Thesis Mochammad Rizky Diprasetya
description
Transcript of Thesis Mochammad Rizky Diprasetya
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MODELING, SIMULATING, AND ANALYZING QUARTER CAR PASSIVE
AND ACTIVE SUSPENSION USING MATLAB SIMULINK AND V-REP
SIMULATOR
By
Mochammad Rizky Diprasetya S
11111034
BACHELORS DEGREE in
MECHANICAL ENGINEERING MECHATRONICS CONCENTRATION FACULTY OF ENGINEERING AND INFORMATION TECHNOLOGY
SWISS GERMAN UNIVERSITY
EduTown BSD City
Tangerang 15339
Indonesia
June 2015
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STATEMENT BY THE AUTHOR
I hereby declare that this submission is my own work and to the best of my knowledge,
it contains no material previously published or written by another person, nor material
which to a substantial extent has been accepted for the award of any other degree or
diploma at any educational institution, except where due acknowledgement is made in
the thesis.
Mochammad Rizky Diprasetya S
____________________________________________
Student
Date
Approved by:
Kirina Boediardjo, ST, M.Sc
____________________________________________
Thesis Advisor
Date
Yunita Umniyati, PhD
____________________________________________
Thesis Co-Advisor
Date
Dr. Ir. Gembong Baskoro, M.Sc
____________________________________________
Dean
Date
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ABSTRACT
MODELING, SIMULATING, AND ANALYZING QUARTER CAR PASSIVE AND
ACTIVE SUSPENSION USING MATLAB SIMULINK AND V-REP SIMULATOR
By
Mochammad Rizky Diprasetya S
Kirina Boediardjo, ST, M.Sc, Advisor
Yunita Umniyati, PhD, Co-Advisor
SWISS GERMAN UNIVERSITY
This thesis works purposes are to develop a mathematical model for quarter car passive
and active suspension, to simulate its behavior, and analyze the difference of both
suspension. The active suspension use PID controller to control the displacement of the
car in vertical direction caused by different road profile. The displacement has to be
small, these condition are achieved by manipulating the force actuator inside the
suspension. The force actuator manipulated according to the output of controller
system. V-REP simulator is used to visualize the physical simulation. The physical
model of suspension developed in V-REP. The result from V-REP compared with the
result from MATLAB Simulink.
Keywords: Quarter Car Suspension, MATLAB, Simulink, V-REP Simulator, Full Car
Suspension, PID, LQR.
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Copyright 2015
by Mochammad Rizky Diprasetya S
All rights reserved
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DEDICATION
I dedicate this works for my parents, my friends, and myself.
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ACKNOWLEDGEMENTS
I thank All for His constant blessing and supports during this thesis development.
I would like to thank to my parents for their patience in these past four years.
I would like to thank Kirina Boediardjo, ST, M.Sc for her support throughout the
development of the thesis, time, and her guidance in building the system model
schematic during development of this thesis.
I would like to thank to Yunita Umniyati, PhD for her inputs, advice, and support for
this thesis.
Last but not least I thank all of my beloved friends who have supported me from the
very beginning until the very end of this thesis development.
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TABLE OF CONTENTS
Page
STATEMENT BY THE AUTHOR ................................................................................ 2
ABSTRACT ................................................................................................................... 3
DEDICATION ............................................................................................................... 5
ACKNOWLEDGEMENTS ........................................................................................... 6
TABLE OF CONTENTS ............................................................................................... 7
LIST OF FIGURES ..................................................................................................... 10
LIST OF TABLES ....................................................................................................... 13
CHAPTER 1 - INTRODUCTION ............................................................................... 14
1.1 Background ............................................................................................................ 14
1.2 Thesis Purpose ....................................................................................................... 16
1.3 Objective ................................................................................................................ 16
1.4 Hypothesis.............................................................................................................. 16
1.5 Thesis Scope .......................................................................................................... 17
1.6 Thesis Limitations .................................................................................................. 17
1.6 Thesis Organization ............................................................................................... 17
CHAPTER 2 - LITERATURE REVIEW .................................................................... 19
2.1 Theoretical Perspectives ........................................................................................ 19
2.1.1 Suspension System.............................................................................................. 19
2.1.2 Spring in Suspension System .............................................................................. 21
2.1.3 Damper in Suspension System............................................................................ 22
2.1.4 Force Actuator in Suspension System................................................................. 23
2.1.5 Hydraulic Actuator .............................................................................................. 23
2.1.6 Electromagnetic Actuator .................................................................................... 24
2.1.7 PID Control ......................................................................................................... 25
2.1.8 Linear Quadratic Regulation (LQR) ................................................................... 27
2.1.9 MATLAB Simulink ............................................................................................ 28
2.1.10 V-REP Simulator ............................................................................................... 29
2.1 Previous Studies ..................................................................................................... 31
2.1.1 Modeling, Simulating, and Analyzing an Overhead Crane Using MATLAB
Simulink and V-REP Simulator [5] .............................................................................. 31
CHAPTER 3 RESEARCH METHODS ................................................................... 33
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3.1 Simulation Model Methodology ............................................................................ 33
3.2 Mathematical Model .............................................................................................. 34
3.2.1 Mathematical Model of Quarter Car Passive Suspension................................... 35
3.2.2 Mathematical Model of Quarter Car Active Suspension .................................... 38
3.2.3 Mathematical Model of Full Car Passive Suspension ........................................ 41
3.2.4 Mathematical Model of Full Car Active Suspension .......................................... 45
3.3 Control Design ....................................................................................................... 49
3.3.1 Proposed PID Control Design ............................................................................. 50
3.3.2 Linear Quadratic Regulator................................................................................. 51
3.4 Mechanical Part ..................................................................................................... 52
3.4.1 Model Parameter ................................................................................................. 56
3.4.2 Road Disturbance Environment .......................................................................... 57
CHAPTER 4 RESULTS AND DISCUSSIONS ....................................................... 60
4.1 System Model Result ............................................................................................. 60
4.1.1 Quarter Car Suspension System.......................................................................... 61
4.1.1.1 Passive Suspension System Response Analysis .............................................. 63
4.1.1.1.1 Impulse Input ................................................................................................ 63
4.1.1.1.2 Step Input ...................................................................................................... 66
4.1.2 Full Car Suspension System ............................................................................... 71
4.1.2.1 Passive Suspension System Response Analysis .............................................. 76
4.1.2.1.1 Impulse Input ................................................................................................ 77
4.1.2.1.2 Step Input ...................................................................................................... 78
4.3 Control Analysis..................................................................................................... 79
4.3.1 PID Controller for Quarter Car Suspension ........................................................ 79
4.3.2 PID Controller for Full Car Suspension.............................................................. 81
4.3.3 LQR Controller for Quarter Car Suspension ...................................................... 83
4.3.4 LQR Controller for Full Car Suspension ............................................................ 83
4.4 Quarter Car Passive and Active Suspension Comparison ...................................... 83
4.4.1 Comparison with PID Controller for Step Input ................................................. 83
4.4.2 Comparison with PID Controller for Impulse Input ........................................... 86
4.4.3 Comparison with LQR Controller ...................................................................... 88
4.5 Full Car Passive and Active Suspension Comparison for Rear Right Suspension 90
4.5.1 Comparison with PID Controller for Step Input ................................................. 91
4.5.2 Comparison with PID Controller for Impulse Input ........................................... 92
4.6 Full Car Suspension Simulink Model and Full Car Suspension V-REP Comparison
...................................................................................................................................... 92
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4.7 Comparison V-REP and MATLAB Simulink Rear Right Suspension of Car Result
with Impulse Input ....................................................................................................... 95
4.8 Comparison V-REP and MATLAB Simulink V-REP Front Right Suspension of
Car Result with Step Input ......................................................................................... 103
CHAPTER 5 CONCLUSIONS AND RECCOMENDATIONS ............................ 110
5.1 Conclusions .......................................................................................................... 110
5.2 Recommendations ................................................................................................ 111
GLOSSARY ............................................................................................................... 112
REFERENCES .......................................................................................................... 113
APPENDICES ........................................................................................................... 114
APPENDIX A - MATLAB CODE FOR LQR CONTROLLER OF QUARTER CAR
SUSPENSION ........................................................................................................... 115
APPENDIX B - MATLAB CODE FOR LQR CONTROLLER OF FULL CAR
SUSPENSION ........................................................................................................... 117
APPENDIX C STATE SPACE MATRIX OF FULL CAR PASSIVE SUSPENSION.................................................................................................................................... 120
APPENDIX D STATE SPACE MATRIX OF FULL CAR ACTIVE SUSPENSION.................................................................................................................................... 123
CURRICULUM VITAE ............................................................................................ 124
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LIST OF FIGURES
Figures Page
Figure 1.1 Suspension System [1] ............................................................................... 14
Figure 1.2 Road Disturbance (left), Load Disturbance (right)..................................... 15
Figure 2.1 Spring and Damper [1] ............................................................................... 19
Figure 2.2 Controllable Damper [2]............................................................................. 20
Figure 2.3 Active Suspension System with Electromagnetic Motor [2] ...................... 21
Figure 2.4 Spring ......................................................................................................... 22
Figure 2.5 Damper ....................................................................................................... 22
Figure 2.6 Hydraulic Actuator ..................................................................................... 23
Figure 2.7 Electromagnetic actuator in suspension system ......................................... 24
Figure 2.8 Common Process Block ............................................................................. 25
Figure 2.9 PID Controller ............................................................................................ 25
Figure 2.10 System Response Based on Varieties Kp, Ki, Kd .................................... 27
Figure 2.11 Overhead Crane Equation Modelling ....................................................... 31
Figure 2.12 Overhead Crane V-REP Simulator ........................................................... 32
Figure 3.1 Free Body Diagram Quarter Car Passive Suspension ................................ 35
Figure 3.2 Free Body Diagram Quarter Car Active Suspension .................................. 38
Figure 3.3 Full Car Passive Suspension Model ........................................................... 41
Figure 3.4 Full Car Active Suspension Model ............................................................. 45
Figure 3.5 Control System Overview .......................................................................... 50
Figure 3.6 Suspension Distance ................................................................................... 50
Figure 3.7 Control System in MATLAB Simulink ...................................................... 51
Figure 3.8 Overall V-REP Model ................................................................................. 53
Figure 3.9 Double Wishbone Joint Type ...................................................................... 54
Figure 3.10 Adjusting Suspension Parameter .............................................................. 54
Figure 3.11 Revolute Joint from Wheel to Arm ........................................................... 55
Figure 3.12 Revolute Joint from Arm to Chassis ......................................................... 56
Figure 3.13 Speed Bump Model .................................................................................. 58
Figure 3.14 Step Input Road Disturbance .................................................................... 59
Figure 4.1 Quarter Car Simulink Block ....................................................................... 61
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Figure 4.2 Quarter Car Passive Suspension Simulink Block ....................................... 62
Figure 4.3 Quarter Car Active Suspension Simulink Block ........................................ 62
Figure 4.4 Impulse Input Road Disturbance ................................................................ 63
Figure 4.5 Car Body Displacement Due to Impulse Input ........................................... 64
Figure 4.6 Suspension Travel Due to Impulse Input.................................................... 65
Figure 4.7 Wheel Displacement Due to Impulse Input ................................................ 66
Figure 4.8 Step Input Road Disturbance ...................................................................... 67
Figure 4.9 Suspension Travel Due to Step Input ......................................................... 68
Figure 4.10 Car Body Displacement Due to Step Input .............................................. 69
Figure 4.11 Wheel Displacement Due to Step Input.................................................... 70
Figure 4.12 Full Car Suspension System ..................................................................... 71
Figure 4.13 Pitching Block Diagram ........................................................................... 73
Figure 4.14 Bouncing Block Diagram ......................................................................... 74
Figure 4.15 Rolling Block Diagram............................................................................. 75
Figure 4.16 Theta, Gamma, Body Displacement for Each Suspension ....................... 76
Figure 4.17 Impulse Input Road Disturbance .............................................................. 77
Figure 4.18 Car Body Displacement of Front Right Suspension due to Impulse Input
...................................................................................................................................... 77
Figure 4.19 Step Input Road Disturbance .................................................................... 78
Figure 4.20 Car Body Displacement of Front Right Suspension due to Step Input .... 79
Figure 4.21 PID Controller .......................................................................................... 80
Figure 4.22 Closed-loop System of Suspension System ............................................. 80
Figure 4.23 Control System for Each Suspension ....................................................... 81
Figure 4.24 Inside Control System Block .................................................................... 82
Figure 4.25 Flowchart for LQR Controller using MATLAB Code ............................. 83
Figure 4.26 Comparison Suspension Travel with PID Controller due to Step Input ... 84
Figure 4.27 Comparison Car Body Displacement with PID Controller due to Step
Input ............................................................................................................................. 84
Figure 4.28 Comparison Wheel Displacement with PID Controller due to Step Input
...................................................................................................................................... 85
Figure 4.29 Comparison Wheel Displacement with PID Controller due to Impulse
Input ............................................................................................................................. 86
Figure 4.30 Comparison Car Body Displacement with PID Controller due to Impulse
Input ............................................................................................................................. 86
Figure 4.31 Comparison Suspension Travel with PID Controller due to Impulse Input
...................................................................................................................................... 87
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Figure 4.32 Car Body Displacement LQR Controller ................................................. 88
Figure 4.33 Wheel Displacement with LQR Controller .............................................. 89
Figure 4.34 Suspension Travel with LQR Controller .................................................. 90
Figure 4.35 Comparison Car Body Displacement of Rear Right Suspension with PID
Controller due to Step Input ......................................................................................... 91
Figure 4.36 Comparison Car Body Displacement of Rear Right Suspension with PID
Controller due to Impulse Input ................................................................................... 92
Figure 4.37 Car Body Displacement V-REP Result .................................................... 93
Figure 4.38 Car Body Displacement MATLAB Simulink Result ............................... 94
Figure 4.39 V-REP Result of Rear Right Suspension with Impulse Input .................. 95
Figure 4.40 Position of Car Body before Drive through the Speed Bump .................. 96
Figure 4.41 V-REP Result of Rear Right Suspension First Peak ................................. 96
Figure 4.42 V-REP Result of Rear Right Suspension Bottom after First Peak ........... 97
Figure 4.43 First Overshoot due to Suspension ........................................................... 98
Figure 4.44 MATLAB Simulink Result Rear Right Suspension due to Impulse Input
...................................................................................................................................... 99
Figure 4.45 MATLAB Simulink Result after Goes through the speed bump ............ 100
Figure 4.46 First overshoot due to Suspension .......................................................... 101
Figure 4.47 V-REP Result of Rear Right Suspension with Step Input ...................... 103
Figure 4.48 Position of Car Body before Drive through the Step Input .................... 104
Figure 4.49 V-REP Result of Rear Right Suspension First Peak ............................... 105
Figure 4.50 V-REP Result of Rear Right Suspension Bottom after First Peak ......... 106
Figure 4.51 MATLAB Simulink Result Rear Right Suspension due to Impulse Input
.................................................................................................................................... 107
Figure 4.52 MATLAB Simulink Result Rear Right Suspension due to Impulse Input
the first peak ............................................................................................................... 107
Figure 4.53 MATLAB Simulink Result Rear Right Suspension due to Impulse Input
after the first peak ...................................................................................................... 108
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LIST OF TABLES
Table Page
Table 3.1 State Space Variable 43
Table 3.2 State Space Variable Definition 44
Table 3.3 State Space Variable 48
Table 3.4 State Space Variable Definition 48
Table 3.5 V-REP Model Parameter 56
Table 4.1 List of Parameter for Full Car Suspension System 71
Table 4.2 Result Comparison V-REP Simulator and MATLAB Simulink for Rear
Right Suspension due to Impulse Input 102
Table 4.3 Result Comparison V-REP Simulator and MATLAB Simulink for Rear
Right Suspension due to Step Input 108
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CHAPTER 1 - INTRODUCTION
1.1 Background
The suspension system must upkeep the vehicle, deliver directional control using
handling manoeuvres and deliver actual isolation of person along for the ride and load
disturbance. The criteria of the suspension personally depend on the purpose of the
vehicle. For example, a normal car driver will need a quite soft ride for low to medium
speed handling and at ease drive.
Figure 1.1 Suspension System [1]
There are two main types of disturbances on a vehicle, road and load disturbances. Road
disturbances have the characteristics of large amount in low rate such as mountains
and small amount in high rate such as road bumpiness. Load disturbance formed
by accelerating, braking, and cornering. Thus, a good suspension scheme is required
with altered disturbance dismissal from these disturbances to create hard or soft
suspension hinge on the purpose of the vehicle.
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Figure 1.2 Road Disturbance (left), Load Disturbance (right)
A car suspension system is the instrument that physically isolated the car body from
the wheels of the car. Suspension system attaches the wheel and the vehicle body by
springs, dampers and some links that attaches a vehicle to its wheels. The spring storing
energy affected by the body mass and aids to stabilize the body from the road
disturbance, while damper spent this energy and aids to reduce the oscillation. The
main role of vehicle suspension system is to separate a vehicle body from road
disturbance which is the upright acceleration transferred to the passenger in order to
improve comfort and well-being while driving a car, and in order to maintain constant
road wheel contact to deliver road holding.
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1.2 Thesis Purpose
The purpose of this thesis are first to build the system model of the quarter car passive
and active suspension system, to simulate the behaviour of the suspension system, to
design control system of the suspension system, and to analyse the result whether it can
achieve the desired condition where the car body
1.3 Objective
The main objective of this thesis are:
To model the suspension system in mathematical model
Develop and analyze the mathematical model using MATLAB Simulink
Develop and analyze the physical model using V-REP Simulator
To control the active suspension using PID control and LQR
To analyze the different between using PID control and LQR
To analyze the different between active and passive suspension
1.4 Hypothesis
The main purpose of this thesis are to analyze quarter car passive and active suspension
system, and analyze the difference between them. Determine which suspension has
good performance to achieve comfort and safety while ride a vehicle.
These are the hypothesis:
Passive suspension system will diminish vibration only affected by fix
parameters which is spring and damper. Hence, this suspension system will not
greatly isolated the body of vehicle from road disturbance.
Active suspension system will diminish vibration affected by fix parameters
which is spring and damper, and controllable force actuator. The force actuators
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will add or dissipate energy according to the control scheme. Hence, this
suspension system will greatly isolated the body of vehicle from road
disturbance.
1.5 Thesis Scope
This thesis scope is to model and simulate the behaviour of two car suspensions system,
the one without controller which is passive car suspensions and the one with the
controller that applied after analysing the passive suspensions, which is active car
suspensions. This experiment will be carried out first to determine what model is
suitable for this simulation to be simulated with MATLAB Simulink. To crosscheck
and to visualize the simulation result, a V-REP is used.
1.6 Thesis Limitations
The main coverage of this thesis is on modelling and the simulation of quarter car
passive and active suspension system. Two main software programs are used which are
MATLAB Simulink and V-REP Simulator. MATLAB Simulink is used to simulate the
suspension system model along with the controller design, while V-REP is used to
simulate the visualization of the suspension system in real life and verify the previous
simulation done in numerical simulator software, because this simulator is considered
to be able to simulate any kind of robotic mechanism and movement.
The actuators is not design in dynamic model. Because the mechanism of the actuator
cannot be developed in V-REP Simulator.
1.6 Thesis Organization
Chapter 1 Introduction
This chapter briefly explains about the purpose, scope, and limitation of the thesis
Chapter 2 Literature Review
This chapter discusses about the related work that have been previously done
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Chapter 3 Research Methodology
This chapter explains the steps how the experiment carried out.
Chapter 4 Result and Discussions
This chapter discusses the result and the measurements that have been done.
Chapter 5 Conclusions and Recommendations
This chapter concludes the thesis, and provides with recommendations for further
development of the thesis
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CHAPTER 2 - LITERATURE REVIEW
2.1 Theoretical Perspectives
2.1.1 Suspension System
A suspension is the system of springs, dampers, and some linkage that connects a vehicle
to its wheel. The main function is to minimize the vertical acceleration transmitted
to the passenger which directly provides road comfort. The suspension itself is divided
into three types, which are: passive, semi-active, and active. [2]
A passive suspension system contains springs and dampers where springs store the
energy and damper dissipate the energy. Its factor normally fixed, being chosen to
achieve a certain level of conciliation between road handling, load transport and ride
comfort. Most suspensions in this type can be measured as a spring in parallel with
damper located at most at each corner of the vehicle. [2]
Figure 2.1 Spring and Damper [1]
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A semi-active suspension system contains springs and controllable damper. The
controllers control the level of damping based on control scheme and spontaneously
adjust the damper to the desire levels. Sensors and actuators are added to sense the road
profile for control input.
Figure 2.2 Controllable Damper [2]
An active suspension system contains springs, dampers and force actuator. This
structure has capability to response to the upright changes in the road profile. The force
actuator will add or dissipate energy from the system. The force actuator controlled by
several type of controller determine by the control scheme. In this system passive
mechanisms are amplified by actuators that supply extra forces while pulling down
or pushing up the body masses. This is for achieving the desired level of comfort in
order to diminish the vibrations due to the road differences.
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Figure 2.3 Active Suspension System with Electromagnetic Motor [2]
In this thesis, mathematical model of passive and active suspension system with
hydraulic actuator will be developed using MATLAB Simulink. The simulation will be
developed using V-REP Simulator.
2.1.2 Spring in Suspension System
The main function of spring in suspension application is to store energy produced by the
mass of the body. There are several type of spring rate depends on the vehicle. For race
car, it has heavy spring and for passenger car, it has soft spring. That is because race car
need hard suspension to maximize handling on high speed, but passenger car need soft
suspension to achieve comfortable drive.
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Figure 2.4 Spring
2.1.3 Damper in Suspension System
The main function of damper in suspension application is to dissipate energy caused
by the spring. In semi-active suspension, the damper will vary and controllable. Damper
also known as shock absorber. Consist of piston that have small holes around the piston
that will make the fluid flow through, and the fluid itself which is hydraulic fluid.
Figure 2.5 Damper
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2.1.4 Force Actuator in Suspension System
Force actuators is only exist in active suspension system. The main function of force
actuator is to produce force that will control the system. Mostly, force actuator that used
in suspension system is hydraulic actuator, and electromagnetic actuator.
2.1.5 Hydraulic Actuator
Figure 2.6 Hydraulic Actuator
In the suspension system that use hydraulic as the force actuator, the controller will give
the output to the servo which control the valve. The valve will determine whether the
hydraulic will flow through to give a certain force using the cylinder actuator or the
valve will remain keep the hydraulic in the hydraulic pump.
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2.1.6 Electromagnetic Actuator
Figure 2.7 Electromagnetic actuator in suspension system
In the suspension system that use electromagnetic as the force actuator, the controller
will give the output to the magnetic coil which produce electromagnetic. As the figure
above if the magnetic coil not activated there will be no magnetic field produced on the
magnetic coil. If the magnetic coil activated there will be magnetic field produced the
magnetic coil. The magnetic field will produced force to keep the magnetic coil align
together.
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2.1.7 PID Control
One of the most common forms of closed loop control system is a PID (Proportional,
Integral, and Derivative) controller. A PID controller can be found in all areas where
control is implemented. While desired speed is inserted, the error value as the difference
between the desired speed and the actual speed is calculated by the controller as an
attempt to minimize the error in outputs by adjusting the process control inputs.
Figure 2.8 Common Process Block
Inside the controller block, the error is processed with three coefficient which are:
Proportional, Integral, and Derivative, shown in figure 2.9.
Figure 2.9 PID Controller
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Mathematically, PID algorithm can be described by:
() = () + () +
()
0
Where:
u = control signal
e = error value
Kp = proportional gain
Ki = integral gain
Kd = derivative gain
Here below explained each functions for each coefficient:
Proportional Coefficient
Kp is the constant of proportionality, or simply the gain of the amplifier. The larger the
value of this coefficient, the faster the system output will respond to the error. If it is
too high, the system can no directly stop at the targeted reference input, and this
phenomenon is called overshoot.
Integral Coefficient
Ki is implemented when the steady state error still occurs in the system, so as long as
an error exists, the output of Ki will grow with time, until the reference value is reached.
The usage of integral control can also create overshoot phenomenon, especially when
the proportional coefficient is large.
Derivative Coefficient
Kd is a derivative control of the output of the process. In an ideal process, an error must
be corrected as quickly as possible without overshoot, and this can be achieved if there
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is maximum gain at the beginning and at the end reducing the gain as the error
approaches zero.
In real applications, there are several possible ways of how to tune PID control in the
system, such as: manual tuning with basic knowledge from characteristic of coefficient,
Ziegler-Nichols, and auto tuning software. Here below in figure is the example of the
system output based on several different usages of PID control possibilities.
Figure 2.10 System Response Based on Varieties Kp, Ki, Kd
2.1.8 Linear Quadratic Regulation (LQR)
For comparison purpose, the LQR approach will be utilized. LQR is one of the most
popular control approaches normally been used by many researches in controlling the
active suspension system.
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Consider a state variable feedback regulator for the system as
() = ()
K is the state feedback gain matrix.
The optimization procedure consists of determining the control input U, which
minimizes the performance index. The performance index J represents the performance
characteristic requirement as well as the controller input limitation. The optimal
controller of given system is defined as controller design which minimizes the
following performance index.
= 1
2 (
0
+ )
The matrix gain K is represented by:
= 1
The matrix P must satisfy the reduced-matrix Riccati equation
+ 1 + = 0
Then the feedback regulator U
() = (1)()
= ()
2.1.9 MATLAB Simulink
MATLAB (Matrix Laboratory) is a tool for mathematical computation and visualization,
and usually used in all areas of applied mathematics, in education research at
universities, and in the industry. It is also a programming language, and is one of the
easiest programming languages for writing mathematical programs. With MATLAB,
the user can analyse data and develop algorithm, and create models and application.
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The MATLAB has an add-on called Simulink, which is able to deliver a collaborating
graphical situation for modelling, simulating and analysing dynamic systems. In this
thesis, Simulink will be used as the main software to simulate the mathematical
modelling.
2.1.10 V-REP Simulator
V-REP (Virtual Robot Experimentation Platform) is a 3D robot simulator developed by
Coppelia Robotics. The robot simulator has control architecture that is distributed to
each object/model via an embedded script, a plugin node, a remote API client, or a
custom solution. The language of the controller can have several option: C/C++,
Python, Java, Lua, Matlab, Octave, or Urbi.
The application of V-REP including:
1. Simulation of factory automation systems
2. Remote monitoring
3. Hardware control
4. Fast prototyping and verification
5. Safety monitoring
6. Fast algorithm development
7. Robotics related education
8. Product presentation
There are three types of joint in V-REP that can be used to simulate movements in real
life:
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Revolute: This type of joint can create rotational movement in cylindrical shap,
and the joint which is already connected to the center of the cylinder can be
driven as a motor or just acting as a passive joint. The speed can be adjusted
here.
Spherical: This type of joint has spherical movement, this provides rotational
movement in three axes.
Prismatic: This type of joint has linear movement that can generate back and
forth movement.
V-REP provides several calculation modules to support the operation of the elemental
object:
A forward and inverse kinematics module is used in solving kinematics calculation for
various mechanism. The module uses the damped least squares pseudo inverse method.
The inverse kinematics module is particularly helpful when dealing with manipulators.
Dynamic or physics module that allow the dynamic simulation of rigid bodies done
with the Bullet Physics Library.
A path planning module is based on random tree algorithm.
A collision detection module that can occur during one scene and this can allow the
user to calculate minimum distance between two points or bodies. Both these modules
use the method of Oriented Bounding Boxes.
To control the simulation, V-REP uses script methodology, and there two types: main
script and child script. Main script as the main scene and child script is responsible for
scene object.
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2.1 Previous Studies
2.1.1 Modeling, Simulating, and Analyzing an Overhead Crane Using MATLAB
Simulink and V-REP Simulator [5]
The main purpose of this review to modeling the mathematical equation of overhead
crane using MATLAB Simulink and mechanical simulation using V-REP Simulator.
There will be only the methodology that is going to be discussed which are modeling
and simulating methodology.
The modeling methodology which are:
Identify the problem to determine the problem and the behavior of the system.
To know what is the real life condition.
Formulate the problem searching and deriving a suitable mathematical
equation that is as similar as possible to the real life condition.
Develop a model convert the mathematical equation to simulation
The MATLAB Simulink of this review thesis shown in figure 2.11.
Figure 2.11 Overhead Crane Equation Modelling
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The modeling methodology which are:
Select an appropriate design the simulation will be done in numerical
computation simulation software and mechanical simulation. The mechanical
simulation will be conducted to get a better visualization and realistic condition.
Establish the scenario for the simulation consider what input that will be given
and any certain limitation must be defined.
Perform simulation and observe the result.
The V-REP Simulator of this review thesis shown in figure 2.12.
Figure 2.12 Overhead Crane V-REP Simulator
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CHAPTER 3 RESEARCH METHODS
3.1 Simulation Model Methodology
This thesis will be carried out by two main steps. Modeling part and simulation part.
Modeling Part
Identify the system At very beginning of the project, system identification is
important. To know what is the real condition of the system, the stiffness of
suspension car, the mass of body car, the mass of suspension car, the mass of
unspring part, and the damper
Formulate the mathematical equation Deriving a suitable mathematical
equation that is related to the real condition.
Develop a model After finding the right mathematical equation, the next step
is to convert the equation into simulation.
Simulation Part
Select appropriate design The simulation will be done in numerical
computation simulation software which is MATLAB, and control will be
developed into the system. After the system response is achieved, the
mechanical simulation will be conducted to get better visualization of the
system behavior and realistic condition.
Establish simulation scenario Consider what input will be given into the
system and any certain limitation must be defined.
Perform simulation and observe the result.
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3.2 Mathematical Model
Before designing a system model, mathematical equation is needed to be observed. Two
basic method to obtain the equation of suspension system are use the second Newton
Law and Hookes Law. The second Newton Law states that the net force on an object
is equal to the rate of change of its linear momentum p in an inertial reference frame:
=
=
()
The second law can also be stated in terms of an objects acceleration. Since Newtons
second law is only valid for constant-mass system, mass can be taken outside. Thus,
=
=
Where F = Force (N), m = Mass (Kg), and a = Acceleration (m/s2).
And the Hookes Law is a principle of physics that states that the force F needed to
extend or compress a spring by some distance X is proportional to that distance. That
is:
=
Where F = Force (N), k = Stiffness of spring (N/m), and X = displacement of the spring
(m).
First the free body diagram is developed for the passive and active suspension system.
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3.2.1 Mathematical Model of Quarter Car Passive Suspension
The free body diagram of quarter car passive suspension system is shown in figure
3.1.
Figure 3.1 Free Body Diagram Quarter Car Passive Suspension
M1 = Mass of the wheel (Kg)
M2 = Mass of the car body (Kg)
r = Road disturbance
Xw = Wheel displacement (m)
Xs = Card body displacement (m)
Ka = Stiffness of car body spring (N/m)
Kt = Stiffness of tire (N/m)
Ca = Damper (Ns/m)
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From the figure 3.1 obtain mathematical equation using the second Newton Law, where
positive sign for every upward direction and negative sign for every downward
direction.
For M1
= 1
( ) + ( ) + ( ) = 1
=( ) + ( ) + ( )
1
(3.1)
For M2
= 2
( ) ( ) = 2
=
( ) ( )
2
(3.2)
From the equations 3.1 and 3.2 above, let the state variable are:
1 =
2 =
3 =
4 =
where
= Suspension Travel (m)
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= Car Body Velocity (m/s)
= Car Body Acceleration (m/s2)
= Wheel Deflection (m)
= Wheel Velocity (m/s)
= Wheel Acceleration (m/s2)
Therefore in state space equation, equations 3.1 and 3.2 can be written as:
() = () + ()
Where
1 = 2 4
2 =
3 = 4
4 =
Rewrite equations 3.1 and 3.2 into matrix form
[ 1234]
=
[
0 1 0 12
2
02
0 0 0 11
1
1
1 ]
[
1234
] + [
00
10
] (3.3)
State-space matrix equation 3.3 above is developed because the matrix will be given
into the LQR function. The LQR function use state-space matrix to calculate the gain
for the LQR controller output.
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3.2.2 Mathematical Model of Quarter Car Active Suspension
The free body diagram of quarter car active suspension shown below in figure 3.2.
Figure 3.2 Free Body Diagram Quarter Car Active Suspension
M1 = Mass of the wheel (Kg)
M2 = Mass of the car body (Kg)
r = Road disturbance (m)
Ua = Force Actuator (N)
Xw = Wheel displacement (m)
Xs = Card body displacement (m)
Ka = Stiffness of car body spring (N/m)
Kt = Stiffness of tire (N/m)
Ca = Damper (Ns/m)
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From the figure 3.2 obtain mathematical equation, where positive sign for every upward
direction and negative sign for every downward direction.
For M1
= 1
( ) + ( ) + ( ) = 1
=( ) + ( ) + ( )
1
(3.4)
For M2
= 2
( ) ( ) = 2
=
( ) ( ) + 2
(3.5)
From the equations 3.4 and 3.5 above, let the state variable are:
1 =
2 =
3 =
4 =
where
= Suspension Travel (m)
= Car Body Velocity (m/s)
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= Car Body Acceleration (m/s2)
= Wheel Deflection (m)
= Wheel Velocity (m/s)
= Wheel Acceleration (m/s2)
Therefore in state space equation, equations 3.4 and 3.5 can be written as:
() = () + () + ()
Rewrite the equations 3.4 and 3.5 into matrix form
[ 1234]
=
[
0 1 0 12
2
02
0 0 0 11
1
1
1 ]
[
1234
] +
[ 01
201
1]
+ [
00
10
] (3.6)
State-space matrix equation 3.6 above is developed because the matrix will be given
into the LQR function. The LQR function use state-space matrix to calculate the gain
for the LQR controller output.
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3.2.3 Mathematical Model of Full Car Passive Suspension
The free body diagram of full car passive suspension system shown below in figure 3.3.
Figure 3.3 Full Car Passive Suspension Model
In full car model the car body is free to heave roll and pitch. The suspension system
connects the car body to the four wheel which are front-left, front-right, rear-left, and
rear-right. They are free to bounce vertically with respect to the car body.
For rolling motion of the car body
= (1 1) + (2 2) (3 3)
+ (4 4) (1 1) + (2 2)
(3 3) + (4 4)
For pitching motion of the car body
= (1 1) (2 2) + (3 3) + (4 4)
(1 1) (2 2) + (3 3) + (4
4)
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For bouncing of the car body
= (1 1) (2 2) (3 3) (4 4)
(1 1) (2 2) (3 3) (4 4)
And also for each side of wheel motion (vertical direction)
1 = (1 1) + (1 1) 1 + 1
2 = (2 2) + (2 2) 2 + 2
3 = (3 3) + (3 3) 3 + 3
4 = (4 4) + (4 4) 4 + 4
For Zs1
1 = + +
1 = + +
For Zs2
2 = + +
2 = + +
For Zs3
3 = + +
3 = + +
For Zs4
4 = + +
4 = + +
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where
ms = mass of the car body (Kg)
muf and mur = front and rear mass of the wheel (Kg)
Ip and Ir = pitch and roll of moment inertia (Kg m2)
Zs = car body displacement (m)
Zs1, Zs2, Zs3, and Zs4 = car body displacement for each corner (m)
Zu1, Zu2, Zu3, and Zu4 = wheel displacement for each corner (m)
a = distance from center of the car body to front wheel (m)
b = distance from center of the car body to rear wheel (m)
Cf and Cr = front and rear damping (Nm/s)
Kf and Kr = stiffness of front and rear car body spring (N/m)
Ktf and Ktr = stiffness tire (N/m)
The state variables of the system are shown in Table 3.1 and the definition of each state
variable is given in Table 3.2.
Table 3.1 State Space Variable
1 = 8 =
2 = 9 =
3 = 10 =
4 = 1 11 = 1
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5 = 2 12 = 2
6 = 3 13 = 3
7 = 4 14 = 4
Table 3.2 State Space Variable Definition
Variable Definition
Roll angle
Roll rate
Pitch angle
Pitch rate
Vertical displacement
Vertical velocity
1 Vertical displacement of front right wheel
1 Vertical velocity of front right wheel
2 Vertical displacement of front left wheel
2 Vertical velocity of front left wheel
3 Vertical displacement of rear right wheel
3 Vertical velocity of rear right wheel
4 Vertical displacement of rear left wheel
4 Vertical velocity of rear left wheel
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State-space matrix for Full Car Passive Suspension is shown in Appendix C.
3.2.4 Mathematical Model of Full Car Active Suspension
The free body diagram of full active suspension shown in figure 3.4
Figure 3.4 Full Car Active Suspension Model
In full car model the car body is free to heave roll and pitch. The suspension system
connects the car body to the four wheel which are front-left, front-right, rear-left, and
rear-right. They are free to bounce vertically with respect to the car body.
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For rolling motion of the car body
= (1 1) + (2 2) (3 3)
+ (4 4) (1 1) + (2 2)
(3 3) + (4 4) + 1 2 + 3 4
For pitching motion of the car body
= (1 1) (2 2) + (3 3) + (4 4)
(1 1) (2 2) + (3 3)
+ (4 4) + 1 + 2 3 4
For bouncing of the car body
= (1 1) (2 2) (3 3) (4 4)
(1 1) (2 2) (3 3) (4 4)
+ 1 + 2 + 3 + 4
And also for each side of wheel motion (vertical direction)
1 = (1 1) + (1 1) 1 1 + 1
2 = (2 2) + (2 2) 2 2 + 2
3 = (3 3) + (3 3) 3 3 + 3
4 = (4 4) + (4 4) 4 4 + 4
For Zs1
1 = + +
1 = + +
For Zs2
2 = + +
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2 = + +
For Zs3
3 = + +
3 = + +
For Zs4
4 = + +
4 = + +
where
ms = mass of the car body (Kg)
muf and mur = front and rear mass of the wheel (Kg)
Ip and Ir = pitch and roll of moment inertia (Kg m2)
Zs = car body displacement (m)
Zs1, Zs2, Zs3, and Zs4 = car body displacement for each corner (m)
Zu1, Zu2, Zu3, and Zu4 = wheel displacement for each corner (m)
a = distance from center of the car body to front wheel (m)
b = distance from center of the car body to rear wheel (m)
Cf and Cr = front and rear damping (Nm/s)
Kf and Kr = stiffness of front and rear car body spring (N/m)
Ktf and Ktr = stiffness tire (N/m)
U1 and U2 = front right and left force actuator
U3 and U4 = rear right and left force actuator
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The state variables of the system are shown in Table 3.3 and the definition of each state
variable is given in Table 3.4
Table 3.3 State Space Variable
1 = 8 =
2 = 9 =
3 = 10 =
4 = 1 11 = 1
5 = 2 12 = 2
6 = 3 13 = 3
7 = 4 14 = 4
Table 3.4 State Space Variable Definition
Variable Definition
Roll angle
Roll rate
Pitch angle
Pitch rate
Vertical displacement
Vertical velocity
1 Vertical displacement of front right wheel
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1 Vertical velocity of front right wheel
2 Vertical displacement of front left wheel
2 Vertical velocity of front left wheel
3 Vertical displacement of rear right wheel
3 Vertical velocity of rear right wheel
4 Vertical displacement of rear left wheel
4 Vertical velocity of rear left wheel
State-space matrix for Full Car Passive Suspension is shown in Appendix D.
3.3 Control Design
In optimal control, it attempts to find the suitable controller that can provide best
performance for the system. Control design is a very importance part for active
suspension system. The controller will give better compromise between ride comfort
and vehicle handling. Nowadays there a lot of various controller that provided in
suspension system.
The controller generates forces to control output parameters such as body displacement,
wheel displacement, and suspension travel. Take the suspension travel as a feedback
parameter to the controller. It will keep the distance between body and wheel at normal
position which is when there is no disturbance.
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3.3.1 Proposed PID Control Design
Figure 3.5 Control System Overview
The overview of control design shown in figure 3.5, on the control design, the controller
control the suspension travel between car body and wheel. The controller maintain the
suspension travel to keep the distance between them, below in figure 3.6, for more detail
Figure 3.6 Suspension Distance
In figure 3.7 shown the control design in Simulink based on the overview control
design.
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Figure 3.7 Control System in MATLAB Simulink
Suspension travel displacement reference is used as main input to be compared to actual
suspension travel displacement; the output of the suspension travel displacement
control will give displacement reference that will be fed to displacement to force
converter. The converter will give force output to the suspension system. There will be
a road disturbance given to the suspension system. It will affect the suspension travel
displacement.
3.3.2 Linear Quadratic Regulator
The LQR approach of vehicle suspension control is widely used in background of many
studies in vehicle suspension control. It has been use in a simple quarter car model,
half-car, and also in full car model. The strength of LQR approach is that in using it the
factors of the performance index can be weighted according to the designers desires or
other constraints.
In this thesis, the Q and R value will be determined by using trial and error until the
system achieve its best performance. Using function lqr() in MATLAB, the Q and R
will produce the K value. The K value will be the new value of the state space variable.
In this thesis, Q is 4 x 4 identity matrix and R is a single value.
= [
1000 0 0 00 1000 0 00 0 1000 00 0 0 1000
]
= 0.03
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State space formed by A, B, C, and D matrix. Rewrite these matrix and their relationship
as:
() = () + ()
() = () + ()
Matrix A is the main suspension system matrix, and relates how the current state affects
the state change . Matrix B is the control matrix, and determines how the system input
affects the state change. Matrix C is the output matrix, and determines the relationship
between the system state and the system output. Matrix D is the feed-forward matrix,
in this thesis the system is not use feed-forward controller which means matrix D is
zero matrix.
After determine the Matrix A, B, C, and D, MATLAB will calculate the gain of matrix
K using lqr() function. Then the matrix K will produce new value of matrix A:
= ( )
Matrix B, C, and D will be the same as previous value. Then the new state space will
be constructed with the new value of matrix.
3.4 Mechanical Part
Type of suspension system that is taken as main design on this thesis is double wishbone
suspension which commonly used by every car and it is more stable then single
wishbone. The design is completely designed using V-REP Simulator.
The main frame is created by cuboid shape, the wheel is created by cylindrical shape,
the joint is created by revolute joint which can generate force in rotational direction,
and the suspension is created by prismatic joint which can generate force in translational
direction.
The overall V-REP model shown in figure 3.8
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Figure 3.8 Overall V-REP Model
The model consist of four wheels, 4 suspensions and the main chassis. All designed
using cuboid, sphere and cylindrical in V-REP Simulator.
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Figure 3.9 Double Wishbone Joint Type
In figure 3.9 shown the wheel is connected to the chassis by double wishbone joint type
which have two arm, lower and upper arm.
Figure 3.10 Adjusting Suspension Parameter
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In figure 3.10 shown that the lower arm connected to the chassis by suspension. The
stiffness suspension (K) and damper coefficient (C) can be adjust in the joint dynamic
properties. The joint can act like a suspension which contain spring and damper.
Figure 3.11 Revolute Joint from Wheel to Arm
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Figure 3.12 Revolute Joint from Arm to Chassis
The lower and upper arm connected to the wheel using revolute joint, shown in figure
3.11. The lower and upper arm also connected to the chassis using revolute joint, shown
in figure 3.12.
3.4.1 Model Parameter
Below in Table 3.5, shown the parameter of the car and the suspension system.
Table 3.5 V-REP Model Parameter
Parameter Value
Distance from center to front (m) 0.3104 m
Distance from center to rear (m) 0.3104 m
Front suspension stiffness (N/m) 36927 N/m
Rear suspension stiffness (N/m) 36927 N/m
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Front damper coefficient (N/ms) 4000 N/ms
Rear damper coefficient (N/ms) 4000 N/ms
Mass of the chassis (Kg) 500 Kg
Mass of the wheel (Kg) 60 Kg
Distance from center to the right (m) 0.1527 m
Distance from center to the left (m) 0.1527 m
3.4.2 Road Disturbance Environment
There are two different type of road disturbance, impulse and step input. The impulse
input is act like speed bump. In figur3 3.13 shown the speed bump created by V-REP
Simulator using cylindrical shape.
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Figure 3.13 Speed Bump Model
For the step input can be seen in figure 3.14. The car will go through the speed bump
and obtain the suspension travel, wheel displacement and car body displacement.
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Figure 3.14 Step Input Road Disturbance
The car will go through the step input road disturbance and obtain the suspension travel,
wheel displacement, and car body displacement. Those data will be shown in graph in
V-REP. Below in figure 3.15 is the example of using graph in V-REP.
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CHAPTER 4 RESULTS AND DISCUSSIONS
4.1 System Model Result
The complete system model that is done can be shown in figure 4.1. The system model
is divided into two parts: passive suspension, and active suspension. Adjustable
parameter are located on the outer side of block.
In passive suspension systems block in figure 4.1. Both equation 4.1 and 4.2 are
generated to get desired output which is suspension travel, wheel displacement, and car
body displacement.
=
( ) + ( ) + ( )
1 (4.1)
=( ) ( )
2 (4.2)
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4.1.1 Quarter Car Suspension System
Figure 4.1 Quarter Car Simulink Block
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Figure 4.2 Quarter Car Passive Suspension Simulink Block
Figure 4.3 Quarter Car Active Suspension Simulink Block
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4.1.1.1 Passive Suspension System Response Analysis
4.1.1.1.1 Impulse Input
Below are the parameters that are assumed:
Mass of car body (M1) = 290 Kg
Mass of wheel (M2) = 60 Kg
Damper (Ca) = 1000 Ns/m
Stiffness of car body spring (Ka) = 16812 N/m
Stiffness of tire (Kt) = 190000 N/m
The first experiment is to give an impulse road input 0.3 m at t = 1 s, can be seen in
figure 4.4.
Figure 4.4 Impulse Input Road Disturbance
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The system response shown in figure 4.5 is showing the movement of the car body due
to impulse input. It occurs oscillating movement to the car body and create
uncomfortable movement for the passenger.
Figure 4.5 Car Body Displacement Due to Impulse Input
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In the figure 4.6, it can be seen the suspension travel due to impulse input. The
suspension travel calculate the travel of the car body minus by wheel displacement. The
negative value means that the wheel displacement act first due to the input, and after
that the car body move because of the suspension create force due to the displacement
of the wheel to the car body.
Figure 4.6 Suspension Travel Due to Impulse Input
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In figure 4.7, it can be seen that the wheel displacement occur due to impulse input.
Figure 4.7 Wheel Displacement Due to Impulse Input
4.1.1.1.2 Step Input
The parameters are still the same with the previous experiment using impulse input.
The second experiment is to give a step road input = 0.3 m at t = 1 s, can be seen in
figure 4.8.
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Figure 4.8 Step Input Road Disturbance
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In figure 4.9, it can be seen that the suspension travel occur due to impulse input. The
suspension travel calculate the travel of the car body minus by wheel displacement. The
negative value means that the wheel displacement act first due to the input, and after
that the car body move because of the suspension create force due to the displacement
of the wheel to the car body.
Figure 4.9 Suspension Travel Due to Step Input
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In figure 4.10, it can be seen that the car body displacement occur due to impulse input.
The car body displacement is oscilating
Figure 4.10 Car Body Displacement Due to Step Input
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In figure 4.11, it can be seen that the wheel displacement occur due to impulse input.
Figure 4.11 Wheel Displacement Due to Step Input
In figure 4.9, 4.10, and 4.11 shown that the suspension is not stable because the
suspension produced oscillating output due to the road disturbance which are step input
and impulse input. The suspension need controller to make it more stable.
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4.1.2 Full Car Suspension System
Figure 4.12 Full Car Suspension System
Table 4.1 List of Parameter for Full Car Suspension System
Parameter Value
Front distance from center to right and left (Tf) 0.1527 m
Rear distance from center to right and left (Tr) 0.1527 m
Front Damper Coefficient 4000 Ns/m
Rear Damper Coefficient 4000 Ns/m
Front Spring Stiffness 36927 N/m
Rear Spring Stiffness 36927 N/m
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Front Tire Stiffness 190000 N/m
Rear Tire Stiffness 190000 N/m
Distance from center to front 0.3104 m
Distance from center to rear 0.3104 m
Mass of front wheel 60 Kg
Mass of rear wheel 60 Kg
Mass of the car body 500 Kg
The full car suspension consist of three type of motion, pitching, rolling and bouncing.
For each motion has its own block diagram in Simulink.
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The pitching block diagram shown in figure 4.13
Figure 4.13 Pitching Block Diagram
This pitching motion give output in theta which is the pitching angle.
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The bouncing motion block diagram shown in figure 4.14
Figure 4.14 Bouncing Block Diagram
This bouncing motion give output in car body displacement.
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The rolling motion block diagram shown in figure 4.15
Figure 4.15 Rolling Block Diagram
This rolling motion give output in gamma which is the rolling angle.
Gamma, theta, and car body displacement are all combined together for each
suspension. Each suspension affected by those output. In figure 4.16 below, shown the
input that affected one of the suspension.
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Figure 4.16 Theta, Gamma, Body Displacement for Each Suspension
4.1.2.1 Passive Suspension System Response Analysis
The output that reported only the front right suspension.
Below are the parameters that are assumed:
Mass of car body (Ms) = 1160 Kg
Mass of wheel (Mu) = 60 Kg
Damper (Ca) = 1000 Ns/m
Stiffness of car body spring (Ka) = 16812 N/m
Stiffness of tire (Kt) = 190000 N/m
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4.1.2.1.1 Impulse Input
The experiment is to give an impulse road input 0.3 m to the front right suspension, can
be seen in figure 4.17
Figure 4.17 Impulse Input Road Disturbance
Figure 4.18 Car Body Displacement of Front Right Suspension due to Impulse Input
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In figure 4.18 above shown that the car body displacement of front right suspension due
to impulse input. The car body displacement oscillates which makes uncomfortable
driving.
4.1.2.1.2 Step Input
The experiment is to give an step road input 0.3 m to the front right suspension, can be
seen in figure 4.19
Figure 4.19 Step Input Road Disturbance
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Figure 4.20 Car Body Displacement of Front Right Suspension due to Step Input
In figure 4.20 above shown that the car body displacement of front right suspension due
to step input. The car body displacement oscillates which makes uncomfortable driving.
4.3 Control Analysis
To prevent the oscillation of the car body due to road disturbance, controller is required.
There are two different types of controllers as previously discussed in chapter 3, PID
controller and linear quadratic regulation. Comparison is done in this chapter to see
which method is more effective.
4.3.1 PID Controller for Quarter Car Suspension
The system is needed to be controlled. The system of the suspension system input is
reference distance between car body and wheel, and the output is force which will be
produced by the force generator. The PID controller for suspension system transfer
function can be formed into block diagram below in figure 4.21:
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Figure 4.21 PID Controller
The PID controller control the suspension travel. The output from the PID controller is
in meter unit. In the controller there is a converter from displacement to force. The
displacement derevatived by twice convert it into acceleration. The acceleration
multiply by the mass of car and wheel and make it into force unit. The force fed into
the suspension. The closed loop system can be shown below in figure 4.22.
Figure 4.22 Closed-loop System of Suspension System
The Kp, and Ki determined by trial and error. The trial and error been done until the
suspension achieve better performance which the suspension travel has small
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oscillation. The car body displacement also achieve its better performance than passive
suspension as suspension travel.
After the controller is developed, this controller is integrated with the system.
4.3.2 PID Controller for Full Car Suspension
The difference between PID controller for full car suspension and quarter car
suspension is, full car suspension have three motion as mentioned in the previous
chapter which are bouncing, rolling and pitching. Each motion affected by the
suspension of each corner of the car. The force that produced by the PID controller fed
to the bouncing, rolling and pitching system. The reference is the same which is the
suspension travel of each suspension. Each suspension produced its own controlled
force.
Figure 4.23 Control System for Each Suspension
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In figure 4.23 above shown the control system of full car suspension. Inside of the
control system box can be shown in figure. Each suspension has its own PID controller
which produce force with reference of its suspension travel.
Figure 4.24 Inside Control System Block
Then the value of force applied to the bouncing, rolling and pitching system.
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4.3.3 LQR Controller for Quarter Car Suspension
Figure 4.25 Flowchart for LQR Controller using MATLAB Code
In figure 4.25 shown the flowchart of the LQR Controller. LQR controller for quarter
car suspension is developed using MATLAB code. The code can be shown in Appendix
A.
4.3.4 LQR Controller for Full Car Suspension
The flowchart is the same with the flowchart LQR Controller for Quarter Car
Suspension, the flowchart can be seen in figure 4.25. The LQR controller for full car
suspension is developed using MATLAB code. The code can be shown in Appendix B.
4.4 Quarter Car Passive and Active Suspension Comparison
4.4.1 Comparison with PID Controller for Step Input
After using manual tuning for the PID Controller, the better performance achieved
when the Kp = 0.01, and Ki = 4.
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Figure 4.26 Comparison Suspension Travel with PID Controller due to Step Input
Figure 4.27 Comparison Car Body Displacement with PID Controller due to Step Input
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Figure 4.28 Comparison Wheel Displacement with PID Controller due to Step Input
Overall from figure 4.26, 4.27, and 4.28 shown that the PID controller makes the
suspension travel, wheel displacement and car body displacement smooth movement
than the passive suspension which produced oscillating movement due to step input.
The comfortable driving achieved with this controller.
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4.4.2 Comparison with PID Controller for Impulse Input
Figure 4.29 Comparison Wheel Displacement with PID Controller due to Impulse Input
Figure 4.30 Comparison Car Body Displacement with PID Controller due to Impulse
Input
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Figure 4.31 Comparison Suspension Travel with PID Controller due to Impulse Input
Overall from figure 4.29, 4.30, and 4.31 shown that the PID controller makes the
suspension travel, wheel displacement and car body displacement smooth movement
than the passive suspension which produced oscillating movement due to impulse input.
The comfortable driving achieved with this controller.