
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00003395/00001
Material Information
 Title:
 Adaptive control of robotic manipulators
 Creator:
 Tosunoglu, L. Sabri
 Publication Date:
 1986
 Language:
 English
 Physical Description:
 viii, 257 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Adaptive control ( jstor )
Coordinate systems ( jstor ) Inertia ( jstor ) Kinematics ( jstor ) Kinetic energy ( jstor ) Matrices ( jstor ) Parametric models ( jstor ) Robotics ( jstor ) Simulations ( jstor ) Velocity ( jstor ) Manipulators (Mechanism) ( lcsh ) Robotics ( lcsh ) City of Gainesville ( local )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1986.
 Bibliography:
 Includes bibliographical references (leaves 250256).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by L. Sabri Tosunoglu.
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 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AEJ6175 ( NOTIS ) 15167090 ( OCLC )

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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 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)

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 Ãœ
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 3Link,
Spatial Manipulator 139
6.2 Numerical Solution of the
Lyapunov Equation 183
6.3 Simulations on the 6Link,
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 nlink, 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.
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.
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 openloop 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 Systemâ€”though not completely successful yetâ€”is
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.
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 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
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
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
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 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. [37], Paul et al.
7
[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 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
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 flexiÂ¬
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, 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,
10
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 [58]. 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 nlinkâ€”here also n degreeoffreedomâ€”manipulator,
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
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 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.
12
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
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 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.
14
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.
15
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
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 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,
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 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
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 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
19
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
20
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 be represented by combinations of revolute
and prismatic joints. Hence, an m degreeoffreedom
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 rigidlink 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.
(jl), 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 degreeoffreedom 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 ndimensional generalized
joint variable vector Â£ 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 0 will then be given by
24
8 =
1s2<Ã>3 â€¢â€¢â€¢ 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 linkâ€”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 Â± 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,
bodyfixed reference frame F. is also attached to each link
i(i)  (i)  (j))
Ð·. Frame F. is defined by its basis vectors ju^ ,u2 ,u3 r
^ (j) . , . . , ~ ( j ) . , ~
Ð¸, is chosen coincident with r. and u0J with s.;
1 j 3 j
j â€” 1,2,..â€¢,n â€¢
If a vector Ã¡ 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 groundfixed Fq frame. Now, it is important to note
that the unit vectors r^ and s^ expressed in their bodyfixed
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 Ã¡ be a given vector. Again, a^ and a will
represent expressions of Ã¡ 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
â€”1 3 3 3 3 3 *â€¢
given by s^ xr^ and using Equation (2.1), it can be shown
that transformation is given by
T. =
D
r .
â€œ3
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 31
COS0 .
3
cosaâ€¢ i
â€¢ sin0.
31
3
sina. .
â€¢ sin0 .
31
3
(2.5)
and
28
s . = T. ,
3 Dl
sina
j1
cosa . ,
Dl
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
â€¢ 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 â€˜vc + 2vc â€¢ V
m c
+ (fip/0 >Â¡1 â€¢ 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 â€¢ v + ~
2 c c 2
A . . A
m
(Vo xp) â€™ uP/o xp) ^
or
(2.11)
1 A A 1
KE=Tmv *v + T
2 c c 2
m
p â€¢ ft /A â€¢ ft /A â€¢ p dm
^p/0 ^p/0
(2.12)
where ft is a dyadic formed by the components of w such
that
(q)
ft = T Â£ â€¢ , â€¢ 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 â€¢ v
2 c c
if
P * Si /n â€¢ fi /n â€¢ p dm
Jm %
p/0 ^p/0
(2.14)
On the other hand, it can be shown that
/N /\.
ÃÃ /A * Ã2 ,A  ~ (w /A â€¢ 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 â€¢ r
% %
/V A
= r. Then
_ j_ /N /N 1 ^
KE = â– =â€¢ m v â€¢ v + co /n
2 c c 2 p/0
/V /V
(p â€¢ p I  pp) dm
m ^
co
p/0
(2.16)
/s /\ 1 ^ /s
KE = t m v â€¢ v + co /A â€¢ J â€¢ 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 â€¢ 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
(Â£TÂ£  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 â€”c â€”c 2 â€”p/0 â€”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 threedimensional column vector
â€”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^,
threedimensional 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 nlink 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 â€”>F^. If a is
a vector undergoing these rotations, then
34
a(0> = T.a(1)
â€” 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 [rkl *kl + 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/Â°i
(i)
VÂ°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 (rki 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 zÂ«
c. /0. c. 0 . , .
Â± j i j k=3+l
l , (rki Â£ki
+ 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 vectormatrix form as
v
= v(0)
:. â€”c .
i i
G co
c . â€”
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 "
â€”j * â€”c^/0jy j < i and joint revolute
, j
 s .
3
, otherwise
(2.31)
37
where () denotes a threedimensional null column vector.
3 xn
For an nlink RRPRP... arm, G eR for example, will take
c4
the form
% = Â«Â£c4/0l =2*5c4/02 â€”3 14xzc4/04 Â»â€¢â€¢â€¢Â»]
Thomas and Tesar defined these positiondependent terms
[Gc>]. as translational firstorder 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 = â€œI/O + w2/l + â€¢** + Wil/i2 + â€œi/i1
(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 â€”
(2.34)
Rewriting Equation (2.32) in vectormatrix form
(0) v ;
to . /n = (A) . = > Ã“ . S.
i/o i/o Â¿
(2.35)
or
= G. a)
â€”l/O i â€”
(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)
â€”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
â€”l/O i â€”
(2.40)
where the j*"*1 column of G.^ e R^xn is now defined by
[G.(l) ] . =
i 3
tT Â£j, j < i and i*"*1 joint revolute
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 = l Â¿m. v(0)T v<0)
. L. 2 i â€”c . â€”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 â€” 4
n
y [ m. G1 G +G.(l)TJ. G}1J (â– w
.L. i c. c. 1 11 â€”
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 (jÃº 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 / â€¢ â€¢ â€¢ r 9n/w^,0)2 , â€¢ . . ,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 . â€¢ 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)
â€”i c. â€” â€”i i â€”
i
where Gc^ and GÂ¡_ 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  Ãœ
1G=i]k + â€”i
â€œVk
(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
Ã“W,
n
= I
j=k
m
Ã³ 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 (il)^ 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 Â£ [GcA â€œ Yk wk + T1
j=k
60k (2.55)
Thus, related generalized force will be
n T
}k = J, mj 3a [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 â€œk + 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. â€œi
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 â€œk + TJ
(2.66)
Definina
3A
D* = â€”PkÃ.
ijk 30.
3 A
1 P
13
2 30,
(2.67)
where D* = [D?.^ Â£ Rnxnxn, equations of motion are given
by
Apk. â€œi + Dijk â€œi â€œj * 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. JÂ£ <â€œVk * "Viâ€™
(2.70)
47
where
[H ] ,
[Gâ€ž ]
CÂ£ iij 80i CÂ£ j
(2.71)
[H ] . .
CÂ£
i x ("j x V/0
X/ J
s . X (s. X z
J 1 câ€ž/0.
'Â£ i
S â€¢ X s .
1 1
s . x s .
1 â€”J
) , i < j < Â£ ;
i,j revolute
) , j < i < Â£ ;
i,j revolute
, j < i < Â£ ;
i prismatic,
j revolute
, i < j < Â£ ;
i revolute,
j prismatic
, otherwise
(2.72)
'Vio = sir â€˜V
(2.73)
'Vio 
s.xsj, i< Â£; i, j revolute
, otherwise
(2.74)
48
[G ], is given by Equation (2.31) and [G0], by Equation
cÂ£ K X, K
(2.37). and Hc^ are called secondorder 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.. â€œi =  2T Dk Ãœ â€˜ â€œk + gk + Tk
^ki
k = 1,2,...,n (2.76)
or
T
Ã¼) Dj. â€”
[y] a) + g + t
(2.77)
D co
n â€”
where
AP = yÂ®1 Dk = Dk(Â®>
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
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.
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 Selftuning Regulators (Parameter Estimation Techniques).
Block diagram representations of these schemes are given in
Figures 3.23.4.
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 Selftuning Regulator
54
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
T T T
Defining the state vector x = (0 , w ) where
P P â€œP
0 e Rn and a> e Rn are the generalized relative joint
P â€”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 Â£ R
(3.2)
Â£pi " Vt!'
%>2 â€œ 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 PPl 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)
â€”p â€”pi â€”p2 p â€”p2 p â€”pi â€”p2 â€”p2
f = f (x . , x 0)
P P Pi p2
â€”p2 Dl(pl} â€”p2
x% D (x . ) x 0
â€”p2 n â€”pl â€”p 2
E R
n
(3.9)
F = F (x . , x 0)
P P pl p2
P T
x D. (x . )
â€”p2 1 â€”pl
x  D (x . )
â€”p2 n pl
Â£ 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 ndegreeoffreedom
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 equationsâ€”Equation
(3.1)â€”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 =
â€”r
1
l
x +
â€”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
â€”r
system
... / T T > T â€ž2n
x = x (t) = (x , , x â€ž) e R
â€”r â€”r â€”rl â€”r2
x , = 0 (t) Â£ Rn, x ~ = to (t) Â£ Rn
â€”rl â€”r ' â€”r2 â€” r
,T T ,T
â€”r = dt = â€”rl' â€”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.
2r â€”rl râ€”rl r â€” rl â€”rl
G = G (x ,) e Rnxn, g (x .) Â£ Rn
r r â€”rl ^r â€”rl
f (x . , x~) = F x ~ = F (x . , x â€ž) x
â€”r â€”rl â€”r2' râ€”r2 r â€”rl â€”r2 â€”r2
f = f (x . , x 0) = 
â€”r â€”r â€”rl â€”r2
T
x _
â€”r 2
D1(irl>
â€¢
x 0
â€”r2
T
x _
â€”r 2
â€¢
D (x . )
n â€”rl
X t
â€”r2
e R
(3.13)
(3.14)
(3.15)
â€¢ 6 â€¢ f
(3.16)
(3.17)
(3.18)
(3.19)
58
F = F (x , , x ~) = 
r r â€”r 1 â€”r 2
x 0 D, (x , )
â€”r2 1 â€”rl
X% D (x , )
â€”r2 n â€”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 â€”rl r r â€”rl â€”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,
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
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 Â¿(t; Xg, tg)  II < Â£ for all t < tQ
The norm  â€¢  represents the Euclidean norm.
(Â£_( 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 cÂ£(t; x Q, tg)
starting sufficiently close to x
converges to x^ as t â€”â–º Â°Â°. In
other words, there exists a real
number y(tg) > 0 such that
I xQ  xe 1 < y (tQ) implies
lim I Â£(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
â€žn
r e R
(i) x is stable
â€”e
(ii) lim 1 $(t; xQ, tQ)  x^ II = 0
t â€”> 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) â€”â–º 00 as  x l â€”â–º Â°Â° for all t
, . , â€¢ av , , .
(Ãv) V = (x, t)
3V
at
+ (grad V) Â£(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) â€”as I x  â€”â–º00
(iv) V = (x) < 0 for all x ^ Â£,
â€žn
x e R
Corollary 3.2: In Corollary 3.1, condition (iv)
may be replaced by
(iva) V(x) < 0 for all x ^ 0^, x e Rn
(ivb) 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, timeinvariant, free dynamic
systems.
Theorem 3.2; The equilibrium state x^ of a
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,
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 openloop control law.
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 u^ = 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
â€”p â€”p â€”p â€”r â€”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
â€” â€” â€”r â€”p
2n
T t.T , T
Â£ = (ej_, e2) = (xrl
T T T.T
x . , x â€ž  x 0)
â€”pi â€”r2 p2
e e Rn, eâ€ž 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 â€”r2
K22
)
(3.32)
where
K
V
is part of the controller yet to be designed
I<2 e 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:
Ã© = Ae + Bz  BA1 u" (3.33)
 P P
66
where
"0
I
D 
, B =
_K1
K2_
I
, â€ž2nx2n â€ž 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 â€”p2 r r
z e Rn, u" Â£ Rn
p
It should be noted that the part of the controller
Up requires only the online 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 online.
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
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 Â¡^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.
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,
â€”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}
â€”p p i â€” â€”
(3.43)
where diagtsgn (v^)] is an nxn diagonal matrix with
diagonal elements sgn (v^), i = l,2,...,n,
r â€¢ â€¢ â€¢ /
69
b = sup { A g + A u }
 r\ <" r r1
0 < X . < 2 7T
Pf 1
u . e U
r, i
x 1; * â€¢ * f 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 . e U,
â€”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 Â£ 13 pi 13 u
(3.46)
where (a..)â€ž 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 , â€žâ– < 2tt
pi, Â£
Â£=1,...,n
â€¢ ,n
(3.48)
k
K
k:
T
k e Rn (3.49)
where constant positive definite K* e Rnxn tQ Â£e chosen
so that
D.
i
and
xT *
â€” ~ K. x t
p2 i â€”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
â€”p2 i â€”p2 â€”p2 i â€”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)xl2
mm â€” ~ max â€”
(3.53)
n
for any x e Rn, where  x  = Â£ x. .
i=l 1
Using Theorem 3.3,
2 ^ _T ^ â€ž ii2
X â€¢ (K*)  x 0 r < x% K* x  < X (K?) I x 0
min i " p2 11 â€”p2 i â€”p2 max i " â€”p2
(3.54)
X . (D!) I x  2 < xT_ D! x ~ < X (D!) I! x 0 2
mm i 11 p2 " â€”p2 i p2 max i 11p2 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)Â£ and (xp2 D! xp2)u, and xÂ£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 â€” p2 i â€”p2
Xp2 Â¥â– 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
â€”p p P â€”P u u â€”r
(3.64)
75
where
Kp = lGp : Fp] (3.65)
AKp = [RÂ± 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)â€”(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
â€¢t
â– 0
T 1 T T
(v A R,v) (x_â€œ, SZ'x , ) dr
â€” p 1â€” â€”pi 1â€”pi
+ 2
(*p2S2Ãp2)dT
+ 2
â€¢t
0
m _â– ) T T
(v A^xR,v)(u siu )dx
â€” p 3â€” â€”3â€”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' = A1 [(g  g ) + (f  f )]
P P P P P
+ (Aâ€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 + 2HÂ®II Hpll II B^' II (376)
where
X . (Q) is the smallest eigenvalue of Q
mm
 â€¢ 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 â€™
(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 â€”*â– 0 . It should also be noted that frequent
updating of g , f , and A_ will affect  z' â€”* 0, hence
â€”p â€”p p â€” Illa. X
 e â€”* 0. At steady state, e = 0, control will take
11 â€”11 max J â€” â€”
the form
u' (t) = u (t) (3.83)
â€”p â€”r
and
79
or
zâ€™ = 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 â€”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
online measurements of plant joint displacements x^ and
the velocities x
â€”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 Â£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 Â£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 lefthand side of Equation (3.86);
otherwise it is understood that I e Rnxn
81
s is the complex variable,
and Kj Â£ R
If and are diagonal matrices
Kx = diag [K1;i], K2 = diag [K2 Â±] (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 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 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 closedloop 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 closedloop
system
x = Ax  Bw
(3.89)
v = Cx
(3.90)
w
f (v, t)
(3.91)
where
B e R
2nxn
C Â£ R
nx2n
A, B, and C are timeinvariant,
f(â€¢) 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) = <Â£ (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 , tj^)
T
v w dt
(3.93)
for all t^ > tg.
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 â€”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)â€”(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 errordriven system equations, Equation
(3.33), closedloop system equations are given by
e = Ae + Bz" (3.98)
where
z"
(3.99)
Â£ 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 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
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 zâ€ 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) 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
2nx2nâ€œ
(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 Â£ 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
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, 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
â€”3.
, T T T . T / ~> t
la  (Â§al' a2' â€”a3 (3109)
where
subscript a is used throughout in this section
to denote the augmented system,
92
â€”a Â£ r3R' â€”al' â€”a2' and aa3 Â£ R"'
al
= eÂ± (3.110)
â€”a2
= e2 (3.111)
and are as defined in
Equations (3.27)(3.28)
also defining
e , =
â€”a 3
= I e . (3.112)
â€”al
e is given by
cl
e , =
â€”a 3
 Ie ,(t) dt (3.113)
â€”al
The control u denotes the plant input and has the
form
u
ap
= u' + u" (3.114)
â€”ap â€”ap
where is now given by
ul =
ap
! Ap(ArlGrirl + â€˜ KlÂ®al
K2a2 " K3a3^ (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
errordriven system equations are obtained as follows:
e = Ae + B z"
â€”a aâ€”a aâ€”
where
â€œ0
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. â€”
and (3.99), respectively
Due to poor estimation of manipulator parameters
in the reference model, closedloop 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
righthand 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 ^ Â£ due to parameter
e _ = 0
â€”a2 â€”
K , e , + K~e0 + K)e! = z"
al â€”al a2 â€”a2 a3 â€”a3 â€”
ss
(3.119)
e . = 0
â€”al â€”
Assuming that the selected K e Rnxn nonsingU]_arf (he
a. 3
equilibrium state is given by
e . = 0
â€”al â€”
e t
â€”a 2
= 0
e
â€”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
â€”2 = â€”
K1 1 + K2 2 =
(3.121)
ss
which in turn gives
= KI1 f
ss
(3.122)
â€”2 = â€”
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
â€”a
v = B P e
â€”cl cL cl â€”cl
(3.124)
Controllers u^ in Section 3.3.2 are valid for the augmented
system since u^ = ut . The closedloop 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 ~ â€” K , +4r K .
a2 s al 2 a3
s
(3.125)
where I is the identity matrix; its order is 3n on the
lefthand 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 timeinvariant part of the errordriven
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 cij/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 â€ž . > 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
groundfixed reference frame. In the literature, hand
coordinates, taskoriented 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
groundfixed 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
Â£, 8, and Â£. Keeping in mind that the frame Fn defined by
basis vectors (ujn^ , u^1^ , uin^ } has been fixed to the
hand, the Euler angles Â£, 8, and Â£ 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 Â£. Let the rotated frame F^ be
denoted by F' with basis vectors (uin^ , u^n^ , uin^ } after
n 1 Â¿ .i
/n (n) '
the rotation. Then, F' is rotated by 8 about u. to
n 1 1
100
r /s (n) " ~ (n) " /v (n) " â€¢,
obtain with basis vectors {u , ; , u^ }.
Finally, F" is rotated about bv Â£ to obtain F . These
J n 3 1 n
successive rotations are illustrated in Figure 4.1. Now,
. r/,v(n) ~ (n) /v (n) *Â»
the basis vectors {u^ , Uj , U3 } of Fn after undergoing
the above rotations will have the following representations
{ujjl, Ujj2' in frame F0 (subscript H denotes the hand)
HI
cosÂ£ cosÂ£  sin? cosB sinÂ£
sinÂ£ cosÂ£ + cos? cosB sinÂ£
sinB sinÂ£
H2
cos? sinÂ£  sine cosB cosÂ£
sine sinÂ£ + cose cosB cosÂ£
sinB cosÂ£
(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 â€¢ uH ^ > 0, i = 1,2,3. In Equation (4.2) subscripts
â€”tipi â€”^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
6link, 6 degreeoffreedom spatial (n = 6) and 3link,
3 degreeoffreedom 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 trkl rkl +
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
= VÂ°}
(4.4)
mi
where zâ€ž e R and 0 e R . In this chapter n is still used
â€”n â€”
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 6link spatial manipulators
m^ = 3
m2 = 3 (4.5)
m = m^ + m2 = 6 and
for 3link planar manipulators
m^ = 2
m2 = 1 (4.6)
m = m^ + m2 = 3
Hand orientation is given through the orientation of
the basis vectors {un^, 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 â€”
(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(Â£) = UH
(4.9)
2
will yield set of m^ nonlinear, coupled algebraic equations
of which only m2 are independent.
Note that for the 3link planar manipulator only one
Euler angle, say e, is needed to specify the hand
orientation. Substituting 8=0 and Â£ = 0, Uâ€ž(Â£, 8/ Â£) will
n
take the familiar form
O'
cose sine
0
sin? cose
0
0
1
mi xmi
For this case Uâ€ž(e) e R x x, m, = 2, will be defined as
n 1
cosÂ£ 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* Â£ Rm is the specified hand position and
orientation expressed in frame Fq
In general, fÂ° maps Rn into Rm. If Râ„¢ 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
fÂ® ^ of Rm (or Râ„¢) 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 Â¡Â¿
n
(4.13)
â€”H/0 = Gn Hi
(4.14)
â€¢ n th mixn
where w = Q_ e R , j column of GHn e R is defined by
[GHn]j ~
s. x zâ€ž /n , if j*"*1 joint is revolute
â€”D â€”1tv u j
s .
(â€œ3
, if j*"*1 joint is prismatic
(4.15)
Zjj/0 . is 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 â€œ (â€”H' â€”H/0>
(4.17)
mi m2 * m
vR e R , â€”H/0 e ^ , x2 e R with m = m^ + m2,
J = JUj) =
=H <5lÂ»'
n
G (x, )
n â€”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(fÂ° (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(fÂ° (x*)) x2 (4.22)
n_1
CH (Â£ (x*))
nn
nl
Gn (Â£ (xj))
nmxn ,. Â«..
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^)
Â¿2
where the (i, j
)^ element
of j (x^)
â€žmxn .
Â£ K is
n 3 J â€¢ .
[j
y i:
=1 3xl,k
x2,k
i =
1,2,â€¢â€¢â€¢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
__ _ / \ Â«jinxrn. Â« to
Jâ€” J^ (x^) Â£ R , k â€” 1,2,...,m
(4.24)
(4.25)
(4.26)
108
J ' â€”2 ^ ~
T T
2 J1
2
â€”2
m
e R
mxm
(4.27)
Given joint displacements x^, hand velocity x* > an<^
acceleration x^, corresponding joint velocities x2 and joint
accelerations Â¿2 can be solved from Equations (4.16) and
(4.24), respectively, provided that the Jacobian J (x^) is
nonsingular.
T1 , > *
x2 = J (x ) x2
(4.28)
2
J_1(x1) x*  J1 (x^) j (xr x*) J 1 (x^) x*
where
(4.29)
J (xj_, xip =
xf J 1T uÂ±) J1 (xÂ±)
*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 o1 4.
J = J 1 (f (x*)) (4.33)
nl
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 singularityfree 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 singularityfree 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 â€ž
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 Â£. Similarly, subscript
r is reserved for the "reference" model parameters.
Referring to Equation (4.35),
x* = (x*T x*T)T
p lpl' â€”p2
(4.36)
â€”pi Â£ ^m' â€”p2 e as <^e^:'ne<^ before,
u* e Rn is the control vector,
P
A _1 (x*.) = A1 (fÂ° (^pl)} e R
P â€œPi P  ^
(4.37)
G* (x*p) Â£ Rnxm is defined symbolically such that
SLp (*pl> = Sp (*pl> 5pl
(4.38)
and
2* (x* ) = 2 (fÂ°
sp pi â€”p 
1
>
(4.39)
where g (â€¢) isdefined by Equation (3.6), hold. Similarly,
P
F* (x* , x*_) e Rnxm is so defined that
lp â€”pi â€”p2
f* (x* , x*.,) = FÃ (x . , x ) x%
lp pi â€”p2 lp pi' â€”p2 â€”p2
(4.40)
113
holds; f*p (x^1, x*2) is given
by
Â£lP (^r â€”p2^ = ip<Â£Â° (^pi> Jp"1 (pij p2>
(4.41)
f (â€¢) is defined by Equation (3.9), and
P
2p
T _*lT _* T*l
x ~ J J . J
â€”p2 P Pi P
T *â€”IT T* T*l
x â€ž J J  J
p2 p pm p
e R
mxm
(4.42)
where
V1
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, closedform 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^) = xÂ£ numerically for joint displacements Xp^<
then calculate Jp, J ^, and finally compute Ap, Gp, F^,
and F2pâ€™ 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
â€”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 ,
â€”rl
 X*
pi
e* = e*(t)
*
= X t
â€”r2
*
â€œ x 0
â€”p2
3 = I I
â€”3 = 1 1
dt
(4.45)
(4.46)
e* = (e*T, e*T, e*T)T e R3m
(4.47)
T ,, â€¢ * * ' * "
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*
â€”p p p 1 r r râ€”rl r r lr 2r â€”r2 1â€”1
 K2e*  K3e*] (4.49)
augmented errordriven system equations will be obtained
subtracting Equation (4.35) from Equation (4.44) and
substituting Equations (4.48)(4.49) as
â€¢ *
e
. * *
A e
B*z*
â€ž*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 â€”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
â€”p p p â€”p u u â€”r
(4.53)
K* = [8* : F* + A*J*_1F* ]
P P IP P P 2pJ
(4.54)
AK* = tR1I*(S1^1)T â€¢ R2v*(S2xÂ£2)T]
(4.55)
[a*j;1j*a
(4.56)
AK* = [R3v*(S3u*)T] (4.57)
( â€¢) 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* = â€”Q*
(4.58)
* if 3mx3m . ^ .
Q e R and Q >0
Choosing a scalar Lyapunov function of the form
â€¢t
TT* , * . Â« *T_* . n
V(e,t)=e Pe+2
, *T_*,*â€”1_ *.
n (v J A R.v )
0 â€” p p 1â€”
/ *T T * \ j
(ipisispi) dT
+ 2
0..
/ *T * *â„¢1 . *T T â˜… xt
(V JpAp S2Z ) <5p2S25p2)d
+ 2
, aT.,*,*!,. *> , *Tâ€ž *, ,
(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
â€¢ * , *, *T * * . ~ *T *i
V(e)=e Qe+2v z
(4.60)
where
118
= J*A*_I[(Â¿* 2*) + (f* f* )]
P P ^P ^P â€œIP â€œIP
(*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* â€”r
It can be shown that bound on e will be
X (P*)
â€ž *n s' o max
â€” 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 closedloop system satisfying V(e*, t) > 0
â€¢ *
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 offline 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 servocontrolled manipulators
solve this equation online 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 online 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 online computation; but considering that
the inverse problem is already solved online 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 online 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 (zerolink lengths, link offsets,
and twist angles which are mostly 0Â° or 90Â° 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 nlink,
th.
n degreeoffreedom manipulator, and that the k actuator
th t h.
is mounted on the (k1) 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, timeinvariant 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
Â°k :
Coefficient of viscous friction referred
to output shaft
Â«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 jÂ£, 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 thirdorder, linear,
timeinvariant model can be used without loss of generality.
In matrixvector form Equation (5.1) is given
by
EQe_ + E2^ + E1Â®. + 1 = E3Ã
(5.3)
â€¢ â€¢
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 timeinvariant,
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)
â€¢ _ 1
x2 = A (Xx) [(x(x^/ â€”2 ^ ^ 1â€”1 ~ ^2â€”2 3^Â£3 ^
(5.8)
125
x, = E.xÂ» + Ecx0 + Lu
â€”3 4â€”2 5â€”3 â€”
(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' â€”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 + Â£
+ ElZl + E2l2
Differentiating Equations (5.12) (5.13) and (5.
respect to time
Â¿1 = Â¿!
i2 = Â¿2
Â¿3 = 4'
and substituting Equations (5.18) and x^ = y^,
with Equation (5.17) into Equations (5.7)(5.9)
system equations become
Â¿i  12
Â¿2 = y3
Z3 = A"1{(E1E5)y1 + (E2E5 + E3E4 f
+ (e5A  Ã  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] Â£ 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 = Â£
+
n
9A_
â€˜ij Jj. y2,k 3Yli
11
k
_d_
dt
V*l>
(5.23)
n 32.ÃZ].)
 " Y2,k 9ylfk
n
Â£Â» 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 vectormatrix form
and given for the plant as follows:
128
0 I
o
I
o
1
Â¿P=
0 0
I
ip+
0
1 1
râ€”
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 + E5pâ€”p ^^p 1 ^
G2p(ip*ip2 = (E2pE5p +E3pE4p "Elp'3Â£p2
+ E5pVV  i(ipl>
G3p(ip)Zp3 =
(5.26)
State vector is as defined by Equations (5.11)â€”(514);
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:
Ã© = Ae + Bz  A'V V
  P P P
(5.28)
In this section, e, z, A, and B are defined as
Â£ = Yr  Â£
, e e R
3n
e . =
D
â€”r j
^pj
, j
= 1,2,3;
~0
I
0â€œ
~0 "
A =
0
0
I
, B =
0
K1
K2
K3
I
,n
(5.29)
(5.30)
â€ž â€ž3nx3n _ â€ž3nxn
A e K , B e k
J 11
z =  y (G. y . ) + A N u
DP â€œPD 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 â€”r
are now given by
Kp  'Glp = G2p 5 G3p]
ZSKp = [Rl v(Slypl)T : R2v(S2j^2)T : RjVlSj^)1]
(5.32)
Ku â€˜ 'VA'
4KU = [R4i(s4!ir)T]
T 3 nx 3 n â€¢
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 Â£
j=l
T â€”1 â€”1 T T
(v A N R.v)(y .S.y .)dr
 P P D ^PD
+ 2
(vTA ^R.v)(uTS.u.)dx
â€” p p 4â€” â€”r 4^4
(5.33)
global asymptotic stability of the closedloop 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
â€¢ â€¢ â€¢
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, timeinvariant
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 Ã, 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 thirdorder system
representation of actuator dynamics of the previous section
reduces to a secondorder 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 â€žnxn
and N' e K are
nxn
are
E^ = diag +
*k
1
(5.36)
N'1 = diag rkKTk Â±
Rk
133
Coupled system equations are obtained if Equation (5.4) is
substituted in Equation (5.35)
(5.37)
x2 = A1 (x^) ig.^) +f (xr x2)  E^  EÂ£x2 +N'_1u}
A is the same as in Equation (5.10); x^ = Q_, x2 = Â£ 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 (Fâ€ž  E' )
p p 2p
x +
P
AV1
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
â€”i â€”n â€”pi â€”i
n
Â£3  lÂ£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 â€¢
P P
1 [ (G E, )x . + (F E' )x .]  T
r r lr â€”rl r 2r â€”r2 .L
j = l
K .e .
33
(5.40)
the augmented errordriven system equations become
where
e = Ae + Bz'  A ' ^u"
 P P P
(5.41)
'0
0
0
â€œ0~
A =
râ€”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 = IRjVÃSjX^)1 : R2v(s2xp2)T]
K = [N'A A1N,_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;
â€¢ â€¢ â€¢
measurement of accelerations and evaluations of A, f_, and g
are not required. The solution of the errordriven 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'
â€” 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
VAX11/750 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 fourthorder, modified predictorcorrector
method. Inclusion of the disturbance rejection feature is
left optional; the user can select the desired option.
Although the program is capable of simulating nlink
manipulators, 3 and 6link 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 3link manipulator and the British system
in the 6link 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; Rv(Sx .) ,
^ ^ D j P J
T
j = 1,2 and Rv(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 3Link, Spatial Manipulator
The 3link manipulator used in the examples is
depicted in Figure 6.1 and the related plant parameters are
given in Tables 6.16.3.
Figure 6.1 Illustration of the 3Link Manipulator
Reference model manipulator and actuator parameters that
are different from the plant are listed in Tables 6.46.5.
The first simulation includes the disturbance rejection
feature, i.e., the integral feedback is activated. Hence,
the system order is 9 for the 3link manipulator; K^, Kj,
3x3
and Â£ R in Equation (5.42) are chosen diagonal
Table 6.1 Kinematic Parameters of the 3Link 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 3Link 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 handfixed reference frame.
141
Table 6.3 Actuator Parameters (Plant)
Actuator No
: 1
2
3
Jk
(103kg.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
Â«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
(103kg.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 errordriven 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 6.26.4 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 6.56.7, and 200 Hz, Figures 6.86.10.
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 6.5bc, 6.6c)
deteriorated compared to Figures 6.2bc, 6.3c (60 Hz) and
Figures 6.8bc, 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.Â£4 1.65 Â£.06 Â£.48 Â£.83 3.30 3.71
Figure 6.2a Joint 1â€”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 2â€”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 3â€”Displacement vs Time
147
Vel. I (l/s)
0.00 0.41 0.83 1.E4 1.65 E.06 E.48 E.89 3.30 3.71
Time (sec)
Figure 6.3a Joint 1â€”Velocity vs Time
148
Vel. Â£ (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 2â€”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 3â€”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 1â€”Input Voltage vs Time
151
Inp.Volt. 2 (v)
Ti me (sec)
Figure 6.4b Actuator 2â€”Input Voltage vs Time
152
Inp.Volt. 3 (v)
Time (sec)
Figure 6.4c Actuator 3â€”Input Voltage vs Time
153
Disp. 1 (ded)
Time (sec)
Figure 6.5a Joint 1â€”Displacement vs Time
154
Disp. 2 (ded)
Ti me (sec)
Figure 6.5b Joint 2â€”Displacement vs Time
155
Disp. 3 (ded)
0.00 0.4S 0.84 1.87 1.69 8.11 8.53 8.96 3.38 3.80
Time (see)
Figure 6.5c Joint 3â€”Displacement vs Time
156
Vel. I (l/s)
Time (sec)
Figure 6.6a Joint 1â€”Velocity vs Time
157
Vel. 2 (1/s)
Time (see)
Figure 6.6b Joint 2â€”Velocity vs Time
158
Vel. 3 (l/s)
Ti me (5bc)
Figure 6.6c Joint 3â€”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 1â€”Input Voltage vs Time
3.38 3
Time (sec)
160
Inp.Volt. 2 (v)
Ti me Ã sec)
Figure 6.7b Actuator 2â€”Input Voltage vs Time
161
Inp.Volt. 3 (v)
Time (sec)
Figure 6.7c Actuator 3â€”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 1â€”Displacement vs Time
163
Disp. 2 (ded)
Ti me (sec)
Figure 6.8b Joint 2â€”Displacement vs Time
164
Disp. 3 (ded)
Ti me (sec)
Figure 6.8c Joint 3â€”Displacement vs Time
165
Vel. L (l/s)
Time (sec)
Figure 6.9a Joint 1â€”Velocity vs Time
166
Vel. E (L/s)
Time (sec)
Figure 6.9b Joint 2â€”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 3â€”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 1â€”Input Voltage vs Time
Time (sec)
169
Inp.Volt. P (v)
0.00 0.^1 0.8E 1.24 1.65 2.06 2.47 2.08 3.29 3.70
Time (sec)
Figure 6.10b Actuator 2â€”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 3â€”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 6.116.13. 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 1â€”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 2â€”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 3â€”Displacement vs Time
176
Vel. I (l/s)
0.00 0.42 0.85 1.27 1.70 2.12 2.54 2.97 3.39 3.82
Time (sec)
Figure 6.12a Joint 1â€”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 2â€”Velocity vs Time
178
Vel. 3 (1/s)
0.00 0.4E 0.85 1.27 1.70 E.1E E.51* Â£.9? 3.39 3.8E
Time (see)
Figure 6.12c Joint 3â€”Velocity vs Time
179
Inp.Volt. 1 (v)
Ti me (sec)
82
Figure 6.13a Actuator 1â€”Input Voltage vs Time
180
Inp.VÃ¼lt. 2 (v)
Ti me (see)
Figure 6.13b Actuator 2â€”Input Voltage vs Time
181
Inp.Volt. 3 (v)
0.00 0.4E 0.85 1.E7 1.70 Â£.1Â£ 8.5^ Â£.97 3.39 3.8Â£
Time (sec)
Figure 6.13c Actuator 3â€”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 â€”] are assumed to be of dimension kxk. Expanding
the above equation and writing in matrixvector 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
â€¢ â€¢ â€¢
ak2I
â€¢
â€¢
â€¢  â€¢
alk1
a2kI
. . .
aT + akkI
184
I represents the identity matrix of order k,
P = [P1X P12 Pik P21 â€¢** pkk]
a = c312 â€¢** qik q21 *â€¢* 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 offline. The solution is obtained by
means of Gauss elimination with complete pivoting.
6.3' Simulations on the 6Link,
Spatial Industrial Manipulator
The 6link, spatial industrial manipulator,
Cincinnati Milacron T3776, 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 T3776. The order of magnitude of these
parameters are representative of DC motors [13, 58], but
otherwise arbitrary.
135
Figure 6.14 Cincinnati Milacron T3776 Industrial Robot
186
Table 6.6 Kinematic Parameters of the 6Link 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 handfixed reference frame.
187
Table 6.8 Actuator Parameters (Plant)
Actuator No :
1
2
3
4
5
6
Jk
(103lbm.ft2) :
100.00
50.00
50.00
30.00
30.00
20.00
Â°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 (103lbm.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 T3776 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 errordriven 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. = diagdO3), 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
6.156.16 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 6.186.20. 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 1â€”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 2â€”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 3â€”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 4â€”Displacement vs Time
193
Disp. 5 (ded)
Time (sec)
00
Figure 6.15e Joint 5â€”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 6â€”Displacement vs Time
195
Vel. I (i/s)
Ti me (sec)
00
Figure 6.16a Joint 1â€”Velocity vs Time
196
Vel. 2 (l/s)
0.00 0.22 0.44 0.67 0.89 1.11 1.33 1.55 1.78 E.00
Time (sec)
Figure 6.16b Joint 2â€”Velocity vs Time
197
Vel. 3 (L/s)
Time (sec)
00
Figure 6.16c Joint 3â€”Velocity vs Time
198
Vel. 4 ( L/s)
0.00 0.82 0.44 0.67 0.09 1.11 1.33 1.55 1.78 8.00
Ti me (see)
Figure 6.16d Joint 4â€”Velocity vs Time
199
Vel. 5 (L/s)
Time (sec)
00
Figure 6.16e Joint 5â€”Velocity vs Time
200
Vel. 6 (l/s)
0.00 0.EE 0.44 0.67 0.89 1.11 1.33 1.55 1.78 E.00
Time (sec)
Figure 6.16f Joint 6â€”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 1â€”Input Voltage vs Time
202
Inp.Volt. 2 (v)
Time (sec)
Figure 6.17b Actuator 2â€”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 3â€”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 4â€”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 5â€”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 6â€”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 1â€”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 2â€”Displacement vs Time
603
Disp. 3 (ded)
0.00 0.22 0.44 0.67 0.89 1.11 1.33 1.55 1.78 2.00
Ti me (sec)
Figure 6.18c Joint 3â€”Displacement vs Time
210
Disp. 4 (ded)
Time (sec)
Figure 6.18d Joint 4â€”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 5â€”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 Â£.00
Time (sec)
Figure 6.18f Joint 6â€”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 1â€”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 2â€”Velocity vs Time
Time (sec)
215
Vel. 3 (L/s)
Ti me I see)
Figure 6.19c Joint 3â€”Velocity vs Time
216
Vel. t (l/s)
Time (sec)
Figure 6.19d Joint 4â€”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 5â€”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 6â€”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 Â£.00
Time (sec)
Figure 6.20a Actuator 1â€”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 2â€”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 3â€”.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 4â€”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 5â€”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 6â€”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 T3776â€”Figures 6.21
6.23â€”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 1â€”Displacement vs Time
227
Disp, 2 (ded)
Time (sec)
Figure 6.21b Joint 2â€”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 3â€”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 4â€”Displacement vs Time
230
Disp. 5 (ded)
Ti me (sec)
Figure 6.21e Joint 5â€”Displacement vs Time
231
Disp. 6 (deÃ¡)
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 6â€”Displacement vs Time
232
Vel. I (l/s)
Figure 6.22a Joint 1â€”Velocity vs Time
Time (sec)
233
Vel. 2 (l/s)
Time (sec)
00
Figure 6.22b Joint 2â€”Velocity vs Time
234
Vel. 3 (1/s)
Time (see)
Figure 6.22c Joint 3â€”Velocity vs Time
235
Vel. T (l/s)
Time (sec)
Figure 6.22d Joint 4â€”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 5â€”Velocity vs Time
237
Vel. & (L/s)
Ti me (sec)
Figure 6.22f Joint 6â€”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 1â€”Input Voltage vs Time
239
i
Inp.Volt. 2 (v)
Ti me (see)
Figure 6.23b Actuator 2â€”Input Voltage vs Time
240
Inp.Volt. 3 (v)
0.00 0.E2 0,â€˜tâ€˜t 0.67 0.89 1.11 1.33 1.56 1.78 E.00
Time (sec)
Figure 6.23c Actuator 3â€”Input Voltage vs Time
241
Inp.Volt. 4 (v)
Time (sec)
Figure 6.23d Actuator 4â€”Input Voltage vs Time
242
Inp.Volt. 5 (v)
Time (sec)
Figure 6.23e Actuator 5â€”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 6â€”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 secondorder, timeinvariant 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 nlink,
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, firstorder
ordinary differential equations for an nlink, n degreeof
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. Errordriven
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 closedloop 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 thirdorder, linear, timeinvariant 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 secondorder, linear, timeinvariant
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 nlink robot manipulators powered with n actuators.
Examples on 3 and 6link, 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 online 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
Controlâ€”A Survey," AutomÃ¡tica, Vol. 19, pp. 471486,
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. 143151, September 1983.
[4] Book, W. J., MaizzaNeto, 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. 424431,
December 1975.
[5] Brady, M., Hollerbach, J. M., Johnson, T. L.,
LozanoPerez, 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 NonRedundant Manipulators," Mechanism
and Machine Theory, Vol. 16, pp. 8191, 1981.
[8] Dubowsky, S., and DesForges, D. T., "The Application
of ModelReferenced Adaptive Control to Robotic
Manipulators," ASME J. Dynamic Systems, Measurement
and Control, Vol. 101, pp. 193200, 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
PolePlacement for Industrial Robots and Manipulators,"
Int. J. Robotics Research, Vol. 1, pp. 6578, 1982.
[12] Golla, D. F., Garg, S. C., and Hughes, P. C., "Linear
StateFeedback Control of Manipulators," Mechanism
and Machine Theory, Vol. 16, pp. 93103, 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. 279281, 1983.
[15] Hang, C. C., and Parks, P. C., "Comparative Studies of
Model Reference Adaptive Control Systems," IEEE T,
Automatic Control, Vol. 18, pp. 419428, October
1973.
[16] Hemami, H., and Camana, P. C., "Nonlinear Feedback in
Simple Locomotion Systems," IEEE T. Automatic Control,
Vol. 21, pp. 855859, 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. 535542, 1981.
[18] Hewit, J. R., Hanafi, A., and Wright, F. W., "Optimal
Trajectory Control of Robotic Manipulators,"
Mechanism and Machine Theory, Vol. 19, pp. 267273,
1984.
[19] Hollerbach, J. M., "A Recursive Formulation of
Lagrangian Manipulator Dynamics," IEEE T. Systems,
Man and Cybernetics, Vol. 10, pp. 730736, November
1980.
[20] Hooker, W. W., and Margulies, G., "The Dynamical
Attitude Equations for an NBody Satellite," J.
Astronautical Sciences, Vol. 12, pp. 123128, 1965.
[21] Horowitz, R., and Tomizuka, M., "An Adaptive Control
Scheme for Mechanical Manipulatorsâ€”Compensation of
Nonlinearity and Decoupling Control," ASME Paper No.
8 OWA/DSC6, 198 0 .
252
[22] Kahn, M. E., and Roth, B., "The NearMinimum Time
Control of OpenLoop 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. 371393, 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. 188193, June 1982.
[25] Koivo, A. J., and Guo, T. H., "Adaptive Linear
Controller for Robotic Manipulators," IEEE T.
Automatic Control, Vol. 28, pp. 162171, February 1983 .
[26] Landau, I. D., "A Hyperstability Criterion for Model
Reference Adaptive Control Systems," IEEE T.
Automatic Control, Vol. 14, pp. 552555, October 1969.
[27] Landau, I. D., "A Generalization of the Hyperstability
Conditions for Model Reference Adaptive Systems,"
IEEE T. Automatic Control, Vol. 17, pp. 246247,
April 1972.
[28] Landau, I. D., and Courtil, B., "Adaptive Model
Following Systems for Flight Control and Simulation,"
J. Aircraft, Vol. 9, pp. 668674, 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. 134142, 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.
868871, 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. 330335,
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. 142151, June
1981.
[34] Luh, J. Y. S., Walker, M. W., and Paul, R. P. C.,
"OnLine Computational Scheme for Manipulators,"
ASME J. Dynamic Systems, Measurement and Control,
Vol. 102, pp. 6976, 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. 468474, June 1980.
[36] Noble, B., and Daniel, J. W., Applied Linear Algebra,
Second Edition, PrenticeHall International, Inc.,
Englewood Cliffs, New Jersey, 1977.
[37] Orin, D. E., McGhee, R. B., Vukobratovic, M., and
Hartoch, G., "Kinematic and Kinetic Analysis of
OpenChain Linkages Utilizing NewtonEuler Methods,"
Mathematical Biosciences, Vol. 43, pp. 107130,
February 1979.
[38] Paul., R., "Modeling, Trajectory Calculation and
Servoing of a ComputerControlled 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, TREE7825, 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. 449455, 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. 4955, 1981.
Popov, V. M., "The Solution of a New Stability
Problem for Controlled Systems," Automation and
Remote Control, Vol. 24, pp. 123, 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. 179188, 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. 547556, May 1983.
[46] Stepanenko, Y., and Vukobratovic, M., "Dynamics of
Articulated OpenLoop Active Mechanisms," J.
Mathematical Biosciences, Vol. 28, pp. 137170, 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. 108118, June 1981.
[48] Stoten, D. P., "The Adaptive Control of Manipulator
Arms," Mechanism and Machine Theory, Vol. 18, pp.
283288, 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 6720112, May 1981.
[50] Takegaki, M., and Arimoto, S., "An Adaptive Trajectory
Control of Manipulators," Int. J. Control, Vol. 34,
pp. 219230., 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. 218228,
September 1982.
[54] Tosunoglu, L. S., "PowerTime 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. 118 , 1978 .
[57] Vukobratovic, M. , and Potkonjak, V., Dynamics of
Manipulation Robots, SpringerVerlag, Berlin,
Heidelberg, 1982.
[58] Vukobratovic, M. , and Stokic, D., Control of
Manipulation Robots, SpringerVerlag, Berlin,
Heidelberg, 1982.
[59] Waldron, K. J., "Geometrically Based Manipulator
Rate Control Algorithms," Mechanism and Machine Theory,
Vol. 17, pp. 379385, 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. 205211, September 1982.
[61] Wander, J., "RealTime 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.
ManMachine Systems, Vol. 10, pp. 4753, 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. 303309, December 1972.
[65] Whitney, D. E., "Force Feedback Control of Manipulator
Fine Motions," ASME J. Dynamic Systems, Measurement
and Control, Vol. 99, pp. 9197, June 1977.
[66] Whyte, H. D., "Practical Adaptive Control of Actuated
Spatial Mechanisms," Proc. IEEE Conference on
Robotics and Automation, St. Louis, Missouri, pp.
650655, 1985.
[67] Yuan, J. S. C., "Dynamic Decoupling of a Remote
Manipulator System," IEEE T. Automatic Control, Vol.
23, pp. 713717, 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. 101109,
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
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This dissertation was submitted to the Graduate Faculty of
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May 1986
Dean, Graduate School
UNIVERSITY OF FLORIDA
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