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.