1. 1 SVY 207: Lecture 6 Point Positioning –By now you should understand: F How receiver knows GPS satellite coordinates F How receiver produces pseudoranges –Aim of this lecture: F To understand how to estimate position and error, given pseudorange measurements and satellite coordinates

2. 2 Point Positioning u Pseudorange observation model u Observation equations u Linearised observation model u Least squares estimation u What are “good” conditions? –Dilution of Precision (DOP) –Mission Planning Software

3. 3 What is being measured?  Autocorrelation: shift bits to match replica  received signal  Actual pseudorange  T  T s ) c (TTs)(TTs) Received signal, driven by satellite clock T s Replica signal, driven by receiver clock T Antenna Satellite clock, T s Transmitted signal Receiver clock T

4. 4 Observation Model  Actual observation P r s  T r  T s ) c –at receiver r F whose clock reads T r when signal is received –from satellite s F whose clock read T s when transmitted F c is the speed of light (in a vacuum)

5. 5 Observation Model u Modelled observation  denote clock times by T, and true times by t P r s  T r  T s )c  t r  t r  t s  t s )c  t r  t s )c  c  t r  c  t s  r s  c  t r  c  t s –where  r s range from receiver to satellite  t r receiver clock error  t s satellite clock error cspeed of light = 299792458 m/s F this model is simplified –since it assumes the speed of light in the atmosphere is c

6. 6 Observation model P r s  r s  c  t r  c  t s –Pythagoras Theorem:  r s  ( (u s  u r ) 2  v s  v r ) 2  w s  w r ) 2 ) ½ –Knowns: Navigation message gives us F satellite position (u s, v s, w s )  satellite clock error  t s can be used to correct P r s –4 unknowns: F receiver position (u r, v r, w r )  receiver clock error  t r

7. 7 Observation Equations u Consider: –4 satellites in view of receiver “A” F Receiver index: r = A F Satellite index: s = 1, 2, 3, 4 P A 1  u 1  u A ) 2  v 1  v A ) 2  w 1  w A ) 2 ) ½  c  t A P A 2  u 2  u A ) 2  v 2  v A ) 2  w 2  w A ) 2 ) ½  c  t A P A 3  u 3  u A ) 2  v 3  v A ) 2  w 3  w A ) 2 ) ½  c  t A P A 4  u 4  u A ) 2  v 4  v A ) 2  w 4  w A ) 2 ) ½  c  t A

8. 8 Linearised Observation Model - Consider observation P, a function of parameters u, v, w,... - Let P 0 denote P computed using provisional values u 0, v 0, w 0,... - Taylor expansion is:

9. 9 Design Matrix A : Example u Pseudorange to satellite 1 is:  Partial derivative w.r.t. station clock  t? u Partial derivative w.r.t. station coordinate v? F hint: use the chain rule

10. 10 Solution

11. 11 Least Squares Solution u Linearised model: b  Ax  where  noise b  column matrix of observations  observed P  computed P 0 x  column matrix of parameters  adjustment to provisional values: (u-u 0 ), (v-v 0 ), etc. A  square design matrix (partial derivatives  P/  u, etc.) u Least squares solution:

12. 12 Covariance Matrix and Errors F Interpretation example: –If pseudorange observation errors are  metres error in u coordinate would be  u  metres –Off diagonal elements indicate degree of correlation If  uv is negative, this means that a positive error in u will probably be accompanied by a negative error in v –Elements  u,  v,  uv define an “error ellipse”

13. 13 Local Coordinate Errors –Errors in local topocentric coordinates F Applications need East, North, Height errors F Also, Height, tends to have largest error F Define transformation matrix G which takes a small relative vector in geocentric system into local topocentric system:

14. 14 Local Coordinate Errors –This is the geocentric covariance matrix –But require topocentric covariance matrix F Use the law of “propagation of errors”

15. 15 Dilution of Precision F VDOP: Vertical Dilution of Precision F HDOP: Horizontal Dilution of Precision F PDOP: Position Dilution of Precision F TDOP: Time Dilution of Precision F GDOP: Geometric Dilution of Precision

16. 16 Dilution of Precision –Interpretation examples:   metres error in observations –gives  HDOP metres error in horizontal position –gives  TDOP seconds error in receiver clock time F Good geometry –Small DOP is good, large DOP is bad –Values of HDOP and VDOP should be less than 5. –Values of 2 is typical if there are 5 or more satellites in view –Small DOP’s achieved if satellites are well spread out –If two satellites pass close in the sky you effectively loose a satellite in this calculation. if only 4 satellites in view, DOP can get too large

17. 17 Mission Planning u Measures of good conditions: –Satellite geometry F Is a function of geographic location F Number of satellites visible F Dilution of precision –Elevation mask F elevation cutoff in receiver F elevation cutoff in software F physical obstructions –Low multipathing environment