ADAPTIVE MODEL-FOLLOWING 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 Model-Following 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 model-following

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 Two-Link

Manipulator............................23

3.4.1 Description of the Physical Model......23

3.4.2 The Kinetic and Potential Energy of a

Two-Link Manipulator.................23

3.4.3 The Dynamical Equation of Motion

of a Two-Link 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 Model-Following Control (PMFJ...38

4.5 Adaptive Model-Following 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 Single-Link Manipulator Case.............57

5.2.1 Results............................... 60

5.3 Two-Link Manipulator Case................68

5.3.1 Results................................69

VI. CONCLUSIONS...................................87

BIBLIOGRAPHY

89

FIGURES

Figure

2.1 Self-Tuning Regulator (STR)....................7

2.2 Model Reference Adaptive Control (MRAC)........8

3.1 The General Coordinates of a N-Link

Manipulator.................................12

3.2 A Two Degree-of-Freedom Manipulator...........24

3.3 Parameters of the i-th Link...................25

4.1 Linear Model-Following Control System

(Perfect Model-Following Control)...........39

4.2 Adaptive Model-Following Control (AMFC) with

Parameter Adaptation........................43

4.3 Adaptive Model-Following Control (AMFC) with

Signal Synthesis Adaptation.................43

4.4 Equivalent Feedback System....................46

4.5 Equivalent Feedback Representation of

Adaptive Model-Following 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 real-time.

2

In this study, an adaptive control scheme is con-

sidered for controlling the robot manipulator. This

approach is based on adaptive model-following 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 well-known 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 end-effector 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

Newton-Euler 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 end-effector). 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 self-tuning 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 Self-Tuning Regulator (STR)

8

Figure 2.2 Model Reference Adaptive Control (MRAC)

Koivo [9] used self-tuning 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 n-link manipula-

tor is linearized with respect to an operating point (sys-

tem equilibrium state). Finally, the equation of motion of

a two-link 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 x-y

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 sum-of-products form

*,= Xa/J,g)1(e,e)*Xc,1g;,(8) (3.2)

-l t-1 t-1

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 N-Link 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

j-1

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

7-1

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

7-1

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 n-link 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

t-1 60j

(3.23)

e-0(Â§.-6.>+Â£(iÂ£) ^60

and

(?(0o-h60.0o + 60)(0o + 60) = Q(0o.0o)0o + Q(0o.0o)60

and also

(3.24)

G(0o.60) = G(0o)+ e,T ) 60

I-cT

.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-

CSIgs-O-<=

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 = o-Â§o = o-0o = 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

Q0-Qi-Gl-G2-o

and therefore, the linearized equation of motion is

Mo60 + Go60 = bU

(3.32)

23

3.4 The Dynamic Equation of a Two-Link Manipulator

3.4.1 Description of the Physical Model

A simple two-degree-of-freedom 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 Two-Link

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 Degree-of-Freedom 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

= (3-51)

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 + rfl-l^2^9

3.4.3 The Dynamical Equation of Motion

of a Two-Link Manipulator

As mentioned earlier the kinetic energy of the

two-link 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 f-2C2sin02 -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

two-link 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

continuous-time adaptive model-following control (AMFC)

algorithm proposed by Landau [12] is considered for a class

of nonlinear time-varying 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 time-invariant 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 model-following control

is discussed, which guarantee the existence of a solution

to a linear model-following control. In Section 4.5 and

4.6, an adaptive model-following control and adaptation

laws are presented respectively. The controller consists

of two parts, a linear part which is responsible for

achieving perfect model-following, and an adaptive part

responsible for compensating and modifying any deviation

of the error of the overall system from that corresponding

to the linear model-following.

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 two-link 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 real-time 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

2-n 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

well-behaved linear uncoupled system, a reference model

described for each degree-of-freedom by a linear second

order time-invariant differential equation is chosen.

With this choice the desired time domain performance char-

acteristics, such as rise-time, overshoot, and damping can

be specified with a minimum number of parameters. The

linear, second order differential equation for each

degree-of-freedom 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 Model-Following Control (PMF)

The design problem of linear model-following con-

trol (perfect model-following) 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 = Xm-Xp) and its derivative asymptotically tend

to be zero.

Consider the linear model-following 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 Model-Following Control System

40

In order to achieve perfect model-following, 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)

+ (Bm-BpKu)r*
The Erzberger conditions can be obtained by impos-*

ing e = X X =o and e = X X n = o.

(Am-Ap + BpKp)Xp{t) + (Bm-BpKu)rSt) = o (4.16)

Which implies

(Am-Ap) + 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, (Am-Ap)] = 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+piAm-Ap) (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

(I-BpB+pXAm-Ap) = 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 Model-Following 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

model-following 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

model-following control is used in which it is assumed that

a solution for perfect model-following 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 real-time 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 signal-synthesis adaptation will be considered.

The parallel AMFC system with signal-synthesis

adaptation Figure 4.3 is described by the following equa-

tions .

The reference model:

43

Figure 4.2 Adaptive Model-Following Control

(AMFC) with Parameters Adaptation

Figure 4.3 Adaptive Model-Following 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(0-fipt/(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 model-following

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 model-following" 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

time-invariant 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 time-invariant block in the

feedforward path described by the state equations:

K(O = 0Â£(O

(4.28)

(4.29)

O

W1

LINEAR

TIME-INVARIANT

W

NONUNEAR

TIME-VARYING

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 time-varying 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

Kalman-Yakubovitch-Popov'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 model-following 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) + (Am-Ap)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) + *PKP-SpAtfp(e,i)]Â£p(0

+ [Bm~ BpAKu(e,t)- BpK

= Ame(0 + Bp[B+p(Am- Ap) + Kp-AKp(e,t)]Xp(t)

+ Bp[B+pBm-AKu(e,t)-Ku]r(t)

e(t) = Ame(t)~ Bp{[AKp(e,t)-B;(Am- Ap)~ Kp]Xp(0

+ [AKu(e,t) + Ku-B+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 model-following 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.o-c0)Xp(0

+ (AA:u(GI0-C1)r(0 (4.43)

Where

C0 = B+p
Ci = B*pBm-Ku (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 time-varying 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

steady-state error of the manipulator position and veloc-

ity. Matrices 2 and ip2 denote a nonlinear time-varying

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

D-BlP

(4.51)

54

Where P is a positive definite matrix solution of the Lya-

punov equation

PAm*AJmP--H (4.52)

Where H is a symmetric positive definite matrix

Then Equation (4.50) becomes,

Z(.s)-BlP(.sl-Amy'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)(Akp-C0)Xpdx

+ J V(x)(Afcll-C1)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(Am-Ap) + 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^dx-c^dt

t-1 j.l J 0 \J 0 J

1 ml n

|XX

^ i-l 1

f'a,Xpldx-C,] -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 single-link manipulator and

a two-link manipulator.

5.2 Single-Link Manipulator Case

The equation of motion for the single-link 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 model-following) 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 model-following control (perfect

model-following) 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 single-link 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 model-following 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 two-link 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 model-following 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 two-link 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 model-following 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 model-following 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

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124-131, 1977.

3. Whitney, D.E. "Resolved Motion Rate Control for Manip-

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Systemsf June 1969, Vol. 10, No. 2.

4. Whitney, D.E. "The Mathematics of Coordinated Control

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5. Luh, J.Y.S., Walker, M.W. and Paul, R.P.C. "Resolved

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3.

6 Paul, R. Robot Manipulators: Mathematicsr Programming

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