Erroneous measurements in multisensor navigation systems must be detected and isolated. A recursive estimator can find fast growing errors; a least squares batch estimator can find slow growing errors. This process is called fault detection. A protection radius can be calculated as a function of time for a given location. This protection radius can be used to guarantee the integrity of the navigation data. Fault isolation can be accomplished using either a snapshot method or by examining the history of the fault detection statistics

Bernath, Greg

English

NASA Technical Reports Server

FAULT DETECTION AND ISOLATION
N93"22568
Greg Bernath
Ohio University
Athens, Ohio
SUMMARY
Erroneous measurements in multisensor navigation systems must be detected and
isolated. A recursive estimator can find fast growing errors; a least squares batch
estimator can find slow growing errors. This process is called fault detection. A
protection radius can be calculated as a function of time for a given location. This
protection radius can be used to guarantee the integrity of the navigation data. Fault
isolation can be accomplished using either a snapshot method or by examining the
history of the fault detection statistics.
BACKGROUND
The objective of fault detection and isolation is to use inconsistencies in
redundant sensor measurement data to detect and isolate sensor malfunctions. One
criterion for determining whether a system can be used as a sole means of navigation is
the percentage of time that the position error is greater than a given maximum. If a
given single measurement is in error, it will cause the navigation solution to be in error,
possibly outside the allowable error threshold. Outside sources may not be able to
broadcast in a timely manner that the signal is in error. For instance, if a single GPS
satellite starts to broadcast erroneous signals, it could be from 15 minutes to several
hours before this is broadcast in the satellite health data. Therefore, it is imperative for
Fault Detection and Isolation (FDI) algorithms to be able to detect and isolate
instrument errors using only data from the instruments themselves.
Previous work in FDI has mainly centered around inertial navigation systems
(refs. 2 through 4). However, FDI can be implemented in any multisensor navigation
system with redundant measurements. Current work is focusing on satellite navigation
using the Global Positioning System (GPS), along with hybrid systems such as GPS
integrated with the Long Range Navigation System (Loran-C) or GPS integrated with an
Inertial Reference System (IRS) (ref. 6). FDI used specifically with GPS is also known
as Receiver Autonomous Integrity Monitoring (RAIM) (ref. 7).
Kalman filters are becoming standard as part of the navigation solution in most
GPS receivers. The Kalman filter can look at the difference between a predicted state
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estimate and the actual measured state, and declare a fault if this difference is too large.
This works well for detecting step errors or fast growing ramp errors. However, this will
not detect a slow growing ramp error, such as might be caused by a GPS satellite clock
drift. To detect slow growing errors, the Kalman filter algorithm must be run in parallel
with a least squares estimator algorithm. The least- squares algorithm requires at least
one redundant measurement for fault detection, and at least two redundant
measurements for isolation.
PARITY SPACE AND ESTIMATION SPACE
Estimation space contains the actual horizontal measurement error and the alarm
threshold for a given positioning error. However, actual positions and actual errors
cannot be known given that the only measurement data is coming from imperfect
sensors. Therefore, the work of detecting and isolating errors is done in parity space.
Parity space is a mathematical tool where measurement noise and biases are used to
create a parity vector. The parity vector determines a detection statistic, d, which is
compared to a detection threshold, TD, in order to determine whether an alarm
condition exists. The parity vector is also used as a tool for fault isolation.
Errors and biases in parity space and estimation space are related, but it is not a
one to one correspondence. The exact correspondence will be determined by
measurement geometries. For instance, with a good geometry, a large measurement
error (parity space) will result in only a small position error (estimation space). The
reverse can also be true. Figure 1 illustrates two different slow growing ramp errors
plotted in parity space versus estimation space. In case I, the detection threshold is
crossed before the alarm threshold is reached, yielding a false alarm. As the error
continues to grow, the alarm threshold is crossed, turning it into a correct fault
detection. In case II, the alarm threshold is crossed before the detection threshold is
reached, resulting in a missed detection. As the error continues to grow, the detection
threshold is crossed, turning it into a correct fault detection. An ideal algorithm would
minimize both the number of false alarms and missed detections.
LEAST SQUARES ESTIMATOR ALGORITHM
In a least-squares approach to fault detection, the relationship between the
measurements and the user state (position) is given by:
= H._. (1)
where: 2_ = measurement vector (n-by-l)
H = data matrix (n-by-m)
B_ = user state vector (m-by-l)
2Lis a vector containing n measurements, one from each instrument. In the case
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of using only GPS satellites, it would consist of the pseudoranges. 13 is the m-element
user state vector, consisting of the user position coordinates and other navigation state
elements such as clock offset with respect to GPS time. H is an n-by-m matrix which
relates the measurements to the user states.
There are three possible cases:
1) n < m : Underdetermined system
2) n = m : Exactly determined system
3) n > m : Overdetermined system
In the underdetermined case, a navigation solution is not possible. In the
exactly-determined case, a navigation solution is possible, but fault detection is not.
Algorithms for managing the redundant measurements in an overdetermined
system form the basis of fault detection. A parity equation can be derived from
equation (1), starting with a mathematical manipulation called the QR factorization on
the data matrix H (ref. 4):
H = QR (2)
H is factored into an n-by-n orthogonal matrix Q (Q'rQ = I) and an n-by-m upper
triangular matrix R. R contains (n-m) rows of zeros along the bottom, due to the n-m
redundant measurements in H. Substituting QR for H in equation (1) gives:
= QRfl.
Q T_ = Q a'QR__
QT_ = R_
Next partition R into an m-by-m upper triangular matrix U and (n-m) rows of zeros,
denoted by 0. Similarly, partition QT into Qt (m-by-n) and Q2 ((n-m)-by-n rows).
The least squares navigation state solution is:
.0. = U-lQt_
The parity equation is:
(3)
(4)
(5)
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Q:z = o (6)
Tile measurement vector ): contains noise (_e) and measurement biases (b_). If )_ is
replaced by (_ - e - b_), the 0 in equation (6) can be replaced by the parity vector I2.
P = Q2_: - Q2 _ - Q2 b
p = -Q2 R - Q2 b
(7)
Thus, a parity vector will be determined by the noise and bias errors. From the parity
vector, it can be determined which instrument is in error and whether or not to raise an
alarm.
PARITY SPACE AND DETECTION PROBABILITIES
Consider a situation with one redundant measurement. In this case, the parity
vector will be reduced to a scalar, and the detection statistic reduces to the absolute
value of the scalar. In the ease where no measurement bias exists, figure 2 shows the
distribution of the parity scalar. Since there is no bias error, position error is definitely
under the alarm threshold and the system is either in normal operation or a false alarm
exists. The probability of a false alarm (PFA) is obtained by integrating the areas outside
of To. For noise having a normal distribution (generally a good assumption), this
integral is a standard Gaussian function.
Figure 3 illustrates the case where a large measurement bias exists, making the
position error larger than the alarm threshold. In this case the system is either correctly
detecting a fault or a missed detection is present. The probability of a missed detection
(PMD) is the integral of the area inside TD. Again, if Gaussian noise is assumed, this is a
standard Gaussian function.
PROTECTION RADIUS
The above example uses a detection threshold, measurement noise, and
measurement bias error as parameters to find PFA and PMD. Accuracy requirements are
stated in a form like "the probability of exceeding 100 meters accuracy is no greater than
0.05". In order to compare FDI results with such specifications, it helps to rearrange the
procedure. This means using the above parameters to determine the protection radius,
which is the largest horizontal position error that is guaranteed to be detected with the
required probabilities of alarm and missed detection. If all parameters are kept
constant, the protection radius will vary only as a function of satellite geometry.
One way of finding the protection radius is to use all satellites in view and go
through the full parity space/parity vector derivation. Another possibility is to take only
7O
the best 5 satellites (geometry wise) in view and use only those for the calculation. The
second method has the advantage that, since there is only one redundant measurement,
the parity vector is reduced to a scalar and the algorithm is simpler. Also, a receiver
would only be required to track five satellites.
A comparison of the two methods is shown in figures 4 and 5. These are plots of
the protection radius as satellite positions change over one day at a single location.
Many different locations were tested (with simulated satellite data) worldwide; figure 4
shows the best case result, figure 5 the worst case. As expected, the all-in-view method
always gives a better result than the best-of-five.
FAULT ISOLATION
In the case with one redundant measurement, faults can be detected but not
isolated. Isolation requires two or more redundant measurements. Consider the case of
two redundant measurements. The parity vector now has two elements, making parity
space two dimensional. Each measurement in parity space can be represented by an axis
extended radially outward from the origin, in both (positive and negative) directions.
The exact orientation of each axis depends on satellite geometry; thus each axis will
rotate over time. The basic premise of fault isolation in parity space is that a bias error
in measurement i will lie along measurement axis i in parity space. A growing bias error
will move outward along a line parallel to the measurement axis.
Figure 6 illustrates a case of flight test data consisting of 4 GPS and 3 Loran-C
measurements. There are 5 unknowns, which are the 3 user coordinates, the user clock
offset from GPS time, and the user clock offset from Loran-C time. This leaves 2
redundant measurements, so parity space is 2 dimensional and it contains 7 axes, 1 for
each measurement. The irregular plots are the traces over time of the end of the parity
vector, as a slow growing ramp error is being artificially injected into each of the
measurements. Each growing error runs parallel to the parity axis for that measurement.
Measurement 2 seems to violate this rule, but what could not be shown on one graph is
that axis 2 rotated significantly (as compared to the others) over the course of the data
run; thus, the plot for instrument 2 ends up curved as it attempts to stay parallel with
the axis as it rotates. Also note that the actual unaltered measurements contained a bias
on instrument 4, as shown by the tracks originating a small distance out on axis 4.
The simplest method for isolating the faulty instrument is the "snapshot" batch
estimator. This method, upon detecting a fault, looks for the parity axis that is closest to
the parity vector, and assumes that the measurement belonging to that axis is faulty.
The history method, on the other hand, determines which axis is parallel to the track
traced out by the parity vector over time. The history method is more accurate, but
requires more computation, has a built in delay and must restart if any measurements
are lost.
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CONCLUSIONS
A fault detection algorithm for a multisensor navigation system has been
presented. A protection radius has been calculated using the all-in-view and the best-of-
five methods, with the all-in-view proving significantly superior. Fault isolation methods
have also been shown, using both snapshot and continuous tracing methods. Efforts
continue on exploring the tradeoffs between algorithms, with the goal of determining
whether minimum navigation system requirements can be satisfied using fault detection
and isolation.
REFERENCES
1. Kline, P. A. and Van Graas, F.: Fault Detection and Isolation for Muitisensor
Navigation Systems. NASA Conference Publication 3131, Joint University Program for
Air Transportation Research 1990-1991.
2. Daly, K. C., Gai, E. G., and Harisson, J. V.: Generalized Likelihood Test for FDI in
Redundant Sensor Configurations. Journal of Guidance and Control, Vol. 2, Jan.-Feb.
1979.
3. Gai, E. G., Adams, M. B., and Walker, B. K.: Determination of Failure Thresholds in
Hybrid Navigation. IEEE Transactions on Aerospace and Electronic Systems, Vol.
AES-12, Nov. 1976.
4. Morrell, F. R., Bailey, M. L., and Motyka, P. R.: Flight Demonstration of
Redundancy Management Algorithms for a Skewed Array of Inertial Sensors.
Proceedings of the AIAA/AHS/ASEE Aircraft Design and Operations Meeting, AIAA
Paper 88-4434, Atlanta, GA, September 7-9, 1988.
5. Golub, G. H., and Van Loan, C. F.: Matrix Computations, Second Edition. The
Johns Hopkins University Press, Baltimore, MD, 1989.
6. Brenner, M.: Implementation of a RAIM Monitor in a GPS Receiver and an
Integrated GPS/IRS. Proceedings of the Third International Satellite Division Meeting
of the Institute of Navigation, September 1990.
7. Van Graas, F. and Farrell, J. L.: Receiver Autonomous Integrity Monitoring
(RAIM): Techniques, Performance and Potential. Proceedings of the 47th Annual
Meeting of the Institute of Navigation, Williamsburg, VA, June 10-12, 1991.
72
I(b I
c_ _ Correct
eu _ Missed Fault
o Detection _ Detec
'I__ Threshold
I
Detection
Threshold
Detection Statistic
Parity Space
Figure 1. Two slowly growing measurement errors plotted in parity space
versus estimation space.
73
AV
0.012
0.01
0.008
0.006
0.004
0.002
I' i
i
__l , !00 -150 -100 1-50 0 501 100 150 200
-T D T D
Parity Scalar (p)
Figure 2.
0.012
0.01
0.008
"-" 0.006
0.004
0.002
Probability density function for the parity scalar in the absence of a
measurement bias error.
i , I
i
{_I , , , "000 -150 -100 L50 0 5 ! 100 150 200
I I
"T D T D
Parity Scalar (p)
Figure 3. Probability density function for the parity scalar in the presence of a
measurement bias error.
74
1400
1200
1000
800
,_
= 600
o 400
200
........... Best-of-five
All-in-view
II
JI II I I
II It l I
i II 'l I I ,i |
J II '_1 I / I/_. t I I V I j I'qI _,ll I I l l l,,. I I, /_II tl_ It /_ "_" I I. | l I t. '_ I _ I - \ _ I ,I|A
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06 5 1'0 15 2'0
Time in hours
25
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.=_
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o
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L_
Figure 4. Best case protection radius (location: $3 W80).
1400
1200
1000
........... Best-of-five
All-in-view
t_
6001- i_! | I_I
400
200
°o _
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t_
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l
i
| l tI
, I| . q IH ;I
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10 15 2'0
Time in hours
Figure 5. Worst case protection radius (location: N36 E140).
25
75
¢q
1500
1000
5O0
0
-500
-1000 [
"150_00
Axis 3
!Axis 7
i ."
Error _
Err0r 7
Error 2
Error 5
o1000 -500 0 500
Axis1
Error 1
......Axis 5
I
1000 1500
P1
Figure 6. Parity vector trajectories as a function of time.
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Fault detection and isolation

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