GPS Review and RAIM

7/25/2019 GPS Review and RAIM

1/14

8

2.2 GPS [4]

GPS

n GPS n

:

2 2 21

1 1 1 1

2 2 2 2

22 2 2

2 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

R R RR

R R R R

n

nRR n R n R n

X X Y Y Z Z eCdtP

eCdtP X X Y Y Z Z

eCdtP X X Y Y Z Z

(2.1)

:iR

P i

( , , ) :i i i

X Y Z i

( , , ) :R R R

X Y Z

:dt

:i

e i.

:C; 1,2 ,i n=

(2.1)

0 0 0

( , , )X Y Z

:


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9

0 1 0 1 0 1

1 1 1

1 1

10

0 2 0 2 0 32 2

2 2 3 20

0

0 0 0

1

1

1

R R R

R R

R R

R R R

n n

nR R

n n n

n n n

R R R

X X Y Y Z Z

eX XPX X Y Y Z Z

eY YP

Z Z

eCdtPX X Y Y Z Z

(2.2)

2 2 20 1 0 1 0 1

( ) ( ) ( )iR

X X Y Y Z Z 1,2 ,i n=

(2.2)

Y HX e (2.3)

1 1

2 2

1 1

R R

R R

n n

R Rn n

P

PY

P

1

2

1n

n

e

ee

e

1n

0 1 0 1 0 1

1 1 1

0 2 0 2 0 2

2 2 2

0 0 0

4

1

1

1

R R R

R R R

n n n

n n n

R R R n

X X Y Y Z Z

X X Y Y Z Z

H

X X Y Y Z Z

4n

0

0

0

4 1

X X

Y YX

Z Z

Cdt

4 1


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10

(Least-Square Method)Y HX e

-1( )T TLSX H H H Y

(2.4)


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11

GPS

;

(Integrity)

(Receiver Autonomous Integrity Monitoring ; RAIM);

GPS(Supplemental)

(Pseudoranges)

GPSGPS(Primary)

GPS

RAIM

FDE(Fault Detection & Exclusion)


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(RAIM)

(Parity Method) GPS

3.1 RTCA (MOPS)[5][6]

RTCA(Radio Technical Commission for Aeronautics)

RTCA 1991 GPS

(Supplement Navigation System)

(Minimum Operational Performance Standards ; MOPS) 3.1

RTCA DO-208MOPS

2.0 0.002/ 30 0.999

1.0 0.002/ 10 0.999

RNAV

0.3 0.002/ 10 0.999

3.l GPS

:

(Alarm Limit)

GPS


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13

(Failure)

(Maximum Allowable Alarm Rate) GPS

(Time to Alarm)

(Detection Probability)=1-(Miss

Detection Probability)

SA 3.1

3.2

RAIM

RAIM

RAIM

(Snapshot)


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14

3.2.1 (Least Square Residuals Method)

(2.3)(2.4)Y

1( )T TLSY H X H H H H Y

(3.1)

Y w :

1( )

T Tw Y Y I H H H H Y

(3.2)

w RAIM

(Sum of Squared Errors ; SSE

T

SSE w w

(3.3)

SSE:

1.SSE GPS

;(Threshold)

2.e

SSE

;


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3.e

SSE n-4

(Chi-Square Distribution)

SSE n-4

:

3.1[7]

X n

(Probability Density Function):

/ 2

( / 2) 1 / 22 if 0

( ) ( / 2) 0 if 0

n

n xx e x

f x nx

1

0( ) x tt e dt

3.1[7]

1 2, ,

nX X X

2

1

nK

K

XY

Y n


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16

3.2[7]

1 2, ,

nX X X 1,2, ,k n

k

X k

k

1 2, ,

na a a

1 1 2 2 n nY a X a X a X

Y Y 2

Y:

1 1 2 2Y n na a a

2 2 2 2 2 2 2

1 1 2 2Y n na a a

(2.3)(3.2)

1

1

1 1

1

( )

( ) ( )

( ) ( )

( )

T T

T T

T T T T

T T

w I H H H H Y

I H H H H HX e

HX H H H H HX I H H H H e

I H H H H e

(3.4)

1( )T TQ H H H H (3.5)

2 1 1

1

( ) ( )

( ) =

T T T T

T T

Q H H H H H H H H

H H H H Q

(3.6)

2 2( ) ( )( ) 2 2

I Q I Q I Q I Q Q I Q Q

I Q

(3.7)

(3.6)(3.7)QI Q n n (Idempotent

Matrix

( ) ( ) 4Trace Q rank Q

( ) 4Trace I Q n


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n n (Orthogonal Matrix) K Q

n n n-4 1 0

(1, 1,0,0,0,0)TI Q K diag K (3.8)

1 2

( , , )Tn

u u u u Ke (3.3)(3.4)

TSSE w w

( ) ( )T Te I Q I Q e

( )T T Te I Q Q Q Q e

( )Te I Q e

(1, 1,0,0,0,0)T Te K diag Ke

(1, 1,0,0,0,0)Tu diag u

2 2 2 2

1 2 5 4n nu u u u (3.9)

e 2

11 12 1 1

21 22 2 2

1 2

n

n

n n nn n

K K K e

K K K eu Ke

K K K e

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

n n nn n n

K e K e K e u

K e K e K e u

K e K e K e u

K 1,2, ,i n


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18

2 2 2

1 21

i i inK K K

3.2u

2(3.9) 2

2 2 2

1 2 4

2

nSSE u u u

(3.10)

(1) (2)

5 132 176

6 102 133

7 90 108

8 82 99

9 77 91

10 73 87

(1): 1/15000 (RTCA DO-208)

(2): 1/3000000 (RTCA DO-229)

3.2 /( 4)SSE n

3.1 2SSE n-4

SSE n-4

/( 4)SSE n (Decision Variable);

/( 4)SSE n

3.1 =33 m /( 4)SSE n

(Alarm)


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3.2.2 (Parity Method)[2]

(2.3) 1( )T TLSX H H H Y

4nH ( ) 4rank H Gram-schmidt ( 4)n nP

0PH 4

T

nPP I n n

1( )

T TH H H

U

P

(3.11)

( )rank U n UnY 4(State Space) n-4(Parity Space):

1( )

T T

LSH H H X

UY Y

P p

(3.12)

( 4) 1n p p(Parity Vector)P 1

( )PY P HX e PHX Pe Pe (3.13)

( )rank U n

( )

| |

T T

T T

H H H

U H P H P P

1 1( ) ( )T T T T T

T

H H H H H H H P

PH PP

4

4

0

0 n

n

I

I


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20

1 | TU H P T

S P P (3.14)

1

1

( )

|

T T

T

n

H H

I U U H PP

1( )T T TH H H P P

1( )T TH H H S

1( )T Tn

S I H H H H (3.15)

(3.2)(3.15)

1 1( ) ( )T

T T T T T T w w Y I H H H H I H H H H Y

1 1( ) ( )

T T T T T Y I H H H H I H H H H Y

1 1 1 1( ) ( ) ( ) ( )

T T T T T T T T T Y I H H H H H H H H H H H H H H H H Y

1

( )T T T

Y I H H H H Y

T T T T Y SY Y P PY p p

p

T Tp w w SSE (3.16)

w(3.16) Tp

SSE

PH QR

Normalized QR Decomposition[32]


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21

4 4n n n nH Q R

P TQ n-4

3.2.3 FDE(Fault Detection & Exclusion) [2]

FDE GPS

(Integrity Monitoring)FDE

(Detection)(Exclusion)

(Least Square

Residual Method)(Parity Method)

:

(1) T Tp w w SSE= = ((3.16))

(Sum of Squared Errors ; SSE)

(2)(3.11)P QR

(Normalized QR Decomposition)[32]H QR= TQ Q I=R

T TQ H Q QR R= = R n-4

0P TQ n-4 0PH =

(3) ( )PY P HX B e

YPPY p

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GPS Review and RAIM