Introduction to Global Navigation Satellite System (GNSS) Module: 2

Transcription

1 Introduction to Global Navigation Satellite System (GNSS) Module: 2 Dinesh Manandhar Center for Spatial Information Science The University of Tokyo Contact Information: dinesh@iis.u-tokyo.ac.jp Slide : 1

2 Satellite Orbits Navigation Data Format Position Computation Output Data Formats Module 2: Course Contents Slide : 2

3 Orbital Mechanics Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problem concerning the motion of spacecraft. A core discipline within space mission design, control, and operation. Celestial mechanics treats the orbital dynamics of natural astronomical bodies such as star systems, planets, and moons. Sputnik-1 The first artificial Earth satellite launched by the Soviet Union in Slide : 3

4 History There is little distinction between orbital and celestial mechanics. The fundamental techniques are the same. Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws of planetary motion in 165. Slide : 4

5 Kepler s Laws of Planet Motion The orbit of every planet is an ellipse with the Sun at one of the two foci (plural of focus). A line joining a planet and the sun sweeps out equal area during equal intervals of time. The square of the orbital period of planet is proportional to the cube of the semi-major axis of its orbit. -a Sun B Planet A a C D Slide : 5

6 Kepler Orbit Kepler orbit can be uniquely defined by six parameters know as Keplerian elements. Semi-major axis (a) Eccentricity (e) Inclination (i) Right ascension of the ascending node (RAAN) (Ω) Argument of perigee (ω) True anomaly (v: Greek letters nu) Slide : 6

7 Orbital Plane The shape of an elliptic orbit can be defined by the semi-major axis and eccentricity. y The satellite position in the orbital plane can be defined by true anomaly. a v x Focus (Earth) Apogee Perigee Slide : 7

8 Equatorial Coordinate System The most common coordinate frame for describing satellite orbits is the geocentric equatorial coordinate system, which is also called an Earth-Centered Inertial (ECI) coordinate system. z North pole Equatorial plane Vernal equinox The direction of the Sun as seen from Earth at the beginning of spring time. x O The Origin at the center of the Earth y The fundamental plane is consisting of the projection of the Earth s equator Slide : 8

9 Orientation of the Orbital Plane z North Orbital Plane to ECI r ECI = R z Ω R x i R z ω r x Ω ω i Perigee y Vernal Equinox Slide : 9

10 x Rotation around the z-axis R z x θ Rotation Matrices z z cos sin sin cos 1 y y Around the x-axis R x 1 cos sin Around the y-axis R y cos sin 1 sin cos sin cos Slide : 1

11 Typical GPS Orbit 26,56 km semi-major axis (2,2 km altitude) The orbital period is approximately 12 hours Less than.1 eccentricity (near circular) 55 degree inclination 6 orbital planes with at least 4 satellites in each plane The ascending nodes of the orbital planes are separated by 6 degree Slide : 11

12 GPS YUMA ALMANAC FILE ID Health Eccentricity Time of applicability PRN ID of SV GPS Yuma Almanac File = usable This shows the amount of the orbit deviation from circular (orbit). It is the distance between the foci divided by the length of the semi-major axis (our orbits are very circular). The number of seconds in the orbit when the almanac was generated. Kind of a time tag. Orbital Inclination Rate of Right Ascension SQRT(A) Square Root of Semi-Major Axis The angle to which the SV orbit meets the equator (GPS is at approximately 55 degrees). Roughly, the SV's orbit will not rise above approximately 55 degrees latitude. The number is part of an equation: # = π/18 = the true inclination. Rate of change in the measurement of the angle of right ascension as defined in the Right Ascension mnemonic. This is defined as the measurement from the center of the orbit to either the point of apogee or the point of perigee. Right Ascension at Time of Almanac (TOA) Right Ascension is an angular measurement from the vernal equinox ((OMEGA) ). Argument of Perigee Mean Anomaly Af Af1 An angular measurement along the orbital path measured from the ascending node to the point of perigee, measured in the direction of the SV's motion. Angle (arc) traveled past the longitude of ascending node (value = ±18 degrees). If the value exceeds 18 degrees, subtract 36 degrees to find the mean anomaly. When the SV has passed perigee and heading towards apogee, the mean anomaly is positive. After the point of apogee, the mean anomaly value will be negative to the point of perigee. SV clock bias in seconds. SV clock drift in seconds per seconds. Af2 GPS week ( 123), every 7 days since 1999 August 22. GPS Week Slide : 12

13 Example of Yuma Almanac File for GPS ******** Week 887 almanac for PRN-1 ******** ID : 1 Health : Eccentricity : E-2 Time of Applicability(s) : Orbital Inclination(rad) : Rate of Right Ascen(r/s) : E-8 SQRT(A) (m 1/2) : Right Ascen at Week(rad) : E+1 Argument of Perigee(rad) : Mean Anom(rad) : E+1 Af(s) : E-4 Af1(s/s) :.E+ Week : Slide : 13

14 Perturbation Forces Satellite orbit will be an ellipse only if treating each of satellite and Earth as a point mass. In reality, Earth s gravitational field is not a point mass. Main force acting on GNSS satellites is Earth s central gravitational force, but there are many other significant perturbations. Non sphericity of the Earth s gravitational potential Third body effect Direct attraction of Moon and Sun Solar radiation pressure Impact on the satellite surfaces of photons emitted by the Sun Slide : 14

15 Accelerations Acting on GNSS Satellites Term Acceleration [m/s 2 ] Earth s central gravity.56 Flatness of the Earth (J2) Other gravity Moon and Sun Solar Radiation Pressure 1-7 Effects of SRP on GNSS satellite position: 5~1 m Slide : 15

16 Satellite orbit in Navigation Message Broadcast ephemeris Kepler orbit parameters and satellite clock corrections 9 orbit perturbation corrections parameters 2 m satellite position accuracy for 2 hours Each GNSS satellite broadcasts only its own ephemeris data Almanac Kepler orbit parameters and satellite clock corrections Less accurate but valid for up to several months Each GNSS satellite broadcasts almanac data for all satellites in the constellation Slide : 16

17 GPS L1C/A Signal NAV MSG Slide : 17

18 Navigation Message, Sub-frame 1 Slide : 18

19 GPS L1C/A Signal NAV MSG, Sub-frame 2 Slide : 19

20 GPS L1C/A Signal NAV MSG, Sub-frame 3 Slide : 2

21 GPS L1C/A Signal NAV MSG, Sub-frame 4 Page 1,6,11,16,21 Slide : 21

22 GPS L1C/A Signal NAV MSG, Sub-frame 4 Page 12,19,2,22,23,24 Slide : 22

23 GPS L1C/A Signal NAV MSG, Sub-frame 4, Page 14, 15 Slide : 23

24 GPS L1C/A Signal NAV MSG, Sub-frame 4, Page 17 Slide : 24

25 GPS L1C/A Signal NAV MSG, Sub-frame 5 Slide : 25

26 Coordinate System Slide : 26

27 Geodetic Coordinate System Satellite Ellipsoid Surface Semi Major Axis Semi Minor Axis Pole User at P(x, y, z) Normal Vector to Ellipsoid at Point P Geodetic Latitude at P Equator Semi Major Axis Geodetic Longitude at P Slide : 27

28 ECEF (Earth Centered, Earth Fixed) ECEF Coordinate System is expressed by assuming the center of the earth coordinate as (,, ) P (X, Y, Z) (,, ) Equator Slide : 28

29 Semi-Minor Axis Geodetic Datum: Geometric Earth Model Pole Semi-Major Axis WGS-84 Geodetic Datum Ellipsoidal Parameters Semi-Minor Axis, b = m Semi-Major Axis, a = m Flattening, f = (a-b)/a = 1/ First Eccentricity Square = e^2 = 2f-f^2 = Equator Slide : 29

30 Coordinate Conversion from ECEF to Geodetic and vice versa Geodetic Latitude, Longitude & Height to ECEF (X, Y, Z) X = Y = N + h cos φ cos λ N + h cos φ sin λ Z = N 1 e 2 + h sin φ φ = Latitude λ = Longitude H = Height above Ellipsoid ECEF (X, Y, Z) to Geodetic Latitude, Longitude & Height φ=atan Z+e2 b sin 3 θ p e 2 acos 3 θ λ=atan2 Y, X h = P cos φ N φ P = x 2 + y 2 θ = atan N φ = a Za Pb 1 e 2 sin 2 φ Slide : 3

31 Topographic, Ellipsoidal & Geoid Height Topographic Surface Ellipsoidal Surface N h H Geoid Surface MSL Topographic Height (H) = Ellipsoidal Height (h) - Geoid Height (N) Slide : 31

32 Position Output Slide : 32

33 Pseudorange equation Perfect World: r c t R t S Real World: Receiver clock error Tropospheric delay Multipath r c S t t I T M R Satellite clock error Ionospheric delay Thermal noise Simplify r c t R t S Slide : 33

34 Range Equation Satellite position at signal transmission time: S S S x, y, z Receiver position at signal reception time: x, y, z r S 2 S x x y y 2 z z S 2 S S x, y, z S x, y, z Slide : 34

35 Pseudorange model S 2 S 2 S 2 S x x y y z z c t t R r Given satellite position & clock in navigation message Unknown receiver position & clock Estimate optimal solution to minimize the error Slide : 35

36 Nonlinear Optimization Problem We have n simultaneous nonlinear equations from n pseudorange observations. We need at least 4 independent observations in order to determine 4 unknown parameters. In general, even a single nonlinear equation cannot be solved without some iterative method by generating a sequence of approximate solutions. Slide : 36

37 Newton-Raphson Method Find successively better approximation x x satisfying y of a nonlinear equation y f (x. ) y y y f (x) y x f x x f y y y x x x x x Slide : 37

38 Newton-Raphson Algorithm Initial Guess x x Solve the equation for x y f x f x x y y: Residual x Update Repeat until a sufficiently x x x small value x is reached. Slide : 38

39 Slide : 39 Pseudorange Equation S S S S t c t c z z y y x x r b b z y x f,,, For given observation Linearize around the initial solution Obtain the update,,, b z y x b z y x,,,

40 Slide : 4 Linearization Partial derivatives with respect to each unknown parameter: Linearized pseudorange residual equation:, r x x x f S, r y y y f S, r z z z f S 1 b f b z r z z y r y y x r x x b z y x f S S S,,,

41 Slide : 41 Vector Description b z y x r z z r y y r x x S S S 1 h x We need at least 4 linearly independent equations in order to determine 4 unknown parameters.

42 Simultaneous equations 1 2 N 1 h 2 h x N h 1 2 N N 4 y H e N 1 N 4 41 N 1 Residual vector y Hx e Observation matrix State vector Slide : 42

43 Least squares problem For given y Hx, efind x to xˆ minimize. Performance Index: J e T e y T y Hx y Hx T y y T T y y 2x Hx x T H T T H y x T T y x H T T Hx H T Hx e T e Find x J xˆ to minimize J x xxˆ Slide : 43

44 Least squares solution Partial derivatives of a scalar function w.r.t the state vector: f (1) T T f x a x x a then x a f (2) T f x x Ax then 2Ax x for all symmetric matrix A J T T Find x xˆ to satisfy 2H y 2H Hx x xˆ T 1 T H H H y Slide : 44

45 GNSS Positioning Calculation Initial Guess x Even the north pole is a good initial guess! Solve the equation for xˆ Pseudorange xˆ T 1 T H H H y Satellite position Satellite clock t S S S x, y, z S Update x x x xˆ Generally takes a few iterations. Slide : 45

46 Error Budget The positioning accuracy depends on the magnitude of error in the individual pseudorange measurement. Source Error DGPS Satellite orbit error 1 ~2m Satellite clock error 1 m Ionospheric delay 4~1 m Can be minimized to Tropospheric delay 1~2 m <1m Thermal noise Multipath 1 m 1m or more Can t be removed Slide : 46

47 Error sources V = 4km/s Ionospheric delay Satellite position error Ephemeris Tropospheric delay 1km Thermal noise 2km 5km Multipath Satellite clock error Receiver clock error Slide : 47

48 Satellite geometry and positioning error The positioning accuracy also depends on the geometric configuration of the satellites. Bad geometry Good geometry Slide : 48

49 Slide : 49 Dilution of precision (DOP) b z y x T g g g g 1 H H G z y x g g g PDOP g b TDOP b z y x g g g g GDOP Position DOP: Time DOP: Geometric DOP:

50 DOP and positioning accuracy Accuracy of any measurement is proportionately dependent on the DOP value. This means that if DOP value doubles, the resulting position error increases by a factor of two. y s s y =s YDOP x s x =s XDOP Slide : 5

51 Data Formats: NMEA, RINEX Slide : 51

52 NMEA Data Format NMEA National Marine Electronics Association format to share position, velocity, satellite visibility and many other formats ASCII file with pre-defined headers For example $GP for GPS Related Data $GPGSV for GPS Satellite Visibility $GN is used for GNSS Slide : 52

53 NMEA Data Format NMEA: National Marine Electronics Association GGA - Fix data which provide 3D location and accuracy data. $GPGGA,123519,487.38,N,1131.,E,1,8,.9,545.4,M,46.9,M,,*47 Where: GGA Global Positioning System Fix Data Fix taken at 12:35:19 UTC ,N Latitude 48 deg 7.38' N 1131.,E Longitude 11 deg 31.' E 1 Fix quality: = invalid, 1 = GPS fix (SPS), 2 = DGPS fix, 3 = PPS fix, 4 = Real Time Kinematic 5 = Float RTK 6 = estimated (dead reckoning) (2.3 feature) 7 = Manual input mode 8 = Simulation mode 8 Number of satellites being tracked.9 Horizontal dilution of position 545.4,M Altitude, Meters, above mean sea level 46.9,M Height of geoid (mean sea level) above WGS84 ellipsoid (empty field) time in seconds since last DGPS update (empty field) DGPS station ID number *47 the checksum data, always begins with * Slide : 53

54 RINEX Data Format Receiver Independent Exchange Format Basically Two File Types *.*N file for Satellite and Ephemeris Related data *.*O file for Signal Observation Data like Pseudorange, Carrier Phase, Doppler, SNR etc Slide : 54

55 Example of RINEX *.*N file 2.12 N RINEX VERSION / TYPERDE MCS UTC PGM / RUN BY / DATEGPSA D D D D-9 IONOSPHERIC CORRGPSB D D D D+3 IONOSPHERIC CORRGPUT D D TIME SYSTEM CORR 17 LEAP SECONDS END OF HEADER D D-12.D+ 5.D D D D D D D D D D D D D D D D D-1 1.D D+3.D+ 2.D+.D D-9 5.D D+5 4.D+ Slide : 55

56 Example of RINEX *.*q file 2.12 N J RINEX VERSION / TYPERDE MCS UTC PGM / RUN BY / DATEQZSA D D-7 3.8D D-7 IONOSPHERIC CORRQZSB 1.854D D D D+5 IONOSPHERIC CORRQZUT D D TIME SYSTEM CORR 17 LEAP SECONDS END OF HEADERJ D D-11.D+ 9.5D D D D D D D D D D D D D D D D D-1 2.D D+3.D+ 2.D+ 1.D D-9 9.5D D+5 2.D+ Slide : 56

57 Example of RINEX *.*O File 3.3 OBSERVATION DATA M (MIXED) RINEX VERSION / TYPENetR9 5.1 Receiver Operator 3-AUG-16 :: PGM / RUN BY / DATECREF1 MARKER NAMEGEODETIC MARKER TYPE OBS AGENCY OBSERVER / AGENCY 5536R512 Trimble NetR9 5.1 REC # / TYPE / VERS TRM NONE ANT # / TYPE... APPROX POSITION XYZ.1.. ANTENNA: DELTA H/E/NG 9 C1C L1C S1C C2W L2W S2W C2X L2X S2X SYS / # / OBS TYPESR 9 C1C L1C S1C C1P L1P S1P C2C L2C S2C SYS / # / OBS TYPESE 9 C1X L1X S1X C7X L7X S7X C8X L8X S8X SYS / # / OBS TYPES 1. INTERVAL GPS TIME OF FIRST OBSL2C CARRIER PHASE MEASUREMENTS: PHASE SHIFTS REMOVED COMMENT L2C PHASE MATCHES L2 P PHASE COMMENT GLONASS C/A & P PHASE MATCH: PHASE SHIFTS REMOVED COMMENT GIOVE-A if present is mapped to satellite ID 51 COMMENT GIOVE-B if present is mapped to satellite ID 52 COMMENT DBHZ SIGNAL STRENGTH UNIT END OF HEADER > G G G G R E E R R G R G G G R > Slide : 57