Chapter 1

Introduction

1.1 Motivation

The motivation for the research presented in this book is to continue thedevelopment of the field of non-linear control theory for mechanical sys­tems. The development of robots having an autonomous and complexbehavior such as the adaptation to environment changes and uncertain­ties, planification and execution strategies without human interventionand the learning ability to improve performances is one of the ultimategoals in research of robotics. The achievement of such machines couldhave a major impact in many fields such as production, stocking andsupervision of dangerous waste, construction and the robotic hollow,inspection, teleoperation, maintenance of satellites, autonomous vehi­des, etc. It clearly appears that most of the problems to be solved toreach such goals imply control problems. The development of controltechniques is a vital objective for the realization and the creation ofintelligent robots.

This book presents the application of modern non-linear systemstheory to control some important classes of underactuated mechanicalsystems. In the eighties, the control of robot manipulators was exten­sively studied. Several control strategies based on passivity, Lyapunovtheory, feedback linearization, etc . have been developed for the fullyactuated case, i.e. systems with the same number of actuators as de­grees of freedom. The techniques developed for fully actuated robotsdo not apply directly to the case of underactuated non-linear mechani­cal systems. Underactuated mechanical systems or vehicles are systemswith fewer independent control actuators than degrees of freedom to becontrolled.

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I. Fantoni et al., Non-linear Control for Underactuated Mechanical Systems

© Springer-Verlag London 2002

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Inverted pendulum

CHAPTER 1. INTRODUCTION

Pendubot

Rotati onal inverted

pendulum

Planar manipul ator

Inert ia wheel One revolute and two prismatic joints

planar manipul ator

Figure 1.1: Examples of underactuated mechani cal systems

1.1. MOTIVATION

Hovercraft

Planar vertical take-off and landing aircraft(PVTOL)

Figure 1.2: Examples of underactuated mechanical systems

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In the last few years, there has been major interest in developingstabilizing algorithms for underactuated mechanical systems. The needfor underactuated algorithms arises in many practical situations, someof which are enumerated below . The interest comes from the need tostabilize systems like ships, underwater vehicles, helicopters, aircraft,airships, hovercrafts, satellites, walking robots, etc., which may be un­deractuated by design. Actuators are expensive and/or heavy and aretherefore sometimes avoided in a system design. Other systems mayalso become underactuated due to actuator failure.

Most models of mechanical systems such as robotic manipulators arebuilt on the assumption that the individual links or members are rigid.This is a correct approximation in some cases while in others it is not.If we take the more realistic non-rigid dynamics into account then allsuch models are essentially underactuated.

In aircraft or space applications, equipment weight is of paramountimportance. Space bound rackets have limited rocket payloads, a size­able portion of which constitutes automatie contral equipment. Thishas fueled research into underactuated mechanical systems. The idea issimply to reduce the weight of any robotic manipulator by reducing thenumber of motors, which are often the most heavy and unwieldy parts.

4 CHAPTER 1. INTRODUCTION

Free floating mobile robots like those used in space and ocean ap­plications often have robotic manipulators mounted on them. Exam­ples include collecting sampIes in submersible vehicles and performingmaintenance operations or debris retrieval in space robots. Close to theobjective where the desired task has to be performed, the primary pro­pelling devices are usually switched off so that the platform is free float­ing. Under such conditions, when the robotics manipulator is moved,the law of conservation dictates that the platform itself will move. Sothe system becomes underactuated as a new degree of freedom is addedand the normal control algorithms do not work.

More specific applications also exist such as in the arena of navaloperations, where fuel and rat ions are supplied to a naval vessel froma supply ship using a swinging crane. Under a high sea state, therelative motion of the two vessels becomes important enough to hindersignificantly even the mundane task of loading and unloading. In spaceapplications, if any one of the motors of a multilinked rabotic armmalfunction while the arm is extended, the only solution is to jettisonthe whole assembly. Unless, that is, an underactuated algorithm canbe utilized to retrieve it.

Underactuation may be due to an actuator failure. A hardware so­lution to actuator failures may be achieved by equipping the vehicIewith redundant actuators. The software option is, on the other hand,a cost-reducing alternative, since it consists of changing to a contrallaw that controls the vehicle using only the remaining actuators, whenan actuator failure is detected. Furthermore, the software solution isweight economical compared to the hardware solution and this can beimportant in space and underwater applications. Moreover, cost andweight considerat ions can motivate constructors to create underactu­ated vehicles .

It is thus clear that there is a need to develop new control techniquesapplicable to underactuated non-linear mechanical systems.

The research is focused towards obtaining control algorithms for gen­eral underactuated non-linear mechanical systems. Since this generalobjective is difficult to accomplish, we are also int erested in stabiliz­ing particular classes of mechanical systems. To simplify the task, re­searchers have been studying simple mechanical systems. Some of thesesystems represent academic benchmarks and are part of a standardcontral laboratory like the inverted pendulum, the rotational invertedpendulum, the pendubot, the planar manipulator with springs betweenlinks, the pendulum driven by a spinning wheel, the ball and beam,

1.1. MOTIVATION 5

and the PVTOL (planar vertical take-off and landing) aircraft. In spiteof the fact that they are simple mechanical plants, they represent achallenge to the non-linear control community.

The study of the PVTOL aircraft is also important because it repre­sents a simplified model of a helicopter in the lateral (or longitudinal)axis. Indeed, if we consider a helicopter for which the yaw and pitch(or roll) angles are fixed, the resulting system is similar to the PV­TOL aircraft. Developing control strategies for the PVTOL aircraftwill normally be useful in the control design for helicopters. Chapters13 through 15 are devoted to modelling and control of helicopter models.

An important remark is the fact that the research on underactu­ated systems is an extension of the research on non-holonomic systems.Indeed, non-holonomic systems have constraints on the velocity andonly kinematic equations of the system are considered. Underactuatedsystems have constraints on the acceleration, and both kinematics anddynamics have to be considered in the control design.

In this book, we will give a detailed presentation of the control ofsome well-known underactuated non-linear mechanical systems. Thereader will find detailed steps to obtain the Euler-Lagrangian models asweIl as various control laws obtained using different approaches basedon Lyapunov theory, passivity, feedback linearization, etc. Real ap­plications will illustrate the performance of the algorithms on severalexperimental platforms.

We have neglected friction in the various inverted pendulum systemsstudied in this book. The convergence analysis proposed deals only withthe ideal case in which friction is zero. However, experimental resultshave shown that the proposed controller performs appropriately whenthe frictional terms are small.

In most of the examples that will be presented, a passivity- or energy­based approach is used in the design of stabilizing controllers. In fact,the passivity-based control technique is standard and has extensivelybeen studied and used in the control community. Indeed, Willems [122Jand Hill & Moylan [37, 38, 39, 40J provided some gener al not ions in thetheory of dissipative systems, in particular the small-gain and passivitytheorems.

Takegaki & Arimoto [113J developed the idea of stabilizing mechan­ical systems by reshaping the potential energy via feedback and byadding damping. This is one of the starting points of "passivity-basedcontrol" .

Byrnes et al. [16J in 1991, derived cond itions under which a non-

6 CHAPTER 1. INTRODUCTION

linear system can be rendered passive via smooth state feedback andthey extended a number of stabilization schemes for global asymptoticstabilization of certain classes of non-linear systems.

Nijmeijer & Van der Schaft [75] described system theoretic proper­ties and stabilization of standard Hamiltonian control systems. ThenMaschke, Van der Schaft & Breedvelt [68J introduced generalized Hamil­tonian control systems as an important class of passive state space sys­tems and studied the stabilization of such systems. In 1996, Van derSchaft provided in his lecture notes, published by Springer [117], a veryuseful synthesis between classical input-output and closed-loop stabil­ity theory, in particular the small-gain and passivity theorems, andpresented developments in passivity-based and non-linear H oo control.

It is also important to remark that recently the journal "InternationalJournal of Robust and Nonlinear Control" has published a special issueentitled "Control of underactuated nonlinear systems" . In this issue,a paper by Shiriaev et al. [100] deals with the stabilization of theupright position of the inverted pendulum system that we will developin Chapter 3. This paper gives an extension on some global propertiesof the controller that we have proposed in [58J (see also Chapter 3).This clearly shows the great interest that the subject of this book hasalready received from other researchers of the control community.

The last three chapters of this book deal with modelling and controlof helicopters. Helicopters are underactuated systems since they havein general six degrees of freedom (position (x , y , z), pitch, roll and yaw)and only four control inputs (pitching, rolling and yaw moments andthe main rotor thrust) . Chapter 13 deals with a helicopter mounted ona platform such that the aircraft can move only vertically and aroundthe vertical axis. The system has three degrees of freedom and twoinputs. Chapters 14 and 15 deal with helicopters moving freely in athree-dimensional space. Chapter 14 presents a Lagrangian model of thehelicopter while Chapter 15 deals with a Newtonian approach. Controllaws based on passivity and partial feedback linearization are proposedfor each case .

We believe this book will be of great value for Ph.D. students and re­searchers in the areas of non-linear control systems, mechanical systems,robotics and control of helicopters. It will help to acquire the appropri­ate models of the proposed systems and to handle other underactuatedsystems.

1.2. OUTLINE OF THE BOOK

1.2 Outline of the book

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Chapter 2 presents general notions and background theory, which willbe used throughout the book.

1.2.1 Energy-based control approaches for several under­aetuated mechanical systems

Chapters 3 to 10 propose a set of underactuated mechanical systems forwhich we apply an energy-based control approach. For all the systems,we take advantage of their passivity properties in order to establisha control law based on Lyapunov theory. The control objective is tostabilize systems around a desired position. The illustrative examplesare the following:

• Chapter 3 deals with the inverted pendulum system and the de­velopment of an energy-based controllaw. The applicability of themethod is illustrated by means of simulations and exp erimental re­sults performed at the University of Illinois at Urbana-Champaign(USA). This work is also published in [59].

• Chapter 4 is a natural extension of Chapter 3. Indeed , the pro­posed convey-crane system is intensely based on the inverted pen­dulum's equations. Again, an energy-based control approach isproposed in order to st abilize the convey-crane at its lower equi­librium position. This work has been presented in [18].

• The pendubot system is introduced in Chapter 5. Experimentsare also given in order to see the performance of the proposedcontrol law. The presentation of this work can be found in [24] .

• In Chapter 6, the rotational inverted pendulum, often called theFuruta pendulum system, is presented and its energy-based con­trol law is developed .

• Chapter 7 deals with the pendulum driven by a spinning wheel ,i.e. the reaction wheel. Two different approaches are considered.

• Chapter 8 presents planar underactuated manipulators withsprings between links . A simple control law is presented for suchsyste ms.

8 CHAPTER 1. INTRODUCTION

• Chapter 9 introduces a planar robot with two prismatic and onerevolute (PPR) joints. This example has four degrees of freedomwith only three control inputs. An energy-based control law isagain presented.

• Finally, in Chapter 10, we propose a control law for the ball andbeam system acting on the ball instead of the beam.

The main contribution of the above systems is to exploit their pas­sivity properties to develop appropriate control laws. Moreover, rig­orous and complete stability analysis for the closed-loop systems arepresented. Note that the main idea is similar for most of the presentedsystems. Therefore, it can be extended to a larger dass of underactu­ated mechanical systems provided that the control law is adapted toeach particular system.

1.2.2 The hovercraft model, the PVTOL aircraft and thehelicopter

From Chapters 11 to 15, we study systems that have a direct real appli­cation, such as hovercrafts, aircraft and helicopters. Control laws havebeen developed using simplified models of such systems.

• Chapter 11 deals with a simplified model of a ship that can alsobe regarded as a hovercraft model. We propose different controlstrategies based on Lyapunov theory.

• In Chapter 12, we present the model of a planar vertical take-offand landing (PVTOL) aircraft that is a simplified aircraft. Wepropose a control strategy by construction of a Lyapunov functionusing the forwarding technique.

The last three chapters present several methodologies for modellinga helicopter and different control procedures are proposed based onLyapunov theory.

• Chapter 13 introduces a Lagrangian model of a VARIO scalemodel helicopter mounted on a platform and a passivity-basedcontrol strategy. The proposed control strategy is based on theuse of non-linear controllers, which ensure asymptotic tracking ofsuitable desired trajectories.

1.2. OUTLINE OF THE BOOK 9

• In Chapter 14, a Lagrangian model of the dynamics of a simplifiedhelicopter permits a Lyapunov design of a unified path trackingcontrol algorithm.

• Finally, Chapter 15 presents a Newtonian helicopter model and arobust control design based on robust backstepping techniques isproposed. The controllaw design is based on an approximation ofthe system obtained by ignoring the small body forces associatedwith the torque control.