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 ComputerScience

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

c© Massachusetts Institute of Technology 2001. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Electrical Engineering and Computer Science

January 15, 2000

Certified 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 Systemswith Application to Robotics and Aerospace Vehicles

byReza Olfati-Saber

Submitted to the Department of Electrical Engineering and Computer Scienceon January 15, 2000, in partial fulfillment of the

requirements for the degree ofDoctor 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 systemswith 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 inRobotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuatedsystems include flexible-link robots, mobile robots, walking robots, robots on mo-bile platforms, cars, locomotive systems, snake-type and swimming robots, acrobaticrobots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater ve-hicles. Based on recent surveys, control of general underactuated systems is a majoropen 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 calledexternal variables. In this work, I exploit such symmetry properties as a means ofreducing the complexity of control design for underactuated systems. As a result,reduction and nonlinear control of high-order underactuated systems with kineticsymmetry is the main focus of this thesis. By “reduction”, we mean a procedureto 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 bytransforming an underactuated system to a cascade nonlinear system with structuralproperties. If all underactuated systems in a class can be transformed into a specificclass 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 thattransform several classes of underactuated systems, which appear in robotics andaerospace applications, into cascade nonlinear systems with structural properties thatare convenient for control design purposes. The obtained cascade normal forms arethree classes of nonlinear systems, namely, systems in strict feedback form, feedfor-ward form, and nontriangular linear-quadratic form. The names of these three classesare 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 controlledusing existing backstepping and feedforwarding procedures. However, control of the

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nontriangular normal forms is a major open problem. We address this problem forimportant classes of nontriangular systems of interest by introducing a new stabiliza-tion method based on the solutions of fixed-point equations as stabilizing nonlinearstate feedback laws. This controller is obtained via a simple recursive method that isconvenient 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 subsystemin cascade with a linear subsystem. For many classes of underactuated systems, thisreduced nonlinear subsystem is physically meaningful. In fact, the reduced nonlinearsubsystem is itself a Lagrangian system with a well-defined lower-order configura-tion vector. In special cases, this allows construction of Hamiltonian-type Lyapunovfunctions for the nonlinear subsystem. Such Lyapunov functions can be then usedfor robustness analysis of normal forms of underactuated systems with perturbations.The Lagrangian of the reduced nonlinear subsystem is parameterized by the shapevariables (i.e. the complement set of external variables). It turns out that “a controllaw that changes the shape variables” to achieve stabilization of the reduced nonlinearsubsystem, plays the most fundamental role in control of underactuated systems.

The key analytical tools that allow reduction of high-order underactuated systemsusing transformations in explicit forms are “normalized generalized momentums andtheir 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 examplesdo not possess integrable normalized momentums. For this reason, we introduce anew procedure called “momentum decomposition” which uniquely represents a non-integrable momentum as a sum of an integrable momentum term and a non-integrablemomentum-error term. After this decomposition, the reduction methods for the in-tegrable cases can be applied. The normal forms for underactuated systems withnon-integrable momentums are perturbed versions of the normal forms for the inte-grable cases. This perturbation is the time-derivative of the momentum-error andonly appears in the equation of the momentum of the reduced nonlinear subsystem.

Based on some basic properties of underactuated systems as actuation/passivityof 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 inrobotics, aerospace systems, and benchmark systems. In all cases, either new con-trol design methods for open problems are invented, or significant improvements areachieved in terms of the performance of control design compared to the availablemethods. Some of the applications of our theoretical results are as the following: i)trajectory tracking for flexible-link robots, ii) (almost) global exponential trackingof feasible trajectories for an autonomous helicopter, iii) global attitude and posi-tion stabilization for the VTOL aircraft with strong input coupling, iv) automaticcalculation of differentially flat outputs for the VTOL aircraft, v) reduction of thestabilization of a multi-link planar robot underactuated by one to the stabilization ofthe Acrobot, or the Pendubot, vi) semiglobal stabilization of the Rotating Pendulum,the Beam-and-Ball system, using fixed-point state feedback, vii) global stabilizationof the 2D and 3D Cart-Pole systems to an equilibrium point, viii) global asymptotic

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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 result allows reduction of acomplex locomotive system called the snakeboard. Another result is global exponen-tial stabilization of a two-wheeled mobile robot to an equilibrium point which is ε farfrom the origin (ε¿ 1), using a smooth dynamic state feedback.

Thesis Supervisor: Alexandre MegretskiTitle: Associate Professor of Electrical Engineering

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Acknowledgments

First and foremost, I would like to thank my advisor Alexandre Megretski. Alex’sunique 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. Ican never thank him enough for giving me the freedom to choose the theoretical fieldsof my interest and helping me to pursue my goals. I have always valued his advicegreatly.

I would like to thank Munther Dahleh for our exciting discussions during the groupseminars 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 BernardLesieutre who have been a constant source of support and encouragement for mefrom the beginning of my studies at MIT.

The past years at MIT were a wonderful opportunity for me to get engaged ininteresting discussions with several MIT professors and scientists. Sanjoy Mitter hasalways amazed me with his brilliant view of systems and control theory. I am verygrateful for his invaluable advice on many important occasions. I particularly wouldlike to thank Eric Feron for his encouragement and for enthusiastically showing mesimulation results and test models related to my work. I wish to thank DimitriBertsekas for being a great teacher and giving me important advice as my academicadvisor during the past years.

Many thanks goes to Nicola Elia and Ulf Jonsson for our long hours of discussionsand their valuable technical comments. I would also like to thank Emilio Frazzolifor many discussions on helicopters and nonlinear control. I would like to take thisopportunity to thank all my friends at LIDS for their encouragement, support, andthe fun times we shared during our group seminars.

I owe my sincere gratitude to Milena Levak, the Dean of the International StudentsOffice at MIT, who has greatly helped me during very important situations and times.

I am incredibly grateful to my family. My parents, Hassan and Soroor, and mysister, Negin, gave me their unconditional love and support during all these years.While I never had the opportunity to see my family during my five and a half years ofstudies at MIT, my heart has always been with them. They never stopped believingin me and did everything they could to help me achieve my goals.

At last, my most special thanks goes to Manijeh. Her beautiful smile, emotionalsupport, and strong encouragement were the driving force of my efforts. With hercreative mind, she gave me many examples of underactuated systems in nature thatI had not thought of myself. Without her help and enthusiasm, this work certainlywould not have been possible.

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Contents

1 Introduction 141.1 Mechanics and Control Theory . . . . . . . . . . . . . . . . . . . . . 141.2 Nonlinear Control Systems . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 Underactuated Systems . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Highly Nonlinear Systems . . . . . . . . . . . . . . . . . . . . 171.2.3 Cascade Nonlinear Systems . . . . . . . . . . . . . . . . . . . 19

1.3 Statement of the Contributions . . . . . . . . . . . . . . . . . . . . . 201.3.1 Normal Forms for Underactuated Systems . . . . . . . . . . . 221.3.2 Reduction and Control of Low-Order Underactuated Systems . 231.3.3 Reduction and Control of High-Order Underactuated Systems 241.3.4 Reduction and Control of Underactuated Systems with Non-

holonomic Velocity Constraints . . . . . . . . . . . . . . . . . 261.3.5 Applications in Robotics and Aerospace Vehicles . . . . . . . . 271.3.6 Control of Nontriangular Cascade Nonlinear Systems . . . . . 27

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Mechanical Control Systems 282.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Simple Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . 282.3 Fully-actuated Mechanical Systems . . . . . . . . . . . . . . . . . . . 302.4 Underactuated Mechanical Systems . . . . . . . . . . . . . . . . . . . 312.5 Flat Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Nonholonomic Mechanical Systems . . . . . . . . . . . . . . . . . . . 322.7 Symmetry in Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 Normalized Momentums and Integrability . . . . . . . . . . . . . . . 35

3 Normal Forms for Underactuated Systems 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Dynamics of Underactuated Systems . . . . . . . . . . . . . . . . . . 393.3 Examples of Underactuated Systems . . . . . . . . . . . . . . . . . . 39

3.3.1 The Acrobot and the Pendubot . . . . . . . . . . . . . . . . . 403.3.2 The Cart-Pole System and the Rotating Pendulum . . . . . . 413.3.3 The Beam-and-Ball System . . . . . . . . . . . . . . . . . . . 423.3.4 The TORA System . . . . . . . . . . . . . . . . . . . . . . . . 423.3.5 The Inertia-Wheel Pendulum . . . . . . . . . . . . . . . . . . 43

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3.3.6 The VTOL Aircraft . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Collocated Partial Feedback Linearization . . . . . . . . . . . . . . . 453.5 Noncollocated Partial Feedback Linearization . . . . . . . . . . . . . 463.6 Partial Feedback Linearization Under Input Coupling . . . . . . . . . 473.7 Normal Forms for Underactuated Systems . . . . . . . . . . . . . . . 493.8 Classes of Structured Nonlinear Systems . . . . . . . . . . . . . . . . 513.9 Normal Forms for Underactuated Systems with 2 DOF . . . . . . . . 53

4 Reduction and Control of High-Order Underactuated Systems 584.1 Shape Variables and Kinetic Symmetry . . . . . . . . . . . . . . . . . 584.2 Underactuated Systems with Noninteracting Inputs and Integrable Mo-

mentums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.1 Underactuated Systems with Actuated Shape Variables . . . . 604.2.2 Underactuated Systems with Unactuated Shape Variables . . 63

4.3 Underactuated Systems with Input Coupling . . . . . . . . . . . . . . 744.4 Underactuated Systems with Non-integrable Momentums . . . . . . . 804.5 Momentum Decomposition for Underactuated Systems . . . . . . . . 864.6 Classification of Underactuated Systems . . . . . . . . . . . . . . . . 1004.7 Nonlinear Control of the Core Reduced System . . . . . . . . . . . . 106

5 Applications to Robotics and Aerospace Vehicles 1125.1 The Acrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 The TORA System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3 The Inertia-Wheel Pendulum . . . . . . . . . . . . . . . . . . . . . . 1275.4 The Cart-Pole System . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.4.1 Global Stabilization to an Equilibrium Point . . . . . . . . . . 1335.4.2 Aggressive Swing-up of the Inverted Pendulum . . . . . . . . . 1375.4.3 Switching-based Controller . . . . . . . . . . . . . . . . . . . . 139

5.5 The Rotating Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 1405.5.1 Aggressive Swing-up of the Rotating Pendulum . . . . . . . . 143

5.6 The Pendubot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.7 The Beam-and-Ball System . . . . . . . . . . . . . . . . . . . . . . . 1475.8 Control of Multi-link Underactuated Planar Robot Arms . . . . . . . 1495.9 Trajectory Tracking for a Flexible One-Link Robot . . . . . . . . . . 150

5.9.1 Model of a Flexible Link . . . . . . . . . . . . . . . . . . . . . 1515.9.2 The Noncollocated Output . . . . . . . . . . . . . . . . . . . . 1525.9.3 Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.10 Configuration Stabilization for the VTOL Aircraft . . . . . . . . . . . 1575.11 Trajectory Tracking and Stabilization of An Autonomous Helicopter . 161

5.11.1 Dynamic Model of A Helicopter . . . . . . . . . . . . . . . . . 1615.11.2 Input Moment Decoupling for A Helicopter . . . . . . . . . . . 1645.11.3 Nontriangular Normal Form of a Helicopter . . . . . . . . . . 1705.11.4 Feedback Linearization of the Unperturbed Model of the Heli-

copter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.11.5 Desired Control Attitude Rd for Position Tracking . . . . . . . 172

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5.11.6 Asymptotic Tracking of a Desired Attitude Rd ∈ SO(3) . . . . 1735.11.7 Thrust for Exponential Attitude Stabilization . . . . . . . . . 181

6 Reduction and Control of Underactuated Nonholonomic Systems 1836.1 Nonholonomic Systems with Symmetry . . . . . . . . . . . . . . . . . 1846.2 Applications to Nonholonomic Robots . . . . . . . . . . . . . . . . . 187

6.2.1 A Rolling Disk . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.2.2 A Nonholonomic Mobile Robot . . . . . . . . . . . . . . . . . 1896.2.3 A Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906.2.4 The Snakeboard . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.3 Notions of ε-Stabilization and ε-Tracking . . . . . . . . . . . . . . . . 2056.4 Global Exponential ε-Stabilization/Tracking for a Mobile Robot . . . 207

6.4.1 Dynamics of a Mobile Robot in SE(2) . . . . . . . . . . . . . . 2076.4.2 Near-Identity Diffeomorphism . . . . . . . . . . . . . . . . . . 2086.4.3 Control Design for ε-Stabilization/Tracking . . . . . . . . . . 210

7 Control of Nonlinear Systems in Nontriangular Normal Forms 2167.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Structure of Nontriangular Normal Forms of Underactuated Systems 2227.3 Stabilization of Nonlinear Systems in Feedback Form . . . . . . . . . 224

7.3.1 Standard Backstepping Procedure . . . . . . . . . . . . . . . . 2247.3.2 Cascade Backstepping Procedure . . . . . . . . . . . . . . . . 226

7.4 Stabilization of Nonlinear Systems in Nontriangular Forms . . . . . . 2317.4.1 Nontriangular Nonlinear Systems Affine in (ξ1, ξ2) . . . . . . . 2327.4.2 Nontriangular Nonlinear Systems Non-affine in (ξ1, ξ2) . . . . 2427.4.3 Global Existence of Fixed-Point Control Laws . . . . . . . . . 2467.4.4 Slightly Nontriangular Nonlinear Systems . . . . . . . . . . . 2537.4.5 Notion of Partial Semiglobal Asymptotic Stabilization . . . . . 258

7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.5.1 The Cart-Pole System with Small Length/Strong Gravity Effects2597.5.2 The Cart-Pole System with Large Length/Weak Gravity Effects 2627.5.3 The Rotating Pendulum . . . . . . . . . . . . . . . . . . . . . 2637.5.4 The Generalized Beam-and-Ball System . . . . . . . . . . . . 265

8 Conclusions 2688.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2698.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

8.2.1 Hybrid Lagrangian Systems . . . . . . . . . . . . . . . . . . . 2708.2.2 Uncertainty and Robustness . . . . . . . . . . . . . . . . . . . 2728.2.3 Output Feedback Stabilization and Tracking . . . . . . . . . . 2728.2.4 The Effects of Bounded Control Inputs . . . . . . . . . . . . . 2728.2.5 Small Underactuated Mechanical Systems (SUMS) . . . . . . 2728.2.6 Underactuated Systems in Strong/Weak Fields . . . . . . . . . 272

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A Dynamics of Mechanical Systems 274A.1 Underactuated Mechanical Systems with Two DOF and Kinetic Sym-

metry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274A.2 A Three-link Planar Robot . . . . . . . . . . . . . . . . . . . . . . . . 275A.3 Two-link Planar Robots: the Acrobot and the Pendubot . . . . . . . 279A.4 The Beam-and-Ball System . . . . . . . . . . . . . . . . . . . . . . . 280A.5 The Cart-Pole System . . . . . . . . . . . . . . . . . . . . . . . . . . 282A.6 The Rotating Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 283A.7 The 3D Cart-Pole System with Unactuated Roll and Pitch Angles . . 284A.8 An Underactuated Surface Vessel . . . . . . . . . . . . . . . . . . . . 286

B Symbolic Calculation of the Flat Outputs of the Helicopter 289

C Proofs of Theorems in Chapter 7 290C.1 Proof of Theorem 7.4.3: . . . . . . . . . . . . . . . . . . . . . . . . . 290C.2 Proof of the Stability of µ-Subsystem in Theorem 7.4.4 . . . . . . . . 291C.3 Proof of Lemma 7.4.4: . . . . . . . . . . . . . . . . . . . . . . . . . . 291C.4 Proof of Theorem 7.4.4: . . . . . . . . . . . . . . . . . . . . . . . . . 292C.5 Proof of Theorem 7.4.5: . . . . . . . . . . . . . . . . . . . . . . . . . 295

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List of Figures

3-1 (a) The Acrobot, (b) The Pendubot. . . . . . . . . . . . . . . . . . . 403-2 (a) The Cart-Pole system, (b) The Rotating Pendulum. . . . . . . . . 413-3 The Beam-and-Ball System. . . . . . . . . . . . . . . . . . . . . . . . 423-4 The TORA system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433-5 The Inertia Wheel Pendulum . . . . . . . . . . . . . . . . . . . . . . 443-6 The VTOL aircraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4-1 A flexible one-link robot arm. . . . . . . . . . . . . . . . . . . . . . . 734-2 A flying bird. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774-3 A triple-link robot arm with two actuators. . . . . . . . . . . . . . . . 814-4 An inverted pendulum on a moving platform with 4 DOF and 2 controls. 844-5 A triple-link robot arm with actuated shape variables. . . . . . . . . . 904-6 A planar three-link robot arm. . . . . . . . . . . . . . . . . . . . . . . 944-7 Three possible actuation configuration for a planar three-link robot arm. 96

5-1 The Acrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145-2 The function γ(q2) for a = 20/3, b = 4, c = 4/3. . . . . . . . . . . . . . 1155-3 The function α(qr) for m1 = m2 = 1 and l1 = l2 = 1. . . . . . . . . . . 1195-4 Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =

(π/3, 0, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . 1205-5 Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =

(π/3, 0, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . 1205-6 Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =

(π, 0, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . . 1215-7 Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =

(π, 0, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . . 1215-8 Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =

(0, π, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . . 1225-9 Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =

(0, π, 0) and control input p2. . . . . . . . . . . . . . . . . . . . . . . 1225-10 The TORA system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235-11 State trajectories and control input of the TORA system for the ini-

tial condition (q1, q2, p1, p2) = (1, 0, 0, 0): (a) (q1, p1), (b) (q2, p2), (c)(qr, pr), and (d) u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5-12 The Inertia-Wheel Pendulum . . . . . . . . . . . . . . . . . . . . . . 128

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5-13 For the initial condition (π, 0, 0), (a) shows the state trajectories for(q1, q1), (b) shows the state trajectory in (q1, q1) plane, (c) shows q2,and (d) shows the control input τ . . . . . . . . . . . . . . . . . . . . . 131

5-14 For the initial condition (π/2, 0, 0), (a) shows the state trajectories for(q1, q1), (b) shows the state trajectory in (q1, q1) plane, (c) shows q2,and (d) shows the control input τ . . . . . . . . . . . . . . . . . . . . . 132

5-15 The Cart-Pole system. . . . . . . . . . . . . . . . . . . . . . . . . . . 1335-16 (a) State trajectory of the cart-pole system for the initial condition

(5, 0, π/3, 0) and σi(·) = sat1(·), (b) The force F and control input u. 1365-17 (a) State trajectory of the cart-pole system for the initial condition

(5, 0, π/3, 0) and σi(·) = sat0.1(·), (b) The force F and control input u. 1365-18 The dead-zone nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . 1385-19 (a) Aggressive swing-up of an inverted pendulum on a cart from the

downward position (q2, p2) = (π, 0), (b) The path of the pendulum in(q2, p2)-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5-20 (a) State trajectory of the cart-pole system for the initial condition(0, 0, π, 0), (b) The input force F and the trajectory in (q2, p2)-plane. 141

5-21 The Rotating Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 1415-22 (a) Aggressive swing-up of an inverted pendulum on a rotating arm

from the downward position (q2, p2) = (π, 0), (b) Trajectory of thependulum in (q2, p2)-plane. . . . . . . . . . . . . . . . . . . . . . . . . 145

5-23 The Pendubot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455-24 The Beam-and-Ball System. . . . . . . . . . . . . . . . . . . . . . . . 1475-25 A flexible one-link robot arm. . . . . . . . . . . . . . . . . . . . . . . 1515-26 The VTOL aircraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575-27 (a) The state trajectories of the VTOL aircraft, and (b) The path of

the center of mass of the aircraft in (x1, y1)-plane. . . . . . . . . . . . 1605-28 A helicopter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6-1 A vertical rolling disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886-2 A mobile robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906-3 The model of a car. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916-4 The Snakeboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956-5 The geometry of the paths of the rear and front wheels of the car with

a radius of rotation ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . 2026-6 The geometry of the paths of the rear and front wheels and the center

of mass of the snakeboard with a radius of rotation ρ. . . . . . . . . . 2046-7 A near identity transformation with a parameter |λ| ¿ 1. . . . . . . . 2096-8 Trajectories of the nonholonomic mobile robot in (x1, x2)-plane for ini-

tial position x = (2, 3)T (v = 0) and orientation angles θ = 0, π/4, π/2, 3π/4.2116-9 Trajectories of the nonholonomic mobile robot in (x1, x2)-plane for ini-

tial position x = (2, 3)T (v = 0) and orientation angles θ = π, 5π/4, 3π/2, 7π/4.2126-10 Trajectories of the nonholonomic mobile robot in (x1, x2)-plane starting

at x = (2, 3)T (v = 0) with angle θ = π/4 . . . . . . . . . . . . . . . . 213

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6-11 Trajectories of the nonholonomic mobile robot in (x1, x2)-plane forinitial position x = (0, 2)T (v = 0) and initial orientation anglesθ = 0, π/4, π/2, 3π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6-12 Evolution of position and controls for the nonholonomic mobile robotin (x1, x2)-plane starting at x = (0, 2)T (v = 0) with angle θ = 0 . . . 214

6-13 Trace trajectories of a two-wheel nonholonomic robot with initial po-sition (5, 4) and heading angles (a) θ = 0, (b) θ = π/2, (c) θ = π, and(d) θ = 3π/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6-14 Position trajectory tracking for a mobile robot from the initial point(4, 4, π/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

7-1 A C2 smooth sigmoidal function with flat tails with its derivatives. . 2487-2 Sigmoidal ribbon S and compact set K. . . . . . . . . . . . . . . . . . 2507-3 The Cart-Pole System . . . . . . . . . . . . . . . . . . . . . . . . . . 2607-4 The state trajectories of the Cart-Pole system. . . . . . . . . . . . . . 2617-5 The Rotating Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 2637-6 The state trajectories for the Rotating Pendulum. . . . . . . . . . . . 2657-7 The state trajectories of the generalized beam-and-ball system with

initial condition (1, 0,−π/6, 0) for g = −9.8 and ν = 1, 2, 3. . . . . . . 267

A-1 A planar three-link robot arm. . . . . . . . . . . . . . . . . . . . . . . 275A-2 The Beam-and-Ball System. . . . . . . . . . . . . . . . . . . . . . . . 281A-3 The Cart-Pole system. . . . . . . . . . . . . . . . . . . . . . . . . . . 282A-4 The Rotating Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 283A-5 The 3D Cart-Pole system with unactuated roll and pitch angles. . . . 285A-6 An underactuated surface vessel. . . . . . . . . . . . . . . . . . . . . 286

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List of Tables

4.1 Classification of Underactuated Systems . . . . . . . . . . . . . . . . 101

5.1 The Values of F ∗(ε, n0). . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 The Values of τ ∗(ε, n0, 1). . . . . . . . . . . . . . . . . . . . . . . . . . 144

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Chapter 1

Introduction

1.1 Mechanics and Control Theory

Control of mechanical systems is currently among one of the most active fields ofresearch due to the diverse applications of mechanical systems in real-life. Though,the study of mechanical systems goes back to Euler and Lagrange in the 1700’s, it wasnot until 1850’s that mechanical control systems came to the picture in regulation ofsteam engines. During the past century, a series of scientific, industrial, and militaryapplications motivated rigorous analysis and control design for mechanical systems.On the other hand, theoretically challenging nature of analysis of the behavior of non-linear dynamical systems attracted many mathematicians to study control systems.As a result, the efforts of engineers and scientists together led to creation of LinearControl, Optimal Control, Adaptive Control, and Nonlinear Control theories. Morerecently, Robust Control theory was born and added to the above picture because ofan inevitable need to deal with the presence of uncertainties in real-life control sys-tems. Before the creation of most of these theories, humans managed to fulfill theirlong-time dream to travel in the space and land on the moon. This is one of the great-est accomplishments of control scientists and engineers of the first half of the pastcentury. Aerospace Vehicles and Robotics Systems with ability to explore the surfaceof other planets are still among the most complex machines built by humans. Forthe past 50 years, Aerospace and Robotics applications remained as some of the mostinfluential sources of motivation for rigorous analysis and control of both mechanicalsystems and nonlinear systems. All along the way, advances achieved by researchersin Mechanics and Nonlinear Control Theory mutually affected and enhanced eachother. Control of Multi-body Systems [53] in mechanics and rigorous analysis (e.g.controllability and observability) and control design (e.g. stabilization and outputtracking) for nonlinear systems affine in control [36, 65], which shaped the field ofnonlinear control theory, were just two theoretical consequences of this effort.

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1.2 Nonlinear Control Systems

In this section, we provide an overview of the past research on classes of nonlinearcontrol systems and mechanical systems that are relevant to this work. In this thesis,our main focus in is on control design for nonlinear systems that arise from controlof an important and broad class of mechanical systems known as underactuated sys-tems. Underactuated Systems are mechanical control systems with fewer actuators(i.e. controls) than configuration variables. One of the main contributions of thisthesis is explicit transformation of high-order underactuated systems into cascadenonlinear systems with both triangular and nontriangular structural properties. Thistransformation is performed using a global/semiglobal change of coordinates obtainedfrom the Lagrangian of the system in closed-form. After applying this transforma-tion, control of the original (possibly) high-order underactuated system, reduces tocontrol of lower-order nonlinear systems (possibly) non-affine in control. This moti-vated us to develop new control design methods for both cascade nonlinear systemswith nontriangular structures and classes of nonlinear systems non-affine in control,i.e. highly nonlinear systems. As a result, the main body of this thesis involves thefollowing topics:

i) Dynamics, Reduction, and Control of Underactuated Systems

ii) Control of Highly Nonlinear Systems

iii) Nonlinear Control of Cascade Systems with Nontriangular Structures

In the following, we present the state-of-the-art of research in each of the above topics.

1.2.1 Underactuated Systems

Underactuated mechanical systems are systems that have fewer control inputs thanconfiguration variables. Underactuated systems appear in a broad range of appli-cations including Robotics, Aerospace Systems, Marine Systems, Flexible Systems,Mobile Systems, and Locomotive Systems. The “underactuation” property of under-actuated systems is due to the following four reasons: i) dynamics of the system(e.g. aircraft, spacecraft, helicopters, underwater vehicles, locomotive systems with-out wheels), ii) by design for reduction of the cost or some practical purposes (e.g.satellites with two thrusters and flexible-link robots), iii) actuator failure (e.g. in asurface vessel or aircraft), iv) imposed artificially to create complex low-order nonlin-ear systems for the purpose of gaining insight in control of high-order underactuatedsystems (e.g. the Acrobot, the Pendubot, the Beam-and-Ball system, the Cart-Polesystem, the Rotating Pendulum, the TORA system (all are described in chapter 3)).

The main control methods applied to examples of inverted-pendulum type under-actuated systems is based on swing-up of the pendulum from its downward positionand then switching to a balancing controller that is designed using a linearizationtechnique or gain scheduling to balance the pendulum [94]. This includes swing upcontrol using energy-based methods [90] for the Acrobot (i.e. a two-link planar robot

15

with an actuator at the elbow) [86], the Pendubot (i.e. a two-link planar robot withan actuator at the shoulder) [92], the cart-pole system [21] , a triple-link inverted pen-dulum [90], and the rotating pendulum [6]. The balancing controller for the Acrobotusing spline functions can be found in [13, 14].

Due to its complexity, the beam-and-ball has been the focus of study among re-searchers with diverse interests including approximate feedback linearization methodsby Hauser et al. [34], using small nested saturations for stabilization of feedforwardcascade nonlinear systems due to Teel [102] and stabilization by output feedback [103].Moreover, stabilization of the beam-and-ball by construction of Lyapunov functionsis addressed in [80]. This is done based on the original work due to Mazenc and Pralyin [56]. Recently, global stabilization of the beam-and-ball system is achieved in [79].

Passivity-based method is mostly used for swing-up control design of underactu-ated systems with inverted-pendulums [28, 90]. Moreover, a passivity-based approachwas employed for a special example of a pendulum on a cart (i.e. the TORA sys-tem) that was transformed to a cascade form [37]. The main drawback of thesepassivity-based methods is their narrow range of applications. In fact, to the best ofmy knowledge, so far no applications of energy-based methods in control of real-lifeunderactuated systems in Robotics, Aerospace, and Marine applications is known.

The VTOL (vertical take off and landing) aircraft is another example of an under-actuated system that has been extensively used as a test-bed for different methods oftrajectory tracking and configuration stabilization. This includes tracking for slightlynon-minimum phase systems [35] and hybrid/switching based control methods [58].

Exponential stabilization of examples of underwater vehicles and surface vesselsthat are underactuated was achieved in [27] and [75] using appropriate coordinatetransformations and analysis of a time-varing linear system. A similar type result forattitude control of an underactuated spacecraft is given in [62].

The role of second-order nonholonomic constrains in the necessity of the use ofdiscontinuous stabilizing feedback laws for stabilization of underactuated systems isdiscussed in [111]. This is mainly based on the famous condition of Brockett onstabilizability of nonlinear systems using time-invariant continuously differentiablestate feedbacks [15]. In addition to this issue, accessibility of classes of underactuatedmechanical systems has been recently addressed in a recent important work by Rey-hanoglu et al. [78]. This is based on a framework applied to analysis of controllabilityof nonholonomic systems [11] and a famous controllability theorem due to Sussmann[97]. An example of a discontinuous stabilizing feedback for a system with an internalunactuated degree of freedom is given in [77].

Adaptive control [33] and sliding mode control techniques [96] have been alsoapplied to underactuated mechanical systems for rather limited applications.

Flexible-link robots are an important class of underactuated systems that areappropriate for space applications due to their lightweight and fast execution of com-mands. The Euler-Bernoulli model for a flexible arm is an infinite dimensional system[25]. A truncated modal analysis can be used to obtain a finite dimensional state-spacemodel for flexible robots [25, 107]. Trajectory tracking for flexible robots is rathercomplicated and common measurements like the angle of rotation or the position ofthe tip, respectively, lead to a poor performance and non-minimum phase zero dy-

16

namics (see [20, 24] and references therein). In [20], a noncollocated minimum-phaseoutput is proposed based on an analysis of the initial infinite dimensional model andthen a finite-order compensator is deigned for trajectory tracking. Here, we obtaina nonlinear noncollocated minimum-phase output for a flexible one-link robot armfrom a finite-order Euler-Lagrangian equations of the system.

The method of controlled Lagrangians (i.e. applying a control input that preservesthe Lagrangian structure of a mechanical system) has been recently applied to localstabilization of the cart-pole system and the rotating pendulum to an equilibriummanifold [10]. However, so far this method has been unable to stabilize the RotatingPendulum or more general underactuated systems to an equilibrium point.

In addition to more traditional methods, recently, hybrid and switching-basedcontrol methods are finding their way in control of underactuated mechanical systems[7, 41] and bipedal locomotion of walking robots [88, 89, 91].

In conclusion, apart from linearization-based techniques, control of underactuatedmechanical systems has been mainly along the line of stabilization of special exam-ples of cascade nonlinear systems and using energy-based methods combined with asupervisory-based switching control. The state-of-the-art of research in control of un-deractuated systems is currently very far from our goal to find control design methodsfor broad classes of high-order underactuated systems that are effective for Roboticsand Aerospace applications.

1.2.2 Highly Nonlinear Systems

To describe what we mean by “highly nonlinear systems”, first we present an evolutionof control systems from linear systems to nonlinear systems. We start with lineartime-invariant control systems in the form

x = Ax+Bu , y = Cx (1.1)

where x ∈ Rn, y ∈ Rm, u ∈ Rp. Questions regarding controllability, observability,stabilization, and tracking for this system using state or output feedback has beenquite well-understood for a long time. However, adding constraints or further specifi-cations to the description of the system might make the control design for the systemrather complicated. For example, if the system is controllable with bounded controlinputs, is there any static or dynamic state feedback that asymptotically stabilizesthe system? or what if A,B,C are known up to an uncertainty (i.e. A = A0 + ∆where A0 is known and ∆ is unknown but norm-bounded). None of these problemscan be addressed as simple as the control problems for the original linear system (1.1).One can observe that minor deviations from the standard problem of stabilization ofan LTI control system and additional constraints make the system rather complex.As a further step, consider the following linear system in feedback connection with amemoryless nonlinearity φ

x = Ax+Bu , y = φ(Cx) (1.2)

17

where φ(0) = 0 and ∃c2 > c1 ≥ 0 : c1z2 ≤ z · φ(z) ≤ c2z

2, ∀z ∈ R. We call thesesystems slightly nonlinear systems. Control design and analysis for systems in the form(1.2) with different types of nonlinearities led to Absolute Stability theory in 60’s. Aninput-output stability approach or a frequency domain analysis are the dominant toolsin dealing with slightly nonlinear systems. In mid-eighties, this notion of feedbackinterconnection of a linear system and a nonlinearity was generalized to feedbackof an LTI system and an uncertainty (or operator) that has bounded gain. Thisled to Robust Stability theory [23] and more recently Integral Quadratic Constraints(IQC’s) [61] due to Megretski and Rantzer. Though, these methods are successful indealing with linear systems, uncertain linear systems, and slightly nonlinear systems,they are not applicable to truly nonlinear systems (i.e. systems with nonlinear time-evolution of the state) that do not include any basic linear parts. To be more precise,a modification of (1.2) as

x = σ(Ax+Bu) , y = Cx (1.3)

called a recurrent neural network with a saturation-type (i.e. sigmoidal) nonlinearityσ, has no fundamental similarities in terms of controllability and observability to anLTI control system or a slightly nonlinear system. In addition, the linearity of theoutput does not simplify analysis of the system due to the fact that time-evolution ofthe system follows a nonlinear law. Dynamic neural networks are examples of highlynonlinear systems and only recently around mid-nineties conditions for controllability(only the discrete-time case) and observability of these systems were introduced byAlbertini and Sontag [2, 3]. The analysis method employed in the preceding work wasa rather involved time-domain analysis. Obviously, due to the fact that a frequencydomain analysis can only deal with systems that have linear time-evolution of thestate.

A rather standard form for nonlinear systems affine in control in analogy to (1.1)is

x = f(x) + g(x)u , y = h(x) (1.4)

where f, g, h are nonlinear smooth functions. By a highly nonlinear system, we meana system in the following form

x = f(x, u) , y = h(x) (1.5)

where f is a nonlinear function of (x, u) (regardless of linearity or nonlinearity of h(x))such that there exist no diffeomorphism (z, v) = (T1(x), T2(x, u)) and matrices A,Bsatisfying z = Az + Bv. A comprehensive local theory regarding controllability, ob-servability, stabilization, tracking, and disturbance decoupling for nonlinear systemsin (1.4) can be found in [36] (see references therein for further details). The maintools to address these control problems were differential-geometry and Lie theory thatbecame very common in the literature around late 70’s and early 80’s. Though thesemethods were rather successful in local analysis of nonlinear systems affine in controlthey usually fail to work for a global analysis and nonlinear systems that are non-affine in control. Moreover, lie algebraic conditions are not robust to uncertainties in

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f, g, h.Input-to-State Stability (ISS) theory, essentially developed and introduced by Son-

tag [82, 84], combines both Absolute Stability and Robust Stability theories in onefor highly nonlinear systems in the general form (1.5). The main tools in this the-ory for robustness analysis to disturbances are control Lyapunov functions (CLF’s).The problem is that in general it is not easy to construct CLF’s for highly nonlinearsystems.

In many control applications, a global/semiglobal control design and analysis isrequired. In addition, after applying certain nonlinear coordinate transformation tothe dynamics of nonlinear systems affine in control, the transformed system or itssubsystems could be nonlinear systems that are non-affine in control. This motivatedus to consider global/semiglobal stabilization and analysis of highly nonlinear systemsthat arise from the study of underactuated mechanical systems and nonholonomicsystems. This topic is covered in chapters 4, 5, 6, and 7.

1.2.3 Cascade Nonlinear Systems

Cascade nonlinear systems arise in many control applications either naturally aftersome change of coordinates, or due to design of an output feedback or a dynamicstate feedback. In general, they are in the following form

z = f(z, ξ)

ξ = g(ξ, u)(1.6)

The most well-known results for cascade systems are related to nonlinear systems intriangular forms. Namely, nonlinear systems in strict feedback forms and feedforwardforms. The backstepping procedure has proven to be successful for systems in strictfeedback form [57, 36, 40, 80]. Control of feedforward cascade nonlinear systems isalso well-studied. The simplest example of a feedforward nonlinear system is a per-turbed chain of integrators that can be stabilized using small nested saturations [102].Then, more general classes of feedforward systems were either controlled using a non-linear small-gain theorem [101] or construction of Lyapunov functions [56]. However,control of cascade nonlinear systems with nontriangular structures has proven to berather problematic. This is due to a counterexample by Sussmann [98] and existenceof the peaking phenomenon [100]. It turns out that if some coupling terms in thedynamics of cascade nonlinear systems satisfy appropriate growth conditions then itis possible to stabilize them using low-gain or high-gain feedback laws [80], [79]. Inchapter 5, we show that the Rotating Pendulum and the Pendubot are examples ofunderactuated systems that have nontriangular structures and they do not satisfy anyof the aforementioned growth conditions. This suggests developing new stabilizationmethods for cascade nonlinear systems with nontriangular structures. This stabiliza-tion method is one of the main contributions of this work which will be presented inchapter 7.

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1.3 Statement of the Contributions

In this section, we briefly present the main contributions of this thesis. More detailedstatement of the contributions are given in the subsequent sections.

The main focus in this thesis is on nonlinear control of underactuated mechani-cal systems. This is motivated by broad applications of underactuated systems andtheoretically challenging problems that they have to offer. Though, control of me-chanical systems with nonholonomic first-order (or velocity) constraints is very well-studied, there is very little known about control of underactuated systems with non-holonomic second-order (or acceleration) constraints. In fact, based on two recentsurveys [78, 87], control of general underactuated mechanical systems is currentlyconsidered a major open problem.

Our main contribution in this thesis is to provide analytical tools that allow usto translate and solve control problems for broad classes of high-order underactuatedsystems to equivalent control problems for lower-order nonlinear systems. We call thissystematic translation procedure reduction of underactuated systems. Such a reduc-tion process requires an explicit transformation, i.e. a global change of coordinates (ordiffeomorphism) in closed-form. We introduce these explicit change of coordinates forbroad classes of underactuated systems including both real-life examples in roboticsand aerospace applications and benchmark nonlinear systems. Under such a diffeo-morphism, the original underactuated system transforms into a nonlinear system thatis referred to as a normal form. It turns out that underactuated systems with symme-try properties can be reduced to normal forms that are structured cascade nonlinearsystems. These cascade normal forms are nonlinear systems in strict feedback form[57], feedforward form [102], and nontriangular linear-quadratic form [69] (see section3.8 for definitions). This in turn implies that these normal forms can be controlledusing backstepping procedure [57], feedforwarding [102, 101, 56, 80], and fixed-pointstate feedback laws [67] (presented in chapter 7), respectively. Therefore, this thesisincludes a complete package for control design and analysis of both low-order andhigh-order underactuated systems with symmetry by providing

i) Classification of underactuated systems,

ii) Reduction of each class to its normal form using an explicit transformation,

iii) Control design for each reduced normal form (or class).

All three types of aforementioned cascade normal forms are special classes ofByrnes-Isidori normal form with a double-integrator linear part [36]. However, thespecial structure of each type of normal form allows effective use of the existing andrecently developed nonlinear control design methods. This is very important, sinceeven stabilization of the general Byrnes-Isidori normal form with a double-integratorlinear part is currently an open problem in nonlinear control theory.

Nearly all current applications of underactuated systems possess certain kineticsymmetry properties. By kinetic symmetry, we mean invariance of the kinetic energyunder the action of a cyclic group. Of course, the reader should notice that this

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is a different notion than the famous classical symmetry in mechanics [1, 54] whichin a simpler form was familiar to Lagrange. The classical symmetry is defined asthe invariance of the Lagrangian of a system under a symmetry group action. Insimple words, mechanical systems that their inertia matrix is independent of a subsetof configuration variables possess kinetic symmetry property w.r.t. this subset ofvariables. We call this subset of variables external variables. The compliment ofthe set of external variables are called shape variables. Thus, shape variables areconfiguration variables that appear in the inertia matrix of a (simple) mechanicalsystem. The key tools in reduction of high-order underactuated systems are normalizedgeneralized momentums conjugate to external or shape variables together with theirintegrals (see section 2.8 for the definitions). It turns out that actuation or lack ofactuation of the shape variables of an underactuated system and also integrabilityof the normalized momentums play the fundamental role in the structure of thecorresponding normal form of the system. These two properties together with anothertwo basic properties of underactuated systems led us to classification of underactuatedmechanical systems with kinetic symmetry to 8 classes. For each class, we provideits associated normal form, control design method, and examples from real-life andbenchmark nonlinear systems. A surprising result is that the reduced-order subsystemof the normal form, for several classes of underactuated systems is a Lagrangiansystem satisfying forced (or unforced) Euler-Lagrange equations. In other words, Iobtain Byrnes-Isidori normal forms for underactuated systems with zero-dynamicsthat are well-defined lower-order Lagrangian systems. In certain cases, this allows usto use the Hamiltonian/Lagrangian of the reduced system as a Lyapunov function.

Several examples of real-life and benchmark-type high-order underactuated sys-tems possess non-integrable normalized momentums (e.g. a flexible-link robot anda 3D Cart-Pole system). We introduce a method called Momentum Decompositionwhich uniquely decomposes a non-integrable normalized momentum as the summa-tion of an integrable momentum and a non-integrable momentum. This class ofunderactuated systems with non-integrable momentums can be then reduced to cas-cade normal forms plus structured perturbation terms. A possible way to deal withthis problem is to design a nonlinear controller for the nominal unperturbed systemthat renders the desired equilibrium point of the underactuated system asymptoti-cally stable and makes the perturbation term exponentially vanishing (the details areprovided in chapter 4). Then, after performing a Lyapunov-based Robustness Analy-sis, the stability of the perturbed system in closed-loop with the nominal nonlinearcontroller can be verified. A Lyapunov function candidate for this analysis can becalculated via the Lyapunov function of the core reduced system of the unperturbedsystem.

In chapter 5, we apply our theoretical results in reduction and control of high-order underactuated systems to a number of complex underactuated systems. Hereis a list of contributions on these applications:

– Global asymptotic stabilization of the Acrobot to any arbitrary equilibriumpoint of the system including the upright equilibrium using a nonlinear statefeedback.

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– Global asymptotic stabilization of the TORA system using a nonlinear statefeedback.

– Global asymptotic stabilization of the Inertia-Wheel Pendulum using a nonlin-ear state feedback.

– Almost global exponential stabilization of the Cart-Pole system, and the Ro-tating Pendulum to their upright relative equilibrium point.

– Global asymptotic and local exponential stabilization of the 2D and 3D Cart-Pole systems to their upright equilibrium point.

– Semiglobal asymptotic stabilization of the Cart-Pole system using a fixed-pointstate feedback law.

– Partial Semiglobal asymptotic stabilization of the Rotating Pendulum and theBeam-and-Ball system using a fixed-point state feedback law.

– Global exponential stabilization of a nonholonomic two-wheeled mobile robotto a point ε¿ 1 far from the origin using a smooth dynamic state feedback.

– Trajectory tracking for a flexible one-link robot arm using a nonlinear noncol-located output.

– Reduction of stabilization of a planar n-link robot underactuated by one tostabilization of the Acrobot or the Pendubot.

– Global position and attitude stabilization for the VTOL aircraft with stronginput coupling.

– Almost global exponential tracking of the feasible trajectories of an autonomoushelicopter.

– Automatic calculation of differentially flat outputs for the VTOL aircraft as aby-product of the decoupling of the effects of the body torque in the translationaldynamics of the VTOL.

– Automatic calculation of weakly differentially flat outputs for a relatively ac-curate model of an autonomous helicopter as a by-product of the approximatedecoupling of the effects of the body torque in the translational dynamics of thehelicopter.

1.3.1 Normal Forms for Underactuated Systems

Underactuated systems as mechanical systems with fewer inputs than configurationvariables with a nonholonomic second-order (or acceleration) constraint are not fully(i.e. exact) feedback linearizable. In [90], Spong showed that actuated configurationvariables of an underactuated system with non-interacting inputs (defined in section3.4) can be globally linearized using an invertible change of control. This is called

22

collocated partial feedback linearization. This is a important initial step in some ofour results and proves that a broad class of underactuated systems can be partiallylinearized. However, the main difficulty arises after applying this change of control.Since the new control appears both in the actuated and unactuated subsystems of theoriginal underactuated system. In chapter 3, we propose an appropriate structure fora global change of coordinates that decouples these two subsystems w.r.t. the newcontrol. Sufficient conditions are provided for the existence of this diffeomorphism.The condition is a simple algebraic test on the inertia matrix of the system. After ap-plying this transformation, one obtains a cascade normal form which is a special classof Byrnes-Isidori normal form with a double-integrator linear part [36]. The weaknessof this result is that to obtain the change of coordinates, based on Frobenius’s theo-rem, it is required to solve a first-order partial differential equation (PDE). Solvingthis PDE, in turn, reduces to solving a finite number of ODE’s which constitute thecharacterization equations of the PDE. For low-order underactuated systems with 2DOF, this PDE can be explicitly solved. Solving this PDE explicitly for high-orderunderactuated is not a feasible approach. Instead, in chapter 4, we present a newmethod to resolve this issue. The method mainly relays on the use of normalizedgeneralized momentums and their integrals (wherever possible).

Partial feedback linearization is an initial simplifying step for reduction and con-trol of underactuated, regardless of the method used for decoupling of the actuatedand unactuated subsystems. For this purpose, we extended the partial feedback lin-earization result in [90] to two other cases: i) linearization of the dynamics of theunactuated configuration variables which is called noncollocated partial feedback lin-earization, and ii) linearization of actuated configuration variables for underactuatedsystems with input coupling. The former one is used for partial feedback lineariza-tion of underactuated systems with unactuated shape variables (e.g. a flexible-linkrobot or the Rotating Pendulum). The latter one is very useful in aerospace applica-tions, particularly, for decoupling of the effects of the body torque in the translationaldynamics of a VTOL aircraft or a helicopter.

1.3.2 Reduction and Control of Low-Order UnderactuatedSystems

In chapter 3, combination of a partial feedback linearization step and an explicit de-coupling change of coordinates is used for reduction of underactuated systems withtwo degrees of freedom. Derivation of explicit normal forms for low-order underac-tuated systems is important, because we later show that control of special classes ofhigh-order underactuated systems reduces to control of a number of low-order (e.g. 2DOF) systems. As an example, in chapter 4, we prove that stabilization of an n-link(n > 2) planar robot with (n− 1) controls reduces to stabilization of the Acrobot orthe Pendubot regardless of n.

There is a simple way to classify underactuated systems with 2 DOF and a singleshape variable. Either their shape variable is actuated, or it is unactuated. We callthese two classes of underactuated systems Class-I and Class-II systems, respectively.

23

Examples of Class-I systems are the Acrobot, the Inertia-Wheel Pendulum, and theTORA system. We proved that all Class-I underactuated systems can be trans-formed into a cascade nonlinear system in strict feedback form and (possibly) globallyasymptotically stabilized using backstepping procedure. In contrast, all Class-II un-deractuated systems can be globally/semiglobally transformed into cascade systemsin nontriangular quadratic normal form (definition 3.8.5). Examples of Class-II sys-tems include flexible-link robots, the Cart-Pole system, the Rotating Pendulum, thePendubot, and the Beam-and-Ball system. Stabilization of general partially-linearnon-triangular cascade nonlinear systems is currently an open problem [36]. Weaddress this problem for important classes of nontriangular linear-quadratic normalforms and more general nontriangular forms in chapter 7.

We provide sufficient conditions so that under a second change of coordinates,Class-II systems can be reduced to cascade nonlinear systems in feedforward forms.Among all the stated examples, the Cart-Pole system is the only one that falls withinthis class and can be globally asymptotically stabilized using a state feedback in theform of nested saturations due to Teel [102]. In chapter 5, we present detailed controldesign for all the aforementioned examples of underactuated systems with 2 DOF.

1.3.3 Reduction and Control of High-Order UnderactuatedSystems

Most of real-life underactuated systems are high-order systems. An aircraft or a heli-copter has 6 DOF and 4 controls. A flexible n-link robot withm deformation modes ateach link has n(m+1) DOF, n controls, and nm unactuated degrees of freedom (e.g.for n = 2,m = 3 the system has 8 DOF and 2 controls). This motivated us to focus onreduction of high-order underactuated systems (in chapter 4). The kinetic symmetryproperty of underactuated systems and the degree in which the shape variables areactuated are two main factors in reduction of high-order underactuated systems. Ourkey analytical tools here are the normalized generalized momentums conjugate to ac-tuated/unactuated configuration variables and their integrals (wherever possible). Itturns out that the integrability of theses normalized momentums plays a fundamentalrole in the structure of the normal forms for high-order underactuated systems.

An important property of normal forms for high-order underactuated systems isthat they are physically meaningful. To be more precise, consider a class of un-deractuated systems with n degrees of freedom, m actuated shape variables withnon-interacting inputs. Assume the normalized momentums conjugate to the (n−m)external variables of this system are integrable. We call this class of high-order under-actuated systems Class-I systems. We prove that there exists a global diffeomorphismobtained from the Lagrangian of the system that transforms the dynamics of a Class-Iunderactuated system into a reduced-order system with a new configuration vector ofdimension (n−m) and a well-defined Lagrangian function parameterized by the shapevariables which satisfies the unforced Euler-Lagrange equations. This normal formis in fact a cascade nonlinear system in strict feedback form with a zero-dynamics(associated with the shape variables as the output) which is a Lagrangian function

24

itself. We refer to this Lagrangian zero-dynamics system as the core reduced system.In chapter 4, we provide a method for stabilization of this core reduced system using anonlinear static state feedback. As a by-product, we obtain an energy-type Lyapunovfunction for the closed-loop core reduced system. In certain cases, the physical prop-erty of the zero-dynamics subsystem allows using Hamiltonian/Lagrangian of the corereduced system as a valid Lypunov function. These Lyapunov functions can be laterused for robustness analysis of the perturbed normal forms of Class-I underactuatedsystems.

Now, consider the class of underactuated systems with integrable normalized mo-mentums conjugate to shape variables and non-interacting inputs. We call themClass-II underactuated systems. After the reduction process, Class-II underactuatedsystems can be explicitly transformed into nontriangular quadratic normal forms. Inthis case, still the reduced-order subsystem (i.e. the zero-dynamics) is a well-definedLagrangian system (under minor conditions). But it satisfies a forced Euler-Lagrangeequation with a rather complex input force with a nonlinear dependence on both theconfiguration and momentum of the reduced Lagrangian system. The complexity ofthe structure of this input force is one of the main sources of extreme difficulties incontrol design for Class-II underactuated systems.

We provide simple algebraic sufficient conditions on the inertia matrix and po-tential energy of the system such that the nontriangular normal form of a Class-IIunderactuated system can be transformed into a feedforward system using a secondexplicit change of coordinates. The subclass of Class-II systems that satisfies thesesufficient conditions are called Class-III systems.

The second important factor in reduction of underactuated systems, particularlyin aerospace applications and locomotive systems, is the presence of input couplingbetween actuated and unactuated subsystems. For flat underactuated systems in-cluding aircraft and helicopters, the reduction is rather straightforward. In chapter4, we address reduction of non-flat underactuated systems with input coupling. Thisclass of real-life systems can potentially have very high degrees of freedom. An exam-ple is a flying bird as a three-body system with 14 DOF and 8 controls. In general,many locomotive systems in the nature fall within this class. Other examples includea swimming fish and a walking human being. All of these three examples have ac-tuated shape variables and they locomote in their environment using the couplingbetween their translational dynamics and shape variables. Therefore, the presence ofinput coupling is very crucial for the purpose of locomotion of these underactuatedsystems. We call these new classes of non-flat and underactuated systems with inputcoupling Class-IV and Class-V systems, respectively.

For the reduction of Class-I and Class-II underactuated systems, we assumed thenormalized momentums are integrable. A natural question to ask is whether it ispossible to reduce these systems if their normalized momentums are non-integrable.The answer to this question is positive. The integrability property of the normal-ized momentums can be considered as the third most important factor in reductionof high-order underactuated systems. To address reduction underactuated systemswith non-integrable momentums, we introduce a new procedure called MomentumDecomposition. This method uniquely decomposes a non-integrable momentum as

25

the summation of an integrable momentum and a non-integrable momentum. Letus call the underactuated systems with non-integrable normalized momentums corre-sponding to Class-I,II,II, respectively, Class-VI, Class-VII, and Class-VIII systems.We prove that normal forms of Class-VI,VII,VIII underactuated systems with non-integrable momentums are perturbed versions of the normal forms of Class-I,II,IIIsystems with a structured perturbation that only appears in the dynamics of themomentum of the reduced subsystem.

In conclusion, we obtain 8 classes of underactuated systems. Namely, Class-Ithrough Class-VIII systems which are precisely defined with their subclasses in section4.6. The result of classification of underactuated systems is summarized in Table 4.1.For each class, related examples and an appropriate control design method are given.Our main contribution is that the methods for reduction and control of high-ordersystems presented in chapter 4, applies to the majority of important examples ofrobots, aerospace systems, and benchmark nonlinear systems.

1.3.4 Reduction and Control of Underactuated Systems withNonholonomic Velocity Constraints

In chapter 6, we focus on reduction of underactuated systems with kinetic symmetryand nonholonomic velocity constraints. Our main result on reduction of nonholonomicunderactuated systems with symmetry is that under certain assumptions, the reducedsystem can be represented as the cascade of the constraint equation and a lower-orderunderactuated (or fully-actuated) Lagrangian system. If the sum of the number ofcontrols and constraint equations is less than the dimension of the configurationvector, the reduced system is underactuated. Otherwise, it is fully-actuated. Thesnakeboard example (Fig. 6-4) is a mobile robot with 6 degrees of freedom, 3 controls,and 2 nonholonomic contraints [50]. Our reduction method, provides a systematicway for calculation of the reduced system of the snakeboard. Furthermore, we provethat the kinematic model of a snakeboard and a car are diffeomorphic. This impliesthat the snake-board can be transformed into a first-order chained-type nonholonomicsystem (equation 2.8) using an explicit change of coordinates.

Based on Brockett’s result [15], the class of mechanical systems with first-ordernonholonomic constraints can not be stabilized to the origin using a smooth staticstate feedback. An example of this class is a simple two-wheeled mobile robot. We in-troduce a new class of parameterized diffeomorphisms that are called near-identity dif-feomorphisms. An example of a near-identity diffeomorphism is ψ(x; θ, ε) = x+ εv(θ)where v(θ) is a unit vector and for a fix θ and ε, ψ is clearly a diffeomorphism whichis equal to id(x) = x for ε = 0. The point xε = x + εv(θ) is called an ε-nearbypoint of x. We use a near-identity change of coordinates for global exponentialstabilization/tracking of a nonholonomic two-wheeled mobile robot to an ε-nearbypoint/trajectory of a desired position/trajectory using a smooth dynamic state feed-back (ε¿ 1). This motivated us to formally define the notions of ε-stabilization andε-tracking for nonlinear systems.

26

1.3.5 Applications in Robotics and Aerospace Vehicles

In chapter 5, we provide detailed control design for several examples of low-order andhigh-order underactuated systems in robotics and aerospace applications. A list ofsome of the examples and their associated control task is stated in the last part ofthe introduction of section 1.3.

1.3.6 Control of Nontriangular Cascade Nonlinear Systems

Stabilization of nontriangular cascade nonlinear systems is currently considered amajor open problem [36]. In chapter 7, we address this stabilization problem forimportant classes of nontriangular cascade nonlinear systems, particularly, systemsin nontriangular linear-quadratic normal form. This includes normal forms for Class-II, Class-IV, and Class-VII underactuated systems.

1.4 Outline

The outline of this thesis is as follows. Chapter 1, provides an introduction andstatement of the contributions. In chapter 2, we give some background on classes ofmechanical control systems and define some basic notions in mechanics. Chapter 3 isdevoted to the dynamics of underactuated system, several examples of underactuatedsystems, partial feedback linearization techniques, normal forms for underactuatedsystems, and classification of low-order underactuated systems. Our main contribu-tions are presented in chapter 4. This includes reduction, classification, and controlof high-order underactuated systems. In chapter 5, we present detailed control designfor several robotics and aerospace applications. In chapter 6, we address reductionand control of underactuated systems with nonholonomic velocity constraints andsymmetry properties. In chapter 7, we introduce fixed-point state feedback lawsfor stabilization of nontriangular cascade nonlinear systems. In chapter 8, we makeconcluding remarks and state possible directions of future research.

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Chapter 2

Mechanical Control Systems

2.1 Introduction

This chapter provides some background on mechanical control systems. We startby defining simple mechanical control systems and introduce Legendre normal formof a simple mechanical system. Then, we discuss fully-actuated mechanical systemsand their trasformation into a set of double-integrators using a change of control.We define underactuated mechanical systems in contrast to fully-actuated mechani-cal systems and demonstrate why control of underactuated systems is a challengingopen problem. A special class of mechanical systems called flat mechanical systemsis described next. We present a result on controllability limitations of underactuatedflat mechnaical systems in lack of gravity terms based on the Legendre normal formof a flat underactuated system. We briefly review state-of-the-art of research in mod-eling and control of mechnaical systems with nonholonomic velocity (i.e. first-order)constranits and clarify their close relations to underactuated mechanical systems thatcontain nonholonomic acceleration constraints. Symmetry in mechanics plays a cru-cial role in some of the main results of this thesis regarding reduction of underactuatedmechanical systems. Therefore, the role of symmetry and conservation laws in me-chanics and their relation through Noether’s theorem is explained in the final section.

2.2 Simple Lagrangian Systems

In this section, we state forced Euler-Lagrange equations of motion as the model ofcontrol mechanical systems and introduce Legendre normal form for simple mechan-ical systems.Simple mechanical systems are systems that their Lagrangian is in the form of the

difference between a (positive semidefinite) kinetic energy and a potential energy

L(q, q) = K − V =1

2qTM(q)q − V (q)

where q ∈ Q denotes the configuration vector that belongs to an n-dimensional config-uration manifold Q, M(q) is the inertia matrix which is a positive definite symmetric

28

matrix. K is the kinetic energy and V (q) is the potential energy of the system. Letfi(q) : Q → Rn, i = 1, . . . ,m be m linearly independent external forces applied tothe system. The Euler-Lagrange equation for this mechanical control system is as thefollowing

d

dt

∂L∂q− ∂L∂q

= F (q)u (2.1)

where u ∈ Rm and F (q) = (f1(q), . . . , fm(q)) denotes the matrix of external forces.The equations of motion for this mechanical system can be derived as the following

Σjmkj(q)qj + Σi,jΓkij(q)qiqj + gk(q) = eTkF (q)u, k = 1, . . . , n

where ek is the kth standard basis in Rn, gk(q) = ∂qkV (q), and Γkij(q) are calledChristoffel symbols and can be defined as

Γkij(q) =1

2(∂mkj(q)

∂qi+∂mki(q)

∂qj− ∂mij(q)

∂qk).

in the vector form, we have

M(q)q + C(q, q)q +G(q) = F (q)u (2.2)

cij = Σnk=1Γ

ikj(q)qk

is an element of C(q, q). The term C(q, q)q contains two types of terms involvingqiqj that are called Centrifugal terms (i = j) and Coriolis terms (i 6= j). Also,G(q) contains the gravity terms (see [95] for more details and examples). The in-teresting relationship between the matrices M and C is that S0 = M(q) − 2C(q, q)is a skew symmetric matrix. In other words, because M is symmetric, one getsM(q) = C(q, q) + CT (q, q). Taking this property into account and noting that M(q)is a positive definite matrix, the Legendre transform with respect to q

p =∂L∂q

=M(q)q

is invertible and the dynamics of the mechanical system in (2.2) can be rewritten inthe canonical form

q = M−1(q)p

p = −G(q) + CT (q, p)M−1(q)p+ F (q)u(2.3)

where C(q, p) = C(q,M−1p). We call (2.3) Legendre Normal Form of a simple me-chanical system. Defining the variables x1 = q, x2 = p, (2.3) can be rewritten as

x1 = M−1(x1)x2x2 = −G(x1) + xT2Q(x1)x2 + F (x1)u

(2.4)

orx = f(x) + g(x)u (2.5)

29

which is a nonlinear system affine in control with state x = col(x1, x2) and vectorfields

f(x) = col(M−1(x1)x2,−G(x1) + xT2 σ(x1)x2)g(x) = col(0, F (x1))

with the property that f(x) has a mixed linear-quadratic structure w.r.t. x2 andg(x) is independent of x2. Note that σ is a cubic matrix such that xT2Q(x1)x2 ∈ Rn.The main advantage of the form (2.5) is that many analytical tools are availablefor controllability and observability analysis and control design of nonlinear systemsaffine in control [36].

Remark 2.2.1. For mechanical systems, equations (2.2) and (2.3) are two equivalentforms. However, The Legendre normal form is a first-order ODE and (2.2) is a second-order ODE. Later, we will see that from control point of view (2.3) is more appropriateand can be directly used for controllability analysis of a mechanical system.

Remark 2.2.2. Defining the Hamiltonian of a simple Lagrangian system as

H(q, p) =1

2pTM−1(q)p+ V (q)

the Legendre normal form in (2.3) is equivqlent to the following forced Hamiltonianequations

q =∂H(q, p)

∂p

p = −∂H(q, p)

∂q+ F (q)u

However, the Hamiltonian form is not appropriate for direct controllability analysisand control design compared to the Legendre normal form.

In the following sections, we describe several important classes of mechanical con-trol systems.

2.3 Fully-actuated Mechanical Systems

Consider the control mechanical system in (2.1). We call (2.1) a fully-actuated me-chanical system if m = rank F (q) = n, i.e. F (q) is an invertible matrix. For fully-actuated systems the number of control inputs is equal to the dimension of their con-figuration manifold. Therefore, fully-actuated mechanical systems are exact feedbacklinearizable (i.e. they have no zero-dynamics [36]). This can be proved by applyingthe following change of control

u = F (q)−1(M(q)v + C(q, q)q +G(q))

and redefining the variables as x1 = q, x2 = q, to obtain

x1 = x2x2 = v

30

which is clearly a (vector) double integrator. This is why most of problems for fully-actuated mechanical systems can be reduced to equivalent problems for linear systems.This fact suggests that control of mechanical systems is challenging either in thepresence of uncertainties (e.g. parametric uncertainties in M,C,G), or when thenumber of control inputs is less than n which in both cases exact feedback linearizationis not possible. Moreover, under any holonomic or nonholomic constraint (both tobe defined), control and analysis of mechanical systems could potentially be rathercomplicated. The same is true if the control input of a mechanical system is requiredto be bounded.

2.4 Underactuated Mechanical Systems

A control mechanical system with configuration vector q ∈ Q and Lagrangian L(q, q)satisfying the Euler-Lagrange equation

d

dt

∂L∂q− ∂L∂q

= F (q)u (2.6)

is called an Underactuated Mechanical System (UMS) ifm = rankF (q) < n = dim(Q)(see the definitions for equation (2.1)). In other words, underactuted systems aremechanical systems that have fewer actuators than configuration variables. Thisrestriction of the control authority does not allow exact feedback linearization ofunderactuated systems. As a special case, assume F (q) = (0, Im)

T . Then, the first(m− n) equations in (2.6) can be expressed as a second-order dynamic equation

ϕ(q, q, q) = 0 (2.7)

which contains Coriolis, centerifugal, and gravity terms that can be highly nonlinear.If there exists no function h such that h = ϕ(q, q, q), equation (2.7) is called a second-order nonholonomic constraint, or a nonholonomic acceleration constraint (e.g. seesection A.8). We assume all underactuated systems throughout this work have second-order nonholonomic constraints unless otherwise is stated.

For the special case of F (q) = (0, Im)T , we refer to the first (n−m) equations of

(2.6) as the unactuated subsystem and to the last m equations of (2.6) as the actuatedsubsystem. In [90], Spong has shown that the actuated subsystem of a general un-deractuated system with F (q) = (0, Im)

T can be linearized using an invertible changeof control. This procedure is called partial feedback linearization. However, afterpartial linearization, the unactuated subsystem of (2.6) still remains as a nonlinearsystem that is coupled with the linearized actuated subsystem through both the newcontrol input and other nonlinear terms. This highly complicates control design forunderactuated systems. One of the main contributions of this thesis is to decouple theactuated and unactuated subsystem of the underactuated system in (2.6) with respectto the new control (see section 3.7). The main focus of this thesis is control designand analysis of underactuated mechanical systems. A comprehensive treatment ofcontrol of underactuated system will be presented in the subsequent chapters.

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2.5 Flat Mechanical Systems

Mechanical systems that their inertia matrix is constant are called Flat MechanicalSystems. One of the characters of a flat mechanical system is that the Christoffelsymbols associated with their inertia matrix vanishes identically and therefore thematrix C(q, q) vanishes as well. Now, assume that for a mechanical system with inertiamatrix M(q) there exists a diffeomorphism z = ψ(q) such that dψ(q) = M 1/2(q)dq,then we have

z =M 1/2q

and the Lagrangian of the system can be written as

L(z, z) = 1

2zT Iz − V (ψ−1(z))

Thus, M(z) = I is constant and in new coordinates the mechanical system is flat.The Legendre normal form for a flat mechanical system with B(q) = I is as the

followingq = M−1pp = τ −G(q)

This is due to the fact that C(q, p) = C(q,M−1(q)p) = 0. The following propositionreveals a major controllability limitation of flat underactuated mechanical systems.

Proposition 2.5.1. Consider an underactuated flat mechanical system with the grav-ity term G(q) = (g1(q), . . . , gn(q))

T . Suppose for some 1 ≤ i ≤ n, qi is unactuatedand gi(q) = 0, then the system is not controllable/stabilizable.

Proof. Based on the assumptions, pi = 0 and pi is a conserved quantity, i.e. pi(t) =pi(0),∀t ≥ 0. If pi(0) 6= 0 then the system cannot be stabilized to any equilibriumpoint/manifold with p = 0.

2.6 Nonholonomic Mechanical Systems

A Lagrangian mechanical system with m < n velocity constraints

W T (q)q = 0

(W is an m × n matrix) that are non-integrable (i.e. @ h(t) : h = W T (q)q) is calleda Mechanical System with First-order Nonholonomic Constraints. Control of non-holonomic mechanical systems with velocity constraints has been extensively studiedin the recent years by numerous researchers. The history of research in this fieldgoes back to early 1900’s. Formulation of the dynamics of nonholonomic systemscan be found in [64]. Controllability and stabilization of nonholonomic systems isconsidered in [11]. In an important work, Bloch and Crouch addressed modeling ofnonholonomic systems on Riemmanian manifolds [8]. More recently, many advanceshave been made on controllability of nonholonomic systems by Lewis and Murray

32

[49], locomotion and control design for nonholonomic systems by Ostrowski in [74],and reduction of nonholonomic systems with symmetry due to Bloch et al. [9].

Following the line of formulation in [11] and based on [64], the dynamics of non-holonomic systems with velocity constraints can be described as the following:

M(q)q + C(q, q) +G(q) = W (q)λ+B(q)uW T (q)q = 0

where λ ∈ Rm is the vector of Lagrange multipliers.

Remark 2.6.1. In the literature [111], [78], sometimes in contrast to nonholonomicsystems with first-order (or velocity) constraints, underactuated mechanical systemsare called systems with second-order (or acceleration) nonholonomic constraints. Thisis due to the fact that if B(q) = (0, I)T is partitioned according to q = (q1, q2)

T , thenregardless of the control input we have

d

dt(∂L∂q1

)− ∂L∂q1

= 0

orm11(q)q1 +m21(q)q2 + h1(q, q) = 0

where mij’s are block matrices of the inertia matrix associated with q = (q1, q2)T .

However, this terminology is somewhat misleading since the Lagrangian of an under-actuated mechanical system satisfies Euler-Lagrange equations of motions withoutany external differential-algebraic constraints that requires use of Lagrangian mul-tipliers in a variational setting as in [8] for nonholonomic systems with first-orderconstraints.

Control design for broad classes of nonholonomic mechanical systems can be re-duced to control of chained-type systems as the following

x(k1)1 = u1x(k2)2 = u2x(k3)3 = x2u1. . .

x(kn)n = xn−1u1

(2.8)

where ki ≥ 1, i = 1, . . . , n are integers. Mobile robots, car-like vehicles, and multi-trailer systems [104] are examples of chained-form systems in (2.8). For ki = 1, system(2.8) is nonholonomic with first-order constraints. Recently, in a an important work[47], (quasi) exponential stabilization of higher-order chained-type systems was ad-dressed by Laiou and Astolfi based on the original work of Astolfi [4]. A discontinuousdynamic-state feedback was introduced for stabilization. Traditionally, stabilizationof chained-form systems with at most first-order nonholonomic constraints has beendone using sinusoids [63] or homogeneous time-varying state feedback [60], [59] due toM’Closkey and Murray. Later, the theoretical results in [60] were applied to local ex-ponential stabilization of the attitude of a spacecraft with nonholonomic constraints

33

[62] and configuration stabilization of a nonholonomic underwater vehicle [27].

2.7 Symmetry in Mechanics

In this section, we define symmetry in mechanics, the connection between symmetryand conservation laws (i.e. Noether’s theorem), and the role of symmetry in reductionof the dynamics of a mechanical system. Formally, the symmetry in mechanics isdefined as the invariance of the Lagrangian of a system under the action of a left (orright) invariant Lie group. For a complete treatment of symmetry see [54] (also [1]),for definitions of Lie groups, left invariant Lie groups, and the action of Lie groupson manifolds see Warner and Boothby.

Almost all real-life mechanical systems possess certain symmetry properties. Forexample, the Lagrangian of a helicopter, a car, or a Satellite is independent of theirposition. This gives rise to symmetries (i.e. invariance of the Lagrangian) w.r.t.translation.

Here, we only consider symmetry of Lagrangian systems in a Euclidean spaceQ = Rn. We say the Lagrangian L(q, q) is symmetric w.r.t. the configuration variableqi iff

∂L∂qi

= 0, i ∈ 1, . . . , n

Denote the ith generalized momentum by pi as

pi =∂L∂qi

and consider the unforced Euler-Lagrange equations of motion

d

dt

∂L∂qi− ∂L∂qi

= 0

It immediately follows that the symmetry of L w.r.t. qi imples that the ith momentumpi is conserved (i.e. pi = 0) and vice versa. The last theorem in its most general formis called Noether’s theorem [1] which establishes a one-to-one equivalence betweenthe existence of symmetries and conservation laws. Lagrange himself was aware ofthis fact and he called qi an ignorable coordinate. This is due to the fact that thedegree of the second-order equation of motion (i.e. Hamiltonian form) for the me-chanical system reduces by 2. Therefore, the existence of symmetry properties leadsto reduction of the mechanical systems.

For the special case of a simple mechanical systems that its Lagrangian is sym-metric w.r.t. qi, the condition pi = 0 is equivalent to a first-order constraint

W T (q)q = pi(0)

whereW (q) = (mi1(q), . . . ,min(q))

T

34

is the ith row of the inertia matrix M(q). If this constraint is non-integrable, theanalysis of the system reduces to the analysis of a mechanical system with a first-order nonholonomic constraint. Therefore, in addition to the conservation laws, theexistence of symmetries gives rise to holonomic/nonholonomic velocity constraints formechanical systems.

In this thesis, we use a different notion of symmetry called Kinetic Symmetry thatis rather similar to classical symmetry. By Kinetic Symmetry, we mean the kineticenergy of the system is invariant w.r.t. qi, i.e.

∂K

∂qi= 0

The kinetic symmetry is equivqlent to classical symmetry in lack of potential en-ergy. However, the majority of systems that we consider here have nonzero (i.e.non-constant) potential energies and the notions of kinetic symmetry and classicalsymmetry do not coincide. It is important to notice that the existence of kinetic sym-metries (in presence of a potential field) does not lead to the existence of conservedquantities. later, we will see that the fact that the generalized momentums are notconserved has a crucial role in controllability of large classes of underactuated me-chanical systems. In addition, we will show that the existence of kinetic symmetryproperties leads to reduction of underactuated Lagrangian systems regardless of thelack of conserved quantities.

2.8 Normalized Momentums and Integrability

In this work, normalized momentums are the most important analytical tools in reduc-tion of underactuated systems. Consider an underactuated system with configurationvector q = (q1, q2) and Lagrangian

L(q, q) = 1

2

[

q1q2

]T [m11(q) m12(q)m21(q) m22(q)

] [

q1q2

]

− V (q)

The generalized momentum conjugate to qi can obtained via partial Lengendre trans-form

pi :=∂L∂qi

= mi1(q)q1 +mi2(q)q2, i = 1, 2

The normalized momentums conjugate to q1 and q2 w.r.t. q1 are defined as thefollowing

π1 := m−111 (q)p1 = q1 +m−1

11 (q)m12(q)q2 , π2 := m−121 (q)p2 = q1 +m−1

21 (q)m22(q)q2

where π1 is always well-defined and π2 is defined wherever m21(q) is an invertiblesquare matrix. We say π = π(q, q) is an integrable normalized momentum, if thereexists a generalized configuration function h = h(q) such that h = π where the

35

function h is defined ash(q, q) = ∇h(q) · q

We call h(q) the integral of π. Whenever @h : h = π, we say π is non-integrable.

36

Chapter 3

Normal Forms for UnderactuatedSystems

3.1 Introduction

Underactuated mechanical systems appear in a variety of real-life control applicationsincluding Robotics (e.g. lightweight flexible-link robots, nonholonomic mobile robots,and walking robots), Aerospace Vehicles (e.g. helicopters, aircraft, spacecraft, andsatellites), Underwater Vehicles, and Surface Vessels. Due to their broad range ofapplications, control of underactuated systems is extremely important. The restric-tion of the control authority for underactuated systems causes major difficulties incontrol design for these systems. During the past decade, the challenging nature ofanalysis and control of underactuated systems has attracted numerous researcherswith interests in nonlinear control theory, robotics and automation, control of auton-umous vehicles, and control of flexible structures. Control of general underactuatedmechanical systems is currently considered a major open problem based on recentsurveys [78, 87].

The reasons behind the complexity of control design for underactuated systems canbe summarized as follows. Underactuated systems are not fully feedback linearizable.Moreover, many recent and traditional methods of nonlinear control design includingbackstepping [36, 57], forwarding [56, 80, 101, 102], high-gain/low-gain designs [80,79], and sliding mode control [40] are not directly applicable to underactuated systemswith the exception of a few special cases (e.g. the beam-and-ball system and the cart-pole system). This is due to the fact that a method for transforming underactuatedsystems into cascade nonlinear systems with upper/lower triangular or nontriangularstructural properties has not yet been discovered. The classes of nonlinear systemsof interest include Byrnes-Isidori normal forms (equation (3.14)) [36], strict feedbackforms (definition 3.8.2) [36, 57], and feedforward forms (definition 3.8.3) [102, 101, 56].

In this chapter, our main contribution is to obtain the aforementioned cascade nor-mal forms for underactuated systems by applying global/semiglobal nonlinear changeof coordinates. The structural properties of the new normal forms allow effectiveuse of many existing control design methods. In addition, a new class of underac-

37

tuated systems is found that leads to a cascade normal form with a nontriangularquadratic structure (see definition 3.8.5). Due to the nontriangular nature of thisnormal form, backstepping/forwarding procedures [80] are not applicable to it. Sta-bilization of this class of nontriangular nonlinear systems is equivalent to control ofa special Byrnes-Isidori normal form with a double-integrator linear part that is amajor open problem in nonlinear control theory [36, 98, 99]. In chapter 7, we addressimportant special cases of this stabilization problem including the cases that appearin control of underactuated mechanical systems.

To attain insight in dealing with control design for higher-order underactuatedsystems (e.g. a helicopter with 6 DOF and 4 controls), first it is very useful toconsider calculation of cascade normal forms for lower-order underactuated systems.For doing so, we focus on the class of underactuated systems with two degrees offreedom and kinetic symmetry (defined in section 2.7). We prove that for this specialclass of underactuated systems both the decoupling change of coordinates and thecorresponding cascade normal form can be obtained in explicit forms. The structureof the obtained normal forms for underactuated systems with two degrees of freedomnaturally leads to their classification in two classes. The first class includes systemslike the Acrobot, the Inertia Wheel Pendulum, and the TORA system (see section 3.3for the definitions) which can be transformed into cascade nonlinear systems in strictfeedback form (definition 3.8.2). The second class includes the Pendubot, the Cart-Pole system, the Rotating Pendulum, and the Beam-and-Ball system (see section 3.3for the definitions) which can be transformed into a nontriangular quadratic normalform (definition 3.8.5). In addition, the cart-pole system can be transformed into afeedforward form after another transformation (definition 3.8.3).

As an initial step, our main result on normal forms for underactuated systemspartly relays on a method called collocated partial feedback linearization due to Spong[90]. Spong showed that the dynamics of the actuated configuration variables of anyunderactuated mechanical system can be globally linearized by applying a change ofcontrol. This allowed application of passivity-based/energy-based methods to solvingswing-up problem for special examples of underactuated systems that involve invertedpendulums. However, after this change of control, the new control appears both inthe actuated subsystem and the unactuated subsystem of the original system. Thiscoupling w.r.t. the input highly complicates control design for underactuated systems.To resolve this issue, we introduce an appropriate structure for a global change ofcoordinates that decouples the linear actuated and nonlinear unactuated subsystemsw.r.t. the new control. This global change of coordinates transforms the dynamicsof the system into a cascade nonlinear system in Byrnes-Isidori normal form with adouble-integrator linear part [36]. In addition, characterization of the cases wherethis normal form reduces to (strict) feedback/feedforward forms is provided.

We introduce a procedure to linearize the dynamics of the unactuated configura-tion variables using a change of control. This is applicable under the condition thatthe number of unactuated variables is less than or equal to the number of actuatedvariables. We call this procedure noncollocated partial feedback linearization. Thisnew partial linearization method is used in obtaining the nontriangular quadraticnormal form for a class of underactuated systems with kinetic symmetry.

38

The outline of this chapter is as follows. First, we present standard models forunderactuated systems in section 3.2. Then, we provide several examples of underac-tuated systems including both benchmark systems and real-life applications in section3.3. In sections 3.4 and 3.5, we explain collocated and noncollocated partial feedbacklinearization methods. Our main results on normal forms for underactuated systemsare presented in section 3.7. In section 3.9, we focus on normal forms and classificationof underactuated systems with two degrees of freedom and kinetic symmetry.

3.2 Dynamics of Underactuated Systems

The Euler-Lagrange equations of motion for an underactuated system are as thefollowing

d

dt

∂L∂q− ∂L∂q

= F (q)τ

where τ ∈ Rm is the control and F (q) ∈ Rn×m is a non-square matrix of externalforces with m < n and full column rank. Here, m denotes the number of controlinputs that is less than the number of configuration variables n. For the case ofsimple Lagrangian systems, the equations of motion for an underactuated mechanicalsystem can be expressed as the following

M(q)q + C(q, q)q +G(q) = F (q)τ (3.1)

Assuming F (q) = [0, Im]T , the configuration vector can be partitioned as q = (q1, q2) ∈

R(n−m) × Rm according to F (q) where q1 and q2 denote the actuated and unactuatedconfiguration vectors, respectively [90]. After partitioning the inertia matrix M(q)accordingly, the dynamics of an underactuated system takes the form

[

m11(q) m12(q)m21(q) m22(q)

] [

q1q2

]

+

[

h1(q, q)h2(q, q)

]

=

[

0τ

]

(3.2)

with τ ∈ Rm. Due to the lack of control in the first equation of (3.2), it is not possibleto fully linearize this underactuated system using a change of control. However, it isstill possible to partially linearize the system such that the dynamics of q2 transformsinto a double integrator [90]. We demonstrate this procedure in sections 3.4 and 3.5.

3.3 Examples of Underactuated Systems

In this section, we present several examples of underactuated systems. These exam-ples include the Acrobot, the Pendubot, the Cart-Pole system, the Rotating Pen-dulum, the Beam-and-Ball system, the TORA (translational rotational actuator)system, the Inertia-Wheel Pendulum, and the VTOL aircraft (vertical take-off andlanding). All the examples are chosen due to the complexity of their control designand the fact that for analysis and control purposes they are of high interest in theliterature. We briefly introduce each example with its related control design task and

39

later use them as applications of our theoretical results by providing control designand analysis for each case in chapter 5. The state-of-the-art of research in controldesign for each example is provided as well. The inertia matrix for each system isgiven for the future use. The examples of this section only include underactuatedsystems with two or three degrees of freedom. Later in chapter 5, we give exam-ples of higher-order underactuated systems that appear in Robotics and Aerospaceapplications.

3.3.1 The Acrobot and the Pendubot

The Acrobot [13, 86] is a two-link planar robot with revolute joints and one actuatorat the elbow as shown in Figure 3.3.1 (a). The Pendubot shown in Figure 3.3.1 (b) isalso a two-link planar robot with revolute joints and an actuator at the shoulder. The

q

q

l 1

L1

m I

m I2 2

1 1

1

2

x

y

q

q

l 1

L1

m I

m I2 2

1 1

1

2

x

y

(a) (b)

Figure 3-1: (a) The Acrobot, (b) The Pendubot.

Acrobot and the Pendubot graphically seem to be very similar (i.e. the share the exactsame inertia matrix). Though, later we will see that the difference in the location oftheir single actuator causes a major difference in their standard representation (i.e.normal form) and control design. The inertia matrix for both the Acrobot and thePendubot is the same and has the following form

M(q2) =

[

a1 + a2 cos(q2) a3 +12a2 cos(q2)

a3 +12a2 cos(q2) a4

]

where the ai’s are positive constants. A possible control task is to stabilize theupright equilibrium point for the Acrobot or the Pendubot from any initial condition.A more complicated task is to stabilize the Acrobot or the Pendubot to any of theirarbitrary infinite equilibrium points. In the past, the former task has been done usingswinging up these double inverted pendulums from their downward initial positionsand bringing them close to their upright position, then switching to a linear controlleraround the upright equilibrium point [13, 86, 90, 94, 28]. The latter task has neverbeen done before to the best of our knowledge. In this work, we perform both of

40

these tasks for the Acrobot and the former task for the Pendubot using a single statefeedback.

3.3.2 The Cart-Pole System and the Rotating Pendulum

The Cart-Pole system shown in Figure 3.3.2 (a) consists of an inverted pendulum ona cart. Figure 3.3.2 (b) shows the Rotating Pendulum [112, 6] which is an invertedpendulum on a rotating arm. The inertia matrix for the Cart-Pole system is

mF

m , l , I

1

2 2 2

q

q1

2

q1

q2

m2, l2, J2

m1, l1, J1

y

z

x

(a) (b)

Figure 3-2: (a) The Cart-Pole system, (b) The Rotating Pendulum.

M(q2) =

[

a1 a2 cos(q2)a2 cos(q2) a3

]

and for the Rotating Pendulum the inertia matrix is

M(q2) =

[

b1 + c1 sin2(q2) b2 cos(q2)

b2 cos(q2) b3

]

where all the ai, bi, ci > 0 are constants. Clearly, the only difference between the iner-tia matrices of these two mechanical systems is in the first element m11(q2). Anothersimilarity between the Cart-Pole system and the Rotating Pendulum is that bothhave the same form of the potential energy

V (q2) = a4 cos(q2)

with a4 > 0. Swing-up control design for the inverted pendulum in the Cart-Pole sys-tem [21, 90] and the Rotating pendulum [112, 6] has been done by several researchers.The drawback of the existing swing-up design methods is that they are relatively slow.In this work, we introduce aggressive swing-up control designs for both the Cart-Polesystem and the Rotating Pendulum using bounded control inputs. In addition, weaddress asymptotic stabilization of both of these systems to their upright equilibriumpoints using nonlinear state feedback.

41

3.3.3 The Beam-and-Ball System

The Beam-and-Ball system illustrated in Figure 3-3 consists of a beam and a ball onit. The task is to bring the ball to the center of the beam by applying a torque τto the beam. The vertical distance between the center of mass of the ball and thelocation of the external torque (i.e. control) is shown by d. The inertia matrix forthe beam and ball system takes the following form

M(q2) =

[

a1 + a2q22 −a3d

−a3d a4

]

where all the ai’s are positive constants. Due to complexity of this system, stabiliza-

dr

Iτ

m , I

q 1

2

2

1

q

Figure 3-3: The Beam-and-Ball System.

tion and tracking for the beam-and-ball system using state or output feedback hasbeen considered by many researchers [34, 102, 103, 80, 79]. In [34], tracking for thebeam-and-ball system was considered using approximate input-output linearization.Semiglobal stabilization of the beam-and-ball system using small nested saturations(i.e. state feedback) was addressed by Teel in [102]. Also, stabilization of this systemusing output feedback is due to Teel and Praly [103]. In [80], global stabilization ofthe beam-and-ball system with friction was achieved using a numeric version of themethod of construction of Lyapunov functions with cross-terms which is originallydue to Mazenc and Praly [56]. Moreover, global stabilization of the beam-and-ballsystem as a perturbed chain of integrators with nontriangular (definition 3.8.4) ho-mogeneous higher-order perturbations was achieved in [79]. In all the aforementionedworks, the model of the beam-and-ball system is taken from [34] with the assumptionthat d = 0. In this work, we consider aggressive stabilization of the beam-and-ballsystem with d = 0 in chapter 7 and introduce the challenges in dealing with the cased 6= 0.

3.3.4 The TORA System

The TORA (Translational Oscillator with Rotational Actuator) system was first in-troduced in [109]. Figure 3-4 illustrates the TORA system consisting of a translationaloscillating platform with mass m1 that is controlled via a rotational eccentric mass

42

m2. The inertia matrix for the system is in the form

M(q2) =

[

a1 a2 cos(q2)a2 cos(q2) a3

]

where the ai > 0 are constants and the potential energy is given by

V (q1, q2)) =1

2k1q

21 +m2gr cos(q2)

Due to the fact that the TORA system is not fully feedback linearizable an an easy

m

m

r

qk

1

2 2

τI

1 2

q 1

Figure 3-4: The TORA system.

change of coordinates was available to transform it to a cascade nonlinear system [109],it received a tremendous attention by several researchers [37, 80, 38, 39]. In most ofthese works the TORA system was used as a benchmark example for passivity-basedtechniques and stabilization/tracking using output feedback in zero gravity, i.e. g = 0.Here, we are interested in global stabilization of the TORA system in the presence ofgravity, i.e. g 6= 0.

3.3.5 The Inertia-Wheel Pendulum

The Inertia-Wheel Pendulum was first introduced by Spong et al. in [93]. Figure3-5 shows the inertia wheel pendulum that consists of a pendulum with a rotatinguniform inertia-wheel at its end. The pendulum is unactuated and the system hasto be controlled via the rotating wheel. The task is to stabilize the pendulum inits upright equilibrium point while the wheel stops rotating. The specific angle ofrotation of the wheel is not important. The inertia wheel pendulum is the first exampleof a flat underactuated mechanical system with two degrees of freedom and a singleactuator. This is due to the constant inertia matrix of the Inertia-Wheel Pendulum.In [93], an energy-based method is used for the swing-up of the pendulum and thena supervisory-based switching strategy is employed to switch to a stabilizing localnonlinear controller with a relatively large region of attraction. Here, we achieveglobal stabilization of the upright equilibrium point of the inertia wheel pendulumusing a single nonlinear state feedback in analytically explicit form.

43

q

q

I

1

2

2

1L , I 1

τ

Figure 3-5: The Inertia Wheel Pendulum

3.3.6 The VTOL Aircraft

The VTOL aircraft depicted in Figure 3-6 is a simplified planar model of a real verticaltake off and landing plane (e.g. the Harrier). The dynamics of the VTOL aircraft is

uε u

u θ

1

2

2

x

y

Figure 3-6: The VTOL aircraft.

given in [35, 55] as the following

x1 = x2x2 = −u1 sin(θ) + εu2 cos(θ)y1 = y2y2 = u1 cos(θ) + εu2 sin(θ)− gθ = ωω = u2

(3.3)

The VTOL aircraft is an underactuated system with three degrees of freedom and twocontrol inputs. In [35], it is assumed that |ε| is relatively small and the VTOL aircraftis treated as a slightly nonminimum phase system. Then, using approximate input-output linearization, bounded tracking is achieved. In section 5.10, we discuss globalconfiguration stabilization for the VTOL aircraft with strong input coupling (i.e.

44

|ε| À 1). In this case, the system is strongly non-minimum phase and stabilizationor tracking for system (3.3) is a challenging problem.

3.4 Collocated Partial Feedback Linearization

The actuated configuration vector for the underactuated mechanical system in (3.2)is q2. The procedure of linearization of the dynamics of q2 is called collocated partiallinearization which is due to Spong [90]. Spong has shown that all underactuatedsystems in the form (3.2) can be globally partially linearized using a change of control.Here, we restate this result as the following with our own phrasing due to applicationsin this work.

Proposition 3.4.1. (Spong [90]) There exists a global invertible change of control inthe form

τ = α(q)u+ β(q, q)

that partially-linearizes the dynamics of (3.2) as the following

q1 = p1p1 = f0(q, p) + g0(q)uq2 = p2p2 = u

(3.4)

where α(q) is an m×m positive definite symmetric matrix and

g0(q) = −m−111 (q)m12(q)

Proof. From the first line of (3.2), we have

q1 = −m−111 (q)h1(q, q)−m−1

11 (q)m12(q)q2

which proves the expression for g0(q). Substituting this in the second line of (3.2),we get

(m22(q)−m21(q)m−111 (q)m12(q))q2 + h2(q, q)−m−1

11 (q)h1(q, q) = τ

Thus, definingα(q) = m22(q)−m21(q)m

−111 (q)m12(q)

β(q, q) = h2(q, q)−m21(q)m−111 (q)h1(q, q)

and observing that α(q) is positive definite and symmetric finishes the proof.The main property of the underactuated system in (3.4) is that after partial

feedback linearization, the new control u appears in the dynamics of both (q1, p1)-subsystem (i.e. nonlinear subsystem) and (q2, p2)-subsystem (i.e. linear subsystem).This is one of the main sources of the complexity of control design for underactuated

45

systems. In section 3.7, we introduce a global change of coordinates that decouplesthese two subsystems while leaves the linear subsystem invariant.

Remark 3.4.1. Defining x = (q1, p1, q2, p2), the dynamics of (3.4) can be rewritten as

x = f(x) + g(x)u

with obvious definitions of f(x) and g(x). In [78], using explicit expressions of f(x)and g(x), sufficient conditions are provided for small-time local controllability (STLC)[97] of a special class of underactuated mechanical systems in the form (3.4).

3.5 Noncollocated Partial Feedback Linearization

In this section, we present a partial feedback linearization procedure for underactu-ated systems that linearizes the dynamics of the unactuated configuration variables.We show this is possible if the number of the control inputs is greater or equal to thenumber of unactuated configuration variables. We call this procedure noncollocatedpartial feedback linearization. To be more precise, consider the following underactu-ated system

m00(q) m01(q) m02(q)m10(q) m11(q) m12(q)m20(q) m21(q) m22(q)

q0q1q2

+

h0(q, q)h1(q, q)h2(q, q)

=

τ0τ10

(3.5)

where q = (q0, q1, q2) ∈ Rn0 × Rn1 × Rn2 with n1 = n2 = m and n0 = n− 2m ≥ 0.

Remark 3.5.1. For the special case where there are equal number of actuated andunactuated variables n = 2m and n0 = 0. Thus, q0 has dimension zero or q = (q1, q2).

Proposition 3.5.1. Consider the underactuated mechanical system in (3.5). Then,there exists a change of control in the form

τ = α1(q)u+ β1(q, q)

where τ = col(τ0, τ1), u = col(u0, u1) and

u1 = α0(q)u0 + α2(q)u2 + β2(q, q)

that partially linearizes the dynamics of (3.5) as the following

q0 = p0p0 = u0q1 = p1p1 = f0(q, p) + g0(q)u0 + g2(q)u2q2 = p2p2 = u2

(3.6)

46

over the setU = q ∈ Rn| det(m21(q)) 6= 0

wheref0(q, p) = −m−1

21 (q)h2(q, p)g0(q) = −m−1

21 (q)m20(q)g2(q) = −m−1

21 (q)m22(q)

Proof. Apparently, based on collocated feedback linearization (proposition 3.4.1)the dynamics of the actuated configuration vector (q0, q1) can be linearized using achange of control τ = α1(q)u+ β1(q, q) that gives

q0 = u0q1 = u1

Over the set U , m21(q) is an invertible matrix and from the last equation in (3.5), weget

q1 = −m−121 (q)h2(q, q)−m−1

21 (q)m20(q)q0 −m−121 (q)m22(q)q2

Thus, after a second change of control as

u1 = −m−121 h2(q, q)−m−1

21 (q)m20(q)u0 −m−121 (q)m22(q)u2

where u2 is the new control, the result follows.

3.6 Partial Feedback Linearization Under Input

Coupling

Consider the underactuated mechanical control system

d

dt

∂L∂q− ∂L∂q

= F (q)τ

with configuration q ∈ Rn and control τ ∈ Rm and rankF (q) = m < n. Without lossof generality, F (q) can be written as

F (q) =

[

F1(q)F2(q)

]

such that F2(q) is an invertible m×m matrix and q can be decomposed to (q1, q2) ∈Rn−m×Rm according to F (q). By input coupling, we mean F1(q) 6≡ 0 for all q. In thefollowing, we provide the conditions which allow partial feedback linearization for anunderactuated (simple) Lagrangian system.

Proposition 3.6.1. Consider the following underactuated system with input coupling

47

(i.e. F1(q) 6≡ 0, det(F2(q)) 6= 0 for all q)

[

m11(q) m12(q)m21(q) m22(q)

] [

q1q2

]

+

[

h1(q, q)h2(q, q)

]

=

[

F1(q)F2(q)

]

τ (3.7)

and assume the following matrix is invertible for all q

Λ(q) = F2(q)−m21(q)m−111 (q)F1(q) (3.8)

Then, there exists a change of control in the form

τ = α(q)u+ β(q, q)

withα(q) = Λ−1(q)[m22(q)−m21(q)m

−111 (q)m12(q)]

β(q, q) = Λ−1(q)[h2(q, q)−m21(q)m−111 (q)h1(q, q)]

that partially linearizes (3.7) as the following

q1 = p1p1 = f0(q, p) + g0(q)uq2 = p2p2 = u

wheref0(q, p) = m−1

11 (q)[F1(q)β(q, q)− h1(q, q)]g0(q) = m−1

11 (q)[F1(q)α(q)−m12(q)]

Proof. From the first equation in (3.7), we get

q1 = −m−111m12q2 −m−1

11 h1 +m−111 F1

substituting this in the second line of (3.7) gives

(m22 −m21m−111m12)q2 + h2 −m−1

11 h1 = (F2 −m21m−111 F1)τ = Λ(q)τ

Thus, applying τ = α(q)u + β(q, q) with α, β as defined in the question partiallylinearizes the system and the first equation of (3.7) reduces to

q1 = m−111 (q)[F1(q)β(q, q)− h1(q, q)] +m−1

11 (q)[F1(q)α(q)−m12(q)]u

and the result follows.

Remark 3.6.1. The partial feedback linearization procedure in proposition 3.6.1 isparticularly useful for autonomous vehicles in SE(3) like an aircraft or a helicopterwith six degrees of freedom.

48

3.7 Normal Forms for Underactuated Systems

The complexity of control design for underactuated mechanical systems is partly dueto the fact that the control input u appears in the dynamics of both the unactu-ated subsystem (i.e. (q1, p1)-subsystem) and the actuated subsystem (i.e. (q2, p2)-subsystem) in (3.4). In this section, we provide a method to decouple these twosubsystems w.r.t. the control input u using a global change of coordinates. Thefollowing result provides an appropriate structure and sufficient conditions for theexistence of a decoupling change of coordinates.

Theorem 3.7.1. Consider an underactuated mechanical system with an inertia ma-trix M(q) = mij(q); i, j = 1, 2 where q = (q1, q2) and q1 = qi1 ∈ Rn andq2 = qj2 ∈ Rm denote the unactuated and actuated configuration variables, respec-tively. Denote

g0(q) = −m−111 (q)m12(q)

and

g(q) =

[

g0(q)Im×m

]

where g0(q) = (g10(q), . . . , gm0 (q)) with gj0(q) ∈ Rn, j = 1, . . . ,m and Im×m is the

identity matrix. Define the following distribution

∆(q) = spancolumns of g(q)

that has full column rank and thus is globally nonsingular. Then, a necessary andsufficient condition for the distribution ∆(q) to be globally involutive (i.e. completelyintegrable) is that

∂gj0(q)

∂q1gi0(q)−

∂gi0(q)

∂q1gj0(q) +

∂gj0(q)

∂qi2− ∂gi0(q)

∂qj2= 0, ∀i, j = 1, . . . ,m (3.9)

In addition, if condition (3.9) holds, there exists a global change of coordinates givenby

z1 = Φ(q1, q2)z2 = (Dq1Φ(q)) · p1 + (Dq2Φ(q)) · p2ξ1 = q2ξ2 = q2

that transforms the dynamics of the system into the normal form

z1 = z2z2 = f(z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(3.10)

Remark 3.7.1. Normal form (3.10) is a special case of the famous Byrnes-Isidori

49

normal form [36] with a double integrator as the following

z = f(z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(3.11)

Remark 3.7.2. The main advantage of the normal form (3.10) is that the control inputof the actuated subsystem of the original system does not appear in the unactuatedsubsystem. This simplifies control design for this underactuated system by reducingthe control of the original higher order system into control of its lower-order nonlinearunactuated subsystem. This is due to a recent result by the author in [67] on controldesign for systems in normal forms (3.10) and (3.11) which is explained in chapter 7.

Remark 3.7.3. The main disadvantage of theorem 3.7.1 is that it is not always possibleto find Φ(q1, q2) in an analytical explicit form. Later, we show that under kineticsymmetry properties of the system it is possible to calculate Φ(q1, q2) in closed-form.

Proof.(Theorem 3.7.1) Note that ∆(q) globally has a full column rank of m and istherefore a globally nonsingular distribution (see [36] for definitions and notations inthis proof). Calculating the Lie bracket of the ith and jth columns of g(q), we get

[gi(q), gj(q)] =

∂gj0(q)

∂q1

∂gj0(q)

∂q2∂ej∂q1

∂ej∂q2

·[

gi0(q)ei

]

−

∂gi0(q)

∂q1

∂gi0(q)

∂q2∂ei∂q1

∂ei∂q2

·[

gj0(q)ej

]

=

∂gj0(q)

∂q1gi0(q)−

∂gi0(q)

∂q1gj0(q) +

∂gj0(q)

∂q2ei −

∂gi0(q)

∂q2ej

0m×1

which based on condition (3.9) implies [gi(q), gj(q)] = 0 for all i, j, q. Therefore ∆(q)is globally involutive. To prove the converse, assume ∆(q) is globally involutive.Then, for all i, j, q, [gi(q), gj(q)] can be expressed as a linear combination of gk(q)’s.But the lower m× 1 block of [gi(q), gj(q)] is identically zero and linearly independentof ek’s. This means that [gi(q), gj(q)] = 0 and thus condition (3.9) holds. Now, weprove the rest of the theorem. Based on Frobenius theorem (see [36]), because ∆(q)is a globally nonsingular and involutive distribution, the following equation

∂φ

∂qg(q) = 0

has d = (n − m) linearly independent solutions φk(q1, q2), k = 1, . . . , n. DenotingΦ(q1, q2) = (φ1, . . . , φd), Φ satisfies the property

∂Φ

∂q1g0(q) +

∂Φ

∂q2= 0.

50

After applying the change of coordinates

z1 = Φ(q1, q2), z2 = z1

we get

z1 = Φ(q1, q2), z2 =∂Φ

∂q1p1 +

∂Φ

∂q2p2

and because ∂Φ/∂q1 is globally nonsingular (due to the proof of Frobenius theorem),based on implicit mapping theorem, there exists a smooth function Ψ such that

q1 = Ψ(z1, ξ1), p1 =

(

∂Φ

∂q1|q=(Ψ(z1,ξ1),ξ1)

)−1

(z2 − ξ2 ·∂Φ

∂q2|q=(Ψ(z1,ξ1),ξ1))

(we drop the substitution q = (Ψ(z1, ξ1), ξ1) in ∂Φ/∂qi due to the simplicity of nota-tion). Calculating z2 as the following

z2 =∂Φ

∂q1f0(q, p) +

∂2Φ

∂q21(∂Φ

∂q1)−2(z2 −

∂Φ

∂q1ξ2)

2 +∂2Φ

∂q22ξ22 + (

∂Φ

∂q1g0(q)) +

∂Φ

∂q2)u (3.12)

and noting that the coefficient of u in the last equation is identically zero, we get

z2 = f(z, ξ1, ξ2)

where f is the right hand side of (3.12). Therefore, the dynamics of the system innew coordinates is in normal form (3.10) and this finishes the proof.

Here is an important corollary of theorem 3.7.1:

Corollary 3.7.1. All underactuated mechanical systems with a single actuator thatare globally partial feedback linearizable can be transformed into normal form (3.10)using a global invertible change of coordinates.

Proof. For this case m = 1 and i = j = 1. By symmetry w.r.t i and j indices,condition (3.9) globally holds for all q.

3.8 Classes of Structured Nonlinear Systems

In this section, we define important classes of structured nonlinear systems that ap-pear frequently in this thesis.

Definition 3.8.1. (cascade system) We say a nonlinear system is in cascade form ifit has the following structure

z = f(z, ξ)

ξ = g(ξ, u)(3.13)

where f : Rn × Rm → Rn, g : Rm × Rp → Rm, (z, ξ) is the composite state, and uis the control input. When the second equation is a linear time-invariant system, i.e.ξ = Aξ +Bu, (3.13) is called a partially linear cascade nonlinear system.

51

Definition 3.8.2. (feedback form) We say a nonlinear system is in strict feedbackform [57], if it has the following triangular structure

z = f(z, ξ1),

ξ1 = ξ2,. . .

ξm = u,

Definition 3.8.3. (feedforward form) We say a nonlinear system is in feedforwardform [102], if it possesses the following triangular structure

x1 = x2 + ϕ1(x2, . . . , xn, u)x2 = x3 + ϕ2(x3, . . . , xn, u)

. . .xn = u + ϕn(xn, u)

where x ∈ Rn and the ϕi’s are at least quadratic in (x, u).

Definition 3.8.4. (nontriangular form) We say a partially linear cascade nonlinearsystem is in nontriangular form, if it has the following structure

z = f(z, ξ1, ξ2, . . . , ξm),

ξ1 = ξ2,. . .

ξm = u,

(3.14)

where z ∈ Rn and u ∈ Rp. The nonlinear system in (3.14) is also called Byrnes-Isidorinormal form [36].

Definition 3.8.5. (nontriangular linear-qudratic form) We say a cascade nonlinearsystem is in nontriangular linear-quadratic form, if it has the following structure

z1 = µ(z1)z2 + η(ξ1)ξ2z2 = φ(z1, ξ1) + Σ(ξ1, z2, ξ2)

ξ1 = ξ2,

ξ2 = u,

(3.15)

where z1, z2 ∈ Rn, u ∈ Rp, µ(z1) is a positive definite matrix, φ : Rn ×Rp → Rn, andΣ : Rp × Rn × Rp → Rn has the following quadratic form in (z2, ξ2)

Σ(ξ1, z2, ξ2) =

[

z2ξ2

]T

Π(ξ1)

[

z2ξ2

]

where Π = (Π1, . . . ,Πn) is a cubic matrix with elements in Rn×n and for v ∈ Rn,vTΠv := (vTΠ1v, . . . , vTΠnv)T ∈ Rn. If η ≡ 0, (3.15) is called a nontriangular

52

quadratic form, or in case Σ ≡ 0, (3.15) is called nontrinagular linear form. In allcases, the quadratic or linearity properties are implicitly w.r.t. ξ2.

Remark 3.8.1. Clearly, if Σ(ξ1, z2, ξ2) = 0, then the nonlinear system in (3.15) reducesto a strict feedback system. In addition, if µ(z1) = In, f(z1, ξ1) = ξ1, and Π(1,1)(ξ1) =0 (Π(1,1) is the first cubic partition of Π according to (z2, ξ2)), then (3.15) reduces to anonlinear system in strict feedforward form. Thus, the famous feedback/feedforwardtriangular forms are special cases of the quadratic nontriangular form.

Remark 3.8.2. In the special case where z1, z2, ξ1, ξ2 are all scalars, Π(ξ1) in definition3.8.5 is a 2×2 matrix (i.e. not a cubic matrix) which is in fact the case for underactu-ated mechanical systems with two DOF. However, we need the generality of definition3.8.5 for further applications related to normal forms of higher-order underactuatedsystems.

3.9 Normal Forms for Underactuated Systems with

2 DOF

In this section, we introduce explicit cascade normal forms for underactuated me-chanical systems with two degrees of freedom and kinetic symmetry w.r.t. one(actuated or unactuated) configuration variable. The importance of these normalforms is due to the fact that many examples of benchmark problems in nonlin-ear control design including the cart-pole system [101, 56, 94], the beam-and-ballsystem [34, 102, 103, 80, 79], the acrobot [13, 86], the pendubot [92], the rotatingpendulum[6, 10], the inertia-wheel pendulum [93], and the TORA example [41, 37]all fall within the class of underactuated systems with two degrees of freedom andkinetic symmetry. Nevertheless, so far no (global) cascade normal forms are knownfor the majority of these nonlinear systems except for the beam-and-ball system, andthe TORA example. This motivated us to provide a (control-oriented) classificationof underactuated systems with two DOF based on their associated cascade normalforms. These cascade forms are systems in strict feedback form [57], feedforwardform [102], and nontriangular quadratic form [69] (to be defined). The main ben-efit of this classification and the associated normal form is that if the system hasfeedback/feedforward triangular structure (under appropriate conditions) it can beglobally asymptotically stabilized using backstepping [36, 57, 80] or forwarding pro-cedures [102, 101, 56, 80]. The only open ended case is stabilization of systems innontriangular quadratic form that will be addressed in this thesis in chapter 7.

Now, consider an underactuated mechanical system with two DOF and configu-ration vector q = (q1, q2)

T . Assume the inertia matrix of this system only dependson q2, i.e. M = M(q2). Then, the system has kinetic symmetry w.r.t. q1. TheLagrangian of this underactuated system can be expressed as the following

L(q, q) = 1

2qT[

m11(q2) m12(q2)m21(q2) m22(q2)

]

q − V (q) (3.16)

53

Here is our main result on normal forms for underactuated systems with two DOFand kinetic symmetry:

Theorem 3.9.1. All underactuated mechanical systems with two degrees of freedom(q1, q2) and kinetic symmetry w.r.t. q1 have a Lagrangian given by (3.16) and can be(possibly globally) transformed into quadratic nontriangular normal form (3.15) (or(3.10)) using a change of coordinates in analytically explicit form.

The proof of this theorem relays on the corresponding proofs for two cases whereeither q2 is actuated, or q2 is unactuated. This provides a natural classification ofunderactuated systems to two classes as the following.

Definition 3.9.1. (Class-I,II underactuated systems) We call an underactuated sys-tem with Lagrangian (3.16) Class-I underactuated system, iff q2 is actuated. We callit a Class-II underactuated system, iff q2 is unactuated.

The Euler-Lagrange equations of motion for this underactuated system are

d

dt

∂L∂q1− ∂L∂q1

= τ1

d

dt

∂L∂q2− ∂L∂q2

= τ2

(3.17)

where for a Class–I system τ1 = 0 and for a Class–II system τ2 = 0. Equation (3.17)can be rewritten as

m11(q2)q1 + m12(q2)q2 +m′11(q2)q1q2 +m′

12(q2)q22 − g1(q1, q2) = τ1

m21(q2)q1 + m22(q2)q2 − 12m′

11(q2)q21 +

12m′

22(q2)q22 − g2(q1, q2) = τ2

(3.18)

where gi(q1, q2) = −∂V (q)/∂qi, i = 1, 2 and ′ denotes d/dq2. The following propo-sition states that all Class–I underactuated systems can be globally and explicitlytransformed into cascade nonlinear systems in strict feedback form.

Proposition 3.9.1. (Class-I normal form) Consider a Class-I underactuated systemwith two DOF. Then, the following global change of coordinates obtained from theLagrangian of the system

z1 = q1 + γ(q2)

z2 = m11(q2)p1 +m12(q2)p2 =∂L∂q1

ξ1 = q2ξ2 = p2

(3.19)

where

γ(q2) =

∫ q2

0

m−111 (θ)m12(θ)dθ (3.20)

54

transforms the dynamics of the system into a cascade nonlinear system in strict feed-back form

z1 = m−111 (ξ1)z2

z2 = g1(z1 − γ(ξ1), ξ1)ξ1 = ξ2ξ2 = u

(3.21)

where u is the new control from collocated partial feedback linearization and g1(q1, q2) =−∂V (q)/∂q1.

Proof. By definition of z1 and z2, we have z1 = m−111 (q2)z2. Noting that z2 = ∂L/∂q1,

from the first line of (3.17), τ1 = 0 and we get

z2 =∂L∂q1

=∂K

∂q1− ∂V

∂q1= g1(q1, q2)

The last equality follows from kinetic symmetry property of the system w.r.t. q1 (i.e.∂K/∂q1 = 0).

Remark 3.9.1. After a second change of coordinates

y1 = z1, y2 = m−111 (ξ1)z2

the dynamics of the system in (3.21) transforms into normal form (3.10).

Corollary 3.9.1. The Acrobot, the Inertia-Wheel pendulum, and the TORA exampleare all Class–I underactuated systems and can be globally transformed into normalform (3.21) using an explicit global change of coordinates.

Proposition 3.9.2. (Class–II normal form) Consider a Class-II underactuated sys-tem with two DOF. Then, the following change of coordinates obtained from the La-grangian of the system

z1 = q1 + γ(q2)

z2 = m21(q2)p1 +m22(q2)p2 =∂L∂q2

ξ1 = q2ξ2 = p2

(3.22)

where

γ(q2) =

∫ q2

0

m−121 (θ)m22(θ)dθ (3.23)

is defined overU = q2|m21(q2) 6= 0

transforms the dynamics of the system into a cascade nonlinear system in quadraticnontriangular form where u is the new control from noncollocated partial feedbacklinearization.

55

Proof. By definition of z1 and z2, it follows that z1 = z2/m21(q2). From the secondline of (3.16), τ2 = 0 and we get

z2 = ∂L/∂q2= ∂K/∂q2 − ∂V (q)/∂q2 (∂K/∂q2 6≡ 0)= g2(q1, q2) +

12m′

11(q2)p21 +m′

21(q2)p1p2 +12m′

22(q2)p22

Hence, after substituting

q1 = z1 − γ(q2)p1 = (z2 −m22(q2)p2)/m21(q2)

in the last equation, we obtain

z2 = g2(z1 − γ(q2), q2) +m′

11(q2)

2m221(q2)

[z2 −m22(q2)p2]2

+m′

21(q2)

m21(q2)[z2 −m22(q2)p2]p2 +

m′22(q2)

2p22

which gives the following quadratic nontriangular normal form for Class-II underac-tuated systems

z1 = m−121 (q2)z2

z2 = g2(z1 − γ(q2), q2)+

m′11(q2)

2m221(q2)

z22 +

m′21(q2)

m21(q2)− m22(q2)m

′11(q2)

2m221(q2)

z2p2

+

m222(q2)

2m221(q2)

m′11(q2)−

m22(q2)

m21(q2)m′

21(q2) +1

2m′

22(q2)

p22

q2 = p2p2 = u

(3.24)

Corollary 3.9.2. The Pendubot, the Cart-Pole system, the Rotating Pendulum, andthe Beam-and-Ball system are all Class-II underactuated systems with two degrees offreedom and can be transformed into a cascade nonlinear system in quadratic nontri-angular form over the set U = q2| m21(q2) 6= 0 using a change of coordinates inexplicit form.

Proof. The proof is by direct substitution of the elements of the inertia matrix ofeach system in equation (3.24).

Proposition 3.9.3. (feedforward normal form) Consider a Class–II underactuatedsystem with two degrees of freedom over the set U = q2|m21(q2) 6= 0 and assumethe following conditions hold:

i) g2(q1, q2) is independent of q1, i.e. Dq1Dq2V (q) = 0.

ii) m11 is constant.

iii) ψ(q2) = g2(q2)/m21(q2) satisfies ψ′(0) 6= 0.

56

Then, applying the change of coordinates

y1 = z1, y2 = z2/m21(q2)

transforms the normal form (3.24) for Class-II underactuated systems with two de-grees of freedom into the following cascade nonlinear system in feedforward form

z1 = z2

z2 = ψ(q2) +

m′22(q2)

2m21(q2)− m22(q2)

m221(q2)

m′21(q2)

p22

q2 = p2p2 = u

(3.25)

In addition, the origin for this feedforward system can be globally asymptotically sta-bilized using nested saturations [102].

Proof. The normal form (3.25) follows from the proof of proposition 3.9.2 by directcalculation and conditions i), ii). The stabilization using nested saturations followsfrom condition iii) and [102].

Corollary 3.9.3. The Cart-Pole system can be transformed into a cascade nonlinearsystem in feedforward form using the change of coordinates in proposition 3.9.3 andcan be globally asymptotically stabilized to its upright equilibrium point over q2 ∈(−π/2, π/2) using nested saturations.

Proof. For the cart-pole system, g2 = g2(q2) and m11 is constant. In addition,ψ(q2) = g tan(q2) (g is the gravity constant) that implies ψ′(0) = g 6= 0. Thus,the cart-pole system satisfies all three conditions in proposition 3.9.3. Also, becausem21(q2) = a cos(q2) (a > 0 is a constant), the stabilization result over q2 ∈ (−π/2, π/2)follows (see section 5.4 for further details on control design for the cart-pole system).

57

Chapter 4

Reduction and Control ofHigh-Order UnderactuatedSystems

In this chapter, we address reduction of high-order underactuated mechanical systemswith kinetic symmetry. By reduction, we mean control of the original higher-orderunderactuated system reduces to control of a lower-order nonlinear system (we pre-cisely define reduction later on). It turns out that actuation or lack of actuation of asubset of configuration variables that somehow represent the “shape” of a mechanicalsystem plays an important role in reduction of underactuated mechanical systems.The details regarding the notion of “shape variables” for Lagrangian systems and“kinetic symmetry” are discussed in the following section.

Our main contribution in this chapter is to obtain structured cascade normal formsfor higher-order underactuated systems in explicit form and present an appropriatecontrol design method for each normal form. These cascade normal forms have struc-tural properties that allow effective use of existing and recently developed systematicnonlinear control design methods. Types of cascade normal forms that are explic-itly obtained for underactuated systems include nonlinear systems in strict feedbackform, feedforward form, and nontriangular linear/quadratic form (see section 3.8 forthe definitions).

4.1 Shape Variables and Kinetic Symmetry

Consider a simple Lagrangian system with configuration vector q ∈ Q. The variablesthat appear in the inertia matrix of the system are called shape variables. If a con-figuration variable qj does not appear in the inertia matrix i.e. ∂M(q)/∂qj = 0, it iscalled an external variable. This implies that for an external variable qj the followingidentity holds

∂K(q, q)

∂qj= 0

58

In other words,a simple Lagrangian system has kinetic symmetry w.r.t. external vari-ables. A more general way to define this symmetry is via action of a group on amanifold. Let qj ∈ G where G is a cyclic group, e.g. R (a translation group), or S1 (arotation group). Decompose the configuration manifold as Q = B×G. The action ofG on Q is a mapping Φ : Q×G→ Q. For a fix j ∈ 1, . . . , n, let α ∈ G and q ∈ Q,define Φ(q, α) = (q1, . . . , qj + α, . . . , qn). Formally, we call qj an external variable, ifM(q) = M(Φ(q, α)). Let Qx = G1 × . . . × Gn be the configuration manifold of allexternal variables where Gi’s are one-dimensional cyclic groups. Then Q = Qx ×Qs

where Qs is called the shape space, i.e. the configuration manifold of the shape vari-ables. The notion of shape variables or internal variables originally appeared in thecontrol literature by the study of interconnected mechanical systems and multi-bodyproblems [44, 45, 46, 51].

In the classical sense, symmetry in mechanics is defined as the invariance of theLagrangian under the action of a group. This is fundamentally different from theinvariance of the kinetic energy. The classical symmetry gives raise to the existenceof conserved quantities while the kinetic symmetry does not give raise to the conser-vation of the generalized momentums. (unless both τj and ∂V (q)/∂qj ( i.e. the jthgravity term) vanish in which case ∂L/∂qj is a conserved quantity). The fact thatthe generalized momentum pj = ∂L/∂qj is not a conserved quantity in the presenceof the kinetic symmetry plays a vital role in controllability and stabilization of broadclasses of underactuated mechanical systems (see remark 4.2.4).

Throughout this chapter, we consider underactuated mechanical control systemswith kinetic symmetry that have the following Lagrangian

L(q, q) = 1

2qTM(qs)q − V (q) (4.1)

where q = (qx, qs) ∈ Q = Qx×Qs is the configuration vector decomposed to externaland shape variables. The forced Euler-Lagrange equations of motion for this systemis

d

dt

∂L∂qx− ∂L∂q

= Fx(q)τ

d

dt

∂L∂qx− ∂L∂q

= Fs(q)τ

(4.2)

where τ ∈ Rm and F (q) = col(Fx(q), Fs(q)) is the force matrix with the underactua-tion property

rank F (q) = m < n = dim(Q)

In analysis and control design for the underactuated system (4.2), a number ofcases arise depending on whether the shape configuration vector qs is fully-actuated,partially-actuated, or unactuated and the presence or lack of any input couplings dueto the force matrix F (q). This leads to analysis of a finite number of cases. For eachcase, we provide a method for reduction of the given underactuated system and globaltransformation of the system into a cascade normal form. Eventually, this leads toclassification of underactuated control systems based on four basic properties (to be

59

mentioned later). It turns out that there are eight different classes of underactuatedsystems and three types of obtained normal forms. Namely, cascade nonlinear systemsin strict feedback form, strict feedforward form, and nontriangular linear-quadraticform.

4.2 Underactuated Systems with Noninteracting

Inputs and Integrable Momentums

In this section, we address reduction of high-order underactuated systems with in-tegrable normalized momentums for the noninteracting input case. There could bethree cases: i) underactuated systems with fully-actuated shape variables, ii) under-actuated systems with unactuated shape variables, and iii) underactuated systemswith partially-actuated shape variables. We address reduction of cases i) and ii) inthe following. Case iii) is rather similar to Case ii) and will not be discussed in thissection. Later, we present reduction of underactuated systems with non-integrablemomentums in case iii).

4.2.1 Underactuated Systems with Actuated Shape Variables

First, we consider underactuated mechanical systems in (4.2) with fully-actuatedshape variables and noninteracting inputs i.e. Fx(q) = 0, Fs(q) = Im.

Theorem 4.2.1. Consider an underactuated system with kinetic symmetry w.r.t. qxand fully-actuated shape variables qs

mxx(qs)qx +mxs(qs)qs + hx(q, q) = 0msx(qs)qx +mss(qs)qs + hs(q, q) = τ

(4.3)

Let τ = α(qs)u + β(q, q) be the collocated partially linearizing change of control for(4.3). Assume all the elements of

ω = m−1xx (qs)mxs(qs)dqs (4.4)

are exact one-forms and let ω = dγ(qs). Then, there exists a global change of coordi-nates (i.e. diffeomorphism) obtained from the Lagrangian of the system

qr = qx + γ(qs) =: Φ(qx, qs)

pr = mxx(qs)qx +mxs(qs)qs =∂L∂qx

(4.5)

that transforms the dynamics of the underactuated system in (4.3) into a cascade

60

normal form in strict feedback form

qr = m−1r (qs)pr

pr = gr(qr, qs)qs = psps = u

(4.6)

In addition, the (qr, pr)-subsystem is a Lagrangian system with configuration vectorqr that belongs to the reduced manifold Qr = Φ(Qx ×Qs) with reduced Lagrangian

Lr(qr, qr, qs) =1

2qTr mr(qs)qr − Vr(qr, qs) (4.7)

that satisfies the unforced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= 0

with

mr(qs) := mxx(qs)

Vr(qr, qs) := V (qr − γ(qs), qs)gr(qr, qs) := −∂Vr(qr, qs)/∂qr

Proof. By definition of qr, pr, we have

qr = m−1xx (qs)pr = m−1

r (qs)pr

From the first line of Euler-Lagrange equation in (4.2), we get

pr =∂L∂qx

=∂K

∂qx− ∂V (qx, qs)

∂qx=: gx(qx, qr)

due to kinetic symmetry w.r.t. qx, ∂K/∂qx = 0. But

gx(qx, qs) = −∂V (qr − γ(qs), qs)

∂qr

∂qr∂qx

= gr(qr, qs)× In−m = gr(qr, qs)

and the second line of (4.6) follows. By direct calculation

pr =∂Lr∂qr

, pr =∂Lr∂qr

which proves the reduced Lagrangian Lr satisfies the unforced Euler-Lagrange equa-tion.

61

Remark 4.2.1. Stabilization of the reduced system

qr = m−1r (qs)pr

pr = gr(qr, qs)(4.8)

is addressed in section 4.7 of this chapter.

Remark 4.2.2. After renaming the variables in (4.6) as (z1, z2) = (qr, pr) and (ξ1, ξ2) =(qs, ps), one obtains

z = f(z, ξ1)

ξ1 = ξ2ξ2 = u

(4.9)

which is a special case of Byrnes-Isidori normal form [36] associated to the outputξ1 = qs which has global uniform relative degree two and the following Lagrangianzero-dynamics

z1 = m−1r (ξ1)z2

z2 = gx(z1 − γ(ξ1), ξ1)Remark 4.2.3. Based on standard backstepping procedure, control design for thenonlinear system in (4.3) reduces to control design for its (qr, pr)-subsystem withcontrol input qs. For this reason, we call (qr, pr)-subsystem the reduced system withreduced configuration vector qr. Explicit transformation of the system into the nor-mal form (4.6) reduces control of the original underactuated nonlinear system withn second-order subsystems to control of the reduced system with (n − m) second-order subsystems. This is a tremendous reduction in complexity of control design for(possibly) high-order underactuated systems.

Remark 4.2.4. If the potential energy V (q) is independent of the external variable qx ,i.e. ∂V (qx, qs)/∂qx = 0, then gr = 0 and the generalized momentum pr is a conservedquantity. Therefore, the nonlinear system in (4.6) is not controllable or stabilizableto any equilibrium points for initial conditions with pr(0) 6= 0. For example, neitherthe Acrobot, nor the Pendubot are controllable/stabilizable in lack of gravity.

The fact that the reduced system with configuration vector qr is a simple La-grangian system that satisfies the Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= 0

without any input forces means that the system must be controlled via its potentialforce (or energy) that is parameterized by qs. Therefore the shape vector qs plays therole of the control input for the dynamics of the reduced system in Qr. In addition,the reduced system is a double-integrator gradient system parameterized with qs as

qr = m−1r (qs)pr

pr = −∂Vr(qr, qs)∂qr

62

which admits a reduced Hamiltonian parameterized by qs as the following

Hr(qr, pr, qs) =1

2pTrm

−1r (qs)pr + Vr(qr, qs) (4.10)

that satisfies

Hr =∂Hr(qr, pr, qs)

∂qsps (4.11)

Based on (4.11) the reduced Hamiltonian Hr is not necessarily a conserved quantity.To see this, first we need the following definition.

Definition 4.2.1. (vector sigmoidal function) ~σ(x) : Rn → Rn is called a vectorsigmoidal function if it satisfies the following properties.

i) ∃L > 0 : ∀x ∈ Rn, ‖~σ(x)‖ ≤ L.

ii) σ(0) = 0 and x~σ(x) > 0,∀x 6= 0.

iii) yT (~σ(x) − ~σ(x + y)) < 0,∀y 6= 0 . Examples of vector sigmoidal functions are~σ(x) = (σ1(x1), . . . , σn(xn))

T where σi are one-dimensional sigmoidal functionslike arctan(z) and tanh(z).

Now, consider the following cases:

i) ps = 0, or ps satisfying ∇qsHr · ps = 0 conserves the energy Hr.

ii) ps = −~σ(∇qsHr) decreases the energy of the reduced system.

iii) ps = ~σ(∇qsHr) increases the energy of the reduced system.

In other words, the energy is not necessarily conserved for the reduced system in theparameterized Hamiltonian form

qr =∂Hr(qr, pr, qs)

∂pr

pr = −∂Hr(qr, pr, qs)

∂qr

Remark 4.2.5. The fact that the total energy of the reduced system can be increased ordecreased is useful for swing-up control design and stabilization of an underactuatedsystem to an equilibrium manifold/point, respectively.

4.2.2 Underactuated Systems with Unactuated Shape Vari-ables

In this section, we address reduction of underactuated mechanical systems in (4.2)that all of their shape variables are unactuated, have equal number of external andshape variables, and their inputs are noninteracting, i.e. Fx(q) = Im, Fs(q) = 0.

63

Theorem 4.2.2. Consider an underactuated mechanical system with fully-actuatedexternal variables qx and unactuated shape variable qs and assume dim(Qx) = dim(Qs) =m. The Euler-Lagrange equations of motion for this system is as the following

mxx(qs)qx +mxs(qs)qs + hx(q, q) = τmsx(qs)qx +mss(qs)qs + hs(q, q) = 0

(4.12)

Let τ = α(qs)u + β(q, q) be the noncollocated partially linearizing change of controlfor (4.12) over

U = qs ∈ Qs | det(mxs(qs)) 6= 0Assume all the elements of

ω = m−1sx (qs)mss(qs)dqs

are exact one-forms over U and let ω = dγ(qs). Then, there exists a global change ofcoordinates (i.e. diffeomorphism) obtained from the Lagrangian of the system

qr = qx + γ(qs)

pr = msx(qs)qx +mss(qs)qs =∂L∂qs

(4.13)

that transforms the dynamics of the underactuated system (4.12) into a cascade non-linear system in nontriangular quadratic form

qr = m−1r (qs)pr

pr = gr(qr, qs) + Σ(qs, pr, ps)qs = psps = u

(4.14)

where Σ = ∇qsK (K is the kinetic energy) is a quadratic form in (pr, ps)

Σ(qs, pr, ps) =

[

prps

]T

Π(qs)

[

prps

]

with a cubic weight matrix Π(qs) and

mr(qs) := msx(qs)gr(qr, qs) := − [∇qsVr(qx, qs)]qx=qr−γ(qs)

In addition, if V (q) = V (qx), then gr ≡ 0 and the (qr, pr)-subsystem is a Lagrangiansystem with configuration vector qr and reduced Lagrangian

Lr(qr, qr, qs) =1

2qTr mr(qs)qr

64

that satisfies the forced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= Σ(qs, pr, ps)

Proof. By definition of qr and pr, we have qr = m−1r (qs)pr. Calculating pr, we get

pr =d

dt

∂L∂qs

=∂L∂qs

= −∂V (qx, qs)

∂qs+∂K

∂qs= gr(qr, qs) + Σ(qs, pr, ps)

Noting that ∂Lr/∂qr = 0, one obtains

d

dt

∂Lr∂qr− ∂Lr∂qr

= pr = Σ

and the forced Euler-Lagrange equation in the question follows. It remains to calculateK(qs, pr, ps) and show it has a quadratic form with a cubic weight matrix. For doingso, note that

[

qxqs

]

= W (qs)

[

prps

]

where W is an n× n matrix given by

W (qs) =

[

m−1sx (qs) −m−1

sx (qs)mss(qs)0 Im

]

Thus

K(qs, pr, ps) =1

2

[

prps

]T

N(qs)

[

prps

]

withN(qs) =W T (qs)M(qs)W (qs)

Setting the ith layer of Π(qs) to dN(qs)/dqis (i.e. element-wise derivative of N(qs)

w.r.t. the ith component of qs) finishes the proof.

Example 4.2.1. Consider an underactuated system with configuration vector q =(x1, x2, θ1, θ2) that belongs to Q = R × R × S1 × S1. The system has the followingLagrangian and (x1, x2) are actuated with external forces F = (F1, F2)

T

L =1

2

x1x2θ1θ2

T

1 0 cos θ1 00 1 0 cos θ2

cos θ1 0 1 00 cos θ2 0 1

x1x2θ1θ2

− cos θ1 cos θ2

65

Apparently, qs = (θ1, θ2) is the vector of shape variables and qx = (x1, x2) is the vectorof actuated external variables of this system. We have

msx(qs) =

[

cos θ1 00 cos θ2

]

and det(msx) = cos θ1 cos θ2. Thus, over U = R2× (−π/2, π/2)2, msx(qs) is invertibleand the system can be partially linearized using a noncollocated partially linearizingchange of control F = α(qs)u+ β(qs, q). Moreover, the vector of one-forms

ω = m−1sx (qs)mssdqs =

1

cos θ10

01

cos θ2

[

dθ1dθ2

]

has exact elements ω1 = (1/ cos θ1)dθ1 and ω2 = (1/ cos θ2)dθ2 which means

γ(θ1, θ2) = (γ0(θ1), γ0(θ2))T

with

γ0(θ) =

∫ θ

0

ds

cos s= log

(

1 + tan(θ/2)

1− tan(θ/2)

)

, θ ∈ (−π/2, π/2)

Now, based on theorem 4.2.2 after applying the change of coordinates

qr =

[

x1 + γ0(θ1)x2 + γ0(θ2)

]

, pr =

[

cos θ1 00 cos θ2

] [

x1x2

]

+

[

θ1θ2

]

the dynamics of the system transforms into normal form (4.14).

Before presenting our next result, we need the following definition and lemma.

Definition 4.2.2. We say a square matrix function m(x) : Rn → Rn2has differen-

tially symmetric rows, if the ith row of the matrix mi∗(x) satisfies

∂m(x)

∂xi=∂mi∗(x)

∂x(4.15)

for i = 1, . . . , n.

Remark 4.2.6. Apparently, in the scalar case with n = 1, the condition of the dif-ferential symmetry of rows trivially holds. This is already used in reduction of aplanar Cart-Pole system in corollary 3.9.3. In addition, any linear combination of aconstant matrix and diagonal matrices in the form diag(f1(x1), . . . , fn(xn)) where fi’sare scalar functions has differentially symmetric rows as well. The following matrixis an example of a non-diagonal matrix with differentially symmetric rows

m(x1, x2) =

[

cos x1 sinx1 sinx20 cosx1 cos x2

]

66

where x1, x2 ∈ R.

Lemma 4.2.1. Suppose m(x) : Rn → Rn2is a matrix with differentially symmetric

rows. Let x(t) = v(t), w(t) ∈ Rn be functions of time. Then, the following identityholds

m(x, x)w(t) = ∇xv(t)Tm(x)w(t)x=x(t) (4.16)

where m is the element-wise time-derivative of the matrix m(x(t)).

Proof. By definition of m, we have

m(x, x)w(t) = col(m1∗w(t), . . . , mn∗w(t))

= col(xT∇xm1∗(x)w(t), . . . , xT∇xmn∗(x)w(t))

= col(xT∇x1m(x)w(t), . . . , xT∇xnm(x)w(t))

= ∇xv(t)Tm(x)w(t)x=x(t)

Proposition 4.2.1. Assume all the conditions in theorem 4.2.2 hold. In addition,the underactuated system (4.12) satisfies the following conditions

i) mxx(qs) is constant.

ii) msx(qs) has differentially symmetric rows, i.e.

∂msx(qs)

∂qis=∂mi∗

sx

∂qs, i = 1, . . . ,m

where mi∗sx(qs) is the ith row of msx(qs).

iii) V (q) = V (qs).

Then, applying the change of coordinates

z1 = qr, z2 = m−1r (qs)pr

(where (qr, pr) are defined in (4.13)) transforms the original system (4.12) into acascade system in feedforward form as the following

z1 = z2z2 = ψ(qs) + pTs Π(qs)psqs = psps = u

(4.17)

where Π(qs) is a cubic matrix and ψ : Qs → Rm is defined as

ψ(qs) = −m−1r (qs)∇qsV (qs)

Moreover, if ψ(0) = 0 and ψ(qs) has an invertible Jacobian ∇qsψ(qs) at qs = 0, theorigin for (4.17) (and (4.12)) can be globally asymptotically and locally exponentiallystabilized over U using a state feedback in explicit form as nested saturations.

67

Proof. We prove that under assumptions in the question pr only contains a quadraticform in ps plus gr(qs) = −∇qsV (qs) and is independent of pr. For doing so, recallthat

Σ(qs, pr, ps) = ∇qsK

in equation (4.14). Since mxx is constant and

K =1

2pTxmxxpx + pTsmsx(qs)px +

1

2pTsmss(qs)ps

we get

∇qsK = ∇qspTsmsx(qs)px+∇qs1

2pTsmss(qs)ps

Clearly, only the first term depends on px (and thus pr). Based on lemma 4.2.1,condition ii) implies msx(qs) satisfies the following property

msxpx = ∇qspTsmsx(qs)px (4.18)

Notice that the time derivative in msx is taken element-wise. Calculating z2, weobtain

z2 =d

dtm−1

sx (qs)pr +m−1sx (qs)pr

but m−1sx (qs)msx(qs) = Im and after differentiating both sides w.r.t. time, we get

d

dtm−1sx (qs) = −m−1

sx (qs)msxm−1sx (qs)

Therefore, z2 can be expressed as

z2 = m−1sx (qs)−msx[px +m−1

sx (qs)mss(qs)ps] + pr

orz2 = m−1

sx (qs)[−msxpx +∇qspTsmsx(qs)px − msxm−1sx (qs)mss(qs)ps

+ gr(qs) +∇qs12pTsmss(qs)ps]= m−1

sx (qs)gr(qs) + pTs Π(qs)ps

where Π(qs) satisfies

−m−1sx (qs)msxm

−1sx (qs)mss(qs)ps +m−1

sx (qs)∇qs1

2pTsmss(qs)ps = pTs Π(qs)ps

and the feedforward normal form in (4.17) follows. The stabilization of feedforwardsystems with higher-order perturbations using nested saturations is due Teel [102].To obtain the stabilizing state feedback law explicitly, define

z3 = qs, z4 = ps

where zi ∈ Rm and letA = [∇qsψ(qs)]qs=0

68

be the invertible matrix in the question. The dynamics of (4.17) can be written as

z1 = z2z2 = Az3 + ϕ(z3, z4)z3 = z4z4 = u

where ϕ(z3, z4) = zT4 Π(z3)z4 + ψ(z3) − Az3 is quadratic in (z3, z4). Following [102],define the change of coordinates and control

η1 = A−1z1 + 2A−1z2 + 2z3 + z4η2 = A−1z2 + z3 + z4w = z3 + z4 + u

we getη1 = η2 + w + 2A−1ϕ(z3, z4)η2 = w

Settingw = −~σ1(η2 + ~σ2(η1))

oru = −z3 − z4 + ~σ1(η2 + ~σ2(η1))

globally asymptotically stabilizes the origin for this system where ~σi’s are satura-tion functions that operate component-wise on an m-dimensional vector and havesufficiently small thresholds and magnitudes (see [102] for further details).

A restriction of underactuated systems with unactuated shape variables whichare considered in this section is that they have equal number of external and shapevariables. This is in fact not necessary, if one uses a collocated partially lineariz-ing feedback that linearizes the dynamics of external variables, or a noncollocatednonlinear combination of qx, qs. The following result provides a global normal formfor a class of underactuated systems with unactuated shape variables. This classis important particularly due to its application in tracking control for flexible-linkrobots.

Theorem 4.2.3. Consider an underactuated mechanical system with fully-actuatedexternal variables qx ∈ Rn−m and unactuated shape variable qs ∈ Rm which is aug-mented with an integrator at the input and satisfies the following equations of motion

mxx(qs)qx +mxs(qs)qs + hx(q, q) = τmsx(qs)qx +mss(qs)qs + hs(q, q) = Fd(qs, qs)

τ = v(4.19)

where Fd(qs, qs) is an internal damping force acting on shape variables (e.g. Fd =−D(qs)qs with D(qs) = DT (qs) ≥ 0). Assume the potential energy of the system onlydepends on shape variables, i.e. V (q) = V (qs). Suppose all the elements of the vector

ω = m−1xx (qs)mxs(qs)dqs

69

are exact one-forms and let ω = dγ(qs). Then, there exists a nonlinear output

y = h(q) = qx + γ(qs)

which has global relative degree 3 w.r.t. v so that after applying the global change ofcoordinates

z1 = qsz2 = psξ1 = y = qx + γ(qs)ξ2 = y = qx +m−1

xx (qs)mxs(qs)qsξ3 = y = qx +m−1

xx (qs)mxs(qs)qs +ddtm−1

xx (qs)mxs(qs)qs

(4.20)

the system in (4.19) transforms into Byrnes-Isidori normal form with a triple-integrator

z1 = z2z2 = f(z, ξ1, ξ2, ξ3)

ξ1 = ξ2ξ2 = ξ3ξ3 = u

(4.21)

where

f(z, ξ1, ξ2, ξ3) =M−1s (z1)[g(z1) + Fd(z1, z2) + Σ(z1, z2, ξ2) +msx(z1)ξ3]

andMs(qs) = mss(qs)−msx(qs)m

−1xx (qs)mxs(qs)

g(qs) = −∇V (qs)

Also, Σ is quadratic in (z2, ξ2) with a cubic weight matrix Π(z1) and

u = m−1xx (qs)v + β(q, q, τ)

is an invertible change of control in v. In addition, the zero-dynamics associated withthe output y = h(q) is a simple Lagrangian system with reduced shape Lagrangian

Ls(qs, qs) =1

2qTsMs(qs)qs − V (qs)

that satisfies the forced Euler-Lagrange equation

d

dt

∂Ls∂qs− ∂Ls∂qs

= Fd(qs, qs)

Moreover, if the following conditions hold

i) V (qs) is a positive definite proper function with V (0) = 0.

ii) pTs Fd(qs, ps) < 0 for all qs ∈ Rm, ps 6= 0.

70

Then, the Lagrangian zero-dynamics given by

qs = M−1s (qs)πs

πs = −∇V (qs) + Fd(qs, qs)

is globally minimum-phase with a reduced Hamiltonian

Hs(qs, πs) =1

2πTsM

−1s (qs)πs + V (qs)

which is a valid Lyapunov function for the zero-dynamics, i.e. Hs is smooth positivedefinite and proper function satisfying Hs ≤ 0 (where πs is the generalized momentum∂Ls/∂qs of the reduced system conjugate to qs).

Proof. By definition of ξ2, we have

ξ2 = ξ3 = m−1xx (qs)(τ − hx(qs, q)) +

d

dtm−1

xx (qs)mxs(qs)qs

or simplyξ3 = m−1

xx (qs)τ + β1(qs, q)

which means ξ3 is an invertible change of variable in τ . Differentiating the lastequation in time gives ξ3 = u with a new control u as defined in the question. Toprove the equation of z2 in (4.21), let us substitute for qx from the definition of ξ3 inthe second line of (4.19) to get

Ms(qs)qs +msx(qs)ξ3 −msx(qs)d

dtm−1

xx (qs)mxs(qs)qs + hs = Fd

or

qs =M−1s (qs)[−msx(qs)ξ3 +msx(qs)

d

dtm−1

xx (qs)mxs(qs)qs + Fd(qs, qs)− hs(q, q)]

that is in the form of second line of (4.21). To obtain the specific structure of f(z, ξ)in (4.21), we need to calculate hs more explicitly as

hs(qs, q) = msxqx + mssqs −∇qsK − gs(qs)

where gs(qs) = −∇V (qs) and ∇qsK is quadratic in (px, ps) = (qx, qs). But

[

pxps

]

=

[

I −m−1xx (qs)mxs(qs)

0 I

] [

ξ2z2

]

thus after substituting for (px, ps) in ∇qsK, we obtain

∇qsK =

[

ξ2z2

]

ΠK(qs)

[

ξ2z2

]

71

where ΠK(qs) is a cubic matrix. Defining

Σ :=

[

ξ2z2

]

Πk(qs)

[

ξ2z2

]

+msx(qs)d

dtm−1

xx (qs)mxs(qs)qs − msxqx − mssqs

or

Σ :=

[

ξ2z2

]

Πk(qs)

[

ξ2z2

]

− msxξ2 + Msqs

gives the desired structure of f(z, ξ) in (4.21) (the last two terms are both quadraticin (z2, ξ2)). To obtain the equations of the zero-dynamics, let ξ1 = ξ2 = ξ3 ≡ 0 andnotice that

ξ2 = qx +m−1xx (qs)mss(qs)qs = 0

is a holonomic velocity constraint and under such a constraint the new Lagrangianof the system can be expressed as

Ls = L(qs, qx, qs) + λT1 (qx + γ(qs)) + λT2 (qx +m−1xx (qs)mss(qs)qs)

where λ1, λ2 ∈ Rn−m are the vectors of Lagrangian multipliers. Under the constraintsy = 0 and qx = −m−1

xx (qs)mxs(qs)qs, Ls can be expressed as

Ls =1

2qTs

[

−m−1xx (qs)mxs(qs)

Im

] [

mxx(qs) mxs(qs)msx(qs) mss(qs)

] [

−m−1xx (qs)mxs(qs)

Im

]

qs−V (qs)

that after simplification takes the form

Ls(qs, qs) =1

2qTs (mss(qs)−msx(qs)m

−1xx (qs)mxs(qs))qs − V (qs)

or

Ls(qs, qs) =1

2qTsMs(qs)qs − V (qs)

Therefore, the zero-dynamics is a simple Lagrangian system with inertia matrixMs(qs) and its Lagrangian Ls satisfies the Euler-Lagrange equation with input damp-ing force Fd(qs, qs). The equation of the zero-dynamics in shape space can be equiv-alently written as

Ms(qs)qs + Cs(qs, qs)qs = g(qs) + Fd(qs, qs)

where Ms = Cs(qs, qs) + CT (qs, qs). Apparently, Hs by definition is smooth properpositive definite function. Calculating Hs, we get

Hs = qTsMs(qs)qs +12qTs Ms(qs)qs + qTs ∇V (qs)

= −qTs Cqs + 12qTs (C + CT )qs + qsg(qs) + qTs ∇V (qs) + qTs Fd(qs, qs)

= qTs Fd(qs, qs) +12qTs (C

T − C)qs

72

but CT − C is a skew-symmetric matrix and for all qs = ps 6= 0

pTs (CT − C)ps = [pTs (C

T − C)ps]T = −pTs (CT − C)ps = 0

ThusHs = pTs Fd(qs, ps) < 0

for all ps 6= 0. Based on LaSalle’s invariance principle [40], all the solutions ofthe zero-dynamics system asymptotically converge to the largest invariant set in(qs, ps) | Hs = 0 = (qs, ps) | ps = 0 which is equal to 0. Therefore, the ori-gin is globally asymptotically stable for the zero-dynamics, i.e. the zero-dynamicsis globally minimum-phase, and Hs is a valid Lyapunov function for the Lagrangianzero-dynamics.

Example 4.2.2. (Flexible link-robots) Consider a flexible one-link robot arm witha single actuator at the joint θ as shown in Figure 4-1. Modeling the link as an

w

M J

x

y

θα

LL

X

Y

Figure 4-1: A flexible one-link robot arm.

Euler-Bernoulli beam and using truncated modal analysis with m modes that haveamplitudes δ = (δ1, . . . , δm)

T [25], the dynamics of this flexible link is an underactu-ated system as the following [25]

[

mθθ(δ) mθδ

mδθ mδδ

] [

θ

δ

]

+

[

hθ(δ, δ, θ)

hδ(δ, θ)

]

=

[

τ

−Dδ

]

with a scalar control τ . The external variable qx = θ is actuated and the m-dimensional shape variable qs = δ is unactuated but has natural springs and damperswith constants kj, dj > 0 at each deformation mode such that

V (δ) =1

2δTkδ, Fd(δ) = −Dδ

with k = diag(k1, . . . , km) and D = diag(d1, . . . , dm). This flexible one-link arm isan example of a high-order underactuated system with kinetic symmetry and manyunactuated shape variables. In section 5.9, based on theorem 4.2.3, we prove thatthere exists a nonlinear noncollocated minimum-phase output in the form

y = θ + σ(cT δ)

73

that has global relative degree 3 w.r.t. the control v of the augmented system with anintegrator τ = v. Here, c ∈ Rm is a constant and σ(·) is a scalar sigmoidal function(see [71] for details on trajectory tracking control design).

4.3 Underactuated Systems with Input Coupling

In this section, we focus on reduction of a class of underactuated systems with in-put coupling. By input coupling, we mean there exists no permutation of rows ofthe force (or input) matrix F (q) with full column rank m that transforms it intocol(F1(q), F2(q)) such that F1 ≡ 0 and F2 is an m ×m invertible matrix. In the dy-namics of aerospace vehicles like aircraft and helicopter, the effect of the body torqueappears in the translational dynamics of the vehicle. This motivated us to considereliminating this coupling effect for the class of underactuated systems with kineticsymmetry and input coupling via a global change of coordinates that decouples thedynamics of the external variables and shape variables w.r.t. to the control appliedto the shape variables. The VTOL aircraft has a constant inertia matrix, while wewill see in chapter 5 that the inertia matrix of an aircraft or a helicopter depends onthe pitch and roll angles and is not constant.

On the other hand, in the nature, a flying bird, a swimming fish, and a walkinghuman being are all examples of mechanical systems that their position is unactuatedand their locomotion is due to the changes in their physical shape (see [74, 9, 49,104] for simpler examples). Moreover, the body of all three examples contains fully-actuated joints. In other words, a bird, a fish, and a human are examples of non-flat underactuated systems with actuated shape variables and unactuated externalvariables. Roughly speaking, the non-flatness property of these systems is due to thefact that any changes in the shape variables of any of them affects the physical shapeand thus their corresponding inertia matrix. Therefore, for the sake of generalityof our analysis, we first consider the class of non-flat underactuated systems withfully-actuated shape variables and input coupling.

Theorem 4.3.1. Consider an underactuated system with kinetic symmetry w.r.t. qxand fully-actuated shape variables. Assume the inertia matrix of this system is blockdiagonal, i.e. M(qs) = diag(mxx(qs),mss(qs)), and the force matrix is independent ofqx, i.e. F (q) = F (qs). The dynamics of the system can be expressed in the form

mxx(qs)qx + hx(q, q) = Fx(qs)τmss(qs)qs + hs(q, q) = Fs(qs)τ (4.22)

where τ ∈ Rm and Fs(qs) is an invertible m×m matrix. Due to mxs(qs) = 0, system(4.22) is globally partially feedback linearizable using an invertible change of control

τ = F−1s (qs)[mss(qs)u+ hs(q, q)]

74

Suppose all the elements of

ω = m−1xx (qs)Fx(qs)F

−1s (qs)mss(qs)dqs

are exact one-forms and let ω = dγ(qs). Then, the following global change of coordi-nates (i.e. diffeomorphism)

qr = qx − γ(qs)pr = mxx(qs)px − Fx(qs)F−1s (qs)mss(qs)ps

(4.23)

with (px, ps) = (qx, qs) transforms (4.22) into the following cascade system in nontri-angular quadratic normal form

qr = m−1r (qs)pr

pr = gr(qr, qs) + Σ(qs, pr, ps)qs = psps = u

(4.24)

In addition, the (qr, pr)-subsystem is a Lagrangian system with configuration vectorqr and the reduced Lagrangian parameterized by qs

Lr(qr, qr, qs) =1

2qTr mr(qs)qr − Vr(qr, qs)

that satisfies the following forced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= −Fx(qs)F−1s (qs)gs(qr + γ(qs), qs) + Σ(qs, pr, ps)

wheremr(qs) := mxx(qs)

Vr(qr, qs) := V (qr + γ(qs), qs)gr(qr, qs) := [gx(qx, qs)− Fx(qs)F−1s (qs)gs(qx, qs)]qx=qr+γ(qs)gx(qx, qs) := −∇qxV (qx, qs)gs(qx, qs) := −∇qsV (qx, qs)

and

Σ(qs, pr, ps) =

[

prps

]T

Π(qs)

[

prps

]

is a quadratic form in (pr, ps) with a cubic weight matrix Π(qs).

Proof. Consider the following generalized momentums

πx :=∂L∂qx

= mxx(qs)qx, πs :=∂L∂qs

= mss(qs)qs

By definition of qr, pr, we have qr = m−1r (qs)pr and

pr = πx − Fx(qs)F−1s (qs)πs

75

Based on Euler-Lagrange equation, πx, πs satisfy

πx =∂L∂qx

+ Fx(qs)τ, πs =∂L∂qs

+ Fs(qs)τ

Thus, we have

pr = πx − Fx(qs)F−1s (qs)πs −d

dtFx(qs)F−1s (qs)πs

=∂L∂qx− Fx(qs)F−1s (qs)

∂L∂qs− pTs πF (qs)ps

where πF (qs) is a cubic matrix with (n−m) layers that are m×m matrices satisfyingthe relation

pTs πF (qs)ps =d

dtFx(qs)F−1s (qs)mxx(qs)ps

Note that τ is eliminated from the equation of pr. By direct calculation, we have

∂L∂qx

= gx(qx, qs),∂L∂qs

= gs(qx, qs) +∂K

∂qs

Thus, noting that ∂K/∂qs is a vector quadratic form in (px, ps) with a cubic weightmatrix as a function of qs and setting

gr(qr, qs) =[

gx(qx, qs)− Fx(qs)F−1s (qs)gs(qx, qs)]

qx=qr+γ(qs)

it follows thatpr = gr(qr, qs) + Σ(qs, pr, ps)

where

Σ(qs, pr, ps) = −Fx(qs)F−1s (qs)∂K

∂qs− pTs πF (qs)ps

which proves equation (4.24). To obtain an explicit expression for Σ, let the cubicmatrix πK(qs) satisfy

Fx(qs)F−1s (qs)

∂K

∂qs=

[

pxps

]T

πK(qs)

[

pxps

]

and note that each layer of πK(qs) is block diagonal according to the partition ofq = (qx, qs) (because M(qs) is block diagonal). By definition of pr, we get

[

pxps

]

= W (qs)

[

prps

]

where W is an n× n matrix given by

W (qs) =

[

m−1xx (qs) m−1

xx (qs)Fx(qs)F−1s (qs)mss(qs)

0 Im

]

76

This meansΠ(qs) = −W T (qs)πK(qs)W (qs)− diag(0, πF (qs))

where the last term is a cubic matrix with block diagonal layers of 0 (as n layers of0n−m) and πF (qs). The part that the reduced Lagrangian satisfies the forced Euler-Lagrange equation given in the question follows from the equation of pr and theidentity

∂Lr∂qr

=

[

∂L∂qx

]

qx=qr+γ(qs)

· ∂qx∂qr

= [gx(qx, qs)]qx=qr+γ(qs)

Example 4.3.1. (A Flying Bird) Figure 4-2 shows the simplified model of a flyingbird. The position of the center of the body of the bird and the orientation of the bodyis denoted by (x,R0) ∈ R3 × SO(3). In addition, each wing has its own orientationRi ∈ SO(3), i = 1, 2 which is fully-actuated. The orientation of the tail is parame-terized by a pitch angle φ3 and roll angle θ3 which both are actuated. Therefore, thisbird is an underactuated system with 14 DOF and 8 control inputs. It is clear that if

φθ

x , R

R

R 2

0

1

3

3

Figure 4-2: A flying bird.

R0, R1, R2, φ3, θ3 are fixed, then the physical shape of a bird does not change undertranslation of the center of mass of its body. This means that the vector of externalvariables is qx = x and the vector of shape variables is qs = (R0, R1, R2, φ3, θ3). Dueto the variable physical shape of the bird, the inertia matrix for this system is notconstant and depends on qs = (R0, R1, R2, φ3, θ3).

The following result provides classes of underactuated systems in theorem 4.3.1that can be globally transformed into nonlinear systems in feedforward form. Inaddition, sufficient conditions are given such that these feedforward forms are globallyasymptotically stabilizable using a state feedback in explicit form.

Proposition 4.3.1. Assume all the conditions in theorem 4.3.1 hold and in addition

i) mr = mxx is constant.

ii) V (q) = kT0 qx + V (qs) where k0 is a constant vector.

77

Then, the global change of coordinates in (4.23) transforms the dynamics of the un-deractuated system (4.22) into a cascade nonlinear system in feedforward form as thefollowing

qr = m−1r pr

pr = gr(qs) + pTs Π(qs)psqs = psps = u

(4.25)

where Π(qs) is a cubic matrix and

gr(qs) = −k0 + Fx(qs)F−1s (qs)∇qsV (qs)

Moreover, if the following conditions are satisfied

iii) dim(qx) = dim(qs).

iv) gr(0) = 0 and the Jacobian matrix ∇qsgr(qs) is invertible at qs = 0.

Then, the origin (qr, pr, qs, ps) = 0 for the nonlinear system in (4.25) can be globallyasymptotically stabilized using a state feedback in explicit form as nested saturations.

Proof. Following the proof of theorem 4.3.1, because mxx is constant ∂K/∂qs is aquadratic form in ps and Σ takes the following form

Σ = Σ(qs, ps) = pTs π(qs)ps

which is independent of pr. In addition, we have

gr(qr, qs) = gr(qs) = −k0 − Fx(qs)F−1s (qs)gs(qs)

and (4.25) holds. The result on stabilization of the origin follows from the methodof nested saturations for feedforward systems due to Teel [102]. (see the proof ofproposition 4.2.1 for further details on control design).

Theorem 4.3.2. Consider a flat underactuated system with configuration vector q =(qx, qs) and a force matrix that is independent of qx. Assume the potential energy ofthe system is independent of qs, i.e. V (q) = V (qx). The Euler-Lagrange equations ofmotion for this system are as the following

mxxqx − gx(qx) = Fr(qs)τr + Fx(qs)τmssqs = Fs(qs)τ

(4.26)

where gx(qx) = −∇qxV (qx), τr ∈ R, Fr(qs) : Rm → Rn−m is a unit vector that is ontoover a unit ball in Rn−m, τ ∈ Rm, mxx and mss are constant, and Fs(qs) is an m×minvertible matrix. Let

τ = Fs(qs)−1mssu

be the partially linearizing change of control. Assume all the elements of

ω = m−1xxFx(qs)F

−1s (qs)mssdqs

78

are exact one-forms and let ω = dγ(qs). Then, the following global change of coordi-nates

qr = qx − γ(qs)pr = mxxpx − Fx(qs)F−1s (qs)mssps

(4.27)

with (px, ps) = (qx, qs) transforms the dynamics of (4.26) into the following form

qr = m−1r pr

pr = gr(qr + γ(qs))− pTs πF (qs)ps + Fr(qs)τrqs = psps = u

(4.28)

where πF (qs) is a cubic matrix satisfying

d

dtFx(qs)F−1s (qs)mssps = pTs πF (qs)ps

In addition, the (qr, pr)-subsystem is a flat Lagrangian system with configuration vec-tor qr and reduced Lagrangian

Lr(qr, qr, qs) =1

2qTr mrqr − V (qr + γ(qs))

satisfying the forced Euler-Lagrangian equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= −pTs πF (qs)ps + Fr(qs)τr

where mr = mxx. Moreover, if the following conditions hold:

i) pTs πF (qs)ps = (pTsQps)Fr(qs) with Q ∈ Rm2.

ii) V (qx) is a linear function.

Then, the reduced Lagrangian system is a fully-actuated flat system that satisfies

d

dt

∂Lr∂qr− ∂Lr∂qr

= Fr(qs)τr

whereτr = τr − pTsQps

is the new scalar control.

Proof. The proof is by direct calculation.

Example 4.3.2. (VTOL aircraft) The VTOL aircraft discussed in section 3.3.6 sat-isfies all the conditions of proposition 4.3.2 and after applying a global change ofcoordinates that decouples the position and orientation dynamics of the VTOL air-craft w.r.t. the input torque u2 in equation (5.41), the reduced dynamics of theposition is a fully-actuated flat mechanical system. The configuration of the VTOL

79

aircraft can be globally asymptotically stabilized to the origin using a smooth staticstate feedback (see [70], or section 5.10 for more details on control design procedure).Later on, we will show a similar result applies to an autonomous helicopter.

Corollary 4.3.1. Assume all the conditions in theorem 4.3.2 hold. In addition,suppose Fx, Fs are constant and τr = 0. Then, the nonlinear system in (4.28) is instrict feedback form.

Example 4.3.3. (Inertia-Wheel Pendulum) The Inertia-Wheel Pendulum satisfies allthe conditions of the preceding corollary and can be globally asymptotically stabilizedusing a state feedback in explicit form (see section 5.3 for more details).

4.4 Underactuated Systems with Non-integrable

Momentums

In this section, we consider the class of underactuated systems with kinetic symmetrythat their normalized momentums are not-integrable. By normalized momentums, wemean

πx = qx +m−1xx (qs)mxs(qs)qs = m−1

xx (qs)∂L∂qx

(4.29)

πs = qx +m−1sx (qs)mss(qs)qs = m−1

sx (qs)∂L∂qs

(4.30)

where πx, πs are the normalized momentums conjugate to qx, qs, respectively. Thenormalized momentum πx (or πs) is called non-integrable, if @h = h(qx, qs) : h =πx (or πs). It turns out that in this case, underactuated systems can be reduced tokinematic systems driven by the shape velocity vector ps as the control input. First,we consider underactuated systems with actuated shape variables and non-integrablenormalized momentums.

Theorem 4.4.1. Consider the underactuated system in (4.3) and suppose all theassumptions in theorem (4.2.1) hold except for the exactness property of the elementsof ω. In other words, assume given

µ(qs) = m−1xx (qs)mxs(qs)

µ(qs)ps is not integrable. Then, after applying the global change of coordinates

qr = qxpr = mxx(qs)px +mxs(qs)ps

with (px, ps) = (qx, qs), the dynamics of the system transforms into the following

80

nontriangular normal form

qr = m−1r (qs)pr − µ(qs)ps

pr = gr(qr, qs)qs = psps = u

(4.31)

that is affine in the shape velocity ps as the control input of (qr, pr, qs)-subsystem.

Proof. The equation of pr follows from the proof of theorem (4.2.1) and the firstequation in (4.31) is a direct result of the definition of pr.

Remark 4.4.1. Defining x = (qr, pr, qs) and v = ps, the dynamics of the (qr, pr, qs)-subsystem can be written as a nonlinear system affine in control

x = f(x) + g(x)v

where g(x) = col(−µ(qs), Im). This system has m (i.e. number of control inputs) firstorder differential equations less than the original system and both the control designand controllability analysis of this system can be carried out in a lower-order spaceR3m instead of the original space R4m.

Remark 4.4.2. Recall that in the integrable case for underactuated systems withactuated shape variables, it is possible to use backstepping procedure due to strictfeedback form of the transformed system. Here, due to nontriangular structure of thenormal form in (4.31) the backstepping procedure is not applicable. Control designfor (4.31) will be treated later in chapter 7.

Example 4.4.1. (three-link planar robot) Consider a planar three-link robot armwith revolute joints (q1, q2, q3) and two actuators at q2, q3 as shown in Figure 4-3.The dynamics of this robot is given in section A.2 (Appendix A). It can be shown

q

q

q

1

2

3

x

y

Figure 4-3: A triple-link robot arm with two actuators.

that the inertia matrix of this triple-link robot has the following structure

M(q) =M(q2, q3) =

m11(q2, q3) m12(q2, q3) m13(q2, q3)m21(q2, q3) m22(q3) m23(q3)m31(q2, q3) m32(q3) m33

81

Thus, qx = q1 is the external variable and qs = (q2, q3) is the vector of shape variables.Moreover, both shape variables are actuated. However, by direct calculation, it canbe shown that the normalized momentum

π1 = q1 +m12(q2, q3)

m11(q2, q3)q2 +

m13(q2, q3)

m11(q2, q3)q3

is non-integrable (see lemma A.2.1). Therefore, based on theorem 4.4.1, a three-linkrobot can be transformed into a nontriangular kinematic normal form with ps =(q2, q3) as the input.

Now, we present the case of underactuated systems with actuated external vari-ables and non-integrable normalized momentums.

Theorem 4.4.2. Consider the underactuated mechanical system in (4.12) and sup-pose all the assumptions in theorem (4.2.2) hold except for the exactness property ofthe elements of ω. In other words, assume given

µ(qs) = m−1sx (qs)mss(qs)

µ(qs)ps is non-integrable. Then, after applying the change of coordinates

qr = qxpr = msx(qs)px +mss(qs)ps

with (px, ps) = (qx, qs), the dynamics of the system transforms into the followingnontriangular linear-quadratic normal form

qr = m−1r (qs)pr − µ(qs)ps

pr = gr(qr, qs) + Σ(qs, pr, ps)qs = psps = u

(4.32)

overU = qs ∈ Qs | det(msx(qs) 6= 0

with ps as the control input for the (qr, pr, qs)-subsystem where

mr(qs) := msx(qs)gr(qr, qs) := ∇qsV (qr, qs)

Σ(qs, pr, ps) := ∇qsK

Proof. The proof for the equation of pr in (4.32) is the same as in theorem 4.2.2. Bydefinition of pr, we have

qx = m−1sx (qs)pr −m−1

sx (qs)mss(qs)ps = m−1r (qs)pr − µ(qs)ps

and the result follows. Based on the structure of normal form (4.32), it is clear that

82

the (qr, pr, qs)-subsystem is kinematic (i.e. has a control input ps that is the shapevelocity vector).

Remark 4.4.3. The normal form in (4.32) is very similar to a normal form previouslyfound for Lagrangian systems with nonholonomic velocity constraints and classicalsymmetry due to Bloch et al. [9]. However, the formulation of theorem 4.4.2 isfundamentally different from the setup studied in [9].

Proposition 4.4.1. Assume all the assumptions in theorem 4.4.2 hold. In addition,the underactuated system (4.12) satisfies the following conditions

i) mxx(qs) is constant.

ii) msx(qs) has differentially symmetric rows, i.e.

∂msx(qs)

∂qis=∂mi∗

sx

∂qs, i = 1, . . . ,m

where mi∗sx(qs) is the ith row of msx(qs).

iii) V (q) = V (qs).

iv) µ(qs) is analytic at qs = 0 (element-wise).

Then, applying the change of coordinates

z1 = qr + µ(0)qs, z2 = m−1r (qs)pr

(where (qr, pr) are defined in (4.13)) transforms the underactuated system (4.12) intoa cascade system in feedforward form as the following

z1 = z2 + ϕ1(qs, ps)z2 = ψ(qs) + pTs Π(qs)psqs = psps = u

(4.33)

where ϕ1(qs, ps) = (µ(0) − µ(qs))ps is at least quadratic in (qs, ps), Π(qs) is a cubicmatrix, and ψ : Qs → Rm is defined as

ψ(qs) = −m−1r (qs)∇qsV (qs)

Moreover, if ψ(0) = 0 and ψ(qs) has an invertible Jacobian ∇qsψ(qs) at qs = 0, theorigin for (4.33) can be globally asymptotically and locally exponentially stabilized overU using a state feedback in the form of nested saturations.

Proof. Following the proofs of proposition 4.2.1 and theorem 4.4.2, after applyingthe change of coordinates

z1 = qr, z2 = m−1r (qs)pr

83

we obtain˙z1 = z2 − µ(qs)psz2 = ψ(qs) + pTs π(qs)psqs = psps = u

Noting that µ(qs) is element-wise analytic at qs = 0 and

µ(qs)ps = µ(0)ps − ϕ1(qs, ps)

whereϕ1(qs, ps) = O(‖(qs, ps)‖2)

by setting z1 = z1 + µ(0)qs the result follows. The rest of the proof is rather similarto the proof of proposition 4.2.1.

Example 4.4.2. (3D Cart-Pole System) Consider an inverted pendulum mountedon a moving platform via an unactuated 2 DOF joint (θ, φ) with two external forcesas shown in Figure 4-4. The angles θ, φ denote rotation around x2-axis and x1-axis,respectively. Let (x1, x2) denote the position of the center of mass of the platformM .

F

F

2

1

x3

x1

2x

M

m

a

φθ

l

φ

Figure 4-4: An inverted pendulum on a moving platform with 4 DOF and 2 controls.

Then, the kinetic energy of the 3D Cart-Pole system (given in section A.7) is in theform

K =1

2

x1x2θ

φ

T

M +m 0 ml cos θ 00 M +m −ml sin θ sinφ ml cos θ cosφ

ml cos θ −ml sin θ sinφ ml2 00 ml cos θ cosφ 0 ml2 cos2 θ

x1x2θ

φ

(4.34)Clearly, the inertia matrix for this system only depends on (θ, φ). Therefore, (θ, φ)are the shape variables and (x1, x2) are the external variables of the 3D Cart-Polesystem. The system is an underactuated system with four DOF (x1, x2, θ, φ) and two

84

actuated external variables (x1, x2). For the matrix

msx(qs) =

[

ml cos θ −ml sin θ sinφ0 ml cos θ cosφ

]

we havedet(msx(qs)) = m2l2 cos2 θ cosφ

Thus, msx is nonsingular over U = R2 × (−π/2, π/2)2. The vector of one-forms isω = µ(qs)ps where

µ(qs) = m−1sx (qs)mss(qs) = l

1

cos θ

sin θ sinφ

cosφ

0cos θ

cosφ

Since the following one-forms are not exact

ω1 =1

cos θdθ +

sin θ sinφ

cosφdφ

ω2 =cos θ

cosφdφ

Theorem 4.2.2 is not applicable and the normalized momentum πs of the 3D Cart-Polesystem is non-integrable. On the other hand, the dynamics of the 3D Cart-Pole systemsatisfies all four conditions of proposition 4.4.1. Therefore, it can be transformed intoa feedforward system over U .

Proposition 4.4.2. The 3D Cart-Pole system (in example 4.4.2) can be transformedinto a nonlinear system in feedforward form using the change of coordinates (in 4.4.1)

z1 = qx + lqs , z2 = m−1sx (qs)pr

In addition, the origin for the system can be globally asymptotically and locally expo-nentially stabilized using nested saturations.

Proof. Transformation into a feedforward system is already proven in example 4.4.2.To prove the stabilization using nested saturations, we need to prove

ψ(qs) = −m−1sx (qs)∇qsV (qs) =

g

cosφ

[

sin θsinφ

]

has an invertible Jacobian at qs = 0. Calculating this Jacobian, we have

∂ψ

∂(θ, φ)= g

cos θ

cosφ

sin θ sinφ

cos2 φ

01

cos2 φ

(θ,φ)=0

= gI2×2 6= 0

85

which is clearly nonsingular and the stabilization result follows.

4.5 Momentum Decomposition for Underactuated

Systems

In this section, we consider reduction of underactuated systems with non-integrablenormalized momentums. We introduce a method to decompose a non-integrablenormalized momentum as an integrable momentum term called locked momentumand a non-integrable term called error momentum. We call this procedureMomentumDecomposition. This decomposition is unique for a fix choice of a locked configuration(to be defined later). This procedure is described in the following.

Consider an underactuated mechanical system with kinetic symmetry and con-figuration vector (qx, qs) where qx and qs denote the vector of external and shapevariables, respectively. The normalized momentums for this system are given by

πx := m−1xx (qs)

∂L∂qx

= qx +m−1xx (qs)mxs(qs)qs (4.35)

πs := m−1sx (qs)

∂L∂qs

= qx +m−1sx (qs)mss(qs)qs (4.36)

which both can be expressed as the following

π = qx + µ(qs)qs (4.37)

Assume π is non-integrable, i.e. @h : h = π. Our goal is to find an integrableapproximation of π which allows us to apply our previous reduction procedures forunderactuated systems with integrable normalized momentums to the present case.We refer to µ(qs) as the shape inertia matrix. The following theorem demonstrateshow every non-integrable normalized momentum can be decomposed as an explicitlyintegrable momentum plus a non-integrable momentum term.

Theorem 4.5.1. (Momentum Decomposition) Let π = qx+µ(qs)qs be a non-integrablenormalized momentum of a mechanical control system with kinetic symmetry w.r.t.qx (η(qs) is a d × m matrix). Denote the shape vector by qs = (q1s , . . . , q

ms )

T andthe normalized momentum by π = (π1, . . . , πd)T . Fix qs and define the locked shapeinertia matrix µl(qs) by its elements

µlij(qjs) = [µij(qs)]qks=qks ;k=1,...,m;k 6=j (4.38)

and the locked momentum as

πl = qx + µl(qs)qs (4.39)

Then, the locked momentum is integrable in an explicit form

qr = qx + γ(qs) (4.40)

86

i.e. qr = πl, where γ(qs) = (γ1(qs), . . . , γd(qs))T and

γi(qs) = Σmj=1

∫ qjs

0

µlij(θj)dθj , i = 1, . . . , d (4.41)

In addition, denotingµe(qs) := µ(qs)− µl(qs) (4.42)

the normalized error momentum can be defined as

πe := µe(qs)qs (4.43)

and the non-integrable momentum π can be uniquely decomposed as

π = πl + πe (4.44)

where πl is integrable and πe is non-integrable. Moreover, πe is independent of (qx, qx)and vanishes at qs = qs.

Proof. The ith element of π can be expressed as

πi = qix + Σmj=1µij(qs)q

js

= qix + Σmj=1[µ

lij(q

js) + µeij(qs)]q

js

= qix + Σmj=1µ

lij(q

js)q

js + Σm

j=1µeij(qs)q

js

orπ = πl + πe

But the ith element of πl is

qix + Σmj=1µ

lij(q

js)q

js = qix + γi

Thus, setting qr = qx + γ(qs), qr = πl and πl is explicitly integrable. Since πe =ηe(qs)qs, it is independent of (qx, qx). Also, by definition of µl(qs), µ

l(qs) = µ(qs) andtherefore both µe(qs) and π

e vanish at qs = qs.In the following, we apply the momentum decomposition method introduced in

theorem 4.5.1 to underactuated systems with kinetic symmetry for two cases: i)underactuated systems with actuated shape variables, and ii) underactuated systemswith unactuated shape variables.

Theorem 4.5.2. Consider the following underactuated system with kinetic symmetryand actuated shape variables

mxx(qs)qx +mxs(qs)qs + hx(q, q) = 0msx(qs)qx +mss(qs)qs + hs(q, q) = τ

(4.45)

Let τ = α(qs)u+ β(q, q) be the global collocated partially linearizing change of controlfor (4.45). Assume the following normalized momentum with µx(qs) = m−1

xx (qs)mxs(qs)

87

is non-integrableπx = qx + µx(qs)qs

Letπx = πlx + πex

be the momentum decomposition of πx as an integrable locked momentum πlx = qx +µlx(qs)qs associated with the locked shape qs = qs and a non-integrable normalizederror momentum πex. Then, there exists a function γ(qs) in explicit form satisfying∇qsγ(qs) = µlx(qs) such that the global change of coordinates

qr = qx + γ(qs)pr = mxx(qs)(qx + µlx(qs)qs)

(4.46)

transforms the dynamics of the underactuated system in (4.45) into a cascade non-linear system in strict feedback form

qr = m−1r (qs)pr

pr = gr(qr, qs) + δqs = psps = u

(4.47)

with a perturbation δ = −pe where pe = mr(qs)πex is the error momentum and

mr(qs) := mxx(qs)Vr(qr, qs) := V (qx, qs)|qx=qr−γ(qs)gr(qr, qx) := −∇qrVr(qr, qs)

The perturbation δ is in the form

δ = mr(qs)µex(qs)u+ pTs Πδ(qs)ps

which is affine in u, quadratic in ps with cubic weight matrix Πδ(qs), and vanishesuniformly in u at (qs, ps) = (qs, 0). In addition, the reduced system is a Lagrangiansystem with reduced parameterized Lagrangian

Lr(qr, qr, qs) =1

2qTr mr(qs)qr − Vr(qr, qs)

that satisfies the forced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= δ (4.48)

Proof. By definition of qr, pr, and mr(qs), qr = m−1r (qs)pr. Due to kinetic symmetry,

we have∂L∂qx

= −∇qxV (qx, qs)

88

which means in new coordinates

∂Lr∂qr

= gr(qr, qs)

On the other hand, by definition of πx and based on momentum decomposition the-orem, we have

∂L∂qx

= mr(qs)πx = mr(qs)(πlx + πex) = pr + pe

Hence

pr + pe =d

dt

∂L∂qx

=∂L∂qx

= gr(qr, qs)

orpr = gr(qr, qs)− pe

The forced Euler-Lagrange equation with input δ = −pe follows from the last equa-tion.

Remark 4.5.1. Theorem 4.5.2 shows that control design for the original underactuatedsystem with a non-integrable normalized momentum reduces to control design forsystem (4.47) which is in strict feedback form plus a nonlinear perturbation δ. Onepossible way to stabilize the nonlinear system in (4.47) is to design a controller forthe unperturbed system which reduces to control of the (qr, pr)-subsystem combinedwith a backstepping stage. Then, perform a Lyapunov-based robustness analysisw.r.t. the nonlinear perturbation δ which vanishes at a desired equilibrium pointwith shape configuration and velocity (qs, ps) = (qs, 0).

Remark 4.5.2. If the normalized momentum πx is integrable, then πe ≡ 0 whichmeans δ, pe ≡ 0 and thus theorem 4.5.2 reduces to theorem 4.2.1 stated earlier forunderactuated systems with actuated shape variables and integrable normalized mo-mentums.

Remark 4.5.3. The main difference between normal forms (4.47) and (4.31) for under-actuated systems with non-integrable normalized momentums is that in the formerone, the perturbation term δ appears in the equation of pr and a reduced Lagrangiansystem with configuration vector qr and input force δ is identifiable (as in theorem4.5.2), but in the latter case, the perturbation µ(qs)ps appears in the equation of qrand no Lagrangian reduced system can be identified.

Example 4.5.1. (three-link planar robot with actuated shape variables) Considerthe three-link planar robot arm with two actuated shape variables (q2, q3) as shownin Figure 4-5 (this system was also discussed earlier in example 4.4.1). The dynamicsof the actuated shape variables of this system can be globally linearized using acollocated partially-linearizing change of control as the following

q2 = u2, q3 = u3

The normalized momentum of this system conjugate to q1 is non-integrable (see lemma

89

A.2.1). Note that for this three-link robot

µ(qs) = m−111 (q2, q3)

[

m12(q2, q3) m13(q2, q3)]

After locking q2, q3 at (q2, q3) = (0, 0), the locked approximation of π1 can be calcu-

q

q

q

1

2

3

x

y

Figure 4-5: A triple-link robot arm with actuated shape variables.

lated as

πl1 = q1 +m12(q2, 0)

m11(q2, 0)q2 +

m13(0, q3)

m11(0, q3)q3

which gives explicit forms for γ1 and γ2 as the following

γ1(q2) =

∫ q2

0

m12(s, 0)

m11(s, 0)ds, γ2(q3) =

∫ q3

0

m12(0, s)

m11(0, s)ds

After applying the change of coordinates

qr = q1 + γ(q2) + γ(q3)

pr = m11(q2, q3)[p1 +m12(q2, 0)

m11(q2, 0)p2 +

m13(0, q3)

m11(0, q3)p3]

the system transforms into normal form (4.47). Calculating the error momentumpe = mr(qs)π

e1, we get

pe = (m12(q2, q3)−m12(q2, 0))q2 + (m13(q2, q3)−m13(0, q3))q3

and thus δ = −pe is given by

δ = −(m12(q2, q3)−m12(q2, 0))u2 − (m13(q2, q3)−m13(0, q3))u3 + λ(q2, q2, q3, q3)

where δ can be made exponentially vanishing by applying a linear control u3 =K3(q3, p3) = −c1q3 − c2p3 with c1, c2 > 0. Keeping this in mind, the dynamics of thereduced system for the three-link robot can be rewritten as

qr = m−1r (q2, 0)pr + ϕ1

pr = gr(qr, q2, 0) + ϕ2

90

where the nonlinear perturbations ϕ1, ϕ2 are given by

ϕ1 = (m−1r (q2, q3)−mr(q2, 0))pr, ϕ2 = (gr(qr, q2, q3)− gr(qr, q2, 0)) + δ

Both ϕ1 and ϕ2 vanish exponentially as (q3, p3) converges exponentially fast to zero.In other words, control of the three-link planar robot which is a sixth order nonlinearsystem reduces to stabilization of the second-order unperturbed reduced system

qr = m−1r (q2, 0)pr

pr = gr(qr, q2, 0)

But this normal form is exactly the same as the reduced system of an Acrobot withconfiguration vector (q1, q2) and the following inertia matrix and potential energy

MAcro(q2) =

[

m11(q2, 0) m12(q2, 0)m21(q2, 0) m22(0)

]

, VAcro(qr, q2) = [V (q1, q2, 0)]q1=qr−γ1(q2)

The shape variable of this Acrobot is q2 which is actuated. Since the Acrobot canbe globally asymptotically stabilized to its upright equilibrium point using a smoothstate feedback u2 = K2(qr, pr, q2, p2) [68] (under minor conditions), after a ratherelementary perturbation analysis it follows that the origin for a three-link planarrobot can be globally asymptotically stabilized using (static) state feedback lawsu2 = K2(qr, pr, q2, p2) and u3 = K3(q3, p3).

Remark 4.5.4. The result of example 4.5.1 is in agreement with our intuition that ifone sets the joint angle q3 identically to zero, a three-link planar robot graphicallylooks the same as an Acrobot with an actuated shape variable q2. Keep in mind thatthis choice is not unique and one could have started by exponential stabilization of(q2, p2)-subsystem to zero and then stabilize an Acrobot with an external variable q1and shape variable q3.

Next, we consider the case of underactuated systems with partially-actuated shapevariables and non-integrable momentums.

Theorem 4.5.3. Consider the following underactuated system with kinetic symmetryand partially-actuated shape variables qs = col(qs1 , qs2) ∈ Qs1 ×Qs2

mxx(qs)qx + mxs1(qs)qs1 + mxs2(qs)qs2 + hx(q, q) = τ1ms1x(qs)qx + ms1s1(qs)qs1 + ms1s2(qs)qs2 + hs1(q, q) = 0ms2x(qs)qx + ms2s1(qs)qs1 + ms2s2(qs)qs2 + hs2(q, q) = τ2

(4.49)

Assume dim(Qx) = dim(Qs1) and dim(Qs2) ≥ 0. Let τ = α(qs)u + β(q, q) be thenoncollocated partially linearizing change of control for (4.49) over

U = qs ∈ Qs | det(ms1x(qs)) 6= 0

which linearizes the dynamics of the shape variables as qs = u. Assume the following

91

normalized momentum is non-integrable

πs1 = qx + µs1(qs)qs

withµs1(qs) = m−1

s1x(qs)

[

ms1s1(qs) ms1s2(qs)]

Letπs1 = πls1 + πes1

be the momentum decomposition of πs1 as an integrable locked momentum πls1 =qx+µ

ls1(qs)qs associated with the locked shape qs = qs and a non-integrable normalized

error momentum πes1. Then, there exists a function γ(qs) in explicit form satisfying∇qsγ(qs) = µls1(qs) such that the following change of coordinates over U

qr = qx + γ(qs)pr = ms1x(qs)(qx + µls1(qs)qs)

(4.50)

transforms the dynamics of the underactuated system in (4.49) into a cascade non-linear system in nontriangular quadratic normal form

qr = m−1r (qs1 , qs2)pr

pr = gr(qr, qs1 , qs2) + Σ(qs1 , qs2 , pr, ps) + δqs1 = ps1ps1 = u1qs2 = ps2ps2 = u2

(4.51)

with a perturbation δ = −pe where pe = mr(qs)πes1is the error momentum and

mr(qs) := ms1x(qs)Vr(qr, qs) := V (qx, qs)|qx=qr−γ(qs)gr(qr, qx) := −∇qs1

Vr(qr, qs1 , qs2)Σ(qs1 , qs2 , pr, ps) := ∇qs1

K

The perturbation δ is in the form

δ = mr(qs)µes1(qs)u+ pTs Πδ(qs)ps

which is affine in u, quadratic in ps with a cubic weigh matrix Πδ(qs) , and vanishesuniformly in u at (qs, ps) = (qs, 0). In addition, if V (q) = V (qx), the reduced systemis a Lagrangian system itself with reduced parameterized Lagrangian

Lr(qr, qr, qs) =1

2qTr mr(qs)qr

92

that satisfies the forced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= Σ(qs, pr, ps) + δ (4.52)

Proof. By definition of qr and pr, qr = m−1s1x

(qs)pr. From the second line of (4.49),we have

d

dt

∂L∂qs1

=∂L∂qs1

= −∇qs1V (qx, qs) +∇qs1

K

which after substituting qx = qr − γ(qs) and qx = m−1s1x

(qs)pr − µls1(qs)ps, one obtains

d

dt

∂L∂qs1

= gr(qr, qs) + Σ(qs, pr, ps)

Based on momentum decomposition theorem

∂L∂qs1

= mr(qs)(πls1+ πes1) = pr + pe

Hence pr = gr(qr, qs) +Σ(qs, pr, ps)− pe. The forced Euler-Lagrange equation followsfrom the last equation in a straightforward fashion.

Remark 4.5.5. Assuming qs2 = 0, the normal form in (4.51) can be rewritten as thefollowing

qr = m−1r (qs1 , 0)pr + ϕ1

pr = gr(qr, qs1 , 0) + Σ(qs1 , 0, pr, ps1 , 0) + ϕ2

qs1 = ps1ps1 = u1qs2 = ps2ps2 = u2

(4.53)

where

ϕ1 = (m−1r (qs1 , qs2)−mr(qs1 , 0))pr

ϕ2 = gr(qr, qs1 , qs2)− gr(qr, qs1 , 0) + Σ(qs1 , qs2 , pr, ps1 , ps2)− Σ(qs1 , 0, pr, ps1 , 0) + δ

where ϕ1 and the first two terms of ϕ2 vanish at (qs2 , ps2) = 0. If δ vanishes at(qs2 , qs2 , u2) = 0, then both nonlinear perturbations ϕ1, ϕ2 can be made exponentiallyvanishing over compact domains by applying the linear control u2 = K2(qs2 , ps2) =−c1qs2 − c2ps2 with c1, c2 > 0. This reduces the control design for the overall systemin (4.51) to control of its (qr, pr, qs1)-subsystem with control input ps1

qr = m−1r (qs1 , 0)pr

pr = gr(qr, qs1 , 0) + Σ(qs1 , 0, pr, ps1 , 0)qs1 = ps1

(4.54)

Later, we show that under certain conditions, stabilization of the last nonlinear system

93

reduces to stabilization of the following reduced system

qr = m−1r (qs1 , 0)pr

pr = gr(qr, qs1 , 0)(4.55)

with control input qs1 . Since qr and qs1 have the same dimensions, this stabilizationproblem can be addressed (see section 4.7 of this chapter).

Example 4.5.2. (three-link planar robot) In this example, we consider a three-linkplanar robot as shown in Figure 4-6 with two controls. Both the external variable θ1and one of the two shape variables θ2, θ3 are actuated. Notice that this is differentthan the case discussed in example 4.5.1 where both shape variables are actuated.The inertia matrix and dynamics of this robot are given in section A.2. The inertia

x

y

l L1

1

θ

θ

3

2

1

θ

, m , 1

I1

τ

τ3

2

τ 1

Figure 4-6: A planar three-link robot arm.

matrix of this robot has the following structure

M =M(θ2, θ3) =

m11(θ2, θ3) m12(θ2, θ3) m13(θ2, θ3)m21(θ2, θ3) m22(θ3) m23(θ3)m31(θ2, θ3) m32(θ3) m33

All of the normalized momentums of this three-link planar robot are non-integrable(see lemma A.2.1). In the following, we consider two cases: i) θ2 is unactuated, andii) θ3 is unactuated. In each case, based on theorem 4.5.3 and using momentumdecomposition procedure we obtain reduced normal forms of the system.

Case (i): θ2 is unactuated. Set (qx, qs1 , qs2) = (θ1, θ2, θ3). Assuming m12(θ2, θ3) >0 for all θ2, θ3, using a global noncollocated partially linearizing feedback law, thedynamics of the shape variables can be transformed into

θ2 = ω2

ω2 = u1θ3 = ω3

ω3 = u2

94

Then, calculating the normalized momentum conjugate to θ2, we get

πθ2 = θ1 +m22(θ3)

m21(θ2, θ3)θ2 +

m23(θ3)

m21(θ2, θ3)θ3

Thus

µθ2(θ2, θ3) =

[

m22(θ3)

m21(θ2, θ3)

m23(θ3)

m21(θ2, θ3)

]

Now, according to the locked shape (θ2, θ3) = (0, 0), the locked shape inertia matrixµlθ2 is given by

µlθ2(θ2, θ3) =

[

m22(0)

m21(θ2, 0)

m23(θ3)

m21(0, θ3)

]

Defining γ1 and γ2 as the integral of the elements of µlθ2

γ1(θ2) =

∫ θ2

0

m22(0)

m21(s, 0)ds, γ2(θ3) =

∫ θ3

0

m23(0)

m21(0, s)ds,

and applying the global change of coordinates

θr = θ1 + γ1(θ2) + γ2(θ3)

ωr = m12(θ2, θ3)(ω1 +m22(0)

m21(θ2, 0)ω2 +

m23(θ3)

m21(0, θ3)ω3)

the dynamics of the three-link robot with unactuated shape variable θ2 transformsinto the following nontriangular quadratic normal form

θr = m12(θ2, θ3)ωrωr = −gr(θr, θ2, θ3) + Σ(θ2, θ3, ωr, ω2, ω3)− peθ2 = ω2

ω2 = u1θ3 = ω3

ω3 = u2

(4.56)

where the error momentum pe is given by

pe = (m22(θ3)

m21(θ2, θ3)− m22(0)

m21(θ2, 0))ω2 + (

m23(θ3)

m21(θ2, θ3)− m23(θ3)

m21(0, θ3))ω3

Clearly, pe vanishes at (θ3, ω3) = (0, 0). This implies the perturbation δ = −pevanishes at (θ3, ω3, u2) = (0, 0, 0) as well. Following the procedure in remark 4.5.5,stabilization of (4.56) reduces to stabilization of its fourth-order subsystem as

θr = m12(θ2, 0)ωrωr = −gr(θr, θ2, 0) + Σ(θ2, 0, ωr, ω2, 0)

θ2 = ω2

ω2 = u1

(4.57)

95

which is exactly the normal form for the Pendubot with inertia matrix and potentialenergy

MPen(q1) =

[

m11(θ2, 0) m12(θ2, 0)m21(θ2, 0) m22(0)

]

, VPen(θ1, θ2) = V (θ1, θ2, 0)

and an unactuated shape variable θ2. Later, we show that the Pendubot can bestabilized using Fixed Point Controllers [67].

Case (ii): θ3 is unactuated. Take (qx, qs1 , qs2) = (θ1, θ3, θ2). After a similar argu-ment to Case (i), it can be shown that the control of the overall system reduces tocontrol of the Pendubot with inertia matrix and potential energy

MPen(q1) =

[

m11(0, θ3) m13(0, θ3)m31(0, θ3) m33

]

, VPen(θ1, θ3) = V (θ1, 0, θ3)

Notice that this Pendubot has an unactuated shape variable θ3. In this case, the per-turbations ϕ1, ϕ2 in (4.53) vanish exponentially over a compact domain by applyingthe linear feedback u2 = K2(θ2, ω2) = −c1θ2 − c2ω2 with c1, c2 > 0.

Remark 4.5.6. In conclusion, based on Figure 4-7, control of the middle robot reducesto control of the Acrobot in example 4.5.1 and control of the other two robots reducesto control of the Pendubot as discussed in Cases (i) and (ii) of example 4.5.2. Thisfact can be intuitively understood by the following graphical procedure without anycalculations.

Figure 4-7: Three possible actuation configuration for a planar three-link robot arm.

Graphical Reduction Procedure for an n-link underactuated planar robot:

1) Set j = n.

2) If j is the last actuated joint, QUIT.

3) If joint j is actuated, lock the joint angle θj = 0 and merge the jth link and the(j − 1)th link as one. Set j := j − 1 and GOTO step 2.

4) If joint j is unactuated, set j := j − 1 and GOTO step 2.

96

This algorithm ends up with the Acrobot or the Pendubot as the final underactuatedsystem with a single actuator and two remaining links.

Proposition 4.5.1. Consider the 3D Cart-Pole system with configuration vector(qx, qs) = (x1, x2, θ, φ) and unactuated shape variables (θ, φ). Then, the followingchange of coordinates over U = R2 × (−π/2, π/2)2

z1 = x+ γ(θ, φ)

z2 = x+ (lθ/ cos θ, lφ/cosφ)T(4.58)

with γ(θ, φ) = (γ0(θ), γ0(φ))T and γ0 defined as

γ0(θ) =

∫ θ

0

l

cos sds = l log

(

1 + tan(θ/2)

1− tan(θ/2)

)

, θ ∈ (−π/2, π/2)

transforms the dynamics of the 3D Cart-Pole System into a cascade nonlinear systemin feedforward form

z1 = z2z2 = ψ(qs) + pTs Π(qs)ps + ϕ(qs)uqs = psps = u

(4.59)

whereψ(qs) = −m−1

sx (qs)∇qsV (qs)

has a nonsingular Jacobian at qs = 0, Π(qs) is a cubic matrix, and ϕ(qs) = −µes(qs)satisfies ϕ(0) = 02×2 and σmax(ϕ(qs)) = O(|qs|2). In addition, the origin for (4.59)can be globally asymptotically and locally exponentially stabilized using a state feedbackin the form of nested saturations.

Proof. We use the method of momentum decomposition to reduce the dynamicsof the 3D Cart-Pole system. This system has equal number of external and shapevariables. Thus, following the notation of theorem 4.5.3, dim(Qs2) = 0 and theexternal and shape configuration vectors of the 3D Cart-Pole system are qx = (x1, x2)and qs = qs1 = (θ, φ), respectively. From the kinetic energy of the 3D Cart-Polesystem in (4.34), the block matrices msx(qs) and mss(qs) for this system can beidentified as

msx(qs) =

[

ml cos θ −ml sin θ sinφ0 ml cos θ cosφ

]

, mss(qs) =

[

ml2 00 ml2 cos2 θ

]

In example 4.4.2, it was shown that the normalized momentum

πs = qx + µs(qs)qs

97

is non-integrable over U = R2 × (−π/2, π/2)2 where

µs(qs) = m−1sx (qs)mss(qs) =

l

cos θ

l sin θ sinφ

cosφ

0l cos θ

cosφ

Hence, after choosing the lock configuration (θ, φ) = (0, 0) and setting φ = 0 andθ = 0, respectively, in the first and second columns of µs, the normalized lockedshape inertia and error inertia for the system can be obtained as

µls(θ, φ) =

l

cos θ0

0l

cosφ

, µes(θ, φ) =

0l sin θ sinφ

cosφ

0l(cos θ − 1)

cosφ

Take

γ0(z) =

∫ z

0

l

cos sds , z ∈ (π/2, π/2)

Then, γ(θ, φ) = (γ0(θ), γ0(φ))T satisfies ∇qsγ(qs) = µls(qs). The change of coordinates

in (4.50) can be explicitly calculated as

qr = x+ γ(θ, φ)pr = msx(θ, φ)(x+ µls(θ, φ)qs)

(4.60)

After simplification, we obtain

q1r = x1 + γ0(θ)

p1r = ml cos(θ)x1 −ml sin(θ) sin(φ)x2 +ml2θ −ml2 sin(θ) tan(φ)φq2r = x2 + γ0(φ)

p2r = ml cos(θ) cos(φ)x2 +ml2 cos(θ)φ

(4.61)

where qr = (q1r , q2r)T is the configuration vector of the reduced system and pr =

(p1r, p2r)T is its conjugate momentum. The nonlinear change of coordinates in (4.61)

transforms the dynamics of the 3D Cart-Pole system into

qr = m−1r (θ, φ)pr

pr = gr(θ, φ) + Σ(qs, pr, ps) + δ

θ = ω1

ω1 = u1φ = ω2

ω2 = u2

(4.62)

where mr = msx and

gr(θ, φ) = −∇qsV (θ, φ) =

[

mgl sin θ cosφmgl cos θ sinφ

]

98

Notice that mxx is constant for this system and Σ is linear in qx, thus

Σ(qs, pr, ps) = ∇qsK = ∇qsqTs msx(qs)qx+∇qs1

2qTs mss(qs)qs

Following the line of proposition 4.4.1, apply a second change of coordinates as thefollowing

z1 = qrz2 = m−1

r (θ, φ)pr(4.63)

We prove that the last change of coordinates transforms the dynamics of the 3DCart-Pole system into the feedforward normal form in (4.59). First, observe that

z1 = x+ γ(θ, φ)z2 = x+ µls(θ, φ)qs

(4.64)

and therefore z1 = z2. In addition, we have

∂L∂qs

= msx(qs)(qx + µs(qs)qs)

= msx(qs)(qx + µls(qs)qs + µes(qs)qs)

= msx(qs)z2 + pe

wherepe = msx(qs)µ

es(qs)qs

But from Euler-Lagrange equation

d

dt

∂L∂qs

=∂L∂qs

= gr(qs) + Σ

and we obtain

msx(qs)z2 = gr(qs) + Σ− msxz2 − pe= gr(qs) +∇qs

1

2qTs mss(qs)qs

+ ∇qsqTs msx(qs)qx − msxqx − msxµls(qs)qs − pe

But similar to condition ii) of proposition 4.4.1, msx(qs) block of the inertia matrixof the 3D Cart-Pole system has differentially symmetric rows that implies

∇qsqTs msx(qs)qx = msxqx

Therefore, we obtain the following equations for the z-subsystem

z1 = z2z2 = ψ(qs) + Σ(qs, ps) + ϕ(qs)u

(4.65)

99

where

ψ(qs) = m−1sx (qs)gr(qs)

Σ(qs, ps) = m−1sx (qs)[∇qs

1

2qTs mss(qs)qs − msxµ

ls(qs)qs −

d

dtmsx(qs)µ

es(qs)qs]

ϕ(qs) = −µes(qs)

and Σ(qs, ps) = pTs Π(qs)ps is quadratic in ps. Notice that from the explicit expressionof µes(qs), ϕ(0) = 02×2 and σmax(ϕ(qs)) = O(|(θ, φ)|2). The stabilization result usingnested saturations [102] follows from the fact that Σ is quadratic and [∇qsψ(qs)]qs=0 =gI2×2 is non-singular (where g is the gravity constant).

4.6 Classification of Underactuated Systems

In this section, we classify underactuated systems based on possession or lack of someor all of the following four properties:

(a) Actuated shape variables

(b) Non-interacting inputs (i.e. lack of input coupling)

(c) Integrable normalized momentums

(d) Extra required conditions (as described in the definition of each class in thefollowing).

This leads at most to sixteen possible classes. However, due to redundancy and thefact that some of these cases never have appeared in applications, we only focus ona subset of these sixteen classes that frequently appear in control of underactuatedmechanical systems. This subset includes eight different classes.

Note: In case an underactuated system is flat (i.e. has constant inertia matrix), wecall the variables that appear in the force matrix F (q) shape variables.

Table 4.1 provides definition and examples of different classes of underactuatedsystems. The following remarks have to be taken into account regarding Table 4.1:

– a “Yes” or a “No” means the corresponding property holds or does not hold,respectively.

– “Q”, “L”, and “LQ” in front of “Nontriangular” refer to “Quadratic”, “Linear”,and “Linear-Quadratic” structure of the normal form with respect to ps (i.e.shape velocity).

– a “Yes” for property (d) means that certain extra conditions are required thatcan be found in the description of the corresponding class.

100

– a controlled VTOL aircraft means the dynamics of the VTOL in closed loopwith a feedback u1 = u1(x, y, x, y) and a single remaining control u2.

Table 4.1: Classification of Underactuated SystemsClass (a) (b) (c) (d) Normal Form Control Design example(s)

I Yes Yes Yes No Strict Feedback BacksteppingAcrobot (2D,1C), TORA,Inertia-Wheel Pendulum

IIa No Yes Yes No Nontriangular Q Fixed-Point LawRotating PendulumPendubot, Beam-n-Ball

IIb No Yes Yes Yes Nontriangular L Fixed-Point Law ≈Flexible-link RobotsIII No Yes Yes Yes Feedforward Nested Saturations 2D Cart-Pole (2D,1C)

IVa Yes No Yes No Nontriangular Q Fixed-Point Law Controlled VTOL (3D,1C)

IVb Yes No Yes Yes Nontriangular Q Fixed-Point Law Controlled VTOL (3D,1C)

V Yes No Yes Yes Strict Feedback BacksteppingVTOL aircraft (3D,2C))≈Aircraft (6D,4C)≈Helicopter (6D,4C)

VIa Yes Yes No No Nontriangular L Fixed-Point Law Three-link arm (3D,2C)

VIb Yes Yes No Yes Feedback+∆ Backstepping Three-link arm (3D,2C)

VIIa No Yes No No Nontriangular LQ Fixed-Point Law Three-link arm (3D,2C)

VIIb No Yes No Yes Nontriangular+∆ Fixed-Point Law Three-link arm (3D,2C)

VIII No Yes No Yes Feedforward Nested Saturations 3D Cart-Pole (4D,2C)

In the following, we rewrite the normal form corresponding to each class of under-actuated systems in table 4.1 by partitioning the state vector as col(z, ξ) ∈ Rn ×Rr.Throughout this section, (z1, z2, ξ1, ξ2) denotes (qr, pr, qs, ps) unless otherwise is stated.Furthermore, the term f0 is due to the potential energy of the system. Here is thedescription of each class:

• Class-I: underactuated systems with actuated shape variables, integrable mo-mentums, and non-interacting inputs. From equation (4.6), the normal form ofClass-I underactuated systems can be rewritten as the following

z1 = N(ξ1)z2z2 = f0(z1, ξ1)

ξ1 = ξ2ξ2 = u

(4.66)

where N(ξ1) is a symmetric and positive definite matrix. System (4.66) is instrict feedback form with a vector field non-affine in ξ1.

• Class-IIa: underactuated systems with unactuated shape variables, integrablenormalized momentums, non-interacting inputs, and equal number of exter-nal and shape variables. From equation (4.14), the normal form of Class-IIa

101

underactuated systems can be rewritten as the following

z1 = z2z2 = f0(z1, ξ1) + zT2 g11(ξ1)z2 + zT2 g12(ξ1)ξ2 + ξT2 g22(ξ1)ξ2ξ1 = ξ2ξ2 = u

(4.67)

where z2 = m−1r (qs) and gij’s are cubic matrices with the property g12 = gT21.

System (4.67) is in nontriangular quadratic form with a vector field non-affinein (ξ1, ξ2).

• Class-IIb: a subclass of Class-IIa underactuated systems with unequal numberof external and shape variables. From equation (4.21), the normal form of Class-IIb underactuated systems can be rewritten as the following

z1 = z2z2 = f0(z1) + fd(z1, z2) + zT2 g11(ξ1)z2 + zT2 g12(ξ1)ξ2 + ξT2 g22(ξ1)ξ2 + g3(z1)ξ3ξ1 = ξ2ξ2 = ξ3ξ3 = u

(4.68)where z, ξ are defined in (4.20) and Fd is a force due to dissipation satisfyingzT2 fd(z1, z2) < 0,∀z1,∀z2 6= 0. The gij’s are cubic matrices with the propertyg12 = gT21. System (4.68) is in nontriangular form with a vector field non-affinein (ξ1, ξ2) and affine in ξ3.

• Class-III: a subclass of Class-IIa underactuated systems satisfying conditionsof proposition 4.2.1. From equation (4.17) the normal form of Class-III under-actuated systems can be rewritten as

z1 = z2z2 = f0(ξ1) + ξT2 g(ξ1)ξ2ξ1 = ξ2ξ2 = u

(4.69)

where z2 = m−1r (qs)pr and g(ξ) is a cubic matrix. System (4.69) is in strict

feedforward form with a vector field non-affine in (ξ1, ξ2).

• Class-IVa: underactuated systems with actuated shape variables, integrablemomentums, and input coupling. From equation (4.24), taking z2 = m−1

r (qs)pr,the normal form for Class-IVa underactuated systems can be written as

z1 = z2z2 = f0(z1, ξ1) + zT2 g11(ξ1)z2 + zT2 g12(ξ1)ξ2 + ξT2 g22(ξ1)ξ2ξ1 = ξ2ξ2 = u

(4.70)

102

The gij’s are cubic matrices with the property g12 = gT21. System (4.70) is innontriangular quadratic form with a vector field non-affine in (ξ1, ξ2). Noticethat normal forms (4.70) and (4.67) represent the same class of cascade nonlinearsystems, but two different classes of underactuated systems.

• Class-IVb: a subclass of Class-IVa underactuated systems with constant mxx

and V (q) = V (qs). From equation (4.25), the normal form for Class-IVb un-deractuated systems can be written as

z1 = z2z2 = f0(ξ1) + ξT2 g(ξ1)ξ2ξ1 = ξ2ξ2 = u

(4.71)

where z2 = m−1r pr and g(ξ1) is a cubic matrix. System (4.71) is in strict feed-

forward form with a vector field non-affine in (ξ1, ξ2).

• Class-Va: underactuated systems with actuated shape variables, integrablemomentums, input coupling, and an extra control in the dynamics of the exter-nal variables with the property that their mxx is constant and V (q) = V (qx).From equation (4.28), the normal form for Class-Va underactuated systems canbe written as

z1 = z2z2 = f0(z1, ξ1) + g0(ξ1)T + ξT2 g(ξ1)ξ2ξ1 = ξ2ξ2 = u

(4.72)

where z2 = m−1r pr, g(ξ1) is a cubic matrix, g0(ξ1) : Rp → Rn is defined by

g0(ξ1) :=1

mxx

R(ξ1)~e, (4.73)

~e is a fixed unit vector in Rn and R(ξ1) ∈ SO(n) is a rotation matrix in Rn. Thecontrol T ∈ R (or thrust) is the magnitude of a force applied to the dynamicsof the vector of external variables, qx ∈ Rn, in the direction ~e. Assuming thatT = K(z, ξ1), system (4.72) is in nontriangular quadratic normal form with avector field non-affine in (ξ1, ξ2).

• Class-Vb: a subclass of Class-Va underactuated systems with the propertythat there exists a positive definite and symmetric matrix Q(ξ1) ∈ Rp×p suchthat

ξT2 g(ξ1)ξ2 = [ξT2 Q(ξ1)ξ2]g0(ξ1) (4.74)

103

and V (qx) is a linear function. From theorem 4.3.2 and equation (4.72), thenormal form for Class-Vb underactuated systems can be given as

z1 = z2z2 = f0 + g0(ξ1)T

ξ1 = ξ2ξ2 = u

(4.75)

where f0 ∈ Rn is a constant vector and T in a new control defined as

T = T + ξT2 Q(ξ1)ξ2

Assuming that the state feedback T = K(z, ξ1) globally asymptotically stabi-lizes the equilibrium z = z0 for the z-subsystem of (4.75), the dynamics of (4.75)is in strict feedback form with a vector field non-affine in ξ1.

• Class VIa: underactuated systems with actuated shape variables, non-integrablenormalized momentums, and non-interacting inputs. From equation (4.31), thenormal form for Class-VIa underactuated systems can be expressed as

z1 = N(ξ1)z2 + g0(ξ1)ξ2z2 = f0(z1, ξ1)

ξ1 = ξ2ξ2 = u

(4.76)

System (4.76) is in nontriangular linear normal form with a vector field affinein ξ2. Note that @h = h(ξ1) : Rp → Rn : Dh(ξ1) = g0(ξ1).

• Class VIb: a subclass of Class-VIa with the property that there exists a parti-tion of qs = col(qs1 , qs2) with an n-dimensional vector qs1 such that the perturba-tion δ in (4.47) vanishes identically at (qs2 , ps2 , u2) = 0 where qs2 = ps2 , ps2 = u2.Let us partition ξi and u as col(ηi, µi), i = 1, 2 and col(u1, u2), respectively.Then, from equation (4.47), the normal form for Class-VIb underactuated sys-tems can be obtained as

z1 = N(η1)z2 + ϕ1(z2, η1, µ1)z2 = f0(z1, η1) + ϕ2(z1, η, µ, u)η1 = η2η2 = u1µ1 = µ2µ2 = u2

(4.77)

where N(η1) = m−1r (qs1 , 0) and the perturbation terms ϕ1, ϕ2 vanish identically

at (µ1, µ2, u2) = 0. The (z, η)-subsystem of (4.77) is a perturbation of a strictfeedback form.

104

• Class VIIa: underactuated systems with unactuated shape variables, non-integrable normalized momentums, and non-interacting inputs. From equation(4.32), the normal form for Class-VIIa underactuated systems can be given as

z1 = z2 + g0(ξ1)ξ2z2 = f0(z1, ξ1) + zT2 g11(ξ1)z2 + zT2 g12(ξ1)ξ2 + ξT2 g22(ξ1)ξ2ξ1 = ξ2ξ1 = u

(4.78)

where z2 = m−1r (qs)pr and the gij’s are cubic matrices satisfying g12 = gT21.

System (4.78) is in nontriangular linear-quadratic normal form with a vectorfield non-affine in (ξ1, ξ2).

• Class VIIb: underactuated systems with partially unactuated shape variables,non-integrable normalized momentums, and non-interacting inputs. In addi-tion, there exists a partition of qs = col(qs1 , qs2) with an n-dimensional vectorqs1 such that the perturbation δ in (4.51) vanishes identically at (qs2 , ps2 , u2) = 0.Following the notation used in the definition of Class-VIb, based on equation(4.51), the normal form for Class-VIIb underactuated systems can be writtenas

z1 = z2 + ϕ1(z2, η1, µ1)z2 = f0(z1, η1) + zT2 g11(η1)z2 + zT2 g12(η1)η2 + ηT2 g22(η1)η2 + ϕ2(z, η, µ, u)η1 = η2η2 = u1µ1 = µ2µ2 = u2

(4.79)where the gij’s are cubic matrices and ϕ1, ϕ2 vanish identically at (µ1, µ2, u2) =0. The (z, η)-subsystem of (4.79) is a perturbation of a nontriangular quadraticnormal form (see the normal form of Class-IIa).

• Class VIII: a subclass of Class-VIIa underactuated systems satisfying the con-ditions of proposition 4.4.1. From equation (4.33), the normal form for Class-VIII underactuated systems can be expressed as

z1 = z2 + ϕ1(ξ1, ξ2)z2 = f0(ξ1) + ξT2 g(ξ1)ξ2ξ1 = ξ2ξ1 = u

(4.80)

wherez1 = qr + µ(0)qs, z2 = m−1

r (qs)pr

and ϕ1 ∼ O(|(ξ1, ξ2)|2) as defined in proposition (4.4.1). System (4.80) is instrict feedforward form with a vector field non-affine in (ξ1, ξ2).

Let us make the following definition.

105

Definition 4.6.1. (core reduced system) Define the core reduced system as the dy-namics of the z-subsystem without the following types of terms:

i) terms containing ξ2,

ii) terms containing z2 in the equation of z2,

iii) high-order perturbation terms ϕi.

The following corollary creates a connection in terms of the dynamics of the z-subsystem among all the above classes of underactuated system.

Corollary 4.6.1. According to definition 4.6.1, for all the aforementioned classes ofunderactuated mechanical systems, the core reduced system is in the normal form

z1 = N(v)z2z2 = f(z1, v)

(4.81)

where v is the control input for (4.81) and N(v) is a positive definite symmetricmatrix. (Note that for several classes N(v) = I).

Global asymptotic stabilization of nonlinear systems in normal form (4.81) isaddressed in theorem 4.7.1 for the case where z1, z2, v ∈ Rn.

Remark 4.6.1. Based on proposition 5.11.4. The normal form of the first-level ap-proximate model of a helicopter is in the form

z = f(z) + g1(ξ1)T + g2(ξ1, ξ2, ε) + g3(ξ1, ε)u

ξ1 = ξ2ξ2 = u

(4.82)

where z = col(x, v) ∈ R6, ξ1 = col(φ, θ, ψ) (i.e. three Euler angles), ε = (ε1, ε2, ε3)T ∈

R3, and g2, g3 are linear with respect to the matrix

E(ε) =

0 ε2 0ε1 0 ε30 0 0

; ε1 > 0, ε2 < 0, , ε3 > 0

Thus, g2, g3 vanish identically at ε = (0, 0, 0). The vector field of (4.82) is quadraticin ξ2 and affine in control u. Under the assumption that ε = (0, 0, 0), (4.82) reducesto the normal form of a Class-VIb underactuated system in (4.75).

4.7 Nonlinear Control of the Core Reduced Sys-

tem

The fundamental property of the three basic cascade normal forms for underactu-ated systems is that the unforced and unperturbed reduced system for all classes of

106

underactuated systems in section 4.6 is the same and is given by

qr = m−1r (qs)pr

pr = gr(qr, qs)(4.83)

We refer to (4.83) as the Core Reduced System (also, see definition 4.6.1). This sectionis devoted to stabilization of the core reduced system for all underactuated systemsdiscussed in this chapter. The main property of the core reduced system is that itis a simple Lagrangian system without any external force. Thus, the shape vector qsplays the role of the control input for the core reduced system. Here is our main resulton global asymptotic stabilization of the core reduced system (4.83). (For simplicityof the notation, in the following theorem the subscript “r” is dropped).

Theorem 4.7.1. Consider the following nonlinear system non-affine in control

q = N(u)pp = g(q, u)

(4.84)

where q, p, u ∈ Rn, g(q, u) : Rn × Rn → Rn is a smooth function with g(0, 0) = 0,N(u) is an invertible matrix for all u, and M(u) = N−1(u) is a positive definite andsymmetric inertia matrix. Suppose there exists an isolated root u = α(q) of g(q, u) = 0with the property α(0) = 0 such that

det

(

∂g

∂u(q, α(q))

)

6= 0

Setψ(q, v) := [M(u)g(q, u)]u=α(q)+v

Then, for all q ∈ Rn, w = ψ(q, v) is a local diffeomorphism around a neighborhoodof v = 0. Assume there exists an open ball Br(0) around w = 0 and a functionβ : Rn × Rn → Rn such that

ψ(q, β(q, w)) = w, ∀w ∈ Br(0) ⊂ Rn

uniformly in q. Let ~σ(x) : Rn → Rn denote the vector sigmoidal function

~σ(x) =x

(1 + ‖x‖2) 12

Then, for all c1, c2 ∈ (0, r/2], the static state feedback u = K(q, p) defined as thefollowing

u = K(q, p) := α(q) + β(q, w),w = Kb(q, p) := −c1~σ(q)− c2~σ(p); (4.85)

globally asymptotically stabilizes the origin (q, p) = (0, 0) for the nonlinear system in(4.84).

107

Proof. Calculating ∇vψ(q, v) at v = 0 (or u = α(q)), we get

∇vψ(q, v) =M(α(q)) · [∇vg(q, α(q) + v)]v=0

By assumption, both M(α(q)) and ∇ug(q, α(q)) are invertible matrices. This meansthat ∇vψ(q, v) is invertible at v = 0 and thus ψ(q, v) is a local diffeomorphism for allq. Consider the invertible change of control w = ψ(q, v) with an inverse v = β(q, w)defined over Rn ×Br(0) so that ψ(q, β(q, w)) = w. Then, the dynamics of (4.84) canbe written as

q = N(u)pp = N(u)w

(4.86)

with u = α(q) + β(q, w). Fix c1, c2 ∈ (0, r/2]. We have

‖w‖ = ‖Kb(q, p)‖ ≤ c1‖~σ(q)‖+ c2‖~σ(p)‖ < c1 + c2 ≤ r

and for all (q, p), Kb(q, p) ∈ Br(0). Applying the state feedback w = Kb(q, p) in thequestion, the closed-loop system in (4.86) takes the following form

q = N(u)pp = −c1N(u)~σ(q)− c2N(u)~σ(p)

(4.87)

with u = K(q, p) (defined in (4.85)). Consider the following positive definite Lya-punov function candidate

H(q, p) = c1φ(q) +1

2‖p‖2

where the positive semidefinite potential function φ(q) : Rn → R≥0 is defined as

φ(q) = (1 + ‖q‖2) 12 − 1

and satisfies the property (∇φ(q))T = ~σ(q). Calculating H along the solutions of theclosed-loop system (4.87), we obtain

H =∂H(q, p)

∂q· q + ∂H(q, p)

∂p· p

= c1~σ(q)TN(u)p− c1pTN(u)~σ(q)− c2pTN(u)~σ(p), (NT = N)

= −c2pTN(u)~σ(p) < 0, ∀p 6= 0

The last inequality is due to positive definiteness of the term

pTN(u)~σ(p) =pTN(u)p

(1 + ‖p‖2) 12

≥ 0

for p 6= 0 and all u, particularly, u = K(q, p). Based on LaSalle’s invariance principle[40], all the solutions of the closed-loop system converge to the largest invariant setΩ in (q, p) : H = 0 = (q, 0). But for system (4.87), Ω = (0, 0). Therefore,

108

the origin (q, p) = (0, 0) is globally asymptotically stable for (4.87) (and (4.84)).Furthermore, H(q, p) is a smooth positive definite and proper Lyapunov function forthe system.

Definition 4.7.1. (bi-saturated PD controller) We call w = Kb(q, p) in equation(4.85), a bi-saturated PD controller.

Corollary 4.7.1. Assume all the conditions in theorem 4.7.1 hold and let

w = kb(q, p) := −c0~σ(q + p), c0 ∈ (0, r]

Then, (q, p) = (0, 0) is GAS for (4.84).

Proof. We prove that the following smooth and positive definite function

H(q, p) = c0φ(q + p) +1

2‖p‖2

is a valid Lyapunov function for the closed-loop system

q = N(u)pp = −c0N(u)~σ(q + p)

(4.88)

We have

H = c0~σ(q + p)T (q + p)− c0pTN(u)~σ(q + p)

= −c0‖~σ(q + p)‖2

Thus, all the solutions converge to the largest invariant set Ω in (q, p) : q + p = 0.By inspection, Ω = (0, 0) and the origin is GAS for (4.88) and thus (4.84).

The following result shows that under further structure in the dynamics of thereduced system it is possible to find a static state feedback in closed-form that globallyasymptotically stabilizes the origin for the reduced system.

Theorem 4.7.2. Consider the reduced system in (4.83) and assume that dim(Qr) =dim(Qs) = d and the reduced inertia matrix is diagonal, i.e.

mr(qs) = diag(mr1(qs), . . . ,mrd(qs))

In addition, assume that (4.83) is in the following form

qjr = m−1rj (q

js)p

jr

pjr = grj(qjr , q

js)

(4.89)

for j = 1, . . . , d. Let qjs = αj(qjr) be an isolate root of grj(qr, qs) for j = 1, . . . , d, i.e.

grj(qr, αj(qjr)) = 0,

[

∂grj

∂qjs

]

qjs=αj(qjr)

> 0

109

Then, the following state feedback

qs = α(qr)− ~σ(qr + pr)

globally asymptotically stabilizes the equilibrium point (qr, pr) = (0, 0) of (4.89) where

α(qr) = (α1(q1r), . . . , αd(q

dr ))

T , ~σ(z) = (σ1(z1), . . . , σn(zi))

and all σi’s are one dimensional sigmoidal functions.

The proof of this theorem relays on the following theorem for the one-dimensionalcase.

Theorem 4.7.3. Consider the following system

q = p,p = f(p) + g(q, u),

(4.90)

where f and g are C1 functions with f(0) = 0 and g(0, 0) = 0, f is decreasing(f ′ ≤ 0), and q, p, u ∈ R. Suppose that zero is not a critical value for g(q, u) and

∂g(q, u)

∂u6= 0

on the manifoldM = Ker(g) = (q, u) ∈ R2 : g(q, u) = 0

Then, g(q, u) has an isolated root α(q) such that

g(q, α(q)) = 0

and there exists a continuously differentiable state feedback law in the following form

u = α(q)− σ(q + p), (4.91)

that globally asymptotically stabilizes the origin for (4.90) (σ is a sigmoidal function).

Proof. Because zero is not a critical value for g, M = g−1(0) = Ker(g) is a onedimensional differentiable manifold in R2. In other words, M is a two-dimensionalcurve in plane that divides R2 into two connected regions:

A+ = (q, u) ∈ R2 : u > α(q)

andA− = (q, u) ∈ R2 : u < α(q).

But gu(q, u) 6= 0 on M , based on implicit mapping theorem it follows that g(q, u) = 0has an isolated root u = α(q) that is a C1 function and M can be parameterized as

M = (q, α(q)) : q ∈ R.

110

Also, on M either gu(q, u) > 0 or gu(q, u) < 0 (otherwise, at some point on Mgu(q, u) = 0). Without loss of generality assume

∂g(q, u)

∂u|u = α(q) > 0

by continuity of gu(q, u) there exists a tubular neighborhood of M

Nδ(M) = (q, u) ∈ R2 : |u− α(q)| < δ, δ > 0

such that gu(q, u) > 0 on Nδ(M). Suppose |σ(s)| < δ,∀s where σ is a sigmoidalfunction and δ > 0. Then, given

u = k(q, p) = α(q)− σ(q + p),

(q, k(q, p)) ∈ Nδ(M),∀q, p and therefore

gu(q, k(q, p)) > 0

for all q, p. Now, consider the closed loop system (4.90) with control input u = k(q, p)as the following

q = p,p = f(p) + g(q, α(q)− σ(q + p)),

(4.92)

we prove that the origin (q, p) = (0, 0) is globally asymptotically stable for (4.92).Define

h(q, p) = −f(p)− g(q, α(q)− σ(q + p)),

and

H(q) =

∫ q

0

h(s, 0)ds

Noting that sh(s, 0) > 0,∀s 6= 0, H(q) is positive semidefinite with H(0) = 0 andH(q) > 0,∀q 6= 0. Consider the following Lyapunov function candidate for (4.92)

V (q, p) = H(q) +1

2p2,

we haveV = p(h(q, 0)− h(q, p)),

buthp(q, p) = −f ′(p) + σ′(q + p) · gu(q, k(q, p)) > 0,

therefore, V ≤ 0,∀q, p. But in V (q, p) = 0 or (q, p) : p = 0 the only invariant setis the origin (q, p) = (0, 0). From LaSalle’s invariance principle, it follows that theorigin is globally asymptotically stable for (4.92) and the result follows.

111

Chapter 5

Applications to Robotics andAerospace Vehicles

In this chapter, detailed control design procedures are provided for several challeng-ing examples of underactuated mechanical systems with kinetic symmetry. In eachcase, distinctive comparisons are made between our approach and available controlmethods (if any) that clarify the advantages of the reduction procedures introducedin chapter 4.

All the examples presented in this section are physically meaningful systems. Theexamples are chosen either based on the importance of their applications in roboticsand aerospace, or due to their challenging nature and significance as benchmarksystems in nonlinear control theory. The systems of the former class are namely aflexible-link robot, a multi-link underactuated arm, the VTOL aircraft, and an au-tonomous helicopter. The latter category include the Acrobot, the Inertia-WheelPendulum, the TORA system, the Pendubot, the Cart-Pole system, the RotatingPendulum, and the Beam-and-Ball system. The aforementioned examples of under-actuated systems include both high-order and low-order systems. In this section, theexamples appear from low-order systems to high-order systems.

Our control design approach for each example consists of three steps:

i) Reduction to a cascade nonlinear system with structural properties.

ii) Control of the low-order nonlinear part of the cascade system.

iii) Reconstruction of the controller for the overall system using the controller ofthe nonlinear part obtained in step ii).

For nonlinear systems in strict feedback form, step iii) is known as the standardbackstepping procedure. The third step is a rather routine procedure and the maindifficulty in control design for an underactuated system is passing through steps i)and ii). These three steps are precisely demonstrated for each example.

Some of our main results include global asymptotic stabilization of the Acrobot,the Cart-Pole system, the Inertia-Wheel Pendulum, a three-link planar pendulumwith actuated shape variables, and the VTOL aircraft. In addition, (almost) global

112

exponential output tracking/stabilization of feasible trajectories of an autonomoushelicopter is achieved. The closest available result to this work is global asymptoticoutput tracking in [31].

Among the aforementioned nonlinear control problems, some of them are known tobe open in the literature. As an example, global stabilization of the Acrobot (Figure5-1) using a smooth static state feedback in analytically explicit form has been anopen problem in nonlinear control for near a decade. One of our contributions isto address this problem. More precisely, we present a nonlinear state feedback inclosed-form that globally asymptotically stabilizes any arbitrary equilibrium point ofthe Acrobot including the upright position. The main available result on asymptoticstabilization of the Acrobot to the upright position can be found in [13, 86]. Themethod in [13] consists of two stages: i) a slow swing-up of the Acrobot using apassivity-based approach that brings it close to the upright position, ii) switchingto a balancing linear controller that locally asymptotically stabilizes the pendulumaround its upright equilibrium point. The main drawback of this two-stage method isthe poor performance of both swing-up and balancing controllers in terms of time (i.e.both controllers are slow). This is due to the fact that a gain-scheduled controllers hasto keep the state of the system close to a set of equilibrium points. However, since thevector field of a closed-loop system vanishes at the set of equilibrium points, the statemoves slowly close to this set of equilibrium points. This is the fundamental weaknessof the gain scheduling method. The nonlinear state feedback that we use does notrequire a swing-up stage and has a large region of attraction (i.e. R4), therefore thenonlinear controller does not need to keep the state close to the set of equilibriumpoints. As a result, from the simulation results, the speed of convergence to thestate to zero, is significantly better than a switching-based controller that uses a slowswing-up and balancing. Traditionally, a rather similar two-stage switching-basedapproach has been applied to control of the Pendubot [92, 90], the Cart-Pole system[94, 28], and the Rotating Pendulum [112, 6]. From now on, we avoid repeating theweaknesses of the passivity-based swing-up methods and the gain-scheduling approachin nonlinear control, for each of these examples throughout this section.

Another contribution is that we introduce aggressive swing-up control designmethods for benchmark-type underactuated systems that involve inverted-pendulums.Namely, the Cart-Pole system, and the Rotating Pendulum. Formally, a swing-upproblem is equivalent to stabilization of a system to a relative equilibrium point.By aggressive swing-up control, we mean (almost) exponential stabilization of aninverted-pendulum system to an equilibrium manifold corresponding to the uprightposition of the pendulum.

5.1 The Acrobot

The Acrobot is a two-link revolute planar robot with one actuator at the elbow, asshown in Figure 5-1. In this section, we provide a detailed control design procedurefor global asymptotic stabilization of the Acrobot to any of its arbitrary equilibriumpoints, particularly, the upright equilibrium point. The inertia matrix for the Acrobot

113

q

q

l 1

L1

m I

m I2 2

1 1

1

2

x

y

Figure 5-1: The Acrobot

is given by (see section A.3)

m11(q2) = a+ b cos(q2)m12(q2) = c+ (b/2) cos(q2)m22(q2) = c

witha = m1l

21 +m2(L

21 + l22) + I1 + I2

b = 2m2L1l2c = m2l

22 + I2

where qi, mi, Li, li, and Ii denote the angle, the mass, the length, the length of thecenter of mass, and the inertia of the ith link, respectively. Clearly, q2 is the shapevariable of the Acrobot. Since this shape variable is actuated the Acrobot is a Class-Iunderactuated system. Also, the potential energy for the Acrobot is as the following

V (q1, q2) = (m1l1 +m2L1)g0 cos(q1) +m2l2g0 cos(q1 + q2)

which depends on both q1, q2. Thus, the Acrobot does not possess symmetry in theclassical sense, but it possesses kinetic symmetry with respect to q1. This showsthat kinetic symmetry is a less restrictive property than classical symmetry as theinvariance of the Lagrangian under symmetry group actions.

Based on theorem 4.2.1 (or proposition 3.9.1), the following global change ofcoordinates

qr = q1 + γ(q2)pr = m11(q2)p1 +m12(q2)p2

transforms the dynamics of the Acrobot to a cascade nonlinear system in strict feed-back form

qr = pr/m11(q2)pr = gr(qr, q2)q2 = p2p2 = u

(5.1)

114

where gr(qr, q2) = −∇q1V (q1, q2)|q1=qr−γ(q2) is given by

gr(qr, q2) = (m1l1 +m2L1)g0 sin(qr − γ(q2)) +m2l2g0 sin(qr − γ(q2) + q2) (5.2)

and the function γ(q2) can be explicitly calculated as the following

γ(q2) =

∫ q2

0

m12(s)

m11(s)ds =

q22

+(2c− a)√a2 − b2

arctan(

√

a− ba+ b

tan(q22)) (5.3)

for q2 ∈ [−π, π]. The function γ(q2) is plotted in Figure 5-2. Observe that γ(q2) isa nonlinear function for (2c − a) 6= 0. The special case of 2c − a = 0 that gives alinear change of coordinates had been previously known. Stabilization of the Acrobot

−8 −6 −4 −2 0 2 4 6 8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

rad

gam

ma

Figure 5-2: The function γ(q2) for a = 20/3, b = 4, c = 4/3.

reduces to stabilization of the following second-order nonlinear system non-affine incontrol

qr = m−111 (v)pr

pr = gr(qr, v)(5.4)

with control input v = q2. This stabilization problem is extremely complicated evenfor a second-order system due to the highly nonlinear way that the control v appearsin the dynamics of (5.4). Here is our main result for the Acrobot:

Proposition 5.1.1. Assume the parameters of the Acrobot satisfy either of the fol-lowing conditions

i) A < B.

ii) B < ρA, a− 2c > 0, and c− b/2 ≥ 0.

115

where

A = m1l1 +m2L1

B = m2l2

ρ =ψ(π)

1− ψ(π) <1

2

ψ(q2) =m12(q2)

m11(q2)

Then, there exists a state feedback in the form

v = Kr(qr, pr) = α(qr)− σ(qr + pr)

that globally asymptotically stabilizes the origin for the reduced nonlinear system ofthe Acrobot in (5.4) where α(qr) is a smooth function that satisfies

gr(qr, α(qr)) = 0

and σ(·) is a scalar sigmoidal function. In addition, there exists a state feedback inexplicit form

u = K(qr, pr, qs, ps)

that globally asymptotically stabilizes the origin (qr, pr, qs, ps) = 0 for the compositedynamics of the Acrobot in (5.1).

Proof. Based on theorems 4.7.2 and 4.7.3, if gr(qr, v) = 0 has an isolated rootv = α(qr) that satisfies

det

(

∂gr(qr, v)

∂v

)∣

∣

∣

∣

v=α(qr)

6= 0

then, there exists a static state feedback in the explicit form

v = Kr(qr, pr) = α(qr)± σ(qr + pr)

which globally asymptotically stabilizes any equilibrium point (qr, 0) of the corereduced system (5.4) (the sign ± in state feedback v = Kr depends on whether∇vgr(qr, α(qr)) < 0, or > 0, respectively). In order to find the conditions under whichthe state feedback Kr exists, it suffices to determine a set of sufficient conditions thatguarantees the following equations have no solutions in v

gr(qr, v) = 0∇vgr(qr, v) = 0

(5.5)

116

This set of nonlinear equations can be rewritten as

A sin(qr − γ(v)) +B sin(qr − γ(v) + v) = 0

Am12(v)

m11(v)cos(qr − γ(v)) +B(

m12(v)

m11(v)− 1) cos(qr − γ(v) + v) = 0

(5.6)

whereA = (m1l1 +m2L1), B = m2l2

Setting q1 = qr − γ(v), from the property sin2(q1 + v) + cos2(q1 + v) = 1, it followsthat

(

A sin(q1)

B

)2

+

(

Am12 cos(q1)

B(m11 −m12)

)2

= 1

Keep in mind that a > b and c > b/2 imply m11(q2),m12(q2) > 0 for all q2. Solvingfor sin2(q1) from the last equation, we obtain

sin2(q1) =B2(m11 −m12)

2 − A2m212

A2(m11 −m12)2 − A2m212

(5.7)

Thus, if A < B the right hand side of (5.7) is greater than 1 and the left hand side of(5.7) is less than or equal to 1. This means the equations in (5.5) have no solutionsin v. On the other hand, (5.7) can be rewritten as

sin2(q1) =[B(m11 −m12)− Am12][B(m11 −m12) + Am12]

A2m11(m11 − 2m12)

By assumption, m12(q2) ≥ c− b/2 ≥ 0. Assuming A ≥ B, we get

B(m11 −m12) + Am12 = (A−B)m12 +Bm11 > 0

Also, m11 − 2m12 = a− 2c > 0. Thus, by taking

B(m11 −m12)− Am12 < 0

the right hand side of the last equation is negative, while the left hand side is positive.In other words, the equations in (5.5) have no real solutions in v. The last conditioncan be expressed as

B

A+B<m12(v)

m11(v)=: ψ(v) (5.8)

It is elementary to show that ψ(v) takes its minimum at v = ±π which is equal to

ψ(π) =c− b/2a− b

Note that ψ(q2) ≤ m12/(a−2c+2m12) < 1/2 for all q2. Therefore, if B < ψ(π)(A+B),

117

or

B <ψ(π)

1− ψ(π)A

where ρ := ψ(π)1−ψ(π)

< 1, equation (5.7) has no real solution in q1 (and thus in v). Thisfinishes the proof of the first part. The existence and closed-form expression for thestate feedback

u = K(qr, pr, qs, ps)

follows from the backstepping procedure.

Remark 5.1.1. Assume both links of the Acrobot have the same uniform mass density.If m1 = m2 and l2 > 3l1, then A < B and condition i) in proposition 5.1.1 is satisfied.Also, if l2 > 1.5l1, then c − b/2 > 0. Let us take m1 = λm2 and l2 = 2l1. Then,a − 2c > 0, and ψ(π) = 5/2(λ + 1). This gives ρ = 5/(2λ − 3) and A,B can becalculated as the following

A = m1l1 +m2L1 = (λ+ 2)m2l1, B = 2m2l1

Therefore, the condition B < ρ(λ)A is satisfied for all λ > λ∗ = 1.5 and condition ii)of proposition 5.1.1 holds.

Remark 5.1.2. To find the isolated root v = α(qr) of gr(qr, v) = 0, let us differentiatethe last equation with respect to parameter s ∈ R. We get

∇qrgr(qr, v) ·dqrds

+∇vgr(qr, v) ·dv

ds= 0

or

dqrds

= 1

dv

ds= −∇qrgr(qr, v)

∇vgr(qr, v)

The solution of this ODE with zero initial conditions both positive and negative intime gives a parameterization (qr, α(qr)) of the isolated root of gr(qr, v) = 0.

For the Acrobot, it is possible to solve the equation gr(qr, v) = 0 in qr explicitly asa function of v. This gives a numeric look up table of pairs (qr, v) as the solution ofequation gr(qr, v) = 0. To show this, notice that gr(qr, v) = 0 is equivalent to

A sin(q1) +B sin(q1 + v) = 0

or(A+B cos(v)) sin(q1) +B sin(v) cos(q1) = 0

Hence, the last equation can be solved in q1 as

q1 = − arctan

(

B sin(v)

A+B cos(v)

)

118

−4 −3 −2 −1 0 1 2 3 4−40

−30

−20

−10

0

10

20

30

40

qr (rad)

α(q r)

(rad

)

Figure 5-3: The function α(qr) for m1 = m2 = 1 and l1 = l2 = 1.

but q1 = qr − γ(v), thus

qr = γ(v)− arctan

(

B sin(v)

A+B cos(v)

)

=: φ(v)

In other words, solving the equation gr(qr, v) = 0 in v is equivalent to inverting thefunction φ(v) which is available in closed-form. This inversion can be numerically doneusing a piece-wise linear approximation, splines, neural networks, or other convenientcurve fitting approaches. In this work, we use a piece-wise linear approximationmethod constructed from a look up table of pairs (φ(v), v) as the solution of equationgr(qr, v) = 0. Figures 5-4 through 5-9 show simulation results for the Acrobot forthe choice of parameters m1 = m2 = 1, l1 = l2 = 1, and I1 = I2 = 1/3 (the massdensities of both links are assumed to be uniform and equal). The simulation resultsdemonstrate effective stabilization of the upright position for the Acrobot from initialconditions with large deviations of q1, q2 from zero.

119

0 1 2 3 4−0.5

0

0.5

1

1.5

time (sec)

q1 (

rad)

0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

time (sec)

q1 (

rad)

0 1 2 3 4−8

−6

−4

−2

0

2

4

time (sec)

q2 (

rad)

0 20 40 60 80 100−8

−6

−4

−2

0

time (sec)

q2 (

rad)

Figure 5-4: Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =(π/3, 0, 0) and control input p2.

0 1 2 3 4−15

−10

−5

0

5

time (sec)

p1 (

rad/

sec)

0 20 40 60 80 100−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time (sec)

p1 (

rad)

0 1 2 3 4−10

0

10

20

30

40

50

time (sec)

p2 (

rad)

0 20 40 60 80 100−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

p2 (

rad/

sec)

Figure 5-5: Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =(π/3, 0, 0) and control input p2.

120

0 1 2 3 4 5−1

0

1

2

3

4

time (sec)

q1 (

rad)

0 20 40 60 80 100−0.6

−0.4

−0.2

0

0.2

0.4

time (sec)

q1 (

rad)

0 1 2 3 4 5−15

−10

−5

0

5

10

15

20

time (sec)

q2 (

rad)

0 20 40 60 80 100−12

−10

−8

−6

−4

−2

0

time (sec)

q2 (

rad)

Figure 5-6: Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =(π, 0, 0) and control input p2.

0 1 2 3 4 5−60

−40

−20

0

20

40

60

time (sec)

p1 (

rad/

sec)

0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

p1 (

rad)

0 1 2 3 4 5−50

0

50

100

150

200

time (sec)

p2 (

rad)

0 20 40 60 80 100−0.5

0

0.5

1

1.5

2

time (sec)

p2 (

rad/

sec)

Figure 5-7: Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =(π, 0, 0) and control input p2.

121

0 0.05 0.1 0.15 0.2−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

time (sec)

q1 (

rad)

0 20 40 60 80 100−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

time (sec)

q1 (

rad)

0 0.05 0.1 0.15 0.23

3.05

3.1

3.15

3.2

time (sec)

q2 (

rad)

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

time (sec)

q2 (

rad)

Figure 5-8: Trajectory of (q1, q2) for the Acrobot with initial condition (q1, q2, p1) =(0, π, 0) and control input p2.

0 0.05 0.1 0.15 0.2−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

time (sec)

p1 (

rad/

sec)

0 20 40 60 80 100−0.2

−0.15

−0.1

−0.05

0

0.05

time (sec)

p1 (

rad)

0 0.05 0.1 0.15 0.2−1.2

−1

−0.8

−0.6

−0.4

−0.2

time (sec)

p2 (

rad)

0 20 40 60 80 100−0.4

−0.3

−0.2

−0.1

0

time (sec)

p2 (

rad/

sec)

Figure 5-9: Trajectory of (p1, p2) for the Acrobot with initial condition (q1, q2, p1) =(0, π, 0) and control input p2.

122

5.2 The TORA System

The TORA1 system depicted in Figure 5-10 consists of a translational oscillating plat-form which is controlled via a rotational eccentric mass. The TORA system has been

m

m

r

qk

1

2 2

τI

1 2

q 1

Figure 5-10: The TORA system.

mainly a benchmark example for passivity-based approaches in control of nonlinearsystems and stabilization methods for nonlinear cascade systems [109, 37, 80, 38, 39].Global stabilization of the TORA system using state feedback and backstepping pro-cedure has been known due to Wan et al. [109]. More recently, global output trackingfor the TORA system was addressed in [39]. Here, we provide a new globally asymp-totically stabilizing state feedback for the TORA system with a physically meaningfulLyapunov function. The advantage of using a Lagrangian-based change of coordinatesis that the reduced system (i.e. zero-dynamics) has a meaningful Lagrangian struc-ture which simplifies the control design. It turns out that for the TORA system, thereduced system is an undamped one-dimensional mass-spring system.

The Lagrangian of the TORA system is as the following

L(q, q) =1

2m1q

21 +

1

2m2[(q1 + r cos(q2)q2)

2 + (r sin(q2)q2)2]

+1

2(m2r

2 + I2)q22 − V (q1, q2)

(5.9)

where the potential energy is equal to

V (q1, q2) =1

2k1q

21 +m2gr cos(q2)

The Lagrangian of the TORA system can be rewritten as

L(q, q) = 1

2

[

q1q2

]T [m1 +m2 m2r cos(q2)m2r cos(q2) m2r

2 + I2

] [

q1q2

]

− V (q1, q2) (5.10)

From the equation of the Lagrangian in (5.10), it is clear that q2 is the shape variable

1Translational Oscillator with Rotational Actuator

123

and q1 is the external variable of the TORA system. Since q2 is actuated, the TORAsystem is a Class-I underactuated system that can be globally transformed into acascade system in strict feedback form. The Euler-Lagrange equations of motion forthe TORA system are given by

(m1 +m2)q1 +m2r cos(q2)q2 −m2r sin(q2)q22 + k1q1 = 0

m2r cos(q2)q1 + (m2r2 + I2)q2 +m2gr sin(q2) = τ

(5.11)

After collocated partial feedback linearization using a change of control

τ = α(q2)u+ β(q, q)

with

α(q2) = (m2r22 + I2)−

(m2r cos(q2))2

(m1 +m2)> 0 , ∀q2 ∈ [−π, π]

the dynamics of the shape variable q2 reduces to

q2 = p2, p2 = u

The normalized momentum conjugate to q1 is

pr =∂L∂q1

= (m1 +m2)q1 +m2r cos(q2)q2

Observe that both pr and the normalized momentum πr = m−111 pr are trivially inte-

grable. The function γ(q2) can be calculated explicitly as

γ(q2) =

∫ q2

0

m12(s)

m11

ds =m2r sin(q2)

m1 +m2

Thus, based on theorem 4.2.1, the global change of coordinates

qr = q1 + (m2r sin(q2))/(m1 +m2)pr = (m1 +m2)p1 +m2r cos(q2)p2

(5.12)

transforms the dynamics of the TORA system into a cascade nonlinear system instrict feedback form

qr = (m1 +m2)−1pr

pr = −k1qr + k1γ(q2)q2 = p2p2 = u

(5.13)

The reduced system with Lagrangian

Lr(qr, qr) =1

2(m1 +m2)q

2r −

1

2k1q

2r

124

satisfies the one-dimensional forced Euler-Lagrange equation

d

dt

∂Lr∂qr− ∂Lr∂qr

= k1γ(q2)

with a nonlinear input force Fr(q2) = k1γ(q2) which depends on the shape variable q2.If q2 ≡ 0, the reduced system is an oscillator with frequency ω0 =

√

k1/(m1 +m2)(rad/sec). Using backstepping procedure, stabilization of the overall system reducesto the stabilization of the reduced undamped mass-spring system with reduced La-grangian Lr(qr, qr).

Proposition 5.2.1. Consider the TORA system in cascade form (5.13). Then, thefollowing statements hold:

i) There exists a bounded state feedback

q2 = Kr(qr, pr) = −σ(c1qr + c2pr) , c1, c2 > 0 (5.14)

that globally asymptotically stabilizes the origin (qr, pr) = 0 for reduced systemof the TORA system in (5.13) where σ denotes a sigmoidal function satisfying|σ(·)| ≤ π/2.

ii) The reduced system has a well-defined new Hamiltonian function Hr(qr, pr) that

is decreasing by time, i.e. ˙Hr ≤ 0. In addition, Hr(qr, pr) is a valid Lyapunovfunction for the reduced system. In other words, the closed-loop reduced systemwith controller Kr(qr, pr) is a passive Hamiltonian system.

iii) There exists a state feedback in explicit form

u = K(qr, pr, q2, p2)

that globally asymptotically stabilizes the origin for the composite system in(5.13).

Proof. We prove parts i) and ii) simultaneously and then part iii) follows fromthe first two parts. First, observe that if σ(x) is a sigmoidal function satisfying|σ(x)| ≤ π/2 for all x, then sin(σ(x)) and thus γ(σ(s)) is a sigmoidal function as well.Define the amended potential function

ψ(qr) =

∫ qr

0

k1γ(σ(c1s))ds =

∫ qr

0

k1m2r sin(σ(c1s))

m1 +m2

ds

and notice that ψ(qr) is positive definite function. Now, consider the following newHamiltonian Hr as a candidate Lyapunov function

Hr(qr, pr) = Hr(qr, pr) + ψ(qr)

125

where Hr is the (original) Hamiltonian of the reduced system given by

Hr(qr, pr) =1

2(m1 +m2)

−1p2r +1

2k1q

2r

Since the input force is not identically zero, the Hamiltonian Hr is not a conserved

quantity. Calculating ˙Hr along the solutions of the closed-loop reduced system withcontroller Kr(qr, pr) gives

˙Hr = Hr + qrψ′(qr)

= (m1 +m2)pr[−γ(σ(c1qr + c2pr)) + γ(σ(c1qr))]= (m1 +m2)pr[σ(c1qr)− σ(c1qr + c2pr)] ≤ 0

where σ(x) = γ(σ(x)) is a sigmoidal function. From the last equation, ˙Hr < 0for all pr 6= 0 and based on LaSalle’s invariance principle, all the solutions of theclosed-loop system converge to the largest invariant set Ω with pr = 0. In this case,Ω = (0, 0) and the origin (qr, pr) = (0, 0) is globally asymptotically stable for thereduced system. Apparently, H(qr, pr) is a valid Hamiltonian for the closed-loopsystem with configuration qr, a new potential energy

V (qr) =1

2k1q

2r + ψ(qr)

and a new nonlinear damping force

Fd(qr, pr) = γ(σ(c1qr))− γ(σ(c1qr + c2pr))

which satisfies prFd(qr, pr) ≤ 0.

Remark 5.2.1. The controller obtained in [109] is equivalent to using the followingstate feedback

q2 = Kr(pr) = −σ(c2pr)which is a globally asymptotically stabilizing feedback for the reduced system. Thisfeedback is a special case of the state feedback in part i) of proposition 5.2.1 withc1 = 0. In this special case, the amended potential ψ(qr) is identically zero.

Figure 5-11 shows the simulation results for the TORA system with the choice ofparameters m1 = 10, m2 = 1, k1 = 5, r = 1 and I = 1. The controller Kr is in theform

Kr(qr, pr) = −a tanh(c1qr + c2pr)

with a = π/2 and c1 = c2 = 1. Figure 5-11 (c), clearly shows that the reduced systemdemonstrates a solution close to the trajectories of position and velocity correspondingto a damped mass-spring system with a linear damping force.

126

0 10 20 30 40 50 60−1.5

−1

−0.5

0

0.5

1

time (sec)

q 1,p1

q1

p1

0 10 20 30 40 50 60−10

−8

−6

−4

−2

0

2

4

6

8

time (sec)

q 2,p2

q2

p2

(a) (b)

0 10 20 30 40 50 60−8

−6

−4

−2

0

2

4

6

8

time (sec)

q r,pr

qr

pr

0 10 20 30 40 50 60−3

−2

−1

0

1

2

3

4

time (sec)

u (c

ontr

ol)

(c) (d)

Figure 5-11: State trajectories and control input of the TORA system for the initialcondition (q1, q2, p1, p2) = (1, 0, 0, 0): (a) (q1, p1), (b) (q2, p2), (c) (qr, pr), and (d) u.

5.3 The Inertia-Wheel Pendulum

The Inertia-Wheel Pendulum is a planar inverted pendulum with a revolving wheelat the end, as shown in Figure 5-12. The wheel is actuated and the joint of the pen-dulum at the base is unactuated. The inertial-wheel pendulum was first introducedby Spong et al. in [93] where a supervisory hybrid/switching control strategy is ap-plied to asymptotic stabilization of the inertia-wheel pendulum around its uprightequilibrium point. First, a passivity-based controller [28] swings up the pendulum.Then, a balancing controller that is obtained by Jacobian linearization or (local) exactfeedback linearization stabilizes the pendulum around its upright position.

Here, we show that the inertia-wheel pendulum is an example of a Class-I underac-tuated system. Based on theorem 4.2.1, the dynamics of the inertia wheel pendulumcan be transformed into a cascade nonlinear system in strict feedback form; using aglobal change of coordinates in an explicit form. Then, global asymptotic stabiliza-tion of the upright equilibrium point can be achieved using the standard backsteppingprocedure [36, 57, 80].

127

q

q

I

1

2

2

1L , I 1

τ

Figure 5-12: The Inertia-Wheel Pendulum

The Lagrangian of the inertia wheel pendulum is as the following

L(q, q) = 1

2qTMq − V (q1)

where q = (q1, q2)T ∈ R2 denotes the configuration vector of the IWP and M is a

constant inertia matrix with elements

m11 = m1l21 +m2L

21 + I1 + I2

m12 = m21 = m22 = I2

The potential energy of the IWP [93] is given by

V (q1) = (m1l1 +m2L1)g0 cos(q1) =: m0 cos(q1)

The Euler-Lagrange equations of motion for the inertia-wheel pendulum are as thefollowing

d

dt(∂L∂q1

)− ∂L∂q1

= 0

d

dt(∂L∂q2

)− ∂L∂q2

= τ

orm11q1 +m12q2 = g1(q1)m21q1 +m22q2 = τ

(5.15)

where g1(q1) = m0 sin(q1).Due to the invariance of the kinetic energy of the inertia-wheel pendulum under the

group action (q1, q2) 7→ (q1 + α, q2 + β) for (α, β) ∈ R2, the inertia matrix is constantand all the Christoffel Symbols associated with M vanish. Thus, the inertia-wheelpendulum is a flat underactuated mechanical system with two degrees of freedomand kinetic symmetry w.r.t. both degrees of freedom. As a consequence, the inertia-

128

wheel pendulum is a special case of underactuated systems with symmetry propertiesconsidered in [68] and can be globally asymptotically stabilized.

Based on theorem 4.2.1, the global change of coordinates

z1 = ∂L/∂q1 = m11q1 +m12q2z2 = q1z3 = q2

transforms the dynamics of the inertia-wheel pendulum into a strict feedback system

z1 = m0 sin(z2)z2 = (z1 −m12z3)/m11

z3 = u(5.16)

where the new control u and the torque τ are related through the expression

τ = (m22 −m21m12/m11)u+ (m21m0/m11) sin(q1)

Note that q2 does not play any important role in the dynamics of the IWP and isignored as an state variable (also, q2 does not appear in L(q, q)).Remark 5.3.1. Clearly, the nonlinear system in (5.16) is exact feedback linearizablewith the choice of the output y = z1 over |z2| < π/2 (i.e. |q1| < π/2). This result hasbeen previously obtained in [93] using a standard method from [36].

Here is our main result for the inertia wheel pendulum:

Proposition 5.3.1. There exists a nonlinear state feedback law (in explicit form) thatglobally asymptotically stabilizes (z1, z2, z3) = 0 (i.e. the upright equilibrium point)for the Inertia Wheel Pendulum in (5.16).

Proof. Consider the scalar nonlinear system

z1 = m0 sin(z2)

with control input z2. The following state feedback

z2 = k1(z1) := c0σ1(c1z1) ; −π/2 < c0 < 0, c1 > 0

(where σ1(s) = tanh(s) is a sigmoidal function) globally asymptotically stabilizesz1 = 0 and V (z1) = z21 is a valid Lyapunov function for the z1-subsystem in (5.16).This is due to the fact that

σ(z1) = −m0 sin(c0σ1(c1z1))

is a sigmoidal function that satisfies σ(0) = 0, σ′(0) > 0, and sσ(s) > 0,∀s 6= 0. Now,define the following change of coordinates and control

µ1 = z2 − k1(z1), µ2 = µ1, v = µ2

129

which can be explicitly expressed as

µ1 = z2 − k1(z1)µ2 = (z1 −m12z3)/m11 − k1v = (m0/m11) sin(z2)− (m12/m11)u− k1

(5.17)

where by the assumption that σ1(s) = tanh(s), we have

k1 = c0c1m0 sin(z2)(1− σ1(c1z1)2)k1 = c0c1m0(1− σ1(c1z1)2)(cos(z2)(z1/m11 −m12z3/m11)− 2c1m0σ1(c1z1) sin(z2)

2)

In new coordinates, the dynamics of the inertia wheel pendulum in (5.16) can bewritten in the following cascade form

z1 = m0 sin(k1(z1) + µ1)µ1 = µ2µ2 = v

(5.18)

The state feedbackv = −c2µ1 − c3µ2; c2, c3 > 0 (5.19)

globally exponentially stabilizes (µ1, µ2) = 0 for the µ-subsystem of (5.18). Due tothe fact that for any exponentially vanishing function µ1(t), the solution of the z1-subsystem in (5.18) is uniformly bounded (the proof for this boundedness is ratherelementary and is omitted), the origin (z1, µ1, µ2) = 0 for the closed loop cascadesystem in (5.18) is globally asymptotically and locally exponentially stable due to atheorem by Sontag [83]. The overall nonlinear state feedback u = k(z1, z2, z3) can becalculated as the following

u =m11

m12

(c2µ1 + c3µ2 +m0

m11

sin(z2)− k1)

which is an explicit function of the state (z1, z2, z3).

Remark 5.3.2. Apparently, using standard backstepping procedure in [36] (chapter9), a Lyapunov function V (z) and a globally stabilizing feedback law for (5.16) canbe obtained by explicit use of V1(z1) = z21 and propagation of the time derivative ofV1 along the solutions of (5.16). However, this only complicates the calculation of thefinal stabilizing state feedback law and there is no need to do so in this special case.

Now, we present simulation results for global stabilization of the inertia-wheelpendulum to its upright equilibrium point. For comparison purposes, we used thesame numeric values for the parameters of the dynamics of the IWP as in [93]. Theseparameters are as follows: m11 = 4.83 × 10−3, m12 = m21 = m22 = 32 × 10−6,m = 38.7× 10−3, g0 = 9.8, and m0 = mg0.

The state trajectories and the control input of the inertia-wheel pendulum forthe downward and vertical initial positions of the pendulum are shown in Figures5-13 and 5-14, respectively. These simulation results demonstrate that the nonlinearcontroller aggressively stabilizes the pendulum to its up-right position. The values of

130

the controller parameters for the simulation are c0 = −π/10, c1 = .03, c2 = 16, c3 = 8and the maximal applied torque was τmax = 0.6 Nm.

0 1 2 3 4 5 6−8

−6

−4

−2

0

2

4

sec

rad,

rad

/sec

q1

dq1 / dt

−0.5 0 0.5 1 1.5 2 2.5 3 3.5−7

−6

−5

−4

−3

−2

−1

0

1

q1

d q 1 /

dt

(a) (b)

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000

sec

d q 2 /

dt (

rad/

sec)

0 1 2 3 4 5 6−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

sec

tau

(Nm

) (c

ontr

ol)

(c) (d)

Figure 5-13: For the initial condition (π, 0, 0), (a) shows the state trajectories for(q1, q1), (b) shows the state trajectory in (q1, q1) plane, (c) shows q2, and (d) showsthe control input τ .

131

0 1 2 3 4 5 6−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

sec

rad,

rad

/sec

q1

dq1 / dt

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

q1

d q 1 /

dt

(a) (b)

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000

7000

sec

d q 2 /

dt (

rad/

sec)

0 1 2 3 4 5 6−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sec

tau

(Nm

) (c

ontr

ol)

(c) (d)

Figure 5-14: For the initial condition (π/2, 0, 0), (a) shows the state trajectories for(q1, q1), (b) shows the state trajectory in (q1, q1) plane, (c) shows q2, and (d) showsthe control input τ .

5.4 The Cart-Pole System

The Cart-Pole system consists of a cart and an inverted-pendulum on it, as shownin Figure 5-15. A horizontal force F as the control input is applied along q1 and thejoint q2 is unactuated. The inertia matrix of the cart-pole system is given by (seesection A.5)

m11 = m1 +m2

m12(q2) = m21(q2) = m2l2 cos q2m22 = m2l

22 + I2

where m1 is the mass of the cart and m2, l2, I2 are the mass, length of the center ofmass, and the inertia of the pendulum, respectively. Also, the potential energy of thecart-pole system is in the form

V (q2) = m2gl2 cos(q2)

132

mF

m , l , I

1

2 2 2

q

q1

2

Figure 5-15: The Cart-Pole system.

The Euler-Lagrange equations of motion for the cart-pole system can be expressed asthe following

(m1 +m2)q1 +m2l2 cos(q2)q2 −m2l2 sin(q2)q22 = F

m2l2 cos(q2)q1 + (m2l22 + I2)q2 −m2gl2 sin(q2) = 0

Apparently, the inertia matrix only depends on q2 and thus q1 is the external variableand q2 is the shape variable of the cart-pole system. Since the shape variable q2 isunactuated, the cart-pole system falls within the category of Class–II underactuatedsystems.

5.4.1 Global Stabilization to an Equilibrium Point

In this section, we introduce a state feedback that globally asymptotically stabilizesthe upright equilibrium point of the cart-pole system in the upper half-plane. First,we need to perform noncollocated feedback linearization on the dynamics of the cart-pole system. Notice that

(m12(q2)−m11m22

m21(q2))q2 + (m1 +m2)g tan(q2)−m2l2 sin(q2)q

22 = F

and the coefficient of q2 is negative thus applying the feedback law

F = (m12(q2)−m11m22

m21(q2))u+ (m1 +m2)g tan(q2)−m2l2 sin(q2)q

22 (5.20)

reduces the dynamics of q2 to a double integrator as

q2 = p2, p2 = u

where u is the new control. Based on theorem 4.2.2, the following change of coordi-nates

qr = q1 + γ(q2)pr = m21(q2)p1 +m22p2

133

where

γ(q2) =

∫ q2

0

m22

m21(θ)dθ =

(m2l22 + I2)

m2l2log

(

1 + tan(q2/2)

1− tan(q2/2)

)

, q2 ∈ (−π/2, π/2)

transforms the dynamics of the cart-pole system into

qr =1

m21(qs)pr

pr = gr(qs) +1

m21(qs)

dm21(qs)

dqsprps −

m22

m21(qs)

dm21(qs)

dqsp2s

qs = ps

ps = u

where qs = q2 and gr(qs) = m2gl2 sin(qs). Clearly, the equation of pr is a quadraticform is (pr, ps) and the reason that the coefficient of p2r is identically zero is thatm11 isconstant for the cart-pole system. This property guarantees that the cart-pole systemis a nonlinear system in strict feedforward form (see [102]). The (strict) feedforwardproperty of the cart-pole system has been used for semiglobal stabilization of thesystem above the half-plane by Teel [101] as an application of nonlinear small-gaintheorem. In [56], Mazenc and Praly constructed a Lyapunov function for the cart-polesystem (after extremely lengthy and complicated algebraic calculations) that provesexistence of a globally stabilizing state feedback law for the cart-pole system abovethe half-plane. Here is our main result for the cart-pole system:

Proposition 5.4.1. There exists an state feedback law in the form of nested sat-urations that globally asymptotically stabilizes the origin (q1, p1, q2, p2) = 0 for thecart-pole system over the upper half-plane.

Proof. The proof relays on the aforementioned change of coordinates and furthersimplification of the reduced system and a famous result due to Teel in [101]. Theequation of pr can be simplified as

pr = − tan(qs)prps +m22 tan(qs)p2s +m2gl2 sin(qs)

Applying the following change of coordinates

z1 = qrz2 = pr/m21(qs)

we getz1 = z2

z2 = tan(qs)(g +m2l

22 + I2

m2l2 cos(qs)p2s)

To map the upper half-plane to R, we use another change of coordinates and control

134

as the following

ξ1 = tan(qs)ξ2 = (1 + tan(qs)

2)psv = (1 + tan(qs)

2)u+ 2 tan(qs)(1 + tan(qs)2)ps

This transforms the dynamics of the cart-pole system into

z1 = z2

z2 = ξ1(k1 + k2ξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(5.21)

where k1, k2 are the following positive constants

k1 = g, k2 =m2l

22 + I2m2l2

Observing that |ξ1/(1 + ξ21)32 | < 1, the nonlinear system in (5.21) is in strict feedfor-

ward form and satisfies all the conditions of Teel’s theorem in [102]. Therefore, theorigin (q, p) = 0 for the cart-pole system can be globally asymptotically stabilizedusing small nested saturations in explicit form. The control law can be obtained asthe following. First, based on [102], apply the change of coordinates and control

η1 =1

k1z1 +

2

k1z2 + 2ξ1 + ξ2

η2 =1

k1z2 + ξ1 + ξ2

w = ξ1 + ξ2 + v

Then, the state feedback law

w = −σ2(η2 + σ1(η1))

orv = −ξ1 − ξ2 − σ2(η2 + σ1(η1))

globally asymptotically stabilizes the origin for the cart-pole system where σi’s aresaturation functions with appropriate small thresholds and magnitudes.

Figure 5-16 shows the simulation results for the cart-pole system for the ini-tial state (q1, p1, q2, p2) = (5, 0, π/3, 0). Clearly, the controller with σi(x) = sat1(x)stabilizes the inverted pendulum in its upright position at q1 = 0 after a rathershort time. The parameters are chosen as m1 = m2 = 1, l2 = 0.75, g = 9.8 andsatε(x) = ε · sgn(x) · min1, |x/ε|. However, based on Teel’s theorem the satu-ration functions must have sufficiently small thresholds and magnitudes. To seehow the controller designed based on small nested saturations performs, we used

135

σi = sat0.1(·), i = 1, 2. The thresholds and magnitudes of saturation functions arereduced by a factor of ten compared to the previous case, but the same initial con-dition (5, 0, π/3, 0) is chosen. The results are shown in Figure 5-17. Clearly, thecontroller has a poor performance in terms of the settling time and maximum peakof the position. The settling time increases by an approximate factor of 10 (whichis expected) and the maximum peak of the position has increases by a factor of 4(which is better than expected). In addition, the convergence rate of the position isalmost linear. This reveals a large deterioration of the performance of the controllerfor nested saturations with relatively small magnitudes.

0 5 10 15 20 25−20

−10

0

10

20

30

40

50

time (sec)

pos/

velo

city

q1

p1

0 5 10 15 20 25−1.5

−1

−0.5

0

0.5

1

1.5

time (sec)

angl

e/an

gula

r ve

loci

ty

q2

p2

0 5 10 15 20 25−40

−20

0

20

40

60

time (sec)

For

ce

0 5 10 15 20 25−6

−4

−2

0

2

4

time (sec)

Con

trol

u

(a) (b)

Figure 5-16: (a) State trajectory of the cart-pole system for the initial condition(5, 0, π/3, 0) and σi(·) = sat1(·), (b) The force F and control input u.

0 50 100 150 200 250−50

0

50

100

150

200

time (sec)

pos/

velo

city

q1

p1

0 50 100 150 200 250−1

−0.5

0

0.5

1

1.5

time (sec)

angl

e/an

gula

r ve

loci

ty

q2

p2

0 50 100 150 200 250−20

−10

0

10

20

30

40

50

time (sec)

For

ce

0 50 100 150 200 250−4

−3

−2

−1

0

1

2

time (sec)

Con

trol

u

(a) (b)

Figure 5-17: (a) State trajectory of the cart-pole system for the initial condition(5, 0, π/3, 0) and σi(·) = sat0.1(·), (b) The force F and control input u.

Remark 5.4.1. It is needless to mention that the controller obtained here for globalasymptotic stabilization of the cart-pole system over the upper half-plane is muchsimpler than the one given by Mazenc and Praly [56].

136

In chapter 7, we introduce a different approach to stabilization of cart-pole sys-tem that is more general than the one presented here. This new method applies tononlinear systems that are not necessarily in strict feedforward/feedback forms.

5.4.2 Aggressive Swing-up of the Inverted Pendulum

In this section, we provide a state feedback law for aggressive swing-up of an invertedpendulum on a cart in a single action without swinging the pendulum. In the previoussection, we showed that using noncollocated feedback linearization the dynamics ofq2 can be reduced to a double integrator

q2 = p2, p2 = u

a globally exponentially stabilizing (i.e. aggressive) state feedback that asymptoticallystabilizes (q2, p2) = 0 is given by

u = −c1q2 − c2p2

where c1, c2 > 0 are constants. The main difficulty in stabilizing the angle of thependulum from initial angle of q2 = π to final angle of q2 = 0 is that at some momentof time, the state trajectory of q2 passes through the point q2 = π/2. Based on(5.20), this means F → ∞ due to the division by cos(q2). To avoid this singularity,we saturate the control input F at a maximum level Fmax that is relatively large, i.e.Fmax À 1. In other words, we apply the following approximate control

Fapp(q2, p2) = Fmaxsat(F/Fmax)

where

F (q2, p2) = (m11m22

m21(q2)−m12(q2))(c1q2 + c2p2) + (m1 +m2)g tan(q2)−m2l2 sin(q2)p

22

The difference between F and Fapp

F − Fapp = φ(F )

can be expressed in terms of the famous dead-zone nonlinearity φ shown in Figure5-18. The function φ(F ) is zero over the interval [−Fmax, Fmax] and linear with unitslope outside of this interval. The dynamics of the closed loop system with controlF = Fapp(q2, p2) can be expressed as

q2 = p2

p2 = −c1q2 − c2p2 +m21(q2)

D(q2)φ(F (q2, p2))

(5.22)

where D(q2) = det(M(q2)) > 0. Apparently, whenever |F | < Fmax (i.e. the controllerdoes not saturate) φ(F ) = 0 and (q2, p2) = (0, 0) is exponentially stable as long as the

137

Φ( )F

FF F

max max

Figure 5-18: The dead-zone nonlinearity.

solution passes through an area that the controller saturates, i.e. |Fapp(q2, p2)| = Fmax.The nonlinear feedback F (q2, p2) has a discontinuity along the lines q2 = ±π/2 in the(q2, p2)-plane. This discontinuity property is closely related to the function p2 = α(q2)defined by

α(q2) = −c1c2q2 −

m11m2gl2 sin(q2)

c2D(q2)

In fact, F (q2, p2) is continuous at isolated points p2 = α(±π/2) on the lines q2 = ±π/2.The following result determines the area in which the controller Fapp(q2, p2) saturates.

Proposition 5.4.2. Consider the neighborhood

Nε = (q2, p2) ∈ I × R | |q2 − π/2| ≤ ε, |p2| ≤ n0π, n0 > 0

for some 0 < ε ≤ π/2 and I = [0, π]. Then, there exist F ∗(ε, n0) > 0 in the form

F ∗(ε, n0) =a1 + a2n0sin(ε)

+ a3n20, a1, a2, a3 > 0

and a neighborhood Uε ⊂ Nε of q2 = π/2 such that for all Fmax ≥ F ∗, the controllersaturates (i.e. |Fapp(q2, p2)| = Fmax) over (q2, p2) ⊂ Uε and remains unsaturated overI × [−n0π, n0π] \Nε.

Proof. F (q2, p2) can be rewritten as

F =c2D(q2)(p2 − α(q2))

m2l2 cos(q2)−m2l2 sin(q2)p

22

which is discontinuous along the lines q2 = ±π/2 and p2 6= α(±π/2). We have

|F | ≤ D0|c1q2 + c2p2|+m11m2gl2m2l2

1

| cos(q2)|+m2l2p

22

≤ D0(c1 + c2n0)π + (m1 +m2)m2gl2m2l2

1

| cos(q2)|+m2l2n

20π

2

138

where D0 > 0 is a constant satisfying 0 < D(q2) ≤ D0 := m11 ·m22 and the equality isachieved at q2 = π/2. Since cos(q2) takes its maximum at the boundary of Nε whereq2 = π/2± ε, for all (q2, p2) 6∈ Nε, D(q2) < D0 and we get

|F | < D0(c1 + c2n0)π + (m1 +m2)m2gl2m2l2 sin(ε)

+m2l2n20π

2

Hence, defining

F ∗(ε, n0) =D0(c1 + c2n0)π + (m1 +m2)m2gl2

m2l2 sin(ε)+m2l2n

20π

2

the controller Fapp does not saturate outside of the region Nε. On the other hand,for p2 6= α(π/2), F (q2, p2) → ∞. By definition, for any Fmax ≥ F ∗, there exists aneighborhood Uε ⊂ Nε of the line q2 = π/2 where F (q2, p2) ≥ Fmax and therefore thecontroller saturates in the region Uε. Over the region Nε \ Uε the controller might ormight not saturate. Nε is an upper bound (w.r.t. ⊂) for the maximal region Uε thatthe controller saturates.

The values of F ∗ for some typical choices of ε, n0 are given in Table 5.1. Using somestandard phase plane arguments, one can show that Fapp semiglobally asymptoticallystabilizes (q2, p2) = 0 for a sufficiently large Fmax ≥ F ∗(ε, n0) and the solution expo-nentially converges to the origin over the region (q2, p2) ∈ R2|q2 ∈ [−π/2+ε, π/2−ε].Due to q2 = p2, q2 is strictly decreasing in the region q2p2 < 0 and therefore the so-lution passes through the saturation region Nε once and exponentially converges tothe origin under the condition that c1, c2 > 0 are sufficiently large. The simulation

Table 5.1: The Values of F ∗(ε, n0).

(n0, ε) π/20 π/10 π/5 π/4 π/2

1 454 234 127 107 782 637 337 192 164 125

results for this controller are shown in Figure 5-19. The parameters are chosen asFmax = 100 and c1 = c2 = 4, g = 9.8. The pendulum starts from the downwardposition at q2 = π and the angle of the pendulum (almost) exponentially convergesto zero. The convergence is not truly exponential, due to the fact that F (q2, p2) issaturated for a relatively short period of time. A sample path in (q2, p2)-plane for theinitial condition starting at (q2, p2) = (π, 0) is shown in Figure 5-19 (b).

5.4.3 Switching-based Controller

For the inverted initial condition (0, 0, π, 0) (i.e q2 = π), one can use the controllerFapp to bring the pendulum to the upper half-plane area and switch (say whenever|q2| ≤ π/3) to the globally stabilizing controller that makes use of nested satura-

139

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−3

−2

−1

0

1

2

3

4

time (sec)

angl

e/an

gula

r ve

loci

ty

q2

p2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−100

−50

0

50

100

time (sec)

cont

rol f

orce

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

q2

p 2

(a) (b)

Figure 5-19: (a) Aggressive swing-up of an inverted pendulum on a cart from thedownward position (q2, p2) = (π, 0), (b) The path of the pendulum in (q2, p2)-plane.

tions. This switching-based controller semiglobally asymptotically stabilizes the up-right equilibrium point of the cart-pole system over the whole plane with q2 ∈ [−π, π].The semiglobal nature of this stabilization is due to bounded initial angular velocityand speed required for the former controller that brings the pendulum to the up-per half plane. The simulation results for the overall stabilization of the cart-polesystem for the initial condition (0, 0, π, 0) where the pendulum is in the downwardposition are shown in Figure 5-20. This figure demonstrates that the switching-basedcontroller aggressively stabilizes the origin for the cart-pole system.

5.5 The Rotating Pendulum

The Rotating Pendulum consists of an inverted pendulum on a rotating arm, as shownin Figure 5-21. The elements of the inertia matrix for the rotating pendulum are givenby (see section A.6)

m11(q2) = I1 +m1l21 +m2(L

21 + l22 sin

2(q2))m12(q2) = m21(q2) = m2L1l2 cos(q2)

m22 = I2 +m2l22

The potential energy for this system is

V (q1, q2) = m2gl2 cos(q2)

Clearly, the inertia matrix of the rotating pendulum only depends on q2. Therefore,qx = q1 is the external variable and qs = q2 is the shape variable of the rotatingpendulum. Due to lack of actuation of the shape variable, the rotating pendulum is aClass–II underactuated mechanical system. The Euler-Lagrange equations of motion

140

0 5 10 15 20 25−15

−10

−5

0

5

10

time (sec)

pos/

velo

city

q1

p1

0 5 10 15 20 25−3

−2

−1

0

1

2

3

4

time (sec)

angl

e/an

gula

r ve

loci

ty

q2

p2

0 5 10 15 20 25−100

−50

0

50

100

time (sec)

For

ce

−0.5 0 0.5 1 1.5 2 2.5 3 3.5−2.5

−2

−1.5

−1

−0.5

0

0.5

q2

p 2

(a) (b)

Figure 5-20: (a) State trajectory of the cart-pole system for the initial condition(0, 0, π, 0), (b) The input force F and the trajectory in (q2, p2)-plane.

x

y

z

1

2q

m , L , I

m , L , I2

1 1

2

q1

2

Figure 5-21: The Rotating Pendulum

for the rotating pendulum are given by

m11(q2)q1 +m12(q2)q2 + 2m2l22 sin(q2) cos(q2)q1q2 −m2l2L1 sin(q2)q

22 = τ

m21(q2)q1 +m22(q2)q2 −m2l22 sin(q2) cos(q2)q

21 −m2gl2 sin(q2) = 0

Noting that the second line of this equation can be simplified as

q1 +m22

m21(q2)q2 −

l2L1

sin(q2)q21 −

g

L1

tan(q2) = 0

and applying the change of control

τ = (m12(q2)−m11(q2)m22

m21(q2))u+ 2m2l

22 sin(q2) cos(q2)q1q2 −m2l2L1 sin(q2)q

22

+ m11(q2)l2L1

sin(q2) +m11(q2)g

L1

tan(q2)

141

(which is well-defined for q2 6= π/2) reduces the dynamics of the shape variable q2 to

q2 = p2p2 = u

Based on theorem 4.2.2, we apply the following change of coordinates

qr = q1 + γ(q2)pr = m21(q2)p1 +m22p2

orz1 = qrz2 = pr/m21(q2)

where

γ(q2) =

∫ q2

0

m22

m21(θ)dθ =

(m2l22 + I2)

m2l2L1

log

(

1 + tan(q2/2)

1− tan(q2/2)

)

, q2 ∈ (−π/2, π/2)

that transforms the dynamics of the rotating pendulum into

z1 = z2

z2 =g

L1

tan(q2) +l2L1

sin(q2)(z2 −m22

m21(q2)p2)

2 +m22

m21(q2)tan(q2)p

22

q2 = p2p2 = u

(5.23)

The nonlinear system in (5.23) is neither in feedforward form (due to the terml2 sin(q2)z

22/L1 in the equation of z2 which depends on z2), nor in strict feedback

form. Therefore, direct feedforwarding or backstepping procedures are not applicableto the rotating pendulum. Based on a recent work [10], global/semiglobal asymptoticstabilization of the rotating pendulum to its upright equilibrium point is consideredan open problem. In chapter 7, we will address this problem based on a theory thatis developed in this thesis for asymptotic stabilization of cascade nonlinear systems innontriangular normal forms. To remove the constraint q2 ∈ (−π/2, π/2), we apply thefollowing invertible global change of coordinates and control (similar to the cart-polesystem)

ξ1 = tan(q2)ξ2 = (1 + tan(q2)

2)p2v = (1 + tan(q2)

2)(u+ 2 tan(q2)p2)

This transforms the dynamics of the rotating pendulum in (5.23) into

z1 = z2

z2 = ξ1(g

L1

+l2L1

1√

1 + ξ21(z2 −

m22

m2l2L1

ξ2√

1 + ξ21)2 +

m22

m2l2L1

ξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(5.24)

142

Notice that in the equation of z2 the quantity in the parentheses is strictly positivedefinite. This equation can be simplified further as

z1 = z2

z2 = ξ1(k1 + k2z22

(1 + ξ21)12

− k3z2ξ2

(1 + ξ21)+ k4

ξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(5.25)

where

k1 =g

L1

, k2 =l2L1

, k3 =2m22

m2L21

, k4 =m22

m2l2L1

(1 +m22

m2L21

)

are all positive constants. Observe that ki/k1, i = 2, 3, 4 all vanish as l2/g → 0(i.e. in relatively high gravity). We use this property later for semiglobal asymptoticstabilization of the rotating pendulum in its upright position.

5.5.1 Aggressive Swing-up of the Rotating Pendulum

In this section, we introduce a state feedback that aggressively swings up an invertedpendulum mounted on a rotating arm in a single action without swinging it back andforth (i.e. a passivity-based swing-up as in [6, 112]). The procedure is very similar tothe one applied to the cart-pole system. Thus, we omit the details. Simply, one hasto set

u = −c1q2 − c2p2, c1, c2 > 0

which gives the following input torque

τ =D(q2)

m2l2L1 cos(q2)(c1q2 + c2p2) + 2m2l

22 sin(q2) cos(q2)q1q2 −m2l2L1 sin(q2)q

22

+ m11(q2)l2L1

sin(q2) +m11(q2)g

L1

tan(q2)

where D(q2) = det(M(q2)). Due to division by cos(q2) in the last equation as q2 →π/2, τ(q2, p2, p1) → ∞ for almost every p2. To avoid this singularity, we apply thesaturated torque as the following

τapp(q2, p2, p1) = τmaxsat(τ/τmax)

where τmax > 0 is the maximum allowed torque (to be specified later). One obtainsthe following result which is very similar to proposition 5.4.2 for the cart-pole system.

Proposition 5.5.1. Consider a neighborhood Nε = (q2, p2, p1) ∈ I×R2||q2−π/2| ≤ε, |p2| ≤ n0π, |p1| < n1π, n0 ≥ n1 > 0 for some 0 < ε ≤ π/2 and I = [0, π]. Then,

143

there exist τ ∗(ε, n0, n1) > 0 in the form

τ ∗(ε, n0, n1) =a1 + a2n0sin(ε)

+ a3n20 + a4n0n1 + a5, ai > 0, i = 1, . . . , 5

and a neighborhood Uε ⊂ Nε of q2 = π/2 such that for all τmax ≥ τ ∗, τapp(q2, p2, p1)saturates (i.e. |τapp(q2, p2)| = τmax) over (q2, p2) ⊂ Uε and it remains unsaturatedover I × [−n0π, n0π] \Nε.

Proof. Following the line of proof of proposition 5.4.2, let us define

D0 = D(π/2), M0 = m11(π/2)

and note that for all q2 ∈ [−π, π], D(q2) ≤ D0 and m11(q2) ≤ M0. For 0 < ε ≤ π/2and q2 = π/2− ε, we have

|τ | < D0(c1 + n0c2)π +M0m2gl2m2l2L1 sin(ε)

+ 2m2l22n0n1π

2 +m2l2L1n20π

2 +M0l2L1

Thus, we define τ ∗(ε, n0) as the right hand side of the last equation. In other words,we have

a1 =D0c1 +M0m2gl2

m2l2L1

, a2 =D0c2m2l2L1

, a3 = m2l2L1π2, a4 = 2m2l

22π

2, a5 =M0l2L1

For the choice of parameters m1 = 1, l1 = .5,m2 = 1, l2 = .75, g = 9.8, n1 = 1,the values of τ ∗(ε, n0, 1) are shown in Table 5.2. The simulation results for almost

Table 5.2: The Values of τ ∗(ε, n0, 1).

(n0, ε) π/20 π/10 π/5 π/4 π/2

1 1471 755 406 341 2472 1885 981 541 459 340

exponential swing-up of the rotating pendulum from the downward position (i.e.initial condition (0, 0, π, 0)) to (q2, p2) = (0, 0) are shown in Figure 5-22. Apparently,the overall solution (q2(t), p2(t)) is not exponentially vanishing but it has a relativelylong exponentially vanishing tail–thus the name almost exponential. The maximaltorque is chosen to be τmax = 200.

5.6 The Pendubot

The Pendubot is a two-link revolute planar robot with one actuator at the shoulder,as shown in Figure 5-23. In this section, we provide an explicit global change ofcoordinates that transforms the dynamics of the Pendubot into a cascade nonlinear

144

0 0.5 1 1.5 2 2.5 3−6

−4

−2

0

2

4

time (sec)

q2

p2

0 0.5 1 1.5 2 2.5 3−60

−40

−20

0

20

40

time (sec)

cont

rol (

torq

ue)

−4 −3 −2 −1 0 1 2 3 4−6

−4

−2

0

2

4

6

q2

p 2

(a) (b)

Figure 5-22: (a) Aggressive swing-up of an inverted pendulum on a rotating arm fromthe downward position (q2, p2) = (π, 0), (b) Trajectory of the pendulum in (q2, p2)-plane.

system in nontriangular quadratic normal form. Stabilization of the Pendubot to itsupright equilibrium point using a single (i.e. without switching) static state feedbackis addressed in chapter 7. The inertia matrix for the pendubot is given by (see section

q

q

l 1

L1

m I

m I2 2

1 1

1

2

x

y

Figure 5-23: The Pendubot

A.3)m11(q2) = m1l

21 +m2(L

21 + l22 + 2L1l2 cos(q2)) + I1 + I2

m12(q2) = m21(q2) = m2(l22 + L1l2 cos(q2)) + I2

m22(q2) = m2l22 + I2

where qi, mi, Li, li, and Ii denote angles, masses, lengths, lengths of center of masses,and inertia, respectively. For further use, it is simpler to use the following expressionsfor the inertia matrix

m11(q2) = a+ 2b cos(q2)m12(q2) = m21(q2) = c+ b cos(q2)m22(q2) = c

145

where a, b, c > 0 are given by

a = m1l21 +m2(L

21 + l22) + I1 + I2

b = m2l2L1

c = m2l22 + I2

The potential energy of the pendubot is

V (q1, q2) = (m1l1 +m2L1)g cos(q1) +m2l2g cos(q1 + q2).

The inertia matrix of the pendubot only depends on q2. Thus, q2 is the shape variableof the pendubot. Since the shape variable q2 is unactuated, the pendubot is a Class–IIunderactuated system. The Euler-Lagrange equations of motion for the pendubot areas the following

m11(q2)q1 + m12(q2)q2 − 2b sin(q2)q1q2 − b sin(q2)q22 + g1(q1, q2) = τm21(q2)q1 + m22(q2)q2 + b sin(q2)q

21 + g2(q1, q2) = 0

(5.26)

where the gravity terms are

g1(q1, q2) = −(m1l1 +m2L1)g sin(q1)−m2l2g sin(q1 + q2)g2(q1, q2) = −m2l2g sin(q1 + q2)

Applying the feedback law

τ = (m12 −m11m22

m21

)u− b sin(q2)(m11

m21

p21 + 2p1p2 + p22) + g1(q1, q2)−m11

m21

g2(q1, q2)

reduces the dynamics of q2 to a double integrator

q2 = p2, p2 = u

Applying the change of coordinates

qr = q1 + γ(q2)pr = m11(q2)p1 +m22p2

where

γ(q2) =

∫ q2

0

m22

m21(θ)dθ =

2c√c2 − b2

arctan

(

√

c− bc+ b

tan(q22)

)

; c > b, q2 ∈ [−π, π]

and denoting qs = q2, we get

qr = pr/m21(qs)

pr = m2l2g sin(qr − γ(qs) + qs)− b sin(qs)(pr −m22ps)(pr − (m11 +m22)ps)

m211

qs = psps = u

146

which apparently the equation of pr consists of a quadratic term in (pr, ps) and areduced gravity term

gr(qr, qs) = m2l2g sin(qr − γ(qs) + qs)

as in theorem 4.2.2. To further simplify the reduced dynamics of the pendubot, definethe change of coordinates

z1 = q1 + γ(q2)z2 = pr/m21(qs)

we get

z1 = z2

z2 =m2l2g sin(z1 − γ(q2) + q2)

m21(q2)− b sin(q2)

m21(q2)[(z2 −

m22

m21(q2)p2)

2 +m22

m21(q2)p22]

q2 = p2p2 = u

(5.27)Clearly, the dynamics of the pendubot in (5.27) is neither in strict feedback form, norin strict feedforward form. Therefore, the overall dynamics of the pendubot cannotbe stabilized using famous backstepping and forwarding approaches. Stabilization ofnonlinear systems in nontriangular forms like Pendubot is addressed in chapter 7.

5.7 The Beam-and-Ball System

The beam-and-ball system consists of a beam and a ball with radius r on it (seeFigure 5-24). The task is to bring the ball from any initial position (and speed) tothe origin. Let d ≥ 0 denote the distance between the center of the mass of the balland the beam (d = r in Figure 5-24). In the current literature [34, 102, 80, 79], whatis known as the beam-and-ball system corresponds to the special case where d = 0.We refer to this case as the conventional beam-and-ball system and to the case whered > 0 as the beam-and-ball system. The following treatment applies to the bothcases of d = 0 and d > 0. From section A.4, the elements of the inertia matrix of the

dxr

I 1

θτ

m , I 2

Figure 5-24: The Beam-and-Ball System.

147

beam-and-ball system are given by

m11(x) = I1 +m(x2 + d2)m12 = m21 = −mdm22 = mλ

where I1,m, I2 denote the inertia of the beam, the mass of the ball , and the inertiaof the ball, respectively. Also, λ = 1 + I2/mr

2 > 1. The inertia matrix of thebeam-and-ball system only depends on the position of the ball x. This means thatthe position of the ball is the shape variable of the beam-and-ball system. Since theposition is unactuated, the beam-and-ball system is a Class–II underactuated system.The equations of motion for the beam-and-ball system with d > 0 is as the following

[I1 +m(x2 + d2)]θ − mdx+ 2mxxθ +mg(x cos(θ)− d sin(θ)) = τd−mdθ + mλx−mxθ2 +mg sin(θ) = 0

(5.28)

While the dynamics of the conventional beam-and-ball system with d = 0 is in theform

(I1 +mx2)θ + 2mxxθ +mgx cos(θ) = τ0mλx−mxθ2 +mg sin(θ) = 0

(5.29)

Clearly, θ does not appear in the second equation and after applying

τ0 = (I1 +mx2)u+ 2mxxθ +mgx cos(θ)

after normalizing the units of (x, v) by 1/λ, the dynamics of the conventional beam-and-ball system can be written as

x = vv = −g sin(θ) + xω2

θ = ωω = u

(5.30)

The last equation is exactly the same as the model of the beam-and-ball systemoriginally obtained in [34] and used in [102, 80, 79]. In contrast, consider the beam-and-ball system with d > 0 in (5.28). After applying the following global change ofcoordinates

z1 = −dθ + λxz2 = −dω + λv

(5.31)

and collocated partially linearizing feedback law

τd = (I1 +m(x2 + d2)− md2

λ)u− mdx

λω2 +

mdg sin(θ)

λ+ 2mxvω +mg(x cos(θ)− d sin(θ))

(5.32)

148

the dynamics of the beam-and-ball system with d > 0 transforms into

z1 = z2z2 = −g sin(θ) + (z1 + dθ)ω2

θ = ωω = u

(5.33)

where the units of (z1, z2) are normalized by 1/λ. Equation (5.33) has many similar-ities to the normal form of the conventional beam-and-ball system in (5.30).

5.8 Control of Multi-link Underactuated Planar

Robot Arms

In this section, we discuss reduction of a planar multi-link planar robot arm. Let usstart by a planar three-link robot. The inertia matrix of a three-link arm is as thefollowing (see section A.2)

M(q2, q3) =

m11(q2, q3) m12(q2, q3) m13(q2, q3)m21(q2, q3) m22(q3) m23(q3)m31(q2, q3) m32(q3) m33

Notice that there is an interesting structure in the way the shape variables (q2, q3)appear in the inertia matrix of a three-link robot. The following result shows thatsuch a nested structure in general exists for any n-link planar robot.

Theorem 5.8.1. Consider an n-link planar robot with revolute joints q1, . . . , qn. De-fine the following index sets

Ik := α|α = ij; (k, k) ≤ (i, j) ≤ (n, n), k = 1, . . . , n

Jk := Ik \ Ik+1, k = 1, . . . , n; In+1 = 0

Then ∀α ∈ Jk : mα = mα(qk+1, . . . , qn),∀k = 1, . . . , n−1 and mα = const. for α ∈ Jn.Proof. The proof is by direct calculation using the rotation matrices Ri = R(qi) ∈SO(2) which represent the orientation of each link in R2. Notice that by definitionJk’s are disjoint sets.

Remark 5.8.1. The inertia matrix of an n-link planar robot arm is independent of q1.i.e. qx = q1 and qs = (q2, . . . , qn) are the vector of external variables and the vectorof shape variables for a multi-link planar robot arm, respectively.

In examples 4.5.1 and 4.5.2, we showed that there are three possible actuationconfigurations for a three-link planar robot with two actuators. Furthermore, basedon the method of momentum decomposition, we proved that stabilization of a three-link robot with two actuators reduces to stabilization of the Acrobot, or the Pendubot.In the following, we show that this fact can be generalized to a multi-link planar robotunderactuated by one.

149

Theorem 5.8.2. There exists a change of coordinates in explicit form that transformsthe dynamics of an n-link planar robot (n ≥ 2) with (n−1) actuators into the perturbedreduced normal forms of two-link planar robots (i.e. the Acrobot or the Pendubot) incascade with (n− 2) double-integrators.

Proof. First, assume q1 is not actuated and (n− 1) of shape variables are actuated.Apparently, it is possible to linearize the dynamics of (n − 2) shape variables using(n − 2) control inputs. Only one actuated shape variable, say q2, and q1 are left.Following the line of momentum decomposition procedure in chapter 4, set all otherjoint angles to a constant locked configuration (e.g. zero). The inertia matrix ofthe overall system only depends on q2 and the reduced form for this system is theperturbed normal form of the Acrobot. Now, assume q1 is actuated. Then, (n − 2)controls are remained for (n−1) shape variables. Therefore, the dynamics of all shapevariables except for one, say q2, can be linearized using collocated partial feedbacklinearization. Only an unactuated shape variable q2 and an actuated external variableq1 are left. Thus, by applying momentum decomposition, one obtains the perturbednormal form of the Pendubot in cascade with (n − 2) double-integrators (resultedfrom partial feedback linearization).

Remark 5.8.2. The preceding proof applies to any underactuated (simple) mechanicalsystem with n degrees of freedom, one external variable q1, (n − 1) shape variables(q2, . . . , qn), and (n− 1) control inputs. The special structure of the inertia matrix ofthe n-link robot here plays no major role except for the fact that M is independentof q1.

5.9 Trajectory Tracking for a Flexible One-Link

Robot

In this paper, we introduce a method that provides a nonlinear noncollocated out-put for trajectory tracking of a flexible one-link robot. The link is modeled as afinite-order Lagrangian system obtained from truncated modal analysis. This non-collocated output is derived based on viewing a flexible-link robot arm as an under-actuated mechanical system and then applying an appropriate change of coordinatesthat transforms the system into a cascade nonlinear system with a minimum-phasezero dynamics. The obtained output is the angle of rotation augmented with the(generalized) saturation of the weighted amplitudes of the deflection modes of theflexible link.

Flexible-link robots are finding their way in industrial and space robotics appli-cations due to their lighter weight and faster response time compared to rigid robots.Control of flexible robots has been studied extensively for more than a decade byseveral researchers [12, 18, 25, 81, 110, 107] (see [24] for a recent survey). Despitetheir applications, control of flexible link robots has proven to be rather complicated.The simplest possible flexible arm is a robot with a single flexible link. The dynamicmodel for a flexible-link robot that has been used by almost all of the researchers isthe Euler-Bernoulli model of a beam. This model is a fourth order equation that leads

150

to an originally infinite-dimensional model of a flexible link. The common collocatedoutput for trajectory tracking of flexible one-link robots is the angle of the rotationof the link at the base. The performance of this output measurement turns out to benot satisfactory due to a weak control of the vibrations of the link [20]. This initiatedfinding other noncollocated output measurements like the position of the end-point ofthe link [18]. Based on [85], this choice of the output leads to a non-minimum phasezero-dynamics. In [20], a noncollocated output is proposed as a linear combinationof the angle of rotation and the slope of the beam at its tip. The model used in [20]is an infinite dimensional linear model of a flexible link robot. Then, an approximatefinite order compensator is designed based on [107]. Here, we take a fundamentallydifferent approach to find a noncollocated output. We use a finite-order state-spacemodel of a flexible link derived by truncated modal analysis based on a Lagrangianformalism due to De Luca and Siciliano [25]. This model is naturally in the form ofan underactuated mechanical system with (m + 1) degrees of freedom and a singleactuator where m is the number of deformation modes of the flexible link. Based onour previous results on normal forms for underactuated mechanical systems [68, 66]and theorem 4.2.3, we propose a noncollocated output as the angle of the rotationaugmented with a generalized saturated weighted linear combination of the deforma-tion amplitudes of the link. We prove that the zero-dynamics corresponding to thisoutput is minimum-phase.

5.9.1 Model of a Flexible Link

The flexible link depicted in Figure 5-25 is modeled as an Euler-Bernoulli beam [25]satisfying

EI∂4w(ζ, t)

∂ζ4+ ρAL4∂

2w(ζ, t)

∂t2= 0

where ζ = x/L is the normalized position along the link of length L and A, I, E, ρare physical parameters of the link. By the assumption of separability in time andspace, the deformation of the beam can be expressed as W (ζ, t) = φ(ζ)δ(t). After

w

M J

x

y

θα

LL

X

Y

Figure 5-25: A flexible one-link robot arm.

performing a truncated modal analysis with m modes, the Lagrangian equations of

151

the flexible link can be written as the following [25]

[

m11(δ) m12

m21 m22

] [

θ

δ

]

+

[

c1(θ, δ, δ)

c2(θ, δ)

]

+

[

0

Kδ + F δ

]

=

[

τ0

]

(5.34)

where θ is the angle of rotation, δ ∈ Rm denotes the deformation amplitudes, M(δ) =mij is the inertia matrix that is a positive definite symmetric matrix and K,F arepositive definite matrices as well. The expressions for mij’s are explicitly given in[25]. Here, we only need an explicit form of m11(δ) (the rest of mij’s are constant) as

m11(δ) = k1 + k2(ΦTe δ)

2 (5.35)

where k1 > 0, k2 =ML i.e. payload mass, Φe ∈ Rm is a constant column vector thatis determined by the m eigenfunctions obtained from the modal analysis. In addition,Coriolis and centrifugal terms can be explicitly given as

c1(θ, δ, δ) = 2MLθ(ΦTe δ)(Φ

Te δ)

c2(θ, δ) = −MLθ2(ΦT

e δ)Φe

Apparently, from (5.34), the flexible one-link robot is an underactuated mechanicalsystem with m + 1 degrees of freedom and one control. The important feature of(5.34) is that the system has kinetic symmetry w.r.t. θ. In the next section, we showthat this symmetry property has a crucial role in derivation of a minimum phasenoncollocated output for the flexible one-link robot.

To apply theorem 4.2.3 to the case of the flexible one-link robot in (5.34), all theelements of

ω1 = m11(δ)−1m12dδ

must be exact one-forms. If this holds, then a nonlinear noncollocated output in theform of y = θ+γ(δ) exists that transforms the system into a cascade nonlinear system.However, this is false and the one-form m11(δ)

−1m12dδ is not exact due to the factthat m12 6= λΦT

e for any λ ∈ R. This suggests an alternative way of constructing thedesired output. By inspection of m11(δ) in (5.35), one can observe that the followingmodified one-form

ω2 = m11(δ)−1ΦT

e dδ

is exact and thus the expression

θ +m11(δ)−1ΦT

e δ

is integrable as an output y = θ+ γ(δ). Next, we show that this is indeed the outputwe were seeking.

5.9.2 The Noncollocated Output

In this section, we derive a noncollocated output and prove that its correspondingzero-dynamics is minimum phase. But first we need to make some assumptions and

152

give a lemma.

Notation. For any two vectors x, y ∈ Rm and the positive definite m×m matrix Q,define the following inner product

〈x, y〉Q := xTQy

Assumption 5.9.1. Suppose the following properties hold

i) Φe satisfies 〈m21,m21〉m−122> 〈Φe,m21〉m−1

22.

ii) The matrix Q0 = m21m12 −m21ΦTe is positive definite.

Remark 5.9.1. By the specific structure of the inertia matrix M(δ) for a flexible linkrobot in [25], assumptions (i) and (ii) are very reasonable and can be algebraiclychecked.

Lemma 5.9.1. Suppose Assumption 5.9.1 holds. Then the following matrices arepositive definite and thus invertible

Q1 = m11(δ)− ΦTem

−122m21

Q2 = m22 −m21m−111 Φ

Te

Proof. First, note that the following matrices are symmetric and positive definite

Q3 = m11(δ)−m12m−122m21

Q4 = m22 −m21m−111 (δ)m12

The proof for this can be found in [90]. This is true regardless of how many deforma-tion modes are chosen as long as mij’s are elements of a symmetric positive definiteinertia matrix M . Noting that Q1 can be rewritten as

Q1 = Q3 + (m12 − ΦTe )m

−122m21

by i) in Assumption 5.9.1, the second term is positive and Q3 is positive as well.Thus, Q1 is positive definite. To prove Q2 is positive definite, notice that for anyx ∈ Rm we have

xTQ2x = xTQ4x+ xTm21m11(δ)−1(m12 − ΦT

e )x

orxTQ2x = xTQ4x+m11(δ)

−1xTQ0x

and because by Assumption 5.9.1 Q0 is positive definite and also m11(δ)−1 > 0, Q2 is

a positive definite matrix.Now, we are ready to present our main result for output tracking of a flexible-linkrobot:

153

Theorem 5.9.1. The nonlinear noncollocated output

y = θ + γ(δ) (5.36)

where γ : Rm → R is given by

γ(δ) =1√k1k2

arctan(

√

k1k2

ΦTe δ) (5.37)

has global uniform relative degree 3 w.r.t. u = τ and determines a global change ofcoordinates

z1 = δ, z2 = δ, ξ1 = y, ξ2 = y, ξ3 = y

that transforms the dynamics of a flexible one-link robot augmented with an integratoru = τ into the following partially linear cascade nonlinear system

z1 = z2z2 = f(z, ξ1, ξ2, ξ3)

ξ1 = ξ2ξ2 = ξ3ξ3 = v

(5.38)

with a new controlv = α(δ)u+ β(δ, δ, θ)

In addition, there exists an ε > 0 with the property ε ∝√

σmin(F ) such that overa neighborhood Uε(0) = (z1, z2) : |z1| < ε, |z2| < ε of (z1, z2) = 0, for the zero-dynamics corresponding to this output the origin is locally uniformly asymptoticallystable (i.e. the noncollocated output y is minimum phase).

Proof. By direct calculations, for y = θ + γ(δ) in (5.36), we have

y = θ +ΦTe δ/k2

k1/k2 + (ΦTe δ)

2

= θ +ΦTe δ

k1 + k2(ΦTe δ)

2

= θ +m11(δ)−1(ΦT

e δ)

alsoy = θ +m11(δ)

−1ΦTe δ − 2k2m11(δ)

−2(ΦTe δ)(Φ

Te δ)

2

but from the second line of equation (5.34), δ can be obtained as

δ = −m−122m21θ −m−1

22 (c2 +Kδ + F δ)

154

Thus, combining the last two equations, we get

y = m11(δ)−1(m11(δ)− ΦT

em−122m21)θ

− m11(δ)−1ΦT

em−122 (c2 +Kδ + F δ)

− 2k2m11(δ)−2(ΦT

e δ)(ΦTe δ)

2

On the other hand, after cancelling δ from the first and second lines of (5.34), oneobtains

(m11(δ)−m12m−122m21)θ + c1 −m12m

−122 (c2 +Kδ + F δ) = τ

Hence, takingm11(δ) = (m11(δ)−m12m

−122m21) > 0

we have

y = m11(δ)(m11(δ)− ΦTem

−122m21)m11(δ)

−1 · (τ − c1 +m12m−122 (c2 +Kδ + F δ))

− m11(δ)−1ΦT

em−122 (c2 +Kδ + F δ)− 2k2m11(δ)

−2(ΦTe δ)(Φ

Te δ)

2

This means that applying the change of control

w = (m11(δ)−m12m−122m21) · (m11(δ)− ΦT

em−122m21)

−1m11(δ)τ + β(δ, δ, θ)

(where β can be explicitly calculated from the last equation of y) partially linearizesthe dynamics of the system as

y = w

and therefore the output y has global relative degree 2 w.r.t. τ and global relativedegree 3 w.r.t. v. To determine the stability of the zero-dynamics corresponding toy, let z1 = δ, z2 = δ and set y ≡ 0 (thus, y, y ≡ 0). From the second equation in(5.34), the zero-dynamics system can be determined as

(m22 −m21m11(δ)−1ΦT

e )δ + 2k2m21m11(δ)−2(ΦT

e δ)(ΦTe δ)

2 + c1 +Kδ + F δ = 0

but c1 = −k2θ2(ΦTe δ)Φe and θ = −m11(δ)

−1(ΦTe δ) (because y = 0), thus

c1 = −k2m11(δ)−2(ΦT

e δ)2(ΦT

e δ)Φe

The equations of the zero-dynamics can be expressed as the following

(m22−m21m11(δ)−1ΦT

e )δ+k2m11(δ)−2(ΦT

e δ)(ΦTe δ)

2(2m21−Φe)+Kδ+F δ = 0 (5.39)

DenotingQ(δ) = (m22 −m21m11(δ)

−1ΦTe )

and noting that Q(δ) is a positive definite matrix, the following Lyapunov functionfor the zero-dynamics system can be proposed

V (δ, δ) =1

2δTQ(δ)δ +

1

2δTKδ

155

Calculating V along the solutions of the zero-dynamics in (5.39), we obtain

V = δTQ(δ)δ + δTKδ +1

2δT Qδ

where

Q =∂Q

∂δδ = 2k2m21m11(δ)

−2(ΦTe δ)(Φ

Te δ)Φ

Te

and1

2δT Qδ = k2m11(δ)

−2(m12δ)(ΦTe δ)(Φ

Te δ)

2

HenceV = −δTF δ − k2m11(δ)

−2(ΦTe δ)

2(ΦTe δ) · ((m12δ)− (ΦT

e δ))

and because m11(δ)−2 ≤ k−21 , we get

V ≤ −σmin(F )|δ|2 +k2k21|Φe|3(|m21 − Φe|)|δ|3|δ|

where σmin(F ) is the smallest singular value of F . Denoting

k3 =k2k21|Φe|3|m21 − Φe|

and defining

ε =

√

σmin(F )

k3

for |δ|, |δ| < ε (i.e. over Uε(0)), V ≤ 0. But the largest invariant set in z2 = δ = 0(i.e. V = 0) for (5.39) is (z1, z2) = (0, 0). Therefore, from LaSalle’s invarianceprinciple the origin (δ, δ) = (z1, z2) = 0 is uniformly locally asymptotically stable forthe zero-dynamics system, i.e. the output y is (locally) minimum phase.

Remark 5.9.2. Viewing σmin(F ) as the strength of damping of the flexible link, theresult of the preceding theorem can be interpreted as follows. The higher the strengthof damping, the larger the region of attraction of the origin for the zero-dynamics.

Remark 5.9.3. For a sufficiently small deformation amplitudes δ. The output y in(5.36) can be expressed as a linear combination of the angle of rotation and weighteddeformation amplitudes as

y = θ +1

ML

(ΦTe δ)

which depends on the payload mass ML. This is in agreement with our intuition thatthe payload mass matters in trajectory tracking control design for a flexible arm.

156

5.9.3 Tracking Control

The control for trajectory tracking can be given as the following. Let yd(t) denotethe desired trajectory and define the error functions

e1 = y − yd(t), e2 = y − yd, e3 = y − yd

Thene1 = e2e2 = e3e3 = u− y(3)d

where y(k)d =

dkyd(t)

dtk. Thus, setting

u = y(3)d + cp(y − yd) + cd(y − yd) + ca(y − yd)

with cp, cd, ca < 0, or applying the partial state feedback

u = y(3)d + cp(ξ1 − yd) + cd(ξ2 − yd) + ca(ξ3 − yd) (5.40)

guarantees that (e1, e2, e3)→ 0 as t→∞ and for sufficiently small initial conditions(δ(0), δ(0)) asymptotic output tracking can be achieved [36].

5.10 Configuration Stabilization for the VTOL Air-

craft

In this section, we give a static state feedback law for global configuration stabilizationof the VTOL (vertical take off and landing) aircraft. The simplified dynamics of the

uε u

u θ

1

2

2

x

y

Figure 5-26: The VTOL aircraft.

157

VTOL aircraft, as shown in Figure 5-26, is given in [35, 55] as the following

x1 = x2x2 = −u1 sin(θ) + εu2 cos(θ)y1 = y2y2 = u1 cos(θ) + εu2 sin(θ)− gθ = ωω = u2

(5.41)

where ε 6= 0. Here, θ denotes the roll angle and the plane moves in a vertical (x1, y1)-plane. Clearly, the system has three degrees of freedom and only two actuators–thusit is underactuated. The effect of the body torque u2 appears in the translationaldynamics of the VTOL aircraft by a factor of ε. This captures a general propertyof real aircrafts with 6 DOF that exhibit a similar input coupling effect. The casewith |ε| ¿ 1 resulting in a weak input coupling has been mostly considered in theliterature [35, 55, 80]. Here, we are interested in the strong input coupling case withan arbitrary ε 6= 0. To decouple the effect of u2 in the translational dynamics of theVTOL aircraft, first let rewrite the dynamics of the VTOL aircraft as

mxxqx − gx(qx) = Fr(qs)τr + Fx(qs)τmssqs = Fs(qs)τ

(5.42)

where qx = (x1, y1)T , qs = θ, mxx = I2×2, mss = 1, Fs = 1, and Fr, Fx, gx are given by

Fr(qs) =

[

− sin(θ)cos(θ)

]

, Fx(qs) =

[

ε cos(θ)ε sin(θ)

]

, gx =

[

0−g

]

which clearly shows Fr(qs) is a unit vector. Based on theorem 4.3.2, we need to checkwhether the vector of one-forms ω defined as ω = m−1

xxFx(qs)F−1s (qs)mssdqs has exact

elements or not. By direct calculation, we get

ω = Fx(qs)dqs =

[

ε cos(θ)dθε sin(θ)dθ

]

which has exact elements. Moreover, ω has an exact differential γ(θ) = [ε sin(θ),−ε(cos(θ)−1)]T that satisfies dγ(θ) = ω. Therefore, applying the following global change of co-ordinates

z1 = x1 − ε sin(θ)z2 = x2 − ε cos(θ)ωw1 = y1 + ε(cos(θ)− 1)w2 = y2 − ε sin(θ)ωξ1 = θξ2 = ω

(5.43)

158

transforms the dynamics of the VTOL aircraft into

z1 = z2z2 = − sin(ξ1)u1 =: v1w1 = w2

w2 = cos(ξ1)u1 − g =: v2ξ1 = ξ2ξ2 = u2

(5.44)

and decouples the translational and attitude dynamics. Here, u1 = u1 − εξ22 is thenew control. Due to [35], the system in (5.44) is dynamic feedback linearizable. Inthis work, we take a different approach that makes use of a static state feedback andbackstepping procedure for global configuration stabilization of the VTOL aircraft.Clearly, (z1, w1) are two flat outputs of the VTOL aircraft which are obtained auto-matically as the by-products of the decoupling change of coordinates (see [29, 106, 105]for more details on differential flatness and flat outputs).

Applying the linear state feedback v1 = c1z and bounded feedback v2 = c0σ(c2w)where σ(s) = tanh(s) and 0 < c0 < g, globally asymptotically and locally exponen-tially stabilizes (z, w) = (0, 0) for the (z, w)-subsystem in (5.44) where c1, c2 ∈ R2 arecoefficients of a Hurwitz polynomial (a rather similar choice is made for v1 in [80] butwith ε = 0 which is not the case here). An important point is that, the unboundedchoice of v2 = c2w is not possible due to a singularity that appears later in controldesign. To stabilize the origin for the composite system, we use the backsteppingprocedure as follows. Solving for u1 and tan ξ1 in equation (5.44) gives

u1 = k1(z, w) :=√

v21 + (v2 + g)2

tan ξ1 = k2(z, w) =−v1v2 + g

Then, defining the change of coordinates and control

µ1 = tan ξ1 − k2(z, w)µ2 = (1 + tan2 ξ1)ξ2 − k2u2 = (1 + tan2 ξ1)(u2 + 2 tan ξ1ξ

22)− k2

we getµ1 = µ2µ2 = u2

Thus, applying u2 = −d1µ1 − d2µ2 with d1, d2 > 0 globally exponentially stabilizes(µ1, µ2) = (0, 0) for the µ-subsystem. The dynamics of the closed-loop system is inthe form

η = f(η, µ)µ = Aµ

(5.45)

where η = col(z, w), A is a Hurwitz matrix. Given µ = 0, for η = f(η, 0), η = 0 isglobally asymptotically and locally exponentially stable. It can be shown that for any

159

solution of the µ-subsystem the solution of the η-subsystem is uniformly bounded andthe asymptotic stability of (η, µ) = 0 for the cascade system in (5.45) follows from atheorem due to Sontag in [82]. This guarantees global asymptotic stability and localexponential stability of the origin for the VTOL aircraft. Figure 5-27 (a) shows thetrajectories of the VTOL aircraft from initial condition (2, 3, 4, 1, π/3, 1) with ε = 0.1.The path of the center of mass of the aircraft is shown in Figure 5-27 (b). Based onthe simulation result in Figure 5-27, the controller performs an aggressive maneuverto stabilize the position and attitude of the aircraft. In the sense that, after a rathershort transient period, the states of the system almost exponentially converge to zero.

0 2 4 6 8 10 12−1

0

1

2

3

4

5

time (sec)

x y roll

0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

time (sec)

dx/dt dy/dt d(roll)/dt

−0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

y

(a) (b)

Figure 5-27: (a) The state trajectories of the VTOL aircraft, and (b) The path of thecenter of mass of the aircraft in (x1, y1)-plane.

160

5.11 Trajectory Tracking and Stabilization of An

Autonomous Helicopter

This section is devoted to trajectory tracking and attitude stabilization nonlinear con-trol design for an autonomous helicopter capable of performing aggressive maneuvers.A helicopter is an underactuated mechanical system with 6 DOF (i.e. the positionand the attitude in R3 × SO(3)) and 4 control inputs which includes the thrust ofthe main rotor and 3 control moments in the body frame. As the dynamic modelof the helicopter, we use a model introduced in [43]. The reason for this choice isthat the helicopter model in [43] is rather accurate. In the sense that, both the effectof the body moments on the translational dynamics and the effect of the thrust ofthe main rotor on the attitude dynamics are taken into account. These two mutualeffects complicate control design for the helicopter. To derive the differentially flatoutputs [29, 106, 105] for this helicopter model, the effects of the body moment on thetranslational dynamics are ignored in [42, 43]. We present a method for decouplingof the translational and attitude dynamics of the helicopter. As a by-product, thisdecoupling method provides the differentially flat outputs for the accurate model ofthe helicopter. In addition, the problem of global exponential tracking of feasibletrajectories of a helicopter is addressed.

We use a rotation matrix R ∈ SO(3) to represent the attitude of the helicopter.This way, the singularities of the Euler angle parameterization can be avoided. In thepast, the use of a rotation matrix in SO(3) as the attitude, has proven to be efficientfor attitude control and tracking of satellites [108, 16], spacecraft [17], and helicopters[43, 31].

T

W

gX

YZ

(x,R)

Figure 5-28: A helicopter.

5.11.1 Dynamic Model of A Helicopter

The dynamic model of the helicopter used in this work is based on a model of thehelicopter given in [43]. Let q = (x,R) ∈ R3 × SO(3) denote the configuration ofthe helicopter depicted in Figure 5-28. Here, R denotes that transformation from the

161

body frame to the reference frame and is defined as

R = exp(ψe3) exp(θe2) exp(φe1)

where ei’s are standard basis in R3 and η = (φ, θ, ψ) denotes the vector of three Eulerangles associated with rotations around x, y, z axes. The angles φ, θ, ψ are called roll,pitch, and yaw, respectively. The attitude kinematics is given by

R = Rω

where ω = (ω1, ω2, ω3)T denotes the vector of angular velocities in the body frame

and ω (i.e. ω· = ω × ·) is a skew-symmetric matrix

ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

The relation between η and ω is as

η = Φ(η)ω

where Φ(η) is called the Euler matrix and is given by

Φ(η) =

1 sinφ tan θ cosφ tan θ0 cosφ − sinφ0 sinφ sec θ cosφ sec θ

(5.46)

The matrix Φ(η) has a singularity at θ = ±π/2. For this reason, we use R for thetracking control design.

The equations of motion for the helicopter are as the following

x = vmv = RFbR = RωJω = −ω × Jω + τb

(5.47)

where Fb, τb denote the forces and moments in the body frame and (x, v) denote theposition and the velocity of the center of mass of the helicopter in the reference frame.The forces and moments in (5.47) are in the form

Fb = mgRT e3 − TeT + Tte2τb = Q(T )eT +Dτ

(5.48)

where T is the thrust of the main rotor, Tt is the thrust of the tail rotor, eT is a unit

162

vector along the shaft of the main rotor defined by

eT (a, b) =

sin(a)− cos(a) sin(b)cos(a) cos(b)

(5.49)

with a, b denoting the pitch and roll tilt angles of the tip of the main rotor, τ =(bT, aT, Tt)

T , D = diag(lb, la, lt) is a constant matrix with positive diagonal elements,and

Q(T ) = Q0 +Q1T1.5

with relatively small constants Q0, Q1 > 0.

Assumption 5.11.1. Notice that for (a, b) = (0, 0), eT = e3. In addition, for|a|, |b| ¿ 1, eT can be approximated as

eT ≈

a−b1

= e3 +

a−b0

Since, a, b are small in practice, we use this approximation of eT instead of eT itself.Moreover, we use the following approximation

Q(T )eT ≈ Q(T )e3

This is justified by the fact that if Q0, Q1, a, b,∼ O(ε), then the first two elements ofQ(T )eT are of the order O(ε2) and can be ignored.

The following simplified forces and body moments are directly used in [31] asthe original expressions of forces and moments. For the sake of clarity, we restatethe approximate forces and moments in the body frame under the approximationassumption 5.11.1.

Corollary 5.11.1. Suppose assumption 5.11.1 holds and let u = Dτ ∈ R3. ThenFb, τb take the following forms

Fb = mgRT e3 − Te3 + E · uτb = Q(T )e3 + u

(5.50)

where

E =

0 ε2 0ε1 0 ε30 0 0

; ε1 = 1/lb, ε2 = −1/la, , ε3 = 1/lt

Proof. The proof is elementary and by direct substitution.Based on corollary 5.11.1, the first-level approximate model of the helicopter is in

163

the formx = v

mv = mge3 − (Re3)T +REu

R = RωJω = −ω × Jω +Q(T )e3 + u

(5.51)

Under a further approximation, by setting E = 03×3 in 5.51, one obtains a second-levelapproximate model of the helicopter as

x = vmv = mge3 − (Re3)T

R = RωJω = −ω × Jω +Q(T )e3 + u

(5.52)

where the control input of the attitude dynamics u has no effect on the translationaldynamics. This is the “approximate model” used in [42, 43, 42]. Here, our main goalis to address trajectory tracking for model (5.51) where E 6= 03×3.

5.11.2 Input Moment Decoupling for A Helicopter

Consider the first-level approximate model of the helicopter in (5.51) with inputmoment u. In this section, we seek a differentially flat output [29, 106, 105] in theform y = x + γ(η) such that y is independent of u. If y is independent of u for allη, we call y = x + γ(η) an exact flat output. On the other hand, if y = x + γ(η) isindependent of u at η = η0, we call y = x+ γ(η) a weak flat output. In the following,we show that there exists no exact flat output y = x+ γ(η) for the helicopter model(5.51). Before doing so, let us present the dynamics of the helicopter using the Eulerangle parameterization and identify the shape variables of the helicopter.

The first-level approximate model of the helicopter in equation (5.51) can berewritten as

x = vmv = mge3 − n(η)T +R(η)Euη = Φ(η)ω

Jω = −ω × Jω +Q(T )e3 + u

(5.53)

where n(η) = R(η)e3 and we refer to it as the unit normal vector. R(η) is given by

R(η) =

cosψ − sinψ 0sinψ cosψ 00 0 1

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

1 0 00 cosφ − sinφ0 sinφ cosφ

or equivalently

R(η) =

cos θ cosψ sinφ sin θ cosψ − cosφ sinψ cosφ sin θ cosψ + sinφ sinψcos θ sinψ sinφ sin θ sinψ + cosφ cosψ cosφ sin θ sinψ − sinφ cosψ− sin θ sinφ cos θ cosφ cos θ

164

Define Ψ(η) = Φ−1(η) and note that Ψ(η) can be calculated as

Ψ(η) =

1 0 − sin θ0 cosφ cos θ sinφ0 − sinφ cos θ cosφ

Denote the configuration vector of the helicopter by q = (x, η). The Lagrangian ofthe helicopter is in the form

L(q, q) = 1

2mvTv +

1

2ωTJω +mgx3

which can be rewritten as

L(x, η, x, η) = 1

2

[

xη

]T

·[

mI3 00 Ψ(η)TJΨ(η)

]

·[

xη

]

+mgx3 (5.54)

The inertia matrix of the helicopter is a block diagonal matrix

M(η) = diag(mI3,Ψ(η)TJΨ(η))

Therefore, the Euler angles are the shape variables of a helicopter, or qs = (φ, θ, ψ)and the position qx = x is the vector of external variables of a helicopter. We needthe following result to check whether a flat output exists for a helicopter.

Proposition 5.11.1. The attitude dynamics of the helicopter can be expressed in theform

Ms(η)η + Cs(η, η)η = Ψ(η)T τb (5.55)

with a well-defined positive definite inertia matrix Ms(η) = Ψ(η)TJΨ(η) (for θ 6=±π/2) given by

Ms(η) =

J1 0 −J1sθ0 J2s

2φ + J3c

2φ (J2 − J3)cθcφsφ

−J1sθ (J2 − J3)cθcφsφ J1s2θ + c2θ(J2s

2φ + J3c

2φ)

(5.56)

(“cθ” and “sθ” denote cos(θ) and sin(θ), respectively) and the Coriolis and centrifugalmatrix

Cs(η, η) = Ψ(η)TJΨ(η, η)−ΨT (η)[(Ψ(η)η)∧]JΨ(η) (5.57)

where J = (diag(J1, J2, J3), τb = Q(T )e3 + u denotes the total body moment, and ·∧denotes the hat operation. In addition, for the case J2 6= J3 (i.e. the anti-symmetriccase) the shape variables of the helicopter are φ, θ; while given J2 = J3 (i.e. theyz-symmetric case) the only shape variable of the helicopter is θ.

Proof. Let us first rewrite the attitude dynamics as the following

η = Φ(η, η)ω + Φ(η)ω

= Φ(η, η)Ψ(η)η − Φ(η)J−1ωJω + Φ(η)J−1τb

165

which multiplying both sides of the last equation by ms(η) = Ψ(η)TJΨ(η) gives theattitude dynamics in the question with

Cs(η, η) = Ψ(η)TJΨ(η)Φ(η, η)Ψ(η)−Ψ(η)T [(Ψ(η)η)∧]JΨ(η)

Noting that Ψ = Ψ(η)Φ(η, η)Ψ(η) finishes the proof of the first part. The secondpart of the result for the anti-symmetric case follows from the fact that Ms(η) isindependent of ψ. In addition, for the symmetric case where J2 = J3, the inertiamatrix of the helicopter in (5.56) takes the form

Ms(θ) =

J1 0 −J1sθ0 J2 0

−J1sθ 0 J1s2θ + J2c

2θ

(5.58)

which does not depend on φ, ψ.Due to the fact that the attitude dynamics in (5.55) is a fully-actuated mechanical

system for θ 6= ±π/2, it is exact feedback linearizable. For the sake of completeness,this is stated as the following corollary.

Corollary 5.11.2. The collocated partially linearizing feedback law

τb = JΨ(η)u+ Φ(η)TCs(η, η)η

which is invertible for θ 6= ±π/2, transforms the attitude dynamics in (5.55) into a3-dimensional double-integrator

η = νν = u

with a new control u.

Following the line of theorem 4.3.1 and based on proposition 5.11.1, the dynamicsof the helicopter in (5.53) can be expressed as

mxxqx − gx = Fr(qs)T + Fx(qs)umss(qs)qs + hs(qs, qs) = Fs(qs)Q(T )e3 + Fs(qs)u

(5.59)

where (qx, qs) = (x, η) and

mxx = mI3

mss(qs) = ΨT (η)JΨ(η)

hs(qs, qs) = Cs(η, η)η

Fr(qs) = −n(η)Fx(qs) = R(η)E

Fs(qs) = Ψ(η)T

Based on theorem 4.3.1, if the following vector of coupling one-forms

ωxs = m−1xxFx(qs)F

−1s (qs)mss(qs)dqs

166

which can be rewritten as

ωxs = m−1R(η)EJΨ(η)dη (5.60)

has exact elements. Then, there exists an exact flat output y = x + γ(η) such thatωxs = dγ(η). Without loss of generality assume m = 1. The following result assuresthat such an exact flat output does not exist for a helicopter.

Proposition 5.11.2. There exists no exact flat output y = x+γ(η) for the helicopter,i.e. there exists no local diffeomorphism γ(η) : R3 → R3 such that ωxs = dγ(η).

Proof. Let fi(η) denote the ith column of R(η)EJΨ(η). Then, ωxs has exact elementsif and only if fi’s satisfy the following property

∂fi(η)

∂ηj=∂fj(η)

∂ηi, ∀i, j = 1, 2, 3, i 6= j (5.61)

where (η1, η2, η3) = (φ, θ, ψ). By direct symbolic calculations in MATLAB, it hasbeen shown that there exists no pair (i, j) with i 6= j that satisfies condition (5.61)(see Appendix B for the code that symbolicly generates the difference of the left andright hand side of the equation (5.61)). Therefore, none of the elements of ωxs areexact one forms.

The following proposition, provides the weak flat outputs for the first-level approx-imate model of the helicopter using momentum decomposition procedure in chapter4.

Proposition 5.11.3. Consider the dynamics of the helicopter as the following

x = vmv = mge3 − (R(η)e3)T +R(η)Euη = ν

Ms(η)ν = −Cs(η, ν)ν +Ψ(η)T u

(5.62)

where u = Q(T )e3 + u and it is assumed that EQ(T )e3 ≈ 0. Define the couplinginertia µ(η) as the following

µ(η) = R(η)EJΨ(η) (5.63)

and let π = µ(η)η denote the nonintegrable coupling momentum between the body mo-ments and the translational dynamics of a helicopter. (The nonintegrability propertyof π is due to proposition 5.11.2). Then, using momentum decomposition procedureπ can be represented as

π = πl + πe

where πl = Gl(η)η denotes the integrable locked momentum associated with locked

167

configuration η = 0 which is given by

πl =

0 ε2J2 cos θ −ε3J3 sinψε1J1 cosφ 0 ε3J3 cosψε1J1 sinφ −ε2J2 sin θ 0

φ

θ

ψ

(5.64)

and πe denotes the non-integrable error momentum that vanishes identically at η = η.In addition, there exists a diffeomorphism γ(η) = (γφ(η), γθ(η), γψ(η))

T satisfyingγ(0) = 0 where

γφ(θ, ψ) = ε2J2 sin θ + ε3J3(cosψ − 1)γθ(φ, ψ) = ε1J1 sin(φ) + ε3J3 sinψγψ(φ, θ) = −ε1J1(cosφ− 1) + ε2J2(cos θ − 1)

(5.65)

such that γ = πl. In addition, the following output

y∗ = mx− γ(η) (5.66)

is a weak flat output (i.e. y∗ does not depend on the body moment u at η = 0) thatsatisfies

y∗ = mge3 − (Re3)T + Ce(η, η)η + Ge(η)u

where

Ce(η, η) = Cl(η, η)−Gl(η)M−1s (η)Cs(η, η)

Cl(η, η) = Gl

Ge(η) = G(η)−Gl(η)

Ge(η) = Ge(η)Ψ−1(η)J−1

and both Ge(η) and Ge(η) vanish at η = η.

Proof. To obtain the locked momentum πl = Gl(η)η from the original non-integrablemomentum π = G(η)η, we use the momentum decomposition procedure as the fol-lowing. Let

R1 = exp(φe1), R2 = exp(θe2), R3 = exp(ψe3)

then, R = R3R2R1 and G(η) can be written as

G(η) = R3R2R1EJΨ(η)

Consider the following matrices

G1 = G(η)|η=(φ,0,0), G2 = G(η)|η=(0,θ,0), G3(η)|η=(0,0,ψ)

To find πl, we need to calculate the ith column Ci of Gi for i = 1, 2, 3. This way eachCi is a function of a single variable. Note that Ψ(η) = Ψ(φ, θ). By definition, Gi’s

168

can be rewritten as

G1(φ) = R1EJΨ(φ, 0), G2(θ) = R2EJΨ(0, θ), G3(ψ) = R3EJΨ(0, 0)

Setting Gl(η) = [C1(φ), C2(θ), C3(ψ)] gives the locked momentum πl = Gl(η)η inthe question (see Appendix B for the symbolic calculation of Gl(η)). Clearly, πl isintegrable and γ = Gl(η)η. Calculating y∗, we get

y∗ = mge3 − (Re3)T +REu− γ

where

γ = Glη +Gl(η)η

= Cl(η, η)η + (G(η)−Ge(η))[M−1s (η)Ψ(η)T u−M−1

s (η)Cs(η, η)η]

= [Cl(η, η)−Gl(η)M−1s (η)Cs(η, η)]η + (G(η)−Ge(η))Ψ

−1(η)J−1u

= Ce(η, η)η +REu−Ge(η)Ψ−1(η)J−1u

whereCe(η, η) = Cl(η, η)−Gl(η)M

−1s (η)Cs(η, η)

Therefore, y∗ takes the following form

y∗ = mge3 − (Re3)T − Ce(η, η)η +Ge(η)Ψ−1(η)J−1u

The weak flat property of the output y∗ follows from the fact that Ge(η) vanishes atη = η = 0.

Corollary 5.11.3. After applying the change of coordinates

z1 = y∗, z2 = y∗

the dynamics of the helicopter in (5.62) transforms into

z1 = z2z2 = mge3 − (Re3)T +∆(η, ν, u)η = ν

Ms(η)ν = −Cs(η, ν)ν +Ψ(η)T u

(5.67)

where the perturbation ∆ is given by

∆(η, ν, u) = −Ce(η, ν)ν + Ge(η)u

In addition, assume ∃L1, L2 > 0 : σmax(Ce(η, ν)) ≤ L1‖ν‖, σmax(Ge(η)) ≤ L2‖η‖,∀η, ν.Let u = Ka(z, η, ν) be a state feedback such that (η(t), ν(t), u(t)) is exponentiallyvanishing with a rate σ1 > 0. Then, ∆(t) is exponentially vanishing with a rateσ2 = 2σ1 > 0.

Proof. The proof is straightforward and is omitted.

169

Remark 5.11.1. Based on the preceding corollary, one can solve the tracking problemfor the unperturbed system with ∆ = 0 and then perform a Lyapunov-based robuststability analysis for the overall system with a perturbation ∆.

5.11.3 Nontriangular Normal Form of a Helicopter

The following proposition provides a non-triangular normal form for the first-levelapproximate model of a helicopter. This normal form will be used later in chapter7 for stabilization of the helicopter without decoupling or ignoring the effect of theinput moment on the translational dynamics.

Proposition 5.11.4. Define the state variables and control

z = col(x, v), ξ1 = η, ξ2 = ν, w = u (5.68)

and assume EQ(T )e3 ≈ 03. Then, the first-level approximate model of the helicoptercan be expressed in the following nontriangular normal form quadratic in ξ2 and affinein w

z = f(z) + g1(ξ1)T + g2(ξ1, ξ2, ε) + g3(ξ1, ε)w

ξ1 = ξ2ξ2 = w

(5.69)

where ε = (ε1, ε2, ε3)T and f, g1, g2, g3 are defined as the following

f(z) =

[

vge3

]

(5.70)

g1(ξ1) =

[

03−m−1(R(η)e3)

]

(5.71)

g2(ξ1, ξ2, ε) =

[

03R(η)E(ε)Φ(η)TCs(η, ν)ν

]

(5.72)

g3(ξ1, ε) =

[

03R(η)E(ε)JΨ(η)

]

(5.73)

with the property that g2, g3 vanish at ε = (0, 0, 0)T .

Proof. The proof is by direct calculation and follows from proposition 5.11.1 andcorollary 5.11.2.

5.11.4 Feedback Linearization of the Unperturbed Model ofthe Helicopter

The dynamics of the unperturbed model of the helicopter is in the following form

x = vmv = mge3 − (Re3)T

R = RωJω = −ω × Jω + τb

(5.74)

170

Both the translational and the attitude dynamics of the model of the helicopter in(5.74) can be linearized as

x = vv = w

R = Rωω = τ

(5.75)

where w ∈ R3 is the control of the (x, v)-subsystem and τ ∈ R3 is the new control forthe attitude dynamics. The relation between τb and τ is given by

τb = Jτ + ω × Jω (5.76)

Based on equation (5.75), the relation between the new control w, the thrust T , andthe normal vector n = Re3 is as the following

mge3 − nT = mw

From the last equation, one can solve for T and n as

T = m‖ge3 − w‖n =

ge3 − w‖ge3 − w‖

(5.77)

Assume that the controller for the translational dynamics is a state feedback

w = k(x, v)

Then, the feedback law w = k(x, v) defines both T and n as the following

T (x, v) = m‖ge3 − k(x, v)‖nc(x, v) =

ge3 − k(x, v)‖ge3 − k(x, v)‖

(5.78)

We refer to nc = nc(x, v) as the control normal vector.

Remark 5.11.2. The reader should notice that the normal vector n = Re3 is a functionof the attitude R, while the control normal vector nc(x, v) is a function of the positionand the velocity (for a stabilization problem).

For the stabilization problem, the state feedback k(x, v) is in the form

w = k(x, v) = −cp(x− x0)− cdv , cp, cd > 0

where (x0, 0) is a desired equilibrium point of the translational dynamics. In contrast,for the tracking problem, one is interested in tracking a trajectory xd(·) in R3. Thecontrol law for the tracking case takes the following form

w = k(x, v) = −cp(x− xd)− cd(v − xd) + xd , cp, cd > 0

where x = x − xd and v = v − xd. Thus, for tracking purposes, the control normal

171

vector nc = nc(x, v) is a function of the position and velocity errors. Notice thatthe stabilization problem is a special case of the tracking problem with (xd, xd, xd) =(x0, 0, 0). Therefore, from now on, by nc we mean

nc(x, v) =ge3 − k(x, v)‖ge3 − k(x, v)‖

To achieve asymptotic tracking, the controller for the attitude dynamics must guar-antee that the normal vector n = Re3 asymptotically tracks the control normal vec-tor nc = nc(x, v). However, this does not uniquely determines the desired attitudeR ∈ SO(3) required for such an asymptotic tracking. This problem is resolved in thenext section.

5.11.5 Desired Control Attitude Rd for Position Tracking

The desired control attitude is a rotation matrix Rd ∈ SO(3) satisfying Rde3 = nc.However, this only determines one column of Rd and the other two columns needto be determined from the trajectory xd and (possibly) the heading direction of thevehicle. For doing so, let us denote the three columns of Rd by

fc = Rde1bc = Rde2nc = Rde3

(5.79)

We refer to fc, bc as the control forward vector and the control binormal vector,respectively. This terminology is adopted from the names of the three unit vectors inthe Frenet frame for a curve in R3 [26]. Given a trajectory xd(·) in R3, the control laww = k(x, v) determines nc. To specify Rd, we need to determine fd which is a unitvector pointing out towards the nose of the vehicle. Let td(t) denote a unit vectortangent to xd(t). Then, the tangent vector td, given by

td =vd‖vd‖

is defined almost everywhere (a.e.) for vd = xd 6= 0. Depending on whether nc ‖ tdand/or td ‖ e3, or not three situations might arise.

i) nc · td 6= ±1: In this case, there is well-defined (nc, td)-plane. We define fc as aunit vector in the (nc, td)-plane orthogonal to nc such that fc · td > 0. Therefore

fc =td − (nc · td)nc√

1− (nc · td)2, nc · td 6= ±1

(where · denotes the inner product of two vectors).

ii) nc · td = ±1 and e3 · td 6= ±1: These two conditions imply that nc ∦ e3, or|nc · e3| < 1. For this case, we define fc as a unit vector in the (nc, e3)-plane

172

such that fc ⊥ nc. Thus

fc = −sgn(nc · td) · sgn(td · e3)e3 − (nc · e3)nc√

1− (nc · e3)2, nc · td = ±1, e3 · td 6= ±1

iii) nc · td = ±1 and e3 · td = ±1: This is equivalent to the case where nc ‖ td ‖ e3and the vehicle moves in the vertical direction. For this case, we set

fc = sgn(nc · td)e1

(e1 in the last equation can be replaced by any other pre-determined unit vectorwhich defines a desired heading direction of the vehicle).

Given fc and nc, the control binormal vector is defined as

bc = fc × nc

Now, the three unit vectors fd, bd, nd determine the desired control attitude Rd asRd = [fc, bc, nc].

5.11.6 Asymptotic Tracking of a Desired Attitude Rd ∈ SO(3)

To do attitude tracking, we need to define a meaningful distance between two differentattitudes in SO(3). Let R = [f, b, n], Rd = [fc, bc, nc] denote the current and desiredattitudes, respectively. We define the Frenet distance between R,Rd ∈ SO(3) as

δF (R,Rd) = ‖f − fc‖2 + ‖b− bc‖2 + ‖n− nc‖2

On the other hand, there is a well-known function ϕ : SO(3)→ R≥0 defined as

ϕ(R0) :=1

2tr(I −R0)

which has similar properties to a norm [16] (“tr” denotes the trace operator oversquare matrices). In fact, ϕ satisfies the following triangular-type inequality:

ϕ(R1RT2 ) + ϕ(R2R

T3 ) ≥ ϕ(R1R

T3 ), ∀R1, R2, R3 ∈ SO(3)

Defining the distance δ asδ(R1, R2) := ϕ(R1R

T2 ),

the triangular inequality over SO(3) takes a more familiar form

δ(R1, R2) + δ(R2, R3) ≥ δ(R1, R3)

In addition, based on Rodrigues’s formula, the rotation by θ around a unit vector kis given by

R(k, θ) = I + sin θk+ (1− cos θ)k2 (5.80)

173

Thusϕ(R(k, θ)) = 1− cos(θ) ≤ 2

In other words, ϕ : SO(3)→ [0, 2]. We prove that the Frenet distance δF (R,Rd) andϕ(RRT

d ) are equivalent upto a constant multiplicative factor. As a distance measurebetween two SO(3) matrices R and Rd, ϕ(RR

Td ) has been effectively used for satellite

attitude control [16] and helicopter motion planning [31].

Lemma 5.11.1. Let

δ(R,Rd) := ϕ(RRTd ) =

1

2tr(I −RRT

d )

Then δ and δF satisfy the following identity

δF (R,Rd) = 4δ(R,Rd)

Proof. We have

δF (R,Rd) =∑3

i=1 ‖(R−Rd)ei‖2=

∑3i=1 e

Ti (R−Rd)

T (R−Rd)ei=

∑3i=1 e

Ti (2I −RRT

d −RTdR)ei

= tr(I −RRTd ) + tr(I − (RRT

d )T )

= 2tr(I −RRTd ) = 4δ(R,Rd)

From now on, we only use δ(R,Rd) as a distance measure instead of the Frenetdistance δF . Note that δ(R,Rd) = 0 ⇔ R = Rd. Since δ ≥ 0, the main idea inattitude tracking is to design ω, as the control input for the attitude kinematics, suchthat δ < 0,∀δ > 0. Before presenting our control design for attitude tracking, weneed to define some notations. Let 〈A,B〉 denote the inner product of two 3 × 3matrices A,B defined as

〈A,B〉 := 1

2tr(ATB)

Then, for two skew-symmetric matrices α1, α2, we have

〈α1, α2〉 = α1 · α2, 〈α, α〉 = ‖α‖22

where · denotes the inner product of vectors in R3. Notice that any matrix A can bedecomposed as

A = skew(A) + sym(A)

where

skew(A) =A− AT

2, sym(A) =

A+ AT

2

Keeping this in mind, the main property of the distance δ(R,Rd) is given by thefollowing lemma [16].

Lemma 5.11.2. Let ωd = (RTd Rd)

∨, then δ(R,Rd) =1

2tr(I − RRT

d ) satisfies the

174

following propertyδ = skew(RRT

d )∨ · (ω − ωd)

(·∨ over skew-symmetric matrices denotes the inverse of the hat operation ·).

Proof. By direct calculation, we have

δ = −1

2tr(

d

dt(RRT

d ))

= −1

2tr(RωRT

d +RωTd RTd )

= −1

2tr(R(ω − ωd)RT

d )

= −1

2tr(RT

dR(ω − ωd))

= −1

2tr((skew(RT

dR) + sym(RTdR))(ω − ωd))

= −1

2tr(skew(RT

dR)(ω − ωd))

=1

2tr(skew(RRT

d )T (ω − ωd))

= 〈skew(RTdR), (ω − ωd)〉

= skew(RTdR)

∨ · (ω − ωd)

and the result follows.The following proposition is a direct result of lemma 5.11.2.

Proposition 5.11.5. The state feedback law

ω = KR(R,Rd, ωd) := ωd − c0skew(RTdR)

∨, c0 > 0

(almost) globally asymptotically stabilizes the equilibrium δ = 0 of

δ = skew(RTdR)

∨ · (ω − ωd)

for all δ(0) 6= 2. In other words, ω = KR renders R = Rd for

R = Rω

(almost) globally asymptotically stable given that R(0) ∈ SO(3) \ ∂SO(3) where

∂SO(3) = R ∈ SO(3) | δ(R,Rd) = 2

Proof. For the closed-loop system

δ = −c0‖skew(RTdR)

∨‖22 < 0

175

for δ > 0. Now, we need to show that skew(RTdR)

∨ = 0 implies δ = 0. For doing so,observe that skew(RT

dR)∨ = 0 implies RT

dR is a symmetric matrix, or

RTdR = (RT

dR)T ∈ SO(3)

The only symmetric matrices in SO(3) are I and I + 2k2 where k is an arbitraryunit vector in R3 (this follows from Rodrigues’s formula). But RT

dR = I + 2k2

means δ = δ(R,Rd) = 2 and this contradicts the assumption that δ(0) 6= 2. Hence,RTdR = I is the only acceptable solution for skew(RT

dR)∨ = 0 which implies R = Rd,

or δ = δ(R,Rd) = 0. In other words, the only invariant equilibrium is δ = 0 thatproves (almost) global asymptotic stability of δ = 0 for δ(0) < 2. This is equivalentto (almost) global asymptotic tracking of R = Rd for R(0) 6∈ ∂SO(3).

Remark 5.11.3. All the matrices in ∂SO(3) are obtained by rotation of Rd around aunit vector k by θ = π (see (5.80). Therefore, to satisfy R(0) 6∈ ∂SO(3) in propo-sition 5.11.5, one can perturb the initial condition R(0) via multiplying R(0) by therotation matrix R(k, ε) defined in (5.80) that slightly rotates R(0). In practice, dueto measurement noise and round-off errors, it is unnecessary to apply such an initialperturbation by a small rotation angle around k.

We restate the following lemma from [16] based on the notation and applicationof interest in this work.

Lemma 5.11.3. ([16]) For any 0 < ε ≤ 2, there exist positive constants d1, d2 > 0such that ϕ(R0) ≤ 2− ε implies

d1‖skew(R0)∨‖2 ≤ ϕ(R0) ≤ d2‖skew(R0)

∨‖2

This helps us to prove that the state feedback law in proposition 5.11.5 exponen-tially stabilizes R = Rd.

Remark 5.11.4. This result in a more general form on Riemannian manifolds has beenproven by Bullo [16], however it is crucial to emphasize the result for the special caseof SO(3) matrices due to its important applications in attitude control of helicopters,aircrafts, satellites, spacecraft, and underwater vehicles. In addition, in [16] addi-tional dissipation functions are used for attitude tracking/stabilization. Therefore,the control laws obtained here are slightly different and simpler than the ones givenin [16].

Proposition 5.11.6. The state feedback law

ω = KR(R,Rd, ωd) := ωd − c0skew(RTdR)

∨, c0 > 0

(almost) globally exponentially stabilizes R = Rd for

R = Rω

176

for all the initial conditions R(0) ∈ SO(3) \ ∂SO(3) where

∂SO(3) = R ∈ SO(3) : δ(R,Rd) = 2

Proof. For all R(0) ∈ SO(3)\∂SO(3), δ(0) = ϕ(RRTd ) < 2. Setting ε = (2−δ(0))/2,

based on lemma 5.11.3, we have

δ = −c0‖skew(RTdR)

∨‖22 ≤ −c0d1δ = −c0δ

Therefore, δ = 0 is globally exponentially stable for all initial conditions δ(0) ∈[0, 2− ε0], 0 < ε0 ¿ 1.

Remark 5.11.5. In the following, we provide the interpretation of the (almost) globalexponential attitude tracking control law in proposition 5.11.6 in terms of Rodrigues’srotation matrix R(θ,k). For doing so, let

R0(θ,k) := RTdR

Then, δ = ϕ(RTdR) = ϕ(R0(θ,k)) = 1− cos(θ) and

δ = skew(RTdR)

∨ · (ω − ωd)

whereskew(RT

dR)∨ = skew(R0(θ,k))

∨ = sin(θ)k

Therefore, the control law in proposition 5.11.6 can be expressed as

ω = ωd − c0 sin(θ)k

which means δ satisfies the following equation

δ = −c0 sin2(θ) = −c0(1 + cos(θ))δ

If for some 0 < ε ¿ 1, ϕ(R0) ∈ [0, 2 − ε], then |θ| < π and there exist d1, d2 ∈ (0, 2]such that

d1 ≤ 1 + cos(θ) ≤ d2 = 2

As a result, we getδ ≤ −c0d1δ = c0δ

which guarantees (almost) global exponential stability of δ = 0. Apparently, if ini-tially δ(0) = ϕ(R0) = 2, θ = π and δ = 0. The stable solution in this case isδ(t) = 2,∀t > 0 and thus δ does not converge to zero. Furthermore, substitutingδ = 1− cos(θ) in the differential equation of δ gives

θ = − sin(θ), θ ∈ (−π, π)

which has an exponentially stable equilibrium point θ = 0. Two possible Lyapunovfunctions for this system are V = θ or V = 1− cos(θ) = δ. Here, we prefer the latter

177

one due to its direct relation with the distance function δ.

Having a stabilizing state feedback ω = KR, using the backstepping procedure,the controller for the composite system can be obtained in a straightforward way.

Proposition 5.11.7. Consider the attitude dynamics of a helicopter as the following

R = Rω,ω = τ

(5.81)

Then, the static state feedback law

τ = Ka(ω,R, ωd, Rd) := −c1(ω − ωd)− c1c0skew(RTdR)

∨ + KR; c0, c1 > 0 (5.82)

where

KR = ωd − c0skew(RTdRω − ωdRT

dR)∨

ωd = (RTd Rd − ω2

d)∨

achieves (almost) global asymptotic tracking for the solution (Rd, ωd) of the closed-loopsystem. In addition, the following positive definite function

V = 2c0c1δ(R,Rd) +1

2‖ω −KR‖2

is a valid Lyapunov function for the system.

Proof. Calculating V , we get

V = 2c0c1δ − c1(ω − ωd + c0skew(RTdR)

∨) · (ω − ωd + c0skew(RTdR)

∨)

= −c1c20‖skew(RRTd )∨‖22 − c1‖ω − ωd‖22 < 0

for (R,ω) 6= (Rd, ωd), because ‖skew(RTdR)

∨‖ = 0⇔ δ(R,RTd ) = 0. Thus, δ = 0 (i.e.

R = Rd) and ω = ωd are (almost) globally asymptotically stable for the closed-loopsystem, given that δ(0) 6= 2.

Here is our main result for the trajectory tracking/stabilization (in position) foran autonomous helicopter:

Theorem 5.11.1. Consider a trajectory xd(·) as a curve in R3 with an associatedattitude Rd(·). Assume Ld := ‖xd‖∞ < ∞ and denote z = col(x, v). Then, thereexists a closed ball Br = z ∈ R6 | ‖z‖ ≤ r around z = 0 and a finite time t0 > 0such that given the state feedback law

τb = ω × Jω − J [c1(ω − ωd) + c1skew(RTdR)

∨ − ωd + c0skew(RTdRω − ωdRT

dR)∨]

T (x, v) = m‖ge3 − k(x− xd, v − vd)‖(5.83)

the tracking error z(t) in position and velocity, for the closed-loop system, globallyexponentially converges to Br, i.e.

∃t0, λ0, d0 > 0 : minp∈Br‖z(t)− p‖ ≤ d0 exp(−λ0(t− t0)), ∀t ≥ t0

178

and remains in Br thereafter. In addition, the total tracking error (x, v, δ, ω) (almost)globally asymptotically converges to zero, given that δ(0) 6= 2.

The following theorem, introduces a high-gain nonlinear controller for the approx-imate model of the helicopter with T, ω as the input. This controller achieves an(almost) global exponential tracking of feasible trajectories for a helicopter.

Theorem 5.11.2. Consider a trajectory xd(·) as a curve in R3 with an associatedattitude Rd(·). Assume Ld := ‖xd‖∞ <∞ and denote z = col(x, v). Then, there exista finite time t0 > 0 and a positive constant c∗ > 0 such that given the state feedbacklaw

ω = ωd − c0skew(RTdR)

∨, c0 > 0T (x, v) = m‖ge3 − k(x− xd, v − vd)‖ (5.84)

for all c0 > c∗, after time t0, the tracking error (z(t), δ(t)) for the closed-loop system,(almost) globally exponentially converges to zero, given that δ(0) 6= 2.

We need the following lemma, before presenting the proof of the last two theorems.

Lemma 5.11.4. Assume Ld = ‖xd‖∞ < ∞ and denote z = col(x, v). Then, thereexists a closed ball Br = z ∈ R6 : ‖z‖ ≤ r, r > 0 around z = 0 in the form and apositive constant C1 > 0 such that the thrust state feedback T (z) = T (x, v) in (5.83)satisfies the following properties:

i) ‖T (z)‖ < C1‖z‖, ∀z ∈ (R6 \Br),

ii) ‖T (z)‖ ≤ C1r, ∀z ∈ Br.

Proof. Let cm := maxcp, cd > 0. Since k(x, v) = −cpx − cdv + xd and T (x, v) =m‖ge3−k(x, v)‖, the thrust feedback T (z) = T (x, v) satisfies the following inequality

‖T (z)‖ ≤ m(g + Ld) +√2cm‖z‖)

due to ‖x‖+ ‖v‖ ≤√2‖z‖. Setting r = r(cm) := (g + Ld)/(cm

√2), we get

‖T (z)‖ ≤ 1

2C1(r + ‖z‖)

where C1 = 2√2mcm. Thus, for all ‖z‖ > r, ‖T (z) < C1‖z‖ and for all ‖z‖ ≤ r,

‖T (z)‖ ≤ C1r.

Proof.(theorem 5.11.1) Denote x = x−xd and v = v−vd. The closed-loop dynamicsof the helicopter in (5.74) can be written as the following

˙x = v˙v = −ge3 +Rde3T (x, v) + ∆

R = Rωω = u

179

where the perturbation ∆ ∈ R3 is given by

∆ =1

m(R−Rd)e3T (x, v)

Calculating the upper bound of ∆, we get

‖∆‖ ≤ 1

m‖(R−Rd)e3‖‖T (x, v)‖

but ‖(R−Rd)e3‖ ≤ δF (R,Rd) = 4δ(R,Rd) and therefore

‖∆‖ ≤ C2δ(R,Rd)‖T (z)‖

where z = col(x, v) and C2 := 4/m. Defining the new perturbation ∆ = col[0,∆], theclosed-loop translational dynamics in z-coordinates can be written as

z = Az + ∆ (5.85)

where

A =

[

03 I3−cpI3 −cdI3

]

is a Hurwitz matrix and z ∈ R6. Consider the Lyapunov function V (z) = zTPz whereP is the symmetric and positive definite solution of the Lyapunov equation

ATP + PA = −I

For all z 6∈ Br, we have

V = −‖z‖2 + 2zTP ∆

≤ −‖z‖2 + 2λmax(P )C2‖z‖‖T (z)‖‖δ‖< −‖z‖2(1− 2

√2λmax(P )C1C2δ) (by lemma 5.11.4 and the property δ ≥ 0)

On the other hand, δ = 0 is exponentially stable for the attitude dynamics, thus∃t0 > 0 : δ(t) ∈ [0, r0],∀t ≥ t0 with

r0 :=1

4√2λmax(P )C1C2

=1

64cmλmax(P ), cm = maxcp, cd > 0

This implies that for t ≥ t0, V < −0.5V and therefore any solution z(t) outside theball Br exponentially converges to Br and remains in it thereafter. Since with δ = 0,∆ = 0 and z = 0 is globally asymptotically stable for z = Az, from Sontag’s theoremon stability of cascade systems (see [83], or theorem 7.3.3 in chapter 7), (z, δ, ω) = 0 is(almost) globally asymptotically stable for the closed-loop system given that δ(0) 6= 2.

Proof.(theorem 5.11.2) Based on the proof of theorem 5.11.1, after some finite timet0 > 0, the position and velocity tracking error z(t) enters a closed ball Br around

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z = 0 exponentially fast and remains in it thereafter. From lemma 5.11.4, over Br wehave ‖T (z)‖ ≤ C3 := C1r. From proposition 5.11.6, δ ≤ −c0δ where c0 = c0d1 andd1 > 0 is a constant. Defining

W (z, δ) = V (z) +1

2δ2

and keeping in mind that δ ≥ 0, we get

W = −‖z‖2 + 2zTP ∆ + δδ

≤ −‖z‖2 + 2rC1λmax(P )‖z‖δ − c0δ2

≤ −1

2‖z‖2 − (c20 − 2r2C2

1λ2max(P ))δ

2

where the last inequality is obtained from

kab ≤ la2 +k2

4lb2

with a = ‖z‖, b = δ, k = 2rC1λmax(P ), l = 1/2. Notice that C4 := r(cm)C1(cm) doesnot depend on cm and is a constant because

C4 = rC1 =g + Ld√

2cm(2√2mcm) = 2m(g + Ld)

Taking

c0 ≥ c∗0 := (2C24λ

2max(P ) +

1

4λmax(P ))

12

or c0 > c∗ := c∗0/d1, we get

W ≤ −1

2‖z‖2 − 1

4λmax(P )δ2

≤ − 1

2λmax(P )[λmax(P )‖z‖2 +

1

2δ2)

≤ − 1

2λmax(P )W

and therefore W = 0, or (z, δ) = 0 is exponentially stable over Br. This implies(z(t), δ(t)) (almost) globally exponentially converges to zero, after some finite timet0 > 0, given that δ(0) 6= 2.

5.11.7 Thrust for Exponential Attitude Stabilization

For exponential attitude stabilization regardless of how the position in (e1, e2)-planechanges, one has to use a different control law for the thrust T than the one givenbefore (for tracking/stabilization of the position). This situation arises during doing

181

a maneuver from a (normal) hover mode to an (inverted) upside-down hover mode.This is equivalent to having no motions in the e3 direction, thus

eT3 (Re3)T = mg

orT = T (R) =

mg

eT3Re3

However, the denominator in the above ratio vanishes as Re3 passes through a hori-zontal plane where the roll angle is ±π/2 and T →∞. This singularity is inherent inthe dynamics of the helicopter and is impossible to avoid regardless of any parame-terization of the attitude [31]. To avoid this problem, we suggest to apply a saturatedthrust given by

T = Tmaxsat(T/Tmax)

where Tmax À 1 is a pre-specified maximum thrust. This results in a slight dropof the height of the position of the helicopter in the e3 direction during performingthis particular maneuver (or in general, any maneuver passing through θ = π/2 orφ = π/2). Still, the main controller for stabilization of the attitude remains the sameas in proposition 5.11.7 (with ωd = 0). Notice that both T and τ are independent of(x, v) in this case.

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Chapter 6

Reduction and Control ofUnderactuated NonholonomicSystems

This chapter is devoted to reduction and control of underactuated mechanical sys-tems with nonholonomic first-order constraints and symmetry. A variety of real-lifecontrol systems including Car-type Vehicles, Mobile Robots, Surface Vessels, and Au-tonomous Underwater Vehicles (AUV) are examples of underactuated control systemswith nonholonomic velocity constraints. All of these examples of nonholonomic sys-tems possess certain symmetry properties, i.e. their kinetic energy, potential energy,or both are independent some configuration variables. This motivated us to focusmainly on reduction and control of nonholonomic systems with symmetry.

Our main theoretical result is that an underactuated system with first-order non-holonomic constraints and kinetic symmetry can be reduced to a well-defined reduced-order Lagrangian system in cascade with the constraint equation. Depending onwhether the number of control inputs and constraint equations is less than or equalto the number of configuration variables, the reduced Lagrangian system is fully-actuated or underactuated, respectively. We present a number of examples with de-tailed reduction process. Namely, a rolling disk, a mobile robot [5], a dynamic modelof a car, and the snakeboard [50]. Moreover, (almost) global exponential stabilizationand tracking for a two-wheeled mobile robot is provided in great details.

Based on a result due to Brockett [15], there exists no smooth static state feedbackthat asymptotically stabilizes a mechanical system with first-order nonholonomic con-straints to the origin. As a consequence, a two-wheeled nonholonomic mobile robot(Figure 6-2), cannot be asymptotically stabilized to the origin using a smooth feed-back. Due to Astolfi [5], this mobile robot can be stabilized using a discontinuouschange of coordinates and a discontinuous static state feedback. Our contributionis to introduce a smooth dynamic state feedback which achieves global exponentialstabilization/tracking to a point/trajectory that is ε far from the desired equilib-rium/trajectory (ε ¿ 1) for a two-wheeled mobile robot. This is based on the useof a new class of diffeomorphisms parameterized by λ that we call call near-identitydiffeomorphism. A near-identity diffeomorphism is equal to an identity function when-

183

ever λ vanishes. In addition, for all λ, a near-identity diffeomorphism remains withina distance λ of an identity function. The dynamic state feedback is designed suchthat λ(t)→ ε as t→∞ and λ(0) ≈ O(1).

6.1 Nonholonomic Systems with Symmetry

This section is devoted to reduction of underactuated mechanical systems with non-holonomic first-order constraints and symmetry. In the following, we precisely char-acterize a broad class of underactuated nonholonomic systems of interest by givingthree assumptions.

To begin, consider a Lagrangian system with an n-dimensional configuration vec-tor q, force matrix F (q), and m nonholonomic first-order constraints as the following

d

dt

∂L∂q− ∂L∂q

= W T (q)λ+ F (q)τ

W (q)q = 0(6.1)

where λ ∈ Rm is the vector of Lagrange multipliers, τ ∈ Rl is the control input,l = rank(F (q)), and m + l ≤ n. Due to m ≥ 1, l ≤ (n − 1) and the nonholonomicsystem in (6.1) is an underactuated system. The term W T (q)λ in (6.1) representsthe effect of the constraint forces [64, 11, 8, 48]. This is based on the principle ofvirtual work which states “the constraint forces do not work on motions allowed bythe constraints”. The principle of virtual work is an axiom of mechanics.

Remark 6.1.1. The last condition that m + l ≤ n is one of the main conditions thatdistinguishes this work from the result of Bloch et al. in [11] on reduction of Caplygincontrol systems. In that work, it is assumed that m + l ≥ n. Later, we show thisassumption leads to over-actuated or fully-actuated reduced systems.

Assumption 6.1.1. Assume W (q) has full row rank. Then, q can be partitionedas (q1, q2) such that W (q) = (W1(q),W2(q)) where W1(q) is an invertible matrix.Therefore, the constraint equation in (6.1) can be rewritten as

W1(q)q1 +W2(q)q2 = 0 (6.2)

Assumption 6.1.2. Assume M(q), F (q),W (q) are all independent of q1.

Assumption 6.1.3. Assume the potential energy is in the form

V (q) = kT q1 + U(q2) (6.3)

where k ∈ Rn−m is a constant.

Remark 6.1.2. Under Assumption 6.1.3 with k = 0, the notions of kinetic symmetryand classical symmetry coincide. Therefore, by saying “symmetry”, we refer to bothof these notions.

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Remark 6.1.3. Broad classes of mechanical systems with nonholonomic velocity con-straints including mobile robots, car-type vehicles, surface vessels, rolling disk, Caply-gin control systems [64, 11], and the snakeboard system [50, 9, 49, 73, 74] satisfyAssumptions 6.1.1 through 6.1.3. Moreover, all the aforementioned examples of non-holonomic systems are underactuated.

Under assumptions 6.1.1 through 6.1.3, the dynamics of the underactuated non-holonomic system in (6.1) can be expressed as

d

dt

∂L∂q− ∂L∂q

=W T (q2)λ+ F (q2)τ

W1(q2)q1 +W2(q2)q2 = 0(6.4)

Nonholonomic systems in the form (6.4) (with F (q) ≡ 0) are called Caplygin systems[64]. Caplygin control systems in the form (6.4) were first introduced by Bloch etal. in [11]. To eliminate λ from (6.4), one can multiply both sides of the forcedEuler-Lagrange equation in (6.4) by a matrix A(q) that annihilates W T (q) [11], i.e.A(q)W T (q) = 0. For doing so, let us define

w12(q2) = −W−11 (q2)W2(q2)

then q = D(q2)q2 where

D(q2) =

[

w12(q2)In−m

]

By direct calculation, it can be readily shown that A(q2) = DT (q2) annihilatesW T (q2). This eventually leads to the following reduction theorem for underactuatednonholonomic systems with symmetry.

Theorem 6.1.1. Consider the underactuated mechanical control system with non-holonomic constraints and symmetry in (6.4). Then, system (6.4) with (2n + m)first-order equations can be reduced to a system of (2n −m) first-order equations inthe following cascade form

qx = wr(qr)qrMr(qr)qr + Cr(qr, qr)qr +Gr(qr) = Fr(qr)τ

(6.5)

where (qx, qr) = (q1, q2), wr(qr) = w12(q2), and

Mr(qr) = D(q2)TM(q2)D(q2)

Gr(qr) = wTr (qr)k +∇qrU(qr)

Fr(qr) = DT (q2)F (q2)

In addition, if V (q) = U(q2) (or k = 0), the reduced system is a well-defined La-grangian system with configuration vector qr = q2 and Lagrangian function

Lr(qr, qr) =1

2qTrMr(qr)qr − U(qr) (6.6)

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which satisfies a forced Euler-Lagrange equation in cascade with the constraint equa-tion as the following

qx = wr(qr)qrd

dt

∂Lr∂qr− ∂Lr∂qr

= Fr(qr)τ(6.7)

Moreover, if l +m < n (or l +m = n), the reduced system with configuration vectorqr is an underactuated (or a fully-actuated) mechanical control system.

Proof. The forced-Euler-Lagrange equation in (6.4) can be rewritten as

M(q2)q + C(q2, q)q +G(q) = W T (q2)λ+ F (q2)τ (6.8)

where C(q2, q) satisfies M = C + CT and G(q) = ∇qV (q). Multiplying both sides ofthe last equation by A(q2) = DT (q2) eliminates λ and gives

DT (q2)M(q2)q +DT (q2)C(q2, q)q +Gr(q) = Fr(q2)τ (6.9)

where Fr(q2) = DT (q2)F (q2) and

Gr(q) = DT (q2)G(q) =[

wT12(q2) I]

[

∇q1V (q)∇q2V (q)

]

= wT12(q2)k +∇q2U(q2) =: Gr(q2)

On the other hand, q = D(q2)q2 implies that

q = D(q2)q2 + D(q2, q2)q2

Substituting q from the last equation in (6.9), we get

DT (q2)M(q2)D(q2)q2 + [DT (q2)C(q2, q)D(q2) +DT (q2)M(q2)D]q2 +Gr(q2) = Fr(q2)τ

which after renaming qr = q2 can be equivalently rewritten as

Mr(qr)qr + Cr(qr, qr)qr +Gr(qr) = Fr(qr)τ (6.10)

where

Mr(qr) := DT (q2)M(q2)D(q2) (6.11)

Cr(qr, qr) := DT (q2)C(q2, D(q2)q2)D(q2) +DT (q2)M(q2)D(q2, q2) (6.12)

To establish that (6.10) is in fact equivalent to the forced Euler-Lagrange equationfor the reduced system with the Lagrangian function Lr(qr, qr) and the force matrixFr(qr), we need to prove Cr(qr, qr) satisfies Mr = Cr + CT

r . By direct calculation, we

186

have

Mr = DTMD + DTMD +DTMD

= DT (C + CT )D + DTMD +DTMD

= (DTCD +DTMD) + (DTCTD + DTMD)

= Cr + CTr

and the result follows.

Remark 6.1.4. Since the Lagrangian reduced system in (6.7) with the reduced con-figuration vector qr and the reduced Lagrangian Lr is itself an underactuated systemfor m + l < n, its reduction process can be addressed based on the reduction theorydeveloped in Chapter 4 of this thesis. Also, the case of m + l = n is presented incorollary 6.1.1.

Many well-studied cases of nonholonomic systems including the rolling disk, a one-legged hopping robot in free-flying mode [51, 63], a two-wheeled mobile robot [5], anda planar three-link diver [22, 32] are special examples of underactuated nonholonomicsystems considered in theorem 6.1.1 which all possess a fully-actuated reduced system.We formalize this special case in the following corollary (which has also been foundindependently in [11]).

Corollary 6.1.1. Consider the underactuated nonholonomic system in (6.4). Assumethe number of inputs and constraints add up to n, i.e. m + l = n. Suppose that q2is actuated and the system has non-interacting inputs, i.e. F (q) = col(0, F2(q2))where F2(q2) is an invertible matrix. Then, there exists a change of control input thattransforms the dynamics of (6.5) (or (6.4)) into the following nontriangular normalform with a vector double-integrator linear part

z = f(ξ1, ξ2) := wr(ξ1)ξ2ξ1 = ξ2ξ2 = u

(6.13)

Proof. Notice that Fr(qr) = DT (q2)F (q2) = F2(q2) and thus Fr is invertible. Afterrenaming the variables as z = qx, ξ1 = qr, ξ2 = qr and applying the change of control

τ = F−1r (qr)(Mr(qr)u+ Cr(qr, qr)qr +Gr(qr))

we get qr = u and the result follows.

6.2 Applications to Nonholonomic Robots

In this section, we provide applications of the theory developed in the preceding sec-tion for reduction and control of several examples of nonholonomic systems includingCaplygin systems and mobile robots.

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6.2.1 A Rolling Disk

A vertical rolling disk (see Figure 6-1) which is not allowed to slip is an example ofmechanical system with nonholonomic velocity constraints. These constraints can beexpressed as

x = r cos(θ)ϕy = r sin(θ)ϕ

To view the rolling disk as a control system, assume the rotation angle ϕ and the

r ϕ

θx

y

z

(x,y)

Figure 6-1: A vertical rolling disk

heading angle θ of the disk can be controlled using torques τ1 and τ2, respectively.The Lagrangian of the rolling disk with configuration vector q = (x, y, ϕ, θ) is

L =1

2qTMq

with a constant inertia matrix

M = diag(m,m, J1, J2)

where m is the mass of the disk and J1 and J2 are the inertia of the disk. Thus, therolling disk is a flat underactuated mechanical system with actuated variables (ϕ, θ)and unactuated variables (x, y). The constraint equation is in the form W (q)q = 0with

W (q) =

[

1 0 −r cos(θ) 00 1 −r sin(θ) 0

]

Apparently W (q) has full row rank. Setting q1 = (x, y) and q2 = (ϕ, θ), one obtainsW1(q2) = I2×2 which is an invertible matrix. Since the number of controls andconstraints add up to n = 4, based on theorem 6.1.1, the reduced system for thisunderactuated nonholonomic system is a fully actuated system. The configurationof the reduced system is qr = (ϕ, θ) and the overall dynamics of the system can be

188

expressed asx = r cos(θ)ϕy = r sin(θ)ϕϕ = u1θ = u2

(6.14)

where ui = τi/Ji, i = 1, 2. The reduced system of the rolling disk is a vector double-integrator qr = u (u = (u1, u2)

T ). Controllability of the vertical rolling disk has beenaddressed in [11, 49].

Assume the input controls of the system are ϕ = ω1 and θ = ω2. Then, afternormalization of (x, y) units by r, the kinematic model of the rolling disk can bewritten as

x = cos(θ)ω1

y = sin(θ)ω1

θ = ω2

(6.15)

Applying the following change of coordinates and control for θ ∈ (−π/2, π/2)

x1 = xx2 = tan(θ)x3 = yv1 = ω1/ cos(θ)v2 = ω2/(1 + tan2(θ)

the kinematics of the rolling disk in (6.15) transforms into a first-order chained-typenonholonomic system as the following

x1 = v1x2 = v2x3 = x2v1

(6.16)

This is a special case of the general chained-type system in (2.8). Exponential sta-bilization of nonholonomic systems in the chained-form (6.16) using a discontinuousstate feedback is addressed in [4]. Also, a homogeneous time-varying state feedbackis used in [60, 59] for local exponential stabilization of such chained-form systems.

6.2.2 A Nonholonomic Mobile Robot

Consider the mobile robot depicted in Figure 6-2 [5, 4]. The robot has two rollingwheels that can be controlled independently using input torques. It is not difficultto see that the dynamics of this mobile robot is exactly the same as the verticalrolling disk in equation (6.14). Clearly, the same arguments hold for the mobile robotand using a change of coordinates and control the dynamics of this mobile robotcan be transformed into the chained-form system (6.16). Exponential stabilizationof this mobile robot using a discontinuous change of coordinates and a discontinu-ous state feedback (both with singularities at the origin) has been addressed in [5].Other researchers have tried time-varying and discontinuous state feedback or both

189

θ(x,y)

τ 1

x

y

τ 2

Figure 6-2: A mobile robot

to stabilize similar classes of nonholonomic systems [76, 19, 60]. Based on a theoremdue to Brockett [15], systems with first-order nonholonomic constraints cannot beasymptotically stabilized using a C1 smooth state feedback.

We propose a new approach for tracking and stabilization of this mobile robot us-ing a smooth time-varying change of coordinates and a smooth dynamic state feedbacklaw that globally exponentially stabilizes the system to an equilibrium point arbitraryclose to the origin (i.e. within a distance ε of the desired equilibrium for ε¿ 1). Tomake this more precise, we need to define new notions of asymptotic stability andtracking. This is done in section 6.3. The details regarding stabilization and trackingfor the two-wheeled nonholonomic robot are presented in section 6.4.

6.2.3 A Car

In this section, we address reduction of a dynamic model of a car as shown in Fig-ure 6-3. The kinematic model of a car is a more common model of a car that hasbeen studied in the literature [63]. This includes steering using sinusoids [63] andstabilization using discontinuous state feedback of the type used for a two-wheeledmobile robot [4]. Here, we take a different approach that later allows us to applyour reduction result for a car, to locomotion control design for the snakeboard [50].The snakeboard is a complex nonholonomic system with certain similarities to a carand will be discussed in the next section. A dynamic model of a car is an exampleof an underactuated nonholonomic system with five degrees of freedom, two con-trols, and two velocity constraints corresponding to the rear and front wheels. Letq = (x, y, θ, ψ, φ) ∈ SE(2) × S1 × S1 denote the configuration vector of a dynamiccar. The position of the middle of the rear wheels of the car is denoted by (x, y), theorientation of the body of the car is denoted by θ, the angle of rotation of each wheelis denoted by ψ, and the orientation of the front wheels w.r.t. the body is denotedby φ. The velocity constraints for the front and rear wheels are as the following

sin(θ + φ) · ddt(x+ l cos θ)− cos(θ + φ) · d

dt(x+ l sin θ) = 0 (6.17)

sin(θ) · x− cos(θ) · y = 0 (6.18)

190

θ

φ

ψ

x

y (x,y)

Figure 6-3: The model of a car.

or equivalently

sin(θ + φ)x− cos(θ + φ)y − l cosφθ = 0 (6.19)

sin(θ)x− cos(θ)y = 0 (6.20)

These two constraints can be rewritten as W (q)q = 0 where W = (W1,W2) is par-titioned according to qx = (x, y) and qr = (θ, ψ, φ). The matrices W1,W2 are givenby

W1(θ, φ) =

[

sin(θ + φ) − cos(θ + φ)sin θ − cos θ

]

, W2(θ, φ) =

[

−l cosφ 0 00 0 0

]

,

Notice that W1(θ, φ) is not invertible at φ = 0. We have

w12(θ, φ) = −W−11 W2 =

[

ρ1 0 0ρ2 0 0

]

where ρ1, ρ2 are defined as

ρ1(θ, φ) =l cos θ

tanφ

ρ2(θ, φ) =l sin θ

tanφ

(6.21)

Thus, from w12(q), D(q) can be obtained as the following

D(q) = D(θ, φ) =

ρ1 0 0ρ2 0 01 0 00 1 00 0 1

191

Later, we use D(q) to calculate the reduced inertia matrix Mr(qr). The Lagrangianof a dynamic car is as the following

L =1

2m[(x− l

2sin(θ)θ)2 + (y +

l

2cos(θ)θ)2]

+1

2(Jb + 2Jv)θ

2 +1

2[(2Jh)(1 +

1

cos2(φ))]ψ2 +

1

2(2Jv)(θ + φ)2

where m is the mass of the car, Jb is the inertia of the body, Jh is the inertia of eachwheel along a horizontal axes, and Jv is the inertia of each wheel along the verticalaxes. After some simplification, the Lagrangian can be expressed as

L =1

2qT

m 0 −(ml/4) sin θ 0 00 m (ml/4) cos θ 0 0

−(ml/4) sin θ (ml/4) cos θ Jθ 0 2Jv0 0 0 2Jh(2 + tan2(φ)) 00 0 2Jv 0 2Jv

q

(6.22)where Jθ = Jb + 4Jv +ml2/4. The differential-algebraic equations of motion for thedynamic car are as the following

d

dt

∂L∂q− ∂L∂q

= W T (q)

[

λ1λ2

]

+

[

03×2I2×2

] [

τ1τ2

]

W (q)q = 0(6.23)

where λ1, λ2 ∈ R are Lagrange multipliers and τ1, τ2 ∈ R are torques applied tothe rear wheels and the steering wheel, respectively. Based on theorem 6.1.1, thedynamics of the car in (6.23) can be reduced to the cascade of the constraint equationand a reduced-order Lagrangian system with configuration vector qr = (θ, ψ, φ) as thefollowing

Mr(qr)qr + Cr(qr, qr)qr = Fr(qr)τ

where

Mr(qr) = DT (q)M(q)D(q)

Cr(qr, qr) = DTC(q, q)D +DTM(q)D

Fr(qr) = DT (q)F (q)

By direct calculation, we have

Fr(qr) = Fr =

0 01 00 1

192

and

Mr(φ) =

Jθ(φ) 0 2Jv0 2Jh(2 + tan2(φ)) 02Jv 0 2Jv

with

Jθ(φ) = Jθ +m(ρ21 + ρ22)−ρ1ml

2sin(θ) +

ρ2ml

2cos(θ) = Jθ +

ml2

tan2(φ)

Clearly, the reduced Lagrangian system is itself an underactuated system with threedegrees of freedom (θ, ψ, φ) and two controls. In addition, (θ, ψ) are the externalvariables and φ is the shape variable of the car. The dynamics of the actuatedvariables (φ, ψ) of the reduced system can be linearized as

ψ = u1φ = u2

using an explicit collocated change of control in the form

τ = α(φ)u+ β(φ, qr)

where

α(φ) =

2Jh(2 + tan2(φ)) 0

0 2Jv(1−2Jv tan

2(φ)

ml2 + Jθ tan2(φ)

)

is well-defined and positive definite for all φ. From the first constraint equation, onecan solve for θ to get

θ = tan(φ)(cos(θ)x+ sin(θ)y)

From this equation, the overall dynamics of the car can be written as

x = r cos(θ)ψ

y = r sin(θ)ψ

θ =r

ltan(φ)ψ

ψ = u1φ = u2

(6.24)

where r is the radius of the wheel. After normalization of the units of (x, y) by r, thekinematic model of the car can be expressed as

x = cos(θ)ω1

y = sin(θ)ω1

θ =1

ltan(φ)ω1

φ = ω2

(6.25)

193

whereω1 = u1, ω2 = u2

Now, applying the following change of coordinates and control [63]

x1 = x

x2 =tan(φ)

l cos(θ)3

x3 = tan(θ)x4 = yv1 = cos(θ)ω1

v2 =(1 + tan2(φ))

l cos(θ)3ω2 +

3 tan(θ) tan2(φ)

l2 cos(θ)3ω1

(6.26)

transforms the nonholonomic system in (6.25) into a chained-form system as thefollowing

x1 = v1x2 = v2x3 = x2v1x4 = x3v1

(6.27)

The system in (6.27) can be stabilized using a discontinuous state feedback [4, 47],or sinusoid inputs [63].

6.2.4 The Snakeboard

The snakeboard system depicted in Figure 6-4 was first introduced in [50]. In thatwork, an experimental study was performed for locomotion of the snakeboard usingdifferent “gait” settings (i.e. a set of sinusoid inputs). The snakeboard example wasone of the main motivations for developing a general framework for reduction of non-holonomic systems with symmetry on Riemannian manifolds in [9]. Controllabilityand motion planning for the snakeboard example as a control system was also consid-ered in [49] and [73, 74], respectively. Here, we show that the snakeboard system canbe reduced to an underactuated Lagrangian system in cascade with the constraintequation. Then, we use this representation to prove that locomotion control of thesnakeboard is equivalent to stabilization of a kinematic car.

Dynamics and Reduction of the Snakeboard

The configuration vector for the snakeboard is q = (x, y, θ, ψ, φ1, φ2) ∈ R2 × S41 .

The velocity constraints for the front and back wheels of the snakeboard are as thefollowing

d

dt(x+ l cos θ) · sin(θ + φ1)−

d

dt(x+ l sin θ) · cos(θ + φ1) = 0

d

dt(x− l cos θ) · sin(θ − φ2)−

d

dt(x− l sin θ) · cos(θ − φ2) = 0

194

φ

ψ φ

θ

2

1

l

x

y

Figure 6-4: The Snakeboard.

These constraints can be simplified as

sin(θ + φ1)x− cos(θ + φ1)y − l cos(φ1)θ = 0 (6.28)

sin(θ − φ2)x− cos(θ − φ2)y + l cos(φ2)θ = 0 (6.29)

Therefore, the constraints are in the formW (q)q = 0 with the constraint matrixW (q)given by

W (q) =

[

sin(θ + φ1) − cos(θ + φ1) −l cos(φ1) 0 0 0sin(θ − φ2) − cos(θ − φ2) l cos(φ2) 0 0 0

]

Partitioning q as q1 = (x, y) and q2 = (θ, ψ, φ1, φ2) and defining the vector η =(θ, φ1, φ2) (i.e. all the variables that appear in W (q)), the constraint matrix W (q)can be partitioned as (W1,W2) where W1 and W2 are defined as

W1(η) =

[

sin(θ + φ1) − cos(θ + φ1)sin(θ − φ2) − cos(θ − φ2)

]

, W2(η) =

[

−l cos(φ1) 0 0 0l cos(φ2) 0 0 0

]

and W1(η) is invertible except for at isolated points φ1 + φ2 = kπ, k ∈ Z. Assumeφ1 + φ2 6= kπ. By direct calculation, we have

w12(η) = −W−11 W2 =

[

ρ1 0 0 0ρ2 0 0 0

]

where

ρ1 =l cosφ1 cos(θ − φ2) + l cosφ2 cos(θ + φ1)

sin(φ1 + φ2)

ρ2 =l cosφ1 sin(θ − φ2) + l cosφ2 sin(θ + φ1)

sin(φ1 + φ2)

(6.30)

195

Hence

D(q2) = D(η) =

[

w12(η)I4×4

]

=

ρ1 0 0 0ρ2 0 0 01 0 0 00 1 0 00 0 1 00 0 0 1

It is assumed that the torque τ = (τ1, τ2, τ3)T ∈ R3 is applied to variables ψ, φ1, φ2

and the rest of variables are unactuated. The Lagrangian of the snakeboard is in theform

L(q, q) =1

2m(x2 + y2) +

1

2Jbθ

2 +1

2Jr(ψ + θ)2

+1

2Jv(φ1 + θ)2 +

1

2Jv(φ2 − θ)2

where Jb, Jr, and Jv are the moment of inertia of the body, the rotor, and the wheelalong the vertical axis, respectively [50]. This Lagrangian can be rewritten as

L(q, q) = 1

2

xy

θ

ψ

φ1φ2

T

m 0 0 0 0 00 m 0 0 0 00 0 Jθ Jr Jv −Jv0 0 Jr Jr 0 00 0 Jv 0 Jv 00 0 −Jv 0 0 Jv

xy

θ

ψ

φ1φ2

with Jθ = Jb + Jr + 2Jv. Clearly, the inertia matrix M is constant. Thus, theCoriolis and centrifugal matrix C(q, q) associated with M vanishes identically. Thealgebraic-differential equations of motion for this system is as the following

d

dt

∂L∂q− ∂L∂q

=W T (q)

[

λ1λ2

]

+

[

03×3I3×3

]

τ1τ2τ3

W (q)q = 0

(6.31)

which can be expressed as

mx = λ1 sin(θ + φ1) + λ2 sin(θ − φ2)my = −λ1 cos(θ + φ1)− λ2 cos(θ − φ2)

Jθθ + Jrψ + Jvφ1 − Jvφ2 = −λ1l cosφ1 + λ2l cosφ2Jrψ + Jrθ = τ1Jvφ1 + Jvθ = τ2Jvφ2 − Jvθ = τ3

(6.32)

Based on theorem 6.1.1, the reduced Lagrangian system for the snakeboard systemin (6.32) is an underactuated system with configuration vector qr = (θ, ψ, φ1, φ2) as

196

the followingMr(qr)qr + Cr(qr, qr)qr = Fr(qr)τ (6.33)

where

Mr(qr) = Mr(η) = D(η)TMD(η)

Cr(qr, qr) = Cr(η, η) = DT (η)MD(η, η)

Fr(qr) = Fr = DT (q)F (q) =

[

01×3I3×3

]

By direct calculation, we obtain the following inertia matrix

Mr(qr) =

Jθ(φ1, φ2) Jr Jv −JvJr Jr 0 0Jv 0 Jv 0−Jv 0 0 Jv

whereJθ = Jθ +m(ρ21 + ρ22)

Setting b := ρ21 + ρ22, we get

b(φ1, φ2) = l2cos2 φ1 + cos2 φ2 + 2 cosφ1 cosφ2 cos(φ1 + φ2)

sin2(φ1 + φ2)

which means b ≥ 0 does not depend on θ, ψ. Thus, Mr(qr) = Mr(φ1, φ2). In otherwords, (φ1, φ2) are the shape variables and (θ, ψ) are the external variables of thereduced snakeboard. Notice that b(φ1, φ2) has an isolated singularity at (φ1, φ2) = 0(under the assumption that |φi| < π/2; i = 1, 2).

The reduced system in (6.33) has non-interacting control inputs (i.e. Fr =col(F 1

r , F2r ) such that F 1

r = 0). Therefore, using collocated partial-feedback lineariza-tion, the dynamics of all three actuated configuration variables (ψ, φ1, φ2) can belinearized using an explicit change of control in the form

τ = α(φ)u+ β(φ, qr)

where φ = (φ1, φ2) and α(φ) is a positive definite matrix with a well-defined p.d. limitas φ→ 0. After collocated partial feedback linearization, we get

ψ = u1φ1 = u2φ3 = u3

(6.34)

This gives the following asymptotic output tracking result for the snakeboard.

Proposition 6.2.1. Denoting ξ1 = (ψ, φ1, φ2) and ξ2 = (ψ, φ1, φ2), we obtain

ξ1 = ξ2, ξ2 = u

197

with u ∈ R3. Let ξd(t) ∈ R3 be a desired C2 smooth trajectory for the snakeboard.Then, the following partial state and output feedback

u = kp(ξ1 − ξd) + kd(ξ2 − ξd) + ξd, kp, kd < 0

achieves global exponential tracking of the trajectory ξd for the snakeboard.

Proof. Set e1 = ξ1 − ξd and e2 = ξ2 − ξd. Then

e1 = e2, e2 = kpe1 + kde2

and the ei’s vanish exponentially.Proposition 6.2.1 allows exponential tracking of certain “gaits” or sinusoid inputs

(ψ, φ1, φ2) for locomotion of the snakeboard [50]. Here, the contribution is that insteadof algebraic manipulation of the equations of motion (6.32) in [50], we propose asystematic method for reduction and motion generation of a complex example like thesnakeboard which is applicable to higher-order underactuated nonholonomic systemsas well.

Locomotion Control of the Snakeboard

The key to locomotion control of the snakeboard is the invariance of the general-ized momentum conjugate to θ in the Lagrangian reduced system of the snakeboardin equation (6.33). Since, the potential energy of the system is zero and θ is anunactuated variable of the reduced system (6.33), the generalized momentum

πθ =∂Lr∂θ

= Jθ(φ1, φ2)θ + Jrψ + Jvφ1 − Jvφ1

is a conserved quantity. Assuming that the system was initially at rest (i.e. had zeroinitial velocity), πθ = 0,∀t > 0 and

θ = − Jr

Jθ(φ1, φ2)ψ − Jv

Jθ(φ1, φ2)(φ1 − φ2) (6.35)

Based on the following assumption, both the last equation and the expression forρ1, ρ2 in (6.30) can be highly simplified.

Assumption 6.2.1. Assume φ1 = φ2 = φ and |φ| < π/2.

Lemma 6.2.1. Under Assumption 6.2.1, the following equations hold:

ρ1(φ) =l cos(θ)

tan(φ)

ρ2(φ) =l sin(θ)

tan(φ)

θ = − Jr tan(φ)2

ml2 + Jθ tan(φ)2ψ

(6.36)

198

In addition, the constraint equation for the snakeboard is equivalent to the followingnonholonomic constraints

sin(θ + φ)x− cos(θ + φ)y − l cos(φ)θ = 0sin(θ)x− cos(θ)y = 0

(6.37)

Proof. Setting φ1 = φ2 = φ, we get

ρ1 = lcos(φ1) cos(θ − φ2) + cos(φ2) cos(θ + φ1)

sin(φ1 + φ2)

= lcos(φ)[cos(θ − φ) + cos(θ + φ)]

sin(2φ)

= lcos(φ)(2 cos(θ) cos(φ))

2 sin(φ) cos(φ)

=l cos(θ)

tan(φ)

Similarly, one obtains ρ2 = l sin(θ)/ tan(φ). Thus, Jθ can be rewritten as

Jθ(φ) = Jθ +m(ρ21 + ρ22) = Jθ +ml2

tan2(φ)

From this equation, we have

θ = − Jr

Jθ(φ)ψ = − Jr tan

2(φ)

ml2 + Jθ tan2(φ)

ψ

In addition, under Assumption 6.2.1, the velocity constraints for the snakeboard canbe expressed as

sin(θ + φ)x− cos(θ + φ)y − l cos(φ)θ = 0

sin(θ − φ)x− cos(θ − φ)y + l cos(φ)θ = 0

By adding the last two equations, one gets

[sin(θ + φ) + sin(θ − φ)]x− [cos(θ + φ) + cos(θ − φ)]y = 0

which can be simplified as

cos(φ)[sin(θ)x− cos(θ)y] = 0

Due to |φ| < π/2, the last constraint reduces to

sin(θ)x− cos(θ)y = 0

and the result follows.Notice that the velocity constraints in (6.37) are exactly the same as the velocity

199

constraints of a car. Roughly speaking, a velocity constraint in the form

sin(θ)x− cos(θ)y = 0

for the snakeboard, implies that the back half of the snakeboard can be removed andtwo new virtual wheels can be added to the middle of the snakeboard. These virtualwheels must be along the body of the snakeboard with orientation angles θ w.r.t. thehorizontal axis. This connection between the snakeboard and the car suggests thefollowing formal equivalence between the two systems.

Proposition 6.2.2. Let q = (x, y, θ, ψ, φ) ∈ SE(2)×S1×S1 denote the configurationvector of both the snakeboard and the car with one difference. For the snakeboard, ψdenotes the angle of the rotor, while ψ is the angle of the rotation of the rear wheelsin a car. Then, the kinematic model of the snakeboard and the car are the samefor φ 6= 0 (i.e. the diffeomorphism that maps the configuration of the snakeboardto the configuration of the car is an identity map). In addition, both the kinematicsnakeboard and the kinematic car can be transformed into a chained-form system asthe following

x1 = v1x2 = v2x3 = x2v1x4 = x3v1

(6.38)

where the diffeomorphism

(x1, x2, x3, x4, v1, v2) = T (x, y, θ, φ, ωψ, ωφ)

for the snakeboard is given by

x1 = x

x2 =tan(φ)

l cos(θ)3

x3 = tan(θ)x4 = y

v1 = − lJr cos(θ) tan(φ)ml2 + Jθ tan

2(φ)ωψ

v2 =(1 + tan2(φ))

l cos3(θ)ωφ −

3 sin(θ) tan(φ)3Jrl cos(θ)3(ml2 + Jθ tan

2(φ)ωψ

(6.39)

and the dynamics of (ψ, φ) is linear as the following

ψ = ωψφ = ωφωψ = u1ωφ = u2

(6.40)

200

Proof. The dynamics of the snakeboard with φ1 = φ2 = φ can be expressed as

x = ρ1(φ)θ

y = ρ2(φ)θ

θ = − Jr tan2(φ)

ml2 + Jθ tan2(φ)

ψ

ψ = u1φ = u2

(6.41)

where

ρ1(φ) =l cos(θ)

tan(φ), ρ2(φ) =

l sin(θ)

tan(φ)

Defining

w1 = −lJr tan(φ)

ml2 + Jθ tan2(φ)

ψ

the kinematic model of the snakeboard takes the following form

x = cos(θ)w1

y = sin(θ)w1

θ =1

ltan(φ)w1

φ = w2

(6.42)

with control inputs (w1, w2) ∈ R2. Equation (6.42) is exactly the same as the kine-matic model of a car with configuration (x, y, θ, φ) where w1 is the velocity of therear wheels of the car and w2 is the velocity of the steering wheel. Now, applyingthe following change of coordinates and control (which is equivalent to the one in thequestion)

x1 = x

x2 =tan(φ)

l cos(θ)3

x3 = tan(θ)x4 = yv1 = cos(θ)w1

v2 = x2

(6.43)

transforms the kinematic model of the snakeboard into the chained-form nonholo-nomic system in (6.38).

Remark 6.2.1. Proposition 6.2.2 reduces the locomotion control for the snakeboardto the stabilization problem for a car which is a rather well-studied problem [63, 4].This is the first direct evidence between control of a locomotion system and a mobilesystem that are both transformable into chained-form nonholonomic systems. Inother words, this is an example of a case where the locomotion control for a systemreduces to the stabilization of a simple system.

201

Geometric Interpretation of (ρ1, ρ2) and the Singularity in the Kinematicsof the Snakeboard

The functions (ρ1, ρ2) in the kinematics and constraint equations of both the snake-board and the car have a meaningful geometric interpretation that is stated in thefollowing. Consider a car with a virtual rear wheel in the middle of the real rearwheels and a virtual front wheel. The path and geometry of the rear and front vir-tual wheels for a car are shown in Figure 6-5. For the snakeboard, the front wheelsare the same but the rear wheel is a virtual wheel along the body of the snakeboardat the center of mass (x, y). If one fixes the orientation of the front wheel at φ. The

φ

φ

θ

ρ

ρ θ l

O

1

2

ρy

x

(x,y)

Figure 6-5: The geometry of the paths of the rear and front wheels of the car with aradius of rotation ρ.

car rotates around the center of rotation O. The path of both wheels are circles.Let ρ denote the radius of the rotation of the rear wheel located at point p = (x, y).Then, the velocity v of point p is tangent to the circle (O, ρ) at p and is obtained by

v = ρθ (6.44)

Therefore, the vertical and horizontal components of v are

x = ρ cos(θ)θ

y = ρ sin(θ)θ(6.45)

In other words, setting ρ1 = ρ cos θ and ρ2 = ρ sin θ, the components of v satisfy thefollowing equations

x = ρ1θ

y = ρ2θ(6.46)

From Figure 6-5, we have

ρ =l

tan(φ)

202

Thus, ρ1 and ρ2 are given by

ρ1 =l cos(θ)

tan(φ), ρ2 =

l sin(θ)

tan(φ)

This is exactly the same as the definition of ρ1, ρ2 for the car and the snakeboardwith φ1 = φ2 = φ. Notice that ρ uniquely determined by the orientation angle ofthe front wheel φ and does not depend on θ. Apparently, when φ → 0, ρ → ∞ andρ1, ρ2 →∞. This is equivalent to the fact that θ is constant (or θ = 0) and the radiusρ of a straight line is infinity.

Now, let ψ denote the angle of rotation of the rear wheel with unit radius in thecar. Then, v = ρθ = ψ, which means

θ =1

ρψ =

tan(φ)

lψ

This gives the final equations of motion for the kinematic car as

x = cos(θ)vy = sin(θ)v

θ =tan(φ)

lv

φ = w

The last system is obtained without any explicit use of nonholonomic velocity con-straints. However, the fact that the rear and front wheels always move on a circle withradius ρ is equivalent to the assumption that the wheels do not slip. The snakeboarddoes not satisfy the last equation since it does not have a real rotating rear wheel.Instead, it has a rotor with angle of rotation ψ. The invariance of the momentumconjugate to θ in the reduced Lagrangian system of the snakeboard plays the roleof the kinematic equation θ = ψ/ρ in the car. One can observe that defining thecurvature as κ = 1/ρ, we have

θ = κ(φ)ψ

where κ(φ) is the curvature of the circle of rotation of the rear wheel of the carcorresponding to the front wheel angle φ. If the rear wheels of the car rotate with aconstant non-zero velocity ψ, the curvature κ(φ) plays the role of the control inputfor the motion of the group element (x, y, θ) ∈ SE(2).

A rather similar property holds for the snakeboard with arbitrary front wheel andrear wheel angles (φ1, φ2). The geometric interpretation of ρ is shown in Figure 6-6.In this figure, (r1, r2) denote the radii of rotation corresponding to the front and backwheels of the snakeboard. In addition, ρ is the radius of rotation of the center ofmass of the snakeboard. This is due to the midsection theorem in planar geometry.Observe that

ρ2 = a2 + h2

203

θγρ

ρρ

φ

φ

φ

2

2

1

1

1φ

r

2r

1

2

x

y

l

la

h

Oγ − θ

Figure 6-6: The geometry of the paths of the rear and front wheels and the center ofmass of the snakeboard with a radius of rotation ρ.

and the radii of rotation r1, r2 satisfy

r21 = h2 + (l + a)2

r22 = h2 + (l − a)2

Adding the last two equation gives

ρ2 =r21 + r22

2− l2

In other words, if r1, r2 (or φ1, φ2) are constant, then ρ is constant and it is uniquelydetermined by φ1, φ2, l. This means that the geometric path of (x, y) is a circlecentered at O with radius ρ.

An elementary calculation shows that

h =2l

tan(φ1) + tan(φ2)

From h, the expressions of ρ, r1, r2, γ can be readily determined as functions of theshape variables (φ1, φ2). Particularly, one obtains the following formula for ρ(φ1, φ2)

ρ2 = a2 + h2 = (l − h tan(φ2))2 + h2 = l2[

(tan(φ1)− tan(φ2))2 + 4

(tan(φ1) + tan(φ2))2

]

It is rather elementary to prove that the radius of rotation ρ from the last equationis exactly the same as ρ =

√

ρ21 + ρ22 for ρ1, ρ2 defined in equation (6.30).

204

6.3 Notions of ε-Stabilization and ε-Tracking

In this section, we define new stability/tracking notions for nonlinear systems calledε-stability/tracking. Consider the following nonlinear system

x = f(x) (6.47)

where f : Rn → Rn is a C1 smooth function.

Definition 6.3.1. (virtual equilibrium) We say x0 ∈ Rn is a virtual equilibrium of(6.47) within a distance ε > 0, if there exists an equilibrium x∗0 of (6.47) that is withina distance ε from x0, i.e. ∃x∗0 ∈ Rn : f(x∗0) = 0, ‖x∗0−x0‖ ≤ ε. We call x∗0 the adjacentequilibrium to x0.

Definition 6.3.2. (ε-stable virtual equilibrium) We call x0 an (exponentially orasymptotically) ε-stable virtual equilibrium of (6.47), if there exists an adjacent equi-librium x∗0 within a distance ε > 0 from x0 that is (exponentially or asymptotically)stable for (6.47) in the sense of Lyapunov.

We are interested in considering the solutions of (6.47) under specific classes ofparameterized transformations where the notion of ε-stability of solutions is useful.Assume there exists a global diffeomorphism ψ(x, λ) : Rn × Rp → Rn in x that iscontinuous in λ and satisfies ψ(x, λ) = x =⇒ λ = 0. By continuity at λ = 0, wehave

∀ε > 0,∃λ = λ(ε) : ‖ψ(x, λ)− x‖ ≤ ε (6.48)

We call ψ(x, λ) a near identity diffeomorphism for ‖λ‖ ¿ 1. Denote the inverse ofy = ψ(x, λ) by x = φ(y, λ) so that

φ(ψ(x, λ), λ) = x, ψ(φ(y, λ), λ) = y

By definition φ(x, 0) = x and φ(x, λ) = x =⇒ λ = 0. Also, φ(x, λ) is continuousw.r.t. λ. Applying the following change of coordinates

xλ = ψ(x, λ)

one obtains a parameterized nonlinear system that we call λ-system

xλ = f(xλ, λ) (6.49)

where

f(xλ, λ) =

[

∂ψ(x, λ)

∂xf(x)

]

x=φ(xλ,λ)

Apparently, any equilibrium point of (6.47) is an equilibrium point of (6.49) and viceversa.

Definition 6.3.3. (ε-stability under transformation) For any ε > 0, let λ = λ(ε)be such that (6.48) holds. We say x = x0 is (exponentially/asymptotically) ε-stable

205

virtual equilibrium of (6.47) under the transformation ψ(x, λ), iff xλ = x0 is a (ex-ponentially/asymptotically) stable equilibrium of the λ-system (6.49) in the senseof Lyapunov with λ = λ(ε), or equivalently iff x0ε = φ(x0, λ(ε)) is (an) a (exponen-tially/asymptotically) stable equilibrium of (6.47) in the sense of Lyapunov, i.e. x0εis an adjacent equilibrium to x0 for the λ-system.

The following result is a common sense corollary of the definition of ε-stabilitywhich means the original system has an asymptotically stable equilibrium point nearx0 in the usual sense of stability.

Corollary 6.3.1. If x0 is an (exponentially/asymptotically) ε-stable equilibrium of(6.47) under the transformation ψ(x, λ), then there exists an equilibrium point x0ε of(6.47) that is (exponentially/asymptotically) stable equilibrium of (6.47) in the senseof Lyapunov within a distance ε > 0 from x0, i.e. x0ε .

Proof. Let x0ε = φ(x0, λ(ε)), then

‖x0ε − x0‖ = ‖x0ε − ψ(x0ε , λ(ε))‖ < ε

Remark 6.3.1. Two simple examples of ψ(x, λ) are the following

ψ(x, λ) = x+ λ~u, λ ∈ R, ‖~u‖ = 1

andψ(x, λ) = x+ Aλ, x ∈ Rn, λ ∈ Rp, A has full rank

Now, consider a nonlinear control system with a dynamic state feedback

x = f(x, u)

λ = g(x, λ)(6.50)

Let ψ(x, λ) be a global diffeomorphism as before. Applying the change of coordinates

xλ = ψ(x, λ)

we obtain the following λ-system

xλ = f(xλ, λ, u)

λ = g(xλ, λ)(6.51)

where

f(x, λ, u) =

[

∂ψ(x, λ)

∂xf(x, u) +

∂ψ(x, λ)

∂λg(x, λ)

]

x=φ(xλ,λ)

g(xλ, λ) = g(φ(xλ, λ), λ)

Definition 6.3.4. (ε-stabilization) For a fix ε > 0, let λf = λ(ε) satisfy (6.48). Letx0 be a desired equilibrium point of the system with zero control, or f(x0, 0) = 0. We

206

say the dynamic state feedback

u = k(x, λ)

λ = g(x, λ)(6.52)

achieves globally asymptotic ε-stabilization of x0 for (6.50) under the transformationψ(x, λ), iff for the closed-loop λ-system

xλ = f(xλ, λ, k(xλ, λ))

λ = g(xλ, λ)

where k(xλ, λ) = k(φ(xλ, λ), λ), (x0, λf ) is a globally asymptotically stable equilib-

rium in the sense of Lyapunov, or equivalently x0ε = φ(x0, λf ) and λf are globallyasymptotically stable equilibriums of (6.50).

The notion of ε-tracking for a nonlinear control system x = f(x, u) can be definedfor an equivalent transformed control system with dynamic state feedback based onthe notion of ε-stabilization.

Definition 6.3.5. (ε-tracking) Consider the following nonlinear system with input uand output y

x = f(x, u)y = h(x)

(6.53)

We say a control law u achieves asymptotic ε-tracking of a trajectory yd(·) for (6.53)under transformation ψ(x, λ), if there exists a control law u that achieves asymptotictracking of the desired trajectory yλ = yd for

xλ = f(xλ, λ, u)

λ = g(xλ, λ)yλ = h(φ(x, λ))

(6.54)

6.4 Global Exponential ε-Stabilization/Tracking for

a Mobile Robot

In this section, we design a nonlinear controller for global exponential ε-stabilizationand ε-tracking of a nonholonomic mobile robot (Figure 6-2).

6.4.1 Dynamics of a Mobile Robot in SE(2)

In the majority of previous control design strategies for this nonholonomic mobilerobot, the kinematic model of the robot is used and the input torque are then obtainedusing adding an integrator and high-gain control. Here, we directly use the dynamic

207

model of the mobile robot as the following

x1 = cos(θ)v1x2 = sin(θ)v1θ = v2v1 = τ1v2 = τ2

(6.55)

Since this mobile robot moves on a surface, a more appropriate form of representationof the dynamics of the mobile robot is in SE(2). SE(2) is the Special Euclideangroup in R2 with elements g = (R, x) where R is the rotation matrix satisfyingRTR = I, det(R) = 1 (i.e. R ∈ SO(2)) and x ∈ R2. Equivalently g can be defined as

g =

[

R x0 I

]

where the group product g1g2 is matrix multiplication. From (6.55), the dynamics ofthe mobile robot in SE(2) can be expressed as the following

x = (Re1)v1R = Rv2v = τ

(6.56)

where

R =

[

cos(θ) − sin(θ)sin(θ) cos(θ)

]

and the skew-symmetric matrix v2 is defined as follows

v2 =

[

0 −v2v2 0

]

or equivalently the evolution of the group element g in (6.56) can be expressed as

g = g · Ω, v = τ

where Ω ∈ se(2) (i.e. the Lie algebra associated with SE(2))

Ω =

[

v2 v10 0

]

We use the representation in (6.56) as the model of this nonholonomic mobile robot.

6.4.2 Near-Identity Diffeomorphism

Consider the following near identity smooth change of coordinates

z = ψ(x, λ) := x+ λRe1 (6.57)

208

where λ ∈ R, ei is the ith standard basis in R2, and R = R(θ) is the rotation matrixby θ in R2. This transformations is schematically depicted in Figure 6-7. The inverse

x

x

z

z

d

d= λ R e

||d||=|λ |

1

2

1

2 1

θ

Figure 6-7: A near identity transformation with a parameter |λ| ¿ 1.

of this transformation is a near identity diffeomorphism given by

x = φ(z, λ) = z − λRe1

Apparently, for all x ∈ R2 the following property holds

‖ψ(x, λ)− x‖ = ‖λRe1‖ = |λ|

and for all ε > 0, λ = ε achieves the property

‖ψ(x, λ)− x‖ ≤ ε

Notice that these properties of the near identity diffeomorphism ψ(x, λ) are true for(R, x) ∈ SE(n) with n ≥ 2 and are not restricted to the special case of n = 2. Thedynamics of the system in z-coordinates (i.e. λ-system) can be expressed as

z = Rλv + (Rλe1)λ

Rλ = Rλv2(λ, λ)v = τ

(6.58)

where v = (v1, v2)T and Rλ, v2 are given by

Rλ =

[

cos(θ) −λ sin(θ)sin(θ) λ cos(θ)

]

, v2(λ, λ) =

0 −λv21

λv2

λ

λ

Note that for λ 6= 1, Rλ is not a rotation matrix and RTλRλ 6= I. In fact, we have

RTλRλ =

[

1 00 λ2

]

, detRλ = λ

Thus, for λ 6= 0, the matrix Rλ is invertible.

209

6.4.3 Control Design for ε-Stabilization/Tracking

In this section, we present our main ε-stabilization/tracking results for a nonholonomicmobile robot.

Proposition 6.4.1. Any desired position xd ∈ R2 for the nonholonomic mobile robotin (6.56) can be rendered globally exponentially ε-stable using a smooth dynamic statefeedback in an explicit form. Equivalently, this feedback law globally exponentiallystabilizes (i.s.o. Lyapunov) an equilibrium xεd of (6.56) with a distance ε from xd, i.e.‖xεd − xd‖ = ε.

Proof. Fix an ε > 0, and let the dynamics of λ be as

λ = −cλ(λ− λf ), λ(0) = λ0 > ε, λf = ε, cλ > 0

Clearly, λ = λf is exponentially asymptotically stable for this system and λ(t) ≥ε > 0,∀t ≥ 0. The task in the question is to stabilize the position of the mobilerobot to x = xd (the orientation of the mobile robot can be trivially exponentiallystabilized by setting v1 = 0 at x = xd). Thus, we set the desired equilibrium of theλ-system in (6.58) to zd = xd. The goal is to globally exponentially stabilize z = zdfor the λ-system. This renders the equilibrium xεd = φ(xd, ε) of the original systemis (6.56) globally exponentially stable (i.s.o. Lyapunov). But ‖xεd − xd‖ = ε, thusthis achieves global exponential ε-stabilization of xd for the original system (6.56).To render zd globally exponentially stable for the λ-system in (6.56) can be renderedglobally exponentially stable. For doing so, let us calculate z. We have

z = Rλv + Rλv + (Rλe1)λ+ (Rλe1)λ

= Rλτ +Rλv2(λ, λ)v +Rλ(v2(λ, λ)− cλI)e1λ

Thus, defining the change of control

τ = R−1λ u− v2(λ, λ)v − (v2(λ, λ)− cλI)λe1

where λ = −cλ(λ− ε), we getz = u

which is a two-dimensional double-integrator. Therefore, applying

u = −cp(z − zd)− cdz, cp, cd > 0

oru = −cp(z − zd)− cd(Rλv +Rλe1λ)

renders the equilibrium point z = zd of the λ-system globally exponentially stable.The overall smooth dynamic state feedback is in the form

τ = −cpR−1λ (z − zd)− cdv − cdλe1 − v2(λ, λ)v − (v2(λ, λ)− cλI)λe1λ = −cλ(λ− ε)

(6.59)

210

Figures 6-8 and 6-9 show the path of the mobile robot starting at position x =(2, 3)T for the initial orientation angles θ = kπ/4, k = 0, . . . , 7. The trajectories forthe position and input torques (controls) are shown in Figure 6-10

Similar results for trajectories starting at x = (0, 2)T with the initial orientationangles θ = 0, π/4, π/2, 3π/2 are shown in Figure 6-11 (for the complement anglesthe trajectories are symmetric w.r.t. vertical axis). Also, trajectories of the posi-tion and input torques (controls) for initial position x = (0, 2)T ) and orientationθ = 0 are shown in Figure 6-12. These results demonstrate that the controller ef-ficiently stabilizes the origin for this nonholonomic mobile robot. Each trajectoryexponentially converges to a point within a distance ε = 0.01 from the desired equi-librium xd = 0. This is sufficiently close to the origin for all practical purposes.The values of the parameters in all simulations for the mobile robot were chosen ascλ = 1, cp = 1, cd = 2, λ(0) = 0.5, λf = ε = 0.01. The trace trajectories of this non-holonomic robot are shown in Figure 6-13. Based on this figure, graphically, the robotmakes a turn to face the desired destination (i.e. the origin) and goes exponentiallyfast towards the destination.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=0

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=pi/4

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=pi/2

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=3pi/4

x1

x2

Figure 6-8: Trajectories of the nonholonomic mobile robot in (x1, x2)-plane for initialposition x = (2, 3)T (v = 0) and orientation angles θ = 0, π/4, π/2, 3π/4.

Asymptotic ε-tracking of a desired output zd can be obtained based on the follow-ing result.

Proposition 6.4.2. The following dynamic state feedback law achieves asymptotic

211

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=pi

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=5pi/4

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=3pi/2

x1

x2

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5theta=7pi/4

x1x2

Figure 6-9: Trajectories of the nonholonomic mobile robot in (x1, x2)-plane for initialposition x = (2, 3)T (v = 0) and orientation angles θ = π, 5π/4, 3π/2, 7π/4.

ε-tracking for the desired output yd(·) of the nonholonomic mobile robot (6.55)

τ = −cpR−1λ z − cdv − cdλe1 − v2(λ, λ)v − (v2(λ, λ)− cλI)λe1+ R−1λ (cpyd + cdyd + yd)

λ = −cλ(λ− ε)(6.60)

Proof. Set the desired trajectory in z-coordinates to zd = yd. Then, the followingfeedback

u = −cp(z − zd)− cd(z − zd) + zd

guarantees that the error e = z − zd globally asymptotically converges to zero andthe result follows from the relation between τ and u.

Figure 6-14 demonstrates simulation results of the ε-tracking for a nonholonomicrobot starting at position (4, 4) with orientation angle π/2. The desired trajectoryis an ellipse (x, y) = (3 sin t, 4 cos t). Clearly, the robot very quickly converges to anε-neighborhood of the desired trajectory (e.g. ε = 0.01).

212

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

time (sec)

Pos

ition

x1x2

0 2 4 6 8 10 12−30

−20

−10

0

10

20

30

time (sec)

Con

trol

tau1tau2

Figure 6-10: Trajectories of the nonholonomic mobile robot in (x1, x2)-plane startingat x = (2, 3)T (v = 0) with angle θ = π/4

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5theta=0

x1

x2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5theta=pi/4

x1

x2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5theta=pi/2

x1

x2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5theta=3pi/4

x1

x2

Figure 6-11: Trajectories of the nonholonomic mobile robot in (x1, x2)-plane for initialposition x = (0, 2)T (v = 0) and initial orientation angles θ = 0, π/4, π/2, 3π/4.

213

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

time (sec)

Pos

ition

x1x2

0 1 2 3 4 5 6 7 8 9 10−5

0

5

time (sec)

Con

trol

tau1tau2

Figure 6-12: Evolution of position and controls for the nonholonomic mobile robot in(x1, x2)-plane starting at x = (0, 2)T (v = 0) with angle θ = 0

−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

(a) (b)

−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

(c) (d)

Figure 6-13: Trace trajectories of a two-wheel nonholonomic robot with initial posi-tion (5, 4) and heading angles (a) θ = 0, (b) θ = π/2, (c) θ = π, and (d) θ = 3π/2.

214

0 2 4 6 8 10 12 14 16 18−4

−2

0

2

4

6

time (sec)

x

x x

d

0 2 4 6 8 10 12 14 16 18−5

0

5

time (sec)

y

y y

d

Figure 6-14: Position trajectory tracking for a mobile robot from the initial point(4, 4, π/2).

215

Chapter 7

Control of Nonlinear Systems inNontriangular Normal Forms

This chapter is devoted to control of nonlinear systems in normal forms that pos-sess nontriangular structures. Control of nonlinear systems in triangular forms, i.e.(strict) feedback and feedforward forms, has been extensively studied in the pastdecade. However, few results are available that deal with nonlinear systems withnontriangular structures. later, we will be more precise on what we mean by nontri-angular forms and present the challenging problems and stabilization limitations ofnonlinear systems in nontriangular forms. The following notations are convenient inthis chapter.

Notation. GAS stands for globally asymptotically stable.

Notation. ‖ · ‖ = 〈x, x〉 12 denotes the Euclidean norm and ‖ · ‖p denotes the Lp-normin Rn.

Notation. (Lie derivative) Let f(x) : Rn → R and g : Rn → Rn be smooth functions.The directional derivative of f(x) with respect to g(x) is defined as any of the followingequivalent forms

Lgf(x) :=∂f(x)

∂xg(x) = 〈∇f(x), g(x)〉 = Df(x) · g(x) (7.1)

where 〈x, y〉 = xTy denotes the inner product in Rn. Moreover, the kth order Lie

derivative of f w.r.t. g is denoted by L(k)g f and is defined as

L(k)g f(x) := Lg(L

((k−1)g f(x)), k ∈ Z+ (7.2)

with L(0)g f(x) = f(x).

The outline of this chapter is as follows: First, we provide some backgroundon stabilization of nonlinear systems in triangular forms, namely, (strict) feedbackand feedforward forms. Secondly, we focus on control of nonlinear systems withnontriangular structures.

216

7.1 Introduction

In the preceding chapters, we obtained three types of normal forms for underactuatedmechanical systems. Namely, cascade nonlinear systems in strict feedback form, strictfeedforward form, and different variations of nontriangular linear-quadratic forms. Inthe past, effective methods have been developed for stabilization of nonlinear systemsin triangular forms (i.e. feedback/feedforward) [57, 36, 102, 103, 101, 56, 80]. Namely,the backstepping procedure [57, 36, 103] and forwarding methods [102, 101, 56, 80] areanalytic tools for stabilization of nonlinear systems in feedback forms and feedforwardforms, respectively.

The class of nonlinear systems affine in control

x = f(x) + g(x)u, x ∈ Rn, u ∈ Rp (7.3)

is one of the most well-studied types of nonlinear systems. Assume there exists anoutput y = h(x) that has uniform relative degree r ≤ n with respect to the input uover an open neighborhood U of x = x0 ∈ Rn, i.e. there exist α(x), β(x) such thatthe following properties hold over U for a constant r

L(k)g h(x) = 0, ∀k ∈ 0, 1, . . . r − 1

y(r) = α(x)u+ β(x)

where α(x) is an invertible matrix. According to [36], there exists a change of coor-dinates and control in the form

(z, ξ) = Φ(x), v = α(x)u+ β(x)

that transforms the nonlinear system in (7.3) into the Byrnes-Isidori normal form ofdegree r ≤ n

z = F (z, ξ1, ξ2, . . . , ξr)

ξ1 = ξ2· · ·

ξr = v

(7.4)

Here, Φ(x) is a diffeomorphism over U . If U = Rn, then y = h(x) has global uniformrelative degree r and the aforementioned change of coordinates is global. The z-subsystem

z = F (z, ξ)

in (7.4) with ξ = (ξ1, . . . , ξr) is called the zero-dynamics of (7.3) [36]. Given ξ1 =ξ2 = . . . = ξr = 0, if z = 0 is asymptotically stable for the zero-dynamics, then boththe output y = h(x) and the zero-dynamics are called minimum phase.

Under the condition that the vector field F in (7.4) does not depend on (ξ2, . . . , ξr),

217

the nonlinear system in (7.4) reduces to

z = F (z, ξ1)

ξ1 = ξ2· · ·

ξr = v

(7.5)

which is in strict feedback form [36, 57]. In general, if at most one of ξ1, . . . , ξr appearsin f , it is known how to stabilize the composite system in (7.4) under appropriatetechnical conditions (see [36], chapter 9). The importance of the strict feedbacknormal form in (7.5) is due to the existence of the following theorem called standardbackstepping [36].

Theorem 7.1.1. (standard backstepping) Consider the nonlinear system in (7.5) andassume there exists a smooth state feedback ξ1 = α(z) with α(0) = 0 such that for theclosed-loop zero-dynamics

z = F (z, α(z)) (7.6)

the origin z = 0 is globally asymptotically stable (GAS). Define K1(z) := α(z). Then,there exists a state feedback

ξj+1 = Kj+1(z, ξ1, . . . ξj), j = 1, . . . , r (7.7)

in an explicit form where ξr+1 := v that renders the origin (z, ξ1, . . . , ξj) = 0 GAS forthe closed-loop (z, ξ1, . . . , ξj)-subsystem (i.e. jth subsystem) of (7.5). In addition, letV0(z) be a smooth positive definite and proper Lyapunov function associated with theclosed-loop zero-dynamics subsystem satisfying V0(z) := ∇V0(z) · f(z, α(z)) < 0,∀z 6=0. Then, the the Lyapunov function Vj(z, ξ1, . . . , ξj), j = 1, . . . , r associated with thejth subsystem is given by

Vj(z, ξ1, . . . , ξj) = V0(z) +1

2(ξ1 −K1(z))

2 + 1j>1(j)1

2Σji=2(ξi −Ki(z, ξ1, . . . , ξi−1))

2

and satisfies Vj < 0,∀(z, ξ1, . . . , ξj) 6= 0 (1A(·) denotes the indicator function of theset A).

Proof. See remark 7.3.2.In the next section, we provide a constructive method for obtaining both the statefeedback Kj and the Lyapunov function Vj in a recursive manner (this can be foundin [36], also see theorem 7.3.1 here).

In contrast to nonlinear systems in strict feedback forms, the following normalform

z = F (z, ξ1, ξ2)

ξ1 = ξ2ξ2 = v

(7.8)

has a nontriangular structure. Throughout this chapter, we assume F (z, ξ1, ξ2) de-pends on both ξ1, ξ2 unless otherwise is stated. Our main focus in this chapter is

218

on stabilization of nonlinear systems in the form (7.8). Our main motivation is that(almost) all nontriangular normal forms for underactuated systems are special classesof Byrnes-Isidori normal form with a vector double-integrator linear part as in (7.8).In addition, the following nontriangular normal form with a triple-integrator linearpart

z = F (z, ξ1, ξ2, ξ3)

ξ1 = ξ2ξ2 = ξ3ξ3 = v

(7.9)

appeared in the study of flexible-link robots and a helicopter model with the effect ofthe input of the attitude dynamics in the translational dynamics. The actual inputin both cases is ξ3 which is augmented with an integrator at the input.

Remark 7.1.1. Notice that by redefining the state variables and zero-dynamics vectorfield as

ζ := col(z, ξ1), η1 := ξ2, η2 := ξ3, F0 := col(F (z, ξ1, ξ2, ξ3), ξ2)

The dynamics of system (7.9) can be rewritten as the following

ζ = F0(ζ, η1, η2)η1 = η2η2 = v

(7.10)

which is a Byrnes-Isidori normal form of degree 2 (the same as (7.8)). Roughlyspeaking, this implies that certain stabilization problems for the class of nonlinearsystems in (7.9) with a triple-integrator linear part reduce to equivalent problems forthe class of nonlinear systems in (7.8) with a (vector) double-integrator linear part.

Stabilization of Byrnes-Isidori normal forms with a double-integrator linear partas in (7.8) is currently considered a major open problem [36]. Particular cases ofinterest include normal forms for mechanical systems where F (z, ξ1, ξ2) has quadraticdependence in the velocity ξ2 (e.g. the beam-and-ball system). Our main contribu-tion in this chapter is to introduce an extension of the backstepping procedure forstabilization of important special classes of normal form (7.8). This new backsteppingprocedure is considerably different from standard backstepping procedure that relieson construction of appropriate Lyapunov functions.

Based on standard backstepping procedure, if a controller ξ2 = K1(z, ξ1) existsthat renders the origin GAS for

z = F (z, ξ1, ξ2)

ξ1 = ξ2(7.11)

Then, the existence of a globally stabilizing state feedback v = K2(z, ξ1, ξ2) for thecomposite system in (7.8) is trivial. In other words, the main stabilization problemof interest is stabilization of nonlinear system (7.11) which is non-affine in control ξ2.

Observe that it is nontrivial to formulate a stabilization problem for the nonlinearsystem (7.8) analogous to the one for a cascade system in strict feedback form, unless

219

α(z) ≡ 0. We formulate this problem for the case of α(z) ≡ 0 as the following:

Problem 7.1.1. Consider the nonlinear system in (7.8). Assume that the zero-dynamics of (7.8) is globally minimum phase, i.e. given ξ1 = ξ2 = 0, the origin z = 0is GAS for the zero-dynamics

z = F (z, 0, 0)

Find sufficient conditions such that there exists a state feedback v = k(z, ξ1, ξ2) thatglobally asymptotically stabilizes (z, ξ1, ξ2) = 0 for (7.8).

Due to a counter-example by Sussmann [98], there exists a third-order nonlinearsystem in the form (7.8) which is not semiglobally stabilizable. This major stabiliza-tion limitation of nonlinear systems in nontriangular forms as (7.8) is because of thepeaking phenomenon that was introduced by Sussmann and Kokotovic [99]. The prob-lem is that driving the state (ξ1, ξ2) exponentially fast to zero using a linear partialstate feedback does not necessarily asymptotically stabilize the composite system in(7.8). Such a partial state feedback might cause the z-subsystem to have finite escapetime. In [99], growth conditions are provided on F (z, ξ1, ξ2) for global stabilization of(7.8) using partial state feedback. The problem is that these growth conditions arerather restrictive. Nevertheless, the work in [99] is one possible solution to problem7.1.1.

In many applications (e.g. tracking for flexible-link robots), the zero-dynamics isnot globally minimum phase. In this thesis, we are mainly interested in stabilizationproblems for normal form (7.8) when the zero-dynamics is not minimum phase. Apossible way to formulate a stabilization problem for normal form (7.8) analogous tothe nature of the one addressed using backstepping procedure for systems in strictfeedback form is the following problem.

Problem 7.1.2. Consider the nonlinear system in (7.8). Assume there exists asmooth state feedback law ξ1 = α(z) with α(0) = 0 such that for the zero-dynamicslocked at ξ2 = 0

z = F (z, α(z), 0)

z = 0 is globally asymptotically stable. Find sufficient conditions such that there existstate feedback laws ξ2 = k1(z, ξ1) and u = k2(z, ξ1, ξ2) which asymptotically stabilizethe origin for the (z, ξ1)-subsystem and the composite system (7.8), respectively.

A third possible stabilization problem is as follows:

Problem 7.1.3. Consider the nonlinear system in (7.8). Assume there exist smoothstate feedback laws ξ1 = α(z) and ξ2 = β(z) satisfying the following conditions:

i) α(0) = 0 and β(0) = 0.

ii) Forz = f(z, α(z), 0)

z = 0 is GAS.

220

iii) (ξ1, ξ2) = (α(z), β(z)) is an invariant manifold for (7.8), i.e. α(z), β(z) satisfythe equation

β(z) =∂α(z)

∂zf(z, α(z), β(z))

iv) Forz = f(z, α(z), β(z))

the origin z = 0 is GAS.

Find sufficient conditions such that there exist state feedback laws ξ2 = k1(z, ξ1) andu = k2(z, ξ1, ξ2) which semiglobally/globally asymptotically stabilize the origin for the(z, ξ1)-subsystem and the composite system (7.8), respectively.

Remark 7.1.2. Based on condition iii) in problem 7.1.3, ξ2 = β(z) is a fixed point ofequation

ξ2 =∂α(z)

∂zf(z, α(z), ξ2)

Thus, α(z), β(z) are not arbitrary controllers. The state feedback α(z) must be chosensuch that it both stabilizes the zero-dynamics locked at ξ2 = 0 and the preceding fixed-point equation must be well-posed (i.e. have a solution). Both of these issues areaddressed later in this chapter.

A common example of nonlinear systems with feedforward structures is a chain ofintegrators with higher-order perturbations as

x1 = x2 + p1(x2, x3, . . . , xn, u)x2 = x3 + p2(x3, . . . , xn, u)

· · ·xn = u+ pn(xn, u)

(7.12)

System (7.12) can be rewritten as

x = Ax+Bu+ p(x, u) (7.13)

(with obvious definitions of A,B, p(x, u)) where the perturbation terms pi are at leastquadratic with respect to their arguments [102]. Nonlinear systems in feedforwardforms can have more general forms as the following

x1 = f1(x2, x3, . . . , xn, u)x2 = f2(x3, . . . , xn, u)

· · ·xn = fn(xn, u)

(7.14)

with an apparent upper triangular structure w.r.t. variables (x1, x2, . . . , xn, u) [56].Feedforward systems of types (7.12) and (7.14) are studied by Teel and Mazenc-Praly.Similar to systems in feedback forms recursive procedures called feedforwarding aredeveloped for stabilization of feedforward systems [102, 101, 56]. The main draw back

221

of all existing results on stabilization of feedforward systems is the low-gain natureof controllers that leads to relatively poor performance of control laws obtained fromfeedforwarding. Here, the performance measure is the settling time of the solutionsof the closed-loop system. The controllers obtained using backstepping procedure donot suffer from this drawback.

7.2 Structure of Nontriangular Normal Forms of

Underactuated Systems

The classification of underactuated mechanical systems presented in section 4.6 allowsus to identify the structure of the vector field f in (7.8) (or (7.9)) for underactuatedsystems in nontriangular forms and feedforward forms.

Proposition 7.2.1. All the normal forms for classes of underactuated systems innontriangular/feedforward forms with a double-integrator linear part (minus the per-turbation terms ϕi) are special classes of the following normal form

z1 = z2 + g0(ξ1)ξ2z2 = f0(z1, ξ1) + zT2 g11(ξ1)z2 + zT2 g12(ξ1)ξ2 + ξT2 g22(ξ1)ξ2ξ1 = ξ2ξ2 = u

(7.15)

where z1, z2 ∈ Rn, ξ1, ξ2, u ∈ Rm. In addition, the functions g0 : Rm → Rn×n,f0(z1, ξ1) : Rn × Rm → Rn and cubic matrices g11 : Rm → Rn×n×n, g12 : Rm →Rn×m×n, and g22 : Rm → Rm×m×n are all smooth and g0(0) = 0.

Proof. The proof is by the definition of each class in section 4.6.

Remark 7.2.1. Equation (7.15) can be rewritten in the form

z = F (z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(7.16)

whereF (z, ξ1, ξ2) = f(z1, ξ1) + L(ξ1)ξ2 + (z2, ξ2)Q(ξ1)(z

T2 , ξ

T2 )

T (7.17)

has a linear-quadratic structure with respect to ξ2 with a cubic matrix Q(ξ1).

Remark 7.2.2. Global stabilization of the core reduced system

z1 = z2z2 = f0(z1, ξ1)

(7.18)

is addressed in section 4.7.

The following special classes of normal form (7.15) are important due to theirapplication in control of underactuated systems:

222

i) g11, g22 = 0: nonlinear system (7.15) is in nontriangular form with a vector fieldaffine in ξ2.

ii) f0(z1, ξ1) = f0(ξ1), g11 = 0: nonlinear system (7.15) is in feedforward form witha vector field quadratic in ξ2.

iii) g0 = 0: nonlinear system (7.15) is in nontriangular form with a vector fieldquadratic in ξ2. Moreover, z1 does not depend on (ξ1, ξ2).

Lemma 7.2.1. Normal form (7.15) can be represented as

z = f(z, ξ1) + g(z, ξ1, ξ2)ξ2ξ1 = ξ2ξ2 = u

(7.19)

where f, g are explicitly given by

f(z, ξ1) =

[

z2f0(z1, ξ1) + zT2 g11(ξ1)z2

]

, g(z, ξ1, ξ2) =

[

g0(ξ1)zT2 g12(ξ1) + ξT2 g22(ξ1)

]

Proof. The proof is by definition of f and g.

Remark 7.2.3. The main condition in problems 7.1.2 and 7.1.3 is that there shouldexist a state feedback ξ1 = α(z) with α(0) = 0 such that for the closed-loop zerodynamics locked at ξ2 = 0

z = F (z, α(z), 0)

the equilibrium z = 0 is globally asymptotically stable. Based on lemma 7.2.1, thisis equivalent to the existence of a globally stabilizing state feedback for z = f(z, ξ1)that can be expressed as

z1 = z2z2 = f0(z1, ξ1) + zT2 g11(ξ1)z2

(7.20)

Apparently, if g11 ≡ 0, then global stabilization of (7.20) is equivalent to globalstabilization of the core reduced system which is addressed in section 4.7. In addition,for the special case where g11 = g11(ξ1, ε) with the property g11(ξ1, 0) = 0, semiglobalstabilization of (7.20) can be achieved (under appropriate technical conditions).

Definition 7.2.1. (slightly nontriangular nonlinear systems) The following subclassof nontriangular nonlinear systems in (7.19)

z = f(z, ξ1) + g(z, ξ1, ξ2, ε)ξ2ξ1 = ξ2ξ2 = u

(7.21)

with the property g(z, ξ1, ξ2, 0) = 0 are called slightly nontriangular nonlinear systems.

223

Control design for slightly nontriangular systems is important due to their appli-cations in aerospace vehicles. As an example, according to proposition 5.11.4, thenormal form of the first-level approximate model of a helicopter is in the form

z = f(z) + g1(ξ1)T + g2(ξ1, ξ2, ε) + g3(ξ1, ε)u

ξ1 = ξ2ξ2 = u

(7.22)

where z = col(x, v) ∈ R6, ξ1 = col(φ, θ, ψ), (i.e. the three Euler angles), ξ2 ∈ R3, andε = (ε1, ε2, ε3)

T ∈ R3. Furthermore, both g2 and g3 vanish at ε = (0, 0, 0). Later, weprovide more examples of slightly nontriangular systems in robotics applications.

7.3 Stabilization of Nonlinear Systems in Feedback

Form

Backstepping procedure is an effective method for global stabilization of nonlinearsystems in strict feedback form. There are two types of backstepping procedure. Onethat makes an explicit use of a Lyapunov function and is referred to as standardbackstepping procedure. The other one is a type of backstepping procedure whichdoes not require any explicit knowledge of a Lyapunov function and we call it cascadebackstepping procedure. Here, we present both of them and clarify the differencesbetween the two methods.

7.3.1 Standard Backstepping Procedure

We start by presenting the standard backstepping procedure for a nonlinear systemaugmented with a single integrator at the input. First, we need to state the followinglemma [36, 40].

Lemma 7.3.1. Assume f(x, y) : Rn × Rm → R is a continuously differentiablefunction. Then, the following identity holds:

f(x, y) = f(x, 0) + g(x, y)y (7.23)

where

g(x, y) =

∫ 1

0

Dyf(x, sy)ds

Proof. Fix (x, y) ∈ Rn+m and let h(s) := f(x, sy) where s ∈ R. We have h′(s) =Dyf(x, sy) · y. Hence

h(1)− h(0) =∫ 1

0

h′(s)ds = (

∫ 1

0

Dyf(x, sy)ds) · y

and the result follows.

224

Remark 7.3.1. An application of lemma 7.3.1 gives

f(x, y + δ) = f(x, y) + g(x, y, δ)δ

The following version of backstepping theorem can be found in [36].

Theorem 7.3.1. (standard backstepping) Consider the following nonlinear system

z = f(z, ξ)

ξ = u(7.24)

where z ∈ Rn, ξ, u ∈ R, and f is a smooth function satisfying f(0, 0) = 0. Assumethere exists a smooth state feedback ξ = α(z) : Rn → R satisfying α(0) = 0 such thatfor

z = f(z, α(z))

z = 0 is GAS. Let V (z) be a smooth positive definite and proper Lyapunov functionassociated with the closed-loop z-subsystem satisfying

DV (z) · f(z, α(z)) < 0, ∀z 6= 0

Then, the following smooth state feedback

u = −c(ξ − α(z)) + ∂α(z)

∂zf(z, ξ1, ξ2)−

∂V (z)

∂z· h(z, µ) (7.25)

renders (z, ξ) = 0 GAS for (7.24) where c > 0 is a constant, µ := ξ − α(z), and

h(z, µ) :=

∫ 1

0

Dξf(z, α(z) + sµ)ds

satisfiesf(z, α(z) + µ) = f(z, α(z)) + h(z, µ)µ

Proof. Consider the following Lyapunov function candidate

W (z, ξ) = V (z) +1

2µ2

Calculating W along the solutions of the closed-loop system in (7.24), we get

W = V + µµ,

=∂V (z)

∂zf(z, α(z) + µ) + (u− ∂α(z)

∂zf(z, ξ1, ξ2))µ,

=∂V (z)

∂zf(z, α(z)) + [u− ∂α(z)

∂zf(z, ξ1, ξ2) +

∂V (z)

∂zh(z, µ)]µ, (lemma 7.3.1)

=∂V (z)

∂zf(z, α(z))− cµ2 < 0

225

for all (z, µ) 6= 0. Therefore, (z, ξ) = 0 is GAS for (7.24) and W (z, ξ) is a validLyapunov function for the composite system.

Remark 7.3.2. Successive application of theorem 7.3.1 proves the statements in the-orem 7.1.1.

In [57], the following theorem which is a special case of theorem 7.3.1 (with az-subsystem affine in ξ) is considered as standard backstepping procedure.

Theorem 7.3.2. ([57]) Consider the following cascade nonlinear system affine in ξ

z = f(z) + g(z)ξ

ξ = u(7.26)

where z ∈ Rn, ξ, u ∈ R, and f, g are smooth functions with f(0) = 0. Let the statefeedback ξ = α(z) with α(0) = 0 render z = 0 GAS for the closed-loop z-subsystem

z = f(z) + g(z)α(z)

Then, the following state feedback

u = −c(ξ − α(z)) + ∂α(z)

∂zf(z, ξ1, ξ2)−

∂V (z)

∂zg(z) (7.27)

renders (z, ξ) = 0 GAS for the composite system (7.26).

Proof. Define f(z, ξ) := f(z) + g(z)ξ. Then

f(z, ξ + µ) = f(z, ξ) + h(z, µ)µ

where h(z, µ) := g(z). Thus, state feedback (7.27) reduces to (7.25) and the resultfollows from theorem 7.3.1.

The proof of standard backstepping procedure automatically provides a Lyapunovfunction W (z, ξ1) for the closed-loop composite system. This is very valuable for(possible) robustness analysis of perturbed nonlinear systems in feedback forms (e.g.normal forms of Class-VI underactuated mechanical systems in chapter 4).

7.3.2 Cascade Backstepping Procedure

The knowledge about the Lyapunov function V (z) for the closed-loop z-subsystemis necessary in the computation of the control law (7.25) for the composite system.Such a knowledge about V (z) is not always available. This motivates developing abackstepping procedure which is not Lyapunov-based. We refer to this procedure thatdoes not require explicit knowledge of any Lyapunov functions as cascade backsteppingprocedure.

The most important tool in developing a non-Lyapunov-based or cascade back-stepping procedure in the following theorem due to Sontag.

226

Theorem 7.3.3. (global stability of cascade systems [83]) Consider the followingcascade nonlinear system

z = f(z, η)η = g(η)

(7.28)

where f : Rn×Rm → Rn and g : Rm → Rm are smooth functions satisfying f(0, 0) = 0and g(0) = 0. Assume the following conditions hold:

i) For z = f(z, 0), z = 0 is globally asymptotically stable.

ii) For η = g(η), η = 0 is globally asymptotically and locally exponentially stable.

iii) For any solution η(t) with initial condition η0 ∈ Rm, the solution of the z-subsystem of (7.28) exists for all t ≥ 0 and is bounded.

Then, (z, η) = 0 is globally asymptotically stable for (7.28).

The main use of theorem 7.3.3 is in stability analysis of the class of cascade systems

x = f(x, η)η = g(η, u)

(7.29)

where a control law u = k(η) is available that render η = 0 GAS for η = g(η, k(η)).In this chapter, we frequently use theorem 7.3.3 in the aforementioned context.

The only non-constructive aspect of theorem 7.3.3 is in its third condition wherethe boundedness of the solutions of the z-subsystem of (7.28) is assumed. One pos-sible assumption to guarantee this boundedness assumption is to assume that thez-subsystem

z = f(z, η)

is input-to-state stable, or simply ISS-stable [82]. Due to Sontag-Wang [84], a Lyapunov-type characterization of the ISS property reveals that the necessary and sufficientcondition for a system to be ISS-stable is that there exist class-K functions α1, α2

and a positive definite smooth ISS Lyapunov function V (z) such that the followingproperty holds:

∂V (z)

∂zf(z, η) < −α1(‖z‖), for ‖z‖ > α2(‖η‖) (7.30)

Condition (7.30) is an extremely restrictive condition which might not hold for largeclasses of nonlinear systems of interest. Since, it essentially ignores the existence ofthe input η. In the sense that over the region ‖z‖ > α2(‖η‖), V < 0 and the solutionof the z-subsystem converges towards the inner level surfaces around the origin z = 0,until the condition ‖z‖ > α2(‖η‖) is violated. However, to guarantee the boundednessof the solution of the z-subsystem it is sufficient that V (z(t)) stays bounded. The timedecay of V (z(t)) is unnecessary. Keeping this in mind, here we provide a constructivecondition that guarantees boundedness of the solutions of the z-subsystem. Thiscondition is less restrictive than the ISS-stability property of the z-subsystem. Thisis influenced by the work of Sepulchre et al. [80] (theorem 4.7, p.129) and generalizesthat result. First, we need to define a class of input disturbances.

227

Definition 7.3.1. (class-C0 functions) Define the following class of functions δ :R≥0 → Rn

C0(l0, l1) := δ(·) | ‖δ‖∞ ≤ l0, ‖δ‖1 ≤ l1, limt→∞

δ(t) = 0 , l0, l1 > 0

where ‖δ‖∞ := supt≥0|δ(t)| and ‖δ‖1 :=∫∞

0|δ(t)|dt denote L∞ and L1 norms of δ(·),

respectively. We say a function δ : t → δ(t) is class-C0 with bounds l0, l1 > 0, ifδ(·) ∈ C0(l0, l1).Theorem 7.3.4. Consider the nonlinear system

z = f(z, η) (7.31)

with a class-C0 input disturbance η(·) ∈ C0(l0, l1). Assume for any l0 > 0, there exista smooth positive definite and proper function V (z), positive constants ρ ≥ 0, k > 0,and λ ∈ (0, 1] such that the following conditions are satisfied:

i) ∇V (z) · f(z, 0) ≤ 0

ii) |∇V (z) · h(z, η)| ≤ kV (z)λ

for all ‖z‖ ≥ ρ and ‖η‖ ≤ l0 where

h(z, η) =

∫ 1

0

Dηf(z, sη)ds

Then, any solution of the system (7.31) is bounded.

Proof. Based on lemma 7.3.1, we have

f(z, η) = f(z, 0) + h(z, η)η

Calculating V gives

V = ∇V (z) · f(z, η)= ∇V (z) · f(z, 0) +∇V (z) · h(z, η)η≤ ∇V (z) · h(z, η)η≤ |∇V (z) · h(z, η)| · ‖η‖≤ kV (z)λ‖η‖

We consider two cases: i) λ ∈ (0, 1) and ii) λ = 1. For λ ∈ (0, 1), over the region‖z‖ ≥ ρ, the last inequality can be rewritten as

1

kV (z(t))−λV (z(t)) ≤ ‖η(t)‖

Integrating both sides of the last inequality w.r.t. t gives

V (z(t))(1−λ) − V (z(0))(1−λ) ≤ k(1− λ)∫ t

0

‖η(t)‖dt ≤ k(1− λ)‖η‖1 ≤ k(1− λ)l1

228

which means z(t) is bounded uniformly in t, because

V (z(t)) ≤ [V (z(0))(1−λ) + k(1− λ)l1]1

1−λ =: L(k, λ, l1, z(0)) (7.32)

and V (z) is a proper function. Thus, K := V −1([0, L(k, λ, l1, z(0)]) is a compact setand z(t) ∈ K,∀t ≥ 0. This implies ‖z(t)‖ < Lz := maxρ, r(K), for all t > 0where r(K) = maxx∈K ‖x‖ denotes the radius of the compact set K. The proof of theboundedness of z(t) for all t > 0 for the case of λ = 1 is very similar and is omitted.The only difference is that the inequality (7.32) is in the form

V (z(t)) ≤ V (z(0)) exp(kl1) (7.33)

Now, we are ready to introduce a non-Lyapunov-based backstepping procedure.

Theorem 7.3.5. (cascade backstepping procedure) Consider the following nonlinearsystem

z = f(z, ξ)

ξ = u(7.34)

Assume there exists a smooth state feedback ξ = α(z) with α(0) = 0 such that for

z = f(z, α(z))

z = 0 is GAS. Let V (z) be a smooth positive definite and proper Lyapunov functionassociates with the closed-loop z-subsystem satisfying

∇V (z) · f(z, α(z)) ≤ 0

In addition, suppose for any l0 > 0, there exist ρ ≥ 0, k > 0, and λ ∈ (0, 1] such that

|∇V (z) · h(z, µ)| ≤ kV (z)λ, ∀(z, µ) : ‖z‖ ≥ ρ, ‖µ‖ ≤ l0 (7.35)

where µ = ξ−α(z) and h(z, µ) =∫ 1

0Dξf(z, α(z)+sµ)ds. Then, the following smooth

state feedback

u = −c(ξ − α(z)) + ∂α(z)

∂zf(z, ξ), c > 0 (7.36)

renders (z, ξ) = 0 GAS for (7.34).

Proof. The closed-loop cascade system in (7.34) with controller (7.36) can be rewrit-ten as

z = f(z, α(z) + µ) =: f(z, µ)µ = −cµ (7.37)

Notice that both µ = 0 is GAS for the µ-subsystem and z = 0 is GAS for z = f(z, 0).On the other hand, the solution of the z-subsystem is bounded according to theorem7.3.4. To see this, note that µ(t) = µ(0) exp(−ct) and thus µ(t) is a class-C0 functionsatisfying ‖µ(t)‖ ≤ l0 := ‖µ(0)‖,∀t > 0 and ‖µ(t)‖1 = 1/c =: l1. But

f(z, α(z) + µ) = f(z, α(z)) + h(z, µ)µ

229

and by the assumption in the question we get

V ≤ kV (z)λ‖µ‖

which guarantees uniform boundedness of the solution z(t) for all t > 0. Based ontheorem 7.3.3, the origin (z, µ) = 0 is GAS for the cascade closed-loop system. Thisimplies (z, ξ) = 0 is GAS for (7.34).

Example 7.3.1. Consider the following nonlinear system in strict feedback form

z1 = z2z2 = σ(ξ)

ξ = u(7.38)

where σ(ξ) is a sigmoidal function that is odd and globally Lipschitz (e.g. tanh(ξ)).The linear state feedback

ξ = α(z) := −k1z1 − k2z2, k1, k2 > 0

renders z = 0 GAS for the z-subsystem. To prove this, let us express the dynamicsof the closed-loop z-subsystem as

z1 = z2z2 = −σ(k1z1 + k2z2)

(7.39)

Define φ(z1, z2) := σ(k1z1 + k2z2) and notice that Dz2φ(z1, z2) > 0 for all z ∈ R2.This means

z2(φ(z1, 0)− φ(z1, z2)) < 0, ∀(z1, z2) ∈ R2 \ (0, 0)

Let us define

ψ(z1) =

∫ z1

0

φ(s, 0)ds =

∫ z1

0

σ(k1s)ds

and consider the following Lyapunov function candidate

V (z1, z2) = ψ(z1) +1

2z22

Calculating V along the solutions of the closed-loop z-subsystem, we get

V = z2(φ(z1, 0)− φ(z1, z2)) < 0, ∀(z1, z2) 6= (0, 0)

and therefore (z1, z2) = 0 is GAS for the z-subsystem. Taking f(z, ξ) := [z2, σ(ξ)]T ,

we havef(z, α(z) + µ) = f(z, α(z)) + h(z, µ)µ

where h(z, µ) = [0, g(z, µ)]T and g(z, µ) =∫ 1

0σ′(α(z) + sµ)ds. Since σ is globally

Lipschitz, there exists an L > 0 such that |g(z, µ)| ≤ L for all (z, µ). Now, we prove

230

that based on theorem 7.3.5, the following nonlinear state feedback

u = −c(ξ − α(z)) + ∂α(z)

∂zf(z, ξ), c > 0

which can be written in an explicit form

u = −c(ξ + k1z1 + k2z2)− k1z2 − k2σ(ξ) (7.40)

renders (z, ξ) = 0 GAS for (7.38). For doing so, we need to show that condition (7.35)holds. We have

|∇V (z) · h(z, µ)| = |z2g(z, µ)| ≤ L|z2| ≤ (2L)V (z)1/2

Therefore, condition (7.35) holds with k = 2L, λ = 1/2 uniformly in µ. This guar-antees boundedness of the solution of the z-subsystem and the GAS property of(z, ξ) = 0 for the composite system. Using standard backstepping procedure, oneobtains a different globally stabilizing control law

u = −c(ξ − α(z)) + ∂α(z)

∂zf(z, ξ)− E(z, ξ), c > 0

with an extra term E(z, ξ) compared to state feedback (7.40) which is given by

E(z, ξ) := ∇V (z) · h(z, µ) =∫ 1

0

z2σ′((1− s)α(z) + sξ)ds

Remark 7.3.3. If the sigmoidal function σ in (7.38) is neither bounded nor globallyLipschitz. Then, we choose the following control law

ξ = α(z) := −σ0(k1z1 + k2z2), k1, k2 > 0

where σ0 is a bounded and globally Lipschitz sigmoidal function. Now, σ(ξ) =σ(σ0(ξ)) is both bounded and globally Lipschitz. The analysis in example 7.3.1 nowcontinues in a similar way with replacing σ by σ.

7.4 Stabilization of Nonlinear Systems in Nontri-

angular Forms

The main class of nontriangular nonlinear systems considered in this chapter are inthe form

z = f(z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(7.41)

231

where z ∈ Rn and f is a smooth function satisfying f(0, 0, 0) = 0. Here, we considerthe single input case with ξ1, ξ2, u ∈ R for the sake of simplicity of notations. Thegeneralization of the obtained results to the case of multiple inputs is trivial. De-pending on whether f(z, ξ1, ξ2) is an affine function of ξ1 and/or ξ2 or none, we breakthe problem of stabilization of (7.41) into a number of cases and address each caseseparately. Throughout this chapter, we put especial emphasis on the structure of fas it appears in the normal forms of underactuated mechanical systems.

7.4.1 Nontriangular Nonlinear Systems Affine in (ξ1, ξ2)

In this section, our goal is to stabilize normal form (7.41) in its simplest possible formgiven as

z = f(z) + g1(z)ξ1 + g2(z)ξ2ξ1 = ξ2ξ2 = u

(7.42)

where (z, ξ1) ∈ Rn×R and ξ2 ∈ R is the control input. The vector fields f(z), g1(z), g2(z)are all smooth and f(0) = 0. Due to the linearity property of the vector filed of the zsubsystem w.r.t. ξ2, normal form (7.42) does not capture the effects of Coriolis andcentrifugal terms in mechanical systems. However, it facilitates explaining the mainnecessary steps that are required to address the stabilization problem of systems innontriangular forms non-affine in (ξ1, ξ2), without excessive technical preliminaries.More importantly, it allows us to show that the method of standard backsteppingand its particular way of construction of Lyapunov functions is not generalizable tononlinear systems in normal form (7.42). In contrast, we provide a cascade backstep-ping procedure for stabilization of nonlinear systems in the form (7.42). Later, wepresent generalization of this non-Lyapunov-based backstepping procedure to classesof normal form (7.41) that are non-affine in (ξ1, ξ2).

First, we demonstrate why the standard backstepping procedure is practicallynot applicable to (7.42). In the sense that, it requires conditions that are extremelyrestrictive.

Proposition 7.4.1. Consider the nonlinear system in (7.42). Assume there exists asmooth state feedback α(z) : Rn → R with α(0) = 0 such that for

z = f(z) + g1(z)α(z) =: f1(z) (7.43)

z = 0 is GAS. Let V (z) be the smooth positive definite and proper Lyapunov functionassociated with (7.43) satisfying Lf1V (z) < 0,∀z 6= 0. Assume α(z) globally satisfiesthe condition Lg2α(z) < 1. Define β(z) as

β(z) :=Lf1α(z)

1− Lg2α(z)

so that ξ2 = β is the fixed-point solution of the equation

ξ2 = ∇α(z) · [f(z) + g1(z)α(z) + g2(z)ξ2]

232

Suppose that Lg2V (z) = 0,∀z ∈ Rn. Then, the following statements hold:

i) For the closed-loop z-subsystem, z = f(z) + g1(z)α(z) + g2(z)β(z) =: f2(z), theorigin z = 0 is GAS.

ii) there exist a smooth state feedback ξ2 = K1(z, ξ1) that renders (z, ξ1) = 0 GASfor the (z, ξ1)-subsystem of (7.42).

iii) there exists a smooth state feedback u = K2(z, ξ1, ξ2) that renders (z, ξ1, ξ2) = 0GAS for the composite system (7.42).

Proof. The proof of i) is trivial due to the assumption Lg2V (z) = 0. This implies

V = Lf2V (z) = Lf1V (z) + β(z)Lg2V (z) = Lf1V (z) < 0

for all z 6= 0 and therefore z = 0 is GAS. To prove ii), first we need to apply a changeof coordinates by setting µ = ξ1 − α(z). The dynamics of the (z, ξ1)-subsystem of(7.42) in new coordinates can be expressed as

z = f2(z) + [g1(z) + A(z)g2(z)]µ1 +B(z)g2(z)µ2µ1 = µ2

(7.44)

where µ2 is the new control and the functions A(z), B(z) : Rn → R are obtained asfollows. By definition of µ2, we have

µ2 = ξ2 −∇α(z) · [f(z) + g1(z)ξ1 + g2(z)ξ2]

= (1− Lg2α(z))ξ2 − Lf1α(z)− µ1Lg1α(z)

This change of variable is globally invertible due to 1−Lg2α(z) > 0. Solving the lastequation for ξ2 gives

ξ2 =Lf1α(z) + Lg1α(z)µ1 + µ2

1− Lg2α(z)= β(z) + A(z)µ1 +B(z)µ2 (7.45)

where

A(z) =Lg1α(z)

1− Lg2α(z), B(z) =

1

1− Lg2α(z)Substituting ξ2 from (7.45) in equation (7.42) gives (7.44). Following the proof thestandard backstepping theorem, let us consider the candidate Lyapunov function

W (z, ξ1) := V (z) +1

2µ21

for the nonlinear system in (7.44). Calculating W , we get

W = V + µ1µ2

= ∇V (z) · [f(z) + g1(z)(α(z) + µ1) + g2(z)(β(z) + A(z)µ1 +B(z)µ2)

= Lf2V (z) + [Lg1V (z) + A(z)Lg2V (z) + µ2]µ1 +B(z)(Lg2V (z))µ2

233

Unlike the case of systems in strict feedback form, for the case of nontriangularnonlinear systems, an extra term δ = B(z)(Lg2V (z))µ2 appears in the expression ofW which is sign indefinite. Unless δ = 0, or Lg2V (z) = 0 for all z, W cannot be madenegative definite. But by assumption, Lg2V (z) = 0 and applying the state feedback

µ2 = −c1µ1 − Lg1V (z), c1 > 0

renders W negative definite for all (z, µ1) 6= 0 because

W = Lf2V (z)− c1µ21

This means that

ξ2 = K1(z, ξ1) := β(z) + [A(z)−B(z)c1](ξ1 − α(z))−B(z)Lg1V (z) (7.46)

renders (z, ξ1) = 0 GAS for the (z, ξ1)-subsystem of (7.42) and W (z, ξ1) is a validLyapunov function for the (z, ξ1)-subsystem. Finally, part iii) follows from part ii)according to the standard backstepping theorem.

Remark 7.4.1. The most restrictive condition in theorem is the assumption thatLg2V (z) = 0 for all z ∈ Rn. However, without this assumption, in the opinion ofthe author, there does not seem to exist any ways to render W negative definitefor all (z, ξ1) 6= 0. This demonstrates the weakness of the particular choice of theLyapunov function used in the standard backstepping procedure, in dealing withnonlinear systems in nontriangular forms.

The following result is the non-Lyapunov-based version of proposition 7.4.1.

Proposition 7.4.2. Assume all the conditions in proposition 7.4.1 hold. In addition,suppose there exist ρ ≥ 0, k > 0, and λ ∈ (0, 1] such that

|Lg1V (z)| ≤ kV (z)λ, ‖z‖ ≥ ρ

Then, the smooth state feedback

ξ2 = K(z, ξ1) = β(z) +G(z)(ξ1 − α(z)) (7.47)

(which does not depend on the Lyapunov function V (z)) renders (z, ξ1) = 0 GAS forthe (z, ξ1)-subsystem of (7.42) where the nonlinear gain G(z) is defined as

G(z) = A(z)− c1B(z), c1 > 0

Proof. Setting µ2 = −c1µ1, from equation (7.44), we have µ1 = −c1µ1 (thus µ1 = 0is globally exponentially stable) and

V = Lf2V + (Lg1V )µ1 +G(z)Lg2V

= Lf2V + (Lg1V )µ1

≤ |Lg1V (z)||µ1|≤ kV (z)λ|µ1|

234

Based on theorem 7.3.4, the solutions of the z-subsystem are bounded. Now, accord-ing to theorem 7.3.3, the origin (z, µ1) = 0 is GAS for the cascade nonlinear system(7.44) which proves the statement in the question.

Now, we would like to present a non-Lyapunov-based backstepping procedurefor stabilization of nontriangular nonlinear system (7.42) which does not require therestrictive condition Lg2V (z) = 0,∀z ∈ Rn. First, we need to make some assumptions.

Assumption 7.4.1. There exists a state feedback α(z) : Rn → R with α(0) = 0satisfying the following conditions:

i) for z = f(z) + g1(z)α(z) =: f1(z), the origin z = 0 is globally asymptoticallystable,

ii) there exists L1 > 0 such that |Lg1α(z)| ≤ L1 for all z ∈ Rn,

iii) there exists L2 > 0 such that Lg2α(z) ≤ 1− 1/L2 for all z ∈ Rn.

The following proposition is the main analytical tool that justifies the use of thecascade backstepping procedure for stabilization of a nontriangular system as (7.42).

Proposition 7.4.3. Consider the nonlinear system in (7.42) and suppose assumption7.4.1 holds. Then, after applying the global change of coordinates µ = Φ(z, ξ) andchange of control defined by

µ1 = ξ1 − α1(z)µ2 = M(z)(ξ2 − α2(z, ξ1))w = M(z)(u− α3(z, ξ1, ξ2))

(7.48)

with M(z) = 1−Lg2α(z) > 1/L2 > 0, the dynamics of the nonlinear system in (7.42)transforms into

z = f0(z) + [g1(z) + A(z)g2(z)]µ1 +B(z)g2(z)µ2µ1 = µ2µ2 = w

(7.49)

where f0(z) := f(z) + g1(z)α1(z) + g2(z)α2(z, α1(z)) and αi’s are defined recursivelyas the following

α1(z) := α(z)

α2(z, ξ1) := [Lfα1(z) + ξ1Lg1α1(z)]/M(z)

α3(z, ξ1, ξ2) := [∂α2

∂z(f(z) + g1(z)ξ1 + g2(z)ξ2) +

∂α2

∂ξ1ξ2]−

M

M[ξ2 − α2(z, ξ1)]

withM(z, ξ1, ξ2) := LfM(z) + ξ1Lg1M(z) + ξ2Lg2M(z)

235

In addition, ξ = Ψ(z, µ), the inverse of Φ, is given in closed-form by

ξ1 = α(z) + µ1ξ2 = β(z) + A(z)µ1 +B(z)µ2

(7.50)

where

β(z) = α2(z, α1(z)), A(z) =Lg1α(z)

M(z), B(z) =

1

M(z)(7.51)

Moreover, A(z), B(z) satisfy the following bounds

|A(z)| ≤ L1L2, 0 < B(z) ≤ L2

for all z ∈ Rn (the constants L1, L2 > 0 are defined in assumption 7.4.1).

Proof. Let us calculate the first two time derivatives of µ1 = ξ1 − α(z). We get

µ2 := µ1 = ξ2 − [Lfα(z) + ξ1Lg1α(z) + ξ2Lg2α(z)]

= (1− Lg2α(z))ξ2 − [Lfα(z) + ξ1Lg1α(z)]

= M(z)(ξ2 −Lfα(z) + ξ1Lg1α(z)

M(z))

Now, according to the definition of α2(z, ξ1)

µ2 =M(z)(ξ2 − α2(z, ξ1))

Denoting w := µ2, w can be calculated as

w = M(z)(u− α2) + M(ξ2 − α2(z, ξ1)

= M(z)(u− α3(z, ξ1, ξ2))

where

α3(z, ξ1, ξ2) := α2(z, ξ1, ξ2)−M

M(ξ2 − α2(z, ξ1))

So far, we proved that the new variables µ1, µ2, w satisfy the chain of integrators

µ1 = µ2, µ2 = w

To calculate the inverse of µ = Φ(z, ξ), we need to solve µ = Φ(z, ξ) in (ξ1, ξ2). Onecan solve for ξ1 as ξ1 = α(z) + µ1. Now, solving for ξ2 from the definition of µ2 gives

ξ2 = α2(z, ξ1) +1

M(z)µ2

= α2(z, α(z) + µ1) +B(z)µ2

= α2(z, α1(z)) + A(z)µ1 +B(z)µ2

which after denoting β(z) = α2(z, α1(z)), it agrees with the definition of β(z) in the

236

question. Substituting ξ1, ξ2 in the first equation of (7.42) shows that

z = f(z) + g1(z)(α(z) + µ1) + g2(z)(β(z) + A(z)µ1 +B(z)µ2)

= f0(z) + [g1(z) + A(z)g2(z)]µ1 +B(z)g2(z)µ2

and this completes the proof of equation (7.49). The bounds for A(z) and B(z) followfrom parts ii) and iii) of assumption 7.4.1.

The following assumption guarantees that the zero-dynamics of the cascade non-linear system in (7.49) with the output y = µ1 is globally minimum phase.

Assumption 7.4.2. Assume for

z = f(z) + g1(z)α(z) + g2(z)β(z) =: f0(z) (7.52)

the origin z = 0 is globally asymptotically stable.

Assumption 7.4.3. Let V (z) be a smooth positive definite and proper Lyapunovfunction satisfying Lf0V (z) ≤ 0. Suppose there exists ρ ≥ 0, k > 0, and λ ∈ (0, 1]such that the following conditions are satisfied

|LgiV (z)| ≤ kV (z)λ, ∀‖z‖ ≥ ρ, i = 1, 2 (7.53)

Here is out main result on global asymptotic stabilization of nontriangular non-linear systems affine in (ξ1, ξ2):

Theorem 7.4.1. Consider the nonlinear system in (7.42). Suppose assumptions7.4.1, 7.4.2, and 7.4.3 hold. Then, the following smooth state feedback in the form ofa PD controller with nonlinear gains

u = α3(z, ξ1, ξ2)−c1

M(z)(ξ1 − α1(z))−

c2M(z)

(ξ2 − α2(z, ξ1)), c1, c2 > 0 (7.54)

that globally asymptotically stabilizes (z, ξ1, ξ2) = 0 for the nontriangular nonlinearsystem (7.42) with M(z) = 1− Lg2α(z).Proof. Based on proposition 7.4.3, nonlinear system (7.42) can be transformed intothe cascade system

z = f0(z) + [g1(z) + A(z)g2(z)]µ1 +B(z)g2(z)µ2 =: F (z, µ1, µ2)µ1 = µ2µ2 = w

(7.55)

such that for z = F (z, 0, 0) = f0(z), z = 0 is GAS. Let us stabilize the µ-subsystemwith a PD controller

w = −c1µ1 − c2µ2, c1, c2 > 0

Then, µ = 0 is globally exponentially stable. Based on theorem 7.3.3, to show that(z, µ) = 0 is globally asymptotically stable for the closed-loop cascade system in

237

(7.55), we need to prove that any solution of the z-subsystem is bounded. For doingso, let us calculate V along the solutions of (7.55). One obtains

V = Lf0V (z) + [Lg1V (z) + A(z)Lg2V (z)]µ1 +B(z)(Lg2V (z))µ2

≤ (|Lg1V (z)|+ |A(z)||Lg2V (z)|)|µ1|+B(z)|Lg2V (z)||µ2|≤ (1 + L1L2 + L2)kV (z)λ‖µ‖ (because |µi| ≤ ‖µ‖, i = 1, 2)

= kV (z)λ‖µ‖

where k := 1 + L1L2 + L2. On the other hand, µ(t) is exponentially vanishing and‖µ(·)‖1 < ∞. Thus, from the proof of theorem 7.3.4, it follows that any solutionof the z-subsystem is bounded. The PD feedback w = −c1µ1 − c2µ2 in the originalcoordinate can be written as

u = α3(z, ξ1, ξ2)−c1

M(z)µ1 −

c2M(z)

µ2

which is a PD controller with nonlinear gains −ci/M(z), i = 1, 2.

Remark 7.4.2. The equation of α3 in the state feedback (7.54) can be expressed in aslightly more explicit way as follows. Define p(z), q(z) as

p(z) :=Lfα(z)

M(z), q(z) :=

Lg1α(z)

M(z)

and observe that they satisfy α2(z, ξ1) = p(z) + ξ1q(z). Then, the right hand side of

α3(z, ξ1, ξ2) = α2 −M

M2µ2

can be explicitly determined as the following

α2 = Lfp(z) + ξ1Lg1q(z) + ξ2(q(z) + Lg2q(z)) (7.56)

M = LfM(z) + ξ1Lg1M(z) + ξ2Lg2M(z) (7.57)

A crucial assumption in theorem 7.4.1 is assumption 7.4.2. In the following, weprovide sufficient conditions on g2(z) such that assumption 7.4.2 holds.

Proposition 7.4.4. Consider the following nonlinear system

z = f(z) + g1(z)ξ1 + g2(z)ξ2

and assume the following conditions hold:

i) there exists a smooth state feedback ξ1 = α(z) such that for

z = f(z) + g1(z)α(z) =: f1(z)

z = 0 is GAS,

238

ii) Lg2α(z) < 1 for all z ∈ Rn,

iii) there exists a function h(z) : Rn → R such that g2(z) = h(z)f1(z).

Let

β(z) :=Lf1α(z)

1− Lg2α(z)Then, for the closed-loop system

z = f(z) + g1(z)α(z) + g2(z)β(z) (7.58)

z = 0 is GAS.

Proof. The closed-loop system with control (ξ1, ξ2) = (α(z), β(z)) can be written as

z =f1(z) + ∆(z)

1− Lg2α(z)(7.59)

where∆(z) := (Lf1α(z))g2(z)− (Lg2α(z))f1(z)

But g2(z) ‖ f1(z), or g2(z) = h(z)f1(z) implies

(Lf1α(z))g2(z) = (Lg2α(z))f1(z), ∀z ∈ Rn

Thus, ∆(z) ≡ 0. This means the closed-loop nonlinear system in (7.59) reduces to

z = S(z)f1(z) (7.60)

with a scaling factor S(z) =: (1 − Lg2α(z))−1 > 0. Since S(z) is a positive definitescalar, z = 0 is GAS for z = S(z)f1(z) and the result follows.

In the following result, we slightly remedy the restrictive condition g2(z) ‖ f1(z)in proposition 7.4.4.

Proposition 7.4.5. Consider the following nonlinear system

z = f(z) + g1(z)ξ1 + g2(z)ξ2

and assume the following conditions hold:

i) there exists a smooth state feedback ξ1 = α(z) such that for

z = f(z) + g1(z)α(z) =: f1(z)

z = 0 is GAS.

ii) Let V (z) be a smooth positive definite and proper Lyapunov function associatedwith z = f1(z) such that ∇V (z)f1(z) ≤ 0. Assume the largest invariant set inΩ = z | ∇V (z)f1(z) = 0 is 0.

239

iii) Lg2α(z) < 1 for all z ∈ Rn,

iv) Assume V (z) satisfies the inequality

(Lf1α(z))(Lg2V (z)) ≤ (Lg2α(z))(Lf1V (z)), ∀z ∈ Rn

where the equality is achieved for g2(z) = h(z)f1(z) with h(z) : Rn → R.

Define

β(z) :=Lf1α(z)

1− Lg2α(z)Then, for the closed-loop system

z = f(z) + g1(z)α(z) + g2(z)β(z) (7.61)

z = 0 is GAS.

Proof. The closed-loop system with control (ξ1, ξ2) = (α(z), β(z)) is in the form

z = S(z)(f1(z) + ∆(z)) (7.62)

where S(z) = (1− Lg2α(z))−1 and

∆(z) = (Lf1α(z))g2(z)− (Lg2α(z))f1(z)

ThusL∆V (z) = (Lf1α(z))(Lg2V (z))− (Lg2α(z))(Lf1V (z)) ≤ 0

which impliesV = S(z)(Lf1V (z) + L∆V (z)) ≤ Lf1V (z) ≤ 0

but the largest invariant set in Ω is 0, therefore z = 0 is GAS for the closed-loopnonlinear system.

Example 7.4.1. Consider the following cascade nonlinear system which is in a non-triangular form affine in (ξ1, ξ2)

z1 = z2 −z2ξ2

2(1 +√

z21 + z22)

z1 = ξ1 +tanh(z1 + z2)ξ2

2(1 +√

z21 + z22)

ξ1 = ξ2

ξ2 = u

We show that there exists a state feedback ξ1 = α(z) which satisfies all the conditionsin theorem 7.4.1. and thus the last nonlinear system can be globally asymptotically

240

stabilized to the origin. Take α(z) = − tanh(z1 + z2). Then, for

z = f1(z) :=

[

z2− tanh(z1 + z2)

]

z = 0 is GAS. This is due to the fact that

V (z) =

∫ z1

0

tanh(s)ds+1

2z22 (7.63)

is a valid positive definite Lyapunov function for z = f1(z). Since

Lg2α(z) =∂α(z)

∂zg2(z) = (1− tanh2(z1 + z2))

(z2 − tanh(z1 + z2))

2(1 + ‖z‖) ,

we get

|Lg2α(z)| ≤ |1− tanh2(z1 + z2)|1 + |z2|

2(1 + ‖z‖) ≤1

2

and thus 1/2 ≤ 1− Lg2α(z) ≤ 3/2 (i.e. L2 = 2 and 1/2 ≤M(z) ≤ 3/2). In addition,due to

|Lg1α(z)| = |∂α(z)

∂z2| = 1− tanh2(z1 + z2) ≤ 1

we obtain L1 = 1. So far, we have proved that assumption 7.4.1 holds. Using propo-sition 7.4.4, we can show that assumption 7.4.2 holds as well. This is a consequenceof the fact that g2(z) satisfies the relation g2(z) = h(z)f1(z) with

h(z) :=−1

2(1 + ‖z‖)

Finally, assumption 7.4.3 can be verified by direct calculation as follows. The Lya-punov function V (z) in equation (7.63) satisfies

|Lg1V (z)| = |∂V (z)

∂z2| = |z2| ≤

√2V (z)

12 , ∀z ∈ R2

and

|Lg2V (z)| = |z2[tanh(z1 + z2)− tanh(z1)]|2(1 + ‖z‖) ≤ |z2| ≤

√2V (z)

12 , ∀z ∈ R2

The last inequality is due to the global Lipschitz property of tanh(x) which implies

| tanh(z1 + z2)− tanh(z1)| ≤ |z2|

Hence for k =√2, λ = 1/2, the following inequalities hold

|LgiV (z)| ≤ kV (z)λ, ∀z ∈ R2, i = 1, 2

241

Therefore, based on theorem 7.4.1, the following PD controller with nonlinear gains

u = α3(z, ξ1, ξ2)−c1

M(z)(ξ1 − α1(z))−

c2M(z)

(ξ2 − α2(z, ξ1)), c1, c2 > 0

globally asymptotically stabilizes the origin (z, ξ1, ξ2) = 0 for the nontriangular non-linear system in the question.

7.4.2 Nontriangular Nonlinear Systems Non-affine in (ξ1, ξ2)

In this section, we generalize our results on cascade backstepping procedure for sta-bilization of nontriangular nonlinear systems affine in (ξ1, ξ2) to cascade nonlinearsystems that their vector fields are non-affine in (ξ1, ξ2). We consider the followingclass of partially-linear cascade nonlinear systems

z = F (z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(7.64)

where F (z, ξ1, ξ2) : Rnz ×R×R→ Rnz is assumed to be a smooth function satisfyingF (0) = 0 and non-affine in (ξ1, ξ2). The function F (z, ξ1, ξ2) can be rewritten as

F (z, ξ1, ξ2) = f(z) + g1(z, ξ1)ξ1 + g2(z, ξ1, ξ2)ξ2 (7.65)

where

f(z) := F (z, 0, 0) (7.66)

g1(z, ξ1) :=

∫ 1

0

(Dξ1F )(z, sξ1, 0)ds (7.67)

g2(z, ξ1, ξ2) :=

∫ 1

0

(Dξ2F )(z, ξ1, sξ2)ds (7.68)

In this section, we provide sufficient conditions for global stabilization of nonlinearsystems in normal form (7.64). Depending on the particular structure of f, g1, g2 anumber of important cases arise that will be discussed later in this chapter. Theimportance of each case is due to its application as the normal form of special classesof mechanical systems of interest. The main emphasis will be on control design fornontriangular normal forms of underactuated systems. Before presenting our mainresult, we need to make some assumptions.

Assumption 7.4.4. Assume there exists a smooth state feedback ξ1 = α(z) withα(0) = 0 that globally asymptotically stabilizes z = 0 for

z = F (z, ξ1, 0) =: f1(z, ξ1) (7.69)

Let V (z) be the smooth positive definite and proper Lyapunov function associatedwith the closed-loop system such that ∇V (z) · f1(z, α(z)) < 0,∀z 6= 0.

242

Assumption 7.4.5. The exists a smooth function β(z) : Rn → R with β(0) = 0 suchthat the following conditions hold:

i) ξ∗2 = β(z) is the unique solution of the following fixed-point equation in ξ2

ξ2 =∂α(z)

∂zF (z, α(z), ξ2)

ii) ∇V (z) · F (z, α(z), β(z)) < 0,∀z 6= 0.

Assumption 7.4.6. Define

ψ(z, ξ1, ξ2) := −c1(ξ1 − α(z)) +∂α(z)

∂zF (z, ξ1, ξ2) (7.70)

and assume there exists an a > 0 such that

∂ψ

∂ξ2=∂α(z)

∂z

∂F (z, ξ1, ξ2)

∂ξ2≤ 1− a, ∀(z, ξ1, ξ2)

This implies

M(z, ξ1, ξ2) := 1− ∂ψ

∂ξ2≥ a > 0, ∀(z, ξ1, ξ2) (7.71)

Assumption 7.4.7. Assume the change of coordinates µ = T (z, ξ) given by

µ1 = ξ1 − α(z)µ2 = ξ2 − ψ(z, ξ1, ξ2)

is a global diffeomorphism for each z ∈ Rn with an inverse ξ = T−1(z, µ) as thefollowing

ξ1 = µ1 + α(z)

ξ2 = φ(z, µ1, µ2)

where φ(z, 0, 0) = β(z).

Now, the following result clarifies our purpose of making the last four assumptions.

Proposition 7.4.6. Consider the nonlinear system in (7.64) and suppose assump-tions 7.4.4 through 7.4.7 hold. Then, the change of coordinates µ = T (z, ξ) and thechange of control given by

µ1 = ξ1 − α(z)µ2 = ξ2 − ψ(z)w = M(z, ξ1, ξ2)(u− γ(z, ξ1, ξ2))

(7.72)

243

is globally invertible where

γ(z, ξ1, ξ2) =Dzψ · F (z, ξ1, ξ2) +Dξ1ψ · ξ2

1−Dξ2ψ(7.73)

In addition, it transforms the dynamics of (7.64) into the following cascade form

z = F0(z, µ1, µ2)µ1 = −c1µ1 + µ2µ2 = w

(7.74)

whereF0(z, µ1, µ2) = F (z, α(z) + µ1, φ(z, µ1, µ2))

Moreover, forz = F0(z, 0, 0) = F (z, α(z), β(z))

z = 0 is GAS.

Proof. Based on assumptions 7.4.6 and 7.4.7, µ = T (z, ξ) is a global diffeomorphismand M(z, ξ1, ξ2) ≥ a > 0. Thus, the change of coordinates and control in (7.74) isglobally invertible. By definitions of µ1 and ψ, we have

µ1 = ξ2 −∂α(z)

∂zF (z, ξ1, ξ2) = −c1µ1 + µ2

and because µ2 = ξ2 − ψ(z, ξ1, ξ2), one gets

µ2 = (1−Dξ2ψ)u− (Dzψ · F (z, ξ1, ξ2) +Dξ1ψ)

= M(z, ξ1, ξ2)(u− γ(z, ξ1, ξ2)) =: w

which means the µ-subsystem has the following linear dynamics

µ1 = −c1µ1 + µ2µ2 = w

global asymptotic stability of z = 0 for

z = F0(z, 0, 0)

follows from part ii) of assumption 7.4.5.

Assumption 7.4.8. Rewrite F0(z, µ1, µ2) as

F0(z, µ1, µ2) = F0(z, 0, 0) + h1(z, µ1)µ1 + h2(z, µ1, µ2)µ2

and assume for any l1, l2 > 0, there exist ρi ≥ 0, ki > 0, λi ∈ (0, 1] for i = 1, 2 such

244

that V (z) satisfies

|∇V (z) · h1(z, µ1)| ≤ k1V (z)λ1 , ‖z‖ ≥ ρ1, |µ1| ≤ l1|∇V (z) · h2(z, µ1, µ2)| ≤ k2V (z)λ2 , ‖z‖ ≥ ρ2, |µ1| ≤ l1, |µ2| ≤ l2

(7.75)

Lemma 7.4.1. For any l > 0, take l1, l2 ≥ l and suppose assumption 7.4.8 holds.Then, the following condition is satisfied

|∇V (z)hi| ≤ kV (z)λ, ‖z‖ ≥ ρ, ‖µ‖ < l, i = 1, 2 (7.76)

with ρ = maxρ1, ρ2 ≥ 0, k = maxk1, k2 > 0, and λ = maxλ1, λ2 ∈ (0, 1].

Proof. Since |µi| ≤ ‖µ‖ for i = 1, 2 and the function ϕ(k, l) = kxλ for x > 0 is anincreasing function of λ (or k) uniformly in k (or λ) the result holds.

We are now ready to present out main result on global stabilization of nontrian-gular nonlinear systems non-affine in (ξ1, ξ2):

Theorem 7.4.2. Consider the nonlinear system in (7.64). Suppose that assumptions7.4.4 through 7.4.8 hold. Then, the following state feedback in explicit form

u = K(z, ξ1, ξ2) := γ(z, ξ1, ξ2)−c2

M(z, ξ1, ξ2)(ξ2 − ψ(z, ξ1, ξ2)), c2 > 0 (7.77)

globally asymptotically stabilizes the origin (z, ξ1, ξ2) = 0 for (7.64).

Proof. According to proposition 7.4.6, after applying the change of coordinates andcontrol in equation (7.72), one obtains cascade system (7.74). The linear µ-subsystemof (7.74) can be globally exponentially stabilized using w = −c2µ2, c2 > 0. The closed-loop µ-subsystem is in the form µ = Aµ where

A =

[

−c1 10 −c2

]

is a Hurwitz matrix with eigenvalues −c1,−c2 < 0. In new coordinates, the controlu can be expressed as the following

u = γ(x)− c2M(x)

(ξ2 − ψ(x))

with x = col(z, ξ1, ξ2) as given in (7.77). Based on assumption 7.4.8 and lemma 7.4.1,any solution of the closed-loop z-subsystem in (7.74) with state feedback w = −c2µ2(or (7.77)) is bounded. Thus, based on theorem 7.3.3, (z, µ) = 0 is GAS for thecascade nonlinear system in (7.74). This implies the origin (z, ξ1, ξ2) = 0 is GAS forthe closed-loop nonlinear system (7.64).

245

7.4.3 Global Existence of Fixed-Point Control Laws

A crucial assumption in the global version of the cascade backstepping procedure intheorem 7.77 is that the fixed point equation in ξ2 of the form

ξ2 =∂α(z)

∂zF (z, α(z), ξ2) =: Ψ(z, ξ2) (7.78)

(see assumption 7.4.5) globally (w.r.t z) has solutions in ξ2. More precisely, thereexists β : Rn0 → R such that

β(z) = Ψ(z, β(z)), ∀z ∈ Rn0

In this section, we provide sufficient conditions for global existence of a solutionξ∗2 = β(z) for the fixed point equation

ξ2 = Ψ(z, ξ2) (7.79)

We refer to the solution β(z) of this fixed point equation as a fixed point control law.Due to the structure of F (z, ξ1, ξ2) in equation (7.15), we focus on global existence offixed-point control laws for nontriangular systems with the following dynamics

z1 = z2z2 = f(z, ξ1, ξ2)

ξ1 = ξ2ξ2 = u

(7.80)

wheref(z, ξ1, ξ2) = f0(z, ξ1) + g0(z, ξ1, ξ2)ξ

22 (7.81)

Before presenting our main result, we need to make some definitions and give somelemmas. Let us focus our attention to the following class of state feedback laws α(z).

Definition 7.4.1. (bounded functions with compact domain of attention) We say afunction α(z) : R2n → R with α(0) = 0 and z = col(z1, z2) is bounded with compactdomain of attention around the origin w.r.t. z2 if there exist a compact neighborhoodK0 ⊂ Rn of z2 = 0 and constants Li > 0, i = 0, 1, 2 such that

i) |α(z)| ≤ L0,∀z ∈ R2n

ii) Dz2α(z) = 0,∀z ∈ Rn × (Rn \ K0)

iii) ‖Dz2α(z)‖ ≤ L1,∀z ∈ R2n

iv) ‖Dz1α(z)‖ ≤ L2,∀z ∈ R2n

Definition 7.4.2. (bounded sigmoidal functions with flat tails) We say σ(x) : R→ Ris a bounded sigmoidal function with a flat tail if the following conditions hold:

246

i) ∃L > 0 : |σ(x)| ≤ L,∀x.

ii) xσ(x) > 0,∀x 6= 0 and σ(0) = 0.

iii) ∃I ⊂ R, 0 ∈ I : σ′(x) = 0,∀x 6∈ I.Clearly, over the region R\ I, σ(x) is constant (which explains the name “flat tail”).

Example 7.4.2. (sigmoidal functions) A nonsmooth example of a sigmoidal functionwith a flat tail is the saturation function sat(x) = sgn(x)min|x|, 1. A C 1 smoothbounded sigmoidal function with flat tails is

σ(x) =

sin(π

2x) x ∈ [−1, 1]

1 x > 1−1 x < −1

(7.82)

or in a more compact form

σ(x) = sin(π

2x) · 1I(x) + (1− 1I(x)) · sgn(x), I = [−1, 1] (7.83)

where 1I(x), sgn(x) denote the indicator function and the sign function, respectively.In general, it is possible to construct a Cr smooth bounded sigmoidal function withflat tails and (relatively) large linear region as

σf (x; a) =

x |x| ≤ asgn(x) · P (|x|) a < |x| ≤ 1sgn(x) |x| > 1

(7.84)

where a ∈ (0, 1) is a constant threshold and P (x) is a polynomial of an appropriateorder. For example, with a = 0.75 and I = [−1, 1] the following 5th order polynomialgives a C2 smooth bounded sigmoidal function with flat tails

P (x) = 768x5 − 3328x4 + 5728x3 − 4896x2 + 2080x− 351 (7.85)

The function σf (x; 0.75) with its first two derivatives is depicted in Figure 7-1. Inthis case, σf (x; 0.75) is strictly increasing over the interval I = [−1, 1].Example 7.4.3. Consider the following double-integrator with a saturation control

z1 = z2z2 = sat(ξ1)

(7.86)

where sat(x) = sgn(x)min|x|, 1. System (7.86) is a special case of the z-subsystemof (7.80) under the assumption that f(z, ξ1, 0) = sat(ξ1). The following state feedback

α(z) = −σ1(z1 + σ2(z2)) (7.87)

globally asymptotically stabilize (z1, z2) = 0 for (7.86) where σ1 is an strictly increas-ing bounded sigmoidal function and σ2 is a bounded sigmoidal function with flat tails

247

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x

σ(x)

0.75 0.8 0.85 0.9 0.95 10.75

0.8

0.85

0.9

0.95

1

x

σ(x)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

2

x

σ’ (x)

−1.5 −1 −0.5 0 0.5 1 1.5−20

−10

0

10

20

x

σ" (x)

Figure 7-1: A C2 smooth sigmoidal function with flat tails with its derivatives.

outside I = [−1, 1]. By definition, α(z) in (7.87) has compact domain of attentionw.r.t. z2 with K0 = I.

Lemma 7.4.2. Consider the nonlinear system in (7.86). Let α(z) be the state feed-back in (7.87) with σ1 and σ2 which are bounded sigmoidal functions with flat tailsoutside I = [−1, 1]. Then, the following statements hold:

i) z = 0 is globally asymptotically stable for (7.86) with state feedback α(z).

ii) there exists a compact neighborhood K ⊂ R2 of z = 0 as the following

K = (z1, z2) ∈ R2 | |z1 + σ2(z2)| ≤ 1, |z2| ≤ 1 ⊂ [−2, 2]× [−1, 1] (7.88)

such that Dz2α(z) = 0 for z 6∈ K.

Proof. Let us define the following global change of coordinates

y1 = z1 + σ2(z2)y2 = z2

(7.89)

In new coordinates, the dynamics of the closed-loop system (7.86) takes the form

y1 = y2 − σ′2(y2)σ1(y1)y2 = −σ1(y1) (7.90)

Define

φ(y1) =

∫ y1

0

σ1(s)ds

248

and consider the following positive definite and smooth Lyapunov function candidate

V (y1, y2) = φ(y1) +1

2y22

which satisfiesV = −σ′2(y2)σ1(y1)2 ≤ 0

We need to find the largest invariant set in Ω = y ∈ R2 | V = 0, or

Ω = y ∈ R2 | |y2| ≥ 1 or y1 = 0

A simple analysis shows that there is no invariant set in the region |y2| > 1 and theonly invariant set for y1 = 0 is (0, 0). Thus, from LaSalle’s invariance principle,y = 0 (or z = 0) is GAS. To prove part ii), notice that Dz2α(z) = σ′2(y2)σ

′1(y1) = 0

for all y ∈ S = y | |y1| > 1, |y2| > 1. Clearly, K is the complement of S in R2. Also,assuming |z1| > 2, we get

|y1| ≥ |z1| − |σ2(z2)| > 1

and therefore Dz2α(z) = 0. This means outside K = [−2, 2] × [−1, 1] ⊃ K, Dz2α(z)vanishes.

The main benefit of lemma 7.4.2 is that outside a compact set K, one can explicitlydetermine the fixed point control law ξ∗2 = β(z) of equation (7.78). Since Dz2α(z) = 0over R2 \ K, we obtain

β(z) =∂α(z)

∂z1z2

and the stability analysis for the closed-loop z-subsystem

z = F (z, α(z), β(z))

reduces to a stability analysis over the sets (R2 \ S) and (S \ K) where the set S iscalled a sigmoidal ribbon defined as the following

S := z ∈ R2 | |z1 + σ2(z2)| ≤ 1

By definition, K ⊂ S. We have

β(z) =

Dz1α(z) · z2 z ∈ (S \ K)0 z ∈ (R2 \ S) (7.91)

Figure 7-2 shows different regions including the sigmoidal ribbon S (light shaded)and the compact set K (dark shaded). The dashed regions in Figure 7-2 show wherethe control law α(z) is constant (i.e. Dα(z) = 0) It remains to find β(z) over thecompact set K. This is done in the following for the general case where z1, z2 ∈ Rn.

Assumption 7.4.9. Suppose that for any compact set 0 ∈ K := Kz2 ×Kξ1 ×Kξ2 ⊂Rn+2, there exist constants ai > 0, bi ≥ 0, i = 1, 2, 3 such that f0, g0, Dξ2f haveincremental linear growth in z1 for all (z2, ξ1, ξ2) ∈ K, i.e.

249

z

z 2

1

1

S

S

-1

-1

1

K

α=+1

α=−1

Figure 7-2: Sigmoidal ribbon S and compact set K.

i) ‖f0(z, ξ1)‖ ≤ a1 + b1‖z1‖

ii) ‖g0(z, ξ1, ξ2)‖ ≤ a2 + b2‖z1‖

iii) ‖Dξ2f(z, ξ1, ξ2)‖ ≤ a3 + b3‖z1‖

The following scaling of the function α(z) is influenced by the work of Sepulchre [79].

Lemma 7.4.3. Let the smooth function α(z) : R2n → R with α(0) = 0 be a boundedfunction that has a compact domain of attention K0 around z2 = 0 with constantsLi > 0, i = 0, 1, 2 satisfying all the conditions in definition 7.4.1. Define

ακ(z) = κ(z1)α(z) (7.92)

where

κ(z1) :=

1 , if bi = 0, i = 1, 2, 31

√

1 + ‖z1‖2, otherwise

Then, ακ(z) is also a bounded function with compact domain of attention K0 aroundz2 = 0 and associated constants Lκ0 = L0, L

κ1 = L1, and L

κ2 = L0 + L2.

Proof. Noting that 0 < κ(z1) ≤ 1, part i) of definition 7.4.1 immediately follows. Toprove properties ii) and iii) of ακ(z) (in the same definition), notice that

Dz2ακ(z) = κ(z1)Dz2α(z)

Thus, Dz2ακ(z) vanishes for all z2 6∈ K0. In addition, ‖Dz2ακ(z)‖ ≤ ‖Dz2α(z)‖ ≤ L1,∀z ∈ R2. To prove the part iv), let us calculate Dz1ακ(z) as

Dz1ακ(z) = (Dz1κ(z1))α(z) + κ(z1)Dz1α(z)

250

but Dz1κ(z1) = z1κ(z1)3 and ‖Dz1κ(z1)‖ ≤ ‖z1κ(z1)‖κ(z1)2 ≤ 1. Hence, we get

‖Dz1ακ(z)‖ ≤ L0 + L2

Theorem 7.4.3. Suppose that Assumption (7.4.9) holds. Let the smooth functionα(z) : R2n → R with α(0) = 0 be a bounded function that has a compact domain ofattention K0 around z2 = 0 with constants Li > 0, i = 0, 1, 2. In addition, let ακ(z)denote the scaled α(z) defined in equation (7.92). Then, there exist L∗i (K0) > 0, i =0, 1, 2 such that for all α(z) with associated constants 0 < Li < L∗i , i = 0, 1, 2, thefollowing fixed-point equation

ξ2 = Dzακ(z) · F (z, ακ(z), ξ2) (7.93)

globally has a unique smooth solution

ξ∗2 = β(z)

with β(0) = 0.

Proof. See section C.1 in Appendix C.

Remark 7.4.3. Based on the contraction mapping theorem, the fixed point ξ∗2 = β(z)in theorem 7.4.3 can be obtained using the following iteration in k

ξ(k+1)2 = Ψ(z, ξ

(k)2 ), ξ

(0)2 = 0 (7.94)

and satisfies β(z) = limk→∞ ξ(k)2 = ξ∗2 .

Proposition 7.4.7. Assume F has the following structure

F (z, ξ1, ξ2) =

[

z2f0(z, ξ1) + g0(z, ξ1)ξ

22

]

Then, the fixed point equation in theorem 7.4.3 reduces to the following quadraticequation in ξ2

ξ2 = Dz1ακ(z) · z2 +Dz2ακ(z) · [f0(z, ακ(z)) + g0(z, ακ(z)]ξ22 (7.95)

so that its solutions can be given in explicit form.

Proof. The fixed point equation in (7.95) can be rewritten as a standard quadraticequation in ξ2 in the form

a(z)ξ22 + bξ2 + c(z) = 0 (7.96)

where

a(z) = Dz2ακ(z)g0(z, ακ(z))

b = −1c(z) = Dz1ακ(z) · z2 +Dz2ακ(z) · f0(z, ακ(z))

251

The condition for global existence of a fixed point solution ξ∗2 = β(z) can be readilyexpressed as ∆(z) = 1− 4a(z)c(z) > 0. Then the solution of the quadratic equation(7.96) in ξ2 with the smallest absolute value [72] is given by

β(z) =2c(z)

1 +√

1− 4a(z)c(z)(7.97)

The solution in (7.97) is the same as the following standard solution of (7.96)

β(z) =

1−√

1− 4a(z)c(z)

2a(z)a(z) 6= 0

c(z) a(z) = 0

Remark 7.4.4. Following the line of proposition 7.4.7, let K be a compact set suchthat Dz2α(z) = 0 for all z 6∈ K (see part (ii) of lemma 7.4.2). Set κ(z) = 1, or letακ(z) = α(z). Then, the existence condition of the fixed point ξ∗2

a(z)c(z) <1

4

has to hold over the compact setK ⊂ Rn0 instead of the whole Rn0 . This tremendouslyreduces the complexity of checking the global existence of a fixed point control law.We demonstrate this in the following. Since K is a compact set, there exists constantsLf0 , Lg0 , Lz2 > 0 such that

‖f0(z, α(z))‖ ≤ Lf0 , ‖g0(z, α(z))‖ ≤ Lg0 , ‖z2‖ ≤ Lz2

for all z ∈ K. Hence, the following condition

|a(z)c(z)| ≤ L1Lg0(L2Lz2 + L1Lf0) <1

4

is satisfied by choosing the constants L1, L2 associated with α(z) according to

0 < L1, L2 < L∗ :=1

2√

Lg0(Lf0 + Lz2)(7.98)

where L∗ can be readily calculated.

Example 7.4.4. (beam-and-ball system) Consider the dynamics of the beam-and-ball system given in equation (5.33). The third-order subsystem of (5.33) is in theform

z1 = z2z2 = −g sin(ξ1) + (z1 + dξ1)ξ

22

ξ1 = ξ2

(7.99)

252

where (ξ1, ξ2) = (θ, ω). Define

f0(z, ξ1) = −g sin(ξ1)g0(z, ξ1) = z1 + dξ1

(7.100)

Clearly, f0, g0 satisfy assumption 7.4.9. Let α(z) = −c0σ2(z1 + σ1(z1)) where σi’s aresigmoidal functions with flat tails outside [−1, 1]. In addition, |σ2(x)| ≤ 1,∀x and0 < c0 < π/2 is a constant (to be determined). One can take c0 to be sufficientlysmall such that condition a(z)c(z) < 1/4 in remark 7.4.4 is satisfied over the compactset K (shown in Figure 7-2). Based on proposition 7.4.7, the state feedback β(z) canbe explicitly and globally obtained from (7.97).

7.4.4 Slightly Nontriangular Nonlinear Systems

In this section, we consider stabilization of a perturbed version of a nonlinear systemin strict feedback form such that the overall system is non-triangular. We call thisclass of systems slightly nontriangular systems.

Theorem 7.4.4. Consider the following nonlinear system

z = f0(z, ξ1) + g0(z, ξ1, ξ2, ε),

ξ1 = ξ2,

ξ2 = u,

(7.101)

where f0, g0 are smooth functions with f0(0) = 0, g0(0) = 0, g0(z, ξ1, 0, ε) = 0,g0(z, ξ1, ξ2, 0) = 0, and ∂g0/∂ξ2 vanishes at ξ2 = 0 (i.e.∃p0 ∈ C1 : g0 = p0(z, ξ1, ξ2, ε)ξ

22ε).

Suppose there exists a state feedback law ξ1 = α1(z) with α1(0) = 0 such that for

z = f0(z, α1(z))

z = 0 is globally asymptotically and locally exponentially stable. Let V (z) be thesmooth positive definite proper Lyapunov function associated with this system satis-fying the property ∇V (z) · f0(z, α1(z)) < 0, ∀z 6= 0. Define

ψ2(z, ξ1, ξ2, ε) = −σ(ξ1 − α1(z)) +∂α1(z)

∂zf(z, ξ1, ξ2, ε)

ψ3(z, ξ1, ξ2, u, ε) = −σ(ξ2 − ψ2) +∂ψ2

∂zf(z, ξ1, ξ2, ε) +

∂ψ2

∂ξ1ξ2 +

∂ψ2

∂ξ2u

(7.102)

where f = f0(z, ξ1) + g0(z, ξ1, ξ2, ε) and σ(·) is a sigmoidal function. In addition,assume for any l0 > 0, there exist k > 0, r0 ≥ 0 and λ ∈ (0, 1] such that

∣

∣

∣

∣

∂V (z)

∂zh0(z, α1(z), δ)

∣

∣

∣

∣

≤ kV (z)λ , ∀|z| > r0,∀|δ| ≤ l0 (7.103)

where h0 is given by

h0(z, ξ1, δ) =

∫ 1

0

∂f0∂ξ1

(z, ξ1 + sδ)ds

253

and satisfiesf0(z, ξ1 + δ) = f0(z, ξ1) + h0(z, ξ1, δ)δ

Then, there exists an ε∗ > 0 such that for all 0 < ε < ε∗ the following hold:

i) The change of coordinates µ = T (z, ξ) as

µ1 = ξ1 − α1(z)µ2 = ξ2 − ψ2(z, ξ1, ξ2, ε)

is invertible over a compact domain Kz×Kξ with a smooth inverse ξ = T−1(z, µ)given by

ξ1 = β1(z, µ1) := µ1 + α1(z)ξ2 = β2(z, µ1, µ2, ε)

with β1(0, 0) = 0 and β2(0, 0, 0, ε) = 0.

ii) The fixed point equations

ξ2 = ψ2(z, ξ1, ξ2, ε)u = ψ3(z, ξ1, ξ2, u, ε)

have unique smooth solutions

ξ∗2 = α2(z, ξ1, ε)u∗ = α3(z, ξ1, ξ2, ε)

with α2(0, 0, ε) = 0 and α3(0, 0, 0, ε) = 0 over a compact domain Kz ×Kξ.

iii) The state feedback ξ2 = α2(z, ξ1, ε) semiglobally asymptotically and locally expo-nentially stabilizes (z, ξ1) = 0 for the (z, ξ1)-subsystem of (7.101).

iv) The state feedback u = α3(z, ξ1, ξ2, ε) semiglobally asymptotically and locallyexponentially stabilizes (z, ξ1, ξ2) = 0 for the composite system in (7.101).

Proof. See section C.4 in Appendix C.The proof of the existence of unique smooth fixed points in theorem 7.4.4 relies

on the following lemma.

Lemma 7.4.4. Consider the following fixed point equation in y

y = φ0(x) + φ1(x, y, ε)ε =: φ(x, y, ε) (7.104)

where φ : Rn×Rm×R→ Rm is continuously differentiable with φ(0, 0, ε) = 0. Then,for any compact set Kx and the interval Iε = [0, ε1], there exists a compact set Kyand an 0 < ε∗ ∈ Iε such that for all 0 < ε < ε∗ the fixed point equation (7.104) has aunique smooth solution

y∗ = α(x, ε)

that belongs to Ky and satisfies α(0, ε) = 0 uniformly in ε.

254

Proof. See section C.3 in Appendix C.The next theorem is a Lyapunov-based fixed point backstepping theorem for sta-

bilization of slightly nontriangular nonlinear systems. First, we need to make a defi-nition.

Definition 7.4.3. (residual of a function w.r.t. its ith argument) Consider a contin-uously differentiable function f(x1, . . . , xm) : Rn1 × . . .× Rnm → Rp. Define

h(x1, . . . , xm, a, δ) =

∫ 1

0

Dxif(x1, . . . , xm)|xi=a+sδ ds

and denoteh(x1, . . . , xm, a, δ) = res(i,a,δ)(f(x1, . . . , xm))

Read res(i,a,δ)(f(x1, . . . , xm)), the residual of f w.r.t. its ith argument with perturba-tion δ around xi = a. Then, the following decomposition of f is always possible

f(x1, . . . , xi + δ, . . . , xm) = f(x1, . . . , xm) + h(x1, . . . , xm, δ)δ

Based on this notation, we have f(x1, x2) = f(x1, 0) + h(x1, x2)x2 with h(x1, x2) =res(2,0,x2)(f(x1, x2)).

Theorem 7.4.5. Consider the perturbed nonlinear system in (7.101) where f0, g0 aresmooth functions satisfying f0(0) = 0, g0(0) = 0, g0(z, ξ1, 0, ε) = 0, and g0(z, ξ1, ξ2, 0) =0 (i.e.∃p0 ∈ C1 : g0 = p0(z, ξ1, ξ2, ε)ξ2ε). Suppose there exists a state feedbackξ1 = α1(z) with α1(0) = 0 such that for

z = f0(z, α1(z))

z = 0 is globally asymptotically and locally exponentially stable. Let V (z) be theLyapunov function for this system with

∂V (z)

∂zf0(z, α1(z)) < 0 , ∀z 6= 0

Decompose f0 as the following

f0(z, ξ1 + δ) = f0(z, ξ1) + h0(z, ξ1, δ)δ

and define

ψ2(z, ξ1, ξ2, ε) = −σ1(ξ1 − α1(z)) +∂α1(z)

∂z(f0(z, ξ1) + g0(z, ξ1, ξ2, ε))

− ∂V (z)

∂zh0(z, α1(z), ξ1 − α1(z))

(7.105)

Then, there exists an ε∗ > 0 such that for all 0 < ε < ε∗ the following hold

i) The fixed point equationξ2 = ψ2(z, ξ1, ξ2, ε) (7.106)

255

for all (z, ξ1) ∈ Kz × Kξ1 has a unique smooth solution ξ∗2 = α2(z, ξ1, ε) withα2(0, 0, ε) = 0.

ii) For a sufficiently large c1 = σ′1(0), the state feedback ξ2 = α2(z, ξ1, ε) semiglob-ally asymptotically stabilizes the origin for the (z, ξ1)-subsystem of (7.101).

iii) Fix an ε > 0 valid for (ii). Then, there exists a static state feedback u =α3(z, ξ1, ξ2, ε) in the following form

u = −σ2(ξ2−α2(z, ε))+∂α2(z, ε)

∂zf2(z, ξ2, ε)−

∂W (z)

∂zh2(z, α2(z, ξ1, ε), ε, ξ2−α2)

(7.107)that semiglobally asymptotically stabilizes the origin for the composite systemin (7.101) for a sufficiently large c2 = σ′2(0) > 0 (σ2 is a sigmoidal function)where

z = col(z, ξ1)

W (z) = V (z) +1

2(ξ1 − α1(z))

2

f2(z, ξ1, ξ2, ε) = col(f, ξ2)

h2(z, ξ2, ε, δ) = res(3,ξ2,δ)(f2(z, ξ1, ξ2, ε))

Proof. See section C.5 in Appendix C.Now, we present some examples of nontriangular nonlinear systems that can be

stabilized using theorems 7.4.4 and 7.4.5.

Example 7.4.5. (perturbed chain of integrators) Consider the following perturbedchain of integrators

x = Ax+Bu+ p(x, u, v)u = v

(7.108)

where p(x, u, v) is smooth and p(x, u, 0) = 0. In addition, A = aij with aij = 1 forj = i + 1 and zero otherwise, and B = (0, . . . , 0, 1)T . Define the following change ofscale and time-scale

zi = ε(n+1−i)xi, i = 1, . . . n

ξ1 = u

ξ2 = v/ε

and denoteζε := (ε−nz1, . . . , ε

−1zn, ξ1, εξ2)

Define a new perturbation p = (p1, . . . pn)T by its elements as the following

pi(z, ξ1, ξ2, ε) := εn−ipi(ζε).

256

In proposition 7.4.8, we show that if the perturbation p satisfies the following growthcondition, then the nonlinear system in (7.108) can be semiglobally asymptoticallystabilized.

Assumption 7.4.10. The perturbation p satisfies the following condition

εn−ipi(ζε)|ε=0 = 0 , ∀(z, ξ1, ξ2) ∈ Rn+2, i = 1, . . . , n

i.e. p(z, ξ1, ξ2, 0) = 0.

Proposition 7.4.8. Suppose Assumption 7.4.10 holds. Then, there exists a staticstate feedback law in the form of a fixed point controller that semiglobally asymptoti-cally stabilizes (x, u) = 0 for (7.108).

Proof. Apply the aforementioned change of scale and time-scale τ = εt. From nowon, (with a little abuse of notation) let (˙) denote d/dτ in new scale and time-scale.The dynamics of the system in new coordinates can be expressed as

z = Az +Bξ1 + p(z, ξ1, ξ2, ε)

ξ1 = ξ2(7.109)

Clearly, p(z, ξ1, 0, ε) = 0 and p(z, ξ1, ξ2, 0) = 0. Thus, system (7.109) is in the form(7.101). Noting that a linear state feedback ξ1 = Kz globally asymptotically stabilizesz = 0 with a quadratic Lyapunov function V (z) = zTPz where P is the solution ofthe Lyapunov equation

(A+BK)TP + P (A+BK) = −I

a semiglobally asymptotically stabilizing feedback for (7.108) can be obtained frompart (ii) of theorem 7.4.5.

Remark 7.4.5. The problem of global asymptotic stabilization of a perturbed chain ofintegrators with nontriangular nonlinear perturbation p in (7.108) that satisfies thecondition

∂p

∂v(x, u, 0) = 0

is addressed in [79] using a state feedback in the form of sum of saturations withrelatively small magnitudes. The work in [79] is an extension of the method of nestedsaturations due to Teel [102]. The main difference between the result of proposition7.4.8 and the work in [79] is that the condition that p(x, u, v) = p(x, u, v)v2 is notneeded in here. The next example clarifies this difference.

Example 7.4.6. Consider the following nonlinear system non-affine in control

x1 = x2 + x1v2 + uv + x2v

x2 = u+ x2v43 + x1v

3 + uv + x1x2uv4

u = v

(7.110)

257

the system is in the form of a chain of two integrators with a perturbation p = (p1, p2)T

where p1(x, u, v) = x1v2 + uv + x2v and p2(x, u, v) = x2v

43 + x1v

3 + uv + x1x2uv4.

This system is not in feedforward form and the perturbation term does not satisfy∂p

∂v(x, u, 0) = 0 for all (x, u). Thus, it can neither be controlled using Teel’s nested

saturations [102] , nor the method in [79]. Applying the change of scale and time-scale

z1 = ε2x1, z2 = εx2, ξ1 = u, ξ2 = v/ε, τ = εt

we getz1 = z2 + εz1ξ

22 + ε2ξ1ξ2 + εz2ξ2

x2 = ξ1 + ε13 z2ξ

432 + εz1ξ

32 + εξ1ξ2 + εz1z2ξ1ξ

42

ξ1 = ξ2

Apparently, this system satisfies all the conditions of proposition 7.4.8 and can besemiglobally stabilized using a static state feedback law in the form of a fixed pointcontroller.

7.4.5 Notion of Partial Semiglobal Asymptotic Stabilization

In this section, we introduce a new notion of stabilizability that is in a sense close tosemiglobal asymptotic stabilizability but rather weaker than that. A reason behinddefining this notion is that it is not always possible to transform any perturbed chainof integrators with an nontriangular structure into the form (7.101) using a change ofscale and time-scale as described in proposition 7.4.8. The following example clarifiesthis issue.

Example 7.4.7. Consider the following nonlinear system in non-triangular form

x1 = x2x2 = η1 + η1x

32η

22

η1 = η2

(7.111)

applying a change of scale and time-scale as

z1 = ε2x1, z2 = εx2, ξ1 = η1, ξ2 = η2/ε, τ = εt

one obtainsz1 = z2

z2 = ξ1 +1

εξ1z

32ξ

22

ξ1 = ξ2

Apparently, as ε → 0 the perturbation term1

εξ1z

32ξ

22 goes unbounded and does not

vanish. A modification of the change of scale and time-scale as the following

z1 = εx1, z2 = x2, ξ1 = η1/ε, ξ2 = η2/ε2, τ = εt

258

resolves this problem and we get

z1 = z2z2 = ξ1 + ε2ξ1z

32ξ

22

ξ1 = ξ2

(7.112)

which is clearly in the form (7.101). Now, the problem is that semiglobal stabilizationof (7.112) does not guarantee semiglobal stabilization of the original system in (7.111).The obtained region of attraction for (7.111) is in the form Kx × Kη1 where Kx islarger than the set of initial conditions x(0) and Kη1 is smaller than the set of initialconditions η1(0) due to the fact that η1 = εξ1 for a relatively small ε > 0.

This motivates us to define a new stabilizability notion that is weaker thansemiglobal stabilizability but still is rather similar to it.

Definition 7.4.4. (partial semiglobal stabilizability) Consider the following system

x1 = f1(x1, x2, u)x2 = f2(x1, x2, u)

(7.113)

We say the origin (x1, x2) = 0 for (7.113) is partially semiglobally asymptoticallystabilizable with respect to x1 if for any compact set of (partial) initial conditionsK1 3 x1(0), there exist a neighborhood A2 of x2 = 0 and a control law u such thatthe closed-loop system in (7.113) has a region of attraction A1 ×A2 with A1 ⊃ K1.

Remark 7.4.6. Apparently, partial semiglobal asymptotic stabilization is strongerthan local asymptotic stabilization.

Now, based on theorem 7.4.5, (x, η1) = 0 is partially semiglobally stabilizablew.r.t. x for (7.111) using a static state feedback.

7.5 Applications

In this section, we present applications of fixed point control laws to stabilization ofa number of challenging benchmark systems in robotics.

7.5.1 The Cart-Pole System with Small Length/Strong Grav-ity Effects

In this section, we analyze the effects of relatively small lengths or large gravityconstants in stabilization of the Cart-Pole system depicted in Figure 7-3. In bothcases, we provide semiglobally stabilizing fixed point state feedback laws for the Cart-Pole system. The cascade normal form for the Cart-Pole system is given in equation

259

mF

m , l , I

1

2 2 2

q

q1

2

Figure 7-3: The Cart-Pole System

(5.21) as the followingz1 = z2

z2 = ξ1(k1 + k2ξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(7.114)

where k1, k2 are the following positive constants

k1 = g ≥ 1, k2 =m2l

22 + I2m2l2

Assume the link l2 has uniform mass density. Thus, I2 = m2l22/3 and k2 =

43l2. Denote

ε = k2/k1 > 0 and apply the following change of scale

y1 =1

k1z1, y2 =

1

k1z2

Since k1 ≥ 1, under this change of scale, semiglobal stabilization in new coordinatesand original coordinates are equivalent (i.e. one implies the other). The dynamics ofthe Cart-Pole system in new scale can be rewritten as

y1 = y2

y2 = ξ1(1 + εξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(7.115)

Under either of the following conditions:

i) g À 1, i.e. relatively high gravity.

ii) l2 ¿ 1, i.e. relatively small lengths.

260

we obtain an ε that is relatively small, i.e. ε = k2/k1 ¿ 1. Clearly, this is inthe form (7.101) and the origin for this system can be semiglobally asymptoticallystabilized according to theorem 7.4.5. Here, we show that theorem 7.4.4 can be alsoused to obtain a stabilizing state feedback for this Cart-Pole system. Setting ξ2 = 0in the y-subsystem, the remaining system is a double-integrator that can be globallyasymptotically stabilized using a bounded control ξ1 = −σ(y1 + y2) (σ is a sigmoidalfunction) with a positive definite Lyapunov function

V (y) = φ(y1) + φ(y1 + y2) + y22

where φ(x) =∫ x

0σ(s)ds and V < 0 for all y 6= 0. To prove the boundedness of the

solutions of the y-subsystem, one can use a simpler Lyapunov function

V1(y) = φ(y1) +1

2y22

Because f0(y, ξ1) = (y2, ξ1)T , f0(y, ξ1 + δ) = f0(y, ξ1) + h0δ with h0 = (0, 1)T . Thus,

we have∂V1(y)

∂yh0 =

∂V1(y)

∂y2= y2

or (independent of δ)

|∂V1(y)∂y

h0| ≤ |y2| <√2V1(y)

12 ,∀y 6= 0

and condition (7.103) of theorem 7.4.4 is satisfied with k =√2, any r0 > 0, and

λ = 12. Thus, a semiglobally stabilizing state feedback that makes no explicit use of

any Lyapunov functions can be obtained using the cascade backstepping procedurein theorem 7.4.4). Figure 7-4, shows the simulation results for the cart-pole systemstarting at the initial state (1, 1, π/3, 0) for the choice of ξ1 = − tanh(y1 + y2).

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

time (sec)

posi

tion/

angl

e

Figure 7-4: The state trajectories of the Cart-Pole system.

261

7.5.2 The Cart-Pole System with Large Length/Weak Grav-ity Effects

Consider the normal form of the Cart-Pole system in (7.114) and apply the followingchange of scale and time-scale given in [67]

y1 = ε2z1, y2 = εz2, η1 = ξ1, η2 = ξ2/ε, w = u/ε2, τ = εt

The dynamics of the system in new coordinates and time-scale is in the form

y1 = y2,

y2 = η1(k1 + k2ε2 η22

(1 + η21)32

),

η1 = η2,η2 = w

(7.116)

Semiglobal stabilization for (7.116) is very similar to the one presented for system(7.115) and will be repeated here.

Now, without loss of generality assume I2 = 0. Thus, k2 = l2. This case has aninteresting physical interpretation. Normalizing the units of y1, y2 by k1 = g as

x1 =1

gy1, x2 =

1

gy2

and setting

ε =

√

k1k2

=

√

g

l2= ω0

(where ω0 is the natural frequency of a pendulum of length l2) we obtain the normalform of the Cart-Pole system with normalized units as

x1 = x2,

x2 = η1(1 +η22

(1 + η21)32

),

η1 = η2,η2 = w

(7.117)

In either of the following cases:

i) g ¿ 1, i.e. relatively weak gravity,

ii) l2 À 1, i.e. relatively large length;

the obtained ε (or the natural frequency ω0) is relatively small, i.e. ω0 = ε¿ 1. Thismeans the solutions in scaled coordinates and time scale are relatively slow. Observethat the conditions of large length/weak gravity are in complete opposition with thecase considered in section 7.5.1 with small length/large gravity effects.

262

7.5.3 The Rotating Pendulum

Consider the Rotating Pendulum as shown in Figure 7-5. The nontriangular normalform for the Rotating Pendulum is given in (5.24) as the following

z1 = z2

z2 = ξ1(g

L1

+l2L1

1√

1 + ξ21(z2 −

m22

m2l2L1

ξ2√

1 + ξ21)2 +

m22

m2l2L1

ξ22

(1 + ξ21)32

)

ξ1 = ξ2ξ2 = v

(7.118)

where m22 = m2l22 + I2. Without loss of generality assume I2 = 0, or m22 = m2l

22

x

y

z

1

2q

m , L , I

m , L , I2

1 1

2

q1

2

Figure 7-5: The Rotating Pendulum

(otherwise, I2 = k0m2l22 with some k0 > 0 and the line of analysis is very similar).

Thus, equation (7.118) can be rewritten as

z1 = z2

z2 = ξ1[k1 + k21

(1 + ξ21)12

(z2 − k2ξ2

(1 + ξ21)12

)2 + k2ξ22

(1 + ξ21)32

]

ξ1 = ξ2ξ2 = v

(7.119)

where

k1 =g

L1

, k2 =l2L1

Remark 7.5.1. Let us define

y1 =1

k1z1, y2 =

1

k1z2, ε =

k2k1

where k2 is constant and k1 À 1. This change of scale transforms (7.119) into the

263

following nonlinear system

y1 = y2

y2 = ξ1[1 +k2ε

1

(1 + ξ21)12

z22 − 2ε2z2ξ2

(1 + ξ21)+ ε(1 + k22)

ξ22

(1 + ξ21)32

]

ξ1 = ξ2ξ2 = v

Apparently, the last equation is not a well-defined slightly nontriangular system. Sinceas ε→ 0, the coefficient of the term containing z22 goes unbounded, i.e. k2/ε→∞.

Now, consider the following change of scale and time scale

y1 = εz1, y2 = z2, η1 = ξ1/ε, η2 = ξ2/ε2, w = v/ε3, τ = εt (7.120)

From (7.119), we obtain

y1 = y2

y2 = η1[k1 +k2

(1 + ε2η21)12

(y2 −ε2k2η2

(1 + ε2η21)12

)2 +ε4k2η

22

(1 + ε2η21)32

]

η1 = η2η2 = w

(7.121)

but

k2

(1 + ε2η21)12

= k2(1 +1

(1 + ε2η21)12

− 1) = k2 −k2ε

2

(1 + ε2η21)12 [1 + (1 + ε2η21)

12 ]

Hencey1 = y2y2 = η1(k1 + k2y

22) + g(y2, η1, η2, ε) + ∆(y2, η1, ε)

η1 = η2η2 = w

(7.122)

where

g(y2, η1, η2, ε) = −2ε2k22y2η1η2(1 + ε2η21)

+ε4k2(1 + k22)η

22

(1 + ε2η21)32

∆(y2, η1, ε) = − ε2η1y22

(1 + ε2η21)12

Notice that g(y2, η1, η2, ε) and ∆(y2, η1, ε) vanish at ε = 0. In addition, the perturba-tion ∆ has an upper bound that vanishes uniformly in η1 due to

|∆(y2, η1, ε)| ≤ εk2y22

The nonlinear system in (7.122) without the perturbation ∆ is in the form (7.101) andsatisfies the conditions of theorem 7.4.5. Thus, it can be semiglobally asymptotically

264

stabilized using the choice of η1 = α(y) = −σ(y1 + y2) (σ is a sigmoidal function).This state feedback renders y = 0 globally asymptotically stable for

y1 = y2y2 = η1(k1 + k2y

22)

(7.123)

The following Lyapunov function

V (y) = φ(y1) + φ(y1 + y2) +1

k2log(1 +

k2

k1y22) (7.124)

with φ(x) =∫ x

0σ(s)ds is a valid Lyapunov function for the closed-loop system in

(7.123) satisfying V < 0 for all y 6= 0. Now, semiglobal asymptotic stabilizationof (7.122) with the high-order perturbation ∆ can be shown using an elementaryargument. It is important to notice that semiglobal asymptotic stabilization of theorigin for (7.122) implies partial semiglobal asymptotic stabilization of the origin forthe dynamics of the Rotating Pendulum in the original coordinates. Figure 7-6 showsthe simulation results for the rotating pendulum with initial condition (π/3, 0, π/4, 0).

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

q1 /

q2

time (sec)

q1q2

Figure 7-6: The state trajectories for the Rotating Pendulum.

7.5.4 The Generalized Beam-and-Ball System

The generalized beam-and-ball system is a partially-linear nontriangular cascade non-linear system in the following form

x1 = x2x2 = g sin(η1) + xν1η

22 , ν ≥ 1

η1 = η2η2 = u

(7.125)

265

for ν = 1, this system is the same as the famous beam-and-ball system that has beenextensively studied and used as an example by many researchers in the past decade[34, 102, 80, 79]. Here, we consider two cases: i) 1 ≤ ν < 3, and ii) 3 ≤ ν <∞. Forthe sake of simplicity assume g = 1.Case (i): Applying the change of scale and time scale

z1 = εx1, z2 = x2, ξ1 = η1/ε, ξ2 = η2/ε2, w = v/ε3, τ = εt

In the new scale, we obtain

z1 = z2z2 = ξ1 + ε2O(|ξ1|3) + ε(3−ν)zν1ξ

22

ξ1 = ξ2ξ2 = v

(7.126)

where

ξ1 + ε2O(|ξ1|3) =sin(εξ1)

ε, ε > 0

Clearly, the origin for system (7.126) can be semiglobally asymptotically stabilizedusing a state feedback in the form of a fixed point controller. This way, the origin forsystem (7.125) can be only partially semiglobally stabilized w.r.t. (x1, x2).Case (ii): It is easy to see that using the change of scale and time-scale

z1 = x1, z2 = x2/ε, ξ1 = η1/ε2, ξ2 = η2/ε

3, w = v/ε4, τ = εt

the dynamics of the system transforms into

z1 = z2z2 = ξ1 + ε5O(|ξ1|3) + ε5zν1ξ

22

ξ1 = ξ2ξ2 = v

(7.127)

and the origin for the system (7.125) can be partially semiglobally stabilized w.r.t.x1. The simulation results for the generalized beam-and-ball system are shown inFigure 7-7.

266

0 5 10 15−0.6

−0.4

−0.2

0

0.2

0.4

angl

e of

the

beam

time (sec)

v=1v=2v=3

0 5 10 15−1

0

1

2

3

4

time (sec)

posi

tion

of th

e ba

ll

v=1v=2v=3

Figure 7-7: The state trajectories of the generalized beam-and-ball system with initialcondition (1, 0,−π/6, 0) for g = −9.8 and ν = 1, 2, 3.

267

Chapter 8

Conclusions

In this chapter, we summarize the results of this thesis and make concluding remarks.This thesis creates a bridge between control of mechanical systems and nonlinear con-trol theory by providing transformations in explicit forms that take an underactuatedmechanical system to a cascade nonlinear system with structural properties. Due tothe special partially-linear cascade structure of the obtained normal forms for under-actuated systems, control design for the original (possibly) high-order underactuatedsystem reduces to control of a low-order nonlinear part of the transformed system.The main focus of this thesis was on identifying broad classes of underactuated sys-tems with kinetic symmetry where such a transformation for them could be found inclosed-form. It turned out that the majority of obtained normal forms for underac-tuated systems are in nontriangular forms. Since control of nonlinear systems withnontriangular structures was an important open problem, a great deal of effort inthis thesis was devoted to the development of control design tools for stabilization ofnontriangular nonlinear systems in chapter 7.

Reduction of underactuated systems with nonholonomic velocity constraints andsymmetry is addressed in chapter 6. As a consequence, the corresponding class of anunderactuated system with nonholonomic velocity constraints is determined by theclass of its reduced system which is a well-defined underactuated (or fully-actuated)mechanical system itself.

The theoretical results of this thesis are applied to challenging control problemsfor a variety of robotics and aerospace systems. Almost for all examples, detailedcontrol design and simulation results are provided. Among these examples are tra-jectory tracking for a flexible-link arm, automatic calculation of the differentiallyflat outputs of the VTOL aircraft, (almost) global asymptotic/exponential trackingof feasible trajectories of an autonomous helicopter, global exponential ε-trackingand stabilization for a two-wheeled mobile robot, global stabilization of the Acrobot,global asymptotic stabilization of the 2D and 3D Cart-Pole systems to an equilib-rium point, and semiglobal asymptotic stabilization of the Rotating Pendulum to anequilibrium point.

268

8.1 Summary

The summary of this thesis is as follows.In chapter 1, an introduction was provided to control of mechanical systems, non-

linear control theory, and underactuated mechanical systems with their applications.In chapter 2, some background on different types of mechanical systems and basic

notions in mechanics are given. These mechanical notions include classical symmetry,kinetic symmetry, shape variables, conservation laws, nonholonomicity, and flatness.

In chapter 3, the dynamics of underactuated mechanical system is defined as theforced Euler-Lagrange equations of motion. Several examples of underactuated sys-tems were presented in chapter 3. For each example, only the inertia matrix and thepotential energy of the system are given. This allows the reader to clearly observethe kinetic symmetry property in each example and identify its corresponding shapevariables. In addition, different methods for partial feedback linearization of underac-tuated mechanical systems are introduced. This includes collocated and noncollocatedfeedback linearization methods as special cases. Finally, the problem of decoupling ofthe dynamics of the actuated and unactuated subsystems of a class of underactuatedsystems after partial feedback linearization is addressed. The decoupling is achievedusing a global change of coordinates so that after transformation, the obtained cascadesystem was in Byrnes-Isidori normal form with a set of double-integrator linear part.This result then was applied to complete classification of underactuated mechanicalsystems with two degrees of freedom and kinetic symmetry to three classes. By “com-plete”, we mean “all” such underactuated systems. It was proven that these threeclasses of underactuated systems could be transformed into three types of cascadenormal forms. Namely, nonlinear systems in strict feedback form, strict feedforwardform, and nontriangular quadratic form. All the transformations are performed usingchange of coordinates in closed-form.

In chapter 4, some of the main contributions of this thesis are presented. Re-duction and classification of high-order underactuated systems is certainly the mostimportant result of this thesis. The number of obtained classes are eight with certainclasses which contain two subclasses. In addition to classification of underactuatedsystems a correspondence is established between classes of underactuated systemsand cascade systems with triangular and nontriangular structural properties. Thiscorrespondence allows control design for the original underactuated system. Themain tools in stabilization of underactuated systems with triangular normal formsare backstepping and forwarding methods. In contrast, the main approach in con-trol of nontriangular nonlinear systems is a fixed-point based backstepping procedurethat is introduced in chapter 7. To deal with reduction of underactuated systemswith non-integrable momentums, we introduced a procedure call momentum decom-position. This procedure represents a non-integrable momentum as the sum of anintegrable momentum and an error momentum term which is non-integrable. Mo-mentum decomposition was applied to reduction of multi-link underactuated robotsand calculation of weakly flat outputs for an accurate model of an autonomous heli-copter in chapter 5. For Class-I,II,III underactuated systems both the transformationsand the obtained normal forms are physically meaningful. This allows construction

269

of physically meaningful Lyapunov functions for stability and robustness analysis ofthese normal forms. The fact that “the core reduced system for (almost) all classes ofunderactuated systems is itself a Lagrangian mechanical system with shape variablesas the control”, creates a unity among different classes of underactuated systems.Global asymptotic stabilization of this core reduced system is addressed in chapter 4.

In chapter 5, several examples of underactuated systems are provided with detailedcontrol design and simulation results. A method is introduced for aggressive swing-upcontrol using bounded input for special classes of underactuated systems that involveinverted pendulums. Furthermore, trajectory tracking for a flexible one-link robotand an autonomous helicopter is addressed in chapter 5. The connection betweendifferential flatness and a decoupling change of coordinates for underactuated systemswith input coupling are demonstrated for the case of the VTOL aircraft and anaccurate model of an autonomous helicopter.

In chapter 6, it was shown that underactuated systems with nonholonomic velocityconstraints and symmetry can be represented as the cascade of the constraint equationand a lower-order underactuated (or fully-actuated) mechanical system. Dependingon the specific class of the obtained reduced subsystem, the associated class of theoriginal nonholonomic system is determined. This framework allowed us to presenta systematic reduction of a snake-type robot called the snake-board. It was demon-strated that the kinematic model of the snake-board with anti-symmetric heading-angles of the front and back wheels is diffeomorphic to the kinematic model of a car.Moreover, the kinematics of the snake-board can be transformed into a first-orderchained-form nonholonomic system. The notations of ε-stabilization and ε-trackingare formally defined in chapter 6. Furthermore, global exponential ε-tracking andε-stabilization is achieved for a two-wheeled mobile robot.

In chapter 7, the problem of stabilization of nonlinear systems in nontriangularnormal forms is addressed for several important special cases. This includes vectorfields that are affine, quadratic, linear-quadratic, and non-affine in variables (ξ1, ξ2).In addition, stabilization of slightly nontriangular systems that are small nonlinearperturbations of systems in strict feedback forms is addressed. This has applicationsin stabilization/tracking for the accurate model of a helicopter, flexible link-robots,and in general underactuated systems with unactuated shape variables and/or inputcoupling.

8.2 Future Work

This section provides potential future directions of research in continuation of thiswork.

8.2.1 Hybrid Lagrangian Systems

From normal form of different classes of underactuated systems, it is clear that theshape variables have a fundamental role as the main control inputs of the reducedsubsystems. For example, in the case of class-I underactuated systems, the reduced

270

system is itself a Lagrangian system where its Lagrangian function is parameterizedwith the shape variables. One potential way to control the system is by switchingbetween a finite set of the values of the shape vector qs. Let Θ := θ1, . . . , θk denotea finite set of shape values θj ∈ Qs, j = 1, . . . , k where Qs denotes the shape manifold.Now, consider the following nonlinear switching system

qr = m−1r (qs)pr

pr = gr(qr, qs)(8.1)

where qs = v(t) and v(t) : R≥0 → Θ is a piece-wise constant control input. A stabi-lizing control v(t) is a switching signal that appropriately puts one of the followingnonlinear dynamics in effect at any time t ≥ 0

qr = m−1r (θj)pr

pr = gr(qr, θj)(8.2)

such that (qr, pr) = 0 is asymptotically stable for (8.1). Setting z = col(qr, pr), thisnonlinear switching system can be rewritten as

z = fj(z), z = col(qr, pr) (8.3)

According to [52], if there exists a common Lyapunov function V (z) such that

∇V (z) · fj(z) < 0, ∀z 6= 0, j ∈ 1, . . . , k

Then, given any arbitrary switching signal v(t) : R≥0 → Θ, z = 0 is globally asymp-totically stable for the closed-loop system in (8.1). The existence of such a commonLyapunov function is an extremely restrictive condition.

Consider a second scenario and assume gr(qr, qs) = gr(qs) and mr(qs) = mr isa constant. In addition, gr(−qs) = −gr(qs) (i.e. gr(0) = 0) and ∃θ0 : gr(θ0) > 0.Setting Θ = 0,±θ0, one possible stabilizing controller for the double-integrator

qr = m−1r pr

pr = u(8.4)

with u = gr(v) is a time-optimal controller that takes values in the set U = 0,±uwhere u = gr(θ0) is a positive constant. Roughly speaking, stabilization using such apiece-wise constant control law is equivalent to stabilization of the reduced system byswitching between a finite set of possible shapes which the system could take. We callthis approach control via changing the shape. The method of “control via changingthe shape” is in complete agreement with natural locomotion in a swimming fish, aflying bird, and a walking human being . In the sense that in all the aforementionedcases, locomotion is achieved via changing the physical shape of the system. A formaland systematic analysis of the method of “changing the shape” for locomotion andcontrol purposes is needed.

271

8.2.2 Uncertainty and Robustness

Control of underactuated systems is rather challenging even in lack of any paramet-ric uncertainties. However, it is important to formulate and address robust stabi-lization/tracking problems for classes of underactuated systems where a satisfactorynonlinear control design is available for them in lack of uncertainties (e.g. Class-Isystems). Normal forms for underactuated systems offer a set of structured nonlinearsystems with a common core reduced system that can be exploited for constructionof Lyapunov functions. The obtained Lyapunov functions could be later employed inrobustness analysis of underactuated systems with parametric uncertainties.

8.2.3 Output Feedback Stabilization and Tracking

The majority of control laws presented in this work are in the form of full state feed-back. The exceptions to this statement include trajectory tracking for an autonomoushelicopter, a mobile robot, and a flexible-link robot. From theoretical point of view, itis important to design nonlinear observers and develop output feedback stabilizationlaws for control of underactuated systems (wherever possible). For example, in thenormal form of Class-I underactuated systems, the equation of the zero-dynamics isglobally linear in the velocity (or momentum) z2. This promises the existence of anoutput stabilization feedback for Class-I underactuated systems (under appropriatetechnical conditions). The same argument applies to Class-III systems and Class-IIsystems without quadratic terms in z2.

8.2.4 The Effects of Bounded Control Inputs

In real-life most of actuators are subject to limitations in their actuation power. Aformal analysis of nonlinear systems with saturated nonlinear state feedback similarto the ones presented in sections 5.4.2 and 5.5.1 has yet to be developed.

8.2.5 Small Underactuated Mechanical Systems (SUMS)

Due to the availability of Micro-electro-mechanical systems (MEMS), it is possibleto build underactuated micro-robots and physically test the effects of the relativelysmall scale of the components of a robot in control design for underactuated systems.This leads to control design for nonlinear systems with slightly nontriangular normalforms. Control of SUMS is very challenging because of both the limitation of theactuation power and the presence of parametric uncertainties that are inevitable indealing with semiconductor devices.

8.2.6 Underactuated Systems in Strong/Weak Fields

The behavior of the zero-dynamics of underactuated systems in a strong potentialfield is dominant with the gravity terms, i.e. any nontriangular normal forms for suchunderactuated systems are slightly nontriangular. In contrast, in zero-gravity the

272

behavior of a Class-I underactuated system is the same as a mechanical system withfirst-order nonholonomic constraints. This suggests a future analysis for possible ex-tensions of some of the existing nonlinear control design approaches for nonholonomicsystems to control methods applicable to underactuated/nonholonomic mechanicalsystems in low-gravity fields. This is of great importance in space exploration appli-cations.

273

Appendix A

Dynamics of Mechanical Systems

A.1 Underactuated Mechanical Systems with Two

DOF and Kinetic Symmetry

In this section, we present the closed-form Euler-Lagrange equations of motion for amechanical system with two degrees of freedom (q1, q2) and kinetic symmetry withrespect to q1. This class of simple Lagrangian systems can be charachterize with thefollowing Lagrangian function

L(q, q) = 1

2

[

q1q2

]T [m11(q2) m12(q2)m21(q2) m22(q2)

] [

q1q2

]

− V (q1, q2) (A.1)

where V (q1, q2) is the potential energy of the system. From forced Euler-Lagrangeequations of motion

d

dt

∂L∂q− ∂L∂q

=

[

F1(q)F2(q)

]

τ (A.2)

where τ ∈ R is the control input and for all q ∈ Q either F1(q) 6= 0, or F2(q) 6= 0. Forthe special cases where F (q) = (1, 0)T or F (q) = (0, 1), we say the system has non-interacting inputs. Otherwise, we say the system has input-coupling. The equtionsof motion in (A.2) can be explicitly written as

m11(q2)q1 +m12(q2)q2 +m′11(q2)q1q2 +m′

12(q2)q22 + g1(q1, q2) = F1(q)τ

m21(q2)q1 +m21(q2)q2 − 12m′

11(q2)q21 +

12m′

22(q2)q22 + g2(q1, q2) = F2(q)τ

(A.3)

where ′ denotes d/dq2 and the gravity terms are given by

gi(q1, q2) =∂V (q1, q2)

∂qi, i = 1, 2

274

A.2 A Three-link Planar Robot

Consider a planar three-link robot arm with revolute joins depicted in Figure A-1.The mass, inertia, length, and length of the center of mass of the ith link is denotedby mi, Ii, Li, li, respectively. To calculate the kinetic energy of this system we followa general method used in [95] which decomposes the kinetic energy as the sum oftranslational and rotational kinetic energies, i.e. K = Kt + Kr. The translationalkinetic energy is given by

Kt =1

2vTmv

where v = (v1, v2, v3)T is the vector of linear velocities of the center of mass of the

links and m = diag(m1,m2,m3). The coordinates of the center of mass of three links

x

y

l L1

1

θ

θ

3

2

1

θ

, m , 1

I1

τ

τ3

2

τ 1

Figure A-1: A planar three-link robot arm.

are as follows

x1 =

[

l1 sin θ1l1 cos θ1

]

x2 =

[

L1 sin θ1 + l2 sin(θ1 + θ2)L1 cos θ1 + l2 cos(θ1 + θ2)

]

x3 =

[

L1 sin θ1 + L2 sin(θ1 + θ2) + l3 sin(θ1 + θ2 + θ3)L1 cos θ1 + L2 cos(θ1 + θ2) + l3 cos(θ1 + θ2 + θ3)

]

275

Thus, denoting θ = (θ1, θ2, θ3)T , we have

v1 = J1θ =

[

l1 cos θ1 0 0−l1 sin θ1 0 0

]

θ1θ2θ3

v2 = J2θ =

[

L1 cos θ1 + l2 cos(θ1 + θ2) l2 cos(θ1 + θ2) 0−L1 sin θ1 − l2 sin(θ1 + θ2) −l2 sin(θ1 + θ2) 0

]

θ1θ2θ3

v3 = J3θ =

[

L1 cos θ1 + L2 cos(θ1 + θ2) + l3 cos(θ1 + θ2 + θ3)−L1 sin θ1 − L2 sin(θ1 + θ2)− l3 sin(θ1 + θ2 + θ3)

L2 cos(θ1 + θ2) + l3 cos(θ1 + θ2 + θ3) l3 cos(θ1 + θ2 + θ3)−L2 sin(θ1 + θ2)− l3 sin(θ1 + θ2 + θ3) −l3 sin(θ1 + θ2 + θ3)

]

θ1θ2θ3

This gives

Kt =1

2θT (JT1 m1J1 + JT2 m2J2 + JT3 m3J3)θ

Now, let us calculate the rotational kinetic energy Kr. Denoting the unit vectororthogonal to the (x, y)-frame by e3 = (0, 0, 1)T , the angular velocities of the links inreference frame are as follows

ω1 = θ1e3ω2 = (θ1 + θ2)e3ω3 = (θ1 + θ2 + θ3)e3

Hence

Kr =1

2ωT1 I1ω1 +

1

2ωT2 I2ω2 +

1

2ωT3 I3ω3 =

1

2θT Iθ

where

I =

I1 + I2 + I3 I2 + I3 I3I2 + I3 I2 + I3 I3I3 I3 I3

Therefore, the total kinetic energy is given by

K =1

2θT (I + JT1 m1J1 + JT2 m2J2 + JT3 m3J3)θ

The inertia matrix of the system takes the following form

M =M(θ2, θ3) =

m11(θ2, θ3) m12(θ2, θ3) m13(θ2, θ3)m21(θ2, θ3) m22(θ3) m23(θ3)m31(θ2, θ3) m32(θ3) m33

276

with elements

m11(θ2, θ3) = I1 + I2 + I3 +m1l21 +m2(L

21 + l22) +m3(L

21 + L2

2 + l23)+ 2(m2L1l2 +m3L1L2) cos θ2+ 2m3L1l3 cos(θ2 + θ3) + 2m3L2l3 cos θ3

m12(θ2, θ3) = m21 = I2 + I3 +m2l22 +m3(L

22 + l23) + (m2L1l2 +m3L1L2) cos θ2

+ 2m3L2l3 cos θ3 +m3L1l3 cos(θ2 + θ3)m13(θ2, θ3) = m31 = I3 +m3l

23 +m3L1l3 cos(θ2 + θ3) +m3L2l3 cos θ3

m22(θ3) = I2 + I3 +m2l22 +m3(L

22 + l23) + 2m3L2l3 cos θ3

m23(θ3) = m32 = I3 +m3l23 +m3L2l3 cos θ3

m33 = I3 +m3l23

(A.4)The potential energy of this three-link robot is given by

V (θ) = (m1l1 +m2L1 +m3L1)g cos θ1 + (m2l2 +m3L2)g cos(θ2 + θ3)+ m3l3g cos(θ1 + θ2 + θ3)

Notice that the potential energy V (θ) is identically zero for a three-link robot in ahorizontal plane. The Euler-Lagrange equations of motion for this system are as thefollowing

d

dt

∂L∂θ− ∂L∂θ

= τ

where τ = (τ1, τ2, τ3)T is the input torque (control) and the Lagrangian L is given by

L(θ, θ) = 1

2θTM(θ2, θ3)θ − V (θ)

Clearly, θ1 is the external variable and θ2, θ3 are the shape variables of this robot.The following lemma is useful and shows that an three-link robot with one or twoactuators is an example of an underactuated system with non-integrable normalizedmomentums. It is important to notice that this property does not hold for a planartwo-link robot and both normalized momentums of the Acrobot (or the Pendubot)are integrable.

Lemma A.2.1. All three normalized momentums

πi = m−1ii (θ)

∂L∂θi

, i = 1, 2, 3

of a planar three-link robot are non-integrable, i.e. 6 ∃h : h = π1.

Proof. First, we prove

π1 = θ1 +m12(θ2, θ3)

m11(θ2, θ3)θ2 +

m13(θ2, θ3)

m11(θ2, θ3)θ3

is non-integrable. For the purpose of contradiction, assume π1 is integrable. Then,

277

π1 − θ1 is trivially integrable and the following one-form must be exact

ω1 =m12(θ2, θ3)

m11(θ2, θ3)dθ2 +

m13(θ2, θ3)

m11(θ2, θ3)dθ3

A sufficient and necessary condition that ω1 is exact is

∂

∂θ3

m12(θ2, θ3)

m11(θ2, θ3)=

∂

∂θ2

m13(θ2, θ3)

m11(θ2, θ3)

for all (θ2, θ3) ∈ S1×S1. A rather lengthy but straightforward calculation shows thatthe last equality does not hold for all θ2, θ3 and therefore π1 is non-integrable. In asimilar way but after a short calculation, it can be shown that π2, π3 are non-integrablenormalized momentums as well.

Proposition A.2.1. A three-link robot in a horizontal plane or zero gravity (i.e.V (θ) = 0) with two actuators at (θ2, θ3), a passive joint θ1, and zero initial velocity isan underactuated mechanical system with a nonholonomic velocity constraint π1 = 0.In addition, there is an invertible change of control

v = α(θ)(τ2, τ3)T + β(θ, θ)

that transforms the dynamics of the system into a driftless nonlinear system aug-mented with an integrator

θ1 = g2(θ2, θ3)u2 + g3(θ2, θ3)u3θ2 = u2θ3 = u3u = v

(A.5)

where u = (u2, u3)T and

gi = −m1i(θ2, θ3)

m11(θ2, θ3), i = 2, 3

Proof. Assume τ1 = 0, then from Euler-Lagrange equations of motion

d

dt

∂L∂θ1

=∂L∂θ1

= 0

Thus m11π1 is a conserved quantity, i.e. m11π1 = 0,∀t ≥ 0. Since m11(θ2, θ3) >0, based on lemma A.2.1, π1 = 0 is a nonholonomic (i.e. non-integrable) velocityconstraint. The normal form in (A.5) is a direct result of collocated partial feedbacklinearization and definition of π1.

Theorem A.2.1. The inertia matrix of an n-link planar robot with n revolute joins

278

(θ1, . . . , θn) (as shown in Figure A-1 for n = 3) has the following structure

M(θ) =

m11(θ2, . . . , θn) m12(θ2, . . . , θn) . . . m1n(θ2, . . . , θn)m21(θ2, . . . , θn) m22(θ3, . . . , θn) . . . m2n(θ3, . . . , θn)...

.... . .

...mn1(θ2, . . . , θn) mn2(θ3, . . . , θn) . . . mnn

In other words, M(θ) = mij(θs, . . . , θn) where s = mini, j + 1 for i, j < n andmnn is constant.

Proof. The proof is by direct calculation similar to the one applied to a three-linkrobot. However, it is easier to prove this by induction. One assumes that the theoremis true for k = 2 and k = n > 2 links and adds an (n+1)th link to the end point of thenth link. It is rather straightforward to prove that the theorem holds for k = n + 1.This proves that the theorem holds for and n ≥ 2.

A.3 Two-link Planar Robots: the Acrobot and the

Pendubot

A two-link planar robot with joint angles θ1, θ2 can be viewed as a special case of athree-link planar robot with l3 = 0 (and thus m3 = I3 = 0)(see Figure A-1). In otherwords, the inertia matrix for a two-link planar robot with revolute joins is given by

m11(θ2) = I1 + I2 +m1l21 +m2(L

21 + l22) + 2m2L1l2 cos(θ2)

m12(θ2) = I2 +m2l22 +m2L1l2 cos(θ2)

m22 = I2 +m2l22

(A.6)

and m21(θ2) = m12(θ2). The potential energy of a two-link robot in a vertical planeis in the form

V (θ1, θ2) = (m1l1 +m2L1)g cos(θ1) +m2l2g cos(θ1 + θ2) (A.7)

Based on equation (A.3), the Euler-Lagrange equations of motion for this two-linkrobot can be expressed as

m11(θ2)θ1 +m12(θ2)θ2 + h1(θ, θ) = F1τ

m21(θ2)θ1 +m22(θ2)θ2 + h2(θ, θ) = F2τ(A.8)

where

h1(θ, θ) = −m2L1l2 sin(θ2)(2θ1θ2 + θ22) + g1(θ1, θ2)

h2(θ, θ) = m2L1l2 sin(θ2)θ21 + g2(θ1, θ2)

g1(θ1, θ2) = −(m1l1 +m2L1)g sin(θ1)−m2gl2 sin(θ1 + θ2)

g2(θ1, θ2) = −m2gl2 sin(θ1 + θ2)

279

Depending on whether θ1 is actuated and θ2 is unactuated, or vice versa; two impor-tant examples of underactuated systems with two degrees of freedom arise.

Definition A.3.1. (Acrobot and Pendubot) The underactuated system in (A.8) iscalled Acrobot if (F1, F2) = (0, 1). It is called Pendubot if (F1, F2) = (1, 0).

Defining the state vector x = (q1, q1, q2, q2), the dynamics of the the system in(A.8) can be expressed as a nonlinear system affine in control

x = f(x) + gi(x)u (A.9)

where u = τ and i = a, p corresspond to the Acrobot and the Pendubot, respectively.Apprently, the drift vector field f(x) is the same for the Acrobot and the Pendubot.However, the control vector fields g1, g2 are different and can be expressed as

ga(q2) =1

D(q2)

0−m12(q2)

0m11(q2)

, gp(q2) =1

D(q2)

0m22(q2)

0−m21(q2)

where D(q2) = det(M(q2)) = m11m22 −m12m21 > 0.

Remark A.3.1. The distribution generated by nested Lie brackets of f(x), ga(x) isnot the same as the distribution generated by f(x), gp(x); unless gp(x) = λga(x) fora constant λ ∈ R. But this means m21 = −λm11 and m22 = −λm12, or equivalentlyD(q2) = 0 which contradicts the property D(q2) > 0. Therefore, the controllabilityproperties of the Acrobot and the Pendubot are different.

A.4 The Beam-and-Ball System

The beam-and-ball system consists of a beam and a ball with radius r on it (seeFigure A-2). Let d ≥ 0 denote the distance between the center of the mass of theball and the beam (d = r in Figure 5-24). In the current literature [34, 102, 80, 79],what is known as the beam-and-ball system corresponds to the special case whered = 0. We refer to this case as the conventional beam-and-ball system and to the casewhere d > 0 as the beam-and-ball system. The following treatment applies to theboth cases of d = 0 and d > 0. Let I1,m, I2 denote the inertia of the beam, the massof the ball , and the inertia of the ball, respectively. The position and velocity of thecenter of mass of the ball is

xc = (x cos(θ)− d sin(θ), x sin(θ) + d cos(θ))vc = (v cos(θ)− x sin(θ)ω − d cos(θ)ω, v sin(θ) + x cos(θ)ω − d sin(θ)ω)

where x = v and θ = ω. Thus, the translational part of the kinetic energy is given by

Ktrans =1

2mv2c =

1

2m(v2 + (x2 + d2)ω2 − 2dωv)

280

dxr

I 1

θτ

m , I 2

Figure A-2: The Beam-and-Ball System.

The rotational kinetic energy of the beam-and-ball system is

Krot =1

2I1ω

2 +1

2I2(

v

r)2

Hence, the total kinetic energy K = Ktrans +Krot equals

K =1

2(I1 +m(x2 + d2))ω2 −mdωv + 1

2λmv2

where

λ = 1 +I2mr2

> 1

is a constant. The potential energy of the beam-and-ball system is given by

V (x, θ) = mg(x sin θ + d cos θ)

Therefore, the Lagrangian of the beam-and-ball system is as the following

L =1

2

[

θx

]T [I1 +m(x2 + d2) −md

−md mλ

] [

θx

]

−mg(x sin(θ) + d cos(θ)) (A.10)

Based on (A.10), the elements of the inertia matrix of the beam-and-ball system are

m11(x) = I1 +m(x2 + d2)m12 = m21 = −mdm22 = mλ

inertia matrix of the beam-and-ball system only depends on the position of the ballx. This means that the position of the ball is the shape variable of the beam-and-ball system which is unactuated. Therefore, the beam-and-ball system is a Class–IIunderactuated system. The Euler-Lagrange equations of motion for the beam-and-ballsystem with d > 0 is given by

[I1 +m(x2 + d2)]θ − mdx+ 2mxxθ +mg(x cos(θ)− d sin(θ)) = τd−mdθ + mλx−mxθ2 +mg sin(θ) = 0

(A.11)

281

While for the conventional beam-and-ball system with d = 0, the Euler-Lagrangeequations take the following form

(I1 +mx2)θ + 2mxxθ +mgx cos(θ) = τ0mλx−mxθ2 +mg sin(θ) = 0

(A.12)

A.5 The Cart-Pole System

The Cart-Pole system depicted in Figure A-3 consists of a pendulum on a cart. Let(q1, q2) denote the configuration vector of the Cart-Pole system. The coordinates of

mF

m , l , I

1

2 2 2

q

q1

2

Figure A-3: The Cart-Pole system.

the center of mass of the cart and pendulum are given by

x1 =

[

q10

]

, x2 =

[

q1 + l2 sin(q2)l2 cos(q2)

]

with linear velocities

x1 =

[

q10

]

, x2 =

[

q1 + l2 cos(q2)q2−l2 sin(q2)q2

]

Thus, the total kinetic energy of the system is

K =1

2m1|x1|2 +

1

2m2|x2|2 +

1

2I2q

22

After simplification, we obtain

K =1

2(m1 +m2)q

21 +m2l2 cos(q2)q1q2 +

1

2(m2l

22 + I2)q

22

The potential energy of the Cart-Pole system is in the form

V (q) = V (q2) = m2gl2 cos(q2)

282

Therefore, the Lagrangian of the Cart-Pole system can be expressed as

L(q, q) = 1

2

[

q1q2

]T [m1 +m2 m2l2 cos(q2)

m2l2 cos(q2) m2l22 + I2

] [

q1q2

]

−m2gl2 cos(q2) (A.13)

and the elements of the inertia matrix for the Cart-Pole system are given by

m11 = m1 +m2

m12(q2) = m21(q2) = m2l2 cos(q2)m22 = m2l

22 + I2

(A.14)

The Euler-Lagrange equations of motion for the Cart-Pole system is in the form

d

dt

∂L∂q− ∂L∂q

=

[

F1

0

]

In other words, the Cart-Pole system is an underactuated system with two DOF andan unactuated shape variable q2.

A.6 The Rotating Pendulum

The Rotating Pendulum consists of an inverted pendulum on a rotating arm as shownin Figure A-4. This system was first introduced in [6]. The coordinates of the center

x

y

z

1

2q

m , L , I

m , L , I2

1 1

2

q1

2

Figure A-4: The Rotating Pendulum.

of mass of both links are given by

x1 =

l1 cos(q1)l1 sin(q1)

0

, x2 =

L1 cos(q1)− l2 sin(q1) sin(q2)L1 sin(q1) + l2 cos(q1) sin(q2)

l2 cos(q2)

283

with linear velocities

x1 =

−l1 sin(q1)q1l1 cos(q1)q1

0

, x2 =

−L1 sin(q1)q1 − l2 cos(q1) sin(q2)q1 − l2 sin(q1) cos(q2)q2L1 cos(q1)q1 − l2 sin(q1) sin(q2)q1 + l2 cos(q1) cos(q2)q2

−l2 sin(q2)q2

The kinetic energy of the Rotating Pendulum is

K =1

2m1|x1|2 +

1

2m2|x2|2 +

1

2I1q

21 +

1

2I2q2

which can be simplified as

K =1

2(m1l

21 + I1 +m2L

21 +m2l

22 sin

2(q2))q21 +m2l2 cos(q2)q1q2 +

1

2(m2l

22 + I2)q

22

The potential energy of the Rotating Pendulum is given by

V (q) = V (q2) = m2gl2 cos(q2)

Thus, the Lagrangian of the Rotating Pendulum can be written as

L =1

2

[

q1q2

]T [m1l

21 + I1 +m2L

21 +m2l

22 sin

2(q2) m2l2 cos(q2)m2l2 cos(q2) m2l

22 + I2

] [

q1q2

]

−m2gl2 cos(q2)

(A.15)Therefore, the elements of the inertia matrix for the Rotating Pendulum are given by

m11(q2) = m1l21 + I1 +m2L

21 +m2l

22 sin

2(q2)m12(q2) = m21(q2) = m2l2 cos(q2)

m22 = m2l22 + I2

(A.16)

The Euler-Lagrange equations of motion for the Rotating Pendulum is as the following

d

dt

∂L∂q− ∂L∂q

=

[

τ10

]

which means the Rotating Pendulum is an underactuated system with two DOF andone unactuated shape variable q2.

A.7 The 3D Cart-Pole System with Unactuated

Roll and Pitch Angles

Consider an inverted pendulum mounted on a moving platform via an unactuated 2DOF joint (θ, φ) (i.e. the roll and pitch angles) with two external forces as shown inFigure A-5. The angles θ, φ denote rotation around x2-axis and x1-axis, respectively.Denote the positions of the centers of mass of M and m as the following by xc1 and

284

F

F

2

1

x3

x1

2x

M

m

a

φθ

l

φ

Figure A-5: The 3D Cart-Pole system with unactuated roll and pitch angles.

xc2 , respectively.

xc1 =

x1x20

, xc2 =

x1 + l sin(θ)x2 + l cos(θ) sin(φ)a+ l cos(θ) cos(φ)

The linear velocities associated with (xc1 , xc2) are given by

xc1 =

x1x20

, xc2 =

x1 + l cos(θ)θ

x2 − l sin(θ) sin(φ)θ + l cos(θ) cos(φ)φ

−l sin(θ) cos(φ)θ − l cos(θ) sin(φ)φ

The kinetic and potential energy of the system are

K =1

2MxTc1xc1 +

1

2mxTc2xc2 , V (q) = mgl cos θ cosφ

The expression for the kinetic energy can be simplified as

K =1

2

x1x2θ

φ

T

M +m 0 ml cos θ 00 M +m −ml sin θ sinφ ml cos θ cosφ

ml cos θ −ml sin θ sinφ ml2 00 ml cos θ cosφ 0 ml2 cos2 θ

x1x2θ

φ

(A.17)Clearly, the inertia matrix for this system only depends on (θ, φ). Therefore, (θ, φ)are the shape variables and (x1, x2) are the external variables of the 3D Cart-Polesystem. The Euler-Lagrange equations of motion for the 3D Cart-Pole system withq = (x1, x2, θ, φ)

T is as the following

d

dt

∂L∂q− ∂L∂q

=

F1

F2

00

285

In other words, the 3D Cart-Pole System is an underactuated system with four DOFand two unactuated shape variables. The equations of motion for the 3D Cart-Polesystem can be explicitly written as

(M +m)x1 +ml cos θθ = ml sin θθ2 + F1

(M +m)x2 −ml sin θ sinφθ +ml cos θ cosφφ = ml cos θ sinφθ2 + 2ml sin θ cosφθφ

+ ml cos θ sinφφ2 + F2

ml cos θx1 −ml sin θ sinφx2 +ml2θ = mgl sin θ cosφ−ml2 sin θ cos θφ2ml cos θ cosφx2 +ml2 cos2 θφ = mgl cos θ sinφ+ 2ml2 sin θ cos θθφ

(A.18)

A.8 An Underactuated Surface Vessel

A surface vessel with no side thruster is illustrated in Figure A-6. Thus, it is anunderactuated system with three degrees of freedom and two actuators. The config-uration vector for this vessel is η = (x, y, θ) where (x, y) denotes the position and θdetermines the orientation of the vessel (i.e. the yaw angle). The velocity vector inthe body frame is ν = (v1, v2, ω) where (v1, v2) are the linear velocities and ω is theangular velocity. The complete dynamics of an underactuated surface vessel can be

θ

Y

X

Figure A-6: An underactuated surface vessel.

found in [30, 111]. Defining the following rotation matrix

R(θ) =

cos(θ) − sin(θ) 0sin(θ) cos(θ) 0

0 0 1

we haveη = R(θ)ν

The kinetic energy of this system is in the form

K =1

2νTJν

286

with J = diag(J1, J2, J3). Thus, the kinetic energy in the reference frame is

K =1

2ηTMη

which means the inertia matrix of a surface vessel is

M = R(θ)JRT (θ) = J

or the surface vessel is a flat underactuated mechanical system. The potential energyof a surface vessel is identically zero. The rotation matrix R satisfies

R = Rω

where ω is the following skew symmetric matrix

ω =

0 ω 0−ω 0 00 0 0

The dynamics of this surface vessel can be expressed as

η = R(θ)ν

Jν = −ωJν −Dν +

τ10τ3

(A.19)

where Dν is a damping term with D = diag(d1, d2, d3) and di > 0. The term ωJνhas the following explicit form

ωJν =

0 −J2v2ω 0J1v1ω 0 0

0 0 0

= C(ν)ν

where

C(ν) =

0 0 J2v20 0 −J1v1

−J2v2 J1v1 0

The overall dynamics of the surface vessel can be rewritten as

η = R(θ)ν

Jν + C(ν)ν +Dν =

τ10τ3

(A.20)

From the second line of the dynamic equation in (A.20), we have

J2v2 + J1v1ω + d2v2 = 0

287

which is a nonholonomic second-order constraint (i.e. a constraint that involvessecond-order time derivatives of the configuration variables) that holds regardlessof the control inputs τ1, τ3.

288

Appendix B

Symbolic Calculation of the FlatOutputs of the Helicopter

This appendix provides a MATLAB code for symbolic calculations required to provethat the vector of one-forms ωxs = R(η)EJΨ(η)dη in proposition 5.11.2 has no ex-act elements. In addition, using momentum decomposition, an exact part of ωxs isobtained that is denoted by C(η)dη in the following.% (a,b,c)=(φ, θ, ψ)=(roll,pitch,yaw) are three Euler angles.

syms a b c;

J1=sym(’J1’);J2=sym(’J2’);J3=sym(’J3’); J=diag([J1,J2,J3]);

e1=sym(’e1’);e2=sym(’e2’);e3=sym(’e3’); E=[0 e2 0; e1 0 e3; 0 0 0];

% A=Ψ(η) is the inverse of the Euler matrix.

A=[1 0 -sin(b); 0 cos(a) cos(b)*sin(a); 0 -sin(a) cos(b)*cos(a)];

R1=[1 0 0; 0 cos(a) -sin(a); 0 sin(a) cos(a)];

R2=[cos(b) 0 sin(b); 0 1 0 ; -sin(b) 0 cos(b)];

R3=[cos(c) -sin(c) 0; sin(c) cos(c) 0; 0 0 1];

R=simplify(R3*R2*R1); G=simplify(R*E*J*A);

% if d1,d2,d3 are not identically zero, G(η)dη is not exact.

d1=simplify(diff(G(:,2),’c’)-diff(G(:,3),’b’));

d2=simplify(diff(G(:,1),’c’)-diff(G(:,3),’a’));

d3=simplify(diff(G(:,1),’b’)-diff(G(:,2),’a’));

% Observation: none of d1,d2,d3 are identically zero

% A=A(a,b); A1=A(a,0); A2=A(0,b); A3=A(0,0);

A1=[1 0 0; 0 cos(a) sin(a); 0 -sin(a) cos(a)];

A2=[1 0 -sin(b); 0 1 0; 0 0 cos(b)];

A3=eye(3);

G1=simplify(R1*E*J*A1); C1=G1(:,1); G2=simplify(R2*E*J*A2); C2=G2(:,2);

G3=simplify(R3*E*J*A3); C3=G3(:,3);

% The obtained locked vector of one-forms ωl = C(η)dη has exact elements

C=[C1 C2 C3];

C = [ 0, cos(b)*e2*J2, -sin(c)*e3*J3] [ cos(a)*e1*J1, 0, cos(c)*e3*J3] [

sin(a)*e1*J1, -sin(b)*e2*J2, 0]

289

Appendix C

Proofs of Theorems in Chapter 7

C.1 Proof of Theorem 7.4.3:

Equation (7.93) can be rewritten as

ξ2 = Dz1ακ(z) · z2 +Dz2ακ(z) · f(z, ακ(z), ξ2) =: ψ(z, ξ2)

Since Dz2ακ(z) = 0,∀z2 6∈ K0, the fixed point equation (7.93) has a (trivial) smoothunique solution as the following

β(z) = Dz1ακ(z) · z2

for all z2 6∈ K0. For the rest of the proof, assume z2 ∈ K0 and set r0 = maxz2∈K0 ‖z2‖.Denote

Kξ1 = B(0, L0) ⊂ RKξ2 = B(0, 1) ⊂ R

(where B(x0, r) = x ∈ Rnx : |x − x0| ≤ r) and set S := Rn × K0 × Kξ2 . Based onAssumption (7.4.9), over S we have

|ψ(z, ξ2)| ≤ ‖Dz1ακ(z)‖ · ‖z2‖+ ‖Dz2ακ(z)‖ · (‖f0(z, ξ1)‖+ ‖g0(z, ξ1, ξ2)‖ · ‖ξ2‖2)≤ (L0 + L2)r0 + L1((a1 + a2)κ(z1) + (b1 + b2)|z1κ(z1)|)≤ (L0 + L2)r0 + L1(a1 + a2 + b1 + b2) ≤ 1

where the last inequality can be achieved by taking

L0, L2 ≤1

3r0−1, L1 ≤

1

3(a1 + a2 + b1 + b2)

−1

This guarantees that ψ(S) ⊆ Kξ2 . Calculating Dξ2ψ(z, ξ2) for all (z, ξ2) ∈ S, we get

|Dξ2ψ(z, ξ2)| ≤ ‖Dz2ακ(z)‖ · |Dξ2f(z, ακ(z), ξ2)|≤ L1(a3κ(z1) + b3‖z1κ(z1)‖)≤ L1(a3 + b3) < 1

290

where the last inequality can be satisfied for L1 < (a3 + b3)−1 that guarantees the

mapping ψ(z, ξ2) is contractive. Thus, after taking

L∗0 = L∗2 =1

3r0−1, L∗1 = min1

3(a1 + a2 + b1 + b2)

−1, (a3 + b3)−1

the result follows from the contraction mapping theorem. The property β(0) = 0 isdue to the uniqueness of the solution ξ∗2 = 0 of (7.93) at z = 0.

C.2 Proof of the Stability of µ-Subsystem in The-

orem 7.4.4

Lemma C.2.1. The origin (µ1, µ2) = 0 for

µ1 = −σ(µ1) + µ2µ2 = −σ(µ2) (C.1)

is globally asymptotically and locally exponentially stable.

Proof. First, in the linearization of this system µ = Aµ, A is a Hurwitz matrixand µ = 0 is locally exponentially stable. On the other hand, µ2 = 0 is globallyasymptotically stable for µ2 = −σ(µ2) (because V2(µ2) = µ22 is a valid Lyapunovfunction that satisfies V2 < 0 for µ2 6= 0). Thus, for any solution µ2(t) of µ2 = −σ(µ2)with µ2(0) ∈ R, |µ2(t)| ≤ |µ2(0)| for all t ≥ 0. Taking V1(µ1) =

12µ21, we have

V1 = −µ1σ(µ1) + µ1µ2≤ −|µ1|σ(|µ1|) + |µ1| · |µ2|≤ −|µ1|(σ(|µ1|)− |µ2(0)|)

Let rµ1 := σ−1(|µ2(0)|) and observe that for |µ1| > rµ1 , V1 < 0 and any solutionstarting out of the closed ball B(0, rµ1), enters B(0, rµ1) after some finite time T0 >0 and remains in it thereafter. Meaning that the solutions of system (C.1) stayuniformly bounded for all t > 0 (with a bound that depends on µ(0)). But givenµ2 = 0 for µ1 = −σ(µ1), µ1 = 0 is GAS. Therefore, based on Sontag’s theorem onstability of cascade nonlinear systems [83], (µ1, µ2) = 0 is GAS for the cascade (C.1).

C.3 Proof of Lemma 7.4.4:

First, we construct Ky. Take r0 = maxx∈Kx |φ0(x)|, Ky = B(0, 2r0), and r1 =max |φ1(x, y, ε)| over Kx × Ky × Iε. We have

|φ(x, y, ε)| ≤ |φ0(x)|+ ε|φ1(x, y, ε)| ≤ r2 := 2r0

291

for all 0 < ε ≤ ε2 := minr0/r1, ε1. Thus, setting Ky = Ky, φ maps Kx × Ky × Iεinto Ky where Iε = [0, ε2]. Define

Ly = max|∂φ1∂y

(x, y, ε)|, 1 ≥ 1

over Kx × Ky × Iε and set ε∗ = minε2, 1/Ly. Then, for all for all 0 < ε < ε∗ thefollowing holds

|∂φ∂y| = ε|∂φ1

∂y| ≤ εLy < 1

over Kx × Ky × (0, ε∗] and the result follows from contraction mapping theorem (orBanach fixed point theorem). In addition, due to uniqueness of the fixed point,α(0, ε) = 0 independent of ε.

C.4 Proof of Theorem 7.4.4:

Noting that ∂ψ3/∂u and ∂ψ2/∂ξ2 are equal and both vanish at ε = 0, the proof of(i) and (ii) follows directly from lemma 7.4.4 with appropriate choices of x, y. Toprove (iii), first we show that α2 locally asymptotically stabilizes the origin (z, ξ1) =0 for the (z, ξ1)-subsystem in (7.101). Then, we prove any solution of the systemstarting in a compact set of initial conditions for a sufficiently small ε > 0 staysuniformly bounded and asymptotically converges to the origin. For doing so, takeµ1 = ξ1 − α1(z), we have µ1 = −σ(µ1) and immediately it follows that µ1 = 0is globally asymptotically and locally exponentially stable for this system (with aLyapunov function V1(µ1) = µ21). The dynamics of the system in new coordinates isa cascade nonlinear system as the following

z = f0(z, α1(z) + µ1) + g0(z, α1(z) + µ1, α2(z, α1(z) + µ1, ε), ε)µ1 = −σ(µ1) (C.2)

orz = f0(z, α1(z)) + h0(z, α1(z), µ1)µ1 + g0(z, µ1, ε)µ1 = −σ(µ1) (C.3)

whereg0(z, µ1, ε) = g0(z, α1(z) + µ1, α2(z, α1(z) + µ1, ε), ε)

Note that g0 can be rewritten as the following

g0(z, ξ1, ξ2, ε) = p0(z, ξ1, ξ2, ε)ξ22ε

where p0 is a smooth function. Thus

∂g0∂z

∣

∣

∣

∣

(z,µ1)=0

=

[

∂p0∂z

α22ε+

∂p0∂ξ1

∂α1(z)

∂zα22 + 2α2

∂α2

∂zp0

]

(z,µ1)=0

= 0

292

where in the last equation ξ1 = α1(z) + µ1, ξ2 = α2(z, ξ1, ε), and α2(0, 0, ε) = 0.Calculating the Jacobian matrix of (C.3) at the origin (z, µ1) = 0, we obtain

Az,µ1 =

∂f0∂z

(z, α1(z)) +∂f0∂ξ1

(z, α1(z))∂α1

∂z+ (

∂h0∂z

+∂h0∂ξ1

∂α1

∂z)µ1 +

∂g0∂z

∗

0 −dσ(µ1)dµ1

(z,µ1)=(0,0)

or

Az,µ1 =

[

Az ∗0 −σ′(0)

]

where

Az =∂f0∂z

(z, α1(z)) +∂f0∂ξ1

(z, α1(z))∂α1(z)

∂z

∣

∣

∣

∣

z=0

is a Hurwitz matrix (by assumption, ξ1 = α1(z) locally exponentially stabilizes z = 0for z = f0(z, ξ1)) and −σ′(0) < 0. Therefore, (z, µ1) = (0, 0) is locally exponentiallystable for (C.3) and there exists a local region of attraction U0 as a small openneighborhood of the origin (z, µ1) = 0 that is forward invariant in time. Now, weshow the solutions of (C.3) starting in a compact set stay uniformly bounded. To doso, consider the following Lyapunov function candidate

W (z, µ1) = V (z) +1

2µ21

Calculating W along the solutions of the closed-loop system, we get

W =∂V (z)

∂zf0(z, α1(z)) +

∂V (z)

∂zh0(z, α1(z), µ1)µ1 +

∂V (z)

∂zg0(z, µ1, ε)− µ1σ(µ1)

Take r1 ≥ r0 to be sufficiently large such that B(0, r1) ⊃ U0. Consider the compactset Ω(c1) = (z, µ1)|W (z, µ1) ≤ c1 (where c1 > 0 to be determined later) withc1 > c2 > 0 such that Ω(c2) ⊃ (K0 ∪ B(0, r1) (Ω(c2) is fixed). Then, there existsε1(c1) > 0 such that for all 0 < ε < ε1(c1), we have

∣

∣

∣

∣

∂V (z)

∂zg0(z, µ1, ε)

∣

∣

∣

∣

< −∂V (z)

∂zf0(z, α1(z)) + µ1σ(µ1)

over Ω(c1) \ U0. Thus, we have

W <∂V (z)

∂zh0(z, α1(z), µ1)µ1(t)

≤∣

∣

∣

∣

∂V (z)

∂zh0(z, α1(z), µ1)

∣

∣

∣

∣

· |µ1(t)|

≤ kV (z)λ|µ1(t)| , ∀|z| > r1,∀t ≥ 0

293

First, assume λ = 1. It follows that

W < kV (z)|µ1(t)| ≤ kW |µ1(t)|

but µ1(t) is an L1 function and ∃l1 > 0 :∫∞

0|µ1(t)|dt ≤ l1. This means that

W (z(t), µ1(t)) < W (z(0), µ1(0)) exp(kl1) =: c3

where c3 > 0 is a constant that only depends on k, l1,W (z(0), µ1(0)). In other words,c3 is independent of ε > 0. Thus, taking a fixed c1 > maxc2, c3 determines acompact set K1 := Ω(c1) ⊃ K0 such that the solutions (z(t), µ1(t)) ∈ K1 for all t ≥ 0.The case 0 < λ < 1 is rather similar, we get

W < kV (z)λ ≤ kW (z, µ1)λ

Thus, c3 can be defined in the following way

d

dt(W (1−λ)) < (1− λ)kµ1(t)

orW (z, µ1) < (W (z(0), µ1(0))

(1−λ) + (1− λ)kl1)1

(1−λ) =: c3(λ) , ∀t > 0

and the boundedness result follows (also, observe that limλ→1− c3(λ) = c3 for λ = 1).It remains to prove that the solutions starting in K0 asymptotically converge to theorigin. Over the compact set K1, take

l2 = max(z,µ1)∈K1

∣

∣

∣

∣

∂V (z)

∂zh0(z, α1(z), µ1)

∣

∣

∣

∣

and

ε0 = max(z,µ1)∈K1\U0

−∂V (z)

∂zf0(z, α1(z)) + µ1σ(µ1) > 0

Then, there exists ε2 > 0 such that for all 0 < ε < ε2∣

∣

∣

∣

∂V (z)

∂zg0(z, µ1, ε)

∣

∣

∣

∣

<ε03

Also, because µ1(t)→ 0 as t→∞, there exists T0 > 0 such that for all t > T0∣

∣

∣

∣

∂V (z)

∂zh0(z, α1(z), µ1)µ1

∣

∣

∣

∣

≤ l2|µ1(t)| <ε03

294

Therefore, over K1 \ U0, we have

W ≤ ∂V (z)

∂zf0(z, α1(z))− µ1σ(µ1) +

∣

∣

∣

∣

∂V (z)

∂zh0(z, α1(z), µ1)µ1

∣

∣

∣

∣

+

∣

∣

∣

∣

∂V (z)

∂zg0(z, µ1, ε)

∣

∣

∣

∣

< −ε0 +ε03+ε03

= −ε03< 0

This means that any solution of the system starting in K0 stays in K1 for all t > 0and enters U0 after some finite time T1 ≥ T0 > 0 and then asymptotically convergesto (z, µ1) = 0, or (z, ξ1) = 0. This finishes the proof of (iii). The proof of (iv) is verysimilar to (iii) and is omitted.

C.5 Proof of Theorem 7.4.5:

The proof of (i) is based on contraction mapping theorem (see lemma 7.4.4). To prove(ii), first we establish local asymptotic stability of the origin (z, ξ1) = 0 for the closedloop (z, ξ1)-subsystem. Over a small open neighborhood of the origin U , fix ε1 > 0such that for all 0 < ε ≤ ε1 (i) holds. Let ξ2 = α2(z, ξ1, ε) be the unique fixed pointof (7.106) over U . Around (z, ξ1) = 0, f0 and g0 can be expressed as

f0(z, α1(z)) = A0z + f0(z)g0(z, ξ1, ξ2, ε) = g0(z, ξ1, ξ2, ε)ξ2ε

where

A0 =

[

∂f0∂z

(z, α1(z)) +∂f0∂ξ1

(z, α1(z))∂α1(z)

∂z

]

z=0

is a Hurwitz matrix by assumption and f0 with all its first order partial derivativesvanish at z = 0. Consider the quadratic Lyapunov function

V0(z) =1

2zTPz

where P is the solution of the following Lyapunov equation

AT0 P + PA0 = −I

Applying the change of coordinates µ1 = ξ1 − α1(z), we show that the candidateLyapunov function

W0(z, µ1) = V0(z) +1

2µ21

295

locally over U is a valid Lyapunov function for a sufficiently large c1 = σ′1(0) > 0.Note that locally σ1(µ1) = c1µ1 +O(|µ1|2). Calculating W0, we have

W0 = −|z|2 + zTP f0(z) + µ1zTPh0(z, α1(z), µ1)

+ zTP g0(z, α1(z) + µ1, α2) · α2(z, α1(z) + µ1, ε)ε

− c1µ21 +O(|µ1|3) + µ1

∂V (z)

∂zh0(z, α1(z), µ1)

We need to determine a local approximation of each term over U in the precedingexpression. The second term zTP f0(z) ∼ O(|z|3), and the third term satisfies

|µ1zTPh0(z, α1(z), µ1)| < l1|µ1| · |z|

for some l1 > 0. Also, α2 vanishes at the origin, thus |α2(z, α1(z) + µ1, ε)| < l2|z| +l3|µ1|. This means that

|zTP g0(z, α1(z) + µ1, α2) · α2(z, α1(z) + µ1, ε)| < l2|z|2 + l3|µ1| · |z|

for some l2, l3 > 0. Also, because ∂V (z)/∂z vanishes at z = 0, there exists l4 > 0such that

|∂V (z)

∂z| ≤ l4|z|

Thus

|µ1∂V (z)

∂zh0(z, α1(z), µ1)| < l4|µ1| · |z|

for some l4 > 0. We obtain

W0 ≤ −|z|2(1− εl2) + (εl3 + l4)|µ1| · |z| − c1|µ1|2 +O(|z|3) +O(|µ1|3)

Take ε2 = min1/3l2, ε1 and a sufficiently small open neighborhood U0 ⊂ U of theorigin in the form U0 = U0(c0) = (z, µ1)|W0(z, µ1) < c0 for a sufficiently smallc0 > 0 such that

O(|z|3) < 1

3|z|2 , O(|µ1|3) < b1|µ1|2

for some b1 > 0 (e.g. b1 = 0.1). Then, for all 0 < ε ≤ ε2, we have

W0 ≤ −1

3|z|2 + (

l33l2

+ l4)|µ1| · |z| − (c1 − b1)|µ1|2

by completing the squares

W0 ≤ −1

3(|z|+ 3

2b2|µ1|)2 − (c1 − b1 −

3

4b22)|µ1|2

296

where b2 = (l3/3l2) + l4. Therefore, by taking

c1 > c∗1 := b1 +3

4b22 > 0

there exists a positive definite matrix Q such that

W0 < −zTQz < 0 , ∀z 6= 0

where z = col(z, µ1) and thus the origin (z, µ1) = 0 is locally exponentially sta-ble. Moreover, U0 is forward invariant in time. To establish semiglobal asymp-totic stabilization of (z, ξ1) = 0, consider the solutions starting in a compact setK0 := K0

z × K0µ13 (z(0), µ1(0)) that includes U0. Take the Lyapunov function

W (z, µ1) = V (z) + 12µ21. Let Ω0 = (z, µ1)|W (z, µ1) ≤ c0 for a sufficiently large

c0 > 0 such that Ω0 ⊃ K0. Fix ε3 ≤ ε2 such that for all 0 < ε < ε3 (i) holds over Ω0

(note that W and thus Ω0 are independent of ε). Calculating W along the solutionsof the closed loop system, we get

W =∂V (z)

∂zf0(z, α1(z)) +

∂V (z)

∂zg0(z, ξ1, ξ2, ε)

+ µ1(ξ2 −∂α1(z)

∂zf0(z, ξ1)−

∂α1(z)

∂zg0(z, ξ1, ξ2, ε)

+∂V (z)

∂zh0(z, α1(z), µ1))

=∂V (z)

∂zf0(z, α1(z))− µ1σ(µ1) +

∂V (z)

∂zg0(z, ξ1, ξ2, ε)

=∂V (z)

∂zf0(z, α1(z))− µ1σ(µ1) +

∂V (z)

∂zg0(z, α1(z) + µ1, α2(z, α1(z) + µ1, ε), ε)

Noting that Ωc0 := Ω0 \ U0 is a compact set and defining

ε0 = minz∈Ωc0

−∂V (z)

∂zf0(z, α1(z)) + µ1σ(µ1) > 0

there exists an ε∗ > 0 such that for all 0 < ε < ε∗, |∂V (z)∂z

g0| < ε0 and W < 0 overΩc

0. Thus, any solution of the system enters U0 after some finite time and exponen-tially converges to the origin. This guarantees semiglobal asymptotic stabilizationof (z, µ1) = 0 and therefore semiglobal asymptotic stabilization of (z, ξ1) = 0. Itremains to prove (iii). This follows from standard backstepping theorem applied tothe composite system (7.101) rewritten in the following form

˙z = f2(z, ξ2, ε)

ξ2 = u

where z = col(z, ξ1) and f2 = col(f, ξ2). Noting that based on (ii), there exists asemiglobally asymptotically stabilizing feedback law ξ2 = α2(z, ε) for the z-subsystem.

297

Then, given the control law (7.107), similar to the argument in the proof of (ii), itcan be shown that there exists a small neighborhood of the origin U0 such that thesolution of the closed loop system is exponentially stable and the positive definiteproper function

Q(z, ξ2, ε) =W (z) +1

2(ξ2 − α2(z, ε))

2

is a valid Lyapunov function for the whole system with the property that Q < 0 overΩ \ U0 where Ω is an appropriate invariant set of the Lyapunov function Q. Thisfinishes the proof.

298

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