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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 7.18 Simulation results for Cart Position Gains Kp=0.8, 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 7.26 Real-time results for Pendulum Angle Gains Kp=450, Ki=45, kd=0.45(Cascaded PID) ...... 69
Figure 7.27 Real-time results for Pendulum Angle Gains Kp=700, 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 7.29 Simulation results for Cart Position Gains Kp=0.6, Ki=0.001, 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 7.35 Simulation results for Pendulum Angle Gains Kp=50, Ki=3000, 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 7.39 Real-time results for Pendulum Angle Gains Kp=450, Ki=70, 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 7.48 Simulation results for Pendulum Angle Gains Kp=50, Ki=3000, kd=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 7.50 Real-time results for Pendulum Angle Gains Kp=450, Ki=45, kd=0.45 (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
e
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
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R
S
M
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Rs
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:
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)
So combined Closed Loop Control Transfer Function:
So combined Closed Loop Control Transfer Function:
=
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