- Email: [email protected]
Trajectory Planning for Ship in Collision Situations at Sea by Evolutionary Computation
Copyright © IFAC Manoeuvring and Control of Marine Craft, Brijuni, Croatia, 1997
TRAJECTORY PLANNING FOR SHIP IN COLLISION SITU ATIONS AT SEA BY EVOLUTIONARY COMPUTATION
Roman Smierzchalski
Department o/Ship Automation, Gdynia Maritime Academy, uf. Morska 83, 81-225 Gdynia, Poland; e-mail: [email protected].
Abstract: A modified version of the EIPN algorithm has been used for computing the optimum trajectory of a ship in a given environment. On the basis of this algorithm, a program estimating a safe trajectory of a ship in collision situation was developed. By taking into account certain boundaries of the manoeuvring region, along with navigation obstacles and other moving ships, the problem of avoiding collisions at sea was reduced to a dynamic optimisation task with static and dynamic constrains. The introduction of a time parameter and moving constrains representing the passed ships is the main distinction of the new system. Sample results of using the program for estimating the ship trajectory in typical navigation situations are given. Keywords: ship control, trajectory planning, genetic algorithms, optimisation problems, multiple-criterion optimisation.
I. INTRODUCTION
The problem of determining the safe trajectory as a task of non-linear programming was formulated by Smierzchalski and Lisowski (1995), Lisowski and Smierzchalski (1995a) in which a kinematic model of the own ship was applied. Another possible approach to this problem is the reduction of the solution space to a finite-dimensional one, by creating a so called digitised matrix of permissible manoeuvres for a given collision situation and a certain time instant by Lisowski and Smierzchalski (1995b). In the digitised set of permissible solutions a set of effective solutions satisfying the Paretooptimum rule is determined taking into consideration an assumed values of the safe passing distance and the course deviation. In (Smierzchalski, 1995, 1996a, 1996b,) the problem of avoiding collisions was formulated as the multi-criterion optimisation task. Three separate criteria were used. A continuous task of the multi-criterion optimisation was reduced to a static multidimensional task. The problem was solved for a static and a moving targets, using the DIDAS-N system (Granat and Wierzbicki, 1995, Kreglewski et af., 1991). The attempt to estimate the safe trajectory
The solution to the problem of avoiding collisions at sea is one of crucial tasks of sea navigation. When determining a safe trajectory for the own ship, we look for a trajectory that is a compromise between the costs of necessary deviation from a given route, or from the optimum route leading to a given steering goal, and the safety of passing all static and dynamic obstacles, strange ship - target. All trajectories which satisfy the condition of avoiding, in a satisfactory way, the risk of collision make a set of permissible trajectories. Therefore the task of determining the safe trajectory is reduced to the choice of an optimum solution, or a subset of optimum solutions from the set of permissible trajectories, taking into account certain cost criteria and the need to satisfy, in any circumstances, the safety conditions. These conditions are, as a rule, defined by the operator, basing on the speed ratio between the ships involved in the passing manoeuvre, as well as the actual visibility, weather conditions, navigation area, manoeuvrability of the ship etc.
129
using genetic algorithms was presented in (Furuhashi et al. , 1996). The collision situation was modelled as a fuzzy process with many inputs. The authors applied a multi-dimensional digitised model of the collision situation in which the rules were defined at each stage of the own ship steering. For selecting the steering rules the authors made use of a fuzzy classifier system.
estimating the safe trajectory, using the C++ language convention. Evolutionary_ algorithmO { int T; I*T - generation counter variable*1
T=O: input_data_foryarametrising_operation_ oC evolutionary_ algorithmO; input_data_for_ defming_environmentO;
Another problem, although in a sense similar to the problem of avoiding collisions at sea, is the problem of steering a mobile robot using the evolutionary algorithms (Lin et al. (1994), Michalewicz (1996), Michalewicz and Xiao (1995), Trojanowski and Michalewicz (1996) , Xiao et al. (1996» . In their papers, the authors present a method of estimating the paths of the robot in a well known environment. Having selected the optimum path, the motion of the robot is controlled in an on-line process along the selected path, including new obstacles. The solution of the well defined problem of planning the robot motion was found using the EPIN (Evolutionary Planner INavigator) adaptation system especially designed for this purpose.
P(T) = creation_ oCinitial_chromosomeyopulationO; building_dynamic_ obstaclesO; population_evaluation ( P(T) ); while ( != terminating_conditionsO ) { T = T + I; Op = operator_selectionO; Par = selection_ofyarents_oryarentO; offspring_creation_ and _ introduction( Op, Par ); building_dynamic_ obstaclesO; population_evaluation ( P(T) ); introduction_ of_new_individuaIO; selection_ oCbest_individual ( P(T) ); }
The main goal of the present paper is to discuss the problem of avoiding collisions at sea, considering it as the evolutionary task of estimating the own ship trajectory in an unsteady environment. Basing on the E/PN planning concept presented by Michalewicz and Xiao, a modified version of the system has been developed which takes into account specific character of the collision avoiding process. The main innovation of the modified versions is the existence of different types of static and moving constraints reflecting the real environment of the moving strange ship - target and its dynamic characteristics as the target of control. In an evolutionary process of search for the optimum trajectory in a collision situation, a time parameter and dynamic constraints were introduced. The dynamic constraints represent moving strange ships - targets being passed by, whose shapes and dimensions depends on assumed safety conditions, namely the safe distance between the passing targets, as well their speed ratio, and bearing.
The multiple iteration process of transforming and creating new individuals, accompanied by the creation of new populations is carried on until the "stop" tests in the program are fulfilled .
3. EVOLUTIONARY PROBLEM TO AVOID COLLISION Formulating the task of estimating the optimum safe trajectory in a well known environment, considered as an evolutionary problem, involves defining the evolutionary steering goal and conditions at which this goal has to be reached.
3. 1
Definition of environment and constraints.
The kinematic model of the ship motion was appl ied for estimating the safe trajectory. The dynamics of the ship is taken into account at the phase of steering down the estimated safe trajectory. The ship sails in the environment with natural constraints (lands, canals, shallow waters) or those resulting from formal regulations (traffic restricted zones, fairways, etc.). It is assumed that these constraints are stationary and that they can be defined by polygons - like the way the electronic maps are created. When sailing in a stationary environment, the own ship meets other sailing strange ships - moving targets. Part of these
2. EVOLUTIONARY ALGORITHM Evolutionary algorithms (Goldberg (1989), Michalewicz (1996» create a class of algorithms with adaptational random search. Their operation bases on probabilistic methods for creating a population of solutions. In the present case of the safe trajectory estimation, they are used for creating the population of passing paths, and finding the optimum solution within this population, i.e. the best possible path with respect to the fitness function . The list below shows an operational concept of the adaptation algorithm
130
targets are a collision threat, while the rest does not influence the safety of sailing of the own ship.
other hand, it is to move safely down the given trajectory - i.e it must avoid navigation obstacles and cannot come to close to other moving targets being passed by. Estimating a ship trajectory in a collision situation is a difficult compromise between the above mentioned necessary deviation from the given course and the safety of sailing. Due to these contrasting conditions, this task is reduced to the multi-criterion optimum planning of the passing path, S, which takes into account the safety and economy of the ship motion. The estimation of the own ship trajectory in the collision situation consists in determining the path, S, from the actual beginning point Pbeg_temp to the actual end point P end_temp . This path has the form of a sequence of elementary line segments s, (i = 1, .., n), linked with each other in turning points (X"Y,) .
The degree of the collision threat with dangerous targets is not constant and depends on the approach parameters : D CPA (Distance at Closest Point of Approach) and TCPA (Time of Closest Point of Approach), as well as on the speed ratio of the both ships, and the aspect and bearing of the target. It as assumed that the dangerous target is the target that has appeared in the area of observation 1 and can cross the estimated course of the own ship at a dangerous distance. In the evolutionary task, the targets threatening with a collision are interpreted as moving dangerous areas having shapes and speeds corresponding to the moving targets determined by the ARP A system. The shapes of these dangerous areas depend of the safety conditions: an assumed 2 DsaJe safe distance , speed ratio, and bearing.
-
- - --
-
- E-nd- - --
- - - -j
""~""'. ~"
In the environment model [see Fig. I and Fig. 2]: • fixed navigation constraints are modelled using convex and concave polygons, • moving targets are modelled as moving hexagons, • the dimensions of the own ship are neglected due to small length of the own ship with respect to maximum length of the areas representing the moving targets.
=135
r
Target 2
/
/
,//
I. ,
Point of COIliSion' t
,
I
I I
II
'"
iljl=oOO
II
Beginning
mOwn Ship
'----- - - - - _ . _- -- - - - - --
---'
Fig. 2. Approaching two moving targets - visible constraint shapes. The actual location of the own ship on the route Ro makes the actual beginning point Pbeg_temp E Ro. The choice of the actual end point, Pend_temp, depends on an assumed sensible horizon, or operator decision. This point, however, should also be located on the route Ro. The path S is assumed a safe path S_safe in the permissible space, limited by
Fig. I. The navigation situation in Dover Straits.
3.2 Planning the own trajectory in a collision situation.
when any of elementary line segments s, (i = I, .. , n) does not cross static constraints 0 _stat, (i =I , .. ,k), and at time instants t determined by current locations of the own ship does not come in contact with moving dangerous areas 0 _dyn(t) I (i =k+ 1,.. ,1) representing the targets. The space Y is named the safe space when
According to transport plans, the own ship should cover a given route Ro in an assumed time. On the 1 The ranges of 5 - 8 N. miles in front of the bow, and 2 - 4 N. miles behind the stem of the ship are assumed. Their actual values depend of the assumed time horizon . 2 The safe distance is selected by the operator depending on the weather conditions, the sailing area, and the speed of the ship.
k
Y= X - U 0 _stat, ,; 1
I
U 0 _dyn(t),
(2)
l;k+1
The set S of paths belonging to the permissible space X is the set S_safe of safe paths when
131
k
I
to D sa!", the width of the dangerous area on each side of the own ship is chosen with the preference of the ship's passage behind the stem of the target, which depends on the course and bearing of the target. Moreover, the dimensions of the restricted dangerous area take into consideration the speed rate between the ships. The ship with higher speed is a bigger collision threat.
{S_safenU O_stat"S_safen U O_dyn(t),}=O ,=1
•
,=k+1
(3) Paths which cross the restricted areas generated by static and dynamic constrains are called unsafe, or dangerous paths.
The steering goal of the evolutionary task of estimating the own ship trajectory in a collision situation is the search ofthe set of safe paths, S_safe, in the permissible space X. with subsequent selection of the optimum path S_opt from the set S_safe with respect to the fitness function determined by the cost of the path.
3.3
- -- -- - - - - -
-.
End. ."\ '. ,2
2: •
Modelling of moving targets in an evolutionary environment
~
(x,.y.l,) ,
Target 1
1,
~,-<"".'-<)_~-<>",~"",>'"",,~<~ ~I • ,
The own ship is assumed to move with a uniform speed 9 along the passing path S from the beginning point (xo ,Yo ) to the end point (x. ,Ye ), and at the initial instant to the motion of the strange ships - targets is defmed as uniform. For each target, its motion is represented by the following parameters: bearing, distance, speed, and course, estimated by the ARPA system. Each paths (individual) is first generated in a random way, and then modified using genetic operators. Next, a set of instantaneous dynamic dangerous areas relating to the targets is attributed to the path by the evolutionary algorithm. The instantaneous locations of these dynamic areas with respect to the passing path depend on time t, determined from the first crossing point (Xint ,Yinr) between the own ship's path S and the trajectory of the target. For example: (see Fig. 3) for the Path 2, it is point (X2,Y2) and dynamic constrain obstacle on time 12' The crossing point (Xint ,Yint) is the point of the biggest collision threat. Having known the length of the line segment from the beginning point, (xo ,Yo), to the crossing point (xmr ,Yinr) and assuming that the own ship will keep moving with the uniform speed 9, it is possible to determine time t which the own ship needs to cover this distance. If the own ship moved with the speed 9 along the path S, then after time 1 it would reach exactly the crossing point with the trajectory of the target. After time t, the instantaneous location of the target with respect to the own ship is modelled by the constraint, being the restriction dangerous area with a hexagonal shape. The detailed shape and dimensions of the hexagon depend on the safety conditions given by the operator.
'I,
(X,.y,t,)
:
(x"y,t,)
,,f'alh 2
Path 3
Path 1 .-"
..
-
1
~ownShi~- ' Beginning
Fig. 3. Crossing paths and dangerous areas attributed to them.
3.4
Population individuals - chromosomes
It was assumed that the passing path S is represented by a single individual - a chromosome. Each individual consists of genes, specified by coordinates (Xi,Yi), being the line segment Si turning points. Linked, the line segments Si determine the passing path S. All individuals in the population have common beginning and end points. The population consists of m individuals. Each individual, apart from the co-ordinates turning point (Xi,Yi) of the particular line segments is attributed the data int, informing whether the next segment of the path is safe and whether the turning point does not enter the restricted dangerous areas. The first population is generated in a random way .. The individuals are reproduced by the genetic operators, this reproduction leading to their modification and possible creation of better offsprings.
3.5
The values assumed in the paper are the following: • the distance in front of the bow which guarantees avoiding the collision is equal to 3 * Pm!. (in practice, safe distance Dsa!. is taken from the range between 0.5 and 3.0 nautical miles), • the similar the distance behind the stem is equal
Individual genetic operators
In their papers (Michalewicz and Xiao et at) suggest using eight genetic operators. Basing on three basic operations, namely the reproduction, crossover, and mutation, they develop modified operators which execute specific functions modifying the passing path
132
- the chromosome. This list of operators includes: soft mutation, mutation, adding a gene, swapping gene locations, crossing, smoothing, deleting a gene and individual repair.
3.6
5. CONCLUSIONS The evolutionary method of estimating the safe and optimum passing path being the own ship' s trajectory in the environment with static and dynamic constraints is a new approach to the problem of avoiding collisions at sea. A number of preliminary tests, presented in the paper, make it possible to formulate the following conclusions: • evolutionary algorithms can be effectively used for solving the problem of avoiding collisions at sea, where the environment is mode led as a set of polygons representing navigation constraints and moving targets, • the task of evolutionary estimation of the own ship trajectory in a collision situation is reduced to an adaptive search for a set of safe paths S_safe in a permissible space X , with subsequent selection of the optimum trajectory with respect to the fitness function, • the strange ship - target is modeled in the evolutionary environment as a dynamical constraint, a moving restriction area having a hexagonal shape. The detailed shape and dimensions of the hexagon depend on safety conditions and parameters of motion entered by the operator. In the search process, instantaneous locations of the dangerous areas are attributed to certain paths.
Individual fitness fun ction
In the population_evaluation(P(T)) procedure, each individual - path is attributed its fitness function, being its value for the examined problem. Such an evaluation of particular solutions in the population makes it possible to recognise and make a distinction between better and worse paths for the actual steering goal in the examined environment. The fitness function must include both the safety conditions of sailing, and the economic conditions resulting from the change in the own ship route . The safety conditions are satisfied for a given path if it does not cross the constraints. Such a path is called a safe path. As it was said, the fitness function takes two forms, alternatively for safe and dangerous paths. For a dangerous path, a number of borders crossings is examined, or the length of the penetration distance inside the restricted dangerous area. In the case of the safe path the quality of the path is estimated using the total cost (Path_Cost) equation as in EPIN system. Path_Cost safe (S) = wd*dist(S)+ ws*smooth(S) +wc *clear (S) where: W d, W S , W c are weight coefficients.
(4)
During the evolutionary selection with respect to the fitness function, the paths are eliminated from the sets of both the safe and unsafe paths.
REFERENCES Furuhashi T, Nakaoka K., Uchikawa Y., (1996). A Study on Classifier System for Finding Control Knowledge of Multi-Input Systems, (F. Herrera, J.L.Verdegay Editors), Genetic Algorithms and Soft Computing, PhisicaVerlang,. Goldberg D.E., (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, Reading, MA,. Granat J., Wierzbicki A. P.; (1995) Multi-Objective Modelling for Engineering Application in Decision Support, Springer-Verlag, Berlin-Heidelberg. Kreglewski K., Granat J., Wierzbicki A. P.; (1991) IAC-DIDAS-N. A Dynamic Interactive Decision Analysis and Support System for Multi-Criteria Analysis of Non-linear Models, v. 4.0. CP-91-010, IIASA, Collaborative Paper of the International Institute for Applied Systems Analysis, Laxenburg, Austria. Lin H.-S., Xiao, 1., Michalewicz Z.; (1994). Evolutionary Algorithm for Path Planning in Mobile Robot Environment", Proc. IEEE Int. Conf. Evolutionary Computation, Orlando,
4. EXEMPLARY PLANNING OF A SAFE TRAJECTORY The operation of the system has been examined for a number of collision situations. The first example (see Fig. 4) represents the moving targets sailing with opposite courses on the right and left sides of the own ship. In the evolutionary process two moving constraints are taken into account. The population consisted of 30 paths, and the changes of paths in the population stopped being recorded after 450 generations. The estimated path secures the passage of the own ship behind the stem of the target I and 2 on the left, and right side, respectively. The next examp les models an environment with static navigational constraints, and with moving targets. This example (see Fig. 5) is specific of sailing in narrow passages. The population consisted of 10 paths, which did not change after I 100 generations and for Fig. 6 after 1000 generations.
133
Smierzchalski R., (1995) The Application of the Dynamic Interactive Decision Analysis System to the Problem of A voiding Collisions at the Sea, (in Polish) I st Conference A wioniki , Jawor, Poland Smierzchalski R; (1996a) The Decision Support System to Design the Safe Manoeuvre A voiding Collision at Sea. ISAS'96, Orlando, USA, . Smierzchalski R. , (1996b) Multi-Criterion Modeling the Collision Situation at Sea for Application in Decision Support. MMAR ' 96, Mi«dzyzdroje, Poland. Trojanowski K. , Michalewicz Z.; Planning Path of Mobil Robot, (in Polish) 1st Conference Evolutionary Algorithms, Murzasichle, Poland, 1996. Xiao J.; Michalewicz Z., Zhang, L., (1996). Evolutionary PlannerlNavigator: Operator Performance and Self-Tuning, Proc. IEEE Int. Conf. Evolutionary Computation, Nagoya, Japan, May, pp. 366-371.
Florida, June, pp .211-216. Lisowski J. ; Smierzchalski R., (1995a) Assigning of Safe and Optimal Trajectory A voiding Collision at Sea. 3rd IF AC Workshop Control Applications in Marine System, TrondheimNorway. Lisowski J.; Smierzchalski R. , (1995b) Methods to Assign the Safe Manoeuvre and Trajectory A voiding Collision at Sea. I st International Conference Marine Technology. Szczecin. Michalewicz Z. ; (1996). Genetic Algorithms + Data structures = Evolution Programs. SprigerVerlang, 3rd edition,. Michalewicz Z. , Xiao J.; (1995). Evaluation of Paths in Evolutionary PlannerlNavigator, Proceedings of the 1995 International Workshop on Biologically Inspired Evolutionary Systems, Tokyo, Japan, May 30-31 , , pp. 45-52. Smierzchalski R., Lisowski J.; (1995) The Process Avoiding Collision at Sea as a Non-Linear Programming Task. 14th International Congress on Cybernetics, Namur-Belgium .
'
.... '
"".
\
=;.~
.
\~
\~
~J
~'"
./
j _ . 'O _
__ ...J~~ Environment
\
T
=
0
T
200
T
=
300
T= 450
Fig. 4. Paths evolution for two targets approached.
T
Environment
=
0
T= 600
T
= 1100
Fig. 5. Path evolution for the case of approaching three moving targets in the presence of static navigation constraints. End. .pHd- S knob
coune-no
/
.'. ...-,~ ...... .. ' .,
.
:
Environment
T = 0
'
......
~ .....
........
. .) /
.:!- '
T= 400
T = 800
T
= 1100
Fig. 6. Path evolution for the case of approaching two moving targets in the presence of static navigation constraints (population 10 paths)
134
TRAJECTORY PLANNING FOR SHIP IN COLLISION SITU ATIONS AT SEA BY EVOLUTIONARY COMPUTATION
Roman Smierzchalski
Department o/Ship Automation, Gdynia Maritime Academy, uf. Morska 83, 81-225 Gdynia, Poland; e-mail: [email protected].
Abstract: A modified version of the EIPN algorithm has been used for computing the optimum trajectory of a ship in a given environment. On the basis of this algorithm, a program estimating a safe trajectory of a ship in collision situation was developed. By taking into account certain boundaries of the manoeuvring region, along with navigation obstacles and other moving ships, the problem of avoiding collisions at sea was reduced to a dynamic optimisation task with static and dynamic constrains. The introduction of a time parameter and moving constrains representing the passed ships is the main distinction of the new system. Sample results of using the program for estimating the ship trajectory in typical navigation situations are given. Keywords: ship control, trajectory planning, genetic algorithms, optimisation problems, multiple-criterion optimisation.
I. INTRODUCTION
The problem of determining the safe trajectory as a task of non-linear programming was formulated by Smierzchalski and Lisowski (1995), Lisowski and Smierzchalski (1995a) in which a kinematic model of the own ship was applied. Another possible approach to this problem is the reduction of the solution space to a finite-dimensional one, by creating a so called digitised matrix of permissible manoeuvres for a given collision situation and a certain time instant by Lisowski and Smierzchalski (1995b). In the digitised set of permissible solutions a set of effective solutions satisfying the Paretooptimum rule is determined taking into consideration an assumed values of the safe passing distance and the course deviation. In (Smierzchalski, 1995, 1996a, 1996b,) the problem of avoiding collisions was formulated as the multi-criterion optimisation task. Three separate criteria were used. A continuous task of the multi-criterion optimisation was reduced to a static multidimensional task. The problem was solved for a static and a moving targets, using the DIDAS-N system (Granat and Wierzbicki, 1995, Kreglewski et af., 1991). The attempt to estimate the safe trajectory
The solution to the problem of avoiding collisions at sea is one of crucial tasks of sea navigation. When determining a safe trajectory for the own ship, we look for a trajectory that is a compromise between the costs of necessary deviation from a given route, or from the optimum route leading to a given steering goal, and the safety of passing all static and dynamic obstacles, strange ship - target. All trajectories which satisfy the condition of avoiding, in a satisfactory way, the risk of collision make a set of permissible trajectories. Therefore the task of determining the safe trajectory is reduced to the choice of an optimum solution, or a subset of optimum solutions from the set of permissible trajectories, taking into account certain cost criteria and the need to satisfy, in any circumstances, the safety conditions. These conditions are, as a rule, defined by the operator, basing on the speed ratio between the ships involved in the passing manoeuvre, as well as the actual visibility, weather conditions, navigation area, manoeuvrability of the ship etc.
129
using genetic algorithms was presented in (Furuhashi et al. , 1996). The collision situation was modelled as a fuzzy process with many inputs. The authors applied a multi-dimensional digitised model of the collision situation in which the rules were defined at each stage of the own ship steering. For selecting the steering rules the authors made use of a fuzzy classifier system.
estimating the safe trajectory, using the C++ language convention. Evolutionary_ algorithmO { int T; I*T - generation counter variable*1
T=O: input_data_foryarametrising_operation_ oC evolutionary_ algorithmO; input_data_for_ defming_environmentO;
Another problem, although in a sense similar to the problem of avoiding collisions at sea, is the problem of steering a mobile robot using the evolutionary algorithms (Lin et al. (1994), Michalewicz (1996), Michalewicz and Xiao (1995), Trojanowski and Michalewicz (1996) , Xiao et al. (1996» . In their papers, the authors present a method of estimating the paths of the robot in a well known environment. Having selected the optimum path, the motion of the robot is controlled in an on-line process along the selected path, including new obstacles. The solution of the well defined problem of planning the robot motion was found using the EPIN (Evolutionary Planner INavigator) adaptation system especially designed for this purpose.
P(T) = creation_ oCinitial_chromosomeyopulationO; building_dynamic_ obstaclesO; population_evaluation ( P(T) ); while ( != terminating_conditionsO ) { T = T + I; Op = operator_selectionO; Par = selection_ofyarents_oryarentO; offspring_creation_ and _ introduction( Op, Par ); building_dynamic_ obstaclesO; population_evaluation ( P(T) ); introduction_ of_new_individuaIO; selection_ oCbest_individual ( P(T) ); }
The main goal of the present paper is to discuss the problem of avoiding collisions at sea, considering it as the evolutionary task of estimating the own ship trajectory in an unsteady environment. Basing on the E/PN planning concept presented by Michalewicz and Xiao, a modified version of the system has been developed which takes into account specific character of the collision avoiding process. The main innovation of the modified versions is the existence of different types of static and moving constraints reflecting the real environment of the moving strange ship - target and its dynamic characteristics as the target of control. In an evolutionary process of search for the optimum trajectory in a collision situation, a time parameter and dynamic constraints were introduced. The dynamic constraints represent moving strange ships - targets being passed by, whose shapes and dimensions depends on assumed safety conditions, namely the safe distance between the passing targets, as well their speed ratio, and bearing.
The multiple iteration process of transforming and creating new individuals, accompanied by the creation of new populations is carried on until the "stop" tests in the program are fulfilled .
3. EVOLUTIONARY PROBLEM TO AVOID COLLISION Formulating the task of estimating the optimum safe trajectory in a well known environment, considered as an evolutionary problem, involves defining the evolutionary steering goal and conditions at which this goal has to be reached.
3. 1
Definition of environment and constraints.
The kinematic model of the ship motion was appl ied for estimating the safe trajectory. The dynamics of the ship is taken into account at the phase of steering down the estimated safe trajectory. The ship sails in the environment with natural constraints (lands, canals, shallow waters) or those resulting from formal regulations (traffic restricted zones, fairways, etc.). It is assumed that these constraints are stationary and that they can be defined by polygons - like the way the electronic maps are created. When sailing in a stationary environment, the own ship meets other sailing strange ships - moving targets. Part of these
2. EVOLUTIONARY ALGORITHM Evolutionary algorithms (Goldberg (1989), Michalewicz (1996» create a class of algorithms with adaptational random search. Their operation bases on probabilistic methods for creating a population of solutions. In the present case of the safe trajectory estimation, they are used for creating the population of passing paths, and finding the optimum solution within this population, i.e. the best possible path with respect to the fitness function . The list below shows an operational concept of the adaptation algorithm
130
targets are a collision threat, while the rest does not influence the safety of sailing of the own ship.
other hand, it is to move safely down the given trajectory - i.e it must avoid navigation obstacles and cannot come to close to other moving targets being passed by. Estimating a ship trajectory in a collision situation is a difficult compromise between the above mentioned necessary deviation from the given course and the safety of sailing. Due to these contrasting conditions, this task is reduced to the multi-criterion optimum planning of the passing path, S, which takes into account the safety and economy of the ship motion. The estimation of the own ship trajectory in the collision situation consists in determining the path, S, from the actual beginning point Pbeg_temp to the actual end point P end_temp . This path has the form of a sequence of elementary line segments s, (i = 1, .., n), linked with each other in turning points (X"Y,) .
The degree of the collision threat with dangerous targets is not constant and depends on the approach parameters : D CPA (Distance at Closest Point of Approach) and TCPA (Time of Closest Point of Approach), as well as on the speed ratio of the both ships, and the aspect and bearing of the target. It as assumed that the dangerous target is the target that has appeared in the area of observation 1 and can cross the estimated course of the own ship at a dangerous distance. In the evolutionary task, the targets threatening with a collision are interpreted as moving dangerous areas having shapes and speeds corresponding to the moving targets determined by the ARP A system. The shapes of these dangerous areas depend of the safety conditions: an assumed 2 DsaJe safe distance , speed ratio, and bearing.
-
- - --
-
- E-nd- - --
- - - -j
""~""'. ~"
In the environment model [see Fig. I and Fig. 2]: • fixed navigation constraints are modelled using convex and concave polygons, • moving targets are modelled as moving hexagons, • the dimensions of the own ship are neglected due to small length of the own ship with respect to maximum length of the areas representing the moving targets.
=135
r
Target 2
/
/
,//
I. ,
Point of COIliSion' t
,
I
I I
II
'"
iljl=oOO
II
Beginning
mOwn Ship
'----- - - - - _ . _- -- - - - - --
---'
Fig. 2. Approaching two moving targets - visible constraint shapes. The actual location of the own ship on the route Ro makes the actual beginning point Pbeg_temp E Ro. The choice of the actual end point, Pend_temp, depends on an assumed sensible horizon, or operator decision. This point, however, should also be located on the route Ro. The path S is assumed a safe path S_safe in the permissible space, limited by
Fig. I. The navigation situation in Dover Straits.
3.2 Planning the own trajectory in a collision situation.
when any of elementary line segments s, (i = I, .. , n) does not cross static constraints 0 _stat, (i =I , .. ,k), and at time instants t determined by current locations of the own ship does not come in contact with moving dangerous areas 0 _dyn(t) I (i =k+ 1,.. ,1) representing the targets. The space Y is named the safe space when
According to transport plans, the own ship should cover a given route Ro in an assumed time. On the 1 The ranges of 5 - 8 N. miles in front of the bow, and 2 - 4 N. miles behind the stem of the ship are assumed. Their actual values depend of the assumed time horizon . 2 The safe distance is selected by the operator depending on the weather conditions, the sailing area, and the speed of the ship.
k
Y= X - U 0 _stat, ,; 1
I
U 0 _dyn(t),
(2)
l;k+1
The set S of paths belonging to the permissible space X is the set S_safe of safe paths when
131
k
I
to D sa!", the width of the dangerous area on each side of the own ship is chosen with the preference of the ship's passage behind the stem of the target, which depends on the course and bearing of the target. Moreover, the dimensions of the restricted dangerous area take into consideration the speed rate between the ships. The ship with higher speed is a bigger collision threat.
{S_safenU O_stat"S_safen U O_dyn(t),}=O ,=1
•
,=k+1
(3) Paths which cross the restricted areas generated by static and dynamic constrains are called unsafe, or dangerous paths.
The steering goal of the evolutionary task of estimating the own ship trajectory in a collision situation is the search ofthe set of safe paths, S_safe, in the permissible space X. with subsequent selection of the optimum path S_opt from the set S_safe with respect to the fitness function determined by the cost of the path.
3.3
- -- -- - - - - -
-.
End. ."\ '. ,2
2: •
Modelling of moving targets in an evolutionary environment
~
(x,.y.l,) ,
Target 1
1,
~,-<"".'-<)_~-<>",~"",>'"",,~<~ ~I • ,
The own ship is assumed to move with a uniform speed 9 along the passing path S from the beginning point (xo ,Yo ) to the end point (x. ,Ye ), and at the initial instant to the motion of the strange ships - targets is defmed as uniform. For each target, its motion is represented by the following parameters: bearing, distance, speed, and course, estimated by the ARPA system. Each paths (individual) is first generated in a random way, and then modified using genetic operators. Next, a set of instantaneous dynamic dangerous areas relating to the targets is attributed to the path by the evolutionary algorithm. The instantaneous locations of these dynamic areas with respect to the passing path depend on time t, determined from the first crossing point (Xint ,Yinr) between the own ship's path S and the trajectory of the target. For example: (see Fig. 3) for the Path 2, it is point (X2,Y2) and dynamic constrain obstacle on time 12' The crossing point (Xint ,Yint) is the point of the biggest collision threat. Having known the length of the line segment from the beginning point, (xo ,Yo), to the crossing point (xmr ,Yinr) and assuming that the own ship will keep moving with the uniform speed 9, it is possible to determine time t which the own ship needs to cover this distance. If the own ship moved with the speed 9 along the path S, then after time 1 it would reach exactly the crossing point with the trajectory of the target. After time t, the instantaneous location of the target with respect to the own ship is modelled by the constraint, being the restriction dangerous area with a hexagonal shape. The detailed shape and dimensions of the hexagon depend on the safety conditions given by the operator.
'I,
(X,.y,t,)
:
(x"y,t,)
,,f'alh 2
Path 3
Path 1 .-"
..
-
1
~ownShi~- ' Beginning
Fig. 3. Crossing paths and dangerous areas attributed to them.
3.4
Population individuals - chromosomes
It was assumed that the passing path S is represented by a single individual - a chromosome. Each individual consists of genes, specified by coordinates (Xi,Yi), being the line segment Si turning points. Linked, the line segments Si determine the passing path S. All individuals in the population have common beginning and end points. The population consists of m individuals. Each individual, apart from the co-ordinates turning point (Xi,Yi) of the particular line segments is attributed the data int, informing whether the next segment of the path is safe and whether the turning point does not enter the restricted dangerous areas. The first population is generated in a random way .. The individuals are reproduced by the genetic operators, this reproduction leading to their modification and possible creation of better offsprings.
3.5
The values assumed in the paper are the following: • the distance in front of the bow which guarantees avoiding the collision is equal to 3 * Pm!. (in practice, safe distance Dsa!. is taken from the range between 0.5 and 3.0 nautical miles), • the similar the distance behind the stem is equal
Individual genetic operators
In their papers (Michalewicz and Xiao et at) suggest using eight genetic operators. Basing on three basic operations, namely the reproduction, crossover, and mutation, they develop modified operators which execute specific functions modifying the passing path
132
- the chromosome. This list of operators includes: soft mutation, mutation, adding a gene, swapping gene locations, crossing, smoothing, deleting a gene and individual repair.
3.6
5. CONCLUSIONS The evolutionary method of estimating the safe and optimum passing path being the own ship' s trajectory in the environment with static and dynamic constraints is a new approach to the problem of avoiding collisions at sea. A number of preliminary tests, presented in the paper, make it possible to formulate the following conclusions: • evolutionary algorithms can be effectively used for solving the problem of avoiding collisions at sea, where the environment is mode led as a set of polygons representing navigation constraints and moving targets, • the task of evolutionary estimation of the own ship trajectory in a collision situation is reduced to an adaptive search for a set of safe paths S_safe in a permissible space X , with subsequent selection of the optimum trajectory with respect to the fitness function, • the strange ship - target is modeled in the evolutionary environment as a dynamical constraint, a moving restriction area having a hexagonal shape. The detailed shape and dimensions of the hexagon depend on safety conditions and parameters of motion entered by the operator. In the search process, instantaneous locations of the dangerous areas are attributed to certain paths.
Individual fitness fun ction
In the population_evaluation(P(T)) procedure, each individual - path is attributed its fitness function, being its value for the examined problem. Such an evaluation of particular solutions in the population makes it possible to recognise and make a distinction between better and worse paths for the actual steering goal in the examined environment. The fitness function must include both the safety conditions of sailing, and the economic conditions resulting from the change in the own ship route . The safety conditions are satisfied for a given path if it does not cross the constraints. Such a path is called a safe path. As it was said, the fitness function takes two forms, alternatively for safe and dangerous paths. For a dangerous path, a number of borders crossings is examined, or the length of the penetration distance inside the restricted dangerous area. In the case of the safe path the quality of the path is estimated using the total cost (Path_Cost) equation as in EPIN system. Path_Cost safe (S) = wd*dist(S)+ ws*smooth(S) +wc *clear (S) where: W d, W S , W c are weight coefficients.
(4)
During the evolutionary selection with respect to the fitness function, the paths are eliminated from the sets of both the safe and unsafe paths.
REFERENCES Furuhashi T, Nakaoka K., Uchikawa Y., (1996). A Study on Classifier System for Finding Control Knowledge of Multi-Input Systems, (F. Herrera, J.L.Verdegay Editors), Genetic Algorithms and Soft Computing, PhisicaVerlang,. Goldberg D.E., (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, Reading, MA,. Granat J., Wierzbicki A. P.; (1995) Multi-Objective Modelling for Engineering Application in Decision Support, Springer-Verlag, Berlin-Heidelberg. Kreglewski K., Granat J., Wierzbicki A. P.; (1991) IAC-DIDAS-N. A Dynamic Interactive Decision Analysis and Support System for Multi-Criteria Analysis of Non-linear Models, v. 4.0. CP-91-010, IIASA, Collaborative Paper of the International Institute for Applied Systems Analysis, Laxenburg, Austria. Lin H.-S., Xiao, 1., Michalewicz Z.; (1994). Evolutionary Algorithm for Path Planning in Mobile Robot Environment", Proc. IEEE Int. Conf. Evolutionary Computation, Orlando,
4. EXEMPLARY PLANNING OF A SAFE TRAJECTORY The operation of the system has been examined for a number of collision situations. The first example (see Fig. 4) represents the moving targets sailing with opposite courses on the right and left sides of the own ship. In the evolutionary process two moving constraints are taken into account. The population consisted of 30 paths, and the changes of paths in the population stopped being recorded after 450 generations. The estimated path secures the passage of the own ship behind the stem of the target I and 2 on the left, and right side, respectively. The next examp les models an environment with static navigational constraints, and with moving targets. This example (see Fig. 5) is specific of sailing in narrow passages. The population consisted of 10 paths, which did not change after I 100 generations and for Fig. 6 after 1000 generations.
133
Smierzchalski R., (1995) The Application of the Dynamic Interactive Decision Analysis System to the Problem of A voiding Collisions at the Sea, (in Polish) I st Conference A wioniki , Jawor, Poland Smierzchalski R; (1996a) The Decision Support System to Design the Safe Manoeuvre A voiding Collision at Sea. ISAS'96, Orlando, USA, . Smierzchalski R. , (1996b) Multi-Criterion Modeling the Collision Situation at Sea for Application in Decision Support. MMAR ' 96, Mi«dzyzdroje, Poland. Trojanowski K. , Michalewicz Z.; Planning Path of Mobil Robot, (in Polish) 1st Conference Evolutionary Algorithms, Murzasichle, Poland, 1996. Xiao J.; Michalewicz Z., Zhang, L., (1996). Evolutionary PlannerlNavigator: Operator Performance and Self-Tuning, Proc. IEEE Int. Conf. Evolutionary Computation, Nagoya, Japan, May, pp. 366-371.
Florida, June, pp .211-216. Lisowski J. ; Smierzchalski R., (1995a) Assigning of Safe and Optimal Trajectory A voiding Collision at Sea. 3rd IF AC Workshop Control Applications in Marine System, TrondheimNorway. Lisowski J.; Smierzchalski R. , (1995b) Methods to Assign the Safe Manoeuvre and Trajectory A voiding Collision at Sea. I st International Conference Marine Technology. Szczecin. Michalewicz Z. ; (1996). Genetic Algorithms + Data structures = Evolution Programs. SprigerVerlang, 3rd edition,. Michalewicz Z. , Xiao J.; (1995). Evaluation of Paths in Evolutionary PlannerlNavigator, Proceedings of the 1995 International Workshop on Biologically Inspired Evolutionary Systems, Tokyo, Japan, May 30-31 , , pp. 45-52. Smierzchalski R., Lisowski J.; (1995) The Process Avoiding Collision at Sea as a Non-Linear Programming Task. 14th International Congress on Cybernetics, Namur-Belgium .
'
.... '
"".
\
=;.~
.
\~
\~
~J
~'"
./
j _ . 'O _
__ ...J~~ Environment
\
T
=
0
T
200
T
=
300
T= 450
Fig. 4. Paths evolution for two targets approached.
T
Environment
=
0
T= 600
T
= 1100
Fig. 5. Path evolution for the case of approaching three moving targets in the presence of static navigation constraints. End. .pHd- S knob
coune-no
/
.'. ...-,~ ...... .. ' .,
.
:
Environment
T = 0
'
......
~ .....
........
. .) /
.:!- '
T= 400
T = 800
T
= 1100
Fig. 6. Path evolution for the case of approaching two moving targets in the presence of static navigation constraints (population 10 paths)
134