Satellite Navigation PVT estimation and integrity€¦ · Satellite Navigation (AE4E08) – Lecture...
Transcript of Satellite Navigation PVT estimation and integrity€¦ · Satellite Navigation (AE4E08) – Lecture...
Satellite Navigation (AE4E08) – Lecture 7
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Satellite Navigation PVT estimation and integrity
AE4E08
Sandra Verhagen
Course 2010 – 2011, lecture 7
Picture: ESA
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Satellite Navigation (AE4E08) – Lecture 7
Today’s topics
• Position, Velocity and Time estimation• Performance measures• RAIM exam material, but not in book!
• Study Sections 6.1, 6.2.1• Read Sections 6.2.2, 6.3
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Satellite Navigation (AE4E08) – Lecture 7
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Recap: Code and Carrier Phase measurements
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Li
Li
sLi L u
i
sLi L u Li
i
fr I T c t tf
fr I T c t t Af
ρρ δ δ ε
δ δ λ εΦ
⎡ ⎤= + + + − +⎣ ⎦
⎡ ⎤Φ = − + + − + +⎣ ⎦
Satellite Navigation (AE4E08) – Lecture 6
Linearization
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( )00
( )0
( )00
( ) ( )00
0 ( )0
( )0
0
( ) 1
1
TT k
k
k
k k T
k
k
x xx
y yy
z zz
b
⎡ ⎤−∂⎡ ⎤ −⎢ ⎥⎢ ⎥ −∂ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥∂ −⎢ ⎥⎢ ⎥ −∂∂ ⎢ ⎥⎢ ⎥ − ⎡ ⎤= = = −⎣ ⎦⎢ ⎥⎢ ⎥∂ ∂ ⎢ ⎥⎢ ⎥ −⎢ ⎥∂⎢ ⎥ −⎢ ⎥−⎢ ⎥∂ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥∂⎣ ⎦ ⎣ ⎦
H(v )x x
H(v )H(v ) x x 1v H(v )
x xH(v )
( )00 0 ...∂
= + − + +∂H(v )y H(v ) v v εv
( ) ( ) ( )k k kb ρρ ε= = − + +y x x %
( ) ( )0 0 0 0( ) k kH bρ= = − +v x x
Satellite Navigation (AE4E08) – Lecture 6
Linearized code observation equations
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(1) (1)
(2) (2)
( ) ( )
( ) 1
( ) 1
( ) 1
T
T
K K T
b ρ
δρ
δρ δδ
δρ
⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥
−⎢ ⎥ ⎢ ⎥ ⎡ ⎤= = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
−⎣ ⎦ ⎣ ⎦
1
1 xδρ ε
1
%M M M
G
0
0
0
0
bbzzyyxx
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
=
b
x
δ
δ
Satellite Navigation (AE4E08) – Lecture 6
Least-squares estimation - BLUE
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linear model ( ) yQAAQAx 1yy
T
Q
11yy
T
xx
−−−=43421
ˆˆ
ˆyyQεAxy ;+=
variance matrix1yyQW −=
See appendix 6.A
Satellite Navigation (AE4E08) – Lecture 6
Least-squares estimation
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linearized model
iteration required, Gauss-Newton method:
; ρρδ= +δρ G v ε Q%
( )
ˆ ˆ
2
ˆˆ
ˆ ˆ
...
ˆ
)
ˆˆ ˆ
ˆ
while
end
vvQ
vv
vv
Q
Qρρ
ρρ
δ η
δ
δ δ
δ
∂∂
−−
−
=
≥
= −
=
=
= ⋅
= +
=
0
0
0H(v )v
1T 1
T 1
0
0
v
v
ρ ρ H(v
G
G Q G
v G Q ρv v vv v
Satellite Navigation (AE4E08) – Lecture 6
Velocity estimation• use position estimates from subsequent epochs (next
lectures)• use Doppler shift observations pseudorange rate
)()()(
)()()()()(
)(
)(kkk
kkkkk
b
TIbbr
ϕε
ρ
&&
&&&&&&
++⋅−=
++−+=
1vv)(kv satellite velocity vector (ECEF),
known from navigation message
user velocity vector (ECEF)v( ) ( ) ( ) ( ) ( )k k k k kb ϕρ ε− ⋅ = − ⋅ + +v 1 1 v &
&&
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry (2D)
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True range: black circleNoise + errors: measured range is in orange areaWidth of orange area depends on σ2 (URE)
2σ
( 2 ) 95.4%trueP ρ ρ σ− ≤ =
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry (2D)
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With 2 observations: ~95% probability that measured position will be in dark orange area
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry
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With 2 observations: ~95% probability that measured position will be in dark orange area
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry
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With 2 observations: ~95% probability that measured position will be in dark orange area
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry
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With 3 observations: ~95% probability that measured position will be in dark orange area
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry
( ) ρQQx 1T
Q
11T
vv
δδδ
ρρρρ−−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ GGGb 43421
ˆˆ
ˆˆ
KI21 σρρ ==− QWIf:
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=== −
2
2
2
2
212ˆˆ
cov
cov
)(
b
z
y
x
T HGG
σσ
σσ
σσvvQ
RMS position error = 332211222 HHHzyx ++=++ σσσσ
influence of satellite geometry on precision = Position Dilution of Precision (PDOP)
Satellite Navigation (AE4E08) – Lecture 6
Satellite geometry
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=== −
2
2
2
2
212ˆˆ
cov
cov
)(
b
z
y
x
T HGG
σσ
σσ
σσvvQ
RMS position error = PDOP⋅=++=++ σσσσσ 332211222 HHHzyx
RMS clock bias error = TDOP⋅== σσσ 442 Hb
RMS overall error =2 2 2 2
11 22 33 44 GDOPx y z b H H H Hσ σ σ σ σ σ+ + + = + + + = ⋅
Accuracy:error in meters + associated statistical valueE.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
Satellite Navigation (AE4E08) – Lecture 6
Performance measuresAccuracy:• 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
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Performance measuresIntegrity:
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 beingraisedTime 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
100 110 120 130 140 150 160 170 180 190 200−500
−400
−300
−200
−100
0
100
200
300
400
500East, North and Up differences − delf001s.04r
Number of epochs
[m]
NorthEastUp
GPS PRN 23 Anomaly, 1 Jan, 2004
at 10 seconds interval
at 18:30 (UT)
3 min.
Not noticed by US for 3 hoursPicked up by EGNOSAlternative: check @receiver
Receiver Autonomous Integrity monitoring (RAIM)
0 60 120 180 240 300 3600
1
2
3
4
5
6
7
8
9
10Local Overall Model teststatistic − delf001s.04r
Number of epochs
α
= 0.01critical value
immediate responseby Overall Model test
up to 14x106
= full hour period
RAIM - Overall model test
RAIM: detect and correct for errors in GPS data @receiverOverall 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)
c
: specifies type of model error: e.g. cycle slip or code outlier
cPQcMDB
QyDcAxyEHQyDAxyEH
AyT
yya
yy
⊥−=∇=
=∇+===
10
0
}{;}{:}{;}{:
λ
external reliability impact of undetected bias, c
, on the unknown parameters: Minimal Detectable Effect (MDE)
if baseline unknowns parameterized as it is also possible to use Minimal Detectable Position Effect (MDPE):
∇
∇=∇ −−− cQAAQAx yyT
yyT 111 )(ˆ
Design computations
TUENb )(=
222 ˆˆˆ UENMDPE ∇+∇+∇=
Satellite Navigation (AE4E08) – Lecture 6
Summary and outlook
• GPS system overview, signals, receivers• GPS measurements and error sources• PVT estimation
Next:Positioning in dynamic environments (Christian Tiberius)
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