ANNUAL of NAVIGATION

Transkrypt

POLISH ACADEMY OF SCIENCES
POLI SH NAVI GATION FORUM
ANNUAL
of
NAVIGATION
Janusz Uriasz
Determination of ship’s safe navigation lane
in the navigational information system
No. 17
2011
Programme Council
Andrzej Banachowicz
Andrzej Felski (Editor)
Marek Grzegorzewski
Jerzy Hajduk
Guenter Hein (Germany)
Michał Holec
Mirosław Jurdziński
Zdzisław Kopacz
David Last (United Kingdom)
Bolesław Mazurkiewicz
Stanisław Oszczak
Boris Rivkin (Russia)
Janusz Śledziński
Frantisek Vejrazka (Czech)
Peter Voersmann (Germany)
Mario Vultaggio (Italy)
Aleksander Walczak
Janusz Zieliński
Reviewers
Reza Ziarati
Andrzej Banachowicz
Executive Editor
Beata Różańska
[email protected]
Address for correspondence
Instytut Nawigacji i Hydrografii Morskiej
Akademia Marynarki Wojennej
ul. Śmidowicza 69
81-103 Gdynia
[email protected]
Published in affiliation
with the Gdańsk Branch of the Polish Academy of Sciences
We are indexed by BAZTECH (www.baztech.icm.edu.pl)
Citation
When citing papers from this volume the following reference should be used:
Author, Title, Annual of Navigation, 2011, No. 17
2
ANNUAL OF NAVIGATION
CONTENTS
List of notations .........................................................................................................5 Abbreviations.............................................................................................................7 Abstract ......................................................................................................................8 Introduction .........................................................................................................9 Historical background........................................................................................11 The present state of the research........................................................................13 The purspose and scope of the study .................................................................15 1. Characteristics of the navigational information system.....................................17 1.1. The navigational system ............................................................................17 1.1.1. Calculation of the fix by the navigational system ..............................19 1.1.2. Estimation of DR position calculated
by the navigational system .................................................................25 1.2. Information system ....................................................................................25 1.2.1. ECDIS — the marine information system .........................................26 1.2.2. On-bridge navigational decision support system ...............................27 1.2.3. Information in the information system...............................................28 2. Presentation of navigation-related information in the navigational
information system ............................................................................................33 2.1. Ship’s contour in the information system ..................................................34 2.1.1. Dilutioned contour .............................................................................34 2.1.2. Determination of ship’s dilutioned contour........................................38 2.2. Cartographic objects in the information system ........................................41 2.2.1. Positioning of a cartographic object in the navigational
information system.............................................................................45 2.2.2. Plotting a cartographic object in the navigational
information system.............................................................................49 2.2.3. Representation of cartographic objects in the navigational
information system.............................................................................51 2.2.4. The cardinality of the cartographic set ...............................................51 2.3. Cartographic zones in the information system...........................................53 3. Determination of ship’s navlane........................................................................61 3.1. Routing.......................................................................................................61 3.1.1. Algorithms of dynamic planning of own ship’s route........................62 3.1.2. Algorithms for determining the other ship’s route.............................64 3.2. Determination of the swept path ................................................................73 3.2.1. Determination of the swept path by analytical methods ....................74 3.2.2. Determination of the swept path by AI methods................................76 17/2011
3
3.3. The navlane................................................................................................79 3.3.1. The routing system .............................................................................79 3.3.2. Ship’s navlane ....................................................................................80 3.3.3. Determination of the safe navlane......................................................82 4. Navigation using the navlane incorporating navigational
information system ............................................................................................85 4.1. Remaining in the navlane...........................................................................85 4.2. Crossing navlanes ......................................................................................90 4.3. The anti-collision function of the navlane .................................................93 Summary............................................................................................................99 References .......................................................................................................103 4
LIST OF NOTATIONS
If not indicated in the text of this work, the symbols have the following meanings:
a
A
A′
A
AD
AR
AS
B
COG
COGa
COGr
CS
ΔCOG
Cρ
d
D
da
da
dD
df
dmin
attribute in the information system
cartographic space set CS representing ship’s contour (silhouette)
complement set of set A
set of attributes of the object in the information system
set of object domain (set) A
set of object arena (set) A
set of buffer zone of an object (set) A
cartographic space set CS representing a navigational danger
course over ground
course over ground according to AIS
course over ground according to radar
cartographic space of the navigation information system
deviation of COG from the preset course (determined by the fairway centre line)
relative course of an object
distance to an object
set of acceptable decisions
distance between own ship and AIS position of another object
distance from the domain end to the limit of unfavourable depths astern of the ship
length of the domain (increased by the buffer zone)
distance from the domain front to the limit of unfavourable depths ahead of the ship
minimum distance between ships, understood as the distance between two closest
points of both ships
dn
length of unfavourable depths area
dr
distance between own ship and radar position of another object
dr
distance from to the domain front to the frame limit ahead of the ship
e
vector of the canonic base
f
navigational position function
utility function defined by the Cartesian product U dz × J
fu
g
decision function
G
Jacobi matrix of navigational position function
h
height over an ellipsoid
H
entropy of information
I
information system
J
set of possible action results
m
dimension of the navigational space
n
dimension of measurement space
N
navigational space
NS
navigational space of the navigation information system
O(xb, yb) polar point
P
planning process
P(xo, yo) tangent point of the anticollision domain of another ship
P(xw, yw) tangent point of the anticollision domain of own ship
Pa
probability of an accident
Pp
covariance matrix of the forecast vector
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Determination of ship’s safe navigation lane in the navigational information system
PS
r
R
Ra
RB
Sa
SA
TB
TBr
TC
TRa
Δt
u
up
Δu
U
Udz
Va,Vr
VE
VN
Vp
Vr
VR
VSOG
Vρ
w
W
W
WPk
x
x
Δx
X
Δy
Y
z
zp
Z
ω
ρxy
σx
σ 2x
σxy
ϕ
λ
γ
6
prohibited space of the navigation information system
underkeel clearance
real space
risk of an accident
relative bearing (angle between the forward part of the centre line of the ship
and a line joining the closest points of both ships)
consequences of an accident
set of buffer points of set A
true bearing at an object
true bearing at the radar position
true course of an object
true bearings on the AIS position
time difference
positional navigational parameter
vector of measured position parameters
vector of forecast increments of navigational position parameters
measurement space
set of possible actions
object speeds according to AIS and radar
speed along the parallel
speed along the meridian
current speed
frame speed
true speed of an object
speed over ground
relative speed of an object
object in the information system
set of points constituting the movement trajectory
set of objects
k waypoint
generalized vector of position coordinates
expected value of random vector x
vector of change in the value of position coordinates
set of navigational data
distance between the ship from the traffic lane centre line
coordinate Y in the Cartesian system
vector of measurements
forecast vector of measurement
coordinate Z in the Cartesian system
rate of turn
correlation coefficient of random variables x and y
standard deviation of random variable x
variance of random variable x
covariance of random variables x and y
latitude
longitude
direction of the reference edge of ship movement trajectory in an anticollision
manoeuvre
ABBREVIATIONS
Abbrev.
ACK
AIS
ARPA
ATA
COG
CPA
CSE
DR
ENC
EPFS
EUT
HSC
IEC
IMO
MMSI
MSC
NMEA
RNC
ROT
SENC
SOG
SOLAS
STCW
STW
TCPA
UTM
VDL
XTE
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Meaning
Acknowledge
Automatic Identification System
Automatic Radar Plotting Aid
Auotmatic Target Acquisition
Course Over Ground
Closest Point of Approach
Course
Dead reckoning
Electronic Navigational Chart
Electronic Position Fixing System
Equipment Under Test
High Speed Craft
International Electrotechnical Commission
International Maritime Organisation
Maritime Mobile Service Identity
Maritime Safety Committee
National Marine Electronic Association Protocol
Raster Navigational Chart
Rate of Turn
System Electronic Navigational Chart
Speed Over Ground
International Convention for Safety of Life at Sea
International Convention on Standards of Training, Certification and
Watchkeeping for Seafarers
Speed Through Water
Time to Closest Point of Approach
Universal Transverse Mercator
VHF Data Link
Cross Track Error
7
ABSTRACT
This study presents a concept of the determination and use of ship’s safe navigation lane in navigational information systems. The ship’s safe navigation lane is
called navlane in this work and is determined from its parameters, chart data, accuracy of position identification and provisions of Collision Regulations.
The work consists of an introduction, four chapters and a summary.
The introduction provides a historical outline to the problem, discusses the
current state of research on the determination of safe movement trajectories, finally
sets the aim and scope of this work.
Chapter 1 presents properties of the marine navigational information system,
which combines a navigational system and an information system. The former enables
estimating of own ship’s position, the latter provides an image of the navigational
situation.
Chapter 2 includes methods of presenting cartographic objects in the navigational information system.
Chapter 3 gives a definition and method of the determination of safe ship
movement lane.
Finally, in the last fourth chapter the reader will find practical use of ship
movement lane ensuring safe navigation, accounting for the impact of the marine
environment, e.g. tides, and collision avoidance.
In the summary, most essential conclusions and remarks are presented.
8
Introduction
The marine navigation of today increasingly uses information tools. This trend
is promoted by the International Maritime Organization implementing the e-navigation
concept, i.e. navigation supported by advancements of telematics [77]. Information
tools have applications in monitoring the processes taking place on board ship (under
way, manoeuvring, cargo and ballast operations etc.) and in ship’s surroundings.
Quick access to information becomes more and more indispensable in safe ship conduct.
On the other hand, excessive informatization and automation in marine navigation
may make some navigators unaware of the limitations of the systems they operate
and processes taking place in these systems. Obviously, both informatization and
automation will be progressing and cannot be given up, therefore the navigator needs
support. The functions of voyage planning and navigation itself can be performed by
the navigation information system. Its most important role will be automatic determination of a ship’s safe lane.
Technologies of information systems, geographic information systems (GIS)
and electronic chart display and information systems (ECDIS) offer new possibilities of performing navigational tasks. These possibilities go so far as to raise hot
discussion at the IMO forum: is the navigator today still the navigating navigator or
just the monitoring navigator [77]?
ECDIS systems are consistently being introduced on ships to comply with
the SOLAS Convention. From 1 July 2018 all merchant vessels with a gross capacity
of more than 10,000 tons and various smaller ships will carry an ECDIS system. The
ECDIS displays on one screen chart information (from data bases), an image of the
surface situation obtained from sensors such as the radar or automatic identification
system (AIS) and position data from GNSS systems. It seems that the ECDIS will
drive away traditional tools of the navigator — printed navigational chart. The fact was
confirmed by the IMO in its revised STCW Convention (Standard of Training, Certification and Watchkeeping for Seafarers) [76], adopted in June 2010. In compliance
with the Convention, all navigators shall be trained in ECDIS operation and usage.
It is of particular importance that the ship personnel are properly qualified and skilled
to handle ECDIS equipment. The amount of information entering the system, a wide
range of options, number of alarms etc. may lead to a wrong perception of the navigational situation displayed on the screen. Unfortunately, such cases took place many
times, resulting in marine accidents [31], [33], [103]. No wonder propositions are
submitted to introduce the S-Mode function for switching the device into an operating
mode that will be the most appropriate from the navigational safety viewpoint.
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For the safety of navigation, another important issue is the extent to which
navigators know and are capable of interpreting the COLREGs. This refers to navigator’s own duties as well as those expected from the encountered vessel. It turns
out in practice that certain rules are repeatedly violated [145]. Counteracting this
problem may involve intensive navigator training or recognition that the regulations
should be amended. Regardless of this, there is a possibility of technological support
to the navigator in interpretation of relevant regulations and in taking decisions
aimed at collision avoidance [101]. The navigator may be provided with information
in a smart manner or provided with smart information [154]. One good solution
proposed herein is the concept of ship’s safe navigation lane — navlane — plotted
on the screen of the navigational information system that enables complying with:
—
—
—
—
10
international recommendations of planning a seagoing ship’s voyage,
requirements and recommendations of the shipowner,
captain’s requirements and standing orders,
provisions of the Collision Regulations.
Historical background
The seemingly simple definition of marine navigation i.e. safe ship’s conduct from the starting point to the final point [5], [21], [35], [80] in practice is difficult. Geodesy and cartography have provided excellent tools for the representation
of real image of the Earth surface — maps [37]. Marine charts contain increasingly
more information and cover the whole world. Cartography today uses modern tools
of geoinformation technology. Charts used by navigators contain both static elements of the Earth surface such as the coast line, depth contour, wreck position, etc.
and periodically changing phenomena, i.e. tides, currents or ice cover. In this way
the navigator has a more complete and reliable image of the environment. His confidence will be full only if, apart from the above data, the navigator knows his position
and its relative reference to navigational dangers. Positioning is based on enhanced
geodesic methods of measurement in GNSS systems [48], [49].
These elements provide a basis for the conduct of navigation understood as
the determination of object position, directions and distances. The safety of navigation is directly measured by the number of navigational accidents. If the position of own
vessel or obstruction is not known exactly, an accident may occur, such as grounding
or collision with an object. In the years 2000–2010 alone the number of reported
navigational accidents involving various ships was 28161 with 8476 fatalities or
missing persons. Of all the ships involved 2038 sank, while 10546 were not fit for
repairs (Fig. 1).
Collisions of two ships are the most crucial from the viewpoint of navigational risk. Every year such collisions rank the third place among causes of total ship
loss (Fig. 1). However, if we take into consideration loss of life and economic
losses, collisions come first. That is why so many actions are taken to minimize the
number of ships’ collisions. One giant step was made by the IMO when it adopted
the International Convention on the Prevention of Collisions at Sea (COLREG) in
1972, enforcement of fitting ships with anticollision equipment (SOLAS Convention,
V/19) and the adoption of the STCW Convention introducing the requirements of
mandatory training of seafarers.
Despite all efforts, effects are far from expected. A large freedom in interpreting Collision Regulations, doubtful principles of agreeing manoueuvres between
ships, differences in training standards in various countries, fatigue, routine actions,
lack of knowledge of what the other ship expects results in different manoeuvres
performed in identical or similar navigational situations. Indication of a limited area
along which the ship is supposed to be steered should reduce the number of variant
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11
decisions. In this way ships’ manoeuvres will be more foreseeable. Navigation will
be safer. A safe lane on the screen of the navigational information system allocated
to one ship will limit the scope of decisions.
Fig. 1. Total number of navigational accidents of sea-going vessels in 2000–2010
and their breakdown by type [96], [97]
12
The present state of the research
Work on determining ship’s safe passage at sea is one of the oldest tasks of navigation. All great discoveries of new lands were brought down to finding safe routes
leading to them. Among the greatest explorers were Bartolomeo Diaz, Christopher
Columbus, Amerigo Vespucci, Piri Reis, Vasco da Gama, Ferdinand Magellan. The
initial part of the route was determined in the proximity of land and visible signs.
Navigation was based on visual observation. This type of navigation is referred to as
terrestrial navigation. As boats started sailing farther from the land, more sophisticated
methods of route determination and positioning were developed. These were first of all
methods that belong to celestial navigation, followed by methods of technical navigation
[5], [34], [51], [67], finally GNSS methods took the lead [35], [48], [49], [50], [82], [90],
[107], [141]. Parallel to advancements in navigation, progress was made in geodesic
measurements and production and reproduction of cartographic maps [2], [18], [37],
[106]. The latest technologies have led to the development of geographic information
systems [22], [95], [97] and electronic charts [40], [55], [56], [58], [68], [169]. In marine
navigation it is the electronic chart display and information system [53], [54], [58],
[72] that may operate as a recorder of voyage data, the so called black box [161].
It is known that during historical expeditions in quest of new sailing routes
navigation helped extend general knowledge. For instance, in 1769 James Cook observed a Venus transit across the face of the Sun [166]. This provided a basis for the
determination of paralax and calculation of the distance between the earth and the Sun.
The determination of the safe navlane call for proper positioning. While determining
a position, we estimate values of navigational parameters and plot a line of position
(LOP). The property of LOP was experimentally discovered by Captain T. H. Sumner
on a sea voyage in 1837 [162]. Records of safe passages were put down in sailing
directions. Among others, they included recommendations for avoiding natural dangers
and obstructions. As time went by, an increasing vessel traffic brought about another
threat — collisions. After one such collision on the River Thames on 3 September
1878 the ship Princess Alice sank, having collided with the Bywell Castle. 600 passengers lost their lives. The event forced authorities to establish and enforce formal
rules of ships’ passing each other. The then applied ‘larboard hand rule’ was soon
eliminated. According to that rule, in an encounter situation in an open area each vessel
to avoid a collision should have turned to starboard. With the situation being differently
assessed on each ship, a collision occurred at times. The Collision Regulations in force
today were adopted by the IMO on 20 October 1972, and entered into force from 15
July 1977 [132], [133]. At present we witness, investigations and search for improving
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13
the existing rules [133], [144], [145], [154]. The large number of collisions worldwide
explains why such work should be continued. The problem of ship’s conduct in terms
of selecting a collision avoiding course and voyage planning is a topic of numerous
studies and research [61], [64], [80], [92], [93], [109], [139], [146], [149], [164]. Due
to the limited accuracy of positioning systems and the size of area relative to the geometric dimensions of the ship the latter was often regarded as a material point. Consequently, in planning manoeuvres the determined trajectory was that of a point representing the ship. This inconvenience was removed by introducing the concept of ship
domain [36], [39], [89], [109], [116], [129], [147], [167], [169] that includes the area
around the ship clear of other objects. Route planning accounted for the ship domain
and its edge. Selecting the right route is then brought down to the problem of optimizing the necessary manoeuvres [108], [149], which, practically, in the ship control
results from decision processes [104], [109]. As best decisions are supposed to be
taken by experts, the expert knowledge of experienced navigators is used in determining
ship’s passage. Such expertise can be applied with the use of artificial intelligence
systems [128], [139], [168]. One example is navigational decision support systems
[80]. In any ship encounter, to determine a safe route the navigator has to correctly
identify and assess the navigatonal situation [17], [84], [109], [131].
We have to do with a different case when determining physical parameters
of waterways, satisfying the conditions of safe passage of ships with characteristic
(maximum) parameters. In defining the ship’s safe manoeuvring area its designers
take into account the area parameters and hydrometeorological conditions [43],
[122], [123]. The safe manoeuvring area, subjected to formal risk assessment [58]
and economical analysis [41] gives rise to the determination of a permanent waterway.
In most cases such procedure is employed in determining permanent waterway in
restricted waters, port approaches, rivers channels and within port areas [45], [46].
In open sea areas and adjoining waters permanent waterways are established in regions where vessel traffic control is recommended. Routing systems are used for this
purpose [60], [61], [62], [64], [69], [74], [80]. Planning a sea route by the technical
navigation equipment used so far comes down to plotting the route from one waypoint
to another [2]. Ship’s deviation from the planned route is actually possible, although
it may be signaled as lateral deviation from the route.
This author proposes the concept of safe navigation lane — navlane — determined ad hoc in navigational information systems. Parameters of the safe navlane
will vary and account for the uncertainty of own ship’s position measurement. The
direction of the navlane will definitely satisfy the requirement of ship’s safe behavior
in compliance with the COLREGs.
14
The purspose and scope of the study
The International Maritime Organization takes actions aimed at enhancing
the safety of navigation and marine environment. The most effective instruments in this
respect are agreements concluded between many countries in the form of international
conventions. Three sea-related conventions are of particular importance for the safety
of navigation:
—
—
—
SOLAS,
COLREG,
STCW.
According to IMO data each of the above conventions applies to approx.
98% or more sea-going vessels. Based on the conventions, mandatory (minimum)
standards have been implemented globally for: vessel construction and equipment,
mutual obligations of ships, seafarers’ training and watchkeeping. Despite all measures
taken, marine accidents causing human and material losses are still reported. In the
majority of cases (approx. 80%) accidents are caused by the human factor. Although
human errors resulting in accidents at sea cannot be eliminated, the number of accidents
can. Collisions, groundings and crashes into obstructions account for about 30% of total
marine accidents [97]. They can be counteracted by skillful ship conduct. Globally,
this can be achieved by changing the traffic organization. The IMO introduces permanent vessel traffic systems in areas of heavy traffic. These systems are charted and
marked with aids to navigation. Traffic separation schemes are one example of such
arrangements. Ships are obliged to use these systems or keep clear of them [74].
The question arises: Is it possible to define ad hoc a traffic system for a single
vessel? The system that, taking into consideration the geometric parameters of the
area, current navigational situation and hydrometeorological conditions would safely
conduct the ship. The relevant lane can be plotted on an electronic navigational
chart. Additionally, it can be marked by virtual aids to navigation [157]. These issues
are the subject of this study.
This work aims at developing a concept of safe navlane determination. The
tool for implementing this task is the navigational information system. Research
done to put the concept into practice included:
—
—
—
assessment of ship’s fairway position accuracy [7],
assessment of a navigational situation on the fairway [109],
assessment of ship encounter situations [113],
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—
—
—
—
—
—
—
—
—
—
—
assessment of characteristic (typical) vessels operating on fixed routes [156],
construction of an expert knowledge base [150],
support to navigator in navigation-related decision making [101],
supplying the navigator with smart information [154],
detection of changes in ship’s speed (velocity) vector [9],
dead reckoning by the navigational information system [12],
application of the domain in navigating in high seas [116],
application of the domain as a criterion of safety assessment [120],
use of buffer zones of cartographic objects in navigation [11],
shape and size of ship’s safety zones [155],
development of standards and ontology of Maritime English and its use in the
information system for effective communication [120], [126].
The methodology of creating safe navlanes and examples of their practical use
are herein presented. Then the properties of the navigational information system are
defined. The system can be divided into the navigational and information parts. The
former estimates ship’s position, the latter displays the present navigational situation.
Further in the study the method of information presentation on the screen of the navigational information system is described, with a focus on the determination of cartographic objects. Besides, the concept of ship’s safe navlane is defined. Also presented
is the method for determining ship’s route and swept path, components of the safe
navlane. Finally, principles of navigation using the safe navlane are set forth.
16
Characteristics of the navigational information system
1. Characteristics of the navigational information system
1.1.
The navigational system
The processes of computing navigational variables from measured navigational parameters take place in the navigational system [82], [12]. Measuring tools
used for the purpose require that the position coordinate are appropriately determined. There are two approaches used to position estimation. The position determined
is based either on measurement results of more than navigational parameter, or by
dead reckoning.
In either case seeking the solution, i.e. calculating the position, we assume
a proper mathematical model. Its choice depends on how well we know the process
considered, required accuracy of coordinates computations and the computing method used. Generally, position determination by the navigational system consists in
identifying its position coordinates in the adopted reference system (e.g. coordinates
for a specific ellipsoid, spherical coordinates for a sphere corresponding to the given
ellipsoid etc.). Practically, position coordinates, as abstract quantities, are not directly measurable. To determines coordinates, physical quantities are first measured
(time, frequency, phase, etc.), then on this basis geometric relationships between the
receiver’s (observer’s) position and the coordinates of aids to navigation, such as
a lighthouse, radionavigation station, satellite etc., Fig. 2).
Fig. 2. A diagram of the process of measuring the depth of a water area
Source: own analysis.
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The geometrical relationship expressing relations between coordinates of aids
to navigation and the measurement point (observer’s position) is called the positional
navigating parameter u (distance, distance difference, pseudorange etc.), while the
relation between the positional navigating parameter and the measurement point in
the in the space considered (coordinate system) is referred to as the navigational
position function f. A general for of this function for geographical coordinates ϕ, λ
can be written as:
u = f (ϕ ,λ ).
(1)
Position determination consists in the identification of the coordinates in an
adopted coordinate (reference) system. Methods of measurements and calculations
of position coordinates may be either direct of iterative. The mutual spatial position
of measuring instruments and aids to navigation (navigational marks) is taken into
account. That is where the distance between them is taken for calculations, a variety
of computing methods can be applied [5]:
a) on the plane from:
•
•
•
•
•
•
•
•
•
•
two or more bearing on other objects,
two or more horizontal angles,
two or more distances,
two or more distance differences,
sums of two or more distances,
bearing and distance,
bearing and horizontal angle,
distance and horizontal angle,
distance and distance difference,
distances difference and sum;
b) on the spherical surface from:
•
•
•
•
•
•
•
18
two own bearings,
two other objects bearing,
two horizontal angles,
two distances,
two distance differences,
two distance sums,
own ship bearing and a distance.
The above lists show that in practice measurements refer to more than one
parameter and they are non-simultaneous measurements, which is due to:
—
—
—
movement of the ship (sensor, receiver),
movement of an aid to navigation (e.g. satellite),
technical conditions (radionavigation station operates in a chain, one-channel
measurement path of the receiver, sequential measurement cycle, asynchronous
measurements from various navigational devices etc.).
In traditional navigation the problem of non-simultaneity of navigational parameters measurements is most often neglected, because it is assumed that associated
errors are small. The exception is when the position is determined from non-simultaneous
lines of position in terrestrial and celestial navigation, when the measurement time
difference is substantial. However, in technical. i.e. automated or integrated navigation, which is very precise, even tiny time differences translate into relatively significant errors of position coordinates or instability of the estimator. That is why algorithms
of navigational data processing should account for the non-simultaneity of measurements, which in terms of technology is not a problem.
1.1.1. Calculation of the fix by the navigational system
1.1.1.1. A fix from simultaneous measurementsof navigational parameters
The basic case of calculating position coordinates in navigation is the determination of coordinated based on navigational parameter measurements regarded
as simultaneous. In the process of determining position coordinates we have a navigational function converting elements of the navigational space into the space of
measurements. This will be written as this relation:
f : Rm ⊃ N → U ⊂ Rn ,
n ≥ m.
(2)
The relation can be expressed as the system of equations:
f1 ( x1 , x2 ,K, xm ) − u1 = 0,
…,
(3)
f n ( x1 , x2 ,K, xm ) − un = 0.
This system of equations in the vector notation will have the form of:
f ( x) − u = 0,
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(4)
19
where:
x = [x1, x2, ..., xm]T — is a generalized vector of position coordinates (state vector),
depending on the adopted system of coordinates, e.g. X, Y, Z,
Δt, ϕ, λ, h etc., x ∈ N,
T
u = [u1, u2, ... , un] — is a vector of measured navigational parameters, u ∈ U.
There are two cases of solving the equations (3). The deterministic case,
where the number of measured navigational parameters is equal to the number of
estimated coordinates, i.e. n = m. In this case the equations (3) are solved by the
Newton method — solving a system of non-linear equations [5]. Then in (k+1)-th
step the vector of position coordinates will have this form:
x ( k +1) = x ( k ) + G −1 ( x ( k ) ) z ( k ) ,
(5)
where:
z — is a vector of measurements; it is the difference between the vector of measured
navigational parameters and the vector of forecast (reckoned) parameters,
which can be written as:
z (k ) = u − f ( x (k ) ) ,
(6)
G — Jacobi matrix of f (navigational position function).
The other case refers to the situation where the number of measured navigational parameters is larger than the number of coordinates being determined (n > m).
Then the equations (3) is solved by the least squares method. In this case in the
(k+1)-th step we obtain an approximation:
[
]
−1
x ( k +1) = x ( k ) + G T ( x ( k ) ) Pu-1G ( x ( k ) ) G T ( x ( k ) ) Pu−1 z ( k ) ,
(7)
where Pu is the matrix of vector covariance of measured position parameters u. The
vector z is determined from the equation (6), while the matrix Pu is the covariance
matrix of the vector z. This vector is the difference of vectors as expressed by the
equation (6), and the vector f(x) — the result of calculations — is determined with
any accuracy, as it is a non-random vector. For this reason we can assume that covariance matrix of the vector u is equal to the covariance matrix of the vector z.
20
Calculations are performed until an assumed accuracy of coordinates is
achieved by the method of subsequent iterations. If the results of iteration of (5) or
(7) are convergent with the real solution x, then the accuracy of calculating the position coordinates is approximately equal to that obtained in the last iteration of the value
of the other/second component (5) or (7). It is often used to determine the accuracy
of the completed iteration. Generally the dead reckoned position or the previous fix
are regarded as the first approximation. In both calculation methods, Newton’s or
least squares, the covariance matrix of the state vector (position coordinates) is calculated from this relation [5]:
(
Px = G T ( x ( k ) ) Pu−1G ( x ( k ) )
)
−1
(8)
.
1.1.1.2. A fix obtained from non-simultaneous measurements of navigational
parameters
The previous section referred to the case of simultaneous measurements,
which in reality practically does not take place. Let us then consider the way of estimating position coordinates and the effect it will have on the final result when
measurement non-simultaneity is taken into account. In the first place, we should
indicate one common moment of performing measurements. We will use the method
of sequential addition of measurements, which consists in forecasting measurement
values for a specific common moment. This approach is similar to known methods
used in terrestrial and celestial navigation, where lines of position are brought down
to one common moment. In m most cases it the moment of the last measurement.
We choose a series of moments/instants t1 < t2 < … < tn, where ti denotes the
moment in time of the i-th measurement of a navigational parameter. For convenience let us agree that we bring down the measurements to the last moment/instant
of measurement (to obtain the present position). The forecast vector of measured
position parameters up will be calculated from this relation:
u p = u + Δu,
(9)
where:
— vector of forecast increments of navigational position parameters values:
n
Δ u = ∑ Δu i
i =1
(10)
,
Δui = eiT ⋅ gradf i ⋅ Δx (i ) ,
17/2011
(11)
21
—
vector of the canonic base of an n-dimensional measurement space (value one occurs
at the i-th position, corresponding to the given coordinate in the measurement space):
[
]
(12)
ei = 0 ,0 ,...,0 ,1,0 ,...,0 ,
—
i
gradient of the i-th navigational function (line, surface or hypersurface of position),
it is the i-th row of the matrix G:
⎡ ∂f ∂f
∂f ⎤
gradf i = ⎢ i , i ,..., i ⎥,
∂xm ⎦
⎣ ∂x1 ∂x2
—
(13)
vector of change in the value of position coordinates that takes place between
the moment of navigational measurement ui and the common moment tn:
[
]
T
Δxi = Δx1i , Δx 2i ,..., Δx mi .
(14)
We can assume that in sufficiently short time periods the navigational position
parameters change linearly. Putting (9) into (6) then into (5) we obtain the formula
for the (k+1)-th approximation of the position coordinates vector in the Newton’s
method:
(15)
( k +1)
(k )
−1
(k )
(k )
=x
x
[
]
+ G ( x ) u + Δu − f ( x ) .
Our procedure is identical in the least squares method. After putting (9) into
(6) and the result of substitution into (7) we obtain:
[
]
[
−1
]
x ( k +1) = x ( k ) + G T ( x ( k ) ) Pu-1G( x ( k ) ) G T ( x ( k ) ) Pu−1 u + Δu − f ( x ( k ) ) . (16)
We obtain the following formulas for the mean value of the vector representing
an increment of position coordinates:
Δx śr( k ) = G -1 ( x ( k ) ) z (pk ) ,
(
Px = G T ( x ( k ) ) Pp−1G ( x ( k ) )
(17)
)
−1
,
(18)
where:
zp — forecast vector of measurement:
z (pk ) = u p − f ( x ( k ) ) ,
22
(19)
Pp — covariance matrix of the forecast vector of measured navigational position
parameters, that according to the formulae (9), (10) and (11) is expressed by
this relation:
n
n
n
Pp = Pu + ∑ PΔui + ∑∑ PΔui Δu j .
i =1
i =1 j =1
i≠ j
(20)
The above relation presents the covariance matrix of the forecast vector of
measurements. Compared to the measurement vector matrix Pp is increased with
these components: covariance matrix of forecast increments of navigational parameter
increments and the covariance matrix of measured navigational parameter increments.
These components are formulated as follows:
• covariance matrix of value increments of navigational position parameters
n
n
i =1
i =1
∑ PΔui = ∑ eiT ⋅ gradf i ⋅ PΔxi (gradf i ) ⋅ ei ,
•
T
(21)
PΔxi — covariance matrix of coordinates increments,
n
•
∑P
i , j =1
i≠ j
Δ u i Δu j
— matrix of covariance between individual increments of measured
values of navigational position parameters
PΔui Δu j = eiT ⋅ gradf i ⋅ PΔxi Δx j (gradf j ) ⋅ e j ,
T
•
(22)
PΔxi Δx j — cross-covariance matrix of coordinates increments.
A general algorithm of position coordinates calculations from non-simultaneous measurements of navigational position parameters is shown in Figure 3
below.
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23
Fig. 3. A general algorithm of position coordinates calculations from non-simultaneous
measurements of navigational position parameters
24
1.1.2. Estimation of DR position calculated by the navigational system
The navigational system can determine a ship’s position using the dead
reckoning method, which is an estimation of the position based on the knowledge
of last fix, the distance covered and directions of ship movement. In practice, accelerations are measured or ship’s speed of movement in time. On this basis increments
of position coordinates along N and E are calculated. In this situation for the generalized
vector of position coordinates x = [φ, λ] the general transformation brings this form:
xt = C0t x 0 .
(23)
The dead reckoning, i.e. the shift of position coordinates, will be possible using the
course over ground:
t
COGt = COG0 + ∫ CO& Gdt .
(24)
0
Coordinate increments can be calculated from this relation:
t
ϕ = ϕ + Δϕ = ϕ + ∫V
dt ,
(25)
λt = λ0 + Δλ = λ0 + ∫V E dt ,
(26)
V N = V SOG ⋅ cos(COG ),
(27)
V E = V SOG ⋅ sin(COG ).
(28)
t
0
0
N
0
t
0
where
1.2.
Information system
The information system is a computer-based system processing/converting
input data into output data by using models and procedures.
Let there be a given set W of objects and a set A of attributes describing
these objects. Consequently, the information system I will be the pair:
I = (W, A),
(29)
where:
A — set of attributes a∈ A: f :b → a ,
{
}
B — set of vectors of the object state,
Bi = [Xi, Si,…, Ri],
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25
Xi = [x1, x2,..., xi]T — generalized vector of position coordinates depending on the
adopted coordinate system (X, Y, Z, Δt, ϕ, λ, h etc.),
Si=[s1, s2,..., si]T — vector of design parameters and manoeuvring characteristics of
an object (vessel),
T
Ri=[r1, r2,..., ri] — vector of navigational risk.
In the information system the composition of a pair of elements can be made
by interaction with the operator or automatically. The existing pairs (w1, a1) and
(w2, a2) will be equal if this condition is satisfied (30):
(w1, a1 ) = (w2 , w2 ) ⇔ w1 = w2 ∧ a1 = a2 .
(30)
1.2.1. ECDIS — the marine information system
The ECDIS (Electronic Chart Display and Information System) is the most
common information system in marine navigation (Fig. 4). This computer-based
system is an information system, meeting the requirements of the SOLAS Convention
(Safety of Life at Sea) V/19 and V/27, presents selected data from a System Electronic
Navigational Chart (SENC) and vessel position information [74], [53].
Fig. 4. A screen of the ECDIS Navi-Sailor system
26
The main function of the system is facilitating route planning and monitoring
by supplying additional navigational information, such as:
— weather information (sensor: meteo station),
— vessel position, speed, direction of movement (sensors: GNSS, gyrocompass,
echosounder, log),
— navigational warnings (sensor: NAVTEX),
— presence of other vessels in vicinity (sensors: ARPA, AIS).
1.2.2. On-bridge navigational decision support system
The research and development in navigation presently tends to focus on information systems capable of processing input data into ordered output information
and analyze it. The analysis aims at developing recommendations for the information system operator. A prototype of such system — decision support system — was
developed at the Maritime University of Szczecin [113]. Its general architecture is
shown in Figure 5.
interaction with operator module
vessels in the area
Voyage data
info on the situation
decisions
data on
vessel traffic
in the area
Detection
trajectory
situational
data
enquiry
Voyage plan,
Navigator’s
preferences,
limitations
Management module
rules, criteria
assessment situation assessment
data on situation, rules
passage plan
trajectory
development of a manoeuvre
Knowledge base: COLREGs; sit. assessment
criteria
Fig. 5. A general architecture of the navigational decision support system
on a sea-going vessel
The system operates in real time. Its function is to observe the ship and its
environment, register navigational data, select, extract, verify and process the data.
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27
The data processing results in showing the navigator information on the identification and assessment of a navigational situation and proposed recommendations (decisions) ensuring safe navigation.
Assuming that D is a set of acceptable decisions and x is a set of navigational data received from sensors, we will write:
D = g ( f (x )),
(31)
where:
f — navigational function,
g — decision function.
The codomain of the decision function g is a set of linguistic recommendations with such values as {turn to starboard, turn to port, stand on, slow down, stop,
reverse the engines, turning circle to starboard, turning circle to port}.
The codomain of the navigational function f is a set defining a navigational
situation, i.e.:
— type of navigation,
— safety (risk) of navigation,
— applicable rule or rules.
The domain of the function f is the state vector that is a sum of own ship vector, radar report and AIS data. For data from the radar, the report will have this form:
x = [TB, d, TC, VR, Cρ, Vρ]T, x ∈ N ∪U ,
(32)
where:
N
— navigational space,
U — measurement space,
TB — true bearing on an object,
d
— distance to the object,
TC — true course of the object,
VR — true speed of the object,
Cρ — relative course of the object,
Vρ — relative speed of the object.
1.2.3. Information in the information system
Information in the navigational information system consists of abstract
quantities (data) describing and forming navigational knowledge. This information
can be presented or transmitted between information systems.
28
The information system itself is a source of information. Provided that it can
display (transmit) n messages with respective probabilities pi, (i = 1,..., n), then the
(weighted) mean number of item of information in the messages from the system
will amount to:
n
⎛ 1 ⎞
H = ∑ p i log 2 ⎜⎜ ⎟⎟ .
i =1
⎝ pi ⎠
(33)
The quantity H is the entropy of the information source (information system).
Navigational knowledge in the system, in turn, is a set of data, facts, rules,
procedures, strategies of behaviour and theories, including guidance for interpretation
and inference. This knowledge will enable the system operator (navigator) performing
the basic task of marine navigation, i.e. safe ship conduct from one waypoint to another.
This also refers to a situation where the operator has incomplete or uncertain information. As for the navigational information system the requirements concerning the
scope of knowledge should cover both procedural knowledge — procedures formulated by experts — and declarative, descriptive knowledge, determined/defined by
sets of facts, statements and rules.
Procedural knowledge, including principles of behaviour, is embodied/contained
in all kinds of rules and regulations.
Declarative knowledge, acquired in the course of studies, training and sea
service, refers to situation analysis and assessment as well as the principles of behaviour.
It includes two basic functions:
a) passage planning based on shipowners’orders — weather routing,
b) ship steering with the simultaneous control of navigational safety level:
• determination of courses and speeds ensuring safety in the present navigational situation, meeting the allocated tasks,
• performance of manoeuvres according to the determined values of courses
and speeds.
In the navigational information system the acquired knowledge is recorded
in forms appropriate for its purpose or manner of use. This knowledge can be represented in a variety of ways, such as:
1. Data base structures. Data bases, allowing to gather data sets and record them
in the way specific for the adopted model, enable efficient data edition, updating,
archivization and further processing. Applications of data bases in navigation
become wider along with the developments in telematic technologies. Voyage
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29
Data Recorder (VDR) systems are good examples of structured data bases. Recorded
events from the voyage may be used in the process of knowledge expansion
(situations and manoeuvres made by navigators). Another example is the idea of
building WEND (Worldwide Electronic Navigational Chart Database). The electronic navigational chart makes up the basic source of knowledge on an area and
essentially supplements navigator’s knowledge. Its data base form enables a choice
of proper layers of vector data for the execution of navigational tasks.
2. Decision rules and trees. The rules represent the knowledge defining conditions for attributing recorded facts to distinguished classes: they define premises,
implications and conclusions. Decision trees perform similar tasks, allowing to
solve a classification task for two or more classes. Both decision rules and trees
constitute the form of knowledge well implemented in expert systems. Decision
trees enable a description of the decision process — inference.
Op
en
Se
a
Ar
ea
NAVIGATION
1
DISTANCE TO
OBJECT
e
cr
In
in
as
g
g
in
as
re
c
De
0
BEARING
0
1
Fig. 6. A decision tree for the domain: navigational situation (0 — safe, 1 — dangerous)
30
3. Decision tables. This very useful method of knowledge representation has a form
of logical decision tables. The decision table contains a description of a decision
situation (SD) that is defined as a set of ordered threes:
(U dz , J , f u ) ,
(34)
where:
Udz — a set of possible actions,
J
— a set of possible action results,
fu
— utility function defined by the Cartesian product U dz × J .
It is well justified to use this form of knowledge representation in marine navigation. It
enables foreseeing the results of the decision made as well, often more importantly,
select proper actions to obtain the intended result.
4. Neural network. Neural networks are mathematical structures able to process
signals. Their pupose is to reproduce processes taking place in the brains of living
organisms. Neural network designers define the network structure. Then the
learning process follows with the aim to make the network operate correctly
with an assumed acceptable maximum error. Neural networks are used in approximation tasks, image recognition, forecasts, selection, optimization etc.
Fig. 7. An assessment of a navigational situation by the neural fuzzy network
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31
5. Algorithms. An algorithm is a convenient and straightforward method of knowledge
representation. It is just a simple method for task solving. Computing algorithms
including optimizing algorithms are important as they represent theoretical knowledge enabling solving specific computing problems, e.g. determination of vessel
encounter or manoeuvre parameters.
6. Recursive procedures. Recursive algorithms (formulas) are often used in defining
navigational knowledge due to their simple notation. These algorithms by definition refer to themselves. Recursive algorithms closely reflect human behaviour
as they allow to present a solution to a part instead of the whole problem. Most
tasks in navigation are solved in the recursive manner, e.g. voyage planning and
execution, planning and execution of a SAR operation, object recognition and
identification. A relevant example shown in Fig. 8 presents the process of ship’s
passage planning described by the formula (35).
Fig. 8. Recursive planning of a passage
⎧ f (WPn → WPn +1 ) ⊗ P (WPn +1 → WPk )
P (WPn → WPk ) = ⎨
f (WPn → WPk )
⎩
for
for
n < k −1
, (35)
n = k −1
where:
P
— planning process,
f
— navigational planning function,
WPk — k-th waypoint.
The passage planning (P) is performed on one interval (n, n + 1) by the planning
function f. The interval range this arbitrarily chosen and can be a time interval, state
of the log, weather forecast, watch, waypoint [80] etc.
32
Presentation of navigation-related information in the navigational information system
2. Presentation of navigation-related information
in the navigational information system
Information is presented on displays of navigational devices. For sea-going
vessels international standards specify requirements for handling, operation, required test results and performance of navigational displays [54]. The requirements
refer to screen appearance, conformity of presented information (e.g. units, terms
used), readability, colours, brightness, symbols, integrity, alarms, indicators, modes
of operation and instructions. Physical property requirements such as the minimum
size of display or resolution are separately defined.
According to MSC 191.8.2.2 and MSC 232.10.2, the minimum working area
of the ECDIS display for presenting a chart for passage monitoring should be 270 mm
x 270 mm. The radar (MSC191.8.2.3) used for the same purpose should have a circular
screen with radius equal to at least:
—
—
—
180 mm for ships with a capacity of less than 500 gross tons,
250 mm for ships with a capacity from 500 to 10.000 gross tons,
320 mm for ships with a capacity over 10.000 gross tons.
The minimum screen resolution (MSC.191/8.4) should be 1280 x 1024 pixels
(or correspondingly, unless otherwise specified in other requirements) [54].
The requirements for the presentation of information in navigational information systems refer to:
—
—
—
—
—
—
—
—
own ship,
cartographic base,
radar images,
tracked objects,
alarms,
AIS,
measurements (e.g. bearings, ranges, depths etc.),
navigational tools (e.g. cursor, VRM, EBL etc.).
The above information are presented in the following forms:
—
—
—
—
text: descriptions, explanations, additional information, labels etc.,
alphanumeric: available on additional alphanumeric displays or separate areas
of the monitor screen,
graphical symbols and signs, icons,
colours: e.g. various operating modes (night, day), or to mark the level of safety,
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33
—
—
—
animations: e.g. blinking,
audio: supplementary to graphical and alphanumeric information, mainly for
alarming,
illumination.
The basic component of the navigational information system is the electronic navigational chart. The chart image displayed on the screen should feature
lines, symbols and text sufficiently large to be distinguishable and interpretable from
the operating distance (Fig. 9), i.e. about 70 cm. At this distance a symbol looked at
should be 4 mm high (i.e. it should be 1.5 times larger than the same symbol on a paper chart). The good practice is to use symbols twice larger. To be readable, symbols
should consist of a minimum number of pixels. A single symbol on the chart should
have about 12 pixels [55]. The display height of a charted object in pixels is calculated dividing the symbol height given in milimetres by 0.312 mm (0.312 mm is the
pixel dimension for the smallest surface area of the displayed chart). It should be
assumed that a text in the navigational information system, like in an ECDIS, should
be readable at one metre distance [51]. An example of a standard symbol: western
buoy of the cardinal marks — a simplified symbol, is given in the Figure below.
Fig. 9. The buoy symbol of the cardinal mark
2.1. Ship’s contour in the information system
2.1.1. Dilutioned contour
According to [54], in the situation where the navigational information system allows to graphically present own ship, the user should have a choice between
the ship’s contour shown to scale or a symbol (Fig. 10).
34
The ship’s contour size or its symbol should be commensurate with its real
dimensions (be sized to scale) or should be 6 mm in diameter, whichever value is
greater. The orientation of the contour (beginning of the speed vector or heading
indicator) should be related with the position of the reference system — usually the
antenna of the GNSS receiver.
Fig. 10. Symbols of own ship according to IEC 61174 and IHO, S-52 (IEC 62288) standards
Most navigational accidents, such as collisions or striking the bottom or
marine structure, take place in restricted waters. In the past, due to low accuracy of
position systems and low precision of information provided by paper charts, the
ship was always regarded as a material point. Today one would ask: Which point
is that? (Fig. 11).
Ship’s dimensions and spatial accuracy of object position were neglected so
the charts were far from perfect. Ship’s dimensions were only taken into account
during harbour manoeuvres, and then almost exclusively visual assessment was in use.
It seems appropriate to take into consideration ship’s geometric dimensions in the
presentation of its contour in the navigational information system. Additionally,
consideration should be given to the accuracy of spatial position determined in
reference to the location of the antenna, with various parameters of normal distributions of position, thus with different matrices of position covariance, and consequently, various parameters of the mean error ellipse and its changing orientation.
The variability of these factors leads to changes in directional errors which results
in changes of the accuracy of determination of ship’s contour [158]. The ship’s
contour increased by directional errors will form a dilutioned ship contour in the
information system [7].
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35
Fig. 11. Does it matter which point representing the whole ship is chosen?
Let us consider a ship with these parameters: L = 54.4 m, B = 12.0 m and determine its dilutioned contour for various confidence levels (68.3% and 95%) and
for different components of position covariance matrix. Example shapes of a dilutioned
ship are given in Figures 12 and 13.
Fig. 12 presents a situation where dilution was calculated for a single directional error (confidence level 68.3%), one-digit mean error in metres and slight positive
correlation (ρ = 0.21).
For comparison, Fig. 13 shows a case where dilution was calculated for doubled
directional error (confidence level 95%) and mean errors amounting to a few metres
similarly to the previous case above, and a strong negative correlation (ρ = –0.83).
In Figure 13 the ship contour is visibly deformed.
36
Fig. 12. A contour of a dilutioned ship for the confidence level 68.3%: σx = 6 m, σy = 8 m,
σxy = 10 m2
Fig. 13. The contour of a dilutioned ship for the confidence level 95%: σx = 6 m, σy = 8 m,
σxy = –40 m2
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37
Fig. 14, in turn, depicts a situation that may occur in real conditions. For instance,
while determining the position of a VTS-conducted ship with the use of the radar
there often occur cases where the accuracy of coordinates is measured in tens, even
hundreds of metres and with a strong correlation. The case presented the covariance
matrix elements were as follows: σx = 100 m, σy = 20 m, σxy = 1900 m2 (confidence
level 95%, correlation coefficient ρ = 0.95).
Fig. 14. The contour of a dilutioned ship for the confidence level 95%: σx = 100 m, σy = 20 m,
σxy = 1900 m2
Navigational parameters are in most cases scalars or two-dimensional vectors.
These parameters are usually obtained by direct or indirect measurements of physical
and geometrical quantities. The ship’s contour, own or other, plotted on the screen
of the navigational information system should account for the assessed accuracy of
measurements. The directional error should be used for this purpose.
2.1.2. Determination of ship’s dilutioned contour
Determining the position of vertex points of ship’s contour in the navigational information system we will make use of the directional error. It enables defining
38
the accuracy of linear objects based on the probability density function of positions
of their points in the special case — external points. In the general case parameters
of the mean error ellipse are calculated from parameters of the density function of
the given random vector probability — elements of the covariance matrix. Therefore, it is advisable to express the directional error as well as the function of these
parameters (36). The distribution of the random vector is determined by its probability
density function. For a two-dimensional normal distribution the function has this
form [106], [7]:
f ( x, y) =
1
2πσ x σ y 1 − ρ xy2
⎧⎪
⎡ (x − x )2
(x − x )( y − y ) + ( y − y )2 ⎤ ⎫⎪, (36)
1
−
exp⎨−
2
ρ
⎢
⎥⎬
xy
2
2
σ xσ y
σ y2 ⎦⎥ ⎪⎭
⎪⎩ 2 1 − ρ xy ⎣⎢ σ x
(
)
where:
x , y — mean values of random variables X, Y,
σx
σy
ρ xy
— standard deviation of the random variable X,
— standard deviation of the random variable Y,
— correlation coefficient of random variables of X and Y.
In navigational interpretation
σx
is the mean error of the coordinate X, i.e. the
directional error along the X axis. Similarly, σ y is the directional error for the variable
Y. Besides, the correlation coefficient of random variables is defined by this relation:
ρ xy =
where
σ xy
,
σ xσ y
(37)
σ xy is the covariance of random variables X, Y.
The covariance matrix of a two-dimensional probability density function
(36) has this form:
⎡ σ x2 σ xy2 ⎤
P=⎢ 2
.
2 ⎥
⎣⎢σ xy σ y ⎥⎦
After rotating the coordinate system by the angle α we obtain a new distribution with other standard deviations. The old coordinates will be expressed by the
new coordinates and the rotation angle by these equations:
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39
x = x′ cosα − y ′ sin α ,
y = x′ sin α + y ′ cosα .
(38)
After transformations of the equation (36) and accounting for (38) we get
the variances and covariance, after the rotation of the coordinate system, expressed
by these formulae:
σ x2′ = σ x2 cos 2 α + σ xy sin 2α + σ y2 sin 2 α ,
(39)
σ y2′ = σ x2 sin 2 α − σ xy sin 2α + σ y2 cos 2 α ,
(40)
σ x′y ′ = −
(
)
1 2
σ x − σ y2 sin 2α + σ xy cos 2α .
2
(41)
If in the above formulae the covariance is expressed via the correlation coefficient and standard deviations (37), then they get another form:
σ x2′ = σ x2 cos 2 α + ρ xyσ xσ y sin 2α + σ y2 sin 2 α ,
(42)
σ y2′ = σ x2 sin 2 α − ρ xyσ xσ y sin 2α + σ y2 cos2 α ,
(43)
σ x′y′ = −
(
)
1 2
σ x − σ y2 sin 2α + ρ xyσ xσ y cos 2α .
2
(44)
The formulae (39)–(41) or (42)–(44) describe the variances and covariance
of the normal distribution after the rotation of the coordinate system.
Standard deviations (mean errors) are determined as the arithmetic root from
the random variable variance. Therefore, the roots of variances (42) and (43) are
directional errors long the new coordinate axes X’, Y’. Then, the formula expressing
the directional error (in the direction determined by the angle α, axis X’) is as follows:
σ α2 = σ x2 cos2 α + σ xy sin 2α + σ y2 sin 2 α
(45)
σ α2 = σ x2 cos 2 α + ρ xyσ xσ y sin 2α + σ y2 sin 2 α
(46)
or
In the special case, where mean errors of the coordinates are equal to the
semi-axes of the mean error ellipse, i.e.
a = σx and b = σy ,
40
then the covariance equals zero, i.e. σxy = 0 (consequently, the correlation coefficient
will also be zero). In such case we obtain the following form of distribution variance
(36) after rotation by the angle α, expressed as the function of the mean error ellipse
parameters:
σ x2′ = a 2cos2α + b 2sin2α,
(47)
σ 2y′ = a 2sin 2α + b 2 cos 2α .
(48)
The directional error, according to (45), will then be expressed by this relation:
σ α2 = a 2cos2α + b 2sin 2α .
(49)
Fig. 15 presents a graph of the directional error in the entire range of angles.
Fig. 15. A graph of the directional error as the function of the angle α
2.2. Cartographic objects in the information system
The standards [55], [58], [68] define the shape, appearance, meaning of the
cartographic marks and symbols charted in the navigational information system. The
standards also refer to graphic symbols used for the presentation of own and other
17/2011
41
vessels. Their size depends on the chart scale, while the shape remains unchanged.
However, in precise navigation (close to dangers) using satellite, radar or laser methods of positioning, precision of measurements should be taken into account (e.g.
position coordinates, distance, bearing etc.) in creating images in combination with
data base information. In this manner cartographic objects can, or rather should be
plotted dynamically, so that their shape will be changeable.
The section 0 shows a method of determining a dilutioned ship contour,
statically accounting for the shape accuracy in all directions. When the ship is in
motion, it will be more appropriate to plot a buffer zone around the ship contour
[11]. The buffer zone is formed as an envelope of the family of directional errors of
the ship contour. The curve of directional errors is elliptical lemniscate of Booth.
The canonical equation of Booth lemniscate is as follows:
(x
2
+ y 2 ) − a 2 x 2 − b2 y 2 = 0,
2
(50)
where:
a — major semi-axis of error ellipse,
b — minor semi-axis of error ellipse.
In a general case we will obtain this relation:
(x
2
+ y 2 ) − σ x2 x 2 − σ y2 y 2 − 2σ xy xy = 0 ,
2
(51)
where:
σ x — error of ship’s position relative to the axis x,
σ y — error of ship’s position relative to the axis y,
σxy — covariance.
Similarly to ship’s buffer zone, we create buffer zones around navigational
dangers — cartographic objects that in terms of geometry are points (e.g. separate
point danger), lines (quays, depth contours, pipelines) and polygons (e.g. prohibited
areas around offshore facilities). Parameters of these objects are stored at HO-ECDB
(Electronic Chart Data Base as supplied by a Hydrographic Office). In this context it is
assumed that the ship has permanent dimensions, and parameters of cartographic objects
are up-to-date, i.e. their dimensions and position are the same as those in the data base.
The buffer zone around a point is shown in Fig. 16, while the buffer zone for
a line is created by parallely shifting the directional error curve (lemniscate of Booth)
along a given line, as depicted in the next Fig. 17.
42
Fig. 16. A buffer zone of a point
Fig. 17. A buffer zone of a line
In the case of a polygon, the lemniscate of Booth is shifted in parallel along
its edge (curve or closed broken lines, as illustrated in Fig. 18.
Geometrically, the ship is a polygon, the one that moves relative to cartographic objects — navigational dangers. That is why ship’s buffer zone changes in
time and space. Its dimensions and orientation depend on the accuracy of position
coordinates, their correlation and ship’s course. The relations between ship’s buffer
zone and navigational dangers are presented in Fig. 19. In every case the safe distance
should be determined as a minimum distance between ship's buffer zone and buffer
zones of navigational dangers.
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43
Fig. 18. A buffer zone of a polygon
Fig. 19. The relation between ship buffer zone and buffer zones of cartographic objects
44
The reader may get an impression that due to the accuracy of modern GNSS
systems and geometrical dimensions of vessels, the size of the buffer zone is so
small in comparison with the ship domain that it can be neglected. This is not true,
however. There are situations where GNSS signal fade away [47], shift of the position etc. In such cases the navigational system is switched into another working
mode (estimation mode, running fix). After prolonged dead reckoning, position errors
accumulate and may be many times larger than ship dimensions (Fig. 20).
Fig. 20. The display of ships in motion in Maritime Volumetric Navigation System,
implemented under the Ariadne project
2.2.1. Positioning of a cartographic object in the navigational information
system
It is said in the previous section that the ship should be regarded as a moving
polygon. The presentation of a ship or ships in the navigational information system
(NIS) will then consist in the proper plotting of polygons (see example in Fig. 21)
[40]. The knowledge of their position accuracy will be crucial for the assessment
of a navigational situation and making relevant navigational or manoeuvring decisions [24].
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45
Fig. 21. Radar echoes on the information system screen.
Above a radar image with detected echoes
Let it be borne in mind that accuracy is defined as the degree of conformity
between the measured or estimated value of a quantity and its real value. Accuracy
is usually interpreted as a statistical measure (or statistical-deterministic) of measuring
errors [5].
One kind of accuracy in reference to radionavigation systems is relative accuracy [34]. This is accuracy with which the user may determine their position relative to another user by the same system and at the same time. The vector of relative
position will be a measure of relative accuracy [11]. Relative accuracy can also be
defined as an accuracy of relative positions, i.e. the vector of coordinates differences
(52). This case is illustrated in Fig. 22, where mean ellipses of coordinate errors of
each position are symbolically marked.
46
Fig. 22. Relative positions of ships
The vector of coordinates difference equals:
⎡ϕ ⎤ ⎡ϕ ⎤ ⎡Δϕ ⎤
Δx = x 2 − x1 = ⎢ 2 ⎥ − ⎢ 1 ⎥ = ⎢ ⎥ .
⎣λ 2 ⎦ ⎣λ1 ⎦ ⎣ Δλ ⎦
(52)
Hence the coordinates difference vector covariance matrix (of relative positions) is equal to:
PΔx = Px2 + Px1 − Px2x1 − PxT2x1 = Px2 + Px1 − Px1x2 − PxT1x2
.
(53)
Individual matrices will have this form:
—
position covariance matrix (ϕ1, λ1)
⎡σ 2
Px1 = ⎢ ϕ1
⎣⎢σ ϕ1λ1
—
(54)
⎤
⎥,
⎦⎥
(55)
1 1
1
position covariance matrix (ϕ2, λ2)
⎡σ2
Px 2 = ⎢ ϕ 2
⎣⎢σ ϕ 2λ2
—
σϕ λ ⎤ ,
⎥
σ λ2 ⎦⎥
σϕ λ
σ λ2
2 2
2
mutual covariance matrix
⎡σ ϕ ϕ
Px1x 2 = ⎢ 1 2
⎣σ λ1ϕ 2
17/2011
σϕ λ
σλλ
⎤
⎥,
2 ⎦
1 2
1
⎡σ ϕ ϕ
PxT1x 2 = ⎢ 1 2
⎣σ ϕ1λ2
σλϕ ⎤ ,
σ λ λ ⎥⎦
1 2
(56)
1 2
47
—
coordinates difference vector covariance matrix:
PΔx
⎡ σ Δ2ϕ
=⎢
⎢⎣σ ΔϕΔλ
σ ΔϕΔλ ⎤
⎥ .
σ Δ2λ ⎥⎦
(57)
Substituting the expressions (54)–(57) into (53) we will obtain the following
form of the coordinates difference vector covariance matrix (relative positions):
⎡
σ ϕ21 + σ ϕ22 − 2σ ϕ1ϕ 2
PΔx = ⎢
⎣⎢σ ϕ1λ1 + σ ϕ 2 λ2 − σ ϕ1λ2 − σ λ1ϕ 2
σϕ λ + σϕ λ −σϕ λ −σ λϕ
σ λ2 + σ λ2 − 2σ λ λ
1 1
2 2
1
1 2
2
1 2
1 2
⎤
⎥.
⎦⎥
(58)
In navigation, while determining a position of other objects relative to own
ship we often apply the polar coordinate system, where a chosen point of own ship is
a system origin, coordinates — distance D and the true bearing α. In such case the
covariance matrix is expressed by this equation:
⎡
2
2
2
2
1 ⎢ Δϕ ⋅ σ Δϕ + Δλ ⋅ σ Δλ + 2Δϕ ⋅ Δλ ⋅ σ ΔϕΔλ
P Dα = 2 ⎢
(Δϕ 2 − Δλ2 )
D ⎢ Δϕ ⋅ Δλ
−
⋅ (σ Δ2ϕ − σ Δ2λ ) +
⋅ σ ΔϕΔλ
⎢⎣
D
D
⎡ σ D2
=⎢
⎣σ Dα
−
⎤
Δϕ ⋅ Δλ
(Δϕ 2 − Δλ2 )
⋅ (σ Δ2ϕ − σ Δ2λ ) +
⋅ σ ΔϕΔλ ⎥
D
D
⎥=
Δλ2 2 Δϕ 2 2
Δϕ ⋅ Δλ
⎥
⋅
+
⋅
−
⋅
σ
σ
2
σ
Δ
ϕ
Δ
λ
Δ
ϕ
Δ
λ
⎥⎦
D2
D2
D2
σ Dα ⎤
⎥
σ α2 ⎦ .
(59)
Therefore, the particular parameters will have this form:
—
error of distance
σD =
—
,
1
Δλ2σ Δ2ϕ + Δϕ 2σ Δ2λ − 2ΔϕΔλσ ΔϕΔλ
2
D
(61)
covariance between the distance and true bearing
σ Dα = −
48
(60)
error of true bearing
σα =
—
1
Δϕ 2σ Δ2ϕ + Δλ2σ Δ2λ + 2ΔϕΔλσ ΔϕΔλ ,
D
1
D2
⎛ Δϕ ⋅ Δλ
⎞
( Δϕ 2 − Δλ 2 )
⎜⎜
⋅ (σ Δ2ϕ − σ Δ2λ ) −
⋅ σ ΔϕΔλ ⎟⎟ .
D
⎝ D
⎠
(62)
2.2.2. Plotting a cartographic object in the navigational information system
The shape and size of a cartographic object is retrieved from the data base
(cartographic base). Except for raster systems, navigational information systems
create vector images, that is sets of points. Their position is defined by vectors originating at the reference point of the system. The position of the point P is determined
by the vector [xp, yp]. This point may be transformed in a number of ways:
•
rotation by the angle α; then its new position P’ will be defined by this expression:
P' = P ⋅ M R ,
(63)
⎡ cos(α ) sin(α ) ⎤
MR = ⎢
⎥
⎣−sin(α ) cos(α )⎦
(64)
where
Fig. 23. Rotation by the angle α
•
scaling:
P' = P ⋅ M S ,
(65)
⎡s 0⎤
MS = ⎢
⎥.
⎢⎣0 s⎥⎦
(66)
where
17/2011
49
Fig. 24. Scaling of a object
•
translation defined by this relation:
P' = M T + P ,
where
[
]
MT = T x ,T y .
(67)
(68)
Fig. 25. Translation of a object
50
In the navigational information system the point P represents the basic, i.e.
smallest, object. Their organized/arranged sets will make up cartographic objects of
more advanced structure — lines or polygons.
2.2.3. Representation of cartographic objects in the navigational information
system
Let us define a set Ω ⊂ R 2 , and call it the cartographic space CS of the navigational system. All points P(x, y) will belong to this set.
CS : ∀x ∧ ∀y P( x, y)∈Ω ,
(69)
where x, y ∈ N.
The sets A, B∈Ω composed of points P(x, y) will form subsets, making up
cartographic objects, e.g. depth contours, wrecks, anchorages, ship contours etc.
Let us now assume that A is a set of points forming a ship contour, while B
is a navigational danger critical for that ship (e.g. shallow water, wreck or another
ship). If there exist such points, it means that a navigational accident has occurred.
(∃x)(∃y) P( x, y)∈ A ∧ P( x, y)∈ B .
(70)
The section 2.2 presents the concept of the buffer zone of a cartographic object. We can say that the buffer zone contains a given cartographic object. Let the set
AS be a buffer zone of the set A, then:
A ⊂ AS ⇔ (∀x)(∀y)(P( x, y)∈ A ⇒ P( x, y)∈ As ) .
(71)
The set AS is created for a cartographic object A by adding a set of points of
the buffer described by the relations (47)–(49), (51). Let the set be denoted as SA.
The two sets are added by the method of Minkowski sum — the sum of vectors as
expressed by this relation:
{
}
AS = A ⊕ S A = P + PS : P∈ A, PS ∈S A .
A
A
(72)
2.2.4. The cardinality of the cartographic set
In navigational considerations (e.g. choice of a safe route) based on an
analysis of cartographic information from the NIS the size of cartographic sets will
be an important factor. The set of cartographic space Ω ⊂ R 2 consists of points P(x, y),
whose position in the space is defined by the vector rr = x, y ∈ R 2 .
[ ]
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51
For the set of space to be useful for solving navigational problems of voyage
planning and ship conduct, the set Ω has to be finite, that is its elements have to be
countable. The size of set Ω is defined by its cardinal number, denoted as K(Ω).
This relation will be true:
x, y ∈ R 2 ⇒ K (∀P( x, y)∈Ω )∈ N .
Naturally, to any set A this relation applies:
A ⊂ Ω ⇔ K ( A)∈ N
and
K ( A) + K ( A' ) = K (Ω) .
If the position of point P in the cartographic space of the NIS is described
by angular coordinates φ, λ, then for the cardinality of set A (φ, λ):
⎧
π π ⎫
ϕ∈ − , ⎪
⎪
A = ⎨P(ϕ ,λ ):
2 2 ⎬.
⎪
λ∈ −π ,π ) ⎪⎭
⎩
(73)
this relation will hold:
A ⊂ Ω ⇔ K ( A)∈ R .
Properties of the set of natural numbers N allow to specify the cardinality of
subset contained in it, which is not true for a set of real numbers R.
It is herein proposed to introduce the concept of cartographic set resolution.
Depending on the water area, such resolution will be maximum 10 m x 10 m (restricted area) or 100 m x 100 m (ocean) or adequately to the system accuracy. Such
approach is justified by regulations enforced by the IMO. In 2003 the IMO implemented Resolution A.953(23), World-Wide Radionavigation System (cancelling
Resolution A.529(13) Accuracy standards for navigation) [78]. The Resolution
defines minimum operational requirements for radionavigation systems (for details
see [50] and SOLAS V/13 [74]). The system should provide:
a)
for navigation in ocean waters:
• accuracy of position determination to 100 metres,
• update of position data at least every 2 seconds;
b)
for navigation in coastal and restricted waters:
• accuracy of position determination to 10 metres.
52
After the separability of the cartographic set is introduced, points P = (ϕ , λ ) ,
estimated by the navigational system, will be replaced by point P' = (ϕ , λ ) of the area
(square) to which they belong P = (ϕ , λ ) . Thus these points will make up a resolution
set ΩR:
⎧max : P∈( A ∪ AS ∪ AD )
,
Ω R ⊂ Ω ∀ P ∃ P ' ⇔ d = OP ' = ⎨
min
:
(
)
P
A
A
A
∉
∪
∪
S
D
⎩
(74)
where:
d
— distance defined by the vector length,
O
— origin of a reference system (e.g. own ship’s GNSS antenna position),
A, AS, AD — sets of own ship contour, buffer zone and domain,
ΩR
— set of points of cartographic space with a given resolution.
Finally we get
A ⊂ Ω
R
⇔ K ( A)∈ N .
2.3. Cartographic zones in the information system
The buffer zone around a ship or other objects, as defined in Section 2.2. of
this study, should be presented in NISs, permanently or if the operator chooses so.
The buffer zone id derived from the uncertainty measurements of position referred
to a local reference system, e.g. a GNSS receiver antenna. Navigating across open or
restricted waters the navigator will intuitively try to keep a certain area around its
ship (extended buffer zone) clear of other vessels. This area is called ship domain
[27], [36], [39], [109], [116], [120], [147], [169].
There exists also the concept of ship area, a wider space including the domain.
Its violation enforces the check whether the ship domain may be violated without
altering own ship’s movement parameters:
A ⊂ AS ⊂ AD ⊂ AR ,
where:
AR — ship arena,
AD — ship domain,
AS — buffer zone,
A — ship contour.
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53
We can write that:
(∃x)(∃y)P( x, y) ∈ A ⇒ P( x, y) ∈ AS ⇒ P( x, y) ∈ AD ⇒ P( x, y) ∈ AR .
(75)
Ψt = 0 [°]
Ψt = 45 [°]
Ψt = 90 [°]
2.5
Dmean
Ψt =135 [ °]
2
CPA mean
1.5
Ψt =180 [ °]
Ψt =225 [ °]
1.5
1
Ψt =315 [ °]
1
0.5
x [Nm]
x [Nm]
Ψt =270 [ °]
0.5
0
0
-0.5
-0.5
-1
-1
-2.5
-2
-1.5
-1
-0.5
0
y [Nm]
0.5
1
1.5
2
2.5
-1.5
-1
-0.5
0
y [Nm]
0.5
1
1.5
Fig. 26. Example ship domains of various types
Fig. 27. Buffer zone, domain, area
54
Flat 2D Romains proposed in the literature have various shapes: circle, rectangle,
ellipse, polygon, or more complex figures (Table 1). They are determined by a variety
of methods:
—
—
—
—
—
statistical — recording of passing distances between ships,
analytical — determination of the domain for assumed values of coefficients
and parameters, e.g. CPA or TCPA, ship’s circulation radius, distance of a crash
stop, decrease of speed in turning etc.,
questionnaire based research,
emprirical research,
artificial intelligence — based on experts involving research combined with
simulations or empirical tests.
Table 1. Chosen 2D ship domains
Term/author
Effective domain/
Fuji and Tanaka
Ship domain/
Goodwin
Shape
Method of determination
statistical (empirical
observations)
circle with step-increase
statistical
of radius for each sector of
(simulations, empirical
navigational lights
observations)
domain and circular arena
statistical, (questionnaire
research)
composition of circle sectors statistical (simulation
research)
semi-ellipse, ellipse
statistical
(empirical observations)
rectangle or ellipse
analytical
ellipse
Domain and arena/
Davis et al
Domain and arena/
Colley et al
Domain/
Coldwell
Dynamic domain/
Wawruch
Relative domain/
hexagon
Śmierzchalski and Weintrit
Domain with fuzzy boundary/ domain with a belt boundary
Zhao et al
Fuzzy domain/
empirical shape
Pietrzykowski
Subjective domain/
Zhu et al
Domain/
Kasyk and Rutkowski
empirical shape
rectangle
analytical
combined methods
artificial intelligence
(simulations and
questionnaires)
artificial intelligence
(questionnaires)
analytical
The size and shape of the ship domain change. It has been proved that the
domain is affected by the type of area, visibility conditions, size and type of ship,
type of cargo carried, navigational risk as well as parameters of another ship in encounter situations (Fig. 28, Fig. 29) [120].
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55
C=315 RB=45 l 100 100
0,45
0,40
C=315 RB=45 l 100 300
0,35
C=315 RB=45 l 300 300
0,30
0,25
0,20
0,15
0,10
0,05
0,00
0
1
2
3
4
5
6
Safe distance [Nm]
Fig. 28. The density function of safe distance (normal distribution) for an encounter of own and
another ship; own ship on course 0°; the other ship on relative bearing 45°, course 315°: a) both
ships 100 m in length; b) own ship 100 m long, the other 300 m; c) both ships 300 m in length
Lo = 100 [m]
Lt = 100 [m]
1.5
Lt = 200 [m]
Lt = 300 [m]
x [Nm]
1
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
y [Nm]
0.5
1
1.5
Fig. 29. Domains of a 100 metre ship in an encounter with ships of different length
(100, 200, and 300 m)
56
There may be different reasons for the intention to keep an area around the
ship or at a certain range of relative bearings clear of other vessels. The most frequent
case is just maintaining the safety of navigation at a preset level, regarded by the
operator as safe. Another case may be the inertia of navigational systems (unable to
promptly correct the disturbances, errors or values of measured parameters differing
from the forecast ones) or manoeuvring limitations of own or the encountered ship.
Such domain is proposed in [140] — the area where the operator is unable to take
action. It identifies the area within which the ship, for any of the above reasons,
cannot take an effective anti-collision manoeuvre.
Fig. 30. The domain where the operator is unable to take action
The shape and size of a domain may be described in linguistic terms: very
safe, safe, or relatively safe distance etc. Using the theory of fuzzy logic [168] in the
work [109] the ship fuzzy domain is proposed, whose shape and size depend on the
degree of membership of the distance on a given relative bearing to the set of safe or
dangerous distances to the target ship. Examples of such domains are shown in Fig. 31.
The above domains should be plotted in the NIS together with the ship contour. The ship domain should be extended by the buffer zone — summing the relevant vectors using the Minkowski method (72). The domains of own and the other
ship should be plotted starting from the origins of reference systems of each vessel:
GNSS antenna position, position obtained by AIS or ARPA, or the position obtained
through the fusion of data from the latter two sources [24].
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57
3
2.5
C= 225 [ °]
2
1.5
γ = 0.9
γ = 0.5
γ = 0.2
1
[Nm]
x
0.5
0
-0.5
-1
-1.5
-2
-3
-2
-1
0
y [Nm]
1
2
3
Fig. 31. Ship fuzzy domains
When AIS transponders are used, an approximate distance of ship identification will be possible provided that both antennas are in line of sight, which can be
expressed by these formulae:
d=
2
∑
i =1
(76)
2hi Ri
and
a 1−e 2 ,
Ri =
1−e 2 sin 2 ϕi
(77)
where:
d
— distance between the antennas,
h
— antenna height,
R
— mean radius of the ellipsoid curvature for the latitude ϕ,
a, e2 — elipsoid constants.
Ship domains practically can be used in the navigational information systems.
Vessel traffic in sea areas should be in compliance with the International Regulations for
Preventing Collisions at Sea (COLREGs). The rules of COLREGs define mutual obligations of ships, principles of behaviour in encounter situations, right of way etc.
58
The current COLREGs were adopted by IMO on 20 October 1972 and entered
into force on 15 July 1977, replacing the regulations from 1960. Almost unchanged,
the COLREGs have been in force to date. Decades after they were enforced, today
these regulations are subject to criticism, accompanied by proposals of amendments
[145]. The main argument for changes is the possibility of using new tools and technologies for the identification and assessment of a navigational situation. An alternative approach is not to amend the existing regulations, but to seek systems/methods
supporting the interpretation of COLREGs [154].
The NIS may perform the function of automatic interpretation of the COLREGs.
As indicated in Section 2.3 the cartographic space Ω of this system enables presentation of cartographic objects in the form of sets. Using certain properties of sets we can
conduct navigation in compliance with the COLREGs and good sea practice.
For instance,
let:
AD — be a set of points P(x, y) of ship domain A,
BD — be a set of points P(x, y) of ship domain B.
The points Pwa, Pwb — will be the tops of domains A and B, i.e. the point lying
at an intersection of domain boundary and forward part of ship centre line, which for
ship AD will be written as:
(∃x)(∃y)Dt ( x, y) ⇔ Dt ( x, y)∈ AD ∧ Dt ( x, y)∈ H ⇔ ODt = max ,
(78)
where:
O
— reference point of the domain (centre), e.g. centre of gravity, centre of
waterplane or the position of the GNSS antenna,
— domain top,
Dt
H(eading) — set of points representing ship’s heading.
Rule 15 of the COLREGs reads: ‘When two power-driven vessels are crossing
so as to involve risk of collision, the vessel which has the other on her own starboard
side shall keep out of the way and shall, if the circumstances of the case admit, avoid
crossing ahead of the other vessel’.
Let us write the sentence:
L ≡ ships A and B are crossing so as to involve a risk of collision, ship B is to starboard
side of ship A, ship B is not being overtaken.
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59
Let us formulate an algorithm (Fig. 32) that will implement the conduct of
ship A in accordance to Rule 15 using information from the NIS, where: C — ship’s
course, Cu — course to avoid a collision, γ — value of course alteration.
START
Current course
C=
Course change
Cu = C
i=1
T
F
τL=1
C= Cu+ C
wi=K(A B)
T
F
C= C‐ Cu
i=i+1
wi>0
F
P(xw,yw) B
T
F
wi ≥ wi‐1
F
C > Cu
Cu= Cu
T
T
Cu= Cu+γ
Cu= Cu
Cu= Cu‐γ
STOP
Fig. 32. Procedure for actions according to Rule 15 of COLREGs
of a cartographic object representing own ship domain
60
Determination of ship’s navlane
3. Determination of ship’s navlane
3.1.
Routing
Planning of a ship’s route, or routing for its presentation in the navigational
information system is part of voyage planning. Ship’s route is determined by the
navigator, who takes into account guidelines and recommendations of the ship’s
owner or operator. The route runs across waypoints, at which the ship alters course.
The number of waypoints along one route may vary from a few to dozens. Routing
may be divided into two basic stages of planning — static and dynamic.
Static planning is generally done before the voyage commences (or underway,
if new guidelines are sent in). Fig. 8 and the expression (35) illustrate the problem of
routing. To plan a safe route, the following factors are considered: position of known
navigational dangers which restrict the freedom of choice: land, shallow water,
wrecks, offshore installations, anchorages etc. Besides, temporary restrictions are
taken into account: currents, tides, closure of special zones. On top of that, the route is
planned in compliance with external recommendations and advice, e.g. established
shipping routes, traffic separation schemes, deep-water routes, waters threatened by
pirates etc.
The basic criteria that are considered include:
—
safe navigation factor, or quantitative assessment of the safety of navigation
expressed by the functional I:
I = F ( B , R , S , M ),
(79)
where:
B — area parameters,
R — vessel parameters,
S — parameters of the position determination system,
M — hydrometeorological parameters,
—
navigational risk factor defined as the product of accident probability and accident
consequences:
Ra = Pa ⋅ S a ,
—
(80)
navigational reliability factor (probability that the vessel in specific conditions
will not be found within the traffic lane allocated for the direction in which this
vessel is moving),
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—
—
—
—
—
—
vessel’s properties,
cargo,
passage time,
passage distance,
fuel consumption,
others.
Dynamic planning, taking place during a voyage and relating to a short period
of time, is aimed at avoiding a collision with another ship. Based on current assessment
of the navigational situation [7], [9], [79], [113], where the presence of other vessels
in the vicinity (their type, speed, geometric parameters etc.) are taken into consideration along with such factors as:
—
—
—
—
—
—
—
risk of collision,
close quarters situation,
visibility,
visibility of lights and aids to navigation,
type of work the ship is engaged in,
traffic intensity,
time available for planning.
The choice of route in dynamic planning, particularly in collision avoidance,
is represented as an optimization problem, which involves the following stages:
—
—
—
—
identify the problem parameters,
identify decision variables,
identify constraints for an acceptable solution,
formulate the objective function.
3.1.1. Algorithms of dynamic planning of own ship’s route
Algorithmic route planning for own ship is considered as a single or multistage problem. In the former case the problem comes down to a single determination
of a manoeuvre that satisfies the assumptions. This can be, for instance, determination of such course that the other ship will be passed at a preset distance [92], [93].
The problem will be written down as:
Y =max {F ( X ) : (D : X ⊂ D )} ,
(81)
where:
D — set of acceptable solutions,
Y — optimal solution,
F — objective function (criterion).
62
The multi-stage optimization problem consists in looking for such control
function uˆ (t ) , defining the optimal trajectory
assume the minimum value:
∧
∧
J ( x(t ), u (t ), t ) =
xˆ (t ), that the quality functional J will
tk
min
u ( t )∈U 0 , x ( t )∈X 0
∫f
0
( x(t ), u (t ), t ) dt ,
(82)
t0
where:
— function of instantaneous losses,
f0
u(t) ∈ U0 — set of acceptable controls,
x(t) ∈ X0 — acceptable space of the trajectory.
The problem may refer to the determination of an optimal trajectory by specifying waypoints and heading angles along the sections joining these points or by
altering rudder and/or engine settings.
In order to solve the problem of collision avoidance algorithms are applied
that work using the following criteria:
—
—
minimum deviation from the planned route,
minimum track.
The first criterion consists in determining such a course that the shift off the
present trajectory will be minimal, which can be written as:
f(xte) = min {f(xte); xte∈Dxte},
(83)
where
Dxte — set of solutions (deviations from the planned route that guarantee collision
avoidance).
The second criterion requires that the chosen collision-avoiding track will be
minimized in terms of extra distance to be covered, expressed in this form:
f(d) = min {f(d); d∈Dd},
(84)
where
Dd — set of solutions (tracks leading to collision avoidance.
In [139] the authors successfully propose the use of genetic algorithms in determining a new route for the ship to avoid a collision. The work [109], in turn, indicates a possibility of using the theory of fuzzy sets in planning an anti-collision route.
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3.1.2. Algorithms for determining the other ship’s route
The determination of a passing route of the encountered ship is of primary
importance in ensuring the safety of navigation. The oldest method to do this is visual
observation, as it allows to roughly estimate whether the other ship is on a collision
course. Apart from observation, the bearing on the other ship is measured at regular
time intervals. If the bearing does not change and the distance is decreasing, it is
considered that the other ship is on a collision course.
t2
t2
t1
t1
∧ TC ∈ K ⇔ TBt 2 = TBt1 + ∫ T&Bdt ≅ TBt1 ∧ d t 2 = d t1 + ∫ d&dt < d t1 → 0 , (85)
TC
where:
TC — true course,
K — set of collision courses,
TB — true bearing,
d — distance to the ship.
t — time.
In the above consideration, the deterministic identification of the other ship’s
course is not important. It is vital, however, to identify whether its course belongs to the
set of collision courses or not. In practice we often use the values of relative courses.
3.1.2.1. Application of the radar for trajectory determination
The radar is a basic shipboard tool for the estimation of the position and trajectory of an encountered ship in a short period of time. The effective estimation is
possible within the radar range, that is expressed by this formula (the result is in metres):
R max = a 4 ( Pi
τ
b
)(
S sk2
λ
2
)(
1 1
)( )(σ ) ,
FT m
(86)
where:
Rmax — maximum range in the open space,
a
— adjustment constant
i
— peak signal power,
T
— pulse duration,
b
— pulse length coefficient,
64
Ssk
λ
F
T
m
σ
— effective antenna area,
— wavelength,
— receiver noise coefficient,
— absolute temperature,
— echo visibility coefficient,
— effective area of object reflection.
The position of objects seen horizontally is estimated by radar. Therefore,
we can assume that the estimation of object position refers to a plane. The navigational parameter obtained from the measurement of a physical quantity (time) is
distance d between the radar antenna and the object at a given angle α of the radar
antenna rotation. If the position of the ship (antenna) is denoted as point P = [xa, ya],
then the position of the point representing the other ship Pr in time t in the navigational information system will be defined as:
Pr = P + ΔP ,
(87)
ΔP = P ⋅ M r ,
(88)
where:
⎡ d ⋅ sin( α )
Mr = ⎢
0
⎣
0
⎤
d ⋅ cos( α ) ⎥⎦ .
(89)
3.1.2.2. Fusion of AIS and radar information in the determination of the other
ship’s passing route
ARPA and AIS systems deliver data on positions of other ships in the vicinity
of our vessel. Due to the different character of position estimation (various methods,
sensors and time) the positions of the same objects obtained from AIS and ARPA
will vary. It is therefore purposeful to make a fusion of data from two different
sources of information for the presentation of a ship's position and its route in the
navigational information system.
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Fig. 33. Flow of information in the data fusion process
Where subsequent positions determined by the AIS and ARPA systems will
form correlated routes , their fusion can be done by statistical methods. The position
determination in the navigational information system will be expressed by these
relations:
—
distance:
d f (i ) =
(
1
σ dr2 (i ) d r (i ) + σ da2 (i ) d a (i )
2
σ (i ) + σ da (i )
2
dr
)
(90)
and distance variance:
−1
⎛
⎞
1
⎟ ,
Var d f (i ) = ⎜⎜ 2
2
⎟
⎝ σ dr (i ) + σ da (i ) ⎠
[
66
]
(91)
—
true bearing:
TB f (i ) =
σ
2
TBr
(
1
2
2
σ TBr
(i )TB r (i ) + σ TBa
(i )TB a (i )
2
(i ) + σ TBa (i )
)
(92)
and true bearing variance:
−1
⎛
⎞
1
⎟ ,
Var TB f (i ) = ⎜⎜ 2
2
⎟
⎝ σ TBr (i ) + σ TBa (i ) ⎠
[
—
]
(93)
speed over ground:
V f (i ) =
(
1
σ Vr2 (i )V r (i ) + σ Va2 (i )V a (i )
σ (i ) + σ Va2 (i )
2
Vr
)
(94)
and speed variance:
−1
⎞
⎛
1
⎟ ,
Var V f (i ) = ⎜⎜ 2
2
⎟
⎝ σ Vr (i ) + σ Va (i ) ⎠
[
—
]
(95)
course over ground:
COG f (i ) =
σ
2
COGr
(
)
1
2
2
σ COGr
(i )COG r (i ) + σ COGa
(i )COG a (i ) (96)
2
(i ) + σ COGa
(i )
and variance of course over ground:
−1
⎞
⎛
1
⎟ ,
Var COG f (i ) = ⎜⎜ 2
2
⎟
⎝ σ COGr (i ) + σ COGa (i ) ⎠
[
where:
i
da , dr
TBa, TBr
Va, Vr
COGa, COGr
]
(97)
— subsequent measurement,
— distance from own ship to AIS and radar positions of the other ship,
— true bearings on AIS and radar positions,
— speeds of the ships measured by AIS and radar,
— courses over ground measured by AIS and radar.
3.1.2.3. Application of the AIS and radar for the determination of the corrected
radar track
Some ships happen not to carry operational devices of the AIS system. In particular, this refers to ships to which the SOLAS Convention does not apply, fishing
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vessels, warships and ships where the AIS equipment has failed. At the same time,
within our ship’s horizontal range there may be ships and aids to navigation (masts,
light vessels, buoys etc.) fitted with AIS transponders. Our ship uses the radar to
simultaneously estimate the positions and speed vectors of all vessels in the vicinity
(Fig. 34). Radar positions of vessels not equipped with the AIS system can be corrected by analyzing the data concerning AIS-carrying vessels (differences in AIS
and radar positions, that are correlated). The tendency to position shifting (correlation) can be determined in given weather conditions (propagation of radar waves).
Then the probable AIS positions can be displayed in the navigational information
system. Let us call these positions the corrected radar track. Similarly, the integrity
of positional information can be examined in the navigational information system on
the basis of AIS and radar positions (on the vessel or at a VTS centre). The relevant
procedure may consist in regularly repeated determination of radar positions of vessels equipped with an AIS transponder and of the radar and AIS positions shift vector
of other vessels. The comparison of radar and AIS positions of a given vessel enables
estimating the system integrity, or consistency of navigational data.
Fig. 34. Vessel positions; source: AIS and radar, or only radar.
Southern Baltic 3.09.2010 — author’s observations
68
The research into the possibility of determining corrected positions was done
in the Southern Baltic on 3 September 2009. The prevailing weather conditions were
good on that day (Table 2). All data were recorded on board the training-research vessel Nawigator XXI, owned by the Maritime University of Szczecin. The ship was
then fitted with the following pieces of equipment:
—
—
—
—
—
—
AIS model: Nauticast X-Pack DS,
GPS model: CSI MiniMax,
GPS model: Koden KGP-913D,
Radar/ARPA model: JMA-5330-12,
Gyrocompass model: Gyro STD22 Anschutz,
Echosounder model: Skipper GDS 101.
Table 2. Weather conditions during the recording of vessel positions
Sea state
Visibility
Atmospheric pressure
Temperature
3/4 (wave height 0.5 to 2.5m)
good, about 10 Nm, no precipitation
1004 hPa
20°C
Passages of several vessels were recorded by radar and AIS during the research work (Fig. 35, Fig. 36).
Fig. 35. The track of the m/v ‘Baltic Skipper’ as recorded by the AIS and ARPA systems
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Fig. 36. The track of the m/v ‘Jolyn’ as recorded by the AIS and ARPA systems
Fig. 37. Change in the distance between AIS and ARPA echoes as the function of time —
m/v ‘Jolyn’
70
When the monitoring commenced, the ‘Jolyn’ was located 7.57 Nautical
miles away on bearing NR=011°. The registration of data was stopped when the
vessel sailed away and was at a distance of 14.3 Nautical miles on bearing
NR=028°. The AIS position shift relative to the radar position is illustrated below.
Fig. 38. The shift of the AIS position relative to the radar position of the m/v Jolyn
(bearing, distance)
The research done allows to state that there is a relation between the radar and
AIS position of other ships. The fact can be effectively used to increase the reliability
of navigational situations presented in navigational information systems.
3.1.2.4. The use of AIS systems for ship’s routing
Artificial intelligence systems find practical applications in determining ship
passage routes. If D is a set of acceptable solutions (routes leading to the destination) planned in accordance with the relation (35), then the AI system will choose
from this set an optimal route, defined by e.g. expression (81).
Let us take a look at one example [139], where the authors used genetic
algorithms for the determination of ship passage route in a collision situation.
If we define the cartographic space CS as a set of points satisfying the relation:
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CS : ∀ x ∧ ∀ y P ( x , y ) ∈ Ω,
then the navigational space NS for route planning (in static planning we neglect dynamical objects such as vessels) will be a subset of cartographic space CS that does
not contain dangerous areas — prohibited space PS for navigation: land, shallow water, sand bars, opposite traffic lanes etc., i.e.:
NS = CS \ PS
.
NS : ∀P ( x , y ){( x , y ) ∈ CS ∧ ( x, y ) ∉ PS }
(98)
The prohibited space PS can be defined as a space that for ship’s safety cannot have a common part with the set of points making up the ship’s contour. To put
it precisely, if underwater navigational dangers exist, the third dimension is taken
into account, i.e. area depth h. The edge bounding the prohibited space will be
termed as safety depth contour hb. For surface objects (or over surface like bridges)
the boundaries of buffer zones of defined areas not accessible for navigation.
If the ship has a draft t, the following applies:
∧
P ( x , y )∈S
h −t ≥ ∑r ,
(99)
where:
h = f (P)
t = g (P) ,
(100)
S
— set of ship’s contour points,
f, g — functions attributing depths and drafts to the points of ship’s contour,
Σr — total underkeel clearance (accounting for static clearance components).
It will be possible to plan a ship’s route between two waypoints belonging
to the space NS, if this statement is true:
∧
∨ (P + M
P ( x , y )∈PN M t > 0
t
∈ NS ) .
(101)
It is understood as the following fact: translation of a point P in the space NS
does not cause this point to move outside the space NS. In route planning these points
will form a structure of chromosomes whose lengths (numbers of genes) correspond
to the number of waypoints, whereas the maximum number of chromosomes is defined.
Further, a population of chromosomes with uniformly distributed lengths in the range
[min...max] is created, where the length 3 may be adopted as min (initial waypoint,
72
final waypoint, intermediate waypoint). The genes (points P) of the space NS, i.e.
their coordinates (x, y) are selected at random.
Chromosomes thus created undergo evolutionary processing (mutation).
Mutation consists in searching the navigational space, and each alternative solution
is compared with each other (the better wins). Eventually, the best solution is
reached (not necessarily the optimal one).
Fig. 39. An example of the evolutionary determination of the best route [139]
3.2.
Determination of the swept path
The swept path in the navigational information system is determined within
an allocated cartographic space of the system defined by the distance between the
extreme points of the moving ship’s contour (waterline) and the planned passage route
(Fig. 40). Swept paths are mostly determined in restricted areas, e.g. ports and its
basins or near quays, etc., in order to ensure safe passage of the ship. Due to uncertainty of the ship’s position, bank effects, yawing or hydrometeorological conditions
the path is wider than the actual ship’s breadth. In practice, parameters of the swept
path in restricted areas are determined for the characteristic ship — the largest ship
that can enter the given area, e.g. Szczecinmax, Panamax, Kamsarmax (or ship’s
parameters are determined in reference to the swept path) [43]. Swept paths can be
determined by empirical methods (synthesis of the results and practical knowledge),
simulations and artificial intelligence methods.
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Fig. 40. Construction of the swept path
3.2.1. Determination of the swept path by analytical methods
Two basic components of the ship’s swept path being determined are considered: rectilinear sections and bends. The following methods may be used for the
determination of their spatial parameters:
—
—
—
—
—
method of three components [43],
Panama Canal method [43],
INM method (Institute of Marine Navigation) [43],
PIANC method [122],
Canadian method [123].
Any of the above methods can be applied in plotting ship's path in the navigational information system. However, all the methods are applicable for straight sections
of the channel, while for bends the three components method does not apply [43].
During the design of fairways certain criteria for the assessment of navigational safety are used. The main such criteria include: underkeel clearance and the
breadth of safe manoeuvring area. It should be noted that all these methods are similar. The differences are in the definition of arguments taken into account.
74
For instance, the INM method defines the breadth of a straight swept path
with this relationship:
d = 2 d n ( 0,95 ) + kB + 2 d r , (102)
where:
dn(0,95) — navigational component of the swept path breadth at the confidence level
0.95 [m],
k
— experimentally defined coefficient:
k = 1.2 – good steerageway mk≤1°,
k = 1.6 – average steerageway 1°<mk≤2°,
k = 1.8 – poor steerageway 2°<mk≤3°,
B
— moulded breadth [m],
— swept path breadth margin [m],
dr
mk
— mean square error of keeping the vessel on a preset course [°].
This formula is used for the determination of the navigational component of
the swept path:
2
(
)
V ⎞
⎛
d n ( 0,95 ) = 1,8 M o2 + τ 2 ⎜ V pp +
⎟ m n2 + m k2 + mα2 , 57
,
3
⎝
⎠
(103)
where:
Mo — circular error of position (confidence level 0.632) [m],
τ — frequency of position determination,
Vpp — mean error of the assessment of current component perpendicular to the swept
path axis [m/s],
V — ship’s speed [m/s],
mn — mean square error of determined course [°],
mk — mean square error of maintaining the ship on preset course [°],
mα — mean square error of determined drift angle [°].
The last term in the formula (102), swept path breadth margin, is calculated
from this relation:
d r = 0,6 B . (104)
However, it seems that in practice the ship’s swept path should be determined in the navigational information system by the PIANC method (Permanent
International Association of Navigational Congress) or its modified version referred
to as the Canadian method. In these methods the arguments needed to calculate the
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swept path breadth are selected by the operator and referred to the breadth or speed
of own vessel. The PIANC method is used to determine the swept path of the ship
for both rectilinear section and bend of the fairway by this formula:
9
d = d m + ∑ d i + d rp + d rl , (105)
i =1
where:
dm — basic manoeuvring speed [m],
di — increase in the sept path due to [m]:
1 — ship’s speed,
2 — cross wind,
3 — crossing current,
4 — longitudinal current,
5 — parameters of the significant wave,
6 — aids to navigation and traffic control systems,
7 — type of bottom,
8 — depth to draft ratio,
9 — type of cargo,
drp — increase of the swept path on starboard side [m],
drl — increase of the swept path on port side [m].
The basic manoeuvring breadth is defined in relation to steerageway:
—
—
—
dm = 1.3B — very good steerageway,
dm = 1.5B — good steerageway,
dm = 1.8B — poor steerageway.
The other values of corrections di, drp, drl are presented in the table worked
out by the PIANC [122].
3.2.2. Determination of the swept path by AI methods
Model-based tests are increasingly used in research on qualitative and quantitative descriptions of vessel movement. Furthermore, models are also used to define
parameters of ship’s swept path by simulation methods [46]. The quality of certain
waterway parameters will depend on the degree of conformity between models and
real objects. Waterways designed by simulation methods utilizing maximum size
ship models are accessible for all vessels operating in the given area. In navigational
practice, the officer conducting the vessel generally manoeuvres the vessel smaller
76
than the maximum size vessel for that restricted area. In open waters the parameters
of own ship are important only in relation to the parameters of the encountered vessel.
In such cases a margin of the swept path is subjectively established on both ship’s
sides, bearing in mind own and the other ship’s parameters. Let us consider whether
in such situations it is possible to objectively determine the swept path of a ship
whose parameters will differ in real time and are dependent on the kind of navigational danger. It is possible if we use artificial intelligence methods (neural networks,
fuzzy-neural networks). These methods are based on acquired expert knowledge.
In [155] various forms of ship’s swept path are presented. For instance, for
overtaking on the fairway the ship’s swept path will be determined between the
shore and the side of the ship being overtaken (Fig. 41). In order to define the swept
path its reference axis is determined in the middle of the space restricted by the
overtaken ship’s side and the fairway limit. For AI systems to learn, specific sets of
data, or facts, have to be collected. One good method is to carry out a series of overtaking manoeuvres for which the safety of own ship is assessed. The assessment
should be done by a group of experts to their own criteria and established scale, e.g.
from very safe to very dangerous. Along with the situation assessment, the parameters describing each situation are recorded. These parameters may include:
Δy
ΔC
— distance from the ship to the swept path reference axis,
— deviation of ship’s heading from the preset course, determined by the centre
line of the fairway,
ω
— ship’s rate of turn,
dmin — minimum distance between the ships,
RB — relative bearing (angle between the fairway centre line and the straight line
joining two closest points of the ships.
Fig. 41. Visualization of an overtaking manoeuvre on the fairway
The gathered data, or facts, are utilized in the construction and learning
process of an artificial neural fuzzy logic network (Fig. 42).
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Fig. 42. The structure of neural fuzzy logic network for 5 input data and 243 inference rules
After the learning process the network is capable of assessing the safety of
own ship’s position on the fairway. The safety assessments affect swept paths for
proper safety levels ranging from ‘very safe’ (0) to ‘very dangerous’ (1) — Fig. 43.
Such path can be plotted on the screen of the navigational information system. The
navigator should manoeuvre within the path appropriate for the selected level of
safety defined in linguistic terms as the safe path or relatively safe path. Such paths
can be created for various manoeuvres and encounter situations. The network can be
taught in real conditions, i.e. while navigating a ship at sea.
Fig. 43. Swept paths for the passing manoeuvre in a restricted area —
various levels of navigational safety
78
3.3.
The navlane
The safe navigation lane — navlane — is an allocated safe waterway within
which the ship navigates freely. The properties of navlane are defined in section 3.3.2.
3.3.1. The routing system
The routing system is defined as a system consisting of one or more routes
or boundaries created to minimize the probability of collision. In navigational practice
routing systems are established to manage the traffic of vessels [61]. These systems
are implemented to increase the safety of navigation in restricted and open waters. In
the former, the establishment of traffic systems is related to natural factors limiting
the choice of the route. Apart from the geometrical shape and size of an area that
make up restrictions in horizontal and vertical planes, other navigational dangers are
likely in restricted waters. Ship routes are determined in restricted areas to run only
where the assumption (99) is satisfied. The width of route is determined by methods
presented in section 3.2. The width of recommended routes often remains unspecified
(choice of track remains at navigator’s discretion), the route being marked by a line
representing its centre — reference line.
In restricted areas routing systems are established in places characterized by
high vessel traffic intensity (e.g. crisscrossing shipping routes) and/or where intensive hydrometeorological phenomena occur (currents, ice).
Although specific parameters of routing systems depend on local conditions,
their determination should take into account the possibility of:
—
—
—
—
—
—
separation of opposite traffic flows,
reduction of collision risks at route intersections,
facilitation of vessel traffic in the given area,
management of traffic in the given area,
reduction of the risk of grounding,
passing by fishing grounds.
The following terms defined by the IMO are used in reference to routing
systems [61], [64], [69], [74]:
—
—
—
—
—
—
traffic separation scheme,
separation zone or line,
traffic lane,
roundabout,
inshore traffic zone,
two-way route,
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—
—
—
—
—
—
recommended route, recommended track,
deep water route,
precautionary area,
area to be avoided,
established direction of traffic flow,
recommended direction of traffic flow.
Fig. 44. An example of traffic separation scheme in the Gulf of Gdansk
Most routing systems to be established are first approved by the IMO.
3.3.2. Ship’s navlane
Routing systems will be helpful in defining a ship’s navlane, as navlanes in
the navigational information system will be either determined by the routing system
or away from it. Generally, it is recommended to use routing systems in compliance
with the established rules, e.g. Rule 10 of the COLREGs, concerning traffic separation schemes [58]. In such circumstances the navlane will be embedded in the routing
system. If the navigator decides not to use the routing system, the relevant navlane
will run outside it.
Definition:
A safe navlane of the ship is a allocated fair waterway within which the ship
can navigate freely.
80
Properties of the navlane are as follows (Fig. 45):
1. The navlane meets two conditions of navigation:
• enables a safe conduct of ship from point A to point B,
• runs along an optimal route.
2. The navlane may meet the requirements of a convex set, i.e. such set for which
all way points and all section of the ship’s route are contained in it.
3. The navlane encompasses the ship’s track.
4. The ship’s track does not have to be the centre line of the navlane,
5. The navlane encompasses the ship’s swept path.
6. The navlane may have a regular shape and constant width.
Fig. 45. Visualization of ship's navlane in the navigational information system
7. The navlane has the following parameters:
• length — l [m] — either the entire length of the navlane or its part between
waypoints, measured along the determined ship’s track,
• width — d [m] — determined at any point of ship’s track at the right angle between the navlane boundaries. The considered widths may be referred to as safe,
minimum, maximum or mean. The navlane width is increased by the buffer zone,
• depth — t [m] — in further considerations the navlane depth may be defined
as safe, minimum, maximum or mean,
• height — h [m] — above the water surface; the considered height may be the
minimum height determined by the minimum air clearance for the ship,
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• direction — α [°] — main direction of the navlane or its part,
• speed — v [knots] — ship’s speed on the navlane, for considerations the various
speeds will be termed minimum, maximum (allowable), recommended, reduced,
economical,
• time — t [hours] — allowed time or period of ship's stay on the navlane or its part.
The example navlane presented in Fig. 46 is that determined for the vessel Pride
of Canterbury. The vessel should have safely manoeuvred within that navlane. However, her movement outside the navlane resulted in a collision with a wreck [103].
Fig. 46. The manoeuvring area of the Pride of Canterbury when she collided with a wreck
3.3.3. Determination of the safe navlane
The safe navlane may be determined in the navigational information system
manually or automatically ad hoc. This process will be based on the cartographic
base of electronic navigational charts (ENC) or raster navigational charts (RNC).
82
The following data will be utilized for the determination of navlane parameters:
1.
2.
3.
4.
5.
Voyage plan.
Minimum distance.
Safety depth contour.
Ship’s domain.
Navlane height.
Fig. 47. Visualization of ship’s navlane
Ad 1. The items of a voyage plan prepared according to IMO recommendations [62]
include information necessary to determine ship’s future route and safe navlane. In
static planning (see 3.1) it will be the port of destination and IMO recommendations
on routing systems, VTS instructions, reporting systems and local regulations.
Ad 2. Minimum distance — the shortest distance between the relative position of
ship’s swept path (see 2.2.1 — vector of relative position) and the safe depth contour
or an isolated danger, or another cartographic object in the system that has to be
safely passed by, e.g. offshore installation [62]. In this study the minimum distance
will be defined as the distance between buffer zones of these objects (see 2.2). The
distance is arbitrarily established, taking into consideration IMO recommendations
and good sea practices.
In the navigational information system based on ENC vector charts the current distances will be calculated to the nodes of the safety contour. Each of these nodes is
defined by the Cartesian coordinates P(xi, yi). For horizontal distances we assume that
calculations are made on the plane surface. The distance of our interest is that of the
waterplane point (buffer zone of the ship) lying closest to a node of the depth contour
(buffer zone of the point). Hence this distance can be expressed by this relation:
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d = min{d i ( j )} ,
(106)
where
d i ( j) =
( x ( j ) − x i ) 2 + ( y ( j ) − y i )2 ,
i = 0,1,..., n; j = 1,2,..., m ,
(107)
where:
n — the number of contour nodes,
m — the number of ship’s waterplane points (ship’s buffer zone).
In case of he use of raster charts there are effective methods for reading out and
handling of graphic information [146], [169].
Ad 3. The safety contour is a depth contour indicated by the operator in the navigational information system from the contours displayed by the system electronic navigational chart (SENC). In navigational systems such as ECIDS the safety contour
is used to differentiate between areas regarded as safe and dangerous for the ship.
The minimum distance is referred to this contour. The user may set the values of
deep contour and shallow contour for the presentation of all depths in the navigational information system.
Ad 4. Ship’s domain: its parameters are taken for the determination of navlane
width, i.e. the navigational width of the waterway that during the ship’s presence
within it is clear of navigational dangers. Therefore, this width and its measure will
be referred to as ship’s domain width (see 2.3). It can be defined deterministically,
in prescriptive or descriptive terms. The width can also be determined on the basis of
the fuzzy domain. Then its values, corresponding to safety levels, can be expressed
linguistically, e.g. large, medium or small width.
Ad 5. Navlane height: this parameter is identified with the elevation of ship’s elements above the water surface (accounting for changes due to the consumption of
fuel and stores, water density etc.). This parameter is necessary to determine the
possibility of passing under such obstructions as bridges, overhead power lines etc.
The navlane can also be determined dynamically while the ship is being
navigated. The dynamic method is aimed at planning and performing anti-collision
manoeuvres in relation to other ships or stationary objects. In such situations additional parameters and data will be utilized, i.e.:
1.
2.
3.
4.
84
Risk of collision or close quarters situation (source: ARPA, AIS).
Closest Point of Approach (CPA) and Time to CPA (TCPA).
Navigational warnings (sources: VTS, GMDSS: NAVTEX, INMARSAT).
Weather forecasts or forecast movement of spills.
Navigation using the navlane incorporating navigational information system
4. Navigation using the navlane incorporating navigational
information system
Navigation in the determined navlane plotted in the navigational information
system fits into the e-navigation concept being implemented by the IMO (MSC,81).
E-navigation is an improved and extended version of navigation supported by available
and new navigational and communications technologies. It is based on current and
reliable information.
Navigators determining ship passages should plot them through the mandatory
routing systems if applicable to their class of ship or type of cargo, as provided by
the SOLAS Convention, Regulation V/10. The same convention obliges navigators
to conduct their ships safely and to avoid dangerous situations (Regulation V/34).
The navigation conducted within a properly laid out navlane satisfies these requirements.
The navlane is an electronic form of the sea route of a vessel.
Work on building marine waterways is also continued with other objectives,
e.g.: sea transport corridors — marine highways or short-sea navigation [136], organization of overall traffic in sensitive areas, e.g. the Marine Electronic Highway in
the Strait of Malacca.
4.1.
Remaining in the navlane
The ship navigating along the determined navlane remains within its boundaries throughout the passage. The navlane is determined according to the principles
set forth in section 3.3.3. Maintaining the ship within the navlane will ensure the
safety of navigation in relation to known stationary obstructions.
Let us denote a set of points forming the navlane as W. This set belongs to
the set CS of the cartographic space Ω in the navigational information system.
Therefore:
W ⊂Ω.
Then the depths h of points belonging to the set W, at instant (t) can be
expressed in this way:
h( x, y, t ) ∈ W (t )
and all own ship drafts in the domain area increased by the buffer zone Zb at moment (t) will be written as:
z ( x, y, t ) ∈ Zb(t ) .
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Navlane depths and ship drafts have to maintain these relations:
1. The set of navlane points contains points of the ship-relating set (arena, domain,
buffer zone):
Zb ⊂ W . (108)
2. The depths of the fairway points at moment t, where the ship is proceeding are
greater than ship’s draft (at other times this condition does not have to be met):
∧
z ( x , y ,t )∈Zb ∧ z ( x , y ,t )∈W
h ( x , y , t ) > z ( x , y , t ) . (109)
If we assume that the environment properties (e.g. water density) and the
condition of the ship (e.g. cargo) do not change in a certain time interval, the ship’s
draft will remain constant. Within that interval, however, the area depth may change
due to periodical phenomena such as tides. This will lead to periodical inaccessibility
or accessibility of some parts the of the navlane. In the latter case the ship will be
able to remain in the given part of the navlane in a defined period of time. Such restrictions for ship’s presence in an area are caused by sea currents (speed, direction),
waves (height, direction, length), wind (speed, direction).
Let us denote this part of the area (navigational space), where favourable
conditions will prevail at moment (t), as a set of points K. Mathematically,
K ⊂Ω,
so depths hk of the set K points at moment (t) will be written in this form:
hk ( x, y, t ) ∈ K (t ) .
We assume that the depth of each point of the set favourable at moment t is
greater than ship’s draft, therefore:
∧
z ( x , y ,t )∈Zb
hk ( x , y , t ) > z ( x , y , t ) . (110)
However, if the assumption is true that there is such a moment for which
there exists a depth of the navlane that is smaller than ship’s draft, written as:
∨
z ( x , y ,t )∈W
h ( x , y , t ) < z ( x , y , t ) , (111)
then navigation in this area can be possible if there occurs a period Δt in which navlane
depths will assume favourable values. Let us write down the function that will attribute
depths to points.
86
Δt = t2 − t1 ,
t2
∫ f ( x, y& , t )dt =h( x, y , t ) ≥ h
k
( x, y , t )
(112)
t1
(the domain is R+, and we are interested only in the depths (moments) for which h>hk).
The navlane point for which the above inequality will be true will be navigable for the ship if Δt is sufficiently large to enable the ship covering a distance
over this point longer than the ship’s domain. In this connection, we will define the
minimum speed that the ship must maintain (Fig. 48). It will have this form:
Vmin =
dD
Δt .
(113)
Fig. 48. Graphic presentation of ship’s possible passage (time and speed)
over the point with periodically favourable depth
In areas where depths are smaller than ship’s draft navigation will be possible
if the tide rises to such a degree that for a period of time the depths in points x, y
exceed the limit of depths favourable for a given ship. The set K of these points will
change in time, which is connected with the direction and rate of tide movement.
This condition has to be met:
Zb ⊂ K ⊂ W . 17/2011
(114)
87
Fig. 49. Visualized possibilities of ship’s passage through an area
with a periodically favourable depth
The set K of favourable depths will create in the navigational information
system an area whose part placed within the navlane will be enclosed in a slot. The
ship’s domain will have to be maintained within the slot. The speed and direction of
the moving slot (and changes of its parameters) will be dependent on the tidal rate
and direction. For the situation presented in Fig. 49 the following denotations are
used:
Vp — current speed,
Vr — slot speed,
Vs — ship’s speed over ground,
dn — length of unfavourable depths area,
df — distance from the domain front to the limit of unfavourable depths ahead of
the ship,
dD — domain length (increased by the buffer zone)
da — distance from the domain end to the limit of unfavourable depths astern of
the ship
dr — distance from the domain front to the slot limit ahead of the ship’s bow.
The slot length is calculated from:
t2
t2
t1
t1
d r = ∫ d& d dt + d D + ∫ d& f dt . 88
(115)
Condition (114) determines following relation for every particular moment (t):
dr ≥ dD
(116)
da ,d f ≥ 0
The speed of slot movement will be essential for estimating the ship’s speed.
Knowing the current vector (speed and direction) we can determine its projection on
the navlane direction, e.g. the centre line.
V r = V p ⋅ cos(θ ) (117)
where
θ — angle between the main direction of the navlane (TC) and the current direction (γ).
In the information system based on the cartographic display in a selected
reference system, e.g. WGS84, we will calculate its speed by defining the speeds
along the meridian Vλ and the parallel Vϕ . To this end we first calculate components of speed vector relative to the bottom: along the meridian VN and along the
parallel VE.
V N = V r ⋅ cos( TC ) ⋅ 1852
V E = V r ⋅ sin( TC ) ⋅ 1852 .
(118)
We take into account parameters of the reference system (M, N — radiuses
curvature in the meridian and prime vertical):
kϕ =
M =
where
1
, M
(
a 1− e
2
)
(1−e 2sin 2ϕ )
3/ 2
, 1
kλ =
N ⋅ cosϕ , N=
where
(119)
a
(1 − e sin ϕ )
2
2
Vϕ = kϕ ⋅ VN .
1
2
.
(120)
Vλ = kλ ⋅VE
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The above speeds will be used to calculate increments of point coordinates
after a period of time Δt, where
Δϕ = Δt ⋅ Vϕ
Δλ = Δt ⋅ V λ
. (121)
Knowing the slot movement speed we can determine a possible range of
ship’s speed changes (more precisely, the projection of the speed over ground vector
on the navlane centre line. The ship’s speed must be sufficient to maintain the ship
domain inside the slot during the whole passage through the area of unfavourable
depths (distance dn). This means that the domain front cannot cross the slot front
limit (da≥0) until it leaves the area of unfavourable depths, i.e. until the ship covers
the total distance df + dn. At the same time the slot end cannot ‘catch’ the end limit of
the ship domain until it leaves the area of unfavourable depths. This will happen
after the ship covers the distance dD + df + dn. In this connection we get this equation:
dD + d f + dn
da + dD + d f + dn
4.2.
⋅ Vr ≤ VSOG ≤
d f + dn
dn
⋅ Vr .
(122)
Crossing navlanes
According to formal (IMO recommendations) and informal principles (good
sea practice) the determined route and navlane of the ship will run clear of known
navigational dangers. While planning a voyage the navigator takes into account areas
of increased navigational risk due to possible encounters where vessel traffic is usually
heavy. Ship encounters cannot be predicted and the officer of the watch conducting
the ship within its established navlane will occasionally meet other vessels. If the
encounter involves two ships, only situations leading to the crossing of navlanes will
be important. Such situation can be expressed as:
Wi ∩ W j ≠ ∅,
(123) where:
Wi, Wj — navlanes of ships i, j.
The joint areas will be further examined whether for (123) there will occur
a situation that this statement is right:
∃(t ) : P( x, y, t ) ∈ ADi ∧ P( x, y, t ) ∈ ADj . 90
(124)
If so, we should assume that there exists or will occur a risk of collision or
close quarters situation. Then the navigator will have to predict action for the given
situation that will be in compliance with the COLREGs. The behaviour of two ships
in an encounter situation is regulated by rules for passing in Part B. Under these
rules the navigator’s ship with mechanical propulsion will have these obligations:
1.
2.
3.
4.
Give way to another ship — Rules 15, 18.
Not to obstruct another vessel to pass — Rules 9, 18.
Take action: alter course or speed — Rules 14, 19.
Stand on: maintain course and speed as the ship having the right of way — Rules 13, 17.
For situations mentioned in points 1 and 2 own ship will take (as far as possible)
the following actions:
⎯ alter the navlane direction so that the expression (124) becomes false (action
under Rules 8, 16),
⎯ alter speed so that the expression (124) becomes false (action under Rules 8, 16),
⎯ alter course and speed so that the expression (124) becomes false (action under
Rules 8, 16),
⎯ if the above actions prove ineffective, the navlane width will get narrower and
the above actions will be taken again,
⎯ if still the above actions remain ineffective, then both ships will be obliged to
take action (action under Rules 8, 16, 17).
For situations in point 3 both ships (unless one of them is being overtaken)
shall take actions, such as:
⎯ alter the navlane direction so that the expression (124) becomes false (actions
provided by Rules 8 and 14 or 19),
⎯ alteration of own ship’s speed such that the expression (124) becomes false
(action under Rules 8 and 14 or 19),
⎯ appropriate alteration of own ship’s course and speed such that the expression
(124) becomes false (action under Rules 8 and 14 or 19),
⎯ if the above actions prove ineffective, the own ship’s navlane will get narrower
and the above actions will remain applicable,
⎯ if still the above actions remain ineffective, the ship will be obliged to stop or
reverse the engines.
For situations mentioned in point 4 own ship has the right of way (stand-on
vessel). Her actions will consist in:
⎯ continuing her intentions, i.e. maintaining her course and speed and proceeding
within the established navlane. At the same time the situation will be observed
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to find out whether the other vessel takes actions to the effect that the expression
(124) will become false (actions provided by Rules 13 and 17),
⎯ giving up intended actions when she recognizes that the other vessel does not
make effective changes and the expression (124) will be false. The stand-on
vessel will take action in compliance with Rule 17,
⎯ obligation to stop executing her intended actions and appropriate alteration of her
navlane (and/or speed), as the actions of the other ship alone will not be effective
and the expression (124) will be true (action under Rule 17).
In each of the above cases of actions referring to points 1 to 4 the navigator
interpreting and applying COLREGs has to account for all navigational dangers and
events and all special circumstances, including own ship’s capabilities. Consequently,
to avoid the direct danger the navigator may act not in compliance with the rules.
It follows from the equation (124) that a vessel in an encounter situation has
to take action. If no other constraints exist, she shall alter her navlane direction
or/and speed. Another issue to be considered by the navigator is the moment when
to start action, i.e. delay Δt for which the inequality (124) will be true. The magnitude of the alteration made will depend on the value of the maximum cardinal number of the set K ( ADi ∩ ADj ) . These magnitudes may be equated with the CPA and
TCPA used in common anti-collision devices.
Table 3. Possible scenarios of actions in a crossing situation
Description of the situation
Action
D
C ≡ Wi ∩ W j ≠
∅
C1 ≡ Wi ∩ W j ≠
∅ ∧ (∀t) ADi
∩ ADj = ∅
C 2 ≡ (∃t ) : P( x, y, t ) ∈ ADi ∧ P( x, y, t ) ∈ ADj
C 3 ≡ (∃Δt ) : (∃ΔCOG ) : Wi ∩ W j ≠ ∅ ∧ ADi ∩ ADj = ∅
C 4 ≡ (∃Δt ) : (∃ΔV ) : Wi ∩ W j ≠ ∅ ∧ ADi ∩ ADj = ∅
C5 ≡ (∃Δt) : (∃ΔCOG) : (∃ΔV) : Wi ∩Wj ≠ ∅ ∧ ADi ∩ ADj = ∅
92
≡ ’Expected manoeuvre’
τC ⇒ ~ τD
τC1 ⇒ ~ τD
τC 2 ⇒ τD
τC 2 ⇒ τC 3 ⇒ τD
{alteration of navlane direction by
angle ΔCOG after Δt}
τC 2 ⇒ τC 4 ⇒ τD
{alteration of ship’s speed by ΔV
after Δt}
τC 2 ⇒ [(τC 3 ⇔ ~ τC 4 ) ⇒ τC 5] ⇒ τD
{alteration of navlane direction and
ship’s speed by, respectively, ΔCOG and
ΔV, after Δt}
4.3.
The anti-collision function of the navlane
The navigator executing the basic task of navigation — safe conduct of the
ship — has to maintain the ship within the fixed navlane. In practice, there will be
encounters with other ships, but only those that may lead to a collision situation will
require special attention. The behaviour of ships is regulated by the Collision Regulations. These define mutual obligations of ships in encounter situations. Let us consider an encounter of two ships, each proceeding along her navlane. We assume that
each ship has information on her own navlane estimated by her own sensors, e.g.
ARPA or received from the AIS system. Regarding ship encounters as the crossing
of their navlanes, we are interested exclusively in situations where their domains are
likely to overlap. Such a case means there will be a risk of collision and preventive
manoeuvres are required. These will be actions as discussed in the foregoing section.
What manoeuvre will be performed depends on the rule applicable to the given passing
situation. The action may be the one defined by the algorithm for executing Rule 15
of COLREGs (Fig. 32) discussed in section 0.
In anti-collision manoeuvres the alteration of course alone, if possible, may
be sufficient. This is actually most frequent case in practice. The speed is mostly
altered as the second option. If the ship, altering her course, is not capable of avoiding
a collision while remaining within the available width of her navlane, the latter will
have to be modified by changing its direction. Regulations as well as good sea practices provide that course alteration should be ‘large enough to be readily apparent to
another vessel’. In practice, to avoid collision the give-way vessel alters her course
and passes astern of the stand-on vessel. Then, following the stand-on vessel, she
returns to her previous course. An analysis of the encounter situation for predicates
contained in Table 3 is presented by the algorithm below (Fig. 50).
The algorithm incorporates processes previously defined for calculations of
ship’s course alteration ΔCOG, speed alteration ΔV, time delay Δt and the emergency
procedure.
Analytical formulas are used for estimating crucial parameters between approaching ships in a collision situation and for planning anti-collision manoeuvres.
Calculations based on these formulas refer to material points [11], regarded as ships.
However, let us note another aspect of the anti-collision problem. The CPA
is the basic criterion for the assessment of a collision situation. The CPA value is the
distance at which the ships will pass each other and whether the other ship will pass
ahead (+CPA) or astern (–CPA) of own ship. Making use of the domain in solving
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the anti-collision problem, the ships, according to the definition of the domain, will
be trying to keep their domains clear of other vessels, but not clear of their domains.
That is why in encounter situations each ship will have a different assessment of the
situation. Therefore, their expectations towards the other ship will be different. The
problem is presented in Fig. 51.
Fig. 50. An algorithm of encounter situation analysis
94
Fig. 51. A graphical presentation of an encounter of ships with different domains
When two ships A and B meet, both will expect the other to take action. In the
encounter of ships C and D, the ship C will not intend to perform any manoeuvres,
expecting the other to make a manoeuvre. The ship D, in turn, will assume that action
has to be taken, most probably by the ship C. Misunderstanding based on mutual
expectations has led to a collision and sinking of many a vessel [31].
Research shows that in encounter situations navigators tend to change their
ship domain. The change depends on the parameters of the other ship [120]. The domain may be subject to dynamic changes then. Although the idea is a positive one,
its execution is not necessarily the right move. It should be borne in mind that the
change of own domain parameters in an encounter situation results from the navigator’s assessment of the situation, which includes the comparison of parameters of the
two ships. Such assessment does not have to be similar to that of the other vessel.
Therefore, it is not the best of solutions.
The safety of navigation calls for using clear understandable criteria of navigational situation assessment. Therefore, in anti-collision manoeuvres real, unchangeable ship domains should be taken into account. Anti-collision manoeuvres such as
giving way should be performed accounting for the two domains. This will be possible when:
a) we adopt a new definition of the domain, i.e. anti-collision domain:
The ship anti-collision domain is an area around the ship clear of other objects
and their domains.
b) vessels will communicate their domain parameters to each other, or
c) they adopt averaged domain parameters.
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The available information technologies today offer possibilities of passing information on own domain to other ships in the horizontal environment. Then manoeuvre
planning will be based on the ship domains.
Let us draw a situation true for (124), as shown in Fig. 52.
Fig. 52. A situation of possible violation of the ship anti-collision domain
In the situation presented above the anti-collision domain will be violated.
The ship will take action to avoid a collision. The manoeuvres may be as follows:
1) such course alteration by ΔCOG that both anti-collision domains will not be violated
[108], where own ship, if possible, will be steered within its navlane, or
2) such alteration of navlane direction that it will lead to safe passing of the other
ship; however, these conditions have to be met:
a) iff the other ship is to pass ahead of own ship, then (Fig. 53):
⎯ change of the navlane direction will be made in reference to the farther
navlane edge (farther from the other ship),
⎯ a new navlane direction will be determined according to this procedure:
• the point of altering the navlane direction will be fixed at the reference
edge O(xb, yb) — this will be the polar point,
96
• from the pole O a new navlane direction of reference edge will be laid
out; the edge of the new direction will contain a tangent to the anticollision domain of the other and/or own ship; the tangent points are
are domian points lying closer to each other,
— alteration of the navlane direction will be continued until the other ship
domain shifts outside the own ship navlane; this action is in line with
good sea practice.
The ship’s navlane boundary edge direction will be determined according to
principles of analytical geometry. If the boundary edge is tangent to the anticollision domain and we denote the points of tangency of own ship as P(xw, yw) while
those of the other ship as P(xo, yo), then the edge equation will have this form:
y − yw =
yo − y w
( x − xw ) .
xo − x w
(125)
If the navlane direction alteration is planned with delay, then the relevant
equation will be based on the polar point. The reference edge equation can always
be expressed by this formula:
y − yb =
yo − yb
( x − xb ) .
xo − xb
(126)
From the above equations we can derive the direction of navlane reference
edge by this relation:
tgγ =
dy .
dx
(127)
As we will be changing the direction in reference to the polar point, it may
be convenient to use the polar form of the straight line equation for the determination of navlane edge direction:
γ = arccos
x o − xb
,
d
y − yb
.
γ = arcsin o
d
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(128)
97
Fig. 53. Visualized alteration of the navlane direction for collision avoidance, for clarity only
the left-hand (portside) reference edge of the navlane is shown
b) if own ship is to pass ahead of the other vessel, then:
— we will make one alteration of the navlane direction; the altered navlane direction should be continued till the complete clearance of the two ships’
domains; We execute the alteration in reference to the navlane edge closer to
the other ship.
98
Summary
Summary
The safety of navigation at sea calls for continued efforts to improve methods
of the conduct of navigation. Both navigator’s competencies and the main task of
navigation, until recently defined as safe conduct of the ship from point A of its
route to point B, are changing [85]. New functions related to maritime transport and
the exploration of the seas in search of resources require a broader look at maritime
navigation. Accordingly, today navigation is identified with the process of planning,
monitoring and control of ship movement. Following that approach and making use
of uptodate technologies, this author proposes the concept of safe navigation lane of
the ship, for which he coined the term navlane. The lane in this case means a specific path within which the ship is navigating. The lane, dynamically determined for
a given ship, is displayed on the screen of the navigational information system. The
concept assumes that the navigator should manoeuvre in his or her navlane throughout
the voyage (deviations are acceptable, but they must be well justified).
The main method of practical navigation, in turn, is the conduct of the ship
along the plotted track line. At the same time the maximum deviation from that line
is not graphically indicated. The deviation from the track line can be arbitrary and is
indicated only by some shipboard systems as an error (cross track error). Similarly,
some routing systems, e.g. recommended route, use the centre line only to indicate
the direction of ship’s movement. By definition, these routes do not have a defined
breadth [61]. In spite of that, deviation will be regarded as an error. When it occurs,
navigators tend to return to the track line as fast as they can or ‘bearing it in mind’
they steer for the nearest waypoint. Consequently, particularly in the close proximity
of the waypoint the present course of the ship may considerably differ from the established direction of the track line or the recommended route direction. In areas
where routing systems are established many ships have their waypoints set in similar
positions [142]. This brings individual ship routes close to each other. As a result,
close quarters situations, even collisions, take place more often [31].
The proposed ship’s navlane, in turn, means a space within which the ship can
manoeuvre. There is no subconscious need to return to the lane centre line. The navlane
comprises the entire planned passage, not only navigationally difficult waters with
routing systems established by the competent administration. The established navlane
parameters account for the ship (dimensions, manoeuvrability, cargo, speed etc.), ship
domain (domains of certain types may account for parameters of encountered ships),
accuracy (errors) of positioning systems. Additional factors taken into consideration in
the determination of navlane in the navigational information system are minimum distance, safety contour, ship’s air draft. The navlane functions are as follows:
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—
—
—
—
—
indicating the ship’s navigational space,
maintaining the ship away from known navigational dangers,
accounting for position error,
determination of an anti-collision space in encounter situations, complying with
the Collision Regulations,
others.
This study does not discuss other benefits of the navlane concept, such as
the determination of ship navlane direction minimizing the negative impact of the
marine environment on the ship: waves, ice etc., and minimizing the negative factors affecting ship’s stability, e.g. ship’s motions (amplitude, period), resonance [23]
or main engine overloading.
It is assumed that ship navlanes will be presented on screens of navigational
information systems of vessels proceeding within their horizontal range and coastal
stations. The technology available today makes such plans feasible. Navigational systems such as GNSS are capable of determining ship’s position with great accuracy.
The AIS system transmits these positions at regular intervals to other ships in the horizontal range. These can have positions of other ships displayed on screens of their
navigational devices. In many cases, however, the spatial dimensions of vessels,
shown as material points, are neglected. This negligence as well as that in determining
their accurate relative positions should not take place. The consequences of such inaccuracy can be dramatic: at the moment of collision of the ships Gdynia and Fu Shan
Hai on 31 May 2003 the ARPA system was indicating a distance of three cables
(about 550 metres) between the ships [30].
Navigational information systems are systematically installed on sea-going
vessels. The International Maritime Organization and its NAV Sub-Committee at
the 54. session introduced amendments to the International Convention on the Safety
of Life at Sea. The installation of the ECDIS system is obligatory and this obligation
for new ships will continue till 2018, when a great majority of sea-going vessels will
carry the system. The IMO is developing a much wider action — strategy of conducting navigation based on the implementation of uptodate technologies. The concept
of e-navigation has been defined: to harmonize and integrate marine means of information. The proper tools based on the basic and more advanced electronic technologies
may improve navigation itself, enhance the safety of shipping and the marine environment protection. The concept of navlane fits well into the idea of e-navigation.
From the practical point of view, it will be useful if the system displays navlanes of
vessels selected by the navigator or, automatically, of vessels involved in a close
quarters situation or risk of collision. The mutual presentation of navlanes will allow
the ships concerned to analyze and assess the situation in order to avoid a collision.
Navlanes presented in navigational information systems will be a development of
the Virtual Aids to Navigation concept [156].
100
Summary
Several marine accidents that have happened recently — groundings, collisions with wrecks or coral reefs — were caused by wrong settings in information
systems or errors in reading out the displayed information [103], [33]. Briefly speaking,
those accidents were caused by misunderstanding of the process automation in navigational devices. It should be noted that according to some research, the degree of
automation in navigation will be increasing. Devices such as ECDIS allow to choose
many options at various levels, e.g. data can be presented in standard or full display.
It turns out that it can be a source of errors (also device errors, e.g. failure to display
isolated dangers). To avoid such shortcomings, this author suggests that, regardless
of the settings of navigational information system devices, the contents of information on the area within ship’s navlane should always be complete (full display). This
also refers to information from other sources: ARPA or AIS.
The use of navlane in navigation will enable assessing and ensuring safety by:
—
—
—
officers of the watch,
VTS operators,
maritime administration in prevention or assessment of collision situations.
The navlane will be of particular importance in:
—
—
—
—
collision avoidance by ships,
presentation of what other ships expect — often ships engaged in activities considerably restricting their manoeuvrability while carrying out survey, search or
underwater operations, expect to have more clear space around them than it
might result from a typical encounter,
navigation in ice affected waters; in such case the navlane might be laid out by
land-based services or co-ordinators such as the icebreaker commander or pilot,
search and rescue operations; from the practical point of view, the navlane
width should not exceed the visibility range (but not less than the domain and
buffer zone widths); with the ship’s passage including the navlane recorded by
the VDR, an analysis of the recording may enable identification or rejection of
potential witnesses to marine accidents.
Marine navigation is mostly based on free choice of the sea route. Routing systems introduced by the IMO do not restrict this freedom much. Vessels are recommended to use them, but there is no obligation. The COLREGs do not regulate these
issues strictly, except for one case where a ship is obliged to maintain her direction of
movement (even in this case there was previously freedom of choice, and the ship may
refrain from following the obligation, if she finds it necessary). Similarly, the concept of
navlane does not restrict navigators in their freedom to choose a sea route. However, it
will introduce transparency and understanding of ships’ mutual behaviour and expectations. Additionally, the most important function of navlane may be the indication of its
direction in compliance with Collision Regulations. The results of author’s research
17/2011
101
show that this function would be extremely helpful these days. Deck officers were asked
to define their own and the other ship’s manoeuvres, in a situation shown in Fig. 54 [154].
Fig. 54. Ships in a collision situation in restricted visibility
Fig. 55. Planning a manoeuvre by own ship and expected from the other ship for the situation
presented in Fig. 54: 1 — stand on, 2 — turn to starboard, 3 — turn to port, 4 — reduce speed,
5 — stop the engines; first value — own ship, second value — the other ship
It follows from the responses (Fig. 55) that almost 70% of the officers expect
actions that do not comply with Rule 19 of COLREGs (I am standing on!), whereas both
ships should have turned to starboard. It seems that actions against the regulations result
from navigators’ decisions being excessively dependent on the indications of navigational
equipment, e.g. radar, not visual observation. The navlane, determined according to the
Collision Regulations, would introduce transparency and better understanding of the regulations. Research aimed at enhancing the safety of navigation is needed. One argument in
its favour is that international Collision Regulations are being neglected. The navlane
concept may change this. Research should be directed towards seeking new technological
solutions as well as possible modification of the existing Collision Regulations.
102
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