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
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. 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
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. 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 3Link,
Spatial Manipulator .
6.2 Numerical Solution of the
Lyapunov Equation .
6.3 Simulations on the 6Link,
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 nlink, 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.
Errordriven 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 steadystate system disturbances. Tracking is achieved
since the errordriven 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 thirdorder,
linear, timeinvariant 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 6link 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 socalled 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
closedloop 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 openloop mechanism is formed. If
each closedloop 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 openloop 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 Systemthough not completely successful yetis
another challenging application area of robotics.
In the analysis of manipulators,basically two
problems are encountered. The first is called the
positioning or pointtopoint pathfollowing 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 degreeoffreedom 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]. Closedform solution
to this problem is not available for a general manipulator.
Duffy instantaneously represents a 6link, serial
manipulator by a 7link, closedloop spatial mechanism with
the addition of a hypothetical link and systematically
solves all possible joint displacements [9]. Paul et al.
obtain closedform 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
pickandplace activities such as spot welding, machine
loading and unloading operations,and can be treated as a
reachingatarget 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 closedloop spatial chains using Lagrange
equations [55]. In the same year, Hooker and Margulies
applied the NewtonEuler formulation to multibody
satellite dynamics [20]. Later, in 1969, Kahn and Roth
were the first to obtain equations of motion specifically
for openloop chains using the Lagrangian approach [22].
Stepanenko and Vukobratovic applied the NewtonEuler method
to robotic mechanisms in 1976 [46].
Even the derivation of closedform dynamic
equations for two 6link 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 NewtonEuler 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 NewtonEuler 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
socalled 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 socalled 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, timeinvariant,
thirdorder models in [7, 13, 58].
1.3.2 Optimal Control of Manipulators
Synthesis of optimal trajectories for a given task
(reachingatarget problem) has been studied by several
researchers. Kahn and Roth [22] presented a suboptimal
numerical solution to the minimumtime problem for a 3link
manipulator. The dynamic model was linearized by neglecting
the second and higherorder terms in the equations of motion,
but the effects of gravity and the velocityrelated terms
were represented by some average values.
The maximum principal has also been employed to
solve the optimal control problem [54, 58]. Powertime
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 6link, 6 degreeoffreedom manipulator) nonlinear,
coupled, firstorder differential equations are obtained for
an nlinkhere also n degreeoffreedommanipulator,
without considering the actuator dynamics. If initial and
terminal conditions are specified for the manipulator, then
a twopoint 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 errordriven equations are written about
the nominal state. The openloop optimal control problem
is then solved using a direct search algorithm. Later,
optimal control is approximated by constantgain, 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 2link
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 endeffector. 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 closedloop 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, closedloop pole assignability for the
centralized and decentralized (independent joint control)
linear feedback control was discussed.
In [47, 58] spatial, nlink manipulators with rigid
links are considered. In general, 6link 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 trialanderror techniques.
Kahn and Roth linearized the dynamic equations of a
2link manipulator and designed a timesuboptimal controller
in [22]. Since the linearized model was only valid
locally, he concluded that average values of the nonlinear
velocityrelated 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 3link, planar manipulator. Yuan [67]
neglected the velocity relatedterms 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 worstcase 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 2link 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,
timeinvariant, and linear set of closedloop dynamic
equations can be obtained [11, 13, 17, 35, 67].
All nonlinear terms in the control expression are
to be calculated offline [11]. Hence, a perfect
manipulator which is "exactly" represented by dynamic
equations and infinite computer precision are assumed
[5]. Online computation of nonlinear terms is proposed
in [17], but the scheme requires (online) inversion of
an n x n nonlinear matrix other than the calculation of all
nonlinear effects. Generation of a lookup table is
suggested in [13], but dimensionality of the problem makes
this approach impractical. This scheme is applied only
to 1 and 2link 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.
Forcefedback control of manipulators is studied in [65].
Proposed diagonal forcefeedback 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 2link 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,
2link 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 secondorder, singleinput, singleoutput system,
neglecting the coupling between system degrees of freedom.
Then, for each degreeoffreedom, position, and velocity
feedback gains are calculated by an algorithm which
minimizes a positive semidefinite 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 taskoriented coordinates [50]. Actuator dynamics
is not included. In this work, an approximate openloop
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 timevarying
(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 3link 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, timeinvariant, 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 1link manipulator.
Lee [30] expressed the dynamics in the
taskoriented coordinates, linearized and then discretized
the equations without including the motor dynamics. All
parameters of the discretized system (216 for 6link
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 timeinvariant, 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
closedloop systems. Implementation of controllers in hand
coordinates and inclusion of actuator dynamics are also
addressed.
The mathematical model of nlink, spatial, serial
manipulators with adjacent links connected by single
degreeoffreedom 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 closedloop
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 thirdorder,
timeinvariant, 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 secondorder,
timeinvariant, 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 3link, spatial and 6link, spatial industrial
(Cincinnati Milacron T3776) 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 nlink, spatial, serial manipulators
are considered. Adjacent links are assumed to be connected
by one degreeoffreedom 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 degreeoffreedom
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 rigidlink 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
(j1), 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 degreeoffreedom 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 ndimensional generalized
joint variable vector e for an n degreeoffreedom robot
manipulator. Consider an n degreeoffreedom arm with its
links connected by revoluteprismaticrevolute...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 linkthe 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,
bodyfixed 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 groundfixed F0 frame. Now, it is important to note
that the unit vectors r. and s. expressed in their bodyfixed
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.
 j1 j1 3
sina. sine.
(2.3)
(2.4)
and
(2.5)
0
s. = T. sinac. j = 2,3,...,n (2.6)
3 j1 j1
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 threedimensional 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.,
threedimensional 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 nlink 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 + [rkl rkl + 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 (rk1 rkl
+ 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+ k1 k1
+ 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 skif
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 vectormatrix 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 threedimensional null column vector.
For an nlink 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 positiondependent terms
[Gci]. as translational firstorder 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 + + i1/i2 +i/i1
(2.32)
i/0 = i s + s2 + ... + ii si + $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 vectormatrix 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 firstorder influence coefficients.
2.3.4 Total Kinetic Energy
Total kinetic energy expression for an nlink
manipulator follows from Equations (2.22) and (2.23)
KE (0)T (0) + 1 (i)T j. (i)1
2 i c c. 2i/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 (il)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 ga [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 secondorder 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
online 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 Selftuning Regulators (Parameter Estimation Techniques).
Block diagram representations of these schemes are given in
Figures 3.23.4.
Gain
Scheduling
I i
Figure 3.2 Block Diagram of Gain Scheduling System
i Model I
i AAdjustment .. 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 Selftuning Regulator
Figure 3.4
All these block diagrams in Figures 3.23.4 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
appl' ppl 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 ndegreeoffreedom
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 equationsEquation
(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 2r2
x = _r(t) e Rn, xr = _r (t) e Rn (3.14)
r r r r
dx (t) TT
S dtr 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 = rrl = 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 = Frrl 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,
timeinvariant 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) dt (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) ise
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(hixI) > 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 xl+ 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
(iva) V(x) < 0 for all x 3 0, x e Rn
(ivb) 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, timeinvariant, free dynamic
systems.
Theorem 3.2: The equilibrium state x of a
e
linear, timeinvariant, 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 openloop control law.
r
This, for example, may be an optimal control law obtained
offline 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, openloop control law ur = ur(t)
does not prove effective as the demand on precise and fast
motion increases. Even today's servocontrolled industrial
manipulators which totally neglect the dynamic coupling
use closedloop 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 rrl r rr2 11 22
(3.32)
where
u" is part of the controller yet to be designed
p
K K2 6 Rnxn are constant matrices to
be selected
errordriven 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 (3.273.30) as follows:
S1 
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 online 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 online.
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 errordriven
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 errordriven 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
openloop 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 1p2
T
x D'x
p2 ip2
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 1pl
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 Ie 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' A1 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
online 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 lefthand 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 timeinvariant part of the errordriven 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 closedloop 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 closedloop
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 timeinvariant,
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 closedloop 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 errordriven system equations, Equation
(3.33), closedloop 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 closedloop 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 closedloop
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 ... A2nB] = ... 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)2n1 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
ncolumns 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 rrl r rr2
 K2e2 K e a3)
2a2 3a3
 K e
1al
(3.115)
and
u" = u"
ap p
(3.110)
(3.111)
(3.116)
