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Making Space for Time: The Special Galilean Group and Its Application to Some Robotics Problems

Jonathan Kelly

TL;DR

Robotics often treats space and time separately and ignores temporal uncertainty. The paper proposes modeling motion with the 10‑dimensional special Galilean group $SGal(3)$, whose Lie algebra and a closed‑form $exp$ map enable a unified representation of spatial and temporal uncertainty via Gaussian perturbations in the tangent space and the $log$ map. Key contributions are (i) a concise description of $SGal(3)$ and its Lie algebra, (ii) a concrete perturbation framework for uncertainty, and (iii) guidance for applying $SGal(3)$ to preintegration, navigation, and calibration, with a focus on temporal delays and their impact on spatial estimates. The work argues that incorporating temporal uncertainty in robotics can improve state estimation and calibration in multisensor systems, with practical relevance to IMU-based fusion and time-stamped measurements.

Abstract

The special Galilean group, usually denoted SGal(3), is a 10-dimensional Lie group whose important subgroups include the special orthogonal group, the special Euclidean group, and the group of extended poses. We briefly describe SGal(3) and its Lie algebra and show how the group structure supports a unified representation of uncertainty in space and time. Our aim is to highlight the potential usefulness of this group for several robotics problems.

Making Space for Time: The Special Galilean Group and Its Application to Some Robotics Problems

TL;DR

Robotics often treats space and time separately and ignores temporal uncertainty. The paper proposes modeling motion with the 10‑dimensional special Galilean group , whose Lie algebra and a closed‑form map enable a unified representation of spatial and temporal uncertainty via Gaussian perturbations in the tangent space and the map. Key contributions are (i) a concise description of and its Lie algebra, (ii) a concrete perturbation framework for uncertainty, and (iii) guidance for applying to preintegration, navigation, and calibration, with a focus on temporal delays and their impact on spatial estimates. The work argues that incorporating temporal uncertainty in robotics can improve state estimation and calibration in multisensor systems, with practical relevance to IMU-based fusion and time-stamped measurements.

Abstract

The special Galilean group, usually denoted SGal(3), is a 10-dimensional Lie group whose important subgroups include the special orthogonal group, the special Euclidean group, and the group of extended poses. We briefly describe SGal(3) and its Lie algebra and show how the group structure supports a unified representation of uncertainty in space and time. Our aim is to highlight the potential usefulness of this group for several robotics problems.
Paper Structure (6 sections, 7 equations, 1 figure)

This paper contains 6 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Matrix Lie Group $\mathrm{SGal}\mathopen{}\mathclose{\left(3\right)$
  3. Uncertainty on $\mathrm{SGal}\mathopen{}\mathclose{\left(3\right)$
  4. Applications
  5. Preintegration
  6. Navigation and Calibration

Figures (1)

  • Figure 1: Visualization of the transformation of (the coordinates of) an event by a randomly right-perturbed element of $\mathrm{SGal}\mathopen{}\mathclose{\left(3\right)$, projected onto the $x$-$y$ plane. The moving reference frame has a nonzero velocity in the positive $x$ direction. Left: Perturbation to $x$ translation and $z$ orientation components only. Middle: Added (small) perturbation in time. Right: Added (larger) perturbation in time. Each plot shows 1,000 samples drawn from a multivariate Gaussian, shaded by temporal offset. The dashed red line is the projection onto the $x$-$y$ plane of the 3$\sigma$ bounds of the transformed Gaussian distribution.