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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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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3

Recap: Code and Carrier Phase measurements

21

12

21

12

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 = − + + − + +⎣ ⎦


Linearization

4

( )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


Linearized code observation equations

5

(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


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


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


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 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 geometry (2D)

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With 2 observations: ~95% probability that measured position will be in dark orange area


Satellite geometry

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Satellite geometry

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Satellite geometry

13



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 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


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 ∇+∇+∇=


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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satellite navigation pvt estimation and integrity€¦ · satellite navigation (ae4e08) – lecture...

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