Actuating Eigenmanifolds of Conservative Mechanical Systems via Bounded or Impulsive Control Actions

Abstract

Eigenmanifolds are two-dimensional submanifolds of the state space, which generalize linear eigenspaces to nonlinear mechanical systems. Initializing a robot on an Eigenmanifold (or driving it there by control) yields hyper-efficient and regular oscillatory behaviors, called modal oscillations. This letter investigates the possibility of transitioning between two modal oscillations without ever leaving the Eigenmanifold. This is essential to generate control inputs that decrease or increase the amplitude of the nonlinear oscillations. First, we prove that this goal can be achieved using bounded inputs only for Eigenmanifolds with unidimensional projection in configuration space. Then, we show that by allowing for impulsive control actions, the problem can be solved for all Eigenmanifolds which self-intersect when projected in configuration space.