Satellite Navigation: Lecture 6 -_error_sources_and_position_estimation
Upcoming SlideShare
Loading in...5
×

Like this? Share it with your network

Share
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
643
On Slideshare
643
From Embeds
0
Number of Embeds
0

Actions

Shares
Downloads
33
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. Satellite Navigation error sources and position estimation Picture: ESAAE4E08Sandra VerhagenCourse 2010 – 2011, lecture 6 1Satellite Navigation (AE4E08) – Lecture 6
  • 2. Today’s topics• Recap: GPS measurements and error sources• Signal propagation errors: troposphere• Multipath• Position estimation• Book: Sections 5.3 – 5.7, 6.1.1 2Satellite Navigation (AE4E08) – Lecture 6
  • 3. Recap: error sources• satellite: • receiver • orbit • clock • clock • instrumental delays • instrumental delays • other• signal path • spoofing • ionosphere • interference • troposphere • multipath 3Satellite Navigation (AE4E08) – Lecture 6
  • 4. Recap: Code and Carrier Phase measurements f 2 ρ Li = r + ⎡δ tu − δ t s ⎤ + ε ρLi I L1 + T + c ⎣ 1 ⎦ f i 2 f12 ⎡δ tu − δ t s ⎤ + λ ALi + ε Φ Li Φ Li = r − 2 I L1 + T + c ⎣ ⎦ fi 4Satellite Navigation (AE4E08) – Lecture 6
  • 5. Signal propagation errors: ionosphere ionosphere ς′ IP : ionospheric pierce point IP hI : mean ionosphere height ς hI RE sin ς ′ = sin ς Earth RE + hI RE 1 I (ς ) = Iz cos ς ′ 5Satellite Navigation (AE4E08) – Lecture 6
  • 6. Signal propagation errors: ionosphere obliquity factor 1 cos ς ′ zenith angle ς 6Satellite Navigation (AE4E08) – Lecture 6
  • 7. Signal propagation errors: ionospherezenith delay mid-latitudes:• 1-3 m at night• 5-15 m mid-afternoonpeak solar cycle near equator:• max. ~36 m 7Satellite Navigation (AE4E08) – Lecture 6
  • 8. Signal propagation errors: ionosphereHow to deal with ionosphere?• apply ionosphere-free combination (dual-frequency receiver required)• apply model (reduction 50 – 70%)• relative positioning (later this course)Satellite Navigation (AE4E08) – Lecture 6
  • 9. Signal propagation errors: troposphere• 9 km (poles) – 16 km (equator)• Dry gases and water vapor• Recall: non-dispersive, i.e. refraction does not depend on frequency• Propagation speed lower than in free space: apparent range is longer (~2.5 – 25 m)• Same phase and group velocities Tρ L1 = Tρ L 2 = TφL1 = TφL 2 = T 9Satellite Navigation (AE4E08) – Lecture 6
  • 10. Signal propagation errors: troposphere R R Refractivity N = ( n − 1) × 10 Δρ = ∫ [ n(l ) − 1] dl = 10 ∫ N (l )dl 6 −6• S S N = Nd + Nw T = 10−6 ∫ N (l )dl =10−6 ∫ [N d (l ) + N w (l )]dl =Td + Tw P N d ≈ 77.64 P : total pressure [mbar] T e T : temperature [K] N w ≈ 3.73 ⋅ 105 e : partial pressure water vapor [mbar] T2 if known refractivity known 10Satellite Navigation (AE4E08) – Lecture 6
  • 11. Signal propagation errors: troposphere tropospheric delay Satellite computed hydrostatic delay T = md (el ) * Tz ,d + mw (el ) * Tz ,w mapping functions Receiver Unknown tropospheric zenith wet delay EarthFigure: H. van der Marel 11 11 Satellite Navigation (AE4E08) – Lecture 6
  • 12. Signal propagation errors: troposphere• Saastamoinen model: zenith dry and wet delays calculated from temperature, pressure and humidity (measurements or standard atmosphere), height and latitude• Hopfield model: dry and wet refractivities calculated• Dry delay in zenith direction 2.3 – 2.6 m at sea level can be predicted with accuracy of few mm’s• Wet delay depends on water vapor profile along path, 0 – 80 cm accuracy of models few cm’s• If no actual meteorological observations available (standard atmosphere applied): total zenith delay error 5 – 10 cm 12Satellite Navigation (AE4E08) – Lecture 6
  • 13. Signal propagation errors: summary ionosphere troposphereheight 50 – 1000 km 0 – 16 kmvariability diurnal, seasonal, solar low cycle (11 yr), solar flareszenith delay meters – tens of meters 2.3 – 2.6 m (sea level) el=30o 1.8 2obliquity factor el=15o 2.5 4 el= 3o 3 10modeling error (zenith) 1 - >10 m 5 – 10 cm (no met. data)dispersive yes no all values are approximate, depending on location and circumstances Satellite Navigation (AE4E08) – Lecture 6
  • 14. Signal propagation errorsHomework exercise:• make plots of the different mapping functions (page 173 Misra and Enge) as function of the elevation angle (ranging from 0 – 90o)• compare them with each other AND with the obliquity factor of the ionosphere delay (slide 22)• try to explain the differences• more details: see assignment on blackboard 14Satellite Navigation (AE4E08) – Lecture 6
  • 15. Multipath• Signal reflected: arrives via two or more paths at the antenna• Reflected signals have different path length and interfere with direct signal• Systematic error (does not average out) • pseudorange error: up to tens of meters • carrier phase error: up to 5 cmSatellite Navigation (AE4E08) – Lecture 6
  • 16. Multipath direct signal reflected signal o phase shift 180Figure: H. van der Marel Satellite Navigation (AE4E08) – Lecture 6
  • 17. Multipath direct signal reflected signal phase shift 180 oFigure: H. van der Marel Satellite Navigation (AE4E08) – Lecture 6
  • 18. MultipathPrimary defense: • careful selection of antenna locations (away from reflectors) not always possible • carefully designed antennas (choke rings; microstrip) no signals from below • signal processing (correlators)Pseudorange multipath can be detected/analyzed by forming a special linear combination of code and carrier phase dataFor GPS: if receiver is static, same multipath pattern repeats after 23h56m (same orbit)Satellite Navigation (AE4E08) – Lecture 6
  • 19. Multipath: example multipath on C/A-code pseudorange PRN28 10 MC PRN28 between –7 and +6 m! Mc = C1 - 4.092*L1 + 3.092*L2 [m] 8 6 4 multipath [m] 2 multipath [m] 0 -2 -4 -6 -8 April 2nd, 2004 14:10-14:30 UT -10 0 200 400 600 800 1000 1200 time [sec.] time [s]Figure: H. van der Marel Satellite Navigation (AE4E08) – Lecture 6
  • 20. GPS error budgetEmpirical values, actual values depend on receiver, atmosphere models, time and location Error source RMS range error [m] satellite clock and ephemeris σ RE / CS = 3 m = SIS URE atmospheric propagation modeling σ RE / P = 5 m receiver noise and multipath σ RE / RNM = 1 m User Range Error (URE) σ URE = 6 m σ URE = σ RE / CS + σ RE / P + σ RE / RNM 2 2 2 Satellite Navigation (AE4E08) – Lecture 6
  • 21. Carrier-smoothing Φ (t ) = r (t ) + c ⋅ (δ t (t ) − δ t (t − τ ) ) + T (t ) − I (t ) + λ ⋅ A + ε i i u i s i i i i Φ (ti ) = ρ IF (ti )precise estimate for change in pseudorange:ΔΦ (ti ) = Φ (ti ) − Φ (ti −1 ) = Δρ IF (ti ) − ΔI (ti ) + Δε Φ (ti ) near zero if epochs close togethercarrier-smoothed pseudorange= weighted average of pseudorange (code) and carrier-derived pseudorange 1 M −1 ρ (ti ) = ρ (ti ) + [ ρ (ti −1 ) + ΔΦ(ti )] M M ρ (t1 ) = ρ (t1 ) 21 Satellite Navigation (AE4E08) – Lecture 6
  • 22. Non-linear observation equations ρ ( k ) = r ( k ) + I ( k ) + T ( k ) + c[δtu − δt ( k ) ] + ε ρk ) ( r (k ) = (x (k ) − x ) + (y 2 (k ) − y ) + (y 2 (k ) − y) 2 = x(k ) − x 22Satellite Navigation (AE4E08) – Lecture 6
  • 23. Non-linear observation equationsCorrected pseudorange: account for satellite clock offset, andcompensate remaining error sourcesNote: increased noise, since corrections/models are not perfect b ρ ( k ) = r ( k ) + cδ tu + ε ρk ) = x( k ) − x + b + ε ρk ) %( %( r (k ) = (x (k ) − x ) + (y 2 (k ) − y ) + (y 2 (k ) − y) 2 = x(k ) − x 23Satellite Navigation (AE4E08) – Lecture 6
  • 24. Linearizationnon-linear model y = H(v) + ε ∂H(v 0 )Taylor series y = H(v 0 ) + ( v − v 0 ) + ... + ε ∂vobserved-minus-computed observations δ y = y − H(v 0 ) linearized model:correction to approximate δ v = v − v0 δ y = Aδ v + εvalues ∂H(v 0 )design matrix A= ∂v 24 Satellite Navigation (AE4E08) – Lecture 6
  • 25. Linearization ∂H(v 0 )y=ρ (k ) = x (k ) − x + b + ερ % (k ) y = H(v 0 ) + ( v − v 0 ) + ... + ε ∂vH ( v 0 ) = ρ0k ) = x( k ) − x 0 + b0 ( Approximations required for: • satellite position at time of transmission t - τ (from ephemeris) problem: τ not precisely known • receiver position at time of reception t • receiver clock error 25Satellite Navigation (AE4E08) – Lecture 6
  • 26. Linearization ∂H(v 0 )y=ρ (k ) = x (k ) − x + b + ερ % (k ) y = H(v 0 ) + ( v − v 0 ) + ... + ε ∂vH ( v 0 ) = ρ0k ) = x( k ) − x 0 + b0 ( T ⎡ ∂H(v 0 ) ⎤ T ⎡ x − x0 (k ) ⎤ ⎢ ∂x ⎥ ⎢− ⎥ ⎢ x( k ) − x0 ⎥ ⎢ ⎥ ⎢ ∂H(v 0 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢− y ( k ) − y0 ⎥∂H(v 0 ) ⎢ ∂y ⎥ ⎢ x( k ) − x0 ⎥ = =⎢ ⎥ = ⎡(−1( k ) )T ⎣ 1⎤ ⎦ ∂v ⎢ ∂H(v ) ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢− z ( k ) − z0 ⎥ ⎢ ∂z ⎥ ⎢ ∂H(v ) ⎥ ⎢ x( k ) − x0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ∂b ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ 1 ⎦ 26Satellite Navigation (AE4E08) – Lecture 6
  • 27. Linearized code observation equations ⎡ δρ (1) ⎤ ⎡ (−1(1) )T 1⎤ ⎢ (2) ⎥ ⎢ ⎥ ⎢ δρ ⎥ ⎢ (−1(2) )T 1⎥ ⎡δ x ⎤ δρ = ⎢ ⎥=⎢ ⎥ ⎢ ⎥ + ερ % ⎢ M ⎥ ⎢ M M ⎥ ⎣δ b ⎦ ⎢ (K ) ⎥ ⎢ (K ) T ⎥ ⎣δρ ⎦ ⎣(−1 ) 1⎦ G 27Satellite Navigation (AE4E08) – Lecture 6
  • 28. Least-squares estimation y = Ax + ε; Q yy x = (A TQ −1 A ) A TQ −1 y −1linear model ˆ yy yy 14243 Q xx variance matrix ˆˆ 28 Satellite Navigation (AE4E08) – Lecture 6
  • 29. Least-squares estimationlinearized model δρ = Gδ v + ε; % Q ρρ v 0 = ... while δv ≥η 2iteration required, Gauss-Newton method: ˆ Qvv ˆˆ δ ρ = ρ − H(v 0 ) ∂H(v 0 ) G= ∂v Qvvˆ = ( G Q ρρ G ) T −1 −1 ˆ δ v = Qvvˆ ⋅ G TQ −1 δ ρ ˆ ˆ ρρ v = v0 + δ v ˆ ˆ v0 = v ˆ end 29 Satellite Navigation (AE4E08) – Lecture 6
  • 30. Summary and outlook• GPS measurements and error sources• Linearized observation equations position estimationNext:Position, Velocity and Time (PVT) estimation 30Satellite Navigation (AE4E08) – Lecture 6