Error-State Kalman Filtering

• Error-State Kalman Filtering is a method that decomposes the system state into a nominal trajectory and a small error-state, enabling robust linearization for nonlinear systems.
• It utilizes additive, invariant, and high-order Taylor expansion variants to improve stability in six-degree-of-freedom motion and complex sensor fusion tasks.
• Empirical studies demonstrate substantial reductions in position and orientation errors, making it effective for applications such as inertial navigation, edge XR, and UAV state estimation.

Error-state Kalman filtering (ESKF) constitutes a principled approach for recursive state estimation of nonlinear systems, particularly in scenarios involving orientation and inertial navigation. Unlike classical Kalman filtering, ESKF estimates the evolution of the error between a nominal (predicted) trajectory and the true system state, enabling robust linearization and enhanced stability—especially for systems with complex rotational or multi-rate structure. ESKF variants, including classical (additive), invariant (multiplicative, Lie group-based), and high-order Taylor expansion-based forms, address the challenges of high-dynamic motion, sensor fusion, global- and body-frame switching, and real-time deployability (Zhong et al., 17 Jul 2025, Han et al., 1 Nov 2025, Ye et al., 2023).

1. System and Error-State Formulation

The ESKF paradigm decomposes the system state $x$ into a nominal trajectory $\bar{x}$ and a small error-state $\delta x$ such that $x \approx \bar{x} \boxplus \delta x$. For six-degree-of-freedom motion in $\mathbb{R}^3 \times S^3$ (with position, velocity, acceleration, quaternion, angular velocity, and angular acceleration), the error-state is defined as: $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$ where $\delta \theta = 2\,\log(q \otimes \bar{q}^{-1})$ is the minimal $\mathrm{SO}(3)$ rotation vector.

For systems evolving on Lie groups (e.g., $SE_2(3)$ for full pose, $SO(3)$ for attitude), the error-state may be defined right- or left-invariantly: $\bar{x}$0 (right-invariant) or $\bar{x}$1 (left-invariant), where $\bar{x}$2 collects all physical states as an element of the matrix Lie group and $\bar{x}$3 is the minimal vector error (Ye et al., 2023).

This separation enables all state updates to be performed via group operations or minimal-coordinate updates, and ensures that linearization remains accurate provided errors are small.

2. High-Order Error-State Dynamics and Discrete-Time Propagation

Classical ESKF employs a first-order Taylor expansion, yielding linearized error-state dynamics: $\bar{x}$4 with $\bar{x}$5 (Jacobian), $\bar{x}$6 process noise mapping, and $\bar{x}$7 white noise. High-order ESKF, as introduced in motion prediction for edge XR, systematically expands the dynamics as a Taylor series in $\bar{x}$8 up to the third order: $\bar{x}$9 where $\delta x$0 and $\delta x$1 are the second and third derivative tensors, respectively (Zhong et al., 17 Jul 2025). The resulting discrete-time update for the $\delta x$2th prediction step employs series-truncated Magnus-Taylor expansions for the state transition and higher-order error-state terms (quantified as $\delta x$3).

Covariance propagation follows: $\delta x$4 where $\delta x$5 is the state transition, $\delta x$6 the process-noise covariance, and higher-order corrections are typically truncated beyond the second or third order for computational tractability.

For right-invariant ESKF on Lie groups, the propagation is characterized by exponential maps: $\delta x$7 enabling consistent scaling and invariance to large initial state errors (Ye et al., 2023).

3. Measurement Update and Covariance Transformation

Measurement updates in ESKF are linearized around the nominal state: $\delta x$8 with $\delta x$9 the Jacobian of $x \approx \bar{x} \boxplus \delta x$0 at $x \approx \bar{x} \boxplus \delta x$1. The innovation adopts a minimal, invariant representation, for example $x \approx \bar{x} \boxplus \delta x$2 for pose and orientation.

The Kalman gain $x \approx \bar{x} \boxplus \delta x$3 and posterior covariance $x \approx \bar{x} \boxplus \delta x$4 are then computed as per standard Kalman filter algorithm, and the correction $x \approx \bar{x} \boxplus \delta x$5 is "injected" back into the nominal state through a retraction (addition for $x \approx \bar{x} \boxplus \delta x$6, group composition for quaternions or general Lie group structure).

Covariance transformation-based ESKF (CT-ESKF) unifies error-state variants by mapping the covariance across different error-state definitions. Let $x \approx \bar{x} \boxplus \delta x$7 relate two error coordinates with $x \approx \bar{x} \boxplus \delta x$8. After each update, one applies: $x \approx \bar{x} \boxplus \delta x$9 to ensure the propagated covariance aligns with the target error-state basis, crucial for robust multi-sensor integration—especially in fusing global-frame (GNSS) and body-frame (ODO) observations (Han et al., 1 Nov 2025).

4. Invariance Principles and Trajectory Independence

Invariant ESKF (InEKF) leverages the group-affine properties of propagation and observation models, yielding error-state dynamics and measurement Jacobians that are independent (or nearly so) of the estimated state trajectory. For $\mathbb{R}^3 \times S^3$0,

$\mathbb{R}^3 \times S^3$1

with most blocks state-independent except for entries related to non-group variables (e.g., biases) (Ye et al., 2023). The result is improved stability, rapid convergence (e.g., $\mathbb{R}^3 \times S^3$210 s after large attitude bias), and immunity to poor initialization as long as group-affine conditions are satisfied.

Analytical results show that, for propagation, EKF, L-InEKF, and R-InEKF all transmit the same information up to the coordinate basis, with differences arising only in Jacobian structure and their interaction with nonlinear measurement models. CT-ESKF enables switching between these bases without switching the core filter, preserving both consistency and information (Han et al., 1 Nov 2025).

5. Implementation Complexity and Real-Time Performance

The computational trade-offs of ESKF frameworks are determined by the order of expansion, state-space dimension, and the use of efficient Lie group operations. High-order ESKF (e.g., PsudoESKF) with $\mathbb{R}^3 \times S^3$3 states carries $\mathbb{R}^3 \times S^3$4 cost for Jacobian calculation, and $\mathbb{R}^3 \times S^3$5 for series expansion or covariance updates. High-order tensors ($\mathbb{R}^3 \times S^3$6, $\mathbb{R}^3 \times S^3$7) induce $\mathbb{R}^3 \times S^3$8 and $\mathbb{R}^3 \times S^3$9 storage requirements, motivating selective truncation (quaternion blocks only or up to $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$0 terms), the use of directional derivative approximations, or neglecting pure cross-terms to balance accuracy and compute demands (Zhong et al., 17 Jul 2025).

Quaternion integration schemes (Zed12/Zed23) are employed to amortize costs of the exponential map, implicitly incorporating third-order corrections.

Empirical benchmarks confirm that per-step latency (sub–1 ms on Apple M1 CPUs) remains well below 10 ms sampling intervals, supporting real-time deployment for high-rate edge XR applications.

6. Comparative Performance and Applications

Extensive empirical studies validate ESKF performance in diverse application settings:

• Edge XR motion prediction: High-order ESKF (PsudoESKF) with third-order expansion achieves a reduction in position error (100 ms horizon, hard motion) from $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$154.1 mm (KF) to $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$217.8 mm (67% reduction), and orientation error from $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$3 (KF) to $$\delta x = \begin{bmatrix} \delta p \ \delta v \ \delta a \ \delta \theta \ \delta \omega \ \delta \alpha \end{bmatrix} \in \mathbb{R}^{15},$$4 (50% reduction). Under 50% packet loss, orientation error is reduced by 49.6% vs. KF (Zhong et al., 17 Jul 2025).
• Multi-sensor navigation: CT-EKF achieves attitude RMSE better than EKF, L-InEKF, or R-InEKF (sub-degree accuracy), even under large initial attitude error or slow IMU propagation (e.g., 2 Hz) (Han et al., 1 Nov 2025).
• UAV full-state estimation: The ES-RIEKF delivers 10% improvement in attitude MAE/RMSE and 4% improvement in position RMSE compared to standard ES-EKF; robustness is maintained during extended GNSS outages (maximum position error within 30 m over a 130 s GNSS denial) (Ye et al., 2023).

A plausible implication is that targeted selection of error basis and order of expansion, coupled with efficient Lie group computation, is essential to obtain robust, accurate, and real-time-capable Kalman filtering for sensor fusion tasks spanning robotics, inertial navigation, and edge compute vision systems.

7. Extensions and Synthesis with Learned or Aerodynamic Models

Contemporary approaches incorporate model-aided and data-driven augmentations, e.g., LSTM networks for drift-free prediction of aerodynamic angles (angle of attack/side-slip) using control-surface deflections and IMU data. These learned predictions integrate as pseudo-measurements within the invariant ESKF framework, reducing reliance on external (e.g., GNSS) sensors, and preserving estimation consistency under denial or degradation (Ye et al., 2023).

Moreover, entropy-based or confidence-driven classifiers can be integrated for motion predictability assessment, enabling dynamic adjustment of filter order, detection of low-predictivity regimes, and informed switching between computationally intensive and lightweight variants (Zhong et al., 17 Jul 2025). This convergence of model-based ESKF and adaptive or learning-aided components suggests a general trend toward hybrid architectures for resilient, high-accuracy navigation and state estimation.

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Key references:

• "Predictability-Aware Motion Prediction for Edge XR via High-Order Error-State Kalman Filtering" (Zhong et al., 17 Jul 2025)
• "CT-ESKF: A General Framework of Covariance Transformation-Based Error-State Kalman Filter" (Han et al., 1 Nov 2025)
• "Semi-Aerodynamic Model Aided Invariant Kalman Filtering for UAV Full-State Estimation" (Ye et al., 2023)