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

Changwu Liu, Yuan Shen

TL;DR

This work tackles simultaneous estimation across multiple frames by introducing the multi-frame group (MFG) denoted $MFG(d,n,m,s,t)$, constructed as a type-II semidirect product of a two-frame group $TFG(d,n,m)$. It develops an automorphism-based approach to classify all linear observed systems on $MFG$, including both process ODEs and algebraic observations. The main results provide explicit frame-wise ODE forms and observation models (via the automorphism structure) and a concrete navigation application: depth-camera inertial odometry with online extrinsics calibration. The framework enables principled, consistent observers for multi-frame navigation and demonstrates improved transient performance through a MFG-IEKF variant compared with traditional EKFs.

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.

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

TL;DR

This work tackles simultaneous estimation across multiple frames by introducing the multi-frame group (MFG) denoted , constructed as a type-II semidirect product of a two-frame group . It develops an automorphism-based approach to classify all linear observed systems on , including both process ODEs and algebraic observations. The main results provide explicit frame-wise ODE forms and observation models (via the automorphism structure) and a concrete navigation application: depth-camera inertial odometry with online extrinsics calibration. The framework enables principled, consistent observers for multi-frame navigation and demonstrates improved transient performance through a MFG-IEKF variant compared with traditional EKFs.

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.

Paper Structure

This paper contains 8 sections, 5 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Multi-Frame Groups
  4. Linear Observed Systems on MFGs
  5. Classification of Group Affine Dynamics on MFGs
  6. Classification of Algebraic Observations on MFGs
  7. Application in Navigation
  8. Conclusion

Key Result

Theorem 1

Every element $\boldsymbol{\chi}$ in a multi-frame group $\text{MFG}(d,n,m,s,t)$ has a matrix group embedding of block-diagonal matrices with $\text{TFG}(d,n,m)$-valued blocks, as: where each ${^{(\cdot)}}\boldsymbol{T}_{(\cdot)}$ takes value in TFG.

Figures (2)

  • Figure 1: Transformation chain structure of MFG. Each node represents a frame. Each arrow is a TFG-valued transformation between two nodes. All transformations are to be estimated simultaneously.
  • Figure 2: Plot of error w.r.t time of MEKF, imperfect IEKF (IEKF) and our proposed MFG-IEKF. Only extrinsics (Camera to IMU) states are demonstrated. 'Rot Err' and 'Trans Err' are short for rotational error $\Vert\log(\hat{\boldsymbol{R}}_c\boldsymbol{R}_c^{-1})\Vert$ and translational error $\Vert\hat{\boldsymbol{p}}_c-\boldsymbol{p}_c\Vert$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1
  • Lemma 1
  • ...and 3 more