Orbit Representation
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1 7.1 Fundamentals 223 For this purpose, code-pseudorange and carrier observations are made of all visible satellites at all monitor stations. The data are corrected for ionospheric and tropospheric delays, for Earth rotation and for relativistic effects. The corrected measurements and carrier-aided smoothed observations are input into the Kalman filter process and are used to estimate the following states (Parkinson et al., 1996, chap. 10): satellite position at epoch, satellite velocity at epoch, three clock parameters per satellite, solar radiation pressure coefficients per satellite, y-axis acceleration bias, two clock parameters per monitor station, and one tropospheric scale factor per monitor station. The estimated perturbations in the elements are used to correct the satellite reference ephemeris and to generate the broadcast ephemerides. In a similar way the satellite clock behavior is predicted and included in the data signal in the form of a second order polynomial. Computation of the satellite trajectories is based on the gravity field parameters and the station coordinates of the World Geodetic System 1984 (WGS 84). In order to improve the accuracy of the ephemeris the WGS 84 station coordinates were replaced by ITRF 91 coordinates in 1994, and by ITRF 94 coordinates in 1996, cf. [2.1.6]. Earth orientation parameters are taken from the IERS Rapid Service [12.4.2]. The process of orbit determination is still based on the technology of the 1980s (Russel, Schaibly, 1980; Swift, 1985) but will be upgraded along with the Accuracy Improvement Initiative (AI I) [7.1.7] Orbit Representation The satellite positions estimated in the Kalman filter process are next represented in the form of Keplerian elements with additional perturbation parameters. Table 7.4 summarizes all parameters that describe the satellite orbit and the state of the satellite clock. The parameters refer to a given reference epoch, t 0e for the ephemeris and t 0c for the clock, and they are based on a four hours curve fit (ICD, 1993). Hence, the representation of the satellite trajectory is achieved through a sequence of different disturbed Keplerian orbits. At present, a fresh data set is broadcasted every two hours, causing small steps between the different overlapping representations. These steps can reach a few decimeters but may be smoothed by suitable approximation techniques, e.g. Chebyshev polynomials [ ]. The parameter set of Table 7.4 is used to compute the satellite time and the satellite coordinates. The unit semicircles can be converted to degrees (multiplication by 180), or to radians (multiplication by π). The first group of parameters is used to
2 224 7 The Global Positioning System (GPS) Table 7.4. Representation of GPS broadcast ephemerides Time parameters t 0e Reference time, ephemeris parameters [s] t 0c Reference time, clock parameters [s] a 0, a 1, a 2 Polynomial coefficients for clock correction (bias [s], drift [s/s], drift-rate (ageing) [s/s 2 ]) IODC Issue of Data, Clock, arbitrary identification number Keplerian parameters A Square root of the semi-major axis [m 1/2 ] e i 0 0 ω M 0 IODE n i C us C uc C is C ic C rs C rc eccentricity [dimensionless] inclination angle at reference time [semicircles] Longitude of ascending node at reference time [semicircles] Argument of perigee [semicircles] Mean anomaly at reference time [semicircles] Issue of Data, Ephemeris, arbitrary identification number Perturbation parameters Mean motion difference from computed value [semicircles/s] Rate of change of right ascension [semicircles/s] Rate of change of inclination [semicircles/s] Amplitude of the sine harmonic correction term to the argument of latitude [rad] Amplitude of the cosine harmonic correction term to the argument of latitude [rad] Amplitude of the sine harmonic correction term to the angle of inclination [rad] Amplitude of the cosine harmonic correction term to the angle of inclination [rad] Amplitude of the sine harmonic correction term to the orbit radius [m] Amplitude of the cosine harmonic correction term to the orbit radius [m] correct satellite time. The second group determines a Keplerian ellipse at the reference epoch, and the third group contains nine perturbation parameters. These are: n secular drift in dω/dt due to the second zonal harmonic (C 20 ); also it absorbs effects of the Sun s and Moon s gravitation and solar radiation pressure over the interval of fit, secular drift in right ascension of the node due to the second zonal harmonic; includes also effects of polar motion, i rate of change of inclination, and C us, C uc short periodic effects of C 20 ; also include higher order effects and C is, C ic short periodic effects of lunar gravitation (during the closest approach C rs, C rc of the space vehicle to the Moon); also absorb further perturbations. Fig explains the Keplerian and the perturbation parameters. Note that the element 0 in the GPS message is not measured from the vernal equinox,, but from the zero meridian, X T. In essence, 0 is not a right ascension angle but a longitude. In recent literature the parameter is therefore designated as longitude of ascending node (LAN).
3 7.1 Fundamentals 225 C uc, C us Z T C rc, C rs satellite C ic, C is reference epoch ν k M n M 0 perigee ω e 0 ω i 0 i A, e equator Y T X T Figure Keplerian and disturbance parameters in the broadcast message Computation of Satellite Time and Satellite Coordinates The GPS system time is characterized by a week number and the number of seconds since the beginning of the current week; the GPS time can hence vary between 0 at the beginning of a week and at the end of a week. The initial GPS epoch is January 5, 1980 at 0 h UTC. This is why the GPS week starts at midnight (Universal Time) between Saturday and Sunday. The GPS week number is included in subframe 1 of the navigation message [ ], and is represented by 10 bits. Hence the largest possible week number is 1023, and at the end of the week with the number 1023 the week number rolls over to zero (cf. ICD-GPS-200C). This event is called the End of Week (EOW) rollover. The first cycle of week numbers ended on August 21, The current second cycle runs from August 22, 1999 until April 6, 2019 (Langley, 1999a). Note that for some purposes a continuous numbering of the GPS week is used (no rollover), e.g. for RINEX data [ ]. The GPS system time is a continuous time scale, and is defined by the weighted mean of the atomic clocks in the monitor stations and the satellites (cf. [7.1.3]). The leap seconds in the UTC time scale, and the drift in the GPS clocks mean that GPS system time and UTC are not identical [2.2.3]. The difference is continuously monitored by the control segment and is broadcast to users in the navigation message. On January 1, 2003, the difference was about 13 seconds (GPS time ahead). Because of constant and irregular frequency errors in the satellite oscillators, the satellite clock readings differ from the GPS system time. The behavior of the individual satellite clocks (rubidium or cesium oscillator) is monitored by the control segment, and predicted in the form of a second degree polynomial. The polynomial coefficients
4 226 7 The Global Positioning System (GPS) are included in the first parameter group of Table 7.4. The individual satellite time, t SV, is corrected to GPS system time, t, using in which t = t SV t SV, (7.3) t SV = a 0 + a 1 (t t 0c ) + a 2 (t t 0c ) 2 ; (7.4) t 0c is the reference epoch for the coefficientsa 0, a 1, a 2. Following the Interface Control Document, the OCS shall control the GPS time to be within one microsecond of UTC (USNO) modulo one second. In practice, GPS time has been kept within about 10 nanoseconds. The term a 0 in the satellite message hence has only a value of a few nanoseconds, (see also [2.2.3]). The time parameter, t, can be substituted in the further calculation by t SV without loss of accuracy. Differentiating (7.4) with respect to time yields an expression for the drift of the satellite clock: t SV = a 1 + 2a 2 (t t 0c ). (7.5) The satellite coordinates X k, Y k, Z k are computed for a given epoch, t, with respect to the Earth-fixed geocentric reference frame X T, Y T, Z T (cf. [2.1.2]). The time, t k, elapsed since the reference epoch, t 0e, is t k = t t 0e. (7.6) A possible change of the week has to be considered. Two constants are required: Also GM = m 3 /s 2 WGS 84 value of the geocentric (7.7) gravitational constant, ω e = rad/s WGS 84 value of the Earth rotation rate. (7.8) π = (exactly). Note that (7.7) is not the refined WGS 84 value for GM from 1994 [2.1.6], but the original WGS 84 value. Actually, the more recent GM value is used for precise prediction of GPS orbits in the OCS whereas the old value is used for the conversion from the predicted Cartesian state vectors into the quasi-keplerian broadcast elements. Hence it should also be used for interpolation purposes to obtain satellite positions at a given epoch. For details on the subject see (NIMA, 2000). Furthermore we use: A = ( A) 2 Semi-major axis, (7.9) GM n 0 = A 3 Computed mean motion, (7.10) n = n 0 + n Corrected mean motion, and (7.11) M k = M 0 + nt k Mean anomaly. (7.12)
5 7.1 Fundamentals 227 Kepler s equation of the eccentric anomaly (3.53), E k = M k + e sin E k, (7.13) is solved by iteration. Because of the very small eccentricity of the GPS orbits (e < 0.001) two steps are usually sufficient: E 0 = M, E i = M + e sin E i 1, i = 1, 2, 3,... (7.14) The satellite coordinates are then obtained, using equations (7.15) to (7.29): cos ν k = cos E k e True anomaly, (7.15) 1 e cos E k 1 e sin ν k = 2 sin E k True anomaly, (7.16) 1 e cos E k k = ν k + ω Argument of latitude, (7.17) δu k = C uc cos 2 k + C us sin 2 k Argument of latitude correction, (7.18) δr k = C rc cos 2 k + C rs sin 2 k Radius correction, (7.19) δi k = C ic cos 2 k + C is sin 2 k Inclination correction, (7.20) u k = k + δu k Corrected argument of latitude, (7.21) r k = A(1 e cos E k ) + δr k Corrected radius, (7.22) i k = i 0 + it k + δi k Corrected inclination, (7.23) X k = r k cos u k Position in the orbital plane, (7.24) Y k = r k sin u k Position in the orbital plane, (7.25) k = 0 + ( ω e )t k ω e t 0e Corrected longitude of (7.26) ascending node, X k = X k cos k Y k sin k cos i k Earth fixed geocentric (7.27) satellite coordinates, Y k = X k sin k + Y k cos k cos i k Earth fixed geocentric (7.28) satellite coordinates, Z k = Y k sin i k Earth fixed geocentric (7.29) satellite coordinates. Equation (7.26) implicitly describes the relation between the vernal equinox,, and the current position of the zero meridian, based on Earth s rotation ω e Structure of the GPS Navigation Data The GPS navigation data, the so-called message, is organized as in Fig The user has to decode the data signal [7.1.4] to access the navigation data. Decoding is executed in the internal receiver processor for on-line navigation purposes. Most
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