Shahab Thesis

7/31/2019 Shahab Thesis

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DISCRETE EVENT MODELING, SIMULATION ANDCONTROL WITH APPLICATION TO SENSOR BASED

INTELLIGENT MOBILE ROBOTICS

BY

Shahab Sheikh-Bahaei

B.S., Electrical Engineering, Isfahan University of Technology, 1999

THESIS

Submitted in Partial Fulfillment of theRequirements for the Degree of

Master of ScienceElectrical Engineering

The University of New MexicoAlbuquerque, New Mexico

December 2003


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To my parents, to whom I owe all that I am today;

to my wife Nasim, to whom I owe my happy life;

and to my wonderful in-laws.

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ACKNOWLEDGEMENTS

I gratefully acknowledge the advice, comments, corrections, encouragements andsupport of my advisor Professor Mo Jamshidi, Director of Center for AutonomousControl Engineering (ACE) and Regents Professor of Electrical and ComputerEngineering at the University of New Mexico.

I am grateful to the members of my committee, ProfessorThomas Caudell, Directorof Visualization Laboratory at High Performance Computing, Education and ResearchCenter and Associate Professor of Electrical and Computer Engineering and ComputerScience, and Professor Chris Smith, Assistant Professor of Electrical and ComputerEngineering, for being in my committee and for their helpful comments and suggestions.Also my special thanks to the fourth, unofficial member of this committee, Dr. NaderVadiee, Director of the UNM-NASA PURSUE Program, for his advice and supportivecomments. My thanks to my uncle, Professor Mansoor Sheik-Bahae, Professor ofPhysics and Astronomy at the University of New Mexico, ProfessorBernard Zeigler ofthe University of Arizona, and the faculty and staff at the University of New MexicoDepartment of Electrical and Computer Engineering. I am also grateful to my colleaguesand friends at UNM, and thankful to the Center for Autonomous Control Engineering(ACE) for financial support of this work1.

And finally, and the most importantly, I gratefully acknowledge the love and spiritualsupport of my wife, Nasim Sassan. This work definitely could not have been

accomplished without her support and patience.

iv1 This work is partially supported by NASA-Ames Research Center Grant number NAG 2-1457


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Discrete Event Modeling, Simulation and

Control with Application to Sensor Based

Intelligent Mobile Robotics

BY


ABSTRACT OF THESIS

Submitted in Partial Fulfillment of theRequirements for the Degree of

Master of ScienceElectrical Engineering

The University of New MexicoAlbuquerque, New Mexico

December 2003


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Discrete Event Modeling, Simulation and Control

with Application to Sensor Based Intelligent Mobile Robotics

by


B.S., Electrical Engineering, Isfahan University of Technology, 1999

M.S., Electrical Engineering, University of New Mexico, 2003

ABSTRACT

This work is focused on modeling, simulation and control of cooperative mobile

robots usingDiscrete Event System Specifications (DEVS), which is a young approach in

modeling and control. Discrete Event approximation of differential equations is also

studied and improved by introducing DEVS-Double Integration method.

Also Fuzzy Logic Control (FLC) and Stochastic Learning Automaton (SLA) are

designed and implemented under DEVS framework. In this work we propose an

Intelligent Discrete Event (I-DEVS) approach to control autonomous agents, i.e. we

introduce a discrete event version of Fuzzy Logic Control via DEVS, which improves the

speed of Fuzzy-Logic Control and makes it more suitable for real-time adaptive control.

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A discrete event simulation environment for modeling and simulation of multi-agent

systems called V-LAB is also partially developed in this research. Finally the simulated

algorithms are implemented on a real commercial robot.

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Contents

LIST OF FIGURES...............................................................................................x

1. Introduction .......................................................................................................... 1

2. DEVS Formalism................................................................................................. 3

2.1 Introduction to Discrete Event Dynamical Systems (DEDS)........................... 3

2.2 Review of DEVS formalism............................................................................. 5

2.2.1 Classic DEVS Atomic Models .............................................................. 5

2.2.2 Classic DEVS Coupled Models............................................................. 7

2.2.3 DEVS Examples .................................................................................... 9

2.3 DEVS&DESS: Combined Discrete Event and Differential Equation Specified Systems..... 13

2.3.1 DEVS Abstraction for Output-Partitioned DEVS&DESS Models ..... 14

2.3.2 DEVS First Order Numerical Integrator ............................................. 17

2.3.3 DEVS Double Integrator ..................................................................... 22

2.4 Discrete Event Control (DEC)........................................................................ 30

2.4.1 Introduction ......................................................................................... 30

2.4.2 Controllability of DEVS Models......................................................... 30

2.5 Intelligent DEVS ............................................................................................ 32

2.5.1 DEVS-Fuzzy Logic ............................................................................. 32

2.5.2 DEVS-SLA.......................................................................................... 38

3. Mobile robotics .................................................................................................. 43

3.1 Introduction .................................................................................................... 43viii


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3.2 Modeling and Controllability of Mobile Robots ............................................ 44

3.3 Motion Control of Mobile Robots .................................................................. 48

3.3.1 Path Planning Methods........................................................................ 49

3.3.2 Obstacle Avoidance............................................................................. 56

3.3.3 Cooperative Robots ............................................................................. 59

3.3.4 Job Evaluation ..................................................................................... 65

4. V-LAB Simulations ........................................................................................ 71

4.1 Simulation Components ................................................................................. 73

4.1.1 Dynamics Model.................................................................................. 73

4.1.2 Collision Model ................................................................................... 74

4.1.3 Terrain Model ...................................................................................... 83

4.1.4 Robot Model ........................................................................................ 84

4.2 Simulation Results.......................................................................................... 85

5. Experiment Results ............................................................................................ 88

6. Conclusions and discussions .............................................................................. 91

7. References .......................................................................................................... 95

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List of Figures

Figure 2.1 A 3-State Machine............................................................................................. 4

Figure 2.2- a) DEVS Behavior,........................................................................................... 7

Figure 2.3. A simple coupled model, composed of three atomic models.......................... 9

Figure 2.4 Graphical notation: (a) external transition......................................................... 9

Figure 2.5. Pulse Generator, an example for a DEVS atomic model ............................... 10

Figure 2.6. A DEVS coupled model example: an integratormade jointly by a Generator

and a Summer............................................................................................................. 11

Figure 2.7- The 3-State Machine in DEVS notation......................................................... 13

Figure 2.8. DEVS representation of DEVS&DESS ......................................................... 16

Figure 2.9-DEVS first order numerical integrator............................................................ 19

Figure 2.10. A DEVS coupled model for Simulation of a free falling mass. ................... 20

Figure 2.11. DEVS implementation of Example 2.5........................................................ 20

Figure 2.12. Simulation of a falling mass using two first order DEVS integrators with

initial velocity v0=20m/s, and initial position x0=100m and Quantum=1. (a) Position

vs. time: DEVS results compared to the exact solution. (b) Position Error vs. time. 21

Figure 2.13 Step responses for the systems of Example2.5 using DEVS integrator. (a)

Step response for system of equation 2.14 (b) Step response for system of equation

2.15 ............................................................................................................................ 22

Figure 2.14. DEVS Double Integrator .............................................................................. 25

Figure 2.15. Determining the time advance value for n-dimensional DEVS double

integrator.................................................................................................................... 26Figure 2.16- Searching the minimum positive root of equation 2.28. .............................. 28

Figure 2.17- Simulation of a falling mass using one DEVS double integrator with initial

velocity v0=20m/s, and initial position x0=100m and Quantum=1. (a) Position vs.

time: DEVS results compared to the exact solution. ................................................. 29

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Figure 2.18- Controllability Definitions: (a) weakly controllable (b) strongly controllable

(c) very strongly controllable path............................................................................. 32

Figure 2.19 A typical consequent membership function (CMF) model ........................... 34

Figure 2.20 A DEVS model for a typical fuzzy rule IF x is A OR y is B THEN z is C35

Figure 2.21. Example of using fuzzy-DEVS to control an inverted pendulum................ 37

Figure 2.22- Learning from action: gaining reward while avoiding penalty. ................... 38

Figure 2.23 Schematic of SLA inside the DEVS Environment.......................................... 39

Figure 2.24. Possible results of an example for SLA_DEVS........................................... 41

Figure 2.25- A mobile automaton learns to reach a goal point using SLA. (a) Action

probabilities vs. time (b) trajectory of the automaton (c) x and y vs. time................ 42

Figure 3.1. Robot Functional Block Diagram................................................................... 43

Figure 3.2- A differential-wheeled mobile robot.............................................................. 44

Figure 3.3. Robot Motion Control .................................................................................... 45

Figure 3.4. Kinematic Control .......................................................................................... 47

Figure 3.5. The system of equation 3.7 for a=1 and b=4;................................................. 51

Figure 3.6. Effect of changing b from 4a down to a......................................................... 52

Figure 3.7. Trajectory of the system of equation 3.7 for different initial conditions........ 52

Figure 3.8. Using time-varying a and b as defined in equation 3.8.................................. 53

Figure 3.9. Membership function used in Fuzzy-Logic Obstacle avoidance ................... 58

Figure 3.10. 3D surfaces generated by the proposed fuzzy logic rule base to perform

obstacle avoidance..................................................................................................... 59

Figure 3.11. Robot trying to push round objects to storage area ..................................... 60

Figure 3.12. A DEVS model for cooperative pushing task .............................................. 61

Figure 3.13. Pushing task performed by a single robot .................................................... 63

Figure 3.14. Pushing task by two robots........................................................................... 65

Figure 3.15. Membership functions for Objective Function, Time and Mission Success

Degree........................................................................................................................ 67

Figure 3.16- Cooperative box-pushing task...................................................................... 70

Figure 4.1Distributing simulation layers using DEVS in V- Lab ................................ 72

Figure 4.2. DEVS coupled model in V-LAB for multi-agent ....................................... 73

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Figure 4.3- Dynamic model contains DEVS-double integrators to simulate moving-

obstacles and robots................................................................................................... 74

Figure 4.4. Examples of colliding polygons: case (a) is more likely to occur than case (b)

................................................................................................................................... 75

Figure 4.5. Evaluation of functionP(x) (equation 4.1) at different areas......................... 76

Figure 4.6. Hard Collision ................................................................................................ 78

Figure 4.7- Three masses pushing each other................................................................... 81

Figure 4.8- Terrain Model is responsible for calculating the distance of the nearest

obstacle to each IR-sensor. ........................................................................................ 83

Figure 4.9- A DEVS coupled model for a differential-drive wheel robot with n infra-red

sensors, one compass and one position sensor .......................................................... 84

Figure 4.10. V-Lab components for autonomous multi-agent simulation implemented in

DEVS......................................................................................................................... 86

Figure 4.11- 3D views from V-Lab simulation ............................................................ 86

Figure 4.12- Path planning among unknown obstacles .................................................... 87

Figure 4.13. Cooperative object pushing .......................................................................... 87

Figure 5.1- Pioneer Robot made by ActivMedia Robotics.............................................. 88

Figure 5.2- Path planning: (a)Left and right velocities (b) Trajectory of the robot (c) State

variables of the robot vs. time (d) View from V-Lab showing the final position ofthe robot and its trajectory......................................................................................... 90

Figure 5.3- Path planning while avoiding obstacles ......................................................... 90

Figure 5.4- Object pushing task. (a) V-Lab real-time view (b) Robot pushing a bottle 91

Figure 5.5- Object pushing task by a single robot ............................................................ 91

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1

1. INTRODUCTION

The field of robotics involves several areas such as control engineering, signal

processing, planning, mechanical engineering, applied mathematics, real-time systems,

software engineering, artificial intelligence and machine learning. Robots perform a wide

variety of tasks in the real world. Due to the large degree of flexibility of the robotic

systems and the complexity of possible task domains, it is important to design control

architectures, which can handle complex systems in unstructured environment. Much

research has been done so far by researchers on solving different robotic problems.

The word Robot refers to two classes of autonomous machines: one is known as

Robotic Arm, and the other one Mobile Robot. Mobile robots, move relative to the

environment they are in, exhibiting some degree of autonomy. Wheeled-Robots and

Walking-Robots (legged-robots) are two basic categories of mobile robots. Applications

of mobile robots are in autonomous moving machines such as: cars, trains, carts and

plains, moreover tasks like painting, cleaning, inspection can be performed by mobile

robots, specially in dangerous, boring or difficult situations where human cant work.

Discrete-event modeling and simulation of mobile robots in a virtual environment is

one of the main aspects of this research. Discrete event modeling and simulation has been

introduced recently as a new tool in modeling, simulation and control.

Work on mathematical foundation of discrete event dynamic modeling and simulation

initiated in the 70s by Zeigler, when DEVS (discrete event system specification) was

introduced as an abstract formalism for discrete event modeling[1]. Because of its system

theoretical basis, DEVS is a general formalism for discrete event dynamical system


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Introduction 2

(DEDS). DEVS can be used to specify systems whose input, state and output trajectories

are piecewise constant [2]. Recently, the Control of Discrete Event Systems is receiving

more attention[3],[4],[5]. Discrete-Event Systems (DES) control theory developed by

Ramadge and Wonham[5], has been used to control several different robotic problems

including manufacturing and assembly tasks, the coordination of mobile robotic agents

[6],[7], a grasping task performed under the supervision of a vision system [8], modeling

dexterous manipulation [9]a hybrid discrete event dynamic systems approach to robot

control [10],[11].

On the other hand, soft computing techniques, e.g. fuzzy logic, genetic algorithms

and neural networks are proven to be useful in control of autonomous agents. Fuzzy logic

control provides a human-like decision making methodology, which has been used

widely in the field of robotics and automation [12]. Design and implementation of fuzzy

logic control under DEVS framework is one of the contributions of this work.

The goal of this thesis is to investigate the power of DEVS in modeling, simulation

and control of sensor-based intelligent mobile robots. First we review the DEVS

formalism in Section 2, also DEVS differential equation solvers, DEVS numerical

integration methods, as well as Fuzzy-DEVS and SLA-DEVS are presented in this

section. In Section 3, modeling, control architecture and controllability of mobile robots,

fuzzy logic based obstacle avoidance, and cooperative mobile robots are discussed. In

Section 4, V-LAB a discrete-event-environment in which mobile robots can be

modeled and simulated is introduced. Simulation results are also presented in this section.

In Section 5, real world experiments results with an ActivMedia Pioneer AT2 mobile

robot are presented.


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3

2. DEVS FORMALISM

2.1 Introduction to Discrete Event Dynamical Systems (DEDS)

Classical dynamic system models, i.e. differential equations based models, have been

developed to describe natural physical systems. Differential Equation System

Specifications (DESS) has been used widely for modeling, simulation and control of

various systems. Since late 20th

century computer has been employed as a basic tool to

simulate physical systems by solving differential equations. Since then the concepts of

discrete time, sampling and difference equations are introduced. Discrete-time models

timing is based on equally spaced time intervals called step-size or sample time. The

accuracy of the model depends on the sample time: smaller sample times provide better

accuracy. But the tradeoff is that the calculation time rises with small sample times.

In contrast, widely developments of man-made systems, such as manufacturing

systems of various types need more general models in order to describe different states

and performances of the systems. Discrete Event Dynamic Systems (DEDS) provides an

explanatory framework for such systems. DEDS can be used to model the event-driven

systems common to man-made systems.

A discrete event system is a dynamic system whose behavior is guided by the

occurrence of discrete events. Although only certain events can occur in any given

situation, the time and the intervals in which they appear is known in general.

Example: A classical example of a discrete event system is a 3-state machine that can

be Idle, Working orBroken-Down. The transition diagram is shown in Figure 2.1


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DEVS Formalism 4

Working

Idle

FIX

BRK

DONE

ON

Broken-Down

Figure 2.1 A 3-State Machine

The initial state is Idle. The event ON denotes the turn-on of the machine, while the

event DONE the completion of a task, BRKa break-down and FIX a completion of

repair.

In response to industrial needs for DEDS models, several simulation packages have

been developed [11]. A discrete event system specification, called DEVS formalism has

been helpful in the development of such software. DEVS is a theoretical formalism for

discrete event dynamical systems (DEDS). It can be used to specify systems whose input,

state and output trajectories are piecewise constant [2]. Several software tools have been

developed for the simulation of DEVS models. The most popular are DEVS-Java [13]

and DEVSim++[14]. In this work we use DEVS-Java to implement our DEVS models.


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DEVS Formalism 5

2.2 Review of DEVS formalism

The DEVS formalism specifies discrete event models in a modular hierarchical form.

In this formalism a discrete event model is decomposed into smaller models, each defines

a discrete event model independently.

2.2.1 Classic DEVS Atomic Models

A 7-tuple specifies the lowest level of the model hierarchy called atomic model. The

following is the classic definition for an atomic DEVS[1]:

DEVS=(X, Y, S, ext, int, , ta)

where

X : is the set of inputs,

Y: is the set of outputs,

S: is the set of sequential states,

ext :Q X S is the external state transition function,

where Q={(s,e) | sS, 0 e < ta(s) } is the total state,

e is the elapsed time since last event.

int : S S is the internal state transition function,

: S Y is the output function,

ta : S +{0,} is the time advance function.

The interpretation of the above elements is as follows: at any given time the system is

at some state, s. If no external event occurs it will remain in the same state for a period of


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DEVS Formalism 6

ta(s) seconds, after which an output equal to the value, (s), is generated and the system

state changed to int(s). In the case that ta(s) is equal to infinity, the system will remain in

the same state, s, until an external event occurs. In other words, if no external event

occurs, system will remain in the same state forever. The first case (ta(s)


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DEVS Formalism 7

X

Classic DEVS

atomic model

OutputInput

t0 t1 t2S (b)s3s1

s0s2

e Out1

Out2

Out3

Parallel DEVS

atomic model

In1

In2

ta(s1)

t0 t1 t2

Y

(c)

t1

(a)

Figure 2.2- a) DEVS Behavior,Figure 2.2- a) DEVS Behavior,

b) Graphical notation forClassic andParallelmodelsb) Graphical notation forClassic andParallelmodels

2.2.2 Classic DEVS Coupled Models2.2.2 Classic DEVS Coupled Models

Basically, a DEVS coupledmodel is composed of DEVS components i.e. atomic and

coupled models, by defining a coupling relation between them.

Basically, a DEVS coupledmodel is composed of DEVS components i.e. atomic and

coupled models, by defining a coupling relation between them.

Here is the definition of DEVS coupled models [1]:Here is the definition of DEVS coupled models [1]:

N=(X,Y,D,{Md|dD},EIC,EOC,IC),N=(X,Y,D,{Md|dD},EIC,EOC,IC),

wherewhere

X={(p,v) | pIPorts, vXp } is the set of input ports and values X={(p,v) | pIPorts, vXp } is the set of input ports and values

Y={(p,v) | pOPorts, vYp } is the set of output ports and values Y={(p,v) | pOPorts, vYp } is the set of output ports and values


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DEVS Formalism 8

D is the set of the component names.

Components are DEVS models, for each dD,

Md=( Xd,Yd,S, ext, int, , ta) is a DEVS (2.1)

with Xd={(p,v)|pIPortsd, vXp }

Yd={(p,v)|pOportsd,vYp}

External Input Coupling connects external inputs to component inputs,

EIC { ( (N, ipN), (d, ipa) ) | ipNIPorts, dD, ipdIPortsd}. ( 2.2)

External Output Coupling connects component outputs to external outputs,

EOC { ( (d, opd), (N, opN) ) | opNOPorts, dD, opdOPortsd}. ( 2.3)

Internal Coupling connects component outputs to component inputs,

IC { ( (a, opa), ( b, ipd) ) | a,b D, opa Oportsa, ipb IPortsb }. ( 2.4)

However, no direct feedback loops are allowed, i.e., no output port of a

component may be connected to an input port of the same component.

Figure 2.3 shows a simple example of a coupled model.


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DEVS FormalismFormalism 9

9

Coupled

Coupled

outin

outin A3

outin A2

outin A1

Figure 2.3. A simple coupled model, composed of three atomic modelsFigure 2.3. A simple coupled model, composed of three atomic models

2.2.3 DEVS Examples2.2.3 DEVS Examples

Graphical notation: Our graphical notation for state transitions is shown in Figure

2.4 where (a) represents an external transition. An input event is specified using ?. For

instance input event in?m means that a message (event) m is input at the input port

in. Figure 2.4(b) denotes an internal transition. An output event is specified using !.

A dotted line represents an internal state transition specified by the internal transition

function. A solid line represents a state transition specified by the external transition

function.

Graphical notation: Our graphical notation for state transitions is shown in Figure

2.4 where (a) represents an external transition. An input event is specified using ?. For

instance input event in?m means that a message (event) m is input at the input port

in. Figure 2.4(b) denotes an internal transition. An output event is specified using !.

A dotted line represents an internal state transition specified by the internal transition

function. A solid line represents a state transition specified by the external transition

function.

si

sj

p?m

si

sjp?m

si

sj

(si) = p!m

si

sjp!m

(a) (b)

Figure 2.4 Graphical notation: (a) external transitionFigure 2.4 Graphical notation: (a) external transition

(b) internal transition(b) internal transition


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DEVS Formalismormalism 10

10

Example 2.1. GeneratorExample 2.1. Generator

As an example consider an atomic model with single input and single output. The

model generates output events equal to 1 or 1, and the frequency of the output events is

proportional to the input value (Figure 2.5). Negative outputs occur when the input is

negative.

As an example consider an atomic model with single input and single output. The

model generates output events equal to 1 or 1, and the frequency of the output events is

proportional to the input value (Figure 2.5). Negative outputs occur when the input is

negative.

X= rate

-1

1

Y

YGenerator

Figure 2.5. Pulse Generator, an example for a DEVS atomic modelFigure 2.5. Pulse Generator, an example for a DEVS atomic model

Here is the formal definition of such a model:Here is the formal definition of such a model:

GEN =(X, Y, S, ext, int, , ta),GEN =(X, Y, S, ext, int, , ta),

X=,X=,

Y={-1,1},Y={-1,1},

S={-1,1},S={-1,1},

ext(s,x,e)=sign(x),ext(s,x,e)=sign(x),

int(s,x)=sign(s),int(s,x)=sign(s),

(s)=s,(s)=s,

0

( , )0

if x

ta s x qif x

x

=

=

where the constant q is a real number called quantum.


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DEVS Formalismmalism 11

11

Example 2.2. SummerExample 2.2. Summer

As another example consider an atomic model which sums up the input value with the

current state, and sends the result immediately.

As another example consider an atomic model which sums up the input value with the

current state, and sends the result immediately.

SUM =(X, Y, S, ext, int, , ta),SUM =(X, Y, S, ext, int, , ta),

X=,X=,

Y=,Y=,

S=,S=,

ext(s,x,e)=s+x,ext(s,x,e)=s+x,

int(s,x)=s,int(s,x)=s,

(s)=s,(s)=s,

ta(s,x)=0.ta(s,x)=0.

Example 2.3. A simple First Order Ordinary Differential Equation solverExample 2.3. A simple First Order Ordinary Differential Equation solver

Using the atomic models mentioned in the two previous examples, in a DEVS couple

model, a simple integrator can be made.

Using the atomic models mentioned in the two previous examples, in a DEVS couple

model, a simple integrator can be made.

Figure 2.6. A DEVS coupled model example: an integratormade jointly by a Generatorand a Summer.Figure 2.6. A DEVS coupled model example: an integratormade jointly by a Generatorand a Summer.

1 2 3

1 20 1 2

0

t t t

t ts x dt x dt x dt+ +

t3

t2t1

x2

x1

x0

Time

S

G

X

SGXSummerGenerator


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DEVS Formalism 12

Example 2.4. Consider the 3-state machine discussed in section 2.1 again. This

system can be described by a DEVS atomic model as follows:

M=(X, Y, S, ext, int, , ta),

X={ON, FIX}

Y=S={Idle, Working, Broken-Down},

if (s = "Broken-down" and x = "FIX")

if (s = "Idle" and x = "ON")

" "( , , )

" "ext

Idles x e

Working

=

int

if (s ="Working" and event "DONE" occurs)

if (s ="Working" and event "BRK" occurs)

otherwise

" "

( ) " "

Idle

s Broken Down

s

=

if s="Working"

otherwise

0( , )ta s x

=

(s)=s.

Figure 2.7 shows the state transition diagram for this atomic model. As shown in the

figure, the model has one input port in and one output port out.


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13

outin

out!BRK

in?FIXin?ON

out!DONE

Down

Broken-Working

Idle

Figure 2.7- The 3-State Machine in DEVS notationFigure 2.7- The 3-State Machine in DEVS notation

2.3 DEVS&DESS: Combined Discrete Event and Differential Equation

Specified Systems

2.3 DEVS&DESS: Combined Discrete Event and Differential Equation

Specified Systems

Since DEVS representation applies to a special subclass of dynamical systems, output-

partitioned DEVS&DESS formalism for hybrid system modeling is introduced by Zeigler

et al [1],[2] as an extension of the DEVS formalism for combined discrete/continuous

modeling and simulation. It provides means of specifying systems with interacting

continuous and discrete trajectories.

Since DEVS representation applies to a special subclass of dynamical systems, output-

partitioned DEVS&DESS formalism for hybrid system modeling is introduced by Zeigler

et al [1],[2] as an extension of the DEVS formalism for combined discrete/continuous

modeling and simulation. It provides means of specifying systems with interacting

continuous and discrete trajectories.

Definition: An output-partitioned DEVS&DESS is a structure [2]:Definition: An output-partitioned DEVS&DESS is a structure [2]:

Op-DEVS&DESS=(X, Y, Q, Sc, , , F)Op-DEVS&DESS=(X, Y, Q, Sc, , , F)

wherewhere

X is the set of input values, which is an arbitrary set. X is the set of input values, which is an arbitrary set.

Q is the set of states. Q is the set of states.

S=S1 S 2 Sn = Rn, i.e., is the continuous state space which is an n-

dimensional vector space.

S=S1 S 2 Sn = Rn, i.e., is the continuous state space which is an n-

dimensional vector space.

={1, 2,, n} is a set of rate of change function with i : S X , ={1, 2,, n} is a set of rate of change function with i : S X ,


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DEVS Formalism 14

with{1,... }

ii n

F F

= U ,1 ,{ ,..., }ii i i mF f f= is the set of threshold values for all

dimensions i,

Y= BF is the output value set which is the crossproduct of all Boolean value

sets of the threshold sensors, ,i b iif i b

,,thr f F

: S Y if the output function which is defined by

(s)=(thrf1,b1(s1), thrf1,b2(s2),, thrf1,n(sn) )

i.e. the output is defined by the values of the threshold sensors.

In the above definition, a continuous system with n-dimensional state space

Sc=S1c S2c Snc=Rn

whose continuous behavior can be modeled by a set of first order

differential equations ={1,2,,n} is considered. Moreover, the system is furnished

with a set of threshold sensors, whose state change only at particular fixed threshold

values fi,bi of continuous state variables si, i.e. x=thr fi,b for all sifi,bi (assuming Boolean valued sensors are used).

2.3.1 DEVS Abstraction for Output-Partitioned DEVS&DESS Models

Since an output partitioned DEV&DESS is a piecewise constant input/output

dynamical system, it has a DEVS representation. The transition functions of a DEVS are

derived from an op-DEVS&DESS in the following way [2]:


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DEVS Formalism 15

1t e

For every total state ((s,x),e) and input segment2:X:

The time advance is the time interval to the next output change, i.e., threshold

crossing, provided a constant input x,

1,(( , )) min{ | ( ) ( ( ( ), ) )}s x

tta s x e s s f STRAJ x d = +

+, ( 2.5)

The internal transition function computes the state at the next threshold

crossing occurring at time t1+ta(s,x)

1

( , )

int ,(( , )) ( ( ( ), ) , )ta s x

s xt

s x s f STRAJ x d x = + , ( 2.6)

The external transition function updates the state at the input event x knowing

that the elapsed time since last event is e.

1

1,(( , ), , ) ( ( ( ), ) , )

t e

ext s xts x e x s f STRAJ x d x

+

= + , ( 2.7)

The output function is inherited from the DEVS&DESS

(( , ))s x s = , ( 2.8)

where STRAJs,x(t): :Q is the state trajectory, and : cf Q X S is the

differential equation representing the continuous behavior of the system, similar to is

(Q is the total state).

2 :A describes a motion through the set A that begins at t 1 and ends at t2, and at every t, (t)describes where the motion is at time t. See [13] for more details.


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DEVS Formalismism 16

16

Figure 2.8 illustrates how DEVS representation of DEVS&DESS works. At time t i+1,

when the external event occurs, ext updates the current state from si to si+1, knowing that

the elapsed time is e. Using the new state (si+1), ta predicts when the continuous system

crosses the next threshold fj+1 (orfj again). Finally, int function calculates the new state

si+2 using ta(s,x), if no external event occurs.

Figure 2.8 illustrates how DEVS representation of DEVS&DESS works. At time t i+1,

when the external event occurs, ext updates the current state from si to si+1, knowing that

the elapsed time is e. Using the new state (si+1), ta predicts when the continuous system

crosses the next threshold fj+1 (orfj again). Finally, int function calculates the new state

si+2 using ta(s,x), if no external event occurs.

ti

e

ta(s,x)

int(s,x)

ext((s,x),e,x )

si

si+1

si+2

jf

1jf +

,( ( ), )t

s xf STRAJ x d

ti+1

Figure 2.8. DEVS representation of DEVS&DESSFigure 2.8. DEVS representation of DEVS&DESS


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DEVS Formalism 17

2.3.2 DEVS First Order Numerical Integrator

DEVS representation of op-DEVS&DESS, leads to first order discrete event

numerical integration [14]. Consider a continuous system with n-dimensional state space,

which is described by differential equation x =r r& . For simplicity we assume that is

piecewise constant: ={i|i is constant in [ti,ti+1) , iZ} . The set of threshold values,

Fi={ fi |fi=q.i, iZ, q} (q is called quantum).

The following atomic DEVS is defined:

DEVS_INT=(X, Y, S, ext, int, , ta),

X= n,

Y= n,

S=n,

( , , )ext s e x s e x = + r r r r

, ( 2.9)

1

0

( , )0

i

i

if x

q e xta s xif x

x

=

=

r r

uuurr rr r

ur( 2.10)

int ( , ) ( , )s x s ta s x x = + r r r r r r

, ( 2.11)

( )s =r r

s , ( 2.12)

Input xX is the value of function, which is generated as external event at the

beginning of each segment. The interpretation of the above transition functions is as

follows (see Figure 2.9):

According to equation 2.7,


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DEVS Formalism 18

r1

1

( , , ) ( )t e

extt

s e x s d s e x +

= + = + r r r r

( 2.13)

where t1 is the moment at which the external event occurs.

The time advance function estimates how long it will take for the system to reach the

next threshold value. If the input is zero, obviously, it will never reach the next threshold;

so the time advance returns Infinity, consequently the atomic model goes into passive

phase.

In the case, which the input is non-zero if:

1) No external event has happened (e=0),

Assuming the rate of change is constant and equal tox, its clear thats=t.x+s0. so

the amount of time required for the system to reachs0+q,is ta=q/x.

2) External event has happened (e>0),

This happens when there is an unexpected external event before the previously

estimated time advance ta is over. Is this case the new input value, x i, is used to

recalculate how much time requires to reach the threshold. Again s=t.x+s0,new , but

so,new=s0,old+e.xold. Substituting s=so,old+q, we get: ta= (q-e.xold)/xnew

Note that this time-advance function works only if one external event occurs

between internal events. In the case of multi-external events ta=qremain/|x|, where

qremain is remaining quantum, calculated at each external event: qremain qremain - e.|x|.

Figure 2.9 illustrates how DEVS numerical integrator works. System starts with an

initial state (zero in this example). At time t0 when (t0)= 0the input x0=0 is generated.

The time advance function calculates t' as the next internal transition time, but before that

happens, changes to 1 at time t1, and input x1 is generated.


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19

3

2

1

S6

S5S4

S3

S2S1

S0

3

2

1

0

x3x1

x0

t'

t6t3t2

t6t3t2

t4

x2t5t1t0

t4 t5t1t0

3q

2q

q

2q

q

Y

S3q

X

(t)

Figure 2.9-DEVS first order numerical integratorFigure 2.9-DEVS first order numerical integrator

This external event will cause an external transition by ext from s0 to s1. Then time

advance function estimates the next threshold crossing, t2, at which an internal transition

from s1 to s2 occurs, and so on. Right before each internal transition an output is

generated by output function . Note that at time t4, where = 0, system goes to passive

phase until t5, at which a new non-zero input is generated.

This external event will cause an external transition by ext from s0 to s1. Then time

advance function estimates the next threshold crossing, t2, at which an internal transition

from s1 to s2 occurs, and so on. Right before each internal transition an output is

generated by output function . Note that at time t4, where = 0, system goes to passive

phase until t5, at which a new non-zero input is generated.

As oppose to conventional discrete time numerical integration methods, which divide

the time axis to equal time-steps, DEVS integrator divides the state space to equal

As oppose to conventional discrete time numerical integration methods, which divide

the time axis to equal time-steps, DEVS integrator divides the state space to equal


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DEVS Formalismsm 20

20

segments (q, 2q, 3q, ). The smaller these segmentations, the higher accuracy we get.

Each segment is equal to the quantum.

segments (q, 2q, 3q, ). The smaller these segmentations, the higher accuracy we get.

Each segment is equal to the quantum.

This model can be used for non-piecewise-constant s, which will generate some

error depending on the value of quantum, i.e. smaller quantum produces smaller error.

Its similar to conventional discrete time numerical integrators whose accuracy depends

on the size of the time-step.

This model can be used for non-piecewise-constant s, which will generate some

error depending on the value of quantum, i.e. smaller quantum produces smaller error.

Its similar to conventional discrete time numerical integrators whose accuracy depends

on the size of the time-step.

Example 2.5. Falling MassExample 2.5. Falling Mass

As an example consider a falling mass m with the initial height h0 and initial velocity

v0. A coupled DEVS containing two cascaded integrators is needed, as shown in Figure

2.10.

As an example consider a falling mass m with the initial height h0 and initial velocity

v0. A coupled DEVS containing two cascaded integrators is needed, as shown in Figure

2.10.

Acceleration PositionVelocityntegrator 2ntegrator 1

Mass m

Figure 2.10. A DEVS coupled model for Simulation of a free falling mass.Figure 2.10. A DEVS coupled model for Simulation of a free falling mass.

Figure 2.11 shows the DEVSJAVA Simulation Viewer, implementing the above

model. Results are presented in Figure 2.12a. Figure 2.12b shows the error generated by

these two cascaded numerical integrators (quantum=1).

Figure 2.11 shows the DEVSJAVA Simulation Viewer, implementing the above

model. Results are presented in Figure 2.12a. Figure 2.12b shows the error generated by

these two cascaded numerical integrators (quantum=1).

Figure 2.11. DEVS implementation of Example 2.5Figure 2.11. DEVS implementation of Example 2.5


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DEVS Formalism 21

(b)(a)

Figure 2.12. Simulation of a falling mass using two first order DEVS integrators with initial velocity v0=20m/s, and

initial position x0=100m and Quantum=1. (a) Position vs. time: DEVS results compared to the exact solution. (b)

Position Error vs. time.

Example 2.6. Linear Systems

Linear systems can be simulated using a single DEVS integrator. Consider the

following two LTI systems.

1 1

2 2

0 1 0

2 2 1

x xu

x x

= +

&

&

( 2.14)

1 1

2 2

0 1 0

1 1 1

x xu

x x

= +

&

& ( 2.15)

The step responses are shown in Figure 2.13. More accurate results can be obtained by

choosing smaller quantum.


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DEVS Formalism 22

Figure 2.13 Step responses for the systems of Example2.5 using DEVS integrator. (a) Step response for

system of equation 2.14 (b) Step response for system of equation 2.15

integrator. (a) Step response for

system of equation 2.14 (b) Step response for system of equation 2.15

(a) (b)

2.3.3 DEVS Double Integrator2.3.3 DEVS Double Integrator

2.3.3.1 Introduction and Motivation2.3.3.1 Introduction and Motivation

2.3.3.1 Introduction and Motivation

By double integrator, we mean a DEVS atomic model, which is equivalent to two

consecutive first order integrators, described in Section 2.3.2, i.e. it integrates the input

signal twice in one shot. The motivation behind defining such a model is:

By double integrator, we mean a DEVS atomic model, which is equivalent to two

consecutive first order integrators, described in Section 2.3.2, i.e. it integrates the input

signal twice in one shot. The motivation behind defining such a model is:

1) The equation of free motion of most natural physical systems is of the same general

form [16]:

1) The equation of free motion of most natural physical systems is of the same general

form [16]:


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DEVS Formalism 23

(2.16)2 0nx x+ =&&

This is the equation of an undamped, simple oscillator. For a mechanical system of

mass m and stiffness k,

n

k

m = . (2.17)

For a rotational system with the single degree of freedom and the moment of inertiaJ

and torsional stiffnessK:

n

k

J = . (2.18)

For a simple linearized pendulum with length land gravitation accelerationg:

n

g

l = . (2.19)

For an electrical LC circuit:

1n

LC = . (2.20)

2) Using one atomic model instead of two, is faster (in terms of computational

complexity), more accurate and is easier to implement when you want to use a large

number of these models.


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DEVS Formalism 24

2.3.3.2 Problem Definition

The goal is to define a DEVS atomic model, which integrates the input signal twice,

i.e. the relation between input and output is:

( ) ( )y t t dtdt= (2.21)

For simplicity, we assume that the input signal, , is one-dimensional piecewise

constant as defined in Section 2.3.2. (Later well show how this model can be extended to

n-dimensional integrator). And again each change in the value of , generates an input

event x=. Also the set of threshold values are supposed to be the same as of Section

2.3.2.

We define theDEVS numerical double integratoras follows:

DEVS_DBL_INT=(X, Y, S, ext, int, , ta),

X=,

Y=,

S=SASI, where SA=, is the actual state; and SI= is the intermediate state.

21( , , ) ( , )2

ext A I I s e x s e s e x s e x = + + + ( 2.22)

2

int

1

( , ) ( ( , ) ( , ) , ( , ) )2A I Is x s ta s x s ta s x x s ta s x x = + + + ( 2.23)

( )s s = ( 2.24)


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DEVS Formalism 25

2

2

1

0 0

0 0

( , ) 1min{ 0 | ( )

21

( )}2

I

remainI

I

A I k

A I k

if x s

qif x s

s

ta s xt s s t x t f

s s t x t f if x+

=

0

=

=

= > + + =

+ + =

( 2.25)

wherefkandfk+1 are the two thresholds, such that: fk


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DEVS Formalismalism 26

26

If the input x is zero but the intermediate state sI is not, then we have a single

integrator similar to what we described in Section 2.3.2.

If the input x is zero but the intermediate state sI is not, then we have a single

integrator similar to what we described in Section 2.3.2.

If the input is not zero, then by integrating this signal twice, we get:If the input is not zero, then by integrating this signal twice, we get:

21

2A Ixdt s s t x t= + + ( 2.26)

which could intersect with the next threshold fk+1 and/or the previous one fk. In this

case ta is equal to the time period to the next threshold crossessing.

Figure 2.14is an example to illustrate how this double integrator works.

n-Dimensional State Space

In this case the time advance function is the time within which the norm of change

of actual state is equal to the quantum i.e., 1 ,A i A ids s s+= =q where sA,i and sA,i+1

are the ith

and i+1stactual states, respectively. See Figure 2.15.

uur uuuur uuur

,A is

uuur

, 1A is +

uuuur

dsuur

Origin

q

q

Figure 2.15. Determining the time advance value forn-dimensionalDEVS double integrator

But,


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DEVS Formalism 27

2

, 1 ,

2

, 1 ,

1( , ) ( , )

2

1( , ) ( , )

2

A i A i I

A A i A i I

s s ta s x s ta s x x

ds s s ta s x s ta s x x

+

+

= + +

= = +

uuuur uuur uur r

uuur uuuur uuur uur r ( 2.27)

by equating |ds|=q, the time-advance function ta(s,x) can be found. For simplicity we

use tinstead ofta(s,x) in the following calculations.

2

2 2

2 24 3 2 2

1

2

1 1( ) ( )

2 2

1( )

4

I

I I

I I

ds q

s t xt q

s t xt s t xt q

x t s x t s t q

=

+ =

+ + =

+ +

uur

uur r

uur r uur r

r uur uur uur

2

0=

( 2.28)

In order to find the time-advance value it is required to solve the above incomplete

fourth-order polynomial equation. To analyze this equation consider:

4 3 2 2( ) 0f t at bt ct q= + + = ( 2.29)

where2 21

0, , 04

I Ia x b s x c s= > = = >r uur r uur

.

We are looking for the minimum positive root of equation 2.29.

This can be done using numerical methods. Looking at the derivative off(t) helps

finding the root faster.


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DEVS Formalism 28

3 2 2( ) 4 3 2 (4 3 2 )f t at bt ct t at bt c = + + = + + ( 2.30)

1,2

3

2

3

( ) 0 8

0

where 9 32

b

tf t a

t

b ac

= = =

=

( 2.31)

Figure 2.16 shows different locations of the roots of equation 2.29. Having this

information we can search for the root in a smaller region.

0

0-0 000

0

0 00

b>0 t'1


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DEVS Formalism 29

Example2.7. Falling Mass

Consider the falling mass mentioned in Example 2.4 again. In Example 2.4 two

cascaded single DEVS integrators were used to simulate a falling mass. Here we simulate

the exact same problem using one DEVS double integrator. Figure 2.17a shows the

DEVS result compared to the exact solution. Figure 2.17b depicts the error generated

(quantum=1, the same quantum as in Example 2.4). As seen in the figure this error is

much less than the error generated by two cascade DEVS integrators.

Figure 2.17- Simulation of a falling mass using one DEVS double integrator with initial velocity v0=20m/s, and initial

position x0=100m and Quantum=1. (a) Position vs. time: DEVS results compared to the exact solution.

(b) Position Error vs. time


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DEVS Formalism 30

2.4 Discrete Event Control (DEC)

2.4.1 Introduction

Ramadge and Wonham [5] were the first control theorists to develop an approach to

design discrete event controllers. They modeled the plant as an automaton and used

language theory to design a controller that forces the plant to perform the desired

behavior. In order to connect means of control to the system, they classified events into

two categories: uncontrollable events, which can be observed, but cannot be prevented

from occurring (e.g. obstacle detected) and controllable events that can be enforced to or

prevented from occurring (e.g. stop or move) [5]. This method has been used by several

researchers to control discrete event models [17],[18].

In this framework a system is said to be controllable when you can reach any desired

state of the system, knowing the current state. Consequently, a controller (supervisor)

exists if the system is controllable.

Zeigler[2] reformulated the Ramadge-Wonham approach to controller design within

the DEVS formalism. He showed that if the plant is strongly controllable (defined

below), then the desired path could be mapped to a DEC using a transformation called

Inverse-DEVS.

2.4.2 Controllability of DEVS Models

Controllability in general means being able to reach any desired final state from any

initial state in finite time. A discrete event specified system is said to be controllable if


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DEVS Formalism 31

there exist a controller that can implement a desired state trajectory[2]. Given a discrete

event plant M=(X, Y, S,ext,int,,ta), let ST be a desired state trajectory.

Definition 1 (weak controllability) A path ST is weakly controllable if a series of

internal and external transitions can take each state on the path to the next state on the

path3

. See Figure 2.18a.

Definition 2 (strong controllability) A path is strongly controllable if either an

internal or external transition can take each state on the path to the next state on the path

(Figure 2.18b). More formally, a path ST is strongly controllable if:

Forall si, si+1 such that si+1 follows si in ST, either:

a) int(si) = si+1, or

b) there is a pair (e,x), such thatext(si,e,x)=si+1 , where 0 < e < ta(si) and x is an

input.

Definition 3 (very strongly controllability)A strongly controllable path ST is a very

strongly controllable if:

For all si, si+1 such that si+1 follows si in ST, either:

a) int(si) = si+1, or

b) there is an input x such thatext(si,e,x)=si+1 for all 0 < e < ta(si).

Figure 2.18 illustrates the above three definitions. Filled circles represent states on the

desired path, ST=( s0, s1, , si, si+1, , sf). Thick lines denote internal or external

transitions that eventually lead one state on the path STto the next.

3 See [2]for a more formal definition forweak controllability.


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DEVS Formalism 32

S0

Si

Si+1

Sf

int

(si)

ext

(si,e,x)

weak

weak

S0

Sk

Si+1

Sf

S0

Si+1 S

f

ta(s0) =

ta(si) =

ta=

Figure 2.18- Controllability Definitions: (a) weakly controllable (b) strongly controllable (c) very strongly controllable

path.

controllable (c) very strongly controllable

path.

(a) (b) (c)

2.5 Intelligent DEVS2.5 Intelligent DEVS

2.5.1 DEVS-Fuzzy Logic2.5.1 DEVS-Fuzzy Logic

A fuzzy logic controller consists of three operations: (1) fuzzification, (2) inference

engine, and (3) defuzzification. The input sensory (crisp or numerical) data are fed into

fuzzy logic rule based system where physical quantities are represented into linguistic

variables with appropriate membership functions. These linguistic variables are then used

in the antecedents (IF-Part) of a set of fuzzy IF-THEN rules within an inference engine

to result in a new set of fuzzy linguistic variables or consequent (THEN-Part) [12]

A fuzzy logic controller consists of three operations: (1) fuzzification, (2) inference

engine, and (3) defuzzification. The input sensory (crisp or numerical) data are fed into

fuzzy logic rule based system where physical quantities are represented into linguistic

variables with appropriate membership functions. These linguistic variables are then used

in the antecedents (IF-Part) of a set of fuzzy IF-THEN rules within an inference engine

to result in a new set of fuzzy linguistic variables or consequent (THEN-Part) [12]

A typical so-called Mamdani type rule can be composed as followsA typical so-called Mamdani type rule can be composed as follows

i

1 1 2 2IF is AND is THEN is , for 1, 2,...,i i ii

1 1 2 2IF is AND is THEN is , for 1, 2,...,i i ix A x A y B i = l

where and are the fuzzy sets representing the iiA1iA2

th-antecedent pairs, and are the

fuzzy sets representing the i

iB

th-consequent, and lis the number of rules.

We defineDEVS-Fuzzifieras follows:

ADEVS-Fuzzifieris a DEVS atomic model:

AMF =(X, Y, S, ext, int, , ta, )

where


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DEVS Formalism 33

The input set, X, is usually a set of real numbers, to be fuzzified.

The output set Y = [0,1], since the output of this atomic model is a fuzzy value.

S is a sequence of fuzzified input values: S={si | si = ext(q,si-1,x) }.

ext(q,s,x)=(x), where is the membership function associated with this fuzzifier.

int(s) and (s) are the identity functions.

ta (s) =0 , since there is no time advance in this model.

(x) is the membership function associated with this fuzzifier.

This atomic model represents an antecedent membership function, while a consequent

membership function can be defined as:

CMF=(X, Y, S, ext, int, , ta, )

where

X is a set of fuzzy values, X=[0,1].

Y is a set of fuzzy sets, Y = {y |y is a fuzzy set}.

S is a set of discrete fuzzy sets (membership functions) 4.

ext(q,s,x)= min( ( ), )ii

a x

a

where is the membership function associated with this model. In fact this function

cuts those values of the fuzzy set (ai)/ai whose membership values are greater than x

(see Figure 2.19).

int(s)=s, (s)=s and ta=0.

4 The notation (ai)/ai denotes a discrete fuzzy set (membership function) [48].


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DEVS Formalism 34

In a typical fuzzy rule, connectives are used to make connection between antecedent

pairs. An AND connective takes the minimum, while an OR connective returns the

maximum value of two pairs [12].

YX

Figure 2.19 A typical consequent membership function (CMF) model

Connectives are defined in DEVS as atomic models:

CNTV=(X1, X2, , Xn, Y, S, ext, int, , ta, )

where

X1== Xn =[0,1] are input sets. (A connective can have n inputs.), Y = S = [0,1],

int(s)=s, (s) = s, ta = 0,

ext(q,s,x1,xn) = min(x1,,xn), in case of AND and,

ext(q,s,x1,xn) = max(x1,,xn), in case of OR.

A fuzzy rule can be composed as a coupled model using the above three atomic

models. For instance, consider the following fuzzy rule: IF x is A OR x is B THEN z is

C. Figure 2.20 shows how this fuzzy rule can be composed using two Fuzzifiers, one

Connective and one CMF(consequent membership function).


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DEVS Formalismlism 35

35

CMF

FZ

B

A

z

y

x

OR

Figure 2.20 A DEVS model for a typical fuzzy rule IF x is A OR y is B THEN z is CFigure 2.20 A DEVS model for a typical fuzzy rule IF x is A OR y is B THEN z is C

In order to fully implement a Fuzzy Logic Controller, a Defuzzifieris needed, which

can be defined in the same way. i.e. takes fuzzy sets as input and calculates the

defuzzified value (see [12] for further details about defuzzification methods).

In order to fully implement a Fuzzy Logic Controller, a Defuzzifieris needed, which

can be defined in the same way. i.e. takes fuzzy sets as input and calculates the

defuzzified value (see [12] for further details about defuzzification methods).

DeFZ==(X1, X2, , Xn, Y, S, ext, int, , ta)DeFZ==(X1, X2, , Xn, Y, S, ext, int, , ta)

wherewhere

Inputs of this atomic model are fuzzy sets so X1=X2==Xn ={x | x is a fuzzy set}.Inputs of this atomic model are fuzzy sets so X1=X2==Xn ={x | x is a fuzzy set}.

Output is a real number, Y = ,S = , int(s) = s, (s) = s, ta = 0,Output is a real number, Y = ,S = , int(s) = s, (s) = s, ta = 0,

ext(q,s,x1,xn) = Center_of_Gravity (x1 x2 xn )ext(q,s,x1,xn) = Center_of_Gravity (x1 x2 xn )

Center_of_Gravity(x) is a function which calculates the defuzzified value of fuzzy set

x using center of gravity method [12]:

Center_of_Gravity(x) is a function which calculates the defuzzified value of fuzzy set

x using center of gravity method [12]:

Center_of_Gravity (x) =Center_of_Gravity (x) = ( )

( )

x x dx

x dx

( 2.32)

Note that the time advance function (ta) is zero for all the above models, since we

dont want to have any delay time in any of the models.

Fuzzy-DEVS, i.e. the discrete event implementation of Fuzzy Logic Control, runs

faster than conventional methods. The reason is that the three operations of the fuzzy


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DEVS Formalism 36

system in Fuzzy-DEVS, are activated only when there is an input event. In other words

the DEVS-Fuzzifiers (see Figure 2.20) are activated only when they receive a new input

event. Consecutively, they only generate output events if the new fuzzified value is

different from the previous one. As a result the connective models are activated only

when there is a change in their inputs, and the story continues. This kind of behavior

saves a lot of unnecessary calculations in a large system. However, for small number of

rules the performance is not significantly improved due to change-detection calculations

performed in Fuzzy-DEVS.

In practice, Fuzzy parameters can be tuned in two ways: first, by means of adding

extra input ports to the DEVS models and sending the parameters to them via input ports.

Second, directly changing the parameters internally, or having parameters accessible

from other models.

The first method makes the Fuzzy-DEVS tunable by connecting it to other DEVS

models, such as GA-DEVS and NN-DEVS, while the second method can be used to tune

the parameters by conventional tuning algorithms such as GA and NN.

Example 2.8. As an example consider the problem of controlling an inverted

pendulum. The linearized model is given by the following equation [19]:

[ ]

0 1 0 0 0

0.4 0 0 0 0.97

0 0 0 1 0

0 0 7.97 0 0.8

1 0 0 0

x u

y x

= +

=

&

( 2.33)


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DEVS Formalism 37

where the four states are , ,x & and . The input to the system is force applied to the

cart. This system can be stabilized using a fuzzy logic controller, with the following three

rules:

x&

IF isPOS and& isPOS THEN Force isNEG

IF isNEG and& isNEG THEN Force isPOS

IF isZEROand& isZEROTHEN Force isZERO

where the membership functions POS, NEG and ZERO are defined in Figure 2.21a. Figure 2.21b

shows the DEVS-Fuzzy Logic Controller. The closed-loop system is shown in Figure 2.21c and

the output () of the system is shown in Figure 2.21d.

0 5 10 15 20 25 30 35-0.5

0

0.5

1

Time(sec)

Theta

ZERONEG POS

a) Membership functions for fuzzy-DEVS b) The fuzzy-DEVS controller

c) The close-loop system d) The output of the system ()

Figure 2.21. Example of using fuzzy-DEVS to control an inverted pendulum


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38

2.5.2 DEVS-SLA2.5.2 DEVS-SLA

2.5.2.1 Introduction to Stochastic Learning Automata (SLA)2.5.2.1 Introduction to Stochastic Learning Automata (SLA)

SLA seeks to model the learning-from-action process from intelligent beings. Namely,

reducing the probability of performing actions, which cause to receive penalty (pain) and

increase the probability of performing actions, which are rewarded from the environment

(pleasure).

SLA seeks to model the learning-from-action process from intelligent beings. Namely,

reducing the probability of performing actions, which cause to receive penalty (pain) and

increase the probability of performing actions, which are rewarded from the environment

(pleasure).

According to a famous remark by a psychologist (Benthams Theory of Value),

nature has placed mankind under the governance of two sovereign masters, pain and

pleasure. It is for them alone to point out what we ought to do, as well as to determine

what we shall do. On the one hand, the standard of right and wrong, on the other the

chain of causes and effects, are fastened to their throne[20], i.e. all human behavior is

motivated by and only by a desire to achieve pleasure and avoid pain. This behavior can

be mathematically formulated and used in control of intelligent systems.

According to a famous remark by a psychologist (Benthams Theory of Value),

nature has placed mankind under the governance of two sovereign masters, pain and

pleasure. It is for them alone to point out what we ought to do, as well as to determine

what we shall do. On the one hand, the standard of right and wrong, on the other the

chain of causes and effects, are fastened to their throne[20], i.e. all human behavior is

motivated by and only by a desire to achieve pleasure and avoid pain. This behavior can

be mathematically formulated and used in control of intelligent systems.

Environment

Automaton

ActionPenalty/Reward

Figure 2.22- Learning from action: gaining reward while avoiding penalty.Figure 2.22- Learning from action: gaining reward while avoiding penalty.

The learning process is as follows:The learning process is as follows:

The system (automata) has the option of choosing one action from n pre-defined

actions, each with a probability of being selected:Pi, i=1...n. (Pi = 1).

The system (automata) has the option of choosing one action from n pre-defined

actions, each with a probability of being selected:Pi, i=1...n. (Pi = 1).


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DEVS Formalism 39

After performing the selected action, the result will be evaluated by one or more

teachers (environment), which will assign penalty or reward to the action based on the

desired objectives of the system.

Next, the system will update the probabilities by increasing (decreasing) the probably

of choosing actions that has received reward (penalty).

This process will be repeated until it converges.

This approach can be used to translate the experience of human to the rover domain.

Human like reasoning is achieved in SLA through the use of a teacher, which penalizes

or rewards a particular action taken by the agent or robot (i.e. using a human expert in

place of the teacher).

Figure 2.23 Schematic of SLA inside the DEVS Environment

2.5.2.2 Stochastic Learning Automata by DEVS

Stochastic Learning Automata (SLA) process can be modeled by DEVS as shown in

Figure 2.23. In this figure, n denotes the number of actions and m is the number of

teachers. Pi denotes the probability of the ith action. Here is a brief description of the


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DEVS Formalism 40

atomic models shown in this figure: The Action-chooser-block chooses an action i

(i=1,2n) based on the probability that the action occurs. The agent (or agents in

multi-agent case) will perform the action and send the result(s) to the teacher(s). Each

teacher (denoted by Tj) assigns a reward orpenalty to the result(s). The summer

block sums up all the rewards and penalties assigned by teachers. If +1,-1 represent

reward and penalty respectively then the output of the summer will be an integerK {-

m, -m+1, , m-1 , m}. The next block updates the probabilities (Pis) based onK.

Example 2.9: As an example, consider a DEVS-SLA model with four actions and

three teachers (i.e. n=4,m=3). Figure 2.24 shows a possible scenario of the simulation.

After receiving the start signal, the Action chooser block changes its state to busy and

after a certain amount of time, generates the output (in this example action 3 has been

chosen), and goes back to the passive state. The Agent performs the action as soon as it

receives the message from Action chooser, changing its state to busy. After the action

is performed, the Agent block sends the result to teachers. As soon as the teachers

receive the result from agent, they go to busy state, evaluating the result. In this example

Teacher 1, assigned a penalty and two other teachers assigned a reward. Summer

collects the penalties and rewards assigned by teachers, and sends the result (in this case

+1) to the Update block. This block increases P3 (since the third action got a reward of

+1) and decreases the others.


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DEVS Formalism 41

Figure 2.24. Possible results of an example for SLA_DEVS

Example 2.10: As another example, consider a mobile automaton (robot) with 8

moving actions: 1.North, 2.East, 3.South, 4.West, 5.North-East, 6.South-East, 7.South-

West and 8.North-West. Starting from some initial position (x0, y0), the automaton learns

to reach a goal position (xg, yg). Two teachers are used: one for X direction and one for Y

direction. Figure 2.25 shows the results of SLA-DEVS for an automaton going from

initial position (10,10) to the goal point (60,80).


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DEVS Formalism 42

y

x

y

x

(a)

P8

P6

P7

P5

P4

P3

P2

P1

(b) (c)

Figure 2.25- A mobile automaton learns to reach a goal point using SLA. (a) Action probabilities vs. time (b) trajectory

of the automaton (c) x and y vs. time.

As figure shown the automaton first learns to perform action number 5 (Moving

toward North-East) until it reaches the desired x=60; after that different actions are

performed to reach the desiredy=80.


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43

3. MOBILE ROBOTICS

3.1 Introduction

Mobile robotics involve several different arias of research such as path and motion

planning, sensor and action planning, sensor fusion, vision, machine learning, artificial

intelligence, pattern and image recognition, environment modeling, autonomous

behaviors, control and automation, multi-agent and cooperative systems. In this work we

focus on modeling, simulation and control of mobile robots, using DEVS and V-Lab5

.

Figure 3.1 shows a closed-loop block diagram for sensor-based mobile robots.

Feedback

External Sensors

(vision, infra-reds, sonar, etc)

Internal Sensors

(encoders,

compasses, etc)

OutsInputsRefs

Figure 3.1. Robot Functional Block Diagram

5 V-Lab is discussed in section 4.


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Mobile Robotics 44

3.2 Modeling and Controllability of Mobile Robotsand Controllability of Mobile Robots

The mobile robot under consideration is a so-called unicycle robot with two

differential-drive wheels on the same axle (Figure 3.2). A robot with this kind of wheel

configuration has a non-linear property and also an underlying non-holonomic property

that adds to the complexity of the motion control problem.

The mobile robot under consideration is a so-called unicycle robot with two

differential-drive wheels on the same axle (Figure 3.2). A robot with this kind of wheel

configuration has a non-linear property and also an underlying non-holonomic property

that adds to the complexity of the motion control problem.

lv

2r

w

l

rv

( , )x y

v

Figure 3.2- A differential-wheeled mobile robotFigure 3.2- A differential-wheeled mobile robot

The locomotion system for such a mobile robot constitutes two parallel driving

wheels, the acceleration of each being controlled by an independent motor. Passive

castors ensure the stability of the platform. The reference point of the robot is the

midpoint of the two wheels; its coordinates, with respect to a fixed frame are denoted by

The locomotion system for such a mobile robot constitutes two parallel driving

wheels, the acceleration of each being controlled by an independent motor. Passive

castors ensure the stability of the platform. The reference point of the robot is the

midpoint of the two wheels; its coordinates, with respect to a fixed frame are denoted by

( , )( , )y ; the main direction of the vehicle is the direction of the driving wheels. With

designating the distance between the driving wheels, the dynamic model is [21]:

l


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Mobile Robotics 45

( )

( )

( ) 1 2

1cos

2

1 0 0sin2 0 0

10 0

1 00 0 10

r l

r l

r l

r

l

v v

xv v

y

v v u ul

v

v

+

+ = + +

&

&

&

&

&

( 3.1)

where u and u are the accelerations of two independent motors. Figure 3.3 shows

the closed-loop feedback control block diagram for such a system.

1 2

Position

CurrentVelocity

Position

VelocityTorqueDesired

VelocityDesiredPosition Volt

Figure 3.3. Robot Motion Control

Figure 3.4 shows how we go from dynamics to kinematics6, assuming the desired

velocity is equal to the actual velocity. This assumption makes it a lot easier to control

this system. In real robots usually PID controllers at the low level control, ensure this

assumption providing the desired velocity. By choosing v and as inputs, the 5-

dimensional system will be reduced to the following 3-dimensional system:

r lv

6 Dynamics is A branch of mechanics that deals with forces and their relation primarily to the motion but sometimes

also to the equilibrium of bodies. (Merrian-Webster Dictionary), while kinematics is a branch of dynamics that

deals with aspects of motion apart from considerations of mass and force(Merrian-Webster Dictionary).


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Mobile Robotics 46

( )

( )

( )

1cos

2

1sin

2

1

r l

r l

r l

x v v

y v v

v vl

= +

= +

=

&

&

&

( 3.2)

where | | and,maxr rv v ,max| |l lv v . By choosing1

(2

r lv v v )= + and1

( r lw v vl

= ) , we get

the kinematic model of equation 3.3.

cos

sin

x v

y v

w

==

=

&&

&

( 3.3)

where is the forward velocity and is the angular velocity. Of course and are

bounded with:

v w v w

min maxv v v and min maxw w w

This system is symmetric without drift and controllable from everywhere [21].

Kinematics is purely differential-geometrical issue independent from the dynamics, assuming the velocity to be the

system input [29].


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Mobile Robotics 47

Velocity=Desired Velocity

Position

I G N O R E

Figure 3.4. Kinematic Controlrol

Its easier to control the linear and angular velocities v, w instead of left and right

velocities; however, in a real robot left and right velocities are required to control the

robot.

Its easier to control the linear and angular velocities v, w instead of left and right

velocities; however, in a real robot left and right velocities are required to control the

robot.

The relation between [The relation between [ ]T

r lv v and [ ]T

v w can be expressed by the following matrix

equation.

1 11

2 2 2

1 11

2

r r

l l

l

v vv v

v vw l

l l

= =

w( 3.4)


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Mobile Robotics 48

Controllability

This model is a non-holonomic7

one, which is always controllable, thus the path

planning problem is always well posed [22],[21].Note that non-holonomic mechanical

systems, either in kinematics or dynamic form, cannot be stabilized at a point by smooth

feedback [22]. The alternatives are time-varying or discontinuous stabilizing feedback.

3.3 Motion Control of Mobile Robots

The motion control of mobile robots among obstacles is classified to three possible

motion tasks as follows [21]:

Point-to-point motion (goal searching): The robot is assigned to reach a

desired goal configuration starting from a given initial configuration, while

avoiding collision with obstacles. This task is sometimes calledPath planning

in the literature.

Path following: The robot must reach and follow a geometric path in the

Cartesian space starting from a given initial configuration (on or off the path).

Trajectory tracking: The robot must reach and follow a trajectory in the

Cartesian space (i.e., a geometric path with an associated timing law) starting

from a given initial configuration.

7 A nonholonomic control system is a system with nonholonomic constraint --a kinematic constraint that cannot be

integrated [22]. For a wheeled mobile robot the nonholonomic constraint is: cos sin 0y x =& & .


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Mobile Robotics 49

3.3.1 Path Planning Methods

Path planning (goal searching) involves two questions: first, is there exists a

collision-free admissible path? Second, how can one compute such a path. The

existence of a path depends on the configuration of the obstacles in the environment and

the initial configuration of robot. Thepath-computation problem has been receiving more

attention by researchers. There are several approaches which can be used for path

planning: Regulation of a nonholonomic dynamic wheeled mobile robot with parametric

modeling uncertainty using lyapunov function [23], a so called virtual vehicle approach

for control of mobile robots [24], game theory approaches [25],[26], a bang-bang control

technique [27], a quasi-continuous output feedback method [28], natural algebraic

structure of chained form system together with sliding mode theory [29], optimal path

planning approaches [30], [31], [32], evolutionary control architecture [16] visibility

graphs [33],[34] and other methods [35],[37],[22],[21].

Problem definition: The problem is simply to determine v and w such that the

system of equation 3.3 steers the robot arbitrarily close to a goal configuration (xg,yg,g).

Our solution: We propose the following control law to steer the robot to a goal

configuration (xg,yg,g).

Consider the system of equation 3.3. The following inputs:


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Mobile Robotics 50

cos( )

sin( )

v a

w b

=

= ( 3.5)

where arctan( )g

g

y y

x

=

, steer the system to the goal point (xg,yg) if a and b are

chosen properly.

After reaching the goal point, robot can turn (in place) to reach the angle gi.e.

0

sin( )g

v

w b

=

= ( 3.6)

Without any loss of generality the goal can be assumed to be at the origin, any other

goal can be moved to the origin be a coordinate transformation..

Using the proposed control law, the system becomes:

cos( )cos( )

cos( )sin( )

sin( )

x a

y a

b

=

=

=

&

&

&

( 3.7)

Here is how one can choose a and b to stabilize the system:

b is associated with the angular velocity and a with the linear velocity of the system.

bsin(-) turns the robot toward the goal direction . The bigger the b, the faster it turns.

On the other hand a determines how fast the robot moves forward. Intuitively, robot

should first turn toward the goal direction and then move forward to the goal.

Consequently, ifb is chosen to be larger than a, this controller law should work. After

reaching the goal position let v=0, and robot turns to reach the goal direction.


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Mobile Robotics 51

Figure 3.5 shows the outcome of this controller, for a=1 and b=4 and initial condition

(1,2,/4).

r a=1 and b=4 and initial condition

(1,2,/4).

(rad)

v,w

x,y

Initial Condition :(1, 2, /4)

y

x

b) State variables vs. timea) Trajectory

Figure 3.5. The system of equation 3.7 for a=1 and b=4;Figure 3.5. The system of equation 3.7 for a=1 and b=4;

It can be seen from the trajectory graph that, robot first goes backward and after a

while goes forward. By using the absolute value forv, robot will always go forward.

It can be seen from the trajectory graph that, robot first goes backward and after a

while goes forward. By using the absolute value forv, robot will always go forward.

Figure 3.6shows the result of changing b for the same system, b varies from 4a to a

and a is equal to 1. In

Figure 3.6shows the result of changing b for the same system, b varies from 4a to a

and a is equal to 1. In

Figure 3.6a the initial direction 0 is equal to /4, and inFigure 3.6a the initial direction 0 is equal to /4, and in

Figure 3.6b, 0 is equal to -/4.Figure 3.6b, 0 is equal to -/4.

Figure 3.7depicts the trajectory of the system for 8 different initial positions.Figure 3.7depicts the trajectory of the system for 8 different initial positions.

b=1

b=1.5

b=2

b=4

b=3

b=1

b=1.5

b=2

b=4b=3

a) 0=/4 b) 0=-/4


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Mobile Robotics 52

Figure 3.6. Effect of changing b from 4a down to a.Figure 3.6. Effect of changing b from 4a down to a.

(6, 0, /4)

(6, -5, /4)(0, -5, /4)(-6, -5, /4)

(-6, 0, /4)

(-6, 5, /4)

(0, 5, /4)

(6, 5, /4)

a=1

b=1

Figure 3.7. Trajectory of the system of equation 3.7 for different initial conditionsFigure 3.7. Trajectory of the system of equation 3.7 for different initial conditions

Parameters a and b can be time varying such that as robot approaches the goal it

slows down, too. For example they can be proportional to the robots distance to the goal.

Parameters a and b can be time varying such that as robot approaches the goal it

slows down, too. For example they can be proportional to the robots distance to the goal.

2 2

( )

2 ( )

( ) ( ) ( )g g

a r t

b r t

r t x x y y

=

=

= +

( 3.8)

Figure 3.8 shows the result for the above control strategy. Note that total energy of

the system, is damping (Figure 3.8b) since both angular and linear velocities w,v decay to

zero.


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Mobile Robotics 53

2 21 ( )2

E Mv Iw= +& &

(b) Total Energy(a) Trajectory

(c) State variables

Figure 3.8. Using time-varying a and b as defined in equation 3.8Figure 3.8. Using time-varying a and b as defined in equation 3.8

Stability8

AnalysisStability8

Analysis

Here we analyze the Lyapunov stability of the system. We find condition fora and b

under which the proposed controller makes the system asymptotic stable. Consider the

following positive definite Lyapunov function:

Here we analyze the Lyapunov stability of the system. We find condition fora and b

under which the proposed controller makes the system asymptotic stable. Consider the

following positive definite Lyapunov function:


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Mobile Robotics 54

2( , , , ) 1 cos( )V r t r = + + ( 3.9)

where rand are polar coordinates of the robot. Obviously this function is a positive

definite one. We need to show that

( , , , )0

V r t

t

( 3.16)

This condition is satisfied ifb is chosen such that:


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Mobile Robotics 56

ab

r > ( 3.17)

Finally,

0, 0, 0a

r bb

> > < 0 (where r0 is the desired distance of the robot to the goal, and can be

arbitrarily small), as a result, a and b, satisfying the following inequality, guarantee the

asymptot

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