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Reduction by Symmetry and Optimal Control with Broken Symmetries on Riemannian Manifolds

Jacob R. Goodman, Leonardo J. Colombo

TL;DR

The paper develops a unified framework for reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces, including scenarios with symmetry-breaking potentials and advected parameters. It derives Euler–Poincaré equations for left-invariant and bi-invariant metrics, and extends the reduction to homogeneous spaces via horizontal lifts, providing constrained variational principles and reconstruction formulas. The authors apply the theory to geodesics on $SO(3)$, the heavy top projected onto $S^2$, obstacle-avoidance problems in both Lie groups and homogeneous spaces, and an optimal-control formulation for left-invariant systems, with concrete robotic-manipulator examples. This framework yields reduced, first-order dynamics on Lie algebras that facilitate efficient numerical integration and scalable trajectory optimization in robotics and control contexts.

Abstract

This paper studies the reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces. We derive the reduced equations of motion in the case of Lie groups endowed with a left-invariant metric, and on Lie groups that admits a bi-invariant metric. We repeated this analysis for Riemannian homogeneous spaces, where we derive the reduced equations by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the underlying Lie group. We study also the case that the underlying Lie group admits a bi-invariant metric, and consider the special case that the homogeneous space is in fact a Riemannian symmetric space. These ideas are applied to geodesics for a rigid body on $SO(3)$ to derive geodesic equations on the dual of its Lie algebra (a vector space), the heavy-top in $SE(3)$ to derive reduced equations of motion on the unit sphere $S^2$, geodesics on $S^2$ as a Riemannian symmetric space endowed with a bi-invariant metric and optimal control problems for applications to robotic manipulators.

Reduction by Symmetry and Optimal Control with Broken Symmetries on Riemannian Manifolds

TL;DR

The paper develops a unified framework for reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces, including scenarios with symmetry-breaking potentials and advected parameters. It derives Euler–Poincaré equations for left-invariant and bi-invariant metrics, and extends the reduction to homogeneous spaces via horizontal lifts, providing constrained variational principles and reconstruction formulas. The authors apply the theory to geodesics on , the heavy top projected onto , obstacle-avoidance problems in both Lie groups and homogeneous spaces, and an optimal-control formulation for left-invariant systems, with concrete robotic-manipulator examples. This framework yields reduced, first-order dynamics on Lie algebras that facilitate efficient numerical integration and scalable trajectory optimization in robotics and control contexts.

Abstract

This paper studies the reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces. We derive the reduced equations of motion in the case of Lie groups endowed with a left-invariant metric, and on Lie groups that admits a bi-invariant metric. We repeated this analysis for Riemannian homogeneous spaces, where we derive the reduced equations by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the underlying Lie group. We study also the case that the underlying Lie group admits a bi-invariant metric, and consider the special case that the homogeneous space is in fact a Riemannian symmetric space. These ideas are applied to geodesics for a rigid body on to derive geodesic equations on the dual of its Lie algebra (a vector space), the heavy-top in to derive reduced equations of motion on the unit sphere , geodesics on as a Riemannian symmetric space endowed with a bi-invariant metric and optimal control problems for applications to robotic manipulators.
Paper Structure (20 sections, 29 theorems, 109 equations)

This paper contains 20 sections, 29 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Riemannian Manifolds
  3. The momentum map
  4. Riemannian Geometry on Lie Groups
  5. Bi-invariant metrics
  6. Euler-Poincaré Reduction by a Lie Group Symmetry
  7. Example: Geodesics for Rigid Body on $\mathbf{\normalfont{\text{SO}}(3)}$
  8. Euler-Poincaré Reduction with Broken Symmetry
  9. Euler-Poincaré reduction with advected parameters
  10. Example: The heavy top
  11. Example: Obstacle avoidance on Lie groups
  12. Euler-Poincaré Reduction with Broken Symmetries on Riemannian Homogeneous Space
  13. Riemannian Homogeneous Spaces
  14. Euler-Poincaré reduction on Riemannian homogeneous spaces
  15. Bi-invariant metrics and Riemannian Symmetric Spaces
  16. ...and 5 more sections

Key Result

Lemma 2.1

If $\Gamma$ is an admissible variation of $\gamma \in \Omega$, then for all $(s, t) \in (-\epsilon, \epsilon) \times [a, b].$

Theorems & Definitions (52)

  • Lemma 2.1: Symmetry Lemma
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proof 1
  • Remark 3.6
  • Corollary 3.7
  • Proof 2
  • ...and 42 more