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
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 distribuacquisi-tion of navigaacquisi-tional informaacquisi-tion, 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:
F(X):X D
) 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 u
t
, determining the optimal trajectory
t x
, that the quality functional J should assume the minimum value:
tk t X t x , U t u f xt ,u t ,t t t t u , t x J 0 0 0 d min , 0 (2) wheref0 – function of instantaneouslosses,
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 i i i t t t f x u x , i = 0, 1, 2, …, k–1 (4)
The equations of state transitions then have this form:
1 2 1 0 0 1 1 1 0 0 1 1 2 0 0 1 , , , , , k k k k k t t t t t t t t t t t t t t t t t u ...,u u ,u x f ... f f u x f x ............... ... u ,u x f f u x f x u x f x (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 , , 0 0 k i X t x U t u d ii t t u t x J (6) where:u(t) – set of allowable control settings,
x(t) – allowable trajectory space,
d(i, i+1) – track covered while transiting from
state x to state ti x ti1
t x – optimal trajectory,
tu – control strategy, defining the opti-mal trajectory.
The control strategy, determining the optimal trajectory consists of a series of control settings:
1 1 0, t,..., tk t u u u u (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 01 (8)
and the fuzzy constraint is a fuzzy set C U with the membership function C:
, R UX :
μC 01 (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 ) (x G x C x X x 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 ) (x* x D X x D (11)
This also applies to a situation where many goals and many constraints exist. Then the fuzzy decision is defined as:
) ( ... ) ( ) ( ) ( ... ) ( ) ( ) ( 2 1 2 1 x x x x x x x Cp C C Gs G G D (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 xhave imposed goals G(x) in each control
stage. The process quality indicator of the multi-stage decision making (control) for k control multi-stages is assumed to be the fuzzy decision [5]:
k k t C G C G C G x D( ) 0 1 1 2 1 0 (13)
described by membership functions:
) ( ) ( ... ) ( ) ( ) ..., , ( 1 1 0 0 1 0 1 1 0 k k k t Gk t Ck t G t C t t t D x u x u x u u (14)
The obtained states xt1,xt2,...,xtk 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:
0 1 0 0 1 0,..., t t max D t ,..., t t t Du uk x u uk x (15)Then the optimal strategy makes up a series of control settings u*: ) ..., , , ( * * * * 1 1 0 ut ut utk u (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,
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: 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
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3. COLREGs 1972, Convention on the International Regulations for Preventing Collisions at Sea, International Maritime Organization.
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Recenzent: prof. dr hab. inż. Bolesław Mazurkiewicz
