Uncertainty in space, time, and motion on the special Galilean group

Abstract

Classical mechanics unfolds within absolute time and Euclidean space, yet our knowledge of where events occur, when they occur, and how motion evolves is inherently uncertain. The special Galilean group provides a natural setting for describing classical spacetime, combining absolute time, Euclidean space, and inertial motion within a single Lie group structure. Although this framework is well known, representing and propagating uncertainty on the group has received comparatively little attention. In this work, we bring together existing results on the structure of the Galilean group and use this unified framework to express uncertainty directly on the group manifold. A main contribution is a compact, closed-form expression for the Galilean group Jacobian, which enables principled uncertainty propagation when composing Galilean transformations. We show that uncertainty in spatial position and orientation, temporal displacement, and inertial motion are intrinsically coupled through the underlying group structure. To illustrate the usefulness of the Galilean framework, we consider the problem of estimating a time-varying transformation between inertial frames from noisy observations collected at distinct instants in time. We show that performing estimation directly on the Galilean group yields substantially more statistically consistent estimates than formulations that treat time independently. Together, these results provide a geometric foundation for reasoning about uncertainty in space, time, and motion in classical mechanics, navigation, and robotics.

Figures (3)

• Figure 1: Galilean spacetime has the structure of a trivial fibre bundle with base space $\mathbb{E} ^{1}$ and fibres $\mathbb{E} ^{3}$. The worldline of a body is a cross-section of the bundle, composed of a sequence of spacetime events (shown as dots). A worldline is straight when the motion is inertial (blue), corresponding to a geodesic in Galilean spacetime. Curved worldlines (red) indicate accelerated motion. Figure inspired by 2005_Penrose_Road.
• Figure 2: Visualization of the transformation of an event by a right-perturbed element of $\mathrm{SGal}(3)$, projected onto the $x$-$y$ plane. Left: perturbation to $x$ translation and $z$ rotation components only. Middle: additional (small) perturbation in time. Right: additional (large) perturbation in time. Points are shaded by time offset, according to the colour bar shown on the right. Temporal uncertainty induces a 'spread' in the spatial uncertainty. Each plot shows 1,200 samples drawn from a multivariate Gaussian.
• Figure 3: Visualization of the problem considered in \ref{['sec:estimation']}. Two spacetime diagrams show inertial frames ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{a} }$ and ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{b} }$ evolving along distinct worldlines, as viewed from a third inertial frame $\underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{g}$ fixed in space and time. Because ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{a} }$ and ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{b} }$ are in relative motion, their spatial separation varies over time (grey dashed lines). Left: observations occur at discrete time indices $n$, where paired measurements of position, orientation, velocity, and local time are obtained at ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{a_n} }\!$ and ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{b_n} }\!$; frames ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{a_3} }\!$ and ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{b_3} }\!$ are highlighted. Right: an alternate perspective. At each time slice (fibre), the spatial frame axes are drawn in red ($x$) and green ($y$); these axes remain fixed in orientation under inertial motion. The estimation task is to find the Galilean transformation $\bm{\mathbf{F}} _{a_{0}b_{0}}$ relating ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{b_0} }$ to ${ \underrightarrow{ \bm{\mathbf{\mathcal{F}}} }_{a_0} }$, along with the inter-measurement time $\scaleobj{0.8}{\Delta}\tau$, using only the paired observations along each worldline.

Theorems & Definitions (3)

• proof
• proof
• proof