
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00001966/00001
Material Information
 Title:
 Adaptive modelfollowing control of a robotic manipulator
 Creator:
 Gheblawi, Usama Ahmed
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 1989
 Language:
 English
 Physical Description:
 vii, 90 leaves : illustrations ; 28 cm
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Science)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Electrical Engineering, CU Denver
 Degree Disciplines:
 Electrical Engineering
 Committee Chair:
 Wall, Edward T.
 Committee Members:
 Bialasiewicz, Jan T.
Majundar, Arun
Subjects
 Subjects / Keywords:
 Manipulators (Mechanism) ( lcsh )
Manipulators (Mechanism) ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaves 8990).
 Thesis:
 Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering, Department of Computer Science and Engineering
 Statement of Responsibility:
 by Usama Ahmed Gheblawi.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 22937066 ( OCLC )
ocm22937066
 Classification:
 LD1190.E54 1989M .G43 ( lcc )

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Full Text 
ADAPTIVE MODELFOLLOWING CONTROL
OF A ROBOTIC MANIPULATOR
by
Usama Ahmed Gheblawi
B.S., University of Colorado, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirement for the degree of
Master of Science
Department of Electrical Engineering and Computer Science
1989
, .. g;
This thesis for the Master of Science degree by
Usama Ahmed Gheblawi
has been approved for the
Department of
Electrical Engineering and Computer Science
by
Arun Majundar
Date
30
11
Gheblawi Ahmed, Usama (M.S., Electrical Engineering)
Adaptive ModelFollowing Control of a Robotic Manipulator
Thesis directed by Professor Edward T. Wall
Abstract The objective of the manipulator con
trol design is to find the appropriate torques that will
drive the manipulator to follow a prespecified trajectory.
This thesis presents the use of adaptive modelfollowing
control (AMFC) technique for the control of a robotic
manipulator that has wide variations in its payload. The
design procedure is simple, effective, and does not require
accurate modeling of the dynamic system. The problem of
perfect asymptotic adaptation is interpreted as a stabil
ity problem. Through the use of hyperstability in conjunc
tion with the properties of the positive dynamic systems,
the largest family of adaptation laws assuring the
stability of the overall adaptive system is obtained, then
the most suitable adaptation law for a specific application
is chosen.
Ill
ACKNOWLEDGEMENTS
This study would not have been possible without
the cooperation and support of many people to whom I will
always be grateful.
I especially wish to express my sincere gratitude
to Professor Edward T. Wall for his cheerful support and
guidance during the preparation of this thesis. I would
like also to thank Dr. Jan Bialasiewicz for his advice,
encouragement, and serving on my committee.
Finally, I wish to thank my family for their
patience, support, and inspiration.
CONTENTS
Chapter
I. INTRODUCTION..................................1
II. REVIEW OF MANIPULATOR CONTROL METHODS.........4
2.1 Introduction.............................4
2.2 Nonadaptive Control Methods..............5
2.3 Adaptive Control Methods...........6
III. MANIPULATOR DYNAMIC EQUATIONS OF MOTION.......10
3.1 Introduction............................10
3.2 Structure of the Manipulator Dynamic
Equation...............................10
3.2.1 The General Dynamic Equation of the
Manipulator..........................10
3.2.2 The Manipulator Mass Inertia Matrix...13
3.2.3 The Centrifugal and Corioles Terms....16
3.2.4 The Gravity Term.......................19
3.3 Linearization of the Dynamical Model....19
3.4 The Dynamic Equation of a TwoLink
Manipulator............................23
3.4.1 Description of the Physical Model......23
3.4.2 The Kinetic and Potential Energy of a
TwoLink Manipulator.................23
3.4.3 The Dynamical Equation of Motion
of a TwoLink Manipulator............30
V
IV. ADAPTIVE CONTROL OF MANIPULATORS..............33
4.1 Introduction.............................33
4.2 The Plant Model..........................34
4.3 The Reference Model......................36
4.4 Perfect ModelFollowing Control (PMFJ...38
4.5 Adaptive ModelFollowing Control
(AMFC)................................41
4.6 Adaptation Mechanism and Stability.......45
4.6.1 Equivalent Feedback Representation
of AMFC..............................46
4.6.2 Hyperstability of the AMFC............53
V. SIMULATION RESULTS............................57
5.1 Introduction.............................57
5.2 SingleLink Manipulator Case.............57
5.2.1 Results............................... 60
5.3 TwoLink Manipulator Case................68
5.3.1 Results................................69
VI. CONCLUSIONS...................................87
BIBLIOGRAPHY
89
FIGURES
Figure
2.1 SelfTuning Regulator (STR)....................7
2.2 Model Reference Adaptive Control (MRAC)........8
3.1 The General Coordinates of a NLink
Manipulator.................................12
3.2 A Two DegreeofFreedom Manipulator...........24
3.3 Parameters of the ith Link...................25
4.1 Linear ModelFollowing Control System
(Perfect ModelFollowing Control)...........39
4.2 Adaptive ModelFollowing Control (AMFC) with
Parameter Adaptation........................43
4.3 Adaptive ModelFollowing Control (AMFC) with
Signal Synthesis Adaptation.................43
4.4 Equivalent Feedback System....................46
4.5 Equivalent Feedback Representation of
Adaptive ModelFollowing Control............51
5.1 Position Response (Case 1)....................62
5.2 Velocity Response (Case 1)....................63
5.3 Position Tracking Error......................64
5.4 Velocity Tracking Error......................65
5.5 Position Response (Case 3)...................66
5.6 Velocity Response (Case 3)...................67
5.7 Position Response (mL = 0.0 Kg, Case 1)......73
5.8 Velocity Response (mL = 0.0 Kg, Case 1)......74
vii
5.9 Joint One Position Response
(mL = 0.0 Kg, Case 2)..>......................75
5.10 Joint One Position Response
(mL = 0.0 Kg, Case 3)........................76
5.11 Joint One Position Error Tracking
(mL = 0.0 Kg)................................77
5.12 Joint Two Position Error Tracking
(ml = 0.0 Kg)................................78
5.13 Joint One Velocity Error Tracking
(mL = 0.0 Kg)............................... 79
5.14 Joint Two Velocity Error Tracking
(mL = 0.0 Kg)................................80
5.15 Position Response (mL = 7 Kg, Case 4)............81
5.16 Velocity Response (mL 7 Kg, Case 4)............82
5.17 Joint One Position Error Tracking
(mL = 7 Kg)..................................83
5.18 Joint Two Position Error Tracking
(m,^ = 7 Kg).................................84
5.19 Joint One Velocity Error Tracking
(m^ = 7 Kg)..................................85
5.20 Joint Two Velocity Error Tracking
(mL = 7 Kg)..................................86
CHAPTER I
INTRODUCTION
Manipulators have been used extensively in hos
tile environments, such as in the nuclear industries, deep
undersea exploration and maintenance operations, space,
and industrial automation. In most of these applications
operating speed and performance indices are relatively
low, and relatively simple control systems have proven ade
quate. With increased demand on manipulators will come the
need for improved manipulator controllers, that can drive
the manipulator more effectively and efficiently than the
conventional controllers (i.e., in terms of operating
speed, use of energy, capability of dealing with various
tasks, accuracy, etc). The contemporary controllers used
by robot manufacturers are largely based on simple conven
tional feedback control techniques that are unable to deal
with complex, uncertain dynamics of the manipulator and its
interaction with other machinery. On the other hand, most
of the proposed advanced controllers are either based on
unrealistic approximation, or On requirements that make it
computationally demanding to implement in realtime.
2
In this study, an adaptive control scheme is con
sidered for controlling the robot manipulator. This
approach is based on adaptive modelfollowing control
systems (AMFC) technique [1], where the design of the adap
tation mechanism is based on the use of the Popov hyperst
ability theory, which assures the asymptotic stability of
the system. The objective of the controller is to cause
the manipulator to follow a given desired performance spe
cified by a reference model.
The control law is composed of two parts. The
first part of the controller is linear in the system state
and the reference input, which assures a perfect model
following control. The second part of the control is pro
duced by an additional adaptive feedback loop, where the
adaptive gains are determined using the hyperstability
theory. The adaptive feedback loop will compensate for
parameter variations, and uncertainties in the dynamical
equations of the manipulator.
The novelty of this control strategy is that the
adaptive controller requires neither accurate modeling of
manipulator dynamics, nor solving the dynamic equations
(which is known to be difficult and computationally demand
ing) And the asymptotic stability of the overall system
can be assured automatically using the hyperstability and
positivity concepts. Thus, eliminating the need for a sep
3
arate stability analysis. Furthermore, through the use of
implicit adaptive control, the need for identification of
manipulator parameters is eliminated.
CHAPTER II
REVIEW OF MANIPULATOR CONTROL METHODS
2.1 Introduction
Manipulators are required to perform two tasks,
namely moving an object (payload) from an initial to a
final position along a prescribed path, and exerting forces
or torques on the manipulated object. In this study, we
will consider only the first task, which is the problem of
moving objects through some prescribed path.
There are a variety of algorithms available for
manipulator control. These algorithms usually assume that
the control structure of the robot has been divided into
two levels. The first level is called path or trajectory
planning, and the second level is called path tracking.
The trajectory planner provides the time history of the
desired positions and velocities, where path tracking is
the process of making the robot's actual positions and
velocities match the desired values provided by the trajec
tory planner. The control schemes can be divided into two
different approaches, nonadaptive and adaptive control.
5
2.2 Nonadaptive Control Methods
Presently there are three wellknown methods that
provide position control of manipulator and are all kine
matically oriented. In resolved motion position control
(RMPC) [2], the desired joint positions are determined
directly by solving the inverse kinematic equations, and
then the related joint velocities and accelerations are
calculated from the positions. Whitney [3,4] proposed a
tracking scheme called the resolved motion rate control
(RMRC), where the linear/angular velocities needed to
maintain the desired endeffector position and orientation
are mapped into joint rates by the inverse Jacobian matrix,
and then the joint accelerations are computed from the
velocities. In resolved motion acceleration control
(RMAC) [5], the corrective cartesian accelerations are
calculated and resolved into joint accelerations, and the
joint positions and rates are measured. All of these
motion control schemes resolve the control into general
ized joint coordinates denoted by q,q, and q. Given
q,q, and q, joint torques of the manipulator dynamics were
obtained by the Lagrangian formulation [ 6 ], and by the
NewtonEuler formulation [ 7 ]. Although recently more
efficient algorithms for computation of dynamics have been
proposed [8,7], these methods still require a considerable
amount of computations during the motion. Furthermore,
these methods may include modeling error (discrepancies
6
between the dynamic model and the actual manipulator dyna
mics)/ thus requiring additional intelligent controllers
to compensate for this error. Also, note that even if
accurate modeling is possible, the dynamic model has to be
a function of the task being performed (i.e., payload and
positions of the endeffector). This fact results in
either very complex dynamic models or inaccurate but sim
plified models; neither of the two is desirable.
2.3 Adaptive Control Methods
Traditionally, control systems have been designed
based on a good understanding of the system to be con
trolled. When knowledge of the system is limited the rela
tive modern issues of robust control, adaptive control, and
learning control become important.
One way to attempt to deal with poor knowledge of
parameters in a control scheme is through techniques that
are generally called adaptive control. The central problem
in the synthesis of adaptive controllers is to prove rig
orously that the resulting overall system is asymptoti
cally stable. Adaptive control is closely related to the
problem of system identification, and generally an
adaptive controller can be viewed as being composed of two
parts; the first part is the identification process which
identifies the parameters of the plant itself, or the
parameters that appear in the controller of the plant. The
second part is the control law which implements a control
law that is in some way a function of the parameters being
identified.
Adaptive control strategies take on many forms,
and some methodologies have become established for design
ing adaptive control systems. Among these methodologies,
the two most widely described are the selftuning regulator
(indirect adaptive control) scheme, which contains sepa
rate identification and control synthesis schemes as shown
in Figure 2.1. And the model reference adaptive control
(direct adaptive control) which merges the identification
and the control synthesis into one scheme. Hence, the
plant parameter estimates are Implied and the controller
parameters are calculated directly as shown in Figure 2.2.
PROCESS
Figure 2.1 SelfTuning Regulator (STR)
8
Figure 2.2 Model Reference Adaptive Control (MRAC)
Koivo [9] used selftuning adaptive scheme to con
trol manipulators, which is composed of a system parameter
identifier and a controller based on the identified system
parameters. No results are given on the effect of payload
variations on the robustness of the controller. Horowitz
and Tomizuka [10] employed an explicit adaptation in which
the manipulator parameters are identified with a double
integrator reference model and then used for adaptive con
trol, compensating for the nonlinearity and decoupling of
the manipulator dynamics. This method requires
computations for both parameter identification and adap
tive control law.
9
An adaptive control scheme using reference model
was also proposed by Dubowsky and Desforges [11], where the
controller is to drive the manipulator to follow the refer
ence model as closely as possible. He employed the steep
est descent method in the adaptation mechanism, and the
stability analysis was done separately using a linearized
model. However, this design method cannot be applied in
general practice since stability analysis for each appli
cation is necessary but difficult to perform.
CHAPTER III
MANIPULATOR DYNAMIC EQUATIONS OF MOTION
3.1 Introduction
A first step in the development of a manipulator
control law, is the derivation of an analytical dynamic
model for the spatial manipulator elements. In this chap
ter, the structure of the dynamic equation of motion is
discussed. The Lagrange method is used in the derivation
of the nonlinear coupled differential equation which
describes the motion of the physical model. Following
this, the general equation of motion of a nlink manipula
tor is linearized with respect to an operating point (sys
tem equilibrium state). Finally, the equation of motion of
a twolink manipulator, which is used as the plant to be
controlled is derived.
3.2 Structure of the Manipulator Dynamic Equation
3.2.1 The General Dynamic Equation of the Manipulator
The general dynamic equations that describe the
motion of a manipulator are presented, along with some
notes on the inherent structure of these equations. The
manipulator is modeled as a set of n moving rigid bodies
11
connected in a serial chain as shown in Figure 3.1. The
bodies are joined together with rotary joints, and the sys
tem is assumed to have a planar motion (i.e., in the xy
plane) which results from the torques applied at various
joints in the plane. The vector equation of such a device
can be written in the following form
M(0)0 + Q(0,0)0 + G(0) = 7 (3.1)
where
M(0) is an n x n mass inertia matrix.
<2(0,0)0 is the centrifugal and coriolis torque vectors.
G(0) is the gravitational torque vector.
0(0 is the joint position vector, with 0 = [0j, 02, , 0]T.
0(0 is the joint velocity vector.
0(0 is the joint acceleration vector.
T is the joint torque vector.
The jfc^ element of Equation (3.1) can be written
as the sumofproducts form
*,= Xa/J,g)1(e,e)*Xc,1g;,(8) (3.2)
l t1 t1
where the ayi,6yi) and cn are parameters formed by products
of such physical quantities as link masses, link inertia
12
X
GRAVITATION
f
Figure 3.1 The General Coordinates
of a NLink Manipulator
13
tensor elements, lengths (i.e., distance to a center of
mass from a joint), and the gravitational acceleration con
stant The f n,qji are functions that embody the dynamic
structure of the motion geometry of the manipulator.
3.2.2 The Manipulator Ma6s Inertia Matrix, M(0)
Intuitively, it should be possible to write the
kinetic energy of a mechanism like a manipulator in qua
dratic form
/CÂ£ = eT/i:(0(O)0 (3.3)
where KE is the total kinetic energy of the dynamical sys
tem, and K(6(t)) a matrix that describes the mass distribu
tion of the manipulator as a function of the joint vector 0.
Each element of AT(0)must have the units of inertia (Kgm2).
Clearly, AT(0)must be positive definite so that the quanti
ties in Equation (3.3) are always positive and represent
energy. Furthermore, AT(0)must be a symmetric matrix.
AT(0)will be termed as the kinetic energy matrix of the
manipulator. Also, it should be clear that it is possible
to describe the potential energy of a manipulator by a sca
lar function of joint position only, say Â£(0) .
Let P E = Â£(0) (3.4)
Where PE is the total potential energy of the system.
14
The Lagrangian of the system is
L = KE PE (3.5)
The dynamic Equation (3.1) of the manipulator is then
derived by using Lagrange's method resulting in
d f 61 A 6L
dtUÂ§(Oj 60(0
(3.6)
Since the potential energy is only a function of joint
position 0(0 then (3.6) becomes
d f Â§KE A 6KE 5/>(0)
d^60(O J 60(0 + 60(0
Now, taking the derivative of Equation (3.3) with respect
to 0(0 gives
6KE
50(0
^(e^(9)e).M(9)e
(3.8)
and the derivative of Equation (3.8) w.r.t. time gives
d f 6KE ) .. .
T,lsm)mKlwKW*
(3.9)
Finally, then differentiating Equation (3.3) w.r.t. 0(0
gives
15
bKE
60
(3.10)
llftTW!)* ,
\2 60 /
Equation (3.10) can also be expressed in the following form
6KE 1 ^
60 2/ri
where e; is the j"1*1 unit vector
Substituting Equations (3.9) and (3.11) into Equation
(3.7) results in the expression
e,.0
60:
0(0
(3.11)
a:(0)0 + /((0)0
l
e>0
,B/C(0)
60/
i 6/>(0) ^
0 + = T
 60(0
(3.12)
Comparing Equations (3.1) and (3.12), it is found by equat
ing coefficients of 0 that
M(0) = K(0) (3.13)
That is, the manipulator mass inertia matrix is the kinetic
energy matrix.
A further important property of the manipulator
mass inertia matrix is that dependence on 0 appears in the
form of the sinusoidal functions sine and cosfne. That is,
in Equation (3.2) the dependence of 0 in the term is in
16
the form of sines and cosines of 0Â£. Since these sinusoidal
functions are bounded for all values of their arguments,
and since they appear only in the numerators of the ele
ments of M(0), it may be concluded that M(0) is bounded for
all 0.
In summary, the properties of M(0) can be stated
as follows:
1) . It is symmetric.
2) . It is positive definite and bounded above and below.
3) . The inverse exists and is positive definite and
bounded.
Property (2) can also be written as
am/
where / is the n x n identity matrix and the ordering is in
the sense of positive definite matrices with o
scalars.
3.2.3 The Centrifugal and Corioles Terms, Q(0,0)0
The centrifugal and coriolis terms can be written
in the following form
/0V,(0)0\
<2(0,0)0 =
(3.14)
\eV(0)0/
17
where the V ( 0 ) are n x n matrices. Clearly then
/eVi(0)\
eV2(e)
<2(0.e)
\0tK3(0)/
Furthermore, by equating terms in 0 in Equations (3
(3.12) there results
Q(0,0)0 = M(0)0^ Â£
6M(0)
60
J A
with the notation
M(0) = Af(0)
at 
n
I
/i
6M(0) .
=0,
60, '
and
then
0y = 0e
M (0)0(0 = X
j1
BM(0) Ta
T70e. 0
60; '
(3.15)
.1) and
(3.16)
(3.17)
(3.18)
substituting Equation (3.18) into (3.16) gives
18
R
<2(9,0)0= X
71
6iW(0) .
0s
bQj
1
2
6M(0)
6 0 y
8
since M(0) is symmetric, then
(3.19)
M(0) = AfT(0)
and
6M(0) f6M(0)V
607 l 60;" J
therefore
Q(0,0)0=X
71
6M(0) T
60y Gj
6M(0) T
60, 6/
>0
(3.20)
Again, dependence on 6 appears only in terms of
sine and cosine functions, so that Q(0,0)0has bounds that
are independent of 0, but increases quadratically with 0.
The properties &f Q(0,0)0may be summarized as:
1) . Quadratic in 0.
2) . Directly related to the time derivative of the mass
inertia matrix as shown in Equation (3.16) .
19
3.2.4 The Gravity Term, G(0)
Let G(0) represents the gravitational torque vec
tor where
sine and cosine functions in the numerators of its elements
so that the bound of G(0) is independent of 0 .
3.3 Linearization of the Dynamical Model
The linearized model of the nonlinear coupled
dynamic equations of the manipulator is considered in this
section. From the previous section, the dynamical equation
of motion for a nlink manipulator was shown to be
(3.21)
Again, dependence on 0 appears Only in terms of
M(0)0 + Q(0,0)0 + G(0) = r
(3.1)
A small perturbation of the torque vector results
in only small deviation from the operating point. Thus,
the perturbed equation of motion are
M(0o + 60)(0o + 60) + Q(0o + 60,0o + 60)(0o + 60)
+ G(0o + 60) = to + 6t
(3.22)
20
Using the Taylor series expansion to linearize
about an operating point q, and ignoring second and higher
order terms of 60 and 60,
A(6M(0)A T
M(0o + 60) = M(0o)+X^^J e^60 +
Therefore,
M(0o + 6e)(0o + 60) = M(0o)0o
v  Be] \ 60 + M(0 )60
I
t1 60j
(3.23)
e0(Â§.6.>+Â£(iÂ£) ^60
and
(?(0oh60.0o + 60)(0o + 60) = Q(0o.0o)0o + Q(0o.0o)60
and also
(3.24)
G(0o.60) = G(0o)+ e,T ) 60
IcT
.6 60,
(3.25)
And the perturbed equations of motion become
21
M(ejeo + Q(eo.ejeo + G(eo;) + M(eo)6e
(3.26)
Next consider the dynamical equation at the operating point
M(0o)0o>Q(0o.0o)0o^G(0o) = i/o (3.27)
and subtracting Equation (3.27) from Equation (3.26), the
linearized equation of motion is
Let
M(0o)60
60
bG T Jr bQ T ir^bMf. T
/ e, + / 0e, + / 0e,
.4160, .4160, 1 ,4i 50,
60 = 6i/
JQ
(3.28)
M(0o) = M(0),= Mo
(t^h
CSIgsO<=
srbM^ T\ ^
.I"?1),0'
22
where M0,Q0,Qi tG0,Gl ,and G2 and are constant n x n
matrices. Therefore, the linearized equation of motion is
A'fo60 + (Qo + Qi)60 + (Go + G1 + G2)60 = 6t/ (3.29)
Equation (3.29) gives an incremental linearized model of
the robot dynamics for small perturbations about the oper
ating point q .
Now let the operating point of the robot (equilib
rium point) be selected as
<2 = <Â§o = oÂ§o = o0o = o> (3.30)
The required torque U_o for this operating point is:
i/o = Mo0o + Qo0o + G(0o) = O (3.31)
Evaluating the matrices Q0, Q i, G x, andC2 at the operating
point results in
Q0QiGlG2o
and therefore, the linearized equation of motion is
Mo60 + Go60 = bU
(3.32)
23
3.4 The Dynamic Equation of a TwoLink Manipulator
3.4.1 Description of the Physical Model
A simple twodegreeoffreedom manipulator, Fig
ure 3.2, is considered in this section. The manipulator is
modeled as two rigid links connected by a frictionless
rotational joint. One end of the manipulator is attached
to the origin of the reference frame in cartesian space by
a rotational joint. The manipulator is assumed to have a
planar motion with gravity acting. Such a manipulator,
although quite simple, is subject to joint torques due to
inertial, centrifugal, coriolis and gravity effects.
Each link has four parameters (see Figure 3.3)
mass, m inertia about the center of mass, Iif the length
of the link, lif and the length of the lowered numbered
joint to the center of mass, r* The mass of joint 1 is rrij
and the mass of the payload is m.,.
3.4.2 The Kinetic and Potential Energy of a TwoLink
Manipulator
The kinetic energy of a rigid body in planar
motion is given by
(3.33)
24
Where m, is the mass, 11 is the moment of inertia about the
center of mass, u, is the velocity of the center of mass,
and ipf is the angular velocity for link i .
The kinetic energy of a point mass (as is the case
for m.j and m,) is given by
(3.34)
Figure 3.2 A Two DegreeofFreedom Manipulator
25
Figure 3.3 Parameters of the it*1 Link
Where m* is the point mass, and u, is the velocity of the
mass then
vf
xt+yt
(3.35)
and
(3.36)
From Figure 3.2, the cartesian coordinates
Xiand yi of the center of mass of link 1 is
= /^sinOt
(3.36)
26
y^rjCOsGj (3.37)
The velocity squared of the center of mass of link 1 is
2 .2 2
vi = x\ + y\
= (Tj0j cos0j)^ + (r ^ 0 j sinGj)2
The angular velocity of link 1 is 1^ = 01 and the moment of inertia is (3.38)
I 1 12 Therefore, the kinetic energy of link 1 is (3.39)
and the potential energy of link 1 becomes (3.40)
PE t = m^gr x cos! (3.41)
The cartesian coordinates x2 and y2 of the point
of mass m.j of joint 1 are
x2 = Z, sinGj (3.42)
CD CO O 0 o 1 II CJ >> (3.43)
(3.43)
27
The squared velocity is
= (Z! 9! cos0!)2 + (ZjGj sinGj)
u2=Z?02 (3.44)
Therefore the kinetic energy is
KE2 = ^mJlZi^ (3.45)
and the potential energy is
PE2 = mjgli cs01 (3.46)
The cartesian coordinates x3 and y3 of the center
of mass of link 2 are
x3 = Z1sin01 + r2sin(01 + 02)
y 3 = Â£! cos0! r2cos(0! + 02)
The velocity squared is
U\ = x\ + yl
=(Z,0, cos! + r20, cos(0j + 02) + r202cos(0i + 02))2
+ (Z10lsin01 + r201sin(01 + 02) + r202sin(01 + 02))2
y2= Z20f+ 2Z1r201(01 + 02)cos02 + r2(01 + 02)2 (3.49)
And the angular velocity of link 2 is
(3.47)
(3.48)
28
tt
ii>2 = 0i + 02 (3.50)
Resulting in a moment of inertia of
= (351)
Therefore, the kinetic energy of link 2 is
tfÂ£3 = !(m2u! + /2 oi>2)
= ^[m.2llQ* + 2012/^28 i Ci + 92)cos62
+ (m2r^/2)(01 + 02)2] (3.52)
and the potential energy of link 2 becomes
PE 3 = m2g llcosQl m2gr2 cos(0! + 02) (3.53)
The cartesian coordinates x4 and y4 of the payload
m2 are
x4 = Zj sin 0t + Z2sin(0 j + 02) (3.54)
y4 = Z i cos0j Z2cos(01 + 02) (3.55)
The velocity squared becomes
29
2 2 *2
N4 = *4 + y4
v\ [Â£!0jcosGj + /20icos(01 + 02) + Z202cos(0! + 02)]2
+ [ Z! 0! sin! + Z201sin(01 + 02)+Z202sin(0i + 02)]2
v24 = Z10f + 2Z1Z201(01 + 02)cos02+Z2(01 + 02)2 (3.56)
Therefore the kinetic energy of the payload is
KZr4 = m2[Z202 + 2Z1Z201(01 + 02) cos 9 2 +Z2(01 + 02)2] (3.57)
And the potential energy is
PE4 = mtg Z j cos! mtg l2cos(0! + 02) (3.58)
The total kinetic and potential energy of the
overall system is the sum of all the kinetic and potential
energies of the different components that make up the com
plete system (i.e.r links 1 and 2, joint 1 and the pay
load) .
It follows that the kinetic energy of the system
is
KE=^KEi (3.59)
i 1
/CF = [C1 + 2C2cos02 + C3]0f
+ [C2cos02 + C3]0102 + ~C302 (3.60)
and the potential energy of the system is
30
PE = Â£ PEi (3.61)
i 1
PE = C4cos0! CgCosCOj + 02) (3.62)
Where the C coefficients are
Cl=miri + ^l + (m/ + m2 + mi)^l
C2 ~ CtTi2r2 + mll2)ll
Cz = (m2rl +12 + mtll)
C4 = [mlrl + (mJ + m2 + ml)ll]g
C 5 = ( ^ 2 ^ 2 + rfll^2^9
3.4.3 The Dynamical Equation of Motion
of a TwoLink Manipulator
As mentioned earlier the kinetic energy of the
twolink manipulator can be expressed in quadratic form as
/fÂ£, = ieTm(0)0
Then placing Equation (3.60) in quadratic form
KE = (QlQz)
Ci + 2C2cos02 + C3 C2cos02 + C3
c2cos02 + c3
(3.63)
Where the matrix m(0) is
31
M(0) =
C! + 2C2 cos02 + C3 C2cos02 + C3
C,c os0
2 + ^3
(3.64)
From Equation (3.20) the centrifugal and coriolis torque
vector takes the form
~6M (0) / 0e . _ 1 rsMw ri
1 1 i CD (O 1 2 L 50/ J
0
Since M(0) is a function of only 02/ then
Q(0,0) =
"6M(0) j ~ 0e2 _ 1 r at6M(0)1
002 2 2 6 0 2
/'2C,sin02 C2sin02>\/,01v\
Q(e.e) 22 2 L (oi)
 C2sin 02
0
0.
(3.65)
lfO\t f2C2sin02 C2sin02'
2uJ(Gi 2\ C,sin0, 0
0 2C2^0! + ^ ]sin02
C201sin02
f
<2(0,0)0 =
2 A
C2 i + sin02 0!sin0
A
7
0
\^2^0i + tr Jsin02
2C2I 0! + Isin02
\
C2 .
0!Sin02 J
(3.66)
The gravitational torque vector G(0) is
/6 PE'
G(0) =
bPE
60
C4sin01 + C5sin(01 + 02)>
C5sin(01 + 02) j
(3.67)
32
Finally, substituting Equations (3.62), (3.64), and (3.65)
into Equation (3.1) the dynamical equation of motion of the
twolink manipulator results
+ 2C2cos02 + C3 C2cosQ 2 + C3'\f&1
C2 cos02 + C3
0,
0
c2\ 0i + ^ jsin02
2C2^01 + Jsin02
_^C20lSin02
C4sin0j + CgSinC! + 02)'
C5sin(01 + 02)
(3.68)
CHAPTER IV
ADAPTIVE CONTROL OF MANIPULATORS
4.1 Introduction
The manipulator control problem is the problem of
finding appropriate torques or forces that will drive the
associated actuators in order that the manipulator may fol
low a specified given trajectory. In this chapter, a
continuoustime adaptive modelfollowing control (AMFC)
algorithm proposed by Landau [12] is considered for a class
of nonlinear timevarying plants. The first property which
a satisfactory adaptive controller design should have is
stability of the overall system. The control scheme con
sidered in this study has the advantage that the global
asymptotic stability of the overall system can be assured
automatically using the hyperstability and positivity con
cepts, thus the need for a separate study of the system
stability is avoided. The design of the AMFC does not
require precision modeling nor the solution of the compli
cated dynamical equation of the system. This allows for
simpler control laws and a significant reduction in
computation time. Furthermore, the design method of the
adaptive controller considered is systematic, provides a
high speed of adaptation, and automatic system stability is
34
assured regardless of the variations in payload and spatial
configuration. However, there are certain limitations and
constraints which will be considered in a later section.
This chapter is organized as follows. In Section
4.2, the manipulator model is presented. In Section 4.3, a
decoupled linear timeinvariant reference model is pres
ented in order for the adaptive controller to force the
manipulator to follow the reference model as closely as
possible. In Section 4.4, perfect modelfollowing control
is discussed, which guarantee the existence of a solution
to a linear modelfollowing control. In Section 4.5 and
4.6, an adaptive modelfollowing control and adaptation
laws are presented respectively. The controller consists
of two parts, a linear part which is responsible for
achieving perfect modelfollowing, and an adaptive part
responsible for compensating and modifying any deviation
of the error of the overall system from that corresponding
to the linear modelfollowing.
4.2 The Plant Model
Consider the manipulator's dynamic equations of
motion which were discussed in Chapter III, Equations (3.1)
and (3.68) for a twolink manipulator.
M(0)0 + Q(0,0)0 + G(0) = T
35
These equations show that the manipulator dynamics are
highly nonlinear coupled functions of positions and velo
cities of the manipulator joints. The nonlinear coupled
characteristics of the manipulator dynamics cause the
design of any controller to be complex and computationally
demanding, thereby making the realtime implementation
quite difficult if not impossible. It is known that the
inertia and the gravity terms are dominant at slower oper
ating speeds, and at faster speeds the centrifugal and
coriolis effects become significant.
Consider the following linearized dynamics equa
tion of the manipulator,
This can be converted into state space representation with
2n dimensional state vector (n is the number of joints)
Mo60 + Go60 = 5t
*p = [*pi *p2]T = [6e 60]T
^p'ip^pCO + flp^/pCO
(4.1)
Where X_p is the plant state vector
and U_ is the plant input vector
where
A
(4.2)
36
and
B
p
(4.3)
In is the n x n identity matrix.
4.3 The Reference Model
A first step in the design of the adaptation mech
anism is the choice of the reference model. In order to
reduce the nonlinear coupled manipulator system to a
wellbehaved linear uncoupled system, a reference model
described for each degreeoffreedom by a linear second
order timeinvariant differential equation is chosen.
With this choice the desired time domain performance char
acteristics, such as risetime, overshoot, and damping can
be specified with a minimum number of parameters. The
linear, second order differential equation for each
degreeoffreedom is of the form
It follows that the reference model equation can be written
in state space representation as:
i = 1,2 n
(4.4)
+ BmrJO
Where X is the model state vector
__m
(4.5)
37
and r is the model input vector
The pair (Am, Bm) should be controllable, and Am a Hurwit
zian (i.e., the reference model is asymptotically stable).
0 /
zI 2%ucl
Where I is n x n identity matrix,
(4.6)
co2/ = diag[oo2, co2,, uj2]
2Â£u}/ = diag[2^! co !, 2Â£2co2,2Â£coJ,
2^ia)i>0
(4.7),
(4.8)
B
m
0
CO2/
nxn
(4.9)
The design objective is to have the plant state
Xp(f), follow closely the reference model state ^m(0* The
error between the states of the reference model and those
of the plant (the generalized state error vector) is
And the final objective is to constrain the error to
approach zero. That is,
lim g(0 = 0
Vt >0
(4.11)
38
4.4 Perfect ModelFollowing Control (PMF)
The design problem of linear modelfollowing con
trol (perfect modelfollowing) is to find the necessary
conditions such that the transfer matrices of the reference
model and that of the controlled plant be identical.
Another way of stating the design problem, is to find the
necessary conditions such that the generalized state error
vector (e = XmXp) and its derivative asymptotically tend
to be zero.
Consider the linear modelfollowing control sys
tem represented in Figure 4.1, which can be described by
the following equations:
The manipulator:
X(t) = ApXp(0 + BpU(0 (4.12)
The reference model:
xman = Amxm(n+Bmrao (4.13)
The manipulator control input:
UiO = KpXp(.t) + KuriO (4.14)
Where Kp is the manipulator feedback gain, Ku is the feed
forward gain, and Ap, Bp, Am, Bm, Kp, and Kur are constant
matrices of appropriate dimensions. The pairs (/1P,5P),
(/lm,Z?p)are stabilizable, furthermore is a stable
matrix.
39
Figure 4.1 Linear ModelFollowing Control System
40
In order to achieve perfect modelfollowing, it is
necessary to find the sufficient conditions for every set
of Am,Bm,Ap, and Bp assuring the existence of solutions
for Ku, and Kp so that the plant state matches (follows)
the model state. Sufficient algebraic conditions related
to the state space representation were established by Erz
berger [13].
Now, substituting Equation (4.14) into Equation
(4.12) , and subtracting Equation (4.12) from Equation
(4.13) there results
Â£(0 = ^m(0^p(0
= AmXm{t) + Bmrjt)~ ApXp(t) + BpKpXp(t)~ BpKur(t)
e(t) = Ame(0 + (Am Ap + BpKp)Xp(t)
+ (BmBpKu)r
The Erzberger conditions can be obtained by impos
ing e = X X =o and e = X X n = o.
(AmAp + BpKp)Xp{t) + (BmBpKu)rSt) = o (4.16)
Which implies
(AmAp) + BpKp = o (4.17)
Bm~ BpKu = o (4.18)
Equations (4.17) and (4.18) can be solved in terms of
Ku, and Kp if and only if
41
rankBp = rank[Bp, (AmAp)] = rank[Bp, Bm]
A class of solutions can be obtained by the use of
the Penrose pseudo inverse of B p, denoted by Bp.
Where
b; = (bJbp)_1bJ
Now, left multiplying Equations (4.17) and (4.18) by Bp
results in
Kp = B+piAmAp) (4.19)
Ku = B*pBm (4.20)
Substituting the values of Kp and Ku in Equations (4.17)
and (4.18) gives the following conditions
(IBpB+pXAmAp) = 0 (4.21)
(/ BpB*p)Bm = 0 (4.22)
The Bp matrix is usually a singular or a rectangular
matrix, and (BPBP */), then Equations (4.21) and (4.22)
imply that (/BPBP) is orthogonal to (/lm/lp) and Bm.
4.5 Adaptive ModelFollowing Control (AMFC)
In spite of the advantages of the linear model
following control systems discussed in Section (4.3), LMFC
does not overcome the difficulties related to the
42
uncertainty in or variations of the manipulator parame
ters. The analysis of the performance of LMFC designs lead
to the conclusion that in order to realize a
modelfollowing control system, which assures the desired
performance in the presence of parameter variations
(and/or poor knowledge about the parameter values), an
adaptive design must be employed. In this case an adaptive
modelfollowing control is used in which it is assumed that
a solution for perfect modelfollowing exists, for any val
ues of the manipulator parameters in a given range of pos
sible variations. The AMFC has the advantage of not
requiring an explicit identification of the manipulator
parameters, and the adaptation laws have an explicit form,
which of course does not require the realtime solution of
a set of linear or nonlinear equations.
Two basic implementations of AMFC are possible:
(1) Parameter adaptation Figure 4.2, in which the control
loop modifies the parameters of the controller.
(2) Signal synthesis adaptation Figure 4.3, where the
control loop modifies the signal applied to the manip
ulator.
Since the two configurations are equivalent, an AMFC system
with signalsynthesis adaptation will be considered.
The parallel AMFC system with signalsynthesis
adaptation Figure 4.3 is described by the following equa
tions .
The reference model:
43
Figure 4.2 Adaptive ModelFollowing Control
(AMFC) with Parameters Adaptation
Figure 4.3 Adaptive ModelFollowing Control
(AMFC) with Signal Synthesis Adaptation
44
Xm(0 = AmXm{0 + Bmr(0 (4.23)
The manipulator:
^p(0 = ^p^p(0 + 5pf/(0 (4.24)
The generalized state error:
Â§(0 = *m(0*p(0 (4.25)
The differential equation of the error e(f) is
+ fimr(0fipt/(0 (4.26)
f/(0 = ^1(0 + i/2(0 (4.27)
Where the manipulator input U_l (f) represents the linear
control which guarantees the perfect modelfollowing
Ul = KpXp(t) + Kur<;0
where Kp and Ku are constant matrices and the manipulator
input (f)is the contribution of the adaptive feedback
loop which is to be determined next using hyperstability
and positivity concepts.
The hypotheses for the design of the adaptive sys
tem are:
45
(1) . Am, Bm, Bp, and/Jp, belong to the class of matrices which
verify the "perfect modelfollowing" conditions,
Equations (4.19) and (4.20) but the values of
Ap and Bp are unknown.
(2) . The parameter values of Ap and Bp are assumed to be
timeinvariant during the adaptation process.
4.6 Adaptation Mechanism and Stability
The AMFC problem is to design an adaptive control
ler that guides the manipulator to follow the reference
model as closely as possible. There are three basic
approaches to the AMFC design problem. The first approach
uses local parametric optimization to derive an adaptive
law, but provides no guarantee on the stability of the
resulting adaptive system, thus requiring a separate sta
bility analysis which is often quite difficult if not
impossible to prove. The work done by Dubowsky [11] is an
example of this approach. The second approach is based on
the use of Lyapunov functions in order to design a stable
adaptive system. However, in practice it is difficult to
find a Lyapunov function with optimum performance. In the
third approach, Landau [14] uses hyperstability and posi
tivity concepts to obtain the largest possible family of
adaptation laws which assures the stability of the AMFC.
Then, an adaptive controller with the best performance can
be selected from these.
46
4.6.1 Equivalent Feedback Representation of AMFC
Theorem 4.1 (Popov Hyperstability) :
Consider a nonlinear time varying feedback system
Figure 4.4, containing a linear timeinvariant block in the
feedforward path described by the state equations:
K(O = 0Â£(O
(4.28)
(4.29)
O
W1
LINEAR
TIMEINVARIANT
W
NONUNEAR
TIMEVARYING
Figure 4.4 Equivalent Feedback System
47
Where e is a n dimentional vector and V, W are m dimentional
vectors, where the pair (A, B) is completely controllable
and the pair (D, A) is completely observable, and the non
linear timevarying block in the feedback path is described
by
= (4.30)
w = f(y (T),t)
0<:X
(4.31)
Where W denotes a functional dependence between W and the
values of V in the interval o
For asymptotic hyperstability of the system, two
conditions must be satisfied
!)
Tl(o.*i)= ftlvT(x)W(x)dx>y
J o ~
Vt, >o
(4.32)
where Yo is a finite positive constant.
2). The transfer matrix of the linear block described by
Equations (3.28) and (4.29)
Z(s) = D(sl A)'1 B
(4.33)
is strictly positive real
48
Lemma 4.1
KalmanYakubovitchPopov's Lemma
The transfer matrix of the system described by
Equations (4.28) and (4.29) is strictly positive if and
only if
Where P is a positive definite matrix solution of the Lya
punov equation
PA + A7 P = H
where H is a symmetric positive definite matrix.
Now, an equivalent feedback representation of the
adaptive modelfollowing control is needed, in order to use
the hyper stability and positivity concepts. Recalling
Equation (4.26)
d = btp
(4.34)
2(t) = Ame(t) + (AmAp)Xp(t)
+ Bmr(t)BpU(t)
(4.26)
The control law U(t') is
(4.35)
Ux(P) = KpXp(t) + Kur_(t)
(4.36)
49
i/2(t) = AA:p(e>0^p(0 + AA:u(ef0r(0
(4.37)
Where Kp and Ku are constant matrices satisfying Equations
(4.19) and (4.20), and the two time varying matrices gener
ated by the adaptation mechanism Atfp(e,f) and AKu(e,t), are
used to assure that the generalized state error e(Â£) goes to
zero asymptotically. Note that the generalized error is
used here as a driving force for the adaptation mechanism.
Combining Equation (3.35) into Equation (3.26),
under the assumption that Erzberger's conditions Equations
(4.21) and (4.22) hold, the result is
e(0^e(0 + [(^fl,^p) + *PKPSpAtfp(e,i)]Â£p(0
+ [Bm~ BpAKu(e,t) BpK
= Ame(0 + Bp[B+p(Am Ap) + KpAKp(e,t)]Xp(t)
+ Bp[B+pBmAKu(e,t)Ku]r(t)
e(t) = Ame(t)~ Bp{[AKp(e,t)B;(Am Ap)~ Kp]Xp(0
+ [AKu(e,t) + KuB+pBm]r(
(4.38)
Let
W_(t) = 1/^0 = (AKp(Q,0 Bp(Am Ap)~ Kp)Xp(t)
+ (AKu(e,t) + Ku BpBm)r(t)
(4.39)
Now Equation (4.38) can be written as
e(t) = Ame(t)BpW(t)
(4.40)
50
Let the output of the linear feedforward block be written
as
Â£(O"0e(O (4.41)
The equivalent feedback representation of the
adaptive modelfollowing control system shown in Figure
4.5, is described by the following equations
e(0 = ^me(0 + 5p^i(0 (4.42)
u(t)0e(O (4.42)
1/(0 =1^(0 = (AKp(g.oc0)Xp(0
+ (AA:u(GI0C1)r(0 (4.43)
Where
C0 = B+p
Ci = B*pBmKu (4.45)
The matrices A/fp(e,0 and A/Cu(e,0 were introduced by Lan
dau [12] as a function of v.
AKp(e,0 = AKp(y,0= P
J 0
AKu(e,0 = AKu(u,0= f
J 0
^(y.t.Odt + iji^y.O
ip1(y,T,0d'c + 1p2(^'0
(4.46)
(4.47)
51
Figure 4.5 Equivalent Feedback Representation of AMFC
(Proportional and Integral Adaptation)
52
In these equations/ the integral and matrices <(>!
and tp! represent a nonlinear timevarying integral rela
tion between the two matrices AKp(v,t) and AKu(v,t) and the
values of K(t) for o
adaptation mechanism. This integral relation assures zero
steadystate error of the manipulator position and veloc
ity. Matrices 2 and ip2 denote a nonlinear timevarying
proportional relation between the two matrices AKp^v.f) and
AKu(v,t) and the values of V(t'). These proportional terms
are introduced at the beginning of the process to acceler
ate the reduction of the error e(Or and they will vanish at
the end of the adaptation process. (That is, the
adaptation process ends when e = o and v = o and therefore
^(O'O0 and i>2(o,0 = o for allf) .
The adaptation law described by Equations (4.46)
and (4.47) can be actually designated by
Where Ff M, G, and N are positive definite matrices, and F
and M are positive semidefinite matrices with appropriate
dimensions. This type of adaptation is called proportional
and integral (PI) adaptation.
(4.48)
(4.49)
0
53
It can be seen from Equation (4.48), (4.49) and
Figure 3.5, that the implementation of the adaptation laws
requires only summers, multipliers, and integrators.
These elements are associated in order to realize canonical
structures of the forms multipliers > PI; amplifier >
multipliers. These basic components of the nonlinear part
of the adaptation mechanism generates the actuator signal
u2(t) from the signal where the signal K(f) is obtained
from the generalized error e(0 through the gain matrix D.
4.6.2 Hyperstability of the AMFC
For the equivalent feedback representation of the
AMFC system described by Equations (4.13), (4.14) and
(4.15) to be asymptotically hyperstable; the proportional
and integral (PI) adaptation law must satisfy Popov's inte
gral inequality Equation (4.5). Furthermore, the transfer
function of the linear feedforward block must be strictly
positive real.
From Equations (4.42) and (4.43), the transfer
matrix is
Z(s) = D(sI Am)~1 Bp (4,50)
A suitable (linear compensator) matrix D can be chosen as
DBlP
(4.51)
54
Where P is a positive definite matrix solution of the Lya
punov equation
PAm*AJmPH (4.52)
Where H is a symmetric positive definite matrix
Then Equation (4.50) becomes,
Z(.s)BlP(.slAmy'B, (4.53)
Which is strictly positive definite, and hence condition
two of Popov's hyperstability is satisfied.
Substituting Equation (4.15) into Popov's inte
gral inequality
S(o.ii)Â£v\x)W(x)dx>y20
(yl is positive finite constant)
tj > o
S(o,f,)fov(x)(AkpC0)Xpdx
+ J V(x)(AfcllC1)rdT (4.54)
J o
Furthermore, substituting Equations (4.48) and
(4.49) into Equation (4.54) there results
55
rdx
?(o.t,)fvT[fgtFV(GXp)TdT;C0 Xpdx
+ fV f* MV(N r)T dx C;
J 0 L Jo
+ fx0'(L'I^DUlc^p)d'c
+ f+1(KTMK)(rTNr)dT>Y;
J o ~ ~ ~ ~
Where
(4.55)
C0 = B*p(AmAp) + Kp
and
C\ = B*pBm Ku
The last two integrals are greater or equal to
zero because F, M, G, and N are at least positive semidefi
nite matrices. For Popov's inequality to hold, it is suf
ficient that each of the first two integrals in inequality
be greater than a negative finite constant as shown in
Equation (4.27) Using the properties of the positive def
inite matrices, the first integral can be written ass
/,=Â£vxidx~co
X pdt
(4.56)
Where
56
F = F\Fl\G = G\G,\v = Flv,^p = G,Xp^F\y C^Gj'C
But the integral Ii, can be expressed also by:
f'e.xjrc.x^dxc^dt
t1 j.l J 0 \J 0 J
1 ml n
XX
^ il 1
f'a,XpldxC,] CCuy
i m l n
>5lI(c)2 C4.S7)
^ i 1 j 1
The second integral in Equation (4.54) also verifies such
an inequality and therefore the Popov integral inequality
is satisfied.
CHAPTER V
SIMULATION RESULTS
5.1 Introduction
In this chapter several simulation cases are given
to illustrate the application of the adaptive model
following control algorithm discussed in Chapter IV. In
order to demonstrate the capability of this technique, it
is applied to the control of a singlelink manipulator and
a twolink manipulator.
5.2 SingleLink Manipulator Case
The equation of motion for the singlelink manipu
lator can be derived from equations (3.40), (3.41), (3.45)
and (3.46) and is given by
where
1 is the length of the link
r is the distance from the joint to the center of mass of
the link
mi is the payload
g is the gravitational constant
(5.1)
58
T(t) is the control input torque
Next, the differential equation is written in
state space representation, and it is assumed that the
angular position and velocity of the manipulator are mea
surable. The state of the manipulator is defined as
*P(0 = [e(0 e(0]T (5.2)
The state space representation of the manipulator is
Xp(n = ApX(t) + BpU(t)
where
^p =
0 1
aCg0 0
0
m:1
M0 = [ mr2 + j^rnl2 + m,l2
G0 = (mr + mll)g
The decoupling property can be achieved by a suit
able choice of the reference model. The parameters of the
model are chosen as Â£ = 1.0, caj = 4, so that a critically
damped response is obtained. With this choice of the ref
erence model, the existence of a solution to the perfect
model matching (linear modelfollowing) is assured.
The reference model is described by
59
Xu(t) = AMXM(t) + BMr(t)
Where
Au
0 1
 16 8
*m =
0
16
The linear modelfollowing control (perfect
modelfollowing) is designed for a nominal value of 5 Kg
(payload), and the reference model and the plant satisfy
the Erzerberger's conditions.
The control law U(t) is given by
f/(t) = KpÂ£p(0 + *ur(0
f FV{GXp)Tdx + FV(GXp)T
. J 0
rh _
f MV'iN r)T dx + MV_(N r)T
_ J 0
Â£pCO
r(0
Simulations were obtained for the following val
ues of the plant parameters, M = 12 Kg, 1 = 0.5 meter and r
=0.25 meter.
The gain matrices are
N = 1
and
0
0.1
60
5.2.1 Results
Three simulation studies were conducted for the
case of a singlelink manipulator, the manipulator is
required to move a 10 Kg payload from the joint angle coor
dinate of (0 radians) to a final position of' (0.7854
radians).
In the first case, only the linear controller is
involved (i.e., there is no adaptation, F = F = M = Ai=0).
This case is considered to compare the performance without
the adaptive feedback loop in the overall control. Figures
(5.1), (5.2), (5.3) and (5.4) show the position and veloc
ity responses, and the evolution of the errors for the lin
ear modelfollowing control system designed for a nominal
value of 5 Kg payload. The position and velocity responses
are very poor and tracking is not accomplished.
In the second case, the positive definite matrix H
is chosen as H = diag [5 1], this value of H will result in
more weight on the position following than the velocity
following. Namely, accuracy in positioning is more empha
sized than that of velocity. Depending upon the applica
tion tasks, relative weights between the accuracies in
position and velocity can be selected. The integral and
proportional gains were set as follows, F = M = 400 and F =
M = 40. The position and velocity tracking improved in
comparison to the first case, which means that the adaptive
61
feedback loop has a great effect on the tracking and there
fore improves the system response. However, there is some
error in the position and velocity responses, see Figures
(5.3) and (5.4).
In third case, the influence of increasing the
adaptive gains is investigated. The integral and propor
tional gains, F, M, F, M, were all increased by a factor of
5. This resulted in a better position and velocity
tracking, where the position and velocity responses are
virtually indistinguishable from the desired position and
velocity responses given by the reference model, see Fig
ures (5.5) and (5.6). Increasing the adaptive gains
resulted in a faster reduction of this error and a faster
speed of adaptation, see Figures (5.3) and (5.4).
Position in Radians
+++: model solid: plant
Figure 5.1 Position Response (Case 1)
o\
to
Velocity in Rad/Sec
1.2
+++: model solid: plant
Time in Seconds
Figure 5.2 Velocity Response (Case 1)
at
Co
Error in Rad/Sec
+ + + : Casel ,: Case2 solid: Case3
Time in Seconds
Figure 5.3 Position Tracking Error
Error in Radians
+ + + : Case! : Case2 solid: Case3
Figure 5.4 Velocity Tracking Error
cn
ui
Position in Radians
: Model solid: plant
Time in Seconds
Figure 5.5 Position Response (Case 3)
cn
at
Velocity in Rad/Sec
: Model solid: plant
Time in Seconds
Figure 5.6 Velocity Response (Case 3)
cn
*]
68
5.3 A two link manipulator case
The equation of motion of the twolink manipulator
is derived in Chapter III and is given by
Mo0(f) + Goe(o = T(O
The state space representation of the manipulator is given
by
where
*P[*PI *P21T
^p = /1p^pC0 + 5pÂ£/(0
0 1
m;1g0 o
The matrices Mq and G0 are
M0 =
G0 =
Cx + 2C2 + C3 C2 + C
C 2 + C 3
c4 + c5 c5
c.
5 Cj
Where the parameters Ci (i = 1.. .5) are defined in Chapter
III.
Again the reference model is chosen such that a
critically damped response is obtained, the parameters are
= %2= 1*0, uo t = uo 2= 5. Therefore, the reference model
is described by
where
0 0 1 0
0 0 0 1
25 0  10 0
0 25 0  10
0 0
0 0
25 0
0 25
With this choice of the reference model, the existence of a
solution to the perfect modelfollowing is assured.
The control law U(t) is given by
U(t) = KpXp(t) + Kur(t)
rh _
I FV(GXp)TdX+ FV(GXp)T
rh _
j MV(Nr)Tdx +MV(Nr)T
J o
xpct)
r(0
Simulations were obtained for the following val
ues of the plant parameters 1^ = I2 = 1.0 meter, r^ = r2 =
0.5 meter, mj = 0.0 Kg, = M2 = 6.0 Kg, and the weighting
matrices 6 is 4 x 4 unit matrix and N is 2 x 2 unit matrix.
5.3.1 Results
A computer simulation study is performed to inves
tigate the quality of performance of the adaptive control
algorithm proposed. A twolink manipulator is considered
here, and required to move from an initial joint angle
70
coordinate of (0.0, 0.0) radians to a final position of
(0.7854, 0.7854) radians. For the first three cases a pay
load of 0.0 Kg is considered, and analysis of the perform
ance of the position and velocity responses and the
evolution of the errors for link 1 and link 2 are
considered. For cases 4,5 and 6, a 7.0 Kg payload is con
sidered and the performance of the position and velocity
responses are analyzed.
Case 1: No adaptation is considered in this case
(i.e., the adaptive gains F = M = /7=jW=0), which means
only the linear modelfollowing controller designed for a
minimal value of 2.0 Kg payload is involved. The perform
ance of the positions and velocities of link 1 and link 2
are very poor and tracking is not accomplished, see Figures
(5.7) , (5.8), (5.13) and (5.14).
Case 2: In this case the positive definite matrix
H is chosen as H = diag [10 10 2 2], this choice of H
results in more weight on the position following than the
velocity following. The integral and proportional gains of
the adaptive feedback loop are, F = M = 1000 and F = M 
4.95. The positions and velocities tracking for link 1 and
link 2 have improved in comparison to the first case. How
ever, the errors are not reduced fast enough. See Figures
(5.8) , (5.11), (5.12), (5.13) and (5.14) for the position
and velocity responses and the evolution of the errors of
link 2.
71
Case 3: Here, the second case is considered again
with the adaptive gains increased by a factor of 5. This
resulted in a better position and velocity tracking, faster
reduction of the errors, and faster adaptation speed. It
is important to note that the ratio between the values of
the proportional gain and the integral gain has an impor
tant influence on the speed of reduction of the model plant
state error. Also the simulation study showed that a
higher ratio leads to a high speed reduction of the error,
but in counterpart, parameter adaptation speed is slower.
The gains are limited by the saturations existing in the
adaptation loop, see Figures (5.10), (5.11), (5.12),
(5.13) and (5.14).
Case 4: A 7.0 Kg payload is considered in this
case, and again no adaptation is involved (F = M = F = M).
Figures (5.15), (5.16), (5.17), (5.18), (5.19) and (5.20)
show that the linear controller alone is not capable of
producing good performance in terms of position and veloc
ity responses.
Case 5: In this case the positive definite matrix
H is chosen as H = diag [6 6 6 6 ]. The integral and the
proportional gains of the adaptive feedback loop are, F = M
= 4000 and F = M = 80. Figures (5.17), (5.18), (5.19) and
(5.20) show that the performance of the position and veloc
ity responses have improved in comparison to case 4. How
ever, the speed of reduction of the state error and the
speed of adaptation are slow.
72
Case 6: Here, the adaptive gains are set as F = M
= 11,800 and F = M = 236 with the H matrix chosen as H =
diag [10 10 2 2], by this choice position tracking is
more emphasized than velocity tracking. Figures (5.17),
(5.18), (5.19) and (5.20) show that the performance of
position response has improved, and the speed of reduction
of the state error and the speed of adaptation also have
improved.
As a whole this simulation exhibits that the pro
posed control method has a potential for high performance
with a very simple structure.
Position in Radians
+ + + : Model solid: Link 1 ,: Link 2
Time in Seconds
Figure 5.7 Position Response (ml = 0.0 Kg, Case 1)
Velocity in Rad/Sec
+++: Model solid: Link 1 ,: Link 2
Time in Seconds
Figure 5.8 Velocity Response (ml = 0.0 Kg, Case 1)
*>>
Position in Radians
solid: Model
Case 2
Time in Seconds
Figure 5.9 Joint One Position Response (ml = 0.0 Kg, Case 2)
j
Ul
Position in Radians
solid: Model ,: Case 3
Time in Seconds
Figure 5.10 Joint One Position Response (ml = 0.0 Kg, Case 3)
Ok
Error in Radians
0.02
(LINK1) ,:Case2 solid: Case3
Figure 5.11 Joint One Position Error Tracking (ml = 0.0 Kg)
vj
Error in Radians
0
(LINK2) ,:Case2 solid: Case3
Time in Seconds
Figure 5.12 Joint Two Position Error Tracking (ml = 0.0 Kg)
GO
Error in Rad/Sec
(LINK1) , + + + : Casel ,: Case2 solid: Case3
Time in Seconds
Figure 5.13 Joint One Velocity Error Tracking (ml = 0.0 Kg)
VO
Error in Rad/Sec
1
(LINK2) , + ++: Casel ,: Case2 solid: Case3
Time in Seconds
Figure 5.14 Joint Two Velocity Error Tracking (ml = 0.0 Kg)
oo
o
Position in Radians
1
solid:Model
:Linkl
.:Link2
Time in Seconds
Figure 5.15 Position Response (ml = 7 Kg, Case 4)
00
Velocity in Rad/Sec
solid:Model ,:Linkl ..:Link2
Figure 5.16 Velocity Response (ml = 7 Kg, Case 4)
oo
to
Error in Radians
+ + + :Case4 ,:Case5 solid:Case6
Figure 5.17 Joint One Position Error Tracking (ml = 7 Kg)
oo
to
Error in Radians
0.8
+ + + :Case4 ,:Case5 solid:Case6
Time in Second
Figure 5.18 Joint Two Position Error Tracking (ml = 7 Kg)
oo
Error in Rad/Sec
(LINK1) :Case5 solid:Case6
Figure 5.19 Joint One Velocity Error Tracking (mi = 7 Kg)
oo
Ul
Error in Rad/Sec
(LINK2):Case5 solid:Case6
Figure 5.20 Joint Two Velocity Error Tracking (m^ = 7 Kg)
co
cri
CHAPTER VI
CONCLUSIONS
The design of a manipulator control system based
on the adaptive modelfollowing control (AMFC) concept has
been presented in this study. The design of the adaptation
mechanism is based on the use of the Popov hyperstability
theory. The controller is continuous and consists of two
parts, the first part is linear in the state and the refer
ence input, where the linear gains are determined using the
known linear portion of the system dynamics. The second
part of the controller is produced by an additional adap
tive feedback loop, where the adaptive gains are determined
by the use of the hyperstability concept.
The design has focused on the following three
important features. First, unlike most other existing
methods, changes in the manipulator payload can be handled
effectively without requiring the payload to be specified
explicitly in the design of the manipulator control system.
In other words, the control system is designed to provide
any desired performance (i.e., accuracy in position and
velocity) equally well for a wide range of payload.
88
Second, the presented design is conceptually and
computationally simpler than most other existing methods
without loss of performance, the implementation of the
adaptation laws does not require the real time solution of
a set of linear or nonlinear equations. Therefore, the aid
of a computer is required only in the design stage for the
computation of the parameters of the adaptation mechanism.
Third, the stability of the manipulator control
ler is guaranteed, leaving control designers free from the
stability problem which is often very difficult to analyze,
if not impossible. Assurance of the manipulator controller
stability cannot be overemphasized, especially in view of
the potential danger of human injury and loss of expensive
equipment as a result of the controller instability.
BIBLIOGRAPHY
1. Landau, I.D. Adaptive Control The Model Reference
Approach. New York: Marcel Dekker, 1979.
2. Paul, R. "The Mathematics of Computer Controlled
Manipulator." Proc. Joint Automat. Contr. Conf. I, pp.
124131, 1977.
3. Whitney, D.E. "Resolved Motion Rate Control for Manip
ulators and Human Prostheses." IEEE Trans. ManMachine
Systemsf June 1969, Vol. 10, No. 2.
4. Whitney, D.E. "The Mathematics of Coordinated Control
of Prosthetic Arms and Manipulators." ASME J. DSMC.
24, Dec. 1972.
5. Luh, J.Y.S., Walker, M.W. and Paul, R.P.C. "Resolved
Acceleration Control of Mechanical Manipulators."
IEEE Trans. Automat. Contr., June 1980, Vol. AC25, No.
3.
6 Paul, R. Robot Manipulators: Mathematicsr Programming
and Control. Cambridge, Mass.: MIT. Press, 1981.
7. Luh, J.Y.S., Walker, M.W. and Paul, R.P.C. "Online
Computational Scheme for Mechanical Manipulators."
ASME J. DSMC 102. June 1980, pp. 6976.
8. Hollerbach, J.M. "A Recursive Lagrangian Formulation
of Manipulator Dynamics and a Comparative Study of
Dynamics Formulation Complexity." IEEE Trans. Syst.
Man. Cybern.f 1980, Vol. II, pp. 730736.
9. Koivo, A. J. and Guo, T.H. "Control of Robotic Manipu
lator with Adaptive Controller." Proc. IEEE Conf. on
Decision and Control, pp. 271276, 1981.
10. Horwitz, R. and Tomizuka, M. "An Adaptive Control
Scheme for Mechanical Manipulators Compensation of
Nonlinearity and Decoupling Control." ASME Journal of
DSMC. June 1986, Vol. 108, pp. 127135.
11. Dubowsky, S. and Desforges, D.T. "The Application of
Model Reference Adaptive Control to Robot Manipula
tors." ASME Journal of DSMCf Sept. 1979, Vol. 101, pp.
193200.
90
12. Landau, I.D. and Courtiol, B. "Design of Multivariable
Adaptive ModelFollowing Control Systems. Automa
tics, 1974, Vol. 10 pp. 483494.
13. Erzberger, H. "Analysis and Design of ModelFollowing
Systems by State Space Techniques." Proc. of Jacc,
1968, pp. 572581.
14. Landau, I.D. "A Hyperstability Criterion for Model
Reference Adaptive Control Systems." IEEE Trans. Aut.
Control AC14. 1969, pp. 552555.

Full Text 
PAGE 1
ADAPTIVE MODELFOLLOWING CONTROL OF A ROBOTIC MANIPULATOR by Usama Ahmed Gheblawi B.S., University of Colorado, 1985 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Master of Science Department of Electrical Engineering and Computer Science 1989
PAGE 2
This thesis for the Master of Science degree by Usama Ahmed Gheblawi has been approved for the Department of Electrical Engineering and Computer Science by Edward T. Wall Jan Bialasiewicz Arun Maj undar '3{) J 1 'KCf
PAGE 3
ii Gheblawi Ahmed, Usama (M. 5., Electrical Engineering) Adaptive ModelFollowing Control of a Robotic Manipulator Thesis directed by Professor Edward T. Wall Abstract The objective of the manipulator con. trol design is to find the appropriate torques that will drive the manipulator to follow a prespecified trajectory. This thesis presents the use of adaptive modelfollowing control (AMFC) technique for the control of a robotic manipulator that has wide variations in its payload. The design procedure is simple, effective, and does not require accurate modeling of the dynamic system. The problem of perfect asymptotic adaptation is interpreted as a stability problem. Through the use of hyperstability in conjunction with the properties of the positive dynamic systems, the largest family of adaptation laws assuring the stability of the overall adaptive system is obtained, then the most suitable adaptation law for a specific application is chosen.
PAGE 4
iii ACKNOWLEDGEMENTS This study would not have been possible without the cooperation and support of many people to whom I will always be grateful. I especially wish to express my sincere gratitude to Professor Edward T. Wall for his cheerful support and guidance during the preparation of this thesis. I would like also to thank Dr. Jan Bialasiewicz for his advice, encouragement, and serving on my committee. Finally, I wish to thank my family for their patience, support, and inspiration.
PAGE 5
CONTENTS Chapter I INTRODUCTION ......... 1 II. REVIEW OF MANIPULATOR CONTROL METHODS ..... 4 2 1 Introduction ............................. 4 2.2 Nonadaptive Control Methods ........ s 2.3 Adaptive Control Methods ............ 6 III. MANIPULATOR DYNAMIC EQUATIONS OF MOTION ... lO 3. 1 Introduction ............................ 10 3.2 Structure of the Manipulator Dynamic Equation .............................. 10 3.2.1 The General Dynamic Equation of the Manipulator ......................... 10 3.2.2 The Manipulator Mass Inertia Matrix . 13 3.2.3 The Centrifugal and Corioles Terms .... 16 3.2.4 The Gravity Term ............ 19 3.3 Linearization of the Dynamical Model .. 19 3.4 The Dynamic Equation of a TwoLink Manipulator ........................... 2 3 3.4.1 Description of the Physical Model ..... 23 3.4.2 The Kinetic and Potential Energy of a TwoLink Manipulator .............. 23 3.4.3 The Dynamical Equation of Motion of a TwoLink Manipulator ... 30
PAGE 6
IV. ADAPTIVE CONTROL OF MANIPULATORS ........... 33 4 1 Introduction ....................... 3 3 4.2 The Plant Model ...................... 34 4.3 The Reference Model .................. 36 4.4 Perfect ModelFollowing Control (PMF) . 38 4.5 Adaptive ModelFollowing Control ( .Am'C) 41 4.6 Adaptation Mechanism and Stability .... 45 4.6.1 Equivalent Feedback Representation of .Am'C 4 6 4.6.2 Hyperstability of the AMFC .......... 53 V. SIMULATION RESULTS .......................... 57 5 1 Introduction .......................... 57 5.2 SingleLink Manipulator Case .......... 57 5.2.1 Results ................................ 60 5.3 TwoLink Manipulator Case .............. 68 5 3 1 Results ............................... 6 9 VI. CONCLUSIONS ................................ 87 BIBLIOGRAPHY ....................... 8 9 v
PAGE 7
FIGURES Figure 2.1 SelfTuning Regulator (STR) ............... 7 2.2 Model Reference Adaptive Control (MRAC) a 3.1 The General Coordinates of a NLink Manipulator ................................. 12 3.2 A Two DegreeofFreedom Manipulator ........ 24 3.3 Parameters of the ith Link .................. 25 4.1 Linear ModelFollowing Control System (Perfect ModelFollowing Control) ...... 39 4.2 Adaptive ModelFollowing Control (AMFC) with Parameter Adaptation ............... 43 4.3 Adaptive ModelFollowing Control (AMFC) with Signal Synthesis Adaptation .............. 43 4.4 Equivalent Feedback System ............... 46 4.5 Equivalent Feedback Representation of Adaptive ModelFollowing Control .......... 51 5.1 Position Response (Case 1) .................... 62 5.2 Velocity Response (Case 1) .................... 63 5.3 Position Tracking Error ....................... 64 5.4 Velocity Tracking Error ....................... 65 5.5 Position Response (Case 3) .................... 66 5.6 Velocity Response (Case 3) .......... 67 5.7 Position Response (ml = 0.0 Kg, Case 1) ....... 7 3 5.8 Velocity Response (ml = o.o Kg, Case 1) ..... 7 4
PAGE 8
vii 5.9 Joint One Position Response (ml = 0.0 Kg, Case 2).l .... 75 5.10 Joint One Position Response (ml = 0.0 Kg, Case 3) ............ 76 5.11 Joint One Position Error Tracking ( mL = 0 0 Kg) ................. 7 7 5.12 Joint Two Position Error Tracking ( ml = 0 0 Kg) . . . . . . . . . . . 7 8 5.13 Joint One Velocity Error Tracking ( ml = 0 0 Kg) . 7 9 5.14 Joint Two Velocity Error Tracking ( ml = 0 0 Kg) . . . . . . . . .. 8 0 5.15 Position Response (mL = 7 Kg, Case 4) ....... 81 5.16 Velocity Response (ml = 7 Kg, Case 4) ..... 82 5.17 Joint One Position Error Tracking (ml = 7 Kg) .......................... 8 3 5.18 Joint Two Position Error Tracking ( mL = 7 Kg) ......... 8 4 5.19 Joint One Velocity Error Tracking ( ml = 7 Kg) . 8 5 5.20 Joint Two Velocity Error Tracking ( ml = 7 Kg) .. 8 6
PAGE 9
CHAPTER I INTRODUCTION Manipulators have been used extensively in hostile environments, such as in the nuclear industries, deep undersea exploration and maintenance operations, space, and industrial automation. In most of these applications operating speed and performance indices are relatively low, and relatively simple control systems have proven adequate. With increased demand on manipulators will come the need for improved manipulator controllers, that can drive the manipulator more effectively and efficiently than the conventional controllers (i.e. in terms of operating speed, use of energy, capability of dealing with various tasks, accuracy, etc) The contemporary controllers used by robot manufacturers are largely based on simple conventional feedback control techniques that are unable to deal with complex, uncertain dynamics of the manipulator and its interaction with other machinery. On the other hand, most of the proposed advanced controllers are either based on unrealistic approximation, or on requirements that make it computationally demanding to implement in realtime.
PAGE 10
2 In this study, an adaptive control scheme is considered for controlling the robot manipulator. This approach is based on adaptive modelfollowing control systems (AMFC) technique [ 1], where the design of the adaptation mechanism is based on the use of the Popov hyperstability theory, which assures the asymptotic stability of the system. The objective of the controller is to cause the manipulator to follow a given desired performance specified by a reference model. The control law is composed of two parts The first part of the controller is linear in the system state and the reference input, which assures a perfect modelfollowing control. The second part of the control is produced by an addi tiona! adaptive feedback loop, where the adaptive gains are determined using the hyperstability theory. The adaptive feedback loop will compensate for parameter variations, and uncertainties in the dynamical equations of the manipulator. The novelty of this control strategy is that the adaptive controller requires neither accurate modeling of manipulator dynamics, nor solving the dynamic equations (which is known to be difficult and computationally demanding) And the asymptotic stability of the overall system can be assured automatically using the hyperstability and positivity concepts. Thus, eliminating the need for a sep
PAGE 11
arate stability analysis. Furthermore, through the use of implicit adaptive control, the need for identification of manipulator parameters is eliminated. 3
PAGE 12
CHAPTER II REVIEW OF MANIPULATOR CONTROL METHODS 2 .1 Introduction Manipulators are required to perform two tasks, namely moving an object (payload) from an initial to a final position along a prescribed path, and exerting forces or torques on the manipulated object. In this study, we will consider only the first task, which is the problem of moving objects through some prescribed path. There are a variety of algorithms available for manipulator control. These algorithms usually assume that the control structure of the robot has been divided into two levels. The first level is called path or trajectory planning, and the second level is called path tracking. The trajectory planner provides the time history of the desired positions and velocities, where path tracking is the process of making the robot's actual positions and velocities match the desired values provided by the trajectory planner. The control schemes can be divided into two different approaches, nonadaptive and adaptive control.
PAGE 13
5 2. 2 Nonadaptive Control Methods Presently there are three wellknown methods that provide position control of manipulator and are all kinematically oriented. In resolved motion position control (RMPC) [2], the desired joint positions are determined directly by solving the inverse kinematic equations, and then the related joint velocities and accelerations are calculated from the positions. Whitney [ 3, 4] proposed a tracking scheme called the resolved motion rate control (RMRC), where the linear/angular velocities needed to maintain the desired endeffector position and orientation are mapped into joint rates by the inverse Jacobian matrix, and then the joint accelerations are computed from the velocities. In resolved motion acceleration control (RMAC) [5], the corrective cartesian accelerations are calculated and resolved into joint accelerations, and the joint positions and rates are measured. All of these motion control schemes resolve the control into generalized joint coordinates denoted by q, cj, and q. Given q, q, and q, joint torques of the manipulator dynamics were obtained by the Lagrangian formulation [ 6] and by the NewtonEuler formulation [ 7 ] Although recently more efficient algorithms for computation of dynamics have been proposed [8,7], these methods still require a considerable amount of computations during the motion. Furthermore, these methods may include modeling error (discrepancies
PAGE 14
between the dynamic model and the actual manipulator dynamics), thus requiring additional controllers to compensate for this error. Also, note that even if accurate modeling is possible, the dynamic model has to be a function of the task being performed (i.e. payload and positions of the endeffector). This fact results in either very complex dynamic models or inaccurate but simplified models; neither of the two is desirable. 2. 3 Adaptive Control Methods 6 Traditionally, control systems have been designed based on a good understanding of the system to be controlled. When knowledge of the system is limited the relative modern issues of robust control, adaptive control, and learning control become important. One way to attempt to deal with poor knowledge of parameters in a control scheme is through techniques that are generally called adaptive control. The central problem in the synthesis of adaptive controllers is to prove rigorously that the resulting overall system is asymptotically stable. Adaptive control is closely related to the problem of system identification, and generally an adaptive controller can be viewed as being composed of two parts; the first part is the identification process which identifies the parameters of the plant itself, or the parameters that appear in the controller of the plant. The
PAGE 15
7 second part is the control law which implements a control law that is in some way a function of the parameters being identified. Adaptive control strategies take on many forms 1 and some methodologies have become established for designing adaptive control systems. Among these methodologies 1 the two most widely described are the selftuning regulator (indirect adaptive control) scheme 1 which contains separate identification and control synthesis schemes as shown in Figure 2 .1. And the model reference adaptive control (direct adaptive control) which merges the identification and the control synthesis into one scheme. Hence 1 the plant parameter estimates are implied and the contJ7oller parameters are calculated directly as shown in Figure 2. 2. PRCCESS REGULATOR ... PARAMETERS RECURSIVE PARAMETER DESIGN .. ESTIMATOR I I I I I I I I r(t) I I .. Yp I I u I REGULATOR PLANT I .. I I I I I I I I ADJUSTABLE SYSTEM I Figure.2.1 SelfTuning Regulator (STR)
PAGE 16
8 REFERENCE xm MODEL e u xp PLANT SIGNAL +1 PARAMETER SYNllfESIS I ADAPTATION ADAPTATION 1 I OAPTATIO ME CHAN IS Figure 2. 2 Model Reference Adaptive Control (MRAC) Koivo [ 91 used selftuning adaptive scheme to control manipulators, which is composed of a system parameter identifier and a controller based on the identified system parameters. No results are given on the effect of payload variations on the robustness of the controller. Horowitz and Tomizuka [ 101 employed an explicit adaptation in which the manipulator parameters are identified with a double integrator reference model and then used for adaptive control, compensating for the nonlinearity and decoupling of the manipulator dynamics This method requires computations for both parameter identification and adaptive control law.
PAGE 17
9 An adaptive control scheme using reference model was also proposed by Dubowsky and Des forges ( 11], where the controller is to drive the manipulator to follow the reference model as closely as possible. He employed the steepest descent method in the adaptation mechanism, and the stability analysis was done separately using a linearized model. However, this design method cannot be applied in general practice since stability analysis for each application is necessary but difficult to perform .
PAGE 18
CHAPTER III MANIPULATOR DYNAMIC EQUATIONS OF MOTION 3 1 Introduction A first step in the development of a manipulator control law, is the derivation of an analytical dynamic model for the spatial manipulator elements. In this chapter, the structure of the dynamic equation of motion is discussed. The Lagrange method is used in the derivation of the nonlinear coupled differential equation which describes the motion of the physical model. Following this, the general equation of motion of a nlink manipulator is linearized with respect to an operating point ( system equilibrium state) Finally, the equation of motion of a twolink manipulator, which is used as the plant to be controlled is derived. 3 2 Structure of the Manipulator Dynamic Equation 3 2 .1 The General Dynamic Equation of the Manipulator The general dynamic equations that describe the motion of a manipulator are presented, along with some notes on the inherent structure of these equations. The manipulator is modeled as a set of n moving rigid bodies
PAGE 19
11 connected in a serial chain as shown in Figure 3. 1. The bodies are joined together with rotary joints, and the system is assumed to have a planar motion (i.e. in the xy plane) which results from the torques applied at various joints in the plane. The vector equation of such a device can be written in the following form (3.1) where is ann x n mass inertia matrix. is the centrifugal and coriolis torque vectors. is the gravitational torque vector. is the joint position vector, with is the joint velocity vector. is the joint acceleration vector. Tis the joint torque vector. The jth element of Equation ( 3.1) can be written as the sumofproducts form (3.2) where the a i', b i&, and c i& are parameters formed by products of such physical quantities as link masses, link inertia
PAGE 20
y GRAVITATION Figure 3 .1 The General Coordinates of a NLink Manipulator 12 X
PAGE 21
13 tensor elements, lengths (i.e. distance to a center of mass from a joint), and the gravitational acceleration constant. The fit, q Jt are functions that embody the dynamic structure of the motion geometry of the manipulator. 3.2.2 The Manipulator Mass Inertia Matrix, Intuitively, it should be possible to write the kinetic energy of a mechanism like a manipulator in quadratic form KE =!aT K(S(t))S 2(3.3) where K E is the total kinetic energy of the dynamical systern, and a matrix that describes the mass distribution of the manipulator as a function of the joint Each. element of must have the units of inertia (Kgm2). Clearly, must be positive definite so that the quantities in Equation ( 3. 3) are always positive and represent energy. Furthermore, must be a symmetric matrix. will be termed as the kinetic energy matrix of the manipulator. Also, it should be clear that it is possible to describe the potential energy of a manipulator by a scalar function of joint position only, say P(S) Let (3.4) Where P E is the total potential energy of the system.
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14 The Lagrangian of the system is L=KEPE (3.5) The dynamic Equation ( 3. 1) of the manipulator is then derived by using Lagrange's method resulting in d ( fiL ) fiL dt = T (3.6) Since the potential energy is only a function of joint position e(t) then ( 3. 6) becomes d ( fiK E) fiK E fiP(S) dt + = T (3.7) Now, taking the derivative of Equation (3.3) with respect to gives fiK E fi ( 1 T 0 ) 0 0=0 e K(8)8 = M(8)8 2(3.8) and the derivative of Equation (3.8) w.r.t. time gives d ( fiK E) 00 0 0 dt (3.9) Finally, then differentiating Equation (3.3) w.r.t. gives
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oKE oe 1. ToK(S). e e 2oel 1. ToK(e) e e 2oen 15 (3.10) Equation ( 3 .10) can also be expressed in the following fonn oKE 1f"'[ ToK(S)J. =L e.e S(t) 2j.1 l_ oej (3.11) where e i is the j th unit vector Substituting Equations (3.9) and (3.11) into Equation (3.7) results in the expression (3.12) Comparing Equations ( 3.1) and ( 3 .12) 1 it is found by equating coefficients of e that = (3.13) That is 1 the manipulator mass inertia matrix is the kinetic energy matrix. A further important property of the manipulator mass inertia matrix is that dependence one appears in the fonn of the sinusoidal functions sine and cosi,ne. That is 1 in Equation ( 3. 2) the dependence of 8 in the term f ii is in
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16 the form of sines and cosines of e 1 Since these sinusoidal functions are bounded for all values of their arguments 1 and since they appear only in the numerators of the elements of it may be concluded that is bounded for all e. In summary, the properties of M(6) can be stated as follows: 1) It is symmetric. 2). It is positive definite and bounded above and below. 3). The inverse exists and is positive definite and bounded. Property ( 2) can also be written as a.m/ n M(6) n whereIn is the n x n identity matrix and the ordering is in the sense of positive definite matrices with a< a.m <13m as scalars. 3.2.3 The Centrifugal and Corioles Terms, The centrifugal and coriolis terms can be written in the following form Q(6,8)8= (3.14)
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where the V f(S) are n x n matrices. Clearly then Q(S,S)= STV1(8) erv 2CS) 17 (3.15) Furthermore I by equating terms in a in Equations ( 3 1) and (3.12) there results with the notation and then dt . T a.= ae. I I .. fBM(S).T. M = /..i 1 I I substituting Equation (3.18) into (3.16) gives (3.16) (3.17) (3.18)
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18 . Ln [oM(8). TJ. Q(8, 8)8 = 8e i 8 1r o8 j 1 [ L e.8 8 2} J 68j (3.19) since is synunetric, then and therefore . {[OM(8). TJ 1 [6M(8). TJT}. Q(8,8)8= L 8e. fie. 8 jl o8j 1 2 B8j 1 (3.20) Again, dependence on a appears only in terms of sine and cosine functions, so that has bounds that are independent of 8, but increases quadratically with e. The properties bf be summarized as: 1) Quadratic in e. 2) Directly related to the time derivative of the mass inertia matrix as shown in Equation ( 3.16).
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19 3.2 .4 The Gravity Term, G(8) Let G(8) represents the gravitational torque veetor where G(S) = 6P(8) 68 (3.21) Again, dependence one appears only in tenns of sine and cosine functions in the numerators of its elements so that the bound of G(8) is independent of 8 3 3 Linearization of the Dynamical Model The linearized model of the nonlinear coupled dynamic equations of the manipulator is considered in this section. From the previous section, the dynamical equation of motion for a nlink manipulator was shown to be (3.1) A small perturbation of the torque vector results in only small deviation from the operating point. Thus, the perturbed equation of motion are M(8 +68)(9 +6S)+Q(8 +68,8 +6S)(S +oS) 0 0 0 0 0 (3.22)
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20 Using the Taylor series expansion to linearize about an operating point q, and ignoring second and higher order tenns of and n (6M(9)) T M(e +69)=M(9 )+ ei69+ 0 0 t. 69tl q Therefore, M(e +68)(8 +oS)=M(8 )S 0 0 0 0 ( f oM (8) .. T) .. + t. + M 1 I q (3.23) and Q(90+oe.a +fia)ca +6a)=Q(e .a )a +Q(e .a )6a 0 0 0 0 0 0 0 (3.24) and also ( n oG ) cce 69) = cce ) + "ei fie 0 0 t. fie. 1 I q (3.25) And the perturbed equations of motion become
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21 M(S )S +Q(S ,9 )S +G(S )+M(S )oS 0 0 0 0 0 0 0 (3.26) Next consider the dynamical equation at the operating point M(S )S +Q(S ,e )S +G(S )=U 0 0 0 0 0 0 0 (3.27) and subtracting Equation ( 3 2 7) from Equation ( 3 2 6) the linearized equation of motion is Let = lq= Mo lq=Qo (3.28)
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22 matrices. Therefore, the linearized equation of motion is (3.29) Equation ( 3. 29) gives an incremental linearized model of the robot dynamics for small perturbations about the operating point q Now let the operating point of the robot ( equilibrium point) be selected as Q={a =o,a =o,a =o} (3.30) 0 0 0 The required torque U o for this operating point is: U =M 8 +Q 8 +G(a )=o (3.31) 0 O_o O_o 0 point results in and therefore, the linearized equation of motion is (3.32)
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23 3 4 Dynamic Equation of a Manipulator 3. 4 .1 Description of the Physical Model A simple twodegreeoffreedom manipulator, Figure 3. 2, is considered in this section. The manipulator is modeled as two rigid links connected by a frictionless rotational joint. One end of the manipulator is attached to the origin of the reference frame in cartesian space by a rotational joint. The manipulator is assumed to have a planar motion with gravity acting. Such a manipulator, although quite simple, is subject to joint torques due to inertial, centrifugal, coriolis gravity effects. Each link has four parameters (see Figure 3. 3) mass, mi, inertia about the center of mass, I ir the length of the link, li, and the length of the lowered numbered joint to the center of mass, r, The mass of joint 1 is mi and the mass of the payload is m1 3. 4. 2 Kinetic and Potential Energy of a Manipulator The kinetic energy of a rigid body in planar motion is given by (3.33)
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24 Where mi is the mass, I, is the moment of inertia about the center of mass, u, is the velocity of the center of mass, and 1lit is the angular velocity for link i The kinetic energy of a point mass (as is the case for m1 and m1 ) is given by 1 2 KE=mv. L 2 L L (x2, y2) x3,y3) mj. '\.9 .,. 2 m2 ".,. '\. .,. '.,. (x4,y4) ml Figure 3 2 A Two Manipulator (3.34) X
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25 joint mi, I joint i+ 1 Figure 3. 3 Parameters of the i th Link Where mi is the point mass, and ui is the velocity of the mass then and From FigUre 3.2, the cartesian coordinates x 1 and y 1 of the center of mass of link 1 is (3.35) (3.36) (3.36)
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26 y 1 =r 1 cos a 1 (3.37) The velocity squared of the center of mass of link 1 is v2=x+y 1 1 1 v2=r2a 1 1 1 (3.38) The angular velocity of link 1 is and the moment of inertia is (3.39) Therefore, the kinetic energy of link 1 is (3.40) and the potential energy of link 1 becomes (3.41) The cartesian coordinates x2 and y2 of the point of mass mi of joint 1 are (3.42) y 2 = L1 cos a 1 (3.43)
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27 The squared velocity is v2=l2a'2 2 1 1 (3.44) Therefore the kinetic energy is (3.45) and the potential energy is (3.46) The cartesian coordinates x3 and y3 of the center of mass of link 2 are (3.47) (3.48) The velocity squared is (3.49) And the angular velocity of link 2 is
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28 (3.50) Resulting in a moment of inertia of (3.51) Therefore, the kinetic energy of link 2 is (3.52) and the potential energy of link 2 becomes (3.53) The cartesian coordinates x 4 and y 4 of the payload m2are (3.54) (3.55) The velocity squared becomes
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29 (3.56) Therefore the kinetic energy of the payload is (3.57) And the potential energy is (3.58) The total kinetic and potential energy of the overall system is the sum of all the kinetic and potential energies of the different components that make up the complete system (i.e. 1 links 1 and 2 1 joint 1 and the payload). It follows that the kinetic energy of the system is 4 KE= L KEi i1 . l 2 + ( C 2 COS 9 2 + C 3] 9 1 9 2 + 2 C 3 9 2 and the potential energy of the system is (3.59) (3.60)
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Where the C coefficients are C4=[m1r1 +(m1+m2+m1)l1]g C 5 = (m2r2 + m1l2)g 3. 4. 3 The Dynamical Equation of Motion of a TwoLink Manipulator 30 (3.61) (3.62) As mentioned earlier the kinetic energy of the twolink manipulator can be expressed in quadratic form as K E =.!.a 1 m(S)S 2Then placing Equation ( 3. 60) in quadratic form [ C1 +2C2cos82+C3 c2cos82+c3 Where the matrix is (3.63)
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31 (3.64) From Equation ( 3. 20) the centrifugal and coriolis torque vector takes the form .. L2 {[oM(e). r] l[oM(e). r]r} Q(e,e)e= ee1 e ee1 e 11 o e 1 2 o 1 Since M(e) is a function of only 921 then (: (2C 2 sin B2 Q(e,e)c e 2 stn 2 1 (0) . (2C 2 sin 92 21 cele2) c e 2 stn 2 The gravitational torque vector is G(e)=oPE = oe (3.65) (3.66) (3.67)
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32 Finally, substituting Equations (3.62), (3.64), and (3.65) into Equation ( 3. 1) the dynamical equation of motion of the twolink manipulator results + ( C1 +2C2cos92+C3 c 2cos92 + c3 C 2cos92 + C 1) c3 a2 (3.68)
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CHAPTER IV ADAPTIVE CONTROL OF MANIPULATORS 4 .1 Introduction The manipulator control problem is the problem of finding appropriate torques or forces that will drive the associated actuators in order that the manipulator may follow a specified given trajectory. In this chapter, a continuoustime adaptive modelfollowing control (AMFC) algorithm proposed by Landau [ 12] is considered for a class of nonlinear timevarying plants. The first property which a satisfactory adaptive controller design should have is stability of the overall system. The control scheme considered in this study has the advantage that the global asymptotic stability of the overall system can be assured automatically using the hyperstability and positivity concepts, thus the need for a separate study of the system stability is avoided. The design of the AMFC does not require precision modeling nor the solution of the complicated dynamical equation of the system. This allows for simpler control laws and a significant reduction in computation time. Furthermore, the design method of the adaptive controller considered is systematic, provides a high speed of adaptation, and automatic system stability is
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34 assured regardless of the variations in payload and spatial configuration. However, there are certain limitations and constraints which will be considered in a later section. This chapter is organized as follows. In Section 4. 2, the manipulator model is presented. In Section 4. 3, a decoupled linear timeinvariant reference model is presented in order for the adaptive controller to force the manipulator to follow the reference model as closely as possible. In Section 4. 4, perfect modelfollowing control is discussed, which guarantee the existence of a solution to a linear modelfollowing control. In Section 4. 5 and 4.6, an adaptive modelfollowing control and laws are presented respectively. The controller consists of two parts, a linear part which is responsible for achieving perfect modelfollowing, and an adaptive part responsible for compensating and modifying any deviation of the error of the overall system from that corresponding to the linear modelfollowing. 4.2 The Plant Model Consider the manipulator's dynamic equations of motion which were discussed in Chapter III, Equations ( 3 .1) and (3.68) for a twolink manipulator.
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35 These equations show that the manipulator dynamics are highly nonlinear coupled functions of positions and velocities of the manipulator joints. The nonlinear coupled characteristics of the manipulator dynamics cause the design of any controller to be complex and computationally demanding, thereby making the realtime implementation quite difficult if not impossible. It is known that the inertia and the gravity terms are dominant at slower operating speeds, and at faster speeds the centrifugal and coriolis effects become significant. Consider the following linearized dynamics equation of the manipulator, This can be converted into state space representation with 2n dimensional state vector ( n is the number of joints) Where X is the plant state vector P and U is the plant input vector P where ( 4.1) ( 4.2)
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36 and (4.3) In is then x n identity matrix. 4.3 The Reference Model A first step in the design of the adaptation mechanism is the choice of the reference model. In order to reduce the nonlinear coupled manipulator system to a wellbehaved linear uncoupled system, a reference model described for each degreeoffreedom by a linear second order timeinvariant differential equation is chosen. With this choice the desired time domain performance characteristics, such as risetime, overshoot, and damping can be specified with a minimum number of parameters. The linear, second order differential equation for each degreeoffreedom is of the form i=l,2,,n ( 4.4) It follows that the reference model equation can be written in state space representation as: X =Am X (t) + B mr(t) m m (4.5) Where X is the model state vector _m
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37 and c_ is the model input vector The pair (Am, B m) should be controllable, and Am a Hurwi tzian ( i o eo the reference model is asymptotically stable) (4.6) Where I is n x n identity matrix, 2 2 2 2] w /=dtag[w1,w2,,wn (4.7), (4.8) (4.9) The design objective is to have the plant state XP(t), follow closely the reference model state Xm(t)o The error between the states of the reference model and those of the plant (the generalized state error vector) is e(t)=X (t)X (t) m P (4.10) And the final objective is to constrain the error to approach zero. That is, lim e(t) = 0 Vt 0 ( 4.11) 1+ CD
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38 4.4 Perfect ModelFollowing Control (PMF) The design problem of linear modelfollowing control (perfect modelfollowing) is to find the necessary conditions such that the transfer matrices of the reference model and that of the controlled plant be identical Another way of stating the design problem, is to find the necessary conditions such that the generalized state error vector (e =X mX p) and its derivative asymptotically tend to be zero. Consider the linear modelfollowing control systern represented in Figure 4 .1, which can be described by the following equations: The manipulator: (4.12) The reference model: (4.13) The manipulator control input: (4.14) Where K P is the manipulator feedback gain, K u is the feedforward gain, and A P 1 B P 1 Am 1 B m 1 K P 1 and K ut are constant matrices of appropriate dimensions. The pairs (API B P) I (Am, B p) are stabilizable, furthermore Am is a stable matrix.
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r Ku + REFERENCE MODEL Xm X Kp Figure 4.1 Linear ModelFollowing Control System 39
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40 In order to achieve perfect modelfollowing, it is necessary to find the sufficient conditions for every set of Am,Bm,Ap, and BP assuring the existence of solutions for K u, and K P so that the plant state matches (follows) the model state. Sufficient algebraic conditions related to the state space representation were established by Erzberger [13]. Now, substituting Equation (4.14) into Equation (4.12), andsubtracting Equation (4.12) from Equation (4.13) there results e(t) = x (t)x (t) m P =Am X m (t) + B mC (t)A px p (t) + B PK px p (t)B PK uC (t) = +(AmAP + B PK p)X P(t) +(BmBpKu)C(t) (4.15) The Erzberger conditions can be obtained by imposing e = X X = o and e = X X = o. _m P _m P (4.16) Which implies ( 4.17) BmBpKu=o (4.18) Equations ( 4.17) and ( 4.18) can be solved in terms of Ku, and KP if and only if
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41 rankB P = rank[B P, (AmA p)] = rank[B P, B mJ A class of solutions can be obtained by the use of the Penrose pseudo inverse of B P' denoted by B;. Where Now, left multiplying Equations ( 4. 17) and ( 4 .18) by B; results in ( 4.19) ( 4.20) Substituting the values of Kp and Ku in Equations ( 4 17) and (4.18) gives the following conditions (4.21) ( 4.22) The B P matrix is usually a singular or a rectangular matrix, and (B PB; "F !), then Equations ( 4. 21) and ( 4. 22) imply that (1BPB;) is orthogonal to (AmAp) and Bm. 4. 5 Adaptive ModelFollowing Control (AMFC) In spite of the advantages of the linear modelfollowing control systems discussed in Section ( 4. 3) LMFC does not overcome the difficulties related to the
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42 uncertainty in or variations of the manipulator parameters. The analysis of the perfonnance of LMFC designs lead to the conclusion that in order to realize a modelfollowing control system, which assures the desired perfonnance in the presence of parameter variations (and/or poor knowledge about the parameter values), an adaptive design must be employed. In this case an adaptive modelfollowing control is used in which it is assumed that a solution for perfect modelfollowing exists, for any values of the manipulator parameters in a given range of possible variations. The AMFC has the advantage of not requiring an explicit identification of the manipulator parameters, and the adaptation laws have an explicit fonn, which of course does not require the realtime solution of a set of linear or nonlinear equations. Two basic implementations of AMFC are possible: ( 1) Parameter adaptation Figure 4. 2, in which the control loop modifies the parameters of the controller. ( 2) Signal synthesis adaptation Figure 4. 3, where the control loop modifies the signal applied to the manipulator. Since the two configurations are equivalent, an .AMFC system with signalsynthesis adaptation will be considered. The parallel AMFC system with signalsynthesis adaptation Figure 4. 3 is described by the following equations. The reference model:
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e(t) r(t Fiqure 4. 2 Adaptive ModelFollowing Control (AMFC) with Parameters Adaptation r Reference Xm 11oe1e1 Aelap1a11on llicllanlam FiquJ::e 4. 3 Adaptive ModelFollowing Control (AMFC) with Signal Synthesis_Adaptation 43
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The manipulator: The generalized state error: e(t)=X (t)X (t) m P The differential equation of the error is = +(AmA p)X /t) + B mCCt)B pU(t) U(t)=U1(t)+U2(t) 44 ( 4.23) (4.24) ( 4.25) ( 4.26) (4.27) Where the manipulator input U 1 ( t) represents the linear control which guarantees the perfect modelfollowing where K P and K u are constant matrices and the manipulator input U 2(t) is the contribution of the adaptive feedback loop which is to be determined next using hyperstability and positivity concepts. The hypotheses for the design of the adaptive systern are:
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45 (1). Am,Bm,Bp,andBp, belong to the class of matrices which verify the "perfect modelfollowing" conditions, Equations (4.19) and (4.20) but the values of Ap and B Pare unknown. (2). The parameter values of Ap and Bp are assumed to be timeinvariant during the adaptation process. 4. 6 Adaptation Mechanism and Stability The AMFC problem is to design an adaptive controller that guides the manipulator to follow the reference model as closely as possible. There are three basic approaches to the AMFC design problem. The first approach uses local parametric optimization to derive an adaptive law, but provides no guarantee on the stability of the resulting adaptive system, thus requiring a separate stability analysis which is often quite difficult if not impossible to prove. The work done by Dubowsky [11] is an example of this approach. The second approach is based on the use of Lyapunov functions in order to design a stable adaptive system. However, in practice it is difficult to find a Lyapunov function with optimum performance. In the third approach, Landau [14] uses hyperstability and positivity concepts to obtain the largest possible family of adaptation laws which assures the stability of the AMFC. Then, an adaptive controller with the best performance can be selected from these.
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46 4. 6.1 Equivalent Feedback Representation of AMFC 4 .1 (Popov Byperstability) : Consider a nonlinear time varyinq feedback system Fiqure 4. 4, containinq a timeinvariant block in the feedforward path described by the state eqilations: W1 w = + BW(t) V(t) = .. LINEAR TIMEINVARIANT NONUNEAR TIME;. VARYING v ... Figure 4. 4 Equivalent Feedback System ( 4.28) ( 4.29)
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47 Where is a n dimentional vector and V, W are m dimentional vectors, where the pair (A, B) is completely controllable and the_pair (D, A) is completely observable, and the nonlinear timevarying block in the feedback path is described by W=W1 W=j(V(r),t) ( 4.30) ( 4.31) Where W denotes a functional dependence between W and the values of V in the interval o 1: t. For asymptotic hyperstability of the system, two conditions must be satisfied 1) TJ(O,t 1)= v t 1 0 where is a finite positive constant. ( 4.32) 2) The transfer matrix of the linear block described by Equations (3.28) and (4.29) Z(s) = D(s/A)1 B ( 4.33) is strictly positive real
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48 Lemma 4.1 KalmanYakubovitchPopov's Lemma The transfer matrix of the system described by Equations ( 4. 28) and ( 4. 29) is strictly positive if and only if ( 4.34) Where P is a positive definite matrix solution of the Lyapunov equation PA+A7P=H where H is a symmetric positive definite matrix. Now, an equivalent feedback representation of the adaptive modelfollowing control is needed, in order to use the hyperstability and positivity concepts. Recalling Equation ( 4. 26) e(t) = + (AmA p)X P (t) +BmC(t)BPU(t) The control law U ( t) is U(t)=U1(t)+U2(t) U1(t)=KPX P(t)+Kur:_(t) (4.26) ( 4.35) ( 4.36)
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49 ( 4.37) Where K P and K u are constant matrices satisfying Equations ( 4 .19) and ( 4. 20), and the two time varying matrices generated by the adaptation mechanism b.K t) and b.K t), are used to assure that the generalized state error goes to zero asymptotically. Note that the generalized error is used here as a driving force for the adaptation mechanism. Combining Equation (3.35) into Equation (3.26), under the assumption that Erzberger' s conditions Equations (4.21) and (4.22) hold, the result is Let e(t) = +[(AmAp) + B PK PB Pb.K t)]X P(t) + [B mB pf::!.K t)B pK u]c_(t) = + B p[B:(AmA p) + K P I::!.K p(e, t)]X P(t) +BP[B;BmI::!.Ku(e,t)Ku]c_(t) e(t) = B P {[I::!.K P (e, t)B; (AmA p)K p]X P (t) W (t) =W 1 (t) = (!::!.K p(e, t)A p)K p)X P (t) +(I::!.Ku(e, t)+ K uB;Bm)c_(t) Now Equation ( 4. 38) can be written as ( 4.38) ( 4.39) ( 4.40)
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50 Let the output of the linear feedforward block be written as V (t) a (4.41) The equivalent feedback representation of the adaptive modelfollowing control system shown in Figure 4.5, is described by the following equations Where e(t) = + B P W 1 (t) = W (t) =W 1 (t) = (b.K p(e, t)Co) X P (t) +(b.Ku(e,t)C1)C(t) Co=B;CAmAp)+KP Cl=B;BmKu ( 4.42) ( 4.42) ( 4.43) ( 4.44) ( 4.45) The matrices b. K P ( e, t) and b. K u ( e, t) were introduced by Landau [ 12 ] as a function of v. it
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w1 .. Bp .. .. D v + l,./" + Am Linear part Nonlinear part .r .. N co , ./' F + '( + ,.........._ ,. L.!..( f+ F c1 X ... M ... .. + "... + M f+ Xp .. G Figure 4. 5 Equivalent Feedback Representation of AMFC (Proportional and Integral Adaptation) 51
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52 In these equations 1 the integral and matrices <1> 1 and 1jJ 1 represent a nonlinear timevarying integral relation between the two matrices and and the values of V(t) t1 which reflects the memory of the adaptation mechanism. This integral relation. assures zero steadystate error of the manipulator position and velocity. Matrices <1>2 and ljJ2 denote a nonlinear timevarying proportional relation between the two matrices and and the values of V(t). These proportional terms are introduced at the beginning of the process to accelerate the reduction of the error e(t)1 and they will vanish at the end of the adaptation process. (That is 1 the adaptation process ends when e = o and v = o and therefore 2(o,t)=o and ljJ2(o,t)=o for alit). The adaptation law described by Equations ( 4. 46) and (4.47) can be actually designated by p(ll_. t) =it Fll_(GX p) T dt + Fll_(GX p) T ( 4.48) ( 4.49) Where F 1 M1 G1 and N are positive definite matrices 1 and F and Mare positive semidefinite matrices with appropriate dimensions. This type of adaptation is called proportional and integral (PI) adaptation.
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53 It can be seen from Equation ( 4. 48), ( 4. 49) and Figure 3. 5, that the implementation of the laws requires only summers, multipliers, and integrators. These elements are associated in order to realize canonical structures of the form: multipliers > PI; amplifier > multipliers. These basic components of the nonlinear part of the adaptation mechanism generates the actuator signal from the signal V(t), where the signal V(t) is obtained from the generalized error ( t) through the gain matrix D. 4.6.2 Byperstability of the AMFC For the equivalent feedback representation of the AMFC system described by Equations ( 4. 13) ( 4. 14) and ( 4 15) to be asymptotically hyperstable; the proportional and integral (PI) adaptation law must satisfy Popov's integral inequality Equation ( 4. 5). Furthermore, the transfer function of the linear feedforward block must be strictly positive real. From Equations ( 4. 42) and ( 4. 43), the transfer matrix is (4,50) A suitable (linear compensator) matrix D can be chosen as (4.51)
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54 Where P is a positive definite matrix solution of the Lyapunov equation PA +ArP=H m m ( 4.52) Where H is a symmetric positive definite matrix Then Equation ( 4. 50) becomes, ( 4.53) Which is strictly positive definite, and hence condition two of Popov's hyperstability is satisfied. Substituting Equation (4.15) into Popov's integral inequality is positive finite constant) ( 4.54) Furthermore, substituting Equations (4.48) and (4.49) into Equation (4.54) there results
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itvr[itFV(GX P)Td1:C0JX Pd,; + i"t vT[i"t M v CN T dec 2 ]cdr + J:1 (VT FV)( x:cx P)dc Where and 55 ( 4.55) The last two integrals are greater or equal to zero because F, M, G, and N are at least positive semidefinite matrices. For Popov's inequality to hold, it is sufficient that each of the first two integrals in inequality be greater than a negative finite constant as shown in Equation ( 4. 27). Using the properties of the positive definite matrices, the first integral can be written as: T T f t [ft J / 1 = o 1 v o vX Pd,;Co X Pdt ( 4.56) Where
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56 But the integral I1, can be expressed also by: ( 4.57) The second integral in Equation ( 4. 54) also verifies such an inequality and therefore the Popov integral inequality is satisfied.
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CHAPTER V SIMULATION RESULTS 5 .1 Introduction In this chapter several simulation cases are given to illustrate the application of the adaptive modelfollowing control algorithm discussed in Chapter IV. In order to demonstrate the capability of this technique, it is applied to the control of a singlelink manipulator and a twolink manipulator. 5. 2 SingleLink Manipulator Case The equation of motion for the singlelink manipulator can be derived from equations (3.40), (3.41), (3.45) and ( 3. 46) and is given by (5.1) where 1 is the length of the link r is the distance from the joint to the center of mass of the link m1 is the payload g is the gravitational constant
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58 ,; ( t) is the control input torque Next, the differential equation is written in state space representation, and it is assumed that the angular position and velocity of the manipulator are measurable. The state of the manipulator is defined as X p(t) = [9(t) S(t)f (5.2) The state space representation of the is where The decoupling property can be achieved by a suitable choice of the reference model. The parameters of the model are chosen 1.0, w = 4, so that a critically damped response is obtained. With this choice of the reference model, the existence of a solution to the perfect model matching (linear modelfollowing) is assured. The reference model is described by
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Where 18] B M = [ 106] The linear modelfollowing control (perfect modelfollowing) is designed for a nominal value of 5 Kg (payload), and the reference model and the plant satisfy the Erzerberger's conditions. The control law U(t) is given by U(t)=KPX 59 +[i'' FV(GX P)7 dc+FV(GX P)7 Jx P(t) + [i11 MV(N c_) 7 de+ MV(N r)7 ]r(t) Simulations were obtained for the following values of the plant parameters, M = 12 Kg, 1 = 0. 5 meter and r = 0 2 5 meter. The gain matrices are N = 1 and
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60 5. 2. 1 Results Three simulation studies were conducted for the case of a singlelink manipulator, the manipulator is required to move a 10 Kg payload from the joint angle coordinate of (0 radians) to a final position of (0.7854 radians). In the first case, only the linear controller is involved (i.e., there is no adaptation, F = F = M = M = 0). This case is considered to compare the performance without the adaptive feedback loop in the overall control. Figures (5.1), (5.2), (5.3) and (5.4) show the position and velocity responses, and the evolution of the errors for the linear modelfollowing control system designed for a nominal value of 5 Kg payload. The position and velocity responses are very poor and tracking is not accomplished. In the second case, the positive definite matrix H is chosen as H = diag [ 5 1] this value of H will result in more weight on the position following than the velocity following. Namely, accuracy in positioning is more emphasized than that of velocity. Depending upon the application tasks, relative weights between the accuracies in position and velocity can be selected. The integral and proportional gains were set as follows, F = M = 400 and F = M = 40. The position and velocity tracking improved in comparison to the first case, which means that the adaptive
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61 feedback loop has a great effect on the tracking and therefore improves the system response. However, there is some error in the position and velocity responses, see Figures (5.3) and (5.4). In third case, the influence of increasing the adaptive gains is investigated. The integral and proportional gains, F, M, F, M, were all increased by a factor of 5. This resulted in a better position and velocity tracking, where the position and velocity responses are virtually indistinguishable from the desired position and velocity responses given by the reference model, see Figures (5.5) and (5.6). Increasing the adaptive gains resulted in a faster reduction of this error and a faster speed of adaptation, see Figures (5.3) and (5.4).
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62 + &0 + + + + = a:l + c.. Ill + .... Q + "'CC = Q C) II) + &0 N Q) Q) rll co = ta .... t.J "'CC + Q s + + + + + + + + Q) e Q) co .... = Eo 0 c. co Q) a: + + = + + + + 0 + + + + co 0 + ++ .4 LO Q) ::s 0" ..I lj
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(J Q) (I) 't:l tU d .... .... (J 0 .... Q) > +++: model solid: plant f 0.8 I+ 0.6 0.4 0.2 O.J+"*+ + + ,+ + ++ + + + + + 0 O.f) f 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.2 Velocity Response (Case 1) 0\ w
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.. "'= ..... Q IIJ I'll CD t) + + + Q + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++ ++ ++ CCI Q Q ++ ++ ++ ++ ++ ++ +++ CCI Q Q Q Q oasjpe(l ( N Q N Q Q 0 0 I &0 "" 10 N 10 Q Q 64 IIJ "'= c Q C.J J1 CIJ 0 J1 = J1 ll'l 0'1 s c .... rf e0 ItS J1 8 c 0 rf .j.l rf IQ 0 Po. t""'' 111 Ql J1 :;j 0'1 rf
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+++: Coset 1 Case2 1 solid: Case3 0.25 A 0.21+ + + + + 0.15j:. + + + In + c::= 0.1 + &U .4 't:l &U P:: 0.05 '*"+ d +f .... .4 :f .... l ......... + ........ 0 : ........ J.. 0 J.. + + + + t 0.051l I v I \ I 0.1 v 0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5. 4 Velocity Tracking Error 0\ ln
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en CIS .... 'd CIS .... r::l 0 .... .... en 0 ll.. : Model solid: J))anl (). 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.5 Position Response (Case 3) 0"1 0"1
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= as c.. '0 .... 0 Ill Q,) '0 0 ::a C\l "4 = 0 co 0 0 N 0 10 C\l C\l 10 "4 "4 10 0 0 0 67 M Q) Ul ra I:.) lr) Q) '0 Ul c: 0 0 c Q,) c.. Cl.l Ul Q) = = .... >. Q,) +J 8 .... 0 e0 ...l \D 11'1 Q) 11 :::::1 0'1 ra..
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68 5. 3 A two link manipulator case The equation of motion of the twolink manipulator is derived in Chapter III and is given by The state space representation of the manipulator is given by where X = [Xp1 P The matrices Mo and G0 are Where the parameters Ci ( i = 1. .. 5) are defined in Chapter III. Again the reference model is chosen such that a critically damped response is obtained, the parameters are 1 = = 1. 0, w 1 = w2 = 5. Therefore, the reference model is described by
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69 X M(t) = AMX M(t) + B MC(t) where 0 l JJ 0 0 0 10 25 0 1] With this choice of the reference model, the existence of a solution to the perfect modelfollowing is assured. The control law U(t) is given by U(t)=KPX /t)+KuC(t) + [Iat1 FV(GX P)T de+ FV(GX P)T ]x P(t) + [J:t. MVCN c/ dr+ MVCN c) T ]cct) Simulations were obtained for the following values of the plant parameters l1 = 12 = 1. 0 meter, r1 = r2 = 0.5 meter, mj = 0.0 Kg, M1 = M2 = 6.0 Kg, and the weighting matrices G is 4 x 4 unit matrix and N is 2 x 2 unit matrix. 5. 3 1 Results A computer simulation study is performed to investigate the quality of performance of the adaptive control algorithm proposed. A twolink manipulator is considered here, and required to move from an initial joint angle
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70 coordinate of ( 0. 0, 0. 0) radians to a final position of ( 0. 7854, 0. 7854) radians. For the first three cases a payload of 0. 0 Kg is considered, and analysis of the performance of the position and velocity responses and the evolution of the errors for link 1 and link 2 are considered. For cases 4, 5 and 6, a 7. 0 Kg payload is considered and the performance of the position and velocity responses are analyzed. Case 1: No adaptation is considered in this case (i.e. the adaptive gains F = M = F= M = 0) which means only the linear modelfollowing controller designed for a minimal value of 2. 0 Kg payload is involved. The performance of the positions and velocities of link 1 and link 2 are very poor and tracking is not accomplished, see Figures ( 5 7 ) ( 5 8 ) ( 5 13 ) and ( 5 14 ) Case 2: In this case the positive definite matrix H is chosen as H = diag [ 10 10 2 2], this choice of H results in more weight on the position following than the velocity following. The integral and proportional gains of the adaptive feedback loop are, F = M = 1000 and F = M = 4. 95. The positions and velocities tracking for link 1 and link 2 have improved in comparison to the first case. However, the errors are not reduced fast enough. See Figures (5.8), (5.11), (5.12), (5.13) and (5.14) for the position and velocity responses and the evolution of the errors of link 2.
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71 Case 3: Here, the second case is considered again with the adaptive gains increased by a factor of 5. This resulted in a better position and velocity tracking, faster reduction of the errors, and faster adaptation speed. It is important to note that the ratio between the values of the proportional gain and the integral gain has an important influence on the speed of reduction of the model plant state error. Also the simulation study showed that a higher ratio leads to a high speed reduction of the error, but in counterpart, parameter adaptation speed is slower. The gains are limited by the saturations existing in the adaptation loop, see Figures (5.10), (5.11), (5.12), (5.13) and (5.14). Case 4: A 7. 0 Kg payload is considered in this case, and again no adaptation is involved ( F = M = F = M) Figures (5.15), (5.16), (5.17), (5.18), (5.19) and (5.20) show that the linear controller alone is not capable of producing good performance in terms of position and velocity responses Case 5: In this case the positive definite matrix H is chosen as H = diag [ 6 6 6 6] The integral and the proportional gains of the adaptive feedback loop are, F = M = 4000 and F = M = 80. Figures ( 5.17), ( 5.18), ( 5.19) and (5.20) show that the performance of the position and velocity responses have improved in comparison to case 4. However, the speed of reduction of the state error and the speed of adaptation are slow.
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72 Case 6 : Here, the adaptive gains are set as F = M = 11,800 and F = M = 236 with the H matrix chosen as H = diag [ 10 10 2 2], by this choice position tracking is more emphasized than velocity tracking. Figures ( 5 17) (5.18), (5.19) and (5.20) show that the performance of position response has improved, and the speed of reduction of the state error and the speed of adaptation also have improved. As a whole this simulation exhibits that the proposed control method has a potential for high performance with a very simple structure.
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73 = I + .. I + I + I + I l + I + I + I + I I + I I + + \ + + N + ..1 I + Q) c + fJl I + ra 1 + I + CJ \ + .. I + 0'1 1 + 1 + I + = In c \ + ., .. \ + c c \ + Q II .:.I I t,) c \ + 41 .... ::3 + &n CIJ Ei \ + c ., + \ 41 Q) + fJl Q \ + e c:: In \ 0 + c.. \ + fJl \\ Q) Q,) c::: "C Q \ c:: ::2 \,\ 0 .4 + '\ ...... .4 + ' fJl + ' 0 ' .. ', r' ., , ., 1.11 .... .. Q) ..... .. ........ J.j .......... ::r .......... 0'1 .. .. rz.. .. .............. = = an C":: N .. 0 .. sueJpea UJ uonJSOd
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4 3.5 3 0 2.5 Q.l [f) ........ 2 "'0 ca .... 1.5 b .... l 0 0 .... Q.l > 0.5 0 0.5 1 0 ,, ... ...... ,' ........ .. ,,/ ...... ' +++: Model I solid: Unk 1 I : Link 2 .... ' ' .. .. .. ,., ______________________ + + + + + + + + ++++++++++++ l 2 3 4 5 6 7 Time in Seconds Figure 5.8 Velocity Response (rnl = 0.0 Kg, Case 1) 8 9 10 ...J
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0.9 0.8 0.7 rn 0.6 ftl .... "tt ftl 0.5 c::l .... 0.4 0 .... ....., .... rn 0 0.3 Cl.. 0.2 0.1 0 0 0.5 1 solid: Model ..: Case 2 ,1.5 2 2.5 3 Time in Seconds 3.5 4 4.5 5 Figure 5. 9 Joint One Position Response ( ml = 0 0 Kg, Case 2) ..... c.n
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.. ... 0 II'J cc 0 cc 0 t":) 0 N 0 0 0 0 0 "' = 0 Q,) C/) s:: .... Q,) s E76 I""') QJ fll ca CJ .. 0\ 0 0 II ..I a QJ fll s:: 0 04 fll QJ l::t:: s:: 0 ..! fll 0 ll4 QJ s:: 0 s:: ..! 0 1J 0 r.n QJ :::l 0\ ..!
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(LINK!) :Case2 solid: Case3 O.OJ 0 Ill 0.01 ca .4 't:! ca ll:: 0.02 .... L. 0 0.03 tz:l 0.04 0.05 r ., 0.5 """"",. \ \ \ \ \ ,, 1 \ '\ .."' .. ... .. .. , \ .. \\ ...... '' \, ............ .. '\ ........ ... ______ ... 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.11 Joint One Position Error Tracking (ml = 0. 0 Kg) .._J .J
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(LINK2) :Ca,se2 solid: Case3 In c::= ell 4 0.01 0.02 0.03 p;: &:I .... .... 0.04 0 .... .... rz:l 0.05 0.06 ./... ; \ \ ........ .. \ , \ ... \ / .. / \ ,/ .. .... _____ ....... ... 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.12 Joint Two Position Error Tracking (ml = 0. 0 Kg) ..1 Q)
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(.) Ill VJ .......... "tt ell 1:1:: ..C (LINKl) +++: Casel : Case2 solid: Case3 0.6, I I I I I I I I I I ++++++ ............ 0.4 ......... ......... ......... ...... + + + + + + + + ... + + 0.2 M .... 0.4 0.6o 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.13 Joint One Velocity Error Tracking (ml = 0. 0 Kg) ..I \D
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(LINK2) +++: Casel : Case2 solid: Case3 0,5 I f\ 0 .. I T1 CJ QJ 0.5 .s 1 \ + + 'tf cG ... +'t. \_ +++++++ 0 . .. 1.5 .., t t Et 2.5 + + + + + 0 0.5 l 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.14 Joint Two Velocity Error Tracking (ml = 0. 0 Kg) CX) 0
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solid:Model :Linkl ... :Link2 0.8 11) 0.6 Cll .... '1:1 Cll .s 0.4 0 .... .... 11) 0 P.. 0.2 .. .... I \ ; I / . \ ; I ' i ., I I . I I ... ,' 0 0.5 l 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5. 15 Position Response (ml = 7 Kg, Case 4) CD .....
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2 I (J Q) Ul ......__ .. (. l "'tf c:l 0.5 .... .... 0 0 : 0 0.5 \. solid:Model :Linkl ..:Link2 ., ........... _____ ... __ \ ,/' \ / I \, / ' .. _,, 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Seconds Figure 5.16 Velocity Response (ml = 7 Kg, Case 4) 0) tJ
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+++:Case4 1 :Case5 I solid:Case6 0 rn 0.05 "C til a ..... a.. e 0.1 a.. a:z:l 0.15 .. ........ '!f: \ '1: + + + + + + + + 0.2L_ __ _L ____ L_ __ _L ____ L_ __ _L ____ L_ __ _L ____ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Second Figure 5.17 Joint One Position Error Tracking (ml = 7 Kg) CD w
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rtl c:= C'CI "'4 't:l C'CI .... a.. 0 a.. a.. +++:Case4 I :Case5 1 solid:Case6 0.6 0.4 + + + + 0.21t + f + f f + [ 0 0 0.5 + + + l ! f f I I I 1 1.5 2 2.5 3 3.5 4 Time in Second +f+tf 4.5 Figure 5.18 Joint Two PositionError Tracking (ml = 7 Kg) 5 (X)
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(J Q) [I) 'l:l Cll t1 .... Lot 0 J.4 J.4 (LINK1) ....:.:Case5 solid:Case6 0.4 0.3 0.2 0.1 0 :f 0.1 I I f \Jf \ t 0.2 Jt t 0.3jj" 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Second Figure 5.19 Joint One Velocity Error Tracking (ml = 7 Kg) (X) U1
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(LINK2) :Case5 solid:Case6 ++ .... .... + + + + + + + C) 41 Cl) ......... 1 + + + + + + + + + ., ell Cl:: Q .... ... 0 0.5 t: rzl (\ 1 't + \ + + To + 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time in Second Figure 5. 20 Joint Two Velocity Error Tracking (ml = 7 Kg) Q) 0\
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CHAPTER VI CONCLUSIONS The design of a manipulator control system based on the adaptive modelfollowing control (AMFC) concept has been presented in this study. The design of the adaptation mechanism is based on the use of the Popov hyperstabili ty theory. The controller is continuous and consists of two parts, the first part is linear in the state and the reference input, where the linear gains are determined using the known linear portion of the system dynamics. The second part of the controller is produced by an additional adaptive feedback loop, where the adaptive gains are determined by the use of the hyperstabili ty concept. The design has focused on the following three important features. First, unlike most other existing methods, changes in the manipulator payload can be handled effectively without requiring the payload to be specified explicitly in the design of the manipulator control system. In other words, the control system is designed to provide any desired performance (i.e., accuracy in position and velocity) equally well for a wide range of payload.
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88 Second, the presented design is conceptually and computationally simpler than most other existing methods without loss of performance, the implementation of the adaptation laws does not require the real time solution of a set of or nonlinear equations. Therefore, the. aid of a computer is required only in the design stage for the computation of the parameters of the adaptation mechanism. Third, the stability of the manipulator controller is guaranteed, leaving control designers free from the stability problem which is often very difficult to analyze, if not impossible. Assurance of the manipulator controller stability cannot be overemphasized, especially in view of the potential danger of human injury and loss of expensive equipment as a result of the controller instability.
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BIBLIOGRAPHY 1. Landau, I D. Adaptive Control The Model Reference Approach. New York: Marcel Dekker, 1979. 2. Paul, R. "The Mathematics of Computer Controlled Manipulator. Proc. Joint Automat. Contr. Con. I, pp. 124131, 1977. 3. Whitney, D. E. "Resolved Motion Rate Control for Manipulators and Human Prostheses. IEEE Trans. ManMachine Systems, June 1969, Vol. 10, No. 2. 4. Whitney, D.E. "The Mathematics of Coordinated Control of Prosthetic Arms and Manipulators." ASME J. DSMC. ll, Dec. 1972.5. Luh, J.Y.S. I Walker, M.W. and Paul, R.P.C. "Resolved Acceleration Control of Mechanical Manipulators. IEEE Trans. Automat. Contr., June 1980, Vol. AC25, No. 3. 6. Paul, R. Robot Manipulators: Mathematics, Programming and Control. Cambridge, Mass.: MIT. Press, 1981. 7. Luh, J.Y.S., Walker, M.w.and Paul, R.P.C. "Online Computational Scheme for Mechanical Manipulators. ASME J. DSMC 102, June 1980, pp. 6976. 8. Hollerbach, J .M. "A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative Study of Dynamics Formulation Complexity." IEEE Trans. Syst. Man. Cybern., 1980, vol. II, pp. 730736. 9 Koi vo, A. J. and Guo, T. H. "Control of Robotic Manipulator with Adaptive Controller. Proc. IEEE Con. on Decision and Control, pp. 271276, 1981. 10. Horwitz, R. and Tomizuka, M. "An Adaptive Control Scheme for Mechanical Manipulators Compensation of Nonlinearity and Decoupling Control." ASME Journal of DSMC, June 1986, Vol. 108, pp. 127135. 11. Dubowsky, S. and Desforges, D.T. "The Application of Model Reference Adaptive Control to Robot Manipulators." ASME Journal of DSMC, Sept. 1979, Vol. 101, pp. 193200.
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90 12. Landau, I.D. and Courtiol, B. "Design of Multivariable Adaptive ModelFollowing Control Automa 1974, Vol. 10 pp. 483494. 13. Erzberger, H. "Analysis and Design of ModelFollowing Systems by State Space Techniques. Proc. of Jacc, 1968, pp. 572581. 14. Landau, I.D. "A Hyperstability Criterion for Model Reference Adaptive Control Systems. IEEE Trans. Aut. Control AC14, 1969, pp. 552555.

