Designing Development and Testing of a PIC Micro-Controller Based Differential Drive Line Tracing...

8/3/2019 Designing Development and Testing of a PIC Micro-Controller Based Differential Drive Line Tracing Vehicle-2 With PID

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Aim

To design, develop and test a PIC based differential drive operated automatic robot.


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Declaration of Originality

We, Krishneel Prasad, Ritesh Kumar, Kirata Iotam and Benjamin Kunal Chand hereby

declare that the report that has been written for this project is our own original workdone with the best of our knowledge. All materials adapted from elsewhere have beenappropriately acknowledged and referenced. The material has not been submittedpreviously, either in whole or in part, for a degree at this or any other institution.

___________________ ____________________

Krishneel Ritesh Prasad Ritesh PrasadStudent Id : s11050297 Student Id : s11048886

___________________ ____________________Kirata Iotam Benjamin Kunal ChandStudent Id : s11045193 Student Id : s11039935

30th November, 2010.


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ACKNOWLEDGEMENT

We would like to take this opportunity to thanks the following people for their help,support and encouragement while doing this project.Firstly we are especially thankful to Imran Jannif for giving us the opportunity to do thisproject. Our family for their understanding by the time we dont spend time with themand finally to all Engineering staffs for their support and encouragement.


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List of FiguresFigure 1.1 PacSea 2Figure 1.2 Pacsea wave model 3Figure 1.4 Pacificsea 4Figure 1.5 Pacsea surf 4Figure 1.6 PacseaLoa 5Figure 1.7 PacSea-Colata model 5Figure 1.8.1 PacSea-MataiAutomatic 1 6Figure 1.8.2 PacSea-MataiAutomatic 2 6Figure 1.8.3 PacSea-MataiManual Robot 6Figure 1.9 PID Block Diagram 7

Figure 2.0 Differential Drive Motor Controller Setup 10

Figure 2.1a IR sensor board 11

Figure 2.1b Shaft encoder and IR sensor unit 11

Figure 2.2 12V Cordless drill motor 12

Figure 2.3 In-house fabricated wooden wheel 13

Figure 2.4 Wheel calculation 13Figures 2.5a Motor controller board 15Figures 2.5b PIC16F877 boot loader board 15Figures 2.5c LM324 comparator PCB board 15Figures 2.5d IR sensor mounting 15

Figure 4.1 SN754410 chip 29Figure 4.2 schematic of sn754410 30Figure 4.3 Pin connection for sn754410 31Figure 4.4 Cordless drill motor 32

Figure 4.5 shows the testing for sn754410 chip 33

Figure 4.5.1 Graph of current, voltage and rpm for sn754410 chip 34

Figure 4.6 differential drive 35

Figure 4.7 A generic H-bridge 35

Figure 4.8 One output on the SN754410 chip 36Figure 4.9 Shows the PWM signal 37

Figure 5.1 Heds 5540 a06 39Figure 5.2 Hardware setup for oktec shaft encoder 41


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Figure 5.3 16 slots aluminum plate 41Figure 5.4 The shaft encoder between two ir sensors, shaft encoder circuit 42Figure 5.5 The hardware for the comparator circuit 43Figure 5.6 A smooth square wave signal from the comparator output 44

Figure 6.1 Graph of Motor Speed versus time at 35% duty cycle 47Figure 6.2 The step response of the Graph of Motor at 35% duty cycle 51Figure 6.3 The graph of Log |1y (t)| vs. time motor at 35% duty cycle 54

Figure 6.4 The graph of Log (1+A*Exp (- x t) y(t)) Log |1y (t)| vs. time at 35% duty cycle 57

Figure 6.5 The Graph of Motor Speed versus time at 50% duty cycle 60

Figure 6.6 The graph of Log |1y (t)| vs. time motor at 50% duty cycle 62

Figure 6.7 The graph of Log (1+A*Exp (- x t) y (t)) Log |1y (t)| vs. time at 50% duty cycle 64

Figure 6.8 Step responses for a proportional controller 69

Figure 6.9 A step response for a PI controller at 0.005 and 80 respectively 70Figure 7.0 A step response for a PI controller at 8.5 and 75 respectively 71

Figure 7.1.1 Proposed mechanical base platform 74Figure: 7.1.2 Uniform holesreducing the weight of the machine 75Figure 7.1.3 Tricycle wheel 76Figure 7.1.4 ABU Robocon 2009 76Figure 7.1.5(a) (b) Proposed wooden wheel 76Figure 7.1.6 Robot arm design 77Figure 7.1.7 Roller arm design 78Figure 7.1.8 Actual view of arm roller 78

Figure 7.1.9 Rubber roller 78Figure 7.8 Stress Analyses 81Figure 7.8.1 Krathong Petal 84

Figure 7.8.2 Krathong Petal 84

Figure 7.8.3 Candle Light Flame 85

Figure 7.8.4 Automatic 1 physical diagram 85

Figure 7.8.5 assembled krathong 87

Figure 7.8.6 Automatic arm description 87

Figure 7.8.7 Auto side view 88

Figure 7.8.8 OSMC motor controller 90

Figure 7.8.9 LDR Sensor Schematic 93

Figure 7.9.0 LDR Sensor 93


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Figure 7.9.1 Force sensor diagram 94Figure 7.9.2 PIC18f6722 Prototype board 95Figure 7.9.3 PIC18F4550 microcontroller 96Figure 7.9.4 Crouzet 82862 001 12 volts DC motor 97Figure 7.9.5 SERVO MOTOR 98

Figure 7.9.6 Battery 99


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List of TablesTable 4.1 Parameters for sn754410 chip 29Table 4.2 Truth table for SN754410 31Table 4.4 No load test for the dc motor 33

Table 6.1 The data recorded for 35 % duty cycle 45-47Table 6.2 The table below shows the step response figures at 35% duty cycle 49-50

Table 6.3 The log(y (t)y ()) data at 35% duty cycle 52-53

Table 6.4 A Log (1 + A*Exp(xt) - y(t)) at 35% duty cycle 55-56

Table 6.5 Calculating y(t)=1-(exp(- x t)) for 50% duty cycle 58-59

Table 6.6 Calculating log |1-y(t)| for 50% duty cycle 60-61

Table 6.7 Calculating Log (1 + A*Exp (xt) - y(t)) at 35% duty cycle 63-64

Table 7.1.0 The comparison of the base unit from previous years robots 73Table 7.1.1 Different masses and forces applied to the robots arm 79

Table 7.1.2 Manual machine path length, amount of weight, spring constant 80

Table 7.2 The summary table of Auto 1s task objects 85Table 7.3 The table shows Auto 2 task object and weight 87Table 7.5.1 Specification for the LDR line tracer sensor 93

Table 7.5.2 A summary of some important features of PIC 18F6722 95


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6.2 Calculation of the Transfer function at 35% duty cycle 456.3 Calculation of the Transfer function at 35% duty cycle 586.4 Comparing the Driver Motor Transfer Function 656.5 Obtaining the Driver Motor Transfer Function through Formula 65

6.7 Selecting the transfer function of the motor by comparing the two methods 68

6.8 Using MATLAB to derive the gain of the PI controller 68

Chapter 7

7.1. Mechanical Aspects of Robocon 2011 Robots 737.1.1 Base Design 1 737.1.2 Mechanical Base Platform 737.1.3 Materials Used 747.1.4 Wheels 757.1.5 Arm Construction 767.1.6 Rubber Rollers 77

7.1.7 Displacement Calculation for Manual Robot 787.2 Stress Analysis Calculation for Manual Robot 817.2.1 Weight Calculation for Manual Robot 827.2.2 Mechanical Design for Automatics Machines. 837.2.3 Distance travelled by the Auto 1 837.2.4 The tasks to be performed Automatic One 837.2.5 Time Taken by Automatic 1 867.3 Automatic Two 877.3.1 Time Taken by Automatic 2 887.3.2Moment Calculation 887.3.3 Weight Calculation for Automatic 2 Robot 89

7.4 Total Weight of All three Machines 907.5 Electronics for Automatic machines 907.5.1 Motor Controller 90

7.5.2 Features of OSMC 91

7.5.3 Sensors 927.5.4 Force Sensors 947.5.5 MICROCONTROLLER 947.5.5.1 PIC18f6722 947.5.5.2 PIC18F4550 USB DEVELOPMENT BOARD 957.5.5.3 VOLTS DC MOTOR 97

7.5.5.4 Motors for the Grippers 987.5.5.5 SERVO MOTOR 987.5.5.6 Battery 997.5.5.7 V 2300 mAH NiMH Rechargeable Battery Pack 100

7.5.5.8 Features and Benefits 100

7.5.5.9 Technical Specifications 100


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Discussion 101

Conclusions and Recommendations 102

References 103

Appendix 104


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Chapter 1

Introduction and Literature Review

Asia-Pacific Robot Contest or ABU Robocon was developed by the Asia-Pacific

Broadcasting Union in 2002, and was joined by many universities and colleges aroundAsia and the Pacific. The main purpose of ABU is to develop students technical skills aswell as designing abilities in developing robots that will be able to compete a given tasksin the battlefield/game ground within a set period of time, which will therefore improveengineering and technology in the region. There are many different robots whichRobocon group had developed in the past. One of the most common robots developed bythe Robocon members is a Line tracing automatic robot, this robot will try to follow awhite track and detecting any obstacles that my come in its way.

Moreover, every year the Robocon group members try to complete at least one automaticrobot and a manual robot. Additionally, the automatic robot with a square base unit has

two motors connect back to back. The speed of each motor must be the same for the twomotors and the two motors must rotate in different directions. Thus, to achieve this, theshaft encoder is introduced in order for the PIC-Microcontroller to synchronize the speedof the motor [1]. In addition, an LDR (Light Dependent Resistor) sensor with a LED(Light Emitting Diode) compiled together to give the ability for the robot to do linetracing. In most cases, the line tracing PCB (printed circuit board) always face downwardin order for the LDR to receive a reflected light from the LED to detect a white or a blacksurface.


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In 2001, the University of the South Pacific joins the competition which was held inTokyo 2002. In the past seven years, the University of the South Pacific has involved in

ABU Robocon contest and below is the details of some robots designed in Robocon inthe past seven years.

1.1 Pacsea

To start off, Pacsea was one of the robotics designed for the competition that was held inBangkok, Thailand in August 2002. The diagram of Pacsea was shown below.

Figure 1.1 PacSea (Source: Msc. Thesis, R.Singh)

Additionally, the theme of the contest is Reach for the Top of Mount Fuji. The robot

must be capable to drop the beach balls into different cylinders at different heights. Inaddition, the task has to be complete by one automatic robot by navigating the ball andplacing the ball in the right place. A PIC16F877 is one major part of Pacsea whichcontrols two relays by switching on and off the wiper window motor; however Pacsea hasno feedback sensor [1]

1.2 Pacsea Wave

In 2003 the game was held in Bangkok, Thailand and the theme of the competition was"Takraw Space Conqueror". To win the contest the robot must shoot a Takraw balls intoall the baskets and the robot will also fill the three nets inside one of the centered basketwith balls [1]. The intelligent part of the robot is chosen as a single PIC16F877 and withPIC16F877 students make the robot slightly intelligent by giving the robot the ability totrace a line [1]. The picture of Pacsea wave is shown below:


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Figure 1.2 Pacsea wave model (Source: Msc. Thesis, R.Singh)

1.3 Pacsea Surge

Furthermore, in 2004 and The Reunion of the Shepherd and the Weaver is the theme ofthe contest. Each team has to carry red boxes (gifts to the god) and the robot will carrythe box across the bridge. To win, the robot has to carry most items within the allocatedtime. Additionally, the central intelligent part of the machine has upgraded toPIC18F8720 from PIC16F877. Moreover, the advantage of using a PIC18F8720 is thatthere are a lot pins (Input/output pins) compared to PIC16F877, it can handle morecomplex game plans. In addition, the type of motor used in Pacsea surge is a wiperwindow and the speed of the motor is generic controlled by a pulse width modulator(PWM) [x] that was sent by PIC18F8720. The diagram of Pacsea surge is as shownbelow [1].

Figure 1.3 Pacsea surge (Source: Msc. Thesis, R.Singh)


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1.6 Pacsea Wave

In 2008, the robocon was held in India, on August 2008. The theme of the contest wasGovinda, Touch the Sky. The robot is required to grab Styrofoam cubes (buffer cubes)

and keep holding on it. The Robot that can hold these cubes until the end of the game willbe declared as the winner of the game. PIC18F6720 is the main intelligent part of therobot.

Figure 1.6 Pacsea - Loa (Source: EE300, K. Chandry)1.7 Pacsea Colata

In 2009, the robotic contest was held in Tokyo, Japan and the theme of the game isTravel Together for Victory Drums. In contrast, the competition which was held inJapan is quite different with the previous competitions and this because there are threerobots need to be designed. And the designed consist of one manual robot and twoautomatic robots. The automatic robot will complete the task by hitting the three victorydrums whereas will give supports.

Figure 1.7 PacSea-Colata model (Source: EE300, K. Chandry)


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1.8 Pacsea Matai

Furthermore, the theme of the robot contest that was held in Cairo, Egypt by the year2010 is Robo-Pharaoh Build Pyramids. The teams need to be accurate and fast. Therewere three robots and this consists of two automatic robots and one Manual robots. The

three robots will work together as a team and complete the missing blocks of threepyramids. The automatic robot will complete the last part of the pyramid without theintervention of human. Additionally the intelligent part of the automatic robots is a singlePIC18F877 with the OSMC as the main controller board. For the manual machines, carrelays are used interchangeably to control the speed of the two wheels.

Figure 1.8.1 PacSea-MataiAutomatic 1

Figure 1.8.2 PacSea-MataiAutomatic 2


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Figure 1.8.3 PacSea-MataiManual Robot

In this year project, students will work together as team and try to complete oneautomatic robot that has the ability of line tracing as well as speed control. The studentsdecide to use three number of PICs microcontroller as a central processing unit of therobot. Out of the three microcontrollers, one is set as a master controller while the othermicrocontrollers are slaves.

PID Control

First of all, PID is one of the most effective close loop speed control for a robot. Inaddition, PID stands for Proportional, Integral and Derivative Control whereas KP standsfor proportional gain, KI stands for Integral gain and KD stands for derivative gain.The diagram below shows how PID is used to control the plant (motor)

Figure 1.9 PID Block Diagram

The controller Input is controlled be the feedback such as if the output is quite high ascompared to the reference input. In addition, the three gains (Kp, Ki and Kd) will takeaction to reduce the error and running the plant as per required.


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Chapter 2

Mechanical Design

2.0 Overview

For this project, our main objective was to implement PID speed control and line trace

the mobile vehicle on a given platform. Since due to lack of time and some foreseen we

could not complete our desired task hence, we discussed with our supervisor with whom

we found out that we have to complete our project using a differential drive motor

controller setup which is the major project for EE313. The figure below illustrates the

following practical setup of the differential drive motor controller setup. Brief detailed

information of each item indicated below is further discussed in this chapter.

12V dc

motor

Motor Controller Boards Boot loader Boards


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Figure 2.0: Differential Drive Motor Controller Setup

The mechanical design of above figure is fairly simple, because this project deals with

PID speed control or synchronising the two driver motors, only few components can be

built to attain this task. Moreover, the differential drive motor controller can be divided

into two halves, the component required to make one half of the differential system is one

12Vdc motor, one PIC16F877,one LM324 comparator , a 12V power supply unit, 16

slots shaft encoder and a frequency divider. For this project, we are suppose to complete

the two halfs and complete the loop of the differential drive system. Since the electronics

of the differential drive setup is mounted on a 15mm wooden board, the positioning of

the PCB boards like boot loader board was not catered. However, the positioning of the

feedback or IR sensor and motor controller board was positioned appropriately.


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a) b)Shaft encoder

Figure 2.1: a) IR sensor board, b) Shaft encoder and IR sensor unit

2.1 Motor

The differential system drive unit is designed using differential steering system which uses

two independently cordless drill DC motor with built in gearbox. The motor is rated at 12V

each and produces a rated current of 1A. The cordless drill motor is also known as the driver

motor because it drives the whole differential system. It is a two terminal device which has a

+ve andve terminal to power up the motor and rotate.

The reasons for choosing this motor:


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1) It is locally available

2) It is cheap and cost around FJ$50.00

3) Capable of generating high torque.

4) And the performance of the motor has been tested in previous Robocon

competitions.

Figure 2.2: 12V Cordless drill motor

The figure shown above is a 12V driver motor which is used in the differential drive system.

2.2 Wheels

The wheels of the differential drive were design out of a timber. The timber was

designed into a wheel shape by using a lathed machine. The wheels of diameter 135mm

was made out from a wood. Two pieces of the timber of fifty millimetres was given to the

technicians to make two wheels of the differential drive steering system. The axle which

connects the shaft encoder to the motor shaft was made out of aluminium. This

aluminium axle is locked onto the chuck of the drill motor. The end of the aluminium

axle which is made rectangular in shape was placed inside the chuck. This was done to

prevent the axle from slipping while in motion. A round collar was made to securely fix

the axle to the wooden wheel. Furthermore, a hole was drilled through the collar and the

axle of the drill motor and a machine screw was used to fix the axle to the collar. A self

tapping screw was used to fix to axle collar to the wooden wheel. The wheel and axle

connections were checked thoroughly. The wooden wheel was sanded and brushed with

sanding sealer. It was also varnish to protect and coat the wood. Rubber glue was then

applied to the circumference of the wheel and the rubber which was cut to shape in the


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lab was placed on top of it and left to dry in order to keep the rubber securely attached to

the wooden wheel. This rubber which is covering the circumference allows the robot to

grip the floor firmly while in motion and to avoid any slips. The rubber was important

because it will allow the robot to grip properly the floor while moving and avoid slips.

Figure 2.2 below shows the in-house design and fabricated wooden wheel which was

used as the steering control for the differential drive system.

Figure 2.3: In-house fabricated wooden wheel

Wheel circumference calculation

The wheels circumference calculation was found using the equation 2.0 stated below

(Equation 2.0)

The radius of the driver was set to be 67.5mm, and diameter of the wheel is 135mm hence

the circumference was found to be:

mC

mmC

C

rC

42.0

424

)5.67(2

2

135mm


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Figure 2.4: Wheel calculation

Advantages of using wooden wheels are:

Can be made of any size [compatible for robots application]

Does not deform in shape under stress

exact length of distance traveled can be determined from the circumference

easier to mount to shaft encoder

it is cheap

It is easily designed by lathing hardwood with appropriate dimensions

2.3 PCB Mountings

This project was entirely base on speed control on the differential drive setup as earlier

mentioned in this chapter. Hence the PCB mounting was done in a logical manner. The pcb

mounting of the motor controller unit was done closest to the driver motor. Since the drill

motor is driven from the motor controller chip and not from the power supply, it was

necessary to mount the motor controller chip closest to the motor. The associated wires

which are connected directly to the motor were connected to the 5mm dual connector.

Moreover, in between the two motor controllers designated for each side, a PCB for LM324

comparator is mounted onto the 15mm thick wooden board. Since the comparator consist of

four in-built similar circuits, the two motor controllers can be connected and share just one

LM324 chip, so it was appropriate to mount it in between the two motor controllers. On the

other hand, the two boot loader board which has PIC16F877 was mounted right next to the

motor controller board. The positioning of the boot loader was done away from the drill

motor. This was done to avoid any disruption in the signals or mechanical damage being

made to the bootloader board. The PIC micro controller board uses RS232 cable to interface

between PC and PIC16F877. The PCB mounting for the IR sensors was mounted such that

the shaft encoder is properly in between the IR sensors and that could have no disturbances

and emitting and receiving the signals. All of the above mentioned PCB was mounted using

a glue gun and a silicon type clip which holds the PCB for at certain height above the

surface. The figure below shows the PCB mounting configuration of the differential drive

setup. The figure below shows zoom-in PCB a component which is mounted on a 15mm

thick wooden board.


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a) b)

c)

d)

Figures 2.5: a) Motor controller board. b) PIC16F877 boot loader board, c) LM324

comparator PCB board, d) IR sensor mounting.

The above figures shows the PCB components which was incorporated in our projects for

the hardware sections. All the PCB etched was done by the school technician and the circuit

and schematic diagrams were drawn in the simulation software. The hardware components

like the connectors, resistors LEDs and duplex wiring wires were purchased form RS


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Company in Australia. The materials for the mechanical parts are added into the part list

which is shown in the Appendix section of this project.


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Chapter 3

Software Implementation

A PIC16F877 is a single compact microcontroller which is widely used in industries tocontrol many applications such as an automatic guided vehicle, temperature control, solar

radiation tracker and many more applications [4].The picture of a PIC16F877A and its corresponding pin layouts

Figure 3.1: A physical diagram of a single PIC16F877 (Source: The MicrochipPIC16F877A,www.solarbotics.com/products/pic16f877a/

Figure 3.2: Aphysical pin layoutof a singlePIC16F877

(Source:PIC16F87XA

memory

organizationtutorial,www.microcontroller.com)
http://www.solarbotics.com/products/pic16f877a/http://www.solarbotics.com/products/pic16f877a/http://www.solarbotics.com/products/pic16f877a/http://www.microcontroller.com/http://www.microcontroller.com/http://www.microcontroller.com/http://www.microcontroller.com/http://www.solarbotics.com/products/pic16f877a/


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Specifications of PIC16F877

High Performance

RSC CPU

Peripheral features Analogue features CMOS

Technology

Operation Speed200 MHz clock

input and 200 nsinstruction cycle

Three timersTimer 0 (8 bit)

Time 1 (16 bit)Timer 2 (8 bit)

8 channels ADCand 10 bit for each

channel

Low-power, highspeed

Flash/EEPROMtechnology

112 KB Flashmemory2944 bytes Datamemory (RAM)2048 bytes ofEEPROM DataMemory

PWM modes:Capture16 bit,resolution is 12.5 nsCompare16 bit,resolution is 200 nsTwo PWM pins foreach PIC

Brown Out Reset Fully static design

The pins arecompatible betweentwo PIC16CXXXand PIC16FXXXmicrocontrollers

Communication:SPIMaster mode andSlave modePSP8 bit withexternal RD, WR andCS controls (40/44 pinonly)

An Analoguecomparator has:-2 analoguecomparators- Programmablevoltage referenceon the chip module-The comparatoroutputs can beaccessed externally

Operating voltagestarts from 2 V to5.5 VLow powerconsumption

Figure 3.3: PIC16F877 specifications (source: Microchip Technology Inc,

http://ww1.microchip.com/downloads/en/devicedoc/39582b.pdf)

3.1 Reasons of using PIC16F877

PIC16F877 is chosen to be used in this project due to its availability in the University ofthe South Pacific and if the chip is faulty it is easy to be replaced. The boot loaderdiagram of PIC16F877 is shown below.
http://ww1.microchip.com/downloads/en/devicedoc/39582b.pdfhttp://ww1.microchip.com/downloads/en/devicedoc/39582b.pdfhttp://ww1.microchip.com/downloads/en/devicedoc/39582b.pdf


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Figure 3.4: The boot loader diagram of PIC16F877 (Source: R. Singh, Lecture Notes)

In figure 3.4, the boot loader is designed in such a way that PIC16F877 can be replacedvery easily when the chip is faulty or the chip is malfunctioned.

3.2 How a PIC microcontroller communicates with a computer

In order for a PIC microcontroller (PIC16F877) to communicate to the computer, an RS-232 serial cable is used to transmit data from the computer to PIC microcontroller andvise versa. In addition, a hex file is sent through from the computer to the PICmicrocontroller and this hex file is created after compiling the PIC C program using PICC compiler.

3.3 The development of PIC communication between three PIC microcontrollers

To begin with, there are many ways to communicate between two or moremicrocontrollers such as using SPI (Serial Peripheral Interface) communication, I2C(Inter-IC) communication and the most common type of communication is a three-buswire configuration.


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In this project, the three-bus wire configuration is used to communicate between PICsmicrocontroller since it is simple and fast. There are three PICs used in thiscommunication and out of these three, one is the master and the other two are the slaves.Moving on, the task of the master is to send a duty cycle (50%) to the two slaves so thatthe two slaves can operate the motors using this initial duty cycle (sent by the master)

respectively. The two motors. And the achieve this, three wires are usedone wire forthe data, one wire for latch and one wire for the clock and these wires are parallel to eachPIC. In addition, this can be illustrated from the diagram shown below.

Figure 3.5 (A 3 bus-wire configuration)

3.4 Programming Development

In order to send a data to each slave, the data has to be packed in a packet and has to bechanged to a binary number for sending purposes. The most challenged part of thecommunication is developing a packet. In this project the data that is send from themaster to the slaves contains the duty cycle and the direction of each motor. Thecalculation of the packet using high level program (PIC C language)

Packet = (dir1*2) + (dir2*4) + (duty1*256) + (duty2*65536); // packet calculation

Parameters

Dir1direction for motor 2Dir2- direction of motor 2

Duty1-duty cycle of motor 1Duty2-duty cycle of motor 2


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Moreover, all the duty cycle and the directions of the motor has to be put in the packet inthe sense that after sending the packet the respect slave will do another calculation toextract the duty cycle and the direction information. In order to send the data from themaster the total packet has to be changed to binary and then send one by one bit to bothslaves. The flow charts shown below explain how the master can communicates to each

respective slave.

Figure 3.6a: Master flow chart

Figure 3.6b: Master flow chart


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From the two figures there are three pins allocated for the master to communicate the twoslaves. The first pin is used for latch and this pin is used to enable a communication, thenext pin is the clock, the clock trigger when will be the data send (for example: the data issend at the rising edge of the clock) and last the pin is for data transmission. In thefollowing flow chart it shows how each slave communicates to the master PIC.

Figure 3.7a: Slave 1 flow chart

Figure 3.7b slave 1 flow chart continued


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Figure 3.7c: Slave 1 flow char continued

Figure 3.7d: slave 1 flow chart continued


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Figure 3.7e: slave 1 flow chart continued

In the main program (Figure 3.6a), the program first initialize all the variables, setup aPWM mode by using - setup_ccp2 (CCP_PWM) and setting up a try state buffer pins.Then the program check if the data has been receive by the master(Figure 3.6 b)this isdone by checking the state of latch in the program (if latch highdata is ready to send,data transmitted, if latch lowdata is not ready for transmission). In addition if the datahas received by the respective slave it will return this value to the main to update thepwm value and thus the motor starts to rotate with this pwm value with the initial dutycycle of 50% (receive by the master).Moreover, the main program always checks the #int_timer1 (Figure 3.7c) and incrementsthe counter by 1 at every time the timer is over flow, the final value of the counter isimportant in calculating the speed of the motor. In addition, the speed is calculated inevery one complete revolution and this shows in the equation shown below.Mesr_rpm = 60 / spr (sprseconds per revolution)

To calculate the time in seconds per revolution is calculated in the equation given below.Spr= ((counter x 0.065536) + (0.000001 x get_timer1 ()))


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The speed calculation is explained in more detailed in the figure 3.7f.

Figure 3.7f: Speed calculation of one wheel

Next, the program will try to compare the calculated speed with required speed by theuser. However the speed are different, the difference between the speed is called the error

and this will then pass to another subroutine called update_pid and this function will tryto correct the error in a minimum number of time (this is shown in figure 3.7d). Theupdate_pid () has three important part such as integral term, derivative term andproportional term. The integral term is just add up the previous error with the currenterror and then multiply it with the integral gain (Ki), however there will the time that thesum of errors exceeds the maximum limit (maximum duty cycle), therefore the sum oferror has to limit within 100 and -100 and this is important in this project. In thederivative term, the previous error has to be subtracted from the current error and thenmultiply with the gain (Kd). For the proportional term, the current error is alwaysmultiply with proportional gain (Kp). (To find the proper gain value for Kp, Ki and Kdeach has to derive the transfer function of the motor and this is done in more detail in

chapter five of this project). The total sum of all the proportional term the integral termand the derivative term is then add up and stored in correction. The correction value isthen limited again within 100 and -100 before sending this value to another subroutinecalled stabilize. In stabilize function; the correction value is then divided with 100 sincethe correction value must be in percentage form. This correction value will update theduty cycle value every time to change the speed of the motor according the referencespeed given from the user.


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3-State Outputs Minimized Power Dissipation Sink/Source Interlock Circuitry Prevents Simultaneous Conduction No Output Glitch During Power Up or Power Down

4.1.1 Other Parameters:Table 4.1 Parameters for sn754410 chip

Output Voltage(Max)(V) 36

Switching Voltage(Max)(V) 36

Peak Output Current(mA) 2000

Drivers Per Package 4

Input Compatibility CMOS, TTL

Delay Time(ns) 800

Operating TemperatureRange(C

-40 to 85

Pin/Package 16PDIP

Fig 4.1 sn754410 chip

4.2 Advantages of using SN754410 Chip

One major advantage of using SN754410 chip is that it is really cheap and readilyavailable. One chip costs about US$2.95 that is approximately FJ$7.00 whereas anOSMC power board is about FJ$400. Thus, if the OSMC board gets bad it has a very

high maintenance cost and is difficult to repair whereas if SN754410 chip goes bad itseasy to replace than to replace an OSMC board.

4.3 Disadvantages of using SN754410 Chip

The disadvantage of using the SN754410 chip is that it heats up very quickly. Chips hadto be in parallel since it provides less output current. Since there are four set of Half H-


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bridge in one chip we had to parallel six Half H-bridges (or three full H-bridges) toincrease the current output. This means to drive one motor we used one and half chipalong with a heavy heat sink and a fan to prevent the chip from burning out.

4.4 Schematic Diagram of SN754410 chips Connection to Motors

Figure 4.2 schematic of sn754410

The chip shows that to run two motors we have to loop three chips to increase the currentoutput. One and half chip is looped together for one motor.

M2M1

R6

1k

R5

1k

R4

1k

R3

1k

R2

1k

R1

1k

+V5V

+

12V

SN745510

P1

P2

P3

P4

P5

P6

P7

P8 P9

P10

P11

P12

P13

P14

P15

P16

U3

SN745510

P1

P2

P3

P4

P5

P6

P7

P8 P9

P10

P11

P12

P13

P14

P15

P16

U2

SN745510

P1

P2

P3

P4

P5

P6

P7

P8 P9

P10

P11

P12

P13

P14

P15

P16

U1


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4.5 Connections to the Chip

Figure 4.3 Pin connection for sn754410

EN 1A 2A FUNCTION

H L H Turn right

H H L Turn left

H L L Fast motor stop

H H H Fast motor stop

L X X Fast motor stop

Table 4.2 Truth table for SN754410

L = Low

H = High

X = dont care

4.5.1 Wiring Diagram for the SN754410

The table 4.3 below shows how the chip is connected to the motor using two H-bridges.

Pin Name Full H-BridgeFunction

Notes

1 1,2EN Motor 1 Enable 0V (LOW) = disable motor5V (HIGH) = enable motor

2 1A Motor 1 Clockwise /Forward

0V (LOW) = ignore5V (HIGH) = apply current in one direction

3 1Y Motor 1 Power(Output)

4 Ground Ground All 4 pins are grounded




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7 2A Motor 1Counterclockwise /Reverse

0V (LOW) = ignore5V (HIGH) = apply current in the otherdirection

8 VCC2 Motor Power Source(Input

attached directly to battery

9 3,4EN Motor 2 Enable 0V (LOW) = disable motor5V (HIGH) = enable motor

10 3A Motor 2 Clockwise /Forward

0V (LOW) = ignore5V (HIGH) = apply current in one direction





15 4A Motor 2Counterclockwise /Reverse

0V (LOW) = ignore5V (HIGH) = apply current in the otherdirection

16 VCC1 IC Logic Power(Input)

Regulated 5v (VCC / VDD)

4.5.1 12VDC Cordless Drill Motor

The Robocon Groups have extensively been using the 12VDC cordless drill motors

mostly because it is very cheap (costing close to $50FJD) compared to other kinds of DC

motors. Hence, we also used the cordless drill motor for our differential drive setup. The

speed of this motor is easily controlled by varying either the armature voltage or field

current. These 12V cordless Drill motors are series-wound DC motors, where the

stationary and rotating coils are wires in series.

Figure 4.4 cordlessdrill motor


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4.6 Testing of SN754410

The chip was tested with a cordless drill motor running in clockwise direction withdifferent voltages and the current output and rpm was tabulated.

Figure 4.5 shows the testing for sn754410 chip

Table 4.4 testing results

Test Number Voltage (V) Current (A) RPM

1 12 1.17 494.1

2 11.5 1.12 480

3 11 1.09 445.8

4 10.5 1.07 424.4

5 10 1.04 410.3

6 9.5 1.03 380.1

7 9 0.99 351.3

8 8.5 0.98 325.3

9 8 0.94 294.6

10 7.5 0.92 215.9

11 7 0.88 246.9

12 6.5 0.87 222.913 6 0.85 195.1

14 5.5 0.83 168.2

15 5 0.79 148.4

16 4.5 0.75 119

17 4 0.71 99.8

18 3.5 0.68 80.9

19 3 0.64 53.5


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20 2.5 0.6 28.8

21 2 0.02 0

Figure 4.5.1 Graph of current, voltage and rpm for sn754410 chip

The graph states that when the voltage decreases the current and the rpm decreasesrespectively. At no load condition the motor can run on a minimum of 2.5 volts. At 2volts the motor stops since the current is very low.

4.7 Differential Drive

The term 'differential' means that robot turning speed is determined by the speeddifference between both wheels, each on either side of your robot [5]. The differentialdrive consists of two wheels independently controlled by two motors. Such anarrangement allows the robot to move forward and backwards, turn in place, and turnwhile moving. As our setup is a replica of a differential drive system and has both of themotor placed opposite direction to each other, one wheel has to move in clockwisedirection and the other has to move in anti-clockwise direction. Since the clockwise andanti-clockwise speeds of the motors are different. The Motors speeds need to besynchronized in order to be at the same speed. The motor speeds are synchronized using

PID control.


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Figure 4.8 One output on the SN754410 chip

However, to control the speed of the DC motors digitally a Pulse Width Modulation(PWM) signals using the PIC microcontroller was used. The micro-controller, PIC

16F877 supplies appropriate PWM to the SN754410 which than controls the appropriatemotors.To control the speed of a DC motor we need a variable voltage DC power source.However if you take a 12V motor and switch on the power to it, the motor will start tospeed up: motors do not respond immediately so it will take a small time to reach fullspeed. If we switch the power off sometime before the motor reaches full speed, then themotor will start to slow down. [2]

If we switch the power on and off quickly enough, the motor will run at some speedbetween zero and full speed. This is exactly what a PWM (pulse width modulation) is [3].The controller switches the motor on in a series of pulses. To control the motor speed it

varies (modulates) the width of the pulses - hence Pulse Width Modulation.

PWM, Pulse Width Modulation, involves taking a square wave and, without changingthe frequency, changing how long in each cycle the wave spends high and low. The ratioof t-high/t-low is the signal's duty cycle. The higher the duty cycle, the higher theaverage voltage on the signal line and the higher the power the signal delivers. In thebasic Pulse Width Modulation (PWM) method, the operating power to the motors isturned on and off to modulate the current to the motor. The ratio of "on" time to "off"time is what calculates the speed of the motor.

Figure 4.9 shows the PWM signal


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4.7.2 Duty Cycle

The use of PWM to drive the DC motors makes implementation easier because of greatercircuit simplicity. Additionally, the use of PWM signals allows one to vary motor speedsimply by changing the Duty Cycle or the Time on part of the PWM signal shown Duty

Cycle can be derived as [4]

%100

onoff

on

TT

TDutyCycle

(Eqn

4.1)Where: - Ton = Time On

: - Toff= Time Off


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Chapter 5

Shaft Encoder5.0 Introduction

The shaft encoder is used to provide feedback from the wheels. the encoder is veryessential in controlling the speed of the automatic vehicle. the encoder provides a signalproduced by the rotation of the wheel is the feedback that is fed back to interrupt port ofthe microprocessor that is port b.The robotic group at the university of the south pacific has used various shaft encoders toget feedback of the motor speed from the wheels. the most common shaft encoders usedby the groups are heds and oktec pair in house built shaft encoder to filter feedback.

5.1 HEDS 5540 a06

The previous year project groups have used HEDS 5540 a06 as their shaft encoder. each

encoder contains a lensed led source, an integrated circuit with detectors and outputcircuitry, and a code wheel which rotates between the emitter and detector IC thisencoder has 1024 counts per revolution and its maximum operating frequency is100Khz.It has three channels (channel i, channel a and channel b).

Figure 5.1 heds 5540 a06features

Supply voltage 5 v _ 10 %Output signal ttl compatible


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Phase shift _ (nominal) 90eLogic state width s min. 45eSignal rise time(Typical at cl = 25 pf, rl = 2.7 k_, 25c) 180 nsSignal fall time

(Typical at cl = 25 pf, rl = 2.7 k_, 25c) 40 nsIndex pulse width (nominal) 90eOperating temperature range -40 ... +100cMoment of inertia of code wheel _ 0.6 gcm2Max. Angular acceleration 250 000 rad s-2output current per channel min. -1 ma, max. 5 ma

5.1.1 Advantages of using HEDS 5540 a06

The major advantages of using this shaft encoder are:

It offers high resolution (that is it gives a high precision).

Enclosed device therefore it is not affected by external light sources.

5.1.2 Disadvantages of using heds 5540 a06

The disadvantage of using this shaft encoder is that it very expensive and maintenance isvery costly. Each HEDS 5540 a06 cost about 114 Australian dollars. Also it has a highresolution which is not needed for our application.HEDS 5540 a06 has 1024 pulses per revolution means it has 512 slots. the wheeldiameter is 140mm so the circumference is:

Circumference = 2r= 2 x x 0.14= 0.44m

Each time the wheel makes a turn it rotates 360 and since there are 1024 pulses perrevolution the angle of each pulse is given as:

=1024

360

= 0.35

Hence, the linear distance that the wheel moves when one of the 1024 pulses passes by isgiven as:

d =360

x circumference

=

360

35.0x 0.44m

= 0.000427m


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Therefore it gives feedback at every 0.35 turn and the linear distance which the wheelmoves is 0.000854m which is very precise and not needed for our application.

5.2 Oktec pair shaft encoder

For our application we have used a oktec pair shaft encoder to filter feedback. oktec pairshaft encoder is an in-house built shaft encoder. Its a 16 slot encoder cut out on 5mmaluminum plate together with a pair of IR phototransistor/photodiode (op140/op550)sensors used to design the shaft encoder. Both have the same emitting and receivingfrequency

Figure 5.2 hardware setup for oktec shaft encoder

Figure 5.3 16 slots aluminum plate

This 16-slot encoder is not capable of differentiating between CW and ACW directionwhich incorporates a comparator circuit designed using lm324 operational amplifier. Thetwo 1k ohm resistors are placed such that the comparator threshold voltage is 2.5v.henceany input signals blow this threshold voltage would produce a logic low output and abovethis threshold voltage would produce a logic high output.


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Figure 5.4 shows the shaft encoder between two ir sensors along with the comparatorcircuit.

5.2.1 Advantages of using oktec pair

The major advantage of using the in-house built shaft encoder is that: Cheaper (as compared to other encoders which are bought).

Since it is built in-house any problems can be easily resolved.

it is suitable for this particular application

A 16 slot shaft encoder generates 32 pulses per revolution. The wheel diameter is 140mmso the circumference is:

r = 70mm = 0.07mCircumference = 2r

= 2 x x 0.07= 0.44m

Each time the wheel makes a turn it rotates 360 and since there are 32 pulses perrevolution the angle of each pulse is given as:

=32

360

= 11.25Hence, the linear distance that the wheel moves when one of the 1024 pulses passes by isgiven as:

d =360

x circumference

=

360

25.11x 0.44m

= 0.01375mTherefore it gives feedback at every 11.25 turn and the linear distance which the wheelmoves is 0.01375m which is applicable for our application.


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5.2.2 Disadvantages of using oktec pair

The major disadvantage of using this shaft encoder is that there could be errors if lightsfrom external sources pass through the shaft encoder. The light can interface with the IR

sensors and cause errors. Therefore, to reduce this error an external cover is needed.

5.3 Comparator

The main purpose comparator circuit (lm324 operational amplifier) is to smoothen theoutput that is coming from the receiving sensor. For the operational amplifier to functiona +5 volts and a ground have to be supplied

5.3.1 Output from the comparator

Figure 5.6 shows the signal given by the ir sensors which is amplified by the comparatorcircuit giving a smooth square wave signal (0v and 5v).

Figure 5.6: Output from the comparator

CHAPTER 6

Figure 5.5 shows the hardware for the comparator circuit


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30984 182301 0.182301 267.1916

29969 212270 0.21227 268.3087

30815 243085 0.243085 267.3582

28845 271930 0.27193 269.4353

32102 304032 0.304032 265.6395

31457 335489 0.335489 266.508

33264 368753 0.368753 264.4407

29894 398647 0.398647 268.268

29857 428504 0.428504 268.4408

30026 458530 0.45853 268.3892

30699 489229 0.489229 267.4059

29103 518332 0.518332 269.2805

32742 551074 0.551074 265.0786

37491 588565 0.588565 259.6178

43586 632151 0.632151 252.9458

4493 636644 0.636644 251.34264818 641462 0.641462 248.0302

48968 690430 0.69043 247.1617

42646 733076 0.733076 253.7438

40656 773732 0.773732 255.7959

41805 815537 0.815537 254.7046

45116 860653 0.860653 251.2373

43568 904221 0.904221 252.8733

43682 947903 0.947903 252.6752

46146 994049 0.994049 250.0615

4441 998490 0.99849 251.954845247 1043737 1.043737 251.0429

47278 1091015 1.091015 248.9596

46483 1137498 1.137498 249.8772

45687 1183185 1.183185 250.5272

4627 1187812 1.187812 250.0365

47763 1235575 1.235575 248.4277

4583 1240158 1.240158 250.3902

46461 1286619 1.286619 249.7409

4846 1291465 1.291465 247.6688

46397 1337862 1.337862 249.864746715 1384577 1.384577 249.4958

48774 1433351 1.433351 247.3829

46806 1480157 1.480157 249.4357

47962 1528119 1.528119 248.2396

48703 1576822 1.576822 247.5217

45985 1622807 1.622807 250.3327


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47455 1670262 1.670262 248.7645

48523 1718785 1.718785 247.5697

48025 1766810 1.76681 248.2663

4781 1771591 1.771591 248.4092

46636 1818227 1.818227 249.6495

49436 1867663 1.867663 246.6781

46862 1914525 1.914525 249.3735

4718 1919243 1.919243 249.1415

49851 1969094 1.969094 246.3236

47496 2016590 2.01659 248.683

47298 2063888 2.063888 248.9286

50898 2114786 2.114786 245.3235

47915 2162701 2.162701 248.2766

49274 2211975 2.211975 246.938

50718 2262693 2.262693 245.3877

49804 2312497 2.312497 246.36249328 2361825 2.361825 246.9177

4996 2366821 2.366821 246.2842

50158 2416979 2.416979 246.0358

49522 2466501 2.466501 246.7643

4959 2471460 2.47146 246.5585

51657 2523117 2.523117 244.4649

48569 2571686 2.571686 247.6177

49159 2620845 2.620845 247.058

51866 2672711 2.672711 244.3066

Step 2: Draw transient response of the graph

From the above data, the transient response was drawn using Microsoft Excel. This graph shows

the transient response of the motor with respect to time.


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Figure 6.1 Shows the Graph of Motor Speed versus time at 35% duty cycle

From the above graph the settling time is assumed to be 0.860653 seconds as shown

on the graph.

ts =0.860653 s

Step 3: Step Response

As referenced from the text Feedback control of dynamic systems[9] the step

response of a motor is given by:

tety

1)( (Eqn 6.1)

Thus, to use the above equation, we need to calculate the value of sigma (). If we

want the settling time to occur when the decaying exponential reaches 1% ~

(0.01),the following equations were derived as shown below.

0

50

100

150

200

250

300

350

0 0.5 1 1.5 2 2.5 3

Graph of Motor Speed Versus Time @ 35% duty

cycle


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57

ns

sn

sn

tW

Wt

tW

tW

e sn

61.4

61.4

)01.0ln(

01.0

Since, wn =

Therefore, we can calculate :

61.4st (Eqn 6.2)

st

61.4 (Eqn 6.3)

Using equation 6.3 and the settling time from step 2, we can now calculate the value of

sigma ().

356.5

860653.0

61.4

61.4

st

Now, we know the value of sigma and can use the equation 6.1 from step 3 to calculate thestep response.

Table 6.2

The table below shows the step response figures at 35% duty cycle.

Cumulative Time (s) - x t Exp(- x t) y(t)

00 1 0

0.001482 -0.007937592 0.992093827 0.007906173

0.017756 -0.095101136 0.909280969 0.090719031

0.03461 -0.18537116 0.830795868 0.169204132

0.062752 -0.336099712 0.714551853 0.2854481470.087872 -0.470642432 0.624600876 0.375399124

0.1154 -0.6180824 0.53897699 0.46102301

0.148382 -0.794733992 0.451701375 0.548298625

0.151317 -0.810453852 0.444656212 0.555343788

0.182301 -0.976404156 0.376663088 0.623336912

0.21227 -1.13691812 0.320806186 0.679193814


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0.243085 -1.30196326 0.271997267 0.728002733

0.27193 -1.45645708 0.233060529 0.766939471

0.304032 -1.628395392 0.196244217 0.803755783

0.335489 -1.796879084 0.165815578 0.834184422

0.368753 -1.975041068 0.138755614 0.861244386

0.398647 -2.135153332 0.118226461 0.881773539

0.428504 -2.295067424 0.1007546 0.8992454

0.45853 -2.45588668 0.085787096 0.914212904

0.489229 -2.620310524 0.072780259 0.927219741

0.518332 -2.776186192 0.062275562 0.937724438

0.551074 -2.951552344 0.05225852 0.94774148

0.588565 -3.15235414 0.042751366 0.957248634

0.632151 -3.385800756 0.033850525 0.966149475

0.636644 -3.409865264 0.033045653 0.966954347

0.641462 -3.435670472 0.032203811 0.967796189

0.69043 -3.69794308 0.024774433 0.9752255670.733076 -3.926355056 0.019715403 0.980284597

0.773732 -4.144108592 0.015857565 0.984142435

0.815537 -4.368016172 0.012676363 0.987323637

0.860653 -4.609657468 0.009955228 0.990044772

0.904221 -4.843007676 0.007883308 0.992116692

0.947903 -5.076968468 0.006238793 0.993761207

0.994049 -5.324126444 0.004872606 0.995127394

0.99849 -5.34791244 0.004758073 0.995241927

1.043737 -5.590255372 0.003734074 0.996265926

1.091015 -5.84347634 0.002898748 0.9971012521.137498 -6.092439288 0.00225989 0.99774011

1.183185 -6.33713886 0.001769357 0.998230643

1.187812 -6.361921072 0.001726048 0.998273952

1.235575 -6.6177397 0.001336448 0.998663552

1.240158 -6.642286248 0.001304042 0.998695958

1.286619 -6.891131364 0.001016763 0.998983237

1.291465 -6.91708654 0.000990712 0.999009288

1.337862 -7.165588872 0.000772724 0.999227276

1.384577 -7.415794412 0.000601674 0.999398326

1.433351 -7.677027956 0.00046335 0.999536651.480157 -7.927720892 0.000360607 0.999639393

1.528119 -8.184605364 0.000278914 0.999721086

1.576822 -8.445458632 0.000214874 0.999785126

1.622807 -8.691754292 0.000167965 0.999832035

1.670262 -8.945923272 0.000130267 0.999869733

1.718785 -9.20581246 0.000100454 0.999899546


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1.76681 -9.46303436 7.76706E-05 0.999922329

1.771591 -9.488641396 7.57069E-05 0.999924293

1.818227 -9.738423812 5.89734E-05 0.999941027

1.867663 -10.00320303 4.52547E-05 0.999954745

1.914525 -10.2541959 3.52095E-05 0.999964791

1.919243 -10.27946551 3.43309E-05 0.999965669

1.969094 -10.54646746 2.62862E-05 0.999973714

2.01659 -10.80085604 2.0382E-05 0.999979618

2.063888 -11.05418413 1.58208E-05 0.999984179

2.114786 -11.32679382 1.20458E-05 0.999987954

2.162701 -11.58342656 9.31927E-06 0.999990681

2.211975 -11.8473381 7.15758E-06 0.999992842

2.262693 -12.11898371 5.45497E-06 0.999994545

2.312497 -12.38573393 4.17777E-06 0.999995822

2.361825 -12.6499347 3.20777E-06 0.999996792

2.366821 -12.67669328 3.12307E-06 0.9999968772.416979 -12.94533952 2.38732E-06 0.999997613

2.466501 -13.21057936 1.83113E-06 0.999998169

2.47146 -13.23713976 1.78313E-06 0.999998217

2.523117 -13.51381465 1.35215E-06 0.999998648

2.571686 -13.77395022 1.04244E-06 0.999998958

2.620845 -14.03724582 8.01127E-07 0.999999199

2.672711 -14.31504012 6.06816E-07 0.999999393

A step response graph was drawn as shown in Figure 6.2.

Figure 6.2 shows the step response of the Graph of Motor at 35% duty cycle


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From the above graph it can be demonstrated that the settling time of 0.904221s

is achieved at 0.992116692 as marked on the graph, at this point the exponential

decay reaches within 1% of the final value. By looking at the step response

graph above, we can assume that it is given by a sum of exponentials, thus we

can write

(Eqn6.4)

Subtracting off the final value and assuming that

- (alpha) is the slowest pole, the following equation can be written as:

(Eqation 6.5)

The slope of the equation 5.5 gives the value of alpha (), and the intercept gives the value of

A. If equation 5.5 is plotted (log graph) it should result in the positive value of A because we

use log(y (t)-y (infinity)). Step 4 shows how A and was calculated.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Graph of y(t) vs Time (s)


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Step 4: Drawing log(y (t)-y ()) vs. time graph and calculating the value of constant Aand alpha ().

Table 6.3

The table below shows the log(y (t)y ()) data at 35% duty cycle.

Cumulative Time (s) 1 - y(t) Log |1 - y(t)|

01 0

0.001482 0.992093827 -0.003447252

0.017756 0.909280969 -0.041301899

0.03461 0.830795868 -0.080505672

0.062752 0.714551853 -0.14596625

0.087872 0.624600876 -0.204397411

0.1154 0.53897699 -0.268429776

0.148382 0.451701375 -0.3451485870.151317 0.444656212 -0.351975636

0.182301 0.376663088 -0.424046937

0.21227 0.320806186 -0.493757266

0.243085 0.271997267 -0.565435459

0.27193 0.233060529 -0.632531273

0.304032 0.196244217 -0.707203133

0.335489 0.165815578 -0.780374671

0.368753 0.138755614 -0.857749437

0.398647 0.118226461 -0.92728531

0.428504 0.1007546 -0.9967351180.45853 0.085787096 -1.066578033

0.489229 0.072780259 -1.137986401

0.518332 0.062275562 -1.205682344

0.551074 0.05225852 -1.281842896

0.588565 0.042751366 -1.369050008

0.632151 0.033850525 -1.470434585

0.636644 0.033045653 -1.480885668

0.641462 0.032203811 -1.492092728

0.69043 0.024774433 -1.605996274

0.733076 0.019715403 -1.7051943350.773732 0.015857565 -1.799763494

0.815537 0.012676363 -1.89700532

0.860653 0.009955228 -2.001948802

0.904221 0.007883308 -2.10329151

0.947903 0.006238793 -2.20489939

0.994049 0.004872606 -2.312238736


62/125

62

0.99849 0.004758073 -2.322568862

1.043737 0.003734074 -2.42781706

1.091015 0.002898748 -2.53778953

1.137498 0.00225989 -2.645912764

1.183185 0.001769357 -2.752184438

1.187812 0.001726048 -2.762947216

1.235575 0.001336448 -2.874047834

1.240158 0.001304042 -2.884708265

1.286619 0.001016763 -2.992780325

1.291465 0.000990712 -3.004052515

1.337862 0.000772724 -3.111975707

1.384577 0.000601674 -3.220638592

1.433351 0.00046335 -3.334090879

1.480157 0.000360607 -3.442965437

1.528119 0.000278914 -3.554528946

1.576822 0.000214874 -3.6678160811.622807 0.000167965 -3.774780927

1.670262 0.000130267 -3.885165113

1.718785 0.000100454 -3.998033553

1.76681 7.76706E-05 -4.109743605

1.771591 7.57069E-05 -4.120864599

1.818227 5.89734E-05 -4.229343724

1.867663 4.52547E-05 -4.344335876

1.914525 3.52095E-05 -4.453340696

1.919243 3.43309E-05 -4.464315147

1.969094 2.62862E-05 -4.5802726232.01659 2.0382E-05 -4.690752178

2.063888 1.58208E-05 -4.800771169

2.114786 1.20458E-05 -4.919164052

2.162701 9.31927E-06 -5.030618235

2.211975 7.15758E-06 -5.145233562

2.262693 5.45497E-06 -5.263207751

2.312497 4.17777E-06 -5.379055901

2.361825 3.20777E-06 -5.493796837

2.366821 3.12307E-06 -5.505417939

2.416979 2.38732E-06 -5.6220895222.466501 1.83113E-06 -5.737281717

2.47146 1.78313E-06 -5.748816754

2.523117 1.35215E-06 -5.868975133

2.571686 1.04244E-06 -5.981950573

2.620845 8.01127E-07 -6.096298401

2.672711 6.06816E-07 -6.216942931


63/125

63

The graph of Log |1y(t)| vs. time was obtained from table 6.3 above.

Figure 6.3 shows the graph of Log |1 y (t)| vs. timemotor at 35% duty cycle

From the graph in Figure 6.3 and equation 6.5, we now calculate the value of A and alpha

().

1

0log10

A

A

Using equation 6.5 and the slope of the graph in figure 6.3 the value of is calculated as

shown:

-7

-6

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2 2.5 3

Graph of Log |1 - y(t)| vs Time (s)


64/125

64

383.5

338.24343.0

768779.1

113708.4

904221.0673.2

103292.2217.64343.0

Step 5: Drawing log Be-t

Vs. time graph and calculating the value of B and.

Table 6.4

Cumulative

Time (s)

- x t Exp(- x t) 1 + A*Exp(-

x t)

1 + A*Exp(-

x t) y(t)

Log (1 + A*Exp(-

x t) y(t))

00 1 2 2 0.301029996

0.001482 -0.007977606 0.992054131 1.992054131 1.984147958 0.297574054

0.017756 -0.095580548 0.908845153 1.908845153 1.818126122 0.259624007

0.03461 -0.18630563 0.830019877 1.830019877 1.660815746 0.220321454

0.062752 -0.337794016 0.71334221 1.71334221 1.427894063 0.154695988

0.087872 -0.473014976 0.623120739 1.623120739 1.247721615 0.0961176990.1154 -0.6211982 0.537300259 1.537300259 1.076277248 0.03192416

0.148382 -0.798740306 0.449895338 1.449895338 0.901596713 -0.04498768

0.151317 -0.814539411 0.442843249 1.442843249 0.887499461 -0.051831902

0.182301 -0.981326283 0.37481366 1.37481366 0.751476748 -0.124084452

0.21227 -1.14264941 0.318972812 1.318972812 0.639778998 -0.193970021

0.243085 -1.308526555 0.270217914 1.270217914 0.542215182 -0.265828327

0.27193 -1.46379919 0.231355639 1.231355639 0.464416167 -0.33309267

0.304032 -1.636604256 0.194639869 1.194639869 0.390884085 -0.407952011

0.335489 -1.805937287 0.164320369 1.164320369 0.330135947 -0.481307185

0.368753 -1.984997399 0.137380971 1.137380971 0.276136585 -0.558876050.398647 -2.145916801 0.116960758 1.116960758 0.235187219 -0.628586283

0.428504 -2.306637032 0.099595626 1.099595626 0.200350226 -0.698210164

0.45853 -2.46826699 0.084731572 1.084731572 0.170518669 -0.768228067

0.489229 -2.633519707 0.071825213 1.071825213 0.144605472 -0.839815271

0.518332 -2.790181156 0.061410088 1.061410088 0.12368565 -0.907680684

0.551074 -2.966431342 0.051486721 1.051486721 0.103745241 -0.984031816


65/125

65

0.588565 -3.168245395 0.042077362 1.042077362 0.084828728 -1.071457046

0.632151 -3.402868833 0.033277665 1.033277665 0.06712819 -1.173095061

0.636644 -3.427054652 0.032482472 1.032482472 0.065528125 -1.183572261

0.641462 -3.452989946 0.03165086 1.03165086 0.063854672 -1.194807324

0.69043 -3.71658469 0.024316876 1.024316876 0.049091309 -1.308995388

0.733076 -3.946148108 0.019329012 1.019329012 0.039044415 -1.408441078

0.773732 -4.164999356 0.015529725 1.015529725 0.03138729 -1.503246178

0.815537 -4.390035671 0.012400287 1.012400287 0.02507665 -1.600730477

0.860653 -4.632895099 0.009726559 1.009726559 0.019681787 -1.70593548

0.904221 -4.867421643 0.007693175 1.007693175 0.015576483 -1.807530583

0.947903 -5.102561849 0.006081148 1.006081148 0.012319941 -1.909391369

0.994049 -5.350965767 0.004743568 1.004743568 0.009616173 -2.016997721

0.99849 -5.37487167 0.004631513 1.004631513 0.009389586 -2.027353535

1.043737 -5.618436271 0.003630313 1.003630313 0.007364388 -2.132863358

1.091015 -5.872933745 0.002814604 1.002814604 0.005713352 -2.243109023

1.137498 -6.123151734 0.002191538 1.002191538 0.004451428 -2.3515006871.183185 -6.369084855 0.001713727 1.001713727 0.003483084 -2.458036027

1.187812 -6.393991996 0.00167157 1.00167157 0.003397618 -2.468825499

1.235575 -6.651100225 0.001292599 1.001292599 0.002629047 -2.58020157

1.240158 -6.675770514 0.001261101 1.001261101 0.002565143 -2.590888422

1.286619 -6.925870077 0.000982048 1.000982048 0.001998811 -2.699228237

1.291465 -6.951956095 0.000956762 1.000956762 0.001947474 -2.710528344

1.337862 -7.201711146 0.000745309 1.000745309 0.001518033 -2.818718733

1.384577 -7.453177991 0.000579597 1.000579597 0.001181271 -2.927650474

1.433351 -7.715728433 0.000445761 1.000445761 0.000909111 -3.041383283

1.480157 -7.967685131 0.00034648 1.00034648 0.000707087 -3.1505268681.528119 -8.225864577 0.000267641 1.000267641 0.000546555 -3.262365868

1.576822 -8.488032826 0.000205918 1.000205918 0.000420792 -3.375932563

1.622807 -8.735570081 0.000160764 1.000160764 0.00032873 -3.483161196

1.670262 -8.991020346 0.000124523 1.000124523 0.00025479 -3.593817426

1.718785 -9.252219655 9.58986E-05 1.000095899 0.000196352 -3.706963848

1.76681 -9.51073823 7.40524E-05 1.000074052 0.000151723 -3.818948846

1.771591 -9.536474353 7.21708E-05 1.000072171 0.000147878 -3.830097202

1.818227 -9.787515941 5.61482E-05 1.000056148 0.000115122 -3.938843129

1.867663 -10.05362993 4.30293E-05 1.000043029 8.8284E-05 -4.054117913

1.914525 -10.30588808 3.34356E-05 1.000033436 6.86451E-05 -4.1633904711.919243 -10.33128507 3.25972E-05 1.000032597 6.6928E-05 -4.174391868

1.969094 -10.599633 2.49252E-05 1.000024925 5.12113E-05 -4.29063395

2.01659 -10.85530397 1.9302E-05 1.000019302 3.9684E-05 -4.401384483

2.063888 -11.1099091 1.49633E-05 1.000014963 3.07841E-05 -4.511673144

2.114786 -11.38389304 1.13773E-05 1.000011377 2.34231E-05 -4.630356026

2.162701 -11.64181948 8.79067E-06 1.000008791 1.81099E-05 -4.742083025


66/125

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2.211975 -11.90706143 6.74262E-06 1.000006743 1.39002E-05 -4.856978716

2.262693 -12.18007642 5.13169E-06 1.000005132 1.05867E-05 -4.975241284

2.312497 -12.44817135 3.92489E-06 1.000003925 8.10266E-06 -5.09137242

2.361825 -12.71370398 3.0096E-06 1.00000301 6.21737E-06 -5.206393442

2.366821 -12.74059744 2.92974E-06 1.00000293 6.05281E-06 -5.218042901

2.416979 -13.01059796 2.2365E-06 1.000002237 4.62382E-06 -5.334999066

2.466501 -13.27717488 1.71315E-06 1.000001713 3.54428E-06 -5.450472041

2.47146 -13.30386918 1.66803E-06 1.000001668 3.45116E-06 -5.462035183

2.523117 -13.58193881 1.2631E-06 1.000001263 2.61525E-06 -5.582486219

2.571686 -13.84338574 9.7251E-07 1.000000973 2.01495E-06 -5.695736629

2.620845 -14.10800864 7.46397E-07 1.000000746 1.54752E-06 -5.810362577

2.672711 -14.38720331 5.64569E-07 1.000000565 1.17139E-06 -5.931300335

Step 5: Also shows how the value of B and () is calculated.Using the data from table 5.4 the graph was plotted as shown in figure 6.4

Figure 6.4 shows the graph of Log (1+A*Exp (- x t) y(t)) Log |1 y (t)| vs. time at

35% duty cycle.

Using the intercepts and the values obtained from the graph in Figure 6.4, thevalue of B and beta () was calculated.

2

30103.0log10

B

B

The graph of Log (1+A*Exp (- x t) y(t)) Log |1y (t)| vs. Time gives a value for

-7

-6

-5

-4

-3

-2

-1

0

1

0 0.5 1 1.5 2 2.5 3

Graph of Log (1 + A*Exp(xt) - y(t)) vs Time (s)


67/125

67

370.5

1636.2

046.54343.0

1484.0312.2

0458.00914.54343.0

Step 6: Calculating the transfer function using the values found in previous steps

Since there is not much difference in the value of (Step 4) and (Step 5), that is

= 5.383 = 5.370

Hence, using equation 6.4 from (Step 3), we get the estimate value of y (t) to be:

370.5383.5 21)(

)()(

eety

BeAeyty

t

tt

Taking the Laplace of the equation gives:

)370.5)(383.5(

907.28889.264)(

)370.5)(383.5(

)383.5(2)370.5()370.5)(383.5()(

370.5

2

383.5

11)(

2

sss

sssy

sss

sssssssy

ssssy

The transfer function of the motor at 35% duty cycle is given as:


68/125

68

)370.5)(383.5(

907.28889.264 2

sss

ssnctionTransferFu

6.3 Calculation of Transfer Function at 50% duty cycle

Table 6.5

Cumulative Time (s) x t Exp (- x t) y(t)=1-(exp(- x t))0 0 1 0

0.002109 0.025755614 0.974573232 0.025426768

0.066574 0.813017666 0.443517657 0.556482343

0.069393 0.84744397 0.428508815 0.571491185

0.134515 1.642729464 0.193451303 0.806548697

0.199853 2.440652801 0.087103971 0.9128960290.204555 2.498074753 0.082243185 0.917756815

0.207 2.52793368 0.079823791 0.920176209

0.210009 2.56468031 0.076943776 0.923056224

0.213404 2.606140865 0.073818872 0.926181128

0.21475 2.62257854 0.07261538 0.92738462

0.215564 2.632519303 0.071897103 0.928102897

0.217154 2.651936765 0.070514511 0.929485489

0.226254 2.763068149 0.063097877 0.936902123

0.226285 2.763446728 0.063073994 0.936926006

0.226469 2.765693781 0.062932423 0.937067577

0.290745 3.550647719 0.02870604 0.971293960.291581 3.560857151 0.028414459 0.971585541

0.292105 3.567256365 0.028233209 0.971766791

0.301493 3.681904874 0.025174974 0.974825026

0.303374 3.704876098 0.024603266 0.975396734

0.312841 3.820489374 0.021917073 0.978082927

0.377494 4.610047327 0.009951347 0.990048653

0.378904 4.627266585 0.009781459 0.990218541

0.444224 5.424970102 0.004405198 0.995594802

0.444323 5.426179114 0.004399875 0.995600125

0.445621 5.442030601 0.00433068 0.99566932

0.454847 5.554700727 0.003869226 0.9961307740.455029 5.556923355 0.003860636 0.996139364

0.519157 6.340069882 0.001764179 0.998235821

0.583444 7.125158155 0.000804606 0.999195394

0.648902 7.92454696 0.000361754 0.999638246

0.657161 8.025407851 0.000327047 0.999672953

0.721467 8.810728156 0.000149125 0.999850875

0.722253 8.820326977 0.0001477 0.9998523


69/125

69

0.730742 8.923996682 0.000133155 0.999866845

0.739284 9.028313636 0.000119965 0.999880035

0.74014 9.038767314 0.000118717 0.999881283

0.748306 9.138492465 0.000107449 0.999892551

0.81336 9.932947526 4.85485E-05 0.999951452

0.814846 9.951094915 4.76754E-05 0.9999523250.823322 10.05460586 4.29873E-05 0.999957013

0.823843 10.06096844 4.27147E-05 0.999957285

0.824817 10.07286316 4.22096E-05 0.99995779

0.833874 10.18346942 3.77899E-05 0.99996221

0.842931 10.29407568 3.38329E-05 0.999966167

0.843167 10.29695776 3.37356E-05 0.999966264

0.843235 10.2977882 3.37076E-05 0.999966292

0.844061 10.30787551 3.33693E-05 0.999966631

0.908684 11.09706709 1.51567E-05 0.999984843

0.90905 11.10153677 1.50891E-05 0.999984911

0.910668 11.12129618 1.47939E-05 0.9999852060.920284 11.23872908 1.31547E-05 0.999986845

0.921276 11.25084362 1.29963E-05 0.999987004

0.923002 11.27192194 1.27253E-05 0.999987275

0.932188 11.38410358 1.13749E-05 0.999988625

0.93351 11.40024816 1.11927E-05 0.999988807

0.933654 11.40200672 1.1173E-05 0.999988827

0.934178 11.40840594 1.11018E-05 0.999988898

0.934982 11.41822458 1.09933E-05 0.999989007

1.000016 12.2124354 4.96829E-06 0.999995032

1.000384 12.2169295 4.94601E-06 0.999995054

1.002262 12.23986409 4.83387E-06 0.9999951661.011672 12.35478127 4.3091E-06 0.999995691

1.012116 12.3602035 4.2858E-06 0.999995714

1.013277 12.37438191 4.22546E-06 0.999995775

1.078511 13.17103517 1.90499E-06 0.999998095

1.0801 13.19044042 1.86838E-06 0.999998132

1.145355 13.98735015 8.42114E-07 0.999999158

1.146102 13.99647269 8.34467E-07 0.999999166


70/125

70

Figure 6.5 Shows the Graph of Motor Speed versus time at 50% duty cycle

Table 6.6

|1-y(t)| log |1-y(t)|

1 0

0.974573232 -0.011185521

0.443517657 -0.353089086

0.428508815 -0.36804024

0.193451303 -0.713428341

0.087103971 -1.059962044

0.082243185 -1.084900081

0.079823791 -1.097867648

0.076943776 -1.113826507

0.073818872 -1.131832597

0.07261538 -1.138971388

0.071897103 -1.143288607

0.070514511 -1.151721503

0.063097877 -1.19998525

0.063073994 -1.200149665

0.062932423 -1.201125548

0.02870604 -1.542026711

0.028414459 -1.546460612

0.028233209 -1.549239755

0.025174974 -1.59903097

0.024603266 -1.609007245

0.021917073 -1.659217453

0.009951347 -2.002118115

0.009781459 -2.009596344

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y(t)

Time(s)

Step Response @ o.5


71/125

71

0.004405198 -2.35603458

0.004399875 -2.356559647

0.00433068 -2.36344386

0.003869226 -2.412375874

0.003860636 -2.413341149

0.001764179 -2.7534573640.000804606 -3.094416869

0.000361754 -3.441587017

0.000327047 -3.485390345

0.000149125 -3.82645062

0.0001477 -3.830619335

0.000133155 -3.875642516

0.000119965 -3.920946793

0.000118717 -3.925486768

0.000107449 -3.968796851

4.85485E-05 -4.3138243

4.76754E-05 -4.321705614.29873E-05 -4.366659843

4.27147E-05 -4.369423075

4.22096E-05 -4.374588887

3.77899E-05 -4.422624575

3.38329E-05 -4.470660262

3.37356E-05 -4.471911937

3.37076E-05 -4.47227259

3.33693E-05 -4.476653453

1.51567E-05 -4.819395003

1.50891E-05 -4.821336161

1.47939E-05 -4.8299175611.31547E-05 -4.880918021

1.29963E-05 -4.8861793

1.27253E-05 -4.895333501

1.13749E-05 -4.944053367

1.11927E-05 -4.951064869

1.1173E-05 -4.951828603

1.11018E-05 -4.954607746

1.09933E-05 -4.958871928

4.96829E-06 -5.303793303

4.94601E-06 -5.305745068

4.83387E-06 -5.3157054324.3091E-06 -5.365613329

4.2858E-06 -5.367968175

4.22546E-06 -5.374125781

1.90499E-06 -5.720107897

1.86838E-06 -5.72853549

8.42114E-07 -6.074628985

8.34467E-07 -6.078590855


72/125

72

Figure 6.6 shows the graph of Log |1 y (t)| vs. timemotor at 50% duty cycle

Table 6.7

x t exp (- x t)1 + A*exp (-

x t)y(t)

1 + A*exp (-

x t) - y(t)

Log (1 +A*exp (- x t)

- y(t))

0 1 2 0 2 0.301029996

-0.02576 0.974573726 1.974573726 0.025426768 1.949146958 0.289844584

-0.813 0.443524743 1.443524743 0.556482343 0.8870424 -0.052055621

-0.84743 0.428515951 1.428515951 0.571491185 0.857024766 -0.067006628

-1.6427 0.193457548 1.193457548 0.806548697 0.386908851 -0.412391335

-2.4406 0.087108149 1.087108149 0.912896029 0.174212121 -0.758921632

-2.49803 0.082247222 1.082247222 0.917756815 0.164490407 -0.783859424

-2.52788 0.079827757 1.079827757 0.920176209 0.159651549 -0.796826864

-2.56463 0.076947654 1.076947654 0.923056224 0.15389143 -0.812785566-2.60609 0.073822653 1.073822653 0.926181128 0.147641524 -0.830791479

-2.62253 0.072619122 1.072619122 0.92738462 0.145234502 -0.837930201

-2.63247 0.071900823 1.071900823 0.928102897 0.143797926 -0.842247377

-2.65188 0.070518186 1.070518186 0.929485489 0.141032696 -0.850680191

-2.76301 0.063101304 1.063101304 0.936902123 0.126199181 -0.898943463

-2.76339 0.06307742 1.06307742 0.936926006 0.126151414 -0.899107876

-2.76564 0.062935844 1.062935844 0.937067577 0.125868266 -0.900083749

y = -5.3037x - 9E-13

-7

-6

-5

-4

-3

-2

-1

0

1

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Log(y(t))

Time(s)

Log |y(t)-1| @0.5


73/125

73

-3.55058 0.028708043 1.028708043 0.97129396 0.057414084 -1.240981563

-3.56079 0.028416447 1.028416447 0.971585541 0.056830906 -1.24541542

-3.56719 0.028235188 1.028235188 0.971766791 0.056468398 -1.248194536

-3.68183 0.025176796 1.025176796 0.974825026 0.05035177 -1.297985261

-3.7048 0.024605057 1.024605057 0.975396734 0.049208323 -1.307961439

-3.82041 0.021918718 1.021918718 0.978082927 0.043835791 -1.358171153-4.60996 0.009952249 1.009952249 0.990048653 0.019903596 -1.701068446

-4.62718 0.009782349 1.009782349 0.990218541 0.019563808 -1.708546601

-5.42486 0.004405668 1.004405668 0.995594802 0.008810865 -2.054981433

-5.42607 0.004400344 1.004400344 0.995600125 0.00880022 -2.055506494

-5.44192 0.004331144 1.004331144 0.99566932 0.008661824 -2.06239064

-5.55459 0.003869649 1.003869649 0.996130774 0.007738875 -2.111322174

-5.55681 0.003861058 1.003861058 0.996139364 0.007721694 -2.112287439

-6.33995 0.001764399 1.001764399 0.998235821 0.003528578 -2.452400312

-7.12502 0.000804718 1.000804718 0.999195394 0.001609324 -2.793356466

-7.92439 0.00036181 1.00036181 0.999638246 0.000723564 -3.140523202

-8.02525 0.000327098 1.000327098 0.999672953 0.000654145 -3.184326099-8.81056 0.00014915 1.00014915 0.999850875 0.000298275 -3.525383023

-8.82015 0.000147726 1.000147726 0.9998523 0.000295426 -3.529551697

-8.92382 0.000133178 1.000133178 0.999866845 0.000266333 -3.574574435

-9.02814 0.000119986 1.000119986 0.999880035 0.000239951 -3.619878268

-9.03859 0.000118738 1.000118738 0.999881283 0.000237455 -3.624418197

-9.13831 0.000107468 1.000107468 0.999892551 0.000214918 -3.667727855

-9.93275 4.8558E-05 1.000048558 0.999951452 9.71065E-05 -4.012751913

-9.9509 4.76847E-05 1.000047685 0.999952325 9.53601E-05 -4.020633147

-10.0544 4.29958E-05 1.000042996 0.999957013 8.59831E-05 -4.065586938

-10.0608 4.27231E-05 1.000042723 0.999957285 8.54378E-05 -4.068350143

-10.0727 4.22179E-05 1.000042218 0.99995779 8.44275E-05 -4.073515904-10.1833 3.77974E-05 1.000037797 0.99996221 7.55873E-05 -4.121551119

-10.2939 3.38398E-05 1.00003384 0.999966167 6.76727E-05 -4.169586335

-10.2968 3.37424E-05 1.000033742 0.999966264 6.7478E-05 -4.170837998

-10.2976 3.37144E-05 1.000033714 0.999966292 6.7422E-05 -4.171198646

-10.3077 3.3376E-05 1.000033376 0.999966631 6.67453E-05 -4.175579466

-11.0968 1.516E-05 1.00001516 0.999984843 3.03167E-05 -4.518317649

-11.1013 1.50924E-05 1.000015092 0.999984911 3.01815E-05 -4.520258787

-11.1211 1.47971E-05 1.000014797 0.999985206 2.9591E-05 -4.528840103

-11.2385 1.31576E-05 1.000013158 0.999986845 2.63124E-05 -4.579840062

-11.2506 1.29992E-05 1.000012999 0.999987004 2.59955E-05 -4.585101289

-11.2717 1.27281E-05 1.000012728 0.999987275 2.54533E-05 -4.5942554

-11.3839 1.13774E-05 1.000011377 0.999988625 2.27523E-05 -4.642974787

-11.4 1.11952E-05 1.000011195 0.999988807 2.23879E-05 -4.649986221

-11.4018 1.11755E-05 1.000011176 0.999988827 2.23486E-05 -4.650749947


74/125

74

-11.4082 1.11043E-05 1.000011104 0.999988898 2.2206E-05 -4.653529063

-11.418 1.09958E-05 1.000010996 0.999989007 2.19891E-05 -4.657793203

-12.2122 4.96948E-06 1.000004969 0.999995032 9.93777E-06 -5.002711188

-12.2167 4.9472E-06 1.000004947 0.999995054 9.89321E-06 -5.004662934

-12.2396 4.83503E-06 1.000004835 0.999995166 9.66889E-06 -5.0146232

-12.3545 4.31015E-06 1.00000431 0.999995691 8.61925E-06 -5.064530606

-12.36 4.28684E-06 1.000004287 0.999995714 8.57264E-06 -5.06688543

-12.3741 4.22649E-06 1.000004226 0.999995775 8.45195E-06 -5.073042975

-13.1708 1.90548E-06 1.000001905 0.999998095 3.81047E-06 -5.419021691

-13.1902 1.86886E-06 1.000001869 0.999998132 3.73724E-06 -5.427449201

-13.9871 8.42346E-07 1.000000842 0.999999158 1.68446E-06 -5.773539294

-13.9962 8.34697E-07 1.000000835 0.999999166 1.66916E-06 -5.777501125

Figure 6.7 shows the graph of Log (1+A*Exp (- x t) y (t)) Log |1 y (t)| vs. time at

50% duty cycle.

From the above tabulated results and graph the value of

= 12.212

= 12.212

y = -5.3037x + 0.301

-7

-6

-5

-4

-3

-2

-1

0

1

-0.2 0 0.2 0.4 0.6 0.8 1

Designing Development and Testing of a PIC Micro-Controller Based Differential Drive Line Tracing...

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