gps review and raim
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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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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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(Least-Square Method)Y HX e
-1( )T TLSX H H H Y
(2.4)
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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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(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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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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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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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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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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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