Research Article Design of Reverse Curves Adapted to the ...downloads.hindawi.com/journals/ace/2016/6503962.pdf · Design of Reverse Curves Adapted to the Satellite Measurements

Research ArticleDesign of Reverse Curves Adapted to the Satellite Measurements

Wladyslaw Koc

Faculty of Civil and Environmental Engineering Gdansk University of Technology Narutowicza 1112 80-233 Gdansk Poland

Correspondence should be addressed to Wladyslaw Koc kocwlpggdapl

Received 26 November 2015 Revised 25 February 2016 Accepted 13 March 2016

Academic Editor Samer Madanat

Copyright copy 2016 Wladyslaw Koc This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper presents a new method for designing railway route in the direction change area adapted to the Mobile SatelliteMeasurements technique The method may be particularly useful in the situations when both tangents cannot be connected inan elementary way using a circular arc with transition curves Thus the only solution would be the application of two circulararcs of opposite curvature signs that is the use of an inverse curve It has been assumed that the design of the geometrical layoutwill take place within an adequate local coordinate system The solution of the design problem takes advantage of a mathematicalnotation and concentrates on the determination of universal equations describing the entire geometrical layoutThis is a sequentialoperation involving successive parts of thementioned layoutThis universal algorithmcanbe easily applied to the computer softwarewhich will allow generating in an automatic way other geometrical layouts Then the choice of the most beneficial variant fromthe point of obtained trains velocities while minimizing the track axis offsets will be held using the optimization techniques Thecurrent designingmethods do not provide such opportunitiesThepresentedmethod has been illustrated by appropriate calculationexamples

1 Introduction

An inspiration for undertaking the analyzed problem isundoubtedly the new technologymdashthe application to the rail-way track satellitemeasurements GPSThe global positioningsystem GPS [1ndash5] enables determining the coordinates ofpoints in a uniform three-dimensional reference systemWGS84 whose origin is placed in the centre of the Earthrsquos massEffective measurements of railway track might be obtainedby the method of Mobile Satellite Measurements elaboratedby a scientific team of the Gdansk University of Technologyand the Naval Academy in Gdynia [6 7] This method refersto a pilot study [8] and involves driving through the testedsection of the route being inspected by the use of antennasinstalled on a travelling rail carriage

The Mobile Satellite Measurements make it possible todetermine the coordinates of the existing railway routeusing the Cartesian coordinate system (which in Polandcorresponds to the national spatial reference system [9]) asdescribed in the works [10ndash13] It would be advantageous ifthe newly designed track axis coordinates were determinedin mentioned system in particular the coordinates are usedin setting out the route in terrain

General principles for design of track geometry wereformulated in the nineteenth century Then their continuousmodification followed which was reflected in the constantlychanging regulations It should be noted however that thedevelopment of computer technology constantly stimulatesthe ongoing search for new solutions presented amongothersin [14ndash17]

The significant precision obtained in the method ofMobile Satellite Measurements in terms of the determinationof coordinates in horizontal plane (with an error in the rangeof several millimeters) [12] inclines to work out new designmethods adapted to the satellite measurement technique [1819] and to new computer-aided programs [20]

The interest of analyzed problem is confirmed by theinvestigations carried out in Europe in 2006ndash2010 as partof the INNOTRACK program [21] The investigations werecoordinated by International Union of Railways (UIC) Over30 participants were engaged in the project including 8leading infrastructure managing directors (among othersfrom UK Germany and France) It appears that on the listof the most frequent problems raised by the infrastructureboard of directors is the ascertainment ldquobad track geometryrdquoThe methods of geometric shaping of tracks that have been

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2016 Article ID 6503962 9 pageshttpdxdoiorg10115520166503962

2 Advances in Civil Engineering

used so far proved to be ineffective Thus there is plenty ofwork to do to improve the present unfavorable situation

The difficulties connected with the geometric track shap-ing in horizontal plane result from the fact that the appliedgeometric elements like straight sections radii of circulararcs and transition curves are very often characterized bylarge lengths and therefore a visual evaluation of the wholesystem using the traditional geodesic techniques becomesineffective The system should be divided into parts andanalyzed individually which causes extra errors

At this point it is essential to note a very importantfact The reason is that the railway project has its owncharacteristics which is in fact generally connected withexisting layout and the regulation of the track axis Insituations relating to regions requiring an alternative routedirection the design will in principle be based on makinga correction of the circular arc radius as well as the type andlength of the transition curves so that the new geometriclayout is most desirable from the rail vehicles kinematic pointof view Simultaneously its position in horizontal plane willnot divert too much from the existing layout

Determining the values of geometrical parameters whichwill ensure meeting these conditions becomes in fact a keyissue Specifying these parameters requires consideration ofmultiple design options andmaking appropriate choice usingoptimization techniques The new method of calculating thecoordinates adapted to the satellite measurements is essentialto generate variants of railway geometrical layout Only afterobtaining the appropriate values of the geometric parametersas a result of this procedure it is possible to use in a rationalway any of the commercial computer programs supportingthe designing process

As proved by the satellite measurements that have beencarried out so far the shape of the railway track in operationcan be so deformed that the determination of the maindirections turns out to be impossible one cannot apply amodel system to the design transition curve-circular arc-transition curveThe only solution in this case is to introducetwo circular arcs of different radius to the geometrical layoutwhich means applying a compound curve [19]

All this however does not cover the whole problem yetBut under some conditions even the use of a compound curvedoes not allow us to connect themajor directions of the routeIn consequence it is necessary to use two inverse arcs in thegeometrical layout A description of the design procedurefor this type of situation can be found in the content of thepaper The presented conception of the designing techniquerelated to a route realignment area creates an opportunity likeother elaborated methods to obtain an analytical solution bythe application of adequate mathematical formulae which ismore convenient for practical usage

2 General Assumptions

The route major directions in the Polish National SpatialReference System can be defined by the following equations[9]

Straight 11198831= 1198601+ 1198611119884

Straight 2

1198832= 1198602+ 1198612119884

In the above equations 1198601and 119860

2are absolute terms in

the expression and 1198611and 119861

2define the slope coefficients of

the two straights The straights have similar slope coefficientvalues and intersect at a distant point (they can also runin parallel to each other) Such a situation justifies theconnection of both straights by reverse curve

The designing procedure will take place within an appro-priate local system of coordinates (LCS) making it possibleto present the course of the route in functional notationThissystem results from the adoption of the coordinates of itsstarting point119874(119884

119874 119883119874) on Straight 1 and making a rotation

of 120573 angleThe problem of obtaining appropriate inclinationsof Straights 1 and 2 in LCS system becomes a crucial questionin thismatterThe inclinationsmust be positive to operate thepositive values of ordinates 119910(119909) and also advantageous in theview of the procedure of determining these ordinates that isneither too big nor too small In this respect the 120573 rotationangle will play a decisive role in it

An assumption is made that within system 119909 119910 Straight1 will pass through the centre of the system with a slope anglebeing equal to 1205874 For the reason that Straight 2 inclinationvalue is similar to Straight 1 it will certainly be placed withininterval (0 1205872) closer to the centre of that (ie inclinationof Straight 1) than to its boundaries A positive inclinationof both straights ensures an analytical description of thewhole geometrical layout by the use of explicit functions 119910(119909)related to the circular arcs

In order to insert an inverse curve between both straightsit is necessary to find such coordinates of point 119874(119884

119874 119883119874)

where Straight 2 can be placed on the right side of Straight 1Knowledge of the point 119874(119884

119874 119883119874) coordinates and the rota-

tion angle 120573 enables mutual points transformation betweenglobal and local coordinate systemThe value of turning angle120573 is determined by the use of the following equation

120573 = 1205931minus120587

4 (1)

where 1205931= atan119861

1for 1198611gt 0 and 120593

1= atan119861

1+ 120587 for

1198611lt 0 To obtain a positive value of angle 120573 from the above

formula the left turn of the system should be made whereasin the case of a negative magnitude a right turn of the systemshould be made

In assuming the coordinates of point 119874(119884119874 119883119874) along

Straight 1 and determining the turning angle 120573 it is possibleto make a transformation of Straights 1 and 2 to the localcoordinate system 119909 119910 The entire geometric system in LCSsystem under consideration is presented in Figure 1

The position of an arbitrary point along the route in localsystem 119909 119910 can be obtained using the following equations[22]

119909 = (119884 minus 119884119874) cos120573 + (119883 minus 119883

119874) sin120573

119910 = minus (119884 minus 119884119874) sin120573 + (119883 minus 119883

119874) cos120573

(2)

Advances in Civil Engineering 3

y1

y2

y3

O

lTC1 lCA1 lCA2lTC2 lTC3

x

y

xK1 xO3xK2xO2xK3

120575

O3

O2

120574

prop1

prop2

K2

K1

K3

x2

x3

x1

ΔyTC2

ΔyTC3

1205874

x2

x3

y2

y3

Figure 1 The geometrical layout under consideration in the local coordinate system

In terms of the local coordinate system Straight 1 isdescribed by

119910(1)

(119909) = 119909 (3)

A general notation of Straight 2 takes the following form

119910(2)

(119909) = 1198862+ 1198872119909 (4)

where

1198862= minus(

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

minus 119884119874) sin120573

+ [(1198602+ 1198612

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

)

minus 119883119874] cos120573

1198872= tan 120575

(5)

Angle 120575 determining the slope of Straight 2 to axis 119909 inthe LCS system is obtained from the formula

120575 = 1205932minus 120573 (6)

where1205932= atan119861

2for1198612gt 0 and120593

2= atan119861

2+120587 for119861

2lt 0

3 Choice of Parameters to Design theGeometric System

The designed geometric system connecting Straight 1 withStraight 2 in the local system of coordinates 119909 119910 is composedof the following components (Figure 1)

(i) First transition curve (TC1) of a specified type oflength 119897

1

(ii) First circular arc (CA1) of radius 1198771and length 119897

1198771

(iii) Second transition curve (TC2) of a specified type oflength 119897

2

(iv) Second circular arc (CA2) of radius 1198772and undefined

length 1198971198772

(v) Third transition curve (TC3) of a specified type oflength 119897

3

Values 1198971 1198771 1198972 1198772 and 119897

3result from the speed analysis

carried out for the designed system Value 1198971198771

is in prin-ciple arbitrary while 119897

1198772is the final magnitude closing up

the whole system However one should take into accountthat the raised problem can be solved when advantage istaken of an appropriate configuration of the abovementionedparameters A key role in this procedure is the characteristicof the curvature used Curve TC1 and arc CA1 have a negativecurvature while arc CA2 and curve TC3 have a positive oneDue to this fact the slope angle of tangent on curve TC1 andarc CA1 is decreasing but on arc CA2 and on curve TC3 it isrising However the following condition should be fulfilled(Figure 1)

120587

4+ Θ (119897

1) minus 1205721+ Θ (119897

2) + 1205722minus Θ (minus119897

3) = 120575 (7)

where Θ(1198971) is slope angle of tangent at the end of TC1 in 119909

1

1199101coordinate system 120572

1is angle of sense CA1 Θ(119897

2) is slope

angle of tangent at the end of TC2 in 11990921199102coordinate system

1205722is angle of sense CA2 andΘ(minus119897

3) is slope angle of tangent

at the end of TC3 in the 1199093 1199103coordinate system

The only element causing a diversity of the curvature signis curve TC2 The curvature along it changes from negative


to positive Simultaneously the value of angle Θ(1198972) is either

positive or negative In the case of 120575 lt 1205874 it would beadvantageous ifΘ(119897

2) lt 0 which corresponds to the situation

when 1198771

lt 1198772 When 120575 gt 1205874 it would be profitable if

Θ(1198972) gt 0 leading to relation 119877

1gt 1198772

4 Transition Curve TC1

The ordinates of transition curve TC1 are determined bythe use of auxiliary system of coordinates 119909

1 1199101(ACS1)

(Figure 1)The course of proceeding takes the following form

(i) Determination of the type of the transition curve(ii) Assumption of the length 119897

1of curve (measured along

the curve) and radius 1198771of the adjacent circular arc

CA1

This in consequence gives the parametric equations 1199091(119897)

1199101(119897) 119897 isin ⟨0 119897

1⟩

For the reason that regardless of the type of the transitioncurve connecting the straight with the circular arc |Θ(119897

1)| =

119897121198771(rad) the tangent value at the end of transition curve

TC1 is defined by formula

1199101015840

1(1198971) = minus tan(

1198971

21198771

) (8)

The next step is to transform curve TC1 to the localsystem of coordinates 119909 119910 (Figure 1) This is accomplishedby a right turn of the axis of system ACS1 through angle1205874 Consequently the following parametric equations areobtained

119909 (119897) = 1199091(119897) cos 120587

4minus 1199101(119897) sin 120587

4

119910 (119897) = 1199091(119897) sin 120587

4+ 1199101(119897) cos 120587

4

119897 isin ⟨0 1198971⟩

(9)

Inserting the final value of parametric 119897 (ie 119897 = 1198971) into (9) it

is possible to obtain coordinates of the end of curve TC1 (iepoint119870

1) 1199091198701

= 1198971198701198751

and 1199101198701 The tangent value 119904

1198701at point

1198701is

1199041198701

= tan(minus1198971

21198771

+120587

4) (10)

5 Circular Arc CA1

The diagram illustrating the position of the circular arc CA1is shown in Figure 2 Assumption is made of the circular arclength 119897

1198771(measured along the arc) Coordinates of point

1198781(1199091198781 1199101198781) the centre of arc CA1 are determined

1199091198781

= 1199091198701

+1199041198701

radic1 + 1199042

1198701

1198771

1199101198781

= 1199101198701

minus1

radic1 + 1199042

1198701

1198771

(11)

y1

yx1

xO

K1

M1

sK1sO2

1205874

R1

R1

xK1 xM1 xO2xS1

S1

O2prop1

prop1

Figure 2 Diagram illustrating the position of circular arc CA1

The equation of circular arc CA1 is as follows

119910 (119909)CA1 = 1199101198781

+ [1198772

1minus (1199091198781

minus 119909)2

]12

119909 isin ⟨1199091198701 1199091198742⟩

(12)

The turning angle of the arc CA1 tangents is

prop1=

1198971198771

1198771

(13)

Inclination 1199041198742

of tangent to arc CA1 at its end at point1198742 is

1199041198742

= tan (atan 1199041198701

minus prop1) (14)

One should now determine the coordinates of point 1198742

the end of circular arc CA1 For this purpose it is necessary tofind the coordinates of point 119872

1(Figure 2) The coordinates

of point 1198742are as follows

1199091198742

= 1199091198721

+tan (prop

12)

radic1 + 1199042

1198742

1198771

1199101198742

= 1199101198721

+1199041198742

tan (prop12)

radic1 + 1199042

1198742

1198771

(15)


The diagram illustrating the position of transition curve TC2is given in Figure 3The ordinates of transition curve TC2 aredetermined by using an auxiliary system of coordinates 119909

2 1199102

(ACS2) The transition curve TC2 connects opposite arcs ofradii 119877

1and 119877

2


y2

y

x

K2

sO2

sK2

O2

x2

ΔyTC2120574

y2

lTC2

x2

Figure 3 Diagram presenting the position of transition curve TC2

The course of proceeding is as follows(i) A linear or nonlinear form of curvature is assumed(ii) The length 119897

2of curve (measured along the curve)

and the radius 1198772of the adjacent circular arc CA2 are

taken into account(iii) Parametric equations 119909

2(119897) 1199102(119897) 119897 isin ⟨0 119897

2⟩ are

obtainedThe examples of solving linear and nonlinear curvature

distribution along the length of a curve have been given in thepaper [23] Other forms of formulae are related to inflexionpoint 119897

0(ie for 119897 isin ⟨0 119897

0⟩) on the curvature diagramwhereas

others refer to the final region 119897 isin ⟨1198970 1198972⟩

The next step is the transformation of curve TC2 to theauxiliary coordinate system 119909

2 1199102(ACS2) (Figure 3) The

position of the system is determined by angle 120574 = atan 1199041198742 If

the slope 1199041198742

= tan(atan 1199041198701

minusprop1) gt 0 then it is necessary to

make a right turn of systemACS2 However if the inclination1199041198742

lt 0 one should make a left turn of system ACS2 Sinceangle 120574 = |atan 119904

1198742| isin ⟨0 1205872⟩ the following expressions are

obtained119909 = 119909

1198742+ 1199092(119897) = 119909

1198742+ 1199092(119897) cos 120574 ∓ 119910

2(119897) sin 120574

119910 = 1199101198742

+ 1199102(119897) = 119910

1198742plusmn 1199092(119897) sin 120574 + 119910

2(119897) cos 120574

119897 isin ⟨0 1198972⟩

(16)

Values 119897TC2 andΔ119910TC2 (Figure 3) are determined from theequations

119897TC2 =10038161003816100381610038161199092 (1198972)

1003816100381610038161003816 =10038161003816100381610038161199092 (1198972) cos 120574 ∓ 119910

2(1198972) sin 120574

1003816100381610038161003816

Δ119910TC2 =10038161003816100381610038161199102 (1198972)

1003816100381610038161003816 =1003816100381610038161003816plusmn1199092 (1198972) sin 120574 + 119910

2(1198972) cos 1205741003816100381610038161003816

(17)

The coordinates of point1198702are as follows

1199091198702

= 1199091198742

+ 119897TC2

1199101198702

= 1199101198742

+ Δ119910TC2(18)

whereas the tangent value 1199041198702

at point 1198702in ACS2 and LCS

system is

1199041198702

= tan [Θ (1198972) plusmn 120574] (19)

x

y

y

3

O3

x3

ΔyTC3

K3

sK3

lTC3

120575

120575

x3

y3

Figure 4 Diagram illustrating the position of transition curve TC3


To determine the position of the end of arc CA2 unknownat this stage of investigation it is necessary to find the valueof tangent 119904

1198703at the end of transition curve TC3 being of

significance in view of arc CA2 A scheme presenting theposition of transition curve TC3 using the auxiliary systemof coordinates 119909

3 1199103(ACS3) is shown in Figure 4

The course of procedure is as follows

(i) The type of transition curve is determined(ii) The length 119897

3of curve (measured along the curve) is

assumed(iii) Parametric equations 119909

3(119897) 1199103(119897) 119897 isin ⟨0 119897

3⟩ are ob-

tained

The next step is the transformation of transition curveTC3 to the auxiliary system of coordinates 119909

3 1199103(ACS3)

whose axes are parallel to the local coordinate system 119909119910 The transfer is possible by a right turn of system ACS3through angle 120575 The parametric equations of curve TC3 insystem ACS3 are as follows

1199093(119897) = 119909

3(119897) cos 120575 minus 119910

3(119897) sin 120575

1199103(119897) = 119909

3(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(20)

Inserting the final value of parameter 119897 (ie 119897 = minus1198973) into

(20) gives values 119897TC3 and Δ119910TC3 The value of tangent 1199041198703

atpoint119870

3in the PUW3 and LCS systems is

1199041198703

= tan(minus1198973

21198772

+ 120575) (21)

The notation of the curve TC3 in the local coordinatesystem calls for the determination of the position of point1198743(1199091198743 1199101198743)


y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2

Figure 5 Effective scheme of the circular arc CA2 position

8 Circular Arc CA2

An effective scheme of the circular arc CA2 position is shownin Figure 5 At this stage of the design procedure the followingdata are known radius 119877

2 coordinates of the outset point

1198702(1199091198702 1199101198702) and the tangent values at the start of 119904

1198702and the

end of 1199041198703

of arc CA2 (position of the end of arc ie the end-point coordinates 119870

3(1199091198703 1199101198703) is unknown) Thus to begin

with the coordinates of point 1198782(1199091198782 1199101198782) the mid-point of

arc CA2 are determined

1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)

The equation for the circular arc CA2 is as follows

119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)

it is possible to determine 1199091198703

and then also 1199101198703

1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|

Figure 6 Idea of the procedure of correcting the coordinates of theorigin of LCS system

The length of arc CA2 projected on 119909-axis amounts to

119897CA2 = 1199091198703

minus 1199091198702 (26)

whereas the turning angle of sense of tangents is

prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)

Hence the length of the circular arc CA2 (measured alongthe arc) follows immediately

1198971198772

= prop21198772 (28)

The knowledge of the position of point 1198703(1199091198703 1199101198703)

makes it possible to determine the coordinates of the end-point 119874

3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)

and the parametric equations of transition curve TC3

119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)

9 Determination of the Right Position ofthe Origin of the Local Coordinate System

To transfer the designed geometric system from local coordi-nate system to the system 2000 one should revise the positionof its initial point along Straight 1 However in general point1198743(1199091198743 1199101198743) denoting the end of the system will not be

lying on Straight 2 but on Straight 3 running parallel to itof equation 119910

(3)= 1198863+ 1198872119909 (Figure 6)

In order to determine the corrected coordinates of theoutset of LCS system it is necessary to find the coordinates ofpoints intersecting Straight 1 with Straight 2 andwith Straight3 Straight 1 intersects Straight 2 at point 119875

12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)


and Straight 2 at point 11987513

11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)

It is intended to shift the start of LCS system (point 119874)along Straight 1 to a new position at point in such a way that11987512

= 11990911987513

within system (and of course 11987512

= 11991011987513

)(Figure 6) This leads to the following general equation

119909= 119910= 11990911987512

minus 11990911987513

(33)

The revised origin of the local coordinate system insystem 2000 has the following coordinates

119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)

The transference of the designed geometric system fromLCS system to system 2000 is carried out by making use ofthe following relations [22]

119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)

10 Calculation Examples

Example 1 (typical casemdashconnection of straights of proximateparallelisms) Consider the following

Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad

Assumed Starting Point

119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981

= 060ms21198972= 160m (linear curvature)

1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)

Figure 7 Visualization of the designed geometric system(Example 1)

1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms

where V is speed of trains ℎ1is cant along arc CA1 ℎ

2is cant

along arc CA2 1198861198981

is unbalanced acceleration along arc CA11198861198982

is unbalanced acceleration along arc CA2 1205951is speed of

acceleration change on transition curve TC1 1205952is speed of



lifting thewheel at the cant transition of curveTC11198912is speed

of lifting the wheel at the cant transition of curve TC2 and 1198913

is speed of lifting thewheel at the cant transition of curveTC3

Correction of the Initial Position of LCS System

119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909

Figure 7 presents the visualization of the designed geo-metrical layout from Example 1 In Table 1 characteristics ofthe principal points of the geometrical layout are given Thecombination of two straight lines close to the parallelism iscarried out on the length of 640359m The length of thecircular arc CA2 closing the entire layout is 119897

1198772= 170359m

It should be noted that the correction of the local coordinatesystem beginning is significant abscissa 119884 about 308m andordinate119883 about 1039m


Table 1 Characteristic of the principal points of the geometricsystem (Example 1)

Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110

The practical applicability to the solution presented inExample 1 (Figure 7) cannot raise any doubts Taking intoconsideration only the calculation technique some problemsarise when parallel straights are joined since it is not possibleunder such circumstances to make a direct correction of thelocal coordinate system position However the computer-aided calculations by the use of the mentioned algorithmmake it possible to solve this problem easily

Example 2 (a case of more diversified gradients of both thestraights) Data relating to Straight 1 of the assumed outsetpoint and the design characteristic are shown as in Example 1(where the obtained length 119897

1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909

Visualization of the designed geometrical layout fromExample 2 is presented in Figure 8 In Table 2 characteristicsof the principal points of the geometrical layout are givenThe combination of two straight lines is carried out on thelength of 55035mThe length of the circular arc CA2 closingthe entire layout is 119897

1198772= 8035m The correction of the local

coordinate system beginning is relatively small abscissa 119884

about 40m and ordinate119883mdash140mHowever at first glance the solution presented in

Example 2 (Figure 8) may raise some doubts It may appearthat inverse curves are of no use in this situation and boththe straights should be connected in an elementary way usinga circular arc with transition curves However in the designprocess one must take account of a need to pass over a fieldobstacle and then the inverse curves can become a sensiblesolution As can be seen the inverse curves may provide analternative even to such an elementary geometric problemwhich is joining two straights by means of a circular arc


Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)


The calculation examples show the correctness of thedeveloped method as well as the opportunity to apply itusing elementary way that is calculation sheets Howeverthis universal algorithm can be easily applied to the computersoftware which will allow generating in an automatic wayby changing the radii of the arcs and the type and lengthof the transition curves another geometrical layout Thenthe choice of the most beneficial variant from the point ofobtained trains velocities while minimizing the track axisoffsets will be held using the optimization techniques Thecurrent designing methods do not provide such opportuni-ties

11 Summing-Up

(i) The application of Mobile Satellite Measurements withantennas installed on a moving rail vehicle makes it possibleto reconstruct the track axis in an absolute reference systemThis creates completely new potentials in the range of railtrack geometric shaping Under conditions of the createdsituation there arises a necessity for working out some newdesign methods

(ii) This paper presents one method more (followingthe studies in [18 19]) relating to the design of a regionof railway track direction change appropriate for MobileSatellite Measurement technique The method may appearto be of particular applicability if both straights of the trackdirection cannot be connected in an elementary way by theuse of a circular arc with transition curves this also concerns


the use of the compound curve Such a situation occurs whenthe connected straights indicate values of the inclinationcoefficient which are very close to each other and intersect ata distant point (theymay also run parallel) In such a situationthe only solution is to introduce to the geometric system twocircular arcs of opposite signs of curvature that is to apply aninverse curve

(iii) The presented conception of the design procedurerelating to the region covering the track direction changeoffers an opportunity to find an analytical solution by the useof appropriatemathematical formulae beingmost friendly inpractical application The design procedure is of a universalnature and creates a possibility for arbitrary acceptance oflengths and radii of circular arcs and differentiation of thetype and lengths of the applied transition curves

(iv) The effects of the application of the analyzed designmethod have been illustrated by exact calculation examplesIts practical applicability cannot cause any doubts Simulta-neously attention has been concentrated on the fact that theinverse curves may provide an alternative even for such anelementary geometric problem which is the connection oftwo straights by using a circular arc In order to implementthe presented procedure it will be indispensable to work outin the near future an appropriate computer-aided techniqueThe computer software will allow generating automaticallyadditional geometrical layoutsThe choice of the best solutionwill be held in the field of optimization The criteria ofoptimizations are the maximum value of the velocity andminimizing the track axis offsets The current designingmethods do not provide such opportunities

Competing Interests

The author declares that he has no competing interests

References

[1] G Seeber Satellite Geodesy Foundations Methods and Applica-tions Walter de Gruyter Berlin Germany 1993

[2] B W Parkinson J J Spilker Jr P Axelrad and P EngeGlobal Positioning System Theory and Applications Volume IIAmerican Institute ofAeronautics andAstronautics RestonVaUSA 1996

[3] M Ferguson GPS Land Navigation Glassford Spokane WashUSA 1997

[4] J Bosy W Graszka and M Leonczyk ldquoASG-EUPOSmdasha mul-tifunctional precise satellite positioning system in PolandrdquoEuropean Journal of Navigation vol 5 no 4 pp 2ndash6 2007

[5] A Leick L Rapoport and D Tatarnikov GPS Satellite Survey-ing John Wiley amp Sons Hoboken NJ USA 4th edition 2015

[6] W Koc and C Specht ldquoApplication of the Polish active GNSSgeodetic network for surveying and design of the railroadrdquo inProceedings of the 1st International Conference on Road and RailInfrastructure (CETRA rsquo10) Opatija Croatia May 2010

[7] C Specht A NowakW Koc and A Jurkowska ldquoApplication ofthe Polish Active Geodetic Network for railway track determi-nationrdquo in Transport Systems and ProcessesmdashMarine Navigationand Safety of Sea Transportation pp 77ndash81 CRC Press LondonUK 2011

[8] T Szwilski R Begley P Dailey and Z Sheng ldquoDetermining railtrack movement trajectories and alignment using HADGPSrdquoin Proceedings of the AREMA Annual Conference Chicago IllUSA October 2003

[9] Council of Ministers of 15 October 2012 on National spatialreference system Dziennik Ustaw pos 1247 2012 (Polish)

[10] W Koc and C Specht ldquoSelected problems of determining thecourse of railway routes by use of GPS network solutionrdquoArchives of Transport vol 23 no 3 pp 303ndash320 2011

[11] W Koc C Specht and P Chrostowski ldquoFinding deformationof the stright rail track by GNSS measurementsrdquo Annual ofNavigation vol 19 no 1 pp 91ndash104 2012

[12] W Koc C Specht P Chrostowski and K Palikowska ldquoTheaccuracy assessment of determining the axis of railway trackbasing on the satellite surveyingrdquo Archives of Transport vol 24no 3 pp 307ndash320 2012

[13] W Koc C Specht and P Chrostowski ldquoThe applicationeffects of continuous satellite measurements of railway linesrdquoin Proceedings of the 12th International Conference amp ExhibitionRailway Engineering Railway Operation Section London UKJuly 2013

[14] M Lindahl Track Geometry for High Speed Railways KTHStockholm Sweden 2001

[15] M H Letts ldquoTrack geometry design testing for transit applica-tionsrdquo Final Report for Transit IDEA Project 41 TransportationResearch Board of the National Academies Washington DCUSA 2007

[16] P Lautala and T DickRailway Alignment Design andGeometryAREMA Lanham Md USA 2010

[17] S Hodas ldquoDesign of railway track for speed and high-speedrailwaysrdquo Proceedia Engineering vol 91 pp 256ndash261 2014

[18] W Koc ldquoDesign of rail-track geometric systems by satellitemeasurementrdquo Journal of Transportation Engineering vol 138no 1 pp 114ndash122 2012

[19] W Koc ldquoDesign of compound curves adapted to the satellitemeasurementsrdquo Archives of Transport vol 34 no 2 pp 37ndash492015

[20] W Koc and P Chrostowski ldquoComputer-aided design of railroadhorizontal arc areas in adapting to satellite measurementsrdquoJournal of Transportation Engineering vol 140 no 3 Article ID04013017 2014

[21] httpwwwinnotracknet[22] G A Korn and T M KornMathematical Handbook for Scien-

tists and Engineers McGraw-Hill New York NY USA 1968[23] W Koc ldquoAnalytical method of modelling the geometric system

of communication routerdquo Mathematical Problems in Engineer-ing vol 2014 Article ID 679817 13 pages 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



used so far proved to be ineffective Thus there is plenty ofwork to do to improve the present unfavorable situation

The difficulties connected with the geometric track shap-ing in horizontal plane result from the fact that the appliedgeometric elements like straight sections radii of circulararcs and transition curves are very often characterized bylarge lengths and therefore a visual evaluation of the wholesystem using the traditional geodesic techniques becomesineffective The system should be divided into parts andanalyzed individually which causes extra errors

At this point it is essential to note a very importantfact The reason is that the railway project has its owncharacteristics which is in fact generally connected withexisting layout and the regulation of the track axis Insituations relating to regions requiring an alternative routedirection the design will in principle be based on makinga correction of the circular arc radius as well as the type andlength of the transition curves so that the new geometriclayout is most desirable from the rail vehicles kinematic pointof view Simultaneously its position in horizontal plane willnot divert too much from the existing layout

Determining the values of geometrical parameters whichwill ensure meeting these conditions becomes in fact a keyissue Specifying these parameters requires consideration ofmultiple design options andmaking appropriate choice usingoptimization techniques The new method of calculating thecoordinates adapted to the satellite measurements is essentialto generate variants of railway geometrical layout Only afterobtaining the appropriate values of the geometric parametersas a result of this procedure it is possible to use in a rationalway any of the commercial computer programs supportingthe designing process

As proved by the satellite measurements that have beencarried out so far the shape of the railway track in operationcan be so deformed that the determination of the maindirections turns out to be impossible one cannot apply amodel system to the design transition curve-circular arc-transition curveThe only solution in this case is to introducetwo circular arcs of different radius to the geometrical layoutwhich means applying a compound curve [19]

All this however does not cover the whole problem yetBut under some conditions even the use of a compound curvedoes not allow us to connect themajor directions of the routeIn consequence it is necessary to use two inverse arcs in thegeometrical layout A description of the design procedurefor this type of situation can be found in the content of thepaper The presented conception of the designing techniquerelated to a route realignment area creates an opportunity likeother elaborated methods to obtain an analytical solution bythe application of adequate mathematical formulae which ismore convenient for practical usage

2 General Assumptions

The route major directions in the Polish National SpatialReference System can be defined by the following equations[9]

Straight 11198831= 1198601+ 1198611119884

Straight 2

1198832= 1198602+ 1198612119884

In the above equations 1198601and 119860

2are absolute terms in

the expression and 1198611and 119861

2define the slope coefficients of

the two straights The straights have similar slope coefficientvalues and intersect at a distant point (they can also runin parallel to each other) Such a situation justifies theconnection of both straights by reverse curve

The designing procedure will take place within an appro-priate local system of coordinates (LCS) making it possibleto present the course of the route in functional notationThissystem results from the adoption of the coordinates of itsstarting point119874(119884

119874 119883119874) on Straight 1 and making a rotation

of 120573 angleThe problem of obtaining appropriate inclinationsof Straights 1 and 2 in LCS system becomes a crucial questionin thismatterThe inclinationsmust be positive to operate thepositive values of ordinates 119910(119909) and also advantageous in theview of the procedure of determining these ordinates that isneither too big nor too small In this respect the 120573 rotationangle will play a decisive role in it

An assumption is made that within system 119909 119910 Straight1 will pass through the centre of the system with a slope anglebeing equal to 1205874 For the reason that Straight 2 inclinationvalue is similar to Straight 1 it will certainly be placed withininterval (0 1205872) closer to the centre of that (ie inclinationof Straight 1) than to its boundaries A positive inclinationof both straights ensures an analytical description of thewhole geometrical layout by the use of explicit functions 119910(119909)related to the circular arcs

In order to insert an inverse curve between both straightsit is necessary to find such coordinates of point 119874(119884

119874 119883119874)

where Straight 2 can be placed on the right side of Straight 1Knowledge of the point 119874(119884

119874 119883119874) coordinates and the rota-

tion angle 120573 enables mutual points transformation betweenglobal and local coordinate systemThe value of turning angle120573 is determined by the use of the following equation

120573 = 1205931minus120587

4 (1)

where 1205931= atan119861

1for 1198611gt 0 and 120593

1= atan119861

1+ 120587 for

1198611lt 0 To obtain a positive value of angle 120573 from the above

formula the left turn of the system should be made whereasin the case of a negative magnitude a right turn of the systemshould be made

In assuming the coordinates of point 119874(119884119874 119883119874) along

Straight 1 and determining the turning angle 120573 it is possibleto make a transformation of Straights 1 and 2 to the localcoordinate system 119909 119910 The entire geometric system in LCSsystem under consideration is presented in Figure 1

The position of an arbitrary point along the route in localsystem 119909 119910 can be obtained using the following equations[22]

119909 = (119884 minus 119884119874) cos120573 + (119883 minus 119883

119874) sin120573

119910 = minus (119884 minus 119884119874) sin120573 + (119883 minus 119883

119874) cos120573

(2)


y1

y2

y3

O


x

y

xK1 xO3xK2xO2xK3

120575

O3

O2

120574

prop1

prop2

K2

K1

K3

x2

x3

x1

ΔyTC2

ΔyTC3

1205874

x2

x3

y2

y3



119910(1)

(119909) = 119909 (3)


119910(2)

(119909) = 1198862+ 1198872119909 (4)

where

1198862= minus(

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

minus 119884119874) sin120573

+ [(1198602+ 1198612

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

)

minus 119883119874] cos120573

1198872= tan 120575

(5)


120575 = 1205932minus 120573 (6)

where1205932= atan119861

2for1198612gt 0 and120593

2= atan119861

2+120587 for119861

2lt 0




1


1198771


2


length 1198971198772


3

Values 1198971 1198771 1198972 1198772 and 119897






120587

4+ Θ (119897

1) minus 1205721+ Θ (119897

2) + 1205722minus Θ (minus119897

3) = 120575 (7)


1



2) is slope










when 1198771



1gt 1198772



1 1199101(ACS1)





CA1


1199101(119897) 119897 isin ⟨0 119897

1⟩


1)| =



1199101015840

1(1198971) = minus tan(

1198971

21198771

) (8)


119909 (119897) = 1199091(119897) cos 120587

4minus 1199101(119897) sin 120587

4

119910 (119897) = 1199091(119897) sin 120587

4+ 1199101(119897) cos 120587

4

119897 isin ⟨0 1198971⟩

(9)



1) 1199091198701

= 1198971198701198751


1198701at point

1198701is

1199041198701

= tan(minus1198971

21198771

+120587

4) (10)

5 Circular Arc CA1




1199091198781

= 1199091198701

+1199041198701

radic1 + 1199042

1198701

1198771

1199101198781

= 1199101198701

minus1

radic1 + 1199042

1198701

1198771

(11)

y1

yx1

xO

K1

M1

sK1sO2

1205874

R1

R1

xK1 xM1 xO2xS1

S1

O2prop1

prop1



119910 (119909)CA1 = 1199101198781

+ [1198772

1minus (1199091198781

minus 119909)2

]12

119909 isin ⟨1199091198701 1199091198742⟩

(12)


prop1=

1198971198771

1198771

(13)



1199041198742

= tan (atan 1199041198701

minus prop1) (14)





1199091198742

= 1199091198721

+tan (prop

12)

radic1 + 1199042

1198742

1198771

1199101198742

= 1199101198721

+1199041198742

tan (prop12)

radic1 + 1199042

1198742

1198771

(15)



2 1199102


1and 119877

2


y2

y

x

K2

sO2

sK2

O2

x2

ΔyTC2120574

y2

lTC2

x2






2(119897) 1199102(119897) 119897 isin ⟨0 119897

2⟩ are



0(ie for 119897 isin ⟨0 119897






the slope 1199041198742

= tan(atan 1199041198701





obtained119909 = 119909

1198742+ 1199092(119897) = 119909

1198742+ 1199092(119897) cos 120574 ∓ 119910

2(119897) sin 120574

119910 = 1199101198742

+ 1199102(119897) = 119910

1198742plusmn 1199092(119897) sin 120574 + 119910

2(119897) cos 120574

119897 isin ⟨0 1198972⟩

(16)


119897TC2 =10038161003816100381610038161199092 (1198972)

1003816100381610038161003816 =10038161003816100381610038161199092 (1198972) cos 120574 ∓ 119910

2(1198972) sin 120574

1003816100381610038161003816

Δ119910TC2 =10038161003816100381610038161199102 (1198972)

1003816100381610038161003816 =1003816100381610038161003816plusmn1199092 (1198972) sin 120574 + 119910

2(1198972) cos 1205741003816100381610038161003816

(17)


1199091198702

= 1199091198742

+ 119897TC2

1199101198702

= 1199101198742

+ Δ119910TC2(18)



system is

1199041198702

= tan [Θ (1198972) plusmn 120574] (19)

x

y

y

3

O3

x3

ΔyTC3

K3

sK3

lTC3

120575

120575

x3

y3











3(119897) 1199103(119897) 119897 isin ⟨0 119897

3⟩ are ob-

tained


3 1199103(ACS3)


1199093(119897) = 119909

3(119897) cos 120575 minus 119910

3(119897) sin 120575

1199103(119897) = 119909

3(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(20)



atpoint119870


1199041198703

= tan(minus1198973

21198772

+ 120575) (21)



y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2


8 Circular Arc CA2




1198702and the

end of 1199041198703





1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)


119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)



1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|



119897CA2 = 1199091198703

minus 1199091198702 (26)


prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)


1198971198772

= prop21198772 (28)



3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)


119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)




(3)= 1198863+ 1198872119909 (Figure 6)


12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)



11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y1

y2

y3

O


x

y

xK1 xO3xK2xO2xK3

120575

O3

O2

120574

prop1

prop2

K2

K1

K3

x2

x3

x1

ΔyTC2

ΔyTC3

1205874

x2

x3

y2

y3



119910(1)

(119909) = 119909 (3)


119910(2)

(119909) = 1198862+ 1198872119909 (4)

where

1198862= minus(

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

minus 119884119874) sin120573

+ [(1198602+ 1198612

119883119874minus [tan (120573 + 1205872) 119884

119874] minus 1198602

1198612minus [tan (120573 + 1205872)]

)

minus 119883119874] cos120573

1198872= tan 120575

(5)


120575 = 1205932minus 120573 (6)

where1205932= atan119861

2for1198612gt 0 and120593

2= atan119861

2+120587 for119861

2lt 0




1


1198771


2


length 1198971198772


3

Values 1198971 1198771 1198972 1198772 and 119897






120587

4+ Θ (119897

1) minus 1205721+ Θ (119897

2) + 1205722minus Θ (minus119897

3) = 120575 (7)


1



2) is slope










when 1198771



1gt 1198772



1 1199101(ACS1)





CA1


1199101(119897) 119897 isin ⟨0 119897

1⟩


1)| =



1199101015840

1(1198971) = minus tan(

1198971

21198771

) (8)


119909 (119897) = 1199091(119897) cos 120587

4minus 1199101(119897) sin 120587

4

119910 (119897) = 1199091(119897) sin 120587

4+ 1199101(119897) cos 120587

4

119897 isin ⟨0 1198971⟩

(9)



1) 1199091198701

= 1198971198701198751


1198701at point

1198701is

1199041198701

= tan(minus1198971

21198771

+120587

4) (10)

5 Circular Arc CA1




1199091198781

= 1199091198701

+1199041198701

radic1 + 1199042

1198701

1198771

1199101198781

= 1199101198701

minus1

radic1 + 1199042

1198701

1198771

(11)

y1

yx1

xO

K1

M1

sK1sO2

1205874

R1

R1

xK1 xM1 xO2xS1

S1

O2prop1

prop1



119910 (119909)CA1 = 1199101198781

+ [1198772

1minus (1199091198781

minus 119909)2

]12

119909 isin ⟨1199091198701 1199091198742⟩

(12)


prop1=

1198971198771

1198771

(13)



1199041198742

= tan (atan 1199041198701

minus prop1) (14)





1199091198742

= 1199091198721

+tan (prop

12)

radic1 + 1199042

1198742

1198771

1199101198742

= 1199101198721

+1199041198742

tan (prop12)

radic1 + 1199042

1198742

1198771

(15)



2 1199102


1and 119877

2


y2

y

x

K2

sO2

sK2

O2

x2

ΔyTC2120574

y2

lTC2

x2






2(119897) 1199102(119897) 119897 isin ⟨0 119897

2⟩ are



0(ie for 119897 isin ⟨0 119897






the slope 1199041198742

= tan(atan 1199041198701





obtained119909 = 119909

1198742+ 1199092(119897) = 119909

1198742+ 1199092(119897) cos 120574 ∓ 119910

2(119897) sin 120574

119910 = 1199101198742

+ 1199102(119897) = 119910

1198742plusmn 1199092(119897) sin 120574 + 119910

2(119897) cos 120574

119897 isin ⟨0 1198972⟩

(16)


119897TC2 =10038161003816100381610038161199092 (1198972)

1003816100381610038161003816 =10038161003816100381610038161199092 (1198972) cos 120574 ∓ 119910

2(1198972) sin 120574

1003816100381610038161003816

Δ119910TC2 =10038161003816100381610038161199102 (1198972)

1003816100381610038161003816 =1003816100381610038161003816plusmn1199092 (1198972) sin 120574 + 119910

2(1198972) cos 1205741003816100381610038161003816

(17)


1199091198702

= 1199091198742

+ 119897TC2

1199101198702

= 1199101198742

+ Δ119910TC2(18)



system is

1199041198702

= tan [Θ (1198972) plusmn 120574] (19)

x

y

y

3

O3

x3

ΔyTC3

K3

sK3

lTC3

120575

120575

x3

y3











3(119897) 1199103(119897) 119897 isin ⟨0 119897

3⟩ are ob-

tained


3 1199103(ACS3)


1199093(119897) = 119909

3(119897) cos 120575 minus 119910

3(119897) sin 120575

1199103(119897) = 119909

3(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(20)



atpoint119870


1199041198703

= tan(minus1198973

21198772

+ 120575) (21)



y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2


8 Circular Arc CA2




1198702and the

end of 1199041198703





1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)


119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)



1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|



119897CA2 = 1199091198703

minus 1199091198702 (26)


prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)


1198971198772

= prop21198772 (28)



3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)


119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)




(3)= 1198863+ 1198872119909 (Figure 6)


12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)



11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













when 1198771



1gt 1198772



1 1199101(ACS1)





CA1


1199101(119897) 119897 isin ⟨0 119897

1⟩


1)| =



1199101015840

1(1198971) = minus tan(

1198971

21198771

) (8)


119909 (119897) = 1199091(119897) cos 120587

4minus 1199101(119897) sin 120587

4

119910 (119897) = 1199091(119897) sin 120587

4+ 1199101(119897) cos 120587

4

119897 isin ⟨0 1198971⟩

(9)



1) 1199091198701

= 1198971198701198751


1198701at point

1198701is

1199041198701

= tan(minus1198971

21198771

+120587

4) (10)

5 Circular Arc CA1




1199091198781

= 1199091198701

+1199041198701

radic1 + 1199042

1198701

1198771

1199101198781

= 1199101198701

minus1

radic1 + 1199042

1198701

1198771

(11)

y1

yx1

xO

K1

M1

sK1sO2

1205874

R1

R1

xK1 xM1 xO2xS1

S1

O2prop1

prop1



119910 (119909)CA1 = 1199101198781

+ [1198772

1minus (1199091198781

minus 119909)2

]12

119909 isin ⟨1199091198701 1199091198742⟩

(12)


prop1=

1198971198771

1198771

(13)



1199041198742

= tan (atan 1199041198701

minus prop1) (14)





1199091198742

= 1199091198721

+tan (prop

12)

radic1 + 1199042

1198742

1198771

1199101198742

= 1199101198721

+1199041198742

tan (prop12)

radic1 + 1199042

1198742

1198771

(15)



2 1199102


1and 119877

2


y2

y

x

K2

sO2

sK2

O2

x2

ΔyTC2120574

y2

lTC2

x2






2(119897) 1199102(119897) 119897 isin ⟨0 119897

2⟩ are



0(ie for 119897 isin ⟨0 119897






the slope 1199041198742

= tan(atan 1199041198701





obtained119909 = 119909

1198742+ 1199092(119897) = 119909

1198742+ 1199092(119897) cos 120574 ∓ 119910

2(119897) sin 120574

119910 = 1199101198742

+ 1199102(119897) = 119910

1198742plusmn 1199092(119897) sin 120574 + 119910

2(119897) cos 120574

119897 isin ⟨0 1198972⟩

(16)


119897TC2 =10038161003816100381610038161199092 (1198972)

1003816100381610038161003816 =10038161003816100381610038161199092 (1198972) cos 120574 ∓ 119910

2(1198972) sin 120574

1003816100381610038161003816

Δ119910TC2 =10038161003816100381610038161199102 (1198972)

1003816100381610038161003816 =1003816100381610038161003816plusmn1199092 (1198972) sin 120574 + 119910

2(1198972) cos 1205741003816100381610038161003816

(17)


1199091198702

= 1199091198742

+ 119897TC2

1199101198702

= 1199101198742

+ Δ119910TC2(18)



system is

1199041198702

= tan [Θ (1198972) plusmn 120574] (19)

x

y

y

3

O3

x3

ΔyTC3

K3

sK3

lTC3

120575

120575

x3

y3











3(119897) 1199103(119897) 119897 isin ⟨0 119897

3⟩ are ob-

tained


3 1199103(ACS3)


1199093(119897) = 119909

3(119897) cos 120575 minus 119910

3(119897) sin 120575

1199103(119897) = 119909

3(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(20)



atpoint119870


1199041198703

= tan(minus1198973

21198772

+ 120575) (21)



y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2


8 Circular Arc CA2




1198702and the

end of 1199041198703





1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)


119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)



1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|



119897CA2 = 1199091198703

minus 1199091198702 (26)


prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)


1198971198772

= prop21198772 (28)



3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)


119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)




(3)= 1198863+ 1198872119909 (Figure 6)


12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)



11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y2

y

x

K2

sO2

sK2

O2

x2

ΔyTC2120574

y2

lTC2

x2






2(119897) 1199102(119897) 119897 isin ⟨0 119897

2⟩ are



0(ie for 119897 isin ⟨0 119897






the slope 1199041198742

= tan(atan 1199041198701





obtained119909 = 119909

1198742+ 1199092(119897) = 119909

1198742+ 1199092(119897) cos 120574 ∓ 119910

2(119897) sin 120574

119910 = 1199101198742

+ 1199102(119897) = 119910

1198742plusmn 1199092(119897) sin 120574 + 119910

2(119897) cos 120574

119897 isin ⟨0 1198972⟩

(16)


119897TC2 =10038161003816100381610038161199092 (1198972)

1003816100381610038161003816 =10038161003816100381610038161199092 (1198972) cos 120574 ∓ 119910

2(1198972) sin 120574

1003816100381610038161003816

Δ119910TC2 =10038161003816100381610038161199102 (1198972)

1003816100381610038161003816 =1003816100381610038161003816plusmn1199092 (1198972) sin 120574 + 119910

2(1198972) cos 1205741003816100381610038161003816

(17)


1199091198702

= 1199091198742

+ 119897TC2

1199101198702

= 1199101198742

+ Δ119910TC2(18)



system is

1199041198702

= tan [Θ (1198972) plusmn 120574] (19)

x

y

y

3

O3

x3

ΔyTC3

K3

sK3

lTC3

120575

120575

x3

y3











3(119897) 1199103(119897) 119897 isin ⟨0 119897

3⟩ are ob-

tained


3 1199103(ACS3)


1199093(119897) = 119909

3(119897) cos 120575 minus 119910

3(119897) sin 120575

1199103(119897) = 119909

3(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(20)



atpoint119870


1199041198703

= tan(minus1198973

21198772

+ 120575) (21)



y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2


8 Circular Arc CA2




1198702and the

end of 1199041198703





1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)


119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)



1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|



119897CA2 = 1199091198703

minus 1199091198702 (26)


prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)


1198971198772

= prop21198772 (28)



3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)


119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)




(3)= 1198863+ 1198872119909 (Figure 6)


12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)



11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y

x

S2

R2

R2sK3

K3

yK2sK2

prop2

prop2

K2

xK3xK2

yS2

xS2


8 Circular Arc CA2




1198702and the

end of 1199041198703





1199091198782

= 1199091198702

minus1199041198702

radic1 + 1199042

1198702

1198772

1199101198782

= 1199101198702

+1

radic1 + 1199042

1198702

1198772

(22)


119910 (119909)CA2 = 1199101198782

minus [1198772

2minus (119909 minus 119909

1198782)2

]12

119909 isin ⟨1199091198702 1199091198703⟩

(23)

From condition

119910 (1199091198703)1015840

CA2 = minus1199091198703

minus 1199091198782

[1198772

2minus (1199091198703

minus 1199091198782)2

]12

= 1199041198703 (24)



1199091198703

= 1199091198782

+1199041198703

radic1 + 1199042

1198703

1198772

1199101198703

= 1199101198782

minus1

radic1 + 1199042

2

1198772

(25)

y

x

|yP12|

|yP13|

P12y12

P13

Straight 1

Straight 2

Straight 3

x

y

yP13

O

O

|xP12

xP12

xP13

||xP13|



119897CA2 = 1199091198703

minus 1199091198702 (26)


prop2=1003816100381610038161003816atan 119904

1198703minus atan 119904

1198702

1003816100381610038161003816 (27)


1198971198772

= prop21198772 (28)



3(1199091198743 1199101198743) (Figure 4)

1199091198743

= 1199091198703

+ 119897TC3

1199101198743

= 1199101198703

+ Δ119910TC2(29)


119909 (119897) = 1199091198743

+ 1199093(119897) = 119909

1198743+ 1199093(119897) cos 120575 minus 119910

3(119897) sin 120575

119910 (119897) = 1199101198743

+ 1199103(119897) = 119909

1198742+ 1199093(119897) sin 120575 + 119910

3(119897) cos 120575

119897 isin ⟨minus1198973 0⟩

(30)




(3)= 1198863+ 1198872119909 (Figure 6)


12

11990911987512

=1198862

1 minus 1198872

= 11991011987512

(31)



11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











11990911987513

=1199101198743

minus 11988721199091198743

1 minus 1198872

= 11991011987513

(32)


= 11990911987513


= 11991011987513


119909= 119910= 11990911987512

minus 11990911987513

(33)


119884= 119884119874+ 119909cos120573 minus 119910

sin120573

119883= 119883119874+ 119909sin120573 + 119910

cos120573

(34)


119884 = 119884+ 119909 cos120573 minus 119910 sin120573

119883 = 119883+ 119909 sin120573 + 119910 cos120573

(35)



Straight 1

1198831= 2787802936485 minus 3365805196119884

1205931= 1859595714 rad

120573 = 1074197551 rad

Straight 2

1198832= 3585111027727 minus 4592528563119884

1205932= 1785194693 rad


119884119874= 649874504911m

119883119874= 600451950986m

Design Data

V = 90 kmh1198971= 90m (clothoid)

1205951= 017ms3

1198911= 278mms

1198771= 500m (119897

1198771= 150m)

ℎ1= 100mm

1198861198981


1205952= 018ms3

minus400

minus300

minus200

minus100

0

100

200

300

400

500

minus200 0 200 400 600 800

y (m

)

x (m)


1198912= 266mms

1198772= 600m (119897

1198772= 170359m)

ℎ2= 70mm

1198861198982

= 058ms21198973= 70m (clothoid)

1205953= 021ms3

1198913= 250mms


2is cant

along arc CA2 1198861198981










119884= 6499053754m

119883= 6003480469m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus257909 + 086126119909


1198772= 170359m




Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

492704 274569 0764 065271198743

546613 319204 0861 07110



1198772= 80350m)

Straight 2

1198832= 10680398947315 minus 1551050124119884

1205932= 1635179667 rad


119884= 6498784841m

119883= 6004385577m

Straight 1

119910(1)

= 119909

Straight 2

119910(2)

= minus14706 + 062832119909







Point 119909 (m) 119910 (m) 119904 Θ (rad)119874 0 0 1000 078541198701

65496 61680 0834 069541198742

193254 139202 0417 039541198702

344580 191579 0386 036871198703

417370 225465 0550 050261198743

477345 261541 0628 05608

minus200

minus100

0

100

200

300

400

minus200 0 200 400 600 800

y (m

)

x (m)



11 Summing-Up







Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Competing Interests


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Design of Reverse Curves Adapted to the ...downloads.hindawi.com/journals/ace/2016/6503962.pdf · Design of Reverse Curves Adapted to the Satellite Measurements