4 navigation systems

Navigation Systems

SOLO HERMELIN

Updated: 08.11.12

1

Table of Content

SOLO

2

Navigation

Navigation SOLO

Definition of Aircraft Present Position and Waypoints

TrueNorth

AircraftLongitudinal

Axis

FromWaypoint 1

ToWaypoint 2

DIS

PresentPosition

GS

WS

WD

Aircraft Steering to Waypoints

1. T-HDG – True Heading2. M-HDG – Magnetic Heading3. T-TK - True Track4. M-TK - Magnetic Track Angle5. TKE – Track Angle Error6. T-DTK – True Desired Track7. XTK – Cross-Track Distance8. DIS – Distance to Destination9. GS - Ground Speed10. WS – Wind Speed11. WD – Wind Direction12. TAS – True Airspeed13. DA – Drift Angle

In order to minimize Fuel, Time and Distances the Aircraft will tend to fly betweenWaypoints, on the Earth Surface, on the Great Circle connecting the Initial and FinalWaypoints, since is the Shortest Distance between two points on a Sphere. During Flight the Aircraft will deviate from the desired flight path (see Figure).Those deviation must be measured and corrected by Steering the Aircraft. The Task of Steering the Aircraft can be performed Manually by the Pilot or by anAutomatic Flight-Control System (AFCS).

Navigation SOLO

VelocityControl

Loop

CommandedAirspeed

HeadingControl

Loop

CommandedHeading

AltitudeControl

Loop

CommandedAltitude

WindComputation

ComputeAircraftVelocityRelativeto Earth

Airspeed

Heading

FlightPath Angle

ComputeLatitude

&Longitude

NorthVelocity

EastVelocity

Lat

Long

Altitude

Open-Loop Guidance Model

WindSpeed

WindDirection

11

1

sTrajectory

Segment Length

Aircraft Steering to Waypoints

CommandedAirspeed

CommandedHeading

CommandedAltitude

Open-LoopGuidnce Model

AutomaticGuidance

&Steering

Computations

Lat

Long

Closed-Loop Guidance Model Block Diagram

WindSpeed

WindDirection

RangeTo Go

Time ofArrival

Computations

CommandedAltitude

CommandedAirspeed

5

Spherical TrigonometrySOLO

Assume three points on a unit radius sphere, defined by the vectors

CBA 1,1,1

A1

B1

C1

B

a

bc

C

A

Laws of Cosines for Spherical Triangle Sides

ab

abc

ca

cab

bc

bca

ˆsinˆsin

ˆcosˆcosˆcosˆcos

ˆsinˆsin


ˆsinˆsin


Law of Sines for Spherical Triangle Sides.

cba

abccba

cba ˆsinˆsinˆsin

ˆcosˆcosˆcos2ˆcosˆcosˆcos1

ˆsin

ˆsinˆsin

ˆsinˆsin

ˆsin 222

The three great circles passing trough those three points define a spherical triangle with

CBA ,,- three spherical triangle

verticescba ˆ,ˆˆ -three spherical triangle side angles

ˆ,ˆˆ - three spherical triangle angles defined by the angles between the tangents to the great circles at the vertices.

6

SOLO

Assume three points on a unit radius sphere, defined by the vectors

CBA 1,1,1

A1

B1

C1

B

a

bc

C

A

Laws of Cosines for Spherical Triangle Sides

The three great circles passing trough those three points define a spherical triangle with

CBA ,,- three spherical triangle

verticescba ˆ,ˆˆ -three spherical triangle side angles

ˆ,ˆˆ - three spherical triangle angles defined by the angles between the tangents to the great circles at the vertices.

ˆsinˆsin


ˆsinˆsin


ˆsinˆsin


c

b

a

Spherical Trigonometry

7

Navigation SOLO

I

Ecuator

1R

2R11,

Ex

Ey

Ez

1

222 ,

,

12 TrajectoryGreat Circcle

1 2

0

Flight on Earth Great Circles

The Shortest Flight Path between two points 1 and 2 on the Earth is on the Great Circles (centered at Earth Center) passing through those points.

1

2111 ,, R

222 ,, R

The Great Circle Distance between two points 1 and 2 is ρ.The average Radius on the Great Circle is a = (R1+R2)/2

a

R – radiusϕ - Latitudeλ - Longitude

kmNmNma 852.11deg/76.60/

8

Navigation SOLO

I

Ecuator

1R

2R11,

Ex

Ey

Ez

1

222 ,

,


1 2

0

1

2


1

2111 ,, R

222 ,, R

The Great Circle Distance between two points 1 and 2 is ρ.

a


212121 cos90sin90sin90cos90cos

/coscos

a

From the Law of Cosines for Spherical Triangles

or

212121 coscoscossinsin/cos a

2121211 coscoscossinsincos a

The Initial Heading Angle ψ0 can be obtained using theLaw of Cosines for Spherical Triangles as follows

a

a

/sincos

/cossinsincos

1

120

2222

22221

coscoscossinsin1cos

coscoscossinsinsinsincos

The Heading Angle ψ from the Present Position (R,ϕ,λ) to Destination Point (R2,ϕ2,λ2)

9

Navigation SOLO

I

Ecuator

1R

2R11,

Ex

Ey

Ez

1

222 ,

,


1 2

0

1

2


The Distance on the Great Circle between two points 1 and 2 is ρ.

1

2111 ,, R

222 ,, RR – radiusϕ - Latitudeλ - Longitude

The Time required to travel along the Great Circle between points 1 and 2 is given by

22

2121211 coscoscossinsincos

yxHoriz

HorizHoriz

VVV

V

a

Vt

2121211 coscoscossinsincos a

10

Navigation SOLO

I

Ecuator

1R

2R

Ex

Ey

Ez

1

2

TrajectoryGreat Circcle

1 2

1R

2R1

2

O

A

B

Ca

b

c

c

Earth Center

North Pole

A

B'B90

P


1

2111 ,, R

222 ,, R

If the Aircraft flies with an Heading Error Δψ we want to calculate the Down Range Error Xd and Cross Range Error Yd, in the Spherical Triangle APB.


Using the Law of Cosines for Spherical Triangle APB we have

aaYd /sin

90sin

/sin

sin

2/sin/sin

/cos/cos/cos0ˆcos 21

90ˆ RRa

aYaX

aYaXaP

dd

ddP

Using the Law of Sines for Spherical Triangle APB we have

aY

aaX

dd /cos

/coscos 1

sin/sinsin 1 aaYd

11

SOLO

Coordinate Systems

1. Heliocentric (Heliocentric) Coordinate System

COORDINATES IN THE SOLAR SYSTEM

Sun

First Dayof Spring

First Dayof Summer

First Dayof Autumn

First Dayof Winter

Vernal EquinoxDirection

X

Y

Z

Venus

Mercury

Moon

Sun at the center of coordinate system (Heliocentric)

Earth plan orbit (Ecliptic) on which Xε and Yε are defined as:• Xε the direction between the Sun to Earth on the First Day of Autumn. This is called Vernal Equinox Direction and points in the direction of constellation Aries (the Ram)

• Zε normal to the Ecliptic in the North hemisphere direction.

• Yε on the Ecliptic and completing the right hand coordinate system.

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SOLO

1 .Heliocentric (Heliocentric) Coordinate System (Continue)


The Earth axis of rotation is tilted relative to Ecliptic and vobbles slightly, in a clockwisedirection opposite to that of the Earth spin, from 22.1° to 24.5° , with a cycle of approximately41,000 years.

G

Gz

Gx

Gy

Ecliptic planenormal

(Ecliptic Pole)

Locus of Lunar plane normal(Lunar Pole)

Lunar Orbital Plane

Earth Orbital Plane (Ecliptic)

Equatorial Plane

Ascending Node

5.2315.5


The Moon’s gravity tends to tilt the Earth’s axis so that it becomes perpendicular to Moon’sOrbit, and to a lesser extent the same is true for the Sun.

This effect is called precession and is produced by the interaction between Earth and Moon.

Sun

First Dayof Spring

First Dayof Summer

First Dayof Autumn

First Dayof Winter


X

Y

Z

Venus

Mercury

Moon

13

SOLO

2. Geocentric-Equatorial Coordinate System


The origin at the center of the Earth .

G

Gz

Gx

Gy

Ecliptic planenormal

(Ecliptic Pole)

Locus of Lunar plane normal(Lunar Pole)

Lunar Orbital Plane

Earth Orbital Plane (Ecliptic)

Equatorial Plane

Ascending Node

5.2315.5


• XG axis on the Equatorial Plane in the vernal equinox direction.

• ZG axis in the direction of North pole.

• YG axis completes the right hand coordinate system.

XG, YG, ZG system is not fixed to the Earth; rather, the geocentric-equatorial frameis non-rotating to the stars (except to the precession of equinoxes) and the Earthturns relative to it.

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SOLO

3. The Right Ascension-Declination System


The Right Ascension-Declination System defines the position of objects in space.

• Celestial Equator that contains the Earth Equatorial Plane.

• The XG, YG, ZG axes are parallel to the Geocentric-Equatorial Plane.

• The origin of the system can be at the Earth origin (geocentric) or at the surface of the Earth (topocentric). Because of he enormous distance of the star the location of the origin doesn’t effect their angular position.

GZ

GX

GYEquatorial

Plane


The fundamental plane is:

The position of a star is defined by two parameters:• right ascension, α, is measured eastward in the plane of the celestial equator from the vernal equinox direction.

• declination,δ, is measured northward from the celestial equator to the line of sight of the object.

15

SOLO

Coordinate Systems

4. The Perifocal Coordinate System


The Perifocal Coordinate System is related to a satellite’s orbit.

• Xω axis in the direction of the orbit Periapsis (direction from the focal point to the point of minimum range of the orbit).

Plane of the Satellite’s Orbit is the fundamental plane with:

GZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

• Zω axis in the direction of (perpendicular to the Satellite’s Orbit and showing the satellite’s movement direction).

vrh

• Yω axis completes the right hand coordinate system.

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SOLO

4. The Perifocal Coordinate System (Continue)


Five independent quantities, called orbital elements,describe size, shape and orientation of an orbit.A sixth element is required to determine the position of the satellite along the orbit at agiven time.

1. a – semi-major axis – a constant defining the size of the coning orbit.

GZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

2. e – eccentricity – a constant defining the shape of the coning orbit.

3. i – inclination – the angle between ZG and the specific angular momentum of the coning orbit . vrh

4. Ω – longitude of the ascending node – the angle, in the Equatorial Plane, betweenthe unit vector and the point where the satellite crosses through the Equatorial Plane in a northerly direction (ascending node) measured counterclockwisewhere viewed from the northern emisphere.

5. ω – argument of the periapsis – the angle, in the plane of the satellite’s orbit,between ascending node and the periapsis point, measured in the direction of satellite’s motion.

6. T – time of periapsis passage – the time when the satellite was at the periapsis.

Classical Orbital Parameters

17

SOLO


COORDINATES IN THE SOLAR SYSTEMGZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

Let find a, e, ω, i, Ω from the initial position and velocity vectors .00 ,vr

1. From the specific angular momentum of the orbit we can findvrh

00 vrh

01 00

hh

vrZ

2. From the specific mechanical energy of an elliptic orbit equation ar

vvE

22 0

00

we obtain 00

0

2 vvr

a

3. i inclination is computed using

GZZi 11cos

2211cos 1

iZZi G

4. The eccentricity vector of a Keplerian trajectory is defined as

Xevvrr

rvve 1

10000

000

from which ee

01

ee

eX

XZY 111

18

SOLO


COORDINATES IN THE SOLAR SYSTEMGZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

Let find a, e, ω, i, Ω from the initial position and velocity vectors (continue).00 ,vr

5. The ascending node (intersection of the equatorial and orbit planes) is given by

01111

111

ZZZZ

ZZN G

G

G

Ω – longitude of the ascending node – is computed using

NX G 11cos

GG ZNX 111sin

cos

sintan 1

6. ω – argument of the periapsis – is computed using

XN 11cos

ZXN 111sin

cos

sintan 1

7. Ө – satellite position from the periapsis – is computed using

rX 11cos

ZrX 111sin

cos

sintan 1

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SOLO



Let find a, e, ω, i, Ω from the initial position and velocity vectors (continue).

GZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

00 ,vr

The rotation matrix from the Perifocal Coordinate System Xε , Yε, Zε to the Geocentric-Equatorial Coordinate System XG, YG, ZG is given by:

100

0cossin

0sincos

cossin0

sincos0

001

100

0cossin

0sincos

313

ii

iiiCG

iii

iii

iii

cossincossinsin

sincoscoscoscossinsinsincoscoscossin

sinsincoscossinsincossincossincoscos

20

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 MassSystem

8.2 Missile Kinematics Model 1 (Spherical Earth)

8.3 Missile Kinematics Model 2 (Spherical Earth)

21

Bz

MV

Bx

ByWy

WzBr

Bp

Wp

BqWqWr

Given an Air Vehicle, 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

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

Coordinate Systems

Table of Content

22

SOLO

Coordinate Systems

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

1 .Inertial System (I)

R

- vehicle position vector

Itd

RdV

- vehicle velocity vector, relative to inertia

IItd

Rd

td

Vda

2

2

- vehicle acceleration vector, relative to inertia

AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE

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SOLO

Coordinate Systems (continue – 2)

2. Earth Center Earth Fixed Coordinate –ECEF-System (E) xE, yE in the equatorial plan with xE pointed to the intersection between the equatorto zero longitude meridian.

The Earth rotates relative to Inertial system I, with the angular velocity

sec/10.292116557.7 5 rad

EIIE zz

11

0

0EC

IE

Rotation Matrix from I to E

100

0cossin

0sincos

3 tt

tt

tC EI


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

24

SOLO

Coordinate Systems (continue – 3)2. Earth Center Earth Fixed Coordinate System (E) (continue – 1)

Vehicle Position ETEI

EIE

I RCRCR

Vehicle Velocity

Vehicle Acceleration

RVRtd

Rd

td

RdV EIE

EI

- vehicle velocity relative to Inertia

Rtd

Rd

td

RdV IE

LE

E

: - vehicle velocity relative to Earth

II

E

I

E

I

Rtd

d

td

VdRV

td

d

td

Vda

RVtd

VdR

td

RdR

td

dV

td

VdEIEEU

U

E

EE

EIU

U

E

IU

0

RVtd

VdRV

td

Vda E

E

EEEU

U

E

22

or

where U is any coordinate system. In our case U = E.


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

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25

SOLO


3 .Earth Fixed Coordinate System (E0)

The origin of the system is fixed on the earth at somegiven 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 Earthsurface, zE0 is pointed down.

100

0cossin

0sincos

sin0cos

010

cos0sin

2/ 00

00

00

00

30200 LongLong

LongLong

LatLat

LatLat

LongLatC EE

00000

00

00000

sinsincoscoscos

0cossin

cossinsincossin

LatLongLatLongLat

LongLong

LatLongLatLongLat

The Angular Velocity of E relative to I is: EIIEIE zz

110 or

0

0

00000

00

00000

000

sin

0

cos

0

0

sinsincoscoscos

0cossin

cossinsincossin

0

0

Lat

Lat

LatLongLatLongLat

LongLong

LatLongLatLongLat

C EE

EIE


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

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26

SOLO

Coordinate Systems (continue – 5)4. Local-Level-Local-North (L) or Navigation Frame

The origin of the LLLN coordinate system is located atthe projection of the center of gravity CG of the vehicleon the Earth surface, with zDown axis pointed down, xNorth, yEast plan parallel to the local level, withxNorth pointed to the local North and yEast pointed tothe 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 LE

LatLongLatLongLat

LongLong

LatLongLatLongLat

sinsincoscoscos

0cossin

cossinsincossin


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

27

SOLO


4. Local-Level-Local-North (L) (continue – 1)

Angular Velocity

IEELIL Angular Velocity of L relative to I

Lat

Lat

LatLongLatLongLat

LongLong

LatLongLatLongLat

C LE

Down

East

NorthL

IE

sin

0

cos

0

0

sinsincoscoscos

0cossin

cossinsincossin

0

0

LatLong

Lat

LatLong

Lat

LongLatLongLatLongLat

LongLong

LatLongLatLongLat

Lat

Long

C LE

Down

East

NorthL

EL

sin

cos

0

0

0

0

sinsincoscoscos

0cossin

cossinsincossin

0

0

0

0

LatLong

Lat

LatLong

DownDown

EastEast

NorthNorthL

IECL

ECLL

IL

sin

cos

Therefore


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

28

SOLO



Vehicle Velocity

Vehicle Velocity relative to I

RVRtd

Rd

td

RdV EIE

EI

HRLatLongLat

LatLongLatLong

LatLatLong

HR

Rtd

RdV EL

L

LE

00

0

0

0cos

cos0sin

sin0

0

0

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

Vtd

Hd

LatHR

V

td

Longd

HR

V

td

Latd

_

0

_

0

_

cos


HeightVehicleHRadiusEarthmRHRR 600 10378135.6

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

29

SOLO



Vehicle Velocity (continue – 1)

We assume that the atmosphere movement (velocity and acceleration) relative to EarthAt the vehicle position (Lat, Long, H) is known. Since the aerodynamic forces on thevehicle are due to vehicle movement relative to atmosphere, let divide the vehiclevelocity in two parts:

WAE VVV

Down

East

NorthL

A

V

V

V

V

- Vehicle Velocity relative to atmosphere

DownW

EastW

NorthW

LW

V

V

V

HLongLatV

_

_

_

,,

- Wind Velocity at vehicle position (known function of time)


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

30

SOLO




Since:

RVtd

VdR

td

d

td

VdRV

td

d

td

Vda EEL

L

E

II

E

I

E

I

2

WAE VVV

WWIL

L

WAAIL

L

A VVtd

VdRVV

td

Vda

Wa

WWEL

L

WAAEL

L

A VVtd

VdRVV

td

Vd 22

HLongLatVHLongLattd

VdHLongLata WEL

L

WW ,,2,,:,,

WAAEL

L

A aRVVtd

Vd

2

where:

is the wind acceleration at vehicle position.


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

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31

SOLO


5 .Body Coordinates (B)

The origin of the Body coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xB pointedto the front of the Air Vehicle, yB pointedtoward the right wing and zB completingthe right-handed Cartesian reference frame.

Bx

Lx

Bz

Ly

LzBy

Rotation Matrix from LLLN to B (Euler Angles):

cccssscsscsc

csccssssccss

ssccc

C BL 321

- azimuth angle

- pitch angle

- roll angle


32

SOLO


5 .Body Coordinates (B) (continue – 1)

Bx

Lx

Bz

Ly

LzBy

Angular Velocity from L to B (Euler Angles):

0

0

0

0

0

0 211

R

Q

PB

LB

0

0

cos0sin

010

sin0cos

cossin0

sincos0

001

0

0

cossin0

sincos0

001

0

0

G

coscossin0

cossincos0

sin01


33

SOLO



Bx

Lx

Bz

Ly

LzBy

Rotation Matrix from LLLN to B (Quaternions):

321

412

143

234

3412

2143

1234

44 3333

BIBLBL

BLBLBL

BLBLBL

BLBLBL

BLBLBLBL

BLBLBIBL

BLBLBLBL

TBLBLBLXBLBLXBL

BL

qqq

qqq

qqq

qqq

qqqq

qqqq

qqqq

qqqIqqIqC

where:

3

2

1

:&4

4

3

2

1

4

3

2

1

BL

BL

BL

BLBL

BLBL

BL

BL

BL

BL

BL

BL

BL

BL

BL

q

q

q

qq

qqor

q

q

q

q

q

q

q

q

q

2sin

2sin

2sin

2cos

2cos

2cos4

BLq

2cos

2sin

2sin

2sin

2cos

2cos1

BLq

2sin

2cos

2sin

2cos

2sin

2cos2

BLq

2sin

2sin

2cos

2cos

2cos

2sin3

BLq


34

SOLO



Bx

Lx

Bz

Ly

LzBy

Rotation Matrix from LLLN to B (Quaternions))continue – 1(

The quaternions are given by the followingdifferential equations:

BL

LIL

BIBBLBLBL

BILBL

BIBBL

BIL

BIBBL

BLBBLBL qqqqqqqqq

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

4

3

2

1

0

0

0

0

2

1

BL

BL

BL

BL

zLzLyLyLxLxL

zLzLxLxLyLyL

yLyLxLxLzLzL

xLxLyLyLzLzL

q

q

q

q

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

or:


35

SOLO



Bx

Lx

Bz

Ly

LzBy

Vehicle Velocity

Vehicle Velocity relative to Earth is divided in:

WAE VVV

w

v

u

V BA

DownW

EastW

NorthW

BL

zW

yW

xW

BW

V

V

V

C

V

V

V

HLongLatV

B

B

B

_

_

_

,,


WWIB

B

WAAIB

B

A

I

VVtd

VdRVV

td

Vd

td

Vda

W

AELALB

B

A

a

RVVtd

Vd

2


Table of Content

36

SOLO


6 .Wind Coordinates (W)

Bx

Lx

Bz

Ly

LzBy

Wz

V

The origin of the Wind coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xW pointedin the direction of the vehicle velocity vectorrelative to air .AV

cos0sin

sinsincossincos

cossinsincoscos

cos0sin

010

sin0cos

100

0cossin

0sincos

23WBC

The Wind coordinate frame is defined by the following two angles:

- angle of attack

- sideslip angle


37

SOLO


6 .Wind Coordinates (W) (continue -1)

Bx

Lx

Bz

Ly

LzBy

Wz

V

Rotation Matrix from L (LLLN) to W is:

- azimuth angle of the trajectory

- pitch angle of the trajectory

Rotation Matrix

32123 BL

WB

WL CCC

The Rotation Matrix from L (LLLN) to W can also be defined by the following Consecutive rotations:

- bank angle of the trajectory

cccssscsscsc

csccssssccss

ssccc

CC WL

WL 321

*1

We defined also the intermediate wind frame W* by:

csscs

cs

ssccc

CWL 032

*


38

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

Angular Velocity of W* relative to LLLN is:

Angular Velocities

cos

sin

0

0

cos0sin

010

sin0cos

0

0

0

0

0

0

2

*

*

**

*

W

W

WW

LW

R

Q

P

Angular Velocity of W relative to LLLN is:

coscossin0

cossincos0

sin01

cos

sin

cossin0

sincos0

001

0

00

0

0

0

0

0 21

W

W

WW

LW

R

Q

P


39

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

We have also:

Angular Velocities (continue – 1)

Down

East

North

WL

WL

LIE

WL

zW

yW

xWW

IE C

Lat

Lat

CC ***

*

*

**

sin

0

cos

Down

East

North

WL

WL

LEL

WL

zW

yW

xWW

EL C

LatLong

Lat

LatLong

CC

***

*

*

**

sin

cos

*

1

sin

0

cosW

IEWL

LIE

WL

zW

yW

xWW

IE

Lat

Lat

CC

*1

sin

cos

WIL

WL

LIL

WL

WIL

LatLong

Lat

LatLong

CC


40

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

The Angular Velocity from I to W is:


DownDown

EastEast

NorthNorth

WL

W

W

WL

ILWL

W

W

WW

ILW

LW

W

W

WW

IW C

R

Q

P

C

R

Q

P

r

q

p

Using the angle of attack α and the sideslip angle β , we can write:

BWBW yz

11

or:

0

0

0

0

3

r

q

p

C

r

q

pWB

W

W

WW

IBW

IWW

BW

but also:

0

0

0

0

3

R

Q

P

C

R

Q

PWB

W

W

WW

LBW

LWW

BW


41

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

We can write:


0

cos

sin

0

0

cos0sin

sinsincossincos

cossinsincoscos

r

q

p

r

q

p

W

W

W

or:

cossin

sinsincossincos

cossinsincoscos

rpr

rqpq

rqpp

W

W

W

This can be rewritten as:

tansincoscos

rpq

q W

Wrrp cossin

cos

sinsincos

tantansincossincossincossincos

W

WW

qrp

qrpqrpp


42

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

We have also:


tansincoscos

RPQ

Q W

WRRP cossin

cos

sinsincos WW

QRPP


43

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

The vehicle velocity was decomposed in:

Vehicle Velocity

WAE VVV

0

0

V

V WA

- vehicle velocity relative to atmosphere

DownW

EastW

NorthW

WL

zW

yW

xW

WW

V

V

V

C

V

V

V

HLongLatV

W

W

W

_

_

_

,,

- wind velocity at velocity position

also

0

0

0

011*

VV

VV WA

WA

DownW

EastW

NorthW

WL

zW

yW

xW

WW

V

V

V

C

V

V

V

HLongLatV

W

W

W

_

_

_

*

*

*

*

* ,,


44

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

The vehicle acceleration in W* coordinates is


WAELALW

W

A

WWIW

W

WAAIW

W

A

I

C

aRVVtd

Vd

VVtd

VdRVV

td

Vd

td

Vda

2*

*

*

*

*

*

from which

*******

*

*

*

2 WW

WA

WWEL

WWA

WLW

W

W

A aVAVtd

Vd

where

RaA

:


45

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

Vehicle Acceleration (continue – 1)

**

*

*

****

****

****

*

*

*

**

**

**

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

where

HR

Lat

Lat

C

a

a

a

A

A

A

A WL

zW

yW

xW

zW

yW

xW

W

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


46

SOLO



Bx

Lx

Bz

Ly

LzBy

Wz

V

Vehicle Acceleration (continue – 2)

Using

cos

sin

*

*

**

*

W

W

WW

LW

R

Q

P

we have

** xWWxW aAV

cos/2 ****

zWzW

yWWyW

V

aA

****

2 yWyWzWWzW

V

aA


Table of Content

47

SOLO

Aerodynamic Forces npp ˆt

nV

ds

wx1

wy1

wz1

tf ˆ

Pressure force

Friction force

WS

WS

A dstfnppF

11

ntonormalplanonVofprojectiont

dstonormaln

ˆˆ

ˆ

airflowingthebyweatedsurfaceVehicleS

SsurfacetheonmNstressforcefrictionf

Ssurfacetheondifferencepressurepp

W

W

W

)/( 2


7. Forces Acting on the Vehicle

48

SOLO

7. Forces Acting on the Vehicle (continue – 1)

Bx

Lx

Bz

Ly

LzBy

Wz

V

WyT

C

L

D

g

Aerodynamic Forces (continue – 1)

L

C

D

F WA

ForceLiftL

ForceSideC

ForceDragD

L

C

D

CSVL

CSVC

CSVD

2

2

2

2

12

12

1

tCoefficienLiftRMC

tCoefficienSideRMC

tCoefficienDragRMC

eL

eC

eD

,,,

,,,

,,,

ityvisdynamic

lengthsticcharacteril

soundofspeedHa

numberynoldslVR

numberMachaVM

e

cos

)(

Re/

/


49

SOLO


Aerodynamic Forces (continue -2)

W

W

W

SfpL

SfpC

SfpD

dswztCwznCS

C

dswytCwynCS

C

dswxtCwxnCS

C

1ˆ1ˆ1

1ˆ1ˆ1

1ˆ1ˆ1

nCq p ˆt

nV

ds

wx1

wy1

wz1

tCq fˆ

Pressure force

Friction force

WS SVq 2

2

1

Wf

Wp

SsurfacetheontcoefficienfrictionV

fC

SsurfacetheontcoefficienpressureV

ppC

2/

2/

2

2

ntonormalplanonVofprojectiont

dstonormaln

ˆˆ

ˆ


50

MomentFriction

S

C

Momentessure

S

CCA

WW

dstRRfdsnRRppM 11

Pr

/

Aerodynamic Moments Relative to C can be divided in Pressure Moments andFriction Moments.

FrictionSkinorFrictionViscous

S

essureNormal

S

A

WW

dstfdsnppF 11

Pr

fp

V

ASALM

Aerodynamic Forces can be divided in Pressure Forces and Friction Forces.


npp ˆt

nV

ds

wx1

wy1

wz1

tf ˆ

Pressure force

Friction force

WS

AERODYNAMIC FORCES AND MOMENTS.

51

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

52

SOLO


BxLx

Bz

Ly

LzBy

Wz

V

Wy

T

C

L

D

gT

T

Thrust

B

B

B

z

y

x

BWB

W

T

T

T

TCT

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


Bx

Lx

Bz

Ly

LzBy

Wz

V

WyT

C

L

D

g

F-35Thrust Vector Control

53

SOLO


Gravitation Acceleration

zgygxg

gg100

0

0

0

010

0

0

0

001

cs

sc

cs

sc

cs

scC EWE

W

gg

cc

cs

sW

2sec/174.322sec/81.90

2

0

00gg ftmg

HR

R


The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth.The Gravitational Potential U (R,ϕ) is given by

,

sin1,2

RUg

PR

aJ

RRU

EE

n n

n

n

μ – The Earth Gravitational Constanta – Mean Equatorial Radius of the EarthR=[xE

2+yE2+zE

2]]/2 is the magnitude of the Geocentric Position Vectorϕ – Geocentric Latitude (sinϕ=zE/R)Jn – Coefficients of Zonal Harmonics of the Earth Potential FunctionPn (sinϕ) – Associated Legendre Polynomials

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

54

SOLO


Gravitation Acceleration


Retaining only the first three terms of theGravitational Potential U (R,ϕ) we obtain:

R

z

R

z

R

z

R

aJ

R

z

R

aJ

Rg

R

y

R

z

R

z

R

aJ

R

z

R

aJ

Rg

R

x

R

z

R

z

R

aJ

R

z

R

aJ

Rg

EEEEz

EEEEy

EEEEx

E

E

E

342638

515

2

31

342638

515

2

31

342638

515

2

31

2

2

4

44

42

22

22

2

2

4

44

42

22

22

2

2

4

44

42

22

22

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

sin

cossin

coscos

R

zR

yR

x

E

E

E

2/1222EEE zyxR

55

SOLO


Force Equations

Bx

Lx

Bz

Ly

LzBy

Wz

V

WyT

C

L

D

g

Air Vehicle Acceleration

WAELALW

W

A

I

C aRVVtd

Vd

td

Vda

2

WAELALW

W

AA aRVV

td

VdamTF

m

2

1 g

Rg

g: Define

ccgm

LT

csgm

CT

sgm

DT

A

A

A

zW

yW

xW

sin

sincos

coscos

cg

sg

m

LTm

CTm

DT

A

A

A

zW

yW

xW

0

sin

sincos

coscos

cossin0

sincos0

001

*

*

*


Table of Content

0

0

T

T B

56

SOLO

23. Local Level Local North (LLLN) Computations for an Ellipsoidal Earth Model

2

2210

20

20

20

5

21

20

60

sin

sin1

sin321

sin1

sec/10292116557.7

sec/051646.0

sec/780333.9

26.298/.1

10378135.6

Ae

e

p

m

e

HR

RLatggg

LatfRR

LatffRR

LatfRR

rad

mg

mg

f

mR

LatHR

V

HR

V

HR

V

Ap

EastDown

Am

NorthEast

Ap

EastNorth

tan

Lat

Lat

Down

East

North

sin

0

cos

DownDownDown

EastEast

NorthNorthNorth

East

North

Lat

LatLong

cos

t

t

dtLatLattLat

dtLongLongtLong

0

0

0

0


I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

SIMULATION EQUATIONS

57

SOLO

Down

East

North

Down

East

North

1 Latcos

1

LatHR

V

HR

V

HR

V

Ap

EastDown

Am

NorthEast

Ap

EastNorth

tan

Long

Lat

Down

East

North

sin

0

cos

Down

East

North

L

V

V

V

V

2

2210

20

20

20

5

21

20

60

sin

sin1

sin321

sin1

sec/10292116557.7

sec/051646.0

sec/780333.9

26.298/.1

10378135.6

Ae

e

p

m

e

HR

RLatggg

LatfRR

LatffRR

LatfRR

rad

mg

mg

f

mR

s

1

s

1

DownDownDown

EastEast

NorthNorthNorth

pR

mR

AH

Long

0Long

Lat

0Lat

Lat

Long

g

Lat g

LOCAL LEVEL LOCAL NORTH COMPUTATIONS

Lat

North DownEast

AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERESIMULATION EQUATIONS

Table of Content

58

SOLO

LBL

BBA

BCG gCT

mF

ma

11

B

BrCrrotorB

IBB

BrCrrotor

BIBC

BIB

BIBCCTCAC

BIB

II

IIMMI

��

��

,,

,,,,1

BCG

TBL

LCG aCa

BIB

BL

LIL

BIBBLBL qqq

2

1

2

1

s

1

CT

CA

M

M

,

,

TBL IqIqC

3434

BIB

BCGa

LCGa

B

BA

T

F

BLC

BLC

s

1 BLqBLq

BLC

s

1 L

ELL

EL

LLCG

LE VRaV

2 s

1 L

EV L

EV

LCGa

BLC

LMR

LEV

LM

BL

BM VCV

Mee

22M

s

1

s

1

LEV

23 WBC

WBC

MV

WEM VVV

LMV

LWV

BIB

BBrotor

BBrotor ,

Missile Kinematics Model 1 in Vector Notation (Spherical Earth)


59

SOLO

rB

rB

r

rotor

B

B

B

zzyzxz

yzyyxy

xzxyxx

BB

BB

BB

B

B

B

zzyzxz

yzyyxy

xzxyxx

zBC

yBC

xBC

zBCA

yBCA

xBCA

zzyzxz

yzyyxy

xzxyxx

B

B

B

q

rI

r

q

p

III

III

III

pq

pr

qr

r

q

p

III

III

III

T

T

T

M

M

M

III

III

III

r

q

p

0

0

0

,

,

,

,

,

,

1

rr ,

B

B

B

r

q

p

s

1

B

B

B

r

q

p

B

B

B

r

q

p

4

3

2

1

0

0

0

0

2

1

4

3

2

1

BL

BL

BL

BL

DownDownBEastEastBNorthNorthB

DownDownBNorthNorthBEastEastB

EastEastBNorthNorthBDownDownB

xLxLByLyLBzLzLB

BL

BL

BL

BL

q

q

q

q

rqp

rpq

qpr

pqr

q

q

q

q

s

1

4

3

2

1

BL

BL

BL

BL

q

q

q

q

4

3

2

1

BL

BL

BL

BL

q

q

q

q

g

C

C

C

T

T

T

mF

F

F

ma

a

a

BL

BL

BL

zB

yB

xB

zBA

yBA

xBA

zB

yB

xB

3,3

3,2

3,111

zBzBA

yByBA

xBxBA

TF

TF

TF

zB

yB

xB

a

a

a

zB

yB

xB

BL

BL

BL

BL

BL

BL

BL

BL

BL

Down

East

North

a

a

a

CCC

CCC

CCC

a

a

a

3,33,23,1

2,32,22,1

1,31,21,1

Down

East

North

a

a

a

BLC

BLC

4

3

2

1

*

1

4

3

2

1

BL

BL

BL

BL

BL

BL

BL

BL

q

q

q

q

qq

q

q

q

q 4

3

2

1

BL

BL

BL

BL

q

q

q

qB

LC

321

412

143

234

3412

2143

1234

BLBLBL

BLBIBL

BLBLBL

BLBLBL

BLBLBLBL

BLBLBLBL

BLBLBLBLB

L

qqq

qqq

qqq

qqq

qqqq

qqqq

qqqq

C

43

2

1

BL

BL

BL

BL

q

q

q

q

Down

East

North

a

a

a

DownW

EastW

NorthW

NorthNorthEastEast

NorthNorthDownDown

EastEastDownDown

Down

East

North

DownE

EastE

NorthE

V

V

V

Lat

Lat

HR

a

a

a

V

V

V

022

202

220

sin

0

cos2

_

_

_

s

1

DownE

EastE

NorthE

V

V

V

cos0sin

sinsincossincos

cossinsincoscosW

BC

WBC

s

1

H

Long

Lat

H

Long

Lat

DownE

p

EastE

m

NorthE

Vtd

Hd

LatHR

V

td

Longd

HR

V

td

Latd

cos

w

v

u

DownM

EastM

NorthM

BL

BL

BL

BL

BL

BL

BL

BL

BL

V

V

V

CCC

CCC

CCC

w

v

u

_

_

_

3,32,31,3

3,22,21,2

3,12,11,1

DownW

EastW

NorthW

DownE

EastE

NorthE

DownM

EastM

NorthM

V

V

V

V

V

V

V

V

V

_

_

_

_

_

_

_

_

_

DownM

EastM

NorthM

V

V

V

DownE

EastE

NorthE

V

V

V

M

M

Vv

uw

wvuV

/sin

/tan1

1

222

MV

DownE

EastE

NorthE

V

V

V

zBCzBCA

yBCAyBCA

xBCxBCA

TM

TM

TM

,,

,,

,,

DownE

EastE

NorthE

V

V

V

LatHR

V

HR

V

HR

V

EastE

NorthE

EastE

Down

East

North

tan0

0

0

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C

*

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat

Lat

Down

East

North

sin

0

cos

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C *

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat

Missile Kinematics Model 1 in Matrix Notation (Spherical Earth)


60

SOLO

LBL

BBA

BCG gCT

mF

ma

11

B

BrCrotorB

IBB

BrCrotor

BIBC

BIB

BIBCCTCA

CB

IBII

IIMMI

,,

,,,,1,

BCG

WB

WCG aCa

BIB

BL

LIL

BIBBLBL qqq

2

1

2

1

s

1

CT

CA

M

M

,

,

TBL IqIqC

3434

BIB

BB

BCGa

WCGa

B

BA

T

F

BLC

BLC W

BC

s

1 BLqBLq

LWa

BLC

IBIWWB zy 11

WM

TWB

BM VCV

L

WB

M

TBL

LE VVCV

s

1

WW

WM

WWWCG

WM

WIW

WM

aVRa

VV

s

1

s

1

WIW

WMV

WMV

WCGa

23 WBC

WIW B

IB

WMV

BLC

WBC

WBC

BMV L

EV L

MR

Mee

22M

s

1

s

1

WMV

LWV

WWa

LW

BL

WB

WW aCCa

L

WEL

L

WLW V

td

Vda

2

LWV

L

W

td

Vd

BBr

BBr

,

Missile Kinematics Model 2 in Vector Notation (Spherical Earth)


61

SOLO

Missile Kinematics Model 2 in Matrix Notation (Spherical Earth)

rB

rB

r

Crrotor

B

B

B

zzyzxz

yzyyxy

xzxyxx

BB

BB

BB

B

B

B

zzyzxz

yzyyxy

xzxyxx

zBCG

A

yBCG

A

xBCG

A

zzyzxz

yzyyxy

xzxyxx

B

B

B

q

rI

r

q

p

III

III

III

pq

pr

qr

r

q

p

III

III

III

M

M

M

III

III

III

r

q

p

,

1

0

0

0

zBCTzBCA

yBCTyBCA

xBCTxBCA

MM

MM

MM

,,

,,

,,

B

B

B

r

q

p

s

1

B

B

B

r

q

p

B

B

B

r

q

p

4

3

2

1

0

0

0

0

2

1

4

3

2

1

BL

BL

BL

BL

DownBEastEastBNorthNorthB

DownBNorthNorthBEastEastB

EastEastBNorthNorthBDownB

NorthNorthBEastEastBDownB

BL

BL

BL

BL

q

q

q

q

rqp

rpq

qpr

pqr

q

q

q

q

Down

Down

Down

Down

s

1

4

3

2

1

BL

BL

BL

BL

q

q

q

q

4

3

2

1

BL

BL

BL

BL

q

q

q

q

g

C

C

C

T

T

T

mF

F

F

ma

a

a

BL

BL

BL

zB

yB

xB

zBA

yBA

xBA

zB

yB

xB

3,3

3,2

3,111

zBzBA

yByBA

xBxBA

TF

TF

TF

zB

yB

xB

a

a

a

zB

yB

xB

WB

WB

WB

WB

WB

WB

WB

WB

WB

zW

yW

xW

a

a

a

CCC

CCC

CCC

a

a

a

3,32,31,3

3,22,21,2

3,12,11,1

zW

yW

xW

a

a

a

zWM

yWWyWW

yWM

zWWzWW

xWWxWM

V

aar

V

aaq

aaV

zW

yW

xW

a

a

a

W

W

M

r

q

Vs

1

B

WBB

WBB

WBW

BW

BBW

BBW

BW

rCqCpCr

qCrCpCq

3,32,31,3

2,2/3,21,2

s

1

cos0sin

sinsincossincos

cossinsincoscos

23W

BC

WBC

WBC

WBC

WBC

BLC

MV

MV

4

3

2

1

*

1

4

3

2

1

BL

BL

BL

BL

BL

BL

BL

BL

q

q

q

q

qq

q

q

q

q 4

3

2

1

BL

BL

BL

BL

q

q

q

qB

LC

321

412

143

234

3412

2143

1234

BLBLBL

BLBIBL

BLBLBL

BLBLBL

BLBLBLBL

BLBLBLBL

BLBLBLBLB

L

qqq

qqq

qqq

qqq

qqqq

qqqq

qqqq

C

43

2

1

BL

BL

BL

BL

q

q

q

q

M

WB

WB

WB

V

C

C

C

w

v

u

3,1

2,1

1,1

w

v

u

s

1

H

Long

Lat

H

Long

Lat

DownW

EastW

NorthW

BL

BL

BL

BL

BL

BL

BL

BL

BL

DownE

EastE

NorthE

V

V

V

w

v

u

CCC

CCC

CCC

V

V

V

_

_

_

_

_

_

3,33,23,1

2,32,22,1

1,31,21,1

DownE

p

EastE

m

NorthE

Vtd

Hd

LatHR

V

td

Longd

HR

V

td

Latd

cos

rr ,

DownE

EastE

NorthE

V

V

V

LatHR

V

HR

V

HR

V

EastE

NorthE

EastE

Down

East

North

tan0

0

0

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C

*

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat

Lat

Down

East

North

sin

0

cos

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C *

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat


SOLO

62

Navigation

Methods of Navigation

• Dead Reckoning (e.g. Inertial Navigation)

• Externally Dependent (e.g. GPS)

• Database Matching (e.g Celestial Navigation, or Terrain Referenced Navigation)

SOLO

63

Navigation

Dead Reckoning Navigation

A Dead Reckoning System uses a Platform Initial Position and Initial VelocityVector and then Computes its Position and Velocity based on Measured orEstimated Velocity Vector and Elapsed Time.

Dead Reckoning Evolution of a Vehicle’s Position Based on Velocity Vector

InitialPosition

CurrentPosition

FinalPosition

0V1

2V2

1V

0

SOLO

64

Navigation

Dead Reckoning Navigation

Historical Development of Inertial Platforms

SOLO

65

Navigation

Dead Reckoning Navigation Based on an Inertial Measurement Unit (IMU)

An Inertial Measurement Unit uses Inertial Sensors (at least three Rateand three Acceleration Sensors). - The Rate Sensors measure the Angular Rates, relative to Inertia, along three orthogonal directions.- The Acceleration Sensors (Accelerometers) measure the Acceleration, relative to Inertia, along the same three orthogonal directions.

The Sensor Case can be attached to a Stabilized Platform (Gimbaled) or Strap to the Vehicle Body.

(b) Strapdown(a) Gimbaled

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66

Navigation

Dead Reckoning Navigation Based on an Inertial Measurement Unit (IMU)

The Gimbaled System can be Local-Level Stabilized or Space-Stabilized

(a) Gimbaled

According to the chosen Azimuth Mechanization the Local-Level can be: - North-Slaved (or North Pointing) - Unipolar - Free Azimuth - Wander Azimuth

SOLO

67

Navigation

Input/Output of an Inertial SystemInputsADC.Angle_of_AttackADC.Mach_NumberADC.Barometric_AltitudeADC.Magnetic_HeadingADC.True_AirspeedINS.Body_Rates (roll, pitch, yaw)INS.Acceleration (lateral, longitudinal, normal)INS.Present_Position (latitude, longitude)INS.True_HeadingINS.Velocity (north, east, vertical)RALT.Radar_Altitude

OutputsINS.Reference_Velocity (north, east, vertical)NAV.AirspeedNAV.Rate_of_Change_AirspeedNAV.Position (latitude, longitude, altitude)NAV.Angle_of_AttackNAV.Attitude (roll, pitch, yaw)NAV.Body_Rates (roll, pitch, yaw)NAV.Flight_Path_AngleNAV.Ground_SpeedNAV.Ground_Track_AngleNAV.Magnetic_VariationNAV.AltitudeNAV.Velocity (north, east, vertical)NAV.Acceleration (lateral, longitudinal, normal)NAV.Wind (direction, magnitude)NAV.Body_to_Earth_TransformNAV.Body_to_Horizon_TransformNAV.Radar_to_Body_TransformNAV.Radar_to_Earth_Transform

SOLO

68

Navigation

Azimuth Gimbal Torque Motor

zGGP

yy

CzGC HkT 1

Rate-GyroTorquer

Pz

Gz

Px

GGHx ,

InputAxis

PickoffAngle

SpinAxis

OutputAxis

Damper

TorqueCommand

AzimuthStabilizedPlatform

Filter &Torque Driver

Platform Stabilization Using Rate-Integrated-Gyros (RIGs)

The only way to keep a Gimbaled Platform in a Desired Angular Position is by controlling its Angular Rate. For this purpose we use a Rate-Integrated-Gyros (RIGs)

Platform Stabilization Around ZP Azimuth Axis

To control the Platform Angular Rate we use:• Rate-Integrated-Gyro (RIG) ZG- Input Axis YP=YG – Output Axis XG – Gyro’s Spin Axis• Azimuth Gimbal Torque Motor• K1 (s) – Filter and Torque Driver

Cz

SOLO

69

Navigation


The Dynamic Equation along Rate-Integrated-Gyro (RIG) Output Axis YP is:

Platform Stabilization Around ZP Azimuth Axis (continue – 1)

GDCzGyG BTTHJ

PP

zGGP

yy

CzGC HkT 1

Rate-GyroTorquer

Pz

Gz

Px

GGHx ,

InputAxis

PickoffAngle

SpinAxis

OutputAxis

Damper

TorqueCommand


PP yG

G

G

DCzGGG H

J

H

TTHsBJss

JG – RIG Moment of Inertia around Output Axisθ – Pickoff Angle - Platform Angular Acceleration around YP Axis - Platform angular Rate around ZP Axis HG – Gyro Angular MomentTC – RIG Torque CommandTD – Disturbance MomentBG - Damping Coefficient

Py

Pz

Tacking Laplace Transform and rearranging:

SOLO

70

Navigation



zGGP

yy

CzGC HkT 1

Rate-GyroTorquer

Pz

Gz

Px

GGHx ,

InputAxis

PickoffAngle

SpinAxis

OutputAxis

Damper

TorqueCommand


PP yG

G

G

DCzGGG H

J

H

TTHsBJss

Define:

Cz

G

C kH

T 1: Angular Rate Command (Δk –Scaling Error)

GG

D

H

T : Gyro Bias

GG

G

B

J : RIG Time Constant

PCP yG

GGzz

GG

G

H

Jk

ssB

Hs

1

1

1

SOLO

71

Navigation



The Pickoff Signal θ, is the Feedback Command to the Azimuth Torque Motor

ssKKsTCz 12

K1(s) - Filter and Torque Driver

fzxxyxzz TTJJJCPPPPPP


zGGP

yy

CzGC HkT 1

Rate-GyroTorquer

Pz

Gz

Px

GGHx ,

InputAxis

PickoffAngle

SpinAxis

OutputAxis

Damper

TorqueCommand



K2 - Torque Motor Gain

The Moment Equation along Platform Z P Axis is:

Czk 1

GCy

G

G

H

J fT

1

1

GG

G

ssB

H

sK1 2K sJPz

1 Pz

PPPP xxyx JJ

DT

SOLO

72

Navigation



DGGxGx

Gy

G

GGz

GGxGx

GGz T

BHsKsKsJsJ

ss

H

Jk

BHsKsKsJsJ

BHsKsK

PP

CC

PP

P /

11

/

/

123

123

1


zGGP

yy

CzGC HkT 1

Rate-GyroTorquer

Pz

Gz

Px

GGHx ,

InputAxis

PickoffAngle

SpinAxis

OutputAxis

Damper

TorqueCommand



Czk 1

GCy

G

G

H

J fT

1

1

GG

G

ssB

H

sK1 2K sJ

Pz

1 Pz

PPPP xxyx JJ

DT

D

Gx

GG

x

yG

GGz

Gx

GG

Gx

GG

z

T

ssJ

BHsKsK

sJ

H

Jk

ssJ

BHsKsK

ssJ

BHsKsK

P

P

CC

P

P

P

1

/1

1

1

1

/1

1

/

21

21

21

or

From the Figure above we obtain:

SOLO

73

Navigation



DGGxGx

Gy

G

GGz

GGxGx

GGz T

BHsKsKsJsJ

ss

H

Jk

BHsKsKsJsJ

BHsKsK

PP

CC

PP

P /

11

/

/

123

123

1

Czk 1

GCy

G

G

H

J fT

1

1

GG

G

ssB

H

sK1 2K sJ

Pz

1 Pz

PPPP xxyx JJ

DT

At Steady-State we obtain:

Ds

GGy

G

GGzz Ts

BHsKKH

Jkt

CCP 01

lim/0

11

We can see that to minimize External Disturbances effect we must have K1(0)K2HG/BG,called “Loop Robustness”, as high as Loop Stability allows. Also we must have HG>>JG in order to minimize the effect of . ThenCy

G

G

H

J

Gzz CPkt 1

Therefore the Misalignment Errors of the Platform are due to Gyros Drift andScaling Error. Both can be measured (off-line) and compensated by Navigation Computer.

SOLO

74

Navigation

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz


Pitch Gimbal Torque Motor

Roll Gimbal Torque Motor

HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G


SOLO

75

Navigation


The Platform is angular isolated from the Aircraft via, at least, three Gimbals. Those Gimbals are, from Aircraft to Platform: - Azimuth (Heading) – Angle ψG

- Pitch – Angle θ - Roll – Angle ϕ The Rotation Matrix from Aircraft to Platform is:

100

0cossin

0sincos

cos0sin

010

sin0cos

cossin0

sincos0

001

321 GG

GG

GPAC

We want to apply Moments on the Platform, related to the Pjckoff Outputsof the Three RIGs mounted on the Platform

z

y

x

z

y

x

P KsK

T

T

T

T

P

P

P

21

The 3 Torque Motors Roll (TR), Pitch (TP) and Heading (TH)are located on Gimbal Axes . PA zyx 1,1,1 '

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz

AzimuthGimbal Torquer

PitchGimbal Torquer


HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G

SOLO

76

Navigation


We want to find the relation between andTR, TP, TH.

PPP zyx TTT ,,

The 3 Torque Motors Roll (TR), Pitch (TP) and Heading (TH)are located on Gimbal Axes . PA zyx 1,1,1 '

PAPPPPPP zHyPxRzzyyxx TTTTTTT 111111 '

PPzy

AAx

P

P

P

GHGPGR

z

y

x

P TTT

T

T

T

T

1

3

1

23

1

123

1

0

0

0

1

0

0

0

1

,'

H

P

R

GG

GG

z

y

x

T

T

T

T

T

T

P

P

P

10sin

0coscossin

0sincoscos

P

P

P

z

y

x

GG

GG

GG

H

P

R

T

T

T

T

T

T

1tansintancos

0cossin

0cos/sincos/cos

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz


PitchGimbal Torquer


HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G

SOLO

77

Navigation


To simplify the implementation the assumption ofsmall Pitch Angle θ is used (see Figure):

P

P

P

z

y

x

GG

GG

GG

H

P

R

T

T

T

T

T

T

1tansintancos

0cossin

0cos/sincos/cos

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz


PitchGimbal Torquer


HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G

P

P

P

z

y

x

GG

GG

H

P

R

T

T

T

T

T

T

100

0cossin

0sincos

z

y

x

z

y

x

P KsK

T

T

T

T

P

P

P

21

where:

SOLO

78

Navigation

zA

yA

xA

Cz

zG

yG

Cy

xGCx

1

NavigationComputer

yA

xA

zA

CxCy

Cz


Pitch Gimbal Torque Motor

Outer RollGimbal

Torque Motor

HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

1

G1

Iner RollGimbal

Torque Motor

SOLO

79

Navigation

Derivation of the IMU Position and Platform Misalignment Error Equations

Cx

Cy

Cz

Px

Pz

Py

x

y

z

xy

z

Platform Misalignment Error Equations

Define:(C)Computer Coordinate System (the computed Platform coordinates)(P) Platform Coordinate System (real Platform coordinates)

The rotation from (C) to (P) is defined bythe three small angles ψx, ψy, ψz as

100

0cossin

0sincos

cos0sin

010

sin0cos

cossin0

sincos0

001

321 zz

zz

yy

yy

xx

xxzyxPCC

0

0

0

100

010

001

1

1

1

100

01

01

10

010

01

10

10

001

xy

xz

yz

xy

xz

yz

z

z

y

y

x

x

0

0

0

:&:

xy

xz

yz

z

y

x

PC IC

SOLO

80

Navigation


Cx

Cy

Cz

Px

Pz

Py

x

y

z

xy

z

Platform Misalignment Error Equations (continue – 1)

Let find the angular rotation vector from C to P

z

zyy

x

xP

CP

0

0

0

0

0

0 321

z

y

x

z

zxy

zyx

zxy

xz

yz

y

y

yx

0

0

1

1

1

0

0

10

010

01

0

0

SOLO

81

Navigation



The command to Platform Torques by the computer (C)are affected by the IMU Gyros errors:- Gyros Scaling Errors- Misalignment of the gyros relative to Platform- Gyros Drift- Gyros Mass-Unbalances

PG

CICG

PIP KI

Platform Rate Commands Vector

33231

23221

13121

G

G

G

G

Kmm

mKm

mmK

KMatrix of Gyros Scaling Errors,Misalignments and Mass-Unbalances

PPP zzyyxx

PG 111

Gyro Drift Vector

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz


PitchGimbal Torquer

RollGimbal Torquer

HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G

Computer Rate Commands VectorCCCCCC zzyyxxIC 111

SOLO

82

Navigation



Let find the angular velocity vector of the Platform (P) relative to the Computer (C):

ICIPCP

CIC

PC

PIP

PIC

PIP

PCP C

PG

CICG

CIC

CIC

PG

CICG KIKI

Using we obtain:

C

ICC

IC

PG

CICG

CIC K

or

zA

yA

xA

Cz

zG

yG

Cy

xGCx

NavigationComputer

yA

xA

zA

Cx

Cy

Cz


PitchGimbal Torquer

RollGimbal Torquer

HeadingOutput

PitchOutput

RollOutput

Resolver

0yV 0xV 0zV

G

z

y

x

z

y

x

G

G

G

z

y

x

xy

xz

yz

z

y

x

C

C

C

CC

CC

CC

Kmm

mKm

mmK

33231

23221

13121

0

0

0

SOLO

83

Navigation


Position Error Equations

- vector representing position from the Earth Center of mass to the Vehicle

r

r

Tr

Iz

Iy

Ixt

t

Ex

Ey

Ez

TrueIndicated

rgAr

- Ideal Accelerometers Measurement VectorA

rrr

Kr

r

Krg

2/33

rg

- Gravity Vector

rrgAArr

For Non-Ideal Accelerometers we have a error between Real Position and ComputedPosition

r

rgrrgAr

SOLO

84

Navigation


Position Error Equations (continue – 1)

rgrrgAr

r

r

Kr

rrrr

Krgrrg

32/3

rr

K

r

rrrr

r

Kr

r

Kr

rrr

K

32332/32

312

rr

Kr

r

r

r

r

r

Kr

r

Kr

r

K

3333

therefore

Arr

r

r

rr

r

Kr

3

3

SOLO

85

Navigation



Define

r

g

r

KS

3:

Maximilian Schuler(1882 – 1972)

SST

2

Shuler Period = 84.4 minutes at Sea Level

Arr

r

r

rrr S

32

SOLO

86

Navigation



Let find the Accelerometer Measurements received by the Navigation Computer (C)

The Accelerometer Errors are related to:- Accelerometers Scaling Errors- Misalignment of the Accelerometers relative to Platform- Accelerometers Biases

PPf

CC bAKIA

Accelerometers Measurement Vector

33231

23221

13121

fff

fff

fff

f

Kmm

mKm

mmK

K Matrix of Accelerometers Scaling Errors and Misalignments

Ideal Accelerometer Measurement Vector

PPPPPP zzyyxxP AAAA 111

PPPPPP zzyyxx

P bbbb 111

Accelerometers Biases Vector

SOLO

87

Navigation



AAA C

: Accelerometers Error Vector

PPPf

PCP

CC

C AIbAKIACAA

We used the relation

IICC PC

CP

11

Finally we obtain

PPf

PC bAKAA

PPf

PS bAKAr

r

r

r

rrr

32

The Position Error Equation is

SOLO

88

Navigation



Let compute Cr

CCIC

CC rrr

Therefore

CCCCCC

CCC

CICC

CCC

CCC

CCCCCC

CCC

zzyyxxIC

CICzyxICzyx

zyxC

zyxzyxC

zyxC

rzyxzyx

zyxr

zyxzyxr

zyxr

111

111111

111:

111111

111

11

In the same way

CCIC

CCIC

CCIC

CIC

CIC

CCIC

CC

rr

rrrr

0

SOLO

89

Navigation



CCIC

CC rrr

Therefore

CCIC

CIC

CIC

CCIC

CC rrrr

2

PPf

P

CCC

CS

CCIC

CIC

CIC

CCIC

C

bAKA

rr

r

r

rrrrr

32 2

Together with the Platform Misalignment Error Equations

PG

CICG

CIC K

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90

Navigation



CCCCCC zzyyxxIC 111

0

0

0

CC

CC

CC

xy

xz

yz

CIC

0

0

0

CC

CC

CC

xy

xz

yz

CIC

22

22

22

0

0

0

0

0

0

CCCCCC

CCCCCC

CCCCCC

CC

CC

CC

CC

CC

CC

yxzyzx

zyzxyx

zxyxzy

xy

xz

yz

xy

xz

yz

CIC

CIC

Czrr 1

z

y

x

z

z

y

x

r

rr

r

rr

CC

CC

21

0

0

33

SOLO

91

Navigation



z

y

x

z

y

x

z

y

x

CCCCCCCC

CCCCCCCC

CCCCCCCC

CC

CC

CC

yxSxzyyzx

xzyzxSzyx

yzxzyxzyS

xy

xz

yz

222

222

222

20

0

0

2

P

P

P

P

P

P

P

P

P

z

y

x

z

y

x

fff

fff

fff

z

y

x

xy

xz

yz

b

b

b

A

A

A

Kmm

mKm

mmK

A

A

A

33231

23221

13121

0

0

0

Position Error Equations

Platform Misalignment Error Equations

z

y

x

z

y

x

G

G

G

z

y

x

xy

xz

yz

z

y

x

C

C

C

CC

CC

CC

Kmm

mKm

mmK

33231

23221

13121

0

0

0

Inertial rotation sensor classification:

Rotation sensorsRotation sensors

GyroscopicGyroscopic

Rate GyrosRate GyrosFree GyrosFree Gyros

Non-GyroscopicNon-Gyroscopic

Vibration Sensors

Vibration Sensors

Rate SensorsRate Sensors Angular accelerometers

Angular accelerometers

DTGDTG RGRGRIGRIGRVGRVG General purpose

General purpose MHDMHDOptic

Sensors

Optic Sensors

RLGRLG IOGIOGFOGFOG Silicon)MEMS(

Silicon)MEMS(

HRGHRG Tuning Fork

Tuning Fork

QuartzQuartz CeramicCeramic

Rate gyro DTG – Dynamically Tuned Gyro

Flex Inversion Cardan joint

95

Main Components of a DTG

Transverse Cut of a DTG

Rate gyro DTG – Dynamically Tuned Gyro

SOLO

96

NavigationInertial Navigation Systems

SOLO

97


98

SOLO

LBL

a

BBA

BCG gCT

mF

ma

B

11 BCG

TBL

LCG aCa

BL

LIL

BIBBLBL qqq

2

1

2

1 TBL IqIqC

3434

BCGa

LCGa

Ba

BLC

BLC

s

1 BLqBLq

BLC

s

1 L

ELL

EL

LLCG

LE VRaV

2 s

1 L

EV L

EV

LCGa

BLC

LMR

LEV

LE

BL

BE VCV

MV

BIB

IMU

Rate GyrosCompensation

AccelerometersCompensation

Rate Gyros

Accelerometers Lg

Strapdown Algorithm (Vector Notation)

Navigation

99

SOLO

IMU

B

B

B

r

q

p

4

3

2

1

0

0

0

0

2

1

4

3

2

1

BL

BL

BL

BL

DownDownBEastEastBNorthNorthB

DownDownBNorthNorthBEastEastB

EastEastBNorthNorthBDownDownB

NorthNorthBEastEastBDownDownB

BL

BL

BL

BL

q

q

q

q

rqp

rpq

qpr

pqr

q

q

q

q

s

1

4

3

2

1

BL

BL

BL

BL

q

q

q

q

4

3

2

1

BL

BL

BL

BL

q

q

q

q

g

C

C

C

T

T

T

mF

F

F

ma

a

a

BL

BL

BL

a

zB

yB

xB

zBA

yBA

xBA

zB

yB

xB

B

3,3

3,2

3,111

zB

yB

xB

a

a

a

zB

yB

xB

BL

BL

BL

BL

BL

BL

BL

BL

BL

Down

East

North

a

a

a

CCC

CCC

CCC

a

a

a

3,33,23,1

2,32,22,1

1,31,21,1

Down

East

North

a

a

a

BLC

BLC

4

3

2

1

*

1

4

3

2

1

BL

BL

BL

BL

BL

BL

BL

BL

q

q

q

q

qq

q

q

q

q 4

3

2

1

BL

BL

BL

BL

q

q

q

qB

LC

321

412

143

234

3412

2143

1234

BLBLBL

BLBIBL

BLBLBL

BLBLBL

BLBLBLBL

BLBLBLBL

BLBLBLBLB

L

qqq

qqq

qqq

qqq

qqqq

qqqq

qqqq

C

43

2

1

BL

BL

BL

BL

q

q

q

q

Down

East

North

a

a

a

DownW

EastW

NorthW

NorthNorthEastEast

NorthNorthDownDown

EastEastDownDown

Down

East

North

DownE

EastE

NorthE

V

V

V

Lat

Lat

HR

a

a

a

V

V

V

022

202

220

sin

0

cos2

_

_

_

s

1

DownE

EastE

NorthE

V

V

V

cos0sin

sinsincossincos

cossinsincoscosW

BC

WBC

s

1

H

Long

Lat

H

Long

Lat

DownE

p

EastE

m

NorthE

Vtd

Hd

LatHR

V

td

Longd

HR

V

td

Latd

cos

w

v

u

DownM

EastM

NorthM

BL

BL

BL

BL

BL

BL

BL

BL

BL

V

V

V

CCC

CCC

CCC

w

v

u

_

_

_

3,32,31,3

3,22,21,2

3,12,11,1

DownW

EastW

NorthW

DownE

EastE

NorthE

DownM

EastM

NorthM

V

V

V

V

V

V

V

V

V

_

_

_

_

_

_

_

_

_

DownM

EastM

NorthM

V

V

V

DownE

EastE

NorthE

V

V

V

M

M

Vv

uw

wvuV

/sin

/tan1

1

222

MV

DownE

EastE

NorthE

V

V

V

DownE

EastE

NorthE

V

V

V

LatHR

V

HR

V

HR

V

EastE

NorthE

EastE

Down

East

North

tan0

0

0

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C

*

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat

Lat

Down

East

North

sin

0

cos

Down

East

North

Down

East

NorthW

L

zW

yW

xW

C *

*

*

*

*

*

*

zW

yW

xW

*

*

*

1

zW

yW

xW

zW

yW

xW

zW

yW

xW

WLC

Lat

Rate GyrosCompensation

Accelerometers

Compensation

Rate Gyros

Accelerometers

Strapdown Algorithm

Navigation

SOLO

100


Magnetic Compass

SOLO

101


Gyrocompass

SOLO

102

NavigationRadar Altimeter

SOLO

103

NavigationExternally Navigation Add Systems

eLORAN LORAN - C

Global Navigation Satelite System (GNSS)

Distance Measuring Equipment (DME)

VHF Omni Directional Radio-Range (VOR) SystemData Base Matching

Terrain Referenced Navigation (TRN)Navigation Multi-Sensor Integration

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104

NavigationGlobal Navigation Satelite System (GNSS)

Satellites of theGPS

GLONASS and GALILEOSystems

Four Satellite Navigation Systems have been designed to give three dimensionalPosition, Velocity and Time data almost enywhere in the world with an accuracy of a few meters• The Global Positioning System, GPS (USA)• The Global Navigation Satellite System , GLONASS (Rusia)• GALILEO (European Union)• COMPASS (China)They all uses the Time of Arrival (range determination) Radio Navigation Systems.

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105


SOLO

106


SOLO

107


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108


Differential GPS Systems (DGPS)

Differential GPS Systems (DGPS) techniques are based on installing one or more Reference Receivers at known locations and the measured and known ranges to the Satellites are broadcast to the other GPS Users in the vicinity. This removes much of the Ranging Errors caused by atmospheric conditions (locally) and Satellite Orbits and Clock Errors (globally).

Global Positioning System (GPS)

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Navigation

A visual example of the GPS constellation in motion with the Earth rotating. Notice how the number of satellites in view from a given point on the Earth's surface, in this example at 45°N, changes with time

The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites. It is maintained by the United States government and is freely accessible to anyone with a GPS receiver.

Ground monitor station used from 1984 to 2007, on display at the Air Force Space & Missile Museum

A GPS receiver calculates its position by precisely timing the signals sent by GPS satellites high above the Earth. Each satellite continually transmits messages that include:• the time the message was transmitted• satellite position at time of message transmission

Global Navigation Satellite System (GNSS)

Global Positioning System

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Navigation

Other satellite navigation systems in use or various states of development include:• GLONASS – Russia's global navigation system. Fully operational worldwide.• GALILEO – a global system being developed by the European Union and other partner countries, planned to be operational by 2014 (and fully deployed by 2019)• BEIDOU – People's Republic of China's regional system, currently limited to Asia and the West Pacific[123]

• COMPASS – People's Republic of China's global system, planned to be operational by 2020.

• IRNSS – India's regional navigation system, planned to be operational by 2012, covering India and Northern Indian Ocean.

• QZSS – Japanese regional system covering Asia and Oceania.

Comparison of GPS, GLONASS, Galileo and Compass (medium earth orbit) satellite navigation system orbits with the International Space Station, Hubble Space Telescope and Iridium constellation orbits, Geostationary Earth Orbit, and the nominal size of the Earth.[121] The Moon's orbit is around 9 times larger (in radius and length) than geostationary orbit

Satellite Position

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Navigation

GZ

GX

GYEquatorial

Plane

Y

Z

X

AscendingNode

Satellite Orbit

PeriapsisDirection


i

N1

A sixth element is required to determine the position of the satellite along the orbit at a given time.

1. a semi-major axis – a constant defining the size of the conic orbit.

2. e, eccentricity – a constant defining the shape of the conic orbit.

3. i, inclination – the angle between Ze and the specific angular momentum of the orbit vrh

4. Ω, longitude of the ascending node – the angle, in the Equatorial Plane, between the unit vector and the point where the satellite crosses trough the Equatorial Plane in a northerly direction

(ascending node) measured counterclockwise where viewed from the northern hemisphere.

5. ω, argument of periapsis – the angle, in the plane of satellite’s orbit, between ascending node and the periapsis point, measured in the direction of the satellite’s motion.

6. T, time of periapsis passage – the time when the satellite was at the periapsis.

GPS Broadcast Ephemerides

SOLO

112

Navigation

113

SOLO KEPLERIAN TRAJECTORIES

Time of Flight on an Elliptic Orbit

From the equation 2rh we can write h

Ad

h

drdt 2

2

where is the area defined by the radius vector as it moves through an angle

2

2

drAd

d

pp

r

focus conicsection

xy

P1

Q1

r1

t1

v

rv tv

d

drAd 2

2

1

periapsis

This proves the 2nd Kepler’s Law that equal area are swept out equal in equal timesby the radius vector.

114


Time of Flight on an Elliptic Orbit (continue – 1)

The period of the orbit depends only on the major axis of the ellipse a.

p

pa

h

eaa

h

ea

h

baT

eap

ph

2/3122/322

21

21

222

or 2/32 aT

The period of an elliptical orbit T is obtained by integrating from Θ= 0 to Θ=2π , and the radius vector sweeps the area of the ellipse A = π a b.

This proves the Kepler’s third law: “the square of the period of a planet orbit is equalTo the cube of its mean distance to the sun”.

115



Let draw an auxiliary circle of radius a, and the same center O as the geometric centerof the ellipse.

x

y

eac

a a

2/121 eab

r

FOCUS

EMPTYFOCUS

c

P1

Q1

a

F

Q

O VS

E

P

Let take any point P on the ellipse withpolar coordinates r,Θ and define the point Q on the circle at the same coordinate x as P.

Eeary

ra

xea

a

xby

Eaa

xay

ellipse

ellipse

circle

sin1sin

sin111

sin12

2

22

2

2

2

2

The angle E of OQ with x axis is called theeccentric anomaly.

aeEarxellipse coscos

116



Let compute

x

y

eac

a a

2/121 eab

r

FOCUS

EMPTYFOCUS

c

P1

Q1

a

F

Q

O VS

E

P

0cos11

sin1sincos1cos

1

2

22

2

EEEeea

EEeaEaEEeaaeEa

xyyxvreah ellipseellipseellipseellpse

We obtain n

aEEe :cos1

3

pttntEetE sin

Integrating this equation gives

Kepler’s Equation

where tp is the time of periapsis ( E (tp) = 0 )

117



From

x

y

eac

a a

2/121 eab

r

FOCUS

EMPTYFOCUS

c

P1

Q1

a

F

Q

O VS

E

P

Eeary

aeEarx

ellipse

ellipse

sin1sin

coscos

2

we have Eea

EeEeaEeaaeEar

cos1

coscos21sin1cos2/1222/12222

Therefore

cos1

sin1sin

cos1

sin1sin

cos1

coscos

cos1

coscos

22

e

eE

Ee

Ee

e

eE

Ee

eE

Ee

Ee

Ee

eEEe

sin1

cos11

sin1

coscos1

sin

cos1

2tan

22

From

2tan

1

1

2tan

E

e

e

or

and are always in the same quadrant.2

2

E

118



We have

x

y

eac

a a

2/121 eab

r

FOCUS

EMPTYFOCUS

c

P1

Q1

a

F

Q

O VS

E

P

Eeary

aeEarx

ellipse

ellipse

sin1sin

coscos

2

and

Eear cos1

The Position Vector of the Satellite is

0

sin1

cos

0

sin

cos

0

11

2 Eea

aeEa

r

r

y

x

q

QyPxq

ellipse

ellipse

Orbit

ellipseellipse

Differentiate in the Orbit Plane

2

222

10

cos

sin

cos10

cos1

sin

0

cos1

sin

0

cos1

sin

e

ane

Ee

anEe

E

EanEe

E

Ee

aeE

qtd

d

Orbit

Orbit


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119

Navigation

The Satellite Position can be computed as follows:

oeisic

rsrc

usuc

oe

oe

ttidotuCuCii

uCuCrr

uCuC

tt

ttnnMMa

n

000

000

000

0

0

3

2sin2cos

2sin2cos

2sin2cos

where:

00

1

0

2tan

1

1tan2

cos1

sin

u

E

e

e

Eear

EeME

y

z

Ver

tical

Equ

inox

Equator

Orbitnode

Pole

Perigee

z

x

i

Satellite

Six Keplerian Elements Define the Satellite Posision (Ω, I, ω, a, e, M0)where M0 = n (t – tP)


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120

Navigation

uur

ur

y

x

q ellipse

ellipse

Orbit

0

sin

cos

0

210

cos

sin

0e

anue

u

y

x

q ellipse

ellipse

Orbit

Orbit

oecoec ttt

0

sin

cos

0

313 ur

ur

iy

x

C

z

y

x

ellipse

ellipse

G

G

u

121


SOLO Navigation


SOLO

122

Navigation

- x, y, z Satellite Coordinate in Geocentric-Equatorial Coordinate System

222 ZzYyXx

- X, Y, Z User Coordinate in Geocentric-Equatorial Coordinate System

Squaring both sides gives

The User to Satellite Range is given by

ZzYyXxzyxZYX

ZzYyXx

r

222222222

2222

2

The four unknown are X, Y, Z, Crr. Satellite position (x,y,z) is calculated from received Satellite Ephemeris Data.Since we have four unknowns we need data from at least four Satellites.

ZzYyXxCrrrzyxr 22222222

where r = Earth RadiusThis is true if (x,y,z) and (X,Y,Z) are measured at the same time. The GPS Satellites clocks are more accurate then the Receiver clock. Let assume that Crr is the range-square bias due to time bias between Receiver GPS and Satellites clocks. Therefore instead of the real Range ρ the Receiver GPS measures the Pseudo-range ρr..


SOLO

123

Navigation


SOLO

124

Navigation

Using data from four Satellites we obtain

444444

224

24

24

24

33333322

32

32

32

3

22222222

22

22

22

2

11111122

12

12

12

1

222

222

222

222

ZzYyXxCrrrzyx

ZzYyXxCrrrzyx

ZzYyXxCrrrzyx

ZzYyXxCrrrzyx

r

r

r

r

or

14

1444

224

24

24

24

223

23

23

23

222

22

22

22

221

21

21

21

444

333

222

111

1222

1222

1222

1222

xxx R

r

r

r

r

PM

rzyx

rzyx

rzyx

rzyx

Crr

Z

Y

X

zyx

zyx

zyx

zyx

141

4414 xxx RM

Crr

Z

Y

X

P


SOLO

125

Navigation


SOLO

126

Navigation


SOLO

127

Navigation

GPS Satellite

GPS ControlStation


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128

Navigation

The key to the system accuracy is the fact that all signal components are controlled by Atomic Clocks.• Block II Satellites have four on-board clocks: two rubidium and two cesium clocks. The long term frequency stability of these clocks reaches a few part in 10-13 and 10-14 over one day.• Block III will use hydrogen masers with stability of 10-14 to 10-15 over one day.

The Fundamental L-Band Frequency of 10.23 MHz is produced from those Clocks.Coherently derived from the Fundamental Frequency are three signals(with in-phase (cos), and quadrature-phase (sin) components): - L1 = 154 x 10.23 MHz = 1575.42 MHz - L2 = 120 x 10.23 MHz = 1227.60 MHz - L3 = 115 x 10.23 MHz = 1176.45 MHz

The in-phase components of L1 signal, is bi-phase modulated by a 50-bps data stream and a pseudorandom code called C/A-code (Coarse Civilian) consisting of a 1023-chip sequence, that has a period of 1 ms and a chipping rate of 1.023 MHz:

signalL

codeompseudorand

ACulation

bpspowercarrier

I ttctdPts

1/

mod50

cos2


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129

Navigation

The quadrature-phase components of L1, L2 and L3 signals, are bi-phase modulated by the 50-bps data stream but a different pseudorandom code called P-code (Precision-code) or Precision Positioning Service (PPS) for US Military use, , that has a period of 1 week and a chipping rate of 10.23 MHz:

signalsLLL

codeompseudorand

Pulation

bpspowercarrier

Q ttptdPts

3,2,1

mod50

sin2


SOLO

130

Navigation

GPS Signal Spectrum

SOLO

131

Navigation


SOLO

132

Navigation

GPS User Segment(GPS Receiver)


SOLO

133

Navigation

GPS User Segment(GPS Receiver)

GPS vs GALILEO

SOLO

134

Navigation

GALILEOGPS

Satelites27 + 324 (32!)

Planes36

Satellite per Plane104 - 7

Plane Spacing120 <60 <

Inclination56 <55 <

Orbit TypeMEO CircularMEO Circular

Orbit Radius29,500 km26,500 km

Period141/4 hour12 hour

Satellite Ground Track Repetition

10 days1 day

Higher GALILEO Orbit coupled with Inclination increase give better coverage at high latitudes.

GPS, GLONASS and GALILEO

SOLO

135

Navigation

SOLO

136

Navigation

GALILEO

GALILEO

SOLO

137

Navigation

GALILEO/ GPS/ GLONASS

GPS Jamming, Anti-Jamming

SOLO

138

Navigation

Rubidium Clocks

SOLO

139

Navigation

Rubidium Clocks

SOLO

140

Navigation

GNSS Status

SOLO

141

Navigation

GPS Status, November 2011

SOLO

142

Navigation

GPS Modernization

SOLO

143

Navigation

GPS III Payload Evolution

SOLO

144

Navigation

GLONASS Constellation, November 2011

SOLO

145

Navigation

GLONASS Modernisation

SOLO

146

Navigation

COMPASS/ BeiDou, November 2011

SOLO

147

Navigation

Quasi Zenith Satellite System (QZSS) - Japan

SOLO

148

Navigation

Indian Regional National Satellite System (IRNSS)

SOLO

149

Navigation

SOLO

150

NavigationDifferential GPS Augmented Systems

SOLO

151

NavigationDifferential GPS Augmented Systems

SOLO

152

Navigation

GNSS Aviation Operational Performance Requirements

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153

Navigation

SOLO

154


LORAN - C

A LORAN receiver measures the Time Difference of arrival between pulses from pairs of stations. This time difference measurement places the Receiver somewhere along a Hyperbolic Line of Position (LOP). The intersection of two or more Hyperbolic LOPs, provided by two or more Time Difference measurement, defines the Receiver’s Position. Accuracies of 150 to 300 m are typical.

LOP from Transmitter Stations (1&2 and 1&3)

LORAN – C (LOng RAnge Navigation) is a Time Difference Of Arrival (TDOA), Low-Frequency Navigation and Timing System originally designed for Ship and Aircraft Navigation.

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155


eLORAN

eLORAN receiver employ Time of Arrival (TOA) position techniques, similar to those used in Satellite Navigation Systems. They track the signals of many LORAN Stations at the same time and use them to make accurate and reliable Position and Timing measurements. It is now possible to obtain absolut accuracies of 8 – 20 m and recover time to 50 ns with new low-cost receivers in areas served by eLORAN.

The Differential eLORAN Concept

Enhanced LORAN , or eLORAN, is an International initiative underway to upgrade the traditional LORAN – C System for modern applications. The infrastructure is being installed in the US, and a variation of eLORAN is already operational in northwest Europe.

A Combined GPS/eLORAN Receiver and Antenna from

Reelektronika

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156


Distance Measuring Equipment (DME)

Aircraft DME Range Determination System

Distance Measuring Equipment (DME) Stations for Aircraft Navigation were developed in the late 1950’s and are still in world-wide use as primary Navigation Aid. The DME Ground Station receive a signal from the User ant transmits it back. The User’s Receiving Equipment measures the total round trip time for the interrogation/replay sequence, which is then halved and converted into a Slant Range between the User’s Aircraft and the DME Station

There are no plans to improve the DME Network, through it is forecast to remain in service for many years. Over time the system will be relegated to a secondary role as a backup to GNSS-based navigation,

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Angle (Bearing Determination)

Determining Bearing to a VOR Station

VHF Omni Directional Radio-Range (VOR) System

The VHF Omni Directional Radio-Range (VOR) System is comp[rised of a serie of Ground-Based Beacons operating in the VHF Band (108 to 118 MHz).A VOR Station transmits a reference carrierFrequency Modulated (FM) with:30 Hz signal from the main antenna.An Amplitude Modulated (AM) carrierelectrically swept around several smallerAntennas surrounding the main Antenna. This rotating patterncreates a 30 Hz Doppler effect onthe Receiver. The Phase Differenceof the two 30 Hz signals gives theUser’s Azimuth with respect to the Northfrom the VOR Site. The Bearing measurementaccuracy of a VOR System is typically on the order of 2 degrees, with a range that extends from 25 to 130 miles.

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TACAN is the MilitaryEnhancement of

VOR/DME

VHF Omni Directional Radio-Range (VOR) System

TACAN (Tactical Air Navigation) is an enhanced VOR/DME System designed forMilitary applications. The VOR component of TACAN, which operates in the UHF spectrum, make use of two-frequency principle, enabling higher bearing accuracies.The DME Component of TACAN operates with the same specifications as civil DME.

The accuracy of the azimuth component is about ±1 degree, while the accuracy of the DME position is ± 0.1 nautical miles. For Military usage a primary drawback is the lack of radio silence caused by Aircraft DME Transmission.

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NavigationData Base Matching

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NavigationTerrain Referenced Navigation (TRN)

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NavigationTerrain Referenced Navigation (TRN)

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Navigation

Navigation Multi-Sensor Integration

Navigation Data

164

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TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

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NavigationWorld Geodetic System (WGS 84)

Geoid - Mean Sea Level of the Earthb

NR

h

N

Vehicle

Ellipsoid

Geoid

Surface

Geometry of the Reference Earth Model

a x

z

y

Reference Ellipsoid – Approximation of Sea Level

Reference Earth Model

h - Vehicle Altitude (the distance from the Vehicle to Ellipsoid along the Normal to EllipsoidRN - the distance from the Ellipsoid Surface along the Normal to Ellipsoid to intersection to yz plane (see Figure)

N - Height of the Geoid above the Reference Ellipsoid

The Reference Ellipsoid was obtained by minimizing the integral of the square ofN over the Earth. Values of N over the Earth have been derived from extensive gravity and satellite measurements. The latest result is the reference Earth Model known as theWorld Geodetic System of 1984 (WGS 84).

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Clairaut's theorem

Clairaut's theorem, published in 1743 by Alexis Claude Clairaut in his Théorie de la figure de la terre, tirée des principes de l'hydrostatique,[1] synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It is a general mathematical law applying to spheroids of revolution. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes.

Clairaut's formula for the acceleration due to gravity g on the surface of a spheroid at latitude φ, was:

where G is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as:

a

baf

:

Alexis Claude Clairaut )1713 – 1765(

2sin

2

51 fmGg

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bNR

h

N

Vehicle

Ellipsoid

Geoid

Surface


a x

z

y


Carlo Somigliana (1860 –1955)

The Theoretical Gravity on the surface of the Ellipsoidis given by the Somigliana Formula (1929)

84

22

2

2222

22

sin1

sin1

sincos

sincos

WGS

epe

e

k

ba

ba

where

1: e

p

a

bk

2

22

:a

bae

- Ellipsoid Eccentricity

a - Ellipsoid Semi-major Axis = 6378137.0 m

b - Ellipsoid Semi-minor Axis = 6356752.314 m

γp – Gravity at the Poles = 983.21849378 cm/s2

γe – Gravity at the Equator = 978.03267714 cm/s2

ϕ – Geodetic Latitude

The Theory of the Equipotential Ellipsoid was first given byP. Pizzetti (1894)

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bNR

h

N

Vehicle

Ellipsoid

Geoid

Surface


a x

z

y


The coordinate origin of WGS 84 is meant to be located at the Earth's center of mass; the error is believed to be less than 2 cm. The WGS 84 meridian of zero longitude is the IERS Reference Meridian. 5.31 arc seconds or 102.5 meters (336.3 ft) east of the Greenwich meridian at the latitude of the Royal Observatory.The WGS 84 datum surface is an oblate spheroid (ellipsoid) with major (transverse) radius a = 6378137 m at the equator and flattening f = 1/298.257223563. The polar semi-minor (conjugate) radius b then equals a times (1−f), or b = 6356752.3142 m. Presently WGS 84 uses the EGM96 (Earth Gravitational Model 1996) Geoid, revised in 2004. This Geoid defines the nominal sea level surface by means of a spherical harmonics series of degree 360 (which provides about 100 km horizontal resolution).[7] The deviations of the EGM96 Geoid from the WGS 84 Reference Ellipsoid range from about −105 m to about +85 m.[8] EGM96 differs from the original WGS 84 Geoid, referred to as EGM84.

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Navigation

The Reference Ellipsoid has the same mass, the same center of mass and the same angular velocity as the real Earth. The Potential U0 on Ellipsoid Surface equals to Potential W0 on the Geoid.

World Geodetic System (WGS 84)


The Equi-potential Ellipsoid furnishes a simple, consistent and uniform reference system for Geodesy, Geophysics and Satellite Navigation. The Normal Gravity Field on the Earth Surface and in Space, is defined in terms of closed formula as a reference for Gravimetry and Satellite Geodesy.

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Geoid product, the 15-minute, worldwide Geoid Height for EGM96 The difference between the Geoid and the Reference Ellipsoid exhibit the following statistics: Mean = - 0.57 m, Standard Deviation = 30.56 mMinimum = -106.99 m, Maximum = 85.39 m

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NavigationWorld Geodetic System (WGS – 84)

bNR

h

N

Vehicle

Ellipsoid

Geoid

Surface


a x

z

yReference Earth Model

ParametersNotationValue

Ellipsoid Semi-major Axisa6.378.137 m

Ellipsoid Flattening (Ellipticity)f1/298.257223563(0.00335281066474)

Second Degree Zonal Harmonic Coefficient of the GeopotentialC2,0-484.16685x10-6

Angular Velocity of the EarthΩ7.292115x10-5 rad/s

The Earth’s Gravitational Constant (Mass of Earth includes Atmosphere)

GM3.986005x1014 m3/s2

Mass of Earth (Includes Atmosphere)M5.9733328x1024 kg

Theoretical (Normal) Gravity at the Equator (on the Ellipsoid)γe9.7803267714 m/s2

Theoretical (Normal) Gravity at the Poles (on the Ellipsoid)γp9.8321863685 m/s2

Mean Value of Theoretical (Normal) Gravityγ9.7976446561 m/s2

Geodetic and Geophysical Parameters of the WGS-84 Ellipsoid

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bNR

h

N

Vehicle

Ellipsoid

Geoid

Surface


a x

z

y

a

baf

:f - Ellipsoid Flattening (Ellipticity)

a - Ellipsoid Semi-major Axis

b - Ellipsoid Semi-minor Axis

e - Ellipsoid Eccentricity 22

222 2: ff

a

bae

211 eafab

Reference Ellipsoid

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NavigationReference Ellipsoid

Ellipse Equation: 12

2

2

2

b

y

a

x

Slope of the Normal to Ellipse:

2

2

tanbx

ay

yd

xd

The Slope of the Geocentric Line to the same point

x

ytan sincos RyRx

Deviation Angle between Geographic and GeodeticAt Ellipsoid Surface

tantan2

2

b

a

tantan

2

21

b

a

b

Vehicle

a

λ – Geographic Latitudeϕ– Geodetic Latitude

Equator

North Pole

Tangentto Ellipsoid

tan1tantan 22

2

ea

b

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Ellipse Equation:

b

Vehicle

a


Equator

North Pole

Tangentto Ellipsoid

x

y

12

2

2

2

b

y

a

x

Slope of the Normal to Ellipse:

2

2

tanbx

ay

yd

xd

The Slope of the Geocentric Line to the same point

x

ytan

1

11

1

1tantan1

tantantan

2

2

2

2

2

2

2

2

2

22

22

2

2

b

a

a

yx

a

x

x

a

b

a

x

y

bx

ayx

y

bx

ay

sincos RyRx

2sin2sin2

tan2sin2

tan

1

2

11

1222

221 f

ba

R

b

ba

a

baR

ba

ba

f

Deviation Angle between Geographic and GeodeticAt Ellipsoid Surface

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For a point at a Height h near the Ellipsoid the value of δ must be corrected: b

h

Vehicle

a

1


Equator

North Pole

Tangentto Ellipsoid

u

u 1

From the Law of Sine we have:

Deviation Angle between Geographic and GeodeticAt Altitude h from Ellipsoid Surface

R

h

hR

huu

11 sin

sin

sin

sin

Since u and δ1 are small: 1R

hu

The corrected value of δ is:

2sin11 11 fR

h

R

hu

Therefore:

2sin1 fR

h

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where λ – Longitude e – Eccentricity = 0.08181919


In Earth Center Earth Fixed Coordinate –ECEF-System (E)the Vehicle Position is given by:

I

0Ex

0Ey

Iz

Northx

EastyDownz

Bx

ByBz

Iy

Ixt

tLong

Lat

0Ez

Ex

Ey

Ez

AV

sin

cossin

coscos

HR

HR

HR

z

y

x

P

M

N

N

E

E

E

E

NhH

e

aRN

2/12 sin1 bNR

h

N

Vehicle

Ellipsoid

Geoid

Surface


a x

z

y

Another variable, used frequently, is the radius of the Ellipsoid referred as the Meridian Radius

2/32

2

sin1

1

e

eaRM

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Let develop the RN and RM:b

a


Equator

North Pole

Tangentto Ellipsoid

x

NR

y


Ellipse Equation: 2222

2

2

2

11 aebb

y

a

x

From this Equation, at any point (x,y) on the Ellipse, we have:

tan

12

2

ay

bx

xd

yd

32

4

32

2222

2

2

2

2

22

2

22

2

2

2 111

ya

b

ya

xbya

a

b

y

x

a

b

y

x

ya

b

xd

yd

y

x

ya

b

xd

yd

From the Ellipse Equation:

2

22

2

2

2222

2

2

2

2

2

2

2

2

cos

sin1

1

1tan1111

e

a

x

ee

a

x

b

a

x

y

a

x

2/122

2

2

2

2/122 sin1

sin1tan

sin1

cos

e

eax

a

by

e

ax

From the Figure above: 2/122 sin1cos e

axRN

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Let develop the RN and RM (continue):b

a


Equator

North Pole

Tangentto Ellipsoid

xNR

y


we have at any point (x,y) on the Ellipse:

tan

12

2

ay

bx

xd

yd 3

22

32

4

2

2 11

y

ea

ya

b

xd

yd

The Radius of Curvature of the Ellipse at the point (x,y) is:

2/322

2

2/322

3323

22

2/3

2

2

2

2/32

sin1

1

sin1

sin1

1

tan1

11

:

e

ea

e

ea

ea

xdyd

xdyd

RM

2/122

2

2

2

2/122 sin1

sin1tan

sin1

cos

e

eax

a

by

e

ax

2/322

2

sin1

1:

e

eaRM

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b

a


Equator

North Pole

Tangentto Ellipsoid

xNR

y


2/322

2

sin1

1:

e

eaRM

2/122 sin1cos e

axRN

a

baf

: 2

2

222 2: ff

a

bae

Using

2222

2/322

2

sin321sin22

3121

sin21

1: ffaffffa

ff

faRM

2222/122

sin31sin22

31

sin21faffa

ff

aRN

2sin321 ffaRM

2sin31 faRN

We used and we neglect f2 terms

!2

11

1

1 nnxn

x n

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The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre

Reference Ellipsoid

Geodetic latitude: the angle between the normal and the equatorial plane. The standard notation in English publications is ϕ

Geocentric latitude: the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude (ϕ) is derived in the above references as

The definition of geodetic (or geographic) and geocentric latitudes

tan1tan 21 e

4 navigation systems

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