geometry of the earth and satellite orbit satellite positioning

Geometry of The Earth and Geometry of The Earth and Satellite OrbitSatellite Orbit

Geometry of The Earth and Geometry of The Earth and Satellite OrbitSatellite Orbit

Satellite PositioningSatellite PositioningSatellite PositioningSatellite Positioning

Geometry of the EarthGeometry of the EarthGeometry of the EarthGeometry of the Earth

CLASSICAL DEFINITION OF GEODESYCLASSICAL DEFINITION OF GEODESY

Geodesy is the science concerned with the study of the shap

e and size of the earth in the geometric sense as well as with t

he form of the equipotential surfaces of the gravity potential.

Friedrich R Helmert (1880)

Geodesy is the science concerned with the study of the shap

e and size of the earth in the geometric sense as well as with t

he form of the equipotential surfaces of the gravity potential.

Friedrich R Helmert (1880)

MODERN DEFINITION OF GEODESYMODERN DEFINITION OF GEODESYMODERN DEFINITION OF GEODESYMODERN DEFINITION OF GEODESY

The relative positions and heights of points on the surface

of the earth

mapping, surveying, charts

Changes in positions with time

deformations and ... navigation

The gravity field of the earth

satellite orbitssatellite orbits

Geodynamics Phenomena

crustal dynamics, sea level

The relative positions and heights of points on the surface

of the earth

mapping, surveying, charts

Changes in positions with time

deformations and ... navigation

The gravity field of the earth

satellite orbitssatellite orbits

Geodynamics Phenomena

crustal dynamics, sea level

CLASSICAL GEOMETRICAL GEODESYCLASSICAL GEOMETRICAL GEODESYCLASSICAL GEOMETRICAL GEODESYCLASSICAL GEOMETRICAL GEODESY

EquipotentialEquipotential


Solid EarthSolid Earth

GeoidGeoid

VerticalVertical

DefinitionsDefinitions

1st Geodetic Problem1st Geodetic Problem

2nd Geodetic Problem2nd Geodetic Problem

Plane Plane Sphere Sphere Ellipsoid Ellipsoid

DefinitionsDefinitions

1st Geodetic Problem1st Geodetic Problem

2nd Geodetic Problem2nd Geodetic Problem

Plane Plane Sphere Sphere Ellipsoid Ellipsoid

pp




GeoidGeoid

Reference EllipsoidReference Ellipsoid

Deviation of VerticalDeviation of Vertical

verticalverticalnormalnormal

CLASSICAL GEOMETRICAL GEODESY (2)CLASSICAL GEOMETRICAL GEODESY (2)CLASSICAL GEOMETRICAL GEODESY (2)CLASSICAL GEOMETRICAL GEODESY (2)

pp

PP

APPROXIMATION OF THE FIGURE OF APPROXIMATION OF THE FIGURE OF THE EARTHTHE EARTH

The solid earthThe solid earth The solid earthThe solid earth

Approximation as a Approximation as a spheresphere Approximation as a Approximation as a spheresphere

Approximation as an Approximation as an ellipsoidellipsoid Approximation as an Approximation as an ellipsoidellipsoid

1st and 2nd Geodetic problems1st and 2nd Geodetic problems

ELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATES

aa = semi-major axis= semi-major axisbb = semi-minor axis= semi-minor axis

ff = flattening= flattening= =

ee22 = eccentricity= eccentricity= =

a - ba - b aa

aa22 - b - b22

aa22

GG

bb

aa

normalnormalpp

PP

EE QQ

ELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATESELLIPSOIDAL (GEODETIC) COORDINATES

11GG 22

GG

pp

PP11

One real point, p Two geodetic points P1, P2

Two geodetic lats, 1G, 2

G

The two minor axes areparallel to each other

One real point, p Two geodetic points P1, P2

Two geodetic lats, 1G, 2

G

The two minor axes areparallel to each other

PP22

THE SIZE OF THE EARTHTHE SIZE OF THE EARTH

Derived from arc measurements Eratosthenes of Alexandria

(276-195 B.C.)

Arc measurements in the meridian Support of early

triangulations(1500...1800 A.D.)

Derived from arc measurements Eratosthenes of Alexandria

(276-195 B.C.)

Arc measurements in the meridian Support of early

triangulations(1500...1800 A.D.)

Equator

Alexandria

SyeneR

SunG

R

N

S

0

SOME REFERENCE ELLIPSOIDSSOME REFERENCE ELLIPSOIDSSOME REFERENCE ELLIPSOIDSSOME REFERENCE ELLIPSOIDS

NameName a (m)a (m) 1/f1/f UsageUsage

Everest (1830)Everest (1830) 63772766377276 300300 IndiaIndia

Airy (1830)Airy (1830) 63765426376542 299299 Great BritainGreat Britain

Clarke (1866)Clarke (1866) 63782066378206 295295 North AmericaNorth America

Clarke (1880)Clarke (1880) 63782496378249 293293 France, AfricaFrance, Africa

International (1924)International (1924) 63783886378388 297297 Europe (ReTrig)Europe (ReTrig)

Krassonsky (1940)Krassonsky (1940) 63782456378245 298298 RussiaRussia

IAG ‘67 (Lucerne)IAG ‘67 (Lucerne) 63781606378160 298.247298.247 IAG RecommendedIAG Recommended

WGS 72WGS 72 63781356378135 298.26298.26 DoD (Doppler)DoD (Doppler)

GRS 80 (Canb)GRS 80 (Canb) 63781376378137 298.257298.257 IAG (Geo Ref Sys)IAG (Geo Ref Sys)

WGS 84WGS 84 63781376378137 298.257298.257 DoD (GPS)DoD (GPS)

NameName a (m)a (m) 1/f1/f UsageUsage

Everest (1830)Everest (1830) 63772766377276 300300 IndiaIndia

Airy (1830)Airy (1830) 63765426376542 299299 Great BritainGreat Britain

Clarke (1866)Clarke (1866) 63782066378206 295295 North AmericaNorth America

Clarke (1880)Clarke (1880) 63782496378249 293293 France, AfricaFrance, Africa

International (1924)International (1924) 63783886378388 297297 Europe (ReTrig)Europe (ReTrig)

Krassonsky (1940)Krassonsky (1940) 63782456378245 298298 RussiaRussia

IAG ‘67 (Lucerne)IAG ‘67 (Lucerne) 63781606378160 298.247298.247 IAG RecommendedIAG Recommended

WGS 72WGS 72 63781356378135 298.26298.26 DoD (Doppler)DoD (Doppler)

GRS 80 (Canb)GRS 80 (Canb) 63781376378137 298.257298.257 IAG (Geo Ref Sys)IAG (Geo Ref Sys)

WGS 84WGS 84 63781376378137 298.257298.257 DoD (GPS)DoD (GPS)

PositioningPositioning NOT NOT SizeSize of Ellipsoid Important of Ellipsoid Important

GEOMETRY OF THE ELLIPSOID (1)GEOMETRY OF THE ELLIPSOID (1)GEOMETRY OF THE ELLIPSOID (1)GEOMETRY OF THE ELLIPSOID (1)

Definition

Semi-major and Semi-minor Axes (a, b) Flattening, 1st and 2nd Eccentricities

Definition

Semi-major and Semi-minor Axes (a, b) Flattening, 1st and 2nd Eccentricities

fa ba

e

a ba

f f22 2

222

22 2

2

2

21a bb

ee

xa

yb

zc

2

2

2

2

2

2 1

GEOMETRY OF THE ELLIPSOID (2)GEOMETRY OF THE ELLIPSOID (2)GEOMETRY OF THE ELLIPSOID (2)GEOMETRY OF THE ELLIPSOID (2)

Geodetic Latitude, G

Meridional and Prime Vertical Sections

Radii of curvature

meridional

prime vertical

Geodetic Latitude, G

Meridional and Prime Vertical Sections

Radii of curvature

meridional

prime vertical

a ee( )

( sin )1

1

2

2 2 3 2

a

e1 2 2 1 2sin

ASTRONOMICAL (TRUE) COORDINATESASTRONOMICAL (TRUE) COORDINATESASTRONOMICAL (TRUE) COORDINATESASTRONOMICAL (TRUE) COORDINATES



VerticalVertical

Equatorial PlaneEquatorial Plane

Axis of RotationAxis of Rotation

9090OO--AA

Meridional PlaneMeridional Plane

Co-latitudeCo-latitude

DEFINITION OF A GEODETIC DATUMDEFINITION OF A GEODETIC DATUMDEFINITION OF A GEODETIC DATUMDEFINITION OF A GEODETIC DATUM

GG

Ellipsoid Size (a & eEllipsoid Size (a & e22)) Ellipsoid Orientation (CIO)Ellipsoid Orientation (CIO) Ellipsoid Position (3)Ellipsoid Position (3) Zero Meridian (BIH, CZM)Zero Meridian (BIH, CZM)

Origin PillarOrigin Pillar

MassMassGeocentreGeocentre

Centre ofCentre ofEllipsoidEllipsoid

CIO AxisCIO Axis

normalnormal

Minor axis of EllipsoidMinor axis of Ellipsoidparallel to CIO Axisparallel to CIO Axis

PP

POSITIONING OF ELLIPSOID & ORIGIN PILLARPOSITIONING OF ELLIPSOID & ORIGIN PILLARPOSITIONING OF ELLIPSOID & ORIGIN PILLARPOSITIONING OF ELLIPSOID & ORIGIN PILLAR

Equip.Equip.

GeoidGeoid

EllipsoidEllipsoid

verticalvertical

normalnormal

H

N

hh

FormulaeFormulae

GG- - AA

= (= (GG - - AA) sin ) sin

hh = N + H= N + H

FormulaeFormulae

GG- - AA

= (= (GG - - AA) sin ) sin

hh = N + H= N + H

AT THE DATUM POINTAT THE DATUM POINTAT THE DATUM POINTAT THE DATUM POINT

GeoidGeoid

EllipsoidEllipsoid

verticalvertical

normalnormal

In GeneralIn General

ooGG == oo

AA

ooGG == oo

AA

hhoo == HHoo (N(No o =0)=0)

But AlsoBut Also

Arbitrary Arbitrary ooGG,,

ooGG,, NN

oo

(eg OSGB ‘70 & ED ‘50)(eg OSGB ‘70 & ED ‘50)

In GeneralIn General

ooGG == oo

AA

ooGG == oo

AA

hhoo == HHoo (N(No o =0)=0)

But AlsoBut Also

Arbitrary Arbitrary ooGG,,

ooGG,, NN

oo

(eg OSGB ‘70 & ED ‘50)(eg OSGB ‘70 & ED ‘50)

}} ObservedObservedOO

ELLIPSOIDAL COORDINATESELLIPSOIDAL COORDINATESELLIPSOIDAL COORDINATESELLIPSOIDAL COORDINATES

ITRFITRF(ex CIO/BIH)(ex CIO/BIH)


GreenwichGreenwich

GG

normalnormal

hhGG

GG

pp

PP

GEODETIC CARTESIAN COORDINATESGEODETIC CARTESIAN COORDINATESGEODETIC CARTESIAN COORDINATESGEODETIC CARTESIAN COORDINATES

OO

SatelliteSatellite


ZZ

YY

XX

GreenwichGreenwichPP

ppITRFITRF(ex CIO/BIH)(ex CIO/BIH)

WGS 84 GEOIDWGS 84 GEOID

- 100 m- 100 m + 100 m+ 100 m+ 50 m+ 50 m0 m0 m- 50 m- 50 m

Reference FramesReference FramesReference FramesReference Frames

ICRS and ITRS ICRS – inertial reference system ITRS – earth fixed reference system

Realization of ITRF a number of stations using VLBI, SLR, and GPS Origin: center of the earth (mass centre and geometric

center), determined by GPS and SLR Orientation: parallel to the CIO, mainly using VLBI

ICRS and ITRS ICRS – inertial reference system ITRS – earth fixed reference system

Realization of ITRF a number of stations using VLBI, SLR, and GPS Origin: center of the earth (mass centre and geometric

center), determined by GPS and SLR Orientation: parallel to the CIO, mainly using VLBI

Orbit DeterminationOrbit DeterminationOrbit DeterminationOrbit Determination

GPS SystemGPS SystemGPS SystemGPS System

Transfer the coordinates of ‘control points’on the earth, through satellites, to users

Before the user position can be determined, the positions of satellites must be known

Sources for satellite orbits from broadcast ephemeris (Keplerian elements) from IGS (coordinates every 15 min) Calculate them with a ground network

Transfer the coordinates of ‘control points’on the earth, through satellites, to users

Before the user position can be determined, the positions of satellites must be known

Sources for satellite orbits from broadcast ephemeris (Keplerian elements) from IGS (coordinates every 15 min) Calculate them with a ground network

GPS SYSTEMGPS SYSTEM

SpaceSegment

UserSegment

ControlSegment

ColoradoSprings

HawaiiKwajaleinAscension IslandDiegoGarcia

Monitor Stations

Orbit Determination – Inverse GPSOrbit Determination – Inverse GPSOrbit Determination – Inverse GPSOrbit Determination – Inverse GPS

A ground network to observe a satellite Four observations

Satellite position Satellite clock error

The same principle as GPS positioning Tested in WAAS testbed Need a global network to cover the earth

A ground network to observe a satellite Four observations

Satellite position Satellite clock error

The same principle as GPS positioning Tested in WAAS testbed Need a global network to cover the earth

Satellite OrbitSatellite OrbitSatellite OrbitSatellite Orbit

Controlled by the force acting on satellite, mainly gravity Consider a point mass

Controlled by the force acting on satellite, mainly gravity Consider a point mass

rr

GMra

orr

GMmF

3

2

M

rf

fea

e

earcos

)1()1(

1122

Keplerian Element of OrbitsKeplerian Element of OrbitsKeplerian Element of OrbitsKeplerian Element of Orbits

a

e

i

Perigee

X

Y

Z

Keplerian Element of OrbitsKeplerian Element of OrbitsKeplerian Element of OrbitsKeplerian Element of Orbits

a - semi-major axis of ellips

e - eccentricity

i - inclination of the orbit

- right ascension of the ascending node

- argument of perigee

T0 - perigee passing time

Computation of Satellite Position at Time tComputation of Satellite Position at Time tComputation of Satellite Position at Time tComputation of Satellite Position at Time t

Calculate position in the orbital plane a, e, , T0

Rotate the coordinates into three dimensional XYZ , i

Calculate position in the orbital plane a, e, , T0

Rotate the coordinates into three dimensional XYZ , i


The mean angular velocity (mean motion) n = 2/P = sqrt(/a3), - GM constant, P - period of orbit

Mean anomalyM = n*(t-T0) =M0 + (t -t0)n

t0 - the time where M = M0

The mean angular velocity (mean motion) n = 2/P = sqrt(/a3), - GM constant, P - period of orbit

Mean anomalyM = n*(t-T0) =M0 + (t -t0)n

t0 - the time where M = M0


The true anomaly v

where E - eccentric anomaly E = M + esinE

The true anomaly v

where E - eccentric anomaly E = M + esinE

)cos

sin1(tan

21

eE

Eev


The argument of latitude = v +

geo-centre radiusr = a(1-ecosE)

Position in Orbit Plane x = rcos y = rsin

The argument of latitude = v +

geo-centre radiusr = a(1-ecosE)

Position in Orbit Plane x = rcos y = rsin


Rotation the plane based on inclination and right ascension of the ascending nodes

X = xcos- ycosisin Y = xsin + ycosicos Z = ysini

Rotation the plane based on inclination and right ascension of the ascending nodes

X = xcos- ycosisin Y = xsin + ycosicos Z = ysini

Orbit PerturbationOrbit PerturbationOrbit PerturbationOrbit Perturbation

The earth is not a point mass There are other forces acting on satellites The true satellite orbit is not a perfect ellipse

Orbit perturbation Analytic solution is difficult

The earth is not a point mass There are other forces acting on satellites The true satellite orbit is not a perfect ellipse

Orbit perturbation Analytic solution is difficult

Ellipse

True orbit

Orbit Determination (Integration Method)Orbit Determination (Integration Method)Orbit Determination (Integration Method)Orbit Determination (Integration Method)

Based on Newton’s Motion LawF = ma

F - force acting on satellite a - acceleration m - mass of satellite

In an inertial frame Velocity and position: integration of acceleration

Based on Newton’s Motion LawF = ma

F - force acting on satellite a - acceleration m - mass of satellite

In an inertial frame Velocity and position: integration of acceleration

t

t

t

t

vdtpp

and

adtvv

00

00

Main force acting on satelliteMain force acting on satelliteMain force acting on satelliteMain force acting on satellite

Gravity of the earth Attraction from the sun, the moon and other planets Tidal effects radiation from the sun drag force of atmosphere …………………………

Gravity of the earth Attraction from the sun, the moon and other planets Tidal effects radiation from the sun drag force of atmosphere …………………………

Earth’s gravity fieldEarth’s gravity fieldEarth’s gravity fieldEarth’s gravity field

As harmonic functions

Acceleration

For Satellite Orbit, 30-100 degree and order Based on the orbit height

As harmonic functions

Acceleration

For Satellite Orbit, 30-100 degree and order Based on the orbit height

UGM

R

a

RP C COSm S SINme

m

n

n

nnm

nm

nm

[ ( ) (sin )( )102

r Ue e

..

Other ForcesOther ForcesOther ForcesOther Forces

Third body (sun, moon, and other planets

Earth Tide

Solar Radiation Pressure

Third body (sun, moon, and other planets

Earth Tide

Solar Radiation Pressure

r Ud d

.. U

GM

r rd

j

j

U kGM a

r rP COSzt

j

j

2

5

3 3 2( )

r C PAm

r r

r rR R

j

j

..

P

Ic

Ar r

u

j

0 2( )

Other ForcesOther ForcesOther ForcesOther Forces

Air Drag

Y bias (particular for GPS satellites) Found an acceleration along the cross track Not fully understood

Others (earth reflection, relativistic effects,…..) Unmodeled forces

Air Drag

Y bias (particular for GPS satellites) Found an acceleration along the cross track Not fully understood

Others (earth reflection, relativistic effects,…..) Unmodeled forces

PIc

Ar r

u

j

0 2( )

r CAm

vva D a r r

..( )

12

tbtar sincos

Initial conditionsInitial conditionsInitial conditionsInitial conditions

Velocity: V(t) = V0 + integration of a

Position: P(t) = P0 + integration of V

Other coefficients associated with force models Initial conditions d0:

V0, P0, and force model coefficients c

Solving P(t) => solving initial conditions In practice, using numerical integration of the following

equation:

Velocity: V(t) = V0 + integration of a

Position: P(t) = P0 + integration of V

Other coefficients associated with force models Initial conditions d0:

V0, P0, and force model coefficients c

Solving P(t) => solving initial conditions In practice, using numerical integration of the following

equation:

),,,( 0drrtfr

Relation of measurement and initial Relation of measurement and initial conditionsconditions

Relation of measurement and initial Relation of measurement and initial conditionsconditions

Measurements: D = g(x, y, z, t)=g(r,t),

Linearising:

x = g1(P0, V0, c), y = g2(P0,V0,c),z =g3(P0,V0,c)

D=g(g1(P0, V0, c), g2(P0,V0,c),g3(P0,V0,c),t)=h(d0,t)

=> partials

=>L = f1(P0, V0, c)

Measurements: D = g(x, y, z, t)=g(r,t),

Linearising:

x = g1(P0, V0, c), y = g2(P0,V0,c),z =g3(P0,V0,c)

D=g(g1(P0, V0, c), g2(P0,V0,c),g3(P0,V0,c),t)=h(d0,t)

=> partials

=>L = f1(P0, V0, c)

00 d

r

r

D

d

D

Calculation of PartialsCalculation of PartialsCalculation of PartialsCalculation of Partials

Considering:

Let

Finally, numerical integration

Considering:

Let

Finally, numerical integration

r f t r r d.. .

( , , , ) 0 r r d. .

( ) 0

rd

fr

rd

f

r

rd

fd

..

.

.

0 0 0 0

rd0

A tfr

( )

, ,

B tf

r( ) .

, ,

C tfd

( ) 0

.. .

( ) ( ) ( ) A t B t C t

)( 0drr

Computation ProcedureComputation ProcedureComputation ProcedureComputation Procedure

Initial Condition

Orbit Integration and partials

Meas. From ground stns

Solve for new initial conditions

New ini. condition

Three main orbit determination methodsThree main orbit determination methodsThree main orbit determination methodsThree main orbit determination methods

Kinematic Method using measurement only Need a dense global tracking network

Dynamic Method Based on orbit integration Measurements: solving for initial conditions Much less tracking stations

Reduced dynamic method Considering both measurement and force model errors Kalman filtering or other mathematic tools

Kinematic Method using measurement only Need a dense global tracking network

Dynamic Method Based on orbit integration Measurements: solving for initial conditions Much less tracking stations

Reduced dynamic method Considering both measurement and force model errors Kalman filtering or other mathematic tools

Coordinate Frame DifferenceCoordinate Frame DifferenceCoordinate Frame DifferenceCoordinate Frame Difference

Both positions of satellite and receiver are presented in WGS84

Satellite : at time when signal transmitted Receiver: at time when signal receiver Need to put both in the same reference Frame Transformation

Psr =R3(e)Pst

e: earth rotation rate,

: signal travel time

Both positions of satellite and receiver are presented in WGS84

Satellite : at time when signal transmitted Receiver: at time when signal receiver Need to put both in the same reference Frame Transformation

Psr =R3(e)Pst

e: earth rotation rate,

: signal travel time

How to determine signal travel time How to determine signal travel time How to determine signal travel time How to determine signal travel time

Iteration: The approx: = 0.07 s Calculate satellite position,

calculate distance = d/c Repeat the procedure

Using pseudorange measurement

Iteration: The approx: = 0.07 s Calculate satellite position,

calculate distance = d/c Repeat the procedure

Using pseudorange measurement

sRT

sT

sRRRR

RRR

sR

dtc

tt

or

dttdtdtc

dttc

t

dttt

and

dtdtc

0

00

0

0

0 )(

Orbit Information – broadcast ephemerisOrbit Information – broadcast ephemerisOrbit Information – broadcast ephemerisOrbit Information – broadcast ephemeris

Websit: http://www.navcen.uscg.gov/pubs/gps/icd200/icd200cw1234.pdf

Give modified Keplerian elements

Websit: http://www.navcen.uscg.gov/pubs/gps/icd200/icd200cw1234.pdf

Give modified Keplerian elements

Orbit Information – Precise EphemerisOrbit Information – Precise EphemerisOrbit Information – Precise EphemerisOrbit Information – Precise Ephemeris

International GNSS Services Global GNSS tracking network Managed by the IGS board Including a number of data centres and processing cen

ters GNSS orbit and related products

– Orbit and clock errors– Ionosphere– Troposphere– Station coordinates and velocity– Earth rotation parameters

International GNSS Services Global GNSS tracking network Managed by the IGS board Including a number of data centres and processing cen

ters GNSS orbit and related products

– Orbit and clock errors– Ionosphere– Troposphere– Station coordinates and velocity– Earth rotation parameters

Accuracy Latency Updates Sample Interval

GPS Satellite Ephemerides/Satellite & Station Clocks

broadcastorbits ~160 cm

real time -- dailySat. clocks ~7 ns

Ultra-Rapid (predicted half)

orbits ~10 cmreal time four times daily 15 min

Sat. clocks ~5 ns

Ultra-Rapid (observed half)

orbits <5 cm3 hours four times daily 15 min

Sat. clocks ~0.2 ns

Rapidorbits <5 cm

17 hours daily15 min

Sat. & Stn. clocks 0.1 ns 5 min

Finalorbits <5 cm

~13 days weekly15 min

Sat. & Stn. clocks <0.1 ns 5 min

Assignment SubmissionAssignment SubmissionAssignment SubmissionAssignment Submission

Two Reports: Week 7 and week 9

Final Report Week 14

Two Reports: Week 7 and week 9

Final Report Week 14

geometry of the earth and satellite orbit satellite positioning

Documents

geodetic latitude

p2two geodetic lats

ptwo geodetic points

otherf1gf2gpp1p2the

positions of satellites

earth mass centre

heights of points

geometric center