Navigation Mathematics Tuesday 5 Feb 2013 NMT EE 570 Location and Navigation Theory amp Practice Slide 1 of 12 Navigation Mathematics Earth surface and Gravity Earth Modeling Tuesday 5 Feb 2013 ID: 627206
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Slide1
EE 570: Location and Navigation: Theory & Practice
Navigation Mathematics
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
Slide
1
of 12Slide2
Navigation Mathematics :
Earth surface and
Gravity - Earth Modeling
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
The Earth can be modeled as an oblate spheroid
A circular cross section when view from
the polar axis (top view)
An elliptical cross-section when viewedperpendicular to the polar axis (side view)This ellipsoid (i.e. oblate spheroid) is an approximation to the “geoid”The geoid is a gravitational equipotential surface which “best” fits (in a least square sense) the mean sea level
Ratio exaggerated
www.nrcan.gc.ca
Slide
2
of 12
Max variation
betw
. ellipsoid
and
geoid
is +3
to
-51
meters.Slide3
Navigation Mathematics :
Earth surface and
Gravity - Earth Modeling
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
WGS 84
provides
a model
of the Earth’s geoidMore recently replaced by EGM2008The equatorial radius The polar
radius
Eccentricity of the ellipsoidFlattening of the ellipsoid
Equator
Slide
3
of 12Slide4
Navigation Mathematics :
Earth surface and
Gravity - Earth Modeling
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
We can define a position “near” the Earth’s surface in terms of latitude, longitude, and height
Geocentric latitude intersects the
center of mass of the Earth
Geodetic latitude (L) is the angle betweenthe normal to the ellipsoid andthe equatorial plane
Equatorial
Plane
Geodetic
Latitude
Geocentric
Latitude
Reference
Ellipsoid
Surface
Normal
Slide
4
of 12Slide5
Navigation Mathematics :
Earth surface and
Gravity - Earth Modeling
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
The longitude (
) is the angle from the
x-axis of the ECEF frame to the projection of onto the equatorial plane.The geodetic (or ellipsoidal) height (h) is thedistance along the normal from
the ellipsoid to the body
longitude
Equatorial
Plane
Geodetic
Latitude
Reference
Ellipsoid
Geodetic
height
Slide
5
of 12Slide6
Navigation Mathematics :
Earth surface and
Gravity
- Earth Modeling
R
E
= Transverse radius of curvature
L
b
h
b
R
E
z
e
X
e
/Ye planeTransverse radius of curvatureThe radius of curvature for east-west motionThe meridian radius of curvature
Tuesday 29 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
x
e
y
e
Disk of constant Latitude (
L
b
)
Slide
6
of 12Slide7
Navigation Mathematics :
Earth surface and
Gravity - Earth Modeling
R
E
= Transverse radius of curvature
R
E
(1-e
2
)
L
b
h
bR
E
z
e
(
RE+hb)Cos(
L
b
)
(R
E
(1-e
2
)+
h
b
)Sin(
L
b
)
x
e
y
e
Disk of constant Latitude (
L
b
)
(
R
E
+h
b
)Cos(
L
b
)
(
R
E
+h
b
)Cos(
L
b
)Sin(
b
)
(
R
E
+h
b
)Cos(
L
b
)Cos(
b
)
X
e
/Y
e
plane
Curvilinear to ECEF coordinates
Tuesday 29 Jan 2013
NMT EE 570: Location and Navigation: Theory & Practice
Slide
7
of 12Slide8
Navigation Mathematics :
Earth surface and
Gravity - Gravity Models
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
Specific Force (
)
Non-Gravitational force per unit mass (units of acceleration)
Accelerometers measure specific forceSpecific force sensed when stationary(wrt Earth) is referred to as the acceleration due to
gravity (
Actually, the reaction to this forceGravitational force () is a result of mass attraction
The gravitational mass attraction force is different from the acceleration due to gravity
Slide 8 of 12Slide9
Navigation Mathematics :
Earth surface and
Gravity - Gravity Models
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
Relationship between specific force, inertial acceleration, and gravitational attraction
When stationary on the surface of the Earth
Recall case 1: A fixed point in a rotating frame
Considering frame {0} to be the {i} frame, {1} = {e}, and {2} ={b} givesCoordinatizing in the
e-frame gives
Fixed point in a rotating frame and
Slide
9
of 12Slide10
Navigation Mathematics :
Earth surface and
Gravity - Gravity Models
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
Thus, when stationary on the surface of the Earth the acceleration is due
to Centrifugal
force
Therefore, the acceleration due to gravity is
Slide
10
of 12Slide11
Navigation Mathematics :
Earth surface and
Gravity - Gravity Models
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
Now,
and hence,
, and thus
The WGS 84 model of acceleration due to gravity (on the ellipsoid) can
be approximated by (
Somigliana
model
)
Slide
11
of 12Slide12
Navigation Mathematics :
Earth surface and
Gravity - Gravity Models
Tuesday 5 Feb 2013
NMT EE 570: Location and Navigation: Theory & Practice
On
March
17, 2002
NASA launched the Gravity Recovery and Climate Experiment (GRACE) which led to the development of some of the most precise Earth gravity modelsNASA's Grace Gravity Model
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12 of 12