Fault Free Integrity of Mid-Level Voting
for Triplex Carrier Phase Differential
GPS Solutions
G. Nathan Green – The University of Texas at Austin
Martin King – NAVAIR
Dr. Todd Humphreys – The University of Texas at Austin
Preliminary Definitions
Error
Distribution
Integrity
Risk
Impact of Correlation on Integrity Risk
Independent
Triplex with
MLV
Completely
Correlated
Triplex with
MLV
Partially
Correlated
Triplex with
MLV
Mid-Level Voting (MLV) Integrity
Monitor
𝑋1 , 𝑋2 , 𝑋3
𝑋(1) ≤ 𝑋
𝑋
≤𝑋
2
≜𝑋
2
𝑅𝑀𝐿𝑉 = 𝑃 𝑋
2
2
𝑅𝑀𝐿𝑉 =
𝑗
𝑉𝑗
−2
𝑘
𝑉𝑘
3
−𝑥
> 𝐴𝐿
𝑓𝑋 𝑥 𝑑𝑥
𝑓𝑋 𝑥 𝑑𝑥
Correlation-Agnostic Integrity
• Without accounting for
correlation, two good
solutions can protect one
bad one.

RMLV  1  P X 1  AL & X  3  AL
• However, the additional
protection is minimal.
Ri  R j  Rk
Low Risk
Solution
Selected
Solution
• In order to protect the
selected solution without
knowing the correlation, the
integrity allocations must be
reduced by half.

Triplex Float CDGPS Solution
 bˆ 
1
T 1
xˆ i      Hi i Hi  HTi i1 yi  Si yi
 Nˆ 
 float ,triplex
T

ˆ
ˆ
ˆ
ˆ
ˆ
ˆ


 Cov   b1 N1 b 2 N 2 b 3 N 3  


T
T
T





y ,1
y ,2,1
y ,3,1 S1 0 0 
S1 0 0 
  0 S 2 0   y ,2,1  y ,2 Ty ,3,2   0 S2 0 

 0 0 S3   



 y ,3,1 y ,3,2 y ,3   0 0 S3 
 S1 y ,1S1T S1Ty ,2,1S 2T S1Ty ,3,1S3T 
 S 2  y ,2,1S1T S 2  y ,2 S 2T S 2 Ty ,3,2 S3T 


T
T
T
 S3 y ,3,1S1 S3 y ,3,2S 2 S3 y ,3S 3 
• Correlations stem from
common atmospheric
errors and the use of
shared reference
receivers
• This covariance can be
evaluated in the MLV
risk expression and
trigger an alert if RMLV
exceeds IRspec
Simplex Fixed, Position Domain Integrity,
and Almost Fixed Solutions
Fixed
Solution
EPIC
Approximates
Fixed PDF
Central, Correct
Fix Mode
Biased,
Incorrect Fix
Modes
GERAFS uses
correct fix plus
a bound with
largest bias
Fixed CDGPS Solution Covariance
• Each simplex solution is
conditioned on its own
particular set of
ambiguities Simplex
• The joint triplex
covariance is
constructed by careful
conditioning
Covariance
 float ,triplex
 b1
 N b
 11
b2b1


  N 2b1
 b b
 3 1
 N 3b1
Triplex Cross
Covariance
 N1
b2 N1
 N 2 N1
b3N1
 N 3 N1
b2
 N 2b2  N 2
b3b2 b3N 2  b3
 N 3b2  N 3N 2  N 3b3  N 3








 b1|N1


 B ,triplex  b2b1|N 2 N1  b2 |N 2


 b3b1|N 3N1 b3b2 |N 3N 2  b3 |N 3 
b jbi |N j Ni  b jbi
1
  Nibi    Ni  Ni N j    Nibi 

 N jbi   N j Ni  N j    N jbi 

 

 
T
Fixed CDGPS Solution Integrity
R fixed ,triplex  1  P  CF1  CF2  CF3 
 P  CF1  CF2  CF3 

 RMLV AL, 0,  B ,triplex
P
CFi
i

 P



CFi   min i PCFi

 
i
R fixed ,MLV  1   PCFi
 
i

 min i PCFi  RMLV AL, 0,  B ,triplex

MLV Triplex GERAFS
• Similar to fixed solution,
GERAFS has three risk
components
 𝑃 𝑁=𝑁
 𝑃 𝑁 ∈ 𝐸𝑃𝐼𝑅
 𝑃 𝑁 ∉ 𝐸𝑃𝐼𝑅 ∪ 𝑁
• The relationship among
these possibilities for
triplex solutions is
unknown
• Use bounds similar to
fixed solution
• Not appropriate for
EPIC since the joint
distribution is unknown
PAFi  PCFi  PEPIRi
RGERAFS ,MLV,V  1   PAFi
 
 min  P 

i

 min i PCFi  RMLV AL, 0,  BV ,triplex
i
EPIRi

 RMLV ,GERAFS AL, IFB,  BV ,triplex







Simulation Methodology
• World wide daily
average availability




Grid of locations
24 hours for SV motion
Availability of Integrity
Availability of Accuracy

AGP
 # t | IRGP  t   IRspec 

 avg   spec

# t

0
 avg   spec

• Error models
 Noise, correlated
multipath, Iono + Tropo,
and position domain
errors
• Simulate three solution
types
 Simplex float
 Simplex GERAFS with
float backup
 Triplex GERAFS with
triplex float backup
• Compare availability vs
accuracy for varied alert
limits
GERAFS
Design Modifications
• Design Modification
 Use MLV for float
integrity, but not
fixed integrity
 Compute GERAFS PL
as though simplex,
but gain accuracy
benefit from MLV
solution
Alternate
GERAFS 1AB
GERAFS 2AB
GERAFS 3AB
Compute PL
Compute PL
Compute PL
Mid-Level Vote
Compute Accuracy
Modified Triplex
Results
Discussion
• Triplex MLV provides
significant benefit to
the float solution when
compared to simplex.
• GERAFS fixing
probabilities are not
aided by triplex MLV,
but accuracy can be
improved
• System continuity may
preclude taking
integrity credit for MLV
 Loss of single receiver
eliminates MLV benefit
for float solution
 Requires additional
airborne receivers to
claim benefit