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Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles

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Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2001 © Massachusetts Institute of Technology 2001. All rights reserved. Author ................................................... Department of Electrical Engineering and Computer Science January 15, 2000 C ertified by.......................................................... Alexandre Megretski, Associate Professor of Electrical Engineering Thesis Supervisor Accepted by............................................ Arthur C. Smith Chairman, Department Committee on Graduate Students Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science on January 15, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract This thesis is devoted to nonlinear control, reduction, and classification of underac- tuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactu- ated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mo- bile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater ve- hicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e. their kinetic energy does not depend on a subset of configuration variables called external variables. In this work, I exploit such symmetry properties as a means of reducing the complexity of control design for underactuated systems. As a result, reduction and nonlinear control of high-order underactuated systems with kinetic symmetry is the main focus of this thesis. By "reduction", we mean a procedure to reduce control design for the original underactuated system to control of a lower- order nonlinear or mechanical system. One way to achieve such a reduction is by transforming an underactuated system to a cascade nonlinear system with structural properties. If all underactuated systems in a class can be transformed into a specific class of nonlinear systems, we refer to the transformed systems as the "normal form" of the corresponding class of underactuated systems. Our main contribution is to find explicit change of coordinates and control that transform several classes of underactuated systems, which appear in robotics and aerospace applications, 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, feedfor- ward form, and nontriangular linear-quadratic form. The names of these three classes are due to the particular lower-triangular, upper-triangular, and nontriangular struc- ture in which the state variables appear in the dynamics of the corresponding nonlin- ear systems. The triangular normal forms of underactuated systems can be controlled using existing backstepping and feedforwarding procedures. However, control of the 2 nontriangular normal forms is a major open problem. We address this problem for important classes of nontriangular systems of interest by introducing a new stabiliza- tion method based on the solutions of fixed-point equations as stabilizing nonlinear state feedback laws. This controller is obtained via a simple recursive method that is convenient for implementation. For special classes of nontriangular nonlinear systems, such fixed-point equations can be solved explicitly. As a result of the reduction process, one obtains a reduced nonlinear subsystem in cascade with a linear subsystem. For many classes of underactuated systems, this reduced nonlinear subsystem is physically meaningful. In fact, the reduced nonlinear subsystem is itself a Lagrangian system with a well-defined lower-order configura- tion vector. In special cases, this allows construction of Hamiltonian-type Lyapunov functions for the nonlinear subsystem. Such Lyapunov functions can be then used for robustness analysis of normal forms of underactuated systems with perturbations. The Lagrangian of the reduced nonlinear subsystem is parameterized by the shape variables (i.e. the complement set of external variables). It turns out that "a control law that changes the shape variables" to achieve stabilization of the reduced nonlinear subsystem, plays the most fundamental role in control of underactuated systems. The key analytical tools that allow reduction of high-order underactuated systems using transformations in explicit forms are "normalized generalized momentums and their integrals" (whenever integrable). Both of them can be obtained from the La- grangian of the system. The difficulty is that many real-life and benchmark examples do not possess integrable normalized momentums. For this reason, we introduce a new procedure called "momentum decomposition" which uniquely represents a non- integrable momentum as a sum of an integrable momentum term and a non-integrable momentum-error term. After this decomposition, the reduction methods for the in- tegrable cases can be applied. The normal forms for underactuated systems with :non-integrable momentums are perturbed versions of the normal forms for the inte- grable cases. This perturbation is the time-derivative of the momentum-error and only appears in the equation of the momentum of the reduced nonlinear subsystem. Based on some basic properties of underactuated systems as actuation/passivity of shape variables, integrability/non-integrability of appropriate normalized momen- tums, and presence/lack of input coupling; I managed to classify underactuated sys- tems to 8 classes. Examples of these 8 classes cover almost all major applications in robotics, aerospace systems, and benchmark systems. In all cases, either new con- trol design methods for open problems are invented, or significant improvements are achieved in terms of the performance of control design compared to the available methods. Some of the applications of our theoretical results are as the following: i) trajectory tracking for flexible-link robots, ii) (almost) global exponential tracking of feasible trajectories for an autonomous helicopter, iii) global attitude and posi- tion stabilization for the VTOL aircraft with strong input coupling, iv) automatic calculation of differentially flat outputs for the VTOL aircraft, v) reduction of the stabilization of a multi-link planar robot underactuated by one to the stabilization of the Acrobot, or the Pendubot, vi) semiglobal stabilization of the Rotating Pendulum, the Beam-and-Ball system, using fixed-point state feedback, vii) global stabilization of the 2D and 3D Cart-Pole systems to an equilibrium point, viii) global asymptotic 3 stabilization of the Acrobot and the Inertia-Wheel Pendulum. For underactuated systems with nonholonomic velocity constrains and symmetry, we obtained normal forms as the cascade of the constraint equation and a reduced- order Lagrangian control system which is underactuated or fully-actuated. This de- pends on whether the sum of the number of constraints and controls is less than, or equal to the number of configuration variables. This resalt allows reduction of a complex locomotive system called the snakeboard. Another result is global exponen- tial stabilization of a two-wheeled mobile robot to an equilibrium point which is E far from the origin (E < 1), using a smooth dynamic state feedback. Thesis Supervisor: Alexandre Megretski Title: Associate Professor of Electrical Engineering 4 Acknowledgments First and foremost, I would like to thank my advisor Alexandre Megretski. Alex's unique view of control theory and his mathematical discipline taught me to see non- linear control from a new perspective that enhanced my understanding of what con- stitutes an acceptable solution for certain fundamental problems in control theory. I can never thank him enough for giving me the freedom to choose the theoretical fields of my interest and helping me to pursue my goals. I have always valued his advice greatly. I would like to thank Munther Dahleh for our exciting discussions during the group seminars and for giving me many useful comments and advice on my work. I wish to give my very special thanks to both George Verghese and Bernard Lesieutre who have been a constant source of support and encouragement for me from the beginning of my studies at MIT. The past years at MIT were a wonderful opportunity for me to get engaged in interesting discussions with several MIT professors and scientists. Sanjoy Mitter has always amazed me with his brilliant view of systems and control theory. I am very grateful for his invaluable advice on many important occasions. I particularly would like to thank Eric Feron for his encouragement and
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