Nonlinear modeling and tracking control of a single-link micro manipulator using controlled Lagrangian method

In this paper a dynamical model and the governing equations of motion of a micro-cantilever beam with rotating joint as an application of under-actuated systems will be developed. The model is based on the geometrically nonlinear equations of motion of the microbeam, employing strain gradient elasticity theory. The Rayleigh-Ritz method is used to discretize partial differential equations to obtain a set of nonlinear ordinary differential equations of motion. Then, a controlled Lagrangian method as a robust procedure for controller design is employed to achieve an acceptable tracking on the hinge's angle of rotation of a micro-cantilever beam while undesirable vibration of the under-actuated flexible variable is damped. A stability analysis of the closed-loop system is also discussed. The advantages of the controlled Lagrangian method lie in its ability to find the Lyapunov function to prove the stability of the system and its capability of handling under-actuated devices. The performance of the designed control scheme is illustrated through several numerical simulation results and some comparisons are made in various situations.