A Quick Guide for the Iterated Extended Kalman Filter on Manifolds

TL;DR

This paper provides a simplified, beginner-friendly overview of the iterated extended Kalman filter (IEKF) on manifolds, emphasizing key derivations and practical implementation details for state estimation on complex geometric spaces.

Contribution

It offers a concise, accessible reference for implementing the IEKF on manifolds, clarifying derivations and correcting common errors in existing descriptions.

Findings

Simplified derivation of IEKF on manifolds using basic matrix calculus

Highlights key steps for practical implementation

Corrects errors in previous IEKF manifold descriptions

Abstract

The extended Kalman filter (EKF) is a common state estimation method for discrete nonlinear systems. It recursively executes the propagation step as time goes by and the update step when a set of measurements arrives. In the update step, the EKF linearizes the measurement function only once. In contrast, the iterated EKF (IEKF) refines the state in the update step by iteratively solving a least squares problem. The IEKF has been extended to work with state variables on manifolds which have differentiable $\boxplus$ and $\boxminus$ operators, including Lie groups. However, existing descriptions are often long, deep, and even with errors. This note provides a quick reference for the IEKF on manifolds, using freshman-level matrix calculus. Besides the bare-bone equations, we highlight the key steps in deriving them.