UNIVERSITY OF NAIROBI DEPARTMENT OF …eie.uonbi.ac.ke/sites/default/files/cae/engineering/eie/DIGITAL... · 3.1 LEGO MINDSTORMS LINE FOLLOWER ROBOT DESIGN ... Simulink Line Tracking

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UNIVERSITY OF NAIROBI

DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING

DIGITAL CONTROL OF A LINE FOLLOWING ROBOT

PROJECT INDEX: PRJ 75

NAME: MUINDI DENNIS MUTHEKE D.

REG NO: F17/2454/2008

SUPERVISOR: PROF. J. M. MBUTHIA

EXAMINER: MR. MUSAU

Project report submitted in partial fulfillment of the requirement for the award of the Degree of: Bachelor of Science in Electrical and Electronic Engineering of the University of Nairobi.

Date of Submission:

24TH

APRIL 2015

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DECLARATION OF ORIGINALITY

I understand what plagiarism is and I am aware of the university policy in this regard.

1. I declare that this final year project report is my original work and has not been submitted elsewhere for examination, award of a degree or publication. Where other people’s work or my own work has been used, this has properly been acknowledged and referenced in accordance with the University of Nairobi’s requirements.

2. I have not sought or used the services of any professional agencies to produce this work.

3. I have not allowed, and shall not allow anyone to copy my work with the intention of passing it off as his/her own work.

4. I understand that any false claim in respect of this work shall result in disciplinary action, in accordance with University anti-plagiarism policy.

SIGNATURE: ………………………………………………………………………………………

DATE: ………………………………………………………………………………………………..

NAME OF STUDENT: Muindi Dennis Mutheke D REGISTRATION NUMBER: F17/2454/2008 COLLEGE: Architecture and Engineering

FACULTY/ SCHOOL/ INSTITUTE: Engineering DEPARTMENT: Electrical and Information Engineering COURSE NAME: Bachelor of Science in Electrical & Electronic Engineering TITLE OF WORK: Digital Control of a Line Following Robot

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CERTIFICATION

This project report has been submitted to the Department of Electrical & Information Engineering, University of Nairobi with my approval as supervisor.

Prof. J. Mwangi Mbuthia

SIGNATURE: ………………………………………………………………………………………

DATE: ……………………………………………………………………………………………….

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DEDICATION

I dedicate this project to the young robotics enthusiasts in Kenya. The robotic age is now upon us.

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ACKNOWLEDGEMENTS

With a grateful heart, I would like to acknowledge all members of staff at the Department of Electrical and Information Engineering, University of Nairobi for their immense contributions in my academic endeavours. Thank you for everything you have done to me, day after day, year after year. I wish to express profound gratitude to my supervisor, Prof. J. M. Mbuthia for providing very vital assistance in acquisition of the Lego Mindstorms kit. I appreciate his timely review and thoughtful assessment of the project progress.

Fellow engineering students, you have been my steadfast companions in this scholarly journey.

I will forever treasure you for the unflinching support you gave me.

Special thanks to Daniele Benedettelli, Laurens Valk, Terry Griffin. You greatly nurtured my

interest in Lego Mindstorms robots.

To the Supreme Creator and Sustainer of the universe, your intelligent design concepts fill me

with wonder. I appreciate and value your grace, goodness and bountiful blessings in my life.

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Table of Contents

DECLARATION OF ORIGINALITY ............................................................................................................................. II

CERTIFICATION ..................................................................................................................................................... III

DEDICATION ......................................................................................................................................................... IV

ACKNOWLEDGEMENTS .......................................................................................................................................... V

TABLE OF CONTENTS ............................................................................................................................................. VI

LIST OF FIGURES .................................................................................................................................................. VIII

LIST OF TABLES ..................................................................................................................................................... IX

ABSTRACT .............................................................................................................................................................. X

CHAPTER 1 ............................................................................................................................................................. 1

1. INTRODUCTION .............................................................................................................................................. 1

1.1 DIGITAL CONTROL AND ROBOTICS: ........................................................................................................... 1

1.2 PROBLEM STATEMENT ................................................................................................................................ 1

1.3 OBJECTIVES ................................................................................................................................................... 1

1.3.1 OVERALL OBJECTIVE .......................................................................................................................................... 1

1.3.2 SPECIFIC OBJECTIVES ........................................................................................................................................................................ 1

PROJECT JUSTIFICATION ................................................................................................................................... 2

1.5 PROJECT SCOPE ............................................................................................................................................ 2

1.6 REPORT ORGANIZATION ............................................................................................................................. 2

CHAPTER 2 ............................................................................................................................................................. 4

2. LITERATURE REVIEW ......................................................................................................................................... 4

2.1 DIGITAL CONTROL SYSTEMS ............................................................................................................................ 4

2.1.1 INTRODUCTION .............................................................................................................................................. 4

2.1.2 WHY DIGITAL CONTROL? .................................................................................................................................. 5

2.1.3 TYPES OF CONTROLLERS............................................................................................................................... 6

2.2 LEGO MINDSTORMS EV3................................................................................................................................... 7

2.2.1 PARTS OF THE LINE FOLLOWING ROBOT .................................................................................................... 8

2.2.1.1. THE EV3 BRICK ........................................................................................................................................................................... 8 2.2.1.2. POWER SUPPLY .......................................................................................................................................... 8

2.2.1.3. CONNECTOR CABLES ................................................................................................................................. 9

2.2.1.4. BEAMS, CONNECTOR PEGS, AXLES AND BUSHES .................................................................................... 9

2.2.1.5. CROSS BLOCKS AND ANGLE ELEMENTS ................................................................................................. 10

2.2.1.6. WHEELS AND TIRES ................................................................................................................................. 10

2.2.1.7. COLOUR SENSOR ....................................................................................................................................... 10

2.2.1.8. SERVO MOTORS ................................................................................................................................... 11

2.2.1.8.1. MODELLING OF A DC SERVO MOTOR ................................................................................................... 12

2.2.1.8.2 DC SERVO MOTOR ENCODERS .................................................................................................................. 14

2.3 APPLICATIONS OF A LINE FOLLOWING ROBOT ........................................................................................... 15

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CHAPTER 3 ........................................................................................................................................................... 16

3. METHODOLOGY ............................................................................................................................................ 16

3.1 LEGO MINDSTORMS LINE FOLLOWER ROBOT DESIGN .......................................................................... 16

3.2. STUDY OF THE LEGO MINDSTORMS EV3 MOTOR .................................................................................. 17

3.2.1. MOTOR SPECIFICATIONS ............................................................................................................................ 17

3.2.2 EV3 SERVO MOTOR IMPLEMENTATION ON MATLAB:......................................................................... 17

3.2.3:S-DOMAIN EV3 SERVO MOTOR TRANSFER FUNCTION: .................................................................................. 18

3.2.4: Z-DOMAIN EV3 SERVO MOTOR TRANSFER FUNCTION: ................................................................................. 20

3.3 LINE FOLLOWING ROBOT ALGORITHMS ................................................................................................. 22

3.4: DIGITAL CONTROLLER DESIGN: ............................................................................................................... 23

3.4.1. PROPOSED CONTROLLER DESIGN: ............................................................................................................ 23

3.4.2: IMPLEMENTATION OF LINE FOLLOWING CONTROL ALGORITHM FOR THE LEGO MINDSTORMS EV3

HARDWARE: ............................................................................................................................................................ 24

CHAPTER 4 ........................................................................................................................................................... 26

4. RESULTS AND ANALYSIS .............................................................................................................................. 26

4.1: EV3 LARGE MOTOR CHARACTERISTICS .................................................................................................. 26

4.2: PID TUNING ................................................................................................................................................ 27

4.3: FREQUENCY RESPONSE ............................................................................................................................. 32

4.4.GENERAL ANALYSIS .................................................................................................................................... 33

CHAPTER 5 ........................................................................................................................................................... 35

5. CONCLUSION AND RECOMMENDATIONS.................................................................................................... 35

5..1 CONCLUSION .................................................................................................................................................... 35

5.2 RECOMMENDATIONS FOR IMPLEMENTATION ............................................................................................ 35

5.3 RECOMMENDATIONS FOR FURTHER WORK ................................................................................................ 35

6 REFERENCES ................................................................................................................................................... 36

APPENDIX A ....................................................................................................................................................... 38

A.1 MATLAB CODE FOR CHAPTER 3 ..................................................................................................................... 38

A.2 MATLAB CODE FOR CHAPTER 4 ..................................................................................................................... 40

APPENDIX B ......................................................................................................................................................... 42

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LIST OF FIGURES Fig 2-1: General configuration of a digital control system………………………………………………………………………4

Fig 2-2: Feedback loop with PD control …………………………………………………………………………….……..6

Fig 2-3: Feedback loop with PI control……………………………………………………………………………………………………7

Fig 2-4: Feedback loop with PID control……………………………………………………………………………………7

Fig 2-5: Front and back views of the EV3 Brick. The back is the 9V battery ……………………………..8

Fig 2-6: Connector cables …………………………………………………………………………………………………….9

Fig 2-7: A collection of Lego beams, axles, connector pegs and bushes…………………………………..9

Fig 2-8: A collection of Cross Blocks and Angle Elements……………………………………………………....10

Fig 2-9: Lego wheels ……………………………………………………………………………………………………………..10

Fig 2-10: EV3 Colour sensor and the measuring ranges in the ‘Colour’ and ‘Reflected Light

Intensity’ modes……………………………………………………………………………………………………………………..11

Fig 2-11: Large servo motor …………………………………………………………………………………………………………………12

Fig.2-12: Model of a servo motor………………………………………………………………………………………………………….12

Fig 2-12: Individual blocks and complete representation of DC servo motor ………………………………………14

Fig. 2-13: Motor encoder …………………………………………………………………………………………………………………….15

Fig. 3-1: Line Following Robot ………………………………………………………………………………………………………………16

Fig.3-2: Motor implementation on Simulink …………………………………………………………………………………………17

Fig.3-3: Step response of the s-domain servo motor transfer function …………………………………………………18

Fig. 3-4: Root Locus of the s-domain servo motor transfer function …………………………………………………….19

Fig. 3-5: Bode plot of the s-domain servo motor transfer function……………………………………………………….19

Fig 3-6: Step response of the z domain servo motor transfer function………………………………………………….20

Fig 3-7: Root Locus of the z-domain servo motor transfer function………………………………………………………21

Fig 3-8: Bode plot of the z-domain servo motor transfer function…………………………………………………………21

Fig 3-9: Basic line following approach……………………………………………………………………………………………………22

Fig 3-10: Simulink Line Tracking Program with PID Controller………………………………………………….24

Fig 4.1: Graph of rotation speed against applied voltage ……………………………………………………………………26

Fig 4.2: Step response: Kp =1 ………………………………………………………………………………………………….27

Fig 4.3: Step response: Kp =5 ………………………………………………………………………………………………….28

Fig 4.4: Step Response: Kp = 2.4 …………………………………………………………………………………………………………..29

Fig 4.5: Step response: Kp =2.4, Ki =0.01, Kd =0 ……………………………………………………………………………………30

Fig 4.6: Step Response: Kp =2.4, Ki = 0.01, Kd = 0.1 ………………………………………………………………………….31

Fig 4.7: Various frequency plots for the compensated system during tuning (Kp = 5) ………………………….32

Fig 4.8: Various frequency plots for the compensated system (Kp =2.4, Ki=0.01, Kd =0.1)………33

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LIST OF TABLES

Table 3-1: Operational specifications ……………………………………………………………………………………………………17

Table 3-2: Lego Mindstorms EV3 large motor characteristics ……………………………………………………………….17

Table 4-1: EV3 Motor Load Charactersitics……………………………………………………………………………………………26

Table 4-2: Effects of Increasing PID Parameters……………………………………………………………………………………33

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ABSTRACT

Design of mobile robots has become an increasingly growing trend in the technology of modern

times. They are very attractive engineering systems, not only because of many interesting

theoretical aspects concerning kinematics, intelligent behavior and autonomy, but also because

of applicability in many human activities.

A typical example is the line following robot (LFR). In order for a LFR to function effectively, it

must demonstrate excellent line tracking control. This is achieved by having accurate and

responsive control algorithms as well as high – precision colour sensor systems.

This project proposes a system to show that good line tracking performance can be achieved

with moderately simple digital control algorithms. The platform used is a differentially-driven

wheeled robot constructed using the Lego Mindstorms components.

The simulation models are presented and analyzed using MATLAB/SIMULINK. The main

programming environment is the EV3 Software.

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CHAPTER 1

1. INTRODUCTION

1.1 DIGITAL CONTROL AND ROBOTICS:

Robotics is a branch of engineering that involves the conception, design, manufacture and

operation of robots. The field overlaps with electronics, computer science, artificial intelligence,

mechatronics, nanotechnology, bioengineering and control engineering. Robots are

mechatronic engineering products, capable of acting autonomously while implementing

assigned behaviors in various physical environments. The developed use of robots in many

areas makes the fundamental understanding of them fundamental.

In recent years there has been a rapid increase in the use of digital controllers in control

system. It has become routinely practicable to design very complicated digital controllers and to

carry out the extensive calculations required for their design. The current adoption of digital

rather than analog control in robotics is due to the genuine advantages found in working with

digital signals rather than continuous time signals.

1.2 PROBLEM STATEMENT

Line following robots need to adapt accurately, faster, efficiently and cheaply to changing

operating conditions. The drawbacks prominent in analog controllers reduce their suitability in

robotics. Hence, the necessity for digital controllers which provide better performance

capabilities.

1.3 OBJECTIVES

1.3.1 OVERALL OBJECTIVE

To design and implement a digital controller for a line following robot.

1.3.2 SPECIFIC OBJECTIVES

i. To design and build a line following robot using LEGO MINDSTORMS components

ii. To develop suitable line following algorithms

iii. To study the performance and challenges of the line following robot

iv. To set the desired performance criteria

v. To design a P.I.D digital controller to ensure satisfactory control of the system

vi. To implement the whole control algorithm on EV3 hardware.

vii. To study the performance of the compensated system

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1.4 PROJECT JUSTIFICATION

The use of analog controllers in control engineering poses problems such as limited accuracy,

susceptibility to noise and drift of power supply, cost ineffectiveness and less flexibility. Digital

control systems are more suitable for modern control systems because of reduced cost, noise

immunity and speed.

The LEGO MINDSTORMS is a powerful robotics kit that contains software and hardware to

create customizable and programmable robots.

Simulations will be done using MATLAB (R2014a).MATLAB (Matrix-laboratory) is a high-level

technical computing environment suitable for interactive debugging, visualization and tuning of

the controller parameters. It provides very strong mathematical and numerical support for the

implementation of advanced algorithms. The SIMULINK models can also be used to program

and run algorithms on LEGO MINDSTORMS robots.

The digital control algorithms will be programmed in EV3 software. The language’s graphical

interface allows the user to build simple programs and then easily and intuitively develop

complex algorithms.

1.5 PROJECT SCOPE

The focus of the project will be on:

The hardware development of the line following robot and study of actuator

mechanisms.

The design and analysis of requirements for satisfactory performance.

The design of the digital controller and implementation of control algorithms.

1.6 REPORT ORGANIZATION

This project report consists of six chapters.

Chapter 1 gives an introductory overview of the project and states the objectives, project

scope, problem statement, project justification and organization.

Chapter 2 provides literature review of all salient topics concerning control of the line following

robot. This includes digital control systems, types of controllers, parts of the robot and its

application.

Chapter 3 includes the project methodology and explains the proposed design of the robot and

digital control algorithms.

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Chapter 4 presents the analysis of the results of the project.

Chapter 5 deals with the overall conclusion, suggestions and recommendations for further

work.

Chapter 6 outlines the references cited in this project report.

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CHAPTER 2

2. LITERATURE REVIEW

2.1 DIGITAL CONTROL SYSTEMS

2.1.1 INTRODUCTION

A control system is an interconnection of components in which the output quantity is

controlled by varying the input quantity to perform a certain objective. Block diagrams are used

together with the transfer functions to describe the cause-and-effect relationships throughout

the system.

Digital control system refers to a discrete data control system in which a digital computer or

controller is used as one of the elements. A digital controller uses digital signals which are

defined only at discrete instants of time and have quantized amplitudes. An analog controller

uses analog signals which are defined over a continuous range of time and whose amplitudes

can assume a continuous range of values.

Fig 2-1: General configuration of a digital control system

To control a physical system or process using a digital controller, the controller must receive

measurements from the system, process them, and then send control signals to the actuator. A

sensor is needed to monitor the controlled variable for feedback control.

A device that converts digital signals to analog signals is called a digital-to-analog (D/A)

converter. Digital-to-analog conversion is simple and effectively instantaneous. Properly

weighted voltages are summed together to yield the analog output.

Conversely, a device that converts analog signals to digital signals is called an analog-to-digital

(A/D) converter. Analog-to-digital conversion is a two-step process and is not instantaneous.

There is a delay between the input analog voltage and the output digital code. The analog signal

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is first converted to a sampled signal at periodic intervals, held over the sampling interval, and

then converted to a sequence of binary numbers, the digital signal.

The sampling rate must be at least twice the bandwidth of the signal, or else there will be

distortion. This minimum sampling frequency is called the Nyquist sampling rate. Samples are

held over before being digitized because the (A/D) conversion occurs via digital counter, which

takes time to reach the correct digital number. Hence the constant analog voltage must be

present during the conversion process. A zero-order sample-and-hold (z.o.h) device yields a

staircase approximation to the analog system. Higher-order devices, such as a first-order hold,

generate more complex and more accurate wave shapes between the samples.

The z-transform is the principal analytical tool for analyzing sampled-data systems, and is

analogous to the Laplace transform for continuous systems. It allows sampled waveform to be

represented at the sampling instants.

2.1.2 WHY DIGITAL CONTROL?

Digital control offers distinct advantages over analog control that explain its popularity.

Speed: The increase in processing speed of computer hardware over the ages has made it

possible to sample and process control signals at very high speeds. Because the sampling

period can be made very small, digital controllers achieve performance that is essentially

the same as that based on continuous monitoring of the controlled variable.

Cost: Advances in very large-scale integration (VLSI) technology have made it possible to

manufacture better, faster, and more reliable integrated circuits. Decrease in the cost of

digital circuitry has made the use of digital controllers more economical even for small, low-

cost applications.

Accuracy: Digital signals are represented in term of zeros and ones with typically 12 bits or

more to represent a single number. This involves a very small error as compared to analog

signals, where noise and power supply drift are present.

Flexibility: An analog controller is difficult to modify or redesign once implemented in

hardware. A digital controller is implemented in firmware or software and its modification is

possible without a complete replacement of the original controller. Furthermore, the

structure of the digital controller need not follow one of the simple forms that are typically

used in analog control. More complex controller structures involve a few extra arithmetic

operations and are easily reliable.

Implementation errors. Digital processing of control signals involves addition and

multiplication by stored numerical values. The errors that result from digital representation

and arithmetic are negligible. By contrast, the processing of analog signals is performed

using components such as resistors and capacitors with actual values that vary significantly

from the nominal design values.

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2.1.3 TYPES OF CONTROLLERS

Proportional Controller: The actuating error signal for the control signal is proportional to

the error signal. The effect on the performance of a system is that it improves the steady-

state tracking accuracy, disturbance signal rejection and relative stability of the system. It

also increases the open-loop gain of the system which results in reducing the sensitivity of

the system. Increasing the gain to very large values may lead to instability of the system.

Proportional-Plus-Derivative (PD) Controller: the actuating error signal consists of

proportional error signal added to derivative of error signal. This increases the damping

ratio of the system and so the peak overshoot is reduced. Rise time and settling time are

reduced. Bandwidth is increased while the gain margin and phase margin are improved. PD

controller improves transient response.

Fig 2-2: Feedback loop with PD control

Proportional-Plus-Integral (PI) Controller: The actuating error signal consists of

proportional error signal added to the integral of error signal. It increases the order of the

system by one, which results in the reduction or removal of the steady state error. The

system becomes less stable. A properly designed PI controller decreases bandwidth but it

still improves the gain margin and phase margin.

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Fig 2-3: Feedback loop with PI control

Proportional-Plus-Integral-Plus-Derivative (PID) Controller: This controller has the

actuating signal consisting of proportional error signal added with derivative and derivative

and integral of error signal. The combined PID control action has the advantage of each of

the three individual control actions.

Fig 2-4: Feedback loop with PID control.

2.2 LEGO MINDSTORMS EV3

The Lego Mindstorms EV3 robotics system is the third generation robot in the Lego Mindstorms

robotics line. It was released as a successor platform to the second generation Lego

Mindstorms NXT 2.0 robot.

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2.2.1 PARTS OF THE LINE FOLLOWING ROBOT

2.2.1.1. THE EV3 BRICK

The EV3 brick is an intelligent programmable microcomputer that controls the motors and

sensors, executes written programs and commands, and provides wireless communication

through Wi-Fi and Bluetooth.

A breakdown of the technical specifications of the EV3 Brick is provided in the following list:

300 MHz, ARM9 Main Microprocessor with Linux-based operating system

On-board program storage including 16 MB of Flash memory and 64 MB RAM Main

Memory

User interface with three-colour, 6 backlit buttons that indicates the brick’s active state

LCD Monochrome Display, 178 X 128 pixels

SD-Micro card reader for 32 GB of expanded memory

4 input ports (for sensors) for data acquisition of up to 1000 samples/second

4 output ports (for motors) for execution of commands

Computer-to-brick communication through on-board USB, or external Bluetooth or Wi-

Fi connection

High-quality speaker

2.2.1.2. POWER SUPPLY

Power supply provides the energy to drive the controller, sensors and actuators. The 2050

mAh Lithium-ion EV3 Rechargeable DC battery pack is the preferred power supply option.

Six AA batteries can also be used to power the robot electronic parts.

Fig 2-5: Front and back views of the EV3 Brick. The back is the 9V battery.

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2.2.1.3. CONNECTOR CABLES

They link motor and sensors to the EV3 Brick.

Fig 2-6: Connector cables

2.2.1.4. BEAMS, CONNECTOR PEGS, AXLES AND BUSHES

Beams are structural elements which carry transverse loads of the robot components. Lego

beams have through-holes on the sides for connector pegs and axles.

Axles are designed to transfer rotational motion, for example from a motor shaft to a wheel.

They can be used to hold structures together.

Bushes prevent the axles from passing through a hole or a cross-hole.

Pins without friction turn smoothly and freely and are used to connect moving beams. Pins with

friction have ridges that increase friction and are great for building stable robotic structures.

Fig 2-7: A collection of Lego beams, axles, connector pegs and bushes

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2.2.1.5. CROSS BLOCKS AND ANGLE ELEMENTS

Cross blocks link beams together in perpendicular manner. Angle elements are designed to link

axles together so that they fit at certain angles.

Fig 2-8: A collection of Cross Blocks and Angle Elements

2.2.1.6. WHEELS AND TIRES

The wheels are used for locomotion

Fig 2-9: Lego wheels.

2.2.1.7. COLOUR SENSOR

The colour sensor is capable of detecting colour contrast between adjacent surfaces, providing

the robot with a form of vision. It can detect seven different colours: red, green, blue, yellow,

black, and white. It supports a light sensor mode, enabling it to detect light intensity reflected

from a surface. It outputs a value from 0 (dark) to 100(bright). For the best accuracy, the sensor

must be held at a right angle, close to the surface being detected.

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I. Front and back views of the colour sensor

II. (a) About 45 degrees (effective measurable range) (b) About 53 mm (c) About 54 mm (d)

About 88 mm. The light blue area is effective measuring range, though gray area is apt

to cause misrecognition.

III. (a) About 53 degrees (b) About 53 mm (c) About 71 mm

Fig 2-10: EV3 Colour sensor and the measuring ranges in the ‘Colour’ and ‘Reflected Light

Intensity’ modes.

2.2.1.8. SERVO MOTORS

The dc servomotors are the actuators used to convert an electrical signal (control voltage)

applied to them into an angular displacement of the shaft. They use tacho feedback for precise

control to within one-degree of accuracy and have built-in rotation sensors.

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Fig 2-11: Large servo motor

2.2.1.8.1. MODELLING OF A DC SERVO MOTOR

The physical model of a DC servo motor is given in Fig.2-12.

Fig.2-12: Model of a servo motor

The armature winding has a resistance and inductance . The field is produced either by a

permanent magnet or the field winding is separately excited and supplied with constant voltage

so that the field current is a constant.

When the armature is supplied with a DC voltage of , the armature rotates and produces a

back e.m.f . The armature current depends on the difference of and . The armature

has a moment of inertia , and frictional coefficient . The angular displacement of the motor

is .

The torque produced by the motor is given by,

, where is the motor torque constant. …. (2.1)

The back e.m.f is proportional to the speed of the motor and hence

, where is the back e.m.f constant. ….. (2.2)

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The differential equation representing the electrical system is given by,

….. (2.3)

Taking Laplace transform of equations (2.1), (2.2) and (2.3),

….. (2.4)

….. (2.5)

….. (2.6)

….. (2.7)

The mathematical model of the mechanical system is given by,

….. (2.8)

Taking Laplace transform of equation (2.8),

….. (2.9)

Therefore,

.... (2.10)

[ .... (2.11)

The block diagram representation of the armature controlled DC servo motor is developed in

steps, as shown in Fig.3-3.

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Fig 2-12: Individual blocks and complete representation of DC servo motor

2.2.1.8.2 DC SERVO MOTOR ENCODERS

A rotary encoder (shaft encoder), is an electro-mechanical device that converts the angular

position or motion of a shaft or axle to an analog or digital code. Incremental rotary encoders

are used to determine position and velocity. This can be either angular or linear motion. They

employ two outputs which are called quadrature outputs, as they are out of phase.

These signals are decoded to produce a count up pulse or countdown pulse. By measuring the

frequency of the square of wave oscillation, the rotational velocity can be computed.

The motor speed is controlled by pulse width modulation (PWM), which works by driving the

motor with a variable duty cycle square wave. This effectively turns the motor on and off. The

longer it is on, the more torque it will generate and the faster it will go.

Lego EV3 optical encoders measure the cumulative rotation of an EV3 motor in degrees. Driving

forward increments the measurement and vice versa.

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Fig. 2-13: Motor encoder

2.3 APPLICATIONS OF A LINE FOLLOWING ROBOT

Industrial Applications: These robots can be used as automated equipment carriers in

industries replacing traditional conveyer belts.

Automobile applications: These robots can also be used as automatic cars running on roads

with embedded magnets.

Domestic applications: These can also be used at homes for domestic purposes like floor

cleaning etc.

Guidance applications: These can be used in public places like shopping malls, museums etc.

to provide path guidance.

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CHAPTER 3

3. METHODOLOGY

The design of the digital controller for a line following robotic was done in stages.

Design and construction of a robot that can follow a line using Lego Mindstorms EV3.

Development of line following algorithms.

Study of the response of the LFR.

Design of a P.I.D digital controller to ensure satisfactory control of the system

Implementation of the whole control algorithm on EV3 hardware

Study of the performance of the compensated system

3.1 LEGO MINDSTORMS LINE FOLLOWER ROBOT DESIGN

A line follower is a mobile robot which is able to follow a visible line on a surface consisting of

contrasting colours.

To build and run the robot, the required hardware included; Lego EV3 brick, power supply, 2

large servo motors, a set of wheels, colour sensor, connector cables, beams, axles, bushes and

pins. The Lego parts were assembled, according to the instructions in Appendix B.

The EV3 brick formed part of the chassis, equipped with wheels. The servo motors were used to

drive the two front wheels. Two rear small castor wheels supported the robot.

The robot had a colour sensor mounted at the front end to identify the line. It was centered between the two front wheels, which were separated by a distance of 7.4 cm. It was designed to follow an oval track made of black electrical tape (18 mm wide) on a white surface.

Fig. 3-1: Line Following Robot

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3.2. STUDY OF THE LEGO MINDSTORMS EV3 MOTOR

3.2.1. MOTOR SPECIFICATIONS

Nominal Voltage 7.2V or 9V

Rotation Speed at no load 160 – 170 rpm

Running Torque 0.20 N-m

Stall Torque 0.40 N-m

Table 3-1: Operational specifications

Lego Mindstorms has not published the EV3 motor’s electromechanical characteristics. The

table below shows the proposed parameters used in this project.

Motor Parameter Unit Value

Torque constant, N.m/A 0.2

Back e.m.f constant, V.s/rad 0.5

Armature resistance, Ω 5

Armature inductance, H 0.005

Viscous damping coefficient, B N.m/rad.s 0.0006

Rotor inertia coefficient, J N.m. 0.001

Table 3-2: Lego Mindstorms EV3 large motor characteristics

3.2.2 EV3 SERVO MOTOR IMPLEMENTATION ON MATLAB:

Fig.3-2: Motor implementation on Simulink

For various input voltages, the response of the motor model under constant torque was

simulated and the results tabulated.

18

3.2.3: S-DOMAIN EV3 SERVO MOTOR TRANSFER FUNCTION:

The continuous time transfer function of the dc servo motor is

………..3.1

…………..3.2

The step response for 9V input is shown below

Fig.3-3: Step response of the s-domain servo motor transfer function

Observations: Settling time = 0.187 seconds Final value = 17.4 Rise time = 0.104 seconds Overshoot (%) = 0

Step Response

Time (seconds)

Am

plitu

de

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

14

16

18

System: untitled1

Settling time (seconds): 0.187

System: untitled1

Peak amplitude: >= 17.4

Overshoot (%): 0

At time (seconds): > 0.3System: untitled1

Rise time (seconds): 0.104

19

The root locus plot is illustrated below.

Fig. 3-4: Root Locus of the s-domain servo motor transfer function

Observation: All closed-loop poles are on the left-half of the s-plane. The system is stable.

The frequency response is as shown below.

Fig. 3-5: Bode plot of the s-domain servo motor transfer function

-1200 -1000 -800 -600 -400 -200 0 200-300

-200

-100

0

100

200

300Root Locus

Real Axis (seconds-1)

Imag

inar

y A

xis

(sec

onds

-1)

-150

-100

-50

0

50

Mag

nitu

de (d

B)

100

101

102

103

104

105

-180

-135

-90

-45

0

Pha

se (d

eg)

Bode Diagram

Gm = Inf dB (at Inf rad/s) , Pm = 119 deg (at 35 rad/s)

Frequency (rad/s)

20

Observations: Gain margin is infinite

Phase margin is

3.2.4: Z-DOMAIN EV3 SERVO MOTOR TRANSFER FUNCTION:

The selected sampling time Ts, is 0.01 seconds. G(s) in cascade with zero-order hold can be converted to G(z) to yield the following transfer function:

Fig 3-6: Step response of the z domain servo motor transfer function

Observations:

Rise time = 0.105 seconds

Settling time = 0.187 seconds

0vershoot (%) =0

Peak amplitude = 17.5

Step Response

Time (seconds)

Am

plitu

de

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2

4

6

8

10

12

14

16

18

System: untitled1


System: untitled1


System: untitled1

Peak amplitude: >= 17.5

Overshoot (%): 0

At time (seconds): > 0.35

21

Fig 3-7: Root Locus of the z-domain servo motor transfer function.

Observation: All poles of the open-loop transfer function are inside the unit circle on the z-plane.

Therefore, the system is stable.

Fig 3-8: Bode plot of the z-domain servo motor transfer function

Observations: Gain margin is 15.6dB

Phase margin is

-1.5 -1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Root Locus

Real Axis

Imag

inar

y A

xis

-20

-10

0

10

Mag

nitu

de (d

B)

100

101

102

103

-180

-135

-90

-45

0

Pha

se (d

eg)

Bode Diagram

Gm = 15.6 dB (at 314 rad/s) , Pm = 109 deg (at 35 rad/s)

Frequency (rad/s)

22

3.3 LINE FOLLOWING ROBOT ALGORITHMS

Line following works by using the colour sensor (in reflected light intensity mode) to read the

changes in the reflected light levels along the edge of a dark and light surface. The Reflected

light Intensity is measured as a percentage from 0% (very low reflectivity) to 100% (very high

reflectivity). More light is reflected from a white surface compared to the black surface.

Depending on the light sensor value, the motors are directed to vary the speed.

In a program white and black values are defined using a threshold value. Threshold is the

average of the sensor value with the sensor on the black line and one found on the white area.

Different measurements for black and white depend on factors such as the light level in the

room, the robot’s battery level, and the type of surface used.

The light sensor will read the light value. Then the robot can be programmed such that if the

sensor sees black, which is when the sensor value is less than the threshold, the robot should

turn right, else it should turn left.

Fig 3-9: Basic line following approach

The robot will be started. It will then be set to move forward. It will be made to steer

right until it detects the line edge.

Once the sensor sees black, the robot will continue to go forward while turning left

gradually.

Whenever the sensor will see white (i.e. the robot leaving the line), the robot will turn

to the right until the sensor finds black again.

The sequence then will be repeated in a loop, unless the robot is stopped.

The simple line following algorithm does not control the robot to follow a line reliably. This is

because it is choosing one of two actions each time it loops; steer left or steer right.

23

3.4: DIGITAL CONTROLLER DESIGN: A robot without a controller will oscillate a lot about the line, leading to more consumption of battery power, less speed and following the line less efficiently.

When designing a line following robot, the transient response specifications are defined as: Rise time: It is how fast the robot will try to get back to the line after it has drifted off.

Overshoot: The distance past the line edge the robot will tend to go as it is responding to an

error. The amount of overshoot indicates the relative stability of the system.

Steady-state error: the offset from the line as the robot follows a long straight line.

Settling time: The time the LFR will take to settle down when it encounters a turn.

The performance criteria were stipulated as follows: Constant speed of 0.1m/s to be maintained despite the presence of turns.

Steady-state error: Less than 2%

Settling time of less than 0.1 seconds

Overshoot (%) of less than 1.0

Finite phase margin

The robot controller to be designed was to be modified until the transient response met was satisfactory.

3.4.1. PROPOSED CONTROLLER DESIGN:

The proposed controller was a Proportional-plus-Integral-plus-Derivative (PID) digital controller.

The PID controller would control the position of the robot with quick response time and minimize the overshoot.

The proportional part would determine the magnitude of turn required to correct the error sensed.

The integral part would improve the steady state error (proportional offset) which increases while the robot is not on the line. The derivative part would measure the deviation from the path and minimize overshoot. It would reduce the oscillating effect about the line. The derivative control is used to provide anticipative action.

24

3.4.2: IMPLEMENTATION OF LINE FOLLOWING CONTROL ALGORITHM FOR THE LEGO

MINDSTORMS EV3 HARDWARE:

Fig 3-10: Simulink Line Tracking Program with PID Controller

Sensors and Motors contain blocks that interface with the EV3 hardware. Actual Speed Values

block uses the values from each motor encoder to calculate the position and velocity of the

robot. Desired Velocity takes the user-provided Velocity (m/s) and converts it into the desired

state values for the velocity controller. Desired Light takes the Color sensor's white and black

values to choose an appropriate reference value for the light. Velocity Control has the PID

controller implementation to control the forward velocity. Line Tracking Controller has the PID

controller implementation to control the turning.

However, to download and run a line tracking Simulink model on the LEGO MINDSTORMS EV3 robot , EV3 Wi-Fi Dongle or USB Ethernet Adaptor, and Wi-Fi Router were required to set up a network connection between EV3 brick and host computer. The line following program was then written in EV3 software programming language. The black and white light intensity values were calibrated accordingly for the robot and the track. Using the provided USB cable, the program was downloaded and run on the robot. PID parameter tuning was done experimentally to achieve smoother line tracking.

25

Fig 3-11: EV3 Software Line Following Program with PID Controller

26

CHAPTER 4

4. RESULTS AND ANALYSIS

4.1: EV3 LARGE MOTOR CHARACTERISTICS

Input Voltage

Torque Rotation speed

Current Mechanical

power Electrical power

Efficiency

4.5 V 17.3 N.cm 24 rpm 0.69 A 0.43 W 3.10 W 14 %

6.0 V 17.3 N.cm 51 rpm 0.69 A 0.92 W 4.14 W 22 %

7.5 V 17.3 N.cm 78rpm 0.69 A 1.41 W 5.17 W 27 %

9.0 V 17.3 N.cm 105 rpm 0.69 A 1.90 W 6.21 W 31 %

10.5 V 17.3 N.cm 132 rpm 0.69 A 2.39 W 7.24 W 33 %

12.0 V 17.3 N.cm 153 rpm 0.69 A 2.77 W 8.28 W 33 %

Table 4-1: EV3 Motor Load Characteristics

From the table, the linear relationship between power level and EV3 large motor speed was noticeable.

Fig 4.1: Graph of rotation speed against applied voltage The rotation speed of the EV3 large motor was proportional to the input voltage.

0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12 14

Ro

tati

on

Sp

ee

d(r

pm

)

Input Voltage (V)

27

4.2: PID TUNING

A. For Kp=1, Ki=0, Kd=0.

Fig 4.2: Step response: Kp =1 Observations:

Rise time = 0.0112 seconds Settling time = 0.0203 seconds Final value = 0.875

Step Response

Time (seconds)

Am

plit

ude

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

System: T


System: T


System: T

Final value: 0.875

28

B. For Kp=5. Ki=0, Kd=0.

Fig 4.3: Step response: Kp =5

Observations:

Rise time = 0.0021 seconds Settling time = 0.00686 seconds Final value = 0.972 Overshoot (%) = 10.6 Peak amplitude = 1.08

Step Response

Time (seconds)

Am

plit

ude

0 0.005 0.01 0.0150

0.2

0.4

0.6

0.8

1

1.2

1.4

System: T

Final value: 0.972

System: T


System: T

Peak amplitude: 1.08

Overshoot (%): 10.6

At time (seconds): 0.00451

System: T


29

C. For Kp=2.4, Ki=0, Kd=0.

Fig 4.4: Step Response: Kp = 2.4

Observations:

Rise time = 0.00425 seconds Settling time = 0.00657 Final value = 0.94 Overshoot (%) = 0.988 Peak amplitude = 0.953

Step Response

Time (seconds)

Am

plitu

de

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1System: T

Final value: 0.944

System: T


Overshoot (%): 0.988


System: T


System: T


30

D. For Kp=2.4. Ki=0.01, Kd=0.

Fig 4.5: Step response: Kp =2.4, Ki =0.01, Kd =0

Observations:


Step Response

Time (seconds)

Am

plitu

de

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System: T


System: T


System: T




System: T

Final value: 0.944

31

E. For Kp=2.4. Ki=0.01, Kd=0.1.

Fig 4.6: Step Response: Kp =2.4, Ki = 0.01, Kd = 0.1

Observations:


Step Response

Time (seconds)

Am

plitu

de

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System: T

Rise time (seconds): 0.00425 System: T




System: T


32

4.3: FREQUENCY RESPONSE

A. For Kp=5,Ki=0, Kd=0.

Fig 4.7: Various frequency plots for the compensated system during tuning (Kp = 5)

Observations

The root locus exhibited complex closed loop poles. Both the phase and gain margin were infinite. The system was stable

33

B. For Kp=2.4, Ki=0.01, Kd=0.1.

Fig 4.8: Various frequency plots for the compensated system (Kp =2.4, Ki=0.01, Kd =0.1)

Observations

The root locus closed-loop poles changed from complex to real. The system was still stable The infinite gain margin showed inherent stability

4.4. GENERAL ANALYSIS The effects of each of controller parameters, Kp, Kd, and Ki on the line following robot are summarized in the table below.

Parameter Rise Time Overshoot Settling Time Steady-state Error

Kp Decrease Increase Small change Decrease

Ki Decrease Increase Increase Eliminate

Kd Small change Decrease Increase No change

Table 4-2: Effects of Increasing PID Parameters

34

The difficulty of tuning increased with the number of parameters that were to be adjusted. To

observe the response that resulted from the tuning adjustments, it was necessary to wait for

several minutes. This made the tuning by trial-and-error a tedious and time-consuming task.

In practice, the stability of a mathematical model was not sufficient to guarantee acceptable

system performance or even to guarantee the stability of the physical system that the model

represented. This is because of the approximate nature of mathematical models.

The main problems associated with the implementation of digital control are related to the

effects of quantization and sampling.

i. Quantization effects: The conversion of signals from analog into digital form and vice-versa

is performed by electronic devices of finite resolution. A device of n-bit resolution has

quantization levels. Here, the analog signals get tied to these finite number of quantization

levels in the process of conversion to digital form. Therefore, by the sheer act of conversion

a valuable part of information about the signal is lost. Furthermore, any computer

employed as a real-time controller must perform all the necessary calculations with limited

precision, thus introducing a truncation error after each arithmetic operation has been

performed. As the computational accuracy is normally much higher than the resolution of

real converters, a further truncation must take place before the computed data is converted

into the analog form.

ii. Sampling effects: The sampling theorem ensures that there is no loss of information due to

sampling and the continuous time signal can be completely recovered from its samples

using an ideal low-pass filter. There are, however, two problems associated with the use of

this theorem in practical control systems. Real signals are not band-limited and hence strict

bandwidth limits are not defined. The ideal-low pass filter, needed for the distortionless

reconstruction of continuous-time signals from its samples, is not physically realizable.

Practical devices, such as the D/A converters, introduce distortions. Thus the process of

sampling and reconstruction also affects the amount of information available to the control

computer, and degrades control system.

The advantages of digital control outweigh its implementation problems for most of the applications.

35

CHAPTER 5

5. CONCLUSION AND RECOMMENDATIONS

5.1 CONCLUSION

A line following robot was designed and built using LEGO MINDSTORMS EV3 components.

Digital control algorithms were developed. The advantages and limitations of implementing the

digital control on different software were studied. The effectiveness of using PID controller for

optimum line tracking was demonstrated by inspecting the movement pattern of the robot

while following the track. To obtain the desired control response, Kp, Ki and Kd were

successfully determined by tuning.

5.2 RECOMMENDATIONS FOR IMPLEMENTATION

For easy navigation on various terrains, additional features will be required. Replacement of

tires with treads should be considered. Walking robots have better mobility than wheeled

robots.

Greater efficiency can be obtained if a control algorithm for obstacle avoidance is also

included. Detecting obstacles ensures the reliable and safe movement of the robot.

A convenient and economical alternative to the AA batteries is the EV3 Rechargeable

Battery. It can be recharged while inbuilt in a model, eliminating the problem of

disassembling and reassembling a robot to replace batteries. It also weighs less.

Programming languages such as C and Java can also be used to implement control

algorithms on EV3 hardware. RobotC is a C-based cross-robotics programming platform

while LeJOS allows Lego Mindstorms robots to be programmed in the Java programming

language.

To provide a valuable and highly interactive insight into the real-time internal working of

the model, Simulink is appropriate.

5.3 RECOMMENDATIONS FOR FURTHER WORK

Designing a controller required the kinematic modeling, which is a much less accurate depiction

of the actual system. The influence of the robot’s dynamics should not be neglected. Dynamic

modeling takes into account such factors such as the robot’s weight, center of gravity,

cornering stiffness, tire slippage and aerodynamic resistance. There may be complexity

encountered while deriving and working with the necessary equations and measuring some of

the parameters. However, it gives a highly accurate portrayal of the robot’s behavior.

There has been much research in the area of control theory and many modern controllers have been developed as a result. The same study of the line following robot may be done using fuzzy logic controllers, neural networks and adaptive controllers.

36

CHAPTER 6

6 REFERENCES

Daniele Benedetelli, “The Lego Mindstorms EV3 Laboratory”, No Starch Press Inc., 2014

Laurens Valk, “The Lego Mindstorms EV3 Discovery Book”, No Starch Press Inc., 2014

Terry Griffin, “The Art of Lego Mindstorms EV3 Programming”, No Starch Press Inc., 2014

Mark Rollins, “Beginning Lego Mindstorms EV3”, APress, 2014

William S. Levie, “Control System Fundamentals”, CRC Press, 1999

Jagan, N.C., “Control Systems”, BS Publications, 2008, Second Edition

Norman S. Nise, Control systems Engineering, Sixth Edition, John Wiley& Sons, Inc, 2011

Katsuhiko Ogata, “Discrete time control systems”, Prentice-Hall Inc. A Pearson Education Company, Upper Saddle River, New Jersey 07458, 1995 Kuo, B.C., “Digital Control Systems”, Saunders, 1992 R. C. Dorf and R. H. Bishop, “Modern Control Systems”, Pearson Education, Inc., Upper Saddle River, New Jersey 07458, Twelfth Edition, 2011 Jones, J., Flynn, A., Seiger, B., “Mobile Robots: From Inspiration to Implementation, A.K Peters,

Welleslay MA, 1999, Second Edition

Thomas Brӓunl, “Embedded Robotics: Mobile Robot Design and Applications with Embedded

Systems”, Springer-Verlag Berlin Heidelberg, Third Edition

Chen, C.T., Analog and Digital Control System Design, Saunders-HBJ, 1993

M. Sami Fadali, Antonio Visioli, “Digital Control Engineering: Analysis and Design”,

Academic Press- an imprint of Elsevier 30 Corporate Drive, Suite Burlington, MA 01803, 2003,

Second Edition

Lego Mindstorms EV3 [online] Available FTP: http://www.lego.com/

Lego Mindstorms EV3 – Wikipedia [online] Available FTP:

http://www.en.wikipedia.org/wiki/Lego_Mindstorms_EV3/

Lego Mindstorms - Wikipedia [online] Available FTP:

http://www.en.wikipedia.org/wiki/Lego_Mindstorms

http://www.lego.com/

http://www.en.wikipedia.org/wiki/Lego_Mindstorms_EV3/

http://www.en.wikipedia.org/wiki/Lego_Mindstorms

37

Encoders [online] Available FTP: http://www.brickengineer.com/pages/2008/09/05/lego-nxt-

motor-wiring/

EV3 Color Sensor [online] Available FTP: http://www.cdstem.wordpress.com//2014/05/09/

EV3 Large Motor [online] Available FTP: http://www.afrel.co.jp.en/archives/96

Mathworks Support for Lego Mindstorms EV3 [online] Available FTP:

http://www.mathworks.com.academia/lego_mindstorms_ev3_software/

Control Tutorial for MATLAB and SIMULINK [online] Available

FTP:http://www.ctms.engin.umich.edu/

Line tracking [online] Available FTP:

http://www.mathworks.com/help/supportpkg/legomindstormsev3/examples/line-

tracking.html

Robotics [online] Available FTP: http://www.whatis.techtarget.com/definition/robotics

How to create a line following robot using Mindstorms [online] Available FTP:

http://www.thetechnicgear.com/2014/03/how_to_create_line_following_robot_using_mindst

orms/

PID Controller for Lego Mindstorms Robots [online] Available FTP:

http://www.inpharmix.com/jps/PID_Controller_For_Lego_Mindstorms_Robots.html.

Lego 9V Technic Motors Compared Characteristics [online] Available FTP:

http://www.philohome.com/motors/

Susmita Das, AyanChakraborty, Jayanta Kumar Ray, SoumyenduBhattacharjee, Biswarup

Neogi : Study on Different Tuning Approach with Incorporation of Simulation Aspect for Z-N

(Ziegler-Nichols) Rules.ijsrp , Vol. 2, Issue 8, August 2012. ISSN 2250 - 3153.

http://www.ijsrp.org/research-paper-0.812/ijsrp-p0823.pdf

http://www.brickengineer.com/pages/2008/09/05/lego-nxt-motor-wiring/

http://www.brickengineer.com/pages/2008/09/05/lego-nxt-motor-wiring/

http://www.cdstem.wordpress.com/2014/05/09/

http://www.afrel.co.jp.en/archives/96

http://www.mathworks.com.academia/lego_mindstorms_ev3_software/

http://www.ctms.engin.umich.edu/

http://www.mathworks.com/help/supportpkg/legomindstormsev3/examples/line-tracking.html

http://www.mathworks.com/help/supportpkg/legomindstormsev3/examples/line-tracking.html

http://www.whatis.techtarget.com/definition/robotics

http://www.thetechnicgear.com/2014/03/how_to_create_line_following_robot_using_mindstorms/

http://www.thetechnicgear.com/2014/03/how_to_create_line_following_robot_using_mindstorms/

http://www.inpharmix.com/jps/PID_Controller_For_Lego_Mindstorms_Robots.html

http://www.philohome.com/motors/

http://www.ijsrp.org/research-paper-0.812/ijsrp-p0823.pdf

38

APPENDIX A

A.1 MATLAB CODE FOR CHAPTER 3 '(ch3) Fig3.3' %Display label

num =0.2; %Define denominator of G(s)

den =[0.000005 0.005003 0.103]; %Define denominator of G(s)

Gs =tf(num,den); %Create LTI transfer function

step(9*Gs) %Create step response

title('Step Response') %Add title to the graph

grid on %Add grid

'(ch3) Fig3.4' %Display label

clear %Clear previous variables from workspace

clf %Clear graphs




rlocus(Gs) %Plot root locus

title('Root LOcus') %Add title to the plot

grid on %Add grid

pause



clf %Clear graphs




margin(Gs) %Display Bode plot

title('Bode Plot') %Add title to the plot

grid on %Add grid

pause

'(ch3) Transformation using the Zero-order hold' %Display label




Ts =0.01 %Define sampling time

Gz =c2d(Gs,Ts,'zoh') %Convert G(s)in cascade with z.o.h to

%G(z)

pause


num =[0 0.3337 0.0345]; %Define numerator of G(z)

den =[1 -0.8104 0.00004513]; %Define denominator of G(z)

Gz =tf(num,den) %Form G(z)as a LTI transfer function

step(9*Gz) %Generate step response

title('Step Response') %Add title to the graph

grid on %Add grid

pause

39

(ch3) Fig3.7' %Display label


clf %Clear graphs



Gz =tf(num,den) %Form Gz as a LTI tranfer function and

display

rlocus(Gz) %Plot rootlocus

title('Root Locus') %Add title to the plot

grid on %Add grid

pause



clf %Clear graphs



Gz =tf(num,den) %Form Gz as a LTI tranfer function and

display

margin(Gz) %Display Bode plot

title('Bode Plot') %Add title to the plot

grid on %Add grid

pause

40

A.2 MATLAB CODE FOR CHAPTER 4 '(ch4) Fig4.2' %Display label


clf %Clear graphs




Gp =0.4*9*Gs)

Kp =1; %Proportional gain

Ki =0; %Integral gain

Kd =0; %Derivative gain

C =pid(Kp,Ki,Kd)

T =feedback(C*Gp,1)

step(T) %Generate step response

title('Step Response') %Add title to the plot

grid on %Add grid

pause



clf %Clear graphs




Gp =0.4*9*Gs)

Kp =5; %Proportional gain



C =pid(Kp,Ki,Kd)

T =feedback(C*Gp,1)



grid on %Add grid

pause



clf %Clear graphs




Gp =0.4*9*Gs)

Kp =2.4; %Proportional gain



C =pid(Kp,Ki,Kd)

T =feedback(C*Gp,1)



grid on %Add grid

pause

41



clf %Clear graphs




Gp =0.4*9*Gs)


Ki =0.01; %Integral gain


C =pid(Kp,Ki,Kd)

T =feedback(C*Gp,1)



grid on %Add grid

pause



clf %Clear graphs




Gp =0.4*9*Gs)


Ki =0.01; %Integral gain

Kd =0.1; %Derivative gain

C =pid(Kp,Ki,Kd)

T =feedback(C*Gp,1)



grid on %Add grid

pause

42

APPENDIX B: BUILDING THE LINE FOLLOWING ROBOT

43

44

45

46

47

48

Documents

UNIVERSITY OF NAIROBI DEPARTMENT OF …eie.uonbi.ac.ke/sites/default/files/cae/engineering/eie/DIGITAL... · 3.1 LEGO MINDSTORMS LINE FOLLOWER ROBOT DESIGN ... Simulink Line Tracking