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UPC gAGE research g roup of A stronomy and Ge omatics EGNOS TUTORIAL Research g roup of A stronomy and GE omatics (gAGE/UPC) Universitat Politcnica de Catalunya e-mail: jaume@mat.upc.es http://gage1.upc.es 1 Manuel Hernndez-Pajares, J.

  1. UPC gAGE research g roup of A stronomy and Ge omatics http://www.sunspotcycle.com/ http://gage1.upc.es / With SA set to Zero, the dominant error is now the error associated with the Ionosphere. – The ionosphere can add a significant amount of error to a user's position solution – Based on several factors: • geographic location • time of day • time with respect to the solar cycle (11 years). 20 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  2. SBAS Concept UPC gAGE research g roup of A stronomy and Ge omatics The pseurorange error is split in its components. • Clock error • Ephemeris error • Ionospheric error • Local errors (troposphere, multipath, receiver noise) Uses a network of receivers to cover broad geographic area 21 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  3. Error Mitigation UPC gAGE research g roup of A stronomy and Ge omatics Error GBAS SBAS component Satellite clock Estimation and Removal each Ephemeris Common Mode error component Ionosphere Differencing Troposphere Fixed Model Multipath and Carrier Smoothing by user Receiver Noise 22 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  4. UPC GEO gAGE research g roup of A stronomy and Ge omatics Integrity GPS-like Differential (Use / signals corrections Don't Use) + ACCURACY + SAFETY + AVAILABILITY + CONTINUITY 23 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  5. The European Geoestationary Navigation UPC Overlay SERVICE (EGNOS) gAGE research g roup of A stronomy and Ge omatics 24 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  6. What EGNOS is? UPC gAGE research g roup of A stronomy and Ge omatics • EGNOS is the European component of a Satellite Based Augmentation to GPS and GLONAS. • EGNOS is being developed under the responsibility of a tripartite group: – The European Space Agency (ESA) – The European Organization for the Safety of Air Navigation (EUROCONTROL) – The Commission of the European Union. 25 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  7. UPC Three existing SBAS Systems gAGE research g roup of A stronomy and Ge omatics WAAS EGNOS MSAS 26 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  8. ECAC Area UPC (ECAC: European Civil Aviation Conference) gAGE research g roup of A stronomy and Ge omatics 70 Artemis IOR AOR-E 60 50 Latitude (°) 40 30 20 -40 -30 -20 -10 0 10 20 30 40 Longitude (°) 27 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  9. EGNOS AOC Architecture UPC (AOC: Avanced Operational Capability) gAGE research g roup of A stronomy and Ge omatics GEO GLONASS AOR-E GPS IOR ARTEMIS RIMS NLES (x 7) EWAN MCC 1 MCC 2 MCC 3 MCC 4 PACF ASQF 28 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  10. EGNOS AOC GROUND NETWORK TOPOLOGY UPC gAGE research g roup of A stronomy and Ge omatics RIMS for a good GEO RANGING RIMS in ECAC NLES MCCs 29 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  11. RIMS MCC NLES ASQF PACF UPC 75 gAGE research g roup of A stronomy and Ge omatics 70 TRO MMK 65 RKK FER TRD 60 STK SPT GLG ALB 55 LON RST CCV CRK 50 FRK ECAC TLS ZRH 45 SOF SBT TBL SDC MAD ROM 40 LSB MAL PDM KON ACR CTN 35 MAD DJA TAV 30 MMT CNR1 25 -30 -20 -10 0 10 20 30 40 50 30 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  12. Ranging & Integrity Monitoring Station UPC (RIMS) 33 RIMS in EGNOS +1 gAGE research g roup of A stronomy and Ge omatics specific RIMS for UTC time GPS GLONASS GEO L1 / L2 L1 L1 Antenna Local Maintenance Equipment & SYNC Local Maintenance Pre- amplifier Core RIMS Operator Computer DATA Receiver RIMS DATA RIMS Atomic Receiver SET Power Supply Clock & Air conditioning FEE RIMS EWAN Core SET CPF CCF 31 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  13. Master Control Center (MCC) UPC gAGE research g roup of A stronomy and Ge omatics • MCC is Subdivided into • MCC is Subdivided into – CCF (Central Control Facility) – CCF (Central Control Facility) • Monitoring and control EGNOS G/S • Monitoring and control EGNOS G/S • Mission Monitoring and archive • Mission Monitoring and archive • ATC I/F • ATC I/F – CPF (Central Processing Facility) – CPF (Central Processing Facility) • Provides EGNOS WAD corrections • Provides EGNOS WAD corrections • Ensures the Integrity of the EGNOS users • Ensures the Integrity of the EGNOS users • Utilises independent RIMS channels for checking of corrections • Utilises independent RIMS channels for checking of corrections • Real time software system developed to high software standards • Real time software system developed to high software standards • 4 MCCs will be implemented in EGNOS • 4 MCCs will be implemented in EGNOS 32 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  14. Master Control Center (MCC) UPC gAGE research g roup of A stronomy and Ge omatics Check Processing Check EWAN FEE CPF Central Processing Facility CCF Central Control Facility EWAN FEE Monitor Monitor ISDN Ground Archive ATC I/f mission Segment PACF 33 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  15. CCF VIEWS UPC gAGE research g roup of A stronomy and Ge omatics CCF: Global Accuracy display 34 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  16. Navigation Land Earth Station UPC (NLES) gAGE research g roup of A stronomy and Ge omatics Uplink the EGNOS message with GEO GPS F DN the GEO ranging signal to GEO. F UP GEO L1 L1/L2 GEO L1 Frequencies • Generate GPS-like signal and Band F UP F DN in MHz transmit it to GEO transponder. INMARSAT 3 C 6455.42 3630.42 RF • Maintaining synchronization of ARTEMIS Ku 13875 12748 the message with GPS time. F UP-R Integrity F DN CAL DN L1 CAL L1 F UP Box ON/OFF RF Adapter Monitoring & Conrol F UP-R L1 F DN L1 70.42 MHz L1 CAL L1 10 MHz Distribution 10 MHz Distribution Core Long Frequency Frequency Receiver 1 PPS Loop Standard Standard Rx message message Offset 1 PPS GEO Time Tx message Core EWAN CPF Ground Communication Computer CCF FEE GPS / GEOs / GLONASS phase and code 1 PPS Network raw measurements 35 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  17. Geostationary satellite Broadcast Areas UPC (GBA) gAGE research g roup of A stronomy and Ge omatics 80 60 40 20 0 AOR -E IOR -20 (15.5°W) (65.5°E) INMARSAT INMARSAT -40 Artemis PRN120 PRN131 21.5 E (15°E) -60 -80 -150 -100 -50 100 150 0 50 36 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  18. EGNOS Wide Area Network UPC (EWAN) gAGE research g roup of A stronomy and Ge omatics � Function � Links all EGNOS components RIMS RIMS RIMS RIMS � Link types: RIMS NLES1 RIMS � MCC-MCC MCC1 MCC2 � High capacity � EWAN's backbone NLES4 NLES2 � MCC-NLES MCC4 MCC3 � Ensures link with GEO's RIMS NLES3 RIMS � MCC-RIMS RIMS RIMS RIMS RIMS � Frame Relay or VSAT 37 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  19. E GNOS System Test Bed (ESTB) E GNOS System Test Bed (ESTB) UPC Ready for Application Demonstrations Ready for Application Demonstrations gAGE research g roup of A stronomy and Ge omatics AOR-E IOR ESTB Reference station ESTB Processing Facility NLES Kourou (French Guyana) Hartebeeshoek (South Africa) European European Commission Commission 38 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  20. The EGNOS System Test Bed UPC (ESTB) gAGE research g roup of A stronomy and Ge omatics • Under ESA Contract, European industry has set-up an • Under ESA Contract, European industry has set-up an EGNOS test bed (fully operational since Feb. 2000) . EGNOS test bed (fully operational since Feb. 2000) . The ESTB is a full-scale real-time prototype of the The ESTB is a full-scale real-time prototype of the final EGNOS system. final EGNOS system. • ESTB main objectives are: • ESTB main objectives are: – to have an assessment of the global performance – to have an assessment of the global performance achievable with EGNOS achievable with EGNOS – to analyze in depth specific critical design issues or – to analyze in depth specific critical design issues or trade-off’s between several options trade-off’s between several options – to develop and validate system test methods – to develop and validate system test methods – to demonstrate to the final users the system operation, – to demonstrate to the final users the system operation, – to provide a representative tool for Civil Aviations to – to provide a representative tool for Civil Aviations to build up SBAS practical experience build up SBAS practical experience 39 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  21. EGNOS Operational Milestones UPC gAGE research g roup of A stronomy and Ge omatics Initial trials System Definition Initial Phase Start October 98 January 96 Development/ Validation Validation Operational/ Detailed Design implementation (ESA). certification. 21 months 21 months 9 months 24 months Advanced Operational Capability (AOC) Start July 2000 April 2002 January 2003 Service operative October 98 January 2005 Program ARTES-9 (ESA) 40 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  22. UPC gAGE research g roup of A stronomy and Ge omatics 214 M EUR Switzerland Others Norway 2% 4% 2% France Italy 32% 14% United Kingdom 16% Austria 1% Germany Netherlands 15% 1% Spain Portugal 11% 2% 41 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  23. EGNOS Benefits UPC gAGE research g roup of A stronomy and Ge omatics • Aviation, maritime navigation, Railways. • Road community: car navigation, fleet management, road pricing, autonomous vehicle guidance, etc. • Timing and telecommunications: synchronization of internet nodes; synchronization of mobile base stations, etc. • Agriculture: precision farming, GIS applications, automation of mobile agriculture, etc). • Many others: fishery, search and rescue, land surveying, meteorology, land survey, leisure, etc. 42 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  24. UPC gAGE research g roup of A stronomy and Ge omatics PART II Data Processing 43 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  25. UPC gAGE research g roup of A stronomy and Ge omatics DATA PROCESSING Navigation equations and SBAS Differential Corrections and Integrity 44 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  26. UPC gAGE research g roup of A stronomy and Ge omatics 45 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  27. PSEUDORANGE MODELING UPC gAGE research g roup of A stronomy and Ge omatics j = c ∆ t= c [t rec (T R )-t ems (T S )] P i ∑ ∑ δ = ρ + ⋅ − + δ j j j j P P c ( dt dt ) i i i i where: ∑ δ = + + + + + ε j j j j j rel Trop Ion K TGD i i i i 46 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  28. •Pseudorange modeling UPC ∑ = ρ ρ + ⋅ − + δ j j j j P c ( dt dt ) i i i i k gAGE research g roup of A stronomy and Ge omatics Taylor linearization of ρ : ( ) ( ) ( ) − j − j − j x x y y z z 2 2 2 ρ ρ = j − + − + − j j j j x x y y z z ≈ ρ + i ∆ + i ∆ + i ∆ j x y z 0 0 0 i i i i i i 0 ρ i ρ i ρ i j j j i i i 0 0 0 − − − − − − j j j j j j x x x x y y y y z z z z + ∑ ∆ ∆ ∆ c dt ( i i i i i i = ≈ ρ ρ j + + ∆ ∆ x + + ∆ ∆ y + + ∆ ∆ z + − δ j j x y z j P x y z c dt ( dt ) 0 0 0 0 0 0 i i i i i i i i i io ρ j i ρ j i ρ j i i k ρ ρ ρ j j j 0 i i i io io io 0 0 0 = + ∆ = + ∆ = + ∆ x x x ; y y y ; z z z i io i i io i i io i • Navigation Equations System:   − − − 1 1 1 x x y y z z i i i  1  0 0 0 ∑ ρ ρ ρ  − ρ + − δ  1 1 1 1 1 1 1 P dt   ∑  − ρ + − δ  i i i 1 1 1 1 ∆ P i i dt k   x o o o ∆     x o   i k   i − − − i o ∑ 2 2 2 x x y y z z     − ρ + − δ 2 2 2 2    P dt  ∑ i i i ∆ 2 − ρ 2 + 2 − δ 2 ∆  P dt  = 1 y y 0 0 0 i i k =        o i  k i ρ 2 ρ 2 ρ 2 i  o     i i i  ∆ ........  z  ∆ ........ o o o   z   i     .......... i       ∑ ∑  cdt  − ρ + − δ   n − ρ n + n − δ n P n n dt n n   P dt cdt    i    i k − − − n n n i i k x x y y z z o i   o i i i 1 0 0 0   ρ ρ ρ n n n     i i i o o o 47 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  29. • Navigation Solution: UPC = ⋅ Y G X ∆   x gAGE research g roup of A stronomy and Ge omatics i   ∆ y   = i X   ∆ � z NOTE: ( ) where i −   1 = T T X G W G G W Y cdt   i • NOTE ∆   x i     ∆ − − − y j j j x x y y z z   i = X i i i =  j G , , , 1  0 0 0   ∆ z ρ ρ ρ j j j   i     i i i o o o cdt   i ∆   N i [ ]   ∆ E G = j j j j j j cos El cos Az , cos El sin Az , sin El , 1   = i X i i i i i   ∆ U i   cdt   i 48 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  30. PROTECTION LEVELS: = ⋅ Y G X UPC � ( ) [ ] = ∆ ∆ ∆ − X N , E , U , cdt 1 = T T X G W G G W Y gAGE research g roup of A stronomy and Ge omatics   2 d d d d NE NV NT   N 2 d d d d   2 P   + − 2 2 2 2 = =  NE EV ET d d d d T SP S E  x y 2 E E d d d d = + + 2 HPL 6.00   d N N NV EV VT     V NE 2 2 2 d d d d       NT ET VT T ( ) -1 T T S = G WG G W = VPL 5.33 d V   w 0 1   ( ) =  − W � 1 = σ + σ + σ + σ 2 2 2 2 w  ε = i i UDRE , i UIRE , | 0 i air , i tropo ,   0 w   N σ K σ -K σ - σ   σ 2 0 1   X ~ N ( 0 , 1 ) σ = σ + σ + σ + σ 2 2 2 2 2 =  � P  i i , flt i UIRE , i air , i tropo , y − > = 7 P ( X 5 . 33 ) 10   σ 2 0   N 49 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  31. Users know the receiver-satellites geometry and can compute bounds on UPC the horizontal and vertical position errors. These bounds are called Protection Levels (HPL and VPL). They provide good confidence (10 -7 /hour probability) that the true position is within a gAGE research g roup of A stronomy and Ge omatics bubble around the computed position. Protection Level True Error P(VPE>VPL) < 10 -7 /sample Tail area Probability -K σ - σ σ K σ N 2 σ ∑ σ = σ + σ + σ + σ 2 2 2 2 2 = 2 VPL K s i i , flt i , UIRE i , air i , tropo V V i i = i 1 GEOMETRY 50 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  32. Fast Corrections Ephemeris + UPC MT1, MT2,5,24, MT6, Long Term Corrections + + MT25, MT7, MT12, gAGE research g roup of A stronomy and Ge omatics UDRE MT9 Clocks + Degradation Param. MT10 IONO Corrections + MT18 IONO GIVE MT26 + Degradation Param. = + − − ρ ρ + + ∆ ∆ + + − + + * * sat sat sat IONO Y C 1 PRC PRC t t dt rel TGD IONO TROP σ = σ σ + σ σ + σ + σ 2 2 2 2 2 2 2 flt flt UIRE UIRE air tropo 51 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  33. Fast and Long-Term Correction Degradation UPC ( )  σ ε ε ε ε 2 + + + + = , if RSS 0 ( MT 10)  σ UDRE rrc er =  fc ltc 2 UDRE gAGE research g roup of A stronomy and Ge omatics i flt , σ ε ε ε ε  + + + + = 2 2 2 2 2 , if RSS 1 ( MT 10)  UDRE fc rrc ltc er UDRE MT25 ( ) 2 − + t t t ε u = lat a fc 2 MT2-6,24 t , v orv ltc 0 1  −  ε t tltc IODF UDRE , = C floor   MT7 i ltc v , 0 ltc v , 0 I =   t t ltc v , 0 u of a I t ≠ i fc i , lat ( when IODF 3 ) < < +  0 , if t t t I  ε 0 0 ltc v _ 1 =  { } ltc v , 1 MT10 + − − − C C max 0, t t t , t I , otherwise   lts lsb _ ltc v _ 1 0 0 ltc v _ 1 B , C , C ,  0 rrc ltc lsb _ ltc v _ 1 Neither fast nor long term corrections ε =  have time out for precision approach I , C , I , er C  Otherwise ltc v _ 1 ltc v _ 0 ltc v _ 0 er C , RSS , ≠ ≠ 3 3 IODF , IODF IODF , IODF = er UDRE  current previous 0 , 1 − current previous ( IODF IODF )  mod2 ε current previous C , C , =  ( ) ( ) a I rrc B iono ramp _ iono step _ + − ≠ t t , 1 fc −  rrc ( IODF IODF ) ∆  t 4 of mod2 current previous I , RSS iono iono 52 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  34. Degradation of Ionospheric Corrections UPC 1 −   2   2 σ = σ 2 2 2 R cos E F =  −  F 1 e  gAGE research g roup of A stronomy and Ge omatics pp +  R h  UIRE pp UIVE     e I N MT10 ( ) ∑ σ 2 = σ 2 = W x , y , N 4 or 3 UIVE n pp pp n ionogrid , B , C , C , = n 1 rrc ltc lsb _ ltc v _ 1 I , C , I , σ ε ( ) ltc v _ 1 ltc v _ 0 ltc v _ 0  2 + = , if RSS 0 ( MT 10)  σ iono =  GIVE iono 2 C , RSS , ionogrid σ ε er UDRE + = 2 2  , if RSS 1 ( MT 10)  iono GIVE iono C , C , iono ramp _ iono step _ I , RSS iono iono ( ) − ε t tiono ( ) = + − MT26 C floor C t t iono iono step _ I iono ramp _ iono iono t GIVE iono , i 53 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  35. SBAS Differential Corrections and Integrity: UPC The RTCA/MOPS gAGE research g roup of A stronomy and Ge omatics 54 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  36. Message Format UPC gAGE research g roup of A stronomy and Ge omatics DIRECTION OF DATA FLOW FROM SATELLITE; MOST SIGNIFICANT BIT (MSB) TRANSMITTED FIRST 250 BITS - 1 SECOND REPEAT FOR 12 IODP (2 BITS) REPEAT FOR 12 MORE SATELLITES MORE SATELLITES UDREI PRC 212-BIT DATA FIELD f 24-BITS 13 12-BIT FAST CORRECTIONS 13 4-BIT UDREI s PARITY IODF (2 BITS) 6-BIT MESSAGE TYPE IDENTIFIER (= 2, 3, 4 & 5) 8-BIT PREAMBLE OF 24 BITS TOTAL IN 3 CONTIGUOUS BLOCKS The corrections, even for individual satellites are distributed across several individual messages. • 250 bits • One Message per second • All messages have identical format 55 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  37. EGNOS Broadcast Messages (ICAO SARPS) UPC Don't use this SBAS signal for anything (for SBAS testing) MSG 0 gAGE research g roup of A stronomy and Ge omatics PRN Mask assignments, set up to 51 of 210 bits MSG 1 Fast corrections MSG 2 to 5 Integrity information MSG 6 MSG 7 Fast correction degradation factor MSG 8 Reserved for future messages MSG 9 GEO navigation message ( X , Y , Z , time, etc.) Degradation Parameters MSG 10 Reserved for future messages MSG 11 SBAS Network Time/UTC offset parameters MSG 12 Reserved for future messages MSG 13 to 16 Many Message Types GEO satellite almanacs MSG 17 Coordinated Through MSG 18 Ionospheric grid point masks Issues Data (IOD) MSG 19 to 23 Reserved for future messages MSG 24 Mixed fast corrections/long term satellite error corrections Long term satellite error corrections MSG 25 Ionospheric delay corrections MSG 26 SBAS outside service volume degradation MSG 27 Reserved for future messages MSG 28 to 61 MSG 62 Internal Test Message MSG 63 Null Message 56 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  38. Fast Corrections Ephemeris + UPC MT1, MT2,5,24, MT6, Long Term Corrections + + MT25, MT7, MT12, gAGE research g roup of A stronomy and Ge omatics UDRE MT9 Clocks + Degradation Param. MT10 IONO Corrections + MT18 IONO GIVE MT26 + Degradation Param. = + − − ρ ρ + + ∆ ∆ + + − + + * * sat sat sat IONO Y C 1 PRC PRC t t dt rel TGD IONO TROP σ = σ σ + σ σ + σ + σ 2 2 2 2 2 2 2 flt flt UIRE UIRE air tropo 57 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  39. Issues of Data (IOD) UPC gAGE research g roup of A stronomy and Ge omatics No IOD: IOD Fast IODE Long-term k GPS k repeat msg, Corrections Corrections Ephemeris small (2 - 5, 24) (25) changes IODC k IODP IODF IODG j GPS k IODP Clock PRN Integrity GLONASS Mask Information DATA (1) (6) IODS IODP Acceleration Service Information Message (7) (27) Ionospheric Ionospheric Mask Corrections IODI (18) (26) 58 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  40. Message Time-Outs: UPC Users can operate even when missing Messages gAGE research g roup of A stronomy and Ge omatics • Prevents Use of Very Old Data • Confidence Degrades When Data is Lost • IODF: Detect Missing Fast Corrections 1 second System Latency Last bit of message: The Correction is tof: Time of applicability estimated by the tof+1sec (1st bit of message) master station Correction time-Out 59 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  41. Associated Maximum En Route, Precision UPC Data Message Update Interval Terminal, NPA Approach Types (seconds) Timeout (seconds) Timeout (seconds) WAAS in Test Mode 0 6 N/A N/A gAGE research g roup of A stronomy and Ge omatics PRN Mask 1 60 None None UDREI 2-6, 24 6 18 12 Fast Corrections 2-5, 24 60 (*) (*) Long Term 24, 25 120 360 240 Corrections GEO Nav. Data 9 120 360 240 Fast Correction 7 120 360 240 Degradation Weighting Factors 8 120 240 240 Degradation 10 120 360 240 Parameters Ionospheric Grid 18 300 None None Mask Ionospheric 26 300 600 600 Corrections UTC Timing Data 12 300 None None Almanac Data 17 300 None None (*) Fast Correction Time-Out intervals are given in MT7 [between 12 to 120 sec] 60 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  42. PRN MASK (MT01) UPC gAGE research g roup of A stronomy and Ge omatics 1 2 3 4 5 6 . 38 . 120 . 210 Bit No Value 0 1 0 1 1 0 1 1 0 GPS GPS GPS GLONASS AORE PRN PRN 2 PRN 4 PRN 5 Slot 1 PRN 120 PRN mask 1 2 3 21 29 Number Each MT01 contains its associated IODP Assignment Up to 51 satellites in 210 slots. PRN Slot 1-37 GPS/GPS Reserved Note: Each Correction set in 38-61 GLONASS MT 2-5,5,6,7,24,25 its 62-119 Future GNSS characterized by its PRN-Mask 120-138 GEO/SBAS number, between 1 to 51. 139-210 Future GNSS/GEO/SBAS/Pseudolites 61 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  43. Fast Corrections (MT2-5,24) UPC gAGE research g roup of A stronomy and Ge omatics • Primarily Removes SA – Common to ALL users – Up to 13 Satellites Per Message – Pseudorange Correction /confidence Bound – Range Rate Formed by Differencing – UDRE degrades Over Time • Acceleration Term in MT 7 • Reset when new Message Received 62 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  44. Fast Corrections (MT2-5) UPC gAGE research g roup of A stronomy and Ge omatics = + − PRC ( t ( ) PRC t ) PRC RRC ( t t ) = + − ρ + ∆ + * sat sat Y C 1 PRC PRC t dt n n n − PRC PRC + − + + rel TGD IONO TROP = RRC n o n − t t n o DIRECTION OF DATA FLOW FROM SATELLITE; MOST SIGNIFICANT BIT (MSB) TRANSMITTED FIRST 250 BITS - 1 SECOND REPEAT FOR 12 IODP (2 BITS) REPEAT FOR 12 MORE SATELLITES MORE SATELLITES UDREI PRC f 24-BITS 13 12-BIT FAST CORRECTIONS 13 4-BIT UDREI s PARITY IODF (2 BITS) 6-BIT MESSAGE TYPE IDENTIFIER (= 2, 3, 4 & 5) 8-BIT PREAMBLE OF 24 BITS TOTAL IN 3 CONTIGUOUS BLOCKS [ ] ) RSS UDRE = ( 0 MT 10 σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , UDRE fc rrc ltc er 63 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  45. Message Type 6 UPC DIRECTION OF DATA FLOW FROM SATELLITE; MOST SIGNIFICANT BIT (MSB) TRANSMITTED FIRST gAGE research g roup of A stronomy and Ge omatics 250 BITS - 1 SECOND IODF 2 , IODF 3 , IODF 4 & IODF 5 (2 BITS EACH) 24-BITS 51 4-BIT UDREI s PARITY 6-BIT MESSAGE TYPE IDENTIFIER (= 6) 8-BIT PREAMBLE OF 24 BITS TOTAL IN 3 CONTIGUOUS BLOCKS Alarm conditions are • Serves Two Purposes indicated with IODF=3 – Alarm for Multiple Satellites • Includes UDREs for all 51 Satellites – Update UDRE in Between Fast Corrections • More efficient Use of Bandwidth • The receipt of MT6 with matching IODF<3 is equivalent to another reception of last fast correction. 64 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  46. Evaluation of UDREI UPC σ 2 Comment: UDREI i UDRE i Meters i,UDRE Meters 2 gAGE research g roup of A stronomy and Ge omatics • With SA=off, the FC can be sent 0 0.75 0.0520 less frequently than 6sec, but it is 1 1.0 0.0924 still necessary to update the 2 1.25 0.1444 “integrity status (UDREs)” at the 3 1.75 0.2830 high rate. 4 2.25 0.4678 • Prec. App: UDRE time-out =12sec 5 3.0 0.8315 FC time-out between 12 -120 sec. 6 3.75 1.2992 7 4.5 1.8709 8 5.25 2.5465 9 6.0 3.3260 The MOPS (RTCA Do 229A) 2.1.1.5.2, 10 7.5 5.1968 establish the satellites deselecting for: - UDRE=14 (not monitored) 11 15.0 20.7870 and 12 50.0 230.9661 - UDRE=15 (don’t use) 13 150.0 2078.695 2.1.4.7.1: In addition, for 14 Not Monitored Not Monitored Precision Approach: UDRE<11. 15 Do Not Use Do Not Use 65 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  47. UDRE degradation: The fast correction was estimated UDRE degradation: The fast correction was estimated UPC by master station at some previous time t n -t lat by master station at some previous time t n -t lat gAGE research g roup of A stronomy and Ge omatics 1 sec I fc (MT 7) t lat (MT 7) t n: Time of applicability The Correction is estimated by the (1st bit of message) master station σ σ ε ε ε ε FC Correction = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er time-Out Acceleration Term Lost Fast Correction Term ≠ ( when IODF 3 ) (Always included)   a I B ( ) a ( )   fc ε ≡ + − rrc t t 2 ε ≡ − + t t t   rrc n − 4 t t fc UDRE lat   2 n o 66 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  48. UPC gAGE research g roup of A stronomy and Ge omatics EXAMPLE 1 (From WAAS MOPS: Practical Examples. Todd Walter) 67 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  49. σ 2 σ 2 ε 2 ε 2 Time Last Last Last UDRE flt fc rrc (sec) Mess. Mess Mess. PRC(t) RRC n Error UPC Time (s) PRC n IODF n t (m) (m) tn (m) 0 -1 0.500 0 - - - - - - gAGE research g roup of A stronomy and Ge omatics 3 -1 0.500 0 - - - - - - 6 5 -2.125 1 -2.563 -0.4375 0.505 0.0957 0.0924 0.0033 0 9 5 -2.125 1 -3.875 -0.4375 1.045 0.1141 0.0924 0.0217 0 12 11 -3.125 2 -3.292 -0.1667 -0.140 0.0957 0.0924 0.0033 0 15 11 -3.125 2 -3.792 -0.1667 -0.171 0.1141 0.0924 0.0217 0 18 17 -4.000 0 -4.146 -0.1458 0.238 0.0957 0.0924 0.0033 0 21 17 -4.000 0 -4.583 -0.1458 0.373 0.1141 0.0924 0.0217 0 24 17 -4.000 0 -5.021 -0.1458 0.893 0.1699 0.0924 0.0775 0 27 17 -4.000 0 -5.458 -0.1458 1.584 0.2954 0.0924 0.2030 0 30 29 -3.500 2 -3.458 0.0417 0.008 0.0964 0.0924 0.0033 0.00069 33 29 -3.500 2 -3.333 -0.0417 0.479 0.1252 0.0924 0.0217 0.01111 36 35 -2.750 0 -2.625 0.1250 0.537 0.0957 0.0924 0.0033 0 39 35 -2.750 0 -2.250 0.1250 1.100 0.1141 0.0924 0.0217 0 FC update rate 6 sec Param Mes. Type value Param Mes. Type value σ 2 MT 2 0.0924 t lat (sec) MT 7 4 UDRE (m2) a (mm/s2) MT 7 4.60 B rrc (m) MT10 0.15 I fc (sec) MT 7 12 RSS UDRE MT10 1 68 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  50. σ 2 σ 2 ε 2 ε 2 Time Last Last Last UDRE flt fc rrc (sec) Mess. Mess Mess. PRC(t) RRC n Error UPC Time (s) PRC n IODF n t (m) (m) tn (m) 0 -1 0.500 0 - - - - - - gAGE research g roup of A stronomy and Ge omatics 3 -1 0.500 0 - - - - - - 6 5 -2.125 1 -2.563 -0.4375 0.505 0.0957 0.0924 0.0033 0 9 5 -2.125 1 -3.875 -0.4375 1.045 0.1141 0.0924 0.0217 0 12 11 -3.125 2 -3.292 -0.1667 -0.140 0.0957 0.0924 0.0033 0 15 11 -3.125 2 -3.792 -0.1667 -0.171 0.1141 0.0924 0.0217 0 18 17 -4.000 0 -4.146 -0.1458 0.238 0.0957 0.0924 0.0033 0 21 17 -4.000 0 -4.583 -0.1458 0.373 0.1141 0.0924 0.0217 0 24 17 -4.000 0 -5.021 -0.1458 0.893 0.1699 0.0924 0.0775 0 27 17 -4.000 0 -5.458 -0.1458 1.584 0.2954 0.0924 0.2030 0 30 29 -3.500 2 -3.458 0.0417 0.008 0.0964 0.0924 0.0033 0.00069 33 29 -3.500 2 -3.333 -0.0417 0.479 0.1252 0.0924 0.0217 0.01111 36 35 -2.750 0 -2.625 0.1250 0.537 0.0957 0.0924 0.0033 0 39 35 -2.750 0 -2.250 0.1250 1.100 0.1141 0.0924 0.0217 0 σ σ ε ε ε ε = + − PRC ( t ) = PRC + RRC ( t − t ) PRC ( t ) PRC RRC ( t t ) 2 = 2 + 2 + 2 + 2 + 2 n n n i flt , U D R E fc rrc ltc er n n n − − PRC PRC PRC PRC = = n o RRC RRC n o a   a a I ( ) B n ( ) − ( ) n t − t 2 ε ≡ − + 2 t t ε ≡ − +   t t t ε ≡ fc + − t t t rrc t t   n o n o fc n lat fc n lat rrc n 2 − 2 4 t t   n o − RRC Time Out Ifc= 12sec ε rrc =0 when no mess. are missed − ∆ = − > PRC Time Out t t t I UDRE Time-Out n o fc − > + t t I 1 t-t n >13 − > ∆ t t 8 t n fc n 69 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  51. σ 2 σ 2 ε 2 ε 2 Time Last Last Last UDRE flt fc rrc (sec) Mess. Mess Mess. PRC(t) RRC n Error UPC Time (s) PRC n IODF n t (m) (m) tn (m) 0 -1 0.500 0 - - - - - - gAGE research g roup of A stronomy and Ge omatics 3 -1 0.500 0 - - - - - - 6 5 -2.125 1 -2.563 -0.4375 0.505 0.0957 0.0924 0.0033 0 9 5 -2.125 1 -3.875 -0.4375 1.045 0.1141 0.0924 0.0217 0 12 11 -3.125 2 -3.292 -0.1667 -0.140 0.0957 0.0924 0.0033 0 15 11 -3.125 2 -3.792 -0.1667 -0.171 0.1141 0.0924 0.0217 0 18 17 -4.000 0 -4.146 -0.1458 0.238 0.0957 0.0924 0.0033 0 21 17 -4.000 0 -4.583 -0.1458 0.373 0.1141 0.0924 0.0217 0 24 17 -4.000 0 -5.021 -0.1458 0.893 0.1699 0.0924 0.0775 0 27 17 -4.000 0 -5.458 -0.1458 1.584 0.2954 0.0924 0.2030 0 30 29 -3.500 2 -3.458 0.0417 0.008 0.0964 0.0924 0.0033 0.00069 33 29 -3.500 2 -3.333 -0.0417 0.479 0.1252 0.0924 0.0217 0.01111 36 35 -2.750 0 -2.625 0.1250 0.537 0.0957 0.0924 0.0033 0 39 35 -2.750 0 -2.250 0.1250 1.100 0.1141 0.0924 0.0217 0 σ σ ε ε ε ε = + − PRC ( t ) PRC RRC ( t t ) 2 = 2 + 2 + 2 + 2 + 2 n n n i flt , U D R E fc rrc ltc er − PRC PRC = RRC n o     a a a I I B B ( ) n ( ( ) ) − t t 2 ε ≡ − +     ε ε ≡ ≡ fc fc + + − − t t t rrc rrc t t t t     n o fc n lat rrc rrc n n − − 2 4 t t 4 t t     n n o o − RRC Time Out Ifc= 12sec ε rrc =0 when no mess. are missed − ∆ = − > PRC Time Out t t t I UDRE Time-Out n o fc − > + t t I 1 t-t n >13 − > ∆ t t 8 t n fc n 70 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  52. σ 2 σ 2 ε 2 ε 2 Time Last Last Last UDRE flt fc rrc (sec) Mess. Mess Mess. PRC(t) RRC n Error UPC Time (s) PRC n IODF n t (m) (m) tn (m) 0 -1 0.500 0 - - - - - - gAGE research g roup of A stronomy and Ge omatics 3 -1 0.500 0 - - - - - - The IODEs are not in 6 5 -2.125 1 -2.563 -0.4375 0.505 0.0957 0.0924 0.0033 0 sequence and the user is aware that a FC is missing. 9 5 -2.125 1 -3.875 -0.4375 1.045 0.1141 0.0924 0.0217 0 RRC is formed using IODFs 12 11 -3.125 2 -3.292 -0.1667 -0.140 0.0957 0.0924 0.0033 0 out of sequence => ε 2 15 11 -3.125 2 -3.792 -0.1667 -0.171 rrc 0.1141 0.0924 0.0217 0 18 17 -4.000 0 -4.146 -0.1458 0.238 0.0957 0.0924 0.0033 0 21 17 -4.000 0 -4.583 -0.1458 0.373 0.1141 0.0924 0.0217 0 24 17 -4.000 0 -5.021 -0.1458 0.893 0.1699 0.0924 0.0775 0 27 17 -4.000 0 -5.458 -0.1458 1.584 0.2954 0.0924 0.2030 0 30 29 -3.500 2 -3.458 0.0417 0.008 0.0964 0.0924 0.0033 0.00069 33 29 -3.500 2 -3.333 -0.0417 0.479 0.1252 0.0924 0.0217 0.01111 36 35 -2.750 0 -2.625 0.1250 0.537 0.0957 0.0924 0.0033 0 39 35 -2.750 0 -2.250 0.1250 1.100 0.1141 0.0924 0.0217 0 σ σ ε ε ε ε = + − PRC ( t ) PRC RRC ( t t ) 2 = 2 + 2 + 2 + 2 + 2 n n n i flt , U D R E fc rrc ltc er − PRC PRC = RRC n o   a a I B ( ) n ( ) − t t 2 ε ≡ − +   ε ≡ fc + − t t t rrc t t   n o fc n lat rrc n − 2 4 t t   n o − RRC Time Out Ifc= 12sec ε rrc =0 when no mess. are missed − ∆ = − > PRC Time Out t t t I UDRE Time-Out n o fc − > + t t I 1 t-t n >13 − > ∆ t t 8 t n fc n 71 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  53. UPC gAGE research g roup of A stronomy and Ge omatics EXAMPLE 2 (From WAAS MOPS: Practical Examples. Todd Walter) 72 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  54. σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er a ( ) ε ≡ − + 2 t t t Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 fc UDRE lat UPC 2 flt UDRE fc rrc (sec) Time (s) Type IODF   a I B ( ) 0 -1 2 0 -1 0.052014 0.0520 0.000014 0 ε ≡  fc +  − rrc t t   rrc − n 4 t t   6 5 6 0 5 0.052014 0.0520 0.000014 0 n o gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 18 17 6 0 17 0.092414 0.0924 0.000014 0 ε rrc =0 when no 24 23 6 0 23 0.092414 0.0924 0.000014 0 messages are missed 30 29 2 1 29 0.052014 0.0520 0.000014 0 36 35 6 1 35 0.052014 0.0520 0.000014 0 42 41 6 1 41 0.052014 0.0520 0.000014 0 48 47 - - 41 0.052329 0.0520 0.000329 0 54 53 6 1 53 0.092414 0.0924 0.000014 0 60 59 - - 53 0.092729 0.0924 0.000329 0 Param Mes. value 66 65 6 2 53 0.094297 0.0924 0.001897 0 Type 72 71 6 2 53 0.092696 0.0924 0.006296 0 σ 2 0.0520 78 77 6 2 53 0.108310 0.0924 0.015910 0 UDRE MT 2,6 0.0924 84 83 6 2 53 0.430000 0.0924 0.337600 0 (m2) a 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 MT 7 0.30 (mm/s2) 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 I fc (sec) MT 7 66 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 t l (sec) MT 7 4 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 B rrc (m) MT10 0.15 120 119 2 1 119 0.052014 0.0520 0.000014 0 126 125 6 1 125 0.052014 0.0520 0.000014 0 RSS UDRE MT10 1 132 131 6 1 131 0.052014 0.0520 0.000014 0 138 137 6 3 119 0.092696 0.0924 0.006296 0 MT 2 update rate 30 sec 144 143 6 3 119 0.108310 0.0924 0.015910 0 MT 6 update rate 6 sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 73 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  55. σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er a ( ) ε ≡ − + 2 t t t Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 fc UDRE lat UPC 2 flt UDRE fc rrc t (s) t2, t6 (s) Type IODF     a a I I B B ( ( ) ) 0 -1 2 0 -1 0.052014 0.0520 0.000014 0 ε ε ≡ ≡   fc fc + +   − − rrc rrc t t t t     rrc rrc − − n n 4 4 t t t t     6 5 6 0 5 0.052014 0.0520 0.000014 0 n n o 0 gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 18 17 6 0 17 0.092414 0.0924 0.000014 0 The receipt of MT 6 (with 24 23 6 0 23 0.092414 0.0924 0.000014 0 IODF<3) is equivalent 30 29 2 1 29 0.052014 0.0520 0.000014 0 to another reception of 36 35 6 1 35 0.052014 0.0520 0.000014 0 last fast correction 42 41 6 1 41 0.052014 0.0520 0.000014 0 48 47 - - 41 0.052329 0.0520 0.000329 0 UDRE Time-Out 54 53 6 1 53 0.092414 0.0924 0.000014 0 t-t UDRE >13 60 59 - - 53 0.092729 0.0924 0.000329 0 66 65 6 2 53 0.094297 0.0924 0.001897 0 72 71 6 2 53 0.092696 0.0924 0.006296 0 = + − PRC ( t ) PRC RRC ( t t ) n n n 78 77 6 2 53 0.108310 0.0924 0.015910 0 − PRC PRC 84 83 6 2 53 0.430000 0.0924 0.337600 0 = RRC n o 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 n − t t 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 n o 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 − RRC Time Out 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 ∆ = − > t t t I 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 n o fc 120 119 2 1 119 0.052014 0.0520 0.000014 0 − > ∆ t t 8 t n 126 125 6 1 125 0.052014 0.0520 0.000014 0 − PRC Time Out 132 131 6 1 131 0.052014 0.0520 0.000014 0 − > + 138 137 6 3 119 0.092696 0.0924 0.006296 0 t t I 1 UDRE fc 144 143 6 3 119 0.108310 0.0924 0.015910 0 Ifc= 66sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 74 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  56. σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er a ( ) ε ≡ − + 2 t t t Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 fc UDRE lat UPC 2 flt UDRE fc rrc t (s) t2, t6 (s) Type IODF   a I B ( ) 0 -1 2 0 -1 0.052014 0.0520 0.000014 0 ε ≡  fc +  − rrc t t   rrc − n 4 t t   6 5 6 0 5 0.052014 0.0520 0.000014 0 n o gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 18 17 6 0 17 0.092414 0.0924 0.000014 0 MT 6 is missed and 24 23 6 0 23 0.092414 0.0924 0.000014 0 t UDRE remains at 41s, 30 29 2 1 29 0.052014 0.0520 0.000014 0 inflating the ε 2 fc term 36 35 6 1 35 0.052014 0.0520 0.000014 0 42 41 6 1 41 0.052014 0.0520 0.000014 0 UDRE Time-Out 48 47 - - 41 0.052329 0.0520 0.000329 0 t-t UDRE >13 54 53 6 1 53 0.092414 0.0924 0.000014 0 60 59 - - 53 0.092729 0.0924 0.000329 0 66 65 6 2 53 0.094297 0.0924 0.001897 0 72 71 6 2 53 0.092696 0.0924 0.006296 0 = + − PRC ( t ) PRC RRC ( t t ) n n n 78 77 6 2 53 0.108310 0.0924 0.015910 0 − PRC PRC 84 83 6 2 53 0.430000 0.0924 0.337600 0 = RRC n o 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 n − t t 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 n o 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 − RRC Time Out 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 ∆ = − > t t t I 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 n o fc 120 119 2 1 119 0.052014 0.0520 0.000014 0 − > ∆ t t 8 t n 126 125 6 1 125 0.052014 0.0520 0.000014 0 − PRC Time Out 132 131 6 1 131 0.052014 0.0520 0.000014 0 − > + 138 137 6 3 119 0.092696 0.0924 0.006296 0 t t I 1 UDRE fc 144 143 6 3 119 0.108310 0.0924 0.015910 0 Ifc= 66sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 75 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  57. σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er a ( ) ε ≡ − + 2 t t t Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 fc UDRE lat UPC 2 flt UDRE fc rrc t (s) t2, t6 (s) Type IODF   a I B ( ) 0 -1 2 0 -1 0.052014 0.0520 0.000014 0 ε ≡  fc +  − rrc t t   rrc − n 4 t t   6 5 6 0 5 0.052014 0.0520 0.000014 0 n o gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 18 17 6 0 17 0.092414 0.0924 0.000014 0 IODF out of sequence, 24 23 6 0 23 0.092414 0.0924 0.000014 0 t UDRE will remain at 53s 30 29 2 1 29 0.052014 0.0520 0.000014 0 until the receip of the next fast correction 36 35 6 1 35 0.052014 0.0520 0.000014 0 42 41 6 1 41 0.052014 0.0520 0.000014 0 UDRE Time-Out 48 47 - - 41 0.052329 0.0520 0.000329 0 54 53 6 1 53 0.092414 0.0924 0.000014 0 t-t 6 >13 60 59 - - 53 0.092729 0.0924 0.000329 0 66 65 6 2 53 0.094297 0.0924 0.001897 0 72 71 6 2 53 0.092696 0.0924 0.006296 0 = + − PRC ( t ) PRC RRC ( t t ) n n n 78 77 6 2 53 0.108310 0.0924 0.015910 0 − PRC PRC 84 83 6 2 53 0.430000 0.0924 0.337600 0 = RRC n o 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 n − t t 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 n o 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 − RRC Time Out 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 ∆ = − > t t t I 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 n o fc NOTE: Because the IODF was not set to 3 and the user has NOTE: Because the IODF was not set to 3 and the user has 120 119 2 1 119 0.052014 0.0520 0.000014 0 − > ∆ t t 8 t not missed 4 messages in row, they know that the service not missed 4 messages in row, they know that the service n 126 125 6 1 125 0.052014 0.0520 0.000014 0 provided is monitoring this and other combinations of old provided is monitoring this and other combinations of old − PRC Time Out 132 131 6 1 131 0.052014 0.0520 0.000014 0 data and it is save for them to continue it use. data and it is save for them to continue it use. − > + 138 137 6 3 119 0.092696 0.0924 0.006296 0 t t I 1 Thence, it is using t-t 6 >13, instead of t-t UDRE >13 for Thence, it is using t-t 6 >13, instead of t-t UDRE >13 for UDRE fc 144 143 6 3 119 0.108310 0.0924 0.015910 0 UDRE Time-Out . UDRE Time-Out . Ifc= 66sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 76 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  58. σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , U D R E fc rrc ltc er a ( ) ε ≡ − + 2 t t t Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 fc UDRE lat UPC 2 flt UDRE fc rrc t (s) t2, t6 (s) Type IODF   a I B ( ) 0 -1 2 0 -1 0.052014 0.0520 0.000014 0 ε ≡  fc +  − rrc t t   rrc − n 4 t t   6 5 6 0 5 0.052014 0.0520 0.000014 0 n o gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 The user is aware of a FC is 18 17 6 0 17 0.092414 0.0924 0.000014 0 missing. 24 23 6 0 23 0.092414 0.0924 0.000014 0 RRC is formed using IODFs 30 29 2 1 29 0.052014 0.0520 0.000014 0 out of sequence => ε 2 36 35 6 1 35 0.052014 0.0520 0.000014 0 rrc 42 41 6 1 41 0.052014 0.0520 0.000014 0 = + − PRC ( t ) PRC RRC ( t t ) 48 47 - - 47 0.052329 0.0520 0.000329 0 n n n 54 53 6 1 53 0.092414 0.0924 0.000014 0 − PRC PRC = RRC n o 60 59 - - 53 0.092729 0.0924 0.000329 0 n − t t 66 65 6 2 53 0.094297 0.0924 0.001897 0 n o 72 71 6 2 53 0.092696 0.0924 0.006296 0 78 77 6 2 53 0.108310 0.0924 0.015910 0 − RRC Time Out 84 83 6 2 53 0.430000 0.0924 0.337600 0 ∆ = − > t t t I 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 n o fc − > ∆ 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 t t 8 t n 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 − PRC Time Out 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 − > + t t I 1 120 119 2 1 119 0.052014 0.0520 0.000014 0 UDRE fc 126 125 6 1 125 0.052014 0.0520 0.000014 0 UDRE Time-Out 132 131 6 1 131 0.052014 0.0520 0.000014 0 t-t UDRE >13 138 137 6 3 119 0.092696 0.0924 0.006296 0 144 143 6 3 119 0.108310 0.0924 0.015910 0 Ifc= 66sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 77 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  59. σ σ ε ε ε ε = + + + + a t 2 2 2 2 2 2 ( ) 2 ε ≡ − + i flt , U D R E fc rrc ltc er 119 t fc lat 2 Time Last Mes Last Mess Last Mes t UDRE σ 2 σ 2 ε 2 ε 2 UPC flt UDRE fc rrc t (s) t2, t6 (s) Type IODF = ∆ − ≠ IO DF 3; if t I / 2 0 fc 0 -1 2 0 -1 0.052014 0.0520 0.000014 0   ∆ − a t I / 2 B ( )  fc  ε ≡ + − rrc t 119 6 5 6 0 5 0.052014 0.0520 0.000014 0 rrc   ∆ 2 t   gAGE research g roup of A stronomy and Ge omatics 12 11 6 0 11 0.052014 0.0520 0.000014 0 Alarm Condition (IODF=3) 18 17 6 0 17 0.092414 0.0924 0.000014 0 t UDRE backs to 119 (the last 24 23 6 0 23 0.092414 0.0924 0.000014 0 received Fast Correction) 30 29 2 1 29 0.052014 0.0520 0.000014 0 36 35 6 1 35 0.052014 0.0520 0.000014 0 UDRE Time-Out 42 41 6 1 41 0.052014 0.0520 0.000014 0 t-t 6 >13 48 47 - - 47 0.052329 0.0520 0.000329 0 54 53 6 1 53 0.092414 0.0924 0.000014 0 60 59 - - 53 0.092729 0.0924 0.000329 0 = + − PRC ( t ) PRC RRC ( t t ) n n n 66 65 6 2 53 0.094297 0.0924 0.001897 0 NOTE: − PRC PRC 72 71 6 2 53 0.092696 0.0924 0.006296 0 When IODF in either message is set to 3, then = RRC n o 78 77 6 2 53 0.108310 0.0924 0.015910 0 n − t UDRE =t n and is not updated to the time of MT6 t t 84 83 6 2 53 0.430000 0.0924 0.337600 0 n o 90 89 2 0 89 0.052074 0.0520 0.000014 0.000060 96 95 6 0 95 0.054732 0.0520 0.000014 0.002720 − RRC Time Out 102 101 6 0 101 0.061394 0.0520 0.000014 0.009380 ∆ = − > t t t I 108 107 6 0 107 0.122814 0.0924 0.000014 0.030400 n o fc 114 113 6 0 113 0.027104 0.0924 0.000014 0.034690 − > ∆ t t 8 t n 120 119 2 1 119 0.052014 0.0520 0.000014 0 − PRC Time Out 126 125 6 1 125 0.052014 0.0520 0.000014 0 − > + t 119 I 1 132 131 6 1 131 0.052014 0.0520 0.000014 0 fc 138 137 6 3 119 0.092696 0.0924 0.006296 0 144 143 6 3 119 0.108310 0.0924 0.015910 0 Ifc= 66sec 150 149 2 2 149 0.052014 0.0520 0.000014 0 78 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  60. Long-Term Corrections (MT25, 24) UPC gAGE research g roup of A stronomy and Ge omatics • Primarily Correct Ephemeris Errors – Also removes Slowly Varying Clock - And discontinuities in Broadcast Ephemeris – Separate Degradation Factors for Lost Messages • For GEO are contained in MT9 79 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  61. Long-Term Corrections (MT25) UPC DIRECTION OF DATA FLOW FROM SATELLITE; MOST SIGNIFICANT BIT (MSB) TRANSMITTED FIRST gAGE research g roup of A stronomy and Ge omatics 250 BITS - 1 SECOND VELOCITY CODE = 0 IODP (2 BITS) 24-BITS a f0 δ x δ y δ z δ x δ y δ z a f0 PARITY S SECOND HALF OF MESSAGE ISSUE OF DATA; SEE [1] PRN MASK NUMBER 6-BIT MESSAGE TYPE IDENTIFIER (= 25) 8-BIT PREAMBLE OF 24 BITS IN 3 CONTIGUOUS BLOCKS S = SPARE (1-BIT) δ δ δ δ    �      x x   x   x ( t ) x x 0 0 GPS / GLONASS             ( ) δ = δ + δ − = + δ � y ( t ) y y t t y y y             0 0 0 GPS / GLONASS             δ δ δ δ z ( t ) z z � z z z             0 0 GPS / GLONASS ( ) δ = δ + δ − + + δ δ = + δ a t ( t ) a a t t a dt dt t fG 0 f 0 f 1 0 fG 0 GPS / GLONASS GLONASS (MT12) = + − ρ ρ + + − + + * sat Y C 1 PRC * dt rel TGD IONO TROP sat dt 80 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  62. GEO Coordinates and clock (MT 9) UPC DIRECTION OF DATA FLOW FROM SATELLITE; MOST SIGNIFICANT BIT (MSB) TRANSMITTED FIRST gAGE research g roup of A stronomy and Ge omatics 250 BITS - 1 SECOND 24-BITS .. . .. .. a Gf1 . . t 0 Z Y X G Y Z G X Y X Z a Gf0 PARITY * G G G G G G G ISSUE OF DATA, SEQUENCING BETWEEN 0 AND 255 6-BIT MESSAGE TYPE IDENTIFIER (= 9) 8-BIT PREAMBLE OF 24 BITS TOTAL IN 3 CONTIGUOUS BLOCKS *ACCURACY EXPONENT; SEE SECTION 2.5.3 OF [1]    �   � �   x ( t )  x x x G G G     1     ( ) ( ) 2 = + − + − � � � y ( t ) y y t t y t t         G G 0 G 0 2         z t � � � ( ) z z z         G G G ( ) = δ + δ − dt ( t ) a a t t Gf 0 Gf 1 0 81 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  63. LONG-TERM DEGRADATION PARAMETER UPC (GPS/GLONASS/GEO) gAGE research g roup of A stronomy and Ge omatics MT10 MT10 • GPS/GLONASS • vcode=1 (MT25, 24) if t 0 < t < t 0 + I ltc _ v 1 0  ε ltc =  ( ) C ltc _ lsb + C ltc _ v 1 max 0, t 0 − t , t − t 0 − I ltc _ v 1 otherwise  • vcode=0 (MT25, 24)  −  t t ε = C ltc   ltc ltc _ v 0 I     ltc _ v 0 • GEO  t 0 < t < t 0 + I geo 0 ε ltc =  ( ) C geo _ lsb + C geo _ v max 0, t 0 − t , t − t 0 − I geo otherwise  σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , UDRE fc rrc ltc er 82 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  64. En Route Through NPA Degradation UPC gAGE research g roup of A stronomy and Ge omatics • For Precision Approach a user is only allowed to miss one of any particular message. However, the user can still operate in less stringent phases of flight even if they have missed two or any particular fast or slow correction messages.  0 Neither fast nor long term corrections ε =  have time out for precision approach er C  Otherwise er σ σ ε ε ε ε = + + + + 2 2 2 2 2 2 i flt , UDRE fc rrc ltc er 83 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  65. Fast Corrections Ephemeris + UPC MT1, MT2,5,24, MT6, Long Term Corrections + + MT25, MT7, MT12, gAGE research g roup of A stronomy and Ge omatics UDRE MT9 Clocks + Degradation Param. MT10 IONO Corrections + MT18 IONO GIVE MT26 + Degradation Param. = + − − ρ ρ + + ∆ ∆ + + − + + * * sat sat sat IONO Y C 1 PRC PRC t t dt rel TGD IONO TROP σ = σ σ + σ σ + σ + σ 2 2 2 2 2 2 2 flt flt UIRE UIRE air tropo 84 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  66. Ionospheric Corrections (MT26) UPC gAGE research g roup of A stronomy and Ge omatics • Only Requiered for Precission Approach – Grid of Vertical Ionospheric Corrections – Users Select 3 o 4 IGPs that Surrounding IPP • 5ºx5º or 10ºx10º for 55º<Lat<55º • Only 10ºx10º for 55º<|Lat|<85º • Circular regions for |Lat|>85º – Vertical Correction and UIVE Interpoled to IPP – Both Converted to Slant by Obliquity Factor •IGP: Ionospheric Grid Point •IPP: Ionospheric Pierce Point 85 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  67. GLOBAL IGP GRID UPC 0 W180 W20 E60 W140 W100 W60 E20 E100 E140 gAGE research g roup of A stronomy and Ge omatics N85 N75 N65 N55 N50 0 0 1 2 4 3 5 6 7 8 S50 S55 S65 S75 Figure 17. Predefined Global IGP Grid S85 86 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  68. IGP MASK Message (MT18) UPC 250 BITS - 1 SECOND gAGE research g roup of A stronomy and Ge omatics 1-SPARE BIT 24-BITS PARITY 201-BIT MASK FIELD 2-BIT ISSUE OF DATA (IODI) BAND NUMBER (4 BITS) NO. OF BANDS (4 BITS) 6-BIT MESSAGE TYPE IDENTIFIER (18) 8-BIT PREAMBLE OF 24 BITS TOTAL IN 3 CONTIGUOUS BLOCKS IGP=201 (suported) IGP=125 (suported) IGP=119 Band 4 Band 5 Band 3 (not supported) 87 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  69. IONOSPHERIC DELAYS and BOUNDS (MT26) UPC D IR E C T IO N O F D A T A F L O W F R O M S A T E L L IT E ; M O S T S IG N IF IC A N T B IT (M S B ) T R A N S M IT T E D F IR S T 2 5 0 B IT S - 1 S E C O N D gAGE research g roup of A stronomy and Ge omatics R E P E A T F O R 1 4 M O R E G R ID P O IN T S 2 4 -B IT S P A R IT Y 7 9 1 0 1 1 1 2 1 3 1 4 1 5 2 3 4 5 6 8 S G IV E I (4 B IT S ) IO D I IG P V E R T IC A L D E L A Y (9 B IT S ) B L O C K ID (4 B IT S ) B A N D N U M B E R (4 B IT S ) 6 -B IT M E S S A G E T Y P E ID E N T IF IE R (= 2 6 ) 8 -B IT P R E A M B L E O F 2 4 B IT S T O T A L IN 3 C O N T IG U O U S B L O C K S S = S P A R E (7 B IT S ) GIVEI (number) BN=3 & IGP=178 VD BI=1 (color) Pos=10 BN=3 IGP=125 BI=0 Pos=5 88 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  70. ESTB Sep 12 th 2002 UPC gAGE research g roup of A stronomy and Ge omatics 89 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  71. UPC IONOSPHERIC PIERCE POINTS (IPP) gAGE research g roup of A stronomy and Ge omatics IPPs trajectories IPP Slant Delay Vertical Delay Ionospheric Layer (350 Km in height) 90 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  72. IGPs Selection Rules UPC gAGE research g roup of A stronomy and Ge omatics • Four Distinct Grid Regions – First look for Surrounding Square Cell – Else Seek Surrounding Triangular Cell – If Neither Available for 5ºx5º look at 10ºx10º – From 75º to 85º Interpolate Using Virtual IGPs – No correction possible if Not Surrounded 91 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  73. |Lat| <= 55 UPC gAGE research g roup of A stronomy and Ge omatics 92 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  74. The selection of Interpolation Grid points UPC |Lat| <= 55 gAGE research g roup of A stronomy and Ge omatics Supported IGP Not Supported 1st 2nd IPP 3th 4th 93 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  75. The selection of Interpolation Grid points UPC 55<|Lat| <= 75 gAGE research g roup of A stronomy and Ge omatics Supported IGP 1st Not Supported IPP 2nd 94 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  76. The selection of Interpolation Grid points UPC 75<|Lat| <= 85 gAGE research g roup of A stronomy and Ge omatics Linear interpoled IGP 95 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  77. The selection of Interpolation Grid points UPC 85<|Lat| gAGE research g roup of A stronomy and Ge omatics 2 3 1 W E 3 2 1 3 W E 4 96 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  78. Ionospheric Delay Interpotation UPC W = xy = − y W ( 1 x ) y 1 gAGE research g roup of A stronomy and Ge omatics 2 τ v2 τ v1 φ 2 λ − λ ipp 1 = x USER'S IPP τ vpp ( φ pp , λ pp ) ∆ λ ∆λ pp = λ pp - λ 1 φ − φ ipp 1 = y ∆φ pp = φ pp - φ 1 ∆ φ φ 1 x τ v3 τ v4 = − − λ 2 λ 1 W ( 1 x )( 1 y ) = − W x ( 1 y ) 3 4 MT26 ( ) 4 ∑ τ λ φ = τ , W ( x , y ) τ vpp pp pp i pp pp vi vi = i 1 1 − ( ) ( )   2 2   R cos E = − τ λ φ = − ⋅ τ λ φ  e  IC , F , = − F 1   pp   +   R h i spp pp pp pp vpp pp pp e I   97 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  79. Ionospheric Delay Interpotation UPC 1 = W 0 W = y y gAGE research g roup of A stronomy and Ge omatics 2 τ v1 φ 2 λ − λ = ipp 1 USER'S IPP x τ vpp ( φ pp , λ pp ) ∆ λ ∆λ pp = λ pp - λ 1 φ − φ = ipp 1 y ∆φ pp = φ pp - φ 1 ∆ φ φ x τ v2 τ v3 1 = 1 − − W x y λ 2 λ 1 W = x 3 4 MT26 ( ) 4 ∑ τ λ φ = τ , W ( x , y ) τ vpp pp pp i pp pp vi vi = i 1 1 − ( ) ( )   2 2   R cos E = − τ λ φ = − ⋅ τ λ φ  e  IC , F , = − F 1   pp   +   R h i spp pp pp pp vpp pp pp e I   98 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  80. Ionospheric Delay Interpotation UPC = − W ( 1 x ) y 2 gAGE research g roup of A stronomy and Ge omatics 2 3 1 = − − W = W ( 1 x )( 1 y ) xy 3 1 W φ − 85 º = y ipp 10 º E 4 λ − λ ipp 3 = − + x ( 1 2 y ) y = − W x ( 1 y ) 90 º 4 MT26 ( ) 4 ∑ τ λ φ = τ , W ( x , y ) τ vpp pp pp i pp pp vi vi = i 1 1 − ( ) ( )   2 2   R cos E = − τ λ φ = − ⋅ τ λ φ  e  IC , F , = − F 1   pp   +   R h i spp pp pp pp vpp pp pp e I   99 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

  81. Ionospheric Delay Interpotation UPC W = y 2 gAGE research g roup of A stronomy and Ge omatics 2 1 1 = 3 W 0 = 1 − − W x y W 3 φ − 85 º = y ipp 10 º E 4 λ − λ ipp 3 W = = − + x ( 1 2 y ) y x 4 90 º MT26 ( ) 4 ∑ τ λ φ = τ , W ( x , y ) τ vpp pp pp i pp pp vi vi = i 1 1 − ( ) ( )   2 2   R cos E = − τ λ φ = − ⋅ τ λ φ  e  IC , F , = − F 1   pp   +   R h i spp pp pp pp vpp pp pp e I   100 Manuel Hernández-Pajares, J. Miguel Juan, Jaume Sanz, Xavier Prats, February 2002.

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