1} F_{k-1}^\top + Q$
• Update:
  • Compute measurement Jacobian: $$G_k = \frac{\partial g_k}{\partial\chi}\Bigr|_{\chi=\hat x_{k|k-1}}$$
  • Kalman gain: $$K_k = P_{k|k-1} G_k^\top (G_k P_{k|k-1} G_k^\top + R)^{-1}$$
  • State and covariance updates:
  • $$\hat x_k = \hat x_{k|k-1} + K_k (\zeta_k - g_k(\hat x_{k|k-1}))$$
  • $P_k = (I - K_k G_k) P_{k|k-1}$

This first-order linearization is performed at each step, with covariances representing local uncertainty (Tellex et al., 2018, Titensky et al., 2018).

2. Infinite-Dimensional and Random-Field EKF Extensions

Recent advances extend EKF methodology to systems with infinite-dimensional measurements, inspired by vision-based localization scenarios in robotics. The state remains finite-dimensional, but observations are modeled as vector fields or images: $$\zeta_k(i) = g_k(\chi_k, i) + v_k(i),\quad i\in\mathbb{R}^d$$ where $g_k$ is $C^1$ in state and $v_k$ is a zero-mean, wide-sense stationary random field with covariance $R(i-i')$. Filter equations generalize to infinite-dimensional operators—measurement Jacobians $G_k(i)$ become mappings from state to pixel space, and updates integrate over the entire measurement domain.

Fourier-transform techniques are used to derive gain kernels and innovation weighting. For camera-based systems, the measurement Jacobian operator $G_k(i)$ becomes: $$G_k(i) = \nabla C(p(\hat\chi_{k|k-1},i))^\top\, J_p(i)$$ with $\nabla C$ the image gradient and $J_p$ the projection Jacobian, providing a system-theoretic justification for the ubiquitous use of image gradients in vision-based EKF pipelines (Varley et al., 23 Sep 2025).

3. EKF on Manifolds and Geometric Filtering

Classical EKF implementations on nonlinear manifolds (e.g., $SO(3)$, $SE(3)$, $SE_2(3)$) relying on naive coordinate charts suffer from non-intrinsic behavior—updates depend on coordinate choice and fail to respect manifold geometry. Geometric EKF variants replace chart-dependent linearizations with constructions involving affine connections, parallel transport, and geodesics:

• Normal coordinates (via exponential and logarithmic map)
• Covariance and noise transport between tangent spaces using parallel transport
• Measurement and process update equations intrinsically defined on the manifold

Implementation ensures that the covariance and mean are always represented in the correct tangent space. Such geometric modifications yield consistent performance, improved transient accuracy, and eliminate spurious confidence effects in highly nonlinear regimes (Ge et al., 2023, Ge et al., 6 Jun 2025, Goodarzi et al., 2016).

4. Robust and Consistent EKF Variants

Outlier-Resilient EKF:

The innovation-saturated EKF (IS-EKF) mitigates the influence of measurement outliers by adaptively clipping the innovation vector using saturation bounds tuned by recursive statistics of recent innovations. This preserves bounded estimation error under arbitrarily large or dense outliers without increasing computational complexity. Theoretical guarantees are provided in both continuous and discrete-time formulations, and empirical results demonstrate strong rejection of GPS faults and sensor anomalies in robotics scenarios (Fang et al., 2019).

Consistency under Symmetries:

Unobservable directions, induced by system symmetries, can result in overconfident and inconsistent EKF estimates, particularly in SLAM and dead-reckoning. Invariant EKF (IEKF) approaches resolve this by selecting state errors compatible with the system's transformation group, ensuring unobservable modes are preserved in the linearized observability matrix and the corresponding covariance. Experimental studies in multi-robot SLAM show up to 50% error reduction compared to standard EKF, matching optimization-based MAP methods (Brossard et al., 2019).

5. Adaptive and Data-Driven EKF Design

Online Process Noise Tuning:

Prediction accuracy and stability of EKF are critically sensitive to the correct setting of process noise covariances $Q_k$. Adaptive approaches include innovation-covariance matching, scaling via innovation statistics, and more recently, direct online learning of scale factors using transformer-based neural networks. The Adaptive Kalman-Informed Transformer (A-KIT) dynamically infers diagonal process noise scalings from recent sensor and process history, optimizing EKF accuracy in complex INS/DVL fusion tasks for underwater vehicles, achieving >49% improvement over static EKF and outperforming state-of-the-art adaptive methods (Cohen et al., 18 Jan 2024).

Sparse Identification of Dynamics (SINDy-EKF):

For nonlinear systems lacking tractable physical models, the EKF can be combined with data-driven discovery of the evolution model via SINDy regression. SINDy-EKF leverages sparse polynomial representations learned from system trajectories, providing analytic Jacobians for fast prediction and filtering. This methodology extends to partially observed systems using time-delay embedding, enabling robust online estimation of both state and parameters and supporting digital twins of complex mechanical structures (Rosafalco et al., 11 Apr 2024).

6. Practical Implementation and Empirical Performance

EKF frameworks now permeate real-time vision-based localization, sensor fusion for drones and underwater vehicles, and uncertainty propagation in deep neural networks. Dense-image EKF filtering, leveraging all pixels and spatial correlations, matches or outperforms sparse-feature methods in high-noise environments. Coordinated multi-sensor fusion in autonomous competition robots (odometry, visual landmarks, physics models) yields near-optimal state fusion and substantial drift reduction (Varley et al., 23 Sep 2025, Kou et al., 2023, Titensky et al., 2018, Balasubramanian et al., 2021, Tellex et al., 2018).

Empirical benchmarks:

• Dense infinite-dimensional EKF reduces mean position MSE by up to an order of magnitude against VINS-MONO in aerial drone flights (Varley et al., 23 Sep 2025).
• Robust IS-EKF maintains bounded error in all outlier regimes for mobile robot localization (Fang et al., 2019).
• Manifold-geometric EKFs halve the transient normalized estimation error compared to classical chart-based filters (Ge et al., 6 Jun 2025).
• Deep learning-guided process noise adaptation yields >49% position RMSE reduction over conventional EKF in underwater navigation case studies (Cohen et al., 18 Jan 2024).

7. Theoretical Guarantees and Limitations

The EKF delivers optimal (minimum error variance) state estimation under local linearization, Gaussian noise, and mild system smoothness assumptions. Infinite-dimensional extensions require square-integrability and invertibility of random-field noise operators; manifold filters must respect affine connection structures for intrinsic propagation and update equations. Filter performance degrades under high nonlinearity or poor model/measurement noise specification; adaptive methods and data-driven dynamics identification improve real-world robustness, but may require substantial representative training data or regularization.

In summary, the extended Kalman filter continues to evolve as a central tool for recursive nonlinear estimation, unifying system-theoretic modeling, geometric state-space methods, adaptive uncertainty quantification, and empirical data-driven approaches—addressing finite- and infinite-dimensional systems, manifold-valued states, measurement outliers, and practical implementation challenges across diverse application domains (Varley et al., 23 Sep 2025, Ge et al., 6 Jun 2025, Fang et al., 2019, Cohen et al., 18 Jan 2024, Rosafalco et al., 11 Apr 2024, Brossard et al., 2019, Goodarzi et al., 2016, Titensky et al., 2018, Kou et al., 2023, Afshar et al., 2022).