Document Sample

A Brief Introduction to the Global Positioning System (GPS) CMPE-118 Lecture 1 OF 28 1 OF 62 Global Positioning System (GPS) • Satellite Navigation system – Multilateration based on one-way ranging signals from 24+ satellites in orbit – Operated by the United States Air Force – Nominal Accuracy • 10 m (Stand Alone) • 1-5 m (Code Differential) • 0.01 m (Carrier Differential) ©2000 by Todd Walter and Per Enge 2 OF 28 2 OF 62 Navigation Terminology • Navigation – Answer the to the question ―Where am I?‖ – Implies the use of some agreed upon coordinate system • Related Terminology – Guidance: Deciding what to do with your navigation information – Control: Orienting yourself/vehicle to follow out the guidance decision. • Area of Study: GNC – Guidance, Navigation, Control 3 OF 28 3 OF 62 Latitude (Parallels) are Longitude (Meridians) are formed by the intersection of formed by the intersection of the surface of the earth with the surface of the earth with a plane parallel to the a plane containing the earths equatorial plane axis. 4 OF 28 4 OF 62 Latitude, Longitude and Altitude • One of many coordinate systems used to described a location on the surface of the earth • Latitude — parallels measured from the Equator. – North is ―+‖ • Longitude — meridians measured from Greenwich Observatory. – East is ―+‖ • Altitude — measured above reference datum: MSL – Normally Up is ―+‖ 5 OF 28 5 OF 62 Stability of Clocks • Clock stability is directly related to Navigation because Earth rotates ~15°/hour. Figure from Hewlett-Packard Application Note 1289: The Science • Difference between of Timekeeping by D. W. Allan, Neil local ―celestial‖ Ashby and Cliff Hodge. time and reference yields Longitude. • Atomic clocks are too big and too expensive for general use. 6 OF 28 6 OF 62 Position Fixing Methods a) Bearing and range (r-q) position fixing (DME-VOR) b) Dual bearing (q-q) position fixing (VOR- VOR) c) Range (r-r) position fixing (DME-DME, GPS) d) Hyperbolic position fixing (LORAN, Omega) From Kyton and Fried, Avionics Navigation Systems, 2nd Ed., pp. 113. 7 OF 28 7 OF 62 r-r Position Fixing (2-D) Assuming you can make the range measurements ri , where i = 1,2,3, then the following three equations can be formed: r12 x - x1 )2 y - y1 )2 r 22 x - x2 )2 y - y2 )2 r32 x - x3 )2 y - y3 )2 8 OF 28 8 OF 62 Fundamentals of Position Fixing • The figure on the previous page raises to important questions: – How do you estimate or measure the ranges? – How do you solve the equations for the unknown x and y? • The range based on measuring the time-of-flight of a RF signal that leaves the transmitter at t = t1 and arrives at the user at t = t2 is given by: r ct2 - t1 ) • In the presence of a clock error, dt (= b/c), the range estimate (or measurement) becomes: r r b ct2 - t1 ) cdt ct2 - t1 ) b ˆ 9 OF 28 9 OF 62 GPS Pseudoranges SV #1 SV #2 As a user located at point X, the true range measurements to the three GPS satellites are: r1 r2 r1True r1 cbu r 2 r 2 cbu True cbu cbu r 3 r 3 cbu True cbu r3 SV #3 Your GPS receiver, however, measures r1, r2 and r3. These range measurement are called pseudoranges. 10 OF 28 10 OF 62 Psueodranges and Satellite Geometry Pseudorange Measurement Resulting Geometry plays a role in Error Position the accuracy of the final Uncertainty solution. Areas 11 OF 28 11 OF 62 GPS Position Fixing 12 OF 28 12 OF 62 Solving Navigation Equations • Solve the r-r equations – Easy and give you insight into the linearization process – GPS navigation equations. • The r-r position fixing system of equations where three independent range measurements are available was given as: r12 x - x1 )2 y - y1 )2 r 22 x - x2 )2 y - y2 )2 r32 x - x3 )2 y - y3 )2 13 OF 28 13 OF 62 Linearization by Expansion r12 x - x1 )2 y - y1 )2 r12 x - x1 )2 y - y1 )2 ˆ ˆ ˆ r 22 x - x2 )2 y - y2 )2 r 22 x - x2 )2 y - y2 )2 ˆ ˆ ˆ r32 x - x3 )2 y - y3 )2 r32 x - x3 )2 y - y3 )2 ˆ ˆ ˆ Exact Equations you Equations the you can or will solve would solve in an ideal world ri ri dri ˆ x x dx ˆ y y dy ˆ 14 OF 28 14 OF 62 Linearization by Expansion (2) For the range measurements, ri2 ri dri )2 ri2 2ridr dr 2 ri2 2ridr (dropped higher order terms) ˆ For the position coordinate x, x - xi )2 x dx - xi )2 x - xi )2 2x - xi )dx ˆ For the position coordinate y, y - yi )2 y dy - yi )2 y - yi )2 2 y - yi )dy ˆ Where, i 1,2, n 15 OF 28 15 OF 62 Linearization by Expansion (3) Taking the difference between the true and estimated values, r i2 - r i2 r i2 - r i2 2 r idr ) -2 r idr -2x - xi )dx - 2 y - yi )dy ˆ Normally you have more equations than unknowns. Thus, you can do a least squares solution. That is, x - x1 ) y - y1 ) r1 dr1 r1 dr x - x2 ) y - y2 ) dx 2 r r2 dy 2 dr n x - xn ) y - yn ) rn rn dr G dx 16 OF 28 16 OF 62 Linearization by Expansion (4) Because we don’t have true ranges, but pseudo-ranges, we augment the G matrix with a column of ones for the time bias. We need at least 3 measurements for the 2-D solution. x - x1 ) y - y1 ) r 1 r1 x - x2 ) 1 dr1 y - y2 ) dx dr 1 2 r2 r2 dy dt drn x - xn ) y - yn ) r rn 1 n dr G dx 17 OF 28 17 OF 62 Least Squares Solution For the moment, without proof, we state that the least squares solution is given by, dx GT G ) GT dr -1 • Algorithm for solving the navigation equation: – 1) Pick an initial guess for x and y – 2) Compute r i for as many measurements as you have ˆ – 3) Form dr i for all measurements and then form G – 4) Solve for dx – 5) Update your initial guesses for x and y as follows: x ( ) x ( -) dx y ( ) y ( -) dy – 6) Repeat until convergence 18 OF 28 18 OF 62 Iterated Solution Numerical Example • Solution is done in MATLAB • Assumes an initial position of [0,0,0] • Walks solution in to the final position • Redraws the range circles at each iteration 19 OF 28 19 OF 62 GPS Signal Structure • GPS broadcasts a modulated carrier on L1 (1575.42 MHz) • Pseudo-Random Noise (PRN) sequence of 1023 ―chips‖ used to spread the signal • PRN is carefully chosen to have unique auto— and cross—correlation properties • All signal components generated from the same 10.23 MHz satellite clock 20 OF 28 20 OF 62 GPS L1 Signal Generation 21 OF 28 21 OF 62 GPS Signal De-Spreading • In order to use the PRN code correlation properties to de-spread the GPS signal, need to recover code down to baseband (no carrier) • Use trigonometric identities to mix down and remove the carrier cos( ) cos( ) cos( ) - sin( ) sin( ) cos( - ) cos( ) cos( ) sin( ) sin( ) 2 cos( ) cos( ) cos( ) cos( - ) 22 OF 28 22 OF 62 Graphical Depiction of De-Spreading 23 OF 28 23 OF 62 PRN Auto- and Cross-Correlation 24 OF 28 24 OF 62 PRN Correlation Example 25 OF 28 25 OF 62 Initial Acquisition Search • Assume 1 channel & 1 ms dwell period • Exhaustive search (if real time) requires: – (32) x (2046) x (20) x 1ms = 1309 seconds • 12 channel assumption requires: – (1309) / 12 = 109 seconds 26 OF 28 26 OF 62 Typical Search Results 27 OF 28 27 OF 62 Things to remember about GPS • Navigation is a hard problem, and only recently has GPS made this easy! • GPS is a r-r system that has precise clocks on board that give you position and your time bias. • PRN signal has correlation properties that allow you to find the signal in the noise even without any knowledge of position. 28 OF 28 28 OF 62 Questions? 29 OF 28 29 OF 62 30 OF 28 30 OF 62 31 OF 28 31 OF 62 Latitude Determination Using Polaris Actual location of Polaris is 89o 05’ The Sky Above Palo Alto on Jan 6, 2002 32 OF 28 32 OF 62 Instruments of Navigation An Astrolabe A Sextant 33 OF 28 33 OF 62 View Through a Sextant Easier to “align” Sun’s (or other celestial body’s) limb with the horizon. 34 OF 28 34 OF 62 Latitude Determination Using the Sun 900 - Sun's Altitude Sun's Declination 35 OF 28 35 OF 62 The Longitude Problem • Celestial map changes because of Earth‘s 15 o/hr (approximately) rotation rate. 36 OF 28 36 OF 62 Longitude Determination • Longitude Determination Methods – Methods based on time • Compare the time between a clocks at the current location and some other reference point. • Requires Stable Clocks – Celestial Methods • Eclipses of Jupiter‘s Moons • Lunar Distance Method 37 OF 28 37 OF 62 Fundamentals of Radionavigation • Radio Frequency (RF) signals emanating from a source or sources. • The generators of the RF signal are at known locations • RF signals are used to determine range or bearing to the known location 38 OF 28 38 OF 62 Classification of Radio Frequencies Name of Band Frequency Range Wavelength Very Low Frequency (VLF) < 30 kHz > 10 km Low Frequency (LF) 30 – 300 kHz 1 - 10 km Medium Frequency (MF) 300 kHz – 3 MHz 100 m – 1 km High Frequency (HF) 3 – 30 MHz 10 – 100 m Very High Frequency (VHF) 30 – 300 MHz 1 – 10 m Ultra High Frequency (UHF) 300 MHz – 3 GHz 10 cm – 1 m Super High Frequency (SHF) 3 – 30 GHz 1 – 10 cm Propagation characteristic of RF signals is a function of their frequency 39 OF 28 39 OF 62 Line of Sight Transmission • VHF (VOR, ILS Localizer) and UHF (ILS Glide Slope, TACAN/DME) are line of sight systems. – Limited Coverage area • LORAN and OMEGA are over the horizon systems – Large coverage area – In the case of Omega, coverage was global • Frequency band in which GPS operates makes it a line of sight system. However, because of the location of the satellites, it is able to cover a large geographic area . 40 OF 28 40 OF 62 INS and Radionavigation Systems Application Navigation System Land Sea Air NDB – Non Directional Beacon X X LORAN – Long RAnge Navigation X X VOR – VHF Omni-directional Range X DME – Distance Measuring Equipment X ILS – Instrument Landing System X MLS – Microwave Landing System X INS – Inertial Navigation System* X X X * INS is not a radionavigation system but is normally used in conjunction with such systems 41 OF 28 41 OF 62 Phases of Flight • The required navigation accuracy and reliability (i.e., integrity, continuity and availability) depend on the phase of flight • Currently, as well as in the past, this meant that an aircraft had to be equipped with various navigation systems. 42 OF 28 42 OF 62 VHF Omni-directional Range (VOR) • Provides Bearing (Y) Information • Operates 112 – 118 MHz • Accuracy 1o to 2 o. • Principles of Operation (Enge et. al. ―Terrestrial Radionavigation, pp. 81) – Transmits 2 Signals • 1st Signal has azimuth dependent phase • 2nd Signal is a reference • D between the phases of signal 1st and 2nd signal is Y 43 OF 28 43 OF 62 Distance Measuring Equipment (DME) • Measures Slant Range (r) • Operates between 962 and 1213 MHz • Based on Radar Principle – Airborne unit sends a pair of pulses – Ground Station receives pulses – After short delay (50 ms) ground station resends the pulses back – Airborne unit receives the signal and calculates range by using the following equation: 1 r c(DT - 50ms) 2 r 44 OF 28 44 OF 62 Instrument Landing System (ILS) • Used extensively during approach and landing to provides vertical and lateral guidance • Principle of Operation – Lateral guidance provided by a signal called the Localizer (108-112 MHz) – Vertical guidance provided by another signal called the Glide Slope (329-335 MHz) • Distance along the approach path provided by marker beacons (75 MHz) 45 OF 28 45 OF 62 Time Scales • Sidereal Time – Based on the An Earth time required by Earth to Apparent complete one revolution about its axis relative to distant stars. Solar Day • Apparent Solar Day - Time required for Earth to complete one revolution with respect to the sun • Mean Solar Time - Same as apparent solar day except it is Sun based on – Hypothetical earth – Rotating in a circular orbit around the sun. – Axis of rotation perpendicular to the orbital plane – Same as Greenwich Mean Time Earth’s (GMT) Orbit 46 OF 28 46 OF 62 Universal & Atomic Time • Universal Time (UT) – Time based on astronomical observations – UT0 – Mean Solar Time measured at the prime meridian – UT1 – UT0 Corrected for Earth‘s irregular spin rate and polar motion • International Atomic Time (TAI) – Based on Ce-133 Atom • Coordinated Universal Time (UTC) – Set to agree with UT1 on January 1, 1958. – Leap seconds introduced to keep it within 0.9 seconds of UT1 47 OF 28 47 OF 62 GPS Time • GPS Time (GPST) – A continuous time scale (no leap seconds) – Based on Cesium and Rubidium standards – ‗Steered‘ to be within fractions of a microsecond modulo one second from UTC • Thus GPST-UTC = whole number of seconds + a fraction of a microsecond. • GPS time information transmitted by the satellites include – GPS second of the week - 604,800 seconds per week – GPS week number – 1024 weeks per epoch 48 OF 28 48 OF 62 GPS Time (2) • GPS satellites carry atomic clocks – Rubidium and/or Cesium frequency standards – Satellite clocks monitored by MCS • Clock bias is modeled as a quadratic dt a f 0 a f 1 (t - t0c ) a f 2 (t - toc )2 Dtr • Parameters of the Quadratic are uploaded to Satellites which in turn broadcasts them as the navigation message – Sub-frame 1 of the navigation message • Clock correction term Dtr takes into account relativistic effects – Account for speed and location in the gravitation potential of the clocks – Net effect results in satellite clocks gaining ~38.4 msec per day – Compensated for by setting the satellite fundamental frequency of 10.23 MHz 0.00455 Hz lower. 49 OF 28 49 OF 62 GPS Coordinate Frames • Inertial Frame of Reference – Defined to be a non-accelerating or rotating coordinate frame of reference – e.g., Earth Centered Inertial (ECI) – Required for analysis of satellite motion, inertial navigation, etc. – Not convenient for terrestrial navigation • Coordinate systems you will mostly encounter in GPS are – Earth Centered Earth Fixed (ECEF) – East-North-Up (ENU) – Geodetic Coordinates • Other coordinate systems used in navigation – North-East-Down (NED) – used widely in aircraft navigation, guidance and control applications – Wander-Azimuth 50 OF 28 50 OF 62 Coordinate Frame Relationships • Geodetic coordinates (f, l, h) to ECEF a 6378137 m e 0.08181919 a N 1 - e sin(f ) ) 2 x N h ) cos(f ) cos(l ) y N h ) cos(f ) sin(l ) ) ) z N 1 - e 2 h sin(f ) • ECEF to Geodetic coordinates – Iterative algorithm – See Wgsxyz2lla.m in toolbox 51 OF 28 51 OF 62 Geometry of Earth (1) • Crude Approximation – A sphere – R0 = 6378.137 km – A spherical model is only good for ―back of the envelope‖ type of calculations – Need a more precise model for navigation applications (especially inertial navigation) • A more accurate model is an ellipsoid – Parameters of the mathematical ellipsoid are defined in WGS-84 52 OF 28 52 OF 62 Geometry of Earth (2) • Topographic Surface – Shape assumed by Earth‘s crust. – Very complicated shape not amenable to mathematical modeling • Geoid – An equipotential surface of Earth's gravity field which best fits, in a least squares sense, global Mean Sea Level (MSL). • Reference Ellipsoid – Mathematical fit to the geoid that happens to be an ellipsoid of revolution and minimizes the mean-square deviation of local gravity and the normal to the ellipsoid 53 OF 28 53 OF 62 WGS-84 Reference Ellipsoid • Some geometric facts about the WGS-84 Reference Ellipsoid – Semi-major axis ( a ) = 6378137 m – Semi-minor axis ( b ) = 6356752 m – Flattening ( f ) = 1-(b/a) = 1/(298.25722) – Eccentricity ( e ) = [f(2-f)]1/2 = 0.081819191 • Given the WGS-84 Ellipsoid parameters, the following are derived quantities: – RNS = a 1 f 3 sin 2 (f ) - 2 )) – REW = a 1 f sin 2 (f ) ) f f’ f = Geodetic Latitude f’ = Geocentric Latitude where f f’ tan(f ' ) (1 - f ) 2 tan(f ) 54 OF 28 54 OF 62 Geoidal Heights 55 OF 28 55 OF 62 56 OF 28 56 OF 62 Orbital Mechanics • Kepler‘s Law – Based on observations made by Tycho Brahe (1546-1601) • First Law: Each planet revolves around the Sun in an elliptical path, with the Sun occupying one of the foci of the ellipse. • Second Law: The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time. • Third Law: The squares of the planets' orbital periods are proportional to the cubes of the semi-major axes of their orbits. • Explanation came later – Isaac Newton (1642-1727) – Universal Law of Gravitation, where combined with his second law leads to GM E mS r F - 2 , r rS - rE r r G( M E mS ) r r GM E r r m r 0 r r3 r3 r3 57 OF 28 57 OF 62 Six Keplerian Elements • Recast the two-body equation of motion. • Characterize orbital ellipse – Semi-major Axis (A) – Eccentricity (e) • Characterize orbit‘s orientation in space – Inclination (i) – Right Ascension of the Ascending Node (W) • Characterize ellipse‘s orientation in orbital plane – Argument of Perigee (w) • Position of the satellite in the orbit – True anomaly (n) • Sometimes it is convenient to sum n and w to form a new variable called argument of latitude 58 OF 28 58 OF 62 GPS Orbital Parameters • Perturbed Orbits - quasi-Keplerian 15 Figure from Bate, Mueller and White, element set Fundamentals of Astrodynamics (1971), pp. 156 – Non-central gravitational force – gravitational fields of the sun and moon – solar pressure • Additional 9 parameters – Three to account for the rate of changes: • Right Ascension of the Ascending Node (W-dot) • Inclination (i-dot) • Mean motion (n-dot) – Three pairs (6 parameter total) to correct • Argument of latitude • Orbit radius • Inclination angle Pertubative Torque caused by Earth’s Equatorial Bulge 59 OF 28 59 OF 62 GPS Constellation and Orbits • Nominal Constellation – 24 Satellites. – At present more than 24 satellites on orbit. • Semi-major axis – 26,560 km • Eccentricity – less than 0.01 • Period – approximately 11 h 58 min • Six orbital planes – Planes designated A through F – Inclination of 550 relative to the equatorial plane – RAAN, W, for the six orbital planes separated by 600. – Four Satellites per orbital plane. Satellites in a given orbital plane are distributed unevenly to minimize the impact of a single satellite failure. 60 OF 28 60 OF 62 GPS Ephemeris Calculation • Compute the satellites position in the orbital coordinate frame – Solve Kepler's equation ( E = M + e sin E ) for eccentric anomaly at epoch k, Ek. • Solution requires iteration if orbit is non-circular – Compute the true anomaly, nk – Compute the argument of latitude Fk – Use Fk to compute the corrections for argument of latitude, radius and inclination then apply the computed corrections. – Compute the x and y coordinates (xk’ and yk’) of the satellite in it‘s orbit. • Covert the computed xk’ and yk’ position into ECEF coordinates – Compute the correction for the longitude of the ascending node. – Apply the correction to the longitude of the ascending node. – Compute the ECEF coordinates 61 OF 28 61 OF 62 GPS Almanac • A subset of clock and ephemeris parameters. – Limited to seven parameters and the associated reference time (toe) • Square root of semi-major axis ((A)1/2) • Eccentricity (e) • Inclination (i) • Longitude of ascending node (W0) • Rate of right ascension (W-dot) • Argument of perigee (w) • Mean anomaly (M) – Reduced precision – Allows determining approximate position of satellites • All satellites broadcast almanac data for all other satellites in the constellation – Sub-frames 4 and 5 of the navigation message – Updated less frequently than the ephemeris parameters in sub-frames 2 and 3. 62 OF 28 62 OF 62

DOCUMENT INFO

Shared By:

Tags:

Stats:

views: | 205 |

posted: | 9/14/2010 |

language: | English |

pages: | 62 |

OTHER DOCS BY hafsakhan21

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.