Classification of Linear Observed Systems on Multi-Frame Groups via Automorphisms

TL;DR

This paper introduces a systematic classification of linear observed systems on multi-frame groups using automorphisms, enabling better understanding and design of observer algorithms for complex navigation problems.

Contribution

It proposes a multi-frame group structure via semi-direct product and classifies all possible linear observed systems on these groups through automorphism analysis.

Findings

Unified framework for multi-frame group systems

Complete classification of linear observed systems on these groups

Application to depth-camera inertial odometry with online calibration

Abstract

Many navigation problems can be formulated as observer design on linear observed systems with a two-frame group structure, on which an invariant filter can be implemented with guaranteed consistency and stability. It's still unclear how this could be generalized to simultaneous estimation of the poses of multiple frames and the general forms of the linear observed systems involving multiple frames remain unknown. In this letter, we propose a multi-frame group structure by semi-direct product using the two-frame group as building blocks, covering all natural extensions. More importantly, we give a systematic direct calculation to classify all possible forms of linear observed systems including process ODEs and algebraic observations on such multi-frame group through its automorphism structure, in comparison to the existing classification on two-frame groups relying on ingenious construction. Depth-camera inertial odometry with online extrinsics calibration is provided as an application.