Lecture 2 Satellite orbits and clocks computation and accuracy

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1 Lecture 2 Satellite orbits and clocks computation and accuracy Contact: jaume.sanz@upc.edu Web site: 1

2 Authorship statement The authorship of this material and the Intellectual Property Rights are owned by J. Sanz Subirana and J.M. Juan Zornoza. These slides can be obtained either from the server or Any partial reproduction should be previously authorized by the authors, clearly referring to the slides used. This authorship statement must be kept intact and unchanged at all times. 5 March

3 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 3

4 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 4

5 The GPS navigation message provides pseudo-keplerian elements to compute satellite coordinates 5

6 (X, Y, Z, Vx, Vy, Vz) (a, e, i, W, w, V) 6 values are needed (x,y,z,vx,vy,vz) to provide the position and velocity of a body. They can be map into the six Keplerian elements (a, e, i, W, w, V ), which provides the natural representation of the orbit! 6

7 (a, e, i, W, w, V) orbit shape orbit orientation position in the orbit a ae V x focus perigee 7

8 True anomaly V(t) t T 0 2p n P, ae, V(t) M Perigee Fictitious body moving at velocity n=2p/p=constant Mean anomaly M(t) T 0 : time of passage by satellite s perigee 2p M ( t) n( t T ) ; n P E( t) M ( t) esin E( t) e E( t) Vt ( ) 2arctan tan 1 e 2 a 8

9 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 9

10 Due to the non-spherical nature of gravitational potential, the attraction of the Sun and Moon, the solar radiation pressure, etc., the true satellite path deviates from the elliptic orbit. At any time an elliptical orbit tangent to the true path can be defined. This is the osculating orbit, whose Keplerian elements vary with time t : a(t),e(t),i(t),w(t),w(t),v(t) True path Instantaneous elliptic tangent (osculating) orbit. 10

11 Different magnitudes of perturbation and their effects on GPS orbits GLONASS Broadcast orbit integration terms 11

12 Central body Central body + J2 Central body + J2 + SM 12

13 Central body Central body + J2 Central body + J2 + SM 13

14 Central body Central body + J2 Central body + J2 + SM 14

15 Calculation of osculating orbital elements from position and velocity (rv2osc.f) 15

16 Calculation of position and velocity from osculating orbital elements (osc2rv.f) Where: 16

17 Exercise: Orbital elements variation: File eci contains the precise position and velocities of GPS satellites every 5 minutes for October 18th, 1995 in a Earth-Centred Inertial system (ECI) [from JPL/NASA server: ftp://sideshow.jpl.nasa.gov/pub/gipsy_products] a) Use program rv2osc to compute the instantaneous orbital elements for each epoch in the file. That is: b) Plot the orbital elements in function of time to show their variation: a(t),e(t),i(t),w (t),w(t),v(t) c) Compare with the broadcast orbital elements Solution: a) cat eci rv2osc> orb.dat b) See the following plots (X, Y, Z, Vx, Vy, Vz) (a, e, i, W, w, V ) 17

18 Semi-major axis Eccentricity Inclination Ascending node 18

19 Argument of Perigee Mean Anomaly 19

20 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 20

21 GPS navigation message One Master Frame includes All 25 pages of Subframes #4 and #5 25 x 30s = 12.5 min 25 pages Subframes #4 and #5 have 25 pages 21

22 Subframe 1 contains information about the parameters to be applied to satellite clock status for its correction. These values are polynomial coefficients that allow time onboard to be converted to GPS time. The subframe also contains information on satellite health condition. Subframes 2 and 3 contain satellite ephemerides. Subframe 4 provides ionospheric model parameters (in order to adjust for ionospheric refraction), UTC information, part of the almanac, and indications whether the A/S is activated or not (which transforms the P code into encrypted Y code). Subframe 5 contains data from the almanac and on constellation status. It allows rapid identification of the satellite from which the signal comes. A total of 25 frames are needed to complete the almanac. 22

23 Ephemeris in navigation message In order to calculate WGS84 satellite coordinates, you should apply de following algorithm [GPS/SPS-SS, table 2-15] (see in the book FORTRAN subroutine orbit.f) 23

24 RINEX ephemeris file 24

25 3.1. Computation of satellite coordinates from navigation message (orbit.f) Computation of t k time since ephemerids reference epoch t oe (t and t oe are given in GPS seconds of week): Computation of mean anomaly M k for t k, Iterative resolution of Kepler s equation in order to compute eccentric anomaly E k : Calculation of true anomaly v k : Computation of latitude argument u k from perigee argument W, true anomaly v k and corrections c uc and c us : t t t k oe M k M 0 n t 3 a M E esin E v k k k k arctan 2 1 e sin cos E k E e k k u w v c cos 2 w v c sin 2 w v k k uc k us k 25

26 Computation of radial distance r k, taking into consideration corrections c rc and c rs : Calculation of orbital plane inclination i k from inclination i o at reference epoch t oe and corrections c ic and c is : Computation of ascending node longitude W k (Greenwich), from longitude W 0 at start of GPS week, corrected from apparent variation of sidereal time at Greenwich between start of week and and reference time t k =t-t oe, and also corrected from change of ascending node longitude since reference epoch t oe. Calculation of coordinates in CTS system, applying three rotations (around u k, i k, W k ) : 1 2cos cos 2w sin 2w r a E c v c v k k rc k rs k w w i i0 it c cos 2 v c sin 2 v k k ic k is k W W W w k 0 E k E oe t w t X k rk Y R ( W ) R ( i ) R ( u ) 0 k 3 k 1 k 3 k Z k 0 26

27 Along Track gage t Orbit.f Nav. message (ephemeris) Conventional Terrestrial Reference System (TRS): Earth Centered, Earth-Fixed (ECEF) the reference system rotates with Earth. (x,y,z) [TRS] X Greenwich z Radial y 27

28 Reference values precise IGS orbits Reference values precise IGS orbits Reference values precise IGS orbits Discrepancy between CODE and IGS combined product Antenna Phase Centre Correction has been applied 28

29 Zoom Broadcast Orbit Updates 29

30 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 30

31 3.2 Computation of satellite coordinates from precise products. Precise orbits for GPS satellites can be found on the International GNSS Service (IGS) server Orbits are given by (x,y,z) coordinates with a sampling rate of 15 minutes. The satellite coordinates between epochs can be computed by polynomial interpolation. A 10th-order polynomial is enough for a centimetre level of accuracy with 15 min data. 31

32 IGS orbit and clock products (for PPP): Discrepancy between CODE and IGS combined product. 32

33 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 33

34 GPS Satellite Clock computation: Broadcast message PRN 2 NAVIGATION DATA GPS RINEX VERSION / TYPE srx/v BAI 95/10/19 03:18:35 PGM / RUN BY / DATE CASA COMMENT COMMENT END OF HEADER D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D+00 dt sat =a 0 + a 1 (t-t 0 ) + a 2 (t-t 0 ) 2 t0 YY MM DD H M S a0 a1 a2 34

35 Computation of satellite clocks from precise products Precise clocks for GPS satellites can be found on the International GNSS Service (IGS) server They are providing precise orbits and clock files with a sampling rate of 15 min, as well as precise clock files with a sample rate of 5 min and 30 s in SP3 format. Some centres also provide GPS satellite clocks with a 5 s sampling rate, like the les obtained from the Crustal Dynamics Data Information System (CDDIS) site. Stable clocks with a sampling rate of 30 s or higher can be interpolated with a first-order polynomial to a few centimetres of accuracy. Clocks with a lower sampling rate should not be interpolated, because clocks evolve as random walk processes. 35

36 Reference values precise IGS clocks SA=off 36

37 Zoom Broadcast Orbit & Clock Updates 37

38 Precise Clock Interpolation: 300s samples 38

39 Precise Clock Interpolation: 30s samples 39

40 Selective Availability (S/A): Intentional degradation of satellite clocks and broadcast ephemeris. (from 25 March, 1990) Instantaneous Error (meters) GPS Before and After S/A was switched off Colorado Springs, Colorado Horizontal Error (meters) Vertical Error (meters) Time of Day (Hours UTC) ANALYSIS NOTES 2 May Data taken from Overlook PAN Monitor Station, equipped with Trimble SVeeSix Receiver - Single Frequency Civil Receiver - Four Satellite Position Solution at Surveyed Benchmark - Data presented is raw, no smoothing or editing S/A was switched off at 2nd May 2000 and Permanently removed in

41 SA=on/off 41

42 42

43 43

44 IGS Precise orbit and clock products: RMS accuracy, latency and sampling 44

45 Contents 1. Elliptic orbit: Keplerian elements. 2. Perturbed Keplerian orbits: Osculating orbit. 3. GPS satellite coordinates computation and accuracy 3.1. From Broadcast Navigation Message From precise products. 4. GPS Satellite clock computation and accuracy 4.1. From Broadcast Navigation Message From precise products. 5. Geographic decorrelation of ephemeris errors. 45

46 Ephemeris Errors and Geographic decorrelation True position Reference Station Satellite location error R ref. e ρ ε ref ref ε ρ user user user User Position from broadcast ephemeris Differential range error due to satellite obit error ρ ρ user ε ε user ref ref A conservative bound: b e with a baseline b 20km 20 1 e e

47 Satellite location error e Range error from CREU and EBRE ρuser ref ρ ε user ε ρ ε user user Differential range error from between CREU and EBRE ref 288 km of baseline 47

48 Range error from CREU and EBRE CREU Absolute positioning ε ρ user user Differential range error from between CREU and EBRE ρuser ref ε ε ρ user ref CREU-EBRE Differential positioning 288 km of baseline 48

49 By Miguel Juan Zornoza a ( ) / 2 : hyperboloid semiaxis B b / 2 : focal length where a b / 2cos( ) A Note: in this 3D problem is NOT the elevation of ray. Differential range error produced by an orbit error parallel to vector Let e e e ( ) 2 B A a a a 2 e 2 e bsin e e e e 1 bsin e û Note: ε ρe û ε Thence: Errors over the hyperboloid (i.e.. B A ctt ) will not produce differential range errors. The highest error is given by the vector, orthogonal to the hyperboloid and over the plain containing the baseline vector and the LoS vector. ˆb û Note: Being the baseline b much smaller than the distance to the satellite, we can assume that the LoS vectors from A and B receives are essentially identical to. That is, B A ˆ ˆ T T ˆ u ρˆ bρˆ b ρˆ ρˆ - ρˆ ρˆ b Note: u sin uˆ T T ˆ ˆ ˆ ˆ Ibˆ ρ ρ bˆ I ρ ρ bˆ bsin T T ε uˆ ε sin uˆ T ε I ρˆ ρˆ Note: being uˆ a vector orthogonal to the LoS ρˆ, thence, e ε T uˆ T b Where: J. Sanz Subirana, JM. Juan Zornoza, M. Hernández-Pajares, ˆρ b b = b bˆ is the baseline vector

50 ORBIT TEST : Broadcast orbits Along-track Error (PRN17) PRN17: Doy=077, Transm. time: sec E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 diff EPH.dat.org EPHcuc_x0.dat < E E E E+03 > E E E E J. Sanz Subirana, JM. Juan Zornoza, M. Hernández-Pajares, 50

51 Orbit error rov1 11 km 15.2 km 19.7 km rov3 ε 39.3 km 31.3 km rov2 Range error Differential range error Baseline: b=31.3km ε I ρˆ ρˆ J. Sanz Subirana, JM. Juan Zornoza, M. Hernández-Pajares, T T b 51

52 Exercise: Justify that clock errors completely cancel in differential positioning. 52

53 ERRORS on the Signal Space Segment Errors: Clock errors Ephemeris errors Propagation Errors Ionospheric delay Tropospheric delay Local Errors Multipath Receiver noise Common Strong spatial correlation Weak spatial correlation No spatial correlation 53

54 References [RD-1] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 1: Fundamentals and Algorithms. ESA TM- 23/1. ESA Communications, [RD-2] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS Data processing. Volume 2: Laboratory Exercises. ESA TM-23/2. ESA Communications, [RD-3] Pratap Misra, Per Enge. Global Positioning System. Signals, Measurements, and Performance. Ganga Jamuna Press, [RD-4] B. Hofmann-Wellenhof et al. GPS, Theory and Practice. Springer-Verlag. Wien, New York,

55 Thank you 55

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