Adaptive control of robotic manipulators

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Adaptive control of robotic manipulators
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viii, 257 leaves : ill. ; 28 cm.
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Tosunoglu, L. Sabri
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Manipulators (Mechanism)   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Includes bibliographical references (leaves 250-256).
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by L. Sabri Tosunoglu.
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Typescript.
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Vita.

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ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS


By

L. SABRI TOSUNOGLU
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA


1986

















ACKNOWLEDGMENTS


The author wishes to express his gratitude to his

committee chairman, Dr. Delbert Tesar, for his guidance,

supervision, and encouragement throughout the development

of this work. In this respect, sincere appreciation goes

to his committee cochairman and the Director of the Center

for Intelligent Machines and Robotics (CIMAR), Dr. Joseph

Duffy, and the committee members, Dr. Roger A. Gater,

Dr. Gary K. Matthew, Dr. George N. Sandor, and Dr. Ralph

G. Selfridge. Working with Dr. Roger A. Gater was very

pleasant and gave the author invaluable experiences.

Financial and moral support of the Fulbright

Commission and its administrators is greatly appreciated.

Sincere thanks are also due to dear friends at CIMAR whose

support and friendship made his studies pleasant throughout

the years. Sofia Kohli also deserves credit for her

professionalism, patience, and excellent typing.



















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . .* ..*

ABSTRACT . . . . .

CHAPTER


1 INTRODUCTION AND BACKGROUND . .

1.1 Manipulator Description and
Related Problems . .

1.2 Dynamics Background . .

1.3 Previous Work on the Control of
Manipulators . . .

1.3.1 Hierarchical Control
Stages . . .

1.3.2 Optimal Control of
Manipulators . .

1.3.3 Control Schemes Using
Linearization Techniques

1.3.4 Nonlinearity Compensation
Methods . .

1.3.5 Adaptive Control of
Manipulators . .

1.4 Purpose and Organization of
Present Work . . .

2 SYSTEM DYNAMICS . . .

2.1 System Description . .

2.2 Kinematic Representation of
Manipulators . . .


*


iii


Page

* ii

* vii






. 1


. 5


. 7














. 721
. 9






. 13


. 15


. 18

. 21

. 21


. 23













CHAPTER


2.3 Kinetic Energy of Manipulators

2.3.1 Kinetic Energy of a
Rigid Body . .

2.3.2 Absolute Linear Velocities
of the Center of Gravities

2.3.3 Absolute Angular Velocities
of Links . . .

2.3.4 Total Kinetic Energy .

2.4 Equations of Motion . .

2.4.1 Generalized Forces .

2.4.2 Lagrange Equations .

TIVE CONTROL OF MANIPULATORS . .

3.1 Definition of Adaptive Control

3.2 State Equations of the Plant
and the Reference Model . .

3.2.1 Plant State Equations .

3.2.2 Reference Model State
Equations . .

3.3 Design of Control Laws via the
Second Method of Lyapunov .

3.3.1 Definitions of Stability
and the Second Method of
Lyapunov . . .

3.3.2 Adaptive Control Laws .

3.3.2.1 Controller
structure 1 .

3.3.2.2 Controller
structure 2 .


Page

. 28


. 28


. 33


. 37

. 39

. 40

. 41

. 44

. 50

50


. 54

. 54


. 56


S. 58



. 58

. 64


. 68


. 68


3 ADAP












CHAPTER

3.3.2.3 Controller
structure 3 .

3.3.2.4 Controller
structure 4 .

3.3.3 Uniqueness of the Solution
of the Lyapunov Equation .

3.4 Connection with the Hyperstability
Theory . . . .

3.5 Controllability and Observability
of the (A,B) and (C,A) Pairs .

3.6 Disturbance Rejection . .

4 ADAPTIVE CONTROL OF MANIPULATORS IN
HAND COORDINATES . . . .

4.1 Position and Orientation of
the Hand .. . ..

4.2 Kinematic Relations between the
Joint and the Operational Spaces

4.2.1 Relations on the Hand
Configuration . .

4.2.2 Relations on Hand Velocity
and Acceleration . .

4.2.3 Singular Configurations

4.3 System Equations in Hand
Coordinates . . .

4.3.1 Plant Equations . .

4.3.2 Reference Model Equations

4.4 Adaptive Control Law with
Disturbance Rejection . .

4.5 Implementation of the
Controller . . .


Page


. 73


. 74


. 80


. 81


. 87

S. 89


. 98


. 99


. 101


104

109


111

111

114


114


118













CHAPTER

5


REFERENCES . . . . .

BIOGRAPHICAL SKETCH . . . .


Page


ADAPTIVE CONTROL OF MANIPULATORS
INCLUDING ACTUATOR DYNAMICS . .

5.1 System Dynamics Including
Actuator Dynamics . .

5.1.1 Actuator Dynamics .

5.1.2 System Equations .

5.2 Nonlinear State Transformation

5.3 Adaptive Controller . .

5.4 Simplified Actuator Dynamics .

5.4.1 System Dynamics .

5.4.2 Adaptive Controller with
Disturbance Rejection
Feature . .

EXAMPLE SIMULATIONS . . .

6.1 Simulations on the 3-Link,
Spatial Manipulator . .

6.2 Numerical Solution of the
Lyapunov Equation . .

6.3 Simulations on the 6-Link,
Spatial Industrial Manipulator

CONCLUSION . . . .


6


7


121


121

121

124

125

128

131

131


133

136


139


183


184

246

250

257


. .
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS

By

L. Sabri Tosunoglu

May 1986

Chairman : Delbert Tesar
Cochairman: Joseph Duffy
Major Department: Mechanical Engineering

Currently industrial robot manipulators operate

slowly to avoid dynamic interactions between links.

Typically each joint is controlled independently and system

stability and precision are maintained at the expense of

underutilizing these systems. As a result, productivity is

limited, and more importantly, the lack of reliability has

hindered investment and wider industrial use. This work

addresses the adaptive control of spatial, serial

manipulators. Centralized adaptive controllers which

yield globally asymptotically stable systems are designed

via the second method of Lyapunov. Actuator dynamics is

also included in the system model.

Lagrange equations are used in deriving dynamic

equations for n-link, spatial robot manipulators which are

modeled with rigid links connected by either revolute or


vii












prismatic pairs. Although manipulators may exhibit

structural flexibility, the rigid link assumption is

justified, because control of manipulators needs to be

understood precisely before flexibility is included. The

plant, which represents the actual manipulator, and the

reference model, representing the ideal robot, are both

expressed as distinct, nonlinear, coupled systems.

Error-driven system dynamics is then written and adaptive

controllers which assure global asymptotic stability of the

system are given utilizing the second method of Lyapunov.

It is shown that these control laws also lead to

asymptotically hyperstable systems.

Integral feedback is introduced to compensate for

the steady-state system disturbances. Tracking is achieved

since the error-driven system is used in deriving the

controllers. Manipulator dynamics is expressed in hand

coordinates and an adaptive controller is suggested for

this model. Actuator dynamics, modeled as third-order,

linear, time-invariant systems, is coupled with manipulator

dynamics and a nonlinear state transformation is introduced

to facilitate the controller design. Later, simplified

actuator dynamics is presented and the adaptive controller

design and disturbance rejection feature are extended for

this system. Adaptive controllers are implemented on the

computer, and numerical examples on 3- and 6-link spatial,

industrial manipulators are presented.


viii
















CHAPTER 1
INTRODUCTION AND BACKGROUND


In this chapter, manipulator description and general

problems associated with this class of systems are addressed

and the previous work in this area is briefly reviewed.

The review mainly concentrates on the dynamics development

and control of manipulators. After an introduction to

general control stages, background on the lowest level

control, the so-called executive level, is presented. This

presentation, in turn, groups the previous work under

optimal control, control schemes utilizing linearization

techniques, nonlinearity compensation methods, and adaptive

control of manipulators.


1.1 Manipulator Description and Related Problems

A robotic manipulator is defined as a system of

closed-loop linkages connected in series by kinematic

joints which allow relative motion of the two linkage

systems they connect. One end of the chain is fixed to a

support while the other end is free to move about in the

space. In this way an open-loop mechanism is formed. If

each closed-loop linkage system consists of a single link,

then a simple serial manipulator will be obtained.












Currently, most industrial manipulators are serial arms due

to their simpler design and analysis.

A robotic manipulator system is defined as a

programmable, multifunction manipulator designed to move

material, parts, tools or specialized devices through

variable programmed motions for the performance of a variety

of tasks without human intervention. In the literature, the

terms robotic manipulator, mechanical arm, manipulator,

artificial arm, robotic arm and open-loop articulated chain

are used interchangeably.

Manipulators find numerous practical applications in

industry [5, 51, 581* and their use is justified mainly for

their dedication on repetitive jobs and for their flexibility

against hard automation. Tesar et al. detail the handling

of radioactive material via robotics implementation to a fuel

fabrication plant in [52]. Positioning/recovery of

satellites in space with the NASA Space Shuttle Remote

Manipulator System-though not completely successful yet-is

another challenging application area of robotics.

In the analysis of manipulators,basically two

problems are encountered. The first is called the

positioning or point-to-point path-following problem and

can be stated as follows: Given the desired position and


*Numbers within brackets indicate references at the end
of this text.












orientation of the free end of the manipulator, i.e., hand

(or gripper) of the manipulator, find the joint positions

which will bring the hand to the desired position and

orientation. This kinematics problem involves a nonlinear

correspondence (not a mapping) of the Cartesian space to

the manipulator joint space.

If a serial manipulator is modeled with n rigid

links and n one degree-of-freedom joints, then the dimension

of its joint space will be n. In Cartesian space, six

independent coordinates are needed to describe uniquely the

position and orientation of a rigid body. Now, for n = 6, a

finite number of solutions can be obtained in the joint

space except at singular points [49]. Closed-form solution

to this problem is not available for a general manipulator.

Duffy instantaneously represents a 6-link, serial

manipulator by a 7-link, closed-loop spatial mechanism with

the addition of a hypothetical link and systematically

solves all possible joint displacements [9]. Paul et al.

obtain closed-form solution for the Puma arm (Unimate 600

Robot) [40]; their method is not general, but applicable

to some industrial manipulators. In practice, however,

some industrial arms make use of iterative methods even in

real time.

When n < 6, joint space cannot span the Cartesian

space. In general, the gripper cannot take the specified












position and orientation. And finally, if n > 6, the

manipulator will be called redundant. In this case,

infinitely many solutions may be obtained and this feature

lends the current problem to optimization (e.g., see [31]).

Whitney was the first to map hand command rates

(linear and angular hand velocities) into joint displacement

rates, known as coordinated control or resolved rate control

[63]. This transformation is possible as long as the

Jacobian (defined in Chapter 4, Section 4.2.2) is

nonsingular. If the Jacobian is singular, the manipulator

is then said to be in a special configuration. In these

cases, there is not a unique set of finite joint velocities

to attain the prescribed hand velocity. In today's practice,

however, special configurations of industrial manipulators

are mostly ignored. Later, related work concentrated on

the derivation of efficient algorithms [41, 59].

The second problem includes dynamic analysis and

control of manipulators and can be stated as follows: Find

the structure of the controller and the inputs which will

bring the manipulator to the desired position and

orientation from its present configuration. If optimization

is introduced with respect to some criterion to improve

the system performance, then it is called an optimal control

problem.

Basic tasks performed by industrial manipulators can

be classified in two groups. The first group tasks include












pick-and-place activities such as spot welding, machine

loading and unloading operations,and can be treated as a

reaching-a-target problem. In this problem initial and

terminal positions are specified, but the path followed

between these two configurations is in general of no

importance except for obstacle avoidance. Optimization can

be introduced to synthesize optimal control and obtain

corresponding optimal paths. Typically, minimization of

time, energy, input power, etc., or any combination of

these indices will improve manipulator performance with

respect to these criteria. The second group tasks include

continuous welding processes, metal cutting, spray painting,

automatic assembly operations, etc. and require tracking

(contouring) of a specified path. The present work

basically considers the tracking problem.


1.2 Dynamics Background

If the manipulator is to be moved very slowly, no

significant dynamic forces will act on the system. However,

if rapid motions are required, dynamic interactions between

the links can no longer be neglected. Currently servo-

controlled industrial manipulators ignore such interactions

and use local (decentralized) linear feedback to control the

position of each joint independently. At higher speeds the

system response to this type of control deteriorates












significantly, even instability can be induced. Hence,

dynamic effects have to be included in the mathematical

model and compensated for to obtain smooth and accurate

response. This has been the main motivation for researchers

to work on the dynamics of manipulators for almost 20 years.

In 1965, Uicker was the first to derive dynamic equations

of general closed-loop spatial chains using Lagrange

equations [55]. In the same year, Hooker and Margulies

applied the Newton-Euler formulation to multi-body

satellite dynamics [20]. Later, in 1969, Kahn and Roth

were the first to obtain equations of motion specifically

for open-loop chains using the Lagrangian approach [22].

Stepanenko and Vukobratovic applied the Newton-Euler method

to robotic mechanisms in 1976 [46].

Even the derivation of closed-form dynamic

equations for two 6-link manipulators was considered to

be an achievement in the field, as referenced in [64].

Since these equations are highly nonlinear, coupled, and

contain a relatively large number of terms, later work

concentrated on computer implementation and numerical

construction of dynamic equations. Then, solutions to both

forward and inverse problems were obtained numerically on

digital computers. Since then numerous techniques have

been developed to find efficient algorithms.

Hollerbach derived recursive relations based on the

Lagrangian approach [19]. Orin et al. [371, Paul et al.












[39], and Luh et al. [34] gave efficient algorithms using

the Newton-Euler formulation. Thomas and Tesar introduced

kinematic influence coefficients in their derivation [53].

In a series of papers [37, 43, 46, 56],Vukobratovic et al.

derived the dynamic equations using different methods.

Later, Vukobratovic gathered this work in [57]. Walker and

Orin compared the computational efficiency of four

algorithms in forming the equations of motion (for dynamic

simulation) using the recursive Newton-Euler formulation

[60]. Featherstone used screw theory in the derivation of

dynamic equations and gave various algorithms for the forward

and inverse problems [10].

The main goal in these studies wasto compute the

dynamic effects in real time. Efficient software coupled

with the revolutionary developments in microprocessors,

today, almost achieved this goal. Use of array processors

in real time dynamics evaluation was studied in [61].


1.3 Previous Work on the Control of Manipulators

1.3.1 Hierarchical Control Stages

In the next stage, questions concerning the control

of manipulators are raised. The following control levels

are frequently mentioned in the literature [45, 58]:

1. Obstacle Avoidance and Decision Making

2. Strategical Level

3. Tactical Level

4. Executive Level












Obstacle Avoidance and Decision Making, or the

so-called highest level control, basically lends itself to

Artificial Intelligence. Here, the ultimate goal is to

reproduce and build human intuition, reasoning, and reaction

into machines. Although that goal has not been achieved yet,

limited subproblems have been solved mostly with the use of

vision systems and sensor technology. Currently, the

human himself has to make almost all intelligent decisions

to operate industrial manipulators. The Strategical Level

receives information from the first level and generates

consistent elementary hand movements, whereas the motion of

each degree of freedom of the manipulator is decided for each

given elementary motion in the Tactical Level. The

Executive Level, in turn, executes the Tactical Level

commands.

It should be noted that the second and third control

levels involve only the kinematics of manipulators and that

it is at the fourth level that all dynamic effects are taken

into account in the control of manipulators. In the following

review, the lowest level of control, the so-called Executive

Level, is considered.

Position control of serial manipulators is studied

in a variety of ways. Due to the complex structure of the

system dynamics, most approaches assume rigid links,

although some manipulators may exhibit structural flexi-

bility. The rigid link assumption is justified, because












the dynamics and control of rigid manipulators need to be

understood precisely before the flexible case can be solved

[12, 58]. Also, external disturbances are almost always

neglected. Actuator dynamics is usually not taken into

account; rather, actuators are represented by their

effective torques/forces acting at each joint. These

torques/forces may be generated by electrical, hydraulic,

or pneumatic motors; however, in all cases they cannot be

assigned instantaneously; thus such models are not

physically realizable.

Very few works in the literature include actuator

dynamics in the mathematical model. In [38], actuator

torques are assumed to be instantaneously controllable, but

approximation curves are used to account for the loading

effects and friction of the actuators. Electric and

hydraulic motors are represented by linear, time-invariant,

third-order models in [7, 13, 58].


1.3.2 Optimal Control of Manipulators

Synthesis of optimal trajectories for a given task

(reaching-a-target problem) has been studied by several

researchers. Kahn and Roth [22] presented a suboptimal

numerical solution to the minimum-time problem for a 3-link

manipulator. The dynamic model was linearized by neglecting

the second- and higher-order terms in the equations of motion,












but the effects of gravity- and the velocity-related terms

were represented by some average values.

The maximum principal has also been employed to

solve the optimal control problem [54, 58]. Power-time

optimal trajectories are determined in [54], whereas the

quadratic performance index is chosen in [581. Unfortunately,

this method is hampered mainly because of the dimensionality

of the problem. With the introduction of 2ncostate variables,

4n (24 for 6-link, 6 degree-of-freedom manipulator) nonlinear,

coupled, first-order differential equations are obtained for

an n-link-here also n degree-of-freedom-manipulator,

without considering the actuator dynamics. If initial and

terminal conditions are specified for the manipulator, then

a two-point boundary value problem will result. The

solution to this problem, even on a digital computer, is

quite difficult to obtain. An interesting feature in [54]

is that a numerical scheme is proposed to obtain optimal

solutions for different initial conditions.

In [18], a quadratic performance index is chosen in

terms of the input torques and the error from a given

nominal state. Dynamic equations of manipulators are not

linearized, but error-driven equations are written about

the nominal state. The open-loop optimal control problem

is then solved using a direct search algorithm. Later,

optimal control is approximated by constant-gain, linear












state feedback resulting with suboptimal control. The

proposed feedback controller is invalid, however, if the

deviation of the manipulator state from the given nominal

state is large. This method is applied to a 2-link

manipulator.

Optimum velocity distribution along a prescribed

straight path is studied using dynamic programming [24].

Several optimum path planning algorithms are developed for

the manipulator end-effector. Typically, total traveling

time is minimized while satisfying the velocity and

acceleration constraints [32, 33, 39]. Actually this is a

kinematics problem and since the geometric path is specified

in advance, it does not solve the optimal positioning

problem.


1.3.3 Control Schemes Using Linearization Techniques

For the closed-loop control of manipulators,

linearization of manipulator dynamics has been examined by

several authors. In this approach, typically, dynamic

equations are linearized about a nominal point and a control

law is designed for the linearized system. But numerical

simulations show that such linearizations are valid locally

and even stability of the system cannot be assured as the

state leaves the nominal point about which linearization

has been conducted.












Golla et al. [12] neglected the gravity effects

and external disturbances, and linearized the dynamic

equations. Then, closed-loop pole assignability for the

centralized and decentralized (independent joint control)

linear feedback control was discussed.

In [47, 58] spatial, n-link manipulators with rigid

links are considered. In general, 6-link manipulators are

treated, but some examples use n = 3 which is termed as
"minimal manipulator configuration" within the text [58].

Most approaches make use of the linearized system dynamics.

Independent joint control (local control) with constant

gain feedback and optimal linear controllers are designed

for the linearized system. Force feedback is also

introduced in addition to the local control when coupling

between the links is "strong" (global control). However,

numerical results for example problems show mixed success

and depend on numerical trial-and-error techniques.

Kahn and Roth linearized the dynamic equations of a

2-link manipulator and designed a time-suboptimal controller

in [22]. Since the linearized model was only valid

locally, he concluded that average values of the nonlinear

velocity-related terms and gravity effects had to be added

to the model to guarantee suboptimality.

Whitehead, in his work [62], also linearized the

manipulator dynamics and discretized the resulting equations












sequentially at nominal points along a specified state

trajectory. Then, linear state feedback control was applied

to each linearized system along the trajectory. An

interesting aspect of this work was the inclusion of the

disturbance rejection feature in the formulation. Later, a

numerical feedback gain interpolation scheme was proposed

and applied to a 3-link, planar manipulator. Yuan [67]

neglected the velocity related-terms and the gravity loads,

and then linearized the remaining terms in the equations of

motion. Later, he proposed a feedforward decoupling

compensator for the resulting linearized system.

In general, once the manipulator dynamics is

linearized, all the powerful tools of linear control theory

are available to design various controllers. However,

since almost all practical applications require large

(and/or fast) motions, as opposed to infinitesimal movements

of manipulators, linear system treatment of robotic devices

cannot provide general solutions. Even a global stability

analysis cannot be conducted. If the worst-case design

is employed for some special manipulators, this in turn

will result with the use of unnecessarily large actuators,

hence, waste of power.


1.3.4 Nonlinearity Compensation Methods

Another approach in the literature uses nonlinearity

compensation to linearize and decouple the dynamic equations.












Such compensation is first used in [16] for the linearization

of 2-link planar manipulator dynamics. In this method,

typically, the control vector is so chosen that all

nonlinearities in the equations are canceled. Obviously,

under this assumption and with the proper selection of

constant gain matrices, a completely decoupled,

time-invariant, and linear set of closed-loop dynamic

equations can be obtained [11, 13, 17, 35, 67].

All nonlinear terms in the control expression are

to be calculated off-line [11]. Hence, a perfect

manipulator which is "exactly" represented by dynamic

equations and infinite computer precision are assumed

[5]. On-line computation of nonlinear terms is proposed

in [17], but the scheme requires (on-line) inversion of

an n x n nonlinear matrix other than the calculation of all

nonlinear effects. Generation of a look-up table is

suggested in [13], but dimensionality of the problem makes

this approach impractical. This scheme is applied only

to 1- and 2-link planar manipulators in [13].

Again, since the stability analysis of the resulting

locally linearized system is not sufficient for the global

stability of the actual, nonlinear system, these approaches

do not provide general solutions to the manipulator control

problem.












Several other controllers have also been designed.

Force-fedback control of manipulators is studied in [65].

Proposed diagonal force-feedback gain matrix uses the

measured forces and generates modified command signals.

This method is simple for implementation, but gains must be

selected for each given task and affect the stability of

the overall system. Variable structure theory is used in

the control of 2-link manipulators [68]. However, the

variable structure controller produces an undesirable,

discontinuous feedback signal which changes sign rapidly.

Centralized and decentralized feedback control of a flexible,

2-link planar manipulator is examined in [4].


1.3.5 Adaptive Control of Manipulators

Although the work on adaptive control theory goes

back to the early 1950s, application to robotic manipulators

is first suggested in the late 1970s. Since then a variety

of different algorithms has been proposed. Dubowsky and

DesForges designed a model reference adaptive controller

[8]. In their formulation, each servomechanism is modeled

as second-order, single-input, single-output system,

neglecting the coupling between system degrees of freedom.

Then, for each degree-of-freedom, position, and velocity

feedback gains are calculated by an algorithm which

minimizes a positive semi-definite error function utilizing












the steepest descent method. Stability is investigated for

the uncoupled, linearized system model.

Takegaki and Arimoto proposed an adaptive control

method to track desired trajectories which were described

in the task-oriented coordinates [50]. Actuator dynamics

is not included. In this work, an approximate open-loop

control law is derived. Then, an adaptive controller is

suggested which compensates gravity terms, calculates the

Jacobian and the variable gains, but does not require the

calculation of manipulator dynamics explicitly. However,

nonlinear, state variable dependent terms in the manipulator

dynamic equations are assumed to be slowly time-varying

(actually assumed constant through the adaptation process)

and hence manipulator hand velocity is sufficiently slow.

Although this assumption is frequently made in several other

works [1, 8, 21, 48, 66], it contradicts the premise, i.e.,

control of manipulators undergoing fast movements.

In [21] adaptive control of a 3-link manipulator is

studied. Gravity effects and the mass and inertia of the

first link are neglected. Also, actuator dynamics is not

considered. Each nonlinear term in the dynamic equations

is identified a priori, treated as unknown, and estimated

by the adaptation algorithm. Then, the manipulator is

forced to behave like a linear, time-invariant, decoupled

system. For the modeled system and the designed controller,












stability analysis is given via Popov's hyperstability

theory [26, 27, 28, 42]. Recently, Anex and Hubbard

experimentally implemented this algorithm with some

modifications [1]. System response to high speed movements

is not tested, but practical problems encountered during

the implementation are addressed in detail.

Balestrino et al. developed an adaptive controller

which produces discontinuous control signals [3]. This

feature is rather undesirable, since it causes chattering.

Actuator dynamics is not included in the formulation.

Stability analysis is presented using hyperstability

theory. Stoten [48] formulated the adaptive control

problem and constructed an algorithm closely following the

procedures in [291. Manipulator parameters are assumed to

be constant during the adaptation process and the algorithm

is simulated only for a 1-link manipulator.

Lee [30] expressed the dynamics in the

task-oriented coordinates, linearized and then discretized

the equations without including the motor dynamics. All

parameters of the discretized system (216 for 6-link

manipulator) are estimated at each sampling time using a

recursive least squares parameter identification algorithm.

Optimal control is then suggested for the identified system.

Stability analysis is not given in this work. The main












drawback in this adaptive control scheme is the large number

of the parameters to be identified. In general, all

estimation methods are poorly conditioned if the models

are overparameterized [2]; here the whole model is

parameterized. Koivo and Guo also used recursive parameter

estimation in [25].


1.4 Purpose and Organization of Present Work

In this work, trajectory tracking of serial, spatial

manipulators is studied. The plant (manipulator) and the

reference model, which represents the ideal manipulator,

are both described by nonlinear, coupled system equations,

and the plant is forced to behave like the reference model.

This is achieved via the second method of Lyapunov, and it

is shown that the proposed controller structures are

adaptive. All the previous works known to the author

typically choose a time-invariant, decoupled, linear system

to represent the reference model, and force the nonlinear

plant to act like the linear reference model.

Due to the nonlinear and coupled nature of the

manipulator dynamics, most of the works fail to supply a

sound stability analysis in studying the dynamic control

of manipulators. Design of controllers in this study is

based on the global asymptotic stability of the resulting

closed-loop systems. Implementation of controllers in hand












coordinates and inclusion of actuator dynamics are also

addressed.

The mathematical model of n-link, spatial, serial

manipulators with adjacent links connected by single

degree-of-freedom revolute or prismatic joint pairs is

presented in Chapter 2. Dynamic equations are derived

using the Lagrange equations. Various definitions of

adaptive control are reviewed, and the design of adaptive

control laws utilizing the second method of Lyapunov is

given in Chapter 3. Basic definitions of stability and the

main theorems concerning the second method of Lyapunov are

also included in this chapter to maintain continuity.

Following a brief introduction to hyperstability, it is

shown that the globally asymptotically stable closed-loop

systems are also asymptotically hyperstable.

In Chapter 4, manipulator dynamics is expressed in

hand coordinates and an adaptive controller is proposed for

this system. As pointed out earlier, inclusion of actuator

dynamics is essential in application, since actuator

torques cannot be assigned instantaneously. Actuator

dynamics is coupled with the manipulator dynamics in

Chapter 5. Each actuator is represented by a third-order,

time-invariant, linear system and the coupled system

equations are formed. Then, a nonlinear state

transformation is introduced to facilitate the controller












design. Simplified actuator dynamics is also introduced

which modeled each actuator as a second-order,

time-invariant, linear system. It is shown that the

controllers given in Chapter 3 can be extended for these

systems. A disturbance rejection feature is also added

through integral feedback.

Chapter 6 presents the computer simulations

performed on 3-link, spatial and 6-link, spatial industrial

(Cincinnati Milacron T3-776) manipulators. Effects of poor

manipulator parameter estimations, controller implementation

delays, measurement delays and the integral feedback on

system response are illustrated. Finally, the conclusions

derived from this work are summarized in Chapter 7.
















CHAPTER 2
SYSTEM DYNAMICS


2.1 System Description

In this study n-link, spatial, serial manipulators

are considered. Adjacent links are assumed to be connected

by one degree-of-freedom rotational, revolute or

translational, prismatic joints. This assumption is not

restrictive, since most kinematic pairs with higher degrees

of freedom can berepresented by combinations of revolute

and prismatic joints. Hence, an m degree-of-freedom

kinematic pair may be represented by mI revolute and m2

prismatic joints, where m = m1 + m2.

The mathematical model also assumes that the

manipulator is composed of rigid links. Actually,

manipulators operating under various payloads and external

forces experience structural deflection. In addition,

transient phenomena such as system shocks introduce

vibrations in the small which are low magnitude, oscillatory

deformations about the mean motion equilibrium.

However, inclusion of deflection effects in the

formulation increases themodel dimensionality and further

complicates the system dynamics. It should be noted that the

dynamic equations of rigid-link manipulator models are












highly nonlinear, coupled, and contain a relatively large

number of terms and that currently industrial manipulators

completely ignore the nonlinear and coupling effects in

their control schemes. Hence, here the rationale is first

to understand precisely and solve the control problem for

manipulators with rigid links and then include deformations

in the formulation in later steps. Also, possible backlash

at joints and connecting gear systems are not included in

the mathematical model.

Link j is powered by an actuator mounted on link

(j-1), j = 1,2,...,n. Here the 0th link is the ground or the

support to which the manipulator is secured, the n link

is the outermost link in the chain which will be called

the hand or gripper of the manipulator. Initially actuator

dynamics is omitted and the effects of actuators are

represented by their resultant torques T. applied by the

(j 1) link on the j link; that is, actuator torques

are considered to be the control variables. Again, this

model is not realizable, since actuator torques cannot be

assigned instantaneously. However, this model is still used

because of its simplicity for the proposed control law

presentation. Later, various actuator models are presented,

their dynamics are coupled with the manipulator dynamics,

and it is shown that the developed control laws can be

extended for this system.












Aside from deformation, which is also payload

dependent, and backlash, most, if not all, currently

available industrial robot arms can be represented with the

proposed manipulator model.


2.2 Kinematic Representation of Manipulators

Associated with each one degree-of-freedom joint i,

joint axis is defined by unit vector s., i = 1,2,...,n.

For revolute joints, joint variable 0i (relative joint

rotation) is measured about s.. Joint variable s. (offset

distance) is measured along s. for prismatic joints.

Obviously, if the kth joint is revolute, then the

corresponding offset distance sk will be constant. In order

to distinguish the joint variables from constant manipulator

parameters, constant offset distances are denoted by double

subscripts skk for all revolute joints. Similarly, if the
th .
m joint is prismatic, relative joint rotation will be

denoted by 0mm which is constant.

In order to represent the joint variables

independent of the manipulator joint sequence, these

variables are compactly given by an n-dimensional generalized

joint variable vector e for an n degree-of-freedom robot

manipulator. Consider an n degree-of-freedom arm with its

links connected by revolute-prismatic-revolute-...-revolute

(RPR...R) joints sequentially. For this arm, generalized

joint variable vector e will then be given by













1= 1s203 T nT


Link j connects the j and (j + 1) joints and

it is identified by its link length r. and the twist angle

a. as depicted in Figure 2.1. Note that according to this

conventionrn can be chosen arbitrarily and an is not defined

for the last link-the hand of the manipulator.






r



s = s
k j+1


S.
-..j


Figure 2.1 Link Parameters r. and aj











In Figure 2.1, s., sk, and r. are unit vectors and

r. is the perpendicular distance between joint axes s. and

sk. Hence, associated with each link j, unit vector r.,

and with each joint j, unit vector s. are defined, where

r j is.

For a manipulator of n links, (n + 1) dextral

reference frames are defined. Manipulator parameters and

reference frames are shown in Figure 2.2. Fixed reference

frame F0 defined by the basis vectors u) ,u 0,u3 () is

attached to the 0 -link, the ground; u3 lying along

Orientation of u0 and 0) is arbitrary. One dextral,

body-fixed reference frame F. is also attached to each link
*(j) A(j) A()
j. Frame F. is defined by its basis vectors 4ul ,u2 ,u3 '
-(j) c k 1(j)
u is chosen coincident with r. and u- with s.;
1 J 3
j = 1,2,...,n.

If a vector a is expressed in the jth reference

frame, its components in this frame will be given by a

column vector a (j). If the superscript (j) is omitted,

i.e., a, it should be understood that the vector is expressed

in the ground-fixed F0 frame. Now, it is important to note

that the unit vectors r. and s. expressed in their body-fixed
J 3
frame F. will have constant representations given by
3

r~j) = (1 0 0)T and s(j) = (0 0 1)T (2.1)
-3 -3















^ (ji)
r., U1
Ju


^S U3
sj, u3


-(j)
2


S11


'22


---^(02


(o)
u1


Figure 2.2 Kinematic Representation of Industrial Manipulator












Let a be a given vector. Again, a(j) and a will

represent expressions of a in frames F. and Fo, respectively.

Transformation relating a() to a is given by


(2.2)


a = Ta (j)


Recognizing that r. = T.rj), s. = T.sj), that uJ is
-3 3 J J- -] 2
given by s. xr. and using Equation (2.1), it can be shown
that transformation T is given by
that transformation T. is given by
J


T. = Fr.


s. xr.
-3 -3


Noting that T1 is given by


T1 =




r. and s. can be
-3 -J


cosO1 -sine9 0

sine1 cose1 0

0 0 1


determined recursively from


cose.
J


r. = T_ cosa sine.
-- j-1 j-1 3


sina. sine.


(2.3)


(2.4)


and


(2.5)












0

s. = T. -sinac. j = 2,3,...,n (2.6)
-3 j-1 j-1

cosa.



The reader is referred to [54] for a detailed treatment of

successive rotations of rigid bodies in space.


2.3 Kinetic Energy of Manipulators

2.3.1 Kinetic Energy of a Rigid Body

Consider a rigid body which is both translating and

rotating. Let F0 be a fixed reference frame defined by the
(0) ^(0) (0)
unit vectors u 1 u2 and u3 Let F be a reference

frame fixed to the body at its center of gravity C. Let
(p) (p) (p)
the unit vectors defining F be u) u2 and u3P)

Reference frames are depicted in Figure 2.3. Let also S be

an arbitrary point of the body. One can write


z = z + p (2.7)
s c


S= + /0 (2.8)


where Wp/0 is the angular velocity of F with respect to F0,

v and v are the linear velocities of the related points.
S C
The kinetic energy (KE) of the body can be expressed

as follows:


1 ^ dm
KE = v v dm


















u3


u(p)
3


(0o)
2


S(0)
u1


Figure 2.3 Reference Frame F Fixed on a Rigid


S(p)
U2














Body


where m is the mass of the body. Kinetic energy can also be

expressed as



KE = [v v + 2v (p x p)
m


+ ( x) (W x p)] dm
p/a p/0 P


(2.9)


Noting that, since C is the center of gravity,


(2.10)


p dm = 0
m











Thus,


KE = v *v + -/ ( x p) (ap/ x p) dm
2 c c 2 p p!0
m


(2.11)


or

1 ^ 1 T
KE =- m v v + p dm
2 c c 2 f p// rp/0


(2.12)


where 0 is a dyadic formed by the components of Wp/0 such

that


(q) 3 (q)
(q e (2.13)
%1j3p/0 k1 ikj k,p/0


where the superscript (q) denotes the components expressed

in an arbitrary frame Fq and


+1, if ikj is a permutation of 123*


ikj = -1, if ikj is a permutation of 321


0, if any two of ikj are equal

T
Note that Q0 = -0 is the transpose of p. Hence,
%p/0 3 p/} 0 ip //0

*{123, 231, 312} is meant.













KE = mv *v p *p dm
2 c c 2 m p/0 "'p/0


(2.14)


On the other hand, it can be shown that


A A A A/
ap/0 ~p/O =-(Wp/0 p/) + p/ p/0


where I is the identity dyadic, i.e., I *r = r.
Ili Al;


(2.15)


Then


1 M^ +1 ^
2 c c 2 p/0


S (p p I pp) dm *ap/0


1 1 ^ ^
KE= m *v +- W J J*
2 c c 2 p/0 p/0


(2.16)




(2.17)


where J is defined as the moment of inertia dyadic, i.e.,


(p p I pp) dm


(2.18)


J = I
% Jm


Note that, since p is fixed in F components of the matrices

P) = J and p = p will be independent of time, and


(pTp I pp ) dm


J- =
Sm


(2.19)












where I is a 3 x3 identity matrix. Furthermore, if the

unit vectors of frame F are along the principal axes, the

matrix J will be diagonal, i.e.,


0 0


(2.20)


where


i = m


(T p 2) dm; i = 1,2,3
*~p~ i;


The kinetic energy of the rigid body can be given as


1 (0)T (0) 1 (p) (p)
KE = 2m y v + p J p/0
2 -c -c f --p/o -p/O


(2.21)


The rigid body described above can be considered to

be the i link of the manipulator, i = 1,2,...,n. Then the

kinetic energy expression for this link becomes


1 (0)T (0) 1 (i)T (i)
KE. =- m. v v + 5 .2 Jw. t. /
i 2 i -c. -c. 2 -i/0 i -i/0
1 1


(2.22)


where


m. is the mass of the i link
1













v(0) is the three-dimensional column vector
1
describing the absolute linear velocity of

the center of gravity of the ith link

expressed in the fixed F0 frame

(i) th
.i/ is the absolute angular velocity of the i
-1/0
th
link, expressed in the i frame F.,

three-dimensional column vector


J. is the 3 x3 inertia matrix of the i link
1
at the center of gravity C. expressed in

the frame F.
1

Total kinetic energy of an n-link manipulator will then be


n
KE = [ KE. (2.23)
i=l 1


Expressions for the absolute linear velocity of the
(0)
center of gravity vci. and the absolute angular velocity
(i)
.i'/ are derived in the following sections.


2.3.2 Absolute Linear Velocities
of the Center of Gravities

Let a manipulator of n links be given displacements

e1,2",..." n. Orientation of the i link, 1 < i < n, can

be considered to be the result of i successive rotations;

the resulting rotation is denoted by Rot: F0 -+Fi. If a is

a vector undergoing these rotations, then












a(0) = T.a(i) (2.24)
I-


where T. is as given by Equation (2.3).

Now, let C. be a fixed point in link i. Position

vector Zci connecting the origin of frame F0 to point C. is

given by



z = s1s + [rk-l rk-l + SkSk] + zc. /0
1 k=2 1 1


(2.25)


where zci/0 is the position vector connecting the origin of

frame F.,0i, to point C., and


z(0) = T. z(i) (2.26)
-c./0. i -c./0i
1 1 1 J


Differentiating Equation (2.25), absolute linear velocity of

point C.,vc., is obtained as follows:


i i
c =j s s + s (rk-1 rk-l




+ s ks ) + c/0 (2.27)












or



= c (s s + s x zc/0 (2.28)
1 j=1 J J j Ci/Oj (2.28)


where Zci/0j is the position vector from the origin 0j of

frame F. to point C. and given by

i
A A A A
z = z z = (r1 r
ci/0j c 3 k=J+ k-1 k-1



+ s k) + z c/0 (2.29)



It is understood that constant offset distance s kk will be

inserted in Equations (2.25), (2.27), and (2.29) for sk-if

the kth joint is revolute. Position vectors defined above

are illustrated in Figure 2.4. It should also be noted that
.th
in Equation (2.28) s. is zero if the j joint is revolute;

4. is zero if it is prismatic. Equation (2.28) can be

represented in vector-matrix form as


(0)
v = v = G W
-c. -c. c. (2.30)
1 1 1


where

dO
= dt



















^ (o)
u3





e


^(o)
u


^ (0)
u 2


u(j)
u2


S(j)
u 1


^ (i)
u3


^ (i)
u2


Figure 2.4


Illustration of Position Vectors


G R3xn, its j column defined by
I


Sj xzci/0j, j
s. j

, otherwise


(2.31)


[Gci1j











where 0 denotes a three-dimensional null column vector.

For an n-link RRPRP... arm, Gc4 R3xn for example, will take

the form


G = s xz / s x z s s xz Q 0 ... 0]
Gc4 [ 1 c 4/01 -2 c 4/02 -3 4 Zc 4/04 0 ... 01



Thomas and Tesar defined these position-dependent terms

[Gci]. as translational first-order influence coefficients

[53].

Now, considering that the arbitrarily chosen point

C. actually represents the center of gravity of the link i,

linear absolute velocity of link i is then given by Equation

(2.28) or Equation (2.30).


2.3.3 Absolute Angular Velocities of Links

Absolute angular velocity of link i is given

by


Wi/0 = /0 + '2/1 + + i-1/i-2 +i/i-1

(2.32)


i/0 = i s + s2 + ... + i-i s-i + $i si

(2.33)


Recalling Equation (2.24), any vector a can be expressed in

frame Fi, provided that its representation in frame F0 and












the related transformation matrix T. are given. The reverse

of this transformation is also always possible, since the

transformation represented by T. is orthogonal. Hence,


a(i) = T a (0)= TT a) (2.34)


Rewriting Equation (2.32) in vector-matrix form



i/ = (0i/ = [ pj s. (2.35)
j=l


or


(0)
Wi = G. w (2.36)


where the jth column of G. eR3xn is defined as
1

sj., j < i and i joint revolute
[G.] = (2.37)
[Gi 0 otherwise


Using Equation (2.34), Wi/0 can also be expressed in frame

F.
1

(i) TT (0) (2.38)
-i/0 1 -i/O


(i) T. s. (2.39)
-i/ j=1












or in more compact form


(i) = G. ) (2.40)
-1/0 1 -


where the j th column of G. E:R3xn is now defined by


T th
TT s., j < i and i joint revolute

i J 0 otherwise


(2.41)


Similar G. matrices are used in [53] and termed as

rotational first-order influence coefficients.


2.3.4 Total Kinetic Energy

Total kinetic energy expression for an n-link

manipulator follows from Equations (2.22) and (2.23)


KE (0)T (0) + 1 (i)T j. (i)1
2 i -c -c. 2-i/0 i -i/0
=1 1 1

(2.42)


Absolute linear velocities of the center of gravities v
-c.
(i) 1
and the absolute angular velocities w-/0 are determined as

linear functions of the generalized joint velocities w within

the previous sections. Substituting Equations (2.30) and

(2.40) into Equation (2.42), the kinetic energy expression

becomes






40





1 T I ni)T (i)
KE = [m.G G + G J G
12 ci c. I 1 --
i=1 1 1

(2.43)


Defining


A T
A n [m. GT Gc
Ip i c C.
P i=1 1 1


+ G(i)T (i)]
1 i 1


Equation (2.43) becomes


1 T
KE = A w_
2 p -


where A = A (6) is an n xn symmetric, positive definite,
P P
generalized inertia matrix of the manipulator [54].


2.4 Equations of Motion

Equations of motion will be derived using the

Lagrange equations which are given by


d IKE 3KE _
dt (r kk k


where


9k, k = 1,2,...,n are the generalized coordinates


dOk
k dt


(2.44)






(2.45)


(2.46)












KE = KE (O,w) = KE (81 62,. 'n, ,W 2,...,In)

is the kinetic energy of the manipulator,

and


Qk is the generalized force associated with

the kth generalized coordinate


Derivation of the generalized force expressions is

given in the following section. Once these expressions for

Qk are obtained, dynamic equations of the manipulator will

directly follow from Equation (2.46).


2.4.1 Generalized Forces

The expressions for generalized forces Qk are derived

by subjecting all generalized coordinates ek to virtual

displacements 6ek and forming the virtual work expression.

The coefficients of 68k 's in this expression constitute

the generalized forces by definition.

Now, let all the externally applied forces acting

on link i be represented by the resultant force f., and

all moments acting on the same link by m.. Here, it will

be assumed that f. acts through point C. in link i. This

point can represent any point in the link, however, for

the current presentation, restriction of point C. to be the

center of gravity of the i link will suffice.

Virtual work 6W done by the force f. and moment m.
1 1


is given by











A A A A
SW = f. *v 6t + m. Wi/0 6t (2.47)
1 C. 1 i/O


where the virtual displacement of link i is W.i/ 6t and that

of point Ci is 6zci = Vci St. Representing vectors in

frame F0, Equation (2.47) becomes


(W = fT G w 6t + mT G. w 6t (2.48)
-1 C. -i 1 -
1


where Gci and Gi are as defined by Equation (2.30) and

Equation (2.36), respectively. Letting 6Wk denote the

resulting virtual work due only to the variation in ek'


6Wk = Qk 5ek (2.49)


and


W = fT [ + mT [Gi] 6 (2.50)
k ci] k -1 k k


where [Gci]k is given by Equation (2.31) and [Gi]k by

Equation (2.37). Hence, generalized force Qk is given by


Q = fT [Gcik + mT [Gk (2.51)


If external effects are represented by gravity loads,

actuator torques, and viscous friction at the joints, then

virtual work 6Wk due to 66k will be











n
6W = k m. ga 6z
k j=k 3 a j,k


a- 566 + T 68
3 k k k k


(2.52)


where


ga : the gravitational acceleration

vector


c.
6z e 6
c k k


(2.53)


Tk : the torque applied on the i link by

the (i-l)th link


- = yk k where Yk is the coefficient of

viscous damping at the k joint and


(2.54)


1 2
S= 1 yi i
i=l


r is the Rayleigh's dissipation function. Similarly,



6Wk = j mn g--a [Gcj k k k k 66k
kk ]ka k- W wk + kJ k


Thus, related generalized force will be


n -
Qk = m [Gcj] k wk k
j=k k


(2.55)


(2.56)











Note that Equations (2.52), (2.55), and (2.56) assume that

the payload is included in the mass of the last link m n.

Payload or any other external effect can be separately

represented in the formulation as given by Equation (2.51).

Defining



g = m g [G ] (2.57)
j=k 3 3


the generalized force Qk becomes


Qk = k 7k 'k + Tk (2.58)


where


k = gk(), k = 1,2,...,n


2.4.2 Lagrange Equations

Total kinetic energy expression inEquation (2.45)

can be written in indical notation, repeating indices

indicating summation over 1 to n.


1
KE = A Wi j (2.59)
Pij


Apij denotes the element (i,j) of the generalized inertia

matrix A Then,


E (A W. 6 + A W. 6. ) (2.60)
awk pij j ik p.. 1 jk












where


F1 if i = k


6ik

0 if i 3 k


KE 1 (A
3k 2 pkj j


(2.61)


(2.62)


+- Aik i)
Pik


Since A is symmetric,
P


-KE A .
W k Pki '


(2.63)


Introducing Equations (2.63), (2.45), and (2.58) into

Equation (2.46)

BA
d .1
(Aki ) i j = gk Yk k + Tk

(2.64)


Noting


d (A
dt pki


W i) = A
Pki


1i + A p i
Apki


where ( ) represents differentiation with respect to time


and












Ak
Pki


3A
Pki
j 3


(2.65)


Equation (2.64) becomes

9A 9A
Pki 1 Pij
A Wi + W. W.
Pki 3 2 k


= gk Yk wk k


(2.66)


Defining


(2.67)


D Pki 1 Pij
ijk 36. 2 36
2 k


where D* =


[D jk] E Rnxnxn, equations of motion are given


A p i + D =
pki 1 jk Wi k gk


- Ykkk + Tk


Now, D* can be replaced by Dijk, D = [Dijk], such that
ijk ijk ijk


Djk i w. = Dijk wi j
ijk 1 j ijk 1 J


holds [53]; D..ijk is defined by
1jk

Dijk = m [H ] [G ]k+ [H]T, J. [GT]k
13 z C i,j c k i 9 Pk


+ [G] T J
i


([G ]k x [G ] j)


(2.68)


(2.69)


(2.70)
















[H I [G ]1.
c ij e. c P


[H ].i, j
C0 1,J3


[H ] -


S x (sj k /0)


i,j revolute


sj x (s. x z ),
-j -C /


j

i,j revolute


Sj x S.
-J -1


i prismatic,

j revolute


s. x s.
-1 -]


i revolute,

j prismatic


, otherwise


[Ge]


s. xs., i< j < ; i,j revolute
-1 -


, otherwise


where


(2.71)


(2.72)


(2.73)


[H ] i,j


(2.74)


, i < j < ;






48





[Gc]k is given by Equation (2.31) and [G ]k by Equation

(2.37). H- and Hck are called second-order rotational and

translational influence coefficients [53]. Again, the

repeated index in Equation (2.70) indicates summation

over 1 to n. Also defining Dk nxn


Dk = [D ijk = [Dijk]; i,j = 1,2,...,n (2.75)


with Dijk as given by Equation (2.70), dynamic equations

finally take the form

T
A e. = w Dk yk ak + gk + Tk


k = 1,2,...,n (2.76)


or

T
WT D1 W

T
WT D. 2




n -
A where + g + (2.77)



w Dw



where


A = A (), Dk = Dk(e)
p p- k Dk~












[y] 6 Rnxn is the diagonal matrix containing

the coefficients of viscous

friction


= g(6) e Rn denotes the equivalent

gravitational torques due to the

mass content of the system as seen

at the joints


T e Rn represents the actuator driving

torques
















CHAPTER 3
ADAPTIVE CONTROL OF MANIPULATORS


3.1 Definition of Adaptive Control

According to Webster's dictionary, to adapt means

"to adjust (oneself) to new circumstances." Adaptive

control, then, in essence, is used to mean a more

sophisticated, flexible control system over the conventional

feedback systems. Such a system will assure high

performance when large and unpredictable variations in the

plant dynamic characteristics occur.

In the literature, however, a common definition of

adaptive control is still missing. Astrom defines adaptive

control as a special type of nonlinear feedback control [2].

Hang and Parks give the definition for model reference

adaptive control as follows:

The desirable dynamic characteristics of the
plant are specified in a reference model and
the input signal or the controllable parameters
of the plant are adjusted, continuously or
discretely, so that its response will duplicate
that of the model as closely as possible. The
identification of the plant dynamic performance
is not necessary and hence a fast adaptation
can be achieved. [15, p. 419]

Landau defines

An adaptive system measures a certain index of
performance using the inputs, the states, and
the outputs of the adjustable system. From the
comparison of the measured index of performance












and a set of given ones, the adaptation mechanism
modifies the parameters of the adjustable system
or generates an auxiliary input in order to
maintain the index of performance close to the
set of given ones. [29, p. 13]

Gusev, Timofeev, et al. [14] include artificial intelligence

and decision making in adaptive control.

In this study adaptive control is defined as

follows:

Definition 3.1: A feedback control system is

adaptive, if gains are selected with the

on-line information of plant outputs and/or

plant state variables along with the nominal

(reference) inputs, nominal outputs and/or

nominal state variables.

This definition is illustrated in Figure 3.1. It

should be noted that the definition given here is in

agreement with the above definitions; it is more specific

than Astrom's and more general than Hang's or Landau's.





U --r. Output
x --- Regulator --- Plant
-r .
ZrIzI


Figure 3.1 Block Diagram Representation of
an Adaptive Control System












Early works on adaptive control, which were

essentially experimental, date back to the 1950s. Later,

advances in the control theory in 1960s and the recent

revolutionary developments in microelectronics matured the

adaptive control theory and its applications considerably

compared to its early stages.

Mainly three approaches are identified in adaptive

control: Gain Scheduling, Model Reference Adaptive Control

and Self-tuning Regulators (Parameter Estimation Techniques).

Block diagram representations of these schemes are given in

Figures 3.2-3.4.


Gain
Scheduling

I i


Figure 3.2 Block Diagram of Gain Scheduling System













i Model I



i A-Adjustment --.. i
I Mechanism




-r
S- Regulator Plant

t



Figure 3.3 Block Diagram of Model Reference Adaptive System





Parameter
I Estimation




Regulator
Design
[ ,


Block Diagram of Self-tuning Regulator


Figure 3.4












All these block diagrams in Figures 3.2-3.4 can be

reduced to the block diagram in Figure 3.1 simply by

shrinking the dotted boxes into the variable regulator in

Figure 3.1.


3.2 State Equations of the Plant
and the Reference Model

3.2.1 Plant State Equations

Defining the state vector x = ( -I- ) where
-p p -p
e c Rn and a e Rn are the generalized relative joint
-p -p
displacement and velocity vectors, respectively, dynamic

equations derived in the previous chapter can be given as

follows:



0 = + u (3.1)
-P 1G -A -1 -
A_ G A F A P



where subscript p stands for "plant," here manipulator

represents the plant,


x = x (t) = ( T x ) R2n (3.2)
-p -p -pl E


xpl = 6p(t), x = 3p(t) (3.3)
-pl p -p2 -p

dx (t) T T T
= -) xT x (3.4)
-p dt -pl -p2












I and 0 denote the n xn identity and null

matrices, respectively


Referring to Equations (2.76) and (2.77),


A = A (x) Rnxn
p p(-pl


S(x pi) = G x p = G (x .)x p
-ap-pl' p-pl p -pl -pl


Gp= Gp (xpl)


f = fp(p x ) = -
-p -p pl -p2









F = F (x, xp ) = -
Fp =p (Xpl, -p2)


SRnxn, gp(pl) E Rn


f (x x ) = F 2 x = F p(x xp2)
-p -pl' ~-p2 p -p2 p -p1 -p2


T
x
-p2



T
x p
_-p2



T
x
-p2



T
-p2


Xp2


D (x ) x
l -pl -p2



D (x ) x
n -pl -p2


(3.7)


(3.8)


Rn


(3.9)


D1 (pl)



Dn (pl)


SRnxn


(3.10)


(3.11)


u = u (t) = T (t) e Rn
-p -p -p


(3.5)


(3.6)












T (t) represents input actuator torques,
-p

n is the number of links of the manipulator

(here also an n-degree-of-freedom

manipulator)


Note that A G and F are not constant; Ap and Gp are
p p p p p
nonlinear functions of the joint variables xil' and

F = Fp (x, x p2). In the formulation, functional

dependencies are not shown for simplicity. Also, G (x p

is not defined explicitly; symbolically, G (x p) is such

that G (xpl)xpI = g holds. External disturbance terms

and the joint friction effects are not shown in the above

formulation.


3.2.2 Reference Model State Equations

Having defined the plant equations-Equation

(3.1)-reference or model state equations which represent

the ideal manipulator and the desired response are given by


0 I 0
x = xr + r (3.12)
A r Ar F



where


subscript r represents the "reference" model

to be followed,












x is the state vector for the reference
--r

system


x = x (t) = (x x ) R2n (3.13)
-r -r -r 1 2-r2


x = _r(t) e Rn, xr = _r (t) e Rn (3.14)
-r -r -r -r

dx (t) TT
S dt-r T *T) (3.15)


Again, referring tothe manipulator dynamic equations, i.e.,

Equations (2.76) and (2.77),


A = A (x r) Rnxn is the generalized

inertia matrix for the reference

system


j(x ) = G x =G (x )x(3.16)
rrl = r-rl = r(rl) -rl (3.16)


Gr = Gr(x) Rnxn, gr(xl ) Rn (3.17)


fr (x r x) = F x = F (xr, ) x (3.18)
-fr -l rr2X = F 2 = Fr-rl r 2 (3 18)


x2 D1(xrl)
f = fr(xr, xr) = Rn

r2 Dn(xrl) -r2_


(3.19)













-r2 D1 rl)

Fr = F(x, x ) = Rn (3.20)

xT D
-r2 Dn


It is important to note that A = A (x r),

G = G (x ) and Fr = Fr (x, x r2) are not constant, but

nonlinear functions of the state vector x In this study,
-r
unlike previous practices, the reference model is

represented by a nonlinear, coupled system, i.e., ideal

manipulator dynamics. All works known to the best

knowledge of the author typically choose a linear, decoupled,

time-invariant system for the reference model and force the

nonlinear system (manipulator) to behave like the chosen

linear system.


3.3 Design of Control Laws via the
Second Method of Lyapunov

3.3.1 Definitions of Stability and the
Second Method of Lyapunov

In this section various definitions of stability

are reviewed. Also, Lyapunov's main theorem concerning

the stability of dynamic systems is given. For a detailed

treatment, the reader is especially referred to the Kalman

and Bertram's work on the subject [23].

Let the dynamics of a free system be described by

the vector differential equation












x = f(x, t), -c < t < + (3.21)


where x Rn is the state vector of the system. Also let

the vector function J(t; x0, to) be a unique solution of

Equation (3.21) which is differentiable with respect to

time t such that it satisfies


(i) 4(t0; x0, to) = x0 (3.22)


(ii) d-t (t; xQ, to) = f(O(t; xQ, to), t) (3.23)


for a fixed initial state x0 and time t0.

A state x is called an equilibrium state of the
-e
free dynamic system in Equation (3.21) if it satisfies


f(x t) = 0, for all t (3.24)


Precise definition of stability is first given by

Lyapunov which is later known as the stability in the sense

of Lyapunov.

Definition 3.2: An equilibrium state x of
-e
the dynamic system in Equation (3.21) is

stable (in the sense of Lyapunov) if for

every real number > 0 there exists a real

number 6(e, to) > 0 such that II x0 xe |e

implies


|I $(t; x0, to) x < e for all t < to

The norm || |represents the Euclidean norm.












In practical applications, the definition of

stability in the sense of Lyapunov does not provide a

sufficient criterion, since it is a local concept and the

magnitude 6 is not known a priori. Stronger definitions of

stability, namely asymptotic stability, asymptotic

stability in the large, and global asymptotic stability,

which are essentially based on the definition of stability

in the sense of Lyapunov with the additional requirements,

are given below. The definition of asymptotic stability

is also due to Lyapunov.

Definition 3.3: An equilibrium state x of
--e

the dynamic system in Equation (3.21) is

asymptotically stable if


(i) It is stable (Definition 3.2)


(ii) Every solution t(t; x0' t0)

starting sufficiently close to x

converges to x as t -> -. In
-e
other words, there exists a real

number p(t0) > 0 such that

1x xII e

lim II1(t; x to) x | = 0
t--*


Definition 3.4: An equilibrium state x of the
dynamic system in Equation (3.21) is-e
dynamic system in Equation (3.21) is












asymptotically stable in the large if

for all x0 restricted to a certain region

r e Rn


(i) x is stable
-e

(ii) lim I| (t; x0, to) x el = 0
t -+-o


Definition 3.5: An equilibrium state x of

the dynamic system in Equation (3.21) is

globally asymptotically stable if the

region r in Definition 3.4 represents the

whole space Rn, i.e., r = Rn.

Lyapunov's main theorem which provides sufficient

conditions for the global asymptotic stability of dynamic

systems and the two corollaries are given below [23].

Theorem 3.1: Consider the free dynamic system


x = f(x, t)


where f(0, t) = 0 for all t. If there

exists a real scalar function V(x, t)

with continuous first partial derivatives

with respect to x and t such that


(i) V(0, t) = 0 for all t


(ii) V(x, t) > a(hix|I) > 0 for all

x 3 0, x e Rn where a(-) is a











real, continuous, nondecreasing

scalar function such that

a(0) = 0


(iii) V(x, t) -- as ||x|l-+- for all t


(iv) dV (x, t) -
dt at


+ (grad V) f(x, t)


< -y (I x |I) < 0


where y(*) is a real, continuous

scalar function such that y(0) = 0

then the equilibrium state x = 0 is globally

asymptotically stable and V(x, t) is a

Lyapunov function for this system.

Corollary 3.1: The equilibrium state

x = 0 of the autonomous dynamic system
--e

x = f(x)


is globally asymptotically stable if there

exists a real scalar function V(x) with

continuous first partial derivatives with

respect to x such that


(i) V(0) = 0


(ii) V(x) > 0 for all x 7 0, x e Rn














(iii) V(x) -+c as I x -+


(iv) V =dV (x) < 0 for all x y 0,



x E Rn


Corollary 3.2: In Corollary 3.1, condition (iv)

may be replaced by


(iv-a) V(x) < 0 for all x 3 0, x e Rn


(iv-b) V(_(t; x0, to)) does not vanish

identically in t > to for any

t and x y 0.


Finally, Lyapunov's following theorem gives the

necessary and sufficient conditions for the (global)

asymptotic stability of linear, time-invariant, free dynamic

systems.

Theorem 3.2: The equilibrium state x of a
--e

linear, time-invariant, free dynamic system


x = Ax (3.25)


is (globally) asymptotically stable if and

only if given any symmetric, positive

definite matrix Q, there exists a symmetric,












positive definite matrix P which is the

unique solution of the matrix equation


AT P + PA = -Q (3.26)


and V = x Px is a Lyapunov function for

the system in Equation (3.25).


3.3.2 Adaptive Control Laws

Plant and the reference model equations are given

by Equations (3.1) and (3.12), respectively. Reference

system control u (t) represents the open-loop control law.
-r
This, for example, may be an optimal control law obtained

off-line through minimization of a performance index.

Due to the error in the initial state, disturbances

acting on the system and the inaccuracies in the

mathematical model such as frictional effects, structural

deflection, and backlash, open-loop control law ur = ur(t)

does not prove effective as the demand on precise and fast

motion increases. Even today's servo-controlled industrial

manipulators which totally neglect the dynamic coupling

use closed-loop control laws.

Now, the aim is to find the structure of the

controller u = u (x (t), x (t), u (t)) such that the
-p -p -p -r -r
desired trajectory is tracked. Defining the error e(t)

between the reference and the plant states












e = e(t) = x (t) x (t) E R2n (3.27)


T TT T T T T T
e = (e e 2) = (x x x x 2) (2.28)


e e R e2 e R" (3.29)
-1 -2


de(t)
e=- (3.30)


and choosing


u = u' + u" (3.31)
-p -p -p


u' = A (A1 G x + A1 F x K, el K2e2)
-p p r r-rl r r-r2 1-1 2-2

(3.32)


where


u" is part of the controller yet to be designed
-p


K K2 6 Rnxn are constant matrices to

be selected


error-driven system equations can be obtained by substituting

Equations (3.31) and (3.32) into Equation (3.1), subtracting

the resulting equation from Equation (3.12) and substituting

Equations (3.27-3.30) as follows:


S-1 -
e = Ae + Bz BA~ u" (3.33)
p -p












where


0 I 0~
A = B = (3.34)
K1 K2



A R2nx2n, B R2nxn


I and 0 are n xn identity and null matrices,

respectively


-1 -= -i
z = -A Gp x A F xp2 + A ur (3.35)


z e R u". e Rn
-p

It should be noted that the part of the controller

u' requires only the on-line calculation of the plant
-p
generalized inertia matrix A = A (x ); other nonlinear
P P -P
terms A- = A (x ),rl Gr = G (x ) and F = F (x ) are

reference model parameters and known a priori for each given
-1
task, i.e., A G and F will not be calculated on-line.
r r r
Various controller structures can be chosen for u"
-P
using the second method of Lyapunov (Theorem 3.1, Corollary

3.1). This method is especially powerful, because it

assures the global asymptotic stability of the error-driven

system, hence the manipulator, without explicit knowledge

of the solutions of the system differential equations. Let












V(e) = eTPe (3.36)


define a real, scalar positive definite function. Using

Equations (3.33) and (3.36),


V(e) = -e Qe + 2v z 2vT A1 u" (3.37)
.. p -p


where


Q e R2nx2n positive definite matrix (Q > 0),


P R nx2n solution of the Lyapunov equation


ATP + PA = -Q (3.38)


and

v = BT P e (3.39)


A discussion on the uniqueness of the solution P of the

Lyapunov equation is given in the following section.

Now, if V(e) < 0 is satisfied, global asymptotic

stability of the error-driven system will then be guaranteed

according to Corollary 3.1. This condition can actually be

replaced by V(e) < 0 in the sense of Corollary 3.2. Also,

V(e) will be a Lyapunov function for the system in Equation

(3.33). Different controller structures are explored below.











3.3.2.1 Controller structure 1

If u" were chosen
-p

u" = A z (3.40)
-p p-

or
II -1
u" = f + A (A u ) (3.41)
-p -p pr -r

where

gp = Gp x p, f = Fp (3.42)


then condition (iv) of Corollary 3.1, V < 0, would be

satisfied. In fact, these choices in Equations (3.40) and

(3.41) correspond to the cancellation of nonlinearities and

can be viewed as the nonlinearity compensation method widely

used in the literature (Chapter 1). However, since this

form of u" assumes exact cancellation of terms a priori,
p
Lyapunov's second method does not guarantee global

asymptotic stability, if cancellations are not exactly

realized.


3.3.2.2 Controller structure 2

Another choice for u" will be
-P

u" = A diag[sgn (v.)] {b + Sk} (3.43)
-p p-


where diag[sgn (v.)] is an n xn diagonal matrix with

diagonal elements sgn (vi), i = 1,2,...,n,












b = sup { -A g + A u }
0 < x < 2 r p r -r
p,1

Ur,i U

i = 1,...,n (3.44)


U is a subset of the set of all possible inputs, within which

open-loop control law u (t) is contained, i.e., u r. U,

i = 1,2,...,n. The generalized inertia matrix A (xpl) is

nonsingular [54], also elements of A A and g are all

bounded, i.e., if


A (x ) = [aij (x pl (3.45)
p -p1 ij -p1


then


(aij) < aij (x ) < (a..) (3.46)


where (a ij) and (a..)u are the lower and upper bounds on

a.ij (x ), 0 < x pl,k < 2w; i,j,k = 1,2,...,n. Similarly,
-1
bounds on the gravity loads g can be given. A u =
-1
A (x (t))u (t) in Equation (3.44) is known for a given
r -rl -r
manipulation task, since it represents the reference.

Referring to Equation (3.43),


S = [s. .] e Rnxn (3.47)


is defined by












s.. = sup
3 0 Xpl, 2
=1, ,n


{ aijl}; i,j = 1,2,...,n


(3.48)


T T *
k = [x K x x K x
-P2 K1 p2 -p2 2 -p2


k e R n


where constant positive definite K* R nxn

so that


T K *
-p2 -p2


T T
x K x I
-p2 n -p2


(3.49)


is to be chosen


-p2 Di -p2


(3.50)


T
SK p2 x > 0 for all x ?
p2 i -p2 -P2


(3.51)


where D., i = 1,...,n is as defined by Equations (2.70) and

(2.75); D. in Equation (3.50) can be replaced by symmetric
1


1 T
D! = 1 (D. + D )
1 2 i i


(3.52)


so that x D! D. x is preserved. Existence
-p2 1 -p2 -p2 p -p2
of positive definite K* is shown using the following theorem
1
[6].

Theorem 3.3: Let M be a symmetric, real matrix

and let min (M) and max (M) be the smallest

and the largest eigenvalues of M,

respectively. Then


and












Smin (M) Ix2 < xT Mx < X max(M)I x!12
mi max -

(3.53)

n
n 2 2
for any x e Rn, where x 2 = \ x2.
i=l

Using Theorem 3.3,


A. (Kx) 2 < T K (K!) x 2
min 11 xp2 112 -p2 i -p2 m< ax I 2 II2

(3.54)


X (D!) x 2 x D! x < X (D!) x2
mmin (D) Ip 2 2 <-p2 1i X2 2 mmax 1 Ip22

(3.55)


Here K* is assumed to be a real, symmetric matrix. If K*
1 1
is not symmetric, then


K*' (K + K*T) (3.56)


must be replaced by K* in Equation (3.54). Also, all
1
entries of D' = D' (x ) are bounded and, in general, D' is
1 1 -pl 1
T
indefinite. Quadratic surfaces x D! x its lower and
-p2 1 -p2i
T D x T *
upper bounds (x D. x ) and (xa D! Xp), and x K. x
-p2 1 -p2 -p2 1 -p2 u -p2 1 -p2
are conceptually represented in Figure 3.5.

If X (K*) is chosen such that
min 1


X (K*) > A (D!)
mm i max 1


(3.57)













xT K*x
-p2 1-p2


T
x D'x
-p2 i-p2


42D ixp2


Figure 3.5


Representation of Quadratic Surfaces


is satisfied, where


ax (D!) =
max I1


sup { .j (D' (x ))
0 4 X < 2 1 -pl
i = 1, ,n


: j = 1,2,...,n}


(3.58)












then


xT K* x > x D! x (3.59)
-p2 i -p2 -p2 i -p2


follows directly from Equations (3.54) and (3.55). In

addition, if XIm (K*) > 0, then xT K. x > 0 for all
mi p2 1 -p2
x 0. That is, symmetric K* E Rnxn is positive definite,
-p2 1
if and onlyif all the eigenvalues of K* are positive [36].
1
One choice for K* which satisfies Equation (3.50)

is


K? = diag[max (D)] (3.60)
1 max 1


where K!, in this example, is a diagonal matrix.

This control described by Equations (3.43)-(3.44),

(3.47)-(3.49) will satisfy Corollary 3.1 and assure the

global asymptotic stability of the manipulator. It should

be noted that b, S, and Ki, i = l,...,n are all constant

matrices, hence its implementation is not computationally

demanding. However, its disadvantage is that the

discontinuous signal due to sgn function will cause

chattering.


3.3.2.3 Controller structure 3

The chattering problem in the above controller will

be alleviated if u" has the form
-p

u" = A Q* v (3.61)
-p p -










where Q* e Rnxn constant, positive definite matrix. In

this case, due to the term in V linear in v(t), i.e., 2vTz,

solution can only be guaranteed to enter a spherical

region containing the origin in the error space [23].

Absolute minimum of V which is not the origin anymore will

lie in this region. In fact, part of the V expression,

V= V'(v)


V = -2vT Q* v + 2vTz (3.62)


will have absolute minimum at


v = (Q*) z (3.63)


In general, this spherical region can be reduced as

the magnitude of u" is increased, which actually translates
-p
into the use of large actuators. This can easily be shown

observing Equation (3.63). Assuming that Q* is the diagonal,

absolute minimum will approach to zero as the magnitudes of

the diagonal elements are increased.

Although this controller eliminates the chattering

problem and is the easiest for implementation, it cannot

completely eliminate the error in the state vector. This

error will be reduced at the expense of installing larger

actuators.


3.3.2.4 Controller structure 4

This controller has the structure


u" = (-K + AK ) x + (Ku + AK ) u(
-p p p -p u u -r


(3.64)












where


Kp = [Gp : Fp] (3.65)



AK = [R1 v (S1 Xp)T : R v (S2 xp2)T] (3.66)



Ku = [A Arl] (3.67)



AKu = [R3 v (S3 ur)T] (3.68)


K and AK e Rnx2n
P P


K and AK s Rnxn


G F and A denote the calculated values

of G F and A given by Equations

(3.6)-(3.7), (3.10), and (2.44),

respectively


R. Rnxn, R. > 0, and (3.69)
1 1


S. e Rnxn, S. > 0, i = 1,2,3; are (3.70)
1 1
to be selected


v is as defined by Equation (3.39)


Let






76




V(e, t) = eTp e + 2 (vTA Rv)(x Sx )dT
0 -p 1- -pl 1-pl




t T -1 T T
+ 2 (vAlR2v) (x 2S'x )dT



+ 2 (vA R ) (urS3ur)dT (3.71)


define a Lyapunov function. Differentiating Equation (3.71)

with respect to time and substituting Equations (3.33),

(3.64)-(3.68), and (3.38) into the resulting expression,

V(e) will be

T T
V(e) = -e Q e + 2v z' (3.72)


where P is the solution of the Lyapunov equation


A P + PA = -Q, Q > 0 (3.73)

and


z = A [(p ) + (f fp)]
p -p -p -p


+ (A- A A A) u (3.74)
r p p r -r

An estimation of the bound of I|e I is given below.

If V(e) is negative outside a closed region r subset

of R2n including the origin of the error space, then all











solutions of Equation (3.33) will enter in this region r

[23]. Substituting Equation (3.39) into Equation (3.72)

T T
V(e) = -e Q e + 2e PB z' (3.75)


V(e) < X in(Q) Iell2 + 21e 11P 111 ||Bz' ii (3.76)


where

x (Q) is the smallest eigenvalue of Q


11* I denotes the Euclidean norm


1 e1l2 = e e (3.77)

SP II = max (P); the largest eigenvalue
of P, since P is positive definite
and symmetric [23]


liz' I = [(z')T z']1/2 (3.78)


Also, recalling Equation (3.34),



Bz' = z' = [T, (z')T (3.79)


where





78




0 denotes then x n null matrix, and


0 E Rn represents the null vector,

11 Bz' I1 = 11 z 1 (3.80)

follows from Equation (3.79). Now, from Equation (3.76),
V(e) < 0 is satisfied for all e satisfying

2 II P I II z'||
IIe 11 > (3.81)
min(Q)

Hence, an upper bound on the error, I e I will be

2 IIP I I z'(3.82)
|ei --ax (3.82)
-emax 4 X sin(Q)
mm

It is clear from Equation (3.82) that this bound on
Il el will be reduced as IIP II is decreased, X in(Q) increased
or I z'0'a --+0. It should also be noted that frequent
max
updating of f and A will affect IIz' -- 0, hence

leimax -+ 0. At steady state, e = 0, control will take
the form

U' (t) = Ur (t) (3.83)
p -r

and

z'- A-1 u" = 0
P -p -












or


z' = 0 (3.84)


hence Equation (3.33) would yield


e = Ae


Controllers presented in this section have the

general form


p = u' + u" (3.85)
-p -p -p


Analysis is given assuming that the calculated A i.e.,
~ -1
A is exact only in the u' part so that A A = I is
p -p p p
satisfied. This assumption is made to facilitate the

analysis. Computer simulations presented later in Chapter 6

did not, however, use this assumption. In the second part

of the controller, i.e., u", calculated terms a, f and

A i.e., f and A are explicitly shown in the

analysis (Controller structure 4). Current arguments with

reference to Equations (3.82) and (3.74) suggest that g
-P
and f may be updated at a slower rate compared to the A .
-p p
This result is important, since especially the calculation

of f in general, requires more computation time compared

to A Although it is clear, the above controllers need the

on-line measurements of plant joint displacements xpl and

the velocities x p2












3.3.3 Uniqueness of the Solution
of the Lyapunov Equation

The Lyapunov equation is given by Equation (3.38).
2nx2n
The uniqueness of its solution P R is guaranteed, if

A e R2nx2n has eigenvalues with negative real parts as

given by the following corollary [6].

Corollary 3.3: If all the eigenvalues of A

have negative real parts, then for any Q

there exists a unique P that satisfies the

matrix equation


A P + PA = -Q


where A, P, and Q e R2nx2n

Recalling Equation (3.34), A is given by


0 I
A =
K1 K2


.2nx2n .
The characteristic equation of A E R2nx2n is


det [sI A] = sn det sI K2 1 K (3.86)


where


I represents a 2n x 2n identity matrix on

the left-hand side of Equation (3.86);

otherwise it is understood that I c Rnxn












s is the complex variable,


K and K2 Rnxn


If K1 and K2 are diagonal matrices


K = diag [K;i ], K2 = diag [K 2;i] (3.87)


where


K and K are the respective diagonal
l;i 2;i
(i,i)th entries of K1 and K2, i = 1,2,...,n


then

n 2
det [sI A] = R (s K .s K ) (3.88)
i=l 2;


that is, the time-invariant part of the error-driven system

(not the manipulator dynamics) will be decoupled. Hence,

referring to Equation (3.88), all the eigenvalues of A will

have negative real parts if K1 ; < 0 and K2; < 0.

Corollary 3.3, then, assures the existence and uniqueness

of the solution of Lyapunov equation.


3.4 Connection with the Hyperstability Theory

In this section, basic definitions and results on

hyperstability are reviewed and it is pointed out that the

globally asymptotically stable closed-loop systems designed












in the previous section (Section 3.3.2) are also

asymptotically hyperstable. It is noted that here only the

necessary results are covered and some definitions are

inserted for clarity. Detailed treatment of the subject

can be found in [29, 42].

The concept of hyperstability is first introduced

by Popov in 1962 [42]. The following definitions of

hyperstability and asymptotic hyperstability are also due

to Popov [29].

Definition 3.6: The closed-loop

system


x = Ax Bw (3.89)


v = Cx (3.90)


w = f(v, t) (3.91)


where


(i) x R2, w R v R A R2nx2n

Be R2nxn, C Rnx2n

A, B, and C are time-invariant,

f(.) cRn is a vector functional

(ii) The pair (A,B) is completely

controllable

(iii) The pair (C,A) is completely

observable











is hyperstable if there exists a positive

constant 6 > 0 and a positive constant

Y0 > 0 such that all the solutions

x(t) = ( (t; x0, to) of Equations (3.89)-

(3.91) satisfy the inequality


|Ix(t) I < 6(||x(0)l + y0) for all t > 0

(3.92)


for any feedback w = f(v, t) satisfying the

Popov integral inequality



n(t tI) = tl w dt > -y2 (3.93)
to
0


for all tI 5 to.

Definition 3.7: The closed-loop system of

Equations (3.89)-(3.91) is asymptotically

hyperstable if

(i) It is hyperstable

(ii) lim x(t) = 0 for all vector
t -.I. -
functionals f(v, t) satisfying the

Popov integral inequality of

Equation (3.93).

Popov's main theorem concerning the asymptotic hyperstability

of the system described in Equations (3.89)-(3.91) and (3.93)

is given below [29].











Theorem 3.4: The necessary and sufficient

condition for the system given by Equations

(3.89)-(3.91) and (3.93) to be

asymptotically hyperstable is as follows:

The transfer matrix


H(s) = C(sI A)-1 B (3.94)


must be a strictly positive real transfer

matrix.

The strictly positive real transfer matrix is defined below.

Definition 3.8: An m x m matrix H(s) of real

rational functions is strictly positive real

if

(i) All elements of H(s) are analytic

in the closed right half plane

Re(s) > 0 (i.e., they do not have

poles in Re(s) > 0)

(ii) The matrix H(jw) + H (-jw) is a

positive definite Hermitian for

all real w.

The following definition gives the definition of the

Hermitian matrix.

Definition 3.9: A matrix function H(s) of the

complex variable s = o + jw is a Hermitian

matrix (or Hermitian) if












H(s) = HT(s*) (3.95)


where the asterisk denotes conjugate.

Finally, the following lemma [29] gives a sufficient

condition for H(s) to be strictly positive real.

Lemma 3.1: The transfer matrix given by

Equation (3.94) is strictly positive real

if there exists a symmetric positive

definite matrix P and a symmetric positive

definite matrix Q such that the system of

equations


A T + PA = -Q (3.96)


C = BTP (3.97)


can be verified.

Recalling the error-driven system equations, Equation

(3.33), closed-loop system equations are given by


e = Ae + Bz" (3.98)


where


z" = z A u" (3.99)
p -p

z is defined by Equation (3.35), A and B are as given by

Equation (3.34). Various controller structures for u" are
-P












given in Section 3.3.2 assuring the global asymptotic

stability of the closed-loop system of Equation (3.98).

Referring to Definition 3.6 and Equation (3.98)


w = -z" (3.100)


The second method of Lyapunov essentially required

that for a positive definite function V(e) = e Pe


V(e) < -e Qe + 2v Tz" (3.101)


is satisfied. Note that Equations (3.38)-(3.39) and (3.98)

are used in obtaining Equation (3.101). If Q is positive

definite, then -Q is negative definite, i.e., -e Qe < 0 for

all e 0. Hence, to satisfy corollary 3.1,


vTz" < 0 (3.102)


is sufficient for the global asymptotic stability of the

system in Equation (3.98).

On the other hand, Theorem 3.4 requires that the

transfer matrix given by Equation (3.94) be strictly positive

real. Lemma 3.1, in turn, requires that positive definite

P which is the solution of the Lyapunov equation, Equation

(3.96), exists and C = B P is satisfied. Noting that Equation

(3.39) defined v = B Pe, both conditions are already required

by the second method of Lyapunov.












However, Theorem 3.4 assumes that the Popov integral

inequality is satisfied. Substituting Equation (3.100) into

Equation (3.93)



-n(t0, t ) = vl z" dt < -y2 (3.103)
0


must hold. But, if vT z" < 0 is satisfied, Equation (3.103)

will also hold. Indeed, Equation (3.103) represents a more

relaxed condition compared to Equation (3.102), but for the

system in Equation (3.98) and z" which is an implicit

function of time, direct use of Popov's condition is not

immediate.

The definition of hyperstability also presumed the

complete controllability and the complete observability of

the pairs (A,B) and (C,A), respectively. These conditions

are checked in the following section.

In view of the above discussions, the closed-loop

system which is globally asymptotically stable will also be

asymptotically hyperstable.


3.5 Controllability and Observability
of the (A,B) and (C,A) Pairs

Definition of hyperstability in the above section

assumed that the pair (A,B) is completely controllable.and

(C,A) is completely observable; A and B are defined in

Equation (3.34). First, for the pair (A,B)












2 I 12nx2n
[B AB A2B ... A2n-B] = ... R2n2
I K2

(3.104)


must have rank 2n for the complete controllability of the

pair (A, B). The controllability matrix, Equation (3.104),

will have full rank 2n, since its first 2n columns will

always span R2n regardless of the choice of matrix

K2 Rnxn. Hence, the pair (A, B) is completely

controllable.

Let P e R 2nx2n, which is the solution of the Lyapunov

equation, be given by




p = (3.105)
P P
L2 31


where Pl. P2, and P3 e Rnxn and P1 and P3 are symmetric.

Then, C e Rnx2n will have the form


C = BTP = [PT PT] (3.106)


For the complete observability of the pair (C, A)



[CT ATC (A )2CT ... (AT)2n-1 C T e R2nx2n2


(3.107)












must have rank 2n. Hence,

T T T
S KP KTP2 +K K P3

[CT AC (A )2C ...] = ...
P P +KTP KT P2 + (KT +KT)P


(3.108)


is supposed to have rank 2n. Since P given by Equation

(3.105) is positive definite, hence nonsingular, first

n-columns of the observability matrix in Equation (3.108)

will be linearly independent. Therefore, a rank of at least

n is assured. Clearly, the rank of this observability

matrix will depend on P2' P3' K and K2. At this stage it

is assumed that P2' P3 of matrix P and the selected K1 and

K2 are such that the (C, A) pair is completely observable.


3.6 Disturbance Rejection

The most important question to be raised of a

control system is its stability. If it is not stable,

neither a reasonable performance can be expected, nor

further demands may be satisfied. As should be clear by

now, in this study, system stability is highly stressed and

actually complete design of the controllers concentrated on

the verification of stability and tracking properties of

the system.












Although stability of a control system is necessary,

it is not sufficient for acceptable system performance.

That is, a stable system may or may not give satisfactory

response. Further demands on a control system other than

the stability will be its ability to track a desired

response, to give acceptable transients and its capability

to reject disturbances. Optimal behavior of the system in

some sense may also be required.

Since global asymptotic stability (also the

asymptotic hyperstability) of the system is assured in the

error space, tracking property is already achieved with

the proposed controllers of Section 3.3.2. Acceptable

transient response will be obtained by the choice of

matrices KI, K2, Q, S., R., i = 1,2,3 as given before.

The main drawback of the designed controllers is

the implicit assumption that the reference model parameters

are exactly the same as that of the actual manipulator.

These parameters include manipulator link lengths, link

offsets, twist angles, link masses, and inertia tensors.

Although close estimations of these constant parameters may

be assumed known a priori, information on their exact

values, in general, will not be available. This

discrepancy will deteriorate the system response. This poor

knowledge of plant parameters, other plant imperfections

which are not represented in the mathematical model,












inaccurate measurement devices, measurement delays, and

delay in the control due to the time required for its

implementation all represent disturbances acting on the

system. If the controller is so designed that under these

disturbances, the plant can still reproduce the desired

response, then the system is said to have the disturbance

rejection feature.

In this section, only an attempt is made to reject

disturbances which will cause steady state error in the

system response through the introduction of integral

feedback. This relatively modest effort, however, greatly

improved the system response under various disturbances

in computer simulations as discussed in Chapter 6. These

simulations basically included the discrepancy in the

manipulator parameters between the reference and the plant

equations, measurement delays, and the delay in control

law implementation.

Let the new state vector e be defined by
-a

T T T T
e = (e e e T3) (3.109)
-a -al -a2 -a3


where


subscript a is used throughout in this section

to denote the augmented system,












R3n Rn
e s1 ea ea 2 ande sR
-a -al ea3


e a2= e2
-a2 -2


eI and e2 are as defined in

Equations (3.27)-(3.28)


also defining


e = -I e a
-a3 -al


(3.112)


ea3 is given by


e3 = -J I al (t) dt


(3.113)


The control u denotes the plant input and has the
-ap


form


u = u' + u"
-ap -ap -ap


(3.114)


where u' is now given by
-ap


pu = Ap(A G x + Ar Fx
-ap p r r-rl r r-r2


- K2e2 K e a3)
2-a2 3-a3


- K e
1-al


(3.115)


and


u" = u"
-ap -p


(3.110)


(3.111)


(3.116)