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Integral Forms in Matrix Lie Groups

Timothy D Barfoot

TL;DR

This work introduces an integral-form framework for matrix Lie group expressions, enabling compact, finite representations of exponentials by exploiting minimal polynomials early and employing recursive integral relations. Building-block functions gamma and Gamma organize the computation, with a fundamental integral relation generalizing the classic $J(oldsymbol{ ho}) = \int_0^1 C( heta \boldsymbol{ ho}) d\theta$ to higher-order blocks, allowing higher $oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol}}}}}}}$ to be generated by simple integration. The method yields blockwise, scalable Jacobians and derivatives for common groups such as $SO(3)$, $SE(3)$, $SE_2(3)$, $SGal(3)$, $Sim(3)$, $SO(2)$, and $SE(2)$, and reproduces known results (e.g., Euler–Rodrigues) while aiding generalization to new matrix Lie groups. The approach offers practical benefits for robotics, computer vision, and graphics by reducing computational effort and clarifying interrelations among Lie-group expressions, with potential extensions to additional substructures and derivatives.

Abstract

Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g., the Euler-Rodrigues formula), which can sometimes be onerous. In this paper, we identify some useful integral forms in matrix Lie group expressions that offer a more streamlined pathway for computing compact analytic results. Moreover, we present some recursive structures in these integral forms that show many of these expressions are interrelated. Key to our approach is that we are able to apply the minimal polynomial for a Lie algebra quite early in the process to keep expressions compact throughout the derivations. With the series approach, the minimal polynomial is usually applied at the end, making it hard to recognize common analytic expressions in the result. We show that our integral method can reproduce several series-derived results from the literature.

Integral Forms in Matrix Lie Groups

TL;DR

This work introduces an integral-form framework for matrix Lie group expressions, enabling compact, finite representations of exponentials by exploiting minimal polynomials early and employing recursive integral relations. Building-block functions gamma and Gamma organize the computation, with a fundamental integral relation generalizing the classic to higher-order blocks, allowing higher to be generated by simple integration. The method yields blockwise, scalable Jacobians and derivatives for common groups such as , , , , , , and , and reproduces known results (e.g., Euler–Rodrigues) while aiding generalization to new matrix Lie groups. The approach offers practical benefits for robotics, computer vision, and graphics by reducing computational effort and clarifying interrelations among Lie-group expressions, with potential extensions to additional substructures and derivatives.

Abstract

Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g., the Euler-Rodrigues formula), which can sometimes be onerous. In this paper, we identify some useful integral forms in matrix Lie group expressions that offer a more streamlined pathway for computing compact analytic results. Moreover, we present some recursive structures in these integral forms that show many of these expressions are interrelated. Key to our approach is that we are able to apply the minimal polynomial for a Lie algebra quite early in the process to keep expressions compact throughout the derivations. With the series approach, the minimal polynomial is usually applied at the end, making it hard to recognize common analytic expressions in the result. We show that our integral method can reproduce several series-derived results from the literature.

Paper Structure

This paper contains 20 sections, 19 theorems, 135 equations, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Common Matrix Lie Groups
  4. Jacobians
  5. Cayley-Hamilton and Minimal Polynomials
  6. Building Blocks
  7. Substructures
  8. Poses, $SE(3)$
  9. Extended Poses, $SE_2(3)$
  10. Galilean Group, $SGal(3)$
  11. Similarity Transformations, $Sim(3)$
  12. Rotations and Poses in the Plane, $SO(2)$ and $SE(2)$
  13. Rotations, $SO(2)$
  14. Poses, $SE(2)$
  15. Derivatives
  16. ...and 5 more sections

Key Result

Lemma 1

Given the building-block functions in eq:bb, we have the following recursive integral relationships:

Theorems & Definitions (34)

  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • ...and 24 more