Global stabilization using LSS-Theorem: Applications to Robotics and Aerospace Vehicles

Underactuated mechanical systems are gaining interest as they can sometimes provide the desired motion or functionality at reduced cost due to their using fewer expensive actuators. The term "underactuated" refers to the fact that such mechanical systems have fewer actuators than degrees of freedom, which makes them very difficult to control. Moreover, underactuated robots have nonlinear dynamics which must be tackled with nonlinear control techniques. Furthermore, control theory for underactuated mechanical systems has been an active area of research for the past 15-20 years. Most of the research has focused on local and global asymptotic stabilization by feedback. Underactuated systems can either possess nonminimum phase or minimum phase characteristics. For minimum phase underactuated systems, the stabilization problem is rather simple and many existing control design methodologies have been proved powerful in providing a solution to this problem. For nonminimum phase underactuated systems, asymptotic stabilization problem has been, and still is, an attractive subject to the researchers in the field of nonlinear control system and theory. In particular, global asymptotic stabilization (GAS) at a desired equilibrium point of such systems by means of a single smooth static or dynamic state feedback law is still largely an open problem in the literature. In this thesis, the problem of GAS via a smooth static state feedback law is addressed for a class of an underactuated nonlinear system that is affine (possibly non affine) in the control, partially feedback linearizable, nonminimum phase and (possibly) has a non-integrable acceleration constraint. The core result of the thesis is formulated through a theorem that the author refers to through this thesis as the Legend of Salah Salman (LSS) Theorem. LSS theorem states the existence of a smooth static state feedback law that globally asymptotically stabilizes the origin of the nonlinear underactuated system that is characterized above. The form of the feedback law is also explicitly provided in the theorem. The author makes a compelling argument of the importance of the characterized class of systems in practical applications and refers to several challenging benchmark examples that have been proposed as test beds for new theories and developments. In this respect, the author studies the application of LSS theorem to benchmark problems of underactuated systems: the inertial wheel pendulum, the TORA, the VTOL aircraft, the beam and ball system, the acrobot, the cart-pole system. The pendubot, the rotational inverted pendulum and other systems are also discussed in this thesis. Theoretical and simulation results show that LSS Theorem significantly simplifies the control design, and provides solutions to some of the most challenging stabilization problems in today's control literature.