In the Name of Allah, the most Beneficent, Most Merciful
Guidance and Navigation of
Aerospace Vehicles
Dr. Muhammad Wasim
Fall 2023
Lecture 2
Guidance of Aerospace
Vehicles
Rocket and
missile
system
Rocket
(Unguided)
Missiles
(Guided)
Tactical (Short
Range)
Strategic
(Long Range)
Ballistic
Cruise
Rocket and missile system:
Rocket and missile system, any of a variety of weapons systems that deliver
explosive warheads to their targets by means of rocket propulsion.
Basic Principle of Rocketry:
Rocket is a general term used broadly to describe a variety of jet-propelled
missiles in which forward motion results from reaction to the rearward ejection
of matter (usually hot gases) at high velocity. The propulsive jet of gases usually
consists of the combustion products of solid or liquid propellants.
Different Types of Propulsion systems in rocketry
In a more restrictive sense, rocket propulsion is a unique member of the family of jetpropulsion engines that includes turbojet, ramjet, scramjet systems.
Non-air breathing Engines (Rocket Propulsion):
The rocket engine is different from these in that the elements of its propulsive jet
(that is, the fuel and oxidizer) are self-contained within the vehicle. Therefore, the
thrust produced is independent of the medium through which the vehicle travels,
making the rocket engine capable of flight beyond the atmosphere or propulsion
underwater. They are capable to speed up to any Mach number.
Air-breathing Engines (Missile Propulsion):
The turbojet, pulse-jet, and ramjet engines, on the other hand, carry only their fuel
and depend on the oxygen content of the air for burning. For this reason, these
varieties of jet engine are called air-breathing and are limited to operation within the
Earth’s atmosphere.
Non-air
breathing
Engines
Air breathing
Engines
Ramjet
Scramjet
Guided Missiles
A guided missile is broadly any military missile that is capable of being guided or directed to a
target after having been launched.
Tactical Missiles:
Tactical guided missiles are shorter-ranged weapons designed for use in the immediate combat
area.
Strategic Missiles:
Long-range, or strategic, guided missiles are of two types, cruise and ballistic. Strategic missiles
usually carry nuclear warheads, while tactical missiles usually carry high explosives.
 Cruise Missiles:
Cruise missiles are powered by air-breathing engines that provide almost continuous
propulsion along a low, level flight path.
 Ballistic Missile:
A ballistic missile is propelled by a rocket engine for only the first part of its flight; for the
rest of the flight the unpowered missile follows an arcing trajectory, small adjustments
being made by its guidance mechanism.
History
There is no reliable early history of the “invention” of rockets. Most historians of rocketry
trace the development to China.
1232:
When the Mongols laid siege to the city of K’ai-feng, capital of Honan province, the Chinese
defenders used weapons that were described as “arrows of flying fire.”
By 1232 the Chinese had discovered black powder (gunpowder) and had learned to use it to
make explosive bombs as well as propulsive charges for rockets.
1241:
In the same century rockets appeared in Europe. There is indication that their first use was by
the Mongols in the Battle of Legnica in 1241.
1249:
The Arabs are reported to have used rockets on the Iberian Peninsula in 1249.
1288:
In 1288 Valencia was attacked by rockets.
1379:
In Italy, rockets are said to have been used by the Paduans (1379) and by the Venetians (1380).
History
Propulsion mechanism:
The propulsive charge was the basic black powder mixture of finely ground carbon (charcoal), potassium nitrate (saltpetre),
and sulfur.
1668:
By 1668, military rockets had increased in size and performance. In that year, a German colonel designed a rocket weighing
132 pounds (60 kilograms); it was constructed of wood and wrapped in glue-soaked sailcloth. It carried a gunpowder charge
weighing 16 pounds.
1792:
There Hyder Ali, prince of Mysore, developed war rockets with an important change: the use of metal cylinders to contain the
combustion powder. Range was perhaps more than a kilometre.
1812:
British have successfully used rockets against America and guarded Baltimore harbour
World War I and after:
Thrust and efficiency was greatly improved by advancement made in propulsion. They have shifted from black powder to
double-base powder (40 percent nitroglycerin, 60 percent nitrocellulose). The improved rockets became the forerunners of
the bazooka of World War II.
1931–32:
Gasoline–oxygen-powered rockets were made by the German Rocket Society.
1931–32:
The technology for a long-range ballistic missile was developed and tested in Germany.
History
World War II:
World War II saw the expenditure of immense resources and talent for the
development of rocket-propelled weapons.
Rockets:
1) Barrage rockets (Germans)
2) The bazooka (US)
Missile:
1) V2 (Germans)
Barrage rockets:
• The Germans began the war
with a lead in this category of
weapon, and their 150millimetre and 210-millimetre
bombardment rockets were
highly effective.
The bazooka (US):
The new rocket, about 20 inches
(50 centimetres) long, 2.36 inches
in diameter, and weighing 3.5
pounds, was fired from a steel
tube that became popularly
known as the bazooka. Designed
chiefly for use against tanks and
fortified positions at short ranges
(up to 600 yards), the bazooka
surprised the Germans when it
was first used in the North African
landings of 1942.
V-2 missile or A-4, German ballistic missile
• Developed by Germany. It was first successfully launched on October 3,
1942, and was fired against Paris on September 6, 1944.
• Two days later the first of more than 1,100 V-2s was fired against Great
Britain (the last on March 27, 1945).
• Belgium was also heavily bombarded. About 5,000 people died in V-2
attacks, and it is estimated that at least 10,000 prisoners from the MittelbauDora concentration camp died when used as forced labour in building V-2s
at the underground Mittelwerk factory.
• After the war, both the United States and the Soviet Union captured large
numbers of V-2s and used them in research that led to the development of
their missile and space exploration programs.
• The V-2 was 14 meters (47 feet) long, weighed 12,700–13,200 kg (28,000–
29,000 pounds) at launching, and developed about 60,000 pounds of thrust,
burning alcohol and liquid oxygen. The payload was about 725 kg (1,600
pounds) of high explosive, horizontal range was about 320 km (200 miles),
and the peak altitude usually reached was roughly 80 km (50 miles).
However, on June 20, 1944, a V-2 reached an altitude of 175 km (109 miles),
making it the first rocket to reach space.
Guided missiles
Missiles
(Guided)
Tactical (Short
Range)
Strategic
(Long Range)
Ballistic
Cruise
Tactical
Missiles
• Guided missiles were a product of post-World War II developments in
electronics, computers, sensors, avionics, and to only a slightly lesser degree,
rocket and turbojet propulsion and aerodynamics.
• Although tactical, or battlefield, guided missiles were designed to perform
many different roles, they were bound together as a class of weapon by
similarities in sensor, guidance, and control systems.
Control:
Control over a missile’s direction was most commonly achieved by the
deflection of aerodynamic surfaces such as tail fins; reaction jets or rockets and
thrust-vectoring were also employed. But it was in their guidance systems that
these missiles gained their distinction, since the ability to make down-course
corrections.
Ways of Guidance
The earliest guided missiles used simple command guidance, but
within 20 years of World War II virtually all guidance systems
contained autopilots or auto stabilization systems, frequently in
combination with memory circuits and sophisticated navigation
sensors and computers. Five basic guidance methods came to be
used, either alone or in combination:
1. command
2. Inertial
3. Active
4. semiactive
5. passive.
Command
Command guidance involved tracking the projectile from the launch site or platform and
transmitting commands by radio, radar, or laser impulses.
Tracking might be accomplished by radar or optical instruments from the launch site or by
radar or television imagery relayed from the missile.
The earliest command-guided air-to-surface and antitank munitions were tracked by eye and
controlled by hand; later the naked eye gave way to enhanced optics and television tracking,
which often operated in the infrared range and issued commands generated automatically by
computerized fire-control systems.
Another early command guidance method was beam riding, in which the missile sensed a
radar beam pointed at the target and automatically corrected back to it. Laser beams were
later used for the same purpose.
Also using a form of command guidance were television-guided missiles, in which a small
television camera mounted in the nose of the weapon beamed a picture of the target back to
an operator who sent commands to keep the target centred in the tracking screen until impact.
A form of command guidance used from the 1980s by the U.S. Patriot surface-to-air system was
called track-via-missile. In this system a radar unit in the missile tracked the target and
transmitted relative bearing and velocity information to the launch site, where control systems
computed the optimal trajectory for intercepting the target and sent appropriate commands
back to the missile.
Inertial
Inertial guidance was installed in long-range ballistic missiles in the 1950s, but, with
advances in miniaturized circuitry, microcomputers, and inertial sensors, it became common
in tactical weapons after the 1970s.
Inertial systems involved the use of small, highly accurate gyroscopic platforms to
continuously determine the position of the missile in space. These provided inputs to
guidance computers, which used the position information in addition to inputs from
accelerometers or integrating circuits to calculate velocity and direction.
The guidance computer, which was programmed with the desired flight path, then
generated commands to maintain the course.
An advantage of inertial guidance was that it required no electronic emissions from the
missile or launch platform that could be picked up by the enemy.
Many antiship missiles and some long-range air-to-air missiles, therefore, used inertial
guidance to reach the general vicinity of their targets and then active radar guidance for
terminal homing.
Passive-homing antiradiation missiles, designed to destroy radar installations, generally
combined inertial guidance with memory-equipped autopilots to maintain their trajectory
toward the target in case the radar stopped transmitting.
Active
With active guidance, the missile would track its target by means of
emissions that it generated itself. Active guidance was commonly used
for terminal homing. Examples were antiship, surface-to-air, and air-toair missiles that used self-contained radar systems to track their
targets. Active guidance had the disadvantage of depending on
emissions that could be tracked or jammed.
Semiactive
Semiactive guidance involved illuminating or designating the target with energy
emitted from a source other than the missile; a seeker in the projectile that was
sensitive to the reflected energy then homed onto the target. Like active guidance,
semiactive guidance was commonly used for terminal homing.
Example:
In the U.S. Hawk and Soviet SA-6 Gainful antiaircraft systems. The missile homed in
on radar emissions transmitted from the launch site and reflected off the target,
measuring the Doppler shift in the reflected emissions to assist in computing the
intercept trajectory. (SA-6 Gainful is a designation given by NATO to the Soviet
missile system. The AIM-7 Sparrow air-to-air missile of the U.S. Air Force used a
similar semiactive radar guidance method.
With semiactive homing the designator or illuminator might be remote from the
launch platform. The U.S. Hellfire antitank missile, for example, used laser
designation by an air or ground observer who could be situated many miles from
the launching helicopter.
Passive
Definition:
Passive guidance systems neither emitted energy nor received commands from an external source; rather, they
“locked” onto an electronic emission coming from the target itself.
The earliest successful passive homing munitions were “heat-seeking” air-to-air missiles that homed onto the
infrared emissions of jet engine exhausts. The first such missile to achieve wide success was the AIM-9
Sidewinder developed by the U.S. Navy in the 1950s. Many later passive homing air-to-air missiles homed onto
ultraviolet radiation as well, using on-board guidance computers and accelerometers to compute optimal
intercept trajectories. Among the most advanced passive homing systems were optically tracking munitions
that could “see” a visual or infrared image in much the same way as the human eye does, memorize it by
means of computer logic, and home onto it. Many passive homing systems required target identification and
lock-on by a human operator prior to launch. With infrared antiaircraft missiles, a successful lock-on was
indicated by an audible tone in the pilot’s or operator’s headset; with television or imaging infrared systems,
the operator or pilot acquired the target on a screen, which relayed data from the missile’s seeker head, and
then locked on manually.
Passive guidance systems benefited enormously from a miniaturization of electronic components and from
advances in seeker-head technology. Small, heat-seeking, shoulder-fired antiaircraft missiles first became a
major factor in land warfare during the final stages of the Vietnam War, with the Soviet SA-7 Grail playing a
major role in neutralizing the South Vietnamese Air Force in the final communist offensive in 1975. Ten years
later the U.S. Stinger and British Blowpipe proved effective against Soviet aircraft and helicopters in
Afghanistan, as did the U.S. Redeye in Central America.
Tactical
Missiles
Antitank and
antiarmor
Air to surface
Air to air
Antiship
Surface to Air
(SAM)
SS-10/SS-11 (French),
TOW (US), UH-1,
Swingfire, Milan, HOT
AGM-12, AGM-45,
AGM-78, AGM-88, AS-7,
AS-8, AS-9
Firebird, AIM-4 Falcon,
AIM-9 Sidewinder, AIM7 Sparrow, AIM-54
Phoenix, AA-1 Alkali
(Soviet)
Hs-293 (Germany), AS-1
Kennel (Soviet), Sea
Skua (British), Harpoon
(US), Tomahawk
SA-1 Guild, SA-6
Gainful, SA-8 Gecko, SA7 Grail, SA-9 Gaskin,
Nike Hercules
Antitank and Antiarmor
• One of the most important categories of guided missile to emerge after World War II was the antitank, or antiarmour,
missile. A logical extension of unguided infantry antitank weapons carrying shaped-charge warheads for penetrating
armour, guided antitank missiles acquired considerably more range and power than their shoulder-fired predecessors. The
tactical flexibility and utility of guided antitank missiles led to their installation on light trucks, on armoured personnel
carriers, and, most important, on antitank helicopters.
• France
S-10/S-11
MILAN & HOT (Equivalent in capability to TOW)
• USA
TOW equipped UH-1
Hellfire
• British
• Swingfire
• Soviets
•
•
•
•
AT-1 Snapper
AT-2 Swatter
AT-3 Sagger
AT-6 Spiral, a Soviet version of TOW and Hellfire (became the principal antiarmour munition of Soviet attack helicopters)
Guidance:
• Electronic commands transmitted along extremely thin wires
• later generations transmitted guidance commands by radio rather than by wire, and semiactive laser designation and
passive infrared homing also became common.
Wire Guided Missile
HellFire
Air-to-surface
• The United States began to deploy tactical air-to-surface guided missiles as a standard
aerial munition in the late 1950s. The first of these was the AGM-12 (for aerial guided
munition) Bullpup, a rocket-powered weapon that employed visual tracking and
radio-transmitted command guidance. The pilot controlled the missile by means of a
small side-mounted joystick and guided it toward the target by observing a small flare
in its tail.
• USA
AGM-45 Shrike (Passively homing onto their radar emissions, no memory circuits and required
continuous emissions for homing, turning off the target radar)
AGM-78 Standard ARM (incorporated memory circuits and could be tuned to any of several frequencies
in flight)
AGM-88 HARM (introduced into service in 1983)
• Soviets
• AS-7 Kerry (Radio-command-guided)
• AS-8 and AS-9
• AS-10 Karen and AS-14 Kedge (television-guided, the last with a range of about 25 miles)
These missiles were fired from tactical fighters such as the MiG-27 Flogger and attack
helicopters such as the Mi-24 Hind and Mi-28 Havoc.AT-2 Swatter
Air-to-Air
• The radar-guided, subsonic Firebird was the first U.S. guided air-to-air missile. Early versions,
which homed onto the infrared emissions from jet engine tailpipes, could approach only
from the target’s rear quadrants. Later versions were fitted with more sophisticated seekers
sensitive to a broader spectrum of radiation. These gave the missile the capability of sensing
exhaust emissions from the side or front of the target aircraft.
• USA
AIM-4 Falcon
AIM-9 Sidewinder
AIM-7 Sparrow
AIM-9L
AIM-54 Phoenix
• Soviets
• AA-1 Alkali (a relatively primitive semiactive radar missile)
• AA-2 Atoll (an infrared missile closely modeled after the Sidewinder)
• AA-3 Anab (a long-range, semiactive radar-homing missile carried by air-defense fighters).
• AA-5 Ash (was a large, medium-range radar-guided missile)
• AA-6 Acrid (was similar to the Anab but larger and with greater range)
• AA-7 Apex, (a Sparrow equivalent)
• AA-8 Aphid (a relatively small missile for close-in use, were introduced during the 1970s)
Antiship
• Despite their different methods of delivery, antiship missiles formed a coherent
class largely because they were designed to penetrate the heavy defenses of
warships.
• USA
Tomahawk
• Soviets
• AS-3 Kangaroo, introduced in 1961 with a range exceeding 400 miles.
• AS-4 Kitchen, a Mach-2 (twice the speed of sound) rocket-powered missile with a range
of about 250 miles
• Rocket-powered Mach-1.5 AS-5 Kelt was first deployed in 1966
• The Mach-3 AS-6 Kingfish, introduced in 1970, could travel 250 miles
Surface-to-Air
• The Guided surface-to-air missiles, or SAMs, were under development when
World War II ended, notably by the Germans, but were not sufficiently
perfected to be used in combat. This changed in the 1950s and ’60s with the
rapid development of sophisticated SAM systems in the Soviet Union, the
United States, Great Britain, and France. With other industrialized nations
following suit, surface-to-air missiles of indigenous design, particularly in the
smaller categories, were fielded by many armies and navies.
• Soviets
• SA-1 Guild
• SA-3 Goa
• SA-5 Gammon
• SA-10 Grumble
Strategic
Missiles
Ballistic Missiles
• Ballistics, science of the propulsion, flight, and impact of
projectiles.
• Ballistic missile, a rocket-propelled self-guided strategicweapons system that follows a ballistic trajectory to deliver
a payload from its launch site to a predetermined target.
• Ballistic missiles can carry conventional high explosives as
well as chemical, biological, or nuclear munitions.
• They can be launched from aircraft, ships, and submarines
in addition to mobile platforms
Ballistic Missile Guidance
Cruise Missile
• Cruise missiles, on the other hand, are powered continuously by airbreathing jet engines and are sustained along a low, level flight path
by aerodynamic lift.
Missile defense system Israel Iron Dome
Tactical
Missile Guidance
Collision Course
Let Target T is moving with a constant velocity 𝑽𝑻 as it does not
change direction and magnitude.
Then we have a pursuer with a constant velocity 𝑽𝒑
A
The line Between the Target T and Pursuer P is a Range vector
From a range vector we establish a target heading angle A.
Now the pursuer need a velocity to collide with target
What will be the magnitude and direction of 𝑽𝒑
Let the pursuer is moving with a fixed velocity 𝑽𝒑 then there
Will be lead angle between the range vector and velocity vector
L
P
T
Collision Course
A
L
P
T
Collision Course
A
L
P
T
Collision Course
A
L
P
T
Collision Course
A
L
P
T
Collision Course
It requires
Constant Target velocity 𝑽𝑻
Constant Pursuer Velocity 𝑽𝒑
A
Lead angle
and a geometric relationship
At this point Differential equations
are not required we can use simple
geometry
L
P
T
• Because the velocities are constant
so we can generate the collision
triangle based on velocity vector
• From the Law of Sines
• Solving for target heading
A
𝑻
T
𝑹 relative
𝑷
Look Angle
velocity
L
P
Look Angle: Lead Angle(when body is aligned with
,Look angle is reference to body axes)
• Collision is geometric for non-maneuvering bodies with collision
triangle similarity and the law of sines.
• Collision for targets having greater heading or speed require the
pursuer to go faster or lead more.
• Greater velocity ratio require less pursuer lead for collision with a
given target heading.
• Greater lead and look pursuers require less velocity ratio for collision
with a given target heading
Qualitatively and Quantitatively Collision
course in terms of Line-of-sight Direction
In collision Triangle the range vector orientation does not
A
changes as time increases.
It means that the line-of-sight direction remain
Constant.
L
P
T
Qualitatively and Quantitatively Collision
course in terms of Line-of-sight Direction
In collision Triangle the range vector orientation does not
A
changes as time increases.
It means that the line-of-sight direction remain
Constant.
Qualitatively parallel Line of sights at
all instances of time
L
P
T
Qualitatively and Quantitatively Collision
course in terms of Line-of-sight Direction
In collision Triangle the range vector orientation does not
A
changes as time increases.
It means that the line-of-sight direction remain
Constant.
Qualitatively parallel Line of sights at
all instances of time
Quantitatively the line-of-sight angle is constant and LOS
rate is zero. (but zero with respect to what frame of
reference?)
L
P
T
Qualitatively and Quantitatively Collision
course in terms of Line-of-sight Direction
In collision Triangle the range vector orientation does not
changes as time increases.
A
It means that the line-of-sight direction remain
T
Constant.
Qualitatively parallel Line of sights at
all instances of time
Quantitatively the line-of-sight angle is constant and LOS
rate is zero. (but zero with respect to what frame of
reference?)
𝜸
L
Flight Path Angle
Angle of the velocity vector 𝑉 with the horizon
P
(LOS angle)
Inertial reference
Qualitatively and Quantitatively Collision
course in terms of Line-of-sight Direction
In collision Triangle the range vector orientation does not
changes as time increases.
It means that the line-of-sight direction remain
Constant.
A
T
Qualitatively parallel Line of sights at
all instances of time
Quantitatively the line-of-sight angle is constant and LOS
rate is zero. (but zero with respect to what frame of
reference?) Collision is ensured when 𝝀 is constant
or 𝝀̇ is zero
L
P
(LOS angle)
Inertial reference
How the pursuer Knows that he is on the collision course?
If the Line-of-sight rate is zero or
Regulating the LOS rate is crucial in maintaining the collision course
Proportional Navigation
• When two bodies are not on a collision course then how we can
enforce them? (Proportional Navigation is a law that enforce it).
• It is implemented as a part of the Homing Loop.
• It is the fundamental of advanced missile guidance. Therefore, its
understanding ProNav is understanding missile guidance.
T
P
Flight Path Angle
Angle of the velocity vector 𝑉 with the horizon
ProPortional navigation
2
1
ProPortional navigation
2
Altitude
1
downrange
ProPortional navigation
𝑉
2
Altitude
Target
1
downrange
ProPortional navigation
𝑛
𝑉
2
Altitude
Target
1
downrange
ProPortional navigation
𝑛
𝑉
2
Altitude
𝛽
Target
1
downrange
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
1
downrange
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
1
downrange
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
𝑛
𝑅
1
downrange
From a guidance point of view, we desire to make the range between missile and target at the expected intercept time as small
as possible (hopefully zero). The point of closest approach of the missile and target is known as the miss distance.
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
𝑅
𝑛
𝐿 + 𝐻𝐸
1
downrange
𝐻𝐸 : This angle represents the initial deviation of the missile from the collision triangle.
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
𝑅
𝑛
𝐿 + 𝐻𝐸
𝜆
1
downrange
ProPortional navigation
𝑛
𝑉
𝛽
2
Target
Altitude
𝑉
𝑅
𝑛
𝐿 + 𝐻𝐸
𝜆
1
downrange
• The closing velocity is defined as the negative rate of change of the
distance from the missile to the target
• Therefore, at the end of the engagement, when the missile and target
are in closest proximity, the sign of will change.
• In the engagement model, the target can maneuver evasively with
acceleration magnitude .
• The angular velocity of the target:
ComPonents of target veloCity
𝑛
𝑉
𝑉
2
= 𝑉 sin 𝛽
𝛽
𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑅
𝑛
𝐿 + 𝐻𝐸
𝜆
1
downrange
target Position ComPonents
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑅
𝑛
𝐿 + 𝐻𝐸
𝜆
1
downrange
missile veloCity ComPonents
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
1
downrange
missile Position ComPonents
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
1
downrange
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑻𝟏
1
downrange
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑹𝑻𝟐
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑻𝟏
1
downrange
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑹𝑻𝟐
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑴𝟏
𝑹𝑻𝟏
1
downrange
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
𝑉
= −𝑉 cos 𝛽
Target
𝑹𝑻𝟐
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑴𝟏
𝑹𝑻𝟏
𝑹𝑴𝟐
1
downrange
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
= −𝑉 cos 𝛽
Target
𝑹𝑻𝑴𝟐 = (𝑹𝑻𝟐 − 𝑹𝑴𝟐 )
𝑉
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝜆
𝑅̇
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑴𝟏
𝑹𝑻𝟏
𝑹𝑴𝟐
1
downrange
𝑹𝑻𝟐
missile target seParation
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
= −𝑉 cos 𝛽
Target
𝑹𝑻𝑴𝟐 = (𝑹𝑻𝟐 − 𝑹𝑴𝟐 )
𝑉
𝑅
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝑅̇
𝜆
𝑹𝑻𝑴𝟏 = (𝑹𝑻𝟏 − 𝑹𝑴𝟏 )
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑴𝟏
𝑹𝑻𝟏
𝑹𝑴𝟐
1
downrange
𝑹𝑻𝟐
CalCulation of line-of-sight angle
𝑛
𝑉
𝑅̇
2
=𝑉
= 𝑉 sin 𝛽
𝛽
𝑅̇
=𝑉
Altitude
= −𝑉 cos 𝛽
𝑅
𝑉
Target
𝑹𝑻𝑴𝟐 = (𝑹𝑻𝟐 − 𝑹𝑴𝟐 )
𝑛
𝐿 + 𝐻𝐸
𝑅̇
=𝑉
𝑉̇
= 𝑉 sin(𝐿 + 𝐻𝐸 + 𝜆)
=𝑎
𝑅̇
𝜆
𝑹𝑻𝑴𝟏 = (𝑹𝑻𝟏 − 𝑹𝑴𝟏 )
=𝑉
𝑉̇
= −𝑉 cos 𝐿 + 𝐻𝐸 + 𝜆
=𝑎
𝑹𝑴𝟏
𝑹𝑻𝟏
𝑹𝑴𝟐
1
downrange
𝑹𝑻𝟐
• Relative Velocity Components
• Line-of-sight angle
𝑑
𝑑
𝑑𝑥
tan 𝑥 =
tan 𝑥
𝑑𝑡
𝑑𝑥
𝑑𝑡
𝑑
𝑑
tan 𝑥 =
tan 𝑥 𝑥̇
𝑑𝑡
𝑑𝑥
• Closing Velocity
• Missile guidance law
• Missile acceleration components in earth coordinates
𝑅
= 𝑅
+𝑅
• line-of-sight angle calculations
Using law of sines
Summary
•
•
•
•
•
•
•
•
(
•
(
)
)
TWO-DIMENSIONAL ENGAGEMENT SIMULATION
• Initializing Parameters
(0,10000)
n=0;
VM = 3000.;
VT = 1000.;
XNT = 0.;
HEDEG = -20.;
XNP = 5.;
RM1 = 0.;
RM2 = 10000.;
RT1 = 40000.;
RT2 = 10000.;
BETA=0.;
2
Altitude
𝐻𝐸 = −20
(40000,10000)
𝑉 = 1000
𝛽=0
Target
1
downrange
TWO-DIMENSIONAL ENGAGEMENT SIMULATION
• Using formulas
VT1=-VT*cos(BETA);
VT2=VT*sin(BETA);
HE=HEDEG/57.3;
T=0.;
S=0.;
RTM1=RT1-RM1;
RTM2=RT2-RM2;
RTM=sqrt(RTM1*RTM1+RTM2*RTM2);
XLAM=atan2(RTM2,RTM1);
XLEAD=asin(VT*sin(BETA+XLAM)/VM);
THET=XLAM+XLEAD;
VM1=-VM*cos(THET+HE);
VM2=VM*sin(THET+HE);
VTM1 = VT1 - VM1;
VTM2 = VT2 - VM2;
VC=-(RTM1*VTM1 + RTM2*VTM2)/RTM;
% Calculation of Line-of-sight angle
% Calculation of lead angle
% summation of Lead and line-of-sight angle
%
% Calculation of closing velocity
TWO-DIMENSIONAL ENGAGEMENT SIMULATION
while VC >= 0
if RTM <1000
H=.0002;
else
H=.01;
end
STEP=1;
FLAG=0;
while STEP <=1
if FLAG==1
STEP=2;
BETA=BETA+H*BETAD;
RT1=RT1+H*VT1;
RT2=RT2+H*VT2;
RM1=RM1+H*VM1;
RM2=RM2+H*VM2;
VM1=VM1+H*AM1;
VM2=VM2+H*AM2;
T=T+H;
end
RTM1=RT1-RM1;
RTM2=RT2-RM2;
RTM=sqrt(RTM1*RTM1+RTM2*RTM2);
VTM1=VT1-VM1;
VTM2=VT2-VM2;
VC=-(RTM1*VTM1+RTM2*VTM2)/RTM;
XLAM=atan2(RTM2,RTM1);
XLAMD=(RTM1*VTM2RTM2*VTM1)/(RTM*RTM);
XNC=XNP*VC*XLAMD;
AM1=-XNC*sin(XLAM);
AM2=XNC*cos(XLAM);
VT1=-VT*cos(BETA);
VT2=VT*sin(BETA);
BETAD=XNT/VT;
FLAG=1;
end
FLAG=0;
S=S+H;
if S >=.09999
S=0.;
n=n+1;
ArrayTN5(n)=T;
ArrayRT1N5(n)=RT1;
ArrayRT2N5(n)=RT2;
ArrayRM1N5(n)=RM1;
ArrayRM2N5(n)=RM2;
ArrayXNCGN5(n)=XNC/32.2;
ArrayRTMN5(n)=RTM;
end
end
Proportional
Navigation
Acceleration Profile
Maneuvering Target
Maneuvering Target acceleration profile
Linearization
2
𝐿 + 𝐻𝐸
𝜆
𝛽
Target
1
downrange
CalCulation of line-of-sight angle
𝑛
𝑉
(180 − 90 − 𝛽)
Target
𝛽
2
Altitude
𝑅
𝑉
𝒀
𝒀𝑻
𝑛
𝑛 𝑠𝑖𝑛(180 − 90 − λ)
𝑛 𝑠𝑖𝑛 90 − 𝜆
𝑛 𝑐𝑜𝑠 𝜆
𝐿 + 𝐻𝐸
𝜆
𝒀𝑴
(180 − 90 − 𝜆)
1
downrange
𝑛 𝑠𝑖𝑛(180 − 90 − 𝛽)
𝑛 𝑠𝑖𝑛 90 − 𝛽
𝑛 𝑐𝑜𝑠𝛽
• Relative Acceleration
For small angles
𝑦̈ = 𝑛 cos 𝛽 − 𝑛 cos(𝜆)
𝑦̈ = 𝑛 − 𝑛
The expression of Line of sight angle can be linearized
𝑦
𝜆=
𝑅
For head on collision the closing velocity can be approximated as
𝑉 =𝑉 +𝑉
For Tail chase
𝑉 =𝑉 −𝑉
Range
𝑅 =𝑉 𝑡 −𝑡
Miss distance
𝑀𝑖𝑠𝑠 = 𝑦(𝑡 )
TWO-DIMENSIONAL ENGAGEMENT SIMULATION Linearized
• Initializing Parameters
(0,10000)
• XNT=0.;
• Y=0.;
• VM=3000.;
2
Altitude
𝐻𝐸 = −20
(40000,10000)
𝑉 = 1000
𝛽=0
Target
• VT=1000.;
• HEDEG=-20.;
• TF=10.;
• XNP=4.;
• YD=-VM*HEDEG/57.3;
• T=0.;
• H=.01;
• S=0.;
• n=0.;
• VC=VM-VT;
1
downrange
TWO-DIMENSIONAL ENGAGEMENT SIMULATION Linearized
while T<=(TF-1e-5)
% YOLD=Y;
% YDOLD=YD;
STEP=1;
FLAG=0;
while STEP<=1
if FLAG==1
STEP=2;
Y=Y+H*YD;
YD=YD+H*YDD;
T=T+H;
end
TGO=TF-T+.00001;
XLAMD=(Y+YD*TGO)/(VC*TGO);
XNC=XNP*VC*XLAMD;
YDD=XNT-XNC;
FLAG=1;
end
FLAG=0;
% Y=.5*(YOLD+Y+H*YD);
% YD=.5*(YDOLD+YD+H*YDD);
S=S+H;
if S>=.0999
S=0.;
n=n+1;
ArrayTL(n)=T;
ArrayY(n)=Y;
ArrayYD(n)=YD;
ArrayXNCG(n)=XNC/32.2;
end
.
Acceleration Profile
Maneuvering Target acceleration profile
Homing Loop
Airframe design
Atmospheric model
Autopilot
Actuator Dynamics
Dynamic model of missile
Aerodynamic model
Aerodynamic coefficients
Guidance
Seeker/Tracker
Line of sight angle calculation
Simulation
results
Results
Assignment (Deadline: 15-11-2024)
• Derive the equations for zero effort miss distance.
• Design the augmented and optimal guidance law and implement it
for engagement simulation. (CLO-02)
• Book
Textbook 1 chapter 8 (page 163- Page 185)