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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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Kinetic energy ( jstor )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Includes bibliographical references (leaves 250-256).
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Typescript.
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Vita.
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by L. Sabri Tosunoglu.

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Full Text















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)




Full Text

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    81,9(56,7< 2) )/25,'$


    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

    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.
    11

    TABLE OF CONTENTS
    Page
    ACKNOWLEDGMENTS Ü
    ABSTRACT vii
    CHAPTER
    1 INTRODUCTION AND BACKGROUND 1
    1.1 Manipulator Description and
    Related Problems 1
    1.2 Dynamics Background 5
    1.3 Previous Work on the Control of
    Manipulators 7
    1.3.1 Hierarchical Control
    Stages 7
    1.3.2 Optimal Control of
    Manipulators 9
    1.3.3 Control Schemes Using
    Linearization Techniques ... 11
    1.3.4 Nonlinearity Compensation
    Methods 13
    1.3.5 Adaptive Control of
    Manipulators 15
    1.4 Purpose and Organization of
    Present Work 18
    2 SYSTEM DYNAMICS 21
    2.1 System Description 21
    2.2 Kinematic Representation of
    Manipulators 23
    iii

    CHAPTER
    Page
    2.3 Kinetic Energy of Manipulators ... 28
    2.3.1 Kinetic Energy of a
    Rigid Body 2 8
    2.3.2 Absolute Linear Velocities
    of the Center of Gravities . . 33
    2.3.3 Absolute Angular Velocities
    of Links 37
    2.3.4 Total Kinetic Energy 39
    2.4 Equations of Motion 4 0
    2.4.1 Generalized Forces 41
    2.4.2 Lagrange Equations 44
    3 ADAPTIVE CONTROL OF MANIPULATORS 50
    3.1 Definition of Adaptive Control ... 50
    3.2 State Equations of the Plant
    and the Reference Model 54
    3.2.1 Plant State Equations .... 54
    3.2.2 Reference Model State
    Equations 56
    3.3 Design of Control Laws via the
    Second Method of Lyapunov 58
    3.3.1 Definitions of Stability
    and the Second Method of
    Lyapunov 58
    3.3.2 Adaptive Control Laws .... 64
    3.3.2.1 Controller
    structure 1 68
    3.3.2.2 Controller
    structure 2 68
    iv

    CHAPTER
    Page
    3.3.2.3 Controller
    structure 3 73
    3.3.2.4 Controller
    structure 4 74
    3.3.3Uniqueness of the Solution
    of the Lyapunov Equation ... 80
    3.4 Connection with the Hyperstability
    Theory 81
    3.5 Controllability and Observability
    of the (A,B) and (C,A) Pairs .... 87
    3.6 Disturbance Rejection 89
    4 ADAPTIVE CONTROL OF MANIPULATORS IN
    HAND COORDINATES 9 8
    4.1 Position and Orientation of
    the Hand 99
    4.2 Kinematic Relations between the
    Joint and the Operational Spaces . . 101
    4.2.1 Relations on the Hand
    Configuration 101
    4.2.2 Relations on Hand Velocity
    and Acceleration 104
    4.2.3 Singular Configurations . . . 109
    4.3 System Equations in Hand
    Coordinates Ill
    4.3.1 Plant Equations Ill
    4.3.2 Reference Model Equations . . 114
    4.4 Adaptive Control Law with
    Disturbance Rejection 114
    4.5 Implementation of the
    Controller 118
    v

    CHAPTER Page
    5 ADAPTIVE CONTROL OF MANIPULATORS
    INCLUDING ACTUATOR DYNAMICS 121
    5.1 System Dynamics Including
    Actuator Dynamics 121
    5.1.1 Actuator Dynamics 121
    5.1.2 System Equations 124
    5.2 Nonlinear State Transformation . . . 125
    5.3 Adaptive Controller 128
    5.4 Simplified Actuator Dynamics .... 131
    5.4.1 System Dynamics 131
    5.4.2 Adaptive Controller with
    Disturbance Rejection
    Feature 133
    6 EXAMPLE SIMULATIONS 136
    6.1 Simulations on the 3-Link,
    Spatial Manipulator 139
    6.2 Numerical Solution of the
    Lyapunov Equation 183
    6.3 Simulations on the 6-Link,
    Spatial Industrial Manipulator . . . 184
    7 CONCLUSION 246
    REFERENCES 250
    BIOGRAPHICAL SKETCH 257
    vi

    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
    Vll

    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.
    1

    2
    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, 58]* 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.

    3
    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

    4
    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

    5
    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

    6
    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. [37], Paul et al.

    7
    [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 vas to 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

    8
    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

    9
    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,

    10
    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 [58]. 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

    11
    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.

    12
    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

    13
    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.

    14
    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.

    15
    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

    16
    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,

    17
    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 [29]. 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

    18
    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

    19
    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

    20
    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 be represented by combinations of revolute
    and prismatic joints. Hence, an m degree-of-freedom
    kinematic pair may be represented by m^ revolute and m2
    prismatic joints, where m = m^ + 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 the model dimensionality and further
    complicates the system dynamics. It should be noted that the
    dynamic equations of rigid-link- manipulator models are
    21

    22
    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
    t h.
    (j-l), j = 1,2,...,n. Here the Cr link is the ground or the
    t h
    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 x . applied by the
    th tli
    (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.

    23
    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 rotation) is measured about s^. Joint variable s^ (offset
    distance) is measured along s^ for prismatic joints.
    t h
    Obviously, if the k joint is revolute, then the
    corresponding offset distance s^ will be constant. In order
    to distinguish the joint variables from constant manipulator
    parameters, constant offset distances are denoted by double
    subscripts s^ for all revolute joints. Similarly, if the
    th
    m joint is prismatic, relative joint rotation will be
    denoted by v/hich 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 £ 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 0 will then be given by

    24
    8 =
    1s2<Í>3 ••• 4>n
    Link j connects the and (j + l)fc joints and
    it is identified by its link length r^ and the twist angle
    as depicted in Figure 2.1. Note that according to this
    conventionr can be chosen arbitrarily and a is not defined
    n n
    for the last link—the hand of the manipulator.
    r
    j
    I
    Figure 2.1 Link Parameters r^ and

    25
    In Figure 2.1, s ^ , s^, and are unit vectors and
    rj is the perpendicular distance between joint axes s^ and
    s,. Hence, associated with each link j, unit vector r.,
    * 1
    and with each joint j, unit vector are defined, where
    /\ - /\
    r . s . .
    D ± 3
    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
    f/\(0) ~ (0) /v
    |U! ,u2 'U
    (0)
    is
    frame FQ defined by the basis vectors
    attached to the 0th link, the ground; lying along s^.
    Orientation of u|^ and is arbitrary. One dextral,
    body-fixed reference frame F. is also attached to each link
    i-(i) - (i) - (j))
    з. Frame F. is defined by its basis vectors ju^ ,u2 ,u3 r
    ^ (j) . , . . , ~ ( j ) . , ~
    и, is chosen coincident with r. and u0J with s.;
    1 j 3 j
    j — 1,2,..•,n •
    If a vector á is expressed in the j^ reference
    frame, its components in this frame will be given by a
    column vector a^ . If the superscript (j) is omitted,
    i.e., a, it should be understood that the vector is expressed
    in the ground-fixed Fq frame. Now, it is important to note
    that the unit vectors r^ and s^ expressed in their body-fixed
    frame F_. will have constant representations given by
    (j) _
    and
    s .
    -J
    (j) _
    (1 0 0)
    (0 0 1)
    (2.1)

    < in
    26
    2.2 Kinematic
    lc Representation of Industrial Manipulator
    Figure
    in >

    27
    Let á be a given vector. Again, a^ and a will
    represent expressions of á in frames F^ and Fq, respectively.
    Transformation relating a
    a = T . a ^ ^
    J-
    (j)
    to a is given by
    (2.2)
    Recognizing that r. = T.ri^ , s. = T.s , that ui^ is
    —1 3 3 3 3 3 *•
    given by s^ xr^ and using Equation (2.1), it can be shown
    that transformation is given by
    T. =
    D
    r .
    “3
    s . x r .
    -3 ~3
    s .
    -3
    (2.3)
    Noting that is given by
    T1 =
    COS0.
    sin0.
    -sin0.
    cost
    (2.4)
    r. and s. can be determined recursively from
    -3 ~3
    r . = T . .
    ~3 3-1
    COS0 .
    3
    cosa• i
    • sin0.
    3-1
    3
    sina. .
    • sin0 .
    3-1
    3
    (2.5)
    and

    28
    s . = T. ,
    -3 D-l
    -sina
    j-1
    cosa . ,
    D-l
    7 j 2,3 , . . . , n
    (2.6)
    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 Fq be a fixed reference frame defined by the
    . (0) WO) .-(0)
    unit vectors u, , u~ , and u-, . Let F be a reference
    12 3 p
    frame fixed to the body at its center of gravity C. Let
    • -i/-** , /\ (P ) /\ (p ) /\ (p )
    the unit vectors defining F^ be u^ , U2 , and u^ .
    Reference frames are depicted in Figure 2.3. Let also S be
    an arbitrary point of the body. One can write
    z
    s
    (2.7)
    V = V + w/r,xp
    s c p/0 ^
    (2.8)
    where go
    p/0
    is the angular velocity
    of F with
    P
    respect to F.
    v , and vc are the linear velocities of the related points.
    The kinetic energy (KE) of the body can be expressed
    as follows:
    KE
    v dm
    s

    29
    u
    (P)
    2
    Figure 2.3 Reference Frame F^
    where m is the mass of the body,
    expressed as
    Fixed on a Rigid Body
    Kinetic energy can also be
    A /\
    KE = 2 tvc ‘vc + 2vc • V m c
    + (fip/0 >¡1 • 5ll dm
    (2.9)
    Noting that, since C is the center of gravity,
    p dm = 0
    (2.10)
    m

    30
    Thus,
    1 1
    KE = ^ i v • v + ~-
    2 c c 2
    A . . A
    m
    (Vo xp) ’ uP/o xp) ^
    or
    (2.11)
    1 A A 1
    KE=T-mv *v + T-
    2 c c 2
    m
    p • ft /A • ft /A • p dm
    ^p/0 ^p/0
    (2.12)
    where ft is a dyadic formed by the components of w such
    that
    (q)
    ft = T £ • , • 00,' *
    iDfP/0 lk: k,
    'Xj
    (q)
    kj ^kiP/O
    (2.13)
    where the superscript (q) denotes the components expressed
    in an arbitrary frame F , and
    +1, if ikj is a permutation of 123*
    = "-1, if ikj is a permutation of 321
    [ 0, if any two of ikj are equal
    Note that ft /n = -ft is the transpose of ft /n. Hence,
    ^p/0 ^p/O %p/0
    *{123, 231, 312} is meant.

    31
    KE =
    m v • v
    2 c c
    -if
    P * Si /n • fi /n • p dm
    Jm %
    p/0 ^p/0
    (2.14)
    On the other hand, it can be shown that
    /N /\.
    ÍÍ /A * Í2 ,A - ~ (w /A • o) /A) I + (*) /A co /A (2.15)
    ^p/0 %p/0 p/0 p/0 ^ p/0 p/0
    where I is the identity dyadic, i.e., I • r
    % %
    /V A
    = r. Then
    _ j_ /N /N 1 ^
    KE = ■=• m v • v + co /n
    2 c c 2 p/0
    /V /V
    (p • p I - pp) dm
    m ^
    co
    p/0
    (2.16)
    /s /\ 1 ^ /s
    KE = t m v • v + co /A • J • a) /A
    2 c c 2 p/0 ^ p/0
    (2.17)
    where J is defined as the moment of inertia dyadic, i.e.,
    r\j
    J =
    % j
    (p • p I - pp) dm
    (2.18)
    m
    Note that, since p is fixed in F , components of the matrices
    P
    = J and p ^ = p will be independent of time, and
    J =
    T T
    (p p I - pp ) dm
    (2.19)
    m

    32
    where I is a 3x3 identity matrix. Furthermore, if the
    unit vectors of frame are along the principal axes, the
    matrix J will be diagonal, i.e.,
    J =
    h 0
    o
    0
    2
    0
    0
    0
    (2.20)
    where
    3i =
    m
    (£T£ - p?) dm; i = 1,2,3
    The kinetic energy of the rigid body can be given as
    1 (0) T (0) 1 (p) _ (d)
    KE = 7T m v v + 7T a) J a)
    2 —c —c 2 —p/0 —p/0
    (2.21)
    The rigid body described above can be considered to
    t h
    be the i link of the manipulator, i = l,2,...,n. Then the
    kinetic energy expression for this link becomes
    KE .
    i
    v(0)T v(0)
    +
    1
    2
    (i) T
    -i/0
    J.
    i
    -i/0
    (2.22)
    where
    m. is the mass of the it^1
    i
    link

    33
    v is the three-dimensional column vector
    —c.
    i
    describing the absolute linear velocity of
    t h
    the center of gravity of the 1 link
    expressed in the fixed Fq frame
    is the absolute angular velocity of the it*1
    th
    link, expressed in the i frame F^,
    three-dimensional column vector
    th
    J. is the 3x3 inertia matrix of the i link
    i
    at the center of gravity CL expressed in
    the frame F.
    i
    Total kinetic energy of an n-link manipulator will then be
    n
    KE = 7 KE. (2.23)
    i=l 1
    Expressions for the absolute linear velocity of the
    center of gravity v^^ and the absolute angular velocity
    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
    8. , 0~, . . . , 0 . Orientation of the ith link, 1 < i < n, can
    1 2 ' n
    be considered to be the result of i successive rotations;
    A
    the resulting rotation is denoted by Rot: Fq —>-F^. If a is
    a vector undergoing these rotations, then

    34
    a(0> = T.a(1)
    — I-
    (2.24)
    where is as given by Equation (2.3).
    Now, let Ch be a fixed point in link i. Position
    vector zCi connecting the origin of frame to point Ch is
    given by
    . = slSl + , K [rk-l *k-l + SkSk] +
    k=2 ~ ~ ~ " ci/0i
    (2.25)
    where z
    c,^q is the position vector connecting the origin of
    frame F^,Ct , to point Ch, and
    (0)
    -i/°i
    (i)
    V°i
    (2.26)
    Differentiating Equation (2.25), absolute linear velocity of
    point Ch,vc^, is obtained as follows:
    v l
    c . = . L ,
    i 3 = 1
    >. s . +

    3 3 3 3
    I (rk-i rk_i
    k= j+1
    + sk sk> + zci/0i
    (2.27)

    35
    or
    = (Sj Sj + i.s. x zr /n )
    j=l
    3 3
    'c. /0 .
    i 3
    (2.28)
    where zc^/q â–  is the position vector from the origin CK of
    frame F^ to point and given by
    Z / /\ z z«
    c. /0. c. 0 . , .
    ± j i j k=3+l
    l , (rk-i £k-i
    + s, s, ) + z /r,
    k k c ./0 .
    3 3
    (2.29)
    It is understood that constant offset distance s^ will be
    inserted in Equations (2.25), (2.27), and (2.29) for s^ - if
    t h
    the k joint is revolute. Position vectors defined above
    are illustrated in Figure 2.4. It should also be noted that
    t h
    in Equation (2.28) s^ is zero if the j joint is revolute;
    is zero if it is prismatic. Equation (2.28) can be
    represented in vector-matrix form as
    v
    = v(0)
    :. —c .
    i i
    G co
    c . —
    i
    (2.30)
    where
    (0 -
    d0
    dt

    36
    u
    (i)
    3
    Figure 2.4 Illustration of Position Vectors
    G s R^Xn, its column defined by
    Ci
    [Gci]j "
    —j * —c^/0jy j < i and joint revolute
    , j - s .
    -3
    , otherwise
    (2.31)

    37
    where () denotes a three-dimensional null column vector.
    3 xn
    For an n-link RRPRP... arm, G eR for example, will take
    c4
    the form
    % = «£c4/0l =2*5c4/02 —3 14xzc4/04 »•••»]
    Thomas and Tesar defined these position-dependent terms
    [Gc>]. as translational first-order influence coefficients
    [53] .
    Now, considering that the arbitrarily chosen point
    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 = “I/O + w2/l + •** + Wi-l/i-2 + “i/i-1
    (2.32)
    ui/0 4*2 s1 + s2 + ... + 4>i_;L s._1 + 4^ s.
    (2.33)
    Recalling Equation (2.24), any vector a can be expressed in
    frame , provided that its representation in frame Fq and

    38
    the related transformation matrix Tt are given. The reverse
    of this transformation is also always possible, since the
    transformation represented by T^ is orthogonal. Hence,
    a
    (i)
    'T1 a(0>
    x —
    (2.34)
    Rewriting Equation (2.32) in vector-matrix form
    (0) v ;
    to . /n = (A) . = > Ó . S.
    -i/o -i/o ¿
    (2.35)
    or
    = G. a)
    —l/O i —
    (2.36)
    where the jt*1 column of G. e R^xn is defined as
    J x
    tGih -
    s_j, j < i and i*" joint revolute
    0 , otherwise
    (2.37)
    Using Equation (2.34), can also be expressed in frame
    F.
    i
    (i)
    -i/0
    = T.
    m(0)
    —i/0
    (2.38)
    *1%
    i
    I
    j = l
    ^ j
    s .
    -3
    (2.39)

    39
    or in more compact form
    ajf/J. = G.(l) ai
    —l/O i —
    (2.40)
    where the j*"*1 column of G.^ e R^xn is now defined by
    [G.(l) ] . =
    i 3
    tT £j, j < i and i*"*1 joint revolute
    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 = l ¿m. v(0)T v<0)
    . L. 2 i —c . —c .
    i=l - i i
    , 1 (i) T M(i)"
    2 -i/0 Ji -i/0
    (2.42)
    Absolute linear velocities of the center of gravities vc
    (i) C^
    and the absolute angular velocities are determined as
    linear functions of the generalized joint velocities co within
    the previous sections. Substituting Equations (2.30) and
    (2.40) into Equation (2.42), the kinetic energy expression
    becomes

    40
    Defining
    1 T
    KE = 2 — 4
    n
    y [ m. G1 G +G.(l)TJ. G}1J (â–  w
    .L. i c. c. 1 11 —
    i=l i i J
    (i)
    (2.43)
    n
    = l [nu G
    i=l 1
    G + G J G 1
    c. c. i i i
    i i
    (2.44)
    Equation (2.43) becomes
    1 T
    KE = 7T- (jú A oj
    2 - p -
    (2.45)
    where = Ap(9_) is an nxn symmetric, positive definite,
    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_
    dt
    3KE
    3 to.
    3KE
    30,
    = Q,
    (2.46)
    where
    9^, k = l,2,...,n are the generalized coordinates
    d0.
    k
    k dt

    41
    KE = KE (0_,w) = KE (0^ , 02 / • • • r 9n/w^,0)2 , • . . ,o>n)
    is the kinetic energy of the manipulator,
    and
    Qk is the generalized force associated with
    th
    the k 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 are derived
    by subjecting all generalized coordinates 0^ to virtual
    displacements 6 0^ and forming the virtual work expression.
    The coefficients of 60^.'s in this expression constitute
    the generalized forces by definition.
    Now, let all the externally applied forces acting
    A
    on link i be represented by the resultant force f., and
    A
    all moments acting on the same link by nr . Here, it will
    be assumed that f^ acts through point Ch in link i. This
    point can represent any point in the link, however, for
    the current presentation, restriction of point Ch to be the
    t h
    center of gravity of the i link will suffice.
    A A
    Virtual work 6W done by the force f^ and moment nr
    is given by

    42
    6W
    A A A
    f . • v 61 + m.
    ic. 1
    i
    A
    wi/0 6t
    (2.47)
    where the virtual displacement of link i is w^yg 6t and that
    of point Ch is 6zc^ = vc^ fit. Representing vectors in
    frame Fg, Equation (2.47) becomes
    6W = fT G a) fit + mT G. w fit (2.48)
    —i c. — —i i —
    i
    where Gc^ and G-¡_ are as defined by Equation (2.30) and
    Equation (2.36), respectively. Letting fiW^ denote the
    resulting virtual work due only to the variation in 0^,
    6Wk = Qk 60k (2.49)
    and
    T
    + m.
    k -i
    [Gi]k
    60,
    (2.50)
    where [GcjJj. is given by Equation (2.31) and [G^]^ by
    Equation (2.37). Hence, generalized force is given by
    Qk - Ü
    1G=i]k + —i
    “Vk
    (2.51)
    If external effects are represented by gravity loads,
    actuator torques, and viscous friction at the joints, then
    virtual work 6W^ due to 6 0^ will be

    43
    ÓW,
    n
    = I
    j=k
    m
    ó z
    cj,6k
    JLL
    3co,
    6 0,
    + t,
    60,
    (2.52)
    where
    g : the gravitational acceleration
    cl
    vector
    5 z
    9zCj
    Cj'0k = 50k
    (2.53)
    th
    Tk : the torque applied on the i link by
    the (i-l)^ link
    3 r
    77 = y, 3u), 'k k 'k
    ^ th
    viscous damping at the k joint and
    r V 1 2
    r = > 7T y. to.
    i=l 2 1 1
    (2.54)
    T is the Rayleigh's dissipation function. Similarly,
    6W. =
    k
    l £ [GcA “ Yk wk + T1
    j=k
    60k (2.55)
    Thus, related generalized force will be
    n T
    }k = J, mj 3-a [Gcj]k Yk wk + TJ
    j=k
    (2.56)

    44
    Note that Equations (2.52), (2.55), and (2.56) assume that
    the payload is included in the mass of the last link mn-
    Payload or any other external effect can be separately
    represented in the formulation as given by Equation (2.51).
    Defining
    n
    9k = X mj & Kc.]k
    3=k J j
    the generalized force becomes
    Qk = gk ~ Yk “k + Tk
    (2.57)
    (2.58)
    where
    gk gk (^_) / ^ 1,2, . . . , n
    2.4.2 Lagrange Equations
    Total kinetic energy expression in Equation (2.45)
    can be written in indical notation, repeating indices
    indicating summation over 1 to n.
    KE
    to. co .
    1 1
    (2.59)
    An.. denotes the element (i,j) of the generalized inertia
    piD
    matrix A . Then,
    P
    oKE
    3wk
    6 ., +
    lk
    co.
    1
    V
    (2.60)

    45
    where
    rl if i = k
    5ik =
    0 if i ? k
    (2.61)
    or
    3KE 1 , _ x
    ~(A (jo + A co. )
    3uk 2 ?kj j pik 1
    (2.62)
    Since Ap is symmetric,
    3KE
    3co,
    = A to.
    Pki 1
    (2.63)
    Introducing Equations (2.63), (2.45), and (2.58) into
    Equation (2.46)
    -rr (A co.)
    dt pki 1
    3 A
    1 Pi j
    2 30. “i
    9k ' + T}
    Noting
    (2.64)
    jtt (A co..) = A co- + A co.
    dt pki 1 pki 1 pki 1
    where ( ) represents differentiation with respect to time
    and

    46
    3A
    ki
    ki
    30 . Wj
    (2.65)
    Equation (2.64) becomes
    oj . +
    Pki 1
    3A_
    ki
    30 .
    3
    3A^
    1 p
    2
    13
    30,
    CO . CO .
    i 3
    = pk - Yk “k + TJ
    (2.66)
    Definina
    3A
    D* = —P-kÍ.
    ijk 30.
    3 A
    1 P
    13
    2 30,
    (2.67)
    where D* = [D?.^ £ Rnxnxn, equations of motion are given
    by
    Apk. “i + Dijk “i “j * 9k - Yk + T,
    (2.68)
    Now, can be replaced by D = [D^j^], such that
    D. ., a), co. = D. ., co. co.
    ijk i j ijk i j
    (2.69)
    holds [53] ; D. is defined by
    13 K
    Dijk - -t [Gc?]k + j, [G,)k
    + "V. J£ <“Vk * "Vi’
    (2.70)

    47
    where
    [H ] ,
    [G„ ]
    C£ iij 80i C£ j
    (2.71)
    [H ] . .
    C£
    -i x ("j x V/0
    X/ J
    s . X (s. X z
    -J -1 -c„/0.
    '£ i
    S • X s .
    -1 -1
    s . x s .
    -1 —J
    ) , i < j < £ ;
    i,j revolute
    ) , j < i < £ ;
    i,j revolute
    , j < i < £ ;
    i prismatic,
    j revolute
    , i < j < £ ;
    i revolute,
    j prismatic
    , otherwise
    (2.72)
    'Vio = sir ‘V
    (2.73)
    'Vio -
    s.xsj, i< £; i, j revolute
    , otherwise
    (2.74)

    48
    [G ], is given by Equation (2.31) and [G0], by Equation
    c£ K X, K
    (2.37). and Hc^ are called second-order rotational and
    translational influence coefficients [53]. Again, the
    repeated index l in Equation (2.70) indicates summation
    over 1 to n. Also defining D, eK
    nxn
    Dk [Dij]k " [Dijk]; 1,j l,2,...,n
    (2.75)
    with D.as given by Equation (2.70), dynamic equations
    1JK
    finally take the form
    Y.. “i = - 2T Dk Ü ‘ “k + gk + Tk
    ^ki
    k = 1,2,...,n (2.76)
    or
    T
    ü) Dj. —
    [y] a) + g + t
    (2.77)
    D co
    n —
    where
    AP = y®1- Dk = Dk(®>

    49
    [y] e
    2.(9.)
    Rnxn is the diagonal matrix containing
    the coefficients of viscous
    friction
    e Rn denotes the equivalent
    gravitational torques due to the
    mass content of the system as seen
    at the joints
    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
    50

    51
    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.
    Figure 3.1 Block Diagram Representation of
    an Adaptive Control System

    52
    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.
    Figure 3.2 Block Diagram of Gain Scheduling System

    53
    Figure 3.3 Block Diagram of Model Reference Adaptive System
    Figure 3.4 Block Diagram of Self-tuning Regulator

    54
    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
    T T T
    Defining the state vector x = (0 , w ) where
    -P -P “P
    0 e 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:
    (3.1)
    where subscript p stands for "plant," here manipulator
    represents the plant,
    ip - ?p -
    , T T v T D2n
    %i' V £ R
    (3.2)
    £pi " Vt!'
    %>2 “ mP
    (3.3)
    )
    T
    d*P(t)
    dt
    (3.4)

    55
    I and 0 denote the nxn identity and null
    matrices, respectively
    Referring to Equations (2.76) and (2.77),
    A = A (x , ) e R
    P P -Pi
    nxn
    (3.5)
    q (x , ) = Gx. = G (x.)x.
    -Pi P-Pl P -Pi -Pi
    (3.6)
    = G (x .) e Rnxn, 2n(x .) e R
    ,n
    P P -Pi
    P -Pi'
    (3.7)
    f (x ., x _) = F x_0 = F (x ,, x ~) x _ (3.8)
    —p —pi —p2 p —p2 p —pi —p2 —p2
    f = f (x . , x 0)
    -P -P -Pi -p2
    —p2 Dl(-pl} —p2
    x% D (x . ) x 0
    —p2 n —pl —p 2
    E R
    n
    (3.9)
    F = F (x . , x 0)
    P P -pl -p2
    P T
    x D. (x . )
    —p2 1 —pl
    x - D (x . )
    —p2 n -pl
    £ R
    nxn
    (3.10)
    = lp(t) e R
    n
    (3.11)

    56
    (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; A and G are
    P P P P P
    nonlinear functions of the joint variables x and
    F = F (x , , x 0) . In the formulation, functional
    P P -Pi -P2
    dependencies are not shown for simplicity. Also, Gp(Xp^)
    is not defined explicitly; symbolically, G (x .) is such
    P P
    that G (x . )x . = g holds. External disturbance terms
    P -Pi -Pi aP
    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
    ur (3.12)
    1
    O
    H
    l
    [
    o
    l
    X =
    —r
    -1
    -l
    x +
    —r
    -1
    A G
    r r
    A F
    r r
    A
    r
    where
    subscript r represents the "reference" model
    to be followed,

    57
    x is the state vector for the reference
    —r
    system
    ... / T T > T „2n
    x = x (t) = (x , , x „) e R
    —r —r —rl —r2
    x , = 0 (t) £ Rn, x ~ = to (t) £ Rn
    —rl —r ' —r2 — r
    ,-T -T ,T
    —r = dt = —rl' —r2
    Again, referring to the manipulator dynamic equations, i
    Equations (2.76) and (2.77),
    Ar = Ar(5.r]_) e Rnxn is the generalized
    inertia matrix for the reference
    system
    g (x.) =Gx. =G (x.) x.
    2-r —rl r—rl r — rl —rl
    G = G (x ,) e Rnxn, g (x .) £ Rn
    r r —rl ^-r —rl
    f (x . , x~) = F x ~ = F (x . , x „) x-
    —r —rl —r2' r—r2 r —rl —r2 —r2
    f = f (x . , x 0) = -
    —r —r —rl —r2
    T
    x _
    —r 2
    D1(irl>
    •
    x 0
    —r2
    T
    x _
    —r 2
    •
    D (x . )
    n —rl
    X t
    —r2
    e R
    (3.13)
    (3.14)
    (3.15)
    • 6 • f
    (3.16)
    (3.17)
    (3.18)
    (3.19)

    58
    F = F (x , , x ~) = -
    r r —r 1 —r 2
    x 0 D, (x , )
    —r2 1 —rl
    X% D (x , )
    —r2 n —rl
    e Rnxn (3.20)
    It is important to note that A = A^ix^),
    G = G (x .) and F = F (x , , x ~) are not constant, but
    r r —rl r r —rl —r2
    nonlinear functions of the state vector x^. In this study,
    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

    59
    X = f (x, t) , -00 < t < +00 (3.21)
    where x e Rn is the state vector of the system. Also let
    the vector function Equation (3.21) which is differentiable with respect to
    time t such that it satisfies
    (i)
    (ii)
    for a fixed initial state Xq and time tg.
    A state x^ is called an equilibrium state of the
    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 of
    the dynamic system in Equation (3.21) is
    stable (in the sense of Lyapunov) if for
    every real number e > 0 there exists a real
    number 6(e, tg) > 0 such that || Xg - x^ || < 5
    implies
    II ¿(t; Xg, tg) - II < £ for all t < tQ
    The norm || • || represents the Euclidean norm.
    (£_( tg ; Xg, tg) = XQ (3.22)
    d^_
    (t; Xg, tg) = f(^(t; Xq, tQ), t) (3.23)

    60
    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 of
    the dynamic system in Equation (3.21) is
    asymptotically stable if
    (i) It is stable (Definition 3.2)
    (ii) Every solution c£(t; x Q, tg)
    starting sufficiently close to x
    converges to x^ as t —► °°. In
    other words, there exists a real
    number y(tg) > 0 such that
    I xQ - xe 1 < y (tQ) implies
    lim I £(t; Xg, tg) - x^JI = 0
    t~*-oo
    Definition 3.4: An equilibrium state x^ of the
    dynamic system in Equation (3.21) is

    61
    asymptotically stable in the large if
    for all Xq restricted to a certain region
    „n
    r e R
    (i) x is stable
    —e
    (ii) lim 1 $(t; xQ, tQ) - x^ II = 0
    t —> oo
    Definition 3.5: An equilibrium state 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 (|| x ||) > 0 for all
    x ^ 0, x e Rn where a(*) is a

    62
    real, continuous, nondecreasing
    scalar function such that
    a(0) = 0
    (iii) V(x, t) —► 00 as || x l| —► °° for all t
    , . , • av , , .
    (ív) V = (x, t)
    3V
    at
    + (grad V) £(x, t)
    < -Y (II x ||) < 0
    where y(*) is a real, continuous
    scalar function such that y(0) = 0
    then the equilibrium state = 0^ is globally
    asymptotically stable and V(x, t) is a
    Lyapunov function for this system.
    Corollary 3.1: The equilibrium state
    = 0^ of the autonomous dynamic system
    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 ^ 0, x e Rn

    63
    (iii) V(x) —as I x || —►00
    (iv) V = (x) < 0 for all x ^ £,
    „n
    x e R
    Corollary 3.2: In Corollary 3.1, condition (iv)
    may be replaced by
    (iv-a) V(x) < 0 for all x ^ 0^, x e Rn
    (iv-b) V( identically in t > tg for any
    tg and ^ 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
    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,

    64
    positive definite matrix P which is the
    unique solution of the matrix equation
    ATP + PA = -Q (3.26)
    T
    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^it) represents the open-loop control law.
    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 u^ = 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

    65
    e = e(t) = x (t) - x (t) e R
    — — —r —p
    2n
    T t.T , T
    £ = (ej_, e2) = (xrl
    T T T.T
    x . , x „ - x 0)
    —pi —r2 -p2
    e e Rn, e„ e Rn
    -1 -2
    de(t)
    (3.27)
    (2.28)
    (3.29)
    (3.30)
    and choosing
    u = u' + u" (3.31)
    -P -P -P
    u'
    -P
    + A 1 F x ^
    r r —r2
    K2-2
    )
    (3.32)
    where
    K
    V
    is part of the controller yet to be designed
    I<2 e 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:
    é = Ae + Bz - BA-1 u" (3.33)
    - P -P

    66
    where
    "0
    I
    D -
    , B =
    _K1
    K2_
    I
    , „2nx2n „ n2nxn
    A e R , B e R
    (3.34)
    I and 0 are nxn identity and null matrices,
    respectively
    z = -A 1 G X , - A 1 F x _ + A 1 u (3.35)
    P P -Pi P P —p2 r -r
    z e Rn, u" £ Rn
    -p
    It should be noted that the part of the controller
    Up requires only the on-line calculation of the plant
    generalized inertia matrix A = A (x ); other nonlinear
    y P P -P
    terms A^.1 = A^ix^), Gr = Gr(xrl) and Fr = Fr(xr) are
    reference model parameters and known a priori for each given
    task, i.e., A^. , G , and F will not be calculated on-line.
    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

    67
    V(e) = eTPe (3.36)
    define a real, scalar positive definite function. Using
    Equations (3.33) and (3.36),
    V(e) = -eTQe + 2vTz_ - 2vT A ^ u" (3.37)
    P P
    where
    Q e j^2nx2n pOSj[tive definite matrix (Q > 0) ,
    P e ¡^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.

    68
    3.3.2.1 Controller structure 1
    If u" were chosen
    -P
    (3.40)
    or
    -1
    (3.41)
    where
    (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}
    —p p i — —
    (3.43)
    where diagtsgn (v^)] is an nxn diagonal matrix with
    diagonal elements sgn (v^), i = l,2,...,n,
    r • • • /

    69
    b = sup { -A g + A u }
    - r\ <" r -r1
    0 < X . < 2 7T
    Pf 1
    u . e U
    r, i
    x 1; * • * f 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 . e U,
    —r r, i
    i = l,2,...,n. The generalized inertia matrix Ap(xp^) is
    nonsingular [54], also elements of A \ A and are all
    P * P
    bounded, i.e., if
    -1
    (iPi> â–  [aij(iPi)!
    (3.45)
    then
    (a..)0 < a..(x .) < (a..)
    13 £ 13 -pi 13 u
    (3.46)
    where (a..)„ and (a..) are the lower and upper bounds on
    a i j (x p 1) / 0 < xpl k < 2tt; i, j , k = l,2,...,n. Similarly,
    bounds on the gravity loads gp can be given. A^ u_r =
    Ar\xr^ (t)) u_r (t) in Equation (3.44) is known for a given
    manipulation task, since it represents the reference.
    Referring to Equation (3.43),
    S = [s. .] e R
    13
    nxn
    (3.47)
    is defined by

    70
    s.. = sup {|a..I}; i,j = 1,2
    -1 0 < x , „■ < 2tt
    pi, £
    £=1,...,n
    • ,n
    (3.48)
    k
    K
    k:
    T
    k e Rn (3.49)
    where constant positive definite K* e Rnxn tQ £e chosen
    so that
    D.
    i
    and
    xT *
    — ~ K. x t
    p2 i —p2
    > 0 for all x f* 0
    -P2
    (3.50)
    (3.51)
    where D^, i = l,...,n is as defined by Equations (2.70) and
    (2.75); in Equation (3.50) can be replaced by symmetric
    D!
    i
    = 2 (Di +
    (3.52)
    T T
    so that x - D! x0=xnD. x - is preserved. Existence
    —p2 i —p2 —p2 i —p2 c
    of positive definite K? is shown using the following theorem
    [6]
    Theorem 3.3: Let M be a symmetric, real matrix
    and let A . (M) and A (M) be the smallest
    mm max
    and the largest eigenvalues of M,
    respectively. Then

    71
    X . (M) || x ||2 < x1 Mx < X (M)||xl|2
    mm — ~ max —
    (3.53)
    n
    for any x e Rn, where || x || = £ x. .
    i=l 1
    Using Theorem 3.3,
    2 ^ _T ^ „ ii2
    X • (K*) || x 0 |r < x% K* x - < X (K?) I x 0
    min i " -p2 11 —p2 i —p2 max i " —p2
    (3.54)
    X . (D!) I x - ||2 < xT_ D! x ~ < X (D!) I! x 0 ||2
    mm i 11 -p2 " —p2 i -p2 max i 11-p2 11
    (3.55)
    Here K* is assumed to be a real, symmetric matrix. If K*
    is not symmetric, then
    K*' = ^ (K* + K*T)
    l 2 i i
    (3.56)
    must be replaced by K* in Equation (3.54). Also, all
    entries of (x ^) are bounded and, in general, is
    T
    indefinite. Quadratic surfaces xp2 Dj xp2, ats lower anc^
    upper bounds (x^2 d! xp2)£ and (xp2 D! xp2)u, and x£2 K* xp2
    are conceptually represented in Figure 3.5.
    If X . (K*) is chosen such that
    mm i
    X . (K.) > \ (D!)
    mm i max i
    (3.57)

    72
    Figure 3.5 Representation of Quadratic Surfaces
    is satisfied, where
    X (D!) = sup (X.(D!(x,))
    max 1 0 < x , . < 2u ^ 1 -Pi
    Pl,l
    d. ~ 1 ^ ^ n
    * j 1,2 , . . . , n}
    (3.58)

    73
    then
    D!
    i
    (3.59)
    follows directly from Equations (3.54) and (3.55). In
    addition, if X (K*) > 0, then xT 0 K* x _ > 0 for all
    mm i — p2 i —p2
    Xp2 ¥■ 0- That is, symmetric K* e Rnxn is positive definite,
    if and only if all the eigenvalues of K* are positive [36].
    One choice for K* which satisfies Equation (3.50)
    is
    K* = diag[X (D!)] (3.60)
    i J max i
    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 , 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
    (3.61)

    74
    where Q* e Rnxn constant, positive definite matrix. In
    this case, due to the term in V linear in v(t), i.e., 2v z,
    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 = j (Q*)_1 z (3.63)
    In general, this spherical region can be reduced as
    the magnitude of u" is increased, which actually translates
    into the use of large actuators. This can easily be shown
    ^|f
    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 + (K + AK ) u
    —p p P —P u u —r
    (3.64)

    75
    where
    Kp = lGp : Fp] (3.65)
    AKp = [R± v (S1 xpl)T : R2 v (S2 xp2)T] (3.66)
    Ku = tAp A"1] (3.67)
    AKu = [R3 v (S3 ur)T] (3.68)
    K and AK e Rnx2n
    P P
    Ku and AK^ e Rnxn
    G , F , and A denote the calculated values
    P P P
    of G , F , and A given by Equations
    P P P
    (3.6)—(3.7), (3.10), and (2.44),
    respectively
    R. e Rnxn, R. > 0, and (3.69)
    l i
    S. e Rnxn, Si > 0, i = 1,2,3; are (3.70)
    to be selected
    v is as defined by Equation (3.39)
    Let

    76
    V(e, t)
    T
    e P e +
    2
    •t
    â–  0
    T -1 T T
    (v A R,v) (x_“, SZ'x , ) dr
    — p 1— —pi 1—pi
    + 2
    (*p2S2Íp2)dT
    + 2
    •t
    0
    m _â– ) T T
    (v A^xR,v)(u siu )dx
    — p 3— —3—IT
    (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
    V(e) = -eT Q e + 2vT z' (3.72)
    where P is the solution of the Lyapunov equation
    ATP + PA = -Q, Q > 0 (3.73)
    and
    z' = A-1 [(g - g ) + (f - f )]
    P -P -P -P -P
    + (A”1 - A"1 Ap A"1) ur (3.74)
    An estimation of the bound of || e || is given below.
    If V(e) is negative outside a closed region r subset
    2 n
    of R including the origin of the error space, then all

    77
    solutions of Equation (3.33) will enter in this region r
    [23]. Substituting Equation (3.39) into Equation (3.72)
    V(e) = -eT Q e + 2eT PB z ' (3.75)
    V(e) < " Xmin(Q) II + 2H®II Hpll II B^' II (3-76)
    where
    X . (Q) is the smallest eigenvalue of Q
    mm
    || • I denotes the Euclidean norm
    II e ||2 = eTe (3.77)
    || P || = the largest eigenvalue
    of P, since P is positive definite
    and symmetric [23]
    = [ (z *) z1]
    1/2
    (3.78)
    Also, recalling Equation (3.34),
    Bz ’
    (z')T]T
    (3.79)
    where

    78
    O denotes then x n null matrix, and
    0 0 Rn represents the null vector,
    I Bz' I = 1 z' i (3.80)
    follows from Equation (3.79). Now, from Equation (3.76),
    V(e) < 0 is satisfied for all e satisfying
    , 2 || P || || z'||
    |e|>-r (3.81)
    Amin(Q)
    Hence, an upper bound on the error, || e || will be
    2 P
    i'L
    ax
    max < A . (Q)
    mm
    (3.82)
    It is clear from Equation (3.82) that this bound on
    || e || will be reduced as || P || is decreased, X . (Q) increased
    or || z * 11 —*■ 0 . It should also be noted that frequent
    updating of g , f , and A_ will affect || z'|| —*- 0, hence
    —p —p p — Illa. X
    || e|| —* 0. At steady state, e = 0, control will take
    11 —11 max J — —
    the form
    u' (t) = u (t) (3.83)
    —p —r
    and

    79
    or
    z’ = 0 (3.84)
    hence Equation (3.33) would yield
    e = A e
    Controllers presented in this section have the
    general form
    u = u1 + u"
    -p -p -p
    (3.85)
    Analysis is given assuming that the calculated A^, i.e.,
    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 g , f , and
    P P P
    A , i.e., g , f , and A , are explicitly shown in the
    P -P -P P
    analysis (Controller structure 4). Current arguments with
    reference to Equations (3.82) and (3.74) suggest that g^
    and fd may be updated at a slower rate compared to the A^.
    This result is important, since especially the calculation
    of f , in general, requires more computation time compared
    to Ap. Although it is clear, the above controllers need the
    on-line measurements of plant joint displacements x^ and
    the velocities x
    —P2

    80
    3.3.3 Uniqueness of the Solution
    of the Lyapunov Equation
    The Lyapunov equation is given by Equation (3.38).
    The uniqueness of its solution P e £2nx2n j_s guaranteed, if
    A e j^2nx2n ^as eigenva]_ues 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
    ATP + PA = -Q
    where A, P, and Q e R2nx2n.
    Recalling Equation (3.34), A is given by
    A =
    The characteristic equation of A e £2nx2n is
    det [si - A] = sn det
    si - K- -
    - K1
    s 1
    (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 e Rnxn

    81
    s is the complex variable,
    and Kj £ R
    If and are diagonal matrices
    Kx = diag [K1;i], K2 = diag [K2 ±] (3.87)
    where
    K,_. and K9>. are the respective diagonal
    (i,i)t*1 entries of and i = 1,2,... ,n
    then
    n 2
    det [si - A] = n (s - K- .s - .) (3.88)
    i=l 1;1
    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 K, . < 0 and K . < 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

    82
    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
    B e R
    2nxn
    C £ R
    nx2n
    A, B, and C are time-invariant,
    f(•) e Rn is a vector functional
    (ii) The pair (A,B) is completely
    controllable
    (iii) The pair (C,A) is completely
    observable

    83
    is hyperstable if there exists a positive
    constant 6 > 0 and a positive constant
    Yq > 0 such that all the solutions
    x(t) = <£ (t; Xq, tg) of Equations (3.89)-
    (3.91) satisfy the inequality
    || x (t) || < 6 (|| x(0)j| + Yq) for all t > 0
    (3.92)
    for any feedback w = f(v, t) satisfying the
    Popov integral inequality
    h (tg , t-j^)
    T
    v w dt
    (3.93)
    for all t^ > tg.
    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 —00
    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].

    84
    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)
    T
    (ii) The matrix H(joj) + 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 = a + jin is a Hermitian
    matrix (or Hermitian) if

    85
    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 hy
    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
    ATP + PA = -Q (3.96)
    C
    T
    B P
    (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"
    (3.99)
    £ is defined by Equation (3.35), A and B are as given by
    Equation (3.34). Various controller structures for u^ are

    86
    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
    T
    that for a positive definite function V(e) = e Pe
    V(e) < -eTQe + 2vTz^' (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
    T
    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
    T
    (3.96), exists and C = B P is satisfied. Noting that Equation
    T
    (3.39) defined v = B Pe, both conditions are already required
    by the second method of Lyapunov.

    87
    However, Theorem 3.4 assumes that the Popov integral
    inequality is satisfied. Substituting Equation (3.100) into
    Equation (3.93)
    -n(tQ, tx) =
    1 vTz" dt < -Yq
    :0
    (3.103)
    must hold. But, if v 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) Pai~rs
    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)

    88
    [B AB A2B ... A2n 1B] =
    ro i
    I K,
    e R
    2nx2n“
    (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 R2nx2n? which is the solution of the Lyapunov
    equation, be given by
    (3.105)
    where P^, P2, and P^ e Rnxn and P^ and P^ are symmetric.
    Then, C e Rnx2n will have the form
    C = BTP = [P2 P^J (3.106)
    For the complete observability of the pair (C, A)
    [CT ATCT (AT)2CT ... (AT)2n 1 CT] e R2nx2n
    (3.107)

    89
    must have rank 2n. Hence
    P
    2
    (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, P^, K^, and . At this stage it
    is assumed that P2, P^ of matrix P and the selected 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.

    90
    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 K^, Kj, 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,

    91
    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
    —3.
    , T T T . T / ~> t
    la - (§al' -a2' —a3 (3-109)
    where
    subscript a is used throughout in this section
    to denote the augmented system,

    92
    —a £ r3R' —al' —a2' and aa3 £ R"'
    -al
    = e± (3.110)
    —a2
    = e2 (3.111)
    and are as defined in
    Equations (3.27)-(3.28)
    also defining
    e , =
    —a 3
    = -I e . (3.112)
    —al
    e is given by
    cl
    e , =
    —a 3
    - Ie ,(t) dt (3.113)
    —al
    The control u denotes the plant input and has the
    form
    u
    -ap
    = u' + u" (3.114)
    —ap —ap
    where is now given by
    ul =
    -ap
    ! Ap(ArlGrirl + ‘ Kl®al
    K2-a2 " K3-a3^ (3.115)
    and
    u" =
    -ap
    = u" (3.116)
    -P
    (3.116)

    93
    where u" is as given for various controllers in Section
    P
    3.3.2. Substituting Equations (3.115)-(3.116) into
    Equation (3.1) and subtracting the resulting equation
    from Equation (3.12), also using Equations (3.110)-
    (3.111) along with Equation (3.112), the augmented
    error-driven system equations are obtained as follows:
    e = Ae + B z"
    —a a—a a—
    where
    “0
    I
    0
    0
    K .
    al
    K 0
    a2
    K ,
    a3
    ' Ba =
    I
    -I
    0
    0
    0
    (3.117)
    (3.118)
    , n3nx3n
    A e R ,
    d
    B
    a
    n3nxn
    e R
    0 e Rnxn null matrix, I e Rnxn identity matrix,
    K . , K t, and K e Rnxn are to be selected,
    al a2' a3
    e and z" are as given by Equations (3.109)
    3. —
    and (3.99), respectively
    Due to poor estimation of manipulator parameters
    in the reference model, closed-loop system signal z" may be
    considered to represent the disturbance. Note that for the

    94
    ss
    controller 4 of Section 3.3.2.4, z" is given by the
    right-hand side of Equation (3.74). At steady state this
    signal z" will be assumed constant, represented by z"
    Note that in general z"
    discrepancies.
    Now, at steady state, equilibrium state is
    determined from
    ss ^ £ due to parameter
    e _ = 0
    —a2 —
    K , e , + K~e0 + K-)e-! = z"
    al —al a2 —a2 a3 —a3 —
    ss
    (3.119)
    -e . = 0
    —al —
    Assuming that the selected K e Rnxn nonsingU]_arf -(-he
    a. 3
    equilibrium state is given by
    e . = 0
    —al —
    e t
    —a 2
    = 0
    e
    —a
    3
    ss
    (3.120)
    Error in the position will thus be completely eliminated.
    The equilibrium state is now checked for the case
    without integral feedback. Recalling Equation (3.98), the
    equilibrium state is given by

    95
    —2 = —
    K1 -1 + K2 -2 =
    (3.121)
    ss
    which in turn gives
    = KI1 f
    ss
    (3.122)
    —2 = —
    for nonsingular . Again, with z_" ss / 0^, the system will
    always have steady state position error.
    It should be noted that for the augmented system,
    the Lyapunov equation is given by
    A P + P A = -Q
    a a a a a
    (3.123)
    where
    A , P , and Q e R
    a a a
    3nx3n
    Also, v is now defined as
    —a
    v = B P e
    —cl cL cl —cl
    (3.124)
    Controllers u^ in Section 3.3.2 are valid for the augmented
    system since u^ = ut . The closed-loop augmented system
    for each case satisfies Corollary 3.1, hence it is also

    96
    globally asymptotically stable. It can be shown that it is
    also asymptotically hyperstable.
    Recalling Corollary 3.3, solution P of Equation
    (3.123) will be unique if all eigenvalues of has negative
    real parts; A and its characteristic equation are given by
    cl
    A =
    a
    Kal Ka2 Ka3
    -I
    det[si - A ]
    3
    n ,
    = s det
    si - K ~ — K , +4r K .
    a2 s al 2 a3
    s
    (3.125)
    where I is the identity matrix; its order is 3n on the
    left-hand side of the equation; otherwise, it is of order n.
    If K ^ e R11X11 is diagonal
    K
    ai
    diag tKai;j]
    where K . . denotes the element (j,j) of diagonal K .,
    cl 1 7 J 31
    i = 1,2,3; j = 1,2,...,n, then
    det[si - A ] =
    a
    n
    n
    j=i
    (S3 - K
    a 2; j
    - K
    al; j
    s + K
    a 3; j
    ,)
    (3.126)

    97
    Again, the time-invariant part of the error-driven
    augmented system will be decoupled if K^, and K ^
    are selected diagonal. This does not, however, mean that
    the manipulator dynamics is decoupled. Forming the Routh
    array for Equation (3.126), K , K _ and K ,
    ax;] IJ ci-j/9
    j = l,2,...,n, must satisfy the following conditions for
    all the roots of Equation (3.126) to have negative real
    parts:
    Ka3;j > 0 <3*127>
    K K „ . > K _ .
    al;j a2;j a3;j

    CHAPTER 4
    ADAPTIVE CONTROL OF MANIPULATORS
    IN HAND COORDINATES
    In this chapter, manipulator dynamics is expressed
    in hand coordinates and an adaptive controller with a
    disturbance rejection feature is given for this system.
    The term hand coordinates is used to mean that the hand
    position and orientation (i.e., configuration) of the
    t h
    manipulator hand (the n link) is expressed in the
    ground-fixed reference frame. In the literature, hand
    coordinates, task-oriented coordinates, operational space,
    and task space are used interchangeably.
    The reason for the representation of system
    dynamics in hand coordinates is that error in hand
    configuration will be directly penalized rather than
    achieving it indirectly through feeding back the joint
    errors. Hence, the rationale is that overall measure of
    error in hand position and orientation will be less when
    equations are expressed in hand coordinates rather than
    expressing them in the joint space. Since for most
    applications, precision of the hand movement has higher
    priority than that of the joints, this approach may yield
    improved end effector response.
    98

    99
    4.1 Position and Orientation of the Hand
    The most useful presentation of the hand position
    is through its Cartesian coordinates expressed in the
    ground-fixed frame Fg defined by its basis vectors
    {u|^ , G<0), Ug0^}. Common practice to define the
    orientation of a rigid body is the use of Euler angles
    £, 8, and £. Keeping in mind that the frame Fn defined by
    basis vectors (ujn^ , u^1^ , uin^ } has been fixed to the
    hand, the Euler angles £, 8, and £ are shown in Figure 4.1.
    Assuming that initially frames FQ and Fn (denoted
    by F^ for the initial position) were coincident, first F^
    is rotated about by £. Let the rotated frame F^ be
    denoted by F' with basis vectors (uin^ , u^n^ , uin^ } after
    n 1 ¿ .i
    /n (n) '
    the rotation. Then, F' is rotated by 8 about u. to
    n 1 1

    100
    r /s (n) " ~ (n) " /v (n) " •,
    obtain with basis vectors {u| , ; , u^ }.
    Finally, F" is rotated about bv £ to obtain F . These
    J n 3 -1 n
    successive rotations are illustrated in Figure 4.1. Now,
    . r/,v(n) ~ (n) /v (n) *»
    the basis vectors {u^ , Uj , U3 } of Fn after undergoing
    the above rotations will have the following representations
    {ujjl, Ujj2' i-n frame F0 (subscript H denotes the hand)
    -HI
    cos£ cos£ - sin? cosB sin£
    sin£ cos£ + cos? cosB sin£
    sinB sin£
    -H2
    -cos? sin£ - sine cosB cos£
    -sine sin£ + cose cosB cos£
    sinB cos£
    (4.1)
    -H3
    sine sinB
    -cose sinB
    cosB
    Hence, nine parameters define the basis vectors of
    Fn of which only three are independent. Various approaches
    exist in the literature to represent hand orientation using
    these parameters. However, for our purposes an expression
    for the orientation error of the hand is needed. As Luh,
    Walker, and Paul suggest in [35],
    5 l <ÃœH i * ÃœH i1
    1=1 p r
    (4.2)

    101
    may be used to represent the orientation error with
    uH 4 • uH ^ > 0, i = 1,2,3. In Equation (4.2) subscripts
    —tipi —^p1
    p and r represent the plant and the reference model,
    respectively, where uH^ is given by Equation (4.1).
    Position error will be given by the difference between plant
    and reference model hand positions.
    4.2 Kinematic Relations between the Joint
    and the Operational Spaces
    4.2.1 Relations on the Hand Configuration
    Following analysis is given for nonredundant
    manipulators. Although it can be extended to redundant
    manipulators, the following treatment is applicable to
    6-link, 6 degree-of-freedom spatial (n = 6) and 3-link,
    3 degree-of-freedom planar (n = 3) manipulators. This
    restriction is valid only for the rest of this chapter.
    Position vector zR originating at the origin of
    Fq and pointing a point H in the hand is given by
    n
    ZH = S1S1 + E trk-l rk-l +
    k=2
    s, s, ]
    k k
    + z
    H/0
    n
    (4.3)
    where
    is
    frame F , 0 ,
    the position vector connecting the origin of
    to point H in the hand and other parameters
    are as defined in Chapter 2. Representation of z^ in Fq,
    will result in m^ nonlinear, coupled algebraic equations in
    terms of the generalized joing displacements 0_

    102
    = V°}
    (4.4)
    mi
    where z„ e R and 0 e R . In this chapter n is still used
    —n —
    to denote the number of links and the variables , m2, m
    are introduced to prevent repetition of referring to n = 3
    and n = 6 separately.
    For 6-link spatial manipulators
    m^ = 3
    m2 = 3 (4.5)
    m = m^ + m2 = 6 and
    for 3-link planar manipulators
    m^ = 2
    m2 = 1 (4.6)
    m = m^ + m2 = 3
    Hand orientation is given through the orientation of
    the basis vectors {u|n^, u.^, u^n^} represented in Fq.
    mixmi
    This, m turn, is given by the rotation matrix Tn e R
    as defined by Equations (2.3)-(2.6)
    T = T (0)
    n n —
    (4.7)
    If the orientation of hand is specified through the Euler
    angles C, 5/ ?, then the basis vectors will be given by

    103
    UHU, B, O [uHl ur2 ur3]
    (4.8)
    where
    Tn(£) = UH (4.9)
    2
    will yield set of m^ nonlinear, coupled algebraic equations
    of which only m2 are independent.
    Note that for the 3-link planar manipulator only one
    Euler angle, say e, is needed to specify the hand
    orientation. Substituting 8=0 and £ = 0, U„(£, 8/ £) will
    n
    take the familiar form
    O'
    cose -sine
    0
    sin? cose
    0
    0
    1
    m-i xm-i
    For this case U„(e) e R x x, m, = 2, will be defined as
    n 1
    cos£ -sine
    (4.10)
    COSC
    sine
    Equation (4.9) .
    Equation (4.4) yielding m^ equations augmented with
    the m2 independent equations of Equation (4.9) will give
    (4.11)

    104
    where
    xx = 0 e Rn
    x* £ Rm is the specified hand position and
    orientation expressed in frame Fq
    In general, f° maps Rn into Rm. If R™ represents a subspace
    of Rm which is identical with the hand's work space, then f^
    maps Rn onto Râ„¢. However, in general, inverse correspondence
    f® ^ of Rm (or R™) to Rn does not constitute a mapping.
    Hence, the forward problem, that is, given x^, finding x*,
    is straightforward and x^ can be determined for any x ^.
    However, the inverse problem, that is, given x*, find x^,
    may or may not have a finite number of solutions. Also
    inverse problem
    *1
    (4.12)
    where superscript -1 here denotes the functional inverse,
    in general, cannot be solved explicitly for x^.
    4.2.2 Relations on Hand Velocity
    and Acceleration
    The absolute linear and angular velocities of the
    hand, vH and Wjwq, resPectively> are given by

    105
    -H = GH ¡¿
    n
    (4.13)
    —H/0 = Gn Hi
    (4.14)
    • n th mixn
    where w = Q_ e R , j column of GHn e R is defined by
    [GHn]j ~
    s. x z„ /n , if j*"*1 joint is revolute
    —D —1tv u j
    s .
    (“3
    , if j*"*1 joint is prismatic
    (4.15)
    Zjj/0 . i-s as given by Equation (2.29) with replaced by H,
    -1th m2xn
    the j column of e R , [Gn]^ is defined by Equation
    (2.37) with i = n. Combining Equations (4.13) and (4.14)
    -2 = J^-l^ -2
    (4.16)
    where
    * T T T
    -2 “ (—H' —H/0>
    (4.17)
    mi m2 * m
    vR e R , —H/0 e ^ , x2 e R with m = m^ + m2,
    J = JUj) =
    =H <5l»'
    n
    G (x, )
    n —1
    s R
    mxn
    (4.18)

    106
    x = (x*, x*)T e R2n, x^ and x2 e Rn (4.19)
    x* = (x*T, x*V e R2m, x^ and x* e Rm (4.20)
    x^ denotes the generalized joint displacements, whereas x*
    represents the position and orientation of the hand in the
    fixed frame Fq.
    The Jacobian J in Equation (4.16) is given in terms
    of joint displacements x^. Introducing Equation (4.12)
    into (4.18)
    J* = J* (xj) = J(f° (x*) ) =
    symbolically,Jacobian J* is expressed in hand coordinates
    x*. Throughout this chapter all functions, when expressed
    in hand coordinates x*, will be denoted by superscript
    asterisk. Hence, Equation (4.16) could be represented by
    x* = J(f° (x*)) x2 (4.22)
    n_1
    CH (£ (x*))
    nn
    n-l
    Gn (£ (xj))
    nmxn ,. «..
    e R (4.21)
    or
    -2
    (4.23)

    Expressions for the hand acceleration is obtained
    differentiating
    Equation (4
    .16) with
    respect to
    * *
    -2
    = J (x,) x2
    + J (x^)
    ¿2
    where the (i, j
    )^ element
    of j (x^)
    „mxn .
    £ K is
    n 3 J • .
    [j
    y i:
    =1 3xl,k
    x2,k
    i =
    1,2,•••fm/
    j = 1,2,
    . . . ,n
    Defining
    Jk =
    3 J
    kl
    30.
    3 J
    kl
    30.
    3 J
    kl
    30.
    n
    3 J
    k2
    30.
    3 J
    k2
    30.
    3 J
    k2
    3 0.
    n
    3 J
    km
    30.
    3 J
    km
    30.
    3 J
    km
    30.
    n
    __ _ / \ «-jinxrn. « to
    J— J^ (x^) £ R , k — 1,2,...,m
    (4.24)
    (4.25)
    (4.26)

    108
    J ' -—2 ^ ~
    T T
    -2 J1
    -2
    —2
    m
    e R
    mxm
    (4.27)
    Given joint displacements x^, hand velocity x* > an<^
    acceleration x^, corresponding joint velocities x2 and joint
    accelerations ¿2 can be solved from Equations (4.16) and
    (4.24), respectively, provided that the Jacobian J (x^) is
    nonsingular.
    T-1 , > *
    x2 = J (x ) x2
    (4.28)
    -2
    J_1(x1) x* - J-1 (x^) j (xr x*) J 1 (x^) x*
    where
    (4.29)
    J (x-j_, xip =
    xf J 1T u±) J1 (x±)
    *T -IT , > T , v
    X* J (x^) J1 (x^)
    e R
    mxm
    (4.30)

    109
    Substituting Equation (4.12) into Equations (4.28)-(4.29),
    x2 and x2 can be determined, given x*, x2, x* from
    -2
    = J
    *-l
    (4.31)
    *-l, *, **
    = J (x^ x2 - J
    -1
    (X*)
    (x*, X*)
    J* 1 (^1^2
    (4.32)
    where
    *-1 -1 o-1 4.
    J = J 1 (f (x*)) (4.33)
    n-l
    j* (x*, x*) = J (f (x*), x*) (4.34)
    4.2.3 Singular Configurations
    The Jacobian given in Equation (4.18) will be
    singular at certain configurations of the manipulator called
    singular configurations. At these configurations the hand's
    mobility locally decreases (i.e., less than m), hence, the
    hand cannot move along or rotate about any given direction
    of the Cartesian space. This is anticipated, since the
    degree of mobility of the hand is the rank of the Jacobian
    and det[J(x^)] = 0 at singular configurations, i.e.,
    rank[J(x^) ] < m.
    Essentially, singularity of Jacobian is a geometry
    problem and the associated singular configurations are the

    110
    property of a given manipulator. Hence, this problem has
    to be addressed first at the design stage of each manipulator.
    That is, elimination of singular configurations as much
    as tolerable by other design requirements through the
    change of kinematic parameters and the identification of
    all remaining singularities are (or should be) part of the
    design process. So far, this aspect is ignored in the
    design of industrial manipulators. This identification will
    define certain subspaces of the manipulator's workspace in
    which manipulator undergoes singular configurations. Once
    these subspaces are identified, the complementary of the
    union of these subspaces in the work space will define
    subspaces, or safe regions, in which manipulator will avoid
    singularities.
    In view of above discussion, singularity avoidance
    which is purely based on geometric considerations need to
    be checked beforehand and the commands which avoid
    singularities should enter the controller. At this point
    it will be assumed that variations in the hand configuration
    in reaching the command configuration also lie in the safe
    region. That is, a singularity-free command which lies
    close to the border of a singularity subspace may cause
    the manipulator to undergo a singularity configuration in
    reaching the command configuration. Above assumption,
    however, requires that the command is so generated and

    Ill
    executed that no singularities are met. It should be noted
    that the variations in hand configurations which are
    required to be singularity free will depend on the system
    transient response as well as the disturbances acting
    on the system.
    Although assumed otherwise, if the hand still
    undergoes a singularity configuration, this level of
    control is not equipped with a remedy. It is not the
    intent of this level controller to avoid singularities,
    but to execute the singularity-free commands. In
    application, if the destination configuration is not
    reached within an anticipated duration (system error cannot
    be reduced to an acceptable precision in a tolerable time
    period), then the controller should activate the emergency
    stop and generate a warning signal to the operator. This
    precaution will be built in and operate whatever the reason
    may be, including the singularity configurations.
    4.3 System Equations in Hand Coordinates
    4.3.1 Plant Equations
    State equations of the plant expressed in hand
    coordinates are obtained substituting Equations (4.12),
    (4.31), and (4.32) into Equation (3.1)
    X* =
    [0] [II „
    mxm raxm
    x* +
    [0] 1
    mxn
    -p
    J*A*-1G* J*A*-1F* +F*
    _P P P P P IP 2p_
    -P
    J*A*-1
    _P P
    (4.35)

    112
    As usual, subscript p is reserved only to denote the
    "plant"; it must not be confused with a counter which is
    always denoted by letters i through £. Similarly, subscript
    r is reserved for the "reference" model parameters.
    Referring to Equation (4.35),
    x* = (x*T x*T)T
    -p l-pl' —p2
    (4.36)
    —pi £ ^m' —p2 e as <^e^:'-ne<^ before,
    u* e Rn is the control vector,
    -P
    A _1 (x*.) = A-1 (f° (^pl)} e R
    P “Pi P - ^
    (4.37)
    G* (x*p) £ Rnxm is defined symbolically such that
    SLp (*pl> = Sp (*pl> 5pl
    (4.38)
    and
    2* (x* ) = 2 (f°
    s-p -pi —p -
    -1
    >
    (4.39)
    where g (•) isdefined by Equation (3.6), hold. Similarly,
    P
    F* (x* , x*_) e Rnxm is so defined that
    lp —pi —p2
    f* (x* , x*.,) = FÍ (x . , x -) x%
    -lp -pi —p2 lp -pi' —p2 —p2
    (4.40)

    113
    holds; f*p (x^1, x*2) is given
    by
    £lP (^r —p2^ = ip<£° (^pi>- Jp"1 (-pij -p2>
    (4.41)
    f (•) is defined by Equation (3.9), and
    -P
    2p
    T _*-lT _* T*-l
    x ~ J J . J
    —p2 P Pi P
    T *—IT T* T*-l
    x „ J J - J
    -p2 p pm p
    e R
    mxm
    (4.42)
    where
    V-1
    r*. = j*. (x*,) = J
    pi pi -pi
    pi
    (4.43)
    Jp^ is as given by Equation (4.26); i = l,2,...,m;
    [0] , and [I] represent mxm and mxn null matrices
    mxn mxm
    and m xm identity matrix, respectively.
    Since (x^) = x^, in general, cannot be inverted
    explicitly, closed-form expressions for J*, A*, G*, F*p,
    and F0 as functions of x cannot be obtained. Given the
    hand configuration x*^ and velocity x*2, one has first to
    solve f0 (x^) = x£ numerically for joint displacements Xp^<
    then calculate Jp, J ^, and finally compute Ap, Gp, F^,
    and F2p’ Although equations are symbolically represented

    114
    in hand coordinates, their implementation still needs the
    joint displacements, i.e., the inverse problem solution.
    This represents added calculation in forming the dynamic
    equations other than the calculation of J and j .
    P P
    4.3.2 Reference Model Equations
    Reference model state equations expressed in hand
    coordinates are given as follows:
    *r =
    [0]
    mxm
    J* A*'1 G*
    r r r
    [I]
    mxm
    J*A*_1F* +F*
    r r lr 2r
    x
    —r
    u* (4.44)
    Definitions of all variables are the same as Equation (4.35);
    this time variables refer to the reference system instead of
    the plant. Equations (4.36) through (4.43) are valid for
    Equation (4.44) when subscript p is replaced by r. The
    reference model produces the desired response for the plant
    to follow. It should be noted that functional dependencies
    are omitted in Equations (4.35) and (4.44).
    4.4 Adaptive Control Law with
    Disturbance Rejection
    Error between the plant and reference model state
    vectors for the augmented system is defined by
    10]
    mxn
    J*A*_1
    r r

    115
    e* = e*(t)
    *
    = x ,
    —rl
    - X*
    -pi
    e* = e*(t)
    *
    = X t
    —r2
    *
    “ x 0
    —p2
    -3 = -I -I
    —3 = 1 -1
    dt
    (4.45)
    (4.46)
    e* = (e*T, e*T, e*T)T e R3m
    (4.47)
    T ,, • * * ' * "
    Letting u = u + u
    y -P -P -P
    (4.48)
    u*' = A*J*_1[ J*A* 1G*x*, + (J*A* 1F* +F* )x*, -K,e*
    —p p p 1 r r r—rl r r lr 2r —r2 1—1
    - K2e* - K3e*] (4.49)
    augmented error-driven system equations will be obtained
    subtracting Equation (4.35) from Equation (4.44) and
    substituting Equations (4.48)-(4.49) as
    • *
    e
    . * *
    A e
    B*z*
    „*t*,*-1 *"
    B J A u
    p p -p
    where
    1
    o
    i
    1
    o
    1
    o
    1
    K1
    K2
    K3
    B* =
    I
    11
    H
    0
    1
    o
    1
    o
    i
    (4.50)
    (4.51)

    116
    K^, K2, and e R111*111 constant matrices to be selected,
    A* e R3mx3m, b* e R3mxn, 0 and I denote the null and
    identity matrices of appropriate dimensions,
    * *-i *
    + J A u
    r r —r
    (4.52)
    Note that subscript a is omitted in this section previously
    used to denote the augmented system. Now, u*" will have
    P
    the following structure
    u = (-K + AK x + (K + AK ) u
    —p p p —p u u —r
    (4.53)
    K* = [8* : F* + A*J*_1F* ]
    P P IP P P 2pJ
    (4.54)
    AK* = tR1I*(S1^1)T • R2v*(S2x£2)T]
    (4.55)
    [a*j;-1j*a
    (4.56)
    AK* = [R3v*(S3u*)T] (4.57)
    ( •) denotes the calculated values, K* and AK* e j^nx2m^
    Ir Jr
    K* and K* e Rnxn, R., S., i = 1,2,3, are as defined by
    U U 11
    Equations (3.69) and (3.70) with the exception that and
    S e Rmxm

    117
    * _*T_* *
    v = B Pe
    P* e ^3mx3m solution of the Lyapunov equation
    A*^p* + p*A* = —Q*
    (4.58)
    * if 3mx3m . ^ .
    Q e R and Q >0
    Choosing a scalar Lyapunov function of the form
    •t
    TT* , * . « *T_* . n
    V(e,t)=e Pe+2
    , *T_*,*—1_ *.
    n (v J A R.v )
    0 — p p 1—
    / *T T * \ j
    (ipisispi) dT
    + 2
    0..
    / *T * *™1 . *T T ★ xt
    (V JpAp S2Z ) <5p2S25p2)d
    + 2
    , aT.,*,*-!,. *> , *T„ *, ,
    (v JpAp R3v )(ur S3ur)dx
    (4.59)
    and using Equations (4.50), (4.52)-(4.57), and (4.58),
    V*(e*) will be obtained
    • * , *, *T * * . ~ *T *i
    V(e)=-e Qe+2v z
    (4.60)
    where

    118
    = J*A*_I[(¿* -2*) + (f* -f* )]
    P P ^P ^P “IP “IP
    (*2p
    - F* )
    * 2p
    x*
    -P2
    + (JA
    r r
    *,*-1 *-l~*~*-l
    -J A ~A"J" ~J *A* 1) u* (4.61)
    P P P P r i* —r
    It can be shown that bound on e will be
    X (P*)
    „ *n s' o max
    — max ^ 5JT
    * 'l
    max
    (4.62)
    As X (P*) is decreased or X . (Q*) is increased or II z*'||
    is reduced through frequent updating of calculated values,
    I e*|| will be reduced. This augmented system will also
    reject constant disturbances at steady state. Similar
    arguments on the zero steady state error can be given as in
    Chapter 3. The closed-loop system satisfying V(e*, t) > 0
    • *
    and V(e ) <0 will be globally asymptotically stable (also
    asymptotically hyperstable).
    4.5 Implementation of the Controller
    When dynamic equations are expressed in joint space,
    information on hand configuration is indirectly supplied by
    the reference model. Each given task in hand coordinates
    will be transformed to the joint space off-line and built
    in the reference system. But, when equations are
    represented in hand coordinates, our immediate concern is
    the hand configuration, not the joint variables. However,
    as mentioned before, given hand configuration, we are unable

    119
    to form dynamic equations directly without reference to
    joint variables. Hence, solution of the inverse problem,
    Equation (4.12), is needed. Actually, this requirement is
    not restrictive, since today's servo-controlled manipulators
    solve this equation on-line mostly using iterative
    techniques.
    It is also interesting to note that currently
    direct measurement of hand configuration is not common, at
    least not feasible enough to equip today's industrial
    manipulators with. When equations are expressed in hand
    coordinates, normally a plant's (manipulator's) hand
    configuration needs to be measured to compute the error.
    However, current implication is that, first, joint variables
    will be measured, which is a common practice, then the
    forward problem, Equation (4.11), will be solved to find
    the "measured" hand configuration. It should be noted
    that system equations expressed in joint space could be
    used and coupled with Equation (4.11) as the output equation.
    Then, however, further development of the controller is
    not immediate.
    Although equations are represented in hand
    coordinates,their implementation requires on-line solution
    of either forward or inverse problem. Forward solution is
    needed if joint displacements are measured, inverse solution

    120
    if hand configuration is measured directly. Of course,
    this means more on-line computation; but considering that
    the inverse problem is already solved on-line on current
    industrial manipulators and that the forward solution is
    straightforward (computationally not demanding), this
    requirement is tolerable for implementation.
    Once the forward or inverse problem is solved,
    implementation of the control does not require any more
    significant computation compared to that of Section 3.3.2,
    Controlled Structure 4, other than J ^. Since the
    Jacobian J is assumed nonsingular, existence of its inverse
    is assured, but on-line computation of J ^ is the major
    drawback of the proposed controller. One remedy to reduce
    computational burden in finding J ^ would be forming J
    symbolically (i.e., each entry of J is explicitly formed
    as a function of the manipulator's kinematic parameters
    and the joint displacements) and then inverting it
    symbolically. Symbolic formulation of various nonlinear
    functions containing a relatively large number of terms
    is studied in [44] . As pointed out in that work, the number
    of terms in J will significantly reduce when special
    manipulator dimensions (zero-link lengths, link offsets,
    and twist angles which are mostly 0° or 90° for industrial
    manipulators) are introduced.

    CHAPTER 5
    ADAPTIVE CONTROL OF MANIPULATORS
    INCLUDING ACTUATOR DYNAMICS
    So far it is assumed that the actuator torques are
    the control variables. Although such a model is easier to
    study, it is not physically realizable, since actuator
    torques cannot be assigned instantaneously. In this
    chapter, manipulator dynamics coupled with the actuator
    dynamics define the system equations. Actuator input
    voltages then become the control variables. An adaptive
    control scheme for this system is also presented.
    5.1 System Dynamics Including
    Actuator Dynamics
    5.1.1 Actuator Dynamics
    It is assumed that n actuators drive an n-link,
    th.
    n degree-of-freedom manipulator, and that the k actuator
    th t h.
    is mounted on the (k-1) link and acts on the k link
    through a gear reduction box. It is further assumed that
    the actuators are permanent magnet DC motors with armature
    current control. Each actuator is modeled as a third order,
    linear, time-invariant system with dynamic equations
    Jk 0k + Dk 0k + Kck 9k + Tk rk KTk 1k
    (5.1)
    Lk ^k + Rk -^k + rk Kvk 9k = uk
    121

    122
    t h
    where the k actuator parameters, k = l,2,...,n, are
    Jk :
    Rotor inertia referred to output shaft
    °k :
    Coefficient of viscous friction referred
    to output shaft
    «ck 5
    Compliance coefficient referred to
    output shaft
    Fk :
    Gear box reduction ratio
    *Tk :
    Actuator torque constant
    Fk :
    Armature inductance
    \ :
    Armature ohmic resistance
    Fvk :
    Actuator back e.m.f. constant
    Tk :
    Joint loading torque
    ek :
    Generalized joint displacement
    Lk :
    Armature current
    uk :
    Actuator input voltage
    If j£, D*, and K*k represent the rotor inertia, viscous
    friction, and compliance coefficients of the actuator,
    respectively, then their values referred to output shaft,
    Jk, Dk, and K
    are given by

    123
    rv Jv
    k k
    4 Di
    r> K*v
    k ck
    (5.2)
    th
    where is the k^ gear box reduction ratio. Although
    Equation (5.1) is given here for permanent magnet DC motors,
    any type of actuator represented by a third-order, linear,
    time-invariant model can be used without loss of generality.
    In matrix-vector form Equation (5.1) is given
    by
    EQe_ + E2^- + E1®. + 1 = E3Í
    (5.3)
    • •
    i - E^0_ - E^i_ = Lu
    where
    Eq = diag[Jk]
    E1 = diag[Kck]
    E2 = diag[Dk]
    E3 = diag[?kETk]
    E4 = diag[-?kKvk/Lk]
    E5 = diag[-Rk/LkJ

    124
    L = diag[l/L^]
    i =
    [il A2
    ... inJ
    u =
    Iul u2
    ... u ]
    n
    E. and L e time-invariant,
    are positive definite except for
    definite; j = 0,1,2,...,5.
    diagonal matrices; E^ and L
    E^ and Ewhich are negative
    5.1.2 System Equations
    Manipulator dynamic equations expressed in joint
    space are given by (see Sections 2.4.2 and 3.2.1)
    T = A (0)9_ - f (0, 0) - 2.(6) (5.4)
    P
    3 n
    Defining the state vector x e R
    i T T T.T ,c
    x = (x^, x2, x3) (5.5)
    x^ = 9_, x2 = 0_, x3 = i (5.6)
    and substituting Equations (5.4) and (5.6) into Equation
    (5.3), new system equations will take the form
    xx = x2 (5.7)
    • _ 1
    x2 = A (Xx) [(x(x^/ —2 ^ ^ 1—1 ~ ^2—2 3^£3 ^
    (5.8)

    125
    x-, = E.x» + Ecx0 + Lu
    —3 4—2 5—3 —
    (5.9)
    where
    (5.10)
    e R
    nxn
    is positive definite, since A^(x^) and EQ
    are both positive definite.
    5.2 Nonlinear State Transformation
    Since Equation (5.9) is the only state equation which
    contains the input vector u, extension of previously given
    control structures to this system is not immediate. However,
    the following state transformation will facilitate the
    controller design. A similar transformation is performed via
    state feedback to decouple and linearize system equations through
    nonlinear term cancellation in [13]. It should be noted
    that here state transformation is not used for this purpose.
    Denoting the new state vector y e R
    3n
    Z = Z2' Z3>
    T T T. T
    (5.11)
    state transformation will be defined by
    Zl =
    (5.12)
    (5.13)
    Z3 = *2' —3}
    (5.14)
    (5.15)
    Equation (5.14) can be written as
    Z3 = Z2' ?i3)
    (5.16)

    126
    or when solved for
    x_3 = q' (y^, y_2 ' Z3)
    where
    2.' = e51 + £
    + ElZl + E2l2
    Differentiating Equations (5.12)- (5.13) and (5.
    respect to time
    ¿1 = ¿!
    i2 = ¿2
    ¿3 = 4' and substituting Equations (5.18) and x^ = y^,
    with Equation (5.17) into Equations (5.7)-(5.9)
    system equations become
    ¿i - 12
    ¿2 = y3
    Z3 = A"1{(E1E5)y1 + (E2E5 + E3E4 -f
    + (e5A - Á - E2)y3 + e5 [2_(y-]
    - giy^) -i.(Z]_> y_2> + N ^
    (5.17)
    ^2}
    + A(i1)y3]
    (5.18)
    17) with
    (5.19)
    x2 = y_2 along
    , transformed
    4^2
    ) + f (y.i* y2} ]
    (5.20)

    127
    where
    N_1 = E3L = diag[rkKTk/LK] £ RnXn
    (5.21)
    diagonal, constant matrix. Also, since and Ej are
    diagonal matrices
    E . E . = E . E .
    ID D i
    E.E.E71 = E.E.eT1 = E.i i,j = 0,1,...,5
    131 D11 D
    Referring to Equation (5.20), A, g_, and f are given as
    follows:
    (5.22)
    A = [Aij1 = £
    +
    n
    9A_
    ‘ij Jj. y2,k 3Yli
    11
    k
    _d_
    dt
    V*l>
    (5.23)
    n 32.ÍZ].)
    - " Y2,k 9ylfk
    n
    £» l
    k=l
    ^2}
    2, k 9y
    l,k
    + y
    9f (yr y2)
    3, k 9y
    2,k
    (5.24)
    .th
    where y. , represents the kV11 element of vector y. e Rn.
    1 f .K 1
    Equation (5.20) is now represented in vector-matrix form
    and given for the plant as follows:

    128
    0 I
    o
    I
    o
    1
    ¿P=
    0 0
    I
    ip+
    0
    -1 -1
    r—
    i
    . -1 -1
    _AP Glp Ap G2p
    A G.
    P 3pJ
    A n
    p p
    u (5.25)
    -P
    where G, , G~ , and G-, e Rnxn be defined such that the
    lp 2p jp
    following hold:
    Glp^p^pl - ElpE5p^pl + E5p—p ^^-p 1 ^
    G2p(ip*ip2 = (E2pE5p +E3pE4p "Elp'3£p2
    + E5pVV - i(ipl>
    G3p(ip)Zp3 = (5.26)
    State vector is as defined by Equations (5.11)—(5-14);
    Up represents the actuator input voltages. Subscript p in
    Equations (5.25)-(5.26) indicates the plant. If subscript p
    is replaced by r, Equations (5.25)- (5.26) will represent
    the reference model state equations.
    5.3 Adaptive Controller
    Actuator input voltage u^ has the form
    u = u' + u"
    -P -P -P

    129
    with
    -1
    u'=NA Y (A G.y.-K.e.)
    -P P P r jr^-rj D~D
    (5.27)
    Following the procedure described in Section 3.3.2, error-
    driven system equations can be obtained as follows:
    é = Ae + Bz - A'V V
    - - P P -P
    (5.28)
    In this section, e, z, A, and B are defined as
    £ = Yr - £
    , e e R
    3n
    e . =
    -D
    —r j
    ^pj
    , j
    = 1,2,3;
    ~0
    I
    0“
    ~0 "
    A =
    0
    0
    I
    , B =
    0
    K1
    K2
    K3
    I
    ,n
    (5.29)
    (5.30)
    „ „3nx3n _ „3nxn
    A e K , B e k
    J -1-1
    z = - y (G. y . ) + A N u
    DP “PD r r -r
    (5.31)
    K , AK e Rnx3n and K , AK e Rnxn of the second part of
    p p u u
    the controller

    130
    = (-Kp + AV Ip + (Ku + AV —r
    are now given by
    Kp - 'Glp = G2p 5 G3p]
    ZSKp = [Rl v(Slypl)T : R2v(S2j^2)T : RjVlSj^)1]
    (5.32)
    Ku ‘ 'VA'
    4KU = [R4i(s4!ir)T]
    T 3 nx 3 n •
    with v = B Pe and P e R is the solution of the Lyapunov
    T1 Iny In
    equation A P + PA = -Q, Q e R , Q > 0, R^, and are
    as defined by Equations (3.69) and (3.70) with i = 1,2,3,4.
    If the following Lyapunov function is used,
    V (e, t) = eTPe +2 £
    j=l
    T —1 —1 T T
    (v A N R.v)(y .S.y .)dr
    - P P D- ^PD
    + 2
    (vTA ^R.v)(uTS.u.)dx
    — p p 4— —r 4-^4
    (5.33)
    global asymptotic stability of the closed-loop system can be
    shown.
    The transformed state vector is composed of joint
    displacements, velocities, and the accelerations. Hence,

    131
    measurement of joint accelerations is the added requirement
    in implementation. Although joint accelerations may be
    measured, it is best avoided because of the relatively high
    noise level in these measurements. An added computational
    • • •
    requirement is the evaluations of A, f, and in Equations
    (5.23)-(5.24) . These computations will slow down the
    updating rate of K in Equation (5.32), hence, error bound
    IT
    I e l^ax increase if computation speed is held constant.
    Otherwise, added computations are not significant, since
    actuator dynamics is represented by linear, time-invariant
    models.
    The system equations, Equation (5.25) can be
    expressed in hand coordinates (see Chapter 4) and/or they
    can be easily augmented to include integral feedback to
    achieve disturbance rejection features. The order of the
    system will rise to 4n (from 3n) if integral feedback is
    added. Simplified actuator dynamics and the corresponding
    system (manipulator + actuators) dynamics which avoids
    acceleration measuresments and the calculations of Á, f,
    and are presented in the following section. Integral
    feedback is also added in that section which is otherwise
    a simplified version of this section.
    5.4 Simplified Actuator Dynamics
    5.4.1 System Dynamics
    Typically, rotor inductivity will be in the order
    -2 -5
    of 10 to 10 henry; hence, the actuator dynamics may be

    132
    simplified approximating L^. - 0 [58] . This significantly
    affects the actuator model. The third-order system
    representation of actuator dynamics of the previous section
    reduces to a second-order system. This eases the analytical
    treatment of the problem.
    Actuator dynamics of Equation (5.1) will now take
    the form
    , K_, K .
    k Tk vk
    R.
    1
    0
    k
    k
    1
    (5.34)
    u.
    k
    R.
    k = 1,2
    where all parameters are as defined in Section 5.1.1, or
    (5.35)
    Eg and E^ are as given by Equation (5.3) and diagonal E2
    j n71 1 „nxn
    and N' e K are
    nxn
    are
    E^ = diag +
    *k
    1
    (5.36)
    N'-1 = diag rkKTk ±
    Rk

    133
    Coupled system equations are obtained if Equation (5.4) is
    substituted in Equation (5.35)
    (5.37)
    x2 = A-1 (x^) ig.^) +f (xr x2) - E-^ - E£x2 +N'_1u}
    A is the same as in Equation (5.10); x^ = Q_, x2 = £ are the
    joint displacement and velocities.
    5.4.2 Adaptive Controller with
    Disturbance Rejection Feature
    The plant equations directly follow from Equation
    (5.37):
    0
    I
    0
    X =
    -p
    A_1(G -E. )
    p p lp
    A"1 (F„ - E' )
    p p 2p
    x +
    -P
    A-V-1
    P P
    (5.38)
    G and F are as defined in Equations (3.6)-(3.10).
    P P
    Similarly, reference model equations will be obtained if
    subscripts p are switched to r in the above equation.

    134
    Letting
    e. = x . - x i = 1,2, e. e R
    —i —n —pi —i
    n
    £3 - -l£i
    -3
    Ie^dt
    (5.39)
    and
    with
    T T T. T
    s (, e^2, s^)
    u = u' + u"
    -p -p -p
    u
    -p
    ' = N' A •
    P P
    1 [ (G -E, )x . + (F -E' )x .] - T
    r r lr —rl r 2r —r2 .L
    j = l
    K .e .
    3-3
    (5.40)
    the augmented error-driven system equations become
    where
    e = Ae + Bz' - A ' ^u"
    - P P -P
    (5.41)
    '0
    0
    0
    “0~
    A =
    r—1
    (N
    K3
    , B =
    I
    -I
    0
    0
    0
    (5.42)
    - -V1(GP -Elp>2pl -AP1{FP -E2p>ip2
    (5.43)

    135
    The second part of the controller, u", will have the form
    P
    as in Equation (3.64) with K^, AK^ e ^nx2n an(3
    AK e R11X11 defined by
    u J
    4Kp = IRjVÍSjX^)1 : R2v(s2xp2)T]
    K = [N'A A-1N,_1]
    u p p r r
    (5.44)
    AKu = [R3v(S3ur)T]
    (~) denotes the calculated or estimated plant parameters.
    Equations (3.39) and (3.73) are valid for this case, but
    A and B as defined by Equation (5.42) should be used.
    The system presented in this section includes
    actuator dynamics, the proposed controller rejects
    steady state disturbances and it is easier to implement;
    • • •
    measurement of accelerations and evaluations of A, f_, and g
    are not required. The solution of the error-driven system
    will enter the spherical region containing the origin of
    error space. Hence, manipulator response will converge
    to the desired response. Bound on the spherical region is
    as given by Equation (3.82) with ||z || replaced by || z'||
    — max max
    z' is as defined by Equation (5.43).

    CHAPTER 6
    EXAMPLE SIMULATIONS
    Proposed adaptive controllers are implemented on
    the computer and system response is obtained under various
    operating conditions. Simulations are conducted on the
    VAX-11/750 system at the Center for Intelligent Machines and
    Robotics (CIMAR), Department of Mechanical Engineering,
    University of Florida. The program, mathematics, library
    and graphics package are developed in FORTRAN 77 and
    supported by the VAX/VMS operating system.
    Manipulator dynamics is coupled with the simplified
    actuator dynamics and the controller structure described
    in Section 4.2 is simulated for various manipulation tasks.
    Plant differential equations are integrated using the
    Hamming's fourth-order, modified predictor-corrector
    method. Inclusion of the disturbance rejection feature is
    left optional; the user can select the desired option.
    Although the program is capable of simulating n-link
    manipulators, 3- and 6-link spatial industrial manipulators
    are used in the examples presented in this chapter. The
    program is developed independent of units; the metric system
    is employed in the 3-link manipulator and the British system
    in the 6-link arm examples.
    136

    137
    Plant parameters composed of manipulator kinematic
    parameters and actuator properties form the input to the
    program. These include manipulator link lengths, link
    offsets, twist angles, link masses, inertias, center of
    gravity locations of each link, actuator rotor inertias,
    coefficients of viscous friction, compliance coefficients,
    gear box reduction ratios, actuator torque constants,
    armature resistances, and back e.m.f. constants. A second
    set of the above parameters (possibly with different
    numerical values) is also input to represent the reference
    model. In fact, the reference model parameters represent
    the closest available estimates of the plant parameters and
    we do not know the exact values of the plant parameters.
    This is simulated via discrepancy in the plant and reference
    model parameters in the computer program.
    Other than the differences in the plant and
    reference model parameters, the following disturbances are also
    introduced. Manipulator initial position is set different
    from the initial position of the reference model. After the
    motion started, an extra payload is added on the manipulator
    hand and the system response is observed while the reference
    model had no information of this payload. Measurement
    delays are simulated using time delays ranging between 0.01
    to 5 ms for different examples. The values of A , G , and
    P P
    F in Equation (5.38) are updated at various frequencies
    ir

    138
    from 10 Hz to 400 Hz in the simulations. Although the
    analytical development assumed that Ap in uV, Equation
    (5.40), is exactly and continuously updated, numerical
    simulations updated at the given frequencies.
    Overall control structure may be considered hybrid
    in the sense that the terms in the controller linear in
    . T
    error and state variables [K.e., i = 1,2,3; R-v(S-x .) ,
    ^ ^ D j P J
    T
    j = 1,2 and R-v(S-.u ) are meant] are supplied continuously
    3 j r
    (analog signal), whereas the nonlinear terms A , G , and F
    P P P
    are updated at the given frequencies only. This latter
    part of the input actually constitutes a train of impulses;
    magnitudes held constant during the entire sampling period
    determined by the input update frequency; hence introducing
    shocks to the system.
    It should be noted that the theoretical development
    did not address all of these disturbances individually and
    even the system stability is not guaranteed under their
    simultaneous action. (Error bound on the system response
    is given before.) Here, numerical simulations test the
    proposed controllers under rather severe conditions. The
    magnitudes of disturbances are chosen arbitrarily. The
    maximum amount of the extra payload, for example, which will
    produce undesirable transients or even induce instability
    is not addressed in this study. Without further research,
    proposed controllers should be extensively experimented

    139
    (via computer and laboratory simulations) if large
    disturbances are expected. However, as the examples below
    reveal, performance of the controllers under the simulated
    disturbances is encouraging.
    6.1 Simulations on the 3-Link, Spatial Manipulator
    The 3-link manipulator used in the examples is
    depicted in Figure 6.1 and the related plant parameters are
    given in Tables 6.1-6.3.
    Figure 6.1 Illustration of the 3-Link Manipulator
    Reference model manipulator and actuator parameters that
    are different from the plant are listed in Tables 6.4-6.5.
    The first simulation includes the disturbance rejection
    feature, i.e., the integral feedback is activated. Hence,
    the system order is 9 for the 3-link manipulator; K^, Kj,
    3x3
    and £ R in Equation (5.42) are chosen diagonal

    Table 6.1 Kinematic Parameters of the 3-Link Manipulator
    (Plant Parameters)
    Joint
    sk
    rk
    ak
    No
    (m)
    (m)
    (deg)
    1
    0.1
    0.6
    90.0
    2
    0.1
    0.5
    0.0
    3
    0.0
    0.4
    -
    Table 6.2 Inertia
    Properties of the 3-Link Manipulator
    (Plant Parameters)
    Link
    Centroid Location*
    Mass
    *1
    Inertia
    h
    No
    (m)
    ( kg)
    (kg.m2;
    about centroid)
    1
    0.20
    0.0
    0.0
    20.0
    0.20
    0.60
    0.60
    2
    0.15
    0.0
    0.0
    10.0
    0.05
    0.20
    0.20
    3
    0.20
    0.0
    0.0
    15.0
    0.03
    0.10
    0.10
    * Expressed in the hand-fixed reference frame.

    141
    Table 6.3 Actuator Parameters (Plant)
    Actuator No
    : 1
    2
    3
    Jk
    (10-3kg.m2)
    : 5.00
    5.00
    1.00
    o|
    X
    (Nm/rad/sec)
    : 0.30
    0.30
    0.25
    7k
    -
    : 30.00
    20.00
    10.00
    «Tk
    (Nm/amp)
    : 0.90
    0.60
    0.25
    *k
    (ohm)
    : 1.00
    1.00
    0.60
    *vk
    (volt/rad/sec)
    : 0.50
    0.50
    0.25
    Table 6
    .4 Reference Model Manipulator Parameters
    Link
    Centroid Location
    Mass
    rk
    No
    (10~3m)
    (kg)
    (m)
    1
    204.55 4.55
    4.55
    22.0
    0.65
    2
    158.33 8.33
    8.33
    12.0
    0.55
    3
    180.00 20.00
    20.00
    25.0
    0.50
    Table
    6.5 Reference Model Actuator Parameters
    Actuator No
    : 1
    2
    3
    Jk
    (10-3kg.m2)
    : 4.00
    4.00
    2.00
    Dk
    (Nm/rad/sec)
    : 0.35
    0.35
    0.30
    K ,
    vk
    (volt/rad/sec)
    : 0.55
    0.45
    0.30

    142
    = diag(-160.5 -160.5 -200.0)
    K2 = diag(-24.5 -24.5 -27.0)
    K3 = diag (189.0 189.0 300.0)
    so that the eigenvalues of the decoupled linear part of
    the error-driven system are located at -1.5, -9.0, -14.0;
    -2.0, -10.0, -15.0 with the first three eigenvalues having
    9x 9
    multiplicity two. Also selecting the Q e R matrix
    diagonal
    Q = diag(2.0)
    the solution P of the Lyapunov equation
    ATP + PA = -Q
    is obtained as
    3x3
    where P, e R , i=l,2,...,6, and P. given by

    143
    P1
    = diag( 8.3431
    8.3431
    9.6266)
    P2
    = diag ( 0.0579
    0.0579
    0.0646)
    P3
    = diag(-8.2905
    -8.2905
    -11.9188)
    P4
    = diag( 0.0432
    0.0432
    0.0394)
    P5
    = diag(-0.0053
    -0.0053
    -0.0033)
    = diag(11.7894
    11.7894
    20.0449)
    The method used in the numerical solution of the Lyapunov
    equation is explained in the following section. and
    3x3
    e R in Equation (5.44) are chosen as follows:
    S1
    = diag(0.5
    0.5
    0.5)
    S2
    = diagd.O
    1.0
    1.0)
    S3
    = diag(0.1
    0.1
    0.1)
    R1
    = diag(0.1
    0.1
    0.1)
    R2
    = diag(2.0
    2.0
    2.0)
    R3
    = diag(0.1
    0.1
    0.1)
    Time delay in measurements is input as 5 ms.
    Initial plant position is set to x ^ = (20.0 60.0 -115.0)T
    deg, whereas the reference model position was x ^
    T
    = (45.0 20.0 -40.0) deg. One second after the motion

    144
    started, 20 kg extra payload is dropped on the manipulator
    hand (plant). Figures 6.2-6.4 illustrate the system
    response and the actuator input voltages when nonlinear
    terms are updated at 60 Hz. The same problem is repeated
    with 10 Hz, Figures 6.5-6.7, and 200 Hz, Figures 6.8-6.10.
    Smooth curves in the displacement and velocity plots
    designate the desired path, whereas the second curve in
    these graphs shows the plant response.
    In all three cases system stability is preserved,
    reference trajectory is tracked, and the steady state error
    is eliminated with the disturbance rejection feature.
    However, with 10 Hz updating rate, response of the second
    and especially the third joints (Figures 6.5b-c, 6.6c)
    deteriorated compared to Figures 6.2b-c, 6.3c (60 Hz) and
    Figures 6.8b-c, 6.9c (200 Hz). As expected, smoother
    actuator input voltage curves are obtained as the update
    rate is increased from 10 Hz to 60 and 200 Hz. Compare,
    for example, Figures 6.7c (10 Hz), 6.4c (60 Hz), and 6.10c
    (200 Hz). The sudden jump in the input voltage curves and
    the deterioration of system response at t = 1 sec is
    because of the addition of extra mass on the manipulator
    hand. Due to the integral feedback action, system response
    converges to the desired path in about 0.2 sec.
    The final simulation is conducted on the same
    manipulator without activating the integral feedback.

    Disp. 1 (ded)
    0.00 0.41 0.83 l.£4 1.65 £.06 £.48 £.83 3.30 3.71
    Figure 6.2a Joint 1—Displacement vs Time
    (r: Reference Model, p: Plant Response)
    Time (sec)
    145

    Disp. 2 (ded)
    0.00 0.41 0.83 1.24 1.85 2.06 2.48 2.89 3.30 3.71
    Time (sec)
    Figure 6.2b Joint 2—Displacement vs Time
    146

    Disp. 3 (ded)
    0.00 0.41 0.83 1.24 1.65 2.06 2.48 2.89 3.30 3.71
    Time (sec)
    Figure 6.2c Joint 3—Displacement vs Time
    147

    Vel. I (l/s)
    0.00 0.4-1 0.83 1.E4 1.65 E.06 E.48 E.89 3.30 3.71
    Time (sec)
    Figure 6.3a Joint 1—Velocity vs Time
    148

    Vel. £ (l/s)
    0.00 0.41 0.83 1.24 1.65 2.06 2.48 2.89 3.30 3.71
    Time (sec)
    Figure 6.3b Joint 2—Velocity vs Time
    149

    Vel. 3 (l/s)
    0.00 0.41 0.83 1.84 1.65 8.06 8.48 8.89 3.30 3.71
    Time (sec)
    Figure 6.3c Joint 3—Velocity vs Time
    150

    Inp.VoIt, 1 (v)
    0.00 0.41 0.83 1.24 1.65 2.08 2.48 2.89 3.30 3.71
    Ti me ( sec)
    Figure 6.4a Actuator 1—Input Voltage vs Time
    151

    Inp.Volt. 2 (v)
    Ti me (sec)
    Figure 6.4b Actuator 2—Input Voltage vs Time
    152

    Inp.Volt. 3 (v)
    Time (sec)
    Figure 6.4c Actuator 3—Input Voltage vs Time
    153

    Disp. 1 (ded)
    Time (sec)
    Figure 6.5a Joint 1—Displacement vs Time
    154

    Disp. 2 (ded)
    Ti me (sec)
    Figure 6.5b Joint 2—Displacement vs Time
    155

    Disp. 3 (ded)
    0.00 0.4-S 0.84 1.87 1.69 8.11 8.53 8.96 3.38 3.80
    Time (see)
    Figure 6.5c Joint 3—Displacement vs Time
    156

    Vel. I (l/s)
    Time (sec)
    Figure 6.6a Joint 1—Velocity vs Time
    157

    Vel. 2 (1/s)
    Time (see)
    Figure 6.6b Joint 2—Velocity vs Time
    158

    Vel. 3 (l/s)
    Ti me (5bc)
    Figure 6.6c Joint 3—Velocity vs Time
    159

    Inp.Volt. 1 (v)
    0.00 0.42 0.84 1.27 1.69 2.11 2.53 2.96
    Figure 6.7a Actuator 1—Input Voltage vs Time
    3.38 3
    Time (sec)
    160

    Inp.Volt. 2 (v)
    Ti me í sec)
    Figure 6.7b Actuator 2—Input Voltage vs Time
    161

    Inp.Volt. 3 (v)
    Time (sec)
    Figure 6.7c Actuator 3—Input Voltage vs Time
    162

    Disp. 1 (ded)
    0.00 0.41 0.82 1.24 1.65 2.06 2.47 2.88 3.29 3.70
    Ti me (sec)
    Figure 6.8a Joint 1—Displacement vs Time
    163

    Disp. 2 (ded)
    Ti me (sec)
    Figure 6.8b Joint 2—Displacement vs Time
    164

    Disp. 3 (ded)
    Ti me (sec)
    Figure 6.8c Joint 3—Displacement vs Time
    165

    Vel. L (l/s)
    Time (sec)
    Figure 6.9a Joint 1—Velocity vs Time
    166

    Vel. E (L/s)
    Time (sec)
    Figure 6.9b Joint 2—Velocity vs Time
    167

    Vel. 3 (L/s)
    0.00 0.41 0.88 1.84 1.65 8.08 8.47 8.88 3.89 3.70
    Time (see)
    Figure 6.9c Joint 3—Velocity vs Time
    168

    Inp.Volt. 1 (v)
    0.00 0.91 0.82 1.24 1.65 2.06 2.97 2.88 3.29 3.70
    Figure 6.10a Actuator 1—Input Voltage vs Time
    Time (sec)
    169

    Inp.Volt. P (v)
    0.00 0.^1 0.8E 1.24- 1.65 2.06 2.4-7 2.08 3.29 3.70
    Time (sec)
    Figure 6.10b Actuator 2—Input Voltage vs Time
    170

    Inp.Volt. 3 (v)
    0.00 0.^1 0.82 1.24^ 1.65 2.06 2.47 2.88 3.29 3.70
    Time (see)
    Figure 6.10c Actuator 3—Input Voltage vs Time
    171

    172
    In this case the system order is 6 and the A and B matrices
    in Equation (5.42) are given by
    0
    I
    0
    A =
    B =
    _K1
    k2
    I
    (6.2)
    where A e and B e R^x^. Adjustment is made on the
    3x3
    and Kj e U matrices so that the dominant system
    eigenvalues are preserved.
    Kx = diag(-13.5 -13.5 -20.0)
    K2 = diag(-10.5 -10.5 -12.0)
    The corresponding eigenvalues are now located at -1.5 and
    -9.0 with multiplicity two and -2.0 and -10.0. The
    nonlinear terms are updated at 60 Hz. In this example,
    and R^, i = 1,2,3, are modified as follows to improve the
    transient response
    = diag(2.0 2.0 2.0)
    52 = diag(2.0 2.0 2.0)
    53 = diag(1.0 1.0 1.0)
    = diag(2.0 2.0 2.0)
    R2 = diag(3.0 3.0 3.0)
    R3 = diag(1.0 1.0 1.0)

    173
    6x6
    Choosing the Q e R matrix as follows
    Q = diag(5.0)
    solution of the Lyapunov equation is given by
    (6.3)
    3x3
    where P^eR , i = 1,2,3, and
    P1 = diag(5.3968 5.3968 5.8750)
    P2 = diag(0.1852 0.1852 0.1250)
    P3 = diag(0.2557 0.2557 0.2188)
    All plant and reference model parameters,
    manipulation task, and the disturbances are kept the same
    as in the previous three simulations. System response and
    input voltages are plotted in Figures 6.11-6.13. The lack
    of integral feedback is best demonstrated by the 45 deg
    steady state offset in the third joint displacement as
    shown in Figure 6.11c. Also more than 9 deg overshoot is
    introduced in the response of this joint. Comparing
    Figure 6.2c (with integral feedback) to Figure 6.11c,
    overall measure of error in system responses can easily be
    assessed. First joint displacement, Figure 6.11a,

    Disp, 1 (ded)
    Time (see)
    Figure 6.11a Joint 1—Displacement vs Time
    174

    Disp. 2 (ded)
    0.00 0AE 0.05 1.E7 1.70 E.1E E.S^ E.37 3.39 3.8S
    Ti me (sec)
    Figure 6.11b Joint 2—Displacement vs Time
    175

    Disp. 3 (ded)
    0.00 0.42 0.85 1.27 1.70 2.12 2.54 2.97 3.39 3.82
    Time (sec)
    Figure 6.11c Joint 3—Displacement vs Time
    176

    Vel. I (l/s)
    0.00 0.4-2 0.85 1.27 1.70 2.12 2.54 2.97 3.39 3.82
    Time (sec)
    Figure 6.12a Joint 1—Velocity vs Time
    177

    Vel. P (l/s)
    0.00 0.4E 0.85 1.27 1.70 2.12 B.St 2.97 3.39 3.82
    Time (sec)
    Figure 6.12b Joint 2—Velocity vs Time
    178

    Vel. 3 (1/s)
    0.00 0.4-E 0.85 1.27 1.70 E.1E E.51* £.9? 3.39 3.8E
    Time (see)
    Figure 6.12c Joint 3—Velocity vs Time
    179

    Inp.Volt. 1 (v)
    Ti me (sec)
    82
    Figure 6.13a Actuator 1—Input Voltage vs Time
    180

    Inp.Vült. 2 (v)
    Ti me (see)
    Figure 6.13b Actuator 2—Input Voltage vs Time
    181

    Inp.Volt. 3 (v)
    0.00 0.4E 0.85 1.E7 1.70 £.1£ 8.5^ £.97 3.39 3.8£
    Time (sec)
    Figure 6.13c Actuator 3—Input Voltage vs Time
    182

    183
    introduces 13 deg overshoot compared to the case with
    integral feedback, Figure 6.2a. Response of the second
    joint has about 15 deg overshoot and steady state error as
    shown in Figure 6.11b. The last example clearly
    demonstrates the improvements obtained in the system
    response when the integral feedback is activated to reject
    disturbances.
    6.2 Numerical Solution of the Lyapunov Equation
    Given A and the positive definite Q, solution P of
    the Lyapunov equation
    ATP + PA = -Q
    is obtained as follows. Here, A = [a..], P = [P..], and
    1J ij '
    Q = [q —] are assumed to be of dimension kxk. Expanding
    the above equation and writing in matrix-vector form
    A*p* = -q* (6.4)
    ,2 ,2 2 2
    where A* e R' x , p* e R , and q* e K are given by
    T
    A + axlI
    a21I
    akix
    a12I
    T
    A ta22I
    • • •
    ak2I
    •
    •
    • | •
    alk1
    a2kI
    . . .
    aT + akkI

    184
    I represents the identity matrix of order k,
    P = [P1X P12 Pik P21 •** pkk]
    a = c312 •** qik q21 *•* qkk]
    Hence, solution of the Lyapunov equation is reduced to the
    solution of simultaneous algebraic equations of Equation
    (6.4). Although numerically more efficient methods exist in
    the literature, this method is used in the simulations, since
    the solution of the Lyapunov equation is required once and
    can be performed off-line. The solution is obtained by
    means of Gauss elimination with complete pivoting.
    6.3' Simulations on the 6-Link,
    Spatial Industrial Manipulator
    The 6-link, spatial industrial manipulator,
    Cincinnati Milacron T3-776, is illustrated in Figure 6.14
    and its kinematic parameters and inertia properties are
    given in Tables 6.6 and 6.7. Actuator parameters are
    presented in Table 6.8 and the reference model actuator
    parameters that are different from the plant parameters are
    listed in Table 6.9. It should be noted that the actuator
    parameters in Tables 6.8 and 6.9 do not represent the
    actuators used in T3-776. The order of magnitude of these
    parameters are representative of DC motors [13, 58], but
    otherwise arbitrary.

    135
    Figure 6.14 Cincinnati Milacron T3-776 Industrial Robot

    186
    Table 6.6 Kinematic Parameters of the 6-Link Manipulator
    (Plant Parameters)
    Joint
    No
    sk
    (in)
    rk
    (in)
    ak
    (deg)
    cp .
    Ymm
    (deg)
    (p
    Ymax
    (deg)
    1
    32.0
    0.0
    90.0
    -135.0
    135.0
    2
    0.0
    44.0
    0.0
    30.0
    117.0
    3
    0.0
    0.0
    90.0
    -45.0
    60.0
    4
    55.0
    0.0
    61.0
    -180.0
    180.0
    5
    0.0
    0.0
    61.0
    -180.0
    180.0
    6
    6.0
    0.0
    -
    -180.0
    180.0
    Table 6.7
    Inertia
    Properties of the 6-
    Link Manipulator
    (Plant
    Parameters)
    Link
    Centroid Location*
    Mass Inertia about
    centroid
    Jl
    h
    ^ 3
    No
    ( in)
    (lbm)
    (103lbm.
    in2)
    1
    0.0
    0.0
    -17.0
    700.0
    0.0
    0.0
    100.0
    2
    20.0
    -1.0
    0.0
    1500.0
    20.0
    180.0
    150.0
    3
    4.0
    -7.0
    0.0
    1000.0 170.0
    26.0
    170.0
    4
    0.0
    0.0
    -20.0
    150.0
    2.0
    2.0
    1.2
    5
    0.0
    0.0
    0.0
    80.0
    0.8
    0.8
    0.3
    6
    0.0
    0.0
    -4.0
    60.0
    0.4
    0.4
    0.2
    * Expressed in the hand-fixed reference frame.

    187
    Table 6.8 Actuator Parameters (Plant)
    Actuator No :
    1
    2
    3
    4
    5
    6
    Jk
    (10-3lbm.ft2) :
    100.00
    50.00
    50.00
    30.00
    30.00
    20.00
    °k
    (lbf.ft/rad/s):
    0.30
    0.30
    0.20
    0.30
    0.25
    0.20
    Fk
    :
    100.00
    100.00
    100.00
    80.00
    30.00
    10.00
    KTk
    (lbf.ft/amp) :
    15.00
    15.00
    10.00
    8.00
    6.00
    6.00
    *k
    (ohm) :
    0.80
    0.80
    0.80
    0.80
    0.70
    0.60
    *vk
    (volt/rad/s) :
    0.50
    0.50
    0.40
    0.30
    0.25
    0.20
    Table 6.9 Actuator Parameters (Reference Model)
    Actuator No :
    1
    2
    3
    4
    5
    6
    Jk (10-3lbm.ft2) :
    150.00
    45.00
    55.00
    25.00
    25.00
    25.00
    Dk (lbf.ft/rad/sec) :
    0.35
    0.35
    0.25
    0.25
    0.30
    0.25

    188
    In this section, three simulations on T3-776 are
    presented. The first two simulations assume that the
    reference model hand carries 5 lbm extra payload throughout
    the motion. Also, an additional 5 lbm payload is added to
    the reference model at t = 0.7 sec increasing the difference
    to 10 lbm. Integral feedback is in effect in these
    simulations, hence the system order is 18. The initial
    reference model position is
    xrl = (-5 100 -25 -90 0 50)T deg
    The initial plant position is set to
    -pi = {0 50 10 50 "40 0)T deg
    so that the differences in joint positions varied between
    5 to 50 deg. Diagonal e R in Equation (5.42), S^,
    1 RyI 8
    in Equation (5.44), i = 1,2,3, Q e R , and the
    solution of the Lyapunov equation, P e ^18x18 are g^ven
    below; , K2, and are so chosen that the eigenvalues
    of A of the error-driven system lie at -1.0, -2.0, -7.0,
    -11.0 each with multiplicity three and at -9.0 with
    multiplicity six.
    Kx = diag(-79 -79 -79 -139 -139 -139)
    K2 = diag(-17 -17 -17 -22 -22 -22)
    K3 = diag(63 63 63 198 198 198)

    189
    S = R. = diagdO-3), i = 1,2,3
    i 1
    Q = diag(l)
    that is Q is chosen 18 xl8 identity matrix and P^,
    j = 1,2,...,6, as given in Equation (6.1), but now of
    dimension 6x6, are given by
    P1 = diag( 1.5154 1.5154 1.5154 2.1289 2.1289 2.1289)
    P2 = diag( 0.0165 0.0165 0.0165 0.0194 0.0194 0.0194)
    P3 = diag(-1.0552 -1.0552 -1.0552 -2.4528 -2.4528 -2.4528)
    P4 = diag( 0.0157 0.0157 0.0157 0.0122 0.0122 0.0122)
    P5 = diag(-0.0040 -0.0040 -0.0040 -0.0013 -0.0013 -0.0013)
    P6 = diag( 1.3543 1.3543 1.3543 4.0255 4.0255 0.0255)
    Measurement delays are taken as 0.01 ms. Figures
    6.15-6.16 give the system response and Figure 6.17 the
    actuator inputs when nonlinear terms are updated at 50 Hz.
    Later, this frequency is increased to 200 Hz and the
    response is plotted in Figures 6.18-6.20. Again, smoother
    response is obtained as the updating frequency is increased.
    It is interesting to note that the response overshoots are
    either reduced in magnitude or completely eliminated as
    the frequency is increased from 50 to 200 Hz. Joint 6
    displacement, for example, has 30.9 deg overshoot with 50 Hz

    Disp. 1 (ded)
    0.00 0.88 0.44 0.67 0.89 1.11 1.33 1.55 1.78 8.00
    Time (sec)
    Figure 6.15a Joint 1—Displacement vs Time
    190

    Disp. 2 (ded)
    0.00 0.22 0.44 0.67 0.09 1.11 1.33 1.55 1.70 0.00
    Time (sec)
    Figure 6.15b Joint 2—Displacement vs Time
    191

    Disp. 3 (ded)
    0.00 0.22 0.44 0.67 0.09 1.11 1.33 1.55 1.78 2.00
    Time (sec)
    Figure 6.15c Joint 3—Displacement vs Time
    192

    Disp. 4 (ded)
    0.00 0.E2 0.44- 0.B7 0.08 1.11 1.33 1.55 1.70 E.00
    Ti me (sec)
    Figure 6.15d Joint 4—Displacement vs Time
    193

    Disp. 5 (ded)
    Time (sec)
    00
    Figure 6.15e Joint 5—Displacement vs Time
    194

    Disp. 6 (ded)
    0.00 0.22 0.44- 0.67 0.89 1.11 1.33 1.S5 1.78 2.00
    Time (sec)
    Figure 6.15f Joint 6—Displacement vs Time
    195

    Vel. I (i/s)
    Ti me (sec)
    00
    Figure 6.16a Joint 1—Velocity vs Time
    196

    Vel. 2 (l/s)
    0.00 0.22 0.4-4- 0.67 0.89 1.11 1.33 1.55 1.78 E.00
    Time (sec)
    Figure 6.16b Joint 2—Velocity vs Time
    197

    Vel. 3 (L/s)
    Time (sec)
    00
    Figure 6.16c Joint 3—Velocity vs Time
    198

    Vel. 4- ( L/s)
    0.00 0.82 0.4-4- 0.67 0.09 1.11 1.33 1.55 1.78 8.00
    Ti me (see)
    Figure 6.16d Joint 4—Velocity vs Time
    199

    Vel. 5 (L/s)
    Time (sec)
    00
    Figure 6.16e Joint 5—Velocity vs Time
    200

    Vel. 6 (l/s)
    0.00 0.EE 0.4-4- 0.67 0.89 1.11 1.33 1.55 1.78 E.00
    Time (sec)
    Figure 6.16f Joint 6—Velocity vs Time
    201

    Inp.Volt. 1 (v)
    0.00 0.22 0.44 0.67 0.09 1.11 1.33 1.55 1.70 2.00
    Ti me (sec)
    Figure 6.17a Actuator 1—Input Voltage vs Time
    202

    Inp.Volt. 2 (v)
    Time (sec)
    Figure 6.17b Actuator 2—Input Voltage vs Time
    203

    Inp.Volt. 3 (v)
    0.00 0.8E 0.44- 0.67 0.89 1.11 1.33 1.55 1.78 8.00
    Ti me (sec)
    Figure 6.17c Actuator 3—Input Voltage vs Time
    204

    Inp.Volt. 4- (v)
    0.00 0.EE 0.H4- 0.67 0.89 1.11 1.33 1.5S 1.78 8.00
    Time (see)
    Figure 6.17d Actuator 4—Input Voltage vs Time
    205

    Inp.Volt. 5 (v)
    0.00 0.22 0.4*+ 0.67 0.89 1.11 1.33 1.55 1.78 2.00
    Time (sec)
    Figure 6.17e Actuator 5—Input Voltage vs Time
    206

    Inp.Volt. 6 (v)
    0.00 0.22 0.Â¥t 0.67 0.83 1.11 1.33 1.55 1.78 2.00
    T i me (5ee)
    Figure 6.17f Actuator 6—Input Voltage vs Time
    207

    Disp. 1 (ded)
    0.00 0.28 0.*t4- 0.67 0.09 1. i 1 1.33 1.55 1.78 2.00
    Time (sec)
    Figure 6.18a Joint 1—Displacement vs Time
    208

    Disp. 2 (ded)
    0.00 0.22 0.ltlt 0.B7 0.89 1.11 1.33 1.55 1.78 2.00
    Time (sec)
    Figure 6.18b Joint 2—Displacement vs Time
    603

    Disp. 3 (ded)
    0.00 0.22 0.4-4- 0.67 0.89 1.11 1.33 1.55 1.78 2.00
    Ti me (sec)
    Figure 6.18c Joint 3—Displacement vs Time
    210

    Disp. 4- (ded)
    Time (sec)
    Figure 6.18d Joint 4—Displacement vs Time
    211

    Disp. 5 (ded)
    0.00 0.22 0.^4- 0.67 0.89 1.11 1.33 1.55 1.78 2.00
    Ti me (sec)
    Figure 6.18e Joint 5—Displacement vs Time
    212

    Disp. 6 (ded)
    0.00 0.22 0.44- 0.67 0.89 1.11 1.33 1.5S 1.78 £.00
    Time (sec)
    Figure 6.18f Joint 6—Displacement vs Time
    213

    Vel. I C1/s)
    0.00 0.22 0.44- 0.67 0.89 1.11 1.33 1.55 1.78 2.00
    Figure 6.19a Joint 1—Velocity vs Time
    Time (sec)
    214

    Vel. 2 (1/s)
    0.00 0.EE 0.44 0.67 0.89 1.11 1.33 1.55 1.78 E.00
    Figure 6.19b Joint 2—Velocity vs Time
    Time (sec)
    215

    Vel. 3 (L/s)
    Ti me I see)
    Figure 6.19c Joint 3—Velocity vs Time
    216

    Vel. t (l/s)
    Time (sec)
    Figure 6.19d Joint 4—Velocity vs Time
    217

    Vel. 5 (l/s)
    0.00 0.22 0.44- 0.67 0.83 1.11 1.33 I.B5 1.78 2.00
    Ti me (sec)
    Figure 6.19e Joint 5—Velocity vs Time
    218

    Vel. 6 (t/s)
    0.00 0.22 0.44- 0.67 0.89 1.11 1.33 1.55 1.78 2.
    Time (see)
    00
    Figure 6.19f Joint 6—Velocity vs Time
    219

    Inp.Volt. 1 (v)
    0.00 0.EE 0.^ 0.67 0.89 1.11 1.33 1.55 1.78 £.00
    Time (sec)
    Figure 6.20a Actuator 1—Input Voltage vs Time
    220

    Inp.Volt. 2 (v)
    0.00 0.22 0.^ 0.67 0.89 1.11 1.33 1.55 1.78 2.00
    Time (sec)
    Figure 6.20b Actuator 2—Input Voltage vs Time
    221

    Inp.Volt. 3 (v)
    0.00 0.22 0.Â¥t 0.67 -0.89 1.11 1.33 1.5S 1.78 2.00
    Time (sec)
    Figure 6.20c Actuator 3—.Input Voltage vs Time
    222

    Inp.Volt. 4- (v)
    0.00 0.22 0.^4 0.67 0.09 1.11 1.33 1.55 1.70 2.00
    Ti me (sec)
    Figure 6.20d Actuator 4—Input Voltage vs Time
    223

    Inp.Volt. 5 (v)
    0.00 0.EE 0.44 0.6? 0.89 1.11 1.33 1.55 1.78 E. 00
    Time (sec)
    Figure 6.20e Actuator 5—Input Voltage vs Time
    224

    Inp.Volt. 6 (v)
    0.00 0.28 0.44 0.67 0.09 l.tl 1.33 1.55 1.70 8.00
    Time (sec)
    Figure 6.20f Actuator 6—Input Voltage vs Time
    225

    226
    updating rate, Figure 6.15f. This overshoot is reduced to
    3 deg with 200 Hz updating as shown in Figure 6.18f. Also
    the joint 5 overshoots of 6.42 and 5.58 deg in Figure
    6.15e (with 50 Hz updating) are completely eliminated when
    the frequency is 200 Hz, Figure 6.18e. Magnitudes of the
    input voltages are also influenced by the frequency change.
    Maximum value of the 6th actuator input voltage is 21.09 v
    (50 Hz) as shown in Figure 6.17f, whereas the same value with
    the increased updating frequency (200 Hz) is reduced to
    9.01 v in Figure 6.20f. Faster updating also produced
    smoother input curves as expected. Compare, for example,
    Figure 6.17 to 6.20.
    The last simulation on T3-776—Figures 6.21-
    6.23—modeled the reference model so that each link had an
    extra payload of 10 lbm. Also at t = 0.7 sec, an extra
    payload of 30 lbm is dropped on the reference model hand.
    In this example, the nonlinear terms are updated at 400 Hz.
    Due to the increased difference between the plant and
    reference model parameters, joint 5 displacement, Figure
    6.21e, introduced 10.78 and 4.63 deg overshoots (which were
    eliminated in Figure 6.18e) in spite of the increased
    updating rate. Joint 6 overshoot in Figure 6.21f is also
    increased to 9.07 from 3.0 deg. Further adjustments on
    , S^, and R^, i = 1,2,3, may reduce the system overshoots
    and improve the overall transient behavior.

    Disp. 1 (ded)
    0.00 0.EE 0.^4- 0.67 0.89 1.11 1.33 1.56 1.78 E.00
    Ti me (sec)
    Figure 6.21a Joint 1—Displacement vs Time
    227

    Disp, 2 (ded)
    Time (sec)
    Figure 6.21b Joint 2—Displacement vs Time
    228

    Disp. 3 (ded)
    0.00 0.28 0.44- 0.67 0.83 1.11 1.33 1.56 1.78 2.00
    T i me (sec)
    Figure 6.21c Joint 3—Displacement vs Time
    229

    Disp. 4- (ded)
    0.00 0.2E 0.44 0.67 0.89 1.11 1.33 1.56 1.78 2.00
    Time (sec)
    Figure 6.21d Joint 4—Displacement vs Time
    230

    Disp. 5 (ded)
    Ti me (sec)
    Figure 6.21e Joint 5—Displacement vs Time
    231

    Disp. 6 (deá)
    0.00 0.22 0.^ 0. 67 0.03 1.11 1.33 1.56 1.70 2.00
    Time (sec)
    Figure 6.21f Joint 6—Displacement vs Time
    232

    Vel. I (l/s)
    Figure 6.22a Joint 1—Velocity vs Time
    Time (sec)
    233

    Vel. 2 (l/s)
    Time (sec)
    00
    Figure 6.22b Joint 2—Velocity vs Time
    234

    Vel. 3 (1/s)
    Time (see)
    Figure 6.22c Joint 3—Velocity vs Time
    235

    Vel. T (l/s)
    Time (sec)
    Figure 6.22d Joint 4—Velocity vs Time
    236

    Vel. 5 (l/s)
    0.00 0.22 0.^ 0.67 0.09 1.11 1.33 1.56 1.70 2.00
    Time (sec)
    Figure 6.22e Joint 5—Velocity vs Time
    237

    Vel. & (L/s)
    Ti me (sec)
    Figure 6.22f Joint 6—Velocity vs Time
    238

    Inp.Volt. 1 (v)
    0.00 0.22 0.4-^ 0.67 0.83 1.11 1.33 1.56 1.78 2.00
    Time (sec)
    Figure 6.23a Actuator 1—Input Voltage vs Time
    239

    i
    Inp.Volt. 2 (v)
    Ti me (see)
    Figure 6.23b Actuator 2—Input Voltage vs Time
    240

    Inp.Volt. 3 (v)
    0.00 0.E2 0,‘t‘t 0.67 0.89 1.11 1.33 1.56 1.78 E.00
    Time (sec)
    Figure 6.23c Actuator 3—Input Voltage vs Time
    241

    Inp.Volt. 4 (v)
    Time (sec)
    Figure 6.23d Actuator 4—Input Voltage vs Time
    242

    Inp.Volt. 5 (v)
    Time (sec)
    Figure 6.23e Actuator 5—Input Voltage vs Time
    243

    Inp.Volt. 6 (v)
    0.00 0.22 0.Vt 0.67 0.89 1.11 1.33 1.56 1.78 2.
    Time (sec)
    00
    Figure 6.23f Actuator 6—Input Voltage vs Time
    244

    245
    In this section, comparisons are provided in an
    attempt to give insight to the system response when several
    parameters (amounts of disturbances) are varied. However,
    it should be kept in mind that the overall system is
    18th order, coupled and nonlinear, and unexpected variations
    in the transient behavior are possible and may not be
    interpreted easily. In all simulations system stability
    is preserved under all the simulated disturbances, the
    manipulator tracked the desired trajectories and steady
    state error is eliminated with the disturbance rejection
    feature.

    CHAPTER 7
    CONCLUSION
    Today's industrial manipulators are built to move
    slowly or the joints are activated one by one to
    avoid dynamic interactions between links. Typically each
    link is modeled as a second-order, time-invariant system
    and the joints are controlled independently. This limited
    practice, however, does not take full advantage of the
    robot technology. Precision remains payload and task
    dependent, even instability may be induced, since a highly
    nonlinear and coupled system is represented by a linear,
    decoupled system and a sound stability analysis is not
    provided.
    This work addresses the tracking problem of spatial,
    serial manipulators modeled with rigid links. Centralized
    adaptive controllers which assure the global asymptotic
    stability of the system are given via the second method of
    Lyapunov. Actuator dynamics is also included in the
    system model. System dynamics is represented in hand
    coordinates and it is shown that the designed controllers
    can be extended for this system.
    The kinetic energy expression for an n-link,
    spatial manipulator is obtained and the Lagrange equations
    246

    247
    are utilized in deriving the dynamic equations. These
    equations form a set of 2n, nonlinear, coupled, first-order
    ordinary differential equations for an n-link, n degree-of-
    freedom arm. In general, they are formed and the forward
    or the inverse problems are solved numerically on digital
    computers.
    The plant, which represents the actual manipulator,
    and the reference model representing the ideal robot are
    both expressed as nonlinear, coupled systems. Error-driven
    system dynamics is then given and the controllers which
    yield globally asymptotically stable systems are designed
    using Lyapunov's second method. It is shown that the
    resulting closed-loop systems are also asymptotically
    hyperstable. Integral feedback is added to compensate for
    the steady state system disturbances. System dynamics
    is expressed in hand coordinates and an adaptive control
    law scheme is proposed for this model. Actuator dynamics,
    modeled as third-order, linear, time-invariant systems,
    is coupled with the manipulator dynamics and a nonlinear
    state transformation is introduced to facilitate the
    controller design. This transformation increased the
    computational requirements and necessitated the measurements
    of joint accelerations. Neglecting armature inductances,
    simplified actuator dynamics is obtained. Each actuator
    is then modeled as a second-order, linear, time-invariant

    248
    system. Joint acceleration measurements and the added
    computations are thus avoided. Adaptive controller design
    and the disturbance rejection feature are applied to this
    system.
    Adaptive controllers are implemented on the computer
    for n-link robot manipulators powered with n actuators.
    Examples on 3- and 6-link, spatial, industrial manipulators
    are presented. Disturbances acting on the plant are
    simulated by the discrepancy in manipulator and actuator
    parameters of the plant and reference model, difference in
    initial positions, measurement delays and the delay in
    control law implementation. In all cases system stability
    is preserved, reference trajectory is tracked, and steady-
    state error is eliminated with the disturbance rejection
    feature.
    The amount of discrepancy between the plant and the
    reference model parameters which will deteriorate the system
    response or even induce instability need to be further
    addressed. Structural flexibility should also be included
    in the dynamic modeling of manipulators. This aspect may
    be omitted until a mature understanding of the control
    problem with rigid body model is established, since
    flexibility further complicates the dynamic equations and
    increases the system dimensionality. Although computer
    simulations indicate the validity of controllers and form

    249
    an inexpensive test base, ultimately experimental
    implementation on actual robots must be realized.
    Advanced controllers call for on-line use of
    computers, but considering that the current industrial
    robots already have computers on board and that
    microcomputer prices are steadily coming down with
    increased memory and faster operations, industrial use of
    these controllers ia feasible if reliable, precise and
    fast operation of manipulators is required. These
    desirable features will force manipulator productivity to
    its full capacity. Although the flexibility of manipulators
    to work in different operations (against hard automation)
    is the key issue to make them attractive, current practices
    concentrate on the dedicated use of robots partly because
    of their slow and unreliable features. With the improved
    operation speed and reliability, flexibility of robots can
    truly be realized. Price increase of the complete robot
    unit (due to increased computer support) will be
    compensated by the increased productivity. Finally, if
    reliability is proven, hesitancy in investment, currently
    the major drawback, will be overcome.

    REFERENCES
    [1] Anex, R. P., and Hubbard, M., "Modeling and Adaptive
    Control of a Mechanical Manipulator," ASME J. Dynamic
    Systems, Measurement and Control, Vol. 106, pp. 211-
    217, September 1984.
    [2] Astrom, K. J., "Theory and Applications of Adaptive
    Control—A Survey," Automática, Vol. 19, pp. 471-486,
    1983.
    [3] Balestrino, A., De Marina, G., and Sciavicco, L.,
    "An Adaptive Model Following Control for Robotic
    Manipulators," ASME J. Dynamic Systems, Measurement
    and Control, Vol. 105, pp. 143-151, September 1983.
    [4] Book, W. J., Maizza-Neto, 0., and Whitney, D. E.,
    "Feedback Control of Two Beam, Two Joint Systems with
    Distributive Flexibility," ASME J. Dynamic Systems,
    Measurement and Control, Vol. 97, pp. 424-431,
    December 1975.
    [5] Brady, M., Hollerbach, J. M., Johnson, T. L.,
    Lozano-Perez, T., and Mason, M. T., Robot Motion:
    Planning and Control, MIT Press, Cambridge,
    Massachusetts, 1982.
    [6] Chen, C. T., Introduction to Linear System Theory,
    Holt, Rinehart, and Winston, Inc., New York, 1970.
    [7] Cvetkovic, V., and Vukobratovic, M., "Contribution to
    Controlling Non-Redundant Manipulators," Mechanism
    and Machine Theory, Vol. 16, pp. 81-91, 1981.
    [8] Dubowsky, S., and DesForges, D. T., "The Application
    of Model-Referenced Adaptive Control to Robotic
    Manipulators," ASME J. Dynamic Systems, Measurement
    and Control, Vol. 101, pp. 193-200, September 1979.
    [9] Duffy, J., Analysis of Mechanisms and Robot
    Manipulators, John Wiley and Sons, Inc., New York,
    1980.
    [10]Featherstone, R., "Robot Dynamics Algorithms," Ph.D.
    Dissertation, University of Edinburgh, United Kingdom,
    1984 .
    250

    251
    [11] Freund, E., "Fast Nonlinear Control with Arbitrary
    Pole-Placement for Industrial Robots and Manipulators,"
    Int. J. Robotics Research, Vol. 1, pp. 65-78, 1982.
    [12] Golla, D. F., Garg, S. C., and Hughes, P. C., "Linear
    State-Feedback Control of Manipulators," Mechanism
    and Machine Theory, Vol. 16, pp. 93-103, 1981.
    [13] Guez, A., "Optimal Control of Robotic Manipulators,"
    Ph.D. Dissertation, Dept, of Electrical Engineering,
    University of Florida, Gainesville, Florida, January 1983.
    [14] Gusev, S. V., Timofeev, A. V., Yakubovich, V. A., and
    Yurevich, E. I., "Algorithms of Adaptive Control of
    Robot Movement," Mechanism and Machine Theory, Vol.
    18, pp. 279-281, 1983.
    [15] Hang, C. C., and Parks, P. C., "Comparative Studies of
    Model Reference Adaptive Control Systems," IEEE T,
    Automatic Control, Vol. 18, pp. 419-428, October
    1973.
    [16] Hemami, H., and Camana, P. C., "Nonlinear Feedback in
    Simple Locomotion Systems," IEEE T. Automatic Control,
    Vol. 21, pp. 855-859, December 1976.
    [17] Hewit, J. R., and Burdess, J. S., "Fast Dynamic
    Decoupled Control for Robotics Using Active Force
    Control," Mechanism and Machine Theory, Vol. 16,
    pp. 535-542, 1981.
    [18] Hewit, J. R., Hanafi, A., and Wright, F. W., "Optimal
    Trajectory Control of Robotic Manipulators,"
    Mechanism and Machine Theory, Vol. 19, pp. 267-273,
    1984.
    [19] Hollerbach, J. M., "A Recursive Formulation of
    Lagrangian Manipulator Dynamics," IEEE T. Systems,
    Man and Cybernetics, Vol. 10, pp. 730-736, November
    1980.
    [20] Hooker, W. W., and Margulies, G., "The Dynamical
    Attitude Equations for an N-Body Satellite," J.
    Astronautical Sciences, Vol. 12, pp. 123-128, 1965.
    [21] Horowitz, R., and Tomizuka, M., "An Adaptive Control
    Scheme for Mechanical Manipulators—Compensation of
    Nonlinearity and Decoupling Control," ASME Paper No.
    8 O-WA/DSC-6, 198 0 .

    252
    [22] Kahn, M. E., and Roth, B., "The Near-Minimum Time
    Control of Open-Loop Articulated Kinematic Chains,"
    Stanford Artificial Intelligence Memo. No. 106,
    December 1969.
    [23] Kalman, R. E., and Bertram, J. E., "Control System
    Analysis and Design via the Second Method of Lyapunov,"
    ASME J. Basic Engineering, Series D, Vol. 82(2),
    pp. 371-393, June 1960.
    [24] Kircanski, M., and Vukobratovic, M., "A Method for
    Optimal Synthesis of Manipulation Robot
    Trajectories," ASME J. Dynamic Systems, Measurement
    and Control, Vol. 104, pp. 188-193, June 1982.
    [25] Koivo, A. J., and Guo, T. H., "Adaptive Linear
    Controller for Robotic Manipulators," IEEE T.
    Automatic Control, Vol. 28, pp. 162-171, February 1983 .
    [26] Landau, I. D., "A Hyperstability Criterion for Model
    Reference Adaptive Control Systems," IEEE T.
    Automatic Control, Vol. 14, pp. 552-555, October 1969.
    [27] Landau, I. D., "A Generalization of the Hyperstability
    Conditions for Model Reference Adaptive Systems,"
    IEEE T. Automatic Control, Vol. 17, pp. 246-247,
    April 1972.
    [28] Landau, I. D., and Courtil, B., "Adaptive Model
    Following Systems for Flight Control and Simulation,"
    J. Aircraft, Vol. 9, pp. 668-674, September 1972.
    [29] Landau, Y. D., Adaptive Control: The Model
    Reference Approach, Marcel Dekker, Inc., New York,
    1979.
    [30] Lee, C. S. G., and Lee, B. H., "Resolved Motion
    Adaptive Control for Mechanical Manipulators," ASME J,
    Dynamic Systems, Measurement and Control, Vol. 106,
    pp. 134-142, June 1984.
    [31] Liegeois, A., "Automatic Supervisory Control of the
    Configuration and Behavior of Multibody Mechanisms,"
    IEEE T. Systems, Man and Cybernetics, Vol. 7, pp.
    868-871, December 1977.
    [32] Lin, C. S., Chang, R. P., and Luh, J., "Formulation
    and Optimization of Cubic Polynomial Joint Trajectories
    for Mechanical Manipulators," Proc. IEEE Conference
    on Decision and Control, Orlando, Florida, pp. 330-335,
    1982.

    253
    [33] Luh, J. Y. S., and Lin, C. S., "Optimum Path Planning
    for Mechanical Manipulators," ASME J. Dynamic Systems,
    Measurement and Control, Vol. 103, pp. 142-151, June
    1981.
    [34] Luh, J. Y. S., Walker, M. W., and Paul, R. P. C.,
    "On-Line Computational Scheme for Manipulators,"
    ASME J. Dynamic Systems, Measurement and Control,
    Vol. 102, pp. 69-76, June 1980.
    [35] Luy, J. Y. S., Walker, M. W., and Paul, R. P. C.,
    "Resolved Acceleration Control of Mechanical
    Manipulators," IEEE T. Automatic Control, Vol. 25,
    pp. 468-474, June 1980.
    [36] Noble, B., and Daniel, J. W., Applied Linear Algebra,
    Second Edition, Prentice-Hall International, Inc.,
    Englewood Cliffs, New Jersey, 1977.
    [37] Orin, D. E., McGhee, R. B., Vukobratovic, M., and
    Hartoch, G., "Kinematic and Kinetic Analysis of
    Open-Chain Linkages Utilizing Newton-Euler Methods,"
    Mathematical Biosciences, Vol. 43, pp. 107-130,
    February 1979.
    [38] Paul., R., "Modeling, Trajectory Calculation and
    Servoing of a Computer-Controlled Arm," Ph.D.
    Dissertation, Stanford University, California, 1972.
    [39] Paul, R. P., Luy, J. Y. S., Bender, J., Berg, E.,
    Brown, R., Remington, M., Walker, M., Lin, C. S., and
    Wu, C. H., "Advanced Industrial Robot Control Systems,"
    School of Electrical Engineering, Purdue University,
    Indiana, TR-EE-78-25, May 1978.
    [40] Paul, R. P., Shimano, B., and Mayer, G. E.,
    "Kinematic Control Equations for Simple Manipulators,"
    IEEE T. Systems, Man and Cybernetics, Vol. 11,
    pp. 449-455, June 1981.
    [41] Popov, E. P., Vereshchagin, A. F., and Minaev, L. N.,
    "Semiautomatic Manipulating Robots Control on the
    Basis of Specialized Calculators," Mechanism and
    Machine Theory, Vol. 16, pp. 49-55, 1981.
    Popov, V. M., "The Solution of a New Stability
    Problem for Controlled Systems," Automation and
    Remote Control, Vol. 24, pp. 1-23, January 1963.
    [42]

    254
    [43] Potkonjak, V. , and Vukobratovic, M., "Contribution of
    the Forming of Computer Methods for Automatic
    Modeling of Spatial Mechanisms Motions," Mechanism and
    Machine Theory, Vol. 14, pp. 179-188, 1979.
    [44] Reischer, M. H., "Symbolic Algebraic Computation of
    Kinematic and Dynamic Mechanism Parameters," Master's
    Thesis, Dept, of Mechanical Engineering, University of
    Florida, Gainesville, Florida, 1985.
    [45] Saridis, G. N., "Intelligent Robotic Control," IEEE
    T. Automatic Control, Vol. 28, pp. 547-556, May 1983.
    [46] Stepanenko, Y., and Vukobratovic, M., "Dynamics of
    Articulated Open-Loop Active Mechanisms," J.
    Mathematical Biosciences, Vol. 28, pp. 137-170, 1976.
    [47] Stokic, D., and Vukobratovic, M., "One Engineering
    Concept of Dynamic Control of Manipulators," ASME J,
    Dynamic Systems, Measurement and Control, Vol. 103,
    pp. 108-118, June 1981.
    [48] Stoten, D. P., "The Adaptive Control of Manipulator
    Arms," Mechanism and Machine Theory, Vol. 18, pp.
    283-288, 1983.
    [49] Sugimoto, K., and Duffy, J., "An Extension of Screw
    Theory with Application to Spatial Mechanisms and
    Robotic Manipulators," Final Report, NSF Grant No.
    ENG 67-20112, May 1981.
    [50] Takegaki, M., and Arimoto, S., "An Adaptive Trajectory
    Control of Manipulators," Int. J. Control, Vol. 34,
    pp. 219-230., 1981.
    [51] Tesar, D., "CIMAR Listing of Proposed Research in
    Intelligent Machines and Robotics," Center
    for Intelligent Machines and Robotics, University of
    Florida, Gainesville, Florida, 1983.
    Tesar, D., Dalton, G. R. , Tosunoglu, L. S., and
    Bryfogle, M., "Assessment for the Design and
    Implementation of Robotics to the Secure Automated
    Fuel Fabrication Plant," Center for Intelligent
    Machines and Robotics, University of Florida,
    Gainesville, Florida, 1983.
    [52]

    255
    [53] Thomas, M., and Tesar, D., "Dynamic Modeling of
    Serial Manipulator Arms," ASME J. Dynamic Systems,
    Measurement and Control, Vol. 104, pp. 218-228,
    September 1982.
    [54] Tosunoglu, L. S., "Power-Time Optimal Control of
    Manipulators," Master's Thesis, Dept, of Mechanical
    Engineering, Middle East Technical University, Ankara,
    Turkey, December 1981.
    [55] Uicker, J. J., "On the Dynamic Analysis of Spatial
    Linkages Using 4 by 4 Matrices," Ph.D. Dissertation,
    Dept, of Mechanical Engineering and Astronautical
    Sciences, Northwestern University, Massachusetts,
    1965.
    [56] Vukobratovic, M. , "Dynamics of Active Articulated
    Mechanisms and Synthesis of Artificial Motion,"
    Mechanism and Machine Theory, Vol. 13, pp. 1-18 , 1978 .
    [57] Vukobratovic, M. , and Potkonjak, V., Dynamics of
    Manipulation Robots, Springer-Verlag, Berlin,
    Heidelberg, 1982.
    [58] Vukobratovic, M. , and Stokic, D., Control of
    Manipulation Robots, Springer-Verlag, Berlin,
    Heidelberg, 1982.
    [59] Waldron, K. J., "Geometrically Based Manipulator
    Rate Control Algorithms," Mechanism and Machine Theory,
    Vol. 17, pp. 379-385, 1982.
    [60] Walker, M. W., and Orin, D. E., "Efficient Dynamic
    Computer Simulation of Robotic Mechanisms," ASME J.
    Dynamic Systems, Measurement and Control, Vol. 105,
    pp. 205-211, September 1982.
    [61] Wander, J., "Real-Time Computation of Influence
    Coefficient Based Dynamic Modeling Matrices for
    Improved Manipulator Control," Master's Thesis, Dept,
    of Mechanical Engineering, University of Florida,
    Gainesville, Florida, 1985.
    Whitehead, M. L., "Control of Serial Manipulators with
    Emphasis on Disturbance Rejection," Master's Thesis,
    Dept, of Electrical Engineering, University of
    Florida, Gainesville, Florida, 1984.
    [62]

    256
    [63] Whitney, D. E., "Resolved Motion Rate Control of
    Manipulators and Human Prostheses," IEEE T.
    Man-Machine Systems, Vol. 10, pp. 47-53, June 1969.
    [64] Whitney, D. E., "The Mathematics of Coordinated
    Control of Prosthetic Arms and Manipulators," ASME J.
    Dynamic Systems, Measurement and Control, Vol. 94,
    pp. 303-309, December 1972.
    [65] Whitney, D. E., "Force Feedback Control of Manipulator
    Fine Motions," ASME J. Dynamic Systems, Measurement
    and Control, Vol. 99, pp. 91-97, June 1977.
    [66] Whyte, H. D., "Practical Adaptive Control of Actuated
    Spatial Mechanisms," Proc. IEEE Conference on
    Robotics and Automation, St. Louis, Missouri, pp.
    650-655, 1985.
    [67] Yuan, J. S. C., "Dynamic Decoupling of a Remote
    Manipulator System," IEEE T. Automatic Control, Vol.
    23, pp. 713-717, August 1978.
    [68] Young, K. D., "Controller Design for a Manipulator
    using Theory of Variable Structure Systems," IEEE T.
    Systems, Man and Cybernetics, Vol. 8, pp. 101-109,
    February 1978.

    BIOGRAPHICAL SKETCH
    L. Sabri Tosunoglu was born in Izmir, Turkey.
    He received B.S. and M.S. degrees in mechanical engineering
    from the Middle East Technical University, Ankara, Turkey.
    He was awarded a Fulbright Fellowship and joined the
    University of Florida's CIMAR to pursue his Ph.D.
    257

    I certify that I have read this study and that in my
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    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
    Mechanical Engineering
    I certify that I have read this study and that in my
    opinion it conforms to acceptable standards of scholarly
    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
    JosephJDu^fy, C
    Professor qf^Meci
    an
    cal Engineering
    I certify that I have read this study and that in my
    opinion it conforms to acceptable standards of scholarly
    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
    . 74"
    Roger/A. Gater
    Associate Professor of
    Mechanical Engineering
    I certify that I have read this study and that in my
    opinion it conforms to acceptable standards of scholarly
    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
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    Associate Professor of
    Mechanical Engineering

    I certify that I have read this study and that in my
    opinion it conforms to acceptable standards of scholarly
    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
    Mechanical Engineering
    I certify that I have read this study and that in my
    opinion it conforms to acceptable standards of scholarly
    presentation and is fully adequate, in scope and quality, as
    a dissertation for the degree of Doctor of Philosophy.
    Ralph G(<3 SelfMdge
    Professor of /dompiiter and
    Information/ Sciences
    This dissertation was submitted to the Graduate Faculty of
    the College of Engineering and to the Graduate School and
    was accepted as partial fulfillment of the requirements for
    the degree of Doctor of Philosophy.
    May 1986
    Dean, Graduate School

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