/
EE 570: Location and Navigation: Theory & Practice EE 570: Location and Navigation: Theory & Practice

EE 570: Location and Navigation: Theory & Practice - PowerPoint Presentation

pasty-toler
pasty-toler . @pasty-toler
Follow
377 views
Uploaded On 2018-02-02

EE 570: Location and Navigation: Theory & Practice - PPT Presentation

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

earth navigation surface gravity navigation earth gravity surface location theory amp practice 570 slide 2013 nmt tuesday mathematics feb

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "EE 570: Location and Navigation: Theory ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

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

Slide

12 of 12