Virtual Holonomic and Nonholonomic Constraints on Lie groups

TL;DR

This paper introduces a geometric framework for virtual holonomic and nonholonomic constraints on Lie groups, enabling systematic control design and motion primitive generation for mechanical systems like UAVs and rigid bodies.

Contribution

It formulates virtual constraints directly on Lie algebras, establishes conditions for feedback enforcement, and generalizes classical reductions to facilitate low-dimensional motion planning.

Findings

Framework applied to quadrotor UAVs and rigid bodies with internal rotors.

Classical control laws recovered as special cases.

Affine constraint-induced motion primitives demonstrated.

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.