EE 570: Location and Navigation: Theory & Practice

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1 EE 57: Location and Navigation: Thory & Practic Navigation Mathmatics Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 1 of 12

Navigation Mathmatics : Earth surfac and Gravity - Earth Modling Th

from th polar axis (top viw) An lliptical cross-sction whn viwd

. oblat sphroid) is an approximation to th goid Th goid is a

th man sa lvl Ratio xaggratd Max variation btw.

2 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling Th Earth can b modld as an oblat sphroid A circular cross sction whn viw from th polar axis (top viw) An lliptical cross-sction whn viwd prpndicular to th polar axis (sid viw) This llipsoid (i.. oblat sphroid) is an approximation to th goid Th goid is a gravitational quipotntial surfac which bst fits (in a last squar sns) th man sa lvl Ratio xaggratd Max variation btw. llipsoid and goid is +3 to -51 mtrs. Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 2 of 12

3 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling WGS 84 provids a modl of th Earth s goid Mor rcntly rplacd by EGM28 Th quatorial radius R = 6,378,137. m Th polar radius R p = 6,356, m Eccntricity of th llipsoid z 2 R p R Flattning of th llipsoid 2 Equator R p R f R R p R 1 / 298 Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 3 of 12

4 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling W can dfin a position nar th Earth s surfac in trms of latitud, longitud, and hight Gocntric latitud intrscts th cntr of mass of th Earth Godtic latitud (L) is th angl btwn th normal to th llipsoid and th quatorial plan z Rfrnc Ellipsoid Surfac Normal Equatorial Plan Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 4 of 12

5 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling Th longitud () is th angl from th x-axis of th ECEF fram to th projction of r b onto th quatorial plan. Th godtic (or llipsoidal) hight (h) is th distanc along th normal from th llipsoid to th body y z r b r b Rfrnc Ellipsoid z x Equatorial Plan Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 5 of 12

6 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling Transvrs radius of curvatur Th radius of curvatur for ast-wst motion Th mridian radius of curvatur y R R E N ( L) R 2 1 sin( L) 1 1 sin R L b 3/2 Disk of constant Latitud (L b ) z R X /Y plan b x R E = Transvrs radius of curvatur Tusday 29 Jan 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 6 of 12

7 Navigation Mathmatics : Earth surfac and Gravity - Earth Modling (R E (1-2 )+h b )Sin(L b ) Curvilinar to ECEF coordinats z (R E +h b )Cos(L b ) x b RE hb Cos( Lb ) Cos( ) b rb yb RE hb Cos( Lb ) Sin( b ) z 2 b RE (1 ) hb Sin( Lb) y X /Y plan b (R E +h b )Cos(L b )Cos( b ) (R E +h b )Cos(L b )Sin( b ) x Disk of constant Latitud (L b ) R E = Transvrs radius of curvatur Tusday 29 Jan 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 7 of 12

8 Navigation Mathmatics : Earth surfac and Gravity - Gravity Modls Spcific Forc (f ib ) Non-Gravitational forc pr unit mass (units of acclration) o Acclromtrs masur spcific forc Spcific forc snsd whn stationary (wrt Earth) is rfrrd to as th acclration du to gravity (g b ) Actually, th raction to this forc Gravitational forc (γ ib ) is a rsult of mass attraction Th gravitational mass attraction forc is diffrnt from th acclration du to gravity Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 8 of 12

9 Navigation Mathmatics : Earth surfac and Gravity - Gravity Modls Rlationship btwn spcific forc, inrtial acclration, and gravitational attraction f Whn stationary on th surfac of th Earth Rcall cas 1: A fixd point in a rotating fram o Considring fram {} to b th {i} fram, {1} = {}, and {2} ={b} givs o Coordinatizing in th -fram givs a ib ib ib r () t r ( t) r ( t) r ( t) 22 ( ) 12 1 ( ) 12 i i i i r ( t) r ( t) ib i i b r ( t) r ( t) ib i i b Fixd point in a rotating fram and = Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 9 of 12

10 Navigation Mathmatics : Earth surfac and Gravity - Gravity Modls Thus, whn stationary on th surfac of th Earth th acclration is du to Cntrifugal forc a ib i i Thrfor, th acclration du to gravity is r b g f r b ib i i b ib v b i i i r b r b ib Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 1 of 12

11 Navigation Mathmatics : Earth surfac and Gravity - Gravity Modls Now, ω i = thus g 1 ω i and hnc, Ω i = 1 1 r 2 b ib i b 1 1 g ω i, and r b ib i i b Th WGS 84 modl of acclration du to gravity (on th llipsoid) can b approximatd by (Somigliana modl) g ( L) sin (L) sin (L) Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 11 of 12

dvlopmnt of som of th most prcis Earth gravity modls NASA's Grac Gravity Modl

12 Navigation Mathmatics : Earth surfac and Gravity - Gravity Modls On March 17, 22 NASA launchd th Gravity Rcovry and Climat Exprimnt (GRACE) which ld to th dvlopmnt of som of th most prcis Earth gravity modls NASA's Grac Gravity Modl Tusday 5 Fb 213 NMT EE 57: Location and Navigation: Thory & Practic Slid 12 of 12

EE 570: Location and Navigation

EE 570: Location and Navigation Navigation Mathematics: Kinematics (Earth Surface & Gravity Models) Aly El-Osery Kevin Wedeward Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA