641
1 INTRODUCTION
In modern navigation, there is a contradiction
betweentheneedforsolving probabilistic problems,
for example, the calculation of the navigation safety
indications,andthelackofthenecessaryinitialdata.
Such initial data are the parameters of navigation
parameters accuracy (NP): the root‐mean‐ square
error(RMSE)m,theexcessE,charact
erizingtheform
of the law of error distributionED, and the
correlation coefficient of latitudeerrors r (Latitude)
andthelongitudeoftheshipʹsposition(Longitude).
Therefore,thetaskofestimationtheINSaccuracy
indicators in the process of operation in the specific
operationcondit
ionsoftheINSbasedontheinternal
signalsoftheIntNSisveryrelevant.
Theseinternalsignalsarethereadingsofthreeor
twoINSthatarepartoftheIntNS.
Theestimationofindicationsduringthevoyageis
conducted with a discreteness of
t=1 or 2 h
diagrams and tables of deviations u between the
samplesy
jiNPfromtheoutputsofthreeINSandtheir
averagevaluey*
i,forexample,forINS‐1:
u
1i=y1i–y*i; (1,а)
y*
i=
3
1
(y1i+y2i+y3i), (1,b)
And similar calculations are made for the
testimonyofothertwoINS.Here,thesymbolsdenote:
j‐istheINSnumber,i‐isthesequencenumberofthe
measurementsorthetimepoint.
Figure 1 shows the diagram of the variation in
latitude production latencies, in conventional units,
and deviations of the same NP for the day of
navigation.
Contemplating the diagram, we find tha
t the
deviations as a whole reflect well the change in the
errors in time. This allows us to assume that by
evasion, it is possible to estimate correctly the
accuracyindicesofNP.
Evaluation of Navigation System Accuracy Indexes for
Deviation Reading from Average Range
A.V.Boykov
A
dmiralMakarovStateUniversityofMaritimeandInlandShipping,Moscow,Russia
V.A.Mikhalskiy&V.N.Ivantsov
TheStateResearchNavigation‐HydrographicInstitute,Saint‐Petersburg,Russia
ABSTRACT:Themethodforestimatingthemeanofsquareerror,kurtosisanderrorcorrelationcoefficientfor
deviations from the average range of three navigation parameter indications from the outputs of three
informationsensorsissubstantiatedanddeveloped.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 4
December 2017
DOI:10.12716/1001.11.04.10
642
Figure 1‐Graphs of errors (solid line) and deviations
(dashedline)latitudeofINS,conventionalunits.
Method of estimation accuracy figure. When we
estimatetheaccuracyindicesbythedeviationsofu
ji
fromtheaverageofthreereadings,У*
i=
3
1
(у1i+y2i+y3i),
y3i),forexample,forthefirstINS‐1atthei‐thtimeof
thedeviationu
1i=y1i–y*/
,
.
Theunknownerrorх1i NPy1 ofthefirstINSat
imomentoftimeequals
х
1i=u1i+y*i
Thiserrorcannotbedetermined,becauseEquation
(2) contains two unknowns, but it is possible for n
values u
1ito determine MSE mx1=my1 of va lues
Y
1[2].In accordance with the theorem on the
variance of the probability theory function [3], the
RMSEm
y1ofthesought‐forvaluey1equals:
m
2
y1=m
2
u1+m
2
y*+2rmu1my*; (3,а)
m
2
u1=
45,1
1
n
n
i
u
1
2
, (3,b)
wheren–isthenumberofdeviations;
(n–1,45)‐the number of deviations, is corrected in
accordancewiththeStudentdistributionlaw;
r=r
u,y*‐deviation correlation coefficient u average
values of NP у*, which can be found from the
relation
r
u,y*=m
2
об/mumy*,
wherem
об‐ RMSEofgeneraldeviationforuand
у*,which, taking into account (1, b), can be taken
equalstom
2
об
3
1
my*.
Wetaker=0atfirst‐orderestimatein(3,a),then
m
2
y1=m
2
u1+m
2
y*.
If we express m
2
y* in terms of (3, a) m
2
y1 as
m
2
y*
=
3
1
my1,,thenwearriveat:
m
2
y1‐0,3m
2
y1=0,67m
2
y1=m
2
u1,
andfinally
m
y1=mu1 67,0/1 =1,23mu1. (5,а).
Now it is possible to determine the correlation
coefficientrpluggedtheobtainedva luesin(4)andto
make the second approximation for the unknown
RMSE (root‐mean‐square error) m
y1 with respect to
(3,a)
my1=mu1
1/0,57
=kmu1=1,32mu1 (5,b)
Kurtosisofdeviationiscalculatedaccordingtothe
formula:
4
1
22
1
1
3
1
()
1, 45
n
in
n
i
u
E
n
u
n
(6)
where n – is the number of deviations and
45,1
1
n
n
i
u
1
2
=m
2
u‐is the root‐mean‐square value
ofdeviation.
Covarianceandcorrelationcoefficientoferrorsby
means of deviations can be computed from the
formulas:
R
lat.longl
n
1
n
ДiШi
uu
1
; (7,а)
R
lat.longl
RШД/mШmД., (7,б)
where
u
lat.iandulong.i‐areINSreadingdeviation
m
lat.иmlong.k‐theroot‐mean‐squarevalueofdeviation
computedfromtheformula(3,b);
n‐the number of deviations in the cycle of
determinationofthecorrelationcoefficient.
Thevaliditycheckingofthemethod.
To check the validity of the method computed
fromformulas(3)‐(5)and(6),theempiricalratioof
the Root‐Mean‐Square Error k*=my2/mu2 was rated
accordingtothedataoftheINSoperatingwithin8
days, indications of which wererecorded every
hour.TheresultswecanfindinTable1below,show
the value of the Root‐Mean‐Square Error m
y
computedfromtheformula(3,b)whereх
2=у2–уэт2is
taken
inplaceofu,andthevalue oftheRoot‐Mean‐
SquareErrorm
u2 ofthesecondINS,whichistheleast
precision device of the above three INS within
twenty‐fourhoursofobservation.
Where у
эт2 is calibration value of Navigation
Parameters.
Inthesametable,thevaluesofkurtosiscalculated
forerrorsanddeviationsarealsogiven.
643
Table1.TheresultsoftheanalysisofthepossibilitytoestimatetheINSaccuracyvaluesaccordingto theNPdeviations
fromthemeanvalues,nominalunits
__________________________________________________________________________________________________
Date ValueLatitudeLongitude
Error Deviation Compar.Error Deviation Compar.
__________________________________________________________________________________________________
25.10  RMSE,n.u. 0,08 0,06 1,32+ 0,17 0,12 1,42+
Kurtosis 2,92,6<1,63,1>
26.10  RMSE,n.u. 0,21 0,16 1,31+ 0,25 0,18 1,39+
Kurtosis‐0,5 0>‐0,7 0,7>
27.10  RMSE,n.u. 0,03 0,02 1,5‐ 0,08 0,09 0,89‐
Kurtosis‐0,9 ‐1,3 <‐
0,4 0,5>
28.10  RMSE,n.u. 0,06 0,05 1,2+ 0,26 0,17 1,53+
Kurtosis 0,71,5>‐0,1 ‐0,4 <
29.10  RMSE,n.u. 0,10 0,08 1,3+ 0,14 0,12 1,17+
Kurtosis‐1,0 ‐1,0 =11,5 3,5<(В)
30.10  RMSE,n.u. 0,15 0,11 1,36‐ 0,26 0,18 1,44
+
Kurtosis‐1,4 ‐1,6 <‐0,1 0,7>
31.10  RMSE,n.u. 0,14 0,11 1,27+ 0,26 0,09 2,89‐
Kurtosis‐1,5 ‐1,5 =‐0,1 ‐0,7 <
01.11  RMSE,n.u. 0,09 0,07 1,29+ 0,30 0,17 1,76‐
Kurtosis 0,20,4>1,0‐0,4 <
__________________________________________________________________________________________________
Weightedaveragevalue 0,124 0,096  K*=1,29 0,226 0,151 k*=1,49
__________________________________________________________________________________________________
The analysis data of the validity checking of the
method is based on the definition of limiting error
y^(P)ofNPinform[2]
y^(P)=K
P1E(P)Khm(Р)my=K^my (8)
where
K
P1E(P)–isthecoefficientofconversionfromtheRMS
tothelimitingerror,whichdependsonthespecified
probabilityPanderrordistributionlaw,determined
bytheupperconfidenceboundaryforthekurtosisE;
K
hm(Р)‐istheCoefficientoftheupperconfidencelimit
forRMS;
m
y–isthevalueofRMSvalueofуerror.
Note: The traditional definition of the limiting
error in the form of y^(P)=KP1(P)my, has two
disadvantages. It contemplates that the errors of all
Navigational Parameters are subjected to Gaussian
LawandRMSispreciselyknownandisnotrandom
variable.Thesehypothesesleadtomisidentificationof
the limiting errors. This attitude was proved 20 or
more years ago, before the well‐known Methodic
appeared[5].
In
thecolumn“comparison”ofTable1isgiventhe
following:
The values of the ratio k*=m
y/mu were obtained
experimentally;
Symbols «+» or «‐» mean, for latitude channel
k*
k=1,32 and for longitude channel k*
k’=1,5
andasaresultofcalculationsthelimitingerrorof
theNPwillnotbegreaterthanthepropervalue,
orviceversa;
Symbols «>», «<», «=», mean that the resulting
ratioofkurtosisoferrorsanddeviationswilllead
to the fact that the limit error of
the NP will be
greaterthanthepropervalue,orless,orequalthe
proper value. When there is a signʺ<ʺ kurtosis
Е
dev.<Еer.coefficient of limiting conversion
K^
dev.<K^er., and limiting error in the calculations
based on the deviations will be determined with
discrepancy.
Analyzing the values given in the Table we can
cometotheconclusion:
Thecoefficientsk*relatedto theweightedmean
values of the RMS m * (of errors)andm * (of
deviations),
havethefollowingvalues:
k*
lat=1,29<k=1.32;k*long=1,49>k=1,32 (9)
For the latitude channel, it is justified to use the
theoreticalcoefficientk=1.32indeterminingtheRMS
bydeviation.Forthelongitudechannel,theuseofthe
factork=1.32canleadtoseriouserrorsincalculation
oflimitingerrors.Theupperconfidenceboundfor
the
RMSratioisassignedbyFisherdistribution.
The Fisher coefficient К
FD for the confidence
probability Pdv = 0.99, of the INS error correlation
interval
К=4... 6 h and the equivalent digits of
independentmeasurementsN=40equalsК
ФЛ=1.69.
In this case, the upper confidence bound for the
coefficient k equals k
h=kКФД=2,23>k*Д=1,49. Thus, the
empiricalvaluek*=1,49canbetakentocomputefrom
theformula(5,b)insteadofthecoefficient1.32.
So, the calculating formulas for the RMS of the
INSnavigationparametersjarethefollowing:
1 Inlatitudem
j=1,32muj,inlongitudeandthecourse
m
j=1,49muj.
2 ThelimitingerroroftheNPmaybefewerthanit
should be, due to out‐of‐true estimation of the
kurtosis, in 44% of cases in the latitude channel
and in 50% in the longitude channel. The
maximum understatement of the limiting error
maybe1%inthe
latitudechanneland48%inthe
longitude channel. The probability of such a
significantunderstatementequals1/16=0.06.
These results point us towards the following
conclusion:
IntheprocessofIntNSoperationitisadvisableto
monitor systematically the RMS values of
coordinates and the courseusing the
deviationsoftheNPfromtheaverageva luesand
comparethemwiththepriorivaluesoftheRMS.
Asaresult,thelargervalue becomesanewpriori
one.IntheabsenceofapriorivaluesoftheRMS,
theirdefinitionbydeviationbecomesmandatory.
644
Itisexpedienttodeterminekurtosisofcoordinates
and course errors on the long time intervals in
caseswhentherearenopriorivaluesofkurtosis.
As the upper confidence bounds to use the
maximumvaluesofthekurtosis’sobtainedwithin
theintervalsoftwenty‐fourorseventy‐two
hours.
2 CONCLUSIONS.
1 Themethodtoestimatethemainaccuracyfigures
of the INS based on the internal signals of the
IntNS on the deviations of the INS indications
from the mean valuesof three was substantiated
anddeveloped.
2 The validity of the method was verified by the
experimentaldata.
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engineeringapplications.‐M.:Science,1988.‐480с.
[4]KondrashikhinV.T. The theory of errors and
its
application to navigational problems.‐М.: Transport,
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