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High-Accuracy Integrated GPS-INS Aircraft Navigation for Landing using Pseudolites and Double-Difference Carrier Phase measurements presented by:

Prof. Hari B Hablani Indian Institute of Technology, Bombay

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

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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 Indian Institute of Technology, Bombay

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Introduction The Global navigation satellite systems are fast becoming preferred means of navigation Satellite based navigation system allows to fly aircraft through time and fuel efficient routes Provides efficient air traffic management The commissioning and maintenance cost of the ground based signalling systems such as instrument landing system can be reduced by relying on GPS-based landing approach Government of India has invested several thousand crores in building space infrastructure in the country. With GAGAN and IRNSS (Indian Regional Navigation Satellite System), air navigation in India will soon become more accurate and efficient Reference: http://www.gps.gov/applications/aviation/

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Problem Description

Provide precise navigation to an aircraft during landing and use PVT estimation 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 integrity beacons Implementation of navigation algorithm using single-difference and double-difference carrier phase measurements Development and simulation of the integrated GPS-INS navigation system Development of the flight control system Receiver autonomous integrity monitoring

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GPS Overview GPS Segments: Space Segment,Control Segment and User Segment Signals: Each satellite transmits two RF signals L1 and L2 fL1 = 1575.42 MHz, and fL2 = 1227.60 MHz

GPS measurements Code phase measurement(Pseudo-range measurement) ρ = rtrue + Iρ(t) + Tρ(t) + c(δtu − δt s ) + ξρ

(1)

Carrier phase measurement φ = λ−1 [r − IΦ + TΦ ] +

c (δtu − δt s ) + N + ξΦ λ

(2)

Where: N is unknown no. of whole cycles, the carrier phase measurement has no information about the no. of whole cycles; this is also called integer ambiguity. σ(ξρ ) = 0.5 m, σ(ξΦ ) = 5 mm Indian Institute of Technology, Bombay

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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 Indian Institute of Technology, Bombay

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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 aircraft landing Centimetre level accuracy can be achieved by using differential carrier phase measurements However, carrier phase measurements require estimation of unknown integer ambiguities

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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 Integrity Beacon signals. These measurements are transmitted to the aircraft. The on-board system consists a GPS receiver,a data link receiver and a computer 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

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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 Indian Institute of Technology, Bombay

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Single-Difference Carrier Phase Measurements

Figure: Vector Geometry

Carrier phase measurements at the user(Aircraft) from i th satellite φiu = Îťâˆ’1 [rui − Iui + Tui ] + c(δtu − δtsi ) + Nui + iφ,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 Indian Institute of Technology, Bombay

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Single-Difference Carrier Phase Measurements(contd..) Carrier phase measurement at the reference station from the same satellite φir = Îťâˆ’1 [rri − Iri + Tri ] + c(δtr − δtsi ) + Nri + iφ,r

(4)

Single-difference carrier phase φiur = φiu − φir φiur = Îťâˆ’1 [(rui −rri )−(Iui −Iri )+(Tui −Tri )]+c(δtu −δtr )+(Nui −Nri )+iφ,u −iφ,r (5) Satellite clock error is eliminated with this differencing. Reference station and user are in proximity, so tropo. delay and iono. delay are eliminated. Nu − Nr is an unknown differential integer We need to estimate this differential integer The estimation process is called integer ambiguity resolution Indian Institute of Technology, Bombay

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

Figure: Geometry of the single difference measurements

−1ir

rur = ru − rr

(6)

i rur

(7)

=

−1ir Xur th

is the unit vector from ref. station to i Νφiur

=

−1ir .Xur

satellite.

i + cδtur + ΝNur + Νiφ,r

(8)

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

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

Carrier phase of the signal from integrity beacons IBj measured at the aircraft Νφju = | − pj + Xur | + c(δtu − δt j ) + ÎťNuj + Îťφ,u

(9)

at the ref. station Νφjr = |pj | + c(δtr − δt j ) + ÎťNrj + Îťφ,r

(10)

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

Îť[φju − φjr ] = | − pj + Xur | − |pj | + c(δtu − δtr ) + Îť(Nu − Nr ) + Îťφ,u − Îťφ,r (11)

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Integer Ambiguity Resolution We define âˆź Xur = Xˆur + Xur

(12)

Xur is the relative vector from user to reference station âˆź Xˆur is initial estimate and Xur is the correction to the estimate The single-difference carrier phase equation for the i th satellite can be written as âˆź i Νφiur + 1ir .Xˆur = −1ir .Xur + cδtur + ÎťNur + Îťiφ,ur

(13)

The single-difference carrier phase equation for the integrity beacons can be written as âˆź j Νφjur − | − pj + Xˆur | + |pj | = −e j .Xur + cδtur + ÎťNur + Îťjφ,ur

the unit vector e j =

(14)

pj − Xˆur |pj − Xˆur |

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Integer Ambiguity Resolution(Contd...)

So, at time tk we have following measurements âˆź i Νδφiur (tk ) = −1ir .Xur (tk ) + Ď„k + ÎťNur + Îťiφ,ur

(15)

âˆź j Νδφjur (tk ) = −e j .Xur (tk ) + Ď„k + ÎťNur + Îťjφ,ur

(16)

−1r Xˆur = Ď Ë†us − Ď rs Νδφur (tk ),−1ir and Îťjφ,ur are known to us âˆź Our aim is to estimate Nur ,Ď„k and correction Xur

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Integer Ambiguity Resolution(Contd...) The unknowns in the eqs. for all GPS satellite and two Integrity Beacon are: ∼ i j Xur (tk ), τk , Nur andNur i=1,2,3...m , m is the no. of visible GPS satellites at time tk j=1,2(Beacons) 1 To solve the above equations Nur is merged with τk and then we determine i 1 remaining Nur − Nur  1   1   1    φ,ur δφur (tk ) 0 −1r (tk ) 1 2   2 2 1 1  Nur − Nur   δφur (tk )   −1r (tk ) 1 φ,ur         .        . . . .           .  ∼     Xur   . . . . (t )   k = m   + λ λ m 1  + λ m       δφm (t ) N − N −1 (t ) 1 τ   k φ,ur k k ur   ur   r   ur  ......    .......   .......   .....       IB  IB   IB1  1  Nur 1 − Nur δφur 1 (tk )  −e 1 (tk ) 1 φ,ur  IB2 1 2 2 −e 2 (tk ) 1 δφIB Nur − Nur IB ur (tk ) φ,ur

(m + 2) ∗ 1

(m + 2) ∗ 4

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Integer Ambiguity Resolution(Contd...) Aircraft collects the single diff. data during the bubble pass The collected data is arranged in the following way  âˆźâˆ—    ˆ  X1âˆźâˆ—  S1 0 0 . I  δφ1 X2  δφ2   0 Sˆ2 0 . I   .           .  + Îť Ν  . =.   âˆźâˆ—   .  .  Xn    ˆ δφn 0 0 0 Sn I ....... N   0 0 ..0 I = 1 0 ..0 I:(m + 2) ∗ (m + 1) 0 0 ..1 Above equation can be solved using least square technique Xˆur is updated in each iteration The above single difference formulation does not cause correlation in the vector  Integer ambiguities are obtained by rounding the float solution to the nearest integer Indian Institute of Technology, Bombay

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Integer Ambiguity Resolution(Contd...)  X1∼∗ X2∼∗  I    .  I    ,Y = .    ∼∗  Xn     ....... I N 

ˆ S1 δφ1 δφ2  0      H = λ  . ,G =.  .  . δφn 0 

0 Sˆ2

0 0

. .

0

0

Sˆn

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 0 ∗ G )−1 ∗ G 0 ∗ H

(17)

The G and H matrices are updated in each iteration Converges in 3-10 iteration Indian Institute of Technology, Bombay

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Integer Ambiguity Resolution(Contd...) Table: Integer Ambiguities

No. 1 2 3 4 5 6 7 8 9

K 1 True Nur − Nur -26093 -62198 -118010 -98252 11727 -91595 26217 2624 2624

k 1 Estimated Nur − Nur -26092.99 -62197.96 -118010.03 -98252.063 11726.96 -91595.09 26216.98 2624.18 2624.18

Error 0.0065 0.03 -0.02 -0.063 -0.032 -.09 -0.019 0.18 0.18

No of measurements:60, σ = 5 mm σ: Standard deviation of the noise No. of visible satellites:8 K=2,3....8(GPS satellite) K=9,10(Pseudolites) Indian Institute of Technology, Bombay

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Double-Difference Measurements The clock bias term δtur is common in the single difference measurements from all satellites at each epoch.This term can be removed by the double-difference Single-difference carrier phase measurement for satellite k and l φkur = φku − φkr k Îťâˆ’1 rur

(18) kφ,ur

(19)

l l φlur = Îťâˆ’1 rur + f .δtur + Nur + lφ,ur

(20)

=

+ f .δtur +

k Nur

+

Double-difference measurement k l φ(kl) ur = φur − φur

=

kl Îťâˆ’1 rur

+

(21) (kl) Nur

+

(kl) φ,ur

(22) (kl)

−1 k l (kl) φ(kl) ur = âˆ’Îť (1r − 1r ).Xur + Nur + φ,ur Indian Institute of Technology, Bombay

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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 ρ2φ I then the covariance of a pair of double-difference measurements is   2 2 1 covφ (dd) = 2ρφ (24) 1 2 Double-difference measurements are correlated Simple rounding the float solution may give false results

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Least-Squares Ambiguity Decorrelation Adjustment(LAMBDA) Decorrelation is achieved by using LAMBDA (Least-Squares Ambiguity Decorrelation Adjustment) technique The estimation of unknown integers carried out in three steps: float solution,decorrelation, and fixed solution y = BδX + AN + 

(25)

Cost function Cw = ||y − BδX − AN − ||2Q −1

(26)

w

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

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

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LAMBDA(contd...) The float solution obtained from eq.[26]   ˆ δX Nˆ

(27)

ˆ Qn = Cov (N)

(28)

where: Nˆ represents the float ambiguity vector Qn is the covariance matrix of the float solution Nˆ The second step consists of min||Nˆ − N||2Q −1

(29)

n

The ambiguities are decorrelated by a Z-transformation. The transformation matrix Z is determined by decomposing Qn in a lower triangular 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 Indian Institute of Technology, Bombay

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LAMBDA(contd..)

ˆ = Z T Nˆ M Qzˆ = Z T QNˆ Z ˆ is the transformed ambiguity vector Where: M Qzˆ is the transformed covariance matrix The minimization process is performed on the transformed ambiguities ˆ − M)T Q −1 (M ˆ − M) ≤ χ2 (M zˆ

(30)

The size of the search ellipsoid depends on the χ The next step is N = Z −T M

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LAMBDA(contd...)   4.97 3.87 Qhat = 3.87 3.01   0.119 ˆ N= 0.157

(31) (32)

Figure: correlated Reference:V.,Sandra GNSS Research Centre, Curtin University Mathematical Geodesy and Positioning, Delft University of Technology Indian Institute of Technology, Bombay

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LAMBDA(contd...) Decorrelated covariance matrix 

0.0865 −0.0364 −0.0364 0.0847   −3 −4 Z= 4 5   0.25 ˆ M= 0.29



Qzhat =

(33) (34) (35)

Figure: Decorrelated Indian Institute of Technology, Bombay

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LAMBDA(contd...)  6.290 5.978 Qhat = 5.978 6.292 0.544 2.340   5.45 Nˆ = 3.10 2.97

 0.544 2.340 6.288

(36)

(37)

The ambiguities are correlated. These are decorrelated by a Z transformation. ˆ = Z T Nˆ M

(38)

T

Qz hat = Z ∗ Qhat ∗ Z

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LAMBDA(cont.)

Figure: Decorrelated Ambiguities

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Simulation and Results

Figure: Flight Trajectory

Reference station position Latitude:0.5704 rad Longitude:1.2956 rad Altitude : 314 m Distance between integrity beacon and reference station: 16 km Distance between integrity beacons:4 km Indian Institute of Technology, Bombay

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Simulation and Results (contd...) Up to 60 Sec, pseudo-range measurements are used to determine the aircraft position after that it switches to the differential carrier phase mode

Figure: Position Error Indian Institute of Technology, Bombay

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Double-Difference

Figure: Position Error

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Conclusion

Simulation results show that centimetre level accuracy can be achieved with carrier phase measurements The flight trajectory is simulated without considering any forces and disturbances The integer ambiguity resolution requires that the user collect data from multiple epochs before a reliable estimate of the ambiguity can be made

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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

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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 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 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 Verhagen,S.,and Li,B.,LAMBDA software package Matlab implementation,version 3.0,Delft University of Technology,pp.14 Agarwal,S.,Hablani,H.B., Precise Positioning Using GPS for Category-III Aircraft Operations Using Smoothed Pseudo range Measurements, Department of Aerospace Engineering Indian Institute of Technology Bombay.

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Thank you

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