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Solo Hermelin

The document outlines the principles of navigation systems for aircraft, detailing the various headings, tracks, and the calculation of deviations from desired flight paths using spherical trigonometry. It also explains the concept of great circles as the shortest distance between two points on Earth and introduces multiple coordinate systems relevant to navigation. Furthermore, it discusses the dynamics of air vehicles within a spherical Earth model and the forces acting on them.

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Navigation Systems SOLO HERMELIN Updated: 08.11.12 1
Table of Content SOLO 2 Navigation
NavigationSOLO Aircraft Steering to Waypoints 1. T-HDG – True Heading 2. M-HDG – Magnetic Heading 3. T-TK - True Track 4. M-TK - Magnetic Track Angle 5. TKE – Track Angle Error 6. T-DTK – True Desired Track 7. XTK – Cross-Track Distance 8. DIS – Distance to Destination 9. GS - Ground Speed 10. WS – Wind Speed 11. WD – Wind Direction 12. TAS – True Airspeed 13. DA – Drift Angle In order to minimize Fuel, Time and Distances the Aircraft will tend to fly between Waypoints, on the Earth Surface, on the Great Circle connecting the Initial and Final Waypoints, 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 an Automatic Flight-Control System (AFCS).
NavigationSOLO Aircraft Steering to Waypoints
5 Spherical TrigonometrySOLO Assume three points on a unit radius sphere, defined by the vectors →→→ CBA 1,1,1 Laws of Cosines for Spherical Triangle Sides ab abc ca cab bc bca ˆsinˆsin ˆcosˆcosˆcos ˆcos ˆsinˆsin ˆcosˆcosˆcosˆcos ˆsinˆsin ˆcosˆcosˆcos ˆcos − = − = − = γ β α Law of Sines for Spherical Triangle Sides. cba abccba cba ˆsinˆsinˆsin ˆcosˆcosˆcos2ˆcosˆcosˆcos1 ˆsin ˆsin ˆsin ˆsin ˆsin ˆsin 222 +−−− === γβα The three great circles passing trough those three points define a spherical triangle with CBA ,, - three spherical triangle vertices cba ˆ,ˆˆ -three spherical triangle side angles γβα 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 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 vertices cba ˆ,ˆˆ -three spherical triangle side angles γβα ˆ,ˆˆ - three spherical triangle angles defined by the angles between the tangents to the great circles at the vertices. βα βαγ αγ αγβ γβ γβα ˆsinˆsin ˆcosˆcosˆcos ˆcos ˆsinˆsin ˆcosˆcosˆcosˆcos ˆsinˆsin ˆcosˆcosˆcos ˆcos + = + = + = c b a Spherical Trigonometry
7 NavigationSOLO 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 2 111 ,, λφR 222 ,, λφR The Great Circle Distance between two points 1 and 2 is ρ. The average Radius on the Great Circle is a = (R1+R2)/2 θρ ⋅= a R – radius - Latitudeϕ λ - Longitude kmNmNma 852.11deg/76.60/ =≈ρ
8 NavigationSOLO Flight on Earth Great Circles 1 2 111 ,, λφR 222 ,, λφR The Great Circle Distance between two points 1 and 2 is ρ. θρ ⋅= a R – radius - Latitudeϕ λ - Longitude ( ) ( ) ( ) ( ) ( ) ( )212121 cos90sin90sin90cos90cos /coscos λλφφφφ ρθ −⋅−⋅−+−⋅−= =  a From the Law of Cosines for Spherical Triangles or ( ) ( )212121 coscoscossinsin/cos λλφφφφρ −⋅⋅+⋅=a ( ){ }212121 1 coscoscossinsincos λλφφφφρ −⋅⋅+⋅⋅= − a The Initial Heading Angle ψ0 can be obtained using the Law of Cosines for Spherical Triangles as follows ( ) ( )a a /sincos /cossinsin cos 1 12 0 ρφ ρφφ ψ ⋅ ⋅− = ( )[ ] ( )[ ]2 222 22221 coscoscossinsin1cos coscoscossinsinsinsin cos λλφφφφφ λλφφφφφφ ψ −⋅⋅+⋅−⋅ −⋅⋅+⋅⋅− = − The Heading Angle ψ from the Present Position (R, ,λ) to Destination Point (Rϕ 2,ϕ2,λ2)
9 NavigationSOLO Flight on Earth Great Circles The Distance on the Great Circle between two points 1 and 2 is ρ. 1 2 111 ,, λφR 222 ,, λφR R – radius - Latitudeϕ λ - Longitude The Time required to travel along the Great Circle between points 1 and 2 is given by ( ){ } 22 212121 1 coscoscossinsincos yxHoriz HorizHoriz VVV V a V t += −⋅⋅+⋅⋅==∆ − λλφφφφ ρ ( ){ }212121 1 coscoscossinsincos λλφφφφρ −⋅⋅+⋅⋅= − a
10 NavigationSOLO Flight on Earth Great Circles 1 2 111 ,, λφR 222 ,, λφR 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. R – radius - Latitudeϕ λ - Longitude Using the Law of Cosines for Spherical Triangle APB we have ( ) ( )aaYd /sin 90sin /sin sin ρ ψ  = ∆ ( ) ( ) ( ) ( ) ( ) 2/sin/sin /cos/cos/cos 0ˆcos 21 90ˆ RR a aYaX aYaXa P dd dd P + = ⋅ ⋅− == = ρ  Using the Law of Sines for Spherical Triangle APB we have ( ) ( )      ⋅= − aY a aX d d /cos /cos cos 1 ρ ( )[ ]ψρ ∆⋅⋅= − sin/sinsin 1 aaYd
11 SOLO Coordinate Systems 1. Heliocentric (Heliocentric) Coordinate System COORDINATES IN THE SOLAR SYSTEM 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.
12 SOLO 1.Heliocentric (Heliocentric) Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM The Earth axis of rotation is tilted relative to Ecliptic and vobbles slightly, in a clockwise direction opposite to that of the Earth spin, from 22.1° to 24.5° , with a cycle of approximately 41,000 years. G Gz Ω Gx Gy Ecliptic plane normal (Ecliptic Pole) Locus of Lunar plane normal (Lunar Pole) Lunar Orbital Plane Earth Orbital Plane (Ecliptic) Equatorial Plane Ascending Node  5.23 15.5 Vernal Equinox Direction The Moon’s gravity tends to tilt the Earth’s axis so that it becomes perpendicular to Moon’s Orbit, 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.
13 SOLO 2. Geocentric-Equatorial Coordinate System COORDINATES IN THE SOLAR SYSTEM The origin at the center of the Earth . G Gz Ω Gx Gy Ecliptic plane normal (Ecliptic Pole) Locus of Lunar plane normal (Lunar Pole) Lunar Orbital Plane Earth Orbital Plane (Ecliptic) Equatorial Plane Ascending Node  5.23 15.5 Vernal Equinox Direction • 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 frame is non-rotating to the stars (except to the precession of equinoxes) and the Earth turns relative to it.
14 SOLO 3. The Right Ascension-Declination System COORDINATES IN THE SOLAR 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 Ω GX GY Equatorial Planeα δ Vernal Equinox Direction 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 COORDINATES IN THE SOLAR 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: • 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.
16 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 a given time. 1. a – semi-major axis – a constant defining the size of the coning orbit. 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, between the unit vector and the point where the satellite crosses through the Equatorial Plane in a northerly direction (ascending node) measured counterclockwise where 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 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 ≠ × = → h h vr Z  ε 2. From the specific mechanical energy of an elliptic orbit equation ar vv E 22 0 00 µµ −=− ⋅ =  we obtain 00 0 2 vv r a  ⋅− = µ µ 3. i inclination is computed using →→ ⋅= GZZi 11cos ε 22 11cos 1 ππ ε ≤≤−      ⋅= →→ − iZZi G 4. The eccentricity vector of a Keplerian trajectory is defined as ( ) → =      ⋅−      −⋅= ε µ µ Xevvrr r vve 1 1 0000 0 00  from which ee  = 01 ≠= → e e e X  ε →→→ ×= εεη XZY 111
18 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 011 11 11 1 ≠×← × × = →→ →→ →→ → ε ε ε ZZ ZZ ZZ N G G G Ω – longitude of the ascending node – is computed using →→ ⋅=Ω NXG 11cos →→→ ⋅      ×=Ω GG ZNX 111sin       Ω Ω =Ω − cos sin tan 1 6. ω – argument of the periapsis – is computed using →→ ⋅= εω XN 11cos →→→ ⋅      ×= εεω ZXN 111sin       = − ω ω ω cos sin tan 1 7. Ө – satellite position from the periapsis – is computed using →→ ⋅=Θ rX 11cos ε →→→ ⋅      ×=Θ εε ZrX 111sin       Θ Θ =Θ − cos sin tan 1
19 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM Let find a, e, ω, i, Ω from the initial position and velocity vectors (continue).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: [ ] [ ] [ ]           ΩΩ− ΩΩ           −          −=Ω= 100 0cossin 0sincos cossin0 sincos0 001 100 0cossin 0sincos 313 ii iiiCG ωω ωω ωε           Ω−Ω Ω+Ω−Ω−Ω− Ω+ΩΩ−Ω = iii iii iii cossincossinsin sincoscoscoscossinsinsincoscoscossin sinsincoscossinsincossincossincoscos ωωωωω ωωωωω
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 Mass System 8.2 Missile Kinematics Model 1 (Spherical Earth) 8.3 Missile Kinematics Model 2 (Spherical Earth)
21 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 Coordinate Systems Table of Content
22 SOLO Coordinate Systems 1.Inertial System (I( R  - vehicle position vector I td Rd V   = - vehicle velocity vector, relative to inertia II td Rd td Vd a 2 2   == - vehicle acceleration vector, relative to inertia AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
23 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 equator to zero longitude meridian. The Earth rotates relative to Inertial system I, with the angular velocity sec/10.292116557.7 5 rad− =Ω EIIE zz  11 Ω=Ω=Ω=←ω ( )           Ω =← 0 0 EC IEω  Rotation Matrix from I to E [ ] ( ) ( ) ( ) ( )           ΩΩ− ΩΩ =Ω= 100 0cossin 0sincos 3 tt tt tCE I AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
24 SOLO Coordinate Systems (continue – 3) 2. Earth Center Earth Fixed Coordinate System (E( (continue – 1( Vehicle Position ( ) ( ) ( ) ( )ETE I EI E I RCRCR  == Vehicle Velocity Vehicle Acceleration RVR td Rd td Rd V EIE EI    ×Ω+=×+== ←ω - vehicle velocity relative to Inertia R td Rd td Rd V IE LE E    ×+== ←ω: - vehicle velocity relative to Earth ( ) ( ) II E I E I R td d td Vd RV td d td Vd a      ×Ω+=×Ω+== ( ) ( )RV td Vd R td Rd R td d V td Vd EIEEU U E EE EIU U E IU              ×Ω×Ω+×             Ω+++=×Ω×Ω+×Ω+× Ω +×+= ← Ω ←←← ω ωωω 0 ( ) ( ) ( )RV td Vd RV td Vd a E E E EEU U E      ×Ω×Ω+×Ω+=×Ω×Ω+×Ω++= ← 22ω or where U is any coordinate system. In our case U = E. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
25 SOLO Coordinate Systems (continue – 4) 3.Earth Fixed Coordinate System (E0( The origin of the system is fixed on the earth at some given 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 Earth surface, zE0 is pointed down. [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =           −           −− − =−−= 100 0cossin 0sincos sin0cos 010 cos0sin 2/ 00 00 00 00 3020 0 LongLong LongLong LatLat LatLat LongLatCE E π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          −−− − −− = 00000 00 00000 sinsincoscoscos 0cossin cossinsincossin LatLongLatLongLat LongLong LatLongLatLongLat The Angular Velocity of E relative to I is: EIIEIE zz  110 Ω=Ω== ←← ωω or ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          Ω− Ω =           Ω          −−− − −− =           Ω =← 0 0 00000 00 00000 00 0 sin 0 cos 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 Lat Lat LatLongLatLongLat LongLong LatLongLatLongLat CE E E IEω  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
26 SOLO Coordinate Systems (continue – 5) 4. Local-Level-Local-North (L) or Navigation Frame The origin of the LLLN coordinate system is located at the projection of the center of gravity CG of the vehicle on the Earth surface, with zDown axis pointed down, xNorth, yEast plan parallel to the local level, with xNorth pointed to the local North and yEast pointed to the 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 LongLatCL E π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          −−− − −− = LatLongLatLongLat LongLong LatLongLatLongLat sinsincoscoscos 0cossin cossinsincossin AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
27 SOLO Coordinate Systems (continue – 6) 4. Local-Level-Local-North (L( (continue – 1) Angular Velocity IEELIL ←←← += ωωω  Angular Velocity of L relative to I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          Ω− Ω =           Ω          − − −− =           Ω =           Ω Ω Ω =← Lat Lat LatLongLatLongLat LongLong LatLongLatLongLat CL E Down East North L IE sin 0 cos 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 ω  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               − −=             −+                       −−− − −− =             −+             =           = • • • • • • • ← LatLong Lat LatLong Lat Long LatLongLatLongLat LongLong LatLongLatLongLat Lat Long CL E Down East North L EL sin cos 0 0 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 0 0 ρ ρ ρ ω  ( ) ( ) ( ) ( ) ( )                         +Ω− −       +Ω =           Ω+ Ω+ Ω+ =+= • • • ←←← LatLong Lat LatLong DownDown EastEast NorthNorth L IEC L ECL L IL sin cos ρ ρ ρ ωωω  Therefore AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
28 SOLO Coordinate Systems (continue – 7) 4. Local-Level-Local-North (L( (continue – 2) Vehicle Velocity Vehicle Velocity relative to I RVR td Rd td Rd V EIE EI    ×Ω+=×+== ←ω ( ) ( ) ( ) ( ) ( ) ( ) ( )          +−               −− − +           +− =×+= •• •• •• ← HR LatLongLat LatLongLatLong LatLatLong HR R td Rd V EL L L E 00 0 0 0cos cos0sin sin0 0 0     ω 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 V td Hd LatHR V td Longd HR V td Latd _ 0 _ 0 _ cos −= + = + = AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE HeightVehicleHRadiusEarthmRHRR =⋅=+= 6 00 10378135.6
29 SOLO Coordinate Systems (continue – 8) 4. Local-Level-Local-North (L( (continue – 3) Vehicle Velocity (continue – 1) We assume that the atmosphere movement (velocity and acceleration) relative to Earth At the vehicle position (Lat, Long, H) is known. Since the aerodynamic forces on the vehicle are due to vehicle movement relative to atmosphere, let divide the vehicle velocity in two parts: WAE VVV  += ( )           = Down East North L A V V V V  - Vehicle Velocity relative to atmosphere ( ) ( )           = DownW EastW NorthW L W V V V HLongLatV _ _ _ ,,  - Wind Velocity at vehicle position (known function of time) AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
30 SOLO Coordinate Systems (continue – 9) 4. Local-Level-Local-North (L( (continue – 4) Vehicle Acceleration Since: ( ) ( ) ( ) ( )RV td Vd R td d td Vd RV td d td Vd a EEL L E II E I E I        ×Ω×Ω+×Ω++=×Ω+=×Ω+== ← 2ω WAE VVV  += ( ) WWIL L W AAIL L A VV td Vd RVV td Vd a      ×Ω+×++×Ω×Ω+×Ω+×+= ←← ωω ( )         Wa WWEL L W AAEL L A VV td Vd RVV td Vd ×Ω+×++×Ω×Ω+×Ω+×+= ←← 22 ωω ( ) ( ) ( ) ( )HLongLatVHLongLat td Vd HLongLata WEL L W W ,,2,,:,,    ×Ω++= ←ω ( ) WAAEL L A aRVV td Vd   +×Ω×Ω+×Ω+×+= ← 2ω where: is the wind acceleration at vehicle position. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
31 SOLO Coordinate Systems (continue – 10) 5.Body Coordinates (B( The origin of the Body coordinate system is located at the instantaneous center of gravity CG of the vehicle, with xB pointed to the front of the Air Vehicle, yB pointed toward the right wing and zB completing the right-handed Cartesian reference frame. Rotation Matrix from LLLN to B (Euler Angles(: [ ] [ ] [ ]           −+ +− − == θφψφψθφψφψθφ θφψφψθφψφψθφ θψθψθ ψθφ cccssscsscsc csccssssccss ssccc CB L 321 ψ - azimuth angle θ - pitch angle φ - roll angle AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
32 SOLO Coordinate Systems (continue – 11) 5.Body Coordinates (B( (continue – 1( ψ θ φ Bx Lx Bz Ly Lz By Angular Velocity from L to B (Euler Angles(: ( ) [ ] [ ] [ ]           +           +           =           =← ψ θφθφ φ ω    0 0 0 0 0 0 211 R Q P B LB                     −           − +                     − +           = ψθθ θθ φφ φφθ φφ φφ φ    0 0 cos0sin 010 sin0cos cossin0 sincos0 001 0 0 cossin0 sincos0 001 0 0 [ ]           =                     − − = ψ θ φ ψ θ φ θφφ θφφ θ       G coscossin0 cossincos0 sin01 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
33 SOLO Coordinate Systems (continue – 12) 5.Body Coordinates (B( (continue – 2( ψ θ φ Bx Lx Bz Ly Lz By Rotation Matrix from LLLN to B (Quaternions(: ( ) [ ][ ] ( ) [ ][ ] { } { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )            −−− − − −           −− −− −− = +×−×−= 321 412 143 234 3412 2143 1234 44 3333 BIBLBL BLBLBL BLBLBL BLBLBL BLBLBLBL BLBLBIBL BLBLBLBL T BLBLBLXBLBLXBL B L qqq qqq qqq qqq qqqq qqqq qqqq qqqIqqIqC  where: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) { } ( ) ( ) ( )          =      =                         =             = 3 2 1 :& 4 4 3 2 1 4 3 2 1 BL BL BL BL BL BL BL BL BL BL BL BL BL BL BL BL q q q q q q qor q q q q q q q q q   ( )                         −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos4 ϕθψϕθψ BLq ( )                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ BLq ( )                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ BLq ( )                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ BLq AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
34 SOLO Coordinate Systems (continue – 13) 5.Body Coordinates (B( (continue – 3( ψ θ φ Bx Lx Bz Ly Lz By Rotation Matrix from LLLN to B (Quaternions( (continue – 1( The quaternions are given by the following differential equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) BL L IL B IBBLBLBL B ILBL B IBBL B IL B IBBL B LBBLBL qqqqqqqqq ⋅−⋅=⋅⋅⋅−⋅=−⋅=⋅= ←←←←←←← ωωωωωωω 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     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                          +Ω−+Ω−+Ω− +Ω+Ω+Ω− +Ω+Ω−+Ω +Ω+Ω+Ω− − 4 3 2 1 0 0 0 0 2 1 BL BL BL BL zLzLyLyLxLxL zLzLxLxLyLyL yLyLxLxLzLzL xLxLyLyLzLzL q q q q ρρρ ρρρ ρρρ ρρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                          +Ω+−+Ω+−+Ω+− +Ω−+Ω−−+Ω+ +Ω−+Ω++Ω−− +Ω−+Ω−−+Ω+ = 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 ρρρ ρρρ ρρρ ρρρ or: AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
35 SOLO Coordinate Systems (continue – 14) 5.Body Coordinates (B( (continue – 4( ψ θ φ Bx Lx Bz Ly Lz By Vehicle Velocity Vehicle Velocity relative to Earth is divided in: WAE VVV  += ( )           = w v u V B A  ( ) ( )           =           = DownW EastW NorthW B L zW yW xW B W V V V C V V V HLongLatV B B B _ _ _ ,,  Vehicle Acceleration ( ) WWIB B W AAIB B A I VV td Vd RVV td Vd td Vd a      ×Ω+×++×Ω×Ω+×Ω+×+== ←← ωω ( ) ( ) W AELALB B A a RVV td Vd    + ×Ω×Ω+×Ω++×+= ←← 2ωω AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
36 SOLO Coordinate Systems (continue – 15) 6.Wind Coordinates (W( The origin of the Wind coordinate system is located at the instantaneous center of gravity CG of the vehicle, with xW pointed in the direction of the vehicle velocity vector relative to air .AV  [ ] [ ]           − −−=           −          −=−= αα βαββα βαββα αα αα ββ ββ αβ cos0sin sinsincossincos cossinsincoscos cos0sin 010 sin0cos 100 0cossin 0sincos 23 W BC The Wind coordinate frame is defined by the following two angles: α - angle of attack β - sideslip angle AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
37 SOLO Coordinate Systems (continue – 16) 6.Wind Coordinates (W( (continue -1( Rotation Matrix from L (LLLN) to W is: χ - azimuth angle of the trajectory γ - pitch angle of the trajectory Rotation Matrix [ ] [ ] [ ] [ ] [ ] 32123 ψθφαβ −== B L W B W L CCC The Rotation Matrix from L (LLLN) to W can also be defined by the following Consecutive rotations: σ - bank angle of the trajectory [ ] [ ] [ ] [ ]           −+ +− − === γσχσχγσχσχγσ γσχσχγσχσχγσ γχγχγ χγσσ cccssscsscsc csccssssccss ssccc CC W L W L 321 * 1 We defined also the intermediate wind frame W* by: [ ] [ ]           − − == γχγχγ χχ γχγχγ χγ csscs cs ssccc CW L 032 * AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
38 SOLO Coordinate Systems (continue – 17) 6.Wind Coordinates (W( (continue -2( Angular Velocity of W* relative to LLLN is: Angular Velocities ( ) [ ]          − =                     − +           =           +           =           =← γχ γ γχ χγγ γγ γ χ γγω cos sin 0 0 cos0sin 010 sin0cos 0 0 0 0 0 0 2 * * * * *         W W W W LW R Q P Angular Velocity of W relative to LLLN is: ( ) [ ] [ ]                     − − =          −           − +           =                     +           +           =           =← χ γ σ γσσ γσσ γ γχ γ γχ σσ σσ σ χ γγσ σ ω           coscossin0 cossincos0 sin01 cos sin cossin0 sincos0 001 0 00 0 0 0 0 0 21 W W W W LW R Q P AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
39 SOLO Coordinate Systems (continue – 18) 6.Wind Coordinates (W( (continue -3( We have also: Angular Velocities (continue – 1( ( ) ( ) ( ) ( )           Ω Ω Ω =           Ω− Ω ==           Ω Ω Ω = ←← Down East North W L W L L IE W L zW yW xW W IE C Lat Lat CC *** * * * * sin 0 cos ωω  ( ) ( ) ( ) ( )           =               − −==           = • • • ←← Down East North W L W L L EL W L zW yW xW W EL C LatLong Lat LatLong CC ρ ρ ρ ω ρ ρ ρ ω *** * * * * sin cos  ( ) ( ) ( ) ( ) [ ] ( )* 1 sin 0 cos W IE W L L IE W L zW yW xW W IE Lat Lat CC ←←← =           Ω− Ω ==           Ω Ω Ω = ωσωω  ( ) ( ) ( ) ( ) [ ] ( )* 1 sin cos W IL W L L IL W L W IL LatLong Lat LatLong CC ← • • • ←← =                         +Ω− −       +Ω == ωσωω  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
40 SOLO Coordinate Systems (continue – 19) 6.Wind Coordinates (W( (continue -4( The Angular Velocity from I to W is: Angular Velocities (continue – 2( ( ) ( ) ( ) ( )           Ω+ Ω+ Ω+ +           =+           =+=           = ←←←← DownDown EastEast NorthNorth W L W W W L IL W L W W W W IL W LW W W W W IW C R Q P C R Q P r q p ρ ρ ρ ωωωω  Using the angle of attack α and the sideslip angle β , we can write: BWBW yz    11 αβω −=← or: ( ) ( ) ( ) [ ]           −           =           −           =−= ←←← 0 0 0 0 3 αβ β ωωω    r q p C r q p W B W W W W IB W IW W BW but also: ( ) ( ) ( ) [ ]           −           =           −           =−= ←←← 0 0 0 0 3 αβ β ωωω    R Q P C R Q P W B W W W W LB W LW W BW AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
41 SOLO Coordinate Systems (continue – 20) 6.Wind Coordinates (W( (continue -5( We can write: Angular Velocities (continue – 3(           −           +                     − −−=           0 cos sin 0 0 cos0sin sinsincossincos cossinsincoscos βα βα βαα βαββα βαββα   r q p r q p W W W or: ( ) ( ) βαα βαβαβα βαβαβα    ++−= −−+−= +−+= cossin sinsincossincos cossinsincoscos rpr rqpq rqpp W W W This can be rewritten as: ( ) βαα β α tansincos cos rp q q W +−−= Wrrp +−= ααβ cossin ( ) ( ) ( )( ) ( ) β βαα ββββααβαβαα cos sinsincos tantansincossincossincossincos W WW qrp qrpqrpp ++ = +++=−++=  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
42 SOLO Coordinate Systems (continue – 21) 6.Wind Coordinates (W( (continue -6( We have also: Angular Velocities (continue – 4( ( ) βαα β α tansincos cos RP Q Q W +−−= WRRP +−= ααβ cossin ( ) β βαα cos sinsincos W W QRP P ++ = AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
43 SOLO Coordinate Systems (continue – 22) 6.Wind Coordinates (W( (continue -7( The vehicle velocity was decomposed in: Vehicle Velocity WAE VVV  += ( )           = 0 0 V V W A  - vehicle velocity relative to atmosphere ( ) ( )           =           = DownW EastW NorthW W L zW yW xW W W V V V C V V V HLongLatV W W W _ _ _ ,,  - wind velocity at velocity position also ( ) [ ] ( ) [ ]           =           −=−= 0 0 0 011 * VV VV W A W A σσ  ( ) ( )           =           = DownW EastW NorthW W L zW yW xW W W V V V C V V V HLongLatV W W W _ _ _ * * * * * ,,  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
44 SOLO Coordinate Systems (continue – 23) 6.Wind Coordinates (W( (continue -8( The vehicle acceleration in W* coordinates is Vehicle Acceleration ( ) ( ) ( ) WAELALW W A WWIW W W AAIW W A I C aRVV td Vd VV td Vd RVV td Vd td Vd a        +×Ω×Ω+×Ω++×+= ×Ω+×++×Ω×Ω+×Ω+×+== ←← ←← 2* * * * * * ωω ωω from which ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )******* * * * 2 W W W A WW EL WW A W LW W W A aVAV td Vd   −×Ω+−=×+         ←← ωω where ( )RaA  ×Ω×Ω−=: AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
45 SOLO Coordinate Systems (continue – 24) 6.Wind Coordinates (W( (continue -9( Vehicle Acceleration (continue – 1( ( ) ( ) ( ) ( ) ( ) ( )           −                     Ω+Ω+− Ω+−Ω+ Ω+Ω+− −           =                     − − − +           ** * * **** **** **** * * * ** ** ** 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 ρρ ρρ ρρ where ( ) ( ) ( ) ( )HR Lat Lat C a a a A A A A W L zW yW xW zW yW xW W +Ω           −           =           = 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 −Ω++=− −Ω+−= −= ρ ρ  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
46 SOLO Coordinate Systems (continue – 25( 6.Wind Coordinates (W) (continue -10) Vehicle Acceleration (continue – 2) Using ( )          − =           =← γχ γ γχ ω cos sin * * * * *     W W W W LW R Q P we have ** xWWxW aAV −= ( ) γρχ cos/2 ** **       Ω+− − = zWzW yWWyW V aA  ( )** ** 2 yWyW zWWzW V aA Ω+− − −= ργ AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
47 SOLO Aerodynamic Forces ( )[ ]∫∫ +−= ∞ WS A dstfnppF  11 ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − ( ) airflowingthebyweatedsurfaceVehicleS SsurfacetheonmNstressforcefrictionf Ssurfacetheondifferencepressurepp W W W − − −−∞ )/( 2 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE 7. Forces Acting on the Vehicle
48 SOLO 7. Forces Acting on the Vehicle (continue – 1) Aerodynamic Forces (continue – 1) ( )           − − − = L C D F W A  ForceLiftL ForceSideC ForceDragD − − − L C D CSVL CSVC CSVD 2 2 2 2 1 2 1 2 1 ρ ρ ρ = = = ( ) ( ) ( ) tCoefficienLiftRMC tCoefficienSideRMC tCoefficienDragRMC eL eC eD − − − βα βα βα ,,, ,,, ,,, ityvisdynamic lengthsticcharacteril soundofspeedHa numberynoldslVR numberMachaVM e cos )( Re/ / − − − −= −= µ µρ AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
49 SOLO 7. Forces Acting on the Vehicle (continue – 2) Aerodynamic Forces (continue -2) ∫∫       ⋅+⋅−= ∫∫       ⋅+⋅−= ∫∫       ⋅+⋅−= ∧∧ ∧∧ ∧∧ W W W S fpL S fpC S fpD dswztCwznC S C dswytCwynC S C dswxtCwxnC S C 1ˆ1ˆ 1 1ˆ1ˆ 1 1ˆ1ˆ 1 Wf Wp Ssurfacetheontcoefficienfriction V f C Ssurfacetheontcoefficienpressure V pp C −= − − = ∞ 2/ 2/ 2 2 ρ ρ ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
50 ( ) ( ) ( )         MomentFriction S C Momentessure S CCA WW dstRRfdsnRRppM ∫∫∫∫∑ ×−+×−−= ∞ 11 Pr / Aerodynamic Moments Relative to C can be divided in Pressure Moments and Friction Moments. ( )       FrictionSkinor FrictionViscous S essureNormal S A WW dstfdsnppF ∫∫∫∫∑ +−= ∞ 11 Pr Aerodynamic Forces can be divided in Pressure Forces and Friction Forces. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE 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 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
52 SOLO 7. Forces Acting on the Vehicle (continue – 3) Thrust ( ) ( )                     − −−== B B B z y x BW B W T T T TCT αα βαββα βαββα 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 σσ σσσ  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE F-35Thrust Vector Control
53 SOLO 7. Forces Acting on the Vehicle (continue – 4) Gravitation Acceleration ( ) ( )                         −           −           − == zg yg xg gg 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ cs sc cs sc cs scC EW E W  ( ) gg          − = γσ γσ γ cc cs s W  2sec/174.322sec/81.9 0 2 0 0 0 gg ftmg HR R == + =           AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth. The Gravitational Potential U (R, ( is given byϕ ( ) ( ) ( ) ( )φ φ µ φ , sin1, 2 RUg P R a J R RU E E n n n n ∇=               −⋅−= ∑ ∞ =  μ – The Earth Gravitational Constant a – Mean Equatorial Radius of the Earth R=[xE 2 +yE 2 +zE 2 ]]/2 is the magnitude of the Geocentric Position Vector – Geocentric Latitude (sin =zϕ ϕ E/R( Jn – Coefficients of Zonal Harmonics of the Earth Potential Function P (sin ( – Associated Legendre Polynomialsϕ
54 SOLO 7. Forces Acting on the Vehicle (continue – 5) Gravitation Acceleration AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Retaining only the first three terms of the Gravitational Potential U (R, ( we obtain:ϕ R z R z R z R a J R z R a J R g R y R z R z R a J R z R a J R g R x R z R z R a J R z R a J R g EEEE z EEEE y EEEE x E E E ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 2 2 4 44 42 22 22 2 2 4 44 42 22 22 2 2 4 44 42 22 22 µ µ µ φ φλ φλ sin cossin coscos = ⋅= ⋅= R z R y R x E E E ( ) 2/1222 EEE zyxR ++=
55 SOLO 7. Forces Acting on the Vehicle (continue – 6) Force Equations Air Vehicle Acceleration ( ) ( ) WAELALW W A I C aRVV td Vd td Vd a    +×Ω×Ω+×Ω++×+== ←← 2ωω ( ) ( ) ( ) WAELALW W A A aRVV td Vd amTF m     +×Ω×Ω+×Ω++×+==++ ←← 2 1 g ωω ( )Rg   ×Ω×Ω−= g:Define                   + −− + −− − − =           γσ α γσ βα γ βα ccg m LT csg m CT sg m DT A A A zW yW xW sin sincos coscos          − +                   −− −− −           −=           γ γ α βα βα σσ σσ cg sg m LT m CT m DT A A A zW yW xW 0 sin sincos coscos cossin0 sincos0 001 * * * AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content ( )           = 0 0 T T B 
56 SOLO 23. Local Level Local North (LLLN( Computations for an Ellipsoidal Earth Model ( ) ( ) ( ) ( ) ( )2 22 10 2 0 2 0 2 0 5 2 1 2 0 6 0 sin sin1 sin321 sin1 sec/10292116557.7 sec/051646.0 sec/780333.9 26.298/.1 10378135.6 Ae e p m e HR RLatgg g LatfRR LatffRR LatfRR rad mg mg f mR + + = +≈ +−≈ −≈ ⋅=Ω = = = ⋅= − Lat HR V HR V HR V Ap East Down Am North East Ap East North tan + −= + −= + = ρ ρ ρ Lat Lat Down East North sin 0 cos Ω−=Ω =Ω Ω=Ω DownDownDown EastEast NorthNorthNorth Ω+= = Ω+= ρφ ρφ ρφ East North Lat Lat Long ρ ρ −= = • • cos ( ) ( ) ∫ ∫ • • += += t t dtLatLattLat dtLongLongtLong 0 0 0 0 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE SIMULATION EQUATIONS
57 SOLO AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE SIMULATION EQUATIONS Table of Content
58 SOLO Missile Kinematics Model 1 in Vector Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
59 SOLO Missile Kinematics Model 1 in Matrix Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
60 SOLO Missile Kinematics Model 2 in Vector Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
61 SOLO Missile Kinematics Model 2 in Matrix Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
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 Velocity Vector and then Computes its Position and Velocity based on Measured or Estimated Velocity Vector and Elapsed Time. Dead Reckoning Evolution of a Vehicle’s Position Based on Velocity Vector
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 Rate and 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
SOLO 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 System Inputs ADC.Angle_of_Attack ADC.Mach_Number ADC.Barometric_Altitude ADC.Magnetic_Heading ADC.True_Airspeed INS.Body_Rates (roll, pitch, yaw( INS.Acceleration (lateral, longitudinal, normal( INS.Present_Position (latitude, longitude( INS.True_Heading INS.Velocity (north, east, vertical( RALT.Radar_Altitude Outputs INS.Reference_Velocity (north, east, vertical( NAV.Airspeed NAV.Rate_of_Change_Airspeed NAV.Position (latitude, longitude, altitude( NAV.Angle_of_Attack NAV.Attitude (roll, pitch, yaw( NAV.Body_Rates (roll, pitch, yaw( NAV.Flight_Path_Angle NAV.Ground_Speed NAV.Ground_Track_Angle NAV.Magnetic_Variation NAV.Altitude NAV.Velocity (north, east, vertical( NAV.Acceleration (lateral, longitudinal, normal( NAV.Wind (direction, magnitude( NAV.Body_to_Earth_Transform NAV.Body_to_Horizon_Transform NAV.Radar_to_Body_Transform NAV.Radar_to_Earth_Transform
SOLO 68 Navigation 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 Czω
SOLO 69 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( The Dynamic Equation along Rate-Integrated-Gyro (RIG( Output Axis YP is: Platform Stabilization Around ZP Azimuth Axis (continue – 1( ( ) θωωθ  GDCzGyG BTTHJ PP −−=++ ( ) ( )       − − −=+ PP y G G G DC zGGG H J H TT HsBJss ωωθ  JG – RIG Moment of Inertia around Output Axis θ – Pickoff Angle - Platform Angular Acceleration around YP Axis - Platform angular Rate around ZP Axis HG – Gyro Angular Moment TC – RIG Torque Command TD – Disturbance Moment BG - Damping Coefficient Pyω Pzω Tacking Laplace Transform and rearranging:
SOLO 70 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 2( ( ) ( )       − − −=+ PP y G G G DC zGGG H J H TT HsBJss ωωθ  Define: ( ) Cz G C k H T ω∆+= 1: Angular Rate Command (Δk –Scaling Error( G G D H T ε=: Gyro Bias G G G B J τ=: RIG Time Constant ( ) ( ) ( )       −−∆+− + −= PCP y G G Gzz GG G H J k ssB H s ωεωω τ θ 1 1 1
SOLO 71 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 3( The Pickoff Signal θ, is the Feedback Command to the Azimuth Torque Motor ( ) ( ) ( )ssKKsT Cz θ12= K1(s( - Filter and Torque Driver ( ) fzxxyxzz TTJJJ CPPPPPP −=−− ωωω K2 - Torque Motor Gain The Moment Equation along Platform ZP Axis is:
SOLO 72 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 4( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D GGxGx G y G G Gz GGxGx GG z T BHsKsKsJsJ ss H J k BHsKsKsJsJ BHsKsK PP CC PP P / 1 1 / / 1 23 1 23 1 ++ + −      −+∆+ ++ = τ τ ωεω τ ω  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D Gx GG x y G G Gz Gx GG Gx GG z T ssJ BH sKsK sJ H J k ssJ BH sKsK ssJ BH sKsK P P CC P P P 1 / 1 1 1 1 / 1 1 / 21 21 21 + + −       −+∆+ + + + = τ ωεω τ τ ω  or From the Figure above we obtain:
SOLO 73 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 5( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D GGxGx G y G G Gz GGxGx GG z T BHsKsKsJsJ ss H J k BHsKsKsJsJ BHsKsK PP CC PP P / 1 1 / / 1 23 1 23 1 ++ + −      −+∆+ ++ = τ τ ωεω τ ω  At Steady-State we obtain: ( ) ( ) ( ) ( ) ( )D s GG y G G Gzz Ts BHsKKH J kt CCP 0 1 lim /0 1 1 → −−+∆+=∞→ ωεωω  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 ω ( ) ( ) Gzz CP kt εωω +∆+≈∞→ 1 Therefore the Misalignment Errors of the Platform are due to Gyros Drift and Scaling Error. Both can be measured (off-line( and compensated by Navigation Computer.
SOLO 74 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs(
SOLO 75 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( 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 G P AC ψψ ψψ θθ θθ φφ φφψθφ We want to apply Moments on the Platform, related to the Pjckoff Outputs of the Three RIGs mounted on the Platform ( ) ( )           =           = z y x z y x P KsK T T T T P P P θ θ θ 21  The 3 Torque Motors Roll (TR(, Pitch (TP( and Heading (TH( are located on Gimbal Axes .PA zyx 1,1,1 ' 
SOLO 76 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( We want to find the relation between and TR, 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 '  ++=++= ( ) [ ] [ ] [ ] ( )  [ ] [ ] ( )  [ ] ( )  P Pzy A Ax 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 , '             −+           −−+           −−−=           = ψθψφθψ                     − − =           H P R GG GG z y x T T T T T T P P P 10sin 0coscossin 0sincoscos θ ψθψ ψθψ                     −=           P P P z y x GG GG GG H P R T T T T T T 1tansintancos 0cossin 0cos/sincos/cos θψθψ ψψ θψθψ
SOLO 77 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( To simplify the implementation the assumption of small 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 θψθψ ψψ θψθψ                     −≈           P P P z y x GG GG H P R T T T T T T 100 0cossin 0sincos ψψ ψψ ( ) ( )           =           = z y x z y x P KsK T T T T P P P θ θ θ 21  where:
SOLO 78 Navigation
SOLO 79 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations 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 by the three small angles ψx, ψy, ψz as [ ] [ ] [ ]           −           −           − == 100 0cossin 0sincos cos0sin 010 sin0cos cossin0 sincos0 001 321 zz zz yy yy xx xxzyx P CC ψψ ψψ ψψ ψψ ψψ ψψψψψ           − − − −           =           − − − ≈           −           −           − ≈ 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 ψψ ψψ ψψ ψψ ψψ ψψ ψ ψ ψ ψ ψ ψ [ ] [ ] [ ]           − − − =×           =×−≈ 0 0 0 :&: xy xz yz z y x P C IC ψψ ψψ ψψ ψ ψ ψ ψ ψψ 
SOLO 80 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 1( Let find the angular rotation vector from C to P ( ) [ ] [ ] [ ] =                               +           +           =← z zyy x x P CP ψ ψψψ ψ ψω     0 0 0 0 0 0 321 ψ ψ ψ ψ ψ ψψψ ψψψ ψψψ ψψ ψψ ψ ψ ψψ           =           ≈           + − =                     − − − +                     − +           ≈ 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 Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 1( 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 ( ) ( ) ( ) ( )P G C ICG P IP KI εωω  ++= ←← Platform Rate Commands Vector           ∆ ∆ ∆ = 33231 23221 13121 G G G G Kmm mKm mmK K Matrix of Gyros Scaling Errors, Misalignments and Mass-Unbalances ( ) PPP zzyyxx P G 111  εεεε ++= Gyro Drift Vector Computer Rate Commands VectorCCCCCC zzyyxxIC 111  ωωωω ++=←
SOLO 82 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 2( Let find the angular velocity vector of the Platform (P( relative to the Computer (C(: ICIPCP ←←← −= ωωω  ( ) ( ) ( ) ( ) ( )C IC P C P IP P IC P IP P CP C ←←←←← −=−= ωωωωω  ( ) ( ) ( ) [ ]{ } ( ) [ ] ( ) ( ) ( )P G C ICG C IC C IC P G C ICG KIKI εωωψωψεωψ  ++×=×−−++= ←←←← Using we obtain:[ ] ( ) ( ) [ ] ψωωψ  ×−=× ←← C IC C IC ( ) [ ] ( ) ( )P G C ICG C IC K εωψωψ  ++×−= ←← or           +                     ∆ ∆ ∆ +                     − − − −=           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 ε ε ε ω ω ω ψ ψ ψ ωω ωω ωω ψ ψ ψ 33231 23221 13121 0 0 0   
SOLO 83 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations - vector representing position from the Earth Center of mass to the Vehicle r  ( )rgAr   += - Ideal Accelerometers Measurement VectorA  ( ) ( ) r rr K r r K rg    2/33 ⋅ −=−= ( )rg  - Gravity Vector ( )rrgAArr   δδδ +++=+ For Non-Ideal Accelerometers we have a error between Real Position and Computed Position r  δ ( ) ( )rgrrgAr   −++= δδδ
SOLO 84 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 1( ( ) ( )rgrrgAr   −++= δδδ ( ) ( ) ( ) ( )[ ] r r K r rrrr K rgrrg    32/3 + +⋅+ −=−+ δδ δ ( ) ( ) r r K r rr rr r K r r K r rrr K     32332/32 31 2 +      ⋅ −+−≈+ ⋅+ −≈ δ δ δ r r K r r r r r r K r r K r r K    3333 +      ⋅+−−≈ δδ therefore Ar r r r r r r K r    δδδδ =            ⋅−+ 33
SOLO 85 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 2( Define r g r K S == 3 :ω Maximilian Schuler (1882 – 1972( S ST ω π2 = Shuler Period = 84.4 minutes at Sea Level Ar r r r r rr S    δδδωδ =            ⋅−+ 32
SOLO 86 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 3( 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 ( ) ( ) ( ) ( )PP f C C bAKIA  δ++= 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 zzyyxx P AAAA 111  ++= ( ) PPPPPP zzyyxx P bbbb 111  δδδδ ++= Accelerometers Biases Vector
SOLO 87 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 4( AAA C  −=:δ Accelerometers Error Vector ( ) ( ) ( ) ( ) ( ) ( ) [ ]( ) ( )PPP f PC P C C C AIbAKIACAA  ×+−++=−= ψδδ We used the relation ( ) [ ]( ) [ ]( )×+≈×−== −− ψψ  IICC P C C P 11 Finally we obtain ( ) [ ] ( ) ( ) ( )PP f PC bAKAA  δψδ ++×−= [ ] ( ) ( ) ( )PP f P S bAKAr r r r r rr    δψδδωδ ++×−=            ⋅−+ 32 The Position Error Equation is
SOLO 88 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 5( Let compute ( )C r δ ( ) ( ) ( ) [ ] ( )CC IC CC rrr   δωδδ ×+= ← Therefore ( ) ( ) ( ) ( ) ( ) CCCCCC CCC C IC C CCC CCC CCCCCC CCC zzyyxxIC C ICzyxICzyx zyx C zyxzyx C zyx C rzyxzyx zyxr zyxzyxr zyxr 111 111111 111: 111111 111 11                   ωωωω δωδδδωδδδ δδδδ δδδδδδδ δδδδ αα ω ++= ×=++×=++ ++= +++++= ++= ← ←← ×= ← In the same way ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( )CC IC CC IC CC IC C IC C IC CC IC CC rr rrrr               δωδω δωωωδωδδ ×+×+         ×         ×++×+= ←← ←←←← 0
SOLO 89 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 6( ( ) ( ) ( ) [ ] ( )CC IC CC rrr   δωδδ ×+= ← Therefore ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ]( ) ( )CC IC C IC C IC CC IC CC rrrr       δωωωδωδδ ××+×+×+= ←←←←2 ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ]( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( )PP f P C CC C S CC IC C IC C IC CC IC C bAKA r r r r r rrrr          δψ δδωδωωωδωδ ++×−=             ⋅−+××+×+×+ ←←←← 32 2 Together with the Platform Misalignment Error Equations ( ) [ ] ( ) ( )P G C ICG C IC K εωψωψ  ++×−= ←←
SOLO 90 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 7( CCCCCC zzyyxxIC 111  ωωωω ++=← ( ) [ ]           − − − =×← 0 0 0 CC CC CC xy xz yz C IC ωω ωω ωω ω  ( ) [ ]           − − − =×← 0 0 0 CC CC CC xy xz yz C IC ωω ωω ωω ω      ( ) [ ] ( ) [ ] ( ) ( ) ( )            +− +− +− =           − − −           − − − =×× ←← 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 C IC C IC ωωωωωω ωωωωωω ωωωωωω ωω ωω ωω ωω ωω ωω ωω  Czrr 1  = ( ) ( ) ( ) ( )           − =           −           =      ⋅− z y x z z y x r r r r r r C C C C δ δ δ δ δ δ δ δδ 21 0 0 33    
SOLO 91 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 8) ( ) ( ) ( )                       +−−+− −+−+ +−+− +                     − − − +           z y x z y x z y x CCCCCCCC CCCCCCCC CCCCCCCC CC CC CC yxSxzyyzx xzyzxSzyx yzxzyxzyS xy xz yz δ δ δ ωωωωωωωωω ωωωωωωωωω ωωωωωωωωω δ δ δ ωω ωω ωω δ δ δ 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 δ δ δ ψψ ψψ ψψ 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 ε ε ε ω ω ω ψ ψ ψ ωω ωω ωω ψ ψ ψ 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
93
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 Navigation Inertial Navigation Systems
SOLO 97 Navigation Inertial Navigation Systems
98 SOLO Strapdown Algorithm (Vector Notation) Navigation
99 SOLO Strapdown Algorithm Navigation
SOLO 100 Navigation Inertial Navigation Systems Magnetic Compass
SOLO 101 Navigation Inertial Navigation Systems Gyrocompass
SOLO 102 Navigation Radar Altimeter
SOLO 103 Navigation Externally Navigation Add Systems eLORAN LORAN - C Global Navigation Satelite System (GNSS) Distance Measuring Equipment (DME) VHF Omni Directional Radio-Range (VOR) System Data Base Matching Terrain Referenced Navigation (TRN) Navigation Multi-Sensor Integration
SOLO 104 Navigation Global Navigation Satelite System (GNSS) Satellites of the GPS GLONASS and GALILEO Systems Four Satellite Navigation Systems have been designed to give three dimensional Position, 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.
SOLO 105 Navigation Global Navigation Satelite System (GNSS)
SOLO 106 Navigation Global Navigation Satelite System (GNSS)
SOLO 107 Navigation Global Navigation Satelite System (GNSS)
SOLO 108 Navigation Global Navigation Satelite System (GNSS) 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) SOLO 109 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 SOLO 110 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 SOLO 111 Navigation GZ GX GY Equatorial Plane εY εZ εX Ascending Node Satellite Orbit Periapsis Direction Vernal Equinox Direction Ω ω i → N1 Θ 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 Θ= 2 rh we can write h Ad h dr dt 2 2 = Θ = where is the area defined by the radius vector as it moves through an angle 2 2 Θ = dr Ad Θd Θ pΘ pΘ−Θ r focus conic section x y → P1 → Q1 → r1 → t1 v  rv tv Θd Θ= drAd 2 2 1 periapsis This proves the 2nd Kepler’s Law that equal area are swept out equal in equal times by the radius vector.
114 SOLO KEPLERIAN TRAJECTORIES 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 ba T eap ph µ ππππ µ 2/3122/322 2 1 2 1 22 2 = −= = − = − == or 2/3 2 aT µ π = 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 equal To the cube of its mean distance to the sun”.
115 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 2) Let draw an auxiliary circle of radius a, and the same center O as the geometric center of the ellipse. x y eac = a a ( ) 2/12 1 eab −= r Θ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Let take any point P on the ellipse with polar coordinates r,Θ and define the point Q on the circle at the same coordinate x as P. Eeary r a x ea a x by Ea a x ay ellipse ellipse circle sin1sin sin111 sin1 2 2 2 2 2 2 2 2 −=Θ=→        Θ=−−=−= =−= The angle E of OQ with x axis is called the eccentric anomaly. aeEarxellipse −=Θ= coscos
116 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 3) Let compute x y eac = a a ( ) 2/12 1 eab −= r Θ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P ( ) ( )( ) ( )( ) ( ) 0cos11 sin1sincos1cos 1 2 22 2 >←−−= −+−−= −=×=−= EEEeea EEeaEaEEeaaeEa xyyxvreah ellipseellipseellipseellpse     µ We obtain ( ) n a EEe ==− :cos1 3 µ ( ) ( ) ( )pttntEetE −=− sin Integrating this equation gives Kepler’s Equation where tp is the time of periapsis ( E (tp) = 0 )
117 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 4) From x y eac = a a ( ) 2/12 1 eab −= rΘ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Eeary aeEarx ellipse ellipse sin1sin coscos 2 −=Θ= −=Θ= we have ( ) ( )[ ] [ ] ( )Eea EeEeaEeaaeEar cos1 coscos21sin1cos 2/1222/12222 −= +−=−+−= Therefore Θ+ Θ− = − − =Θ Θ+ Θ+ = − − =Θ cos1 sin1 sin cos1 sin1 sin cos1 cos cos cos1 cos cos 22 e e E Ee Ee e e E Ee eE ( )( ) Ee Ee Ee eEEe sin1 cos11 sin1 coscos1 sin cos1 2 tan 22 − −+ = − +−− = Θ Θ− = Θ From 2 tan 1 1 2 tan E e e − + = Θ or and are always in the same quadrant.2 Θ 2 E
118 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 5) We have x y eac = a a ( ) 2/12 1 eab −= rΘ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Eeary aeEarx ellipse ellipse sin1sin coscos 2 −=Θ= −=Θ= and ( )Eear cos1−= The Position Vector of the Satellite is ( )             − − =           Θ Θ =           = += 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 1 0 cos sin cos1 0 cos1 sin 0 cos1 sin 0 cos1 sin e an e Ee an Ee E EanEe E Ee aeE q td d Orbit Orbit − ⋅ ⋅           Θ+ Θ− = ⋅− ⋅ ⋅             − − =⋅⋅             − − =             − −− = 
GPS Broadcast Ephemerides SOLO 119 Navigation The Satellite Position can be computed as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )oeisic rsrc usuc oe oe ttidotuCuCii uCuCrr uCuC tt ttnnMM a n −⋅+⋅+⋅+= ⋅+⋅+= ⋅+⋅+= −⋅Ω+Ω=Ω −⋅∆++= = 000 000 000 0 0 3 2sin2cos 2sin2cos 2sin2cosωω µ  where: ( ) Θ+=         ⋅ − + ⋅=Θ ⋅−⋅= ⋅+= − 00 1 0 2 tan 1 1 tan2 cos1 sin ωu E e e Eear EeME Six Keplerian Elements Define the Satellite Posision (Ω, I, ω, a, e, M0) where M0 = n (t – tP)
GPS Broadcast Ephemerides SOLO 120 Navigation ( ) Θ+=           =           = ωuur ur y x q ellipse ellipse Orbit 0 sin cos 0  ( ) 2 1 0 cos sin 0 e an ue u y x q ellipse ellipse Orbit Orbit − ⋅ ⋅           + − =           =    ( ) oecoec ttt ⋅+−⋅=Θ ωω [ ] [ ] [ ]           −−Ω−=           =           0 sin cos 0 313 ur ur iy x C z y x ellipse ellipse G G ωε Θ+= ωu
121 GPS Broadcast Ephemerides SOLO Navigation
Global Positioning System SOLO 122 Navigation - x, y, z Satellite Coordinate in Geocentric-Equatorial Coordinate System ( ) ( ) ( )222 ZzYyXx −+−+−=ρ - 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  ρ 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 ρ where r = Earth Radius This 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..
Global Positioning System SOLO 123 Navigation
Global Positioning System SOLO 124 Navigation Using data from four Satellites we obtain ( ) ( ) ( ) ( ) 444444 22 4 2 4 2 4 2 4 333333 22 3 2 3 2 3 2 3 222222 22 2 2 2 2 2 2 2 111111 22 1 2 1 2 1 2 1 222 222 222 222 ZzYyXxCrrrzyx ZzYyXxCrrrzyx ZzYyXxCrrrzyx ZzYyXxCrrrzyx r r r r ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ρ ρ ρ ρ or ( ) ( ) ( ) ( )      14 1444 22 4 2 4 2 4 2 4 22 3 2 3 2 3 2 3 22 2 2 2 2 2 2 2 22 1 2 1 2 1 2 1 444 333 222 111 1222 1222 1222 1222 x xx R r r r r PM rzyx rzyx rzyx rzyx Crr Z Y X zyx zyx zyx zyx               −++− −++− −++− −++− =                         ⋅−⋅−⋅− ⋅−⋅−⋅− ⋅−⋅−⋅− ⋅−⋅−⋅− ρ ρ ρ ρ 14 1 4414 xxx RM Crr Z Y X P − =             =
Global Positioning System SOLO 125 Navigation
Global Positioning System SOLO 126 Navigation
Global Positioning System SOLO 127 Navigation GPS Satellite GPS Control Station
Global Positioning System SOLO 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 code ompseudorand AC ulation bps power carrier I ttctdPts −− +⋅⋅⋅⋅= 1/ mod 50 cos2 θω
Global Positioning System SOLO 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 code ompseudorand P ulation bps power carrier Q ttptdPts −− +⋅⋅⋅⋅= 3,2,1 mod 50 sin2 θω
Global Positioning System SOLO 130 Navigation
GPS Signal Spectrum SOLO 131 Navigation
Global Positioning System SOLO 132 Navigation GPS User Segment (GPS Receiver)
Global Positioning System SOLO 133 Navigation GPS User Segment (GPS Receiver)
GPS vs GALILEO SOLO 134 Navigation GALILEO GPS Satelites 27 + 3 24 (32!) Planes 3 6 Satellite per Plane 10 4 - 7 Plane Spacing 120 ͦ 60 ͦ Inclination 56 ͦ 55 ͦ Orbit Type MEO Circular MEO Circular Orbit Radius 29,500 km 26,500 km Period 141 /4 hour 12 hour Satellite Ground Track Repetition 10 days 1 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 Navigation Differential GPS Augmented Systems
SOLO 151 Navigation Differential GPS Augmented Systems
SOLO 152 Navigation
GNSS Aviation Operational Performance Requirements SOLO 153 Navigation
SOLO 154 Navigation Externally Navigation Add Systems 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.
SOLO 155 Navigation Externally Navigation Add Systems 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
SOLO 156 Navigation Externally Navigation Add Systems 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,
SOLO 157 Navigation Externally Navigation Add Systems 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 carrier Frequency Modulated (FM) with: 30 Hz signal from the main antenna. An Amplitude Modulated (AM) carrier electrically swept around several smaller Antennas surrounding the main Antenna. This rotating pattern creates a 30 Hz Doppler effect on the Receiver. The Phase Difference of the two 30 Hz signals gives the User’s Azimuth with respect to the North from the VOR Site. The Bearing measurement accuracy of a VOR System is typically on the order of 2 degrees, with a range that extends from 25 to 130 miles.
SOLO 158 Navigation Externally Navigation Add Systems TACAN is the Military Enhancement of VOR/DME VHF Omni Directional Radio-Range (VOR) System TACAN (Tactical Air Navigation) is an enhanced VOR/DME System designed for Military 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.
SOLO 159 Navigation Data Base Matching
SOLO 160 Navigation Terrain Referenced Navigation (TRN)
SOLO 161 Navigation Terrain Referenced Navigation (TRN)
SOLO 162 Navigation Externally Navigation Add Systems
SOLO 163 Navigation Navigation Multi-Sensor Integration Navigation Data
164 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
SOLO 165 Navigation World Geodetic System (WGS 84) Geoid - Mean Sea Level of the Earth Reference Ellipsoid – Approximation of Sea Level Reference Earth Model h - Vehicle Altitude (the distance from the Vehicle to Ellipsoid along the Normal to Ellipsoid RN - 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 of N 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 the World Geodetic System of 1984 (WGS 84).
SOLO 166 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 ba f − =: Alexis Claude Clairaut )1713–1765(             −+= φ2 sin 2 5 1 fmGg
SOLO 167 Navigation World Geodetic System (WGS 84) Reference Earth Model Carlo Somigliana (1860 –1955) The Theoretical Gravity on the surface of the Ellipsoid is given by the Somigliana Formula (1929)    84 22 2 2222 22 sin1 sin1 sincos sincos WGS e pe e k ba ba φ φ γ φφ φγφγ γ − + = + + = where 1: −= e p a b k γ γ 2 22 : a ba e − = - 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 – Geodetic Latitudeϕ The Theory of the Equipotential Ellipsoid was first given by P. Pizzetti (1894)
SOLO 168 Navigation World Geodetic System (WGS 84) Reference Earth Model 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.
SOLO 169 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) Reference Earth Model 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.
SOLO 170 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 m Minimum = -106.99 m, Maximum = 85.39 m
SOLO 171 Navigation World Geodetic System (WGS – 84) Reference Earth Model Parameters Notation Value Ellipsoid Semi-major Axis a 6.378.137 m Ellipsoid Flattening (Ellipticity) f 1/298.257223563 (0.00335281066474) Second Degree Zonal Harmonic Coefficient of the Geopotential C2,0 -484.16685x10-6 Angular Velocity of the Earth Ω 7.292115x10-5 rad/s The Earth’s Gravitational Constant (Mass of Earth includes Atmosphere) GM 3.986005x1014 m3 /s2 Mass of Earth (Includes Atmosphere) M 5.9733328x1024 kg Theoretical (Normal) Gravity at the Equator (on the Ellipsoid) γe 9.7803267714 m/s2 Theoretical (Normal) Gravity at the Poles (on the Ellipsoid) γp 9.8321863685 m/s2 Mean Value of Theoretical (Normal) Gravity γ 9.7976446561 m/s2 Geodetic and Geophysical Parameters of the WGS-84 Ellipsoid
SOLO 172 Navigation World Geodetic System (WGS 84) Reference Earth Model a ba f − =:f - Ellipsoid Flattening (Ellipticity) a - Ellipsoid Semi-major Axis b - Ellipsoid Semi-minor Axis e - Ellipsoid Eccentricity 2 2 22 2 2: ff a ba e −= − = ( ) 2 11 eafab −=−= Reference Ellipsoid
SOLO 173 Navigation Reference Ellipsoid Ellipse Equation: 12 2 2 2 =+ b y a x Slope of the Normal to Ellipse: 2 2 tan bx ay yd xd =−=φ The Slope of the Geocentric Line to the same point x y =λtan λλ sincos RyRx == Deviation Angle between Geographic and Geodetic At Ellipsoid Surface λφ tantan 2 2 b a =       = − λφ tantan 2 2 1 b a ( ) φφλ tan1tantan 2 2 2 e a b −==
SOLO 174 Navigation Reference Ellipsoid Ellipse Equation: λφδ −= 12 2 2 2 =+ b y a x Slope of the Normal to Ellipse: 2 2 tan bx ay yd xd =−=φ The Slope of the Geocentric Line to the same point x y =λtan       −=       −+       − = + − = + − = 1 11 1 1 tantan1 tantan tan 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 ay x y bx ay λφ λφ δ λλ sincos RyRx == ( )  ( ) ( )λλλδ 2sin2sin 2 tan2sin 2 tan 1 2 11 12 22 22 1 f ba R b ba a ba R ba ba f ≈             +       − =            − = ≈≈<< −−  Deviation Angle between Geographic and Geodetic At Ellipsoid Surface
SOLO 175 Navigation Reference Ellipsoid For a point at a Height h near the Ellipsoid the value of δ must be corrected: u−= 1δδ From the Law of Sine we have: Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface ( ) R h hR huu ≈ + ≈= − 11 sin sin sin sin δδπ Since u and δ1 are small: 1δ R h u ≈ The corrected value of δ is: ( )λδδδ 2sin11 11 f R h R h u       −=      −=−= Therefore: ( )λλδλφ 2sin1 f R h       −+=+=
SOLO 176 Navigation World Geodetic System (WGS 84) where λ – Longitude e – Eccentricity = 0.08181919 Reference Earth Model In Earth Center Earth Fixed Coordinate –ECEF-System (E) the Vehicle Position is given by: ( ) ( ) ( ) ( )           + + + =           = φ φλ φλ sin cossin coscos HR HR HR z y x P M N N E E E E  ( ) NhH e a RN += − = 2/12 sin1 φ Another variable, used frequently, is the radius of the Ellipsoid referred as the Meridian Radius ( ) ( ) 2/32 2 sin1 1 φe ea RM − − =
SOLO 177 Navigation Reference Ellipsoid Let develop the RN and RM: Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface Ellipse Equation: ( ) 222 2 2 2 2 11 aeb b y a x −==+ From this Equation, at any point (x,y) on the Ellipse, we have: φtan 1 2 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: ( ) φ φ φ 2 22 2 2 2 222 2 2 2 2 2 2 2 2 cos sin1 1 1 tan1111 e a x e e a x b a x y a x − =      − −+=      += ( ) ( ) ( ) 2/122 2 2 2 2/122 sin1 sin1 tan sin1 cos φ φ φ φ φ e ea x a b y e a x − − ==→ − = From the Figure above: ( ) 2/122 sin1cos φφ e ax RN − ==
SOLO 178 Navigation Reference Ellipsoid Let develop the RN and RM (continue): Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface we have at any point (x,y) on the Ellipse: φtan 1 2 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 tan 1 11 : φφ φφ e ea e ea ea xd yd xd yd RM − − = − − −       + =               + = ( ) ( ) ( ) 2/122 2 2 2 2/122 sin1 sin1 tan sin1 cos φ φ φ φ φ e ea x a b y e a x − − ==→ − = ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − =
SOLO 179 Navigation Reference Ellipsoid Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − = ( ) 2/122 sin1cos φφ e ax RN − == a ba f − =: 2 2 22 2 2: ff a ba e −= − =Using ( ) ( )[ ] ( ) ( ) [ ] ++−≈      +−++−≈ −− − = φφ φ 2222 2/322 2 sin321sin2 2 3 121 sin21 1 : ffaffffa ff fa RM ( )[ ] ( ) [ ]φφ φ 222 2/122 sin31sin2 2 3 1 sin21 faffa ff a RN +≈    +−+≈ −− =  [ ]φ2 sin321 ffaRM +−≈ [ ]φ2 sin31 faRN +≈ We used and we neglect f2 terms ( ) ( ) + − ++= − !2 1 1 1 1 nn xn x n
SOLO 180 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 ( ) isϕ derived in the above references as The definition of geodetic (or geographic) and geocentric latitudes ( ) ( )[ ]φφψ tan1tan 21 e−= −

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4 navigation systems

  • 3. NavigationSOLO Aircraft Steering to Waypoints 1. T-HDG – True Heading 2. M-HDG – Magnetic Heading 3. T-TK - True Track 4. M-TK - Magnetic Track Angle 5. TKE – Track Angle Error 6. T-DTK – True Desired Track 7. XTK – Cross-Track Distance 8. DIS – Distance to Destination 9. GS - Ground Speed 10. WS – Wind Speed 11. WD – Wind Direction 12. TAS – True Airspeed 13. DA – Drift Angle In order to minimize Fuel, Time and Distances the Aircraft will tend to fly between Waypoints, on the Earth Surface, on the Great Circle connecting the Initial and Final Waypoints, 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 an Automatic Flight-Control System (AFCS).
  • 5. 5 Spherical TrigonometrySOLO Assume three points on a unit radius sphere, defined by the vectors →→→ CBA 1,1,1 Laws of Cosines for Spherical Triangle Sides ab abc ca cab bc bca ˆsinˆsin ˆcosˆcosˆcos ˆcos ˆsinˆsin ˆcosˆcosˆcosˆcos ˆsinˆsin ˆcosˆcosˆcos ˆcos − = − = − = γ β α Law of Sines for Spherical Triangle Sides. cba abccba cba ˆsinˆsinˆsin ˆcosˆcosˆcos2ˆcosˆcosˆcos1 ˆsin ˆsin ˆsin ˆsin ˆsin ˆsin 222 +−−− === γβα The three great circles passing trough those three points define a spherical triangle with CBA ,, - three spherical triangle vertices cba ˆ,ˆˆ -three spherical triangle side angles γβα ˆ,ˆˆ - three spherical triangle angles defined by the angles between the tangents to the great circles at the vertices.
  • 6. 6 SOLO Assume three points on a unit radius sphere, defined by the vectors →→→ CBA 1,1,1 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 vertices cba ˆ,ˆˆ -three spherical triangle side angles γβα ˆ,ˆˆ - three spherical triangle angles defined by the angles between the tangents to the great circles at the vertices. βα βαγ αγ αγβ γβ γβα ˆsinˆsin ˆcosˆcosˆcos ˆcos ˆsinˆsin ˆcosˆcosˆcosˆcos ˆsinˆsin ˆcosˆcosˆcos ˆcos + = + = + = c b a Spherical Trigonometry
  • 7. 7 NavigationSOLO 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 2 111 ,, λφR 222 ,, λφR The Great Circle Distance between two points 1 and 2 is ρ. The average Radius on the Great Circle is a = (R1+R2)/2 θρ ⋅= a R – radius - Latitudeϕ λ - Longitude kmNmNma 852.11deg/76.60/ =≈ρ
  • 8. 8 NavigationSOLO Flight on Earth Great Circles 1 2 111 ,, λφR 222 ,, λφR The Great Circle Distance between two points 1 and 2 is ρ. θρ ⋅= a R – radius - Latitudeϕ λ - Longitude ( ) ( ) ( ) ( ) ( ) ( )212121 cos90sin90sin90cos90cos /coscos λλφφφφ ρθ −⋅−⋅−+−⋅−= =  a From the Law of Cosines for Spherical Triangles or ( ) ( )212121 coscoscossinsin/cos λλφφφφρ −⋅⋅+⋅=a ( ){ }212121 1 coscoscossinsincos λλφφφφρ −⋅⋅+⋅⋅= − a The Initial Heading Angle ψ0 can be obtained using the Law of Cosines for Spherical Triangles as follows ( ) ( )a a /sincos /cossinsin cos 1 12 0 ρφ ρφφ ψ ⋅ ⋅− = ( )[ ] ( )[ ]2 222 22221 coscoscossinsin1cos coscoscossinsinsinsin cos λλφφφφφ λλφφφφφφ ψ −⋅⋅+⋅−⋅ −⋅⋅+⋅⋅− = − The Heading Angle ψ from the Present Position (R, ,λ) to Destination Point (Rϕ 2,ϕ2,λ2)
  • 9. 9 NavigationSOLO Flight on Earth Great Circles The Distance on the Great Circle between two points 1 and 2 is ρ. 1 2 111 ,, λφR 222 ,, λφR R – radius - Latitudeϕ λ - Longitude The Time required to travel along the Great Circle between points 1 and 2 is given by ( ){ } 22 212121 1 coscoscossinsincos yxHoriz HorizHoriz VVV V a V t += −⋅⋅+⋅⋅==∆ − λλφφφφ ρ ( ){ }212121 1 coscoscossinsincos λλφφφφρ −⋅⋅+⋅⋅= − a
  • 10. 10 NavigationSOLO Flight on Earth Great Circles 1 2 111 ,, λφR 222 ,, λφR 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. R – radius - Latitudeϕ λ - Longitude Using the Law of Cosines for Spherical Triangle APB we have ( ) ( )aaYd /sin 90sin /sin sin ρ ψ  = ∆ ( ) ( ) ( ) ( ) ( ) 2/sin/sin /cos/cos/cos 0ˆcos 21 90ˆ RR a aYaX aYaXa P dd dd P + = ⋅ ⋅− == = ρ  Using the Law of Sines for Spherical Triangle APB we have ( ) ( )      ⋅= − aY a aX d d /cos /cos cos 1 ρ ( )[ ]ψρ ∆⋅⋅= − sin/sinsin 1 aaYd
  • 11. 11 SOLO Coordinate Systems 1. Heliocentric (Heliocentric) Coordinate System COORDINATES IN THE SOLAR SYSTEM 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.
  • 12. 12 SOLO 1.Heliocentric (Heliocentric) Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM The Earth axis of rotation is tilted relative to Ecliptic and vobbles slightly, in a clockwise direction opposite to that of the Earth spin, from 22.1° to 24.5° , with a cycle of approximately 41,000 years. G Gz Ω Gx Gy Ecliptic plane normal (Ecliptic Pole) Locus of Lunar plane normal (Lunar Pole) Lunar Orbital Plane Earth Orbital Plane (Ecliptic) Equatorial Plane Ascending Node  5.23 15.5 Vernal Equinox Direction The Moon’s gravity tends to tilt the Earth’s axis so that it becomes perpendicular to Moon’s Orbit, 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.
  • 13. 13 SOLO 2. Geocentric-Equatorial Coordinate System COORDINATES IN THE SOLAR SYSTEM The origin at the center of the Earth . G Gz Ω Gx Gy Ecliptic plane normal (Ecliptic Pole) Locus of Lunar plane normal (Lunar Pole) Lunar Orbital Plane Earth Orbital Plane (Ecliptic) Equatorial Plane Ascending Node  5.23 15.5 Vernal Equinox Direction • 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 frame is non-rotating to the stars (except to the precession of equinoxes) and the Earth turns relative to it.
  • 14. 14 SOLO 3. The Right Ascension-Declination System COORDINATES IN THE SOLAR 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 Ω GX GY Equatorial Planeα δ Vernal Equinox Direction 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. 15 SOLO Coordinate Systems 4. The Perifocal Coordinate System COORDINATES IN THE SOLAR 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: • 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.
  • 16. 16 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 a given time. 1. a – semi-major axis – a constant defining the size of the coning orbit. 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, between the unit vector and the point where the satellite crosses through the Equatorial Plane in a northerly direction (ascending node) measured counterclockwise where 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. 17 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 ≠ × = → h h vr Z  ε 2. From the specific mechanical energy of an elliptic orbit equation ar vv E 22 0 00 µµ −=− ⋅ =  we obtain 00 0 2 vv r a  ⋅− = µ µ 3. i inclination is computed using →→ ⋅= GZZi 11cos ε 22 11cos 1 ππ ε ≤≤−      ⋅= →→ − iZZi G 4. The eccentricity vector of a Keplerian trajectory is defined as ( ) → =      ⋅−      −⋅= ε µ µ Xevvrr r vve 1 1 0000 0 00  from which ee  = 01 ≠= → e e e X  ε →→→ ×= εεη XZY 111
  • 18. 18 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM 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 011 11 11 1 ≠×← × × = →→ →→ →→ → ε ε ε ZZ ZZ ZZ N G G G Ω – longitude of the ascending node – is computed using →→ ⋅=Ω NXG 11cos →→→ ⋅      ×=Ω GG ZNX 111sin       Ω Ω =Ω − cos sin tan 1 6. ω – argument of the periapsis – is computed using →→ ⋅= εω XN 11cos →→→ ⋅      ×= εεω ZXN 111sin       = − ω ω ω cos sin tan 1 7. Ө – satellite position from the periapsis – is computed using →→ ⋅=Θ rX 11cos ε →→→ ⋅      ×=Θ εε ZrX 111sin       Θ Θ =Θ − cos sin tan 1
  • 19. 19 SOLO 4. The Perifocal Coordinate System (Continue) COORDINATES IN THE SOLAR SYSTEM Let find a, e, ω, i, Ω from the initial position and velocity vectors (continue).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: [ ] [ ] [ ]           ΩΩ− ΩΩ           −          −=Ω= 100 0cossin 0sincos cossin0 sincos0 001 100 0cossin 0sincos 313 ii iiiCG ωω ωω ωε           Ω−Ω Ω+Ω−Ω−Ω− Ω+ΩΩ−Ω = iii iii iii cossincossinsin sincoscoscoscossinsinsincoscoscossin sinsincoscossinsincossincossincoscos ωωωωω ωωωωω
  • 20. 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 Mass System 8.2 Missile Kinematics Model 1 (Spherical Earth) 8.3 Missile Kinematics Model 2 (Spherical Earth)
  • 21. 21 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 Coordinate Systems Table of Content
  • 22. 22 SOLO Coordinate Systems 1.Inertial System (I( R  - vehicle position vector I td Rd V   = - vehicle velocity vector, relative to inertia II td Rd td Vd a 2 2   == - vehicle acceleration vector, relative to inertia AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 23. 23 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 equator to zero longitude meridian. The Earth rotates relative to Inertial system I, with the angular velocity sec/10.292116557.7 5 rad− =Ω EIIE zz  11 Ω=Ω=Ω=←ω ( )           Ω =← 0 0 EC IEω  Rotation Matrix from I to E [ ] ( ) ( ) ( ) ( )           ΩΩ− ΩΩ =Ω= 100 0cossin 0sincos 3 tt tt tCE I AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 24. 24 SOLO Coordinate Systems (continue – 3) 2. Earth Center Earth Fixed Coordinate System (E( (continue – 1( Vehicle Position ( ) ( ) ( ) ( )ETE I EI E I RCRCR  == Vehicle Velocity Vehicle Acceleration RVR td Rd td Rd V EIE EI    ×Ω+=×+== ←ω - vehicle velocity relative to Inertia R td Rd td Rd V IE LE E    ×+== ←ω: - vehicle velocity relative to Earth ( ) ( ) II E I E I R td d td Vd RV td d td Vd a      ×Ω+=×Ω+== ( ) ( )RV td Vd R td Rd R td d V td Vd EIEEU U E EE EIU U E IU              ×Ω×Ω+×             Ω+++=×Ω×Ω+×Ω+× Ω +×+= ← Ω ←←← ω ωωω 0 ( ) ( ) ( )RV td Vd RV td Vd a E E E EEU U E      ×Ω×Ω+×Ω+=×Ω×Ω+×Ω++= ← 22ω or where U is any coordinate system. In our case U = E. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 25. 25 SOLO Coordinate Systems (continue – 4) 3.Earth Fixed Coordinate System (E0( The origin of the system is fixed on the earth at some given 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 Earth surface, zE0 is pointed down. [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =           −           −− − =−−= 100 0cossin 0sincos sin0cos 010 cos0sin 2/ 00 00 00 00 3020 0 LongLong LongLong LatLat LatLat LongLatCE E π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          −−− − −− = 00000 00 00000 sinsincoscoscos 0cossin cossinsincossin LatLongLatLongLat LongLong LatLongLatLongLat The Angular Velocity of E relative to I is: EIIEIE zz  110 Ω=Ω== ←← ωω or ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          Ω− Ω =           Ω          −−− − −− =           Ω =← 0 0 00000 00 00000 00 0 sin 0 cos 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 Lat Lat LatLongLatLongLat LongLong LatLongLatLongLat CE E E IEω  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 26. 26 SOLO Coordinate Systems (continue – 5) 4. Local-Level-Local-North (L) or Navigation Frame The origin of the LLLN coordinate system is located at the projection of the center of gravity CG of the vehicle on the Earth surface, with zDown axis pointed down, xNorth, yEast plan parallel to the local level, with xNorth pointed to the local North and yEast pointed to the 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 LongLatCL E π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          −−− − −− = LatLongLatLongLat LongLong LatLongLatLongLat sinsincoscoscos 0cossin cossinsincossin AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 27. 27 SOLO Coordinate Systems (continue – 6) 4. Local-Level-Local-North (L( (continue – 1) Angular Velocity IEELIL ←←← += ωωω  Angular Velocity of L relative to I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          Ω− Ω =           Ω          − − −− =           Ω =           Ω Ω Ω =← Lat Lat LatLongLatLongLat LongLong LatLongLatLongLat CL E Down East North L IE sin 0 cos 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 ω  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               − −=             −+                       −−− − −− =             −+             =           = • • • • • • • ← LatLong Lat LatLong Lat Long LatLongLatLongLat LongLong LatLongLatLongLat Lat Long CL E Down East North L EL sin cos 0 0 0 0 sinsincoscoscos 0cossin cossinsincossin 0 0 0 0 ρ ρ ρ ω  ( ) ( ) ( ) ( ) ( )                         +Ω− −       +Ω =           Ω+ Ω+ Ω+ =+= • • • ←←← LatLong Lat LatLong DownDown EastEast NorthNorth L IEC L ECL L IL sin cos ρ ρ ρ ωωω  Therefore AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 28. 28 SOLO Coordinate Systems (continue – 7) 4. Local-Level-Local-North (L( (continue – 2) Vehicle Velocity Vehicle Velocity relative to I RVR td Rd td Rd V EIE EI    ×Ω+=×+== ←ω ( ) ( ) ( ) ( ) ( ) ( ) ( )          +−               −− − +           +− =×+= •• •• •• ← HR LatLongLat LatLongLatLong LatLatLong HR R td Rd V EL L L E 00 0 0 0cos cos0sin sin0 0 0     ω 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 V td Hd LatHR V td Longd HR V td Latd _ 0 _ 0 _ cos −= + = + = AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE HeightVehicleHRadiusEarthmRHRR =⋅=+= 6 00 10378135.6
  • 29. 29 SOLO Coordinate Systems (continue – 8) 4. Local-Level-Local-North (L( (continue – 3) Vehicle Velocity (continue – 1) We assume that the atmosphere movement (velocity and acceleration) relative to Earth At the vehicle position (Lat, Long, H) is known. Since the aerodynamic forces on the vehicle are due to vehicle movement relative to atmosphere, let divide the vehicle velocity in two parts: WAE VVV  += ( )           = Down East North L A V V V V  - Vehicle Velocity relative to atmosphere ( ) ( )           = DownW EastW NorthW L W V V V HLongLatV _ _ _ ,,  - Wind Velocity at vehicle position (known function of time) AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 30. 30 SOLO Coordinate Systems (continue – 9) 4. Local-Level-Local-North (L( (continue – 4) Vehicle Acceleration Since: ( ) ( ) ( ) ( )RV td Vd R td d td Vd RV td d td Vd a EEL L E II E I E I        ×Ω×Ω+×Ω++=×Ω+=×Ω+== ← 2ω WAE VVV  += ( ) WWIL L W AAIL L A VV td Vd RVV td Vd a      ×Ω+×++×Ω×Ω+×Ω+×+= ←← ωω ( )         Wa WWEL L W AAEL L A VV td Vd RVV td Vd ×Ω+×++×Ω×Ω+×Ω+×+= ←← 22 ωω ( ) ( ) ( ) ( )HLongLatVHLongLat td Vd HLongLata WEL L W W ,,2,,:,,    ×Ω++= ←ω ( ) WAAEL L A aRVV td Vd   +×Ω×Ω+×Ω+×+= ← 2ω where: is the wind acceleration at vehicle position. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 31. 31 SOLO Coordinate Systems (continue – 10) 5.Body Coordinates (B( The origin of the Body coordinate system is located at the instantaneous center of gravity CG of the vehicle, with xB pointed to the front of the Air Vehicle, yB pointed toward the right wing and zB completing the right-handed Cartesian reference frame. Rotation Matrix from LLLN to B (Euler Angles(: [ ] [ ] [ ]           −+ +− − == θφψφψθφψφψθφ θφψφψθφψφψθφ θψθψθ ψθφ cccssscsscsc csccssssccss ssccc CB L 321 ψ - azimuth angle θ - pitch angle φ - roll angle AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 32. 32 SOLO Coordinate Systems (continue – 11) 5.Body Coordinates (B( (continue – 1( ψ θ φ Bx Lx Bz Ly Lz By Angular Velocity from L to B (Euler Angles(: ( ) [ ] [ ] [ ]           +           +           =           =← ψ θφθφ φ ω    0 0 0 0 0 0 211 R Q P B LB                     −           − +                     − +           = ψθθ θθ φφ φφθ φφ φφ φ    0 0 cos0sin 010 sin0cos cossin0 sincos0 001 0 0 cossin0 sincos0 001 0 0 [ ]           =                     − − = ψ θ φ ψ θ φ θφφ θφφ θ       G coscossin0 cossincos0 sin01 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 33. 33 SOLO Coordinate Systems (continue – 12) 5.Body Coordinates (B( (continue – 2( ψ θ φ Bx Lx Bz Ly Lz By Rotation Matrix from LLLN to B (Quaternions(: ( ) [ ][ ] ( ) [ ][ ] { } { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )            −−− − − −           −− −− −− = +×−×−= 321 412 143 234 3412 2143 1234 44 3333 BIBLBL BLBLBL BLBLBL BLBLBL BLBLBLBL BLBLBIBL BLBLBLBL T BLBLBLXBLBLXBL B L qqq qqq qqq qqq qqqq qqqq qqqq qqqIqqIqC  where: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) { } ( ) ( ) ( )          =      =                         =             = 3 2 1 :& 4 4 3 2 1 4 3 2 1 BL BL BL BL BL BL BL BL BL BL BL BL BL BL BL BL q q q q q q qor q q q q q q q q q   ( )                         −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos4 ϕθψϕθψ BLq ( )                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ BLq ( )                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ BLq ( )                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ BLq AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 34. 34 SOLO Coordinate Systems (continue – 13) 5.Body Coordinates (B( (continue – 3( ψ θ φ Bx Lx Bz Ly Lz By Rotation Matrix from LLLN to B (Quaternions( (continue – 1( The quaternions are given by the following differential equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) BL L IL B IBBLBLBL B ILBL B IBBL B IL B IBBL B LBBLBL qqqqqqqqq ⋅−⋅=⋅⋅⋅−⋅=−⋅=⋅= ←←←←←←← ωωωωωωω 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     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                          +Ω−+Ω−+Ω− +Ω+Ω+Ω− +Ω+Ω−+Ω +Ω+Ω+Ω− − 4 3 2 1 0 0 0 0 2 1 BL BL BL BL zLzLyLyLxLxL zLzLxLxLyLyL yLyLxLxLzLzL xLxLyLyLzLzL q q q q ρρρ ρρρ ρρρ ρρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                          +Ω+−+Ω+−+Ω+− +Ω−+Ω−−+Ω+ +Ω−+Ω++Ω−− +Ω−+Ω−−+Ω+ = 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 ρρρ ρρρ ρρρ ρρρ or: AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 35. 35 SOLO Coordinate Systems (continue – 14) 5.Body Coordinates (B( (continue – 4( ψ θ φ Bx Lx Bz Ly Lz By Vehicle Velocity Vehicle Velocity relative to Earth is divided in: WAE VVV  += ( )           = w v u V B A  ( ) ( )           =           = DownW EastW NorthW B L zW yW xW B W V V V C V V V HLongLatV B B B _ _ _ ,,  Vehicle Acceleration ( ) WWIB B W AAIB B A I VV td Vd RVV td Vd td Vd a      ×Ω+×++×Ω×Ω+×Ω+×+== ←← ωω ( ) ( ) W AELALB B A a RVV td Vd    + ×Ω×Ω+×Ω++×+= ←← 2ωω AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 36. 36 SOLO Coordinate Systems (continue – 15) 6.Wind Coordinates (W( The origin of the Wind coordinate system is located at the instantaneous center of gravity CG of the vehicle, with xW pointed in the direction of the vehicle velocity vector relative to air .AV  [ ] [ ]           − −−=           −          −=−= αα βαββα βαββα αα αα ββ ββ αβ cos0sin sinsincossincos cossinsincoscos cos0sin 010 sin0cos 100 0cossin 0sincos 23 W BC The Wind coordinate frame is defined by the following two angles: α - angle of attack β - sideslip angle AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 37. 37 SOLO Coordinate Systems (continue – 16) 6.Wind Coordinates (W( (continue -1( Rotation Matrix from L (LLLN) to W is: χ - azimuth angle of the trajectory γ - pitch angle of the trajectory Rotation Matrix [ ] [ ] [ ] [ ] [ ] 32123 ψθφαβ −== B L W B W L CCC The Rotation Matrix from L (LLLN) to W can also be defined by the following Consecutive rotations: σ - bank angle of the trajectory [ ] [ ] [ ] [ ]           −+ +− − === γσχσχγσχσχγσ γσχσχγσχσχγσ γχγχγ χγσσ cccssscsscsc csccssssccss ssccc CC W L W L 321 * 1 We defined also the intermediate wind frame W* by: [ ] [ ]           − − == γχγχγ χχ γχγχγ χγ csscs cs ssccc CW L 032 * AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 38. 38 SOLO Coordinate Systems (continue – 17) 6.Wind Coordinates (W( (continue -2( Angular Velocity of W* relative to LLLN is: Angular Velocities ( ) [ ]          − =                     − +           =           +           =           =← γχ γ γχ χγγ γγ γ χ γγω cos sin 0 0 cos0sin 010 sin0cos 0 0 0 0 0 0 2 * * * * *         W W W W LW R Q P Angular Velocity of W relative to LLLN is: ( ) [ ] [ ]                     − − =          −           − +           =                     +           +           =           =← χ γ σ γσσ γσσ γ γχ γ γχ σσ σσ σ χ γγσ σ ω           coscossin0 cossincos0 sin01 cos sin cossin0 sincos0 001 0 00 0 0 0 0 0 21 W W W W LW R Q P AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 39. 39 SOLO Coordinate Systems (continue – 18) 6.Wind Coordinates (W( (continue -3( We have also: Angular Velocities (continue – 1( ( ) ( ) ( ) ( )           Ω Ω Ω =           Ω− Ω ==           Ω Ω Ω = ←← Down East North W L W L L IE W L zW yW xW W IE C Lat Lat CC *** * * * * sin 0 cos ωω  ( ) ( ) ( ) ( )           =               − −==           = • • • ←← Down East North W L W L L EL W L zW yW xW W EL C LatLong Lat LatLong CC ρ ρ ρ ω ρ ρ ρ ω *** * * * * sin cos  ( ) ( ) ( ) ( ) [ ] ( )* 1 sin 0 cos W IE W L L IE W L zW yW xW W IE Lat Lat CC ←←← =           Ω− Ω ==           Ω Ω Ω = ωσωω  ( ) ( ) ( ) ( ) [ ] ( )* 1 sin cos W IL W L L IL W L W IL LatLong Lat LatLong CC ← • • • ←← =                         +Ω− −       +Ω == ωσωω  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 40. 40 SOLO Coordinate Systems (continue – 19) 6.Wind Coordinates (W( (continue -4( The Angular Velocity from I to W is: Angular Velocities (continue – 2( ( ) ( ) ( ) ( )           Ω+ Ω+ Ω+ +           =+           =+=           = ←←←← DownDown EastEast NorthNorth W L W W W L IL W L W W W W IL W LW W W W W IW C R Q P C R Q P r q p ρ ρ ρ ωωωω  Using the angle of attack α and the sideslip angle β , we can write: BWBW yz    11 αβω −=← or: ( ) ( ) ( ) [ ]           −           =           −           =−= ←←← 0 0 0 0 3 αβ β ωωω    r q p C r q p W B W W W W IB W IW W BW but also: ( ) ( ) ( ) [ ]           −           =           −           =−= ←←← 0 0 0 0 3 αβ β ωωω    R Q P C R Q P W B W W W W LB W LW W BW AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 41. 41 SOLO Coordinate Systems (continue – 20) 6.Wind Coordinates (W( (continue -5( We can write: Angular Velocities (continue – 3(           −           +                     − −−=           0 cos sin 0 0 cos0sin sinsincossincos cossinsincoscos βα βα βαα βαββα βαββα   r q p r q p W W W or: ( ) ( ) βαα βαβαβα βαβαβα    ++−= −−+−= +−+= cossin sinsincossincos cossinsincoscos rpr rqpq rqpp W W W This can be rewritten as: ( ) βαα β α tansincos cos rp q q W +−−= Wrrp +−= ααβ cossin ( ) ( ) ( )( ) ( ) β βαα ββββααβαβαα cos sinsincos tantansincossincossincossincos W WW qrp qrpqrpp ++ = +++=−++=  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 42. 42 SOLO Coordinate Systems (continue – 21) 6.Wind Coordinates (W( (continue -6( We have also: Angular Velocities (continue – 4( ( ) βαα β α tansincos cos RP Q Q W +−−= WRRP +−= ααβ cossin ( ) β βαα cos sinsincos W W QRP P ++ = AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 43. 43 SOLO Coordinate Systems (continue – 22) 6.Wind Coordinates (W( (continue -7( The vehicle velocity was decomposed in: Vehicle Velocity WAE VVV  += ( )           = 0 0 V V W A  - vehicle velocity relative to atmosphere ( ) ( )           =           = DownW EastW NorthW W L zW yW xW W W V V V C V V V HLongLatV W W W _ _ _ ,,  - wind velocity at velocity position also ( ) [ ] ( ) [ ]           =           −=−= 0 0 0 011 * VV VV W A W A σσ  ( ) ( )           =           = DownW EastW NorthW W L zW yW xW W W V V V C V V V HLongLatV W W W _ _ _ * * * * * ,,  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 44. 44 SOLO Coordinate Systems (continue – 23) 6.Wind Coordinates (W( (continue -8( The vehicle acceleration in W* coordinates is Vehicle Acceleration ( ) ( ) ( ) WAELALW W A WWIW W W AAIW W A I C aRVV td Vd VV td Vd RVV td Vd td Vd a        +×Ω×Ω+×Ω++×+= ×Ω+×++×Ω×Ω+×Ω+×+== ←← ←← 2* * * * * * ωω ωω from which ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )******* * * * 2 W W W A WW EL WW A W LW W W A aVAV td Vd   −×Ω+−=×+         ←← ωω where ( )RaA  ×Ω×Ω−=: AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 45. 45 SOLO Coordinate Systems (continue – 24) 6.Wind Coordinates (W( (continue -9( Vehicle Acceleration (continue – 1( ( ) ( ) ( ) ( ) ( ) ( )           −                     Ω+Ω+− Ω+−Ω+ Ω+Ω+− −           =                     − − − +           ** * * **** **** **** * * * ** ** ** 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 ρρ ρρ ρρ where ( ) ( ) ( ) ( )HR Lat Lat C a a a A A A A W L zW yW xW zW yW xW W +Ω           −           =           = 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 −Ω++=− −Ω+−= −= ρ ρ  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 46. 46 SOLO Coordinate Systems (continue – 25( 6.Wind Coordinates (W) (continue -10) Vehicle Acceleration (continue – 2) Using ( )          − =           =← γχ γ γχ ω cos sin * * * * *     W W W W LW R Q P we have ** xWWxW aAV −= ( ) γρχ cos/2 ** **       Ω+− − = zWzW yWWyW V aA  ( )** ** 2 yWyW zWWzW V aA Ω+− − −= ργ AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 47. 47 SOLO Aerodynamic Forces ( )[ ]∫∫ +−= ∞ WS A dstfnppF  11 ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − ( ) airflowingthebyweatedsurfaceVehicleS SsurfacetheonmNstressforcefrictionf Ssurfacetheondifferencepressurepp W W W − − −−∞ )/( 2 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE 7. Forces Acting on the Vehicle
  • 48. 48 SOLO 7. Forces Acting on the Vehicle (continue – 1) Aerodynamic Forces (continue – 1) ( )           − − − = L C D F W A  ForceLiftL ForceSideC ForceDragD − − − L C D CSVL CSVC CSVD 2 2 2 2 1 2 1 2 1 ρ ρ ρ = = = ( ) ( ) ( ) tCoefficienLiftRMC tCoefficienSideRMC tCoefficienDragRMC eL eC eD − − − βα βα βα ,,, ,,, ,,, ityvisdynamic lengthsticcharacteril soundofspeedHa numberynoldslVR numberMachaVM e cos )( Re/ / − − − −= −= µ µρ AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 49. 49 SOLO 7. Forces Acting on the Vehicle (continue – 2) Aerodynamic Forces (continue -2) ∫∫       ⋅+⋅−= ∫∫       ⋅+⋅−= ∫∫       ⋅+⋅−= ∧∧ ∧∧ ∧∧ W W W S fpL S fpC S fpD dswztCwznC S C dswytCwynC S C dswxtCwxnC S C 1ˆ1ˆ 1 1ˆ1ˆ 1 1ˆ1ˆ 1 Wf Wp Ssurfacetheontcoefficienfriction V f C Ssurfacetheontcoefficienpressure V pp C −= − − = ∞ 2/ 2/ 2 2 ρ ρ ntonormalplanonVofprojectiont dstonormaln ˆˆ ˆ  − − AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 50. 50 ( ) ( ) ( )         MomentFriction S C Momentessure S CCA WW dstRRfdsnRRppM ∫∫∫∫∑ ×−+×−−= ∞ 11 Pr / Aerodynamic Moments Relative to C can be divided in Pressure Moments and Friction Moments. ( )       FrictionSkinor FrictionViscous S essureNormal S A WW dstfdsnppF ∫∫∫∫∑ +−= ∞ 11 Pr Aerodynamic Forces can be divided in Pressure Forces and Friction Forces. AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE AERODYNAMIC FORCES AND MOMENTS.
  • 51. 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 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content
  • 52. 52 SOLO 7. Forces Acting on the Vehicle (continue – 3) Thrust ( ) ( )                     − −−== B B B z y x BW B W T T T TCT αα βαββα βαββα 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 σσ σσσ  AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE F-35Thrust Vector Control
  • 53. 53 SOLO 7. Forces Acting on the Vehicle (continue – 4) Gravitation Acceleration ( ) ( )                         −           −           − == zg yg xg gg 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ cs sc cs sc cs scC EW E W  ( ) gg          − = γσ γσ γ cc cs s W  2sec/174.322sec/81.9 0 2 0 0 0 gg ftmg HR R == + =           AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth. The Gravitational Potential U (R, ( is given byϕ ( ) ( ) ( ) ( )φ φ µ φ , sin1, 2 RUg P R a J R RU E E n n n n ∇=               −⋅−= ∑ ∞ =  μ – The Earth Gravitational Constant a – Mean Equatorial Radius of the Earth R=[xE 2 +yE 2 +zE 2 ]]/2 is the magnitude of the Geocentric Position Vector – Geocentric Latitude (sin =zϕ ϕ E/R( Jn – Coefficients of Zonal Harmonics of the Earth Potential Function P (sin ( – Associated Legendre Polynomialsϕ
  • 54. 54 SOLO 7. Forces Acting on the Vehicle (continue – 5) Gravitation Acceleration AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Retaining only the first three terms of the Gravitational Potential U (R, ( we obtain:ϕ R z R z R z R a J R z R a J R g R y R z R z R a J R z R a J R g R x R z R z R a J R z R a J R g EEEE z EEEE y EEEE x E E E ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= ⋅                 +⋅−⋅      ⋅−        −⋅      ⋅−⋅−= 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 2 2 4 44 42 22 22 2 2 4 44 42 22 22 2 2 4 44 42 22 22 µ µ µ φ φλ φλ sin cossin coscos = ⋅= ⋅= R z R y R x E E E ( ) 2/1222 EEE zyxR ++=
  • 55. 55 SOLO 7. Forces Acting on the Vehicle (continue – 6) Force Equations Air Vehicle Acceleration ( ) ( ) WAELALW W A I C aRVV td Vd td Vd a    +×Ω×Ω+×Ω++×+== ←← 2ωω ( ) ( ) ( ) WAELALW W A A aRVV td Vd amTF m     +×Ω×Ω+×Ω++×+==++ ←← 2 1 g ωω ( )Rg   ×Ω×Ω−= g:Define                   + −− + −− − − =           γσ α γσ βα γ βα ccg m LT csg m CT sg m DT A A A zW yW xW sin sincos coscos          − +                   −− −− −           −=           γ γ α βα βα σσ σσ cg sg m LT m CT m DT A A A zW yW xW 0 sin sincos coscos cossin0 sincos0 001 * * * AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE Table of Content ( )           = 0 0 T T B 
  • 56. 56 SOLO 23. Local Level Local North (LLLN( Computations for an Ellipsoidal Earth Model ( ) ( ) ( ) ( ) ( )2 22 10 2 0 2 0 2 0 5 2 1 2 0 6 0 sin sin1 sin321 sin1 sec/10292116557.7 sec/051646.0 sec/780333.9 26.298/.1 10378135.6 Ae e p m e HR RLatgg g LatfRR LatffRR LatfRR rad mg mg f mR + + = +≈ +−≈ −≈ ⋅=Ω = = = ⋅= − Lat HR V HR V HR V Ap East Down Am North East Ap East North tan + −= + −= + = ρ ρ ρ Lat Lat Down East North sin 0 cos Ω−=Ω =Ω Ω=Ω DownDownDown EastEast NorthNorthNorth Ω+= = Ω+= ρφ ρφ ρφ East North Lat Lat Long ρ ρ −= = • • cos ( ) ( ) ∫ ∫ • • += += t t dtLatLattLat dtLongLongtLong 0 0 0 0 AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE SIMULATION EQUATIONS
  • 57. 57 SOLO AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE SIMULATION EQUATIONS Table of Content
  • 58. 58 SOLO Missile Kinematics Model 1 in Vector Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 59. 59 SOLO Missile Kinematics Model 1 in Matrix Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 60. 60 SOLO Missile Kinematics Model 2 in Vector Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 61. 61 SOLO Missile Kinematics Model 2 in Matrix Notation (Spherical Earth( AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
  • 62. 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(
  • 63. SOLO 63 Navigation Dead Reckoning Navigation A Dead Reckoning System uses a Platform Initial Position and Initial Velocity Vector and then Computes its Position and Velocity based on Measured or Estimated Velocity Vector and Elapsed Time. Dead Reckoning Evolution of a Vehicle’s Position Based on Velocity Vector
  • 64. SOLO 64 Navigation Dead Reckoning Navigation Historical Development of Inertial Platforms
  • 65. SOLO 65 Navigation Dead Reckoning Navigation Based on an Inertial Measurement Unit (IMU( An Inertial Measurement Unit uses Inertial Sensors (at least three Rate and 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
  • 66. SOLO 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
  • 67. SOLO 67 Navigation Input/Output of an Inertial System Inputs ADC.Angle_of_Attack ADC.Mach_Number ADC.Barometric_Altitude ADC.Magnetic_Heading ADC.True_Airspeed INS.Body_Rates (roll, pitch, yaw( INS.Acceleration (lateral, longitudinal, normal( INS.Present_Position (latitude, longitude( INS.True_Heading INS.Velocity (north, east, vertical( RALT.Radar_Altitude Outputs INS.Reference_Velocity (north, east, vertical( NAV.Airspeed NAV.Rate_of_Change_Airspeed NAV.Position (latitude, longitude, altitude( NAV.Angle_of_Attack NAV.Attitude (roll, pitch, yaw( NAV.Body_Rates (roll, pitch, yaw( NAV.Flight_Path_Angle NAV.Ground_Speed NAV.Ground_Track_Angle NAV.Magnetic_Variation NAV.Altitude NAV.Velocity (north, east, vertical( NAV.Acceleration (lateral, longitudinal, normal( NAV.Wind (direction, magnitude( NAV.Body_to_Earth_Transform NAV.Body_to_Horizon_Transform NAV.Radar_to_Body_Transform NAV.Radar_to_Earth_Transform
  • 68. SOLO 68 Navigation 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 Czω
  • 69. SOLO 69 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( The Dynamic Equation along Rate-Integrated-Gyro (RIG( Output Axis YP is: Platform Stabilization Around ZP Azimuth Axis (continue – 1( ( ) θωωθ  GDCzGyG BTTHJ PP −−=++ ( ) ( )       − − −=+ PP y G G G DC zGGG H J H TT HsBJss ωωθ  JG – RIG Moment of Inertia around Output Axis θ – Pickoff Angle - Platform Angular Acceleration around YP Axis - Platform angular Rate around ZP Axis HG – Gyro Angular Moment TC – RIG Torque Command TD – Disturbance Moment BG - Damping Coefficient Pyω Pzω Tacking Laplace Transform and rearranging:
  • 70. SOLO 70 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 2( ( ) ( )       − − −=+ PP y G G G DC zGGG H J H TT HsBJss ωωθ  Define: ( ) Cz G C k H T ω∆+= 1: Angular Rate Command (Δk –Scaling Error( G G D H T ε=: Gyro Bias G G G B J τ=: RIG Time Constant ( ) ( ) ( )       −−∆+− + −= PCP y G G Gzz GG G H J k ssB H s ωεωω τ θ 1 1 1
  • 71. SOLO 71 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 3( The Pickoff Signal θ, is the Feedback Command to the Azimuth Torque Motor ( ) ( ) ( )ssKKsT Cz θ12= K1(s( - Filter and Torque Driver ( ) fzxxyxzz TTJJJ CPPPPPP −=−− ωωω K2 - Torque Motor Gain The Moment Equation along Platform ZP Axis is:
  • 72. SOLO 72 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 4( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D GGxGx G y G G Gz GGxGx GG z T BHsKsKsJsJ ss H J k BHsKsKsJsJ BHsKsK PP CC PP P / 1 1 / / 1 23 1 23 1 ++ + −      −+∆+ ++ = τ τ ωεω τ ω  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D Gx GG x y G G Gz Gx GG Gx GG z T ssJ BH sKsK sJ H J k ssJ BH sKsK ssJ BH sKsK P P CC P P P 1 / 1 1 1 1 / 1 1 / 21 21 21 + + −       −+∆+ + + + = τ ωεω τ τ ω  or From the Figure above we obtain:
  • 73. SOLO 73 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( Platform Stabilization Around ZP Azimuth Axis (continue – 5( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D GGxGx G y G G Gz GGxGx GG z T BHsKsKsJsJ ss H J k BHsKsKsJsJ BHsKsK PP CC PP P / 1 1 / / 1 23 1 23 1 ++ + −      −+∆+ ++ = τ τ ωεω τ ω  At Steady-State we obtain: ( ) ( ) ( ) ( ) ( )D s GG y G G Gzz Ts BHsKKH J kt CCP 0 1 lim /0 1 1 → −−+∆+=∞→ ωεωω  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 ω ( ) ( ) Gzz CP kt εωω +∆+≈∞→ 1 Therefore the Misalignment Errors of the Platform are due to Gyros Drift and Scaling Error. Both can be measured (off-line( and compensated by Navigation Computer.
  • 74. SOLO 74 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs(
  • 75. SOLO 75 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( 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 G P AC ψψ ψψ θθ θθ φφ φφψθφ We want to apply Moments on the Platform, related to the Pjckoff Outputs of the Three RIGs mounted on the Platform ( ) ( )           =           = z y x z y x P KsK T T T T P P P θ θ θ 21  The 3 Torque Motors Roll (TR(, Pitch (TP( and Heading (TH( are located on Gimbal Axes .PA zyx 1,1,1 ' 
  • 76. SOLO 76 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( We want to find the relation between and TR, 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 '  ++=++= ( ) [ ] [ ] [ ] ( )  [ ] [ ] ( )  [ ] ( )  P Pzy A Ax 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 , '             −+           −−+           −−−=           = ψθψφθψ                     − − =           H P R GG GG z y x T T T T T T P P P 10sin 0coscossin 0sincoscos θ ψθψ ψθψ                     −=           P P P z y x GG GG GG H P R T T T T T T 1tansintancos 0cossin 0cos/sincos/cos θψθψ ψψ θψθψ
  • 77. SOLO 77 Navigation Platform Stabilization Using Rate-Integrated-Gyros (RIGs( To simplify the implementation the assumption of small 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 θψθψ ψψ θψθψ                     −≈           P P P z y x GG GG H P R T T T T T T 100 0cossin 0sincos ψψ ψψ ( ) ( )           =           = z y x z y x P KsK T T T T P P P θ θ θ 21  where:
  • 79. SOLO 79 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations 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 by the three small angles ψx, ψy, ψz as [ ] [ ] [ ]           −           −           − == 100 0cossin 0sincos cos0sin 010 sin0cos cossin0 sincos0 001 321 zz zz yy yy xx xxzyx P CC ψψ ψψ ψψ ψψ ψψ ψψψψψ           − − − −           =           − − − ≈           −           −           − ≈ 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 ψψ ψψ ψψ ψψ ψψ ψψ ψ ψ ψ ψ ψ ψ [ ] [ ] [ ]           − − − =×           =×−≈ 0 0 0 :&: xy xz yz z y x P C IC ψψ ψψ ψψ ψ ψ ψ ψ ψψ 
  • 80. SOLO 80 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 1( Let find the angular rotation vector from C to P ( ) [ ] [ ] [ ] =                               +           +           =← z zyy x x P CP ψ ψψψ ψ ψω     0 0 0 0 0 0 321 ψ ψ ψ ψ ψ ψψψ ψψψ ψψψ ψψ ψψ ψ ψ ψψ           =           ≈           + − =                     − − − +                     − +           ≈ z y x z zxy zyx zxy xz yz y y yx 0 0 1 1 1 0 0 10 010 01 0 0
  • 81. SOLO 81 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 1( 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 ( ) ( ) ( ) ( )P G C ICG P IP KI εωω  ++= ←← Platform Rate Commands Vector           ∆ ∆ ∆ = 33231 23221 13121 G G G G Kmm mKm mmK K Matrix of Gyros Scaling Errors, Misalignments and Mass-Unbalances ( ) PPP zzyyxx P G 111  εεεε ++= Gyro Drift Vector Computer Rate Commands VectorCCCCCC zzyyxxIC 111  ωωωω ++=←
  • 82. SOLO 82 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Platform Misalignment Error Equations (continue – 2( Let find the angular velocity vector of the Platform (P( relative to the Computer (C(: ICIPCP ←←← −= ωωω  ( ) ( ) ( ) ( ) ( )C IC P C P IP P IC P IP P CP C ←←←←← −=−= ωωωωω  ( ) ( ) ( ) [ ]{ } ( ) [ ] ( ) ( ) ( )P G C ICG C IC C IC P G C ICG KIKI εωωψωψεωψ  ++×=×−−++= ←←←← Using we obtain:[ ] ( ) ( ) [ ] ψωωψ  ×−=× ←← C IC C IC ( ) [ ] ( ) ( )P G C ICG C IC K εωψωψ  ++×−= ←← or           +                     ∆ ∆ ∆ +                     − − − −=           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 ε ε ε ω ω ω ψ ψ ψ ωω ωω ωω ψ ψ ψ 33231 23221 13121 0 0 0   
  • 83. SOLO 83 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations - vector representing position from the Earth Center of mass to the Vehicle r  ( )rgAr   += - Ideal Accelerometers Measurement VectorA  ( ) ( ) r rr K r r K rg    2/33 ⋅ −=−= ( )rg  - Gravity Vector ( )rrgAArr   δδδ +++=+ For Non-Ideal Accelerometers we have a error between Real Position and Computed Position r  δ ( ) ( )rgrrgAr   −++= δδδ
  • 84. SOLO 84 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 1( ( ) ( )rgrrgAr   −++= δδδ ( ) ( ) ( ) ( )[ ] r r K r rrrr K rgrrg    32/3 + +⋅+ −=−+ δδ δ ( ) ( ) r r K r rr rr r K r r K r rrr K     32332/32 31 2 +      ⋅ −+−≈+ ⋅+ −≈ δ δ δ r r K r r r r r r K r r K r r K    3333 +      ⋅+−−≈ δδ therefore Ar r r r r r r K r    δδδδ =            ⋅−+ 33
  • 85. SOLO 85 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 2( Define r g r K S == 3 :ω Maximilian Schuler (1882 – 1972( S ST ω π2 = Shuler Period = 84.4 minutes at Sea Level Ar r r r r rr S    δδδωδ =            ⋅−+ 32
  • 86. SOLO 86 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 3( 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 ( ) ( ) ( ) ( )PP f C C bAKIA  δ++= 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 zzyyxx P AAAA 111  ++= ( ) PPPPPP zzyyxx P bbbb 111  δδδδ ++= Accelerometers Biases Vector
  • 87. SOLO 87 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 4( AAA C  −=:δ Accelerometers Error Vector ( ) ( ) ( ) ( ) ( ) ( ) [ ]( ) ( )PPP f PC P C C C AIbAKIACAA  ×+−++=−= ψδδ We used the relation ( ) [ ]( ) [ ]( )×+≈×−== −− ψψ  IICC P C C P 11 Finally we obtain ( ) [ ] ( ) ( ) ( )PP f PC bAKAA  δψδ ++×−= [ ] ( ) ( ) ( )PP f P S bAKAr r r r r rr    δψδδωδ ++×−=            ⋅−+ 32 The Position Error Equation is
  • 88. SOLO 88 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 5( Let compute ( )C r δ ( ) ( ) ( ) [ ] ( )CC IC CC rrr   δωδδ ×+= ← Therefore ( ) ( ) ( ) ( ) ( ) CCCCCC CCC C IC C CCC CCC CCCCCC CCC zzyyxxIC C ICzyxICzyx zyx C zyxzyx C zyx C rzyxzyx zyxr zyxzyxr zyxr 111 111111 111: 111111 111 11                   ωωωω δωδδδωδδδ δδδδ δδδδδδδ δδδδ αα ω ++= ×=++×=++ ++= +++++= ++= ← ←← ×= ← In the same way ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( )CC IC CC IC CC IC C IC C IC CC IC CC rr rrrr               δωδω δωωωδωδδ ×+×+         ×         ×++×+= ←← ←←←← 0
  • 89. SOLO 89 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 6( ( ) ( ) ( ) [ ] ( )CC IC CC rrr   δωδδ ×+= ← Therefore ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ]( ) ( )CC IC C IC C IC CC IC CC rrrr       δωωωδωδδ ××+×+×+= ←←←←2 ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ]( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( )PP f P C CC C S CC IC C IC C IC CC IC C bAKA r r r r r rrrr          δψ δδωδωωωδωδ ++×−=             ⋅−+××+×+×+ ←←←← 32 2 Together with the Platform Misalignment Error Equations ( ) [ ] ( ) ( )P G C ICG C IC K εωψωψ  ++×−= ←←
  • 90. SOLO 90 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 7( CCCCCC zzyyxxIC 111  ωωωω ++=← ( ) [ ]           − − − =×← 0 0 0 CC CC CC xy xz yz C IC ωω ωω ωω ω  ( ) [ ]           − − − =×← 0 0 0 CC CC CC xy xz yz C IC ωω ωω ωω ω      ( ) [ ] ( ) [ ] ( ) ( ) ( )            +− +− +− =           − − −           − − − =×× ←← 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 C IC C IC ωωωωωω ωωωωωω ωωωωωω ωω ωω ωω ωω ωω ωω ωω  Czrr 1  = ( ) ( ) ( ) ( )           − =           −           =      ⋅− z y x z z y x r r r r r r C C C C δ δ δ δ δ δ δ δδ 21 0 0 33    
  • 91. SOLO 91 Navigation Derivation of the IMU Position and Platform Misalignment Error Equations Position Error Equations (continue – 8) ( ) ( ) ( )                       +−−+− −+−+ +−+− +                     − − − +           z y x z y x z y x CCCCCCCC CCCCCCCC CCCCCCCC CC CC CC yxSxzyyzx xzyzxSzyx yzxzyxzyS xy xz yz δ δ δ ωωωωωωωωω ωωωωωωωωω ωωωωωωωωω δ δ δ ωω ωω ωω δ δ δ 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 δ δ δ ψψ ψψ ψψ 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 ε ε ε ω ω ω ψ ψ ψ ωω ωω ωω ψ ψ ψ 33231 23221 13121 0 0 0   
  • 92. 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
  • 93. 93
  • 94. Rate gyro DTG – Dynamically Tuned Gyro Flex Inversion Cardan joint
  • 95. 95 Main Components of a DTG Transverse Cut of a DTG Rate gyro DTG – Dynamically Tuned Gyro
  • 98. 98 SOLO Strapdown Algorithm (Vector Notation) Navigation
  • 103. SOLO 103 Navigation Externally Navigation Add Systems eLORAN LORAN - C Global Navigation Satelite System (GNSS) Distance Measuring Equipment (DME) VHF Omni Directional Radio-Range (VOR) System Data Base Matching Terrain Referenced Navigation (TRN) Navigation Multi-Sensor Integration
  • 104. SOLO 104 Navigation Global Navigation Satelite System (GNSS) Satellites of the GPS GLONASS and GALILEO Systems Four Satellite Navigation Systems have been designed to give three dimensional Position, 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.
  • 108. SOLO 108 Navigation Global Navigation Satelite System (GNSS) 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).
  • 109. Global Positioning System (GPS) SOLO 109 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)
  • 110. Global Positioning System SOLO 110 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
  • 111. Satellite Position SOLO 111 Navigation GZ GX GY Equatorial Plane εY εZ εX Ascending Node Satellite Orbit Periapsis Direction Vernal Equinox Direction Ω ω i → N1 Θ 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.
  • 113. 113 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit From the equation Θ= 2 rh we can write h Ad h dr dt 2 2 = Θ = where is the area defined by the radius vector as it moves through an angle 2 2 Θ = dr Ad Θd Θ pΘ pΘ−Θ r focus conic section x y → P1 → Q1 → r1 → t1 v  rv tv Θd Θ= drAd 2 2 1 periapsis This proves the 2nd Kepler’s Law that equal area are swept out equal in equal times by the radius vector.
  • 114. 114 SOLO KEPLERIAN TRAJECTORIES 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 ba T eap ph µ ππππ µ 2/3122/322 2 1 2 1 22 2 = −= = − = − == or 2/3 2 aT µ π = 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 equal To the cube of its mean distance to the sun”.
  • 115. 115 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 2) Let draw an auxiliary circle of radius a, and the same center O as the geometric center of the ellipse. x y eac = a a ( ) 2/12 1 eab −= r Θ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Let take any point P on the ellipse with polar coordinates r,Θ and define the point Q on the circle at the same coordinate x as P. Eeary r a x ea a x by Ea a x ay ellipse ellipse circle sin1sin sin111 sin1 2 2 2 2 2 2 2 2 −=Θ=→        Θ=−−=−= =−= The angle E of OQ with x axis is called the eccentric anomaly. aeEarxellipse −=Θ= coscos
  • 116. 116 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 3) Let compute x y eac = a a ( ) 2/12 1 eab −= r Θ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P ( ) ( )( ) ( )( ) ( ) 0cos11 sin1sincos1cos 1 2 22 2 >←−−= −+−−= −=×=−= EEEeea EEeaEaEEeaaeEa xyyxvreah ellipseellipseellipseellpse     µ We obtain ( ) n a EEe ==− :cos1 3 µ ( ) ( ) ( )pttntEetE −=− sin Integrating this equation gives Kepler’s Equation where tp is the time of periapsis ( E (tp) = 0 )
  • 117. 117 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 4) From x y eac = a a ( ) 2/12 1 eab −= rΘ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Eeary aeEarx ellipse ellipse sin1sin coscos 2 −=Θ= −=Θ= we have ( ) ( )[ ] [ ] ( )Eea EeEeaEeaaeEar cos1 coscos21sin1cos 2/1222/12222 −= +−=−+−= Therefore Θ+ Θ− = − − =Θ Θ+ Θ+ = − − =Θ cos1 sin1 sin cos1 sin1 sin cos1 cos cos cos1 cos cos 22 e e E Ee Ee e e E Ee eE ( )( ) Ee Ee Ee eEEe sin1 cos11 sin1 coscos1 sin cos1 2 tan 22 − −+ = − +−− = Θ Θ− = Θ From 2 tan 1 1 2 tan E e e − + = Θ or and are always in the same quadrant.2 Θ 2 E
  • 118. 118 SOLO KEPLERIAN TRAJECTORIES Time of Flight on an Elliptic Orbit (continue – 5) We have x y eac = a a ( ) 2/12 1 eab −= rΘ FOCUS EMPTY FOCUS c → P1 → Q1 a F Q O VS E P Eeary aeEarx ellipse ellipse sin1sin coscos 2 −=Θ= −=Θ= and ( )Eear cos1−= The Position Vector of the Satellite is ( )             − − =           Θ Θ =           = += 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 1 0 cos sin cos1 0 cos1 sin 0 cos1 sin 0 cos1 sin e an e Ee an Ee E EanEe E Ee aeE q td d Orbit Orbit − ⋅ ⋅           Θ+ Θ− = ⋅− ⋅ ⋅             − − =⋅⋅             − − =             − −− = 
  • 119. GPS Broadcast Ephemerides SOLO 119 Navigation The Satellite Position can be computed as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )oeisic rsrc usuc oe oe ttidotuCuCii uCuCrr uCuC tt ttnnMM a n −⋅+⋅+⋅+= ⋅+⋅+= ⋅+⋅+= −⋅Ω+Ω=Ω −⋅∆++= = 000 000 000 0 0 3 2sin2cos 2sin2cos 2sin2cosωω µ  where: ( ) Θ+=         ⋅ − + ⋅=Θ ⋅−⋅= ⋅+= − 00 1 0 2 tan 1 1 tan2 cos1 sin ωu E e e Eear EeME Six Keplerian Elements Define the Satellite Posision (Ω, I, ω, a, e, M0) where M0 = n (t – tP)
  • 120. GPS Broadcast Ephemerides SOLO 120 Navigation ( ) Θ+=           =           = ωuur ur y x q ellipse ellipse Orbit 0 sin cos 0  ( ) 2 1 0 cos sin 0 e an ue u y x q ellipse ellipse Orbit Orbit − ⋅ ⋅           + − =           =    ( ) oecoec ttt ⋅+−⋅=Θ ωω [ ] [ ] [ ]           −−Ω−=           =           0 sin cos 0 313 ur ur iy x C z y x ellipse ellipse G G ωε Θ+= ωu
  • 122. Global Positioning System SOLO 122 Navigation - x, y, z Satellite Coordinate in Geocentric-Equatorial Coordinate System ( ) ( ) ( )222 ZzYyXx −+−+−=ρ - 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  ρ 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 ρ where r = Earth Radius This 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..
  • 124. Global Positioning System SOLO 124 Navigation Using data from four Satellites we obtain ( ) ( ) ( ) ( ) 444444 22 4 2 4 2 4 2 4 333333 22 3 2 3 2 3 2 3 222222 22 2 2 2 2 2 2 2 111111 22 1 2 1 2 1 2 1 222 222 222 222 ZzYyXxCrrrzyx ZzYyXxCrrrzyx ZzYyXxCrrrzyx ZzYyXxCrrrzyx r r r r ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ⋅⋅−⋅⋅−⋅⋅−=−++− ρ ρ ρ ρ or ( ) ( ) ( ) ( )      14 1444 22 4 2 4 2 4 2 4 22 3 2 3 2 3 2 3 22 2 2 2 2 2 2 2 22 1 2 1 2 1 2 1 444 333 222 111 1222 1222 1222 1222 x xx R r r r r PM rzyx rzyx rzyx rzyx Crr Z Y X zyx zyx zyx zyx               −++− −++− −++− −++− =                         ⋅−⋅−⋅− ⋅−⋅−⋅− ⋅−⋅−⋅− ⋅−⋅−⋅− ρ ρ ρ ρ 14 1 4414 xxx RM Crr Z Y X P − =             =
  • 127. Global Positioning System SOLO 127 Navigation GPS Satellite GPS Control Station
  • 128. Global Positioning System SOLO 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 code ompseudorand AC ulation bps power carrier I ttctdPts −− +⋅⋅⋅⋅= 1/ mod 50 cos2 θω
  • 129. Global Positioning System SOLO 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 code ompseudorand P ulation bps power carrier Q ttptdPts −− +⋅⋅⋅⋅= 3,2,1 mod 50 sin2 θω
  • 134. GPS vs GALILEO SOLO 134 Navigation GALILEO GPS Satelites 27 + 3 24 (32!) Planes 3 6 Satellite per Plane 10 4 - 7 Plane Spacing 120 ͦ 60 ͦ Inclination 56 ͦ 55 ͦ Orbit Type MEO Circular MEO Circular Orbit Radius 29,500 km 26,500 km Period 141 /4 hour 12 hour Satellite Ground Track Repetition 10 days 1 day Higher GALILEO Orbit coupled with Inclination increase give better coverage at high latitudes.
  • 135. GPS, GLONASS and GALILEO SOLO 135 Navigation
  • 142. GPS Status, November 2011 SOLO 142 Navigation
  • 144. GPS III Payload Evolution SOLO 144 Navigation
  • 145. GLONASS Constellation, November 2011 SOLO 145 Navigation
  • 147. COMPASS/ BeiDou, November 2011 SOLO 147 Navigation
  • 148. Quasi Zenith Satellite System (QZSS) - Japan SOLO 148 Navigation
  • 149. Indian Regional National Satellite System (IRNSS) SOLO 149 Navigation
  • 153. GNSS Aviation Operational Performance Requirements SOLO 153 Navigation
  • 154. SOLO 154 Navigation Externally Navigation Add Systems 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.
  • 155. SOLO 155 Navigation Externally Navigation Add Systems 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
  • 156. SOLO 156 Navigation Externally Navigation Add Systems 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,
  • 157. SOLO 157 Navigation Externally Navigation Add Systems 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 carrier Frequency Modulated (FM) with: 30 Hz signal from the main antenna. An Amplitude Modulated (AM) carrier electrically swept around several smaller Antennas surrounding the main Antenna. This rotating pattern creates a 30 Hz Doppler effect on the Receiver. The Phase Difference of the two 30 Hz signals gives the User’s Azimuth with respect to the North from the VOR Site. The Bearing measurement accuracy of a VOR System is typically on the order of 2 degrees, with a range that extends from 25 to 130 miles.
  • 158. SOLO 158 Navigation Externally Navigation Add Systems TACAN is the Military Enhancement of VOR/DME VHF Omni Directional Radio-Range (VOR) System TACAN (Tactical Air Navigation) is an enhanced VOR/DME System designed for Military 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.
  • 164. 164 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
  • 165. SOLO 165 Navigation World Geodetic System (WGS 84) Geoid - Mean Sea Level of the Earth Reference Ellipsoid – Approximation of Sea Level Reference Earth Model h - Vehicle Altitude (the distance from the Vehicle to Ellipsoid along the Normal to Ellipsoid RN - 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 of N 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 the World Geodetic System of 1984 (WGS 84).
  • 166. SOLO 166 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 ba f − =: Alexis Claude Clairaut )1713–1765(             −+= φ2 sin 2 5 1 fmGg
  • 167. SOLO 167 Navigation World Geodetic System (WGS 84) Reference Earth Model Carlo Somigliana (1860 –1955) The Theoretical Gravity on the surface of the Ellipsoid is given by the Somigliana Formula (1929)    84 22 2 2222 22 sin1 sin1 sincos sincos WGS e pe e k ba ba φ φ γ φφ φγφγ γ − + = + + = where 1: −= e p a b k γ γ 2 22 : a ba e − = - 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 – Geodetic Latitudeϕ The Theory of the Equipotential Ellipsoid was first given by P. Pizzetti (1894)
  • 168. SOLO 168 Navigation World Geodetic System (WGS 84) Reference Earth Model 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.
  • 169. SOLO 169 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) Reference Earth Model 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.
  • 170. SOLO 170 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 m Minimum = -106.99 m, Maximum = 85.39 m
  • 171. SOLO 171 Navigation World Geodetic System (WGS – 84) Reference Earth Model Parameters Notation Value Ellipsoid Semi-major Axis a 6.378.137 m Ellipsoid Flattening (Ellipticity) f 1/298.257223563 (0.00335281066474) Second Degree Zonal Harmonic Coefficient of the Geopotential C2,0 -484.16685x10-6 Angular Velocity of the Earth Ω 7.292115x10-5 rad/s The Earth’s Gravitational Constant (Mass of Earth includes Atmosphere) GM 3.986005x1014 m3 /s2 Mass of Earth (Includes Atmosphere) M 5.9733328x1024 kg Theoretical (Normal) Gravity at the Equator (on the Ellipsoid) γe 9.7803267714 m/s2 Theoretical (Normal) Gravity at the Poles (on the Ellipsoid) γp 9.8321863685 m/s2 Mean Value of Theoretical (Normal) Gravity γ 9.7976446561 m/s2 Geodetic and Geophysical Parameters of the WGS-84 Ellipsoid
  • 172. SOLO 172 Navigation World Geodetic System (WGS 84) Reference Earth Model a ba f − =:f - Ellipsoid Flattening (Ellipticity) a - Ellipsoid Semi-major Axis b - Ellipsoid Semi-minor Axis e - Ellipsoid Eccentricity 2 2 22 2 2: ff a ba e −= − = ( ) 2 11 eafab −=−= Reference Ellipsoid
  • 173. SOLO 173 Navigation Reference Ellipsoid Ellipse Equation: 12 2 2 2 =+ b y a x Slope of the Normal to Ellipse: 2 2 tan bx ay yd xd =−=φ The Slope of the Geocentric Line to the same point x y =λtan λλ sincos RyRx == Deviation Angle between Geographic and Geodetic At Ellipsoid Surface λφ tantan 2 2 b a =       = − λφ tantan 2 2 1 b a ( ) φφλ tan1tantan 2 2 2 e a b −==
  • 174. SOLO 174 Navigation Reference Ellipsoid Ellipse Equation: λφδ −= 12 2 2 2 =+ b y a x Slope of the Normal to Ellipse: 2 2 tan bx ay yd xd =−=φ The Slope of the Geocentric Line to the same point x y =λtan       −=       −+       − = + − = + − = 1 11 1 1 tantan1 tantan tan 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 ay x y bx ay λφ λφ δ λλ sincos RyRx == ( )  ( ) ( )λλλδ 2sin2sin 2 tan2sin 2 tan 1 2 11 12 22 22 1 f ba R b ba a ba R ba ba f ≈             +       − =            − = ≈≈<< −−  Deviation Angle between Geographic and Geodetic At Ellipsoid Surface
  • 175. SOLO 175 Navigation Reference Ellipsoid For a point at a Height h near the Ellipsoid the value of δ must be corrected: u−= 1δδ From the Law of Sine we have: Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface ( ) R h hR huu ≈ + ≈= − 11 sin sin sin sin δδπ Since u and δ1 are small: 1δ R h u ≈ The corrected value of δ is: ( )λδδδ 2sin11 11 f R h R h u       −=      −=−= Therefore: ( )λλδλφ 2sin1 f R h       −+=+=
  • 176. SOLO 176 Navigation World Geodetic System (WGS 84) where λ – Longitude e – Eccentricity = 0.08181919 Reference Earth Model In Earth Center Earth Fixed Coordinate –ECEF-System (E) the Vehicle Position is given by: ( ) ( ) ( ) ( )           + + + =           = φ φλ φλ sin cossin coscos HR HR HR z y x P M N N E E E E  ( ) NhH e a RN += − = 2/12 sin1 φ Another variable, used frequently, is the radius of the Ellipsoid referred as the Meridian Radius ( ) ( ) 2/32 2 sin1 1 φe ea RM − − =
  • 177. SOLO 177 Navigation Reference Ellipsoid Let develop the RN and RM: Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface Ellipse Equation: ( ) 222 2 2 2 2 11 aeb b y a x −==+ From this Equation, at any point (x,y) on the Ellipse, we have: φtan 1 2 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: ( ) φ φ φ 2 22 2 2 2 222 2 2 2 2 2 2 2 2 cos sin1 1 1 tan1111 e a x e e a x b a x y a x − =      − −+=      += ( ) ( ) ( ) 2/122 2 2 2 2/122 sin1 sin1 tan sin1 cos φ φ φ φ φ e ea x a b y e a x − − ==→ − = From the Figure above: ( ) 2/122 sin1cos φφ e ax RN − ==
  • 178. SOLO 178 Navigation Reference Ellipsoid Let develop the RN and RM (continue): Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface we have at any point (x,y) on the Ellipse: φtan 1 2 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 tan 1 11 : φφ φφ e ea e ea ea xd yd xd yd RM − − = − − −       + =               + = ( ) ( ) ( ) 2/122 2 2 2 2/122 sin1 sin1 tan sin1 cos φ φ φ φ φ e ea x a b y e a x − − ==→ − = ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − =
  • 179. SOLO 179 Navigation Reference Ellipsoid Deviation Angle between Geographic and Geodetic At Altitude h from Ellipsoid Surface ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − = ( ) 2/122 sin1cos φφ e ax RN − == a ba f − =: 2 2 22 2 2: ff a ba e −= − =Using ( ) ( )[ ] ( ) ( ) [ ] ++−≈      +−++−≈ −− − = φφ φ 2222 2/322 2 sin321sin2 2 3 121 sin21 1 : ffaffffa ff fa RM ( )[ ] ( ) [ ]φφ φ 222 2/122 sin31sin2 2 3 1 sin21 faffa ff a RN +≈    +−+≈ −− =  [ ]φ2 sin321 ffaRM +−≈ [ ]φ2 sin31 faRN +≈ We used and we neglect f2 terms ( ) ( ) + − ++= − !2 1 1 1 1 nn xn x n
  • 180. SOLO 180 Navigation World Geodetic System (WGS 84) Reference Earth Model 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 ( ) isϕ derived in the above references as The definition of geodetic (or geographic) and geocentric latitudes ( ) ( )[ ]φφψ tan1tan 21 e−= −

Editor's Notes

  • #4: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
  • #5: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
  • #6: Solo Hermelin, “Spherical Trigonometry”,
  • #7: Solo Hermelin, “Spherical Trigonometry”,
  • #8: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
  • #9: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
  • #10: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
  • #11: George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
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  • #64: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #65: Ian Moir, Allan Seabridge, “Military Avionics Systems”, John Wiley &amp; Sons, LTD., 2006
  • #94: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #97: M.S. Grewal, L.R. Weill, A.P. Anrews, “Inertial Positioning Systems, Inertial Navigation and Integration”, John Wiley &amp; Sons, 2001
  • #98: M.S. Grewal, L.R. Weill, A.P. Anrews, “Inertial Positioning Systems, Inertial Navigation and Integration”, John Wiley &amp; Sons, 2001
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  • #102: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #103: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #104: M.S. Grewal, L.R. Weill, A.P. Anrews, “Inertial Positioning Systems, Inertial Navigation and Integration”, John Wiley &amp; Sons, 2001
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  • #106: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #107: “GPS Compedium”,
  • #108: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #109: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
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  • #120: G. Xu, “GPS, Theory, Algorithms and Applications”, 2nd Edition, Springer, 2003, 2007
  • #121: G. Xu, “GPS, Theory, Algorithms and Applications”, 2nd Edition, Springer, 2003, 2007
  • #122: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #123: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #124: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #125: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #126: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #127: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
  • #128: R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
  • #129: B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins, “GPS Theory and Practice”, 2nd Ed., Springer –Verlag,1993 Guochang Xu, “GPS Theory, Algorithms and Applications”, 2nd Ed., Springer-Verlag, 2003, 2007 R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
  • #130: B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins, “GPS Theory and Practice”, 2nd Ed., Springer –Verlag,1993 Guochang Xu, “GPS Theory, Algorithms and Applications”, 2nd Ed., Springer-Verlag, 2003, 2007 M.S. Grewal, L.R. Weill, A.P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, John Wiley &amp; Sons, 2001 R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
  • #131: R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
  • #132: M. S. Grewal, L. R. Weill, A. P. Andrews, “Global Positioning Systems, Inertial Navigation and Integration”, 2001, John Wiley &amp; Sons
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  • #135: R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
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  • #137: R. Prasad, M. Ruggieri, “Applied Satellite Navigation Using GPS, Galileo and Augmented Systems”, Artech House, 2005
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  • #143: Terry More, “GNSS Status and Future Developments”, The University of Nottingham, 1911
  • #144: Terry More, “GNSS Status and Future Developments”, The University of Nottingham, 1911
  • #145: Terry More, “GNSS Status and Future Developments”, The University of Nottingham, 1911
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  • #158: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #159: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
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  • #161: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #162: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
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  • #164: “Basic Guide to Advanced Navigation”, NATO Research and Technology Organisation Publication, SET-114/RTG-65,
  • #166: Averil B. Chatfield, “Fundamentals of High Accuracy Inertial Navigation”, Progress in Astronautics and Aeronautics 174, 1997 George M. Siouris, “Aerospace Avionics System”, Academic Press, 1993, Appendix B
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