S´eminaire de recherche TAMCIC `a l`ENST de Bretagne, 16 f´evrier

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S´eminaire de recherche TAMCIC `a l`ENST de Bretagne, 16 f´evrier
Advanced RAIM FDE algorithms for safe
navigation
Igor Nikiforov
LM2S/ICD/UTT, FRE CNRS 2848
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 1/41
Plan
Quality measures and integrity
Measurement model and working hypotheses
Protection zone and positioning failure
Criteria
Principle of conventional RAIM FDE algorithms
Principle of advanced RAIM FDE algorithms
Scenario of simulation : APV-I
List of RAIM algorithms
Some results
Interpretation of results
Conclusion and perspectives
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 2/41
Quality measures and integrity
The recent researches show that the detection and
exclusion of the GPS navigation message degradation
is crucially important for many transportation
systems. It is proposed to “encourage all the
transportation modes to give attention to autonomous
integrity monitoring of GPS signals..." [1].
[1] “Vulnerability Assessment of the Transportation Infrastructure Relying on the
GPS", John A. Volpe National Transportation System Center, U.S. Department of
Transportation, August 2001
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Quality measures and integrity
Navigation satellites
Ionospheric and tropospheric effects
Jamming
Spoofing
Meaconing
INS, baro, terrain correlation
Local effects
Multiple satellite
signal corruption
Odometer
DGPS, Loran-C
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Quality measures and integrity
Navigation satellites
d
i
=
r̂
i
kX
= i −X
d
uk
i +
2
ctˆ
r
+
ξˆ
Fault
i
Vertical error
True position X
Estimated position X̂
Horizontal error
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Quality measures and integrity
Navigation satellites
Xi
Fault
Railway track


 x = φ(l)
y = ϕ(l)


z = η(l)
S(b
l)
S(b
l)
C(l)
Xu=S(l)
Train
terval
ction In
C(l)
rote
rack P
T
e
h
t
Along
True locomotive position
Estimated position = acceptable error
Estimated position = positioning failure
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Measurement model and working hypotheses
Model of GPS


 r1 = d1 (x, y, z) + c tr1 + ξ1
..
..
..
.. ,
.
.
.
.


rn = dn (x, y, z) + c tr1 + ξn
where di (x, y, z) = kXi − Xu k2 , ξi ∼ N (0, σi2 ).
Model of Galileo


 rn+1 = dn+1 (x, y, z) + c tr2 +
..
..
..
.
.
.


rn+m = dn+m (x, y, z) + c tr2 +
ξ1
.. ,
.
ξn
where tr1 − tr2 = known constant.
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 7/41
Measurement model and working hypotheses
Barometric altimeter
hb = z + ξb , ξi ∼ N (0, σb2 ).
GPS/baro measurement equations with faults
Yk ≃ H0 (Xk − X0k ) + Ξk + Υl (k, k0 ), Ξk ∼ N (0, Σ)
where
Υl (k, k0 ) =
(
0 if k < k0
Υl if k ≥ k0
is a fault and k0 is the fault onset time.
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Positioning error autocorrelation functions : correlation time constant
varies from 138 sec to 833 sec
1.2
longitude
latitude
altitude
Autocorrelation Rxx (τ )
1
0.8
0.6
0.4
0.2
0
-0.2
-300
-200
-100
0
100
200
300
Delay τ (min)
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 9/41
UERE (cm) of different types of channels (extracted
from [Have03])
Channel \ elevation◦
5
10
15
20
30
40
50
60
90
GPS current L1
844
770
709
659
581
529
496
476
460
GPS II L1/L5
227
186
171
164
159
157
156
156
156
GPS III L1/L5
190
136
115
104
96
93
92
91
91
GPS/SBAS L1
207
166
144
127
105
91
83
78
73
GPS/SBAS L1/L5
179
121
96
84
73
69
67
67
66
Galileo E1/E5
188
134
113
102
93
90
89
88
88
Galileo/iono SBAS E1
216
177
156
142
122
110
104
100
96
GEO L1
252
219
202
191
176
168
164
162
159
GEO L1/L5
240
193
175
167
161
159
159
159
158
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Protection zone
Satellites (GPS and/or Galileo)
Bias
z }| {
Y = HX + ξ + Υ
True position
Protection zone
Acceptable error
Positioning Failure
2VAL
HAL
Density of acceptable errors
Density of nominal errors
Density of positioning failure
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Horizontal positioning failures
600
Ellipses of uncertainty
PF
400
PF
ŷ − y
200
0
PF
Acceptable error
PF
True position
-200
PF
ϕ
Fault directions
-400
-600
-600
Positioning failure (PF)
-400
-200
0
200
400
600
x̂ − x
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Vertical positioning failures
Failure zone
f (ẑ − z)
VAL
Υ 6= 0
VAL
Υ=0
Υ 6= 0
ẑ − z
density of acceptable errors
density of nominal errors
density of positioning failure
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 13/41
Q. : How to define a fault that leads to a positioning failure ?
A. : a fault Υ is considered as a horizontal positioning failure if
(1 − pf )
z
max
no fault: Υ=0
}|
{
P0 (∃t : k0 ≤ t ≤ k0 + m − 1, kXht − X̂ht k2 > HAL)
k+m−1≤k0 ≤k
a fault: Υ6=0
z
}|
{
+pf PΥ (kXhk − X̂hk k2 > HAL) > pr ,
A fault Υ is considered as a vertical positioning failure if
(1 − pf )
z
max
no fault: Υ=0
}|
{
P0 (∃t : k0 ≤ t ≤ k0 + m − 1, |zt − ẑt | > VAL)
k+m−1≤k0 ≤k
a fault: Υ6=0
z
}|
{
+pf PΥ (|zk − ẑk | > VAL) > pr ,
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Criteria : RAIM FDE algorithm is a pair (N, ν)
Probability of false detection (exclusion is possible) :
α′ (N, ν, k) = P0 ({k ≤ N ≤ mα + k − 1} ∩ {ν > 0}) ,
Probability of false alert (exclusion is impossible or failed
exclusion) :
α(N, ν, k) = P0 ({k ≤ N ≤ mα + k − 1} ∩ {ν = 0}) ;
Probability of missed alert :
γ(N, ν) =
l
max Pk0 (N −k0 +1 > mτ )+
1≤l≤n(k0 )
Plk0
({k0 ≤ N ≤ mτ + k0 − 1} ∩ {ν 6= l}∩{ν > 0})
Probability of failed exclusion :
ω(N,ν,k0) =
max Plk0({k0 ≤N≤ mτ +k0 −1}∩{ν = 0})
1≤l≤n(k0 )
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Principle of conventional RAIM FDE algorithms
Unconstrained Generalized Likelihood Ratio Test (GLRT) =
conventional snapshot Least Squares (LS)-based RAIM FDE :


 H0 if
where
fW Υ
b (Z)
j
max1≤j≤n f0 (Z)
<h
δ(Ye ) =
fW Υ
(Z)
b
j

 Hν if ν=arg max1≤j≤n
f0 (Z) ≥ h
e Ye = Σ− 12 Y
Z = W Ye = W ξe [+W Υ],
is the parity vector, fθ (Z) is the density of N (θ, In−4 ) and the
nonzero element of the estimated vector
is given by
b j = (0, . . . , 0, υ
Υ
bj /σj , 0, . . . , 0)T
υbj = arg min kZ −
W Υj k22
WjT Z
= T
Wj Wj
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Principle of advanced RAIM FDE algorithms (1/2)
Constrained GLRT = advanced RAIM FDE :

fW Υ
e (Z)

max1≤l≤n f e l (Z) < h
 H0 if
W Υ0
δ(Ye ) =
fW Υ
(Z)
e

 Hν if ν=arg max1≤l≤n f l (Z) ≥ h
e
W Υ0
(
2 )
υl where υ
el = arg max {fW Υl (Z)} = arg min
,
Z − Wl σ
|υl |≥bl
l 2
|υl | ≥ bl
| {z }
NEW!
(e
υj , j) = arg max max
1≤i≤n |υi |≤ai
= arg min
n
fW Υ
b i (Z)
min
i | ≤ ai
| {z }
1≤i≤n |υ
NEW!
o
(
2 )
Z − Wi υi σi Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
2
– p. 17/41
Principle of advanced RAIM FDE algorithms (2/2)
z3
600
-200
Acceptable error
es
e_
j
-400
-200
0
200
400
hy
po
th
Positioning failure (PF)
z1
-600
-600
j
es
e_
th
hypothese_0k
Fault directions
-400
j
e_
llit
hypothese_0j
True position
PF
sa
b_j
a_j
PF
-a_j
-b_j
z2
ese_k
0
te
d_
k0
d_j0
hy
po
satellit
PF
hypoth
ŷ − y
200
e_k
ese_k
hypoth
PF
d_k
j
400
error
d_
PF
Parity space (case of Z ∈ IR3 )
Z = W Ye
600
x̂ − x
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Scenario of simulation : APV-I
Parameter of simulation model
Interval of latitudes
Latitudinal step
Interval of longitudes
Longitudinal step
Simulation period
Baroaltimeter standard deviation
Pseudo-range/baro noise AR coefficient
Elevation masking angle
HAL/VAL (APV-I)
Probability of a single channel failure/h
Required risk per approach
Required probability of false alert per approach
Required probability of missed alert
Time to alert
RAIM sampling period
Value
[−80◦ ; 80◦ ]
20◦
[0◦ ; 360◦ ]
36◦
24/72 h
19 m
0.998
5◦
40/50 m
10−4
10−7
2 · 10−5
10−3 /failure
10 sec
1 sec
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Grid points spread over the earth surface
δφ
φ0
λ0
δλ
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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List of tested RAIM algorithms
Standard snapshot LS based RAIM algorithm
Snapshot constrained GLRT based RAIM algorithm
Sequential constrained GLRT based RAIM
algorithm
Standard snapshot LS based RAIM with
Pre-Filtering (PF)
Snapshot constrained GLRT based RAIM with PF
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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GPS SBAS L1/L5 : availability of detection and
exclusion
Type of RAIM
Detection availability
Exclusion availability
Min.
Max.
Mean
Min.
Max.
Mean
Standard snapshot LS-based
0.86
0.99
0.94
0.24
0.70
0.50
Standard snapshot LS-based with baro
0.88
1.00
0.95
0.26
0.75
0.54
Snapshot constrained GLR-based
0.91
1.00
0.98
0.39
0.85
0.65
Sequential constrained GLR-based
0.91
1.00
0.98
0.39
0.87
0.67
Standard snapshot LS-based with PF
0.97
1.00
1.00
0.73
1.00
0.91
Snapshot constrained GLR-based with PF
0.98
1.00
1.00
0.81
1.00
0.95
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
100
200
Longitude
300
0
Snapshot constrained GLR: detection
200
Longitude
300
Snapshot constrained GLR: exclusion
−50
Latitude
−50
Latitude
100
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
200
Longitude
0.6
300
0.8
1
– p. 23/41
Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
100
200
Longitude
300
0
Sequential constrained GLR: detection
200
Longitude
300
Sequential constrained GLR: exclusion
−50
Latitude
−50
Latitude
100
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
200
Longitude
0.6
300
0.8
1
– p. 24/41
Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
50
0
100
200
Longitude
300
0
Conventional RAIM with pre−filtering: detection
100
200
Longitude
300
Conventional RAIM with pre−filtering: exclusion
−50
Latitude
−50
Latitude
0
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
200
Longitude
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
0.6
300
0.8
1
– p. 25/41
Conventional RAIM/GPS SBAS L1/L5: detection
Conventional RAIM/GPS SBAS L1/L5: exclusion
−50
Latitude
Latitude
−50
0
50
50
0
100
200
Longitude
300
0
Conventional RAIM/GPS SBAS L1/L5+baro: detection
100
200
Longitude
300
Conventional RAIM/GPS SBAS L1/L5+baro: exclusion
−50
Latitude
−50
Latitude
0
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
200
Longitude
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
0.6
300
0.8
1
– p. 26/41
Galileo E1/E5 : availability of detection and exclusion
Type of RAIM
Detection availability
Exclusion availability
Min.
Max.
Mean
Min.
Max.
Mean
Standard snapshot LS-based
0.87
1.00
0.97
0.61
1.00
0.82
Standard snapshot LS-based with baro
0.87
1.00
0.97
0.61
1.00
0.83
Snapshot constrained GLR-based
0.89
1.00
0.97
0.65
1.00
0.88
Sequential constrained GLR-based
0.89
1.00
0.97
0.65
1.00
0.89
Standard snapshot LS-based with PF
0.97
1.00
1.00
0.71
1.00
0.92
Snapshot constrained GLR-based with PF
0.97
1.00
1.00
0.71
1.00
0.92
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
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Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
100
200
Longitude
300
0
Snapshot constrained GLR: detection
200
Longitude
300
Snapshot constrained GLR: exclusion
−50
Latitude
−50
Latitude
100
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
200
Longitude
0.6
300
0.8
1
– p. 28/41
Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
50
0
100
200
Longitude
300
0
Conventional RAIM with baroaltimeter: detection
100
200
Longitude
300
Conventional RAIM with baroaltimeter: exclusion
−50
Latitude
−50
Latitude
0
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
200
Longitude
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
0.6
300
0.8
1
– p. 29/41
Snapshot constrained GLR: detection
Snapshot constrained GLR: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
100
200
Longitude
300
0
Sequential constrained GLR: detection
200
Longitude
300
Sequential constrained GLR: exclusion
−50
Latitude
−50
Latitude
100
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
200
Longitude
0.6
300
0.8
1
– p. 30/41
Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
100
200
Longitude
300
0
Conventional RAIM with PF: detection
200
Longitude
300
Conventional RAIM with PF: exclusion
−50
Latitude
−50
Latitude
100
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
200
Longitude
0.6
300
0.8
1
– p. 31/41
Snapshot constrained GLR: detection
Snapshot constrained GLR: exclusion
−50
Latitude
Latitude
−50
0
50
50
0
100
200
Longitude
300
0
Snapshot constrained GLR with PF: detection
100
200
Longitude
300
Snapshot constrained GLR with PF: exclusion
−50
Latitude
−50
Latitude
0
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
200
Longitude
0.4
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
0.6
300
0.8
1
– p. 32/41
GPS SBAS L1/L5 and Galileo E1/E5 augmented by
baroaltimeter : availability of detection and exclusion
Conventional RAIM: detection
Conventional RAIM: exclusion
−50
Latitude
Latitude
−50
0
50
0
50
0
0.2
100
0.4
200
Longitude
0.6
300
0.8
0
1
0.2
100
0.4
200
Longitude
0.6
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
300
0.8
1
– p. 33/41
Interpretation of results : pre-filtering
100
Positioning Failure
density of positioning failure
Positioning Failure
60
80
40
threshold
residual (m)
residual (m)
60
40
20
0
-20
-40
-60
density of positioning failure
20
threshold
0
-20
density of nominal errors
threshold
failure onset time
threshold
density of nominal errors
failure onset time
-40
-80
-100
50
100
150
200
250
300
350
400
-60
50
Elapsed time (s)
z
without PF
}|
SNRk0 = m SNRk>k0 =
r
{
1−a
m
1+a
100
150
200
250
300
350
Elapsed time (s)
400
with PF
}|
{ z GPS
}| {
m
a = 0.998
SNRk0 = √
1 − a2
z
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 34/41
Interpretation of results : constrained GLR based
RAIM
Satellites (GPS and/or Galileo)
Bias
z }| {
Y = HX + ξ + Υ
Probability of detection
1
Fault direction
H−
1
H0
Failure density
H+
1
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 35/41
General conclusion
As expected, the improvement in the quality of the pseudo-range
measurements proposed by the new constellations seriously improves the
availability of the detection and exclusion functions of RAIM.
The proposed new RAIM algorithms perform better than the standard
existing RAIM algorithms. The improvement is in the range of a few percent
of availability gain in most of the cases tested.
A new filtering technique introduced in this study seriously improves the
RAIM availability. However this improvement should be further validated for
constellations with dual frequency measurements since the filtering
characteristics designed in this study were based only on single frequency
measurements.
Extensive simulations were conducted during this study to assess the
performance of several new RAIM algorithms for different constellations
(e.g. existing GPS L1, Galileo, GPS modernized + Galileo, etc...). In these
simulations, the potential of RAIM to support APV I performance level was
assessed.
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 36/41
The results of simulations (1/2)
1. In the case of GPS L1 constellation, the RAIM availability to support APV I
performance level is close to zero. This confirms that expecting to support
APV I operation with the existing GPS signals is unrealistic.
2. In the case of GPS + Galileo combined constellations, the availability of the
tested RAIM algorithms is 100%, even for the demanding fault detection and
exclusion function. This shows that for future receivers such as currently
designed by standardisation bodies, combining GPS and Galileo
constellations with dual frequency measurements on both constellations,
the RAIM provided integrity is a promising technique with a high level of
performance.
3. In the case of Galileo constellation alone the availability of the RAIM
detection function for the advanced algorithms with pre-filtering is close to
100%. The availability of the exclusion function may reach 100% for some
regions (mainly polar and equatorial regions). However, in the case of
Galileo constellation alone, the availability of the RAIM with pre-filtering at
mid-latitudes (e.g. Europe, USA) for APV I is limited, with approximately 74%
of availability for the fault detection and exclusion function.
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 37/41
The results of simulations (2/2)
3.1 The case of APV II has not been investigated in this study, but lower
availability figures would be expected due to the more stringent
requirements for the vertical guidance.
3.2 Therefore, a Ground Integrity Channel seems to be required for Galileo if
integrity is aimed to be provided worldwide with an availability close to 100%
for APV I and II performance levels.
3.3 In this context, the RAIM algorithms could be used with an acceptable
availability as a backup/complementary integrity scheme to the main Ground
Integrity Channel system integrity.
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 38/41
Perspectives
To continue the comparison between two constrained GLR-based RAIM
algorithms (snapshot and sequential) in the case of ramp biases with
different slopes. It is also interesting to compare all the above mentioned
algorithm for other modes of flight.
To study the robustness of the proposed RAIM algorithms with respect to
the autoregressive coefficient of the pseudo-range (barometric) noise model.
The presented simulation is based on the idealistic hypothesis that there are
no systematic errors in pseudo-distances before the fault onset time. It will
be useful to study the robustness of the proposed RAIM algorithms with
respect to non-zero systematic errors in pseudo-distances, which do not
lead to a positioning failure. Also, the impact of multipath phenomena and
other local effects on the proposed RAIM algorithms should be studied to
estimate the variation of their statistical performances.
The presented simulation is based on the hypothesis that the Galileo time
bias with respect to the GPS time is a known constant. It is necessary also
to compare the above mentioned RAIM schemes in the case of the unknown
time bias.
To extend the advanced RAIM algorithms to the integrity monitoring of GNSS
channels augmented by additional sensors (INS, terrain correlation, etc.)
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 39/41
Multi-sensor navigation systems integrity monitoring
GPS
Loran-C
INS
baro
Terrain-matching
Jamming
Spoofing
Meaconing
Faults
Altimeter radar
Terrain profile
DGPS, Loran-C
Geoid
altitude
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 40/41
INS/GPS/terrain correlation
Sensor fault
accelerometer
ωa
R
h
hr
Terrain profile
az
φ(t, t0 )
terrain
radio altim. baro altimeter
correlation
+
hs
+
+
ξb
ξr
Multi-sensor integrated navigation system
R
R
v0
+
−
−
+
ĥ
h0
h − hs
z̃r
z̃b
LS/Kalman filter
za
ẑ
+
ĥ
Terrain profile
Geoid
Local altitude hs
Séminaire de recherche TAMCIC à l’ENST de Bretagne, 16 février 2006
– p. 41/41

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