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

Jonathan Kelly

cs.ROSep 22, 2024

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

Authors: Jonathan Kelly

Organizations: Space & Terrestrial Autonomous Robotic Systems (STARS) Laboratory, University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, Canada

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.

Figures & tables

Panel of Figure 1 — Fig. 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.

Lie Group Methods

Lie group methods use transformation groups and their Lie algebras to model rotations, rigid motions, symmetries, and other structured dynamics.

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FAST-Sync: Fast Group Synchronization for any Matrix Lie Group

Shane Holmes, Yiran Luo, Firat Taxpulat, David M. Rosen+1

Northeastern University, Boston, USA · School of Interactive Computing, Georgia Institute of Technology, Atlanta, GA, USA

Open paper AskPDF arXiv

Group synchronization (GS) is the problem of estimating a set of

N

unknown elements

g_1,\ldots, g_N \in \mathcal{G}

in a group

\mathcal{G}

, given noisy measurements of a subset of their pairwise ratios

g_i^{-1} g_j

. GS problems lie at the core of many state estimation tasks in robotics and computer vision, including 3D vision, robotic mapping, inertial navigation, and molecular reconstruction. Unfortunately, GS problems are typically both high-dimensional and non-convex, and therefore hard to solve in general. In this paper, we present Fast-Sync, a fast linear approximation method for GS that is suitable for initializing local manifold-based optimizers or certifiable global methods. Our approach generalizes chordal initialization to arbitrary matrix Lie groups, and additionally proposes two new key algorithmic enhancements: we show how to exploit both the Kronecker-product structure in the problem data matrix and the topology of the synchronization graph to improve speed, scalability, and accuracy. Experimental evaluation across several GS tasks demonstrates that Fast-Sync provides high-quality initializations that enable local optimizers to efficiently recover globally optimal GS solutions, achieving high success rates even with considerable measurement noise.

Abstract

Lie Group Methods

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