Optimization of the ship movement trajectory in the...

116 Scientific Journals 18(90)

Scientific Journals Zeszyty Naukowe Maritime University of Szczecin Akademia Morska w Szczecinie

2009, 18(90) pp. 116–122 2009, 18(90) s. 116–122

Optimization of the ship movement trajectory in the navigational decision support system

Optymalizacja trajektorii ruchu statków w nawigacyjnych systemach wspomagania decyzji

Zbigniew Pietrzykowski, Janusz Magaj

Akademia Morska w Szczecinie, Wydział Nawigacyjny, Instytut Nawigacji Morskiej 70-500 Szczecin, ul. Wały Chrobrego 1–2

Key words: decision support, optimization of ship movement trajectory, safety of navigation

Abstract Ship collisions belong to major hazards to the safety of navigation. This article presents a prototype

navigational decision support system aimed at assisting the navigator in the process of ship conduct.

The problem of choosing the right track in ship encounter situations and optimization criteria are considered.

The authors present algorithms for the optimization of ship‟s route in collision situations, implemented in the

navigational decision support system being developed. The description of the system operation intended to

determine optimized ship movement trajectories is based on simulated ship encounters.

Słowa kluczowe: wspomaganie decyzji, optymalizacja trajektorii ruchu statków, bezpieczeństwo nawigacji

Abstrakt Kolizje statków należą do największych zagrożeń bezpieczeństwa nawigacji. W artykule przedstawiono pro-

totyp systemu wspomagania decyzji nawigacyjnych, mającego na celu wspieranie nawigatora w procesie kie-

rowania statkiem. Rozważono również problem wyboru właściwej trasy statku w sytuacjach spotkaniowych,

jak i kryteria optymalizacji trasy. Autorzy zaprezentowali algorytm optymalizacji trasy statku w sytuacjach

kolizyjnych, realizowany w nawigacyjnym systemie wspomagania decyzji. Opis funkcjonowania systemu,

mającego na celu ustalenia zoptymalizowanej trajektorii ruchu statku, oparty został na symulowanych spo-

tkaniach statków.

Formulation of the problem

Shipboard navigational equipment and systems

supply information that helps the navigator to take

decisions in the process of ship conduct. Electronic

navigation charts (ENC) are commonly used for

displaying navigational situations and related

information. These are data bases with standardized

contents, structure and format, containing all chart

information needed for navigation to be safe. The

ENC is part of the Electronic Chart Display and

Information System (ECDIS). The latter is a

navigational information system enabling a display

of selected information from the ENC incorporated

in the system plus data from other navigational

devices and systems.

The design and implementation of navigational

decision support systems is a new major step in

developing information systems on board ships, an

aid to the navigator responsible for safe ship

conduct. These systems, apart from supplying

information, perform other functions connected

with solving navigational situations by generating

solutions and proposing them to the navigator. One

such function is the determination of a safe ship

movement trajectory in the process of collision

avoidance.

The problem of choosing the right route in

collision situations requires that a number of partial

tasks should be solved. These include: situation

analysis and assessment, and the determination

of a ship trajectory that will ensure the safe passing


Zeszyty Naukowe 18(90) 117

of a vessel or vessels. The latter task is an example

of optimization problems. Its constraints are due

to the specific character of the area, ship‟s

manoeuvrability, regulations in force and other

factors. It is also important to take account of safety

criteria as well as economy-based criteria that

navigators apply while choosing the right route.

tasks of the navigational decision support system

The basic tasks of the navigational decision

support system are as follows: automatic acquisi-

tion and distribution of navigational information,

analysis of the navigational situation, solving

a collision situation and interaction with the

navigator.

In particular, the system should enable:

acquisition, fusion and integration of naviga-

tional data available on board,

display of a navigational situation,

analysis of the navigational situation taking

account of navigators‟ criteria,

signalling dangerous situations and showing the

current level of navigational safety,

planning a manoeuvre or manoeuvres and

movement trajectory in collision situations,

display of the proposed manoeuvre or

manoeuvres,

possibility of explaining (justifying) the deve-

loped solution.

The system should also store data for future

reproduction and analysis of ship movement

processes.

The systems in question are real time systems.

Their task is to observe the vessel and the

environment, register navigational information,

select, extract, verify and process that information.

The processed information presented to the navigator

concerns the identification and assessment of

a navigational situation and the proposed solutions

(decisions) ensuring safe navigation.

The general architecture of the navigational

decision support system developed by a team of the

Maritime University of Szczecin [1] is shown in

figure 1.

The key functions of the system are the

acquisition and distribution of navigational infor-

mation, analysis and assessment of a navigational

situation, solving collision situations and inter-

action with the navigator.

Information on the parameters of navigator‟s

own ship movement and that of other vessels is

fundamental for effective solutions to collision

situations. With this objective in view, algorithms

for navigational data acquisition, selection and

integration have been developed. The range and

accuracy of this information is of vital importance

for the navigator (situation assessment for making

decisions), and for the automatic process of solving

collision situations, i.e. optimization of the ship

movement trajectory.

Optimization of ship movement trajectory in a collision situation

How to alter the ship‟s course, especially when

a collision has to be avoided, can be presented as an

optimization problem [2]. The following steps have

to be made to formulate an optimization problem:

establish the problem parameters,

establish decision variables,

define limitations to an acceptable solution,

formulate the function of the criterion (objective

function) for the achievement of the objective.

The problem can be considered as a single- or

multi-stage control. In the former case the problem

comes down to a single choice that takes account of

preset criteria for the selection of ship‟s route. For

instance, the course alteration may be such that

other vessels will be passed at a preset safe distance

and the increased route length or time will be

minimized. This problem can be expressed with

this relation:

DF(X):X)F(X max (1)

where:

D – set of acceptable solutions of the optimi-

zation problem,

Fig. 1. Architecture of the navigational support system of a sea

going vessel

Rys. 1. Architektura systemu wspomagania decyzji nawigacyj-

nych na statku morskim



X – solution,

X*

– optimized solution.

F – objective function (criterion).

As to the multi-stage control problem, the

optimization task consists in finding such a control

function tu

, determining the optimal trajectory

tx

, that the quality functional J should assume

the minimum value:

kt

tXtx,Utu

tt,tu,txfttu,txJ

0

00

dmin, 0 (2)

where

f0 – function of instantaneous losses,

u(t) U0 – set of acceptable control settings,

x(t) X0 – allowable trajectory space.

The problem may be aimed at establishing the

optimal trajectory by defining ship‟s turn points and

headings to proceed along the sections set by the

turn points or rudder and/or engine settings at

chosen moments of time.

In both cases the formulation of route choice

criteria is essential.

Criteria of the choice of route

The trajectory defined by the navigator and

related manoeuvres have to be effective, lawful [3],

timely and noticeable. The manoeuvres should be

feasible and should satisfy economical criteria such

as the minimum route and time lengthening and

fuel consumption.

The effective manoeuvre is one that puts the

ship on a trajectory enabling safe passing of

obstructions and navigational dangers.

The timely and noticeable manoeuvre means the

ship is being steered in a manner allowing other

traffic participants to take notice of it, assess the

manoeuvre effectiveness and will not force the

vessels in vicinity to take extra actions, which may

be induced by a late manoeuvre or insignificant

alterations of ship‟s course or speed the other

navigators will fail to notice.

The possibility of a given manoeuvre to be

performed depends on the allowable range of

control variables, ship‟s manoeuvrability and

present values of the ship state vector.

The economical aspect of the manoeuvre trans-

lates into the minimization of costs related with the

mentioned lengthened route and time, fuel, etc.

As to the criteria of safety assessment – how

safe it is to pass other vessels or objects – the

following can be distinguished [2]:

closest point of approach CPAL and time to the

closest point of approach TCPAL,

ship domain,

safety/risk level indicators.

CPAL is a widely used criterion for navigational

situation assessment, implemented in automatic

radar plotting aid (ARPA). The criterion assumes

that the navigator defines a minimum, i.e. safe

distance, at which other vessels or objects will pass

(CPAL). An additional criterion – TCPA – its

minimum value to be exact, is also defined by the

navigator.

The ship„s domain is an area around the vessel

that the navigator should keep clear of other

objects. Another ship‟s entry into the danger zone –

ship domain – is interpreted as a threat to

navigational safety [4]. Two- and three-dimensional

domains are proposed in the literature on the

subject. The former, describing an area around the

ship, may have various shapes: circle, rectangle,

ellipsis, polygon or more complex planar figures.

There are criteria incorporating at the same time

the two quantities: CPAL and TCPAL. One example

is the risk level indicator known as Kearon

coefficient.

It is possible to take into account uncertainties in

the safety assessment by using, e.g. fuzzy logic,

which enables expressing the safety level in

linguistic terms used by the human: „safe‟, „barely

safe‟, „dangerous‟ etc. Crisp values, e.g. the

measured distance x are attributed a degree of

membership (x) 0, 1. This means that apart

from the membership (1) or lack of membership

(0), as set forth in the classical sets theory, there

may exist partial membership. This includes the

fuzzy closest point of approach and ship fuzzy

domain.

The former of the fuzzy criteria of safety

assessment implies that a tolerance interval is

accepted CPALmin, CPALmax (CPALmin CPAL

CPALmax), and any CPA value is attributed its

degree of membership to the set „fuzzy closest

point of approach‟. The ship fuzzy domain is

similarly interpreted.

Both the fuzzy CPA and ship fuzzy domain are

criteria incorporating imprecise judgment characte-

ristic of the human.

According to commonly adopted principles of

good sea practice the moment of starting a collision

avoiding manoeuvre is defined on the basis of the

ship encounter phase or/and TCPAL. In the former

case it is the interval of distance to another vessel

(object) within which action (manoeuvre) should be

taken. The navigator may also define the moment



of starting the manoeuvre as the TCPALmin by

specifying the minimum time to the closest point of

approach. While deciding on the moment of

starting the manoeuvre the navigator has to allow

for the time in which the ship responds (e.g. course

alteration or speed reduction). The criterion for the

timely manoeuvre performance can also be

described using the tools of the fuzzy sets theory.

The criterion of noticeable manoeuvre can be

formulated as follows: when the course is altered, it

is recommended that the change could be visible to

others. This means that the course alteration should

be possibly close to the suggested one, recom-

mended by experienced navigators and stated in the

literature on ship manoeuvring. The above criterion

can be described, like the fuzzy CPA and ship

fuzzy domain, using the fuzzy sets theory.

Lengthened route. Most frequently, in

optimization problems concerning the choice of

route such factors as lengthening of the track and

time, fuel etc. make up an element of the objective

function (quality indicator) and are minimized. The

lengthened route can also be described using the

value of shift from the original trajectory.

Assuming that the minimum and maximum

trajectory shifts accepted by the navigator are

known, we can describe the lengthened route using

a relevant membership function of the fuzzy set.

The presented criteria may be applied in

optimization problems of ship movement trajectory

in the process of single or multiple decision making

(single- and multi-stage control).

Multi-stage control

Dynamic programming is one of the standard

methods of dynamic optimization, used in multi-

stage problems of decision making and control.

The optimal ship control in terms of an

established control quality indicator is determined

using Bellman principle of optimality. The

principle defines the fundamental property of

optimal policy, which says that regardless of the

initial state and initial decision, the remaining

decisions must constitute an optimal policy with

regard to the state resulting from the first decision.

Consequently, the calculations start from the final

stage and follow backward to the initial stage. It has

been proved [5] that the process of ship collision

avoidance satisfied the duality conditions,

therefore, using the optimality principle we can

commence calculations from the initial stage and

continue to the final stage.

For a given finite space of states X = {x1, ... xn}

and finite space of control settings U ={u1, ... um},

transitions of states in subsequent k stages of

control are determined by this function:

f: X U X (3)

such that:

),(1 iii ttt uxfx

, i = 0, 1, 2, …, k–1 (4)

The equations of state transitions then have this

form:

12100

11

100112

001

,

,

,,

,

kk

kkk

ttttt

ttt

tttttt

ttt

u...,uu,uxf.....ff

uxfx.....................................................

u,uxffuxfx

uxfx

(5)

Attempting at the possible least lengthening of

track with a vessel proceeding at constant speed,

may be a control quality indicator (optimality

criterion), yielding the time-optimal control.

The area of collision risk is assumed to be

a safety criterion. The above constraints imposed

on state variables (safety criterion, COLREGs) and

control settings are taken into account by checking

whether the state variable has not exceeded the

constraints at each considered node r by rejecting

the nodes where excess values have been detected.

Accounting for the constraints we can determine

allowable control settings (U0) and allowable states

(X0).

For instance, the quality indicator (2) in the

route minimization problem will assume this form:

1

0,

1,min,,00

k

iXtxUtu

iidttutxJ (6)

where:

u(t) – set of allowable control settings,

x(t) – allowable trajectory space,

d(i, i+1) – track covered while transiting from

state itx to state

1itx

tx

– optimal trajectory,

tu

– control strategy, defining the opti-

mal trajectory.

The control strategy, determining the optimal

trajectory consists of a series of control settings:

110,...,,

kttt uuuu (7)

Respectively, the starting point is adopted to be

the ship‟s current position described by its state

vector, while the final point is:

fixed final point of the trajectory (control with

the fixed final point), mostly in restricted areas,



fixed final course typically corresponding to the

original course, mostly in open sea areas.

The above problem can be solved by dynamic

programming methods, the bound-and-branch

method or using the graph theory tools [6].

Multi-stage control in a fuzzy environment

In consideration of the fact that there occur

inaccuracies and uncertainties (imprecisions) in

defining the goals and constraints in the problem of

choosing the optimal trajectory of ship movement

in real conditions, we can choose the multi-stage

decision making (control) in a fuzzy environment

[7] as an alternative to the classical approach of

dynamic optimization.

The concept of fuzzy environment is understood

as the ordered four G,C,D,U (G – fuzzy goal, C –

fuzzy constraints, D – fuzzy decision, U – set of

decisions) [8]. The fuzzy goal is defined as a fuzzy

set G U with the membership function G:

R,UX:μG 10 (8)

and the fuzzy constraint is a fuzzy set C U with

the membership function C:

R,UX:μC 10 (9)

For a given finite space of states X = {x1, ... xn}

and a finite space of control settings U ={u1, ... um}

the state transitions in subsequent control stages k

are determined by the function (4), (5).

In the case of decision making in a fuzzy

environment, i.e. with the constraint C and goal G,

described, respectively, by the membership func-

tions C(x) and G(x) the fuzzy decision D is found

from this relationship:

))(),((min)( xxx CGXx

D

(10)

It is assumed that the optimal decision is

a decision maximizing the degree of membership in

the set of fuzzy decision D:

))((max)( * xx DXx

D

(11)

This also applies to a situation where many

goals and many constraints exist. Then the fuzzy

decision is defined as:

)(.....)()(

)(.....)()()(

21

21

xxx

xxxx

CpCC

GsGGD

(12)

where:

p – number of goals,

s – number of constraints.

The control process for the space of states X and

of control settings U consists in selecting control

settings u with imposed constraints C(x), while the

states x have imposed goals G(x) in each control

stage. The process quality indicator of the multi-

stage decision making (control) for k control stages

is assumed to be the fuzzy decision [5]:

kkt GCGCGCxD 12110)(0

(13)

described by membership functions:

)()(

...)()()...,,(

1

10010

1

10

kk

k

tGktCk

tGtCtttD

xu

xuxuu

(14)

The obtained states kttt xxx ...,,,

21 are determined

by the subsequent application of the equations of

state transitions (5).

The problem of multi-stage control in a fuzzy

environment is formulated as follows:

010010

...,,max...,, tttDtttD xuuxuukk

(15)

Then the optimal strategy makes up a series of

control settings u*:

)...,,,(****

110

kttt uuuu (16)

Like in the multi-stage control discussed in

Chapter 5, the above problem can be solved by

dynamic programming methods, by the branch and

bound method, or by using the graph theory.

The trajectory optimization in the navigational decision support system

The algorithm for the determination of safe ship

movement trajectory, as presented in Chapter 3, has

been implemented as a module in the navigational

decision support system (see Chapter 2). The

module operates using data from data acquisition

and integration modules (positions, courses and

speeds of own and other ships). The system data,

formulated by the navigator, make up a separate

group. These data include, among others, current

visibility conditions, criteria of navigational

situation assessment, navigator‟s preferences

concerning the choice of optimization algorithm,

constraints connected with economical criteria of

the choice of route (extended track and time), as

well as the time Top of starting the manoeuvre,

enabling the navigator taking a decision and

performing (starting) the manouevre.

The optimization results – proposed safe

trajectories – are presented graphically on the

operator‟s/navigator‟s interface screen. An ENC is



employed for the visualization of the situation and

proposed solutions. The display on the navigational

chart includes one trajectory described by the turn

points with their positions, times to reach these

points and courses required to proceed along the

determined trajectory. The navigator sees the

trajectory that is in compliance with international

collision regulations. The functionality of the chart

allows, on navigator‟s request, to display other

acceptable trajectories that the system works out.

Figure 2 shows a navigational situation in the

form it is presented to the navigator on an ENC.

According to the regulations, the navigator‟s ship

(in good visibility conditions) is obliged to give

way to the ship on the starboard beam. The

suggested manoeuvre may threaten the other ships,

in relation to which the first ship has the right of

way. The safe trajectory has been determined by the

multi-stage control method for the preset

CPAL = 0.5 Nm.

Fig. 2. Ship encounter situation no 1; multi-stage control;

Top = 60 s; CPAL = 0.5 Nm

Rys. 2. Sytuacja spotkaniowa statków nr 1; kontrola wieloeta-

powa – multistage control; Top = 60 s, CPAL = 0,5 Mm

Figure 3 shows a ship movement trajectory

determined by the method of multi-stage control in

a fuzzy environment. The solution takes account of

the tolerance interval for the CPA value, described

by function of membership to the fuzzy set of safe

CPA.

Fig. 3. Ship encounter situation no 1; multi-stage control

in fuzzy environment; Top = 60 s, CPALmin = 0.5 Nm,

CPALmax = 3 Nm

Rys. 3. Sytuacja spotkaniowa statków nr 1; kontrola wieloeta-

powa w środowisku rozmytym; Top = 6 s, CPAL = 0,5 Mm,

CPAlmax = 3 Mm

Under the regulations in force, the navigator‟s

ship should perform a manoeuvre to starboard or

alter speed. If such a solution does not exist, the

system releases such information and proposes to

change the criteria for choosing a new route or to

consider a manoeuvre to port (fig. 4 ).

Fig. 4. Ship encounter situation no 2; multi-stage control;

Top = 180 s, CPALmin = 1 Nm, CPALmax =3 Nm

Rys. 4. Sytuacja spotkaniowa nr 2; kontrola wieloetapowa;

Top = 180 s, CPALmin = 1 Mm, CPALmax = 3 Mm

The change of parameters describing navigator‟s

preferences in the process of defining a safe

trajectory will start computational procedures (figs.

5 and 6).

Fig. 5. Ship encounter situation no 3; multi-stage control in

fuzzy environment; Top = 180 s, CPALmin = 0.5 Nm,

CPALmax =3 Nm

Rys. 5. Sytuacja spotkaniowa nr 3; kontrola wieloetapowa

w środowisku rozmytym; Top = 180 s, CPALmin = 0,5 Mn,

CPALmax = 3 Mm

Fig. 6. Ship encounter situation no 4; multi-stage control

in fuzzy environment; Top = 60 s, CPALmin =1 Nm,

CPALmax = 3 Nm

Rys. 6. Sytuacja spotkaniowa nr 4; kontrola wieloetapowa

w środowisku rozmytym; Top = 60 s, CPALmin = 1 Mm,

CPALmax = 3 Mm



Summary

Avoiding collision situations has been

considered as one of the most important

navigational functions of the decision support

system, besides the acquisition and distribution of

navigational information, analysis and assessment

of navigational situations, and interaction with the

navigator.

The problem of the choice of route in ship

encounters is presented as an optimization problem.

The relevant criteria of the choice of route have

been characterized. Then two types of algorithms

for ship route optimization in collision situations

have been presented: multi-stage control and multi-

stage control in a fuzzy environment.

The latter algorithm provides an alternative

to the classical approach of dynamic optimization.

It takes account of typically human inaccuracies

and uncertainties in formulating goals and

constraints. The algorithm enables finding a solu-

tion which compromises contradictory goals

occurring in an optimization problem.

Based on the mentioned algorithms, the

discussed navigational decision support system has

proved operational providing safe optimized

trajectories in selected ship encounter situations.

References

1. PIETRZYKOWSKI Z., MAGAJ J., CHOMSKI J.: Model of

navigational decision support system on a sea-going vessel

m. Scientific papers AM Szczecin no. 13(85), Szczecin

2008, 65–73.

2. PIETRZYKOWSKI Z.: Modelling of Decision Processes in

Sea-Going Ship Movement Control. Studies No 43,

Maritime University of Szczecin, 2004.

3. COLREGs 1972, Convention on the International

Regulations for Preventing Collisions at Sea, International

Maritime Organization.

4. PIETRZYKOWSKI Z., URIASZ J.: The ship domain – a

criterion of navigational safety assessment in an open sea

area. The Journal of Navigation, 62, The Royal Institute of

Navigation, Cambridge 2009, 93–108.

5. PIETRZYKOWSKI Z.: Fuzzy Control in Solving collision

Situations at Sea, Computational Intelligence: Methods and

Applications. Eds. L. Rutkowski, R. Tadeusiewicz, L.A.

Zadeh, J. Żurada, AOW EXIT, Warszawa 2008, 103–111.

6. DEO N.: The Theory of Graphs and its Application in

Technology and Computer Science. PWN, Warszawa

1980.

7. KACPRZYK J.: Multi-stage fuzzy control. WNT, Warszawa

2001.

8. BELLMAN R.E., ZADEH L.A.: Decision making in a fuzzy

environment. Management Science, No 17, 1970.

Recenzent:

prof. dr hab. inż. Bolesław Mazurkiewicz

Akademia Morska w Szczecinie

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