SIMPLIFIED ALGORITHM FOR A CONSTANT SPEED MISSILE
ABDUL JAAFAR BIN ABDUL HALIM
A project report submitted in partial fulfillment of the
requirements for the award of the degree of Master
of Engineering Electrical-Electronics
and Telecommunication
(Electronic Warfare)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iv
To my honourable lecturer and supervisor, Dr Shahrum Shah Bin Abdullah,
My beloved mother, Fatimah Binti Latiff,
My beloved wife, Nor Afizah Binti Sukiman,
My beautiful daughter, Nurul Nafeesa Qisteena and Nurul Nadia Aneesa,
Major Shariffuddin (RSD),
My MELA1CKA collegues, Maj Shahbani (TUDM), Maj Saiful Nubli (TUDM),
Maj Amir Rudin (RSD), Maj Mohd Fauzan (RSD), Lt Rahmat (TLDM)
and Lt Sabri (TLDM)
Thank you for all your motivation and advise.
v
ACKNOWLEDGEMENTS
In preparing this thesis, I was in contact with many people and academicians.
They have contributed towards my understanding and thoughts. In particular, I wish
to express my sincere appreciation to my main thesis supervisor, Dr Shahrum Shah
Bin Abdullah for his encouragement, guidance, critics, advice, motivation and
friendship during the development and completion of the thesis. Without his support
and interest, this thesis would not have been the same as presented here.
My sincere appreciations also extend to all my colleagues and others who
have provided assistance at various occasions. Their views and tips are useful indeed.
Unfortunately, it is not possible to list all of them in this limited space. I am grateful
to all my family members. I would like to thank my computer for not giving me any
serious problems during the writing of this thesis. I would like to thank and say a
prayer to God For His Divine help.
vi
ABSTRACT
Disputes and unstable status of todays world must be tackled wisely and as
those irresponsible parties may decide to use missile as their option weapon. This
matter should be in top priority as the number of the terrorist attacks keep increasing
from time to time and the technology knowledge of terrorist should not be look down
upon. A suitable missile defense system for the Malaysian Army is very vital for the
success of a mission. In this project, the development process of this system is
customized to suit the Malaysian Army’s needs and its framework is suggested to be
used for all the development projects that will be involving the military in Malaysia.
The missile technology is still young in this region and it is a very vital technology
for any countries to develop themselves for their own defense purposes. In this
century, the sources of threat from other countries and terrorist groups towards our
country have increased tremendously. The objective of the project is to realized the
movement of a constant speed missile in a 2D simulation form. The simulation was
performed using MATLAB 6.5 software with certain assumption and constraints
such as the location of the enemy/target is stationary, no environmental factor such
as gravitational, aerodynamics or propulsion of the missile to be considered and the
missile itself has a constant speed and have a limited turning point. This thesis
hopefully will initiate a basic concept in guiding a missile toward its target. The
project was developed generally and it can be extends in terms of problem and
application other investigation for future works.
vii
ABSTRAK
Pergolakan di sekitar dunia pada masa kini mesti diberi perhatian yang
sewajarnya dalam menghadapi sebarang kemungkinan serangan dan ancaman peluru
berpandu. Perkara ini dipandang serius dengan peningkatan jumlah pergerakan yang
telah dilakukan oleh pihak penganas dan tahap teknologi yang diperolehi oleh
mereka. Satu sistem pertahanan peluru berpandu untuk Tentera Darat Malaysia
(TDM) adalah amat penting bagi membolehkan satu misi mencapai objektifnya.
Kajian projek ini telah membangunkan satu sistem pertahanan peluru berpandu yang
sesuai untuk digunakan di dalam Malaysia. Sistem pertahanan peluru berpandu ini
dibangunkan dengan menggunakan kesemua proses pembangunan produk yang
dicadangkan untuk kegunaan TDM. Teknologi peluru berpandu ini masih boleh
diterokai dengan lebih mendalam kerana ianya merupakan satu proses pertahanan
yang kritikal dan penting di dalam pertahanan Malaysia. Dalam abad ini, punca
ancaman daripada negara lain dan juga kumpulan pengganas terhadap negara kita
telah meningkat secara mendadak. Salah satu daripada perkara yang telah meningkat
kepentingannya secara global adalah sistem pertahanan peluru berpandu. Objektif
projek ini adalah untuk merealisasikan pergerakan sebuah peluru berpandu dengan
pecutan tetap dalam bentuk simulasi 2D. Simulasi ini dilakukan dengan
menggunakan perisian MATLAB 6.5 dengan beberapa anggapan dan kekangan
seperti lokasi musuh/sasaran adalah tetap, tidak mengambilkira faktor semulajadi
seperti graviti, aerodinamik atau daya tujahan bahan api peluru berpandu serta ianya
mempunyai pecutan yang tetap dan kebolehan membelok yang terhad. Tesis ini
diharapkan dapat memulakan konsep asas dalam pengenalan peluru berpandu ke
sasaran yang dikehendaki. Projek ini dibangunkan secara am dan ianya boleh
dikembangkan dalam bentuk permasalahan dan aplikasi pada masa akan datang.
viii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iv
ACKNOWLEDGEMENT
v
ABSTRACT
vi
ABSTRAK
vii
TABLE OF CONTENT
viii
LIST OF TABLES
xi
LIST OF FIGURES
xii
INTRODUCTION
1
1.1
Introduction
1
1.2
Algorithm Definition
1
1.3
Electronic Warfare Definition
2
1.3.1
Electronic Support (ES)
2
1.3.2
Electronic Attack (EA)
2
1.3.3
Electronic Protect (EP)
3
1.4
Objective
3
1.5
Scope Of Work
3
1.6
Outline Of The Thesis
4
ix
2
LITERATURE REVIEW
6
2.1
Introduction To Guided Weapon
6
2.2
Overview Of Existing Path Planning Technique
7
2.2.1
7
2.3
Close Loop System
12
2.4
Homing Guidance
12
2.5
Basic Trajectory Analysis
13
2.5.1
Kinematics
14
2.5.2
Turn Rate And Lateral Acceleration
14
2.6
3
4
Kinematics/ Relative Geometry
The Concept
15
METHODOLOGY
18
3.1
Introduction
18
3.2
Problem Depiction
18
3.3
The Execution of Geometrical Analysis
21
3.4
The Execution of MATLAB Simulation
24
3.5
Conclusion
26
RESULTS
27
4.1
Introduction
27
4.2
Result Of The Geometrical Analysis
27
4.3
Result Of The MATLAB Simulation
30
4.4
Comparison Of Geometrical Analysis
And MATLAB Simulation
4.5
33
Result by Varying The Parameters in
MATLAB Simulation
33
4.5.1
Change The Initial Direction To 0 Degree
34
4.5.2
Change The Initial Direction To
180 Degree
4.5.3
36
Change The Initial Direction To
270 Degree
38
x
4.5.4
Change The Initial Direction To
45 Degree
40
4.5.5 Change The Initial Position (-2 -2)
42
4.5.6
Change The Initial Distance 10 cm
44
4.5.7
Change The Initial Distance 5 cm
46
4.5.8
Change The Steering Constant
-15 Degree
4.5.9
48
Change The Steering Constant
-30 Degree
4.5.10 Change The Steering Constant 20 Degree
50
52
4.5.11 Change The Steering Constant 20 Degree
5
And The Target Angle To 140 Degree
54
4.5.12 Change The Target Angle 140 Degree
56
4.5.13 Change The Velocity Constant 0.5 cm/s
58
4.5.14 Change The Velocity Constant 4 cm/s
60
CONCLUSION
62
5.1
Introduction
62
5.2
Conclusion
62
5.3
Future Development
63
REFERENCES
64
xi
LIST OF TABLES
TABLE NO
TITLE
PAGES
3.1
Determined parameters for initial work
20
3.2
Definition of parameters used in MATLAB
25
4.1
The result of calculation with mathematical equation
28
4.2
MATLAB simulation result
33
4.3
Result comparison using geometrical analysis and
MATLAB simulation
4.4
34
The changes in parameters when initial direction= 00
36
4.5
0
The changes in parameters when initial direction= 180
38
4.6
The changes in parameters when initial direction= 2700
40
4.7
The changes in parameters when initial direction= 450
42
4.8
The changes in parameters when initial position (-2,-2)
44
4.9
The changes in parameters when target distances 10 cm
46
4.10
The changes in parameters when target distances 5 cm
48
4.11
The changes in parameters when steer ability -15 degree
50
4.12
The changes in parameters when steer ability -30 degree
52
4.13
The changes in parameters when steer ability 20 degree
54
4.14
The changes in parameters when steer ability 20 degree
and target angle 140 degree
56
4.15
The changes in parameters when target angle 140 degree
58
4.16
The changes in parameters when velocity
constant 0.5 cm/s
4.17
60
The changes in parameters when velocity
constant 4 cm/s
62
xii
LIST OF FIGURES
FIGURE NO
TITLE
PAGES
1.1
Flowchart represent the scope of work
4
2.1
List of the wavelength main uses in guided weapon
7
2.2
Pursuer/Target scenario contains in the change in
time of the vector from the pursuer to the target
8
2.3
Kinematics equations
10
2.4
Guidance loop
11
2.5
The close loop system
12
2.6
Basic trajectory analysis
13
2.7
Basic idea in forming the algorithm
16
2.8
Enhancement of parameter in Figure 2.7
16
3.1
Illustration of the original problem
19
3.2
The visual depiction of the original problem
19
3.3
The visual depiction in plotting execution
21
3.4
The visual depiction in MATLAB simulation execution
24
3.5
Flowchart of the MATLAB program
26
4.1
MATLAB program used in order to guide the missile
to its target
32
4.2
Missile movement using MATLAB simulation
32
4.3
Missile simulation when initial direction is change
to 0 degree
4.4
Missile simulation when initial direction is change
to 180 degree
4.5
35
37
Missile simulation when initial direction is change
to 270 degree
39
xiii
4.6
Missile simulation when initial direction is change
to 45 degree
4.7
41
Missile simulation when changes initial
position (-2-,2)
43
4.8
Missile simulation when target distance 10 cm
45
4.9
Missile simulation when target distance 5 cm
47
4.10
Missile simulation when changes steer
ability -15 degree
4.11
Missile simulation when changes steer
ability -30 degree
4.12
51
Missile simulation when changes steer
ability 20 degree
4.13
49
53
Missile simulation when change steer ability 20 degree
and target angle 140 degree
55
4.14
Missile simulation when target angle 140 degree
57
4.15
Missile simulation when velocity constant 0.5 cm/s
59
4.16
Missile simulation when velocity constant 4 cm/s
61
CHAPTER 1
INTRODUCTION
1.1
Introduction
This thesis was written for the purpose in presenting an algorithm that can
define a movement of a constant speed missile until it reach its desired target or
location. This will involve some of the navigation, guidance and control processing
topic which are the guide line in developing the algorithm. In electronic warfare
(EW) system, the determination a location of a target location is one of the
fundamental function.
1.2
Algorithm Definition
Algorithm is a finite set of well define instruction in accomplishing a task
with the given initial state until it reach it end state. There are various way in
classifying an algorithm. In terms of implementation, an algorithm can be define in
mean of recursion/iteration, logical, serial/parallel, deterministic/non deterministic,
and exact/approximate. Sometimes it can also be classified by design paradigm such
as devide and conquer, dynamic programming, greedy method, linear programming,
reduction, search and enumeration and probabilistic and heuristic paradigm [1].
2
1.3
Electronic Warfare Definition
EW is military action to exploit the electromagnetic spectrum which
encompasses the interception, identification and location of electromagnetic
emissions. The employment of electromagnetic energy to reduce or prevent hostile
use of the electromagnetic spectrum and actions to ensure its effective use by
friendly forces [2]. Electronic warfare has three main omponents: electronic support,
electronic attack, and electronic protection.
1.3.1
Electronic Support (ES)
Electronic support (ES) is the passive use of the electromagnetic spectrum to
gain intelligence about other parties on the battlefield in order to find, identify, locate
and intercept potential threats or targets. This intelligence, known as ELINT, might
be used directly by fire-control systems for artillery or air strike orders, for
mobilization of friendly forces to a specific location or objective on the battlefield, or
as the basis of electronic attack or electronic protection actions. Because ES is
conducted passively, it can be performed without the enemy ever knowing it. Its
counterpart, SIGINT, is continuously performed by most of the world's countries in
order to gain intelligence derived from other parties' electronic equipment and
tactics. An older term for ES is electronic support measures (ESM)[3].
1.3.2
Electronic attack (EA)
Electronic attack (EA) is the active or passive use of the electromagnetic
spectrum to deny its use by an adversary.Active EA includes such activities as
jamming, deception, active cancellation, and EMP use. Passive EA includes such
activities as the use of chaff, towed decoys, balloons, radar reflectors, winged
decoys, and stealth. EA operations can be detected by an adversary due to their
active transmissions. Many modern EA techniques are considered to be highly
classified. An older term for EA is electronic countermeasures (ECM)[3].
3
1.3.3
Electronic Protection (EP)
Active EP includes such activities as technical modifications to radio
equipment (such as frequency-hopping spread spectrum). Passive EP includes such
activities as the education of operators (enforcing strict discipline) and modified
battlefield tactics or operations. Older terms for EP include electronic protective
measures (EPM) and electronic counter countermeasures (ECCM)[3].
1.4
Objective
The main objective of this project is to realize the movement of a missile
towards the desired destination in a simulation form. In order to accomplish the
project, the following sub-objectives are necessary to be achieved before hand.
1.4.1
To implement and possibly improve the existing mathematical model of path
planning for constant speed missile.
1.4.2
To develop simulations for the movement of a constant speed object
representing a constant speed missile
1.4.3
To implement existing guidance algorithm that allowing the constant speed
missile reaching the desired destination from its initial position.
1.5 Scope of Work
The scope of work is to clearly define the specific field of the research and
ensure that the entire content of this thesis is confined to the scope. It is begins with
the implementation or possibly an improved implementation of existing
mathematical model on a missile for the following specifications such as it has
constant speed at all times with velocity of v(t) and limited swerving ability of α(t).
4
The next step is to model a simulation to represent the missile as per the
specifications. Then the missile is visualized in a simulation to move from its initial
position to the desired destination. Finally, the validation of the approach
effectiveness is carried out in between the targeted and simulation destination. The
scope of work can be described in terms of flowchart as per the following Figure
1.1.
Literature Review
Mathematical equation for the
implementation of constant turning
ability, α (t )
Mathematical equation for the
implementation of constant
speed v(t)
Guidance algorithm for the
control mechanism
Integrated mathematical modeling of path planning
for constant speed missile
Simulation analysis using MATLAB the movement of robot
from its initial position to the desired destinattion
Comparison of geometrical analysis and simulation
Figure 1.1 : Flowchart represent the scope of work
1.6
Outline Of The Thesis
Chapter 1 in this thesis basically an introduction of all the topics and
discussion which involve in developing the algorithm. It seem to be the overview of
all the subject in the thesis.
Chapter 2 is the literature review which introduces the overview of the
existing algorithm and missile research. The explanation begins with the related
existing work which is found to be related to this project. This chapter is then
described the existing missile system implementation and current interest in the
missile research.
5
Chapter 3 provides the method that is used of this project. It covers the
technical explanation of this project and the implemented technique of mathematical
equations and guidance algorithm for both plotting and simulation.
Chapter 4 deals with the results of the geometrical analysis and MATLAB’s
simulation. The multi-conditions of the execution for MATLAB’s simulation will
also be presented in this topic.
Chapter 5 presents the conclusions of the project as well as some constructive
suggestions for further development and the contribution of this project. As for future
development, some suggestions are highlighted with the basis of the limitation of the
effectiveness mathematical equation and simulation analysis executed in this project.
The aim of the suggestions is no other than the improvement of the study.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction To Guided Weapon
A guided weapon system may be described briefly as a weapon system in
which the warhead is delivered by an unmanned guided vehicle[4]. The word missile
comes from the Latin verb mittere, meaning ‘to send’[5]. Guided weapon is a
weapon system in which the warhead is delivered by an unmanned guided vehicle
[6]. Guided Missile System is a guidance package that attempts to keep the missile
on a course to intercept the target [7].
Sensor
Ultra Violet
SAGW
Visible
SAGW and ATGW
Wavelength
<0.4µm
0.4-0.7µm
guidance
weather
Eyeball surveillance
0.7-1.5µm
Mid IR SAGW
3-5µm
ATGW and SAGW
Only for very short range fuzing or
Target and missile tracking in fair
Near IR SAGW
Far IR
Use
8-14µm
Tail chasing IR homing missiles
Forward hitting IR homing missiles
Imaging homing heads surveillance
Passive detection of aircraft
Lasers ATGW and SAGW
Milimetric Wave ATGW
1-10mm
Active and passive homing
7
Centrimetric Wave Radar
SAGW
Long Wave Radar
Lasers
ATGW and SAGW
Target and missile trackers
1-10cm
Homing head for large missiles
Medium range surveillance
>10cm
Getting too big for Tactical Battle
Narrow Beam Transmitter in IR or
<15µm
optical region fuzing, tracking and
beam riding.
Figure 2.1. List of wavelength main uses in guided weapon [8]
2.2
Overview of Existing Path Planning Technique
In the overview of path planning technique in previous works, we will begin
with the kinematics geometry introduced by Ching Fang Lin [9]. The other path
planning methods will be discussed through out this sub topics.
2.2.1
Kinematics/Relative Geometry
By assuming a pursuer can be treated as a geometric point as in Figure 2.2
and the corresponding guidance is ideal, the character of the pursuer trajectory can be
estimated. The kinematics equation of a pursuer/target scenario contains in the
change in time of the vector from the pursuer to the target. The required accelerations
may be estimated.
The kinematics are characterized by two pairs of kinematics equations in
other practical applications. One of them determine the relative motion of the centers
of mass of pursuer and target in the horizontal plane. The other one will determine
this motion in the vertical plane.
8
Figure 2.1 : Relative Geometry and Parameters for Intercept Guidance [10]
Figure 2.2 Pursuer/target scenario contains in the change in time of the vector
from the pursuer to the target [9]
The geometry of the homing guidance in the horizontal plane is described by
the following relations:
R = VΤ cos(σ true − γ Τ ) − (Vm cos(σ true − γ Τ )
(2.1a)
Rσ true = −VΤ sin(σ true − γ Τ ) − (Vm sin(σ true − γ Τ )
(2.1b)
where, the subscript true in σtrue
is omitted for the next consideration. The
nomenclature is displayed in Figure 2.1 and the variables are:
R = ( x r − γ r )1 / 2
2
2
σ = tan −1
γr
xr
(2.2)
&
The closing velocity Vc =−R of the pursuer and target is determined by
Equation (2.1(a)), while the line-of-sight (LOS) rate σ& is related to the parameters of
the motion the pursuer and target by Equation (2.1(b)). The LOS is defined as an
imaginary straight line from an observer’s eyes to a target [11]. A set of analogous
equations is used for the vertical plane.
9
Differentiation of Equation (2.1(b)) and substitution of Equation (2.1(a))
yield,
Rσ + 2 Rσ = −VΤ sin(σ − γ Τ ) + VΤ cos(σ − γ Τ )γ Τ
− V&m sin(σ − γ ) + VΤ cos(σ − γ )γ&
(2.3)
Because of kinematics Equation (2.1) and (2.3) are nonlinear differential
equations, they render it a difficult process to study and analyze guidance loops. The
analysis can be facilitated by linearizing the kinematics equations relative to the
reference trajectory of the pursuer, using approximation methods. With initial values
given by γ
To
and γ , the angles γT and γ are perturbed during the engagement by γt
0
and γm where γt and γm are very small compared to γTo and γ0 , respectively.
Consequently, the instantaneous angles of the velocity vectors are γT = γT + λt and γ =
γ0 + γm . A particularly simple case, which is of course , linear as well, occurs when
sin σ ≅ σ , cos σ ≅ 1, sin γt ≅ γt , cos γt ≅ 1, sin γm ≅ γ
m
, and cos γm ≅ 1. The LOS
angle in Figure 2.1 can be approximated as :
σ≅
γr
R
≅
γΤ −γm
Vc t g
=
γr
(2.4a)
Vc t g
where
tg= tf − t = time to go and
tf= flight time
For analysis purposes, assuming the intercept geometry is simplified to a
head-on engagement with constant closing velocity, the range is determined from
R ≅ Ro − Vc t ≅ Vc (t f − t ) = Vc t g
(2.4b)
where R =V t .
0
c
f
10
The relative velocity and acceleration are
γ&r = γ&Τ − γ&m = VΤγ t cos γ Τ − Vmγ m cos γ 0
(2.4c)
γ&&r = γ&&Τ − γ&&m = AΤγ − Amγ ≅ VΤ γ&Τ − −Vm γ&
(2.4d)
0
where, ATγ= γ&&Τ = actual target lateral acceleration ≈ VT γ& Τ cos γTo ≈ VT γ&Τ
Amγ= γ&&Τ = actual target lateral acceleration ≈ Vm γ& cos γ o ≈ Vm γ&
(2.4e)
Equations (2.4) are linear equations representing the dynamics of the engagement.
The signal that result from subtracting Amγ from the simulated ATγ is integrated twice
to generate yr . With this information, Equation (2.4a) can then be used to obtain the
true LOS angle σtrue . Assuming VT and Vm are constant, and hence Vc is constant,
differentiating Equation (2.4a) leads to the following formula for the LOS rate σ& :
σ& = Vc γ r / R 2 + γ&r / R
or
Vcσ& = γ r / t g + γ& r / t g
2
(2.5)
Rewriting the equation with the help of the geometry yields
γ&r = Vc t g σ& − Vcσ = γ&Τ − γ&m = VΤγ t cos γ t − Vmγ m cos γ o
o
or in block diagram form shown in the following Figure 2.3.
Figure 2.3 : Kinematics equations[12]
(2.6)
11
Differentiating Equation (2.6) yields
γ&&r = Vct gσ&& − 2Vcσ& = AΤγ − Amγ = VΤγ&t cos γ t − Vmγ& cos γ o
o
(2.7a)
or in more general form,
γ&&r = Rσ&& − 2Vcσ& = AΤγ − Amγ
(2.7b)
By applying the assumption made so far, Equation (2.3) can be reduced to Equation
(2.7b) , while Equation (2.6) can be derived from Equation (2.1). The component of
the system for γ&&r to σ& , based on Equation (2.7a) can be modeled with an integrator
contained in the positive feedback loop as shown in the following Figure 2.4.
Figure 2.4 : Guidance loop [12]
The kinematics element however become s unstable on account of the presence of
positive feedback. That is the output signal σ& grows continuously for a constant
input γ&&r . A negative feedback ( Amγ ) loop is consequently added to the model in
order to counter the effects of the positive feedback. The Guidance Law, Flight
Control System (FCS) and the pursuer dynamics and propulsion are all contained in
this loop. From Equation (2.8), AΤγ is given by
AΤγ = ( R s − 2Vc )σ& + Amγ = Vc (t g s − 2)σ& + Amγ
(2.8)
12
2.3
Close Loop System
Most of the guidance system for missiles is using a Close Loop System or
concept as shown in Figure 2.5. There is some observation instrumentation which
measures the missile behaviour contained in the missile itself or in other platform.
The missile data is fed to a guidance computer which containing the desired path of
the missile to impact successfully to the target. The computer now can determine
what maneuver the missile should do to increase the chances on hitting the target.
The computer again passes the steering instruction to adjust the direction or the
aerodynamic control surfaces. The adjustment now observes by the missile
observation instrumentation and now the loop is complete.
Target
Data
Guidance
Computer
Steering
Orders
Close
Loop
Missile
Data
Missile
Position
Sensor Observing
Missile Behaviour
Figure 2.5 The close loop system [13]
2.4
Homing Guidance
Missiles make measurements for homing guidance with its on-board seekers
that track the line-of-sight (LOS) to the target by measuring the range and range
rate. In command guided missiles, a radar makes the LOS measurement by tracking
the missile and the target from the ground, then uplinks commands to the missile
where the homing guidance laws will be apply. The missile homing guidance law
best suited to intercept is determine by a target’s speed and capability to maneuver,
deliberately or inadvertently.
13
Stationary or slowly moving targets are the easiest to intercept, and guidance
laws for them can use simple seeker measurements of the look angle between the
missile’s center line and the LOS. Such missiles have minimal requirements for
inertial instruments but perform poorly against moving or maneuvering targets. For
faster moving targets and maneuvering targets, commanding missile to the LOS
turning rate is very effective, and most of today’s homing missiles rely on this
measurement[14].
LOS turning rate measurements require accurate rate gyros aboard the missile
to measure and remove body rates and/or a gimbaled seeker head. Combining
estimates of target acceleration with LOS turning rate can reduce miss distance, but
if the target acceleration estimates are inaccurate, they can increase miss distance
relative to what it would have been if the estimate were not used. By taking missile
and target time constants and acceleration limits into account, optimum homing
guidance laws maximize a missile’s effectiveness against maneuvering targets.
2.5
Basic Trajectory Analysis
V
J
α
a
X
P
R
Y
θ
I
O
Figure 2.6 Basic trajectory analysis [15]
14
2.5.1
Kinematics :
Position of P
X = R cos θ
(2.9)
Y = R sin θ
(2.10)
Velocity of P
2.5.2
X’ = V cos α
(2.11)
Y’ = V sin α
(2.12)
Turn Rate and Lateral Acceleration
Differentiate Equation (2.9) and (2.10),
X’
=
Y’
cos θ -sin θ
R’
sin θ
Rθ’
cos θ
(2.13)
Substitute Equation (2.11) and (2.12) into (2.13),
V cos α
V sin α
=
cos θ -sin θ
R’
sin θ
Rθ’
cos θ
(2.14)
Inverting Equation (2.14),
R’
=
V cos (α-θ)
(2.15)
Rθ’
=
V sin (α-θ)
(2.16)
- the velocity component parallel to OP affects the rate of change of OP.
- the velocity component perpendicular to OP affects the rotation rate of OP.
15
From Figure 2.6, the lateral acceleration a = a (-sin θ, cos θ).
Differentiate Equation (2.11) and (2.12)
X’’
=
Y’’
cos α -sin α
V’
sin α cos α
Vα’
(2.17)
Obtain
V’
=
0
(2.18)
Vα’
=
a
(2.19)
- the lateral acceleration changes the turn rate but not the speed.
2.6
The Concept
The basic concept or idea in forming the algorithm for the missile is by assuming
θ(t)=0
(2.20)
v1= v2.
(2.21)
as shown in Figure 2.7. With that assumption, we will obtain
dc= d(t) - (d(t)+1)
where
d(t)
- distance of the target location
θ(t)
- direction relative to the current heading of the missile
v
- velocity of the missile
dc
- distance the missile have move
(2.22)
16
d(t)
v1
Static target
v2
d(t+1)
dc
Figure 2.7 Basic idea in forming the algorithm
By enhancing the parameters from Figure 2.7, we will get the Figure 2.8 as shown
below.
Static target
d(t+1)
θ(t+1)
d(t)
v
θ(t)
v
Figure 2.8 Enhancement of parameter in Figure 2.7
From Figure 2.8, we will assume that θ(t)≠0 and θ(t+1)≠0 where we know the value
for each parameter. To steer the missile;
θc= θ(t+1) - θ(t)
(2.23)
dc= d(t+1) - d(t)
(2.24)
17
The basic algorithm that is form is shown as below:
define θ(t)
define d(t)
while (NOT TARGET)
------------------------------------calculate θ(t+1)
calculate d(t+1)
------------------------------------calculate θc
calculated dc
------------------------------------steer the missile θc
------------------------------------if
θ(t) = θ(t+1)
d(t) = d(t+1)
End
CHAPTER 3
METHODOLOGY
3.1
Introduction
In this chapter will discussed the method used in guiding a constant speed
missile until it reach its target. It will begins with the problem depiction which
described the consideration taken in this project and continue the approach for
geometrical analysis and Matlab’s simulation.
3.2
Problem Depiction
This problem concerns the design of an algorithm to update the current
heading of a missile given the location of a stationary enemy target. Here we will
design a simplified Preset Guidance algorithm for a constant speed missile. Assume
that at time t, the target location is given to the missile consisting of distance, d(t),
and direction relative to the current heading of the missile, θ(t). At the same instance
(time t), the missile will turn towards the target with angle α (α ≤ θ, in general),
resulting in a traveled distance D at t+1. The objective of this project is to form an
algorithm to find the desired destination at time t+1 (i.e. finding d(t+1) and θ(t+1)).
The constraints of the missile are:
•
It moves at a constant speed, v, at all times.
•
It has a limited turning point, α.
19
The visual depiction of the problem is illustrated in Figure 3.1. It is expected
that the algorithm can be used iteratively until the missile reach the target, that is
until d=0.
If time permits, the problem will be investigated again using 3
dimensional coordinate space and using additional constraints.
θ(t+1)
d(t+1)
Target
v
D
d(t)
θ(t)
v
Figure 3.1 Illustration of the original problem
Distance
(cm)
α(t+1) β(t+1)
θ (t + 1)
Target
X
d (t + 1)
ω
d (t )
v (t )
β (t )
θ (t )
Distance
(cm)
Figure 3.2 The visual depiction of the original problem
20
In order to have a guideline for the work to be executed in the plotting and
MATLAB’s simulation, the significant values are determined as per the following
parameters in Table 3.1.
Parameter
Value
Constant of velocity, (v cm/s)
Constant of swerving ability, α(°/s)
1
-20
Initial target distance, d(cm)
7
Initial target angle, θ (°)(degree)
40
Initial position grid
Initial direction, (degree)
(0,0)
90
Table 3.1: Determined parameters for initial work
Note that the consideration for the distance is purposely taken in centimeter
(cm) as it can be applied in plotting in an A4 graph paper.
21
3.3
The Execution of Geometrical Analysis
The approach of mathematical equation for geometrical analysis can be
illustrated in the following Figure 3.3.
β(t+1)
α(t+1)
Distance
(cm)
γ
Target
X
θ (t + 1)
d (t + 1)
ω
y
v (t )
θ (t )
d (t )
x
Distance
(cm)
Figure 3.3 : The visual depiction in plotting execution
From Figure 3.3, the mathematical equation to obtain the value d(t+1) and θ(t+1)
can be derived as follows :
d (t + 1) =
y 2 + (d (t ) − x) 2
(3.1)
where
x = v cos θ (t ) and
y = v sin θ (t )
(3.2)
Thus it can be rewritten as
d (t + 1) = (v sin 2 θ (t )) 2 + (d (t ) − v cos θ (t )) 2
(3.3)
22
if θ (t ) = 0 0 so,
d (t + 1) = (d (t ) − v) 2 = d (t ) − v (proven correct)
From Figure 3.3,
cos ω =
y
d (t + 1)
ω = cos −1
y
d (t + 1)
(3.4)
and
γ = 90 0 − ω
(3.5)
so from Equation 3.5 and Equation 3.2, we will obtain,
θ (t + 1) = θ (t ) − α + γ
= θ (t ) − α + (900 − ω )
⎛ y ⎞
⎟⎟
= θ (t ) − α + 90 0 − cos −1 ⎜⎜
⎝ d (t + 1) ⎠
⎛ v sin θ (t ) ⎞
⎟⎟
⎝ d (t + 1) ⎠
θ (t + 1) = θ (t ) − α + 90 0 − cos −1 ⎜⎜
if θ (t ) = 0 0 so,
θ (t + 1) = θ (t ) − α + 90 0 − cos −1 (0)
= −α + 90 0 − 90 0
= −α
(no turning if θ (t ) = 0 0 )
Variable β will steer the missile left, right or going straight.
β (t − 1) = 0 0
(3.6)
23
At t=0 and the value d(t) and θ(t) is given to the missile
If θ (t ) = 0 then β (t ) = 0 0
(the missile going straight)
If θ (t ) − α ≥ 0 then β (t ) = α (the missile turn left)
If θ (t ) − α ≤ 0 then β (t ) = −α
(the missile turn right)
From Equation 3.3 and Equation 3.6, a general equation can be written as
d (t + n) = (v sin θ + (n + 1)) 2 + (d (t ) + (n + 1)) − v cos θ (t + (n − 1)) 2
(3.7)
and
⎛ v sin −1 θ (t + (n − 1)) ⎞
⎟
d (t + (n − 1)) ⎟⎠
⎝
θ (t + n) = θ (t + (n − 1)) − α + 90 0 − cos −1 ⎜⎜
(3.8)
The consideration of steering command which either to turn left or right and the
stoppage point will be explained in detail in the result of plotting in Chapter 4.
24
3.4
The Execution of MATLAB Simulation
The approach of the mathematical equation for the MATLAB simulation can
be illustrated in the following Figure 3.4.
β(t+1)
α(t+1)
Distance
(cm)
λ (t + 1)
θ (t + 1)
Target
X
y
d (t )
Δx
Target_Position_y
d (t + 1)
x
v
β (t )
Δy
θ (t )
Target_pos_x
Distance
(cm)
Figure 3.4: The visual depiction in MATLAB simulation execution
From Figure 3.4, the mathematical equation to obtain value of d(t+1) and θ(t+1) can
be derived as follows :
d (t + 1) = x 2 + y 2
θ (t + 1) = tan −1
y
x
(3.9)
(3.10)
25
The parameters used in the MATLAB based on Figure 3.4 are defined as follows :
No.
Definition
Parameter
1.
Constant of velocity
2.
Constant of steering ability
3.
Initial target distance
4.
Initial target angle
theta_initial ( θ )
5.
Initial position grid
Init_pos
6.
Initial direction
init_dir
7.
Current direction
8.
Difference of swerving angle and target angle
beta (β)
9.
Steering command (clockwise/anticlockwise
c
v
Alpha ( α )
D
lamda (λ)
direction)
10.
Changes of motion ( x-axis)
∆x
11.
Changes of motion (y-axis)
∆y
12.
Changes of motion (angle)
∆θ
13.
Iteration number
maxtime
Table 3.2 : Definition of parameters used in MATLAB
If the initial position is represented as [ x0 y 0 θ 0 ]
Where,
x0 = initial _position_on_x_axis
y 0 = initial _position_on_y_axis
θ 0 .= initial target angle
So, the consideration of the mobile robot position at t=(t+1) can be written as :
[ x1 y1 θ 1 ] = [ x0 y 0 θ 0 ] + [ Δx0 Δy 0 Δθ 0 ]
26
The flowchart of MATLAB program can be shown in Figure 3.5.
Start
Set the following parameters : v, alpha, d,
theta_init, init_pos, init_dir of the missile
Store the initial positon
Move as of v and alpha
Update new position
Check either d
is within the
0.5cm radius ?
Check if
completion of
iteration ?
Plot the result
End
Figure 3.5 : Flowchart of MATLAB program
3.5
Conclusion
The methodology used for both geometrical analysis and MATAB simulation
are following the method that have been introduced by Chang-Fang Lin[10]. The
difference of the equation that introduced in this project comparing by Chang-Fang
Lin[10] is its describing on a moving target and we used the static target.
CHAPTER 4
RESULTS
4.1
Introduction
This chapter will discussed the result by geometrical/calculation, the result of
MATLAB simulation, the comparison of geometrical analysis and MATLAB
simulation and other simulation using changes in various parameters. This result are
came from the determined parameters from Table 3.1 and to measure the
effectiveness of this parameter in MATLAB simulation, changes of parameter have
been done.
4.2
Result of the Geometrical Analysis
Considering to the Equation 3.7 and Equation 3.8, the guidance algorithm and
error tolerance of a moving missile can be determine by the mathematical equation as
shown in Table 4.1.
Time
Calculation of the mathematical equation
(sec)
t=0
Measurement
by plotting
d (t ) = 7cm
d (t ) = 7cm
β (t ) = 50 0
β (t ) = 50 0
28
d (t + 1) = (1 − sin 50 0 ) 2 + (7 − 1cos 50 0 ) 2 = 6.403cm
t=1
⎛ 1sin 50 0
⎝ 6.403
β (t + 1) = 50 0 − 20 0 + 90 0 − cos −1 ⎜⎜
⎞
⎟⎟ = 36.87 0
⎠
d (t + 1) = 6.40cm
β (t + 1) = 36.5 0
d (t + 2) = (1 − sin 36.87 0 ) 2 + (6.403 − 1 cos 36.87 0 ) 2 = 5.63cm d (t + 2) = 5.65cm
t=2
⎛ 1sin 36.87 0
5.63
⎝
β (t + 1) = 36.87 0 − 20 0 + 90 0 − cos −1 ⎜⎜
⎞
⎟⎟ = 22.98 0
⎠
β (t + 2) = 22 0
d (t + 3) = (1 − sin 22.98 0 ) 2 + (5.63 − 1cos 22.98 0 ) 2 = 4.725cm d (t + 3) = 4.75cm
t=3
⎛ 1sin 22.98 0
β (t + 3) = 22.98 − 20 + 90 − cos ⎜⎜
5.63
⎝
0
0
−1
0
⎞
⎟⎟ = 7.710
⎠
d (t + 4) = (1 − sin 7.710 ) 2 + (4.725 − 1cos 7.710 ) 2 = 3.76cm
t=4
⎛ 1sin 7.710
⎝ 63.76
β (t + 4) = 7.710 − 20 0 + 90 0 − cos −1 ⎜⎜
⎞
⎟⎟ = −10.24 0
⎠
d (t + 5) = (1 − sin( −10.24 0 )) 2 + (3.76 − 1 cos(−10.24 0 )) 2
β (t + 3) = 7.5 0
d (t + 4) = 3.75cm
β (t + 4) = −10 0
d (t + 5) = 2.75cm
= 2.78cm
t=5
β (t ) = 6 0
⎛ 1sin( −10.24 0 ) ⎞
⎟⎟ = 6.08
β (t + 5) = −10.24 − 20 + 90 − cos ⎜⎜
2.78
⎝
⎠
0
t=6
0
−1
0
d (t + 6) = (1 − sin 6.08 0 ) 2 + (2.78 − 1cos 6.08 0 ) 2 = 1.81cm
d (t + 6) = 1.8cm
⎛ 1sin 6.08 0
β (t + 6) = 6.08 − 20 + 90 − cos ⎜⎜
⎝ 1.81
β (t + 6) = −10 0
0
0
0
−1
⎞
⎟⎟ = −10.55 0
⎠
d (t + 7) = (1 − sin(−10.55 0 )) 2 + (1.81 − 1cos(−10.55 0 )) 2 = 0.7 d (t + 7) = 0.75cm
t=7
β (t + 7) = 5 0
⎛ 1sin( −10.55 0 ) ⎞
⎟⎟ = −5.3
0.71
⎝
⎠
β (t + 7) = −10.55 0 − 20 0 + 90 0 − cos −1 ⎜⎜
29
d (t + 8) = (1 − sin(−5.34 0 )) 2 + (0.71 − 1cos(−5.34 0 )) 2 = 0.28c d (t + 8) = 0.25cm
t=8
β (t ) = −5
⎛ 1sin( −5.34 0 ) ⎞
⎟⎟ = −4.77
β (t + 8) = 7 − 5.34 − 20 + 90 − cos ⎜⎜
0.28
⎝
⎠
0
0
0
0
−1
Table 4.1 : The result of calculation with mathematical equation
By referring to the Table 4.1, we will see that at t=4, the value of β (t + 4) is
negative because another constraint need to be included which is known as steering
command in order to determine the turning ability of α (t ) = 20 0 s −1 either to be
turning right or turning left. The consideration has been mentioned earlier referring
to the Equation 3.6.
Referring to the result t=8 from the same table above, the value of d(t+8) is
within the radius of 0.5cm. In order for the missile to determine either to stop or
continue moving , we included another constraint call error tolerance ( γ (t ) ). The
consideration of ( γ (t ) ) as follows :
i)
If d(t) > 0.5cm, then γ (t + 1) = v(t), the missile keep moving.
ii)
If d(t) < 0.5cm, then γ (t + 1) = 0, the missile stop moving.
30
4.3
Result of MATLAB Simulation
In order to guide the missile to its target, the MATLAB program used are as
shown in Figure 4.1.
clear all
clc
v=1
theta_initial=(25/180)*pi
d=10
alpha=(25/180)*pi
init_pos=[0 0]
init_dir=(90/180)*pi
%velocity constant(m/s)
%initial target angle (deg)
%initial target distance (m)
%swerving constant(deg)
%initial position grid
%initial direction(lamda)
target_pos_x = d*cos(theta_initial)
target_pos_y = d*sin(theta_initial)
%initial target position (x-axis)
%initial terget position (y-axis)
initial_position=[init_pos init_dir]
maxtime=50
%initial position & direction
plot(initial_position(1),initial_position(2),'ro')
hold on
grid
c=1
for i=1:maxtime
lamda = initial_position(3)
delta_x=v*cos(lamda)
delta_y=v*sin(lamda)
delta_alpha=c*alpha
%current direction angle
%current target position (x-axis)
%current target position (y-axis)
%swerving direction (clockwise/anticlockwise)
%changes in position
delta_position=[delta_x delta_y delta_alpha]
%current position
initial_position=initial_position+delta_position
%current distance
d_new(i) = sqrt((initial_position(1)-target_pos_x)^2+...
(initial_position(2)-target_pos_y)^2)
%current theta
theta_new(i) = atan((initial_position(2)-target_pos_y)/...
(initial_position(1)-target_pos_x))
%beta=current swerving angle-current target angle
%beta is the controller for the swerving either clockwise or anticlockwise
beta(i) = initial_position(3)-theta_new(i)
if beta(i) < 0
c = -1
else
c=1
end
%swerve anticlockwise
%swerve clockwise
%STOP execution if the movement reach the distance of
%-0.5<d(target)<+0.5 for x-axis or y-axis
if initial_position(1)>target_pos_x-0.5 & initial_position(1)...
<target_pos_x+0.5 & initial_position(2)>target_pos_y-0.5...
initial_position(2)<target_pos_y+0.5
v=0
alpha = 0
end
31
plot(initial_position(1),initial_position(2),'ro',target_pos_x,...
target_pos_y,'rx')
hold on
grid
xlabel('distance(m)')
ylabel('distance(m)')
grid
end
%table of the result
[d_new'; theta_new'; beta']
[d_new' theta_new' beta']
Figure 4.1 : MATLAB Simulation Program
Figure 4.1. MATLAB program used in order to guide the missile to its target,
32
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
distance,
d(cm)
6.4032
5.6352
4.7308
3.7422
2.7638
1.7727
0.81
0.1976
0.1976
0.1976
theta, θ (rad)
theta, θ ( 0)
Value
1
-20
7
40
(0,0)
90
beta, β (rad)
0.5782
33.12
0.6435
0.4715
27.01
0.4011
0.3889
22.28
0.1347
0.353
20.22
-0.1785
0.4173
27.00
0.1063
0.3574
20.47
-0.1829
0.5838
33.45
-0.0602
0.2743
15.72
0.5984
0.2743
15.72
0.5984
0.2743
15.72
0.5984
Table 4.2 : MATLAB simulation result
beta, β ( 0)
36.86
22.98
7.71
-10.23
6.09
-10.48
-3.45
34.28
34.28
34.28
33
4.4
Comparison of Geometrical Analysis and MATLAB Simulation
The comparison of result achieved using geometrical analysis from Table 4.1
and the MATLAB simulation result in Table 4.2 are shown in Table 4.3
beta, β ( 0)
distance, d (cm)
Geometrical
MATLAB
Geometrical
MATLAB
0.6403
6.4032
36.87
36.86
5.63
5.6352
22.98
22.98
4.725
4.7308
7.71
7.71
3.76
3.7422
-10.245
-10.23
2.78
2.7638
6.085
6.09
1.81
1.7727
-10.55
-10.48
0.717
0.81
-5.345
-3.45
0.280
-4.777
0.1976
34.28
Table 4.3 : Result comparison using geometrical analysis and MATLAB simulation
The result shows differences because the decimal places taken account during
geometrical calculation is 3 compare to the simulation result from MATLAB
simulation. We omitted the value beta, β the result from last iteration since the
distances of the missile towards target are beyond the original consideration.
4.5
Result by Varying the Parameter in MATLAB Simulation
As we mentioned earlier, we will achieve result by varying the parameters
involved in the equation using MATLAB simulation. The result may be vary due to
the changes in parameters such as follows:
i)
varying the initial direction
ii)
varying the positional grid
iii)
varying the initial target distance
iv)
varying the constant of turning ability
v)
varying the initial target angle
vi)
varying the constant velocity
34
For every cases, the simulation will be executed using the initial parameter in Table
3.1. All the results of the parameters involved will be shown in a table and figure
accordingly.
4.5.1
Change The Initial Direction To 0 Degree
The result of MATLAB
parameters in Table 4.3.
simulation are shown in Figure 4.3 and the
35
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
40
(0,0)
0
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.2670
5.9291
5.4138
4.7376
3.9228
2.9986
2.0064
1.0439
0.0564
0.0564
0.0564
0.0564
0.0564
0.0564
0.0564
0.0564
0.8009
0.9554
1.1068
1.2524
1.3870
1.4984
1.5492
1.3615
0.6981
0.6981
0.6981
0.6981
0.6981
0.6981
0.6981
0.6981
45.8697
54.7184
63.3895
71.7284
79.4373
85.8175
88.7269
77.9768
39.9821
39.9821
39.9821
39.9821
39.9821
39.9821
39.9821
39.9821
-1.1499
-0.9554
-0.7578
-0.5543
-0.3398
-0.1021
0.1961
0.0348
0.3491
0.3491
0.3491
0.3491
0.3491
0.3491
0.3491
0.3491
-65.8579
-54.7184
-43.4013
-31.7463
-19.4613
-5.8475
11.2312
1.9931
19.9939
19.9939
19.9939
19.9939
19.9939
19.9939
19.9939
19.9939
Table 4.4 : The changes in parameters when initial direction = 00
36
4.5.2
Change The Initial Direction To 180 Degree
The result of MATLAB simulation shown in Figure 4.4 and the parameters
in Table 4.5.
37
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
40
(0,0)
180
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
7.7926
8.4026
8.8004
8.9681
8.8986
8.5949
8.0702
7.3490
6.4693
5.4888
4.4998
3.5218
2.5304
1.5611
0.5624
0.4794
0.4794
0.4794
0.4794
0.4794
0.6156
0.5176
0.4108
0.2998
0.1881
0.0791
-0.0232
-0.1131
-0.1821
-0.2151
-0.1855
-0.2378
-0.1940
-0.3178
-0.2621
0.3088
0.3088
0.3088
0.3088
0.3088
35.2571
29.6444
23.5276
17.1704
10.7730
4.5303
-1.3287
-6.4775
-10.4294
-12.3194
-10.6241
-13.6195
-11.1109
-18.2013
-15.0112
17.6858
17.6858
17.6858
17.6858
17.6858
2.1770
1.9259
1.6836
1.4455
1.2082
0.9681
0.7213
0.4622
0.1821
-0.1339
0.1855
-0.1112
0.1940
-0.0313
0.2621
-0.6579
-0.6579
-0.6579
-0.6579
-0.6579
124.6827
110.3015
96.4244
82.7877
69.1969
55.4457
41.3108
26.4715
10.4294
-7.6688
10.6241
-6.3687
11.1109
-1.7926
15.0112
-37.6797
-37.6797
-37.6797
-37.6797
-37.6797
Table 4.5 : The changes in parameters when initial direction = 1800
38
4.5.3
Change The Initial Direction To 270 Degree
The result of MATLAB simulation shown in Figure 4.5 and the parameters
in Table 4.6.
39
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
40
(0,0)
270
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
7.6811
8.6025
9.6022
10.5546
11.3703
11.9872
12.3650
12.4809
12.3282
11.9158
11.2686
10.4299
9.4649
8.4677
7.5088
6.5099
5.5622
4.5622
3.6368
2.6422
1.6829
0.6831
0.4122
0.4122
0.4122
0.4122
0.4122
0.7980
0.8458
0.8486
0.8184
0.7656
0.6981
0.6221
0.5421
0.4624
0.3872
0.3214
0.2712
0.2448
0.2531
0.2175
0.2241
0.171
0.1702
0.2633
0.2968
0.1628
0.1457
-0.7234
-0.7234
-0.7234
-0.7234
-0.7234
45.7036
48.4413
48.6016
46.8720
43.8480
39.9821
35.6294
31.0475
26.4829
22.1760
18.4075
15.5324
14.0204
14.4957
12.4568
12.8348
9.7936
9.7478
15.0799
16.9985
9.3240
8.3446
-41.4311
-41.4311
-41.4311
-41.4311
-41.4311
3.5653
3.1684
2.8166
2.4978
2.2015
1.9199
1.6469
1.3778
1.1084
0.8345
0.5513
0.2524
-0.0702
0.2705
-0.0429
0.2995
0.0035
-0.3448
-0.0887
0.2268
0.0117
-0.3202
0.8979
0.8979
0.8979
0.8979
0.8979
204.1945
181.4629
161.3144
143.0558
126.0859
109.9579
94.3225
78.9104
63.4811
47.7941
31.5745
14.4556
-4.0205
15.4923
-2.4570
17.1532
0.2005
-19.7476
-5.0801
12.9895
0.6701
-18.3387
51.4252
51.4252
51.4252
51.4252
51.4252
Table 4.6 : The changes in parameters when initial direction = 2700
40
4.5.4
Change The Initial Direction To 45 Degree
The result of MATLAB simulation shown in Figure 4.6 and the parameters
in Table 4.7.
41
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
40
(0,0)
45
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.0044
5.0408
4.0426
3.0949
2.0953
1.1850
0.3093
0.3093
0.3093
0.6836
0.7322
0.7190
0.8093
0.8207
0.5572
-0.2632
-0.2632
-0.2632
39.1516
41.9351
41.1791
46.3508
47.0037
31.9124
-15.0742
-15.0742
-15.0742
-0.2473
0.0532
-0.2827
-0.0239
0.3138
0.2282
0.6995
0.6995
0.6995
-14.1635
3.0469
-16.1910
-1.3688
17.9722
13.0696
40.0623
40.0623
40.0623
Table 4.7 : The changes in parameters when initial direction = 450
42
4.5.5
Change The Initial Position To (-2,-2)
The result of MATLAB simulation shown in Figure 4.7 and the parameters
in Table 4.8.
43
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
40
(-2,-2)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
9.1896
8.3712
7.4206
6.4207
5.4858
4.4870
3.5419
2.5424
1.6196
0.6490
0.3523
0.3523
0.3523
0.6416
0.5760
0.5366
0.5387
0.4789
0.4689
0.5509
0.5617
0.3716
0.1361
0.2454
0.2454
0.2454
36.7462
32.9891
30.7325
30.8528
27.4279
26.8552
31.5515
32.1701
21.2825
7.7948
14.0547
14.0547
14.0547
0.5802
0.2966
-0.0130
0.3340
0.0447
-0.2944
-0.0273
0.3110
0.1520
0.0385
-0.4200
-0.4200
-0.4200
33.2296
16.9871
-0.7445
19.1291
2.5601
-16.8611
-1.5635
17.8118
8.7055
2.2050
-24.0545
-24.0545
-24.0545
table 4.8 : The changes in parameters when initial position (-2,-2)
44
4.5.6
Change The Target Distances 10cm
The result of MATLAB simulation shown in Figure 4.8 and the parameters
in Table 4.9.
45
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
10
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
9.3885
8.5851
7.6432
6.6433
5.7072
4.7082
3.7647
2.7653
1.8374
0.8542
0.1632
0.1632
0.1632
0.6164
0.5501
0.5086
0.5064
0.5635
0.5720
0.4932
0.4822
0.6478
0.7934
1.3000
1.3000
1.3000
35.3029
31.5057
29.1289
29.0029
32.2732
32.7600
28.2469
27.6169
37.1013
45.4402
74.4545
74.4545
74.4545
0.6053
0.3225
0.0150
-0.3319
-0.0399
0.3007
0.0304
-0.3077
-0.1242
0.0793
-0.7764
-0.7764
-0.7764
34.6672
18.4705
0.8591
-19.0088
-2.2852
17.2219
1.7411
-17.6228
-7.1133
4.5417
-44.4665
-44.4665
-44.4665
Table 4.9 : The changes in parameters when target distance 10cm
46
4.5.7
Change The Target Distances 5cm
The result of MATLAB simulation shown in Figure 4.9 and the parameters
in Table 4.10.
47
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
5
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
4.4240
3.7137
2.8904
1.9794
1.0082
0.0128
0.0128
0.0128
0.5241
0.3502
0.1767
0.0041
-0.1648
0.6981
0.6981
0.6981
30.0166
20.0569
10.1201
0.2348
-9.4385
39.9821
39.9821
39.9821
0.6976
0.5224
0.3469
0.1704
-0.0097
-0.5236
-0.5236
-0.5236
39.9535
29.9193
19.8679
9.7593
-0.5555
-29.9880
-29.9880
-29.9880
Table 4.10 : The changes in parameters when target distance 5cm
48
4.5.8
Change The Steering Constant -15 Degree
The result of MATLAB simulation shown in Figure 4.10 and the parameters
in Table 4.11.
49
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-15
5
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.4032
5.6978
4.8962
4.0130
3.0651
2.0743
1.0834
0.1139
0.1139
0.1139
0.5241
0.3502
0.1767
0.0041
-0.1648
0.6981
0.6981
0.6981
0.6981
0.6981
30.0166
20.0569
10.1201
0.2348
-9.4385
39.9821
39.9821
39.9821
39.9821
39.9821
0.7308
0.5864
0.4379
0.2819
0.1110
-0.0974
0.0746
0.5259
0.5259
0.5259
41.8549
33.5847
25.0797
16.1452
6.3573
-5.5784
4.2725
30.1197
30.1197
30.1197
Table 4.11 : The changes in parameters when steer ability -15 degree
50
4.5.9
Change The Steering Constant -30 Degree
The result of MATLAB simulation shown in Figure 4.11 and the parameters
in Table 4.11.
51
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-30
5
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.4032
5.5297
4.5301
3.6782
2.6845
1.8019
0.8097
0.3565
0.3565
0.3565
0.5782
0.4964
0.4904
0.6188
0.6542
0.4400
0.3367
-0.8483
-0.8483
-0.8483
33.1151
28.4302
28.0865
35.4404
37.4678
25.2000
19.2837
-48.5845
-48.5845
-48.5845
0.4690
0.0272
-0.4904
-0.0952
0.3930
0.0836
-0.3367
1.3719
1.3719
1.3719
26.8609
1.5578
-28.0865
-5.4524
22.5082
4.7880
-19.2837
78.5725
78.5725
78.5725
Table 4.12 : The changes in parameters when steer ability -30 degree
52
4.5.10 Change The Steering Constant 20 Degree
The result of MATLAB simulation shown in Figure 4.12 and the parameters
in Table 4.13.
The constant of steer ability is not in the quadrant of the target where target is in the
first quadrant and the steer ability 20 degree is in second quadrant.
53
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
20
5
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.4032
6.2524
6.5957
7.3283
8.2741
9.2735
10.2076
10.9937
11.5753
11.9158
11.9956
11.8103
11.3703
10.7017
9.8483
8.8764
7.8819
7
6.4032
6.2524
0.5782
0.4218
0.2754
0.1775
0.1358
0.1400
0.1766
0.2350
0.3071
0.3872
0.4706
0.5532
0.6307
0.6981
0.7489
0.7741
0.7616
0.6981
0.5782
0.4218
33.1151
24.1576
15.7729
10.1659
7.7776
8.0182
10.1144
13.4591
17.5885
22.1760
26.9525
31.6833
36.1219
39.9821
42.8915
44.3348
43.6189
39.9821
33.1151
24.1576
1.3417
1.8471
2.3426
2.7896
3.1803
3.5252
3.8376
4.1283
4.4053
4.6742
4.9399
5.2064
5.4779
5.7596
6.0579
6.3818
6.7433
7.1558
7.6248
8.1303
76.8428
105.7885
134.1671
159.7680
182.1445
201.8978
219.7898
236.4390
252.3035
267.7042
282.9215
298.1847
313.7343
329.8680
346.9525
365.5031
386.2072
409.8322
436.6931
465.6445
Table 4.13 : The changes in parameters when steer ability 20 degree
54
4.5.11 Change The Steering Constant To 20 Degree and Target Angle To 140
Degree
The result of MATLAB simulation shown in Figure 4.13 and the parameters
in Table 4.14.
55
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
20
5
140
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.4032
5.6352
4.7308
3.7422
2.7638
2.0128
1.9169
2.5600
3.5131
4.5101
5.4393
6.2419
6.8802
7.3283
7.5691
7.5939
7.4019
7
6.4032
5.6352
-0.5782
-0.4715
-0.3889
-0.3530
-0.4173
-0.6981
-1.2105
-1.5579
1.4827
1.5022
-1.5647
-1.4623
-1.3448
-1.2188
-1.0884
-0.9564
-0.8254
-0.6981
-0.5782
-0.4715
-33.1151
-27.0041
-22.2734
-20.2173
-23.8999
-39.9821
-69.3286
-89.2252
84.9183
86.0351
-89.6146
-83.7499
-77.0204
-69.8040
-62.3356
-54.7756
-47.2729
-39.9821
-33.1151
-27.0041
2.4981
2.7405
3.0069
3.3201
3.7334
4.3633
5.2247
5.9212
3.2297
3.5592
6.9752
7.2219
7.4534
7.6765
7.8952
8.1123
8.3303
8.5521
8.7813
9.0236
143.0730
156.9559
172.2134
190.1512
213.8220
249.8981
299.2328
339.1233
184.9737
203.8451
399.4887
413.6179
426.8765
439.6541
452.1796
464.6135
477.0990
489.8021
502.9290
516.8062
Table 4.14 : The changes in parameters when steer ability 20 degree and
target angle 140 degree
56
4.5.12 Change The Target Angle 140 Degree
The result of MATLAB simulation shown in Figure 4.14 and the parameters
in Table 4.15.
57
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
1
-20
7
140
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.4032
6.2524
6.5957
7.3283
8.2741
9.2735
10.2291
11.2272
12.1888
13.1856
14.1512
15.1470
16.1154
17.1102
18.0808
19.0748
20.047
21.0403
22.0138
23.0065
-0.5782
-0.4218
-0.2754
-0.1775
-0.1358
-0.1400
-0.1097
-0.1155
-0.0920
-0.0983
-0.0792
-0.0855
-0.0696
-0.0757
-0.0620
-0.0679
-0.0559
-0.0615
-0.0509
-0.0563
-33.1151
-24.1576
-15.7729
-10.1659
-7.7776
-8.0182
-6.2828
-6.6150
-5.2691
-5.6299
-4.5360
-4.8968
-3.9862
-4.3355
-3.5509
-3.8888
-3.2015
-3.5223
-2.9152
-3.2245
1.7999
1.2945
0.7990
0.3520
-0.0387
0.3145
-0.0648
0.2900
-0.0825
0.2728
-0.0953
0.2601
-0.1050
0.2502
-0.1125
0.2424
-0.1186
0.2361
-0.1236
0.2308
103.0852
74.1395
45.7609
20.1600
-2.2165
18.0123
-3.7113
16.6091
-4.7250
15.6240
-5.4581
14.8966
-6.0136
14.3296
-6.4432
13.8829
-6.7925
13.5221
-7.0789
13.2185
Table 4.15 : The changes in parameters when target angle 140 degree
58
4.5.13 Change The Velocity Constant 0.5 cm/s
The result of MATLAB simulation shown in Figure 4.15 and the parameters
in Table 4.16.
59
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
0.5
-20
7
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
6.6896
6.2776
5.7980
5.2987
4.8224
4.3226
3.8514
3.3515
2.8837
2.3842
1.9121
1.4124
0.9491
0.4573
0.4573
0.4573
0.4573
0.4573
0.4573
0.4573
0.6408
0.5971
0.5737
0.5784
0.5483
0.5512
0.5101
0.5081
0.5649
0.5736
0.4965
0.4868
0.6494
0.7871
0.7871
0.7871
0.7871
0.7871
0.7871
0.7871
36.7004
34.1975
32.8574
33.1265
31.4026
31.5687
29.2148
29.1003
32.3534
32.8516
28.4359
27.8804
37.1929
45.0794
45.0794
45.0794
45.0794
45.0794
45.0794
45.0794
0.5809
0.2755
-0.0501
0.2943
-0.0247
0.3215
0.0135
-0.3336
-0.0413
0.2991
0.0271
-0.3123
-0.1258
0.0856
0.0856
0.0856
0.0856
0.0856
0.0856
0.0856
33.2697
15.7786
-2.8694
16.8554
-1.4146
18.4132
0.7732
-19.1062
-2.3654
17.1303
1.5521
-17.8863
-7.2049
4.9025
4.9025
4.9025
4.9025
4.9025
4.9025
4.9025
Table 4.16 : The changes in parameters when velocity constant 0.5cm/s
60
4.5.14 Change The Velocity Constant 4 cm/s
The result of MATLAB simulation shown in Figure 4.16 and the parameters
in Table 4.17.
61
Parameter
Constant of velocity, v(cm/s)
Constant of swerving ability, α
Initial target distance, d(cm)
Initial target angle, θ (degree)
Initial position grid
Initial direction (degree)
Value
4
-20
7
40
(0,0)
90
distance,
d(cm)
theta, θ
(rad)
theta, θ
(deg)
beta, β (rad)
beta, β (deg)
5.3855
5.1553
6.4816
8.5700
10.8207
14.5118
18.5116
22.3216
26.3138
30.1672
34.1509
38.0263
42.0036
45.8921
49.8644
53.7616
57.73
61.6334
65.5987
69.5067
0.0929
-0.6844
-1.3494
1.3303
0.9852
0.8622
0.8644
0.8045
0.8148
0.7768
0.7880
0.7605
0.7711
0.7498
0.7596
0.7422
0.7512
0.7366
0.7449
0.7322
5.3206
-39.1975
-77.2838
76.1899
56.4251
49.3805
49.5065
46.0759
46.6658
44.4895
45.1309
43.5559
44.1630
42.9431
43.5044
42.5078
43.0233
42.1871
42.6625
41.9351
1.1288
1.5571
1.8730
-1.1558
-0.4616
0.0105
-0.3408
0.0682
-0.2912
0.0959
-0.2644
0.1122
-0.2475
0.1229
-0.2360
0.1304
-0.2276
0.1361
-0.2213
0.1404
64.6495
89.1794
107.2718
-66.1958
-26.4371
0.6014
-19.5185
3.9060
-16.6778
5.4925
-15.1429
6.4260
-14.1750
7.0388
-13.5164
7.4684
-13.0353
7.7948
-12.6745
8.0411
Figure 4.17 : The changes in parameters when velocity constant 4 cm/s
CHAPTER 5
CONCLUSION
5.1
Introduction
This chapter will conclude all the finding through out the development of the
project. Some idea or constructive suggestion for further and contribution to this
project also will be discussed. This idea and suggestion is basically try to improve
the study about the algorithm involve in this project.
5.2
Conclusion
With the limitation that have been mentioned earlier, this project achieved the
objectives and scope of works. The movement of the constant speed missile with the
MATLAB simulation can be implement if it is done in first quadrant. Other than
implementation in this quadrant, several parameter need to be recognized if we need
the missile still obey the rule of movement and guidance. The initial simulation and
calculation with the mathematical equation proved the algorithm is reliable to be
implemented. But the weakness of the algorithm increase if other parameter have
been change and it need future improvement. Some of the parameter that have been
recognized with certain level of critical effect such as :
5.2.1 High priority parameter that need to be improved are initial target angle and
swerving constant because both of it only apply in the first quadrant.
63
5.2.2
Medium priority parameter is the parameter which sometimes applicable or
can be used in the algorithm but involve other parameters such as velocity constant
and swerving constant ( sometimes just applicable in the first quadrant).
5.2.3
Low priority parameter that nee d to be consider are initial post grid, initial
direction (degree), initial target distance and initial target angle.
5.3
Future Development
For future development of this project which can improved the algorithm,
some of the recommendation and suggestion are:
5.3.1
The basic implementation of movement of the missile toward target, the
mathematical equation introduced using Ching-Fang Lin model can be used as a
guideline.
5.3.2
Improved further the algorithm by considering other factors such as the
missile aerodynamics and propulsion, the moving target , 3D dimensional coordinate
system or gravitational forces.
5.3.2 Consider the real information involving the real characteristics and capability
of missile by cooperate with the arm forces in sharing the data and information that
we need.
64
REFERENCES
[1]
Missile Guidance
http://www.aerospaceweb.org/question/weapons/q0187.shtml
[2]
Richard A Poisel,Target Acquisition in Communication Electronic Warfare
Systems, Artech House Inc, 2004 : p 1.
[3]
Electronic Warfare
http://en.wikipedia.org/wiki/Electronic_warfare
[4]
R G Lee, T.K Garlands-Collins, D.E Johnson, E Archer, C Sparkes, G.M
Moss and A.W Mowat, Guided Weapons Third Edition , Brassey’s UK Ltd,
1998 : p. 1
[5]
A.A.M Madkour, M.A Hossain , P.K Dahal Evolution Missile Guidance
Algorithm for Interception of Manouvering Targets in 3D Environment.
[6]
David L. Adamy, Introduction To Electronic Warfare Modelling And
Simulation, Artech House Inc, 2003
[7]
D. Curtis Scheleher, Introduction To Electronic Warfare, Artech House Inc,
1986.
[8]
R G Lee, T.K Garlands-Collins, D.E Johnson, E Archer, C Sparkes, G.M
Moss and A.W Mowat, Guided Weapons Third Edition , Brassey’s UK Ltd,
1998 :
p. 148
[9]
Ching-Fang Lin, Modern Navigation, Guidance and Control Processing,
Prentice Hall Inc., 1991 : p. 34
[10]
Ching-Fang Lin, Modern Navigation, Guidance and Control Processing,
Prentice Hall Inc., 1991 : p. 35
[11]
Military Aviation Webring : GLOSSARY/Abbreviation
http://www.voodoo.cz/hornet /glossary.html
[12]
Ching-Fang Lin, Modern Navigation, Guidance and Control Processing,
Prentice Hall Inc., 1991 : p. 37
65
[13]
Ching-Fang Lin, Modern Navigation, Guidance and Control Processing,
Prentice Hall Inc., 1991 :
[14]
Sergei A.Vakin, Lev N. Shustov, Robert H. Dunwell, Fundemental Of
Electronic Warfare, Artech House Inc, 2001.
[15]
Basic Trajectory Anaysis
http://dynlab.mpe.nus.edu.sg/mpelsb