A cheat sheet for probability distributions of orientational data

TL;DR

The paper provides a compact, engineering-friendly survey of probabilistic models for orientational data across 1–3 DOFs, detailing density expressions, parameter estimation, and sampling for each family, with practical guidance and a Python library rotstats. It argues that antipodal and elliptical-contour models such as ACG, Bingham, Kent, and ESAG offer advantages over naive Euclidean fitting, especially for concentrated data, and shows how tangent-space and Lie-algebra formulations facilitate efficient inference. Diffusion-based formulations and diffusion-equation solutions on $\mathbb{S}^1$, $\mathbb{S}^2$, and $\mathrm{SO}(3)$ present alternative ways to generate PDFs. Through real-data experiments, the authors demonstrate model fitting and sampling capabilities, concluding that ACG often provides the most convenient balance of expressiveness and tractability in engineering and robotics contexts.

Abstract

The need for statistical models of orientations arises in many applications in engineering and computer science. Orientational data appear as sets of angles, unit vectors, rotation matrices or quaternions. In the field of directional statistics, a lot of advances have been made in modelling such types of data. However, only a few of these tools are used in engineering and computer science applications. Hence, this paper aims to serve as a cheat sheet for those probability distributions of orientations. Models for 1-DOF, 2-DOF and 3-DOF orientations are discussed. For each of them, expressions for the density function, fitting to data, and sampling are presented. The paper is written with a compromise between engineering and statistics in terms of notation and terminology. A Python library with functions for some of these models is provided. Using this library, two examples of applications to real data are presented.

Figures (11)

• Figure 1: Parametrisation of orientations produced by 1-DOF, 2-DOF and 3-dOF rotations. For the 3-DOF case, the frame $E$ has origin at the centre of the spherical wrist but is rigidly attached to the end-effector.
• Figure 2: Density of the von Mises distribution for $x\in\mathbb{S}$ and ten equidistant values of $\kappa$ from 0 to 5.
• Figure 3: Effect of concentration parameters. Let $\mathbf{R}=\exp\{(0,0,\pi/4)^{\top}\}$. LEFT: vMF with $\bm{\upmu}=\mathbf{R}(1,0,0)^{\top}$. RIGHT: ESAG with $\bm{\upxi}_3=\mathbf{R}(1,0,0)^{\top}$, $\bm{\upxi}_2=\mathbf{R}(0, -\sqrt{2}/2, \sqrt{2}/2)^{\top}$, $\bm{\upxi}_1=\bm{\upxi}_2\times\bm{\upxi}_3$. This leads to $\psi=\pi/4$
• Figure 4: Frames used in the reparametrisation of $\mathrm{FB}_5$ (left) and ESAG (right) distributions (schematic).
• Figure 5: A $d=3$ example of an ACG distribution with $\mathbf{\Lambda}=\mathrm{diag}(4, 0.25, 0.01)$ i.e. the mean is $(1,0,0)^{\top}$ and the other principal directions are parallel to the $Y-$ and $Z-$axes. LEFT: Generation of contours by intersecting $\mathcal{S}^2$ with the contour ellipsoids of $\mathcal{N}(\mathbf{0},\mathbf{\Lambda})$, showing $(x,y,z)\mathbf{\Lambda}^{-1}(x,y,z)^{\top}=1$ in blue. RIGHT: Simulation of 200 samples.