1 Principles of the Global Positioning System Lecture 03 YUAN Linguo Email: lgyuan@163.com Dept. of Surveying Engineering, Southwest Jiaotong University Principles of the Global Positioning System 2005-3-11 2 2 Principles of the Global Positioning System 2005-3-11 3 Principles of the Global Positioning System 2005-3-11 4 Introduction 3 Principles of the Global Positioning System 2005-3-11 5 Why do We Study Orbit? The applications of GPS depend substantially on knowing the satellite orbits. 1. For single receiver positioning, an orbital error is highly correlated with the positional error. 2. In relative positioning, relative orbital errors are considered to be approximately equal to relative baseline errors. Principles of the Global Positioning System 2005-3-11 6 Orbit Information and SA Technology How to obtain Orbital Information: ? either transmitted by the satellite as part of the broadcast message, or ? can be obtained (typically some days after the observation) from several sources. 4 Principles of the Global Positioning System 2005-3-11 7 Orbit Inf. and SA: ? The activation of SA in the Block II satellites may lead to a degradation of the broadcast orbit up to 50-100 m. Civil Community: ? Since some users need more precise ephemerides, the civil community must generate its own precise satellite ephemerides. Orbit Information and SA Technology Principles of the Global Positioning System 2005-3-11 8 Orbit Description 5 Principles of the Global Positioning System 2005-3-11 9 Keplerian Motion Principles of the Global Positioning System 2005-3-11 10 Artificial Earth Satellite: 6 Principles of the Global Positioning System 2005-3-11 11 Keplerain elements: Orbit plane Node i ω ? ν Z θ 0 Greenwich Vernal equinox Satellite perigee equator i Inclination ? Right Ascension of ascending node ω Argument of perigee ν True anomaly Principles of the Global Positioning System 2005-3-11 12 Keplerain elements in plane a Focus Center of Mass ae Satellite PerigeeApogee b E ν r a semimajor axis b semiminor axis e eccentricity ν True anomaly E Eccentric anomaly M Mean anomaly 7 Principles of the Global Positioning System 2005-3-11 13 The mean angular satellite velocity n Principles of the Global Positioning System 2005-3-11 14 Orbit Representation 8 Principles of the Global Positioning System 2005-3-11 15 Orbit Representation Principles of the Global Positioning System 2005-3-11 16 Orbit Representation 9 Principles of the Global Positioning System 2005-3-11 17 Differential Relations Principles of the Global Positioning System 2005-3-11 18 Perturbed Motion 10 Principles of the Global Positioning System 2005-3-11 19 Keplerian Motion vs. Perturbed Motion Principles of the Global Positioning System 2005-3-11 20 Disturbing Accelerations In reality, many disturbing accelerations act on a satellite and are responsible for the temporal variations of the Keplerian elements. They can be divided into: 11 Principles of the Global Positioning System 2005-3-11 21 Disturbing Accelerations Principles of the Global Positioning System 2005-3-11 22 Disturbing Accelerations z For GPS satellites, altitude is about 20200 km, the indirect effect of solar radiation pressure and air drag may be neglected. z The shape of the satellites is irregular which renders the modeling of direct solar radiation pressure more difficult. Different Satellites are different radiation pressures z The variety of materials used for the satellites has a different heat-absorption which results in additional and complicated perturbing accelerations. z Accelerations may arise from gas leaks in the container of the gas-propellant. 12 Principles of the Global Positioning System 2005-3-11 23 1. Nonsphericity of the Earth z The regression of the node in the equatorial plane z The rotation of the perigee z The variation of the mean anomaly Principles of the Global Positioning System 2005-3-11 24 Perturbation from Flattening J 2 The J 2 perturbation can be computed from the Lagrange planetary equations ? ? = ? 3 2 na e 2 cosi a 2 (1 ? e 2 ) 2 J 2 ?ω = 3 4 na e 2 5 cos 2 i ?1 a 2 (1 ? e 2 ) 2 J 2 ? M = n + 3 4 na e 2 3cos 2 i ?1 a 2 (1 ? e 2 ) 3 J 2 13 Principles of the Global Positioning System 2005-3-11 25 2. Tidal Effects Among all the celestial bodies in the solar system, only the sun and the moon must be considered because the effects of the planets are negligible. ? The maximum of the perturbing acceleration is reached when the three bodies are situated in a straight line. ? Apart from the direct effect of the tide generating bodies, indirect effects due to the tidal deformation of the solid earth and the oceanic tides must be taken into account. ? The model for the indirect effect due to the oceanic tides is more complicated. Principles of the Global Positioning System 2005-3-11 26 3. Solar Radiation Pressure The perturbing acceleration due to the direct solar radiation pressure has two components: 1. The principal component is directed away from the sun. 2. The smaller component acts along the satellite's y-axis. This is an axis orthogonal to both the vector pointing to the sun and the antenna which is nominally directed towards the center of the earth. 14 Principles of the Global Positioning System 2005-3-11 27 3. Solar Radiation Pressure ? The first component is in the order of 10-7 ms-2 ? The second component is called y-bias, and is believed to be caused by a combination of misalignments of the solar panels and thermal radiation along the y-axis. ? The solar radiation pressure which is reflected back from the earth's surface causes an effect called albedo. For GPS, the associated perturbing accelerations are smaller than the y-bias and can be neglected Principles of the Global Positioning System 2005-3-11 28 4. Relativistic Effect 1. The relativistic effect on the satellite orbit is caused by the gravity field of the earth and gives rise to a perturbing acceleration. 2. This effect is smaller than the indirect effects by one order of magnitude. 3. The numerical values of perturbing acceleration results in an order of 3·10 -10 ms -2 15 Principles of the Global Positioning System 2005-3-11 29 Perturbation on orbits and approximate size 10 -9 Albedo radiation 10 -7 Solar radiation ~0Drag 10 -10 Ocean tides 10 -9 Earth tides 5x10 -6 Third body 3x10 -7 Other gravity 5x10 -5 J 2 0.6Central Acceleration (m/sec 2 )Term Principles of the Global Positioning System 2005-3-11 30 Orbit Determination 16 Principles of the Global Positioning System 2005-3-11 31 Orbit Determination: orbital parameters and satellite clock biases. Principles of the Global Positioning System 2005-3-11 32 Orbit Determination Position vector is a function of ranges, whereas the velocity vector is determined by range rates. ? At present, the observations for the orbit determination are performed at terrestrial sites, such as TOPEX/Poseidon. ? The GPS data could also be obtained from orbiting receivers. 17 Principles of the Global Positioning System 2005-3-11 33 Clues of Orbital Determination The actual orbit determination is performed in two steps. 1. A Kepler ellipse is fitted to the observations (theoretically). (1) Initial value problem (2) Boundary value problem 2. This ellipse serves as reference, then is improved by taking into account perturbing accelerations. Add all perturbation parameters into Kepler Orbit (1) Analytical solution (2) Numerical solution Principles of the Global Positioning System 2005-3-11 34 Keplerian Orbit 18 Principles of the Global Positioning System 2005-3-11 35 Initial Value Problem z The derivation of the Keplerian parameters from position and velocity vectors, given at the same epoch and expressed in an equatorial system, is an initial value problem for solving the differential. z Recall that the two given vectors contain six components (six Keplerian parameters). Since both vectors are given at the same epoch, the time parameter is omitted. Principles of the Global Positioning System 2005-3-11 36 Boundary Value Problem 1. It is assumed that two position vectors S(t1) and S(t2) at epochs t1 and t2 are available. 2. Note that position vectors are preferred for orbit determination since they are more accurate than velocity vectors. 3. The given data correspond to boundary values in the solution of the basic second-order differential equation. 19 Principles of the Global Positioning System 2005-3-11 37 Orbit Improvement ? If there are redundant observations, the parameters of an instantaneous Kepler ellipse can be improved because each observed range gives rise to an equation. ? The vector can be expressed as a function of the Keplerian parameters. Thus, it actually contains the differential increments for the six orbital parameters. ? In the past, orbit improvement was often performed in the course of GPS data processing when, in addition to terrestrial position vectors, the increments were determined. The procedure became unstable or even failed for small networks. In the case of orbit relaxation, only three degrees of freedom were assigned to the orbit. Principles of the Global Positioning System 2005-3-11 38 Perturbed Orbit Analytical Solution In order to be suitable for Lagrange's equations, the disturbing (Earth) potential is expressed as a function of the Keplerian parameters. The tidal potential also has a harmonic representation, and thus the tidal perturbations can be analytically modeled. GM V= V r +? 20 Principles of the Global Positioning System 2005-3-11 39 Numerical Solution Principles of the Global Positioning System 2005-3-11 40 Orbit Dissemination 21 Principles of the Global Positioning System 2005-3-11 41 Tracking Networks Objectives and Strategies The official orbit determination for GPS satellites is based on observations at the five monitor stations of the control segment. 1. The broadcast ephemeredes for Block I satellites: ~5m. 2. For the Block II satellites: up to 50-100 m by SA. An orbital accuracy of about 20 cm is required for specific missions such as TOPEX/Poseidon or for investigations which require an accuracy at the level of 10 -9 . Principles of the Global Positioning System 2005-3-11 42 Tracking Networks ? Minimum Number of Sites: in a global network is six, if a configuration is desired where at least two satellites can be tracked simultaneously any time from two sites. ? Global Network and Regional Network: Global Network result in higher accuracy and reliability compared to regional networks. ? Orbit System Tie: The tie of the orbital system to terrestrial reference frames is achieved by the collocation of GPS receivers with VLBI and SLR trackers. ? GPS Site Distribution: The distribution of the GPS sites is essential to achieve the highest accuracy. 22 Principles of the Global Positioning System 2005-3-11 43 A Comparison of Two Distribution of GPS Sites ?The sites are regularly distributed around the globe; ?Each network site is surrounded by a cluster of additional points to facilitate ambiguity resolution Principles of the Global Positioning System 2005-3-11 44 Examples for Global Networks Several networks have been established for orbit determination. ? Regional ? Continental size (the Australian GPS) ? Global networks 1. Global Orbit Tracking Experiment (GOTEX) 2. The Cooperative International GPS Network (CIGNET) 3. In 1990, IAG installed an International GPS Service for Geodynamics (IGS) 23 Principles of the Global Positioning System 2005-3-11 45 Ephemerides Three sets of data are available to determine position and velocity vectors of the satellites in a terrestrial reference frame at any instant: – Almanac data, – Broadcast ephemerides, and – Precise ephemerides Principles of the Global Positioning System 2005-3-11 46 1. Almanac Data Purpose: provide the user with less precise data to facilitate receiver satellite search or for planning tasks e.g., the computation of visibility charts. ? The almanac data are updated at least every six days and are broadcast as part of the satellite message. ? The almanac message essentially contains parameters for the orbit and satellite clock correction terms for all satellites. ? All angles are expressed in semicircles. 24 Principles of the Global Positioning System 2005-3-11 47 2. Broadcast Ephemerides Purposes: to compute a reference orbit for the satellites 1. The broadcast ephemerides are based on observations at the five monitor stations. 2. Additional tracking data are entered into a Kalman filter and the improved orbits are used for extrapolation. 3. The orbital data could be accurate to approximately 5m based on three uploads per day; with a single daily update one might expect an accuracy of 10 m. 4. The Master Control Station is responsible for the computation of the ephemerides and the upload to the satellites. The ephemerides are broadcast (mostly) every hour and should only be used during the prescribed period of approximately four hours to which they refer . Principles of the Global Positioning System 2005-3-11 48 Distribution of Ephemerides ? The broadcast ephemeris is decoded by all GPS receivers and for geodetic receivers the software that converts the receiver binary to an exchange format outputs an ASCII version ? The exchange format: Receiver Independent Exchange format (RINEX) has a standard for the broadcast ephemeris. ? Form [4-char][Day of year][Session].[yy]n e.g. brdc0120.02n 25 Principles of the Global Positioning System 2005-3-11 49 3. Precise Ephemerides 1. The official precise orbits are produced by the NSWC together with the DMA and are based on observed data in the (extended) tracking network. 2. The post-mission orbits are available upon request about four to eight weeks after the observations. 3. The most accurate orbital information is provided by the IGS with a delay of about two weeks. 4. Less accurate information is available about two days after the observations. 5. Currently, IGS data and products are free of charge for all users. Principles of the Global Positioning System 2005-3-11 50 3. Precise Ephemerides The precise ephemerides – satellite positions and – velocities at equidistant epochs. Since 1985, NGS began to distribute precise GPS orbital data. Formats: – the specific ASCII formats SP1 and SP2 –their binary counterparts ECF1 and ECF2. – Later, ECF2 was modified to EF13 format. Typical spacing of the data is 15 minutes. 26 Principles of the Global Positioning System 2005-3-11 51 NGS Format ? Each NGS format consists of a header containing general information (epoch interval, orbit type, etc.) followed by the data section for successive epochs. – The position: kilometer – The velocity: kilometer/second ? NGS formats are described in Remondi (1989, 1991b ). ? NGS provides software to translate orbital files from one format to another. Principles of the Global Positioning System 2005-3-11 52 Web Resources ? IGS -- International GPS Service http://igscb.jpl.nasa.gov/ ? Description of RINEX standard ftp://igscb.jpl.nasa.gov/igscb/data/format/rinex2.txt ? Description of SP3C standard ftp://igscb.jpl.nasa.gov/igscb/data/format/sp3c.txt 27 Principles of the Global Positioning System 2005-3-11 53 Assignment 1. Illustrate 6 Keplerian orbit parameters 2. Please describe in detail the Broadcast emphemerides (RINEX), and Precision emphemerides (SP3C). 3. A broadcast ephemeris file for Jan 16, 2002 is given below. All of the GPS satellites are given. Please compute the satellite position very 30s.