Global Navigation Satellite Systems II AERO4701 Space Engineering 3 Week 4
Last Week Examined the problem of satellite coverage and constellation design Looked at the GPS satellite constellation Overview of GPS operation, satellite orbital model, ephemeris data processing
Overview First Hour GPS Signal, pseudorange and error terms Solving GPS user position and velocity Second Hour GDOP and GPS accuracy quantification Differential GPS
GPS Signals Three types of data sent to ordinary users Almanac data: Satellite status/health, coarse time data Ephemeris: Precise GPS satellite orbital model data used to predict satellite position Coarse/Acquisition Code: Repeating code, unique for each satellite Signal correlator in user receiver determines time of flight of signal and thus range to satellite based on speed of light (pseudorange)
User Clock Bias This is the major source of error in the pseudorange Cheap receivers want to use a small, light-weight clock, has large drift rates Typically clock error can contribute as much as ~10km User clock bias forms a common error on a set of pseudorange measurements from each satellite, thus we can estimate the clock bias while estimating the position of the receiver
Satellite Clock Bias Atomic clock on satellite has very small drift rates (in the order of nano-seconds over a couple of hours) Can potentially cause errors ~2-5m Ground tracking/monitoring sites model clock error for each satellite, describes the error term as a second order polynomial: Where a f0, a f1 and a f2 are ground station updated correction coefficients to the polynomial t gd is the group delay, another ground station tuned parameter trel is the time error due to relativistic effects. This term can be calculated: Where e is orbit eccentricity, and E k is eccentric anomaly of the SV.
Satellite Position Correction for Time of Signal Reception Calculated satellite positions from ephemeris data are at transmit time where as position is computed at the time of signal reception at the receiver Earth has rotated during this time so we need to account for the difference in ECEF reference frames during the transit time Each of the satellite positions it transformed into the ECEF position at the reception time: θ = ω ( ρ'/ c) ie
Atmospheric (Tropospheric and Ionospheric) Errors GPS signals are delayed during passage through the Earth s Troposphere and Ionosphere Causes large errors ~20-40m Can be corrected using two methods: Using broadcast correction parameters (control by ground station) Use dual frequency receivers (L1 and L2) frequencies, pseudorange is computed on each frequency, delay is frequency dependent:
Total Pseudorange Equation
GPS Theory of Positioning 2D example: Imagine a flat world We have known satellite positions and known range to each satellite A and B are two possible user locations, we will choose A because it lies on the surface of the earth, which is where our assumed position is
GPS Theory of Positioning Imagine now our receiver has a clock error (delay) of 0.5 seconds The range now seems to be larger (by the same amount) for each satellite Our location is assumed to be at B instead of A
GPS Theory of Positioning Imagine we have a third satellite If there was no clock error, the intersection of all three circles would be at point A With clock errors we can gradually vary the predicted user clock error until all three circles intersect at point A and the clock bias is then known In the real world (3D) we need one more satellite to locate the user position
Solving User Position and Clock Bias 4 satellites required, four simultaneous equations to solve (x,y,z,cb u ) Solve equations using an iterative method
Solving User Position and Clock Bias: Iterative Method Start off with an initial position Break up the actual state: Δx is a small linearised error Linearise the non-linear equations
Iterative Method: Jacobian Matrix
Iterative Method: Continued Solve for Δx Add Δx back to state x and iterate until Δx -> 0 For most initial conditions, user position and clock bias estimate will converge to the true position Other errors in the pseudorange need to be corrected for first
Summary Pseudorange corrections: satellite clock, Earth rotation for transit time, ionospheric errors Use iterative method to solve 3D user position and receiver clock bias Pseudoranges from at least four satellites
Position Solution with more than Four Satellites In this case we have more equations than unknowns (over-determined system) H matrix becomes [Nx4] where N is the number of satellites We can use a least squares method applying a pseudoinverse to solve for Δx
GPS Satellite Geometry Satellite geometry plays a big part in the accuracy of the final position solution Good Geometry Bad Geometry
Dilution of Precision (DOP) Dilution of precision indicates the scaling relationship between errors in the pseudoranges and the final positioning accuracy Based purely on the geometry of the satellite positions, assuming all satellites have a similar amount of pseudorange error
Multipath Errors Multipath errors occur when the same signal from a satellite arrives at the receiver at two difference times via two different paths due to reflection Common in urban environments with tall buildings Receivers can overcome to some degree by using the shortest path (shortest time difference) signal
Differential GPS (DGPS) Most sources of GPS error come from pseudorange error, these errors are caused by the ionosphere or satellite clock errors These errors tend to be common for two receivers operating in the same local area DGPS uses a base station with a known location to determine what the local are pseudo range errors are by comparing measured pseudoranges to the known range to each satellite (based on known base station location) These pseudorange errors are then communicated to a user receiver and corrected before solving the GPS position Results in much better performance than GPS operating in standalone mode