485

1 INTRODUCTION

Satellite positioning systems are indispensable in

carrying out basic marine navigation tasks.

Navigationisunderstoodasaprocessofconducting 

a vehicle from one point to another in a safe and

effective way in the appropriate physical and

geographical environment (Kopacz, Urbański, 1998;

Czaplewski, Morgaś, Kopacz, 2010; C

zaplewski,

2014). However, these systems are not enough to

solve the problem of accessibility to reliable and

highly accurate information about an object’s

position,inparticularinrelationtonavigationmarks

that are located on land and at sea. This is

particularly important when performing many

navigationtasks as part of special works as well as

submarine navigation tasks. Furthermore, it is not

alwayspossibletopreciselydeterminethepositionof

objects other tha

n vessels when carrying out

hydrographic works by using satellite technology. 

Forexample,itisverydifficulttopreciselyposition

navigationmarksatsea.Itisalsonoteasytolocat

e

thepositionofnavigationalhazards,suchaswrecks,

underwater and above‐water rocksas well asother

naturalandartificialobstructionstonavigation.This

isbecauseitisimpossibleformeasurementteamsor

hydrographicsurveyvesselstodirectlyapproachthe

objects whose positions are being determined.

Additionally,itisincrediblyimport

antfortheproper

performanceoftasksatseathattherelativepositions

of vessels be precisely maintained. This is

particularlyimportantfor:

 hydrographic survey vessels during  bathymetric

measurementsandhydrographicsweepsurveys;

 research vessels, for example, during seabed

exploration;

 warships during the search for submarines,

minesweeping, or the performa

nce of common

tasksinvolvingtheuseofartilleryandrockets.

Development of the IANS

Chain Using a Satellite

System

K.Czaplewski

GdyniaMaritimeUniversity,Gdynia,Poland

ABSTRACT:Thispa

perpresentsideasforcreatinganddevelopingInteractiveNavigationalStructures(IANS)

byusingsatellitesystems.Thepaperpresentsthepossibilityofadaptingthecontemporarymethodsofrobust

estimation that are used in geodesy for the purpose of performing selected marine navigation tasks. IANS

utilisemodern M‐estimation methods.Interactivestructures canassisthumanbeingsincarrying outspecial

sks at sea, for example, in identifying objects at sea and on land without the need to approach them.

Moreover,thesestructurescanalsobeusedintheprocessofnavigationcarriedoutbyunderwatervehiclesor

unmannedwatercraft.Thispaperpresentsthem

athematicalmethodsthatareessentialforcreatingIANS.The

theoreticalassumptionsareillustratedwithanexampleofhowIANScanbeusedwhileperformingatypical

navigationtask.Thepapercloseswithinformationaboutresearchstudieswhichdealwiththepracticalaspects

ofusingthissetofmathematicalmethods.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 8

Number 4

December 2014

DOI:10.12716/1001.08.04.02

486

The above‐mentioned issues and other similar

problems are of vital significance  for maritime

economy and national defence. In order to solve

these problems one must not only use satellite

navigation systems but also terrestrial navigation

(geo‐navigation).

If geo‐navigation methods are to be used today,

theymustbe

developedin terms of techniques and

technology as well as the methods of analysing

measurement results. Currently, classical geo‐

navigation uses procedures that do not meet

accuracy standards. Geo‐navigation should refer to

thewell‐establishedmethodsofmoderngeodesythat

arestillbeingdevelopedinordertobeable

tomeet

these standards. Additionally, it is necessary to

automatecalculationsbecausethiswillallowoneto

implement them in integrated navigation systems.

Such implementation should make it possible to

simultaneouslyusedifferentnavigationsystems(so‐

called combined systems) while employing new

positioning methods. As for data analysis, one

should use

 robust M‐estimation methods whichare

newbutwell‐establishedtheoreticallyandwhichare

currentlybeingdeveloped.Thisapproachtotheissue

ofdeterminingthepositionsofa movingobjectatsea

andastationaryobjectthatisbeingobservedmakes

itpossibletocreatedynamic, interactivenavigational

structures, which

are dealt with in this paper. The

term ‘Interactive Navigational Structures’ (IANS)

(Czaplewski, 2004) should be understood as the

existing systems of navigation ma rks, the other

stationary objects that are observed and craft

carrying out navigation tasks (also submarine

navigation), which allow one to establish the

coordinates of the remaining

 elements of the

structure due to the (geometrical and physical)

interdependencies between them. The fact that the

structureisinteractivemeansthatitcanbeconstantly

alteredbythenavigator(observer).

2 BASICASSUMPTIONS

From a geometric perspective, the Interactive

NavigationalStructureismadeupofasetofpoints















njZ ,...,1:Z

 withknowncoordinates(e.g.



YX ,

) in a known coordinate system as well as

subsets of points that are being determined, i.e.















niP ,...,1:P

 and















nlR ,...,1:R

.

Sets

 canbemadeupofvisualnavigationmarks,

elements of radar navigation systems, radio‐

navigation stations and DGPS (Differential Global

Positioning System) reference stations. Set



consists of a vessel’s positions

 at sea that are

being determined or the positions of groups of

vesselsthatcarryoutaparticulartasktogether(e.g.a

hydrographicsweepsurvey).Subset

 ismadeup

ofpointswhosecoordinatesaredeterminedbasedon

the positions that belong to set

, which may

complementset

 undercertainconditions.

Letusassumethat,atstage



k ,thereisagiven

set





Z  whichisinsufficienttodetermineavessel’s

position at stage







k . Based on a set of points





Z , a vessel’s positions



 are determined, and

thecoordinatesofpoints



 aredeterminedbased

on these positions. Then, a set of points

   















 kkk

RZZ ,

 is available at stage







k ;

thissetcanbesupplementedwithotherelementsof

navigation systems. As for the adopted accuracy

criteria, the IANS chains can take different forms.

GeneralrelationswithinIANSareshowninFigure1.



Figure1. Concept of the Interactive Navigational

Structure’schain(ownwork).

Thesetspresentedabovehavebeenconnectedto

form one observational arrangement, within which

geometricquantitiesaremeasured.

Let us assume that a vessel’s position

 was

determinedbyusingthebearingsanddistancesfrom

point

 totheelementsofset Z ;atthesametime

its position was established by using the DGPS.

Moreover, the bearing and distance between a

vessel’s position and point R belonging to set



were measured. For such a measurement

arrangement, we can create IANS’s basic element

(Fig.2).



Figure2.IANS’sbasicelement(ownwork).

487

It is assumed that the interactive character of

navigationalstructuresmakes it possible to transfer

the elements of set

 to set Z , which requires

creatingaspatialarrangementthatwouldallowone

to control the determinations. The element of IANS

that is presented in Figure 2 does not meet these

requirements. The basic element of the Interactive

Navigational Structure does not ensure reliable

assessmentof theaccuracyoftheposition of points



whosecoordinatesarebeingdetermined,andsucha

structureisnotveryrobusttogrosserrors.Inorder

tosolvetheseproblems,theelementsofIANScanbe

connectedtoform IANS chains. Figure 3 presents a

fragmentofsuchachain.



Figure3.FragmentofanIANSchain(ownwork).

Thenavigationalstructure that is presented here

is not only interactive, but also integrative in

character.Thisisbecauseitcombinesdifferentkinds

of observations from various navigation systems. It

should be assumed that any kinds of connections

between the elements of sets

Z ,,  will be used

when putting IANS into practice. It is also possible

thatobservationswillbefraughtwithsuchlevelsof

error that they will haveto be rejected orthey will

influencethefinal determinationstoalesser extent.

The methodology for establishing an adjustment

problemforeachof

thestagesofdevelopingIANSas

wellasforusingattenuationtoreducetheinfluence

of gross errors on the final determination is

presentedlaterinthepaper.

3 ADJUSTMENTPROBLEMFORTHEBASIC

ELEMENTOFIANSANDTHESOLUTION

By using functional models which are typical of

geodesy (e.g. Wi

śniewski, 2009; Baran, 1999) and

whichhavebeenadoptedforthepurposeofcreating

InteractiveNavigationalStructures,onecanpresenta

robust and decision adjustment problem (1) for the

basicelementofIANS:

  



iiP Ri

DGPS

Pii

iii ii

DGPS DGPS DGPS

DGPS DGPS

PPP

iii

221 21

00 0

221

ˆˆ

functional models

statistic models

min

XXX

VAd Ad L

Vd XX

CQP CP

CQP

 













 













 























xxx xx

 



 



 



DGPS DGPS DGPS

PR PR iii

ii ii

PP P

ii i

ii ii

DR DR T T

ˆˆ

Φ,Φ,

XX XX

XX X

dd dd VPVV P V

























xx x

(1)

As part of the above adjustment problem, these

equivalentcovariancematriceswerepresentedinthe

statistical model:









 and





DGPS





,where:

iii

i xxx

)PV(xTP ,







DGPS

XXX

)PV(XTP ,





 (2)

are equivalent weight matrices that have been

determined based on decision and attenuation

matrices:

))T(V(x)V(xT

T,



))T(V(X)V(XT

DGPS

T,

 (3)

where:













υtυtυtDiag

,...,,



)T(V

 – attenuation

function







)(x)(x)(x

RZi

Diag TTT ,

 –decisionfunction

and





















niiniiZ

ddNNDiag

,1,,1,

,...,,,..., ttttT



)(x



























)(

,...,,,...,

RRi

dRdRNRNRDiag ttttT )(x



Detailedrulesforcreatingdecisionfunctionsand

attenuation functions are described, for example, in

the papers written by Yang (1994), Yang, Song and

Xu (2002), and Wiśniewski (2002), whereas the

methodologyforcreatingadecisionandattenuation

function (3) is described in detail in Czaplewski’s

paper(2004).Inorder

tosimplifyexpression(1),the

followingdesignationscanbeintroduced:

DGPS



















,





















,

DGPS





















,

















,















DGPS

Diag

CCC



,















DGPS

Diag

PPP





488

Thus,thefinaladjustmentproblemisexpressedin

thefollowing,classicalform:



ii i

ioi

DR DR T

iii

minΦ minΦ

VAd L

ddVPV

































 (4)

and the solution is as follows (Baran, 1999;

Wiśniewski,2009):





iii iii

dAPAAPL









 (5)

Therefore, the matrix of adjusted coordinates

takesthefollowingform:

 

ˆˆ

XXd





   





 (6)

with an estimator of the covariance matrix (Yang,

1997):





iii

2T 2 21

ˆˆˆ

oiii o o

XXX

CAPAQP











 (7)

The process of solving the problem which is

expressed as formula (4) has an iterative character.

Thefirst(starting)stepshouldbeimplementedina

classical way, i.e. by using the method of least

squares.

4 DEVELOPINGTHEIANSCHAIN

AsingleelementofIANSonlyallowsoneto

adjusta

given vessel’s position

. Points belonging to



can only be determined without supernumerary

observationsatthisstage.ThisiswhytheIANSchain

shouldbedeveloped.Forthepurposeofpresenting

the development of the Interactive Navigational

Structure’s chain, let us assume a navigational

situation in which a vessel has moved to position

1i

.Thevessel’snewpositionismeasuredrelative

to navigation marks that belong to set

ZZ 

1i

,

measurementsusingtheDGPS toset

ZZ 



DGPS

i 1

 by

using the DGPS, relative to the previously

determinedpoints

RR 

 andrelativetonew(for

point

1i

) points

RR 

1i

. Let us also assume

thattheelementsofthevessel’spathvector(i.e.the

course and the distance travelled) are known. The

navigation arrangement which is analysed in this

paperispresentedinthefigurebelow.



Figure4.ThestageofdevelopingIANSatmoment





Since we assume that also DGPS measurements

aremadeatposition

1i

 (withtheaimofobtaining

coordinates

DGPS

1

 with covariance matrix

DGPS







),wewillwritethefollowing:



 



DGPS

i1 i1

DGPS

Pi1i1

i1 i1 P

i1 i1

ˆˆ

VXX

VdXX

XXd































 (8)

The coordinates

 of position

 have

already been adjusted and they are represented by

estimator





 with covariance matrix

  





XXX

PQC



 inthepresentIANSchain.

However,positions

 and

1i

 are linked to each

otherthroughtheelementsofavessel’spathvector

aswellasindirectly,i.e.throughpointsbelongingto

 which are common to both elements of IANS.

Given the above, estimator



 is not the final

vectorofthecoordinatesofthepreviousposition

.

Let us assume that vector



i

 is the final one.

Then, if we treat the previous vector as a pseudo‐

observation, according to the general principles of

sequentialadjustment,wecanwritethefollowing:

 

  

 

iP iP

i1 i

ii1

ˆˆ

XXV

Vd d

Xd Xd V









 



 





 (9)

When navigating a single object, coordinates





i

 donothavemuchpracticalimportanceforthe

currentposition

1i

 (althoughbasedonthe values

of the vector of adjustments

 one can draw

additional conclusions about the accuracy of

navigation).Theadjustedcoordinatesoftheprevious

position are the result of a joint analysis of all

observations that were available for the current

situation

1i

.Observationsmustbeanalysedjointly

so as to obtain the best possible determinations of

currentcoordinatesthatarerepresentedby





i

.

Vector





i

 of the coordinates of points



which was obtained at position

 will also be

treated as a pseudo‐observation. If we assume that

vector





i

 is a joint representation of the

489

coordinates of position

1i

, we can write the

following:

 

iR iR

i1 i

ˆˆ

XXV

Vd d

Xd Xd V



















(10)

By using the determinations related to the goal

function, which were presented in Czaplewski’s

paper(2004),aswellasthefunctionalandstatistical

modelsthatwereformulatedearlier,onecanpropose

the following robust and decision adjustment

problem:

   



 

i1 i1 P i1 R i R i P i1

i1 i1 i i

DGPS

P i1 i1

i1 i1 i1

i1 i1 i1 i1 i1 i1

PR R P

i1 i

ˆˆ ˆˆ

ˆˆ

XX XX

VAd Ad Ad Ad L

Vd XX

Vdd

CP C



  





 

   





   













x x

xx x

   



DGPS DGPS DGPS DGPS

i1 i1 i1 i1

ii ii

PP PP

ii ii

RR RR

ii ii

i1 i1

21 21

ˆˆ ˆˆ

21 21

ˆˆ ˆˆ

DR DR

min Φ Φ

XX XX

CP CP







 

































   

i1 DGPS i1 i i1 i i1

DR DR DR DR

ˆˆˆˆ

ΦΦΦΦ

XXXX

dddd



















(11)

with equivalent covariance matrices



 PC





which have  replaced the original ma

trices



 PC



.Whenweintroducethesedenotations:

















DGPS

,

 























000

iiii

RRP

AAAA

,























DGPS

 

















DGPS

Diag

XXX

CCCCC

ˆˆ



,

 

















DGPS

Diag

XXX

PPPPP

ˆˆ





wecanwriteproblem(11)inthefollowingform:



i1 i1

i1 i1 X i1

i1 o i1

DR DR

min Φ Φ

VAd L









 















(12)

Thisestimatoristhesolutiontotheproblem:





i1 i1 i1 i1 i1 i1

dAPAAPL







 







(13)

Moreover,





ˆˆ

oi1i1i1

ˆˆ

CC APA







 







(14)

and

oi1i1i1

i+1

VPV













(15)

Thenthevectorofalloftheadjustedcoordinates

takesthefollowingform:



i1 i1

ˆˆ

XXd





































  



























(16)

5 NUMERICALTEST

Inordertoillustratetheabovetheoreticalanalyses,a

numerical test will be presented. This test will

demonstrate the possibility of utilising IANS while

sailingalongthecoastbyusinggeo‐navigation. Let

usassumethatavesselismanoeuvringinthewaters

oftheSzczecinLagoonandthenavigatordetermines

the vessel’s position by using the DGPS and

observing navigation ma

rks on shore. The way in

which the vessel manoeuvres is presented in

Figure5.

490

Figure5. The vessel’s manoeuvring path in the waters of

theSzczecin Lagoon (ownwork prepared by using Navi‐

Sailor4000ECS).

During the test, the vessel’s positions were

determined by using the DGPS



, PP  and they

were also calculated analytically by utilising

observationsofnavigationmarksonland.

Table1.Vessel’spositionsatseaduringthetest.

_______________________________________________

Vessel’s GeographicGauss‐Krüger

positions coordinatescoordinates

[]  []X[m]  Y[m]

_______________________________________________

P 5349.56’N01428.14’E  5966784.67465033.79

P  5346.67’N01427.47’E 5961425.53464255.31

 5345.33’N01423.13’E 5958982.34459467.49

P  5347.40’N01421.60’E 5962837.07457820.13

_______________________________________________

Thevisua

lnavigationmarksthatwereusedinthe

testarepresentedinTable2.

Table 2. Selected navigation marks in the waters of the

SzczecinLagoon.

_______________________________________________

Fixednavigation Geographiccoordinates[][]

marksGauss‐Krügercoordinates X[m]Y[m]

_______________________________________________

ChurchinWarnołęka



S 5341.8’N 01423.10’E

5952434.41459377.85

Churc

hinNoweWarpno



S  5343.50’N01423.10’E

5955587.98459405.13

LeftlightatFairway 5348.40’N01420.30’E

Gateno.1



5964704.43456409.44

_______________________________________________

Table 3 presents the navigational observations

tha

t were made by observing the navigation marks

listedinTable2.

Table3.Opticalbearingstothelandmarks.

_______________________________________________

Vessel’s LandmarksTruebearings[]

positions

_______________________________________________

  ChurchinWarnołęka





S 181.0

ChurchinNoweWarpno



S  245.0

LeftlightatFairwayGateno.1



 332.0

P   ChurchinWarnołęka





S 171.5

ChurchinNoweWarpno



S  257.5

LeftlightatFairwayGateno.1



 323.0

_______________________________________________

Aspartofthistest,thenavigatorhadtodetermine

the coordinates of object T (Fig. 5) apart from

determining the vessel’s positions



. With this

endinview,theopticalbearingstotheobjectwhich

was being observed were determined, one for each

position.Table4presentsthevaluesofthebearings

thatweremeasured.

Table4.TruebearingstomarkT.

_______________________________________________

Vessel’sposition  Truebearing[]

_______________________________________________

P 226.2

P 300.1

028.0



P 084.5

_______________________________________________

The calculat

ions are presented below. The

navigator determined the vessel’s position at P1 as

well as the first bearing  to object T. Then, after

moving to position P2 and changing course, the

navigatordeterminedthevessel’spositionaswellas

another bearing to object T. Next,  the navigator

determinedtheapproximatecoordinatesof objectT

(Table 5) by using the coordinates of the vessel’s

positions(P1andP2)aswellastheobservationstha

t

hadbeenmaderelativetotheobject.

Table5. Approximate coordinates  of object T that were

determinedatpositionP2.

_______________________________________________

Geographiccoordinatesystem=5347.6’N

 =01424.7’E

RectangularcoordinatesystemX=5963178.25

Y=461228.31

_______________________________________________

Thevesselwasoncourse243whenmovingfrom

position P2 to position P3. The distance t

ravelled

betweenthesepositionswas2.9nauticalmiles(5375

m). At position P3 the navigator took bearings to

three navigation marks (Table 3) and to object T

(Table4).Therefore,itbecamepossibletodetermine

the vessel’s position by using the observations tha

t

had been made as well as the first estimate of the

coordinatesofobjectT.

Forthepurposesofthetestthatisdescribedhere,

systems of observation equations are formulated

accordingtothefollowingmodel:







XFXXF

jTPj

TPTT

TPjjjj

TPjiji

jjj

FNRNR

Fdd

FNRKR

FNRNR

















,1,1

where:

 –i‐thbearingtothe n‐th navigation mark at

position

,1

 – vessel’s course between positions

1j

and

,1

 –distancebetweenpositions

1j

 and

 –bearingtoobjectTatposition

491















– coordinates of the vessel at position

„j”















– coordinatesofobjectT

Later in the paper only the determinations of

adjustmentstotheobservationsthatweremadewill

bepresented,aswellastheestimatedcoordinatesof

thevessel’sandobjectT’spositions.

After formulating the system of observation

equations and the adjustment problem, one should

establish, according to the principles of robust

adjust

ment that were described in the first part of

thispaper,whethertheobservationsthatweremade

are fraught with gross errors, which may influence

theresultsoftheadjustment.Inordertodothis,an

adjustmentcovariancematrixisdetermined:



(17)

where:





VPV



– coefficientofvariance

iii

AAPAAPQ



















– cofactor ma trix of

thevectorofadjustmentsV

Moreover, the acceptable range of standardised

adjustmentsisdeterminedbasedontheformula:

ivv

vk;k





(18)

Standardised adjustments make it possible to

determine the value of the attenuation function

whichisanelementofthedecisionandattenuation

matrix described by formula (3). The attenuation

function serves the purpose of obtaining an

equivalentweightmatrixfromformula(2).Therange

of acceptable adjustments depends on the adop

ted

confidence level



. Given that adjustments (

)

assume random values, but in accordance with

normaldistribution,wecanwritethefollowing:



vi v i

Pk v k Pkv k

exp dv

 





  











(19)

where:



C



– standard deviation for the i‐th

adjustment





– standardised adjustment within the

acceptablerange

kkv ,

In navigational pra

ctice, it is assumed that

coefficient



 forconfidencelevel 95,0





.

In this numerical test, analyses were limited to

attenuation function











υtυtυtDiag

,...,,

)T(V

,

whereas decision function









)(x)(x)(x

RZi

Diag TTT ,

was omitted. For the

purposeof navigation tasks, the Danishattenuation

functionisusedmostoften;itisdescribedindetailin

papers written by Hampel et al.(1986),Wiśniewski

(2009),andCzaplewski(2004). Thisfunction canbe

expressedastheformula:





vk,k

for

exp l v k v k, k

for











 





(20)

Figure6.Danishattenuationfunction.

The attenuation function is used to determine

equivalentweights,inaccordancewiththefollowing

formula:





iii

vk,k

for

ptvp

exp l v k p v k, k

for











 







(21)

The values of parameters l and g are selected

experimentallysothatthenumberofiterationsisnot

too large. It is usually assumed that parameters

01.0



 and 2



g . In accordance with

assumptions(2)and(3),anequivalentweightmatrix

isconstructed:







iii

tv p 0 0

0tvp 0

00 tvp

xxx

P T(V )P









































 (22)

After adopting the range of standardised

adjustments

2;2

 and confidence level

%95





, the following values of standardised

adjustments were obtained at the initial

(identification)stage:

492

v 6.8136 v



 ,

v 41.5704 v



 ,

 8955.0



,

v 17.3476 v



 ,

 0979.25



,

 4231.15



,

Onecaneasilynoticethatalloftheobservations,

except for the third bearing, are fraught with gross

error, which results from the assumed range of

standardisedadjustments



v .Therefore,weused

the procedure of robust adjustment, which is

iterative. In order to simplify the presentation of

results, the tables below show the results of robust

adjustmentforposition

.

Forthe purposeof theexample thatisdescribed

here, the bearing to the church in Nowe Warpno



S  wasfraughtwithgrosserror.Thisisconfirmed

by the results presented in Tables 6 and 7. The

bearingwhichwasadoptedforthecalculationsand

which was fraught with error was 245.0, and it

should be 247.0. The robust adjustment process

allowedustoobtain:

1 Estimated

coordinatesofthevessel’sposition:













































11.459463

21.5958991

38.4

87.8

49.459467

34.5958982

ˆˆ

333

X xdX

withanerrorindeterminingtheposition

9.43



.

2 EstimatedcoordinatesofobjectT’sposition:













































43.461228

20.5963178

12.0

05.0

31.461228

25.5963178

ˆˆ

X xdX

withanerrorindeterminingtheposition

3.17



.

The vessel was on course



2.337KR

 when

moving from position

 to position

P . The

distance between these positions was 2.26 nautical

miles(4182m).At position

P  the navigator made

observations relative to the same navigation marks

thatwereusedatposition

 (Table3)andtookthe

fourth bearing to object T (Table 4). Like at the

previous position, it is possible to determine the

coordinatesofthevessel’spositionandtoonceagain

adjustthecoordinatesofobjectT’sposition.Asatthe

previous stage, we will now start identifying

observations

thatarefraughtwith gross errors. The

acceptedrangeofstandardisedadjustmentsandthe

confidence level did not change. The values of the

standardised adjustments to observations made at

position

P  areasfollows:

 11278.0



,

 2361.0



,

 0235.0



,

 0985.0



,

 4352.0



,

 4467.0



,

Table6.Valuesoftheattenuationfunctionforparticularobservationsatposition

.

__________________________________________________________________________________________________

Iteration ParametersoftheValuesoftheattenuationfunctionforparticularobservations

Step  attenuationfunction

l  g



vt 





vt 











vt 











__________________________________________________________________________________________________

10.003 2.010.0091 1111

20.03 2.010.8186 1111

30.12.010.6305 1111

40.82.0111111

__________________________________________________________________________________________________

Table7.Valuesofthestandardisedadjustmentsforobservationsmadeatposition

.

__________________________________________________________________________________________________

Iteration ParametersoftheValuesofthestandardisedadjustmentsforparticularobservations

Step  attenuationfunction

l  g

 

_

__________________________________________________________________________________________________

10.003 2.00.4380 4.5830 0.2288 0.0026 0.3165 0.2425

20.03 2.00.4236 4.1477 0.2313 0.0452 0.2474 0.2061

30.12.00.3996 3.2949 0.2355 0.1163 0.1319 0.1453

40.82.00.3694 1.6858 0.2407 0.2063 0.0144 0.0686

__________________________________________________________________________________________________



493

As one can see above, all of the adjustments are

within the accepted range

2;2

. Therefore,

it is not necessary to use the procedure of robust

adjustment and decision and attenuation function

)V(xT

.Now,thecoordinatesofposition

P  and

of object T’s position can be determined. The

determinations of adjustments at the last stage of

calculationsinthistestareasfollows:

1 Estimatedcoordinatesofthevessel’sposition:











































03.457827

45.5962838

90.6

39.1

13.457820

06.5962837

ˆˆ

444

X xdX

withanerrorindeterminingtheposition

98.24



.

2 EstimatedcoordinatesofobjectT’sposition:













































38.461228

12.5963180

05.0

92.1

43.461228

20.5963178

ˆˆ

X xdX

withanerrorindeterminingtheposition

72.16



.

6 CONCLUSIONS

Themathematicalmethodsthatare proposedinthis

paper allow one to create Interactive Navigational

Structures(IANS),whichcanbehelpfulinperforming

many typical navigational tasks. IANS can be an

invaluable tool that aids in determining the

coordinatesofawreck’s position when the depth of



watermakesitimpossibleformeasurementteamsto

come close to it (Czaplewski, 2004). Moreover, the

methodology forcreatingand developing IANS was

successfully used in radar navigation as a tool that

made the determinations carried out by artificial

neuralnetworksmoreprecise(CzaplewskiandWąż,

2004).

The development

of IANS can act as a

supplementary, analytical method of determining a

vessel’spositionatsea.Letusassumethatavesselis

movingthroughwaterswherethetypicalnavigation

marks are unavailable, but where one can observe

other stationary objects. At the first stage, the

navigator determines the vessel’s positions

 by using

theavailablesystemsofnavigationmarksandatthe

same time makes observations relative to stationary

objects. Observations that are made in accordance

with the principles that are described in this paper

will allow one to determine the coordinates of such

objects and then further use them in

navigation as

alternative navigation marks. It should be

remembered that the quality of determinations will

not be as high as when the coordinates of typical

navigationmarksareused.However,sinceoneneeds

tohavecontinuousinformationaboutthepositionof

one’s vessel when there are no other possibilities of

determining

 this position, the proposal presented in

thispaperisaproperalternative.

Currently, the author and his research team are

carrying out studies that focus on the use of

InteractiveNavigationalStructuresintheVTS(Vessel

Traffic Service) system. A vessel traffic controller

usingtheVTSsystemcaneasilyspecify

theposition

ofavesselwhichhasstateditspositionwhilebeingin

watersthatarecoveredbythissystemifthecontroller

has the appropriate software which uses the

mathematical methods of creating and developing

IANS.

REFERENCES

Baran, W. L. (1999). Teoretyczne podstawy opracowania

wyników pomiarów geodezyjnych. PWN, Warsaw,

Poland.

Czaplewski, K. (2004). Pozycjonowanie z wykorzystaniem

Interaktywnych Struktur Nawigacyjnych.

WydawnictwoAMW,Gdynia,Poland.

Czaplewski K.Morgaś W. and KopaczZ. (2010). National

Systems of Safety and Protection of Navigation.

Scientific Journal of Maritime University of Szczecin,

Szczecin,Poland.



CzaplewskiK.andWążM.(2004).TheUsingoftheNeutral

Networks and Robust Estimation Methods In Radar

Navigation. The Proceedings of 5th Russian Scientific‐

Technical Conference “The Present‐Day State and

Problems of Navigation and Oceanography”, pp. 105‐

115,St.Petersburg,Russia.

Hampel F. R., Ronchetti E. M., Rousseeuw

P.J and Stahel

W.A.(1986). Robust Statistics.The Approach Basedon

Influence Functions. John Wiley & Sons, New York,

USA.

Kopacz Z. and Urbański J. (1998). The Navigation of the

Beginning of the 21st Century. Geodezja i Kartografia,

no.XLVII,issues1‐2,pp.59‐68,Warsaw,Poland.

Wiśniewski

 Z. (2002). Koncepcja opracowania wyników

pomiarów nawigacyjnych. Wydawnictwo AMW,

Gdynia,Poland.

WiśniewskiZ.(2009).RachunekWyrównawczywGeodezji

(zprzykładami).WydawnictwoUWM,Olsztyn,Poland.

Yang Y.  (1994). Robust Estimation for Dependent

Observations. Manuscripta Geodaetica, no. 19,

Germany.

YANG Y. (1997). Estimators of Covariance Matrix AT

Robust Estimation Based

on Influence Functions. ZfV,

Helf4,Germany

Yang Y., Song L. and Xu T. (2002). Robust Estimator for

Correlated Observations Based on Bifactor Equivalent

Weights. Journal of Geodesy nr 76, pp 353 – 358,

Heidelberg,Germany

