Disha aai presentation%20prof%20hablani

High-Accuracy Integrated GPS-INS Aircraft Navigationfor Landing using Pseudolites and Double-Difference

Carrier Phase measurements

presented by:

Prof. Hari B Hablani

Indian Institute of Technology, Bombay

DISHA-2013, NEW DELHIA Seminar by Airport Authority Officers’ Association(India)

Indian Institute of Technology, Bombay AE 797 - I Stage Project November 25, 2013 1 / 35

Overview

1 Introduction

2 Problem Description

3 GPS Overview

4 Precise Positioning using carrier phase measurements

5 Aircraft Landing Using Integrity Beacons

6 Single-Difference Carrier Phase Measurements

7 Integer Ambiguity Resolution

8 Double-Difference Carrier Phase Measurements

9 Least-Squares Ambiguity Decorrelation Adjustment(LAMBDA)

10 Simulation Results

11 Conclusion

12 Future Work


Introduction

The Global navigation satellite systems are fast becoming preferred means ofnavigation

Satellite based navigation system allows to fly aircraft through time and fuelefficient routes

Provides efficient air traffic management

The commissioning and maintenance cost of the ground based signallingsystems such as instrument landing system can be reduced by relying onGPS-based landing approach

Government of India has invested several thousand crores in building spaceinfrastructure in the country. With GAGAN and IRNSS (Indian RegionalNavigation Satellite System), air navigation in India will soon become moreaccurate and efficient

Reference: http://www.gps.gov/applications/aviation/


Problem Description

Provide precise navigation to an aircraft during landing and use PVTestimation to control the aircraft

The research work includes the following

Simulation of an aircraft landing trajectory, presently it is kinematic

Integer ambiguity resolution by achieving geometric diversity using integritybeacons

Implementation of navigation algorithm using single-difference anddouble-difference carrier phase measurements

Development and simulation of the integrated GPS-INS navigation system

Development of the flight control system

Receiver autonomous integrity monitoring


GPS Overview

GPS Segments: Space Segment,Control Segment and User Segment

Signals: Each satellite transmits two RF signals L1 and L2fL1 = 1575.42 MHz, and fL2 = 1227.60 MHz

GPS measurements

Code phase measurement(Pseudo-range measurement)

ρ = rtrue + Iρ(t) + Tρ(t) + c(δtu − δts) + ξρ (1)

Carrier phase measurement

φ = λ−1[r − IΦ + TΦ] +c

λ(δtu − δts) + N + ξΦ (2)

Where: N is unknown no. of whole cycles, the carrier phase measurement hasno information about the no. of whole cycles; this is also called integerambiguity.σ(ξρ) = 0.5 m, σ(ξΦ) = 5 mm


GPS Overview

Errors in measurement

Ionospheric and tropospheric refraction(5 m)

Clock and Ephemeris Errors (3 m)

Multipath Errors (1.5 m)

Receiver noise (0.6 m)

Satellite Geometry

Figure: GDOP

Reference:

http : //electronicdesign.com/site − files/electronicdesign.com/files/archive/autoelectronics.com/telematics/navigations ystems/GPS1.jpg


Precise Positioning using carrier phase measurements

The Pseudo-range measurements provide accuracy around 5-20 meters

This accuracy level is not acceptable in some applications such as aircraftlanding

Centimetre level accuracy can be achieved by using differential carrier phasemeasurements

However, carrier phase measurements require estimation of unknown integerambiguities


Aircraft Landing Using Integrity Beacons

One integrity beacon is placed on each side of the approach path to a runway.

Ground reference station measures the carrier phase of the GPS and IntegrityBeacon signals.

These measurements are transmitted to the aircraft.

The on-board system consists a GPS receiver,a data link receiver and acomputer to execute the navigation algorithm.

Figure: Aircraft landing using Integrity Beacon

Reference: Cohen, C.E., Pervan, B.S., Cobb, H.S., Precision Landing of Aircraft Using Integrity Beacons, in Parkinson, B.W and Spilker Jr. J.J., Global

Positioning System: Theory and Applications Volume II, Vol. 163 Progress In Aeronautics and Astronautics, AIAA, 1996


Aircraft Landing Using Integrity Beacons(contd..)

Figure: Aircraft landing system

Reference: Cohen, C.E., Pervan, B.S., Cobb, H.S., Precision Landing of Aircraft Using Integrity Beacons, in Parkinson, B.W and Spilker Jr. J.J., Global

Positioning System: Theory and Applications Volume II, Vol. 163 Progress In Aeronautics and Astronautics, AIAA, 1996


Single-Difference Carrier Phase Measurements

Figure: Vector Geometry

Carrier phase measurements at the user(Aircraft) from i th satellite

φiu = λ−1[r i

u − I iu + T i

u] + c(δtu − δt is) + N i

u + εiφ,u (3)

Ref.- Cohen, C.E., Pervan, B.S., Cobb, H.S., Precision Landing of Aircraft Using Integrity Beacons, in Parkinson, B.W and Spilker Jr. J.J., Global

Positioning System: Theory and Applications Volume II, Vol. 163 Progress In Aeronautics and Astronautics, AIAA, 1996, pp. 427-459


Single-Difference Carrier Phase Measurements(contd..)

Carrier phase measurement at the reference station from the same satellite

φir = λ−1[r i

r − I ir + T i

r ] + c(δtr − δt is) + N i

r + εiφ,r (4)

Single-difference carrier phase

φiur = φi

u − φir

φiur = λ−1[(r i

u−r ir )−(I i

u−I ir )+(T i

u−T ir )]+c(δtu−δtr )+(N i

u−N ir )+εi

φ,u−εiφ,r (5)

Satellite clock error is eliminated with this differencing.

Reference station and user are in proximity, so tropo. delay and iono. delayare eliminated.

Nu − Nr is an unknown differential integer

We need to estimate this differential integer

The estimation process is called integer ambiguity resolution


Single-Difference Carrier Phase Measurements(Contd...)

Figure: Geometry of the single difference measurements

rur = ru − rr (6)

r iur = −1i

rXur (7)

−1ir is the unit vector from ref. station to i th satellite.

λφiur = −1i

r .Xur + cδtur + λN iur + λεi

φ,r (8)

Reference: Misra, P., and Enge, P., GPS Signals, Measurements, and Performance, Revised 2nd Ed., Ganga-Jamuna Press, 2012


Single-Difference Carrier Phase Measurements(Contd...)

Carrier phase of the signal from integrity beacons IBj

measured at the aircraft

λφju = | − pj + Xur |+ c(δtu − δt j ) + λN j

u + λεφ,u (9)

at the ref. station

λφjr = |pj |+ c(δtr − δt j ) + λN j

r + λεφ,r (10)

Single-difference carrier phase of the signal, received from the integritybeacons.

λ[φju −φj

r ] = | − pj +Xur | − |pj |+ c(δtu − δtr) +λ(Nu −Nr ) +λεφ,u −λεφ,r (11)


Integer Ambiguity Resolution

We defineXur = Xur + X∼ur (12)

Xur is the relative vector from user to reference station

Xur is initial estimate and X∼ur is the correction to the estimate

The single-difference carrier phase equation for the i thsatellite can be written as

λφiur + 1i

r .Xur = −1ir .X∼ur + cδtur + λN i

ur + λεiφ,ur (13)

The single-difference carrier phase equation for the integrity beacons can bewritten as

λφjur − | − pj + Xur |+ |pj | = −e j .X∼ur + cδtur + λN j

ur + λεjφ,ur (14)

the unit vector e j =pj − Xur

|pj − Xur |


Integer Ambiguity Resolution(Contd...)

So, at time tk we have following measurements

λδφiur (tk ) = −1i

r .X∼ur (tk ) + τk + λN i

ur + λεiφ,ur (15)

λδφjur (tk ) = −e j .X∼ur (tk ) + τk + λN j

ur + λεjφ,ur (16)

−1r Xur = ρus − ρrs

λδφur (tk ),−1ir and λεj

φ,ur are known to us

Our aim is to estimate Nur ,τk and correction X∼ur



The unknowns in the eqs. for all GPS satellite and two Integrity Beacon are:

X∼ur (tk ), τk ,NiurandN

jur

i=1,2,3...m , m is the no. of visible GPS satellites at time tk

j=1,2(Beacons)

To solve the above equations N1ur is merged with τk and then we determine

remaining N iur − N1

ur

λ

δφ1ur (tk )

δφ2ur (tk )..

δφmur (tk ).......

δφIB1ur (tk )

δφIB2ur (tk )

=

−11r (tk ) 1

−11r (tk ) 1. .. .

−1mr (tk ) 1.....−e1(tk ) 1−e2(tk ) 1

[X∼ur (tk )τk

]+ λ

0N2

ur − N1ur

.

.Nm

ur − N1ur

.......N IB1

ur − N1ur

N IB2ur − N1

ur

+ λ

ε1φ,ur

ε2φ,ur

.

.εmφ,ur

......

εIB1

φ,ur

εIB2

φ,ur

(m + 2) ∗ 1 (m + 2) ∗ 4 (m + 2) ∗ 1 (m + 2) ∗ 1



Aircraft collects the single diff. data during the bubble pass

The collected data is arranged in the following way

λ

δφ1

δφ2

.

.δφn

=

S1 0 0 . I

0 S2 0 . I..

0 0 0 Sn I

X∼∗1

X∼∗2

.

.X∼∗n

.......N

+ λε

I =

0 0 ..01 0 ..00 0 ..1

I:(m + 2) ∗ (m + 1)

Above equation can be solved using least square technique

Xur is updated in each iteration

The above single difference formulation does not cause correlation in thevector ε

Integer ambiguities are obtained by rounding the float solution to the nearestinteger



H = λ

δφ1

δφ2

.

.δφn

, G =

S1 0 0 . I

0 S2 0 . I..

0 0 0 Sn I

, Y =

X∼∗1

X∼∗2

.

.X∼∗n

.......N

Size of matrices G, H, and Y is:

G : n(m + 2) ∗ (4 ∗ n + m + 1)

H : n(m + 2) ∗ 1

Least square estimation:

Y = (G ′ ∗ G )−1 ∗ G ′ ∗ H (17)

The G and H matrices are updated in each iteration

Converges in 3-10 iteration



Table: Integer Ambiguities

No. True NKur − N1

ur Estimated Nkur − N1

ur Error

1 -26093 -26092.99 0.00652 -62198 -62197.96 0.033 -118010 -118010.03 -0.024 -98252 -98252.063 -0.0635 11727 11726.96 -0.0326 -91595 -91595.09 -.097 26217 26216.98 -0.0198 2624 2624.18 0.189 2624 2624.18 0.18

No of measurements:60, σ = 5 mmσ: Standard deviation of the noiseNo. of visible satellites:8K=2,3....8(GPS satellite)K=9,10(Pseudolites)


Double-Difference Measurements

The clock bias term δtur is common in the single difference measurementsfrom all satellites at each epoch.This term can be removed by thedouble-difference

Single-difference carrier phase measurement for satellite k and l

φkur = φk

u − φkr (18)

= λ−1rkur + f .δtur + Nk

ur + εkφ,ur (19)

φlur = λ−1r l

ur + f .δtur + N lur + εl

φ,ur (20)

Double-difference measurement

φ(kl)ur = φk

ur − φlur (21)

= λ−1rklur + N(kl)

ur + ε(kl)φ,ur (22)

φ(kl)ur = −λ−1(1k

r − 1lr ).Xur + N(kl)

ur + ε(kl)φ,ur (23)


Double-Difference(contd...)

If there are m visible satellites, there will be m-1 double difference equations

If covariance of the carrier phase measurements is ρ2φI then the covariance of

a pair of double-difference measurements is

covφ(dd) = 2ρ2φ

[2 11 2

](24)

Double-difference measurements are correlated

Simple rounding the float solution may give false results


Least-Squares Ambiguity DecorrelationAdjustment(LAMBDA)

Decorrelation is achieved by using LAMBDA (Least-Squares AmbiguityDecorrelation Adjustment) technique

The estimation of unknown integers carried out in three steps: floatsolution,decorrelation, and fixed solution

y = BδX + AN + ε (25)

Cost functionCw = ||y − BδX − AN − ε||2

Q−1w

(26)

Qw :Covariance matrix of the double-difference measurementsy : Double-Difference measurements

Eq[26] is solved by using least square technique to obtain float solution


LAMBDA(contd...)

The float solution obtained from eq.[26][ˆδX

N

](27)

Qn = Cov(N) (28)

where:N represents the float ambiguity vectorQn is the covariance matrix of the float solution N

The second step consists ofmin||N − N||2

Q−1n

(29)

The ambiguities are decorrelated by a Z-transformation.

The transformation matrix Z is determined by decomposing Qn in a lowertriangular matrix L and a diagonal matrix D

Reference:De Jonge,P.,and Tiberius,C., The LAMBDA method for integer ambiguity estimation:implementation aspects, Delft Geodetic Computing Centre

LGR series, No. 12, Delft,Netherlands, 1996


LAMBDA(contd..)

M = ZT N

Qz = ZTQNZ

Where: M is the transformed ambiguity vectorQz is the transformed covariance matrix

The minimization process is performed on the transformed ambiguities

(M −M)TQ−1z (M −M) ≤ χ2 (30)

The size of the search ellipsoid depends on the χ

The next step is

N = Z−TM


LAMBDA(contd...)

Qhat =

[4.97 3.873.87 3.01

](31)

N =

[0.1190.157

](32)

Figure: correlated

Reference:V.,Sandra GNSS Research Centre, Curtin University Mathematical Geodesy and Positioning, Delft University of Technology


LAMBDA(contd...)

Decorrelated covariance matrix

Qzhat =

[0.0865 −0.0364−0.0364 0.0847

](33)

Z =

[−3 −44 5

](34)

M =

[0.250.29

](35)

Figure: Decorrelated


LAMBDA(contd...)

Qhat =

6.290 5.978 0.5445.978 6.292 2.3400.544 2.340 6.288

(36)

N =

5.453.102.97

(37)

The ambiguities are correlated.These are decorrelated by a Z transformation.

M = ZT N (38)

Qzhat = ZT ∗ Qhat ∗ Z

Figure: correlated

Reference:De Jonge,P.,and Tiberius,C., The LAMBDA method for integer ambiguity estimation:implementation aspects, Delft Geodetic Computing Centre

LGR series, No. 12, Delft,Netherlands, 1996


LAMBDA(cont.)

Figure: Decorrelated Ambiguities


Simulation and Results

Figure: Flight Trajectory

Reference station positionLatitude:0.5704 radLongitude:1.2956 radAltitude : 314 m

Distance between integrity beacon and reference station: 16 km

Distance between integrity beacons:4 km


Simulation and Results (contd...)

Up to 60 Sec, pseudo-range measurements are used to determine the aircraftposition after that it switches to the differential carrier phase mode

Figure: Position Error


Double-Difference

Figure: Position Error


Conclusion

Simulation results show that centimetre level accuracy can be achieved withcarrier phase measurements

The flight trajectory is simulated without considering any forces anddisturbances

The integer ambiguity resolution requires that the user collect data frommultiple epochs before a reliable estimate of the ambiguity can be made


Future work

Simulation of an aircraft landing trajectory

An Integrated GPS-INS navigation system will be developed

In the Integrated system INS solution will be updated by the GPS

Development of the flight control system


References

Misra, P., and Enge, P., GPS Signals, Measurements, and Performance, Revised 2nd Ed.,Ganga-Jamuna Press, 2012, pp. 199-280

Cohen, C.E., Pervan, B.S., Cobb, H.S., Precision Landing of Aircraft Using IntegrityBeacons, in Parkinson, B.W and Spilker Jr. J.J., Global Positioning System: Theory andApplications Volume II, Vol. 163 Progress In Aeronautics and Astronautics, AIAA, 1996,pp. 427-459

De Jonge,P.,and Tiberius,C., The LAMBDA method for integer ambiguityestimation:implementation aspects, Delft Geodetic Computing Centre LGR series, No. 12,Delft, Netherlands, 1996

Verhagen,S.,and Li,B.,LAMBDA software package Matlab implementation,version 3.0,DelftUniversity of Technology,pp.14

Agarwal,S.,Hablani,H.B., Precise Positioning Using GPS for Category-III AircraftOperations Using Smoothed Pseudo range Measurements, Department of AerospaceEngineering Indian Institute of Technology Bombay.


Thank you


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