Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

Abstract

On the basis of controlled Lagrangians, a controller design is proposed for underactuated mechanical systems with two degrees of freedom. A new kinetic energy equation (K-equation) independent of the gyroscopic forces is found due to the use of their property. As a result, the necessary and sufficient matching condition comprises the new K-equation and the potential energy equation (P-equation) cascaded, the regular condition, and the explicit gyroscopic forces. Further, for two classes of input decoupled systems that cover the main benchmark examples, the new K-equation, respectively, degenerates from a quasilinear partial differential equation (PDE) into an ordinary differential equation (ODE) under some choice and into a homogeneous linear PDE with two kinds of explicit general solutions. Benefiting from one of the general solutions, the obtained smooth state feedback controller for the Acrobots is of a more general form. Specifically, a constant fixed in a related paper by the system parameters is converted into a controller parameter ranging over an open interval along with some new nonlinear terms involved. Unlike what is mentioned in the related paper, some categories of the Acrobots cannot be stabilized with the existing interconnection and damping assignment passivity based control (IDA-PBC) method. As a contribution, the system can be locally asymptotically stabilized by the selection of the new controller parameter except for only one special case.

Keywords

underactuated mechanical systems / controlled Lagrangians (CL) method / gyroscopic forces / matching condition / stabilization / Acrobot