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

Amanze C. Egere

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.

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

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.
Paper Structure (10 sections, 3 theorems, 57 equations)

This paper contains 10 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. The Group $\mathop{\mathrm{SE}}\nolimits(3)$ and a Compact Phase Space
  4. The Discrete Dynamical System
  5. The Cohomological Equation
  6. Fourier Analysis on $\mathbb{T}^3$ and Peter-Weyl Theory on $\mathop{\mathrm{SO}}\nolimits(3)$
  7. The Reduced Linear System
  8. Orbit Reduction and Obstructions
  9. An Explicit Screw Motion Along the $z$--Axis
  10. Conclusion

Key Result

Lemma 5.1

A necessary condition for solvability of eq:cohomological is

Theorems & Definitions (7)

  • Lemma 5.1
  • proof
  • Proposition 7.1: Reduced transport equation
  • proof
  • Theorem 8.1: Finite-orbit solvability and obstructions
  • proof
  • Remark 8.2