Energy Shaping Control for a Class of Underactuated Euler-Lagrange Systems

Abstract

The paper presents a new energy shaping control design for a class of underactuated Euler-Lagrange systems. Flexible joint robots, Series Elastic Actuators, and Variable Impedance Actuated Robots Albu-Schäffer et al. [2008] belong for example to this class. First, classical PD control with feed-forward compensation is revisited and a novel, straight-forward and general formulation for the stability analysis is given. Lower bound conditions for the gains of this controller motivate the introduction of the new approach, which generalizes results from Albu-Schäffer et al. [2007], Ch. Ott et al. [2008]. For shaping the potential energy, feedback variables based on the collocated states are introduced, which are statically equivalent to the noncollocated state variables. In this way the passivity is ensured while exactly satisfying steady state requirements formulated in terms of the noncollocated states (such as desired equilibrium configuration and desired stiffness). Using the passivity property, a Lyapunov based analysis can be easily carried out for arbitrarily low feedback gains. The controller is augmented by noncollocated feedback to shape the kinetic energy. Experimental results for a Variable Stiffness Robot Grebenstein et al. [2011] validate the proposed controller.