895

1 INTRODUCTION

The satellite system era started in the mid‐1980s,

whenthe American NAVSTAR GPS system became

fully operative. Currently, there are 11 satellite

systems in different parts of the world, with more

under construction, which have had an increasing

impact on the global economy [Czaplewski 2015].

Satellite systems

 have become a widely recognised

andusedtoolinmanytypesofnavigation.Maritime

navigation mainly employs systems for

determination of vessel coordinates, but also for

ensuringsailingsafetyateverystageofaseajourney

[Urbanskietal.2008].Asinsailing,themainjobthat

satellitesystemsdo

inairnavigationistopositionan

aircraft, especially during its take‐off and landing

[Bialyetal.2011].Satellitesystemsareusedinland

navigationwithpopularSatNavreceivers,whichare

sopopularthatitisnotoftenrealisedthatnavigation 

signalscomefromsatellitesystems.Railvehiclesare

another

areainwhichlandnavigationisused.Here,

satellitesystemsareusedinstock‐taking,diagnostics

anddesignworkinrailways.Thegreatestadvantage

of mobile satellite measurements is their ability to

perform measurements in a unified and coherent

system of spatial coordinates [Specht, Koc 2016;

Spechtetal.2019].

Satellitesystemsarealsousedin

otherareasofresearchandpracticalapplications,e.g.

in geodesy. Apart from conventional methods of

determining coordinates for a  point on the globe,

Use of a Least Squares with Conditional Equations

Method in Positioning a Tramway Track in the Gdansk

Agglomeration

K.Czaplewski,C.Specht,P.Dąbrowski&M.Specht

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

Z.Wiśniewski

UniversityofWarmiaandMazury,Olsztyn,Poland

W.Koc,A.Wilk,K.Karwowski,P.Chrostowski&J.Szmagliński

GdańskUniversityofTechnology,Gdańsk,Poland

ABSTRACT: Satellite measurement techniques have been used for many years in different types of human

activity,includingworkrelatedtostakingoutandmakinguseofrailinfra structure.Firstandforemost,satellite

techniquesareappliedtodeterminethetramwaytrackcourseandto

analysethechangesofitspositionduring

itsoperation.

This paper proposes using the least squares with conditional equations method, known in geodesy (LSce).

Whenapplied,thismethodwillallowforimprovementofthefinaldeterminationaccuracy.Thispaperpresents

asimplifiedsolutiontotheLScealignmentproblem.Thesimplification

involvesreplacementoftheparameter

binding equations with equivalent observational equations with properly selected weights.The results

obtained with such a solution were demonstrated with a randomly selected section of a tramway track in

Gdańsk.

Thearticlepresentsthetheoreticalfoundationsofthetestmethod,theexperimentorganisationandthe

results

obtained with MathCad Prime 3.0 software. It also presents the outcome of a study associated with the

executionoftheprojectNoPOIR.04.01.01‐00‐0017/17entitled“Developinganinnovativemethodofprecision

determinationofarailvehicletrajectory”executedbyaconsortiumoftheGdańskUniversityofTechnology

andGdyniaMaritimeUniversity.



http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 13

Number 4

December 2019

DOI:10.12716/1001.13.04.25

896

they are also used‐combined with mathematical

alignmentmethods–toincreasetheaccuracyoffinal

determinations, e.g. [Bakula, Kazmierczak 2017,

Czaplewski, Waz, 2017, Swierczynski, Czaplewski

2015].

For several years, co‐authors of this paper have

been presenting opportunities of using satellite

techniquesinmeasurementsofrailwayinfrastructure

[Koc 2012,

 Specht et al. 2011, Specht et al. 2014].

Researchinthisareaisalsoconductedinotherparts

oftheworld,e.g.[Gikasetal.2008,Chenetal.2015,

2018]. The European Commission has financed

severalprojectsaimedatapplyingsatellitesolutions

for positioning multiple units and the results were

published,amongothers,in[Filipetal.,2001,Urech

etal.,2002,MertensP.,FranckartJ.‐P.,2003].Thereis

also the Positive Train Control (PTC) system in the

USA, which has been used on most American

railways since 2015. It employs the NAVSTAR GPS

systeminboth its basic form and inthe differential

version[Bettsetal.2014].Currently,the Automated

Train Management System (ATMS) is under

construction in Australia; starting in 2020, it will

enable train positioning with high reliability and

accuracy[ACIL2013].However,eachsolutionbased

only on satellite systems faces the problem of

positioning accuracy because of GNSS system

accessibility, disrupted not only for natural reasons

[Czaplewski2018]. Therefore,theauthorsattempted

toadaptmethodsofresultalignment,whicharewell‐

knowningeodesy.Thispaperproposestheuseofthe

least squares with conditional equations method to

increasetheaccuracyofmobilemeasurements,which

canbesupportedbyadditionalprecisionpositioning

of GNSS receiver antennas on measurement

platforms.Theproposedadaptationofthealignment

methodwasverifiedusingameasurementcampaign

carriedoutinGdanskinautumn2018.

2 LEASTSQUARESWITHCONDITIONAL

EQUATIONSMETHODWITHASIMPLIFIED

SOLUTION

There are alignment problems in geodetic

measurements in which the parameters of an

observational equations system must meet extra

conditions (Wisniewski 1985, 2013, 2016). These

conditions in a geodetic network can apply to



coordinates of some points being part of the given

network.Thisisasituationdealtwithinthetaskof

monitoring the position of a tramway track. It is

possible when the distance between measurement

points (antennas of GNSS receivers) are measured

with accuracy sufficient to treat the quantities as

relativelyerror‐free(comparedtootherobservations

in the network). Coordinates of the antennas on a

measurement platform, contained in vector

[,, ]

XXX 

, must be a solution of the

followingsystemofconditionalequations

 (1)

On the other hand, according to the general

principles of determination of geodetic network

coordinates, vector X determination is based on

satellite observations which enable formulating the

followingsystemofobservationalequations

(2)

where

[, , ]

yy 

 is a vector of observation.

[, , ]

vvv 

 denotes a vector of random

observation errors with the covariance matrix







(P – matrix of weights,



 – 

coefficient of variance). With known approximate

coordinates

00 0

[,,]

XXX 

of receiver

antennas, the equation



yv=FX

can then be

replaced with a linearobservational equation of the

followingform:













  











FX y FX X y

XF AyXX

 (3)

The quantity

is a vector of unknown

incrementssuchthat

dXX X

.Thematrix



is a known matrix of coefficients (

()rank r



A ),

whereas





  

00 0

112 2

FyFyFy







  





lFX y

XX X



is a vector of absolute terms. Considering

that

dXX X

, a system of observational

equations can likewise be reduced to a linear form

(1). Developing function





 X

 to the following 

form

897









   



XXX

XXXBX

 (4)

givesthefollowinglinearconditionalequation

d BX 0

 (5)

where:





Taking into account the conditions binding the

parametersbeingdeterminedandapplicationofthe

goalfunctionoftheleastsquaremethodleadstoan

alignment problem of the following form

(Wiśniewski2016):

m() in



























XPv

 (6)

To obtain a strict solution of the problem, it is

necessary to replace the primary goal function

()



 vXPv

 with the following Lagrange

function

2() )(



XBvXPv k

 (7)

wherekisavectorofunknownLagrangemultipliers.

The problem (6) can be solved practically in an

different, numerically simpler way. To this end, the

conditional equation

d BX 0

 must be

replacedwithanequivalentobservationalequation



BX v

 (8)

Sinceitisrequiredthat



v0

,thenthevectorof

fictitious observational errors



 must be assigned

with such a covariance matrix











, that

fictitious residuals



should meet the condition



v0

 (within thecalculationprecisionlimits).For

mutually independent observations, this results in

theadoptionofsufficiently large, diagonal elements

of the matrix of weights



P  (theoretically, these

shouldbeinfinitelylargequantities).Inthismanner,

the problem (6) is replaced with a conventional

alignmentproblemwiththefollowingform

min

)

(in

(

)































































AX l v

Alv

vPv

BX v

XvvPv



 (9)

where:





 





 

 











Vector



 is a combined vector of observation

errorswiththecovariancematrix

221































 (10)

where













*





is a combined matrix of weights (



 –  the

coefficient of variance common to both elements of

model (10)). Problem (9) has a solution as an

estimator of increments

 with the following

form

()



XAPAAPl









 (11)

where:

TTT



APA APA BPB





TTT





APl APl BP





Determination of the covariance matrix for the

estimatorgives

()







CAPA



 (12)

where:

ˆˆ

nwr









vPv



 (13)

Vector

ˆˆ

[]

TTT



vv v



 is a combined vector of

residualsdeterminedfromthefunction



vAXl



 (14)

898

Thefollowingvectorisacoordinateestimatorfor

GNSSreceiverantennas

ˆˆ

dXX X

 (15)

with the covariance matrix

ˆˆ



. Let us note

thatsquaresofmeanerrorsofdeterminedestimators

are the diagonal elements of the matrix, i.e. 

ˆˆ

[]

m 



3 ORGANISATIONOFTHEEXPERIMENT

The measurement campaign was conducted during

the night of 28/29 November 2018 between 11 p.m.

and 4.00 a.m. in Gdansk with a Bombardier NGT6

tramway(Fig.1).



Figure1.BombardierNGT6tramway‐source:www.gait.pl

Two mobile measurement platforms with the

measuring instruments‐GNSS receivers – were

pulledbytherailwayvehicle(Fig.2)Theinstruments

weresuppliedbytwoleadingproducersofgeodetic

instruments.



b/ 

Figure2 a‐Mobile measurement platform, b‐GNSS

receivertracks

Moreover, an inclinometer, an accelerator and a

compass were installed to conduct other

measurements not described in this paper. The

measurements were performed on a 3‐kilometre

route in Gda nsk (Fig. 3). They involved repeated

passageofthemeasurementunitalongasectionwith

variousnumbersofbuildingsalongit,which

affected

theaccessibilityof the satellite systems (Fig. 3).The

mean speed of the unit at which the measurements

wereperformedwas10km/h.



Figure3. Plan of the measurement unit passage‐source:

maps.google.pl

The diagram of the measurement platforms

constructed for the tests is shown in Fig. 4. The

receiversweredeployedinsuchawaythatfourwere

situatedattheverticesofasquare,andthefifthwas

situated at the intersection of its diagonals. The

construction of the mobile measurement platform



enabled designing a geometric measurement

structure in the shape of a square with the sides of

155 cm to 170 cm. The precision placement of the

GNSS receivers in the track axis and above the rail

wasperformedinthelocalsystemwiththeuseofan

electronictotalstation

andaprismaticmirrorplaced

onadedicatedtripodwith anaccuracyof ca.1mm

(rms).



Figure4Adiagramofthemobilemeasurementplatform(1

–rails,2–pointsofforcedcentringofaGNSSreceiver,3–

platformwheels,d

i–distancesbetweentheGNSScentring

points)–source:preparedbytheauthors

899

Thepositiondatawererecordedinrealtimewith

the 1 Hz  frequency during the measurement

campaignfordifferentGNSSreceiverconfigurations:

 positioningwiththeuseofGPScorrectiondata,

 positioning with the use of GPS+GLONASS

correctiondata,

 positioning with the use of GPS+GALILEO

correctiondata.

4 THEMETHODAPPLICATIONFORANTENNA

POSITIONALIGNMENTONA

MEASUREMENTPLATFORM.

The method described in section 2 was applied to

align the satellite measurement results, also taking

into account previous tachymetric measurements.

Thetachymetricmeasurementsallowedforprecision

positioning of the

GNSS receiver antennas  on the

measurement platforms relative to one another and

dimensioningthewholemeasurementsystemonthe

measurement platform. The observation results are

showninFig.5.



Figure5.Distancesbetweenantennasonthemeasurement

platform‐source:preparedbytheauthors

The alignment process was performed for all

kindsofobservations.However,becauseofthelarge

amountofthestudymaterial,onlyitsrandompartis

presented in this paper (Fig.3). 60 seconds of

recording with one type of receivers using

adjustmentfromGPS+GALILEO+GLONASSwere

used for the

 alignment. The least squares with

conditional equations method was applied with the

use of MathCad Prime 3.0 software, enabling

calculationsandexportofthealignmentresultstoa

fileinaformatcompatiblewithArcGISembeddedin

theGISlaboratoryattheDepartmentofGeodesyand

OceanographyoftheMaritime

UniversityinGdynia.

No partial results are presented because of the

largesizeofthemeasurementsample.However,the

final results are shown in Fig. 6. The graphic

presentation of the results was prepared in the

ArcGISprogram.



Figure6.Alignedrecordingresultsforaselectedsectionof

the tramway track loop in Gdansk – source: prepared by

theauthors

The GNSS antenna position, aligned by the least

squares with conditional equations method, is

marked in yellow. The red colour denotes the

antenna position determined by satellite techniques

foronerecordingmoment.

Figure7showsmeanerrorsofpositioncoordinate

estimatorsobtainedfromthecovariancematrix(12).



Figure7. Errors of mean antenna position coordinate

estimators–source:preparedbytheauthors

The errors lie within the interval



0.0004, 0.0053 mm 

.

5 CONCLUSIONS

This paper proposes the non‐standard use of a

methodofaligningobservationresultswhichiswell‐

known in conventional geodesy to calculate the

results of satellite measurements in mobile

measurementcampaigns.Theconstantdevelopment

of satellite techniques, aided by the data processing

methods in geodesy, can bring

 improved quality to

mobile measurements  which require precision

determinationofpointcoordinatesinvariousstudies

and practical implementations of technical

assignments.

The justifiability of the theoretical assumptions

wasconfirmedbyrecordingdatainthemeasurement

campaign described in the paper. The use of the

proposed alignment method should be confirmed

withdatafromothermeasurementcampaigns,which

willbetheobjectofstudiesinthenextstagesofthe

900

projectexecution.Moreover,theteamwilladaptthe

solutions derived from the solutions applied in this

methodandadaptotherpopula rmethodsofaligning

observation results. The work outcome will be

presentedinfuturepublicationsshowing  theresults

of studies obtained as part of the project No

POIR.04.01.01‐00‐0017/17.



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