Nonlinear Control Of A Class Of Underactuated Mechanical Systems

Underactuated mechanical systems (UMS) are mechanical control systems with fewer control inputs than the number of configuration variables. Control of underactuated mechanical systems is currently an active and challenging field of research due to the lack of control inputs and their broad applications in robotics, aerospace vehicles, and transportation vehicles. As a typical underactuated mechanical system, control of underactuated crane system is not only very practical but also helpful to the development of control theory on UMS. This dissertation is devoted to nonlinear control of a class of UMS, which is supported by the fund of Experimental Equipment Exploitation in Harbin Institute of Technology"Practical simulation technology of gantry crane system".Based on the Euler-Lagrange equation for mechanical control system, the important classes of mechanical control systems are discussed, namely, fully-actuated mechanical systems, UMS, flat mechanical systems, and nonholonomic mechanical systems, particularly, dynamics and control problems of UMS are focused. In addition, the dynamical model of three-dimensional crane system is derived by using Lagrange equations, which can be easily simplified into the dynamical models of two-dimensional and one-dimensional crane systems.A procedure to reduce control design for original UMS is developed by considering that almost all real-life mechanical control systems possess kinetic symmetry properties, i.e. their kinetic energy only depends on a subset of configuration variables called shape variables. As a result, several closed-loop changes of coordinates are designed to transform several classes of UMS into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular form. The proposed changes of coordinates are successfully employed to transform the original dynamics of the Translational Oscillator with Rotational Actuator (TORA) system, the Cart-Pole system, and the Pendubot system into strict feedback normal form, feedforward normal form, and nontriangular normal form, respectively. Once the reduced nonlinear cascade system is obtained, the special structural property of the normal forms is considered to the following control design of the UMS. The triangular normal forms of underactuated mechanical systems can be controlled using existing Backstepping procedures and nested saturation scheme. However, there is still no totally effective control design scheme for the nontriangular normal forms. In this work, the problem is addressed by introducing a fixed pointed controller based on the solutions of fixed-pointed equations as stabilizing nonlinear state feedback law. This controller can be obtained via a recursive method that is convenient for implementation, and also the sufficient conditions for global existence of the controller are provided. The stabilization control of the TORA system, the Cart-Pole system, and Beam-and-Ball system is realized by using Backstepping procedure, nested saturation scheme, and fixed pointed controller, repectively, and simulation results demonstrated the feasibility of the proposed approaches.Finally, this dissertation focuses on control design and experiment implement of underactuated crane system. A nonlinear controller design scheme based on partial feedback linearization technique is presented for the three-dimensional and two-dimensional crane systems with various cord length. By choosing the actuated degrees corresponding to the freedom as the outputs, the controller is designed to track the trajectories of outputs while providing internal dynamics stability of unactuated degrees. The analysis of the internal dynamics shows that the stability of the zero dynamics guarantees the stability of the control system. Simulation results are presented to show the feasibility of the presented scheme. To verify its practicability, the nonlinear controller is used to realize the anti-swaying and positioning control of a two-dimensional scaled gantry crane system which can do hoisting and traveling motions simultaneously, and implemental results are given and discussed.