11 fighter aircraft avionics - part iv

Fighter Aircraft Avionics
Part IV
SOLO HERMELIN
Updated: 04.04.13
1

Table of Content
SOLO
2
Introduction
Jet Fighter Generations
Second Generation (1950-1965(
Third Generation (1965-1975(
First generation (1945-1955(
Fourth Generation (1970-2010(
4.5Generation
Fifth Generation (1995 - 2025(
Aircraft Avionics
Third Generation Avionics
Fourth Generation Avionics
4.5Generation Avionics
Fifth Generation Avionics
Cockpit Displays
Communication (internal and external(
Data Entry and Control
Flight Control
Fighter Aircraft

Table of Content (continue – 1(
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Aircraft Propulsion System
Aircraft Flight Performance
Navigation
Earth Atmosphere
Flight Instruments
Power Generation System
Environmental Control System
Aircraft Aerodynamics
Fuel System
Jet Engine
Vertical/Short Take-Off and Landing (VSTOL(
Engine Control System
Flight Management System
Aircraft Flight Control
Aircraft Flight Control Surfaces
Aircraft Flight Control Examples
Fighter
Aircraft
Avionics
I
I

Table of Content (continue – 2(
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4
Equations of Motion of an Air Vehicle in Ellipsoidal Earth Atmosphere
Fighter Aircraft Weapon System
References
Safety Procedures
Tracking Systems
Aircraft Sensors
Airborne Radars
Infrared/Optical Systems
Electronic Warfare
Air-to-Ground Missions
Bombs
Air-to-Surface Missiles (ASM( or Air-to-Ground Missiles (AGM(
Fighter Aircraft Weapon Examples
Air-to-Air Missiles (AAM(
Fighter Gun
Avionics III

Continue from
Part III
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Fighter/attack aircraft can carry a number of items fastened to racks underneath the aircraft.
These items are called ‘‘Stores’’ and include Weapons (Bombs, Rockets, Missiles(, Extra Fuel
Tanks, Extra Sensor Pods, or Decoys (e.g., Chaff to fool radar-guided missiles and Flares to
fool infra-red guided missiles(. The Stores Management System (SMS( manages the
mechanical and electrical connections to weapons and senses their status under control of
the Mission Central Computer (MCC(; thus all weapons are readied via the SMS.
Weapons carried may include Rockets, Bombs (both Ballistic-dumb and Radar, Infra-red, or TV
guided(, and Missiles (which are typically ‘‘Fire and Forget’’ Self-guided using TV video, Laser,
Imaging Infra-red, or Radar Seekers(. Most aircraft also have internal fuselage-mounted
Guns.
Weapon release modes include automatic (AUTO( and Continuously Computed Impact Point
(CCIP( plus special modes for Guided Weapons. In AUTO mode, the MCC controls weapon
release based on computed impact point, current target position, and predicted aircraft position
at release. In CCIP mode, the MCC computes a predicted impact point which is displayed
on the HUD, and the aircrew controls weapon release with the bomb button on the
HOTAS.
Stores

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8
The Aircraft part of the Weapons System is checked for Operability and Safety on the
Ground before the Weapons are Loaded. After the Weapons are Loaded on the Stations
and Power (External or Aircraft Internal( and recognized in the Weapon System
Inventory (Weapon Type and Station( the Weapons Power Bit check the Weapon
Servicibility. This information is displayed to the Avionics.
The Weapons can be loaded on a Fighter Aircraft on the existing External Weapon
Stations or if available on Internal Bay Stations (F-22, F-35( .
When the Aircraft is on the Ground the Weapon
Launching Signal are disabled. In addition, usually
the Weapons are in a Safe Mode.
The Weapons can be Launched only when the
Aircraft is on the Air and the Pilot activated the
MASTER ARM switch. The Launching sequence
can Start after activated the Launch Switch that is
usually located on the Flight Control Stick.
The Launching sequence is defined to assure the
Safety of the Launching Aircraft.
The Weapons System will indicate a Successful or
Unsuccessful Launch and will choose the Next
Weapon to be Launched according to a predefined
sequence.Weapon Management Displays

SOLO Fighter Aircraft Weapon System
The Weapon System advises the Pilot how to Launch the Weapons.
In general from the Third Fighter Generation and up the Aircraft Weapon System
included a Computer that provided Flight Instruction Displays for the Pilot, to
Release Bombs or Launch Missiles (A/A or A/G(.
Target Designation
The Aircrew may designate a Target for A/A or A/G Attack in one of two ways:
by Radar or by HUD/HMD designate.
To designate a target by Radar, the Radar must already be tracking a Target. The Radar
Target is identified as the Target by a Member of the Aircrew pushing the designate
switch on the HOTAS.
To perform a HUD/HMD designation, the Aircrew must first position the HUD/HMD
reticle (on the HUD( using the Target Designator Controller (TDC( Switch on the
HOTAS (the TDC Switch is similar to a Joystick(. Once the HUD Reticle is properly
positioned, the aircrew pushes down on the TDC switch to designate a target.
The MCC must transform the HUD/HMD Reticle position from HUD coordinates to
obtain Range, Azimuth, and Elevation to Target. No matter how the Target was
designated, the HUD/HMD Reticle changes shape to indicate that a Target is
Designated. A Designated Target may be undesignated by pushing the Undesignate
Switch on the HOTAS.

A/G Weapon Selection
Weapon selection includes selecting the type of Weapon, the number to drop, and the
desired spacing on the ground. This is done by the aircrew using the MPD stores display and
Keyset switches. Depending on the type of weapon selected, a default delivery mode is defined
and displayed. At any time prior to weapon release, the aircrew may push the AUTO/CCIP
toggle switch on the Keyset, causing the delivery mode to change from AUTO to CCIP or from
CCIP to AUTO. Weapon-ready determination is also assumed to be part of this function.
Mode Selection
The Pilot may choose between Air-to-Air (A/A( and Air-to-Ground (A/G(
Steering in A/G Mode
Compute the Steering Cues for display on the HUD/HMD and MPD based either on Waypoint
Steering or Target Attack Steering. The MCC can hold a Number of Aircrew-entered Waypoints
(Latitude, Longitude, Altitude( which may be used as Steer-to Points and as Target Designation
Points. The Aircrew may also associate an Offset (Range, Bearing( from the currently selected
Waypoint which is taken into account. Prior to Target Designation, Steering Cues are provided
based on the Currently Selected Waypoint (if any(. After Target Designation, Steering Cues are
provided based on Target Location relative to Aircraft Position

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Air-to-Ground Missions
11
MULTI-COMMAND HANDBOOK 11-F16

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Bombs:
-Dumb (Gravity( Bombs
- Guided (Smart( Bombs
* TV Bombs (Wallay(
* Laser Guided Bombs (Paveway(
* Gliding Bombs with Data Link and IR/Optical Seeker
* Inertial/GPS Bombs (JDAM(
* Inertial/GPS/EO (Spice(
* Small Diameter Bombs
USAF artist rendering of JDAM kits
fitted to Mk 84, BLU-109, Mk 83,
and Mk 82 unguided bombs
GBU-39 Small Diameter Bomb
Armement Air-Sol Modulaire
(Air-to-Ground Modular Weapon(
(AASM(

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Dumb Bombs Delivery
There is the possibility to program visual cues
in the computer of the F-16. Beside waypoints
there are 4 types of cues. These are called
VIP, VRP, PUP and OA’s.
VIP = Visual Initial Point
VRP = Visual Reference Point
PUP = Pull Up Point
OA = Offset Aim
The Bomb Delivery in Type 3 Fighters and up is done by the Weapon Delivery Computer.
The Pilot chooses the Bomb Delivery Mode (TOSS, LAT, CCIP,..( in A/G Mode,
Designates the Ground Target using the Gun Sight or HUD and after this the Weapon
System provides Flight Instruction and Automatically Releases the Bombs.

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Dumb Bombs Delivery (continue – 1(
Pop-Up
This type of delivery can be useful for all static targets.
Think about buildings, bridges, runways and even
vehicles. The ordnance that can be used is the whole
range from low and high drag dumb bombs, cluster and
laser guided bombs.
TOSS (English word for throwing something up in the air(
For a low level ingress we should use a LAT delivery. LAT
stands for Low Altitude TOSS. During this delivery the
bomb will be released upwards. The range will become
greater but the accuracy smaller. Therefore the best type of
bomb used will be a cluster bomb. This is a very nice way to
attack a group of vehicles like a SA-2 or SA-3 site. But also
freefall bombs can be used against large targets.
High Altitude Dive Bombing (HADB(
This delivery should keep the attacker above a
planned altitude and can be used for hitting all
types of static target like buildings, bridges and
vehicles. Any type of bomb can be used. It is also
possible to use missiles like the AGM-65 with this
delivery.

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CCIP (Continuous Computed Impact Point(
The objective of a CCIP delivery is to fly the Aircraft in a manner to arrive at or close
to the Planned Release Parameters (Altitude, Airspeed and Dive Angle( with the CCIP
Cue close to the Intended Aiming Point. When the CCIP Cue superimposes the
Target, the Pickle Button / Trigger should be actuated to initiate Weapons Release /
Firing

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For Dumb Bombs the MCC solves the ballistic trajectory equations of motion.
This is done initially to determine Weapon Time of Fall when the Estimated Time-to-Go to
Release (based on Aircraft Ground Speed and Target Ground Range( is less than one
minute.
Initialization must be repeated if a New Target is Designated. Once initialized, the Weapon
Trajectory must be computed at least every 100 ms. Outputs include Time-to-Go to Release,
Weapon Time of Fall, Down Range Error, and Cross Range Error. When Time-to-Go to
Release falls below ΔT ms. and AUTO delivery mode is selected, Weapon Release is
scheduled.
Thereafter, whenever Time-to-Go to Release is recomputed, Weapon Release is
rescheduled.

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Air-to-Surface Missiles (ASM( or Air-to-Ground Missiles (AGM(
An air-to-surface missile (ASM( or air-to-ground missile (AGM or ATGM( is a missile
designed to be launched from military aircraft (bombers, attack aircraft, fighter
aircraft or other kinds( and strike ground targets on land, at sea, or both. They are
similar to guided glide bombs but to be deemed a missile, they usually contain some
kind of propulsion system. The two most common propulsion systems for air-to-
surface missiles are Rocket Motors and Jet Engines. These also tend to correspond to
the range of the missiles — short and long, respectively. Some Soviet air-to-surface
missiles are powered by Ramjets, giving them both long range and high speed.
AGM-65 Maverick
Electro-optical, Laser, or
Infra-red Guidance Systems
TAURUS KEPD 350
IBN (Image Based Navigation(,
INS (Inertial Navigation System(,
TRN (Terrain Referenced
Navigation( and MIL-GPS
Guidance System
Storm Shadow
Inertial, GPS and TERPROM.
Terminal guidance using imaging
infrared
AGM-158 JASSM
(Joint Air-to-Surface Standoff Missile(
INS/GPS Guidance

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An air-to-air missile (AAM( is a missile fired from an aircraft for the purpose of
destroying another aircraft. AAMs are typically powered by one or more rocket motors,
usually solid fuelled but sometimes liquid fuelled. Ramjet engines, as used on the MBDA
Meteor (currently in development(, are emerging as propulsion that will enable future
medium-range missiles to maintain higher average speed across their engagement
envelope.
Air-to-air missiles are broadly put in two groups. The first consists of missiles designed
to engage opposing aircraft at ranges of less than approximately 20 miles (32 km(, these
are known as short-range or “within visual range” missiles (SRAAMs or WVRAAMs(
and are sometimes called “dogfight” missiles because they emphasize agility rather than
range. These usually use infrared guidance, and are hence also called heat-seeking
missiles. The second group consists of medium- or long-range missiles (MRAAMs or
LRAAMs(, which both fall under the category of beyond visual range missiles
(BVRAAMs(. BVR missiles tend to rely upon some sort of radar guidance, of which there
are many forms, modern ones also using inertial guidance and/or "mid-course updates".
Air-to-Air Missiles (AAM(
A detailed description on the subject can be founded in the Power Point
“Air Combat” Presentation. Here we give a brief summary of the subject.

Air- to-Air missile launch envelope

Kinematics no-escape-zone
Return to Table of Content

01-21
Air-to-Air Missiles Modes of Operation

Lock-On Before Launch
•High agility
•Tight radius turn
•Excellent minimum ranges
Active Homing Phase
• IMU alignment
• Radar slave- full target data
• HMD Slave- partial target data
• Seeker activation
• Target Lock-On
Pre Launch Phase

01-23
2
• Inertial navigation
• Trajectory shaping for maximum range
Midcourse Guidance Phase
• IMU alignment
• Target data transfer
Lock-On After Launch
3
• Seeker activation
• Target Lock-On
• Final homing
Homing Phase
1 Pre Launch Phase

AMRAAM
A/A MISSILES
AMRAAM AIM - 120C-5 Specifications
Length: 12 ft 3.65 m
Diameter: 7 in 17.8 cm
Wing Span: 17.5 in 44.5 cm
Fin Span: 17.6 in 44.7 c
Weight: 356 lb 161.5 kg
Warhead: 45 lb 20.5 Kg
Guidance: Active Radar
Fuzing: Proximity (RF( and Contact
Launcher: Rail and eject
AIM-120C
Rocket motor PN G672798-1 is an enhanced
version with additional 5” (12 cm( of propellant.
Estimation: add 10% (12/140( to obtain
mp ~ 52 kg
Wtot ~ 120,000 N s
AMRAAM AIM-120 Movie

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A-A Missiles Development in RAFAEL
BVRBVR
Short RangeShort Range
PYTHON-4PYTHON-4
PYTHON-3PYTHON-3
SHAFRIR-2SHAFRIR-2
SHAFRIR-1SHAFRIR-1
PYTHON-5PYTHON-5
DERBYDERBY
Rafael Python 5
Promo, Movie
Derby - Beyond Visual Range Air-to-Air Missile, Movie

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Evolution of Air-to-Air Missiles in RAFAEL
PYTHON-4PYTHON-4
1st GENERATION
SHAFRIR-1SHAFRIR-1
2nd GENERATION
SHAFRIR-2SHAFRIR-2
3rd GENERATION
PYTHON-3PYTHON-3
4th GENERATION
SERVICE: SINCE 1993SERVICE SINCE 1978
HITS: OVER 35 A/C
DURING 1982 WAR
SERVICE: 1968-1980
HITS: OVER 100 A/C
DURING 1973 WAR
SERVICE: 1964-1969
0°-(10°)
30°
180°
45°
30°
LEAD/LAG
ANGLE
0°
MAX.
ASPECT
ANGLE
TYPICAL 3rd
GENERATION
MISSILE
LAUNCHER
Short Range
DERBYDERBY
ACTIVR BVR
Dual Range
PYTHON-5
5th GENERATION
Full Sphere IR
Missile
Full Scale Development

2.9
3.
6
Russian Air-to-Air Missiles
RVV-MD, RVV-BD New Generation Russian Air-to-Air Missiles, Movie
Russian Air Power, Movie
Russian Air Force vs USAF (NATO( Comparison, Movie
SU-30SM Intercept with R-77 Missile, Movie
Ukranian A-A Missile ALAMO, R-27, Movie
Return to Table of ContentReturn to Movies Table

People’s Republic of China (PRC) Air-to-Air Missiles
• PL - 1 - PRC version of the Soviet Kaliningrad K-5 (AA-1 Alkali), retired.
• PL - 2 - PRC version of the Soviet Vympel K-13 (AA-2 Atoll), based on AIM-9 Sidewinder, retired.
• PL - 3 - updated version of the PL-2, did not enter service. PL-2, 3
• PL - 5 - updated version of the PL-2, several versions:
• PL - 5A - Semi-Active Radar homing AAM, resembles AIM-9G.
Did not enter service
• PL - 5B - IR version, entered service 1990 to replace PL-2. Limited of boresight.
• PL - 5C - Improved version comparable to AIM-9H or AIM-9L in performance.
• PL - 5E - All-aspect attack version, resembles AIM-9P in appearance.
• PL - 7 - PRC version of the IR-homing French R550 Magic AAM.
Did not enter service.
• PL - 8 - PRC version of the Israeli RAFAEL Python 3.
• PL - 9 - short range IR missile, marked for export. One known improved version PL - 9C.
• PL - 10 - medium-range air-to-air missile. Did not enter service.
PL-5
PL-8
PL-9
PL-7

People’s Republic of China (PRC) Air-to-Air Missiles (continue)
• PL - 11 - Medium Range Air-to-Air Missile (MRAAM), based on the HQ-61C and Italian ASPIDE (AIM-7
technology. Known version include:
PL -11
Length: 3.690 m
Body diameter: 200 mm
Wing span: 1 m
Launch weight: 220 kg
Warhead: HE-fragmentation
Fuze: RF
Guidance: Semi-Active
CW Radar
Propulsion: Solid propellant
Range: 25 km
• PL - 11 - MRAAM with semi-active radar homing, based on the
HQ-61C SAM and ASPIDE seeker technology.
Exported as FD-60.
• PL - 11A - Improved PL-11 with increased range, warhead, and more
effective seeker. The new seeker requires target illumination
only during the last stage, providing a Lock On After Launch
capability.
• PL - 11B - Also known as PL-11AMR, improved PL-11 with AMR-1,
active radar-homing seeker.
• LY - 60 - PL-11, adopted to navy ships for air-defense, sold to Pakistan
but doesn’t appear to be in service with the Chinese Navy.

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F4-Phantom Armament

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F-16

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http://www.freerepublic.com/focus/f-news/2845813/posts
F-15

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F-15C: M61A1 Vulcan Cannon and AIM-9M Sidewinder, Movie

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F-18

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The F/A-18 E/F Super Hornet, with its array of weapons systems, is the world's
most advanced high-performance strike fighter. Designed to operate from aircraft
carriers and land bases, the versatile Super Hornet can undertake virtually any
combat mission.

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F-22 Raptor
http://www.ausairpower.net/APA-Raptor.html
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Internal Weapon
Bay

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Lockheed_Martin_F-35_Lightning_II

F-35 Simulator - AA and AG Modes _ Avionics-1, Movie
Lockheed_Martin_F-35_Lightning_II
42

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Su-32/34

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Su-35

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47

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Fighter Gun

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Performance of Aircraft Cannons in terms of their Employment in Air Combat
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Safety Procedures
Safety of Personal when the Aircraft is on the Ground and when it is in the Air.
Avionics includes Safety Procedures:
Fighter Aircraft on the Ground
In this case the Aircraft Weight is sustained by the Wheels and a Weight-on-Wheels
Switch (WOW) and the Master Arm (MA) Switch are in Safe Mode preventing the
Release/Fire Signals to reach the Weapon Storage
Ground Crew will perform the following:
* Visual Check of the Unpowered Aircraft
* Connect an External Power Generator and will check the Avionics Serviceability
* By pressing WOW Safety-Override and MA=ARM will check the
Weapon Release System.
* Disconnect the External Power Generator and Load the Weapons on Storage
* Install the Weapons External Safety Devices, to be removed before Taxiing to
Take Off. In general, the Weapons have also internal Safety Devices.
* Reconnect External Power Generator, insert the Weapons in the SMS Inventory,
(WOW = Safe) and perform Power On BIT of the Weapons to check their
Serviceability.
* Disconnect the External Power Generator and the Aircraft (already fueled) is
ready to be delivered to the Air Crew.

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Safety Procedures
Avionics includes Safety Procedures (continue – 1):
Fighter Aircraft on the Ground
In this case the Aircraft Weight is sustained by the Wheels and a Weight-on-Wheels
Switch (WOW) and the Master Arm (MA) Switch are in Safe Mode preventing the
Release/Fire Signals to reach the Weapon Storage
Air Crew will perform the following:
* Visual Check of the Unpowered Aircraft
* Start the Engines that provide Internal Power and will check the Avionics
Serviceability (WOW = Safe and MA = Safe)
* Insert the Weapons in the SMS Inventory, and perform Power On BIT of the
Weapons to check their Serviceability.
* Input to Avionics Data necessary for the Mission.
* The Avionics will be in NAV Mode.
* Before Taxiing to Take Off the Ground Crew will remove all Weapons Safety
Devices.
* Pilot will Taxi and Take Off.
* After Landing the Ground Crew will Reinstall Weapons Safety Devices.

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Safety Procedures
Avionics includes Safety Procedures (continue – 2):
Fighter Aircraft in the Air
In this case the Weight-on-Wheels Switch (WOW) is in ARM.
MA = Safe preventing Release/Launch of Weapons.
To operate the Weapons the pilot must put MA = Arm.
The Pilot can switch between the three Operational Modes:
- NAV : Navigation Mode
- A/A: can Launch A/A Missiles and Fire Gun Projectiles
- A/G: can Launch A/G Missile or release Bombs
The Avionics will deliver Safety Warnings due to
- An Aircraft Malfunction
- A Flight Hazard
- Fuel Shortage
In case of a Weapon Release Malfunction the Pilot may:
• Jettison the Weapon
• Perform Safety Procedures at Landing.

59
SOLO AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
1. Inertial System Frame
2. Earth-Center Fixed Coordinate System (E)
3. Earth Fixed Coordinate System (E0)
4. Local-Level-Local-North (L) for a Spherical Earth Model
5. Body Coordinates (B)
6. Wind Coordinates (W)
7. Forces Acting on the Vehicle
8. Simulation
8.1 Summary of the Equation of Motion of a Variable Mass
System
8.2 Missile Kinematics Model 1 (Spherical Earth)
8.3 Missile Kinematics Model 2 (Spherical Earth)

60
Given a missile with a jet engine, we define:
1. Inertial System Frame III zyx ,,
3. Body Coordinates (B) , with the origin at the center of mass.BBB zyx ,,
2. Local-Level-Local-North (L) for a Spherical Earth Model LLL zyx ,,
4. Wind Coordinates (W) , with the origin at the center of mass.WWW zyx ,,
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERESOLO
Coordinate Systems
Table of Content

61
SOLO
Coordinate Systems
1.Inertial System (I(
R

- vehicle position vector
I
td
Rd
V


= - vehicle velocity vector, relative to inertia
II
td
Rd
td
Vd
a 2
2


== - vehicle acceleration vector, relative to inertia
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
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62
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Coordinate Systems (continue – 2)
2. Earth Center Fixed Coordinate System (E(
xE, yE in the equatorial plan with xE pointed to the intersection between the equator
to zero longitude meridian.
The Earth rotates relative to Inertial system I, with the angular velocity
sec/10.292116557.7 5
rad−
=Ω
EIIE zz

11 Ω=Ω=Ω=←ω
( )










Ω
=← 0
0
EC
IEω

Rotation Matrix from I to E
[ ]
( ) ( )
( ) ( )










ΩΩ−
ΩΩ
=Ω=
100
0cossin
0sincos
3 tt
tt
tCE
I

63
SOLO
Coordinate Systems (continue – 3(
2.Earth Fixed Coordinate System (E) (continue – 1)
Vehicle Position ( ) ( )
( ) ( )ETE
I
EI
E
I
RCRCR

==
Vehicle Velocity
Vehicle Acceleration
RVR
td
Rd
td
Rd
V EIE
EI



×Ω+=×+== ←ω - vehicle velocity relative to Inertia
R
td
Rd
td
Rd
V IE
LE
E



×+== ←ω: - vehicle velocity relative to Earth
( ) ( )
II
E
I
E
I
R
td
d
td
Vd
RV
td
d
td
Vd
a





×Ω+=×Ω+==
( ) ( )RV
td
Vd
R
td
Rd
R
td
d
V
td
Vd
EIEEU
U
E
EE
EIU
U
E
IU













×Ω×Ω+×












Ω+++=×Ω×Ω+×Ω+×
Ω
+×+=
←
Ω
←←←
ω
ωωω
0
( ) ( ) ( )RV
td
Vd
RV
td
Vd
a E
E
E
EEU
U
E





×Ω×Ω+×Ω+=×Ω×Ω+×Ω++= ← 22ω
or
where U is any coordinate system. In our case U = E.
Table of Content

64
SOLO
3.Earth Fixed Coordinate System (E0)
The origin of the system is fixed on the earth at some
given point on the Earth surface (topocentric( of
Longitude Long0 and latitude Lat0.
xE0 is pointed to the geodesic North, yE0 is pointed to the East parallel to Earth
surface, zE0 is pointed down.
[ ] [ ]
( ) ( )
( ) ( )
( ) ( )
( ) ( ) =










−










−
−
=−−=
100
0cossin
0sincos
sin0cos
010
cos0sin
2/ 00
00
00
00
3020
0
LongLong
LongLong
LatLat
LatLat
LongLatCE
E π
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )









−−−
−
−−
=
00000
00
00000
sinsincoscoscos
0cossin
cossinsincossin
LatLongLatLongLat
LongLong
LatLongLatLongLat
The Angular Velocity of E relative to I is: EIIEIE zz

110 Ω=Ω== ←← ωω or
( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
( )









Ω−
Ω
=










Ω









−−−
−
−−
=










Ω
=←
0
0
00000
00
00000
00
0
sin
0
cos
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
Lat
Lat
LatLongLatLongLat
LongLong
LatLongLatLongLat
CE
E
E
IEω

Table of Content

65
SOLO
4.Local-Level-Local-North (L)
The origin of the LLLN coordinate system is located at
the projection of the center of gravity CG of the vehicle
on the Earth surface, with zDown axis pointed down,
xNorth, yEast plan parallel to the local level, with
xNorth pointed to the local North and yEast pointed to
the local East. The vehicle is located at:.
Latitude = Lat, Longitude = Long, Height = H
Rotation Matrix from E to L
[ ] [ ]
( ) ( )
( ) ( )
( ) ( )
( ) ( ) =










−










−
−
=−−=
100
0cossin
0sincos
sin0cos
010
cos0sin
2/ 32 LongLong
LongLong
LatLat
LatLat
LongLatC L
E π
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )









−−−
−
−−
=
LatLongLatLongLat
LongLong
LatLongLatLongLat
sinsincoscoscos
0cossin
cossinsincossin

66
SOLO
4.Local-Level-Local-North (L) (continue – 1)
Angular Velocity
IEELIL ←←← += ωωω

Angular Velocity of L relative to I
( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
( )









Ω−
Ω
=










Ω









−
−
−−
=










Ω
=










Ω
Ω
Ω
=←
Lat
Lat
LatLongLatLongLat
LongLong
LatLongLatLongLat
CL
E
Down
East
North
L
IE
sin
0
cos
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
ω

( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
( ) 













−
−=












−+






















−−−
−
−−
=












−+












=










=
•
•
•
•
•
•
•
←
LatLong
Lat
LatLong
Lat
Long
LatLongLatLongLat
LongLong
LatLongLatLongLat
Lat
Long
CL
E
Down
East
North
L
EL
sin
cos
0
0
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
0
0
ρ
ρ
ρ
ω

( ) ( ) ( )
( )
( )
























+Ω−
−






+Ω
=










Ω+
Ω+
Ω+
=+=
•
•
•
←←←
LatLong
Lat
LatLong
DownDown
EastEast
NorthNorth
L
IEC
L
ECL
L
IL
sin
cos
ρ
ρ
ρ
ωωω

Therefore

67
SOLO
Vehicle Velocity
Vehicle Velocity relative to I
RVR
td
Rd
td
Rd
V EIE
EI



×Ω+=×+== ←ω
( )
( )
( )
( ) ( )
( ) ( )









+−














−−
−
+










+−
=×+=
••
••
••
←
HR
LatLongLat
LatLongLatLong
LatLatLong
HR
R
td
Rd
V EL
L
L
E
00
0
0
0cos
cos0sin
sin0
0
0




ω
where is the vehicle velocity relative to Earth.EV

( )
( ) ( )










=














−
+
+
=
•
•
DownE
EastE
NorthE
V
V
V
H
HRLatLong
HRLat
_
_
_
0
0
cos

from which
( )
( ) ( )
DownE
EastE
NorthE
V
td
Hd
LatHR
V
td
Longd
HR
V
td
Latd
_
0
_
0
_
cos
−=
+
=
+
=
HeightVehicleHRadiusEarthmRHRR =⋅=+= 6
00 10378135.6

68
SOLO
Vehicle Velocity (continue – 1(
We assume that the atmosphere movement (velocity and acceleration( relative to Earth
At the vehicle position (Lat, Long, H( is known. Since the aerodynamic forces on the
vehicle are due to vehicle movement relative to atmosphere, let divide the vehicle
velocity in two parts:
WAE VVV

+=
( )










=
Down
East
North
L
A
V
V
V
V

- Vehicle Velocity relative to atmosphere
( )
( )










=
DownW
EastW
NorthW
L
W
V
V
V
HLongLatV
_
_
_
,,

- Wind Velocity at vehicle position
(known function of time(

69
SOLO
Since:
( ) ( ) ( ) ( )RV
td
Vd
R
td
d
td
Vd
RV
td
d
td
Vd
a EEL
L
E
II
E
I
E
I







×Ω×Ω+×Ω++=×Ω+=×Ω+== ← 2ω
WAE VVV

+=
( ) WWIL
L
W
AAIL
L
A
VV
td
Vd
RVV
td
Vd
a





×Ω+×++×Ω×Ω+×Ω+×+= ←← ωω
( )
  





Wa
WWEL
L
W
AAEL
L
A
VV
td
Vd
RVV
td
Vd
×Ω+×++×Ω×Ω+×Ω+×+= ←← 22 ωω
( ) ( ) ( ) ( )HLongLatVHLongLat
td
Vd
HLongLata WEL
L
W
W ,,2,,:,,



×Ω++= ←ω
( ) WAAEL
L
A
aRVV
td
Vd 

+×Ω×Ω+×Ω+×+= ← 2ω
where:
is the wind acceleration at vehicle position.
Table of Content

70
SOLO
5.Body Coordinates (B)
The origin of the Body coordinate system
is located at the instantaneous center of
gravity CG of the vehicle, with xB pointed
to the front of the Air Vehicle, yB pointed
toward the right wing and zB completing
the right-handed Cartesian reference frame.
Rotation Matrix from LLLN to B (Euler Angles):
[ ] [ ] [ ]










−+
+−
−
==
θφψφψθφψφψθφ
θφψφψθφψφψθφ
θψθψθ
ψθφ
cccssscsscsc
csccssssccss
ssccc
CB
L 321
ψ - azimuth angle
θ - pitch angle
φ - roll angle

71
SOLO
5.Body Coordinates (B) (continue – 1) ψ
θ
φ Bx
Lx
Bz
Ly
Lz
By
Angular Velocity from L to B (Euler Angles):
( )
[ ] [ ] [ ]










+










+










=










=←
ψ
θφθφ
φ
ω



0
0
0
0
0
0 211
R
Q
P
B
LB



















 −










−
+




















−
+










=
ψθθ
θθ
φφ
φφθ
φφ
φφ
φ



0
0
cos0sin
010
sin0cos
cossin0
sincos0
001
0
0
cossin0
sincos0
001
0
0
[ ]










=




















−
−
=
ψ
θ
φ
ψ
θ
φ
θφφ
θφφ
θ






G
coscossin0
cossincos0
sin01

72
SOLO
θ
φ Bx
Lx
Bz
Ly
Lz
By
Rotation Matrix from LLLN to B (Quaternions):
( ) [ ][ ] ( ) [ ][ ] { } { }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )











−−−
−
−
−










−−
−−
−−
=
+×−×−=
321
412
143
234
3412
2143
1234
44 3333
BIBLBL
BLBLBL
BLBLBL
BLBLBL
BLBLBLBL
BLBLBIBL
BLBLBLBL
T
BLBLBLXBLBLXBL
B
L
qqq
qqq
qqq
qqq
qqqq
qqqq
qqqq
qqqIqqIqC

where: ( )
( )
( )
( )
( )
( )
( )
( )
{ }
( )
{ }
( )
( )
( )









=





=
























=












=
3
2
1
:&
4
4
3
2
1
4
3
2
1
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
BL
q
q
q
q
q
q
qor
q
q
q
q
q
q
q
q
q


( ) 























−

















=
2
sin
2
sin
2
sin
2
cos
2
cos
2
cos4
ϕθψϕθψ
BLq
( ) 























+

















=
2
cos
2
sin
2
sin
2
sin
2
cos
2
cos1
ϕθψϕθψ
BLq
( ) 























−

















=
2
sin
2
cos
2
sin
2
cos
2
sin
2
cos2
ϕθψϕθψ
BLq
( ) 























+

















=
2
sin
2
sin
2
cos
2
cos
2
cos
2
sin3
ϕθψϕθψ
BLq

73
SOLO
θ
φ Bx
Lx
Bz
Ly
Lz
By
Rotation Matrix from LLLN to B (Quaternions)
(continue – 1)
The quaternions are given by the following
differential equations:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
BL
L
IL
B
IBBLBLBL
B
ILBL
B
IBBL
B
IL
B
IBBL
B
LBBLBL qqqqqqqqq ⋅−⋅=⋅⋅⋅−⋅=−⋅=⋅= ←←←←←←← ωωωωωωω
2
1
2
1
*
2
1
2
1
2
1
2
1
( )
( )
( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 























−−−
−
−
−
=












04321
3412
2143
1234
2
1
4
3
2
1
B
B
B
BLBLBLBL
BLBLBLBL
BLBLBLBL
BLBLBLBL
BL
BL
BL
BL
r
q
p
qqqq
qqqq
qqqq
qqqq
q
q
q
q




( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
( )

























+Ω−+Ω−+Ω−
+Ω+Ω+Ω−
+Ω+Ω−+Ω
+Ω+Ω+Ω−
−
4
3
2
1
0
0
0
0
2
1
BL
BL
BL
BL
zLzLyLyLxLxL
zLzLxLxLyLyL
yLyLxLxLzLzL
xLxLyLyLzLzL
q
q
q
q
ρρρ
ρρρ
ρρρ
ρρρ
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
( )

























+Ω+−+Ω+−+Ω+−
+Ω−+Ω−−+Ω+
+Ω−+Ω++Ω−−
+Ω−+Ω−−+Ω+
=
4
3
2
1
0
0
0
0
2
1
BL
BL
BL
BL
zLzLByLyLBxLxLB
zLzLBxLxLByLyLB
yLyLBxLxLBzLzLB
xLxLByLyLBzLzLB
q
q
q
q
rqp
rpq
qpr
pqr
ρρρ
ρρρ
ρρρ
ρρρ
or:

74
SOLO
θ
φ Bx
Lx
Bz
Ly
Lz
By
Vehicle Velocity
Vehicle Velocity relative to Earth is divided in:
WAE VVV

+=
( )










=
w
v
u
V
B
A
 ( )
( )










=










=
DownW
EastW
NorthW
B
L
zW
yW
xW
B
W
V
V
V
C
V
V
V
HLongLatV
B
B
B
_
_
_
,,

( ) WWIB
B
W
AAIB
B
A
I
VV
td
Vd
RVV
td
Vd
td
Vd
a





×Ω+×++×Ω×Ω+×Ω+×+== ←← ωω
( ) ( )
W
AELALB
B
A
a
RVV
td
Vd



+
×Ω×Ω+×Ω++×+= ←← 2ωω
Table of Content

75
SOLO
6.Wind Coordinates (W)
The origin of the Wind coordinate system
is located at the instantaneous center of
gravity CG of the vehicle, with xW pointed
in the direction of the vehicle velocity vector
relative to air .AV

[ ] [ ]










−
−−=










−









−=−=
αα
βαββα
βαββα
αα
αα
ββ
ββ
αβ
cos0sin
sinsincossincos
cossinsincoscos
cos0sin
010
sin0cos
100
0cossin
0sincos
23
W
BC
The Wind coordinate frame is defined by the following two angles:
α - angle of attack
β - sideslip angle

76
SOLO
6.Wind Coordinates (W) (continue -1)
Rotation Matrix from L (LLLN( to W is:
χ - azimuth angle of the trajectory
γ - pitch angle of the trajectory
Rotation Matrix
[ ] [ ] [ ] [ ] [ ] 32123 ψθφαβ −== B
L
W
B
W
L CCC
The Rotation Matrix from L (LLLN( to W can also be defined by the following
Consecutive rotations:
σ - bank angle of the
trajectory
[ ] [ ] [ ] [ ]










−+
+−
−
===
γσχσχγσχσχγσ
γσχσχγσχσχγσ
γχγχγ
χγσσ
cccssscsscsc
csccssssccss
ssccc
CC W
L
W
L 321
*
1
We defined also the intermediate wind frame W* by:
[ ] [ ]










−
−
==
γχγχγ
χχ
γχγχγ
χγ
csscs
cs
ssccc
CW
L 032
*

77
SOLO
Angular Velocity of W* relative to LLLN is:
Angular Velocities
( )
[ ]









−
=



















 −
+










=










+










=










=←
γχ
γ
γχ
χγγ
γγ
γ
χ
γγω
cos
sin
0
0
cos0sin
010
sin0cos
0
0
0
0
0
0
2
*
*
*
*
*








W
W
W
W
LW
R
Q
P
Angular Velocity of W relative to LLLN is:
( )
[ ] [ ]




















−
−
=









−










−
+










=




















+










+










=










=←
χ
γ
σ
γσσ
γσσ
γ
γχ
γ
γχ
σσ
σσ
σ
χ
γγσ
σ
ω










coscossin0
cossincos0
sin01
cos
sin
cossin0
sincos0
001
0
00
0
0
0
0
0 21
W
W
W
W
LW
R
Q
P

78
SOLO
We have also:
Angular Velocities (continue – 1)
( ) ( )
( )
( ) 









Ω
Ω
Ω
=










Ω−
Ω
==










Ω
Ω
Ω
= ←←
Down
East
North
W
L
W
L
L
IE
W
L
zW
yW
xW
W
IE C
Lat
Lat
CC ***
*
*
*
*
sin
0
cos
ωω

( ) ( )
( )
( )










=














−
−==










=
•
•
•
←←
Down
East
North
W
L
W
L
L
EL
W
L
zW
yW
xW
W
EL C
LatLong
Lat
LatLong
CC
ρ
ρ
ρ
ω
ρ
ρ
ρ
ω ***
*
*
*
*
sin
cos

( ) ( )
( )
( )
[ ] ( )*
1
sin
0
cos
W
IE
W
L
L
IE
W
L
zW
yW
xW
W
IE
Lat
Lat
CC ←←← =










Ω−
Ω
==










Ω
Ω
Ω
= ωσωω

( ) ( )
( )
( )
[ ] ( )*
1
sin
cos
W
IL
W
L
L
IL
W
L
W
IL
LatLong
Lat
LatLong
CC ←
•
•
•
←← =
























+Ω−
−






+Ω
== ωσωω


79
SOLO
The Angular Velocity from I to W is:
( ) ( ) ( ) ( )










Ω+
Ω+
Ω+
+










=+










=+=










= ←←←←
DownDown
EastEast
NorthNorth
W
L
W
W
W
L
IL
W
L
W
W
W
W
IL
W
LW
W
W
W
W
IW C
R
Q
P
C
R
Q
P
r
q
p
ρ
ρ
ρ
ωωωω

Using the angle of attack α and the sideslip angle β , we can write:
BWBW yz



11 αβω −=←
or:
( ) ( ) ( )
[ ]










−










=










−










=−= ←←←
0
0
0
0
3 αβ
β
ωωω 


r
q
p
C
r
q
p
W
B
W
W
W
W
IB
W
IW
W
BW
but also:
( ) ( ) ( )
[ ]










−










=










−










=−= ←←←
0
0
0
0
3 αβ
β
ωωω 


R
Q
P
C
R
Q
P
W
B
W
W
W
W
LB
W
LW
W
BW

80
SOLO
We can write:










−










+




















−
−−=










0
cos
sin
0
0
cos0sin
sinsincossincos
cossinsincoscos
βα
βα
βαα
βαββα
βαββα


r
q
p
r
q
p
W
W
W
or:
( )
( )
βαα
βαβαβα
βαβαβα



++−=
−−+−=
+−+=
cossin
sinsincossincos
cossinsincoscos
rpr
rqpq
rqpp
W
W
W
This can be rewritten as:
( ) βαα
β
α tansincos
cos
rp
q
q W
+−−=
Wrrp +−= ααβ cossin
( ) ( ) ( )( )
( )
β
βαα
ββββααβαβαα
cos
sinsincos
tantansincossincossincossincos
W
WW
qrp
qrpqrpp
++
=
+++=−++= 

81
SOLO
We have also:
( ) βαα
β
α tansincos
cos
RP
Q
Q W
+−−=
WRRP +−= ααβ cossin
( )
β
βαα
cos
sinsincos W
W
QRP
P
++
=

82
SOLO
The vehicle velocity was decomposed in:
Vehicle Velocity
WAE VVV

+=
( )










=
0
0
V
V
W
A

- vehicle velocity relative to atmosphere
( )
( )










=










=
DownW
EastW
NorthW
W
L
zW
yW
xW
W
W
V
V
V
C
V
V
V
HLongLatV
W
W
W
_
_
_
,,

- wind velocity at velocity position
also
( )
[ ] ( )
[ ]










=










−=−=
0
0
0
011
*
VV
VV
W
A
W
A σσ

( )
( )










=










=
DownW
EastW
NorthW
W
L
zW
yW
xW
W
W
V
V
V
C
V
V
V
HLongLatV
W
W
W
_
_
_
*
*
*
*
*
,,


83
SOLO
The vehicle acceleration in W* coordinates is
( )
( ) ( ) WAELALW
W
A
WWIW
W
W
AAIW
W
A
I
C
aRVV
td
Vd
VV
td
Vd
RVV
td
Vd
td
Vd
a







+×Ω×Ω+×Ω++×+=
×Ω+×++×Ω×Ω+×Ω+×+==
←←
←←
2*
*
*
*
*
*
ωω
ωω
from which
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )*******
*
*
*
2
W
W
W
A
WW
EL
WW
A
W
LW
W
W
A
aVAV
td
Vd 

−×Ω+−=×+








←← ωω
where
( )RaA

×Ω×Ω−=:

84
SOLO
Vehicle Acceleration (continue – 1)
( ) ( )
( ) ( )
( ) ( ) 









−




















Ω+Ω+−
Ω+−Ω+
Ω+Ω+−
−










=




















−
−
−
+










**
*
*
****
****
****
*
*
*
**
**
**
0
0
022
202
220
0
0
0
0
0
0
0
zWW
yWW
xWW
xWxWyWyW
xWxWzWzW
yWyWzWzW
zW
yW
xW
WW
WW
WW
a
a
aV
A
A
AV
PQ
PR
QRV
ρρ
ρρ
ρρ
where
( )
( )
( )
( )HR
Lat
Lat
C
a
a
a
A
A
A
A
W
L
zW
yW
xW
zW
yW
xW
W
+Ω










−










=










= 2*
*
*
*
*
*
*
*
sin
0
cos
 - Acceleration due to external forces on the
Air Vehicle in W* coordinates
That gives
( )
( ) *****
*****
**
2
2
zWWyWyWzWW
yWWzWzWyWW
xWWxW
aVAVQ
aVAVR
aAV
−Ω++=−
−Ω+−=
−=
ρ
ρ


85
SOLO
Vehicle Acceleration (continue – 2)
Using ( )









−
=










=←
γχ
γ
γχ
ω
cos
sin
*
*
*
*
*




W
W
W
W
LW
R
Q
P
we have
** xWWxW aAV −=
( ) γρχ cos/2 **
**






Ω+−
−
= zWzW
yWWyW
V
aA

( )**
**
2 yWyW
zWWzW
V
aA
Ω+−
−
−= ργ
Table of Content

86
SOLO
Aerodynamic Forces
( )[ ]∫∫ +−= ∞
WS
A dstfnppF

11
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ

−
−
( )
airflowingthebyweatedsurfaceVehicleS
SsurfacetheonmNstressforcefrictionf
Ssurfacetheondifferencepressurepp
W
W
W
−
−
−−∞
)/( 2
7. Forces Acting on the Vehicle

87
SOLO
7. Forces Acting on the Vehicle (continue – 1)
Aerodynamic Forces (continue – 1)
( )










−
−
−
=
L
C
D
F
W
A

ForceLiftL
ForceSideC
ForceDragD
−
−
−
L
C
D
CSVL
CSVC
CSVD
2
2
2
2
1
2
1
2
1
ρ
ρ
ρ
=
=
=
( )
( )
( ) tCoefficienLiftRMC
tCoefficienSideRMC
tCoefficienDragRMC
eL
eC
eD
−
−
−
βα
βα
βα
,,,
,,,
,,,
ityvisdynamic
lengthsticcharacteril
soundofspeedHa
numberynoldslVR
numberMachaVM
e
cos
)(
Re/
/
−
−
−
−=
−=
µ
µρ

88
SOLO
Aerodynamic Forces (continue -2)
∫∫ 





⋅+⋅−=
∫∫ 





⋅+⋅−=
∫∫ 





⋅+⋅−=
∧∧
∧∧
∧∧
W
W
W
S
fpL
S
fpC
S
fpD
dswztCwznC
S
C
dswytCwynC
S
C
dswxtCwxnC
S
C
1ˆ1ˆ
1
1ˆ1ˆ
1
1ˆ1ˆ
1
Wf
Wp
Ssurfacetheontcoefficienfriction
V
f
C
Ssurfacetheontcoefficienpressure
V
pp
C
−=
−
−
= ∞
2/
2/
2
2
ρ
ρ
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ

−
−

89
( ) ( ) ( )
  

  

MomentFriction
S
C
Momentessure
S
CCA
WW
dstRRfdsnRRppM ∫∫∫∫∑ ×−+×−−= ∞ 11
Pr
/
Aerodynamic Moments Relative to C can be divided in Pressure Moments and
Friction Moments.
( )


  

FrictionSkinor
FrictionViscous
S
essureNormal
S
A
WW
dstfdsnppF ∫∫∫∫∑ +−= ∞ 11
Pr
Aerodynamic Forces can be divided in Pressure Forces and Friction Forces.
AERODYNAMIC FORCES AND MOMENTS.

90
SOLO
( ) ( ) ( )∫∫ −++= ∞
<> iopenS
outflowoutopenflowinflowinopenflow dsnppmVmVT







1:
0
/
0
/ THRUST FORCES
( ) ( ) ( ) ( )[ ]∫∫ −×−+×−−×−= ∞
<> iopenS
OoutflowoutopenflowCoutopeninflowinopenflowCiopenCT dsnppRRmVRRmVRRM







1:
0
/
0
/,
THRUST MOMENTS
RELATIVE TO C
( ) ( )∫∫ −+ ∞
> inopenS
inflowinopenflow dsnppmV




1
00
/
( ) ( )∫∫ −+ ∞
< outopenS
outflowoutopenflow dsnppmV




1
0
/
T

outopenR

iopenR

CR

C
Table of Content
CTM ,


91
SOLO
Thrust
( ) ( )




















−
−−==
B
B
B
z
y
x
BW
B
W
T
T
T
TCT
αα
βαββα
βαββα
cos0sin
sinsincossincos
cossinsincoscos
**

( )
[ ] ( )






















−
==










=
*
*
*
cossin0
sincos0
001
*
1
W
W
W
W
W
W
z
y
x
W
z
y
x
W
T
T
T
T
T
T
T
T
σσ
σσσ


92
SOLO
Gravitation Acceleration
( ) ( )
























−









 −










−
==
zg
yg
xg
gg
100
0
0
0
010
0
0
0
001
χχ
χχ
γγ
γγ
σσ
σσ cs
sc
cs
sc
cs
scC EW
E
W

( )
gg









−
=
γσ
γσ
γ
cc
cs
s
W

2sec/174.322sec/81.9
0
2
0
0
0
gg ftmg
HR
R
==
+
=










The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth.
The Gravitational Potential U (R, ( is given byϕ
( ) ( )
( )
( )φ
φ
µ
φ
,
sin1, 2
RUg
P
R
a
J
R
RU
E
E
n n
n
n
∇=














−⋅−= ∑
∞
=

μ – The Earth Gravitational Constant
a – Mean Equatorial Radius of the Earth
R=[xE
2
+yE
2
+zE
2
]]/2
is the magnitude of
the Geocentric Position Vector
– Geocentric Latitude (sin =zϕ ϕ E/R(
Jn – Coefficients of Zonal Harmonics of the
Earth Potential Function
P (sin ( – Associated Legendre Polynomialsϕ

93
SOLO
Gravitation Acceleration
Retaining only the first three terms of the
Gravitational Potential U (R, ( we obtain:ϕ
R
z
R
z
R
z
R
a
J
R
z
R
a
J
R
g
R
y
R
z
R
z
R
a
J
R
z
R
a
J
R
g
R
x
R
z
R
z
R
a
J
R
z
R
a
J
R
g
EEEE
z
EEEE
y
EEEE
x
E
E
E
⋅
















+⋅−⋅





⋅−







−⋅





⋅−⋅−=
⋅
















+⋅−⋅





⋅−







−⋅





⋅−⋅−=
⋅
















+⋅−⋅





⋅−







−⋅





⋅−⋅−=
34263
8
5
15
2
3
1
34263
8
5
15
2
3
1
34263
8
5
15
2
3
1
2
2
4
44
42
22
22
2
2
4
44
42
22
22
2
2
4
44
42
22
22
µ
µ
µ
φ
φλ
φλ
sin
cossin
coscos
=
⋅=
⋅=
R
z
R
y
R
x
E
E
E
( ) 2/1222
EEE zyxR ++=

94
SOLO
23. Local Level Local North (LLLN) Computations for an Ellipsoidal Earth Model
( )
( )
( )
( )
( )2
22
10
2
0
2
0
2
0
5
2
1
2
0
6
0
sin
sin1
sin321
sin1
sec/10292116557.7
sec/051646.0
sec/780333.9
26.298/.1
10378135.6
Ae
e
p
m
e
HR
RLatgg
g
LateRR
LateeRR
LateRR
rad
mg
mg
e
mR
+
+
=
+=
+−=
−=
⋅=Ω
=
=
=
⋅=
−
Lat
HR
V
HR
V
HR
V
Ap
East
Down
Am
North
East
Ap
East
North
tan
+
−=
+
−=
+
=
ρ
ρ
ρ
Lat
Lat
Down
East
North
sin
0
cos
Ω−=Ω
=Ω
Ω=Ω
DownDownDown
EastEast
NorthNorthNorth
Ω+=
=
Ω+=
ρφ
ρφ
ρφ
East
North
Lat
Lat
Long
ρ
ρ
−=
=
•
•
cos
( )
( ) ∫
∫
•
•
+=
+=
t
t
dtLatLattLat
dtLongLongtLong
0
0
0
0
SIMULATION EQUATIONS

95
SOLO AIR VEHICLE IN ELLIPTICAL EARTH ATMOSPHERE
SIMULATION EQUATIONS
Table of Content

96
SOLO
Force Equations
Air Vehicle Acceleration
( ) ( ) WAELALW
W
A
I
C
aRVV
td
Vd
td
Vd
a



+×Ω×Ω+×Ω++×+== ←← 2ωω
( ) ( ) ( ) WAELALW
W
A
A aRVV
td
Vd
amTF
m



+×Ω×Ω+×Ω++×+==++ ←← 2g
1
ωω
( )Rg
 
×Ω×Ω−= g:where


















+
−−
+
−−
−
−
=










γσ
α
γσ
βα
γ
βα
ccg
m
LT
csg
m
CT
sg
m
DT
A
A
A
zW
yW
xW
sin
sincos
coscos









−
+


















−−
−−
−










−=










γ
γ
α
βα
βα
σσ
σσ
cg
sg
m
LT
m
CT
m
DT
A
A
A
zW
yW
xW
0
sin
sincos
coscos
cossin0
sincos0
001
*
*
*
Table of Content

97
SOLO
( ) ( ) ( ) ( )LB
L
BB
A
B
CG gCT
m
F
m
a

++= ∑
11
( )
[ ] [ ] ( ) ( )
[ ][ ] ( )
{
( ) ( )
[ ] ( )
}B
BrCrrotor
B
IB
B
BrCrrotor
B
IBC
B
IB
B
IBCCTCAC
B
IB
II
IIMMI
←←←
←←←
−
←
⋅×−⋅−
×−−+= ∑
ωωω
ωωωω



,,
,,,,
1
( )
( ) ( )B
CG
TB
L
L
CG aCa

=
( )B
IB←ω

( ) ( )
B L
L
IL
B
IBB LB L qqq ←← −= ωω
2
1
2
1
s
1
CT
CA
M
M
,
,


∑
[ ]{ } [ ]{ } TB
L IqIqC ρρρρ

+×−×−= 3434
( )B
IB←ω
( )B
CGa

( )L
CGa

( )
( )B
B
A
T
F


∑
B
LC
B
LC
s
1 BLqBLq
B
LC
s
1
( ) ( )
( )[ ]( ) ( ) ( )
( ) ( )L
E
LL
EL
LL
CG
L
E VRaV

×Ω+−×Ω×Ω−= ← 2ω s
1
( )L
EV
( )L
EV

( )L
CGa

B
LC
( )L
MR

( )L
EV

( ) ( )L
M
B
L
B
M VCV

=
δξωξςξ Mee

+−−=
2
2
δM

s
1
s
1ξ

ξ

ξ

( )L
EV

[ ] [ ] 23 αβ −=
W
BC
α
β
W
BC
MV
WEM VVV

−=
( )L
MV

( )L
WV

( )B
IB←ω

( ) ( )B
Brotor
B
Brotor ←← ωω

,
Missile Kinematics Model 1 in Vector Notation (Spherical Earth)

98
SOLO
Missile Kinematics Model 1 in Matrix Notation (Spherical Earth)

99
SOLO
Missile Kinematics Model 2 in Vector Notation (Spherical Earth)

100
SOLO
Missile Kinematics Model 2 in Matrix Notation (Spherical Earth)

References
SOLO
101
PHAK Chapter 1 - 17
http://www.gov/library/manuals/aviation/pilot_handbook/media/
George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”,
Academic Press, Inc., 1993
R.P.G. Collinson, “Introduction to Avionics”, Chapman & Hall, Inc., 1996, 1997, 1998
Ian Moir, Allan Seabridge, “Aircraft Systems, Mechanical, Electrical and Avionics
Subsystem Integration”, John Wiley & Sons, Ltd., 3th Ed., 2008
Ian Moir, Allan Seabridge, “Military Avionics Systems”, John Wiley & Sons, LTD.,
2006

References (continue – 1)
SOLO
102
S. Hermelin, “Air Vehicle in Spherical Earth Atmosphere”
S. Hermelin, “Airborne Radar”, Part1, Part2, Example1, Example2
S. Hermelin, “Tracking Systems”
S. Hermelin, “Navigation Systems”
S. Hermelin, “Earth Atmosphere”
S. Hermelin, “Earth Gravitation”
S. Hermelin, “Aircraft Flight Instruments”
S. Hermelin, “Computing Gunsight, HUD and HMS”
S. Hermelin, “Aircraft Flight Performance”
S. Hermelin, “Sensors Systems: Surveillance, Ground Mapping, Target Tracking”
S. Hermelin, “Air-to-Air Combat”

References (continue – 2)
SOLO
103
S. Hermelin, “Spherical Trigonometry”
S. Hermelin, “Modern Aircraft Cutaway”

104
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –
Stanford University
1983 – 1986 PhD AA

105
SOUND WAVES
SOLO
Disturbances propagate by molecular collision, at the sped of sound a,
along a spherical surface centered at the disturbances source position.
The source of disturbances moves with the velocity V.
-when the source moves at subsonic velocity V < a, it will stay inside the
family of spherical sound waves.
-when the source moves at supersonic velocity V > a, it will stay outside the
family of spherical sound waves. These wave fronts form a disturbance
envelope given by two lines tangent to the family of spherical sound waves.
Those lines are called Mach waves, and form an angle μ with the disturbance
source velocity:
a
V
M
M
=





= −
&
1
sin 1
µ

106
SOUND WAVESSOLO
Sound Wave Definition:
∆ p
p
p p
p1
2 1
1
1=
−
<<
ρ ρ ρ2 1
2 1
2 1
= +
= +
= +
∆
∆
∆
p p p
h h h
For weak shocks
u
p
1
2
=
∆
∆ρ
1
1
11
1
1
1
1
1
2
1
2
1
1
uuuuuu
ρ
ρ
ρ
ρρρ
ρ
ρ
ρ ∆
−≅
∆
+
=
∆+
==(C.M.)
( ) ( ) ppuuupuupu ∆++




 ∆
−=+=+ 11
1
11122111
2
11
ρ
ρ
ρρρ(C.L.M.)
Since the changes within the sound wave are small, the flow gradients are small.
Therefore the dissipative effects of friction and thermal conduction are negligible
and since no heat is added the sound wave is isotropic. Since
au =1
s
p
a 





∂
∂
=
ρ
2
valid for all gases

107
SPEED OF SOUND AND MACH NUMBERSOLO
Speed of Sound is given by
0=






∂
∂
=
ds
p
a
ρ
RT
p
C
C
T
dT
R
C
p
T
dT
R
C
d
dp
d
R
T
dT
Cds
p
dp
R
T
dT
Cds
v
p
v
p
ds
v
p
γ
ρ
ρ
ρ
ρ
ρ
===





⇒







=−=
=−=
=00
0
but for an ideal, calorically perfect gas
ρ
γγ
ρ p
RTa
TChPerfectyCaloricall
RTpIdeal
p
==






=
=
The Mach Number is defined as
RT
u
a
u
M
γ
==
∆
1
2
1
1
111
−−






=





=





=
γ
γ
γ
γ
γ
ρ
ρ
a
a
T
T
p
p
The Isentropic Chain:
a
ad
T
Tdd
p
pd
sd
1
2
1
0
−
=
−
==→=
γ
γ
γ
γ
ρ
ρ
γ

108
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (1)
( )
( )
( )
  ( )
12
2
2
2
1
2
1
2
2
22
2
2
1
22
1
2
2
2
2
22
1
1
2
1
12
22
2
11
1
2
2
221
2
11
2211
2
1
2
1
2
1
2
1
*
12
1
2
1
12
1
1
4..
...
..
uu
u
a
u
a
uaa
uaa
au
h
a
u
h
a
EC
uu
u
p
u
p
pupuMLC
uuMC
p
a
−=−

















−
−
+
=
−
−
+
=
→
−
+
=+
−
=+
−
→−=−→



+=+
=
∗
∗
=
γγ
γγ
γγ
γ
γ
γγ
ρρρρ
ρρ ρ
γ
Field Equations:
122
2
2
1
1
2
2
1
2
1
2
1
2
1
uuu
u
a
u
u
a
−=
−
+
+
−
−
−
+ ∗∗
γ
γ
γ
γ
γ
γ
γ
γ
u u a1 2
2
= ∗
u
a
u
a
M M1 2
1 21 1∗ ∗
∗ ∗
= → =
Prandtl’s Relation
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
( )
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
2
1
2
1
1
2
1
2
1
2
1
21
2
1212
2
21
12 +
=
−
−=
+
→−=−
−
+
−+ ∗
∗
uu
a
uuuua
uu
uu
Ludwig Prandtl
(1875-1953)

109
 
G Q= =0 0,
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )[ ]
( )( ) ( )
M
M
M
M
M
M
M
M
M
2
2
2
2
1
1
2
1
2
1
2
1
2
1
2
2
1
1
2
1 1
2
1
1
1 2
1
2 1 2
1 1 1 1 1
1
2
=
+
− −
=
+ − −
=
+
+
− +
− −
=
− +
+ / + − / / + − / + − −
∗
=
∗
∗
∗
γ
γ γ γ
γ
γ
γ
γ
γ
γ γ γ γ γ
or
( )
M
M
M
M
M
H H
A A
2
1
2
1
2
1
2
1
21 2
1 2
1
1
2
1
2
2
1
1
1
2
1
2
1
1
=
+
−
−
−
=
+
+
−
+
+
−
=
=
γ
γ
γ γ
γ
γ
γ
( )
( )
ρ
ρ
γ
γ
2
1
1
2
1
2
1 2
1
2
2 1
2 1
2
1
2
1 2 1
1 2
= = = = =
+
− +
=
∗
∗
A A u
u
u
u u
u
a
M
M
M
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

110
 
G Q= =0 0,
( )
( )
( ) ( )
( )
p
p
u
p
u
u
u
a
M
M
M
M
M M
M
2
1
1
2
1
1
2
1
1
2
1
2
1
2
1
2 1
2
1
2 1
2 1
2
1
2
1
2
1 1 1 1
1 1
1 2
1
1
1 1 2
1
= + −





 = + −






= + −
− +
+





 = +
/ + − / − −
+
ρ
γ
ρ
ρ
γ
γ
γ
γ
γ γ
γ
or
(C.L.M.)
( )
p
p
M2
1
1
2
1
2
1
1= +
+
−
γ
γ
( )
( )
( )
h
h
T
T
p
p
M
M
M
a
a
h C T p RTp
2
1
2
1
2
1
1
2
1
2 1
2
1
2
2
1
1
2
1
1
1 2
1
= = = +
+
−






− +
+
=
= = ρ ρ
ρ
γ
γ
γ
γ
( )
( )
( )
s s
R
T
T
p
p
M
M
M
2 1 2
1
1
2
1
1
1
2
1
1
1
2
1
2
1
1
2
1
1
1 2
1
−
=






















= +
+
−






− +
+
















−
−
− −
ln ln
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
( )
( )
( )
( )
s s
R
M M
M
2 1
1 1
2 1
2 3
2
2 1
2 41
2
2
3 1
1
2
1
1
−
≈
+
− −
+
− +
− << γ
γ
γ
γ
K Shapiro p.125
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

111
STEADY QUASI ONE-DIMENSIONAL FLOWSOLO
STAGNATION CONDITIONS
(C.E.) constuhuh =+=+ 2
22
2
11
2
1
2
1
The stagnation condition 0 is attained by reaching u = 0
2
/
21202
020
2
1
1
1
2
1
2
1
22
1
2
M
TR
u
Tc
u
T
T
c
u
TTuhh
TRa
auM
Rc
pp
Tch p
p
−
+=
−
+=+=→+=→+=
=
=
−
=
=
γ
γ
γ
γγ
γ
Using the Isentropic Chain relation, we obtain:
2
1
0102000
2
1
1 M
p
p
a
a
h
h
T
T −
+=





=





=





==
−
−
γ
ρ
ρ γ
γ
γ
Steady , Adiabatic + Inviscid = Reversible, , ( )
q Q= =0 0, ( )~ ~
τ = 0 ( )
 
G = 0
∂
∂ t
=





0

SOLO
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