Strict Modes Everywhere - Bringing Order into Dynamics of Mechanical Systems by a Potential Compatible with the Geodesic Flow

Abstract

Strict nonlinear normal modes provide very regular families of oscillations within conservative mechanical systems. However, a strict normal mode will generally be an isolated curve within the configuration space of the system. In this paper, we design a potential that will densely fill the configuration space with strict normal modes such that each configuration belongs to one mode and each mode passes through a common point, the equilibrium. As the potential can be realized by (nonlinear) elastic elements it can be used to execute a variety of periodic trajectories very efficiently. Most of the required torques will come from the elastic elements in the system and not from the actuators. We also design a controller stabilizing the system to a desired target mode and a controller performing swing-up and compensating dissipated energy. Finally, we showcase the approach for a two DoF manipulator. The experiments show that the approach performed well for the example system.