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Optimal Compensation of Marine Navigation Sensor Errors
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OPTIMAL COMPENSATION OF MARINE NAVIGATION SENSOR ERRORS
J.
C. McMillan
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Abstract. A low-cost microprocessor-based Marine Integrated Navigation system employing a 14 state Kalman filter and several other innovative and unique features has been designed and developed at the Defence Research Establishment Ottawa for installation on naval and other vessels. The ability of the system to correctly identify and compensate for electromagnetic propagation anomalies and various other deterministic and stochastic errors provides for superior navigation even under severe conditions, as demonstrated during recent sea-trials. In addition to a well designed and robust Kalman filter, the system incorporates comprehensive waypointing functions, a selective multi-sensor capability, and numerous operator conveniences. This paper describes the methods developed for real time identification and compensation of various measurement errors. It also presents sea-trial results showing the effectiveness of these methods.
Keywords. Marine navigation, Integrated system, systems, Parameter estimation, Sea-trial results
INTRODUCTION
filtering,
Adaptive
The solution to this mathematical problem will be useful only if certain assumptions are satisfied. The optimality criterion itself for example involves minimising the error covariance, when it may be more important to mInImlse the maximum error. Existance of the Kalman-Bucy solution requires that various conditions be satisfied. Although different sets of sufficient conditions exist, it is generally assumed that Wand V must be independent, zero mean and uncorrelated in time, that V must have a positive definite covariance matrix, and that X and Z are zero mean and Gaussian. Stability of the solution requires several more technical assumptions as stated by Ka Iman and Bury (1961). For prac t ica 1 reasons the types of stochastic processes used when modelling system errors are generally even further restricted to a few simple one or two parameter stationary processes, such as first order Markov, white noise, or constant biases.
THE LINEAR ESTIMATION PROBLEM Generally speaking, the optimal linear estimation problem, sometimes referred to as a "filtering" problem, is formulated simply as a pair of linear equations describing the time evolution of a state vector X, and its relationship with a measurement vector Z (see for example Gelb, 1974):
F*X + G*W H*X + V
W '" N(O,Q) V 'v N(O,R)
Optimal
As stated in this manner, the problem has been thoroughly solved by Kalman and Bucy (1961) and concise, efficient and stable digital techniques are well known (Gelb, 1974; Bierman, 1977) for implementing the standard "Kalman filter" solution using a microprocessor. Applying this general knowledge to a particular problem however, can still pose some very significant design questions.
In the course of designing and implementing an optimally integrated marine navigation system at the Defence Research Establishment in Ottawa, many such problems were discovered and dealt with. We attempt here to loosely classify these problems and present our methods of dealing with them.
Where
filters,
estimate of X and its covariance. Ideally this error model, or truth model identifies all significant sources of error affecting the quantities to be determined, and represents them as components of the state vector, or of the measurement noise if they are uncorrelated. The filter attempts to estimate the value of each error in the state vector. These errors can then be effectively removed from the system to the extent that the filter can estimate them.
There has commonly been a traditional set of implicit and explicit assumptions made in the formulation and solution of optimal linear estimation problems. It has also been generally understood that for most applications some, if not all, of these assumptions are at best only approximately true. Although the theory of optimal estimation has been expanded to cover more general situations, such as certain types of nonlinearities, coloured noise, and so on, the practicing system designer still encounters problems for which the general theory offers no practical solution.
X Z
Kalman
(3)
(4)
are assumed to be vectors of zero-mean, Gaussian white noise processes. Here the state vector X is to be estimated from the measurement vector Z given the system dynamics matrices (F ,G,Q), the measurement matrices (H,R), and an initial
Furthermore, the mathematical representation of the real problem, as given by equations (1) and (2), is not unique, and there are no general
liHI
le rules for formulating th e best such mathematical statement. Choosing appropriate state and meas urement v ecto rs,
X and Z,
is critical to the
success of the system design. Therefore a thorough knowledge of the part icular problem to
i\lci\lill;111 consisting of a speed-log to measure speed through the wat e r (or relative to the earth if it is bottom tracking) and a gyrocompass to measure bearing relative to north.
be solved is a crucial factor in success, and an
Loran-C
understanding of Optimal although obviously very
giving moderately accurate (one sigma '0 200 metres) hyperbolic lines of position (LOPs) in coastal areas (1000 nmi range). When the position is found by intersecting these LOPs, the error is magnified by what is called the geomet ric di lut ion of prec is ion (GooP) , which can be quite large if the LOPs intersect at a small angle.
Estimation important,
techniqu e, is by no
means sufficient.
In
particular,
integrated
when
navigation
designing system,
an all
optimally sources
of
error must be understood, both qualitatively and quantitativel y (stochastically and det e rministically). On close inspection it becomes clear that many of these errors are not Gaussian,
or
e xponent ially
are
not
continuous,
corre lated
and
not
or
are
not
uncorre lated.
In such cases it becomes necessary to preprocess
many measurements to force them to conform as closely as possible to a stochastic model. Some of these errors are to some extent deterministic
and can therefore be corrected with sufficiently clever software. In this case it is clear that any design which fails to recognize the deterministic nature of a significant error is not tloptimal" in the true sense, even though it may be optimal in the Kalman filter sense. If an error is in fact deterministic, but due to lack of knowledge is assumed to be random, then no matter how sophisticated the estimation technique, it cannot perform as well as the determinist ic correct ion func t ion (since the error covariance would not be zero). The particular example to be dealt with in this paper is a marine navigat ion problem of having to find the best possible, or optimal estimate of position and velocity in real time, given a predetermined set of sensors (those already onboard Canadian forces vessels, or soon to be insta lled). In the course of developing this and other integrated navigation systems we have found that in fact a majority of our effort has been directed toward dealing with deterministic and quasi-deterministic errors. Although a great deal of time and effort went into the initial simulation and design of the suboptimal filter (McMillan, 1980; Liang and McMillan, 1982) in the final real-time system, the filter software is dwarfed by the many special application routines such as
for
specific
error correction,
error detection, data rejection, filter adaptation and so on. In this paper we attempt to describe,
in a systematic
way,
some of these
various nonstandard routines.
is
a
The specific problem was to obtain the optimal estimate of ships true position and velocity (in WGS-72 coordinates) given the following t ypes of navigation systems and sensors:
Speed Log (doppler) Gyrocompass Loran-C Receiver (hyperbolic) Omega Receiver (hyperbolic) Transit Receiver (single channel) Navi gator (manual position fixes) a
fairly
complete
shipboard navigation equipment,
navigation
aid,
Omega is also a fairly popular radio aid for ocean going vessels, giving less accurate hyperbolic LOPs (one sigma '0 2000 metres), but having almost global coverage. More modern Omega and Loran sets will automatically convert these LOPs into Latitude/Longitude, but for optimal integration it 1S still preferable to process the LOPs. Transit is a satellite navigation system consisting of about five satellites in polar orbits transmitting at a stable frequency. The receiver measures the doppler frequency shift giving relative velocity. Knowing the satellite position and the ships velocity (with which the Transit receiver must be supplied), the receiver then solves for the ships position, whenever a Transit satellite passes within range. This happens at irregular intervals of about 90 minutes at low latitudes and more frequently at high latitudes. The Transit pos1t10n fix accuracy depends upon the satellite maximum elevation angle the accuracy of the supplied ship velocity, and the ships latitude. Although this sensor suite was originally chosen for a conventionally operated system it is also well suited for optimal integration, since the same factors of sensor independence and complementarity are desirable in both cases. Indeed the goals of the integrated system are the same as those of a navigator, and an experienced and well trained operator would attempt to perform many of the data smoothing and combining functions that the integrated system performs automatically. Of course the computer has significant advantages in speed and precision, whereas the operator has the advantage of experience, judgement and "gut fee 1 ings". Genera lly these feelings no doubt arise from very real
navigator can look out and see approximately how fast the vessel is moving, whether it is turning etc., and can mentally dead reckon. Although not very precise, this should be a very reliable qualitative check. The operator may also receive useful information about system status in the case of station down-time for maintenance or advance warning of detectable disturbing conditions such as a PCD (polar cap disturbance) affecting certain Omega signals. An integrated system can, and should, attempt to incorporate
possible,
as much
but
set
of
and our
standard
that
is not
aim was
For
this
to build an integration unit sufficiently general so that it could be used with any subset of this equipment and with various brands and models of these sensors. Practically any major vessel will have a dead-reckoning system
judgement
cannot
navigators "gut
is
radio
information, coming from the navigator's personal "navigation sensors". For example the
THE MARINE NAVIGATION PROBLEM
This
popular
by
feelings",
directly
reason
it
and
itself which
available is
experience
reproduce stem
to
important
integrated system include capability as an integral part.
from
the
data
system.
that
operator
as
the
the
input
(lpl illl'" (:()1l1 pellsal i()1l SPECIAL SOFTWARE In this section we classify and briefly describe some of the special software features built into our Marine Integrated Navigation System (MINS). Although these features are in response to, or in anticipation of, problems specific to marine navigation, it is expected that similar types of problems commonly arise in other applicat ions, and that the techniques for dealing with them can be generalized.
transmitter. There is also a problem with reception of Omega signals near the antipodal point, where the "wrong-way path" signals compete with the shortest path signal. Loran-C has its regions to be avoided as well, such as near baseline extensions where the branches of the hyperbolic LOP fold back on themselves. These are simple situations which the navigator should be well aware of, even without being warned, but there are more subtle problems that deserve a navigation warning, such as choosing Omega station pairs (LOPs) that are not linearly independent.
Deterministic (Theoretical) Corrections A significant portion of the error in the radio-aid determined position (Loran and Omega) is due to the fact that the signals do not travel at constant speeds along geodesics. Both Loran and Omega signals are affected by the conductivity of the surface over which they propagate, and Omega in particular is affected by the height of the D region of the ionosphere and the magnetic bearing along the propagation path. These effects are largely deterministic and Can be predicted using known physical laws. The magnetic field and the earth's surface conductivity can be reasonably well modelled. Fluctuations in the ionospheric height are largely diurnal, due to solar illumination so they can also be predicted although not very precisely, especially over the non-illuminated portion. In the case of Omega these nonlinear propagation
"errors" have been stu·died in some depth by the V.S. Coast Guard (Morris and Cha, 1974) and the results are published in the form of Phase Propagation Correction tables (pPCs). For MINS we have implemented in real time the algorithm used in generating these PPC tables. In this way we can obtain even more accurate phase corrections by reducing the spatial and temporal quantization, and by interpolation. These PPCs typically remove about a half of a lane or more of LOP error, which if ignored could easily result in a position error of 5 nmi. or more, depending on the geometric dilution of precision (GDOP) • Similarily the Loran-C phase errors in Canadian coastal waters have been studied and modelled by the Canadian Hydrographic Service (Gray, 1980) and MINS also generates these in real time. These Loran corrections come in the form of Secondary Phase Factors (SPF) which correct for nonlinear " propagation over salt water, and an Additional Secondary Factor (ASF) which corrects for the effect of land-path, if there is any. Combined, these factors typically account for about one microsecond of propagation delay (Wells and Davidson, 1983) in each LOP, which t rans lates into 600 met re s or more of pos it ion error if ignored.
Deterministic Situation Detection (Warnings) There are other situations where one could use information that has a deterministic nature but an unpredictable effect. In these cases it 1S impossible to correct the resulting error, but one can issue a warning and/or ignore the affec ted measurement. For examp le wi t h re spec t to the radio aids, Loran and Omega, there are known (i.e. deterministic) areas where signal reception is unreliable or the geometry is poor. In the case of Omega for example there are near field effects such as modal interference which severly degrades reception within several hundred miles of the
As well as performing all of the above tests, MINS calculates the CDOP for each Loran and Omega LOP, and issues "bad geometry" warnings for those which exceed a certain threshold. It is not necessary to reject measurements in this case because the fi Iter wi 11 automat ically account for bad geometry by deweighting the measurement. These warnings are all intended to alert the navigator to take corrective measures, such as selecting different stations.
Quasi-Deterministic Event Detection/Correction Navigation systems are also plagued by certain events that occur randomly In spacetime, but which have a well defined effect. These can also be detected, and in some cases the adverse effect can be completely removed. Loran-C for example often suffers from what are known as cycle selection errors, or cycle hops. The Loran receiver is attempting to track the third ascending zero crossing on the leading edge of the transmitted pulse. These pulses are modulated at 100 KHz, so there are 10 microseconds between ascending zero crossings. Under certain conditions, described in some detail by Gray (1980), the leading edge of the pulse may be distorted, causing the receiver to track the wrong zero crossing and produce a measurement that is in error by IOn microseconds, where n is some small integer. A single cycle hop results in a lane shift of 3000 metres on the baseline, which is sufficiently large with respect to normal Loran errors to enable detection using a simple test. The measured time delay can then be adjusted by the appropriate multiple of 10 microseconds and passed on to the filter as a good Loran-C measurement. This test is performed on the Loran time-delay measurement of course, rather than on the resulting position fix, otherwise it would not be such a simple test. In MINS there are two phases to the Loran cycle selection test. The primary test, which will detect all cycle selection errors under normal circumstances, uses continuity to detect an error when it first occurs. If for some reason a cycle hop is not detected when it first occurs, then the only chance for detection is when a sufficiently accurate independent position fix is available, such as from Transit, at which time detection is quite certain. Great care is therefore taken to ensure proper Loran cycle selection, since Loran, being the most accurate cant inuous subsystem, has the greatest effect on the filter, and the consequence of error here is quite significant. The Omega system suffers from a similar problem. Omega stations transmit continuously and the receiver measures the phase difference between pairs of signals, leading to a lane ambiguity problem. The Omega lane count must be initialized, after which the receiver should be able to keep track of lane changes. It is
J
C. I\ldvlill,\Il
however possible for a lane slip to occur, resulting in an LOP error of about 8 nmi. or more. The Transit Satellite system also has significant deterministic errors that can be corrected. Since Transit is a doppler positioning system, any error in the ships velocity estimate given to the
Transit
receiver
will cause an error in the resulting position calculation. Moreover, the position error will be a deterministic function of the velocity error. The precise linearized relationship (a 2 x 2 matrix) has been developed at DREO, and although the relationship is not a simple one, the determining factors are available (satellite maximum elevation angle and rising quadrant, ships latitude) and it is well within the capability of the microprocessor to evaluate it. There are two aspects to knowing this relationship. Firstly, if the Transit receiver is fed raw DR velocity measurements, then the filters estimate of the DR velocity error should be used to correct the Transit position fix. Secondly, whether DR or filtered velocity is given to the Transit receiver, the position/velocity error relationship is an important part of the H matrix of equation (2), especially for high or low elevation Transit passes, where the sensitivity is high and strongly directional. Spurious Data Rejection
The
Omega
p.rrors
in
particular
are
not
very
stationary, with sudden ionospheric disturbances (SIDs) causing an almost immediate decrease in ionospheric height, resulting in a noticeable Omega phase jump, followed by a gradual return to a normal level. Polar cap disturbances (PCDs) can also cause a sudden Omega phase error, which is even larger and longer lasting. Morris and Cha (1974) give a description of these phenomenon. To help the filter respond to these unmodelled errors MINS adjusts appropriate elements of the Q matrix whenever the Omega innovation fails a residual test. Part phase
of
the
time-varying
error
covariance
nature in
fact
of
the
seems
Omega to
be
deterministic. One would expect that this error covariance should be larger during periods of rapid phase shift change, such as when the day/night terminator crosses a north-south propagation path. Figure 4.3.4 of the report by Morris and Cha (1974) seems to bear this out, but more general evidence is required. Since the Omega phase shifts (PPCs) are being evaluated by MINS, it would be a simple matter to monitor them for rapid change and adjust the appropriate elements of the Q or the R matrix accord ing ly. Another adaptive feature responds to ship manoeuvers by detecting large short term heading changes. Again the Kalman filter Q matrix is temporarily adjusted to compensate for an error model approximation.
Spurious data performance if vector.
can wreck it is used
havoc with filter to update the state
Most navigation measurements have error
distributions that are approximately Gaussian only under normal conditions. Whenever something goes wrong it is quite possible to obtain measurement errors well in excess of 10 sigma, which the filter is not expecting to ever receive. Fortunately it is usually easy to detect such measurements and discard them.
Although more extensive adaptive techniques are available, (Carew and Belanger, 1973; Chin, 1978) the relatively low level of error model uncertainly in this case does not warrant such drastic measures. It seems that the fine tuning required in this case is still best done using the ad-hoc methods described here. OPERATOR INTERACTION
In MINS we have at against
least
two lines of defence
spurious measurements
from each
sensor.
The first is a " pre-filter" which ensures that the raw measurements satisfy reasonableness tests enforcing physical limits such as a maximum speed, and limits on acceleration in the form of heading and speed continuity tests. The radio aid measurements are also subjected to loose continuity tests before being passed on to the filter. The optimal filter itself 4-sigma residual test on individually, before updat ing If the innovation is not less expected value: 2
then performs a each measurement the state vector. than 4 times its
16* (HPH
(z(t) - H*X(t l t-dt»i
T
+ R). .
1,1
( 5)
Since the navigator has ultimate responsibility and often valuable information, MINS provides the
operator
measurement
Adaptive Schemes deterministic and all of the quasi-deterministic corrections have been applied, and the spurious measurements removed,
Once
the Z vector should then conform as well as possible to the standard stochastic models used by the optimal linear filter. It is still possible however, to adjust the stochastic the
themselves
non-stationary
errors.
in
real
time,
nature
of
to
the
account
substantial
interactive
errors
are
not
corre lated
to
each
other or with the other measurement and process errors (V and W in equations (1) and (2».
then the measurement zi(t) is not filtered.
models
with
capability. The operator can "deselect" any measurement or set of measurements from the gyrocompass, the speedlog, or the seven element Z vector: three Omega LOPs, two Loran LOPs, Transit latitude and longitude. The filter will then ignore any deselected measurements, until they are reactivated by the operator. The operator can also apply an independent position fix by entering a latitude and longitude (WGS-72) along with an estimate of the accuracy of this fix. MINS will then inform the operator of the distance from the filtered position to this fix. If the operator still wishes to apply this fix MINS will integrate these two measurements into the Kalman filter, under the assumption that the latitude and longitude
for
measurement
Of course the operator is also allowed to change the Omega and Loran-C station selection. MINS also provides extensive waypoint and voyage planning capabilities and special routines for converting Omega and Loran-C LOPs into latitude and longitude, and vice versa. PERFORMANCE RESULTS As an example of the importance of implementing such prefiltering schemes, some typical sea
liX:)
Optimal Compensation trial data is presented. Figures to 3 show the ships positions according to Loran-C, Omega and the MINS filter for a 10 hour period, as recorded on June 21, 1984. The Loran-C position shown in Figure 1 are based on two LOPs from an Internav LC204 receiver set to slaves Y and Z of chain 5930, with spurious data rejected and ASF and SPF corrections applied by MINS. Similarly the Omega positions shown in Figure 2 were obtained using the MINS supplied PPCs and spurious data rejection. The Omega position was calculated from 3 LOPs (AC, AD and DF) from a The MINS filter track Magnavox MXll05 receiver. is also shown in Figure 2 for comparison. Figure 3 shows all three t racks for a shorter period to more clearly illustrate some of the details.
Chin, L. (1978). Advances in Adaptive Filtering. Control and Dynamic Systems, Advances in Theory and Applic a tions, Vol. 15. Gelb, A. (1974). Applied Optimal Estimation. the MIT Press, Cambridge, Mas s achusetts. Gray, D.H. (1980). The Preparation of Loran-C Lattices for Canadian Charts. The Canadian Surveyor, Vol. 34, No. 3. Kalman, R.E., and R. Bury (1961). New Results in Linear Filtering and Predicting. ASME Journal of Basic Enginee ring, Vol. 83D. Liang, D.F., and J.C. McMillan (1982). Development Analysis of a Marine Integrated Navigation System. IEEE PLANS 82 Record, pp. 233-245. McMillan, J.C. (1980). A Kalman Filter for Marine Navigation. M Phil. thesis Universit y of Waterloo, Waterloo, Ontario. Morris, P.B., and M.Y. Cha (1974). Omega Propagation Corrections: Background and Computational Algorithm. NTIS report No. ONSOD 01 74. Wells, D.E., and D.A. Davidson (1983). Loran-C Phase Lag Investigation, Technical Report No. 96, Dept. of Surveying Engineering, U. of New Brunswick.
From Figures and 2 it is clear that the positioning ability of these radio aids by themselves is highly unsatisfactory. In Figure 1 the Loran cycle selection errors are quite obvious and quite devastating: about 6 nmi when the typical (and modelled) Loran accuracy is about 0.1 nmi. The Omega errors seen in Figure 2 are incredibly large, up to 65 nmi, and worst of all they are not due to any abnormal signal condition. The large spikes in Figure 2 are obviously not due to a SID or a PCD, and they are not the result of a lane slip either. What has in fact happened is simply that LOP DF was briefly unavailable at these points, and the resulting 2-LOP position fix was drastically different from the 3-LOP fix, because the remaining 2 LOPs (AC and AD) were very nearly parallel, in the direction of the spikes. As seen in Figures 2 and 3, the MINS filter was not adversely affected by any of these radio aid problems. The prefilter functioned flawlessly in removing the Loran cycle selection errors. The Kalman filter did an excellent job of removing the Omega bias and, as seen in Figure 3, the Loran noise as well. The geometry problem that devastated the Omega position did not effect the MINS position at all because MINS processes the LOPs individually rather than the Omega position, and the geometry is implicitly included in the filter error model.
ACKNOWLEDGEMENTS The a uth o r wi s hes to acknowledge the cont ri buti on of Mr. C.A. Maskell t o the sea tr i a l t es tin g of MIN S a nd the post missi on data analys i s , and Mrs. F. Stil es for he r pa t ience on the wo r d proces s o r.
43 .
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2
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T
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~ .\
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1\
U
D
"' CONCLUSIONS
D E
a R
In the case of a marine integrated navigation system, using standard navigation sensors and subsystems, it was found that very serious errors quite conunonly occur, which do not conform to the standard stochastic error models used in Kalman filter design. It has been shown that once these errors are understood, a well designed prefilter can be implemented to effectively remove them. The important features of such a prefilter have been systematically described for a particular integrated system, MINS. Some sea trial results have been presented to illustrate the importance of using such a prefilter, and demonstrate the success of the MINS prefilter. REFERENCES Bierman, G.J. (1977). Factorization Methods for Discrete Sequential Estimation. Academic Press, New York. Carew, B., and P.R. Belanger (1973). Identification of Optimum Filter Steady-State Gain for Systems with Unknown Noise Covariances. IEEE Transactions on Automatic Control, Vol. Ac-18.
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43
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LONGITUDE ~
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DEGREES
Fig . 1 . Lo r a n Position .
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43.75
-
43 . 50
Omega
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-
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LONGI TUD E (degrees) Fig. 2 . Omega and MINS positions.
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LONGITU DE (degrees) Fig . 3 . Omega , Lor an and MINS positions.
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