5. Description of the simulation tool
5.3 Introduction into the maritime navigation
Geographic coordinate system – is a coordinate system that enables every location on the Earth to be specified by a set of numbers or letters. The coordinates are often chosen in such a way that one of the numbers represents vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude, longitude and elevation (Pros-Wellenhof and Bernhard 2007).
Latitude – is the angle between the equatorial plane and the straight line that passes through that point and is normal to the surface of a reference ellipsoid which approximates the shape of the Earth (is defined in range[ 90 , 90 ] ).
Longitude – is the angle east or west from a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often improperly called great circles), which converge at the north and south poles (is defined in range [ 180 ,180 ] ).
Figure 29. A graphical representation of a sphere coordinates based system of geographical coordinates
Spherical distance (or the great-circle/orthodromic distance) – is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). Spherical distance between
points ( 1, 1) and ( 2, 2) might be calculated by means of the Haversine formula, which is represented in (5.3.1).
2 1 2 1
2 2
2 1
,
a = sin ( /2) + cos( )cos( )sin ( /2) c = 2atan2( a , (1-a) )
d = R c
, (5.3.1)
where c is the Earth radius equal to 6371 Km.
Haversine formula is incorporated by means of the VBA function, shown in listing A-4 in the appendix.
Bearing – is the angle between a line connecting us ( 1, 1) and another object
2 2
( , ), and a north-south line. Bearing is calculated by means of the following formula:
= atan2( sin().cos(2), cos( )sin(1 2) - sin( )cos(1 2)cos() ). (5.3.2)
This formula is incorporated into Arena by means of VBA function, shown in listing A-5.
Calculation of the destination point’s latitude and longitude ( 2, 2) with the initial coordinates( 1, 1), bearing ( 1, 1) and travelling distance d given is carried out by means of the corresponding formulas:
2 = asin( sin( )cos(d/R) + cos( )sin(d/R)cos( ) )1 1 , (5.3.3) 2= 1 + atan2( sin( )sin(d/R)cos( ), cos(d/R)-sin( )sin( 1 1 2) ). (5.3.4)
These formulas are incorporated by means of VBA functions, represented in listing A-6.
Intersection of two great circles defined by the arcs is a set of geographical pairs of coordinates that define the point where two great circles intersect.
Figure 30. Intersection of two great circles defined by the arcs
A unit vector might be created from the center of the Earth to any point on its surface, say it is defined by De Cart 3d coordinates, say
e{ ,ex ey ez, } {cos( ) cos( ), cos( ) sin( ), sin( )}
. (5.3.5)
Obviously and then might be inverted in the following way:
atan 2( ,ez sqrt ex( 2ey2)), (5.3.6)
atan 2(ey ex, ). (5.3.7)
The unit perpendicular (written in listing A-8) to the plane of any great circle is found with respect to the definition of vector multiplication of a pair of vectors in the following way:
P(e1,e2) = e1 X e2 ,
||e1 X e2||
(5.3.8)
where e1 X e2
is a vector cross-product of a pair of vectors:
e1 X e2 {y z -y z , z x -z x , x y -y x } = {x ,y ,z } 1 2 2 1 1 2 2 1 1 2 1 2 v v v
. (5.3.9)
VBA vector cross-product of a pair of vectors is shown in listing A-9. This cross-product of a pair of vectors in spherical coordinates is done by means of (5.3.10)-(5.3.12). Robust VBA
function for calculation of vector cross product of a pair of vectors, defined in spherical coordinates is shown in listing A-10 in the appendix of this thesis.
1 2 1 2 1 2
1 2 1 2 2 1
x -sin( - ) sin(( + )/2) cos(( - )/2) - - sin( + ) cos(( + )/2)sin(( - )/2)
v
, (5.3.10)
1 2 1 2 1 2
1 2 1 2 1 2
y sin( - ) cos(( + )/2) cos(( - )/2) + + sin( + ) sin(( + )/2) sin(( - )/2)
v
, (5.3.11)
zv cos( )cos(1 2)sin( 2- )1 . (5.3.12)
And ||e1 X e2||
is the length of the vector cross product of the corresponding pair of vectors, which has the beginning at the zero point of the coordinate system (listing A-11):
||e1 X e2||=||{x ,y ,z }||= x +y +zv v v v2 v2 v2
. (5.3.13)
In order to find the coordinates of intersections of great circles one should find the coordinates of the perpendicular to the plane surface, defined by the perpendiculars to the corresponding pair of great circles. In other words:
{ 1,1,1,1},{1,2,1,2} {2,1,2,1},{2,2,2,2} P P e( ( 1,1,e1,2), (P e 2,1,e2,2))
. (5.3.14)
Which then are inverted to the spherical coordinates
Figure 31. Algorithm for finding the intersection point illustration
So, the formal algorithm corresponding to (5.3.14) is simply as follows (listing A-12):
Find a perpendicular to the first circle’s plane
Find a perpendicular to the second circle’s plane
Find both perpendiculars to the plane formed by the perpendiculars to the circles’
planes
Make an inverse transformation of the coordinates of the perpendiculars so as to find coordinates of points of intersection of circles.
Sailing Speed Reduction takes place during sailing of a vessel as a result of exogenous factors such as significant wave height and/or wave directions influence. Gruzinskiy and Khokhlov (1977) suggested a continuous function (5.3.15) for vessel speed loss estimation depending on such parameters as deadweight, initial speed, wave angle and finally significant wave height.
vv0h(0.745 0.245 qw)(1 1.35 10 6Dv0), (5.3.15)
0 (0.745 0.2456 )
(1 1.35 10 (0.745 0.245 ))
w w
v h q
v Dh q
, (5.3.15*)
where v – is a reduced speed of a vessel in knots, v – is a speed of a vessel in calm sea in 0 knots, h – is a significant wave height in meters, q - is a wave angle in radians, isw D- is a deadweight of a vessel.
According to Gruzinskiy and Khokhlov (1977) this formula is applicable for vessels with deadweight changing in range from 4 to 20 kilotons and the speed in range from 9 to 20 knots, the standard error of this formula is said not to exceed 0.5 knots. (5.3.15*) is used to find the initial speed given the reduced one.
The reduced speed is calculated by means of VBA function in the ARENA, represented in listing A-16.
Fuel consumption is usually given for the design speed of a vessel and thus whilst sailing it might well slightly vary as a result of changes of the engine speed of vessels.
Formula (5.3.16) described in Norlund and Gribkovskaia (2013) provides the way to calculate real fuel consumption of a vessel during sailing.
3
( ) ( )0 v FC v FC v
v
, (5.3.16)
where v – is the design speed of a vessel, 0 v – is the engine seed of a vessel measured in the same units as the design speed, FC v – is the fuel consumption corresponding to the ( )0 design speed of a vessel.