Southwest Jiaotong University Principles of the Global Positioning System Lecture 03 YUAN LInguo EmailIgyuan@163.com Dept of Surveying Engineering, Southwest Jiaotong University Fundamental knowledg Quick Overview Kelperian 3 Laws First Law: ellipse, Sun is a focus Second law: the same area in same time Earth Third law: 1_ai Principles of the Global Positioning System 2005-3-11(2
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
Perigee and Apogee: The point of closest approach of the satellite with to the earth' s center of mass is called perigee and the most distant position is the apogee. Nodes: The intersection between the equatorial and the orbital plane with the unit sphere is termed the nodes, where the ascending node defines the northward crossing of the equator. ae E apogee geocenter perigee 電 Principles of the Global Positioning System 2005-3-11(3 Introduction Principles of the Global Positioning System 2005-3-11(4
2 Principles of the Global Positioning System 2005-3-11 3 Principles of the Global Positioning System 2005-3-11 4 Introduction
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 In relative positioning, relative orbital errors are considered to be approximately equal to relative baseline errors 電 Principles of the Global Positioning System 2005-3-11(5 Orbit Information and SA Technology How to obtain orbital information: o either transmitted by the satellite as part of the broadcast message or o can be obtained(typically some days after the observation) from several sources Principles of the Global Positioning System 20053-11(6
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
Orbit Information and SA Technology Orbit Inf and sa: o The activation of sa in the block ii satellites may lead to a degradation of the broadcast orbit up to 50-100m Civil Community Since some users need more precise ephemerides, the civil community must generate its own precise satellite ephemerides 電 Principles of the Global Positioning System 2005-3-11(7 Orbit Description Principles of the Global Positioning System 2005-3-11(8
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
Keplerian Motion Orbital parameters The movement of mass m2 relative to m, is defined by the homogeneous 2nd order differential equation G(m1+m2) y=0 Principles of the Global Positioning System Artificial Earth Satellite. Points Mass: negligible W=GMa=3980005103m3s2 The analytical solution of differential equation leads to the well-known Keplerian motion defined by six orbital parameters The orbital parameters correspond to the six integration constants of the second-order vector equation Principles of the Global Positioning System 2005-3-11(10 5
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:
Keplerain elements: Orbit plane Satellite equinox Node A greenwich Inclination Q2 Right Ascension of ascending node oo Argument of perigee V True anomaly Principles of the Global Positioning System 2005-3-11(11 Keplerain elements in plane Apogee Perigee FocuS Center of Mass a semimajor axIs V True anomaly b semiminor axis E Eccentric anomaly e eccentricity M Mean anomaly Principles of the Global Positioning System 20053-11(12
6 Principles of the Global Positioning System 2005-3-11 11 Keplerain elements: Orbit plane Node i ω Ω ν Z θ0 Greenwich Vernal equinox Satellite perigee equator i Inclination Ω 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 Apogee Perigee b E ν r a semimajor axis b semiminor axis e eccentricity ν True anomaly E Eccentric anomaly M Mean anomaly
The mean angular satellite velocity n an angular satellite velocity n(also known the mean motion) with revolution period P follows from Kepler's Third Law given 2 For gPS orbits, a=26560 kn, so, an orbital period 12 sidereal hours. The ground track of the satellites repeats every sidereal day 電 Principles of the Global Positioning System 2005-3-11(13 Orbit Representation n orbital plane, the position vector r and the velocit tv vector i=drdt (with eccentric true anomaly cos e-e coSy X3=X3 Principles of the Global Positioning System 2005-3-11(14
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
Orbit Representation The transformation of r and r into the equatorial system x o is performed by a rotation matrix X satellite perigee R 3D rotation R. e3=0 LOT orbit &e Principles of the Global Positioning System 2005-3-11(15 Orbit Representation R=R3{-91{-i}R3{-m}=[eg2e3] In order to rotate the system x into the terrestrial system X, an additional rotation through the angle Oo, the transformation matrix, therefore, becomes quired The R=R3O。}3{-2R1{-i}R3{ Orbital Plane Space-fixed Sys. ->Terrestrial Sys. 電 Principles of the Global Positioning System 2005-3-11(16
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
Differentia/ Relations The derivatives of p and p with respect to the six Keplerian parameters are required in one of the The vectors r and i depend only on the parameters a.e. To. whereas the matrix is only a function of the haining parameters a i, s2 The differential relations The meaning? aR 中=Rdn+Rsd+Rsdm R 電 Principles of the Global Positioning System 20053117 Perturbed Motion The Keplerian orbit is a theoretical orbit and does not include actual perturbations The perturbed motion is based on an inhomogeneous differential equation of second order p+“P= For GPS satellites, the acceleration p is at least 10+times attractive force A=? Analytical solution Au =l Principles of the Global Positioning System 2005-3-11
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
Keplerian Motion VS Perturbed Motion The parameters pi are constant They are time dependent Thus, for the position and velocity vector of the perturbed motion. we have P=pt,p, (t)) p=pt,p, (o) 電 Principles of the Global Positioning System 20053-11(19 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 Gravitational. Nonsphericity of the Earth Tidal attraction(Direct and Indirect Non- Solar radiation pressure(direct and indirect gravitational,Air drag Relativistic effect Others(solar wind, magnetic field forces Principles of the Global Positioning System 2005-3-11(20 10
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:
