521
1 INTRODUCTION
Aftertheintroductionofmarinenavigationradarsfor
collisionavoidancepurposes,approachparametersof
tracked objects were determined in a graphical
manner by manual radar plots. At the beginning,
analyticalformulaefordeterminationofmotion and
approach parameters and collision avoidance
manoeuvres were derived in a polar coordinate
system,naturalforradarplots,withinputva
luessuch
as distances, bearings, velocities, courses and their
changes.
The introduction of computer controlled
AutomaticRadarPlotting Aids (ARPAs) hascreated
the need for algorithms for determination of motion
and approach parameters but calculations in such
systemsaresystem‐specificbecausetheyusemainlya
Cartesiancoordinatesystem.Thisiscausedby:
simple equations of motion in a system of
Cartesiancoordinates,
simple esti
mation algorithms for motion
parameters in digital tracking filters (because for
objects travelling with constant velocities and
courses their polar position changes – radial and
angular velocities‐are not constant and in
Cartesiancoordinatesareconstant),
reductionofnumberoftrigonometricandcircular
funct
ions which, when used in numerical
calculations, are connected with longer and less
accuratecalculations.
Publication of such algorithms is very rare‐
Jakševič (1967) and Lord (1968) are two of the very
few that have been published. Only the second has
some derivations and all of them are limit
ed to the
predicted object CPA (Closest Point of Approach)
distanceandthetimeintervaltoitsoccurrence.These
parametersarewell‐establishedapproachparameters
usedincollisionavoidancesystemsfeaturingARPAs
aswellasinmanualradarplots.
Inthi
spaperotherapproachparameterssuchas:
the predicted object distance on course and the
timeintervaltoitsoccurrence,
thepredictedobject distance abeam and the time
intervaltoitsoccurrence,
thepredictedobjectdistanceandthetimeinterval
toitsoccurrence
Approach Parameters in Marine Navigation – Graphical
Interpretations
A.S.Lenart
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Inthispaperapproachparameterswidelyusedcollisionavoidancesystemssuchasthedistanceat
closestpointofapproachandtimetotheclosestpointofapproachandlessknownandusedasthedistanceon
course,thedistanceabeamandanydistanceandtheti
mesintervalstotheiroccurrencesarederived,analyzed
andgraphicallyinterpretedinthecombinedcoordinatesystemforpositionandmotion.Theycanbeusedin
collision avoidance systems and for reversed purposes‐manoeuvring to required approach parameters,
intentionalapproachesandnavaltacticalmanoeuvres.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 3
September 2017
DOI:10.12716/1001.11.03.19
522
arepresentedinanalyticalandgraphicalform.
This paper is mainly a combined and shortened
versionofLenart1999a,1999b,2000a,2000band2010
withemphasistographicalinterpretations.
2 ASSUMPTIONSANDINPUTPARAMETERS
For the purposes of this analysis, own vessel and
extraneous objects of interest are regarded as if the
massofeachobjectwasconcentrated
atapoint.Itwill
be assumed that all moving external objects are
travelling at constant velocity and course. In the
movable plane tangential to the Earth’s surface
CartesiancoordinatessystemOx,Oy(Fig.1),withOy
pointing North, O is the present position of own
vessel.Itisalso
beassumedthatmanualplotsorthe
radar processing and tracking (ARPA) or AIS
(AutomaticIdentificationSystem)hasyielded:
the present relative position of each object of
interestX,Y,
thecomponentsofitstruevelocityVtx,Vtyand/or
thecomponentsofitsrelativevelocityVrx,Vry.
Figure1.Inputparameters
Therelationshipofownandanobject’svelocities
canbedescribedbyequations
V
tx=Vrx+Vx (1)
V
ty=Vry+Vy (2)
2
ty
2
txt
VVV (3)
2
ry
2
rxr
VVV (4)
whereV
x,Vy=ownvelocitycomponents,
V
x=Vsin (5)
V
y=Vcos (6)
2
y
2
x
VVV (7)
where=owncourse(theanglemeasuredclockwise
fromOytoV).
Fromtheabove
V
tx=Vrx+Vsin (8)
V
ty=Vry+Vcos (9)
and
V
rx=Vtx‐Vsin (10)
V
ry=Vty‐Vcos (11)
Ownandanobject’smotionparametersshouldbe
either ground or sea referenced and a drift angle is
assumedtobezero.
3 COMBINEDCOORDINATESSYSTEMFOR
POSITIONANDMOTION
A conventional PPI displays the position of each
object by plotting them in polar (r, ‐
distance,
bearing)orCartesian(x,y)coordinates.Ifweapplya
scaling factorτto the velocity coordinates (V, ) or
(V
x,Vy)suchthat
r=Vτ (12)
= (13)
x=V
xτ (14)
y=V
yτ (15)
thenthepositionandvelocitycoordinatescoupledby
timeτcanbeplottedonacommondisplay.Onsucha
display, besides own velocity vector (V, ) and
positionsofobjects(X,Y), vectorsoftheirstrue (V
tx,
V
ty) or relative motion (Vrx, Vry) canbe plotted in a
coordinatesystemparallelshiftedtothepoint(X,Y).
Thiscorrespondstoequations
x=X+V
txτ (16)
y=Y+V
tyτ (17)
or
x=X+V
rxτ (18)
523
y=Y+V
rxτ (19)
In a graphical interpretation the above equations
mean that vectors of velocity are plotted in this
coordinatessystemofpositionasτ–minutesvectors
ofpredictedmotiondrawnfromthepresentpositions
ofownvesselandobjects.Thefullareaof(V
x,Vy)or
(V, ) is the area our manoeuvres which can be
limited by our maximum velocity V
max – the circle
centredat(0,0)andhavingradiusV
max.
4 EQUATIONSOFRELATIVEMOTION
Therelativepositionofanobject,attimet,isgivenby
X(t)=X+V
rx t (20)
Y(t)=Y+V
ry t (21)
IfD(t)isthedistancetoanobjectattimet,then
t)YVXV(2tVR
)t(Y)t(X)t(D
ryrx
22
r
2
22
(22)
oraftersquaringbothsidesandrearrangements
0)t(DRt)YVXV(2tV
22
ryrx
22
r
(23)
where
22
YXR
(24)
5 EQUATIONOFTRUEMOTION
AsubstitutionofEquations(10)and(11)toequation
of relative motion (23) and rearranging yields
equationoftruemotion
(V
2
+Vt
2
)TD
2
‐2[(X+VtxTD)sin+(Y+VtyTD)cos]VTD
+2(XV
tx+YVty)TD+R
2
‐D
2
=0 (25)
6 CPADISTANCEANDTIME
6.1 EquationsforD
CPAandTCPA
In equation of relative motion (23) the distance
reachesaminimumD
CPA(Lenart1983)
r
rxry
CPA
V
YVXV
D
(26)
andtimetoachieveCPA‐T
CPA
2
r
ryrx
CPA
V
YVXV
T
(27)
6.2 DerivationofEquationV=f(
,DCPA)
FromEquations(26)and(4),bysquaringbothsides
and rearranging terms, we obtain a quadratic
equationinV
ry
(X
2
‐D
2
CPA)V
2
ry‐2XYVrxVry+(Y
2
‐D
2
CPA)V
2
rx=0 (28)
whosesolutionis
V
ry=ADCPAVrx (29)
where
2
CPA
2
2
CPA
2
CPA
DCPA
DX
DRDXY
A
(30)
A substitution of Equations (10, 11) to Equation
(29)andrearrangingyields
Figure2.LinesDCPA=const.andcirclesTCPA=const.
=0.2h,X=Y=5n.m.,Vtx=‐10kt,Vty=10kt
cossinA
B
V
DCPA
DCPA
(31)
where
B
DCPA=ADCPAVtx‐Vty (32)
andrealsolutionsexistif(Equation(30))
524
CPA
DR (33)
Equation (31) gives the velocity V which own
vessel must adopt to achieve the required CPA
distance D
CPA (in respect to the selected object) for
various assumed own courses , but we should
searchforsolution
0V
(34)
andV,forwhich
0T
CPA
(35)
6.3 Graphicalinterpretation
Agraphicalinterpretation ofA
DCPAandBDCPAcanbe
obtained on a plotting in Cartesian coordinates of
own velocity (V
x, Vy) by substituting Equations (5)
and(6)toEquation(31)
V
y=ADCPAVx‐BDCPA (36)
Inthesecoordinatesallpointscorrespondingtoa
given value of D
CPA will lie on two straight lines
havingslopesA
DCPA+andADCPA
(valuesADCPAwith+
or–inthe numerator of Equation (30) respectively),
intersecting inthepoint (V
tx, Vty) and cuttingthe Vy
axis at‐B
DCPA+ and‐BDCPA
(the values of BDCPA
obtained on putting respective values of A
DCPA in
Equation(32)).
Inthecombinedcoordinatesystem(Equations(12‐
15))Equations(32)and(36)transformrespectivelyto
cossinA
B
r
DCPA
DCPA
(37)
and
y=A
DCPAx‐BDCPAτ (38)
Figure2illustratesafamilyoflines(36)or(37)and
(38)forvariousD
CPAforanexemplaryobject.
6.4 CollisionThreatParametersArea
WecanuseEquations(36)or(38)and(30),(32)witha
substitutionof
D
CPA=DS (39)
whereD
S=assumedsafevalueofDCPA(thresholdsset
by the system’s operator), to draw lines V(D
CPA=DS).
Forthatpartoftheareaboundedbytheselinesand
withinwhichT
CPA>0,ownvessel’smotionparameters
are leading to a threat of collision. This region is
namedtheCollisionThreatParametersArea–CTPA
(Lenart1983)anewradardisplayandplottechnique.
ThesizeandthepositionofCTPAareindependentof
ownvessel’smotionparameters.Ifwealsoplotown
vessel’svelocityvector(actualorsimulated)andthis
terminates inside the CTPA then there is a collision
threat. Any manoeuvre, by change of course and/or
velocity,whichdeflectstheend of this vector outof
theCTPAisapossiblemeansof avoiding the given
threat.
6.5 Derivationofequation
V=f(
,TCPA)
A substitution of Equations (4), (10) and (11) to
Equation(27)givesaquadraticequationinV
T
CPAV
2
‐[(X+2VtxTCPA)sin+(Y+2VtyTCPA)cos]V
+(V
t
2
TCPA+XVtx+YVty)=0 (40)
whosesolutionis
TCPA
2
TCPATCPA
TCPATCPA
C)cosBsinA(
cosBsinAV
(41)
where
CPA
txTCPA
T2
X
VA
(42)
CPA
tyTCPA
T2
Y
VB
(43)
CPA
tytx
2
tTCPA
T
YVXV
VC
(44)
Realsolutionsexistif
(
TCPA
2
TCPATCPA
C)cosBsinA
(45)
Equation(41)canyielduptotwovelocities
V 0
whichownvesselmustadopttoachievetherequired
timetoCPA‐T
CPA(inrespect to theselectedobject)
forvariousassumedowncourses.
6.6 Graphicalinterpretation
A graphical interpretation of solutions given by
Equation (41) can be obtained in Cartesian
coordinates of own velocity (V
x, Vy) substituting
Equations(5–7)toEquation(41)
2
CPA
2
TCPAy
2
TCPAx
T2
R
)BV()AV(
(46)
The above equation reveals that the locus of
points,forwhichT
CPAisaconstant,isacirclecentred
at(A
TCPA,BTCPA)andhavingradius
)T2/(R
CPA
.
Figure 2 illustrates a family of circles for various
valuesofT
CPA≥0foranexemplaryobject.
525
Transformation of Equation (46) to (x, y)
coordinates(Equations(14‐15))yields
2
CPA
2
TCPA
2
TCPA
T2
R
)By()Ax(
(47)
6.7 Derivationofequation
=g(V,DCPA)
Ifwesearchforowncoursewhichwillleadtothe
requiredCPAdistanceD
CPAatanassumedownspeed
Vthenwecangetaninversefunction=g(V,D
CPA)to
the function V=f(, D
CPA) by a substitution to
Equation(31)thetrigonometricidentities
2
tan1
2
tan2
sin
2
(48)
2
tan1
2
tan1
cos
2
2
(49)
whichwillresultinequation
0)BV(
2
tanVA2
2
tan)BV(
DCPA
DCPA
2
DCPA
(50)
anditssolution
VB
BV)1A(VA
2
tan
DCPA
2
DCPA
22
DCPADCPA
(51)
Realsolutionsexistif
1A
B
V
2
DCPA
2
DCPA
2
and
RD
CPA
(52)
andEquation(51)cangiveuptofourowncourses,
whichwillleadtotherequiredCPAdistanceD
CPAat
an assumed own speed V if they additionally fulfil
Condition (35). Graphically these solutions are the
intersectionpointsoflinesV(D
CPA=const.)withacircle
V=const.(acirclecentredat(0,0)andhavingradius
V).
7 DISTANCEANDTIMEONCOURSE
7.1 EquationsforD
candTDc
ThepredictedobjectdistanceoncourseDc(Figure1)
and the time interval to its occurrence T
Dc are
sometimes used as additional criteria for collision
threat.TheyareusedinsomeARPAsforcalculation
ofBCR‐thebowcrossingrangeandBCT‐thebow
crossing time. These parameters are given by
equations(Lenart1999b)
cosVsinV
YVXV
D
rxry
rxry
c
(53)
cosVsinV
sinYcosX
T
rxry
Dc
(54)
D
c>0meansthatanobjectwillcrossthecourseof
own vessel ahead and D
c<0 that an object will cross
thecourseastern.Interpretation of the signof T
Dcis
similartoT
DCPA–forTDc<0Dchas taken place inthe
past.
7.2 DerivationofequationV=f(
,Dc)
AsubstitutionofEquations(10)and(11)toEquation
(53)andrearrangingyields
cossinA
VVA
V
Dc
tytxDc
(55)
where
sinDX
cosDY
A
c
c
Dc
(56)
Figure3.LinesDc=const.andTDc=const.
=0.2h,X=Y=5n.m.,Vtx=‐10kt,Vty=10kt
526
Equation(57)issimilartoEquation(31)with(32)
but A
Dc is dependent on . Equation (57) gives the
speedV,whichownvesselmustadopttoachievethe
required distance on course D
c (in respect to the
selected object) for various assumed own courses ,
butweshouldsearchforsolution
0V
(57)
andforwhich
0T
Dc
(58)
Condition(58)meansthattheapproachoncourse
isatpresentorwillbeinthefuture,notinthepast.
7.3 DerivationofEquation
=g(TDc)
SubstitutingEquations(10)and(11)toEquation(54)
resultsinEquation
cosVsinV
cosXsinY
T
txty
Dc
(59)
Thisequationrevealsthatthetimetodistanceon
course T
Dc is independent of own velocity V.
Thereforefromtheabove
Dcty
Dctx
TVY
TVX
tan
(60)
and Equation (60) gives own course , which will
leadtotherequiredtimetodistanceoncourseT
Dc.
7.4 Graphicalinterpretation
A graphical interpretation of solutions given by
Equation (60) can be obtained in Cartesian
coordinatesofownvelocity(V
x,Vy)
x
Dcty
Dctx
y
V
TVY
TVX
V
(61)
andthelocusofpoints,forwhichT
Dcisaconstant,is
a straight line crossing the origin of coordinates.
Figure 3 illustrates a family of lines (53) for various
required D
c and a family of straight lines (61) for
variousvaluesofT
Dc≥0foranexemplaryobject.
7.5 SignofD
c
It can proved (Lenart 2010) that if for a given own
course exists own velocity V(D
CPA=0)>0 with
T
DCPA(DCPA=0)>0 then for V>V(DCPA=0) an object will
passastern(D
c<0),andforV<V(DCPA=0)anobjectwill
pass ahead (D
c>0). This sign of Dc is illustrated in
Figure2.
8 DISTANCEANDTIMEABEAM
8.1 EquationsforD
abandTDab
ThepredictedobjectdistanceabeamD
abandthetime
interval to its occurrence T
Dab are sometimes used
additional criteria for collision threat. These
parametersaregivenbyequations(Lenart2000a)
cosVsinV
YVXV
D
ryrx
rxry
ab
(62)
cosVsinV
cosYsinX
T
ryrx
Dab
(63)
D
ab>0 means that an object will be abeam on the
starboardsideofownvessel,andD
ab<0thatanobject
willbeabeamon the port side. Interpretation of the
signT
DabissimilartoTDCPA–forTDab<0Dabhastaken
placeinthepast.
8.2 DerivationofequationV=f(
,Dab)
SubstitutingEquations(10)and(11)toEquation(62)
andrearrangingyields
cossinA
VVA
V
Dab
tytxDab
(64)
where
cosDX
sinDY
A
ab
ab
Dab
(65)
Equation(64)issimilartoEquation(31)with(32)
but A
Dab is dependent on . Equation (64) gives the
velocityV,whichownvessel must adopt to achieve
the required distance abeam D
ab (in respect to the
selected object) for various assumed own courses ,
butweshouldsearchforsolution
V 0 (66)
andV,forwhich
0T
ab
(67)
Condition(67) meansthattheapproachabeam is
atpresentorwillbeinthefuture,notinthepast.
8.3 DerivationofequationV=f(
,TDab)
SubstitutingEquations(10)and(11)toEquation(63)
resultsinEquation
VcosVsinV
cosYsinX
T
tytx
Dab
(68)
hence
527
cosBsinAV
TDabTDab
(69)
where
Dab
txTDab
T
X
VA
(70)
Dab
tyTDab
T
Y
VB
(71)
Equation(69)canyieldthevelocity
V 0
,
which
ownvesselmustadopttoachievetherequiredtimeto
the distance abeam T
Dab (in respect to the selected
object)forvariousassumedowncourses.
8.4 Graphicalinterpretation
A graphical interpretation of solutions given by
Equation (69) can be obtained in Cartesian
coordinates of own velocity (V
x, Vy) substituting
Equations(5)through(7)toEquation(69)
2
TDab
2
1
2
TDab
2
1
2
TDab
2
1
y
2
TDab
2
1
x
BA
)BV()AV(
(72)
The above equation reveals that the locus of
points,forwhichT
Dabisaconstant,isacirclecentred
at
11
22
(,)
TDab TDab
AB
and crossing the origin of a
coordinatessystem.
Figure4illustratesafamilyoflines(64)forvarious
required D
ab and a family of circles (72) for various
valuesofT
Dab≥0foranexemplaryobject.
Figure4.LinesDab=const.andcirclesTDab=const.
=0.2h,X=Y=5n.m.,Vtx=‐10kt,Vty=10kt
8.5 SignofDab
Itcanbeproved(Lenart2010)thatthesignofformula
under the modulus in Equation (26) is the sign
oppositetothesignofthedistanceabeamD
ab,ifown
course is equal to bearing to an object (when T
Dab>0
andT
CPA>0).
9 DISTANCEANDTIME
9.1 DerivationofequationT
D=f(Vr,D)
SolvingaquadraticequationofrelativemotioninT
D
(23)weobtain(fort=T
D)
2
r
2
rxry
2
rryrx
D
V
)YVXV()DV()YVXV(
T
(73)
or(Equations(26)and(27))
r
2
CPA
2
CPAD
V
DD
TT
(74)
andrealsolutionsexistif
CPA
DD (75)
Equation(73)or(74)givestimeT
Dtoachievethe
distanceDtotheselectedobject.
9.2 Timetosafedistance
SinceinEquation(73)or(74)Dcanbeanydistance,
we can substitute D=D
S (as in Section 6.5) and this
timecanbenamedthetimetosafedistanceandhave
beenproposed analyzed and applied to detection of
dangerousobjectsandtodisplaythepossibleevasive
manoeuvres (accurate Predicted Areas of Danger
insteadoftheirgeometricalapproximations)inLenart
(2015).
9.3 Derivationof
equationV=f(
,D,TD)
SolvingaquadraticequationinV(25)weobtain
VTd
2
VTdVTd
VTdVTd
C)cosBsinA(
cosBsinAV
(76)
where
D
txVTd
T
X
VA
(77)
D
tyVTd
T
Y
VB
(78)
528
2
D
22
D
tytx
2
tVTd
T
DR
T
)YVXV(2
VC
(79)
Realsolutionsexistif
(
VTd
2
VTdVTd
C)cosBsinA
(80)
Equation (76) can yield up to two own velocities
0V
,whichownvesselmustadopttoachievethe
required distance D at the required time TD (in
respect to the selected object) for various assumed
owncourses.
9.4 Graphicalinterpretation
A graphical interpretation of solutions given by
Equation (76) can be obtained in Cartesian
coordinates of own velocity
(Vx, Vy) substituting
Equations(5–7,24)toEquation(25)
2
D
2
VTdy
2
VTdx
T
D
)BV()AV(
(81)
Theaboveequationrevealsthatthelocusofpoints
forwhichDandT
Dareconstantsisacirclecentredat
(A
VTd,BVTd)andhavingradius DT
D
/ .
Figure 5 illustrates a family of circles (81) for
variousrequiredDandT
Dforanexemplaryobjectas
well as, for comparison, circles V(T
CPA=const.)
(Equation(47))andthelineV(D
CPA=0)(Equation(36)).
Figure5.CirclesD,TD=const.
=0.2h,X=Y=5n.m.,Vtx=‐10kt,Vty=10kt
It should be noted from Equations (73) and (74)
thattherecanexisttwotimesofapproachatdistance
D: shorter‐approach at the point A (Figure. 1) and
longer‐approach at the point B. If only the earlier
(thefirst)approachisinterestingforus,then,forthis
time condition
TCPA>TD for selected own motion
parameters V, should be fulfilled. This criterion
fulfilpointsofcircleV(D,T
D),whichlieinsideacircle
V(T
CPA=const.)forthesametime(markedinFigure5
bythethickerline).
Graphically solutions of Equation (76) are the
intersectionpointsofthecircleV=f(D,T
D)withaline
ofanassumedowncourse.
10 TIMETOMANOEUVRE
It has to be emphasized that the manoeuvres
calculatedinthepreviousSectionsarekinematicand
should be undertaken immediately. If we require to
have the time lapse t for calculations, for the
decisiontoinitiate
amanoeuvreandfortheexecution
of the calculated manoeuvre then (X, Y) in the
previous equations should be replaced by (X
t, Y
t)
respectively,givenbyequations
X
t=X+Vrxt (82)
Y
t=Y+Vryt (83)
11 CONCLUSIONS
Formulae for such approach parameters as the
predictedobjectCPAdistance,thedistanceoncourse,
the distance abeam, any distance and the times
intervals to their occurrences in a Cartesian
coordinatessystemhavebeenderived,analyzedand
graphically interpreted in the combined coordinate
systemforposition
andmotion.
More than 80 directly applicable formulae for
collision avoidance and quite reversed purposes‐
manoeuvring to required approach parameters,
intentionalapproachesandnavaltacticalmanoeuvres
havebeenprovided–almostallofthemarederived
fromonebasicequationofrelativemotion.
Theintroductionsuchauxiliaryparametersasthe
distance
on course and the distance abeam, apart
from the main approach parameter‐the distance to
CPA,makespossible:
aresignationfromassumption(Section2)thatthe
mass of each object was concentrated at a point
which can have significance when distances are
comparabletoobjects’dimensions,
more complete
analysis of the main parameters
(e.g.conclusionsinSections7.5,8.5and9.2).
Interpretationandplottingofderivedformulaein
the combined coordinate system of position and
motionenabletheirapplicationsaswellincomputer
controlled radar systems as in manual radar plots –
some of them are very simple in
manual plots as it
hasbeenshowninLenart(1983).
529
Itmustbeemphasizedthatowingtothefactthat
in the derived formulae trigonometric and inverse
trigonometric functions of extraneous objects’
parametersarenotused,computercalculationscanbe
fasterandmoreaccurate.
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