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

The paper presents an impedance controller for a class of underactuated Euler-Lagrange systems based on energy shaping. For shaping the potential energy, feedback variables are introduced, which are functions of the collocated state variables only, but which are statically equivalent to the noncollocated state variables. In this way the passivity of the system can be ensured, while exactly satisfying all steady state requirements formulated in terms of the noncollocated states (such as desired stiffness and desired equilibrium configuration). The method is constructive, allowing the direct formulation of the controller and of the corresponding energy function. The controller additionally uses the noncollocated feedback to shape the kinetic energy. Under some conditions on the potential energy of the plant, the closed loop system can then be seen as a feedback interconnection of passive systems and is proven to be asymptotically stable. Experimental results for a flexible joint robot validate the proposed controller.