Fuzzy Control for an under-actuated Robotic

Fuzzy Control for an under-actuated Robotic

Manipirlator: Pendubot

Xiao Qing Ma

A Thesis

in

The Department

of

Mechanical Engineering

Presented in Partiai Fulfïhent of the Requirements

For the Degree of Master of Appiied Science at

Concordia University

Montreal, Quebec, Canada

Aupst 200 1

O Xao Qing Ma, 2001

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Abstract

Fuzzy Control of an Under-actuated Ro bo tic Maoipulator:

Pendubot

Xiao Qing Ma

Control of under-actuated mechanicd systerns (robots) represents an important cIass

of control problem. This thesis studies severai relateci control problems associated with

an under-actuated robot, Pendubot, h m the point view of fuPy logic control. To swing

up the Pendubot fiom a rest position to the upright confi-don, a fuzzy aigorithm is

proposed fiom non-complete sets of linguistic d e s that Iink some mechanism states to

the sign of a singe conno1 action. Therein, a simpiifïed Tnikamoto's reasoning method

and quasi-hear-mean aggregating operators are used to denve and anaiyze the controlIer

input-out rnappings. in order to balance the Pendubot at the unstable upright top

configuration after swinghg up, another simple fuzzy controiier is derived according to

its joint states. This combining fuzzy aigorithm for swin-eing-up m d baiancing is

successfuiiy applied to the Pendubot. This thesis also investigates the case t b t the

Pendubot tracks a desired signai and a corresponding fuzzy scheme is proposed, which

combines the iinear reguiator theory with the Takagi-Sugeno h z q methodolog. The

stability and stability conditions for this fuzq scheme are analyzed. Numericd

simuiations for ail the above controüers are carried out to validate the theoreticai anaiysis

by using SIMULIMC F ï y , the hardware experiments in the Pendubot have

successfully been conducted in the Robotics and Mechatronics Laboratory.

.-- Ill

Acknowledgements

The author dedicates this thesis to the thesis supervisor Associate Professor Dr. Chun-

Yi Su, for his invaluable guidance and consistent encouragement through out the course

of this research and looks forward for his guidance in a i i future endeavors.

The author would like to express her appreciation to h k Z. Cai, whose enthusiasm to

build and set up the whole system and àelp me perfom the experiments played a

significant roIe in the deveIopment of this project.

The author wodd Iike to express her sincere thanks to Mr. Z. L. Liu for the useful

discussions with the author.

1 would like to thank my entire f d y for their love and support throughout my

graduate school days. They and especiaily my husband, KangZe Zheng, gave me the

encouragement to compIete this work

Table of Contents

List of Figures .-. v u

Chapter 1

1.1

13

1.3

1.4

Chspter 2

2.1

2.2

2.3

2.4

2.5

Chapter 3

3.1

3 .?

3.3

3 -4

Chapter 4

4.1

4.2

Introduction

Overview

Objectives

Thesis Organization

Contribution

Basic Knowledge

Introduction of Robot

Review of Robouc Control

Fu;n Logic Conml and Reasoning

Introduction of Digitai Conml

Description of Pendubot

Pendubot Mode1

The Dynamics

introduction of Identification

The Equilibrium ManifoId

Controiiability

Swing Up Control

A Two-Rule-Based Fuzzy ControiIer

A Two-Rule-Si@-Based Strategy for the Pendubot

Chapter 5

Chapter 6

Chapter 7

7.1

7.2

Chapter 8

8.1

8 -2

8.3

8.4

8 -5

Chapter 9

9.1

9 3

9.3

9.4

Andysis of the Controt Law

Bdancing Control

System Analysis

Conml Law

Shbüity Anaiysis of Fuzy Trajectory Tncking Controiier

Fuzzy Control Aigorithm

Stabiiity Adys i s

Combining the Controllers

Combining in the Simulation

Combining in the implementation

Simulation

Simulation Pendubot Model

Swing-Up Simulation Mode[

The Balancidg Simulation Model

The Final SimuIation Mode1

Simuiation Resuits

8.51 The Simuiation Resuits for the Continuou System

8.5.2 The Simdation ResuIts for the Discrete System

Red-time Experimentd Tests

introduction

Flow Chart

Experimentai Results for Swinging-up and BaIancing

Photographs of Experiments

Chapter 10 Conclusions and Recommendrtions for Future Work

10.1 Discussion

10.2 Conclusions

10.3 Recornmendations for Future Work

References

Appendü A Linearized equations

List of Figures

Figure 2.1

Figure 2 3

Figure 23

Figure 2.4

Figure 2.5

Figure 3.1

F i p 3.2

Fi_- 3.3

Figure 4.1

Figure 5.1

F i p 5.1

Fi-- 5.3

F i p e 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Fi_- 8.5

Figure 8.6

Figure 8.7

Fuzzy System

Function tanh(t)

Structures for a digital control Ioop

Front and side perspective drciwings of the Pmdubot

Pictorial of the Pendubot's interface with its conmuer

Coordinate description of the Pendubot

Equiiibrium configuration

Uncontrollable configurations

ïhe two situations

The BaIancing corifiguration at the top position

The link one on the right side arouud the equiIr'brïum point

The iink one on the left side around the equiIr'bnum point

S wing-up and baiancing controI

Subsystem 1: Pendubot mode1

Subsystem 2: swing-Up controllet

Subsystem 3: baiancing controller

Swinghg up simdation

Balancing simulation

Subsystem 4: switching block

Figure 8.8 The whole simulation system

Figure 8.9 Discrete-tirne Pendubot madel

F i p 8.10 TheOutputofq,

Figure 8.1 1 The Output of q,

Figure 8.12 The Output of torque r

Figure 8.1 3 The angular velocity for one

Figure 8-14 The angular velocity for Iiak two

Figure 8.15 The Output of s oust swinghg up)

Figure 8.16 The Output of q, (just mvinging up)

Figure 8.1 7 The Output of q2 (jus swinghg up)

Figure 8.18 The Output of q,

Figure 8.1 9 The Output of q,

Figure 8.20 The Output of torque r

Figure 8.2 1 The angular veiocity for Iink one

Figure 8.22 The angdar veiocity for hk two

Figure 8.23 The Output of r Cjust swinghg up)

Figure 8.24 The Output of q, oust swinghg up)

Figure 8.2s The Output of q, Gust swinging up)

How Chart

Figure 9.1 Effect of torque

Figure 9.2 Effect of ag i e for Link one

Figure 9 3 Effect of angie for Iink two

Figure 9.4 Effect of mgdar vetocity for iink one

F i m m 9.5 Effect of angular velocity for link two

Figure 9.6 The Pendubot at the rest configuration

Figure 9.7 BaIancing at the top configuration

Figure 9.8 The Pendubot on the left side

Figure 9.9 The Pendubot on the right side

Chapter 1

Introduction

In ment yevs the control of underdctuated mechicai systerns has arûacted

p w i n g attention and is a topic of great interest [2][3][1 I][I5][10]. As yet, there is no

comprehensive theory for underactuated robot manipdators, Le., manipdators having

fewer acntators than degrees of W o m . Examples of such synems are ilIustrated, for

examp[e in [2] [16][7[ 11 [5] [a. Frequentiy cited appIications inciude saving weight and

energy by using fewer actuators and gaining M t tolerance to actuator failure.

The challenge of s o h g control probiems associated with this class of systems WU

stimuiate new results in robot control theory in the years ro corne, interest in studying the

under-actuated mechanical systems is aiso motivated by their role as a class of strongly

nonlinear .stems where cornpiex i n t d dynamics, nonhoIonomic behavior, and lack of

feedback IinearizabiIity are ofien exhiiited, for which traditionai nonluiear control

mettiods are insriffcient and new approaches mus be developed

Tho@ dynamics of the under-actuated mechanical systems is well uuderstood, the

difficulty of the control problem for under-actuaîed mechanism is obviously due to the

reduced dimension of the input space. The literame on the conm~l of under-actuated

systems is rnahiy ment [2] 141 [5] [13] [12], and the discussion mainiy focuses on two-

degeesf-fieedom exarnples [3] 1151 [a] [59]. Eririier work that deais with control of

under-actuated robotic systems is described in [jq [Sq. Under-actuated mechanicd

systems have also k e n investigated h m nonholonimc constraint point of view [13] [lq,

where, for instance, Oriolo and Nakamura [57] and Wichlund [58] established the

conditions for partid integrability of secondsrder nonholonomic constrahts and

discussed control problems.

WhiIe some interesthg techniques and fesults have been presented in the mentioned

above publications, the conmi of such systems stiii remains an open problem. For

example, most of the control schemes mentioned above either faiIed to provide a

thorou& analysis of the overd system stability or assumed that gravitation forces didn't

act on the passive joints. Fusthemore, the precise knowkdge of dynamic mode1 is

generalIy required.

F u z q bgic conmi techniques were origindy advocated by Zadeh and Mamdani as a

means of both coiiecting human knowiedge and experience and dealing with

unceaainties in the control process, It has become a very popular tooI in control

engineering [8][7I[lO][lI]. Fuzzy control systems have the advantages that no formai

mathematicai modelr are needed and the systern uncertainties c m be coped wilh.

Therefore, aimine at the mentioned above control prublems for such under-actuated

mechanid nonhear systems, this thesis wiII investigate severai reIated conml

pmblems, associated with an under-actuated robot-Pendubof fiom the point view of the

fuzzy bgic control.

The Pendubot (PenduIum robot) is a benchmark system for under-actuated robot

manipulators, coasisthg of a double penddum with an actuator at only the second joint.

Using the Pendubot, one can rnainly investigate the set-point regdation, inctuding

swingkg up and balancing, and trajectory tracking.

The pmblem of swinghg the Pendubot h m the open' loop stable coofiguration

q, = - K I 2. q, = O to any of the inverted equilibria is an interesting problem because of

the strong nonlinearity and dynamic coupling between the W. Many dserent control

algorithm could bave been used to perfonn the swing up. in the classic control field

Spong [2] and Block (1) applied smaii partial feedback linearization to swing up the nvo

k h m hanging straight down to standing straight up. Yoshida [6] used energy-based

methods to complete the swing-up control of an inverted penduIum. in addition, some

mearchers proposed the robust adaptive control algorithm or hybrid algorithni to control

the two-link under-actuated robots [3] [SI. In the fuzzy control field, Sanchez [15] and

Vidolov [IO] designed a MIMO fuzzy PD controller to swing up an underactuated robot

Sousa and Madrid [I Il- proposed a control strategy based on genetic algorithm and

Takagi-Sugeno fuzzy methodoIogy to control the Pendubot- However, the genetic

aigorithm is not easy to implement because it depends on the number of d e s . Some

other fuzzy controt schemes are just implemented in the simdation system, and they

didn't give the experimentai resuits.

Once both links are m g up, the problem of baIancing the Peadubot about an open

loop unstabIe equilibrium should be investigated. ïhe Pendubot possesses infinitdy

many confi-pations at which it can be baianced Spong [2] and BIock [I l proposed a

baiancing controller, using iinear quadratic optimal control theory to bdance the

Pendubot at top or mid configurations. Furthemore, the robust daptive control

algorithm or hybrid algorithm has &O been proposed, but they are not fidiy general,

Besides the above-mentioned probIem associated with the Pendubot, trajectory

tracking for nonlinear systerns aIso can be achieved in the Pendubot, Berkemeier [4]

denved a surprising set of exact trajectories of the nonIinear equations of motion. which

involve inverted periodic motions. Moceover, Begovich and Sanchez [I4] combined

Iinear regulator theory with the Takagi-Sugeno fuzzy methodology a new algorithm to

get a nover approach to achieve trajectory tracking for the Pendubot. However, they

didn't anaiyze the stability of the whole control system. in this thesis, nodinear system

trajectory tracking has been analyzed and impiemented in red tirne in the Pendubot.

1.2 Objectives

The objective of this research is to develop a di@ controiier based on fuzzy logical,

approximate reasoning, and quasi-iinear-mean aggegating operators, to swing up and

balance the imder-actuated Robot - the Pendubot, Noniinear system trâjectory tracking

wiil ais0 be anaiyzed, based on the linear regufator theory and the Take-Sugeno (TS)

fuzzy methodology. important issues for the under-actuated robotic controiier design are

to develop the fuzzy logic aigorithms and impiement them, The motivation is to make an

effort tomrds applying fuzzy controlling methods to the actual robot 1 will build a

completed and feasible simularion mode1 and mite a digital controuer program.

The Pendubot is Uistalled in the Robotics and Mecbtronics Labontory, led by Dr%

It is a typicai under-actuated penddum robot which is an electro-mechanicd system

consisting of two rigid links interconnected by revohte joints.

Chapter 2 rnainly coiisists of ttiree main parts. The £îrst part is a brief introduction to

robotic control. The second part is a description of fuPy logic theory and fuzn control.

The third part is a description of the Pendubot

Chapter 3 discusses the dynamics of the Pendubot, introduces the equilibrium

manifold and identxcation of the iink inertia parameters, and discusses the controllability

of this system. in this chapter? I provide the mathemaricd mode1 bat c m be used to

derive various controiiers and to simulate the response of the system. It serves as the base

for the whole research.

Chapter 4 denves a two-de-based fuzzy controI scheme fiom a simpiified

Tsukamoto's reasoning method and quasi-hez-mean qgegathg operators to swing-up

the Pendubot h m its rest position. The fuzzy scheme is derived fiom non-comptete sets

of linpuistic d e s thrit Link some mechanism states to the sign of a singie control action.

At the end of this chapter, I analyze the coatrd law and give the principie of control

parameters' tuning.

Chapter 5 dispIays a fuPy control dgorithm for baiancing the Pendubot at unstable

open-loop equilibriurn states. This scheme is aiso obtained based on the fuzzy mie base,

but contrary with the d g - u p ' s d e base, because their control gods are dierent.

Chapter 6 anaiyzes the stabiiity of a fuzzy scherne for üajectory tracking of the

nonhear system: Pendubot. There are two steps: the first one is the stability anaiysis of

the fuPy control systern, consisting of a nonlinear plant and a fùzzy controller, the

second one is to give the condition of applying Iinecu regdation theory in the nodinar

system that consists of each local Iinear system by using the fupy methodology.

Chapter 7 displays the switching aigorithm both in the simulation and in the

experiment since the mode1 of the Pendubot in the simulation is different from the actual

model. These ciifferences display not only in the parameter adjusting but aiso in the

controller combining. This chapter wiIl point out ail these detaüs.

Chapter 8 is the simuIation of the whole control process using the SIMULINK. 1 start

h m the Pendubot mode1 founding, and then buiId the swing-up controller and balancing

controiier respectively, haliy comect them with the switching controUer. In this chapter,

I also e.xpIain the tuning of controllers and give the simulation results for bath

continuou-tirne and discrete-time systems.

Chapter 9 presents the implementai method and resuits. I design a digital controller

according to the fuzzy controt scheme (given in the above chapters) using Mimsofi C

7.0 Ianguage. 1 present the design ciue through flow chart and give the implementai

results using digitai photopphs and the response of joints.

Surnmary, coaclusions, the recommendations for future work, and reference are

presented in the next parts. Fmaily, Appendk A gives the linearization equations used in

Chapter 6.

2.4 Contributions

The approach in this thesis is partiy sirniIar to the work of [1][16][14], but there are

significant diierences as well, which are exactiy my contributions.

1) A simpiifïed Tsukamoto's reasoning method and quasi-Iinear-mean aggregating

opentors are used to derive and anaiyze the swing-up control input-output

rnappings. I appiy this fuzzy controi scheme to swing up the Pendubot h m the

rest position (q ,= -812 , q,=O) to the unstable top position

(4, q* =O)-

2) According to the Pendubot's joints states and fuzy control rule, 1 design another

fuzzy controller to balance the Pendubot at the top position. Through a l o g i d

svitching algorithm, 1 combine the swing-up controller with this baiancing

controiler.

3) Based on the deveIoped T-S fuzzy trajectory tracking scheme on the baiancing

Pendubot, 1 give the stability analysis of this TakagiSugeno fuzzy scheme;

4) 1 have combined the fuPy swing-up controlIer with the T-S fuzzy trajectory-

tracking controller to generate a novel control scherne. 1 implemented it in the

real tirne Pendubot;

5) Al1 of the above are impternented in the Pendubot Mode1 P-3 in our laboratory,

and the simulations of 1) and 2) are achieved using SIMLILIPX.

Chapter 2

Basic KnowIedge

2.1 Introduction of Robot

As many books reIated to robots are introduced robots are basicalIy positionhg and

hancilhg devices [34]. A usefd robot is one that is able to control its movement and the

force it applies to its environment [35]. Presentiy dif3erent aspects of robotics research

are cmied out by e.qerts in various fields that are r n e c ~ c a i manipulation, locomotion.

cornputer contml, and artifïciai intelIigence. The major devant fieIds are rnechanics,

control theory, and cornputer science. Ornitting the accessorid parts, robot actuaiiy is the

mdti-Iink mechanics device. Thus, my resemh object is limited to this device and how

to contml it.

To controi robot requires the knowledge of a mathematicai model and of sorne sort of

intelligence to act on the model. The mathematicai model is obrained h m the basic

physicd laws governing the robot's dynamics. intelligence requires sensory capabiiities

and means for acting and teacting to the sensed variables, Therefore, there are two main

steps for simple induda1 robot: the 6rst step is about the mechanics of mechanical

manipuIators and its dynamics; the second step is to appiy the control theory to moriift

the actions and reactions of the robot to different stimuli. The partidar controiier used

wiii depend on the complexity of the mathematical model, the application at tan& the

availabie resources, and a host of other criteria

Step 1: Modeiing

For servo-control design purpom, and to design better con~ollers, it is necessary to

reveal the dynarnic behavior of the robot via a mathematicai model obtained from basic

physicai Iaws. I begin the development with the generd Lapnge equations of motion

[38]. Consider then Lagrange's equations for a conservative system as given by

where q is an n-vector of genenlized coordinates q,, r is an n-vector of generalized

force r,, and the Lagrangian L is the diffe~nce between the kinetic and potentid

energies

L = K - P (2.2)

It can then be show [34] that the robot dynamics are given by

D(q)q + C(q, 4) + + eh) + rd = f (2.3)

where D(q) is a symmetric, positive-dennite inertia matrix, C(q,q) is a vector

containing the effects of the Coriolis and centripetai toque, G(q) is an n-vector of

grave toques, F(q) represents fiiction, ana rd represents extemai disturbances.

Step 2: Control

A usefiil robot is one whose üajectory in its workspace may be specified, and the

forces it exerts on its environment may be controtled, Looking back at equation (2.3), my

control objectives wiI1 d l y fdl into one of the foilowing categories:

1. Motion Con~o l . Given a desired trajectory, specified by the vector time functions

qd ( t ) and q, (t) , design and implement a controlIer whose output s(t) wiU drive the

a d ûajectory { q(t), q(t ) ) to { q, (t) , qd ( t ) } asymptoticdly. Note tbat the desired

üajectory is usually specified in the task space and some ~reprocessing is required to

obtain a desired joint space tnjectory. The alternative wodd be to obtain the

dynamics of the robot in the task space where the desired üajectoty is specified- The

Fine motion control is then accomplished using precision movements of the end-

effector.

2. Force Conrrol. Whén the robot cornes in contact with its environ ment^ the contact

forces and reactions of the robot need to be reguiated. The force control requkments

are specifTed by a desired force vector &(t) in the task space. ï h s force vector is

the üajectory in the force space that the endeffector should follow, when a force

controik is propaly designed.

3. Motion and Force Conh-ol. In some cases, the robot is required to follow a desired

motion trajectory while exerting a certain force on i t . environment. In this case, both

previous control objectives are combined to design a suitable controller.

2.2 Review of Robotic Control

From equation (2.3), 1 know the dynamic mode1 of a robot is described as a set of

hi-dy nonlinear md coupled differential equation. That mems the robot system is a

compiicated n o f i e u system. There are many control methods for such a nodinear

system. I wiI1 discuss them as following.

Firstly, a common approach is to linearize the dyamics and apply linear conml

theol. However. the operating region of a linearized design is limitecl; so the method of

partial feedback linearization applying to the nonlinear systems is very limited [1][2]p9].

Secondly, the deveiopments in the theory of geometric nonhem control provide

powef i methods for controuer design for a Iarge cIass of noniinear qstems [29][40].

However, it requires the exact knowiedge of the system, which is seldom met for mbotic

system. W l y , to overcome this drawback, the adaptive control schemes have k e n

used for the robotic systems [41][42]. The adaptive control approach is only applicable to

the fidi-actuated robots, but for the under-actuated dynamic systems [G], it remains an

open problem. Fourthiy, with development of the fuzzy IoPjc and fuzzy conmI, more and

more cesearchers would- Like to apply this control method to ded wiîh the uncertain

nonlinear systw. For exampIe, Marcio developed a genetic algorithm to coatrol the

under-actuated dynamic systerns [f 11; Edgar proposeci a fiizzy PD scheme to swing-up

the Pendubot [151; Edgar and Ofelia ais0 developed a nodinea. systern trajectocy

tncking scheme [14]; Micheal [473 extended the ciassicd Lyapunov synthesis method to

the domain of computing with words to fonn a adaptive fiizzy controiler.

From the above survey on the robotic control systern, particularly for the under-

actuated robot or dynamic systems, we can see bar the fmy controI approach may be

more suitabte for the cornplex uncertain nonlinear systems. Next, 1 will introduce the

fuzzy Io& and fuzzy control.

The ability to control a system in uncertainty or h o w n environments is one of the

most important chancteristics of any interligent controt system. Fuzzy Uiference

procedures are becoming, therefore, inçreasingiy crucial to the process of managing

uncenaiaty. Fuzzy sets theory provides a qstematic fiamework for deaihg with different

mes of unceTtainty within a singe conceptuai h e w o r k [Ml.

A fuu_v controller works simila- to conventionai controllers [4q. Tt takes a set of

input values, performs some caiculations, and generates a set of output vaiues. Figure 2.1

illustrates a typicd fiizzy system. According to this fi_preII 1 wiU iutroduce the steps of

des img a fuzzy controiier.

Cnsp inputs Fded inputs Fuzzy conclusions Crisp outputs

l 7 01 Li

I i y l

i - I Mechanisrn I

uZ i F d c a t i o n Defwzification Y2 7.

I ! c. i t

f ! ! ! i !

Un i Rule-Base ! f , Ym

-:

Figure 2.1 Fuzzy system

Step 1: Fuzzification

Fupitication is a mapping £iom the observed input to the fuzzy sets defined in the

comsponding universes of discourse [45]. Therefore* fuzzification is a mnpping h m

U E R " to the unit hypernibe [O, lIM. Fuzq sets were inmduced by Zadeh to

incorponte concepts such as vapueness into cornputer ystems [48]. In conventional set

theory an eiement either belongs to a set or does not, but in normal hurnan decision

making this sharp distinction does not normdy exist

A f 5 z q set "Ay is charrctexized by a membership function, whose elements x of X a

r d member p,(x) is in the interval [O, 11, and pA(.r) representing the _ d e of

mernbership of element x in the fuzzy set A. Here? ttre membership function can be

chosen &ou& many ways, including mathematic function, self-defined, or a simple

refletion relation.

I shodd especidly out thac in our swing-up control algoritha the f k q sets are

given by:

- e-' where tanh(t) = , whose curve is illustrated in the Fi-me 2.2 Thus this

et +e-'

dennition is coincident with the definition of fii7zy set.

Figure 2.2 Function tanh(t)

Step 2: RuleBase

A fuzn d e is defined as a Iinguist logic relation between the input and output. It is

sïmply expressed as:

IF ...... THEN ......

Rule base means a i i mies for the controUing system, which consists of each simple d e .

The translation of this d e base system into 2 fÛzzy relation is done by construcMg the

fÙzq relation R, for each d e r, and combining these retations into a singie fiizzy

relation R. This combining of fuzzy d e s into a fuzzy relation is cailed aggregation.

Therefore, in order to express the d e base by mathemathion, we shouId aggregate dI

d e bases (supposeci number=q) by using mathematic method, in the process of

qgregating, the weight A, ( ~ ( t ) ) i = 12, ...q of each firing rule is considered. This weight

depends on the importance of that nriag d e .

Step 3: Inference

Merence is dso cdled fuzn reasoning. Reasoning is the process by which we c m

conclude about new facts from already existing facts and causai links. Therefore, fuz.

reasoning refers to processes by which fiizzy conclusions about fuzy facts are obtained

h m fuzzy mie base or h q relations. In other words, infemce process is a decision-

making logic which determines fuzzy outputs corresponding to W f i e d inputs. with

respect to the hzq des . Thus, we should speciQ kinds of operators such as irnplicatioq

conjunction, aggregation; and reasoning operator.

Step 4: D e f d ~ c a t i o n

A conclusion is obtained fiom each d e for a certain observation, and then these

conclusions are agpgated to obtain one conclusion. The resultant conclusion is a fuzzy

set that can be defumned to obtain a crisp output Hence, defimification produces a

non-fuzzy output or a crisp output, using one of some usuaf methods [4q or particdar

method [20]- The results of deiùzz3cation is the nnal results of the whole fmq system

that is composed of four main parts.

Fuzzy algorithm represents a powerful concept which can be used to provide fllPy

models and fuzzy controllers for controt engineering. Such dgorithms are essentially a

collection of "IF' and ''THEN" statements which use the input/output fuzzy sets, and

relations dehed, to deduce the control signal necessary to achieve the desired response.

2.4 Introduction of Digüai Corttrot

In a typicai conûol system. several control loap are present. Each is designed to

perform some particulru task. The following procedure must take place in a digital

control loop:

S tep 1: M e m system output and compare with the desired value to give an error.

Step 2: Use the error, via a controi taw, to cornpute an actuating signal,

Step 3: AppIy this corrective input to the system.

Step 4: Wait for the next sampIing instant.

Step 5: Go to Step 1.

The Ioop structure which achieves this cm take many forms, but two of the most

common arrangements are shown in Figure 23. In my project, 1 utilized the second

structure (b).

Digital Ref. in T

output

Ib)

Figure 2 3 Structures for a digitai control loop

System

2.5 Description of Pendubot

The Pendubot, short for Pendulum Robot, is an electro-mechanical (or Mechatronic)

system consisting of two rigid links interconnected by revoIute joints. The nrst joint is

actuated by a DC-motor and the second joint is unactuated. Thus the second link may be

thought of as a simple pendulum whose motion can be conmiied by actuation of the fint

link. The Pendubot is sirniIar in spirit to the classicai inverted penduium on a cart

However, the nature of the dynamic couphg between the two links of the Pendubot

resuits in some interesting properties not found in these other devices.

DIA Cornputer A/D

Therefore, the Pendubot is simple robot consisting of two rïgid links. Because it also

is an undsr-actuated rnechanism, it is a nonlinear system. That mens we can control it

using various conml methods suitable for noniinear system. On the other haad as a

benchmark robo~ now 1 just study its motion controIt such as set-point regdation and

Figure 2.4 Front and side perspective drawings of the Pendubot.

The Pendubot is shown schematically in Figure 2.4. The acruated joint is driven by a

hi& toque 90VDC permanent magnet motor without gear reduction. To give joint one

direct drive contrai, we designed the Pendubot to hng off the side of a tabIe couphg

link one directiy to the shaft of the motor. The mount and bearings of the motor are then

the support for the entire system, Link ooe also inciudes the karing housing which

aiiows for the motion of joint two. NeedIe d e r bearings riding on a ground sh& were

used to construct this revolute joint for joint two. The shaft extends out both directions of

the housing alIowing couphg to both Iink two and au optical encader mounted on link

C

one. This opticai encoder produces îhe position feedback of iink two. The design gives

both links MI 360' of motion. Link o i however, cannot continuously rotate due to the

encoder cable for lii?lr two. Links have no constraint on continuous revolutions.

Two links are !4 inch (0.635 cm) thick Iength duminum. Link 1 is 6 inches (1524 cm)

long and is directly coupled to the sh& of a 90V permanent magnet DC motor mounted

on the supporting base. Link 2 is 9 inches (22.86 cm) iong and contains a coupling that

attaches to the shaft of joint two.

The final componeat of the Pendubot's hardware is its controller. See Fi,we 2.5 for a

pictoriai description of the interface between the Pendubot and the controlIer. Two

Dynamics Research Corporation 1250 countdrev resolution optical encoders, one

attached at the elbow joint and the other attached to the motor, and are used as the

feedback mechanism for the joint angies. An Advanced Motion Control's 25A8 Pm1

servo amplifier is used to drive the motor- in the control aigorithm this amplifier can be

thought of as just a gain, in the case of the Pendubot we setup the amplifier in torque

mode and adjused it for a gain of 1 V=I .2Amps.

in an attempt to simpw the controüer for the Pendubot and minimize its cos4 we

implemented our control algorithm using only the microprocessor in our PC Mead of

purchasin- an additional DSP card. We used a 486DX2/50 IBM compatible PC with a

D/A c a d and an encoder interface card. The CIO-DAC-02 w d fiom computer Boards,

Inc. is used for digital to analog conversion and a Dynarnics Research Corporation opticd

encoder card is used to intedice with the optical encoders, Controller timing is provided

by a timer board of Cornputer Boards, hc. that utilizes the 9513 tirner chip. Using the

standard s o h w e Iibrary routines suppiied with the interface cards together with our own

drives we are able to program controI aigorithms directly in Microsoft C 7.0.

Figure 2.5 Pictorid of the Pendubot's interface with its controller

The range of sample period is from 0.001 second to 0.0 16 second. The smple period i

choose is 0.005 second. This control design worked very weii. As we know, the

parameters of the controiler are changing with different sample tirne, and the d e r

sample period, the more simiIar as the continuous time controIler. However, we shodd

realize that we shouldn't onIy pursue the smaii sample penod, A 5 ms sampüng period

was used in my controllers in order to diow cornputer to Save response data whiie the

conaolIer was operating.

Chapter 3

Pendubot Mode1

in this chapter 1 discuss the dynamics of the Pendubot. The nonlinear equations of

motion are described first. This provides a mathematicai mode1 that can be used to denve

various controllers and to simuiate the response. Since the device is a two links robot

(with ody one actuator), its dynamic equations can easiiy be derived using the so-caiied

Euler-Lagrange equations and c m be found in numerous robotics textbooks. In

additional, the inertia parameters of the Pendubot have been provided by the Mechiltronic

Systems, inc.

Figure 3.1 Coordinate description of the Pendubot. 1, is the length of link one, l,, and le, are the distances to the center

of mass of the respective links and q, and q, are the joint andes of the respective links.

3.1 The Dynamics

Consider first a dynamic system defined on a configuration manifoId Q. Let

(q? q) = (ql,..., q", q l , . . . ? q n ) denote Iocd coordinates on the tangent bundIe M=TQ, We

refer to q,q, and Q as the vectors of generaiized coordinates, generalized velocities, and

generaiized acceleration, respectively. Let the system be under the action of m<n, e l ,

independent control forces andior torques, Le., there are fewer conml inputs than degrees

of fkedom. Also let r E Ra denote the vector of control variables. Thus, the actuated

degrees of Eeedom are m , and the unactuated d e p e s of fkeedom are n - m . in o u

device-Pendubot, there are just two links and one motor (see Figure 3.1). Derived h m

equation (3.3). we omit the fnction F(q) , and the external disturbances rd , so the

equations of motion for the Pendubot can be simply written in matrix:

where,

D(q) is symmetric and positive dehite, and

d,l = rn,~:~ + m,(l,' + 1:: + 21,1,, cos q,) + Il + i2

d,, = 4, = ~ ( 1 : ~ + 111,1 cos qJ + 1,

2 dz = mJCz + I2

C(q, q ) hm been chosen nieh that D - 2C is skew symmetric.

Note the O in the vector on the right side of equation 3.1, indicating the absence of

an actuator at the second joint. The symboI r i s the vector of torque applied to the links

and q is the vector of jo&t angle positions.

Other,

4, =(m,l,, +m,l,)gcosq, +mJ,,gcos(q, + q 2 )

4 2 = m , g L cos(% + 92

ml : the totd mass of link one.

1, : the lengh of link one (See Figure 3.1).

ICI : the distance to the center of m a s of Iink 1 (See Figure 3.1).

1, : the moment of inertia of iink one about its centroid.

m, : the total mass of link two.

l2 : the len@ of iink two (See Figure 3.1).

i,, : the distance to the center of m a s of link 2 (See Figure 3 -1).

& : the moment of inertia of iiuk one about its centroid

g : the acceleration of pvity.

From the above equations it is observed ttiat the seven dynamic parameters cm be

grouped into the following five parameters equations

Substituthg these panmeters (3.5) into the equations (3 2)--(3 -4) leaves the folIowing

matrices

8, + 8, + 28, COS q2 B2 + 8, COS qz

8, + 8, cos q, 4

Finaiiyt using the invemrTiIe property of the mass matrix D(q) , the state equations are

given by

[ P i ] = 9~ - DW-Q.M - w-w Qz

3.2 Introduction of Idenl~@cah~on

For a control design that neglects fiction, these 6ve panmeters are al1 needed. There

is no reason to go a step M e r and fhd the individuai parameters since the conwl

eqlliitions can be written with only the five parameters. Mechatronic Systems, hc.

performed on-Iine identification experiments using the HamiItonian based energy

equation method. The fïrst step in ttiis approach is to write the totai energy as the sum of

the kinetic energy and potentiai energy as

where D(q) is the inertia ma& deked in (3.1) and V ( q ) represents the potential

energy due to gravity. The totd energy E is h e m in the inertia parameters and so may

be written as

E = W(q, q)8 (3.12)

where 8 is the pafametrization dehed in (3.5). Using the well-known passivity or skew-

swimetry ProPertY

E = grr (3.13)

we have that, between any two times t = T and t = T + dT

Equation (3.14) cm be used with a standard ieast squares aigorithm to identib the

parameter vector 8 by applying an open Ioop signal r + s(t) and recording 7, q, q over

a given time interval. Carrying out this identification procedure using a step input to

excite the Pendubot resdted in the followhg parameter values,

3.3 The Equilibrium Manifold

Figure 3 2 Equil%rium configuration

U n d e r - a c ~ d mechmicd systems generally have equilibria thai depend on both

their kinematic and d p m i c parameters; see, for example, the Pendubot if the Pendubot

is mounted so that the joint axes are perpendicuiar to gravity, then the= wiii be a

continuum of equiiibrium canfigurations, as shown in Figure 3.2, =ch corresponding to a

constant value, F , of the input toque r . Examining the equation (3.1) we see the

equihïrium configurations are determined by

0,g cos(q, 1 + 4g cos(q, + qq,) = (3.15)

It is easily seen that, applying a constant toque ?, the Pendubot will balance at a

configuration (q, , q,} such that

The nodinear -stem Pendubot may be iinearized about an operating point

(expressed in 3-18]. Aithou& these cont&udons are unstable, the whole system is

controUabIe by using kir& of control aigorithms. The motion range of h k one is 360":

and in the eqdiirium manifold, there exists:

However, we can dso eady undersîand physicaiiy how the linerirized systern becomes

uncontroUabIe at q, = 0, k K as iiiustnted in Figure 3.3. Note that the zero or reference

position for q, is horizontal. As the Pendubot approaches these unstabie and

uncontrollable configurations, the controilability matrix of the linearized approximation

becomes increasingly iU-conditioned

Figure 33 Uncontroiiable configurations

r Actuaiiy, fiom eqyition (3.17), q, =cos-'(-) + We can ako h d if and ody if

4g

7 = 0, q, = O, +K. However, a constant value, 7, of the input toque r is impossible

equd to zero to make the second iink keep upward equiliirium. nius, the configurations

(q, = O7 +a), shown as Figure 33, are mcontroiiable. In addiaonal, the doser to these

uncontroIIable configurations the Pendubot is, the poorer the controiiabiiity of the

Pendubot is.

Chapter 4

Swing Up Control

As introduced by L.Zadeh in its seminal papers [22], fuzzy logic and approxhate

rexoning provide a h e w o r k for fonnalizing intuition and cornmon sense knowledge.

In the field of process control, these basic tools have been used to fomalize experienced

operator knowledge into fuzzy d e bases (FRB) giving rise to fuzzy logic controllers via

the h z q machinery or the fuzzy associative memory model. The FRB proposed in the

literature to describe a mdtipk-input-multipwut (MIMO) plant controUer are

usuaily stated as a coIlection of if"Then d e s such as:

If x, is &', x7is 4', ..., andx,is F,',

Then v,is G{ , v, is q, ..., and v, is G,/

with j=I2, ..., M, the x, and v, denote the linguistic input and output variabLes

respectively, whiie F j J and G,' stand for their Iinguistic qualincations which are

associated with b z y subsets. In such a knowIedge representation, each if-part of a rde

can be viewed as a description of a particular process state to which a collection of

linguisticdly specified control actions is associated Therefore, the number of d e s

depends- directly upon the number of couples {process state, controt actions) an

experienced operator c i enmerate.

DEerent methods have been proposed to establish and justify such FRB: tùpy-

model-based control, verbaikation of expert experience, observational control [El.

However, because they are based on expert knowledge andfor input-output records made

on a plant, such methods are useless when the plant is not physicaily avaiiable nor

manually controlled. On the other hanci, even when a plant is manuaiiy controlled, the

d e verbdization process itseif can be a difncuit task This is due to the existence of

cross-coupüng effects benveen the different variables. Besides these facts. the

irnplementation of a MIMO controller deriving fiom a standard FRB is much more

complicated than the im'plementation of a M I S 0 controllet [ X I . Ever though there are

some theoretical results coaceming the decomposition of muhivariable control d e s [XI

or distributed fuPy control of muhivariable systems [2q, they do not lead to a reduchon

of the size of the FRB.

in this thesis, 1 present how difXerent basic principles of fùzn logic can be used to

cope with the cross-couplhg effects which constructing srnaIl FRB for SIMO and MM0

plants. The approach 1 propose is a set in the context of feasibility studies of controllers

when no experienced operator knowledge is available. Provided they sati*

completeness, srnall FRB are attractive because they lead to easy-to-implement

controilers. To deai with the Iack of experienced operator 1 propose to use qualitative

common sense knowledge about the p h to be controued. This qualitative knowledge

conesponds to few particular process states for which the qualification of the controk

actions is easy to denve-

In this part, a fuzzy logic controller (FLC) was designed by the authos in [18] and

1191 will be described. Under the assurnption of a positive causaiity link between the

comol input and the pl& output, a joint linguistic variable (plant-state) = (E, RE) is

constnicted &t to represent the plant state [27J, the atomic variables E and RE denoting

the error and the rate of change of enor respectively. Based on the qualitative knowledge

of the cause-ef3ect relationship, and because of the lack of the plant dynamics knowledge

ody w o pIant states are stated:

E is Positive And RE is Positive,

and

E is Negative And RE is Negafive.

Using the notion ofjoint variabIe these two propositions muslate to

plant-srare is P,

and

plant-stare k N,

where P and N are respectively defked h m the Cartesian prociuct of the fuPy subsets of

the atomic variable which makes up the joint variable. Then, using the avaiiable

knowledge about the plant, two control d e s are derived which correspond to the two

situations depicted in figure 4.1:

If plant-sate is P Then controI action is P

if plant-state is N Then controI action is N

Set Point

e,(k - 1)

Figure 4.1 The two s i d o n s

In these d e s , the variable "control action" rnay stand either for the control input of

the plant or its rate of change. Note that in these two siNations the si@ of the controi

action can be defined without any ambiguity. The subsets associated with the conclusions

are primarily defmd over the unit interva1 such as:

where f is chosen to be an automorphimi of the unit interval. To guaranty the

completeness of the controller, it is sufficient that the support of each of the four input

fuzn subsets associated with the atomic variables cover the corresponding base set

ene iy . For instance, let X denotes either E or RE, these input nibsets c m be dehed

such as:

P ( x ) + ,W(x) = 1 vx E X (4.3)

XP(x) = (1 + tanh(a,x)) 12 a, > O (4-4)

The degree of Eirïng a,, i = 13 of a d e is defmed via the algebmic product hence a, is

a monotone increrising fiinction of the crisp variables e and re , whiie al is monotone

d d g . The fuay reasoning method is a simpiified version of the Tsukamoto's

method which le& to a crisp control action a :

v=lf-'(a,)-f2'(az)I o = ~ v ; G > O (4.5)

Since f is an automorphism, we conchde that for Ve c O, re < O (e > O, re > 0) the

value of v is negative {v < O}, also v = O o a,e + a,re = O . A relationship between rhis

controiier and classical Pt or PD controiiers can be derived, and results concerning the

stability of closed Ioops systems have been proved for the class of LTI-SISO plant

modeIs and some non-lineac Euler-Lagrange modeIs.

4.2 A Two-RuIeSign-Based Strategy for the Pendubot

From the above section 4.1, we can derive a novel two-de-sign-based f k q control

scheme for the Pendubot. in the Pendubot, the second link may be thought of a s a simple

penddum whose motion can be controiied by the torque applied to the h t joint.

However, the nature of dynamic coupling between the two links resuits in properties not

fotmd m the cm-pole system. Specifïcaüy, in the Pendubot there is a continuum of

equilriurn con6gurations, each corresponding to a constant value of the torque.

The control problern we consider is stated as follows: the Pendubot being in the

downward stable configuration (q, = -RIS; q, = O), the contrai god is to find a

suitable control law that will swing up the Pendubor at the inverted unstable

configuration (q, = ICI 2; q2 = 0) and successflllIy switch to the balance controiler,

The first joint is calIed the shouider (S)- connect with link one, and the other the

eibow (E)- connect wvith link two. Now we separateiy discuss the joint variable of two

joints.

SS represents the shoulder state that is made up with SE and RSE, that is

SS=(SE, RSE). (4-6)

The anguiar shouider error (SE), is defined as

The tate of change of SE (RSE), at tirne t, is defined as

where his the sarnpiing period ( in our experiment, the sampling period used to

approximate the rate of changes of the output variables is set to 0.005s).

ES represents the eibow state that is made up with EE and REE, that is

ES={EE, REEf.

The angular eibow enor (EE), is defhed as:

The rate of change of EE 0, at time t, is dehed as

(4.1 1 )

Not ody for the shoulder state (SS) but &O for the eIbow staa (ES), they are

separarety made up with MO atomic variables the an-dar emr (se or ee ) and its rate of

change (rse or tee) . Each of these atomic variabIes is associated with bvo fuzzy sets that

are Iabeied positive and negaiive.

Let ): denotes one of these atomic variables, then, h m the equations (4.3)-(4.4) in

the section 4.1, for any x E R , its reiated fuPy sets are @ven by:

Now we consider the joint shoulder as an example to derive the fuzzy control Law.

Any red value of se (resp. rse ) is @en a non zero grade of rnembership to botb input

fuzzy se&. A controI strategy for the higbest priority god is srated as:

Rulel: IF "SS is SSP* THEN force r is Positive

Rule 2: IF "SS is SSN" THEN force z is Negative

The fuzzy sets SSP and SSN of SS are defined using a quasi-linear-mean aggregating

opentor [BI, that is the membership grade of any pair (se, rse) to SSP and SSN is given

by :

SP(se, rse) = psSEP(se) + ( 1 - p, ) RSEP(rse) (4.14)

SSX(se, rse) = psSEN(se) i (1 - ps)RSEN(rse) (4.15)

where the weighting parameter ps E [0,1] is dram ikom the unit interval. It is used to

&e more or less importance to one of the two variables. These two des, having - preconditions that cover any range of sensor readings, are given the same meaning: the

more m e the precondition the more positive (resp. negative) the conclusion. Therefore,

accordiig to the equation (4.1)-(4.2) in section 4.1, we use the foIIowing output fuzn

sets as the system output:

where is related to the apptied force, r , by a scaling factor G, such that r = G,C , and

a slight modification of the reasoning method orïginaiiy proposed by Tsukamoto [2O] the

defuzpned out put is simpIy given by:

r(se, rse) = G,c = G, [SSP(se, rse) - SSN(se, me)] (4.1 8 )

Using equations (4.12) - (4.15), we cm get:

S(se, rse) = S(se, , se, )

= SSP(se, rse) - SSN(sez rse)

in the above equation (4.19), se,, + stand for k,se, k+e. The joint variable SS is

reIated to two fuay labels, SSP and SSNI fiom equations (4.12)-(4.19, ive aIso cm write

as:

SSP(se, me) + SSV((se, rse) = 1 V(se, rse) E R' (420)

SSP(se, rse) = p,SEP(se) + (1 - pz ) RSEP(rse) ( 4 3 1 )

SimiIarly, the defition of tbe eibow emir dom to dehe two categories of elbow

states, ESP (positive) and ESN (negative), cortesponding to saes where the second Iink

is falling to the ri& or to ttie Iek Using the same basic membership functions, given by

equation (4.12)-(4- G), and the same kind of aggregation operators (4.14)-(4.23, the

fuzzy sets related to ESP and ESN are such that:

ESP(ee,ree) + ESN(ee, ree) = 1 V(ee, ree) E R'

ESP(ee,ree) = peEEP(ee) + (1 - pe)REEP(ree)

Therefore, same as the derivation of (4. I9), we can ais0 get:

E(ee,ree) = E(ee,, ee,) = BP(ee,ree) - ESN(ee, ree)

= pt tanh(k,ee)' + (1 - pt ) tanh(k,ree)

= Pt t awee , + ( 1 - PJ t d ( e e , )

where eel , ee, stand for k,ee , k,ree .

in order to swing up the Pendubot anci let the two links arrive at the goal position (top)

at the same the? we shodd use the foIbwing swing-up coatrol phase:

IF "SS is SSPn ALSO IF "ES is ESPn 'ïEEN torque is Positive; -

IF "SS is SSN" ALSO IF "ES is ESN" THEN torque is Negative,

Where the Iogical connective "ALSO" is used to expiicitly give more importance to the

joint variable SS than to the joint variable ES in this swing-up controiler. Here "ALSO"

is again a quasi-linear-mean operator, with A. > 0.5 as its parameter. The mie firing

strengths are thus given by:

OS. = ASSP(se,rse) + ( 1 - A) ESP(ee,ree) (4.25)

OSN = ÂSSN(se, rse) + (1 - A) ES1V(ee, ree) (4.26)

where OS is o v e d joint variabie used to describe the q s k m state including the

shouIder joint states and the elbow joint states.

Thus as the mentioned before around equation (4.18), using the already given

reasoning method to combine the heo joint mites, and k y sets for the torque variable,

the control torque is given by:

r = G ~ S P - OSNI

= G[A(SSP(se, rse) - S N ( s e , me)) + ( 1 - R)(ESP(ee, ree) - ESN(ee, ree))]

= G[AS(se,, se,) + (1 - A)E(ee, ,ee,)] (437)

where S(se, , se2), E(ee, , ee,) are defined as equations (4.19) and (4.34).

4.3 Anaiysfs of the Conho1 Law

Both mappings S(q, se,), E(ee,, ee,) are R' + (-II), they are continuous and

nondecreasing with respect to their arguments. The control torque is thus continuous and

bounded. Given (4.19)- (4.24), and (4.23, it is easy to see that using

( S(se,,se,), E(ee,, ee,) ) as a coordinate system, the control torque changes its sign on a

straight 1i.e whose dope is - (Â 4 1 - A)) where, in accordance with the chosen highest

priority goal, R > 0.5 should be chosen caretùlly.

It should be noted that this single fuzzy f D-scheme rrquires a very fine tuning to be

successful in swinghg up and iet both Iinks access to the top unstable configuration. As

we see above, k,, k,& ,k, correspond the two atomic variabIes the anguIar error and its

rate of change in the kternal field, which depend on îhe priariiy control goal. For

instance, for tbe swing up conmller, the priorify control god is to swing link one and

Iink two to the upward inverted configuration, so k,,k, reIated with the anguIar enor

should be set close to 1. On the other hand k,, k+ reIated with the angular velocity of two

links should be set ma& certainly, k,,k, k,,k, E [0,11.

The parameters ps,p, are used to give more or t e s importance to one of the two

variables h m the extemal fietd. In ttris swing-up conwller, both of p3,pC are chosen

0.5. It means the anguIar error and its rate of change are considered as the equai

importance.

Findly, let us study the scding hctor Gin the equation (4.27), whicti is the saine

meanhg as G, in equation (4.18). h this fuzzy contrd aIgorithm, the defuzzification is

very simpIe, which is a slight modification of the reasoning method, shown in (4.27) and

(4. i 8). From the equations (4.1 9) and (4.24), we can reake that:

Because the function ''tauh3 such as tanh(se,) , ta&&) , tanh(ee,) , tanh(ee,) is dram

fTom the interval [-IJ], and the parameters p,, pe dso belong to the unit interval [0,1].

Hence, the abdute value of part [nS(se,,se,) + (1 - A)E(ee, ,ee,)] is less than or e@

to 1. That means the scaling factor G shouid be chose relative bigger. During the process

of our implement. 1 find this factor plays an important pole in whole concrol algorithm.

and its iittle chan_&g dso affects the output.

In the Chqten 8 and 9, when 1 simulate and implement our control aigorithm. 1 cm

adjust the coatroller's parameters based on the above anaiysis. 1 can conchde that there

exists ciifference between the continuous-tirne system and the discrete-the system, and

between the simulation and the actuai experiment. AIthough, the control aigorithm is

same, the tuning is important and different.

Chapter 5

Balancing Control

The conml for balancing the Pendubot is very similar to the classicd cm-pole

inverted pendulurn problem. A baIaucing configrnation is defined to be any configuration

where Ise1 i 0 and [eel E O . The highest priority goal is now to keep the second link

balanced ar the top position. Figure 5.1 illustrates the baiancing configuration at the top

position. in order to irnplement this goal. we design a PD-like fuPy balancing connalier.

The d e base is almost contnry to the swing-up controller's d e base because the goais

of both conuolIers are dif3erent.

Figure 5.1 The bdancing configuration at the top position

Baiancing the Pendubot is sùnilar to the action in acrobaties. in order to keep both

links baiancd at the unstable configuration, the motor should provided the appropriate

torque with the correct diiection and magnitude aithough this torque is too srnaII to close

ta zero. At the top unstable position, there are four possible situations for the two links,

iflustrated in Fi-pre 5.2 and Figure 53.

Figure 5.2 Thé link one on the right side around the equilibrium point

in the Figure 5.2 (a),

q, ~ ~ 1 2 , 5 se=lrl î , -q, > O ,

¶2 <O, ee=lr l2- (q,+q2)>OT

but the control torque has to be sufncientiy negative so that the second link start to Ml to

the le4 tbat is,

rcO.

Next, in the Figure 5.2 (b), the subsequent attempt to keep a balancing configuration wilI

require a positive torque to move both Iinks toward the inverted upright configuration.

Figure 5 3 The link one on the left side around the equilibrium point

Sùnilarly, in the Figuration 5.3 (a),

q, > ~ / 2 , 3 s e = ~ / 2 - q , <O,

but the controI torque has to be sdlicientiy positive so h t the second link start to faII to

the right, that is,

Next, in the Figure 5.3 (b), the subsequwt attempt to keep a baiancinç configuration ~vill

require a negative toque to move both k toward the inverted upright configuration.

5.2 Conid Law

As we defined in the Ckpter 4, the £ht joint is called the shoulder (S), and the other

the elbow (E). The @ar shouIder ermr (SE), is defined as: se = K 12 - q, ; the anguiar

elbow e m r (EE), is defined as: ee = al2 -(qI + q2) . The joint variabIe ES, describing

the elbow state is made up with EE and REE (rate of change of EE), whereas SS

(shoulder state) is made up with SE and RSE. The de£inition of the elbow error allows to

defme two categories of eIbow state, ESP (positive) and ESN (negative), curresponding

to -tes where the second link is f a b g to the right or the left.

in the control system, there are two kinds of inputs: the e m r signals se, ee and theu

rate of change rse, ree ; there is ody one output, the toque s actuated Iink one. in fact,

fiom the Chapter 4,1 can get:

Hence, 1 just put two scaluig factors to hem, that is,

k,,se, k,,rse and k,,ee, k,,ree

which are Iike to the PD controller. M e r the e m r and the rate of change, such as se and

rse , are conespond with each other by the scaling factors k,, , k,, , we can simply depict

the "SSP" and "SSN" as:

IF "se is Positive" AND IIF "rse is Positive" THEN "SS is SSP"

IF "se is Negative" AND IF "rse is Negative" THEN "SS is SSW

Here, "ANDn is the Zadeh's Iogicai "AND. As the d u e s of se and rse are ml

numbers, we cm use mathematic form to express these d e s , such as:

SS = kplse + k , p e ,

which has included both d e s "SSP" and "SSW. Similady, the elbow states can aiso

eqressed as the following:

ES = k,,ee + k , p e

Because isl the baIancing control the highest priority goal is to keep the second link

baianced, the elbow &tes are more important than the çhouider states. Thus, the

following IWO d e s are used to design a fuzzy PD-iike balancing controlier:

IF "ES is E W ALSO IF "SS is SSN" THEN torque is Positive

IF "ES is ESP" ALSO IF "SS is SSP" TBEN toque is negative

where the logicai connective "ALSO" is used to give more importance to the joint elbow

variabIe "ESn than to the joint shouider variable "SS". Here, "ALSO" is again a quasi-

linear-mean operator, with A > 0.5 as its parameter. As the discussion in Chapter 8, 1

chose 0.75 for the A .

ïherefare, the bdmcing control law is:

r=+us+(t-~)sq

= -A(kPlee + S p e ) - (1 - A)(k,,se i k p e )

Chapter 6

Stability Anabsis of Fwzy Trajectory Tracking

ControIler

The paper [14] displays a fuzzy trajectory tracking control scherne, which has been

appIied to the Pendubot in our laboratory by Sheng Cai. This aigorithm is derived h m

combining Iinear regulator theory with the Takagi-Sugeno (T-S) fuzzy methodology.

Zheng Cai illusirate the application of this algoritùm, by forcing the Pendubot to track in

real-time a sinusoiciai signai of significative amplitude, around an unstable equilibrim

point. Therefore, the implernent shows the feasibility of this aigorithm. However, the

author of [I4] failed to provide the stability d y s i s , which is very basic requirement for

the control aigorithm. In this chapter, 1 will andyze its stability.

6.1 Fuyy Control Algorithm

In the chapter 3, the dynamic ma& equation of motion for the Pendubot is $en by

equation (3.1). If the state variables are defmed as equation (3.9), the state equation of the

Pendubot can be written as:

A

x ( r ) = f (x) + g(x)u(t) = F(x(r), Nt), t ) (6.1)

where x = [q, y tj,, %, q2 1' is the state vector and u(t) = [r, O]' is the input vector. The

output is y(t) = qz .

A linear model for a speciflc equiliirium point can be obtaining using Taylor series.

For the Pendubot every equilibrium point must s a t e q, + q2 = al 2. Thus hearization

of equation (6.1) by Taylor series is @en as:

da = F, (t)& + F, (t)& (6.2)

where

x,, u, are the vaiues of x and u for the specific equilibrium point. Refer to Appendix A

for a f d I denvation of these partiai derivation terms.

Delking x := &; A := F (t); B > F, ( r ) , we have

x( t ) = Ax(t) + Bu(t) (6.5)

In the T-S fuzzy methodology 1301 [31], nodineas systems are approximated by a set

of linear local models, which is described by a set of fiizzy "IF-THEN" mies, with fuay

sets in the antecedents and locd Iinear time invariant systems in the consequents. Every i-

th d e of a T-S fuzzy model has the folIowvhg fom:

i" ~ u l e

Il? q ( t ) is M,, ,.-.! z , ( t ) is A4 ,,,..., z q ( t ) is MT

Where i = 1 ,..., r with r the number of de s ; the z,u = 1 ,..., q) are the prernise

variables, which may be functions of the States or another variables; the M,, are the

fuzzy sets; x E Rn is the state vector; u E Rm is the input vector, y E RP is the output

vector; 4, BI , andC, are matrices of adequate dimension. Therefore, the final state and

output of the fuzzy system is inferred as foIIows:

where,

W. ( ~ ( t ) ) = 1, W, (i(t)) E [OJ] for ail i, is a known nonlinear function of x(t) and

y is the membership function of r,(t) in M, .

Now Iet's consider the Iinear reguIator theory (LRT) [29] [?Il. Given a h e a r system:

x(t) = Ax(t) + Bu([) + Po( r )

b ( t ) = SW(I) (6.9)

e(t) = Gr([) + Qw(t)

where w is a vector containing extenial disturbances andor references and e is the

tracking m r . Accordingly, st& feedback and error feedback controiiers c m be

desinged as

u(t) = k.r(f) + Lm([) (6.1 O)

Although Iinear regdation theory cornes fiom hear system, many researchers have

successhlly used it in the nonIinear systern and verifïed its possibility and stability [291

[32]. Here we have lhearized the nonlineiir system (6.1) at each equiiibnum point, so we

can. use LRT and T-S fuPy methadology to control the noniinear system.

For the nonlinear system (Pendubot), we cm Iinearize it at each chosen eguilibrium

point, and then we design tocal control laws based on the LRT, for each local tinear

rnodel. The fuzzy trajectory tracking control dgorithm is described as the folIowings.

iF z, ( t ) is M z,(t) is M ,,,..., zq(r) is MT

Hence, the output of the fuzzy controUer is given as:

33

Stepl:

Let o ( t ) = 0 , the fuPy trajectory tracking control aigoritbm is simpIified a fii;T2y

state feedback closed-loop controf aigorithm [33]. Accordingly, the funy control

system becomes,

w = x-, w1 ( m 4 W + Xml {Y (WM xœl w, (m~ml

= x.l w ' ( W ) A , X ( ~ ) + x-l x.l {W. ( : ( t P 1 W ) ) B , K , x W }

= -1 14 Iw1 ( m w , ( m [ 4 + B,Kl lm

= zml zl ( w Y , ~ , x ( r ) ~ (6.15)

where H,, = A, + B,K,, i, j = 1,2 ,..., r , IV, = w,(s(t)) , and w, = w, ( ~ ( t ) ) .

To investigate the stability of (6.19, 1 empIoy the foilowing Lyapunov functioa in

quadratic form [43],

where (.)' denotes the transpose of a vector or matrix, P E Rn'"' is a symrnetric positive

definite matrix. DSerentiaiing (6.163, we have,

From (6.15) and (6.17), we get,

where O, = -(Hi P + PH,,)

design the membership fiuictions 4 (~(t)) in (6.8) as foilows,

othenvise

(6.20)

If we a s m e Zi-, w,l k ( t ) ' ~ = ~ w,~,x(r)) 1 t 0 , rubstitute the above nvo equatioa for

ml into (6.1 8), we have,

We can conciude that,

EquaIities bold when x(t) = 0.

Therefore, the above derivation has verined the stabüity of the fuzzy state feedback

contro t algorithm.

Step2:

Let w(t) # O , we should show the conditions for the solvability of this nonlinear

system (the Pendubot).

For a linear system déscriid in equation (6.9), the state feedback and enor fkedback

controiiers can be designed as equation (6.10). The Iinear regulator theory is based the

following three assumptions (accordhg to 131 1):

Al: O($) c clos(^+) = {A E c R~[A]Z

.a: the pair (A, B) is stabilizable;

A3: the pairs [C O], and i] ir deiectabk.

The first assumption does not involve a Ioss of generality because asymptoticaIly stabIe

modes in the exosystem do not affect the regdation of the output, The second assumption

is indeed necessary for aqmptotic stabiiization of the cIosed Ioop via either state or emr

feedback. ï h e third one is stmnger than the assumption of detectability of the pair

(C,A), which would be necessary for aymptotic stabilizarion of the closed Ioop via

error feedback.

Suppose Assumptions Al, A2, and A3 hold, and then the linear state feedback

reguiator problem is soIvable if and ordy if there exist matrices ïi and I' which soIve the

Iinear ma& equations

. w = A ~ + B ~ + P (6.22)

CTl+Q=CI (6.23)

As a matter of fact, conditions (6.22) and (6.23) are exactly the conditions which the

matrices II and r should fiilfill so that the folIowing Iinear mappings hold

~ ( t ) = nw(t) (6.24)

~ ( t ) = ru([) (6.29

if and ody if the conditions (6.24) and (6.15) hold, the tracking error, will approach to

zero, e(t ) + O as r -+ m , which can be shown as the follows.

e(t) = Gr(t) + Qw(t)

= Cnw(r) + Qcù(t)

=(CrIsQ)w(t) = O (6.26)

in the fupy mjectury tracking control aigorith, the nonlinear systern Pendubot is

regarded as sweral local linear model around each equilibrium point. The d e s for the

plant and controlIer are expressed in equations (6.1 1)-(6.12). Then, we can write the

tracking emr of the whole system as:

d l ) = z W, (~ [ ) [c ,L(o + QP<O] (6.27)

where w, (z(t)) is defined in (6.18), which is reIated with the weight of i-th Iocal hear

modet. As each local conml law is desigd based on the linear re-dation theory, each

tracking error ei ( t ) + O .if and only if t hnc exist matrices ïI, and T, (for i-th locai bear

system) which solve the linear matrix equations

iI,S = A,n, + B,T, +<

C'nt + Q, = O

That also can be written as:

~ ( t ) = n,o(t)

Substituting to (6.27), we can get the tracking error e(t) = 0.

Chapter 7

Combining the Controllers

The control scheme 1 proposed is to use a s o d e d logic based switching control:

design a f k z q PD-Iike controuer 1 (Chapter 4) to swing up the Iink one and link two to

the top configuration at the almost same t h e , then switch to another fuzzy PD-üke

controller II (Chapter .S) to keep the iinks balanced at the unstable position

(q, = ~ 1 2 , qi = O ) , or switch to another trajectory tracking controiier iii (Chapter 6,

implemented by Zheng Cai) to handle the Pendubot to track the desire trajectory. This

switching aigorithm witl connect each both controller together.

7.1 Combining in the Simulaîion

With both the swing up controiier and the balancing controiier complete, an dgorithm

is needed to comect them. At first, I separatety designed tbe controUers using

SIMULINK, and then designed a switching function to combine them together. tu the

whoie research process, 1 finish the parameters adjusting in the simulation and apply its

resuIts to the Pendubot. h i t i d y when working with only cornputer simulations of the

Pendubot, the controllers were switched by Merent switching functions in the

simulation of the discrete mode1 and the continuous model. Let's show the two switching

function as the followings.

In the simulation of continuous model, the switching function is same as the actuai

control dgorithm, which consists of the e m r between the current states and the goal

states (both links' position in the top configuration, that is, q, =a/2, q, =O).

Observhg the output of the torque, we cm fZnd the controihg results of the simulation

are very smooth and stable. There is no big jump for the torque during the whole

simulation. M e r the switching function was determined, 1 use the muitiport switching

biock to connect the two controiiers and the çwitching function. ï he details about this

part are illustrated in the Chapter 8. The switching funetion is chosen as:

Switching - Funcrion

where

@ l = 4,

421 = q2

& & is the logicai "AND" collation operarion

abs is the function of absolute value.

in the simuiation of the discrete model, the switching between the swing-up controIIer

and the baIancing was determined by the simdation time. When 1 change the model fiom

the continuous mode1 to the discrete model, the parameters of the controiiers has been

changed. However, this kind of controIIer is closed to the actuai digitai controller because

1 chose the same sampling h e . That means the tuning of the parameters is suitable for

the reai digital controIler whose original code is C Prograrn. The mritching time was

determined by observing when the swing-up control had brought the links almost to rest

at the desired equ%'brium position. Moreover, the current simulation time is calcdated by

the "clock" bIock that records the current simdation the . Because this switching method

is sudden and compelling, the curve of the torque dispIays a little dithering in the two

links. AIthough this method worked weii for the simulation but behaved poorly when

reaiized on the actuai Pendubot. The reason is that the simulations are exactly repeatabie

but the actuai runs are susceptible to dif3erent initial conditions and computational noise

making them unable to repeat reliably. In addition, the simuiation mode1 doesn't consider

the fiction of the whole system and other actud factor. Al1 in dl , the switching t h e is

set at 0.7 second, the switching function simply is:

Swithing - Function = 1 + (u[l J > 0.7)

where u[l] is the curren~simdation time.

7.2 Combining in the Impiementution

The foiiowing algorithm was used instead to give the Pendubot more intelligence and

switch the control by watching the States of the system.

if (!catch) ( /* if links have not corne in range to baIance */

If (fabs(x1 k-HALFPI) < -10 (

If (fabs(x3k) < 20) (

F..... /* bdancing controI or trajectory tracking controI */

Ïf(fabs(u) < 6) ( /* ifbaiancing control check is not too l q e */

- pd = 0; /* switch h m swing up control to b a h c e */

catch = 1;

1 /* endif*/

} /* endif*/

1 /* endif */

1 /* endif */

if (pd) ( /* if swing up control is stüi in use */

. . . . . . /* swing up controiier */

1 else {

. . . . . . /* balancing controiler or trajectov tracking controiier */

1 /* endif */

This algorithm wai i for iink one to arrive within 0.1 radians of its equilibrium

position and then checks iink two. If Iink two is also within 020 radians of its

eq&brium position, the baIancing control is calculated. if the control output is less then

6 volts, 9.95 volts being the maximum DAC output for the Pendubot, the conml is

switched to the baiancing mode. Otherwise the Iinks are passing too quickly through the

equilibrium point and the swing up control remains intact.

Another implementation issue that arises in the controller design of the Pendubot is

the approximation of the joint velocities. There is oniy position feedback in the system so

a finite approximation is used to estimate the velocity. To h d the velocity 1 simply used

the finite ciifference method,

x2k = (x 1 k-x 1 old) / SAMPLE; /* xI=q , , x2=q, */

x4k = (xjkdold) / SAMPLE; I* x3=q2 , x4= (ir */

This method creates numericd e m r or noise in the caicuIation of the control effort,

though, due to the finite resolution of the optical encoders. Then 1 simpIy take the

average of the last three velocities heiped to Bter and decrease this noise, shown as the

next program.

/* fiIter vel~city with an average */

x2k = (x2k+x2oId1+~201d2) / 3.0; /* average last 3 velocities to get */

x4k = (x4k+x4ddI+x401d2) / 3.0; /* rid of quantization noise *!

A dither signai was dso needed IO he1p balance the links in the top position. Due to

fxiction and the increased effect of gravity on the links in the top position, the balancing

control was not able to hold the links motionless.

Chapter 8

Simulation

Tbis chapter displays the simulation r e d i s found when simulating the Pendubot with

the SIMULINK 3.0 in MATLAB 6.0-1 start to found the Pendubot mode1 according to

the dynamic equation. The second step is to build the swing-up controiier based on the

fuay swing-up aigorithm (see chapter 4) and adjust the control parameters in order to

reach the desired results. The third step is the same as the second step to get the

simuiation mode1 of balancing controlier. in succession, I combine the above two

controlIers together using the suitable switching function to complete the whole

simdation system.

Figure 8.1 Swing-up and baiancing control

65

As we know, the control task is to bring the Pendubot fiom a straight-down resMg

position to a baianced straight-up verticai position (as show in Figure 8.1). Initial and

goal positions are ( q, = -K / 2, 9, = O ) and ( q, = a / 2, q, = O ), respectively.

8.i SUndation Pendubot Mode1

As we know fiom chapter 3, the dynamic equations of Pendubot conskt of the angies

(ql , q2), angular velocities ( q,, q2), and angular acceleration (9 , ,q2), so 1 shouid derive

the reIationship between them to build up the simulation model. We sirnulate the mode1

(3.1) using SIMLTLINK, and the derivation is shown as the following part.

Recaii the equation (3.9):

1 can cornpute and get

Therefore, connecting the three part (8.2), (8.3), and (8.4), 1 have

Figure 8.2 Subsystem 1: Pendubot mode1

in my simdation model, the function of "qlddot" is expressed as equation (8.9, and

the hmction of "Qddot" is equation (8.6). As far the torquer or T, it depends on the

feedback value computed by the controI law. ïhe real simulation block about the

Pendubot model is illustrated in Figure 8.2, which is dehed as subsystem 1.

8.2 Swing-Up SimuIation Modei

At the begimhg of the simuiating swing-up controiier, 1 just found it based on the

swing-up control Iaw, which is the equation (4.27),

r = G [ X ( s e , , se, ) + (I - R)E(ee, , ee,)] (8-7)

where

W e , 9 se2 1 = Pr t a w e , 1 + (1 - ps 2,) M ( s e 2 ) (8.8)

E(ee, 9 ee2 1 = Pt t a w e e , + (1 - /Je ) tanh(ee2) (8.9)

These fiinctions were introduced in Chapter 4. Here, 1 should deai with the huiings such

as the weight parameters pJ,pt,h, the defimification parameter G , and the scding

factors k, , k,, k3 , k, , which are defïned as:

se, = k,se , se, = k p e ,

ee, = k3ee, ee, = k p e

Firstiy, we chose the suitable scaiing factors k,,k&, k, by triai and error based on

the main goaI is to swing up two links to the top position and let them arrive there at the

same t h e as possible. The important problem that we should pay attention to is that the

angular veIocities (q,,q,) can't be too high when both Links are closed to the goal

position. Because during the balancing the Pendubot, the angular veIocities of Links are

alrnost equd to enor, the switcbg between two controiiers is not UIlStabIe if the anguiar

velocities (q, , q,) at the tirne of switching are very large. Therefore, 1 appIied the above

idea to choose the four factors, which are:

Secondiy, 1 consider the wei& parameters .us, pe, Â after the scaiing factors were

determined. 1 simply set pr, pe = 0.5, which means the importance of angies and angdar

velocities is same. Another parameter A displays the weight of two links during the

swing-up, and it depends on tbe main con~ol goal: swing up the Iink one fîrst, and make

the link two arrive at the goal position at the sarne t h e . Thus Â > 0.5, we change it in the

continuous t h e mode1 simulation and the discrete mode1 simulation.

Finaily, G is related with the acruaI toque applied to the Pendubot. If it is too srnail,

the ünk one will pass the tip position until the link two arrives there, or the accelerations

at the switching t h e is too large. if it is too big, there is not enough energy to swing up

wo links. According to our trial and error, the range of G is about 3 1-36.

Figure 8 3 Subs~stem 2: swing-up controlier

70

Figure 8.3 shows the subsystem 2 in the simulation model, where the function

S(se, , se,) cornes fiom equation (8.8), E(ee, , ee,) h m (8.9), and the function T - swing-

up cornes fiom (8.7).

8.3 The Bafancing Simulation Model

Chapter 5 has given the balancing aigonthm, and now we recall the equation (5.1)

ï' = -A(k,,ee + k,,ree) - ( 1 - A)(k,,se + k,,rse) . (8.10)

Where ee, ree,se,rse are dehed in chapter 4 and in section 8.2 of this chapter. What we

shouid do is to determine the conhot parameters k,, , k,, , k,, . k,, and the weight factor

rt . Because the pnority control goal in the balancing is to keep the unactuated Iink two

upright, the weight 1 of elbow joint is Iarge than 30% and we chose il =0.75. As for tbe

other four controi parameters, we andyzed the data of controller designed by LQR

method in order to decide our control parameters. Finally, 1 have

k,, = 21 -65, kdl = 2.75

kpz = 0.88, kdZ = 4.25

ql-in

b Ku)

b f orque-balanung TJalanung

qldot-in

q2dot-in

Figure 8.4 Subsystem 3: baiancing controller

Figure 8.4 illustrate the simulation block of Mancing controüer, and the function

T-balancing is corne fiom equation (8.10).

8.4 The Final Simulation Mode1

Step 1 :

After completing system modei, swing-up model, and balancing modeI, 1 began with

comecting system mode1 witti swing-up mode1 and get the swùiging-up simulation

results (see Section 8.5). Figure 8.5 shows the d g - u p simulation block.

Figure 8.5 Swinging up simulation

Step 2:

I just adjust the &dancing controIIer by connecting the system model and the

balancing mode[, sbown in Figure 8.6. Because the initiai condition is set based on the

anaiysis of rhe swing-up simuiation r d & , the baIancing simdation resuits are not very

important, Thus we dÏddt show them separately.

Figure 8.6 Balancing simulation

Step 3:

1 design the switching funcrion (see chapter 7) and build a mritching subsystem that is

iiiustrated in Figure 8.7.

(a) Using in the continuous tirne model (b) Using in the discrete time model

Figure 8.7 Subsystem 4: switching block

Step 4:

Connect al1 subsysterns to simulate the swinging-up and balancing the Pendubot.

Figure 8.8 shows the whole simulahg biock. The simulation red ts are in Section 8.5.

Figure 8.8 The whole simuiation system

Step 5:

1 first simtdate the continuous time mode1 and the section 8.5.1 shows the simulation

results. Then 1 replace the htegrator biock of continuous system with the Discrete-Time

Integrator and change the swïtching block Figure 8.9 shows discrete-the system modeI,

where the sarnpiing time is equal to 0.005 seconds, same as the actuai digital controUer.

Of course, we shotdd adjust the conml parameters such as A, G, k etc. Finally, Ive

simdate the discrete-time system and get ttie simulation d t s (se section 8.52).

Figure 8.9 Discrete-tirne Pendubot mode1

8.5 Simulation Resulis

There are two kinds of simulation model: one is the continuous time modeI; the other

is the discrete time model. Thus, we present the results of both of them. Since we use the

digital controller in experiments, the sampIing period is the same as that in the discrete

time simulation mûdel, Their control parameters are aIso approximate. Section 8.5.1

illustrates the simulation results for the continuous system. Section 8.52 iiiustrates the

simulation resuits for discrete system. AI1 results indude the positions of two links,

their anguiar velocities, and the torque r .

8.5. I The Simulafibit Resuiîs for the Continuou Sysîem

Part 1: the swinging up and balancing the Pendubot

q, (radian)

Figure 8.10 The output of q,

Time t (second)


Figure 8.12 The output of torque r

2 t

15

18

5

a

5

-1t

q, (radian l s)

..........................................................................

- .......................... <.............................:............-

t ............... .............. : ............................ : .............-

-.y

.............. ............. i... ........... : ............. : .............. ;

T i e t (second)

t 1 2 3 4 5

T i e t (second)

Figure 8.13 The an-dar velocity for Link one

q2 (radian 1s)

T i e t (second)

Figure 8.14 The an-dar velocity for link two

Part 2: Swinging up the Pendubot

r (N.m)

21 , f

15 - ................. ................................ L...............

............................... .............. ;. ................A..

.................................................... ................. 1

.... : ................. ; ................. L ...............

Tie t (second)

Figure 8.15 The output of r Cjust swinging up)

Time t (second)

Figure 8.16 The output of q, oust swin-ing up)

q2 (radian)

- -

!

-7 1 O

1

1.5 1 1.5 2

Tme t (second)

Figure 8.17 TIie output of qz oust swinghg up)

8.5.2 The Simulatron ResiuZts for îhe Dkweie System

Part 1 : the swinghg up and balancing the Pendubot

q, (radian)

2

-1 5 -. .. - .- . . - . . . .:. . . ... . . , . . . . .: .. .. . . . .. . .. . :. . . .. . .- -.. .-. ;. . . . .. . . . . . . .-

-2 I t 2 3 4 5

T ï e t (second)


q2 (radian)

2 3 4 5

Time t (second)


T i e t (second)

Figure 8.20 The output of torque r

q, (radian l s)

Time t (second)

Figure 8.21 The angular velocity for iink one

Tune t (second)

Figure 822 The anguiar velocity for link two

Part 2: Swinging up the pendubot

1.5 2

T i e t (second)

Figure 833 The output of r (just swinghg up)

q, (radian)

2 T i e t (second)

Figure 824 The output of q, (just swinging up)

q, (radian)

I 1.5 1 1.5 2

T i e t (second)

Figure 8.25 The output of q2 (just swinghg up)

Chapter 9

Real-time Experimental Tests

9.1 Introduction

This chapter presents the reai-time experiment results of the fuzzy controilers, using

the Pendubot Mode1 P-2 as the test machine. The controller is the digitai conmIIer

designed by the algorithms proposed in the above related chapters. In addition, we use a

graphical user interface (GUI), named MSDEVEXE, supplied by Mechaironic Systems,

Inc., to simpl* the task of m i n g controIIers, outputthg the data file.

in our cornputer, we have instdled Microsoft C t t 7.0 in the C:\MSI directory. 1 *te

the C program origind code for our controller, and then we use command "~nsmake

<filename" to compile i t ConsequentIy, 1 get the executive fies named the same with

"fïiename". F i l y , I nin r d tirne controllers in the Widows 3.1 environments. In

Widows 3.1 we mate a PIF file for our controlIers. Use the PIF file editor ta create a pif

file that

1. indicates the name of our executable,

2. NI~S in a fui1 screen DOS box (Display Usage: Fui1 Screen),

3. nms in exclusive mode (Ekecution: Exclusive) and

4. sets as the Start-Up Directory the directory where the executable is located.

In our experiment, 1 adjusted the controlliing parameters in tbe controlling program.

We find there is a little clifference cornparing with those in the simulation. However, this

result show the différence between the actuai Pendubot and the simuIation mode], and

this is aiso discussed in the next chapter.

ïhe experiment is divided to two pm. The £irst part is to design the çwinging-up and

balancing controuer in order to swing up the Pendubot f?om the rest configuration

( q, = -x 12, q', = 0 ) and to baiance it at the unstable position (q, = K 13, q2 = O 1.

The second part is to combine our swing up controller with the trajectory tracking

controIIer (this controller is designed by Zheng Cai). The clifference of switchin;

algorithm between the two parts is just using the different limited value of (q, - x / 2 )

. . . .*.

Section 9.2 is the flow chart of our conrroller. Section 9.3 describes the experirnental

results of the fïrst part; plotted by MdIe according to the output data. Section 9.4

describes the photographs of experiment for two parts.

9.2 FIow Cliart

The next five pages is the flow chart of my controbg program.

Exit and End

t Conuol Circulation

Calcufate raw velocity x2k<xI k-XI old)/SAMPLE;

Filter vchcity with an averag .*(x2k+~ldl+x2oIdtyj.O; x4k(x4k+x401dl+x4oIPy3.0;

f I

\ CacuIate Swingkg up Contm t Toque

Sfunuio W... By quaion (4.19) haion=- By equiuion (4.24)

IF- By qwbn (4.27')

CaIculatt Baiamhg Control Toque By Fiption (5.1)

in Chaptcr 5

output and save the / .-.YI /

Output &a fi& P

9.3 Eiperimentd Ra& for Swngingq and B d ~ c i n g

From the flow chart in the above section 92, we can see that, in each loop, the system

torque ( r ), and cunent Ume (t), and then output these states to the data file. When the

digital controller stops, the system output the data me. Hereby, we can get one data file

after we operate the controller one t h e . This data file is saved as M-file of MA?ZAB.

Finaiiy, we reprogram this data file and run it to pIot the effect of each variable such as

Figure 9.1-9.5 shows the experimental resuIts for swinging-up and balancing.

Comparing with the simuIation resuits of discrete system, shown in Figure 8.1 8-8.22, the

torque of the system is close but not equal to zero; both links actually have a little dither,

and the controller keeps them balancing at the unstable position.

Fignre 9.1 Effect of torque

Figure 9 3 mect of angle for iink two

- Figure 92 Effect of angle for link one

Figure 9.4 Effect of angular velocity for link one

Figure 9.5 Effect of angdar veIocity for link two

Figure 96 iIlustmtes the Pendubot itt the rest configuration. Fimm 9.7 illustrates the

Pendubot balancing at the top UILStab1e configuration. Figure 9.8 and 9.9 illustrate two

positions whiie the Pendubot is tracking a desirrd irajectory.

Figure 9.6 The Pendubot at the rest conti,oution

Figure 9.7 Balancing at the top configuration

94 -

Figure 9.8 The Pendubot on the left side

Figure 9.9 The fendubot on the right side

Chapter 10

Conchsions and Recommendations

for Future Work

The study of under-ainiated mechanisms is a topic of current research in the control

community. These systems are characterized by the fact that they have fewer independent

control actuators than degrees of fixedom to be controiled, Different control schemes,

derived fkom noniinear systems theory 121 [49], classical control techniques [3] [SI, or

intelligent control methodologies [I I ] [501, have been proposed. From the recent research

papers, we can find that the fuPy control method attracts many researchers because of its

robust characteristic.

In brief, the fuzzy control method used in the Pendubot, can be divided into severai

cases. First, the practicai techniques based on genetic algorithms (GA), are introduce for

the automatic tuning [50] [I 11. The second case is to design a fiizzy PD controlIer for the

under-actuated robot swing-up control (151 [SI], and this method can automaticaüy

change the membership fimctions of the fuay PD scherne. The third case is to convert

the noniinear system to a set of linear Iocd system rnodel at each equiiibrium point. based

on the Takagi-Sugeno fuzzy methodology [I4], and then they applied the liear control

method to control the systern based on the fuzzy mie base. Comparing to the above

methods, the thesis presents a two-de-base fiizzy control scheme based on quasi-Iinear-

mean aggregating operator and Tsukamoto's reasoning.

From another point of view, the research to the under-actuated mechanism inchdes

the foliowing aspects:

1 . Identification. There are off-he or on-line, fuzzy method or classicai method,

energy method or optimization methd

2. Swing up and Balonce. This is a set-point regdation, and I can design various

controllers using various control methods, such as passivity or energy method,

heuristic, fuPy logic, partial feedback Iinearization, machine leamhg etc.

3. Switching C o n ~ o l . So-cal1ed Iogic based mvitching control is widely proposed:

the logic condition may be the error of angle or the distance to the goal position

[15]. 1 use the e m r of angle or use the switching time as the switching

parameters.

4. Trajectory Tracking. This is a new study direct for the under-actuated mechanism.

There is a IittIe contribution besides the paper [14] [4].

10.2 Conclusions

in my study, a fùzq scheme for swing-up controI is proposed and implemented in

the actuai under-actuated mechanisni - the Pendubot, Comparing with the previous

partial feedback Iinearization method, I find that, the swing-up speed is faster and the

stability is certainly good. This.method is simpler than the classicai conîrol method;

however, the timing is a Little dficuit for the simulation or the experiment, The

proposai method qresents a new class of fuzzy controls. The real-the

irnplementation also shows the success of the application and feasibiiity of the fiiPy

control method to the nodhear system.

In my simulation, we start with the continuous time systern, and then change it

into the discrete time m m . We conclude that the control parameters are different

for these two control systems although the sampling Ume is 0.005. We also realize

that the smaller the sampIing time is, the more approximate the two different systems

are. Comparing with the actual experimeatd model, the simulation model is ody a

mathematization model that ignores some existin; factors such as friction and

extemal disturbance.

-4nother importance part is the stabiiity analysis of the fupy scheme for the

Pendubot trajectory tracking. 1 verified the possibiiity and vaiidity of a fuzzy state

feedback control scheme and a f b y noniinear regdation control method. The

stability proof gives us the suggestion that we can appIy Iinear control theory and

fuzzy control method to sîudy the noniinear system.

Finaliy, in our experiment, digital computer pIays the role of di8;ta.i contmlier.

From our experiment, 1 conclude that, for controlhg systems with slow dynamics

this wiii be indistinguishable fiom the performance give by a continuous controiier,

-but for systems that respond at higher fiequencies 1 may encounter problems in

achiewig control. Therefore the cycle speed, or the sampling penod, pIays an

-important roIe in the design of the digitai controIIer:

10.3 Recommendations for Future Work

1. Generaluation

The fiizzy contra[ scheme developed in this thesis has been successfùily

applied to the Pendubot, The Pendubot is an example of the under-actuated

mechanics. Therefore, the future work is to generate this fuzzy control method to

general under-actuated system. Furthermore, in order to generalize this controI

method, we should design an automatic method to complete the controller's

parameters tuning; for example, using cornputer program to search a best set of

controI parameters.

As we know, ive start our research from the simulation using SIMCILINK, and we

can also program M-file code to simulate the control system. Furthermore, it is better

for the discrete time system. According to the expianation of function of MATLAB

[53] [54], M-hies can be directly changed to the execute fiIes. Therefore, we cm use

the M-fies to SimuIate the control system and directiy to conml the actual modeI:

the Pendubot.

3. Trajectoy Tracking

For the noniiiear system, trajectory tracking is a Little more dfacult than the set-

point regulation. ïhe research results [32] 1291 for this aspect are limited, and the

application to the Pendubot is vars [I4]. Thesefore, we are inveçtigating to apply

other modem control methods, except the linear regdation theory and T-S fuPy

methodology, to force the Pendubot to ûack a desired trajectory.

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Appendix A: Linearized equations

(A. 10)

Fuzzy Control for an under-actuated Robotic

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