Cohomological Equation for Robotic Screw Motion on the Lie Group SE(3)

TL;DR

This paper analyzes the cohomological equation for screw motions on SE(3), combining Fourier and Peter-Weyl analysis to understand obstructions in robotic rigid-body motions.

Contribution

It introduces a method to reduce the cohomological equation on SE(3) to finite-dimensional systems, revealing explicit resonance conditions for robotic motion analysis.

Findings

Explicit conditions for solvability of the cohomological equation.

Finite-dimensional reduction using Fourier and Peter-Weyl theory.

Identification of resonance phenomena in screw motions.

Abstract

We study the cohomological equation associated with screw motions on the Euclidean motion group SE(3). Working on the smooth manifold M = T^3 x SO(3), we combine Fourier analysis in the translational variables with Peter-Weyl theory on SO(3) to reduce the equation to a family of finite-dimensional linear transport systems along frequency orbits induced by the rotational component. In the case of finite-order rotations, solvability is governed by explicit finite-dimensional linear obstructions encoded by monodromy operators. An explicit screw motion along the z-axis illustrates the resulting resonance conditions. Since rigid motions on SE(3) arise naturally as configuration spaces in robotic kinematics, the results provide a precise description of obstruction phenomena relevant to robotic rigid-body motion.