Satellite Navigation PVT estimation and integrity Picture: ESA AE4E8 Sandra Verhagen Course 1 11, lecture 7 1 Satellite Navigation (AE4E8 Lecture 7
Today s topics Position, Velocity and Time estimation Performance measures RAIM exam material, but not in book! Study Sections 6.1, 6..1 Read Sections 6.., 6.3 Satellite Navigation (AE4E8 Lecture 7
Recap: Code and Carrier Phase measurements f 1 s ρli = r+ I L1 + T + c δtu δt ε ρ f + i f Φ = + + δ δ + λ + ε Φ Li r 1 s I L1 T c tu t f i ALi Li Li 3 Satellite Navigation (AE4E8 Lecture 7
y = ρ = x x + b + % ε ( k ( k ( k ρ H( v = ρ = x x + b ( k ( k Linearization T ( k T H(v x x ( k x x x H(v y y H(v ( 1 ( k y ( k ( k T = = x x = 1 v H(v ( k z z z ( k x x H(v b 1 H(v y = H(v + ( v v +... + ε v 4 Satellite Navigation (AE4E8 Lecture 6
Linearized code observation equations δρ (1 (1 T δρ ( 1 1 ( ( T δρ ( 1 1 δ x = = δb + ε% M M M ( K ( K T δρ ( 1 1 ρ δx δb x x = y y z z = b b G 5 Satellite Navigation (AE4E8 Lecture 6
Least-squares estimation - BLUE y = Ax + ε; Q yy 1 yy 1443 Q linear model ( T 1 x A Q A T 1 = A Q y variance matrix ˆ xx ˆ ˆ yy W = 1 Q yy See appendix 6.A 6 Satellite Navigation (AE4E8 Lecture 6
linearized model Least-squares estimation δρ = Gδ v+ ε% Q ; ρρ iteration required, Gauss-Newton method: v =... while Q vv ˆˆ δvˆ Q vv ˆˆ δρ= ρ H(v G = = H(v v η ( T 1 GQ ρρg 1 T 1 δ vˆ = Qvv ˆˆ G Qρρδρ vˆ = v + δ ˆ v v = ˆ v end 7 Satellite Navigation (AE4E8 Lecture 6
Satellite Navigation (AE4E8 Lecture 6 Velocity estimation use position estimates from subsequent epochs (next lectures use Doppler shift observations pseudorange rate ( ( ( ( ( ( ( ( ( ( k k k k k k k k b T I b b r ε ϕ ρ & & & & & & & & + + = + + + = 1 v v (k v satellite velocity vector (ECEF, known from navigation message user velocity vector (ECEF v ( ( ( ( ( k k k k k b ϕ ρ ε = + + v 1 1 v & & &
Satellite geometry (D True range: black circle Noise + errors: measured range is in orange area Width of orange area depends on (URE P( ρ ρ = 95.4% true 9 Satellite Navigation (AE4E8 Lecture 6
Satellite geometry (D With observations: ~95% probability that measured position will be in dark orange area 1 Satellite Navigation (AE4E8 Lecture 6
Satellite geometry With observations: ~95% probability that measured position will be in dark orange area 11 Satellite Navigation (AE4E8 Lecture 6
Satellite geometry With observations: ~95% probability that measured position will be in dark orange area 1 Satellite Navigation (AE4E8 Lecture 6
Satellite geometry With 3 observations: ~95% probability that measured position will be in dark orange area 13 Satellite Navigation (AE4E8 Lecture 6
Satellite Navigation (AE4E8 Lecture 6 Satellite geometry ( ρ Q Q x 1 T Q 1 1 T vv δ δ δ ρρ ρρ = G G G b 43 14 ˆ ˆ ˆ ˆ I K 1 ρρ = = Q W If: = = = 1 ˆ ˆ cov cov ( b z y x T H G G Q vv RMS position error = 33 11 H H H z y x + + = + + influence of satellite geometry on precision = Position Dilution of Precision (PDOP
Satellite geometry Q vv ˆ ˆ = ( G T G 1 = H x = y cov cov z b RMS position error = RMS clock bias error = x y z 11 33 + + = H + H + H = PDOP b 44 = H = TDOP RMS overall error = + + + = 11 + + 33 + 44 = GDOP x y z b H H H H Satellite Navigation (AE4E8 Lecture 6
Accuracy: Performance measures error in meters + associated statistical value E.g.: difference true horizontal position and the radius of a circle which encloses 95% of the position estimates indicated by the GNSS receiver if measurement would be repeated large number of times, i.e. there is a 95% probability that the measured position will be inside the circle centered at the true position
Performance measures Accuracy: DOP : Dilution of Precision precision geometrical strength (depends on satellite geometry the lower, the better URE : User Range Error depends on satellite ephemeris and clock errors, ionosphere and troposphere, multipath, noise, and other error sources HPL/VPL : Horizontal/Vertical Protection Level horizontal/vertical position is assured to be within region defined by HPL/VPL RMS error : see previous slide = formal precision 17 Satellite Navigation (AE4E8 Lecture 6
Performance measures Integrity: ability of a system to provide timely warnings to users when the system should not be used alarm limit: position error that should result in an alarm being raised Time To Alarm (TTA: time between occurrence of integrity event (position error too large and alarm being raised Continuity: probability that a system will provide the specified level of accuracy and integrity throughout an operation of specified period (assuming accuracy and integrity requirements are met at the start of the operation Availability: percentage of time that system is operating satisfactory, i.e. required accuracy, integrity and continuity can be provided
GPS PRN 3 Anomaly, 1 Jan, 4 5 4 3 East, North and Up differences delf1s.4r North East Up 1 [m] 1 3 4 at 18:3 (UT 3 min. 5 1 11 1 13 14 15 16 17 18 19 Number of epochs Not noticed by US for 3 hours Picked up by EGNOS Alternative: check @receiver at 1 seconds interval
Receiver Autonomous Integrity monitoring (RAIM up to 14x1 6 Local Overall Model teststatistic delf1s.4r 1 9 8 7 6 5 4 3 critical value α =.1 immediate response by Overall Model test 1 6 1 18 4 3 36 Number of epochs = full hour period
RAIM - Overall model test RAIM: detect and correct for errors in GPS data @receiver Overall model test: does provide good model?
Design computations internal reliability size of model error that can just be detected by using the appropriate test statistics: Minimal Detectable Bias (MDB H H a : E{ y} = Ax; : E{ y} = Ax+ c ; MDB= = c T λ 1 Q P y A c D{ y} = Q D{ y} = Q yy yy c : specifies type of model error: e.g. cycle slip or code outlier
Design computations external reliability impact of undetected bias, c, on the unknown parameters: Minimal Detectable Effect (MDE xˆ = ( A Q A A Q T 1 1 T 1 yy yy c if baseline unknowns parameterized as b =( N E U it is also possible to use Minimal Detectable Position Effect (MDPE: T MDPE= Nˆ ˆ ˆ + E + U
Summary and outlook GPS system overview, signals, receivers GPS measurements and error sources PVT estimation Next: Positioning in dynamic environments (Christian Tiberius 4 Satellite Navigation (AE4E8 Lecture 6