BTP Sem2 Final

8/2/2019 BTP Sem2 Final

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Project Code: D06

CONTROL AND STABILITY OF AN INVERTED

PENDULUM SYSTEM:

STRATEGIES AND IMPLEMENTATION

VYOM JAIN MUDIT GOEL VINAYAK SINGH

2007ME10532 2007ME10508 2007ME10531

Supervisor (s)

Prof. S.P.Singh Prof. S.K.Saha

Examiner

Prof. K. Gupta

Department of Mechanical Engineering

IITD


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Certificate

This is to certify that the project on Control and Stability of Inverted

Pendulum is being pursued to my satisfaction and that the goals set upon at the

outset of this endeavour have been worked upon to the best of the students

abilities and resources.

I hereby allow this project to be presented for evaluation and dissertation with

full consent.

Supervisors:

Prof. S.K.Saha Prof. S.P.Singh

Mechanical Engineering Department Mechanical Engineering Department

Indian Institute of Technology, Delhi Indian Institute of Technology, Delhi


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Acknowledgment

We would like to express our sincere gratitude to Prof. S.P. Singh and Prof.

S.K. Saha for giving us the opportunity to work under their supervision. Their

never ending support, close supervision, monitoring and expedient tips helped

us immensely in our work.

We would also like to thank Mr. Madhu (Vibration Research Lab) and Mr.

Jaitley (Mechatronics Lab) for extending their full support towards the

realization of this project.

We would also like to thank Mr. Arun, Ph.D student under Prof. S.K. Saha, for

helping us throughout the semester regarding all the electronics-related issues.

Also, we would like to thank Kamal Gupta and Sanjay Dhakar, members of the

Robotics Club, IIT Delhi for helping us in the manufacturing of motor driver

circuit.

MUDIT GOEL VYOM JAIN VINAYAK SINGH

2007ME10508 2007ME10532 2007ME10531


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Abstract

The work seeks to implement a control strategy to carry out the self-balancing of an

inverted pendulum mounted on a cart moving on a slider and powered by a DC motor. The

project entails both simulation as well as manufacturing. A rig, details of which are

mentioned later, was manufactured to carry out the implementation physically. Selection of

various parts for the system has been done and the selected components have been mentioned

appropriately. CAD models of the various parts manufactured were also developed.

LABVIEW is used to simulate the control strategy. Finally, various control strategies have

been implemented and studied on which included SISO control strategies to balance

pendulum angle and cart position individually. Also cascaded PID logics to balance both cart

position and pendulum angle simultaneously has been implemented.

Keywords: Inverted pendulum, PID, Self-balancing, LABVIEW Simulation


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Contents

Certificate ............................................................................................................................................... iii

Acknowledgment .................................................................................................................................... v

Abstract ................................................................................................................................................. vii

List of Figures ......................................................................................................................................... ix

List of Tables ......................................................................................................................................... xii

Nomenclature ...................................................................................................................................... xiii

Chapter 1. Introduction .................................................................................................................... 1

Chapter 2. Literature Review and Objectives ................................................................................... 3

Chapter 3. System Modelling ............................................................................................................ 6

3.1 Analytical Modelling ............................................................................................................... 6

3.2 Modelling in MATLAB ............................................................................................................ 15

3.3 Modelling in LabVIEW ........................................................................................................... 18

Chapter 4. Parts Selection and Procurement ................................................................................. 23

4.1 Mechanism Comparison ....................................................................................................... 23

4.2 Slider Mechanism .................................................................................................................. 23

4.3 Motor Selection .................................................................................................................... 26

4.4 Encoder Selection ................................................................................................................. 28

4.5 Motor Driver ......................................................................................................................... 28

4.6 Data Acquisition Card (DAQ) ................................................................................................. 29

Chapter 5. Equipment Design and Fabrication ............................................................................... 31

Chapter 6. Developing of Sensing and Actuation of system ........................................................... 43

6.1 Encoder Interfacing ............................................................................................................... 43

6.2 Motor Driver Fabrication ...................................................................................................... 45

Chapter 7. Control Implementation ............................................................................................... 50

7.1 SISO Control of Cart Position(x) ............................................................................................ 50

7.2 SISO Control of Pendulum Angle(

) ...................................................................................... 57

7.3 Cascaded PID logic to control Pendulum Angle and Cart Position ....................................... 65

Chapter 8. Conclusions and Future Scope ...................................................................................... 80

References ............................................................................................................................................ 82

Appendix-A ............................................................................................................................................ 83

Appendix B ............................................................................................................................................ 88

Appendix C ............................................................................................................................................ 93

Appendix D .......................................................................................................................................... 100

Appendix E .......................................................................................................................................... 103


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List of Figures

Figure 1.1: Schematic of an inverted pendulum ..................................................................................... 1

Figure 2.1 Gantt Chart of Part 1 of the work .......................................................................................... 5

Figure 2.2 Gantt Chart of Part2 of work ................................................................................................. 5

Figure 3.1 FBD of the inverted pendulum setup ..................................................................................... 6

Figure 3.2: DC Motor block diagram (Ogata[2008]) ............................................................................... 9

Figure 3.3 Schematic for SISO theta control ......................................................................................... 13

Figure 3.4 Schematic of cascaded PID logics for control ...................................................................... 14

Figure 3.5 Schematic of Parallel PID logics for control ......................................................................... 15

Figure 3.6 - Quadrant-wise division of pendulum angle ....................................................................... 16

Figure 3.7 - Pendulum Angle with time ................................................................................................ 17

Figure 3.8 - Cart Position with time ...................................................................................................... 17

Figure 3.9 - Cart velocity with time ....................................................................................................... 18

Figure 3.10: LabVIEW Code for simulation of control of cart position X of inverted pendulum .......... 19

Figure 3.11: Front Panel of simulation for PID control of only cart position ........................................ 19

Figure 3.12: Block Diagram of simulation for only angle theta control ................................................ 20

Figure 3.13: Front Panel for only theta-control .................................................................................... 20

Figure 3.14: Block Diagram for simulating cascading control of PID logics .......................................... 21

Figure 3.15: Front Panel for simulating cascading of PID logics ........................................................... 22

Figure 4.1 - Igus Toothed Belt Axis ........................................................................................................ 24

Figure 4.2 - Dry LIne - Low Profile Linear Guide System NK-01/02-40 ................................................. 25

Figure 4.3 - Dry Line - Low Profile Linear Guide Systen NK-02-80 ....................................................... 25Figure 4.4 - Moment rating of selected igus slider ............................................................................... 26

Figure 4.5 - Characteristic curves of selected igus slider ...................................................................... 26

Figure 4.6 - RoboKits Motor - selected motor ...................................................................................... 27

Figure 4.7 - Incremental Encoder BI-52S-2500-PU ............................................................................... 28

Figure 4.8 - Motor Driver (www.dimensionengineering.com) ............................................................. 29

Figure 4.9 - DAQ NI-PC 6221 Port Drawing ........................................................................................... 30

Figure 5.1 - Final CAD model ................................................................................................................. 31

Figure 5.2 - Igus Slider ........................................................................................................................... 32

Figure 5.3 - Cart along with the pendulum ........................................................................................... 32

Figure 5.4 - Driving Mechanism ............................................................................................................ 33

Figure 5.5 - Driven Mechanism Assembly ............................................................................................. 33

Figure 5.6 - Stand on the ends of the slider .......................................................................................... 34

Figure 5.7 - Middle Stand ...................................................................................................................... 34

Figure 5.8 : Stands after TIG Welding ................................................................................................... 35

Figure 5.9 : 5mm Aluminium plates ...................................................................................................... 35

Figure 5.10 : Plates used to support the driving and the driven shaft assemblies ............................... 35

Figure 5.11 : Timing belt 3.5m long and 10mm wide ........................................................................... 36

Figure 5.12 : Profile of the timing pulleys ............................................................................................. 36

Figure 5.13 : 8mm roller bearing .......................................................................................................... 36Figure 5.14 : Bearing with housing ....................................................................................................... 37
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Figure 5.15 : Driving Motor with housing ............................................................................................. 37

Figure 5.16 : Encoder along with housing ............................................................................................. 38

Figure 5.17 ; Driving shaft assembly ..................................................................................................... 38

Figure 5.18 : Driven shaft assembly ...................................................................................................... 39

Figure 5.19 : Cart assembly ................................................................................................................... 39

Figure 5.20 : Pendulum ......................................................................................................................... 40

Figure 5.21 : Joint between cart and timing belt .................................................................................. 40

Figure 5.22 : Joint between wooden block and stand .......................................................................... 41

Figure 5.23 : Support for the slider ....................................................................................................... 41

Figure 5.24(a) and (b) : Final setup of inverted pendulum ................................................................... 42

Figure 6.1 : 2 output signals generated by incremental encoder(NI Developer Zone forum) ............. 43

Figure 6.2 : Simplified Counter/Timer Model of DAQ........................................................................... 43

Figure 6.3 : Layout of the connector block of NI-6221 ......................................................................... 44

Figure 6.4 : Block diagram of the code to measure encoder's position ............................................... 44

Figure 6.5: L7805 chip (http://www.mindkits.co.nz/store) .................................................................. 45Figure 6.6: picture showing pinouts of L7805 chip(http://www.mindkits.co.nz/store) ....................... 45

Figure 6.7: ATmega16 chip (http://www.futurlec.com/Atmel/ATMEGA16.shtml) .............................. 46

Figure 6.8: Pinouts of Atmega 16 (Appendix B) .................................................................................... 46

Figure 6.9: Soldering in progress for Atmega 16 Base .......................................................................... 47

Figure 6.10: Soldered Atmega 16 Base, crystal, ISP port and L7805 chip on matrix board .................. 47

Figure 6.11: L298 chip (Appendix C) ..................................................................................................... 48

Figure 6.12: Connections for L298N for driving DC motor (Appendix C) .............................................. 48

Figure 6.13: Soldered L298 chip ............................................................................................................ 49

Figure 6.14: Motor Driver card ............................................................................................................. 49

Figure 7.1: Block Diagram for real time x-control ................................................................................. 51

Figure 7.2 Simulation results for Kp=10, 25,50,100 .............................................................................. 52

Figure 7.3 Real-time results for Kp=100, 150, 300, 500 ........................................................................ 53

Figure 7.4 Simulation results for Ki=100,500,1000,5000 ...................................................................... 54

Figure 7.5 Real-time results for Ki=3000, 30000, 60000, 80000 .......................................................... 55

Figure 7.6 Simulation results for kd=0.001, 0.01, 0.1, 1 ....................................................................... 56

Figure 7.7 Real-time results for Kd=0.003, Kd=0.3, kd=3 ...................................................................... 57

Figure 7.8: Block Diagram for theta-control real-time implementation............................................... 58

Figure 7.9 Simulation results for Kp=40, 75, 200, 500 .......................................................................... 59

Figure 7.10 Real-time results for Kp=200, 450, 700.............................................................................. 60

Figure 7.11 Simulation results for KI=0.005, 0.01, 0.05 ........................................................................ 61

Figure 7.12 Real-time results for Ki=5, 10, 50, 100 ............................................................................... 62

Figure 7.13 Simulation results for Kd=0.002, 0.02, 0.2 ......................................................................... 63

Figure 7.14 Real-time results for Kd=0.35, 0.1, 1 .................................................................................. 64

Figure 7.15: Block Diagram for real-time implementation fo cascading of PID logics ......................... 65

Figure 7.16 Simulation results for Cart Position Gains Kp=0.25, Ki=0.001, kd=0.01(Cascaded PID) .... 66

Figure 7.17 Simulation results for Cart Position Gains Kp=0.6, Ki=0.001, kd=0.01(Cascaded PID) ...... 66


Figure 7.19 Real-time results for Cart Position Gains Kp=0.2, Ki=0, kd=0.001(Cascaded PID) ............. 67

Figure 7.20 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.001(Cascaded PID) ........... 67Figure 7.21 Real-time results for Cart Position Gains Kp=0.7, Ki=0, kd=0.001(Cascaded PID) ............. 67
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Figure 7.22 Simulation results for Pendulum Angle Gains Kp=40, Ki=3000, kd=0(Cascaded PID) ....... 68

Figure 7.23 Simulation results for Pendulum Angle Gains Kp=50, Ki=3000, kd=0(Cascaded PID) ...... 68

Figure 7.24 Simulation results for Pendulum Angle Gains Kp=100, Ki=3000, kd=0(Cascaded PID) ..... 68

Figure 7.25 Real-time results for Pendulum Angle Gains Kp=200, Ki=45, kd=0.45(Cascaded PID) ...... 69



Figure 7.28 Simulation results for Cart Position Gains Kp=0.6, Ki=0.0001, kd=0.01(Cascaded PID) .... 70


Figure 7.30 Simulation results for Cart Position Gains Kp=0.6, Ki=0.01, kd=0.01(Cascaded PID) ........ 70

Figure 7.31 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.001(Cascaded PID) ........... 71

Figure 7.32 Real-time results for Cart Position Gains Kp=0.45, Ki=0.1, kd=0.001(Cascaded PID) ........ 71

Figure 7.33 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.001(Cascaded PID) ........... 71

Figure 7.34 Simulation results for Pendulum Angle Gains Kp=50, Ki=1000, kd=0 (Cascaded PID)....... 72


Figure 7.36 Simulation results for Pendulum Angle Gains Kp=50, Ki=3500, kd=0 (Cascaded PID)....... 72Figure 7.37 Real-time results for Pendulum Angle Gains Kp=450, Ki=10, kd=0.45 (Cascaded PID) ..... 73

Figure 7.38 Real-time results for Pendulum Angle Gains Kp=450, Ki=45, kd=0.45 (Cascaded PID) ..... 73


Figure 7.40 Simulation results for Cart Position Gains Kp=0.6,Ki=0.0005,kd=0.001 (Cascaded PID) ... 74

Figure 7.41 Simulation results for Cart Position Gains Kp=0.6,Ki=0.0005,kd=0.01 (Cascaded PID) ..... 74

Figure 7.42 Simulation results for Cart Position Gains Kp=0.6,Ki=0.0005,kd=0.015 (Cascaded PID) ... 74

Figure 7.43 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.0005 (Cascaded PID) ........ 75

Figure 7.44 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.001 (Cascaded PID) .......... 75

Figure 7.45 Real-time results for Cart Position Gains Kp=0.45, Ki=0, kd=0.005 (Cascaded PID) .......... 75

Figure 7.46 Simulation results for Pendulum Angle Gains Kp=50, Ki=3000,kd=0.01 (Cascaded PID) .. 76

Figure 7.47 Simulation results for Pendulum Angle Gains Kp=50, Ki=3000, kd=0.1 (Cascaded PID).... 76


Figure 7.49 Real-time results for Pendulum Angle Gains Kp=450, Ki=45, kd=0.1 (Cascaded PID) ....... 77


Figure 7.51 Real-time results for Pendulum Angle Gains Kp=450, Ki=45, kd=0.7 (Cascaded PID) ....... 77

Figure 7.52 Settling time variation with change in Kp values for cart position .................................... 78

Figure 7.53 Settling time variations with change in Ki values for cart position .................................... 78

Figure 7.54 Settling time variations with change in Kd values for cart position .................................. 78

Figure 7.55 Settling time variations with change in Kp values for pendulum angle............................. 79

Figure 7.56 Settling time variations with change in Ki values for pendulum angle .............................. 79

Figure 7.57 Settling time variations with change in Kd values for pendulum angle............................. 79


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List of Tables

Table 4.1 Table showing comparison of various mechanism considered ............................................ 23

Table 4.2 Comparative Analysis of various sliders ................................................................................ 24


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Nomenclature

Symbol Meaning

mc Mass of Cart

mp Mass of pendulum

Angle of pendulum from vertical

F Force transferred from the motor to the cart

mb Mass of bob Length of pendulum

Inertia of cart and pendulum about pulley

Inertia of pendulum about hinge point Acceleration due to gravity Total mass of cart, pendulum and bob Friction coefficient DC Motor Torque current constant DC motor armature inductance

DC motor armature resistance

Radius of pulley Cart positionN Reaction force between cart and pendulum in x-direction

P Reaction force between cart and pendulum in y-direction

Kb Motor Back Emf Constant

Tm Torque applied by motor


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Chapter 1. Introduction

Inverted Pendulum designs are not based on obscure or abstract mechanics. The simple

example of a walking man can be considered conceptually equivalent. The inverted

pendulum problem is one of the most important problems in control theory and has been

studied excessively in control literatures. It is well established benchmark problem that

provides many challenging problems to control design. The system is nonlinear and unstable.

Our project deals with the implementation of this idea to balance an inverted pendulum with

respect to a reference (= 0).

In simple understandable terms, it depends on the torque generated by a motor to counteract

any perturbation in the natural inverted position of the pendulum in the non inertial reference

frame of the cart the pendulum is attached to.

According to control purposes of inverted pendulum, the control of inverted pendulum can be

divided into two aspects. The first aspect is the swing-up control of inverted pendulum. The

second aspect is the stabilization of the inverted pendulum. This project mainly focuses on

stabilization of inverted pendulum. Various strategies to control inverted pendulum along

with its base (cart) position are also discussed and implemented.

The stroke length, RPM and torque requirements of the motor were determined using the

cardinal values obtained rigorously (Lagrangian Modeling) and the assumed reasonable

values of parameters like mass of the cart, mass of the pendulum etc.

Figure 1.1: Schematic of an inverted pendulum


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The other portion was the comparison of various designs to implement the idea of an inverted

pendulum. A brief idea of the implementation is also presented to give a sense of the project.

In nut shell, the design consists of a slider mechanism (ideally frictionless), a cart with the

pendulum and an encoder mounted on it, a motor according to the torque and RPM

requirements, a compatible motor driver card, bearings to support the rotating shafts, belt

type mechanism to transfer the power of the motor, an encoder as the input to the control

system and a PID controller.

The applications of the model are aplenty. Any two wheeled drive like skate boards, the

robot torsos, and even unicycles can incorporate this to achieve balancing in one plane.


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Chapter 2. Literature Review and ObjectivesThe literature on control strategies for inverted pendulum can be roughly split into two parts:

local control (i.e. in a region slightly displaced from the desired upright position) and global

control (including swing up say from the bottom most position of the pendulum). While the

local control is essentially linear, global control is grounded in the use of switching logics or

advanced mathematical tools for linearization. The control itself can differ on the basis of the

type of control strategy (PID, fuzzy logic etc) or the mode of implementation (type of

software used, like MATLAB or LABVIEW). Finally, system itself can be different with

respect to the number of pendulums, type of base used (rotary base, moving cart etc).

The basic analytical modelling procedure of an Inverted Pendulum has been shown in

Altas(2005). The Free Body equations of the system are linearized and their Laplace is taken.

Then the author describes the derivation of the transfer function between pendulum angle and

applied force and illustrates elementary state space modelling. Further, he applies basic

Proportional-Integral-Derivative (PID) controller and studies the response by varying the

constants in PID controller.

Saha S.K. (2008) presents a detailed analysis of the Lagragian modelling and focuses on thetechniques for doing so.

Craig(2007) talk about taking sensor inputs in LabVIEW and gives a walk through of the

necessary steps to implements a basic controller.

Dockhorn(2006) describes the steps from data acquisition, simulation of the system,

development of the PID controller circuit, and implementation of the controller applied to a

real system. One of the challenges in the case of cart type inverted pendulum is to prevent the

cart from going off the tracks. This problem is dealt with by adding an offset to the angle

measured. The offset is proportional to the distance of the cart from the centre. Thus cart is

mostly around the centre of the rail.

Ogata(2008) discusses the modelling parameters and modelling procedure for a DC motor.

The relevant literature comprise of transfer function between torque and voltage. This

transfer function is directly used (in this project) in conjunction with the plant (inverted

pendulum) model and the overall transfer function has been developed.


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Vizins(2010)examines various control strategies that can be adopted to control a two degree

of freedom system by just one output. The first strategy is of modifying the effective set-point

of pendulum angle and is similar to that discussed by Dockhorn(2006).The second strategy

comprises of simultaneous control of cart position and pendulum angle by adding the

individual control actions of each input variable, viz. Pendulum angle and cart position. So,

the net control signal or the voltage sent to the motor is algebraic summation of two control

signal.

Lam(2004)describes swing up pendulum control. This paper describes two methods to swing

a pendulum attached to a cart from an initial downwards position to an upright position and

maintain that state. A nonlinear heuristic controller and an energy controller have been

implemented in order to swing the pendulum to an upright position. After the pendulum is

swung up, a linear controller has been implemented to maintain the balanced state. The

heuristic controller outputs a repetitive signal at the appropriate moment and is finely tuned

for the specific experimental setup. The energy controller adds an appropriate amount of

energy into the pendulum system in order to achieve a desired energy state. The linear

controller is a stabilizing controller based on a model linearized around the upright position

and is effective when the cart-pendulum system is near the balanced state.

Objectives of the Project

In order to study the control logics on an inverted pendulum, it was essential to setup a basic

test bench so that the control logics can be applied. The current work mainly focuses on the

practical implementation of inverted pendulum using PID control. So, the objectives of the

work have been defined as the following:-

Simulation of the Inverted Pendulum Dynamics using MATLAB by doing ananalytical modeling of the system before

Designing of the entire setup for implementation of Inverted Pendulum Control Fabricating the setup along with the electrical hardware to actuate the system Interfacing of the setup to the computer using NI-DAQ Card and LabVIEW Implementation of PID control in the form of a stabilized inverted pendulum design Simulating PID control of Inverted Pendulum in LabVIEW


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Work Plan

The figure 2.1 and 2.2 show the work methodology followed in the part 1 and part 2 of the

project respectively.

Figure 2.1 Gantt Chart of Part 1 of the work

Figure 2.2 Gantt Chart of Part2 of work

W1 W2 W3 W4 W1 W2 W3 W4 W5 W1 W2 W3 W4 W1 W2 W3

Literature Review

System Modeling

Simulation on MATLAB

Mechanism Options

Market Survey

Finalizing Mechanisms

Electrical Components Study

SelectionOrdering

CAD

Market Visit

Manufacturing

LABVIEW Study

Simulation Techniques

Report Making

NOVEMBER

GANTT CHART

M

i

n

or

1

M

i

n

or

-

2

M

i

d

S

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m

B

r

e

a

k

AUGUST SEPTEMBER OCTOBER

W 1 W 2 W 3 W 4 W 1 W 2 W 3 W 4 W 2 W 3 W 4 W 1 W 2 W 3 W 4

Learning LabVIEW

Procurement of material

Manufacturing

DAQ Card Repair

Encoder Testing

Setting up of system in lab

Motor Driver card manufacturing

Testing of motor driver card

Report Making

Control Study

Testing of motor

Learning control on Labview

Implementation of SISO control on pendulum angleImplementation of SISO control on cart position

Implementation of MISO control

PID Gains tuning

Report Making

S

E

M

B

RE

A

K

M

I

N

O

R

S

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GANTT CHART

JANUARY FEBRUARY MARCH APRIL


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Chapter 3. System ModellingThe system was modelled and the results were simulated on several platforms. Before

working on any platform, analytical modelling of the system which is done in Section 3.1.

Analytical modelling helps in calculating all the transfer functions required which are then

used to simulate the system on MATLAB and LabVIEW covered under Section 3.2 and

Section 3.3respectively.

3.1 Analytical Modelling

Figure 3.1 FBD of the inverted pendulum setup(Altas[2005])

Starting from basic Free Body Equations as described by Altas(2005), the cart position

function and pendulum angle functions are derived. Figure 3.1 shows the FBD of the

inverted pendulum system.

Putting the values of constants as

Mass of cart, = 620 Mass of pendulum, = 40 Mass of bob, = 135 Length of pendulum,

= 325


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Inertia of cart and pendulum about pulley, = 1.13 1032Inertia of pendulum about hinge point, = 4 1 03 2

= 9.81

/

2

= + mp + mb = 0.79 Frictional drag, = 1 /Motor Torque Constant, = 0.3 /Motor Back Emf Constant, = 0.3 /Motor Inductance,

= 4

Motor armature resistance, = 1.5 Radius of cart, = 0.0325Tm=Torque applied by motor

F = Force applied by motor on cart

V = Voltage supplied to motor

P and N are the reaction forces between cart and pendulum in x and y directions respectively

Free Body Equations:

For cart:

+ = . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 1 ) + + = . . . . . . . . . . . . . . . . . . . . . . . . . . (3 .2)

For pendulum:

= + + + 2 cos + 2 2 sin . . . . . .(3 .3)For Pendulum: Force balance along the pendulum length:

sin

+

cos

+

sin

=

+ 2

+

+

cos

. . . .(3.4)

For Pendulum: Force balance perpendicular to the pendulum length:


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sin cos = . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 5 )Substituting N in equation 3.1

+ + 2 cos + + 2 2

sin =

. . .(3.6)[ + ( + 2)2] + ( + ) sin = + cos . . . . . . . . . . . . . ( 3 . 7 )

On linearizing the above equation, we get

[ + ( + 2)2] + ( + ) = ( + ) . . . . . . . . . . . . . . . . . . ( 3 . 8 ) [

2

+ (

+

)]

+

+ 2

=

. . . . . . . . . . .

. . .(3.9)

On solving for x by taking Laplace Transforms, we have:

= [+ + 22 + 2 () . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 1 0 )On substituting the values in equation 3.10, we get

=

0.004 +

0.04 + 0.27

0.1625

2

0.04 + 0.1350.1625 10

2

(

) . . . . . . . . . . . . . . . . . . . . ( 3 . 1 1 )

= 0.428 102 . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 1 2 )On substituting the values in equation 3.9, we get

1.07 + 0.1752 + 0.052 = 1.245

2 +

0.05

2 =

. . . . . . . . . . . . . . . . .

. . . . . . .(3 .13)

On substituting for x from Equation 3.12, we get

1.245 0.428 102 2 + 0.428 102 0.052 = . . . . . . . . . . . . . . . . . . . . ( 3 . 1 4 ) 0.5322 12.45 + 0.428 10 0.052 = . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 1 5 )

0.4828

3

12.45

10

+ 0.428

2

=

. . . . . .

. . . . . . . . . . . . . . . . . . ( 3 . 1 6 )


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= 0.48283 + 0.4282 12.45 10 . . . . . . . . . . . . . . . . . . .(3.17)

On substituting for from Equation 3.12, we get = 0.428 102 0.48283 + 0.4282 12.45 10 . . . . . . . . . . . . . . . ( 3 . 1 8 )

= (0.4282 10)(0.4284 + 0.4283 12.452 10) . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 1 9 )

Modelling of DC Motor

Figure 3.2: DC Motor block diagram (Ogata[2008])

Motor Torque Constant, = 0.3 Motor Back Emf Constant, = 0.3 Motor Inductance, = 4 Motor Armature Resistance, = 1.5 Radius of pulley,

= 0.0325

Ogata (2008) gave a detailed analysis of the modelling of DC motor. This modelling is

applied here for deriving open loop transfer function for control for rotation angle.

= = . . (3.20) =

(

+

) . . . (3.21)

Substituting the value of F from Equation 3.21 in Equation 3.19, we get:


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0.48283 12.45 10 + 0.4282 = + . (3.22)On simplifying, we get:

=

[0.4824 + 0.482 + 0.4283 + 0.428 12.45 0.4282 12.45 10 (10 + 10) ..(3.23)On substituting the motor constants in equation 3.23:

= 9.23 0.0001934 + 0.7243 + 0.67552 18.68 15.9 . . . (3.24)

The model presented above incorporates the fact that the mechanical system/plant is perfectly

ideal, and does not give desired results. So, further assumptions can be made and the model is

more simplified in order to obtain desired results. As per the real system/rig, the friction

present in the cart is significant and the pendulum motion does not result in counter motion in

cart. Therefore, for the cart position control, the cart and the pendulum can be assumed to

lumped masses. The resulting modelling is presented below.

= + + . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 2 5 )On taking Laplace, we obtain

= + + 2 . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 2 6 ) = 1 + + 2 + . . . . . . . . . . . . . . . . . . . . . . . . .(3.27)

=1

0.792 + . . . . . . . . . . . . . . . . . . . . . . . . . .(3.28)On combining equations 3.28 and 3.12, we get:

= 10.792 + 20.4282 10 . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 2 9 )

=

0.338

3 + 0.428

2

7.9

10

. . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 . 3 0 )


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On substituting for F from equation 3.25 in Equation 3.21, we get:

( + ) = + . . 3.31On taking Laplace Transforms and simplification, we get:

= 3 + + 2 + + . . 3.32On substituting values in equation 3.26, we get

= 0.30.00010273 + 0.0385 + 0.000132 + 0.0487 + 2.77 = 10.0003423 + 0.1282 + 9.4 . . . . . . . . . . . . . . . . . . . . . . (3.33)

Similarly solving for

= . = 20.4282 10 3 + + 2 + + On substituting the values, we get

= 20.4282 10 0.30.0001033 + 0.038 + 0.000132 + 0.0487 + 2.77 . . (3.34)

= 2

0.4282 100.3

0.0001033 + 0.038132 + 2.8187 . . (3.35) = 0.0001464 + 0.05433 + 42 1.271 94 . (3.36)Equations 3.36 and Equation 3.33 are the equations that are used as transfer functions to

model the plant in the simulation.


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Control equations:

For implementing State space control, the requisite modelling has been done below:

Putting values in Equation 3.8 and 3.9, we get:

= 10.64 + 12.34 + 10.64 0.0325

. (3.37) = 2.08 + 25.95 + 2.088

0.0325 . (3.38)

Writing in State-Space notation:

= 0 1 0 0

0 10.64 12.34 00 0 0 10 2.08 25.95 0 + 0

10.640

2.088 . (3.39)


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Now first PID SISO control of pendulum angle is illustrated.

= = = + + . . (3.40)

Let plant Transfer function be H1 for

Let plant Transfer function be H1for

= = 0.0001464 + 0.05433 + 42 1.271 94 . (3.41)

On combining, we get:

=

1 +

=

+ + 0.0001464 + 0.05433 + 42 1.271 941 + + + 0.0001464 + 0.05433 + 42 1.271 94 ..(3.42)

Controller Plant0

Figure 3.3 Schematic for SISO theta control


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Now simultaneous control of pendulum angle and cart position is presented through two

strategies.

Control Strategy 1Cascaded- Offsetting of setpoint of pendulum angle

= = 0.428 102 = 0 = + + . . . (3.43)

So combined Closed Loop Control Transfer Function:


= 1 + . . . (3.43)

Figure 3.4 Schematic of cascaded PID logics for control

Plant H2Controller x

Controller Plant H10

x


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Control Strategy 2Parallel- Addition of control actions of pendulum angle and cart

position

=

= 0.428

10

2

= 0 = + + . (3.44)



=

1 + .

. (3.45)

3.2 Modelling in MATLABThe main purpose of simulating the system on MATLAB was to understand the amount of

force required to move the cart. Also, before actually proceeding to manufacturing, it was

important to calculate the stroke length of the carriage rails.

So, to simulate the actual control algorithm, the force applied on the cart was kept constant.

But only its direction was changed on the basis of the quadrant in which the pendulum was

present and the angular velocity of the pendulum. The convention used for the quadrants has

been shown in the Figure 3.6.

Plant H2Controller x

Controller Plant H10

x

Figure 3.5 Schematic of Parallel PID logics for control


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1. If the pendulum was in the II or III quadrant, the direction of the force will be same asthat of the angular velocity of pendulum, that is, if the angular velocity of pendulum is

clockwise, then a positive force will be applied to the cart.

2. If pendulum lies in I quadrant and the pendulums angular velocity is greater than 0,negative force will be applied to the cart.

3. If the pendulum lies in IV quadrant and the pendulums angular velocity is lesser than0, positive force will be applied to the cart.

Mass of the cart, mass of the pendulum and the force applied by the motor are taken

as inputs. Since it is difficult to solve the equations analytically, hence the equations

are solved numerically using ode45 solver. The complete code used for the purpose is

attached in the appendix-A.

The simulation was done for Mass of Cart=2 Kg and Mass of pendulum=0.5kg.

The results obtained were plotted and the following were results.

Figure 3.6 - Quadrant-wise division of pendulum angle

First

Quadrant

IV

III II

Positive Direction

of Force

Negative Direction

of Force

Horizontal Slider

V

ertical

Second

Quadrant

Fourth

Quadrant

Third

Quadrant


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Figure 3.7 - Pendulum Angle with time

As can be seen from the Figure 3.7, pendulum is brought to the upright position from the

lowermost position within 3 seconds. After the pendulum reaches this position, linear control

will be implemented and the system will be controlled in the upright position. A sharp drop

seen in the above figure is due a change in the angle from 360 degrees to 0.

Figure 3.8 - Cart Position with time

As can be seen from the Figure 3.8, the cart position varies within [-0.6.0.6] m. Hence, a

stroke length of 1.2 m is suitable for our system. So, an igus slider of length 1.5 m is used to

accommodate for the mechanical blockers at the ends of the slider.


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Figure 3.9 - Cart velocity with time

As can be seen from the Figure 3.9, the cart velocity varies within [-2, 2] m/s. Hence, the

motor selected for this purpose has been selected so as that the speed of the motor can

account for this range.

The force used for the simulation was 10N and a frictional force of 5N is also taken into

account. So, in a nutshell, we can say that the motor selected has to meet the torque and the

rpm required. If we select a pulley of diameter 80mm, the motor required should have a

torque of 0.4Nm. To incorporate the account of resistance of air and also to have some safety

limits, the motor required has to have a torque of 1Nm.With this diameter of pulley.

Speed of the motor =260

2 = 4703.3 Modelling in LabVIEW

To implement the control system in real, it was necessary to build the control platform on the

computer. LabVIEW files were created to interface the input and output to the computer via

DAQ cards thereby enabling the user to manipulate the control logic and the gain values

chosen easily for the control.

SISO control on cart position X: Before creating the files to do real-time control,

simulation was done by modelling the plant with a transfer function. Figure 3.10 shows the


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LabVIEW graphical code for simulating an inverted pendulum system. Only feedback of the

cart position X has been taken i.e. only PID control with 1 input has been implemented.

Figure 3.10: LabVIEW Code for simulation of control of cart position X of inverted pendulum

Figure 3.11 shows the front panel of the LabVIEW code shown in Figure 3.10. The panel

provides user the options to change the set point for the cart position and PID gains. These

can be varied before the start of the simulation as well as during run-time.

Figure 3.11: Front Panel of simulation for PID control of only cart position


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SISO Control on angle theta: Here cart position X was not taken into account. The transfer

function for calculation of angle from the motor voltage was used to do the simulation. Figure

3.12shows the block diagram for the simulation of the control of only angle theta.

Figure 3.12: Block Diagram of simulation for only angle theta control

Figure 3.13shows the front panel for theta-control. Here, again like Figure 3.11, user has the

options to carry the gains of the PID logic and set-point.

Figure 3.13: Front Panel for only theta-control


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Cascading of PID logics for cart position x and angle theta:

This logic has been unsuccessfully implemented byDockhorn[2]. In this work, this logic has

been used successfully in simulation.

Since a PID can handle only 1 input and 1 output at a time, it is not possible to give both cart

position and theta as inputs and hence cant give separate gains to each of them. Hence this

technique takes use of the concept of cascading of PID logics. So, first PID logic is

implemented on the cart position and the resultant output is used to adjust the set-point of

theta for the next PID logic. It takes the form

0

=

0 +

+

+

(3.46)

As can be seen in the equation 3.46, the set-point of theta gets adjusted every time cart

position sways away from its set-point. Hence, the only stable position for the system

becomes the point where set-points of both theta and cart position are reached. For example,

if the cart is moving in positive direction to control theta, set-point of theta changes so to

maintain it, cart has to move in opposite direction so as to ensure that cart position doesnt

sway away much from its set-point. Figure 3.14 shows the block diagram of the simulation

for cascading of this technique.

Figure 3.14: Block Diagram for simulating cascading control of PID logics


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Figure 3.15 shows the front panel displaying all the parameters like the current values of

pendulum angle theta, cart position and control action value.

Figure 3.15: Front Panel for simulating cascading of PID logics


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Chapter 4. Parts Selection and ProcurementSystem comprises of many components and hence all of them were dealt individually. This

chapter focuses on all those various components.

4.1 Mechanism ComparisonTable 4.1 shows comparison of various mechanisms considered.

Table 4.1 Table showing comparison of various mechanism considered

ACTUATOR

TYPE/

Parameter

Rack and

Pinion

(Rack

Moving)

Rack and

Pinion

(Pinion

Moving)

Chain Rubber BeltLinear

Actuator

Inertia + ++ +++ +++ ++++

Friction +++ ++ +++ +++ +++

Space

Considerations+ ++ ++ ++ +++

Stroke

Adjustability

Rack+Bench

length to be

increased

Rack+Bench

length to be

increased

Chain+Bench

length to be

increased

Belt+Bench

length to be

increased

No

Adjustibility

Reference: +++ Good ++ Average + Bad

From the above comparison, it was decided to use chain or Belt actuated mechanism.

According to our requirements of stoke length of 1.2 m, on doing the calculations, the length

of the rubber belt comes out as 3.3 m. To prevent backlash error, a module of at least lessthan 6mm should be used. Finally, a timing belt and gear pulleys of module 4mm were used.

4.2 Slider MechanismComparative analysis of various slider mechanism was done and has been shown in the Table

4.2


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Table 4.2 Comparative Analysis of various sliders

SLIDER TYPE/

Parameter

IGUS Sliders Conventional

Rails

Telescopic

Channels

Smoothness +++ ++ +++

Load Bearing

Capacity

+++ ++ +

Length

Available

+++ +++ +

Cost ++ +++ ++

Weight ofCarriage

+++ ++ +

Reference: +++ Good ++ Average + Bad

From the above comparison, it was decided to use IGUS Sliders.

The following parts were considered for the mechanism.

IGUS ZLW Belt Drive

Figure 4.1 shows the sliding drive system considered for the system.

Figure 4.1 - Igus Toothed Belt Axis

Although this belt drive system from IGUS, would have simplified the designing process but,

due to very high cost, it was rejected

IGUS Dryline N Series Guide Systems width 40 mm

Figure 4.2 shows the slider rails considered for the system.


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Figure 4.2 - Dry LIne - Low Profile Linear Guide System NK-01/02-40

Reason for rejection: Since the width of the cart required was about 100 mm,

therefore, we would have to use two of the above sliders in parallel. This would cause

problems of misalignment between the two sliders.

IGUS Dryline N Series Guide Systems width 80 mm

Figure 4.3 shows the slider mechanisms of width 80mm finally selected for the system.

Figure 4.3 - Dry Line - Low Profile Linear Guide Systen NK-02-80

Since the product fulfilled all the purposes and the cost was within the acceptable limit. So,

further analysis for the above slider was done and shown below.

Selection of IGUS Sliders

The IGUS slider with width 80 mm was selected.


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The maximum moment and load carrying capacity of the sliders is shown in the figure 4.4:

Figure 4.4 - Moment rating of selected igus slider

The maximum speeds possible at various loads is shown in the figure 4.5.

Figure 4.5 - Characteristic curves of selected igus slider

The following criteria were satisfied:

1. The maximum vertical load and maximum moment applied is far less than thecapacity of the slider.

2. Since, the applied load is far less than applied, sufficiently high speeds can bereached.

4.3 Motor SelectionThe servo motor has some control circuits and a potentiometer (a variable resistor, aka pot)

that is connected to the output shaft. In the picture above, the pot can be seen on the right side

of the circuit board. This pot allows the control circuitry to monitor the current angle of theservo motor. If the shaft is at the correct angle, then the motor shuts off. If the circuit finds


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that the angle is not correct, it will turn the motor the correct direction until the angle is

correct. The output shaft of the servo is capable of travelling somewhere around 180 degrees.

Usually, its somewhere in the 210 degree range, but it varies by manufacturer. A normal

servo is used to control an angular motion of between 0 and 180 degrees. A normal servo is

mechanically not capable of turning any farther due to a mechanical stop built on to the main

output gear.

To modify a motor so that it can rotate more than 180 degrees, the mechanical stop inside the

motor needs to be removed. By doing this, we lose position control, although speed control is

gained. So, a potentiometer based servo motor cant be used. Hence, we have to used an

optical based servo motor which essentially is a DC motor along with an optical encoder.

Therefore, it was decided to use a DC motor and an encoder separately.

The figure 4.6 shows the motor finally selected and used.

Figure 4.6 - RoboKits Motor - selected motor

Motor specifications:

450RPM 12V DC motors with Metal Gearbox 25000 RPM base motor 6mm shaft diameter Gearbox diameter 37 mm. Motor Diameter 28.5 mm Length 63 mm without shaft Shaft length 15mm

300gm weight 20kgcm torque


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No-load current = 800 mA(Max), Load current = upto 9.5 A(Max) Price: Rs. 900

4.4 Encoder SelectionTwo encoders are required for the application; one to measure the rotation of the motor and

the other one to measure the angle of the pendulum. Since the angle of the pendulum moves

only varies from 0-360 degrees, to know the number of rotations of the pendulum an absolute

encoder has to be used for that purpose. On the other hand, to note the current position of the

cart, it is required to note more than one rotation and hence, an incremental encoder has to be

used.

The encoders have been procured from BTH which is a multinational company and providesservice for its products and hence the parts are standard which can be procured for the project

later on.

An incremental encoder has 4 signals which ultimately calculate the direction of rotation of

the shaft and also the speed of movement. The encoder has been shown in the Figure 4.7.

Figure 4.7 - Incremental Encoder BI-52S-2500-PU

4.5 Motor DriverA motor driver has to be used to control the voltage given to the motor. This motor driver

takes analog inputs provided by the DAQ Card. This motor driver takes analog voltage as

input and then calculates the voltage that has to be supplied to the motor.

Input Voltage Range=0-5V

Input Current Range = 0-30mA

Output Voltage Range= 0-12V


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This has been procured from the Robotics Club, IIT Delhi as this has been tried and tested

successfully by them. This is made by the companyDimensionEngineering and is a standard

part that can be procured later at any stage of the project. The figure 4.8 shows the motor

driver. The data sheet of this has been attached in the Appendix-E.

Figure 4.8 - Motor Driver (www.dimensionengineering.com)

4.6 Data Acquisition Card (DAQ)DAQ Card is required to transfer the reading from the encoders to the computer and then givethe controlled outputs back to the motor. The Data Acquisition card present in the Lab was

studied and checked for compatibility. Two models were available in the lab but the model

relevant for the project has been selected.

Model Number: NI PCI-6221

Specifications:

OUTPUT PORTs: Two 16-bit analog outputs (833 kS/s) INPUT/OUTPUT PORT: 24 digital I/O COUNTER: 32-bit counters

Required Inputs:

Two Angular EncodersDigital Inputs

Required Outputs:One Motor Driver CardAnalog output


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The figure 4.9 shows the port of the DAQ which has 68 pins and all the pins have been

marked for their corresponding functions.

Figure 4.9 - DAQ NI-PC 6221 Port Drawing


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Chapter 5. Equipment Design and FabricationThe Figure 5.1 shows the final CAD Drawing of the complete system.

Figure 5.1 - Final CAD model


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The figure Figure 5.2 shows the CAD Model of the Igus slider that has been ordered.

Figure 5.2 - Igus Slider

After this, the cart along with the pendulum has been designed and the Figure 5.3 shows the

CAD Model for that.

Figure 5.3 - Cart along with the pendulum

This cart has now been attached to the slider. After this, to move the cart along the slider, a

chain and sprocket system has been designed. So, on one side of the slider, we have a drivingmechanism. On the other side, there is a driven mechanism. The driving mechanism consists


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of a motor attached to a shaft which is further connected to an incremental encoder to

measure the rotation of the motor so as to calculate the position of the cart. The assembly is

supported by the bearings which in turn are mounted on a plate. The Figure 5.4 shows the

CAD model of the driving mechanism.

Figure 5.4 - Driving Mechanism

The driven mechanism assembly consists of a idler sprocket supported by 2 bearings on a

plate. The Figure 5.5 shows the CAD Model of the driven mechanism.

Figure 5.5 - Driven Mechanism Assembly

To support the slider, the driving mechanism and the driven mechanism, three stands are

made. Two of the three stands are identical which are at the ends and the Figure 5.6 shows

the CAD model of that stand. The shape of this stand has been made so as to support the


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driving (or driven) mechanism and the slider. Also, provision has been made to accommodate

the chain of the pulley.

Figure 5.6 - Stand on the ends of the slider

The Figure 5.7 shows the CAD Model of the middle stand.

Figure 5.7 - Middle Stand

After procuring the components for the setup like aluminium box channels, shafts, couplers

etc. in the last semester, the manufacturing was started. First of all, the stands were made

which had to support the complete setup. The channels were cut as per the requirements and

were taken to a manufacturer outside IIT where they were welded using TIG Welding. Figure

5.8 shows the welded stands. One fork of the left stand supports the driving motor and the

other stand supports the IGUS Sliders.


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Figure 5.8 : Stands after TIG Welding

Then plates were cut using sheet metal cutter to support the sliders. Figure 5.9 shows the

5mm Aluminium plates used in the setup.

Figure 5.9 : 5mm Aluminium plates

To support the driving shaft assembly and the driven shaft assembly, the 5mm plates were

bent so as to take in account the profile of the gear. Figure 5.10 shows both the plates.

Figure 5.10 : Plates used to support the driving and the driven shaft assemblies


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Further, to drive the system, timing belt was selected. To prevent any mismatch, the timing

pulleys of the same module as the belt were obtained. Figure 5.11 shows the timing belt and

figure 5.12 shows the profile of the timing pulleys thus obtained.

Figure 5.11 : Timing belt 3.5m long and 10mm wide

Figure 5.12 : Profile of the timing pulleys

To drive the system, 8mm shaft was selected and was cut according to the CAD Model. Thus

8mm bearings were selected. Outer diameter of the selected bearings was 16mm. Figure 5.13

shows the selected bearings.

Figure 5.13 : 8mm roller bearing


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To mount the bearings in the system, aluminium housing was made from a 25mm diameter

aluminium rod. The bearings were fitted into the housing with a tight fit. A 2mm strip was

welded tangentially to the rod for base of the housing. Figure 5.14 shows the close-up view of

the bearing fitted into housing.

Figure 5.14 : Bearing with housing

To mount the motor and the encoder on the support, L-shaped housings were made and

connected to them. Figure 5.15 and figure 5.16 shows the motor and the encoder along with

their housings. Two incremental encoders were used and both have the similar housings.

Figure 5.15 : Driving Motor with housing


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Figure 5.16 : Encoder along with housing

To connect the motor and the encoder to the 8mm shaft, muff couplings were used as per the

CAD drawings. The driving assembly was made by assembling the motor with the shaft and

that with the encoder. The shaft was supported by 2 bearings and to level the bearings, the

bearing supports were made. Figure 5.17 shows the driving shaft assembly.

Figure 5.17 ; Driving shaft assembly

Similarly, the driven shaft assembly was made. The shaft was simply mounted using the two

bearings and their stands. Figure 5.18 shows the driven shaft assembly.


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Figure 5.18 : Driven shaft assembly

Now, to make the cart, a 5mm plate was first attached to the Igus slider. The remaining

assembly was fixed over the cart. To hold the pendulum, a shaft was made with provision to

attach the encoder on one side and pendulum on the other side. The shaft along with the

bearing and encoder were supported on the cart plate. The bearing was supported by a

bearing support to level the encoder with the bearing. The complete cart assembly including

the encoder, its housing, shaft, cart, bearing is shown in the figure 5.19.

Figure 5.19 : Cart assembly

The pendulum was made by attaching an 8mm aluminium rod to a steel cylinder. The steel

cylinder was attached as a bob to increase the moment of inertia of the pendulum about the

pivot point so as to reduce the air drag caused while oscillations. Figure 5.20 shows the

pendulum.


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Figure 5.20 : Pendulum

The complete system was mounted on the sliders and the stands. Now to drive the cart, the

motor was attached to the cart using a timing pulley. The belt was wound tightly on the

pulley and the open ends of the belt were fixed to the cart by attaching 2 plates on the cart.

The bolts on the pulley could be tightened to hold the belt in tension. Figure 5.21 shows the

close-up view of the joint between cart and belt.

Figure 5.21 : Joint between cart and timing belt


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The complete setup was assembled and placed in the Vibrations Research Lab. Due to the

uneven heights of all the stands, the setup was not stable. To solve the problem, it was

required that the system be fixed to a heavy base. Hence, a wooden board with the

dimensions of the setup was obtained from the carpentry workshop. The setup was joined to

the setup using 6mm bolts and the joint is shown in the figure 5.22. Further, counter-boring

was done on each of the holes so that the setup rests on the wooden base and not on the heads

of the bolt.

Figure 5.22 : Joint between wooden block and stand

Since, the slider was of the shape of a strip, to prevent bending, it was necessary to provide it

some support. Hence, to increase the stiffness of the slider a 3x1 cross-sectional aluminium

channels was attached. Figure 5.23 shows the arrangement done for the same.

Figure 5.23 : Support for the slider

Finally, the setup was assembled and the figure 5.24(a) and (b) shows the final setup thus

made.


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Figure 5.24(a) and (b) : Final setup of inverted pendulum


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Chapter 6. Developing of Sensing and Actuation of systemTo interface the system with the computer, it was required to interface the encoder(Section

6.1) to the DAQ Cards and the voltage generated by the DAQ Card should be sent to the

motor. But since DAQ Card can only supply a very small current, a motor driver is required

to amplify the signal via a battery. This has been discussed in Section 6.2.

6.1 Encoder InterfacingAn incremental encoder takes 2 signals as inputs one 5V and the other one, ground. It

generates various output signals. For the current purpose, it was required to measure the

angular rotation of the encoder. So, only 2 signals are required one is waveform A and

other is waveform B. Figure 6.1 shows a general output of the 2 waveforms. When

waveform A leads waveform B by a phase of 90 degrees, it implies that the encoder is

moving in clockwise direction, and when it lags, it indicates that encoder is moving in

counter-clockwise direction.

Figure 6.1 : 2 output signals generated by incremental encoder(NI Developer Zone forum)

Hence, the measurement must be made continuously to ensure that the final reading takes

into account this. THE NI-DAQ has 2 counters to do this function. Each counter has a source

and up/down line which automatically calculates the final output. Figure 6.2 illustrates this.

Figure 6.2 : Simplified Counter/Timer Model of DAQ


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The NI-6221 DAQ card has 2 counters and the figure 6.3 shows the pins of connector block

SCB-68 that correspond to the source and up/down line.

Figure 6.3 : Layout of the connector block of NI-6221

As can be seen from the figure 6.3, pin 37 correspond to the source of counter 0 and pin 45

corresponds to auxiliary line of counter 0.To test the counter, a labview code was made to

read from the counter and display the final angle in a numeric indicator. Figure 6.4 shows theblock diagram of the code.

Figure 6.4 : Block diagram of the code to measure encoder's position

The encoder was tested and the code worked fine. The numeric indicator continuously

showed the current position of the encoder as per a reference position given.


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6.2 Motor Driver FabricationA motor driver has to be used to control the voltage given to the motor. This motor driver

takes analog inputs provided by the DAQ Card. This motor driver takes analog voltage as

input and then provides the required amount of voltage, current and energy to the motor.

The whole motor driver circuit is composed of three modules:

1. L7805: The L7806 chip is three-terminal positive regulator package, making it usefulin a wide range of applications. This regulator can provide local on-card regulation,

eliminating the distribution problems associated with single point regulation. Each

chip employs internal current limiting, thermal shut-down and safe area protection,

making it essentially indestructible. If adequate heat sinking is provided, they can

deliver over 1A output current. Although designed primarily as fixed voltage

regulators, these devices can be used with external components to obtain adjustable

voltage and currents. Figure 6.5 shows a typical L7805 chip. Data-sheet of L7805

chip is attached in Appendix-D.

Figure 6.5: L7805 chip (http://www.mindkits.co.nz/store)

This chip was used to generate a regulated supply of +5 Volts to be given to Atmega

16. The pinouts and connections are shown in figure 6.6.

Figure 6.6: picture showing pinouts of L7805 chip(http://www.mindkits.co.nz/store)
http://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5ahttp://www.mindkits.co.nz/store/components-ics-breakout-boards/silicon-chips-ics/5-x-l7805-7805-voltage-regulator-5v-1-5a


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2. Atmega 16: This takes the analog input voltage corresponding to the rpm of themotor from the DAQ card and then calculates the energy to be given to the motor.

The output is in the form of PWM (Pulse Width Modulation) at a high frequency. The

voltage at the output is only +5 Volts and the current is also in milli-amperes. So, this

has to be stepped up to 12 V and the current current ~ 2 Amperes has also to be

provided. The Atmega 16 is shown in Figure 6.7. Data-sheet of AtMega 16 is attached

in Appendix-B.

Figure 6.7: ATmega16 chip (http://www.futurlec.com/Atmel/ATMEGA16.shtml)

The pinouts of this micro-controller are shown in Figure 6.8.

Figure 6.8: Pinouts of Atmega 16 (Appendix B)

The Atmega 16 controller was connected to a base before mounting on the matrix

board. The input signals were given on Pin 40 (Analog Input) and the PWM output is

obtained on pin 19. Figure 6.9 shows the soldering in progress for the base.
http://www.futurlec.com/Atmel/ATMEGA16.shtmlhttp://www.futurlec.com/Atmel/ATMEGA16.shtmlhttp://www.futurlec.com/Atmel/ATMEGA16.shtmlhttp://www.futurlec.com/Atmel/ATMEGA16.shtml


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Figure 6.9: Soldering in progress for Atmega 16 Base

The status of matrix board after soldering of Atmega 16 base, crystal, ISP and L7805

chip is shown in figure 6.10.

Figure 6.10: Soldered Atmega 16 Base, crystal, ISP port and L7805 chip on matrix board

3. L298N: It is a high voltage, high current dual full-bridge driver designed to acceptstandard TTL logic levels and drive inductive loads like DC motors. Figure 6.11

shows a typical L298N chip. Data-sheet of L298N is attached in Appendix-C.


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Figure 6.11: L298 chip (Appendix C)

The L298 chip takes the following inputs:

Power supply: 12 V Logic Supply: 5V Ground PWM (logic form) Direction of rotation (logic Form)The L298 chip gives the output to the motor by two terminals. Figure 6.12 shows all

the pin connections of the L298 chip.

Figure 6.12: Connections for L298N for driving DC motor (Appendix C)


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The L298 chip was soldered onto a base to allow for easy dismounting of the chip. The view

of soldered L298 chip is shown in Figure 6.13.

Figure 6.13: Soldered L298 chip

All the chip connections were made and soldered. Figure 6.14 shows the final motor driver

card.

Figure 6.14: Motor Driver card


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Chapter 7. Control Implementation

After the complete setup has been installed and the necessary electrical connections have

been setup, control implementation was done. To get familiarity with the labVIEW, basic

experiments were done e.g. synchronising counter with the encoder to calculate the final

displacement of the encoder, and taking measurement from a simple potentiometer. As has

been discussed in Section 3.3, before proceeding to the real-time control implementation,

labVIEW codes were created to simulate the plant using values from the analytical model of

the plant. After the simulation was done, real-time implementation was done and the results

were compared. As the trend observed with the change in control logic parameters in

simulation were similar to real-time system, trends in simulation results were used to vary the

parameters in real-time system to obtain the desired characteristics.

As discussed in Section 3.3, first of all SISO controls for cart position and pendulum angle

were applied individually and then cascaded PID logic was used to balance both cart position

and pendulum angle simultaneously. This section focuses on the results obtained from

simulation and real-time system and their comparison with each other.

7.1 SISO Control of Cart Position(x)For real-time implementation, the voltage calculated by the PID logic was supplied to motor

as voltage and values of cart position and angle theta of the pendulum were taken by the

encoders mounted on the plant. Figure 7.1 shows the block diagram of the labVIEW code

used for real-time control.


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Figure 7.1: Block Diagram for real time x-control

The PID logic used consists of 3 parameters Kp, Kd, Ki. To understand the variation and

the effect of each of these on the control characteristics, each of them was varied keeping the

other two fixed. First of all, proportional constant Kpwas varied keeping Kd, Ki constant.

Figure 7.2shows the simulation results for 3 different values of Kp and values of Ki and Kd

have been kept constant at 500 and 0.01 respectively. The set-point of the cart has been set at

0.08m and simulation is done for 10 seconds at a sampling time of 1 ms. Results show the

value of 2 parameters cart position X and control action.


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As can be seen from Figure 7.2,with increase in Kp values from 10 to 100, damping of the

system increases and the settling time decreases monotonously. Also, rise times constantly

decrease. Similar analysis is also done for the real-time system. But since there are unwanted

differences in the modelling and the actual system, the values obtained are different from the

simulation results. Hence, only trends have been focussed at in this work.

Figure 7.3shows the results obtained with the real-time system for change in Kp values from

100-300.

Figure 7.2 Simulation results for Kp=10, 25,50,100


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For real-time system also, damping of the system increases monotonously with increase inKp values. Settling time of the system and rise time also decrease continuously. Hence, the

trends match exactly with the simulation results.

As with Kp, similar operations were performed with Ki and Kd. Figure 7.4 shows the results

obtained for the simulation by varying Ki from 100 to 4000 whereas Kp and Kd have been

kept constant at 25 and 0.01 respectively.

Figure 7.3 Real-time results for Kp=100, 150, 300, 500


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As can be seen from Figure 7.4, damping decreases with increase in Ki and settling time

increases continuously. To compare the results with the real-time system, their results are

displayed in Figure 7.5. Here, Kp and Kd have been kept constant at 300 and 0.003

respectively but Ki has been varied from 3000 to 80000.

Figure 7.4 Simulation results for Ki=100,500,1000,5000


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For the real-time system also, results match with the simulation. With increase in Ki,

damping decreases and settling time increases continuously.

Finally, trends were studied were studied for variation in Kd. Figure 7.6 show variation of

results on changing values of Kd from 0.001 to 1 whereas Kp and Ki have been kept constant

at 25 and 500 respectively.

Figure 7.5 Real-time results for Ki=3000, 30000, 60000, 80000


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Results from the simulations suggest that with increase in Kd, damping decreases and settling

time increases continuously.

Figure 7.7 shows the variation in a real-time system. Values of Kp and Ki have been kept

constant at 300 and 30000 respectively whereas Kd has been varied from 0.003 to 3

respectively.

Figure 7.6 Simulation results for kd=0.001, 0.01, 0.1, 1


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The results show that with increase in values of Kd, damping of the system increases and

settling time does not change significantly. Unlike Kp and Ki, trend of Kd is different from

what is observed in simulation. This signifies that there is feature of the system that has bot

been captured in the simulation models.

7.2 SISO Control of Pendulum Angle(

)

To do real-time control implementation, like Figure 7.1, motor voltage was given to the

motor via DAQ Card and similarly, angle theta and cart position are measured by encoders

and supplied to the labVIEW code via DAQ Cards. Figure 7.8shows the block diagram for

the simulation.

Figure 7.7 Real-time results for Kd=0.003, Kd=0.3, kd=3


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Figure 7.8: Block Diagram for theta-control real-time implementation

Like Section 7.1, in this section also, 3 parameters of the PID logics Kp, Ki, and Kd are

varied to study and observe the trends. Both simulation and the real-time system have been

studied so that any differences, if any, can be commented upon.

Figure 7.9 shows the simulation results for 3 different values of Kp. The set-point of the

angle has been set at 0.057 radians and simulation is done for 2 seconds at a sampling time of

1 ms. Results show the value of 3 parameters cart position X, pendulum angle and controlaction.


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As can be seen from Figure 7.9, with increase in values of Kp from 40 to 500, damping of the

system decreases monotonously. The settling time also decreases continuously.

Figure 7.10 shows the results obtained for the real-time system. Here Ki and Kd have been

kept constant at 5 and 0.35 respectively but Kp is varied from 200 to 700. Since, the position

of the cart is not controlled, hence the cart goes out of the desired limits very soon. So,

pendulum can be kept within limits only for 1-2 seconds without any manual intervention.The graphs have been obtained only for the region where there was no manual intervention.

On start-up, pendulum was left at its swing up position, and unlike simulation, stability of the

pendulum is studied only.

Figure 7.9 Simulation results for Kp=40, 75, 200, 500


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As can be seen from the Figure 7.10, as values of Kp increases, the stability of the pendulum

increases and hence, its tendency to sway away from the centre decreases. But due to noise in

the surroundings, the control action keeps fluctuating and hence theta keeps on oscillating.

The pendulum stays within range only for 2-3 seconds because in its effort to control

pendulum angle, it moves rapidly on the cart and hence crosses the boundaries. Hence,

similar analysis has been done for Ki and Kd.

Figure 7.11 shows the results for the simulation of variation in values of Ki. Kp and Kd have

been kept constant at 75 and 0.002 respectively.

Figure 7.10 Real-time results for Kp=200, 450, 700


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As can be seen from Figure 7.11, as values of Ki increase from 0.005 to 0.1, damping of the

system decreases. Steady state error of the system also increases with time continuously

whereas the settling time decreases continuously.

In real-time system also, results were measured for changing values of Ki from 5 to 100

keeping Kp and Kd constant at 450 and 0.35 respectively. Figure 7.10 show the results

obtained.

Figure 7.11 Simulation results for KI=0.005, 0.01, 0.05


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Figure 7.12 suggest that with increase in values of Ki from 5 to 100, system becomes more

stable and less sensitive to noise. Its tendency to sway away from the centre decreases. This is

explicit from noticing that that oscillations in the system decrease due to noise with increase

in Ki.

Finally, simulation was done by varying values of Kd from 0.002 to 0.2 respectively. Kp and

Ki have been kept constant at 75 and 0.05 respectively. Figures 7.13 show the results

obtained.

Figure 7.12 Real-time results for Ki=5, 10, 50, 100


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Figures 7.13 suggest that as value of Kd increases from 0.002 to 0.2, damping decreases

continuously. Also, settling increases continuously.

Now, similar readings are obtained for the real-time system. The graphs obtained are shown

BTP Sem2 Final

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