Error-State Kalman Filter (ESKF)

• ESKF is a recursive Bayesian estimator that decomposes the state into a nominal trajectory and a linearly modeled error state, improving linearization accuracy.
• It leverages manifold geometry and Lie group operations to ensure statistical and geometric consistency through techniques like parallel transport and retraction.
• Widely applied in robotics and navigation, the ESKF enhances real-time state estimation in systems such as visual-inertial odometry and SLAM.

The error-state Kalman filter (ESKF) is an advanced recursive Bayesian estimator designed for state estimation in nonlinear, high-dimensional, and often manifold-valued dynamic systems. By decomposing the estimation problem into a nominal (nonlinear) state and an error state (modeled and propagated linearly), the ESKF achieves significant improvements in linearization accuracy, robustness, and computational tractability, particularly for systems with rotational states on Lie groups such as SO(3) or SE(3). Geometric variants of the ESKF, including those defined with respect to affine connections or group-invariant error metrics, address issues that arise when naively applying Euclidean-space Kalman filtering to manifold-valued states, ensuring both geometrical and statistical consistency across prediction and update stages.

1. Formal Structure and Theoretical Foundations

An ESKF propagates a nominal or reference state $\hat x$ according to the full nonlinear system dynamics, while the error-state $\delta x$ parameterizes the local deviation from $\hat x$, with the true state given by $x_{\mathrm{true}} = \hat x \boxplus \delta x$ where $\boxplus$ denotes the appropriate composition (e.g., vector addition, quaternion multiplication, or group exponential) depending on the underlying state manifold (Im, 2024). The error-state is initialized and repeatedly reset to zero after each corrector step to maintain linearization validity.

The error-state dynamics are derived from first-order Taylor linearization of the true nonlinear process about the nominal trajectory. For piecewise affine systems, the error-state propagation is given by

$\delta x_{k+1} = F_k \delta x_k + G_k w_k,$

where $F_k$ and $G_k$ are the Jacobians of the nonlinear system dynamics with respect to $\delta x$ and the process noise $w_k$ at the nominal operating point, respectively. The associated covariance $P_{k+1}$ is propagated accordingly, preserving the structure and minimality of local perturbations (Im, 2024, Solà, 2017).

2. Geometric Consistency and Operation on Manifolds

Classical EKF implementations on Euclidean coordinates disregard the intrinsic geometry of Lie groups or smooth manifolds, leading to systematic inconsistencies. The ESKF, when rigorously designed, leverages the minimal geometric requirement of an affine connection to define geodesics, exponential and logarithmic maps, and parallel transport. The error state $\epsilon_k = \xi_k \boxminus \hat \xi_k$ is an element of the tangent space $T_{\hat \xi_k} \mathcal{A}$ at the current estimate on the manifold $\mathcal{A}$ (Ge et al., 2023).

Geometric ESKF frameworks include the following critical corrections:

• Update Step: Measurement noise covariances are parallel transported from their frame at the true (unknown) state to the tangent at the nominal state, ensuring that the statistical fusion of priors and likelihoods occurs in a common vector space (Ge et al., 2023).
• Reset Step: After injecting the mean error correction via geodesic retraction, the a-posteriori covariance must be parallel transported from the old to the new tangent space, preserving the geometric meaning of uncertainty after the state estimate is shifted (Ge et al., 2023).
• Box-Plus/Minus Operators: The box-plus ($\boxplus$) and box-minus ($\boxminus$) operations generalize additive perturbations using manifolds' exponential and logarithmic maps, e.g., $R \boxplus \epsilon = R\,\exp(\epsilon^\wedge)$ for $SO(3)$ (Ge et al., 2023).

This approach generalizes naturally to any manifold admitting an affine connection, including all matrix Lie groups (e.g., $SO(3)$, $SE(3)$) (Ge et al., 2023).

3. Algorithmic Workflow

A canonical ESKF iteration comprises the following sequence (Im, 2024, Ge et al., 2023):

1. Prediction (Propagation):
   • Advance $\hat x$ (nominal state) using the nonlinear system model.
   • Propagate error covariance $P$ using the linearized error dynamics.
2. Measurement Update:
   • Compute the innovation using the measurement model linearized at $\hat x$.
   • Apply the Kalman gain to update the error-state mean and covariance.
   • Parallel transport (if on a manifold) the measurement covariance into the nominal's tangent frame.
   • Inject the corrected error state into the nominal state with manifold retraction ($\boxplus$).
3. Reset:
   • Set $\delta x \coloneqq 0$.
   • Parallel transport (if manifold-valued) the covariance to the new tangent.
   • Optionally apply the Joseph-form for covariance update to ensure numerical stability.

The only nonstandard elements introduced by manifold structure awareness are the two parallel-transport operations on the measurement and state covariances (Ge et al., 2023).

4. Practical Implementation and Advantages

The ESKF has been widely adopted in robotics, visual-inertial odometry, navigation, and aerospace state estimation, including high-rate inertial sensor fusion, visual SLAM, LiDAR SLAM drift compensation, and multi-sensor localization (Asil et al., 19 Dec 2025, Wu et al., 14 May 2025, Parikh et al., 2024, Markovic et al., 2021, Kabiri et al., 2024). Its practical advantages over the conventional EKF include:

• Linearization around Small Error: The error-state is maintained near zero, resulting in more accurate linearization and fewer higher-order artifacts.
• Minimal Parameterization: The error-state for orientation is typically a 3-vector on $SO(3)$, avoiding singularities associated with overparameterized representations.
• Consistency and Stability: Handling the error-state in the correct tangent space reduces the risk of inconsistent covariance growth and estimator divergence (Han et al., 1 Nov 2025).
• Real-Time Performance: ESKF maintains $O(n^2)$ complexity per time step, making it suitable for embedded systems with strict computation constraints (Kabiri et al., 2024).

5. Extensions: Invariant, Transformed, and Hybrid ESKF Variants

Several ESKF variants address additional theoretical and practical limitations:

• Invariant ESKF (InEKF): Defines the error-state via group operations (left or right invariance), leading to filter linearizations independent of the current estimate, enhancing global consistency, and providing provable stability for certain group-affine systems (C. et al., 2024, Ye et al., 2023).
• Transformed ESKF (T-ESKF): Applies a state-dependent linear transformation to the error-state, rendering the unobservable subspace constant (typically in visual-inertial navigation), thereby preventing inconsistency due to linearization point dependence and observability mismatch (Tian et al., 27 Oct 2025).
• Covariance Transformation ESKF (CT-ESKF): Unifies different ESKF formulations by showing that error-state and covariance pairs related via a linear transform yield equivalent filtering statistics; this allows for trajectory-independent linearization even in the presence of both global- and body-frame observations (Han et al., 1 Nov 2025).
• Hybrid Qf-ES-EKF/UKF: Combines ESKF propagation for all states with a computationally efficient UKF refinement restricted to the orientation error block, balancing accuracy and computational load in VIO applications (Asil et al., 19 Dec 2025).

The following table summarizes key properties of several ESKF variants:

Variant Name	Error-State Definition	Key Benefit
Standard ESKF	Local tangent at $\hat x$	Efficient, minimal-param error, practical
InEKF (L/R)	Group-invariant error	Trajectory independence, stability
CT-ESKF	Covariance transform	Unifies trajectory independence
T-ESKF	State-dependent transform	Consistent observability, fast propagation
Qf-ES-EKF/UKF	ESKF + UKF on orientation	Improved accuracy for quaternions

6. Application Domains and Empirical Outcomes

The ESKF is foundational in visual-inertial navigation, SLAM, multi-rate sensor fusion, multi-agent and satellite pose estimation, and mobile robotics. Systematic empirical evaluations demonstrate:

• Enhanced accuracy: Position and rotation RMSEs are systematically reduced in challenging conditions compared to classical EKF and batch-based methods (Asil et al., 19 Dec 2025, Wu et al., 14 May 2025, Kabiri et al., 2024, Wang et al., 2023).
• Real-time performance: Capable of fusing high-rate IMU and lower-rate exteroceptive sensors within tight resource budgets.
• Robustness: Adaptive tuning and outlier rejection (e.g., via Mahalanobis gating or dynamic sensor confidence) further improve filter stability and observability in the presence of environmental degradation or modeling uncertainty (Asil et al., 19 Dec 2025, Parikh et al., 2024).
• Consistency: Extensions such as T-ESKF and CT-ESKF maintain correct observability properties, eliminating false observability in the presence of gauge freedoms (Tian et al., 27 Oct 2025, Han et al., 1 Nov 2025).

7. Limitations, Implementation Notes, and Best Practices

While ESKF achieves theoretical and practical improvements, several limitations and implementation guidelines remain (Parikh et al., 2024, Han et al., 1 Nov 2025, Ge et al., 2023):

• Geometric Corrections: Failure to properly account for manifold structure in covariance update and reset leads to inconsistencies and transient overshoot, especially in pose/attitude estimation (Ge et al., 2023).
• Error-State Initialization: Large initial errors can challenge linearization accuracy; invariant variants or staged initialization strategies may mitigate divergence (Han et al., 1 Nov 2025).
• Sensor Model Calibration: Biases and extrinsic calibrations (e.g., IMU-to-camera, sensor drift) must be included explicitly, often as random-walk states.
• Numerical Stability: Use Joseph-form for covariance update, gating for outlier rejection, and normalization for unit quaternion states (Kabiri et al., 2024, Parikh et al., 2024).
• Parallel Transport and Lie Group Exponential/Logarithm: Implementations must support group exponential/log (e.g., Rodrigues' formula for SO(3)), and efficient/sparse computation of parallel transport when working on high-dimensional manifolds or large landmark maps (Ge et al., 2023).
• Tuning and Adaptation: Process and measurement noise covariances must be tuned to application specifics. Adaptive approaches may be required in environments with non-stationary sensor reliability or model drift (Asil et al., 19 Dec 2025).

For systems on Lie groups, group-affine error propagation and parallel transport are essential to maintain both geometric and statistical consistency (Ge et al., 2023, C. et al., 2024). In mixed-sensor scenarios (e.g., fusing GNSS with body-frame odometry), CT-ESKF enables seamless switching between invariant error bases without loss of statistical equivalence (Han et al., 1 Nov 2025).

---

References:

• (Ge et al., 2023) "A Note on the Extended Kalman Filter on a Manifold"
• (Im, 2024) "Notes on Kalman Filter (KF, EKF, ESKF, IEKF, IESKF)"
• (Han et al., 1 Nov 2025) "CT-ESKF: A General Framework of Covariance Transformation-Based Error-State Kalman Filter"
• (Tian et al., 27 Oct 2025) "T-ESKF: Transformed Error-State Kalman Filter for Consistent Visual-Inertial Navigation"
• (Asil et al., 19 Dec 2025) "Adaptive Covariance and Quaternion-Focused Hybrid Error-State EKF/UKF for Visual-Inertial Odometry"
• (Kabiri et al., 2024) "Graph-Based vs. Error State Kalman Filter-Based Fusion Of 5G And Inertial Data For MAV Indoor Pose Estimation"
• (Parikh et al., 2024) "Pose estimation of CubeSats via sensor fusion and Error-State Extended Kalman Filter"
• (Ye et al., 2023) "Semi-Aerodynamic Model Aided Invariant Kalman Filtering for UAV Full-State Estimation"
• (C. et al., 2024) "Invariant Kalman Filter for Relative Dynamics"
• (Wang et al., 2023) "EDI: ESKF-based Disjoint Initialization for Visual-Inertial SLAM Systems"
• (Solà, 2017) "Quaternion kinematics for the error-state Kalman filter"