Controller design for mechanical systems with underactuation degree one based on controlled Lagrangians method

Abstract

This article extends controller design based on the controlled Lagrangians method from two to n degrees of freedom (DOF) mechanical systems with underactuation degree one. Importantly, a new kinetic energy equation (K-equation) is found from which the gyroscopic forces are separated due to the use of their property. Further, along with them chosen as an explicit solution, the other K-equations are satisfied for any regular controlled kinetic energy. As our main contribution, a sufficient matching condition is obtained which comprises the new K-equation and one P-equation (potential energy equation) cascaded, the regular condition and the explicit gyroscopic forces. Accordingly, the matching condition has an advantage for the fewest partial differential equations (PDEs) to be solved and the explicit gyroscopic forces. With all unknown functions of the controlled kinetic energy, the new K-equation can be simplified further under some choices on these functions and/or assumptions on specific original systems. Finally, taking the advantage, we obtain a nonlinear smooth state feedback control law that achieves local asymptotic stabilisation for a class of 3-DOF Pendubots in a vertical plane.