Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems
2026-07-03 • Robotics
Robotics
AI summaryⓘ
The authors explain different ways to estimate a vehicle's position and motion when moving anywhere on Earth, considering factors like Earth's shape and gravity changes. They compare traditional error-state Kalman filters with newer 'invariant' versions that use math symmetries for more stable error calculations. Depending on how errors are defined and the system setup, different filter types (standard, left-invariant, right-invariant) work better. Their note gives clear formulas for these four filter types to help others understand and use them in global navigation systems.
error-state Kalman filterinvariant Kalman filterLie Group symmetriesnavigation framebody frameglobal navigationstate estimationinertial navigation systemserror propagation
Authors
Antonia Hager, Torleiv H. Bryne
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.