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Virtual Holonomic and Nonholonomic Constraints on Lie groups

A. Anahory Simoes, A. Bloch, L. Colombo, E. Stratoglou

Abstract

This paper develops a geometric framework for virtual constraints on Lie groups, with emphasis on mechanical systems modeled as affine connection systems. Virtual holonomic and virtual nonholonomic constraints, including linear and affine nonholonomic constraints, are formulated directly at the level of the Lie algebra and characterized as feedback--invariant manifolds. For each class of constraint, we establish existence and uniqueness conditions for enforcing feedback laws and show that the resulting closed--loop trajectories evolve as the dynamics of mechanical systems endowed with induced constrained connections, generalizing classical holonomic and nonholonomic reductions. Beyond stabilization, the framework enables the systematic generation of low--dimensional motion primitives on Lie groups by enforcing invariant, possibly affine, manifolds and shaping nontrivial dynamical regimes. The approach is illustrated through representative examples, including quadrotor UAVs and a rigid body with an internal rotor, where classical control laws are recovered as special cases and affine constraint--induced motion primitives are obtained.

Virtual Holonomic and Nonholonomic Constraints on Lie groups

Abstract

This paper develops a geometric framework for virtual constraints on Lie groups, with emphasis on mechanical systems modeled as affine connection systems. Virtual holonomic and virtual nonholonomic constraints, including linear and affine nonholonomic constraints, are formulated directly at the level of the Lie algebra and characterized as feedback--invariant manifolds. For each class of constraint, we establish existence and uniqueness conditions for enforcing feedback laws and show that the resulting closed--loop trajectories evolve as the dynamics of mechanical systems endowed with induced constrained connections, generalizing classical holonomic and nonholonomic reductions. Beyond stabilization, the framework enables the systematic generation of low--dimensional motion primitives on Lie groups by enforcing invariant, possibly affine, manifolds and shaping nontrivial dynamical regimes. The approach is illustrated through representative examples, including quadrotor UAVs and a rigid body with an internal rotor, where classical control laws are recovered as special cases and affine constraint--induced motion primitives are obtained.
Paper Structure (15 sections, 13 theorems, 100 equations, 5 figures)

This paper contains 15 sections, 13 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on Differential Geometry
  3. Background on vector bundles and vector fields
  4. Background on Riemannian Manifolds
  5. Euler--Poincaré equations
  6. Virtual holonomic constraints on Lie groups
  7. Virtual holonomic constraints on Lie Groups
  8. Virtual Holonomic Constraint for a Quadrotor on $\normalfont{\text{SE}}(3)$
  9. Virtual Nonholonomic Constraints on Lie Groups
  10. Nonholonomic mechanical systems
  11. Nonholonomic mechanics on Lie groups
  12. Virtual nonholonomic constraints on Lie groups
  13. Virtual nonholonomic constraint for a quadrotor UAV
  14. Virtual affine nonholonomic constraints on Lie groups
  15. Example: Rigid body with an internal rotor

Key Result

Theorem 1

Let $g:[a,b]\to G$ be a solution of the Euler--Lagrange equations for the left-invariant mechanical Lagrangian $L(g,\dot g)=\tfrac{1}{2}\|\dot g\|^2 - V(g)$, and define the left-trivialized velocity $\xi := g^{-1}\dot g \in \mathfrak{g}$. Then $\xi:[a,b]\to\mathfrak{g}$ satisfies the Euler--Poincaré or, using eq:g-connection-formula,

Figures (5)

  • Figure 1: Geometric visualization of the orthogonal decomposition $\mathfrak{so}(3) = \mathfrak d \oplus \mathfrak f$ on the unit sphere. The axis $\mathfrak d = \mathrm{span}\{e_3\}$ (blue) represents the constraint subspace, while the plane $\mathfrak f = \mathrm{span}\{e_1,e_2\}$ corresponds to its orthogonal complement. The equator (red) and meridians (green) illustrate geodesic curves associated with the induced metric. A vector $\xi$ decomposes as $\xi = \mathfrak P(\xi) + \mathfrak Q(\xi)$ via the orthogonal projectors onto $\mathfrak d$ and $\mathfrak f$, reflecting the projection structure underlying virtual nonholonomic constraints.
  • Figure 2: Time evolution of the rigid--body angular velocity components $(\omega_1,\omega_2,\omega_3)$ under the virtual affine nonholonomic constraint. The closed--loop system exhibits a bounded periodic regime rather than convergence to equilibrium.
  • Figure 3: Evolution of the affine constraint function $\Phi(t)$. The constraint function vanishes, demonstrating exact invariance of the affine constraint manifold under the closed--loop dynamics.
  • Figure 4: Closed--loop trajectory of the angular velocity $(\omega_1,\omega_2,\omega_3)$ in angular--velocity space. The motion evolves along a closed curve, consistent with the periodic regime induced by the virtual affine constraint.
  • Figure 5: Projection of the closed--loop trajectory onto the $(\omega_1,\omega_2)$ plane, illustrating the periodic rotational regime enforced by the affine constraint.

Theorems & Definitions (37)

  • Theorem 1: goodman2022reduction
  • Remark 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 27 more