Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems

Abstract: Global navigation systems require state estimation algorithms that handle Earth's curvature, Earth's rotation, and gravitational variations. These factors can typically be neglected in local navigation algorithms for robots, drones, etc. In classical error-state Kalman Filtering (ESKF) the error state dynamics are trajectory-dependent. Invariant ESKFs utilize Lie Group symmetries to represent the error, which can render error propagation trajectory-independent for group-affine systems. Choosing between a standard filter (where position and velocity errors are defined additively in the navigation frame), a left-invariant filter (where errors are represented in the body frame) and a right-invariant filter (where errors are represented in the navigation/world frame) depends on system dynamics and sensor configuration. This note presents the mathematical formulas for four classical and invariant ESKFs for globally applicable aided inertial navigation systems. It is intended to serve as a systematic reference for comparison and implementation.

• The paper presents four ESKF architectures (geodetic, additive, left-, and right-invariant) that address global navigation challenges through rigorous derivations.
• It details continuous-time error propagation, the use of Lie group tools, and contrasts NED versus ECEF coordinate effects for accurate state estimation.
• Through anchor-based formulations and invariant error modeling, the study offers practical strategies for robust performance in high-dynamic and low-update-rate scenarios.

Error-State Kalman Filter Kinematic Derivations for Global Aided Inertial Navigation

Overview and Motivation

The paper "Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems" (2607.03211) provides a comprehensive and formal treatment of the error-state Kalman filter (ESKF) kinematics required for globally robust aided inertial navigation. It addresses the modeling complexities introduced by planetary-scale phenomena (Earth’s curvature, rotation, and gravitational variation) and emphasizes the necessity of mathematically rigorous filter formulations for numerical stability and estimation consistency in high-dynamic applications such as aerospace and long-range platform navigation.

The main contribution is the systematic derivation, direct comparison, and clarification of four foundational global ESKF structures: classical geodetic (NED), additive ECEF, left-invariant ESKF, and right-invariant ESKF, all accounting for continuous-time error-state propagation, coordinate system selection, and Lie-group-based error modeling.

Preliminaries: Model Structure and Lie Group Framework

Fundamental to global inertial navigation state estimation is the treatment of attitude, position, and velocity in a non-Euclidean geometry appropriate for the Earth's rotating and curved reference frames. The paper unifies notation and provides rigorous definitions for all relevant states, biases, and stochastic disturbances. Two distinct coordinate systems are considered:

• Geodetic/NED (North-East-Down): Suitable for local navigation wherein Earth rotation and curvature can often be neglected.
• ECEF (Earth-Centered, Earth-Fixed): Required for global applications where those effects become significant.

The authors present the continuous-time navigation equations in both systems, accommodating IMU error processes (driven by first-order Gauss-Markov dynamics). They introduce the requisite Lie group tools for invariant error-state propagation: $SO(3)$ for attitude and $SE_2(3)$ for the full pose (attitude, velocity, position). The treatment formalizes the exponential maps and associated Jacobians for both right- and left-invariant error representations, necessary for exact covariance resets and consistency under nonlinear propagation [Sola2021, Barfoot_2024].

Error-State Kalman Filter Formulations

Classical Geodetic (NED) ESKF

This formulation follows the standard procedure of defining all errors (position, velocity, attitude, biases) additively in their respective local NED frames. The attitude update utilizes a left-multiplicative quaternion correction but is linearized for the ESKF. The system Jacobian $F$ encodes the coupling introduced by Earth's rotation and local curvature via meridian and normal radii, and the dependency on trigonometric functions introduces singularities at $\mu = \pm 90^\circ$. The measurement update is inherently well-conditioned for navigation-frame sensors (e.g., GNSS), with trivial Jacobians.

Key limitations include:

• Trajectory-dependent linearization and error-state propagation
• Ill-conditioning near the poles

Additive ESKF in ECEF

Here, position and velocity errors are defined additively in ECEF coordinates, while attitude errors are represented as body-frame vectors on $SO(3)$, also using a right perturbation. This formulation is computationally attractive and straightforward to implement but does not exploit the group-affine structure of the problem. The $F$ matrix maintains trajectory dependence due to the presence of state-dependent cross-terms. Measurement Jacobians for world-frame sensors are trivial, ensuring stable updates, but long propagation intervals or high-dynamics can introduce unmodeled error coupling.

Left-Invariant (LI) $SE_2(3)$ ESKF in ECEF

This filter leverages the left-invariant structure of the inertial navigation equations (inputs expressed in the body frame). Attitude, position, and velocity errors are defined multiplicatively (using group actions) in the body frame, enabling trajectory-independent error-state propagation for group-affine systems. The resulting system matrix $F$ does not depend on the trajectory or the current navigation state, provided bias-velocity coupling is managed. Covariance reset is performed using the full $SE_2(3)$ right Jacobian, ensuring exact consistency under the nonlinear group operations.

Advantages:

• Trajectory-independence of propagation (for ideal cases)
• Improved numerical stability over large propagations or low measurement rates

However, measurement models in navigation frames (e.g., GNSS in ECEF) are more complex, as the Jacobian contains nontrivial attitude-to-position couplings, potentially destabilizing the filter if the attitude estimate is poor.

Right-Invariant (RI) $SE_2(3)$ ESKF in ECEF

This structure defines errors in the navigation/world frame via left-multiplicative error composition. While measurement updates for navigation-frame sensors become structurally simple, the error-state propagation is not group-affine for the body-driven inertial system; thus, $SE_2(3)$0 includes terms proportional to the vehicle's absolute position and velocity. These entries can numerically dominate, leading to ill-conditioned propagation and spurious growth in uncertainty. The authors detail an "anchored" formulation, subtracting a local ECEF anchor to mitigate these issues. Inclusion of the global gravity gradient further increases the structural fidelity and improves attitude observability.

Observations:

• Full group-affinity is violated due to bias coupling and nonconstant gravity
• Anchor-point formulation is essential for numerical tractability over global domains

Filter Implementation and Practical Considerations

The paper emphasizes that in low measurement rate or GPS-denied environments, the choice of error modeling (additive vs invariant) and covariance reset is critical for filter performance. The left-invariant filter is generally preferred for inertial navigation because it aligns with the underlying dynamics and enables invariant error propagation, suppressing trajectory-induced errors in the absence of frequent updates. Conversely, right-invariant and additive ESKF formulations are disadvantaged in high-dynamic or global scenarios due to the aforementioned numerical pathologies.

The authors rigorously derive all Jacobians required for closed-form update and reset operations, and explicitly detail the conditions under which each architecture retains (or loses) its invariance properties. They also clarify how measurement frame selection (body, navigation, ECEF, geodetic) interacts with error-state definitions, impacting both filter observability and numerical conditioning.

Theoretical and Practical Implications

This formalization advances the community's understanding of error-state observability, propagation consistency, and numerical stability in globally scoped inertial navigation. The results reinforce that for group-affine systems, particularly when the system and error structure are aligned, invariant ESKF architectures avoid performance degradation associated with unmodeled trajectory dependencies. The explicit anchor-point treatment for ECEF right-invariant filters is a critical computational detail for implementation in large-scale systems.

While the analysis is targeted towards classical navigation sensors and models, the presented formalism provides a theoretical foundation for future integration with multimodal or learned sensor models (magnetic, visual, quantum) and for adaptive or variable structure filters. The comprehensive derivations and comparison offer a ready reference for implementers of global-scale navigation systems, including GNSS-denied and alternative PNT platforms.

Conclusion

The systematic derivations and rigorous comparison presented in this paper (2607.03211) fully clarify the structure, properties, and numerical considerations of four globally applicable ESKF architectures. The analysis underscores the necessity of aligning error-state definitions with system dynamics and coordinate frames, especially for long-range or low-update-rate navigation. Future work can exploit these foundational derivations to extend error-state filtering to hybrid sensor suites and more general Lie-group-based estimation scenarios, as well as to adaptive and learning-based frameworks for robust global navigation.