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# Satellite Navigation: Lecture 6 -_error_sources_and_position_estimation

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• 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