- Email: [email protected]
Dynamic Optimisation of Ship’s Safe Trajectory in Collision Situations
Copyright @ Ir AC Manoeu vrin g and C ontrol of Mar ine Craft. Aalborg. Denm ark, 2000
DYNAMIC OPTIMISA TION OF SHIP'S SAFE TRAJECTORY IN COLLISION SITUATIONS
16zef Lisowski
Gdynia Maritime Academy. Department of Ship Automation 83 Morska Str.. 81-225 Gdynia. Poland Tel / Fax: + 4858628947 J E-mail: [email protected]
Abstract: This paper presents an application of a method of dynamic programming for the determination of the ship's safe trajectory during passing the encountered objects. A dynamic model of the process with kinematic constraints of state determined by a three-layer artificial neural network has been used for the synthesis of control. Nonlinear activation functions in the first and second layers may be characterised by a tangent curve while the output layer is of a sigmoidal nature. Neural Network Toolbox from the MA TLAB package has been used to model the network. The learning process used an algorithm of a backward propagation of the error with adaptively selected learning step. The considerations have been illustrated with example of computer simulation the PROGNEURAL programme of determination the safe ship's trajectory in situation of passing a big number of the objects encountered. recorded on the ship ' s screen radar in real navigational situation at sea. Copyright © 2000 IF AC Keywords: marine systems. optimal control . dynamic programming. computational methods. neural networks. safety analysis. optimal trajectory. computer simulation.
stationary and game controlling features . In practice methods of selecting a manoeuvre assume the form of relevant controlling algorithms programmed in the microprocessor controller generating the option of the ARP A anti-collision system or of the training simulator found in a laboratory of a maritime academy (Cross. 1994; Lisowski. 1998). The mode of the steering of a ship which is a multi-dimensional and non-linear dynamic object. depends on the range of precision of the information on the current navigational situation and on the adopted model of the process. During the development of the process the following relevant elements are to be considered: equations of kinematics and dynamics of own ship. disturbances generated by the sea wavy motion and the operation of the wind and sea currents.
1. INTRODUCTION The anti-collision ARPA (Automatic Radar Plotting Aids) system ensures automatic monitoring of at least 20 "j" objects encountered. determination of their movement parameters (speed Vj • course \fIj ) and elements of approaching to own ship (D min] = DCPAj - Distance of the Closest Point of Approach. T minj = TCPAj - Time to the Closest Point of Approach) and also the assessment of a risk of collision. However. the range of the function of a standard ARP A system ends up with a simulation of a manoeuvre selected by the navigator (Fig. 1). The problem of the selection of this manoeuvre is very difficult considering a high complexity of the steering process which has dynamic. non-linear. multi-dimensional. non-
379
navigational constraints, strategy of the encountered objects and the function of the purpose of the control.
-.
-. p
-.
0
Z
'P.'i'
!\X
~
ex
v,v x
Ship
n
y
Fig. 2. Ship as an object of steering under collision situations: a - rudder angle, n - rotational speed y
propeller, 'I' - course,
of screw turning
speed, V - speed,
(x, y) -
position,
Z
tjJ - angular
V - acceleration,
- disturbances vector,
P- dynamic parameters vector, 0 - constraints vector (encountered "j" objects). Fig. I. The process of safe passing of own ship with "j" encountered objects.
The identification research conducted with regard to a few types of the cargo vessels under regular operational conditions at various speeds and loading states allows to the following assessment of the values of the above model: TI=5~50 s, T2=IO~IOO s, T3=5~500 s, al=50~IOOO slrad, kl=O,OI70,3 I/s, k2=I~IO m.
A wide variety of models to be adopted directly influences the synthesis of various algorithms of control and afterwards on the effects of safe steering of the movements of own ship.
2. DYNAMIC MODEL OF THE PROCESS WITH KINEMATIC CONSTRAINTS OF ST ATE
2.2 Equations for the process state The description of the ship's dynamics in the form of (I) allows for the following representation of the state equations in a discrete form:
2. / Ship dynamics It is assumes that the movement of the encountered objects, generally the j-th one is rectilinear and regular with a speed Vj and a course 'l'j, with regard to the dynamic features of own ship. Considering a specific nature of the process of ship's steering under collision situations which may be characterised by: high alterations of the course within the range 20° ~ 90° , reduction of speed, not more than by 30%, the model of ship's dynamics may be presented in the form as in Fig. 2. Simplitications with regard to a precise model of the ship's dynamics cover the omission of the drift angle and fall in the ship's speed during the manoeuvre and the adoption of a non-linear mathematical description of the ship's dynamic features in the rudder control system according to Nomoto and the linear one in the control system of the rotational speed of the propeller (Fossen, 1994):
x
= V sin'!'
y
= V cos'!'
Xl.k+1 = xl.k + Xu . ~tk+1 . sin X~. k X2 . k+1 = xa + Xu . ~tk+1 . cos Xu XU+I
= Xu + Xu . ~tk+1
XU-t-I
= Xu + ~ (-x u
-al ' XuI X4.kI+ (2)
I
k,
= Xu + x n.k . ~tk+1
Xn,k+1
= x n.k + T T
I
[-(T2 + T1)x n.k - X",k +
2 1
k2 .
X7,k+1
nm . U 2.k ]~tkT l
= Xu + ~tk-t-I
where:xl=x, X2=y, X3='I', '4='f'
Xs=V,
2.3 Constraints of the control and process state
(I)
• 2
The generated constraints are a result of the necessity to consider physical values characterising the process:
380
Moreover, a criterium of optimisation is taken into consideration in the form of smallest possible way loss for safe passing of the objects, which, at a constant speed of the own ship, leads to the timeoptimal control :
to ensure safe shipping having regard, at the same time, to the recommendations of the regulations on the priority way law:
t ~
I
(5) together with consideration to real navigational constraints::
=
f Vdt
Ik
=V
f dt
(11 )
2.6 Generation of the ship's domains by an artificial neural network NEUROCONSTR
(6)
The area of a collision risk - the ship's domain is defined by the operator of the anti-collision ARPA system on the basis of the information expressed in numbers supplied by the navigational devices and measuring systems installed on board a ship and also according to the operator's subjective impressions concerning the degree of the collision risk in the analysed navigational situation. The adopted ship's domain is represented in the form of a circle, parable, ellipse or hexagon the dimensions of which depends on the relative speed of the object being passed and these are modified on the basis of the answer from an appropriately prepared neural network which assesses the degree of the collision risk. Let's consider a network which has five inputs and one output which aim is to ascribe one among the acceptable values of the responses, with the smallest error possible, to particular input vectors (Hunt, et aI., 1998; Leondes, 1998):
2.4 Initial and final conditions
In practice the process state is controlled on a current basis by the arrangements of the ARPA anti-collision system (radar, gyrocompass, ship's log, angular and rotational speed indicators).The initial conditions of the process are determined by a variety e", described during the initiation of the automatic monitoring of the encountered objects by the ARPA system:
e
Having regard to the final conditions of k two types of control tasks have been adopted: steering with a determined final point of the trajectory (DPP)
y
=r[Wx]
(8)
(12) (13)
which corresponds to the situation of the ship's control in restricted waters, steering with a determined final course in the open waters (OPC)
y = [0 I]
(14)
the following is to be found: (15)
(9) where: Yk - network response, Yek - expected network activation functions of neural network response, layers, Pj - position of "j" object, YJ - speed of the "j" object, Y - speed of own ship, "'wj - relative course of the "j"object, IVw) - relative speed, k index of the time moment. The values of the elements of the Xk vector are provided from the ARPA system, and the Yk values determine the degree of the collision risk through the dimension of domain imputed to "j" object. The one way network has three layers of neurons. The nonlinear activation functions in the first and second layers represent a tangent nature and the output layer represent the sigmoidal nature. The network was modelled with the use of Neural Network Toolbox from the MATLAB package (Fig. 3).
r-
2.5 Index of the control quality
The basic criterium for the ship's control is to ensure safe passing of the vessels, which is considered in the state constraints: (10)
The dependence (10) is determined by the area of the collision hazard the so-called ship's domain and which assumes the form of a circle, parable, ellipse, hexagon, etc. (Colley, et aI., 1983; Oavie, et aI., 1980). The ships' domains may have a permanent or variable shapes generated, for example, by an artificial neural network (Lisowski, et aI., 2(00).
381
generate the optimal strategies from the point of the state resulting from the first decision . It results from this that the calculations using this method are usually initiated from the final stage and then the process goes toward the first one (Bell man and Dreyfus, 1957).
1,
1.
,- -, ~-""--I
<
r r--'---
1,
3. J Bel/man's fun ctional equation
1.
In the work mentioned in (Lisowski, 1981) it has been demonstrated that the process of the collision prevention fulfils the duality conditions, therefore the optimal trajectory of the ship under a collision situation is determined using the optimisation principle and is commenced from the calculation of the first stage and then it is directed toward the final stage. The optimal time for the ship to go through k stages is as follows :
Fig. 3. The structure of the neural network generating the ships' domains. The learning process used the algorithm of the backward propagation of the error with adaptively matched learning step and the momentum.
2.7 The International Regulations for Preventing Collisions at Sea COLREG rules
+ ~k[XI. k' X 2.k ' XI.k+I(XI.k' x:u (x:U _ 1' XU_I
The incorporation of the maritime navigation COLREG rules into the mathematical model is effected in the following manner: - in good visibility conditions the selection of the alternate manoeuvre depends on the angle ~j (Fig. I). At the value of
o $;
~
j
$;
180
(X U - 2 ' U I. k- 2 '
X6,k-1 , (X6.k-2' U U - 2 '~k-2 )~k-I» ' XU+I (x u
' x1,k (X:\k_I' X4.k_1 (XU
~k-2'~k-I )x'i,k (XU
(18)
_ 2 ' Ul.k-2'
_1' X6.k-I(XIl.k -2 ' U U - 2 '
~k-2 )'~k-I » ,X".k (x\k-1' X6.k-1 (XIl.k-2'}
(16)
U2.k-2 · ~k-2 )'~k-I)]
the own ship encounters the object from starboard, whilst for - 180 "
~k-2 ) '~k-I)' X,.k (X,.k_I'
$; ~
1< 0
k=3,4, ...,K The optimal time for the ship to go through the k stages is a function of the system state at the end of the k-\ stage and control (u L k- 2 ' U 2.k-2 ) at the k-2
( 17)
there happens an encounter with the object from port side. According to the COLREG rules the objects which are approaching from starboard must give way to our ship. The conditions (16) and (17) are being implemented into the procedure NEUROCONSTR by adequate formulation of state variables constraints (5). For angle ~j satisfying condition (17) a circle constraint is imposed. For objects appearing on the starboard (16) constraint in the form of a parable, ellipse or hexagon seems the most adequate. - in restricted visibility conditions the starboard rule giving way to the object approaching from starboard does not apply. but the own ship must keep a greater safe approach distance. Ds= 2-3 nm.
stage. The time of going through the k-th stage be determined as follows : L1t k =
~S k (x
where: .6.S k
-
I.k ' X 2.k ' X I. k + I ' X 2. k + I )
~
t k may
( 19)
a distance of the route from point
(x Lk' X 2.k) to point (x I.k +I ' X 2.k+I)· By going from the first stage to the last one the formula (18) determines the Bellman's functional equation for the process of the ship's control by the alteration of the angle of the rudder angle and the rotational speed of the propeller. The constraints for the state variables and the control values generate the NEUROCONSTR procedure in the algorithm for the determination of the safe ship's trajectory. The consideration of the constraints resulting from maintaining safe approaching distance and the recommendations of the way priority law is performed by checking whether the state variables have not exceeded constraints in each of the
3. ALGORITHM OF SAFE CONTROL OF SHIP Determination of the optimal control of the ship in terms of an adopted index of the control quality may be performed by applying Bellman's principle of optimisation. The principle describes the basis feature of the optimal strategy - whatever the initial state and decisions are the remaining decisions must
382
intersections intersections discovered.
considered in which
To2 - time of delay equal approximately to the time constant for the propulsion system main engine - propeller shaft - propeller. Therefore the control at the k-I stage has the following form :
and by rejecting the the excess has been
3.2 Simulation model of the ship's dynamics The dynamic features of the ship during the course alteration by an angle .1.'1' is described in a simplified manner with the use of transfer function : To
G I (s)
k(
'fI. - 'fI._ 1
_
(20)
~t._ 1 -
To' + a
~t._1 -
To'
(23)
s
T.
approximately equal to the time constant of the ship as a course control object, k (
U 2k -
.1.'1'
k,e-T. ,s
-
To2
+ V. _I (24)
I
4. SIMULATION OF THE SAFE SHIP'S TRAJECTORY BY PROGNEURAL PROGRAMME
tm=T o' +-.(21) 'I' Differential equation of the second order describing the ship's behaviour during the change of the speed by Ll Y is approximated with the use of the first order inertion with a delay: = ---"..---
V.-V._I dt k-I
= 1,2, .. ., K - I
results from the static characteristics of the rudder steering. The course manoeuvre delay time is as follows:
dyeS)
To'
UI.k._1 -
where: To' == T, - manoeuvre delay time which is
G 2 (s) =
'fI. - 'fI._ 1 1'fI. - 'fI. 11
~t._1 -
The results of learning by the network were used during the simulation of encountering own ship with other ships. The ships were assigned the domains which aims was to provide data on the areas dangerous for the anti-collision system of own ship. All ships moved along rectilinear courses. At every step of calculations the values of the domains were updated (Fig. 4).
(22)
dn(s) 1+ Tks where: Tk - the time constant of the ship's hull and the mass of the accompanying water,
383
(1)
(2a)
mls ZENIT La Manche Range 12 nm, !!J=2', 60 objects
20
15
10
".
.
".
...
5
1000
2000
)000
5000
1000
7000
1000
lOaD
(2b)
....•• ,.
~V-
0
~
-5
..• ,.
~ ;
-5
0
5
.,. 10
Fig. 4 . Continued over page.
384
1000
2000
SOOO
.000
5000
.000
7000
1000
'000
The correctness of the assessment of the safety of the passing vessels with the use of the networks depends, to a decisive degree, on the correctness of the data used in the process of the network learning. The use of the experience of a few experienced navigators during the learning by the network may lead to the situation when the network acquires their averaged knowledge. The introduction of elements of the artificial intelligence represented by a properly prepared network, to the systems of determining the ship's domain, and as a consequence - her safe trajectory in a collision situation may help less experienced navigator supervising the anti-collision system assist the navigational situation, increase the safety of the anti-collision manoeuvre and accelerate the process of selecting a manoeuvre avoiding the collision.
REFERENCES
·10
, .0
0
1000
2000
SOOO
~OOO
5000
5
.000
7000
(3b)
'00
,
.0
.'
20
'000
0000
8000
8000
,
'.
,
00 '0
10
:
~
120
,
. ~
0 '0
o
·5
Bellman, R.E., and S .E. Dreyfus (1957) . Dynamic programming. Princeton University Press, New York. Colley, BA, Curtis, R.G., and c.r. Stockel (1983). Manoeuvring times, domains and arenas. Journal of Navigation. 36, 324-328. Cross, S.l. (1994). Objective assessment of maritime simulator training. Proc. of the Int. Con! the Development and Implementation of International Maritime Training Standards. Malmo. Davie, P.V., Dove, MJ., and C.T. Stockel (1980). A computer simulation of marine traffic using domains and arenas. Journal of Navigation. 33, 215-222. Fossen, T.1. (1994). Guidance and control of ocean vehicles. 10hn Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapore. Hunt, KJ., Irwin, G.R., and K. Warwick (1998). Neural network engineering in dynamic control systems. Advances in Industrial Control. Springer. Leondes, C.T. (1998) . Control and dynamic systems. Neural network systems techniques and applications, 7, Academic Press. Lisowski, J. (1981). Ship's anti-collision systems. Maritime Publishing, Gdansk (in polish). Lisowski, J. (1998). The application of modern control theory in synthesis of computer simulation programmes of ship's safe manoeuvring for effective training of marine engineers, Global Congr. on Eng. Education. Cracow. Lisowski, J., Rak, A. and W. Czechowicz (2000). Neural network classifier for ship domain assessment. Journal of Mathematics and Computers in Simulation. Elsevier Science B.V., 51, 399-406.
1000
2 000
3000
"000
15000
' 0 00
7000
Fig. 4. Simulation of the safe own ship's trajectory in the situation 60 encountered objects: 1- radar picture of the situation in the La Manche region, 2a- safe trajectory for con-ro 20 000 DWT own ship in good visibility, 3a- safe trajectory for this ship in poor visibility, 2b and 3b- reference rotational speed, 2c and 3c- reference rudder angle.
5. CONCLUSION The neural networks presented in this paper may be used as the elements of the systems for the assessment of the safety of the ships' passing by the introduction of the possibility to make a current correction of the sizes ships' domains. They are able to represent the heuristic knowledge similar to that which is represented by an experienced navigator.
385
DYNAMIC OPTIMISA TION OF SHIP'S SAFE TRAJECTORY IN COLLISION SITUATIONS
16zef Lisowski
Gdynia Maritime Academy. Department of Ship Automation 83 Morska Str.. 81-225 Gdynia. Poland Tel / Fax: + 4858628947 J E-mail: [email protected]
Abstract: This paper presents an application of a method of dynamic programming for the determination of the ship's safe trajectory during passing the encountered objects. A dynamic model of the process with kinematic constraints of state determined by a three-layer artificial neural network has been used for the synthesis of control. Nonlinear activation functions in the first and second layers may be characterised by a tangent curve while the output layer is of a sigmoidal nature. Neural Network Toolbox from the MA TLAB package has been used to model the network. The learning process used an algorithm of a backward propagation of the error with adaptively selected learning step. The considerations have been illustrated with example of computer simulation the PROGNEURAL programme of determination the safe ship's trajectory in situation of passing a big number of the objects encountered. recorded on the ship ' s screen radar in real navigational situation at sea. Copyright © 2000 IF AC Keywords: marine systems. optimal control . dynamic programming. computational methods. neural networks. safety analysis. optimal trajectory. computer simulation.
stationary and game controlling features . In practice methods of selecting a manoeuvre assume the form of relevant controlling algorithms programmed in the microprocessor controller generating the option of the ARP A anti-collision system or of the training simulator found in a laboratory of a maritime academy (Cross. 1994; Lisowski. 1998). The mode of the steering of a ship which is a multi-dimensional and non-linear dynamic object. depends on the range of precision of the information on the current navigational situation and on the adopted model of the process. During the development of the process the following relevant elements are to be considered: equations of kinematics and dynamics of own ship. disturbances generated by the sea wavy motion and the operation of the wind and sea currents.
1. INTRODUCTION The anti-collision ARPA (Automatic Radar Plotting Aids) system ensures automatic monitoring of at least 20 "j" objects encountered. determination of their movement parameters (speed Vj • course \fIj ) and elements of approaching to own ship (D min] = DCPAj - Distance of the Closest Point of Approach. T minj = TCPAj - Time to the Closest Point of Approach) and also the assessment of a risk of collision. However. the range of the function of a standard ARP A system ends up with a simulation of a manoeuvre selected by the navigator (Fig. 1). The problem of the selection of this manoeuvre is very difficult considering a high complexity of the steering process which has dynamic. non-linear. multi-dimensional. non-
379
navigational constraints, strategy of the encountered objects and the function of the purpose of the control.
-.
-. p
-.
0
Z
'P.'i'
!\X
~
ex
v,v x
Ship
n
y
Fig. 2. Ship as an object of steering under collision situations: a - rudder angle, n - rotational speed y
propeller, 'I' - course,
of screw turning
speed, V - speed,
(x, y) -
position,
Z
tjJ - angular
V - acceleration,
- disturbances vector,
P- dynamic parameters vector, 0 - constraints vector (encountered "j" objects). Fig. I. The process of safe passing of own ship with "j" encountered objects.
The identification research conducted with regard to a few types of the cargo vessels under regular operational conditions at various speeds and loading states allows to the following assessment of the values of the above model: TI=5~50 s, T2=IO~IOO s, T3=5~500 s, al=50~IOOO slrad, kl=O,OI70,3 I/s, k2=I~IO m.
A wide variety of models to be adopted directly influences the synthesis of various algorithms of control and afterwards on the effects of safe steering of the movements of own ship.
2. DYNAMIC MODEL OF THE PROCESS WITH KINEMATIC CONSTRAINTS OF ST ATE
2.2 Equations for the process state The description of the ship's dynamics in the form of (I) allows for the following representation of the state equations in a discrete form:
2. / Ship dynamics It is assumes that the movement of the encountered objects, generally the j-th one is rectilinear and regular with a speed Vj and a course 'l'j, with regard to the dynamic features of own ship. Considering a specific nature of the process of ship's steering under collision situations which may be characterised by: high alterations of the course within the range 20° ~ 90° , reduction of speed, not more than by 30%, the model of ship's dynamics may be presented in the form as in Fig. 2. Simplitications with regard to a precise model of the ship's dynamics cover the omission of the drift angle and fall in the ship's speed during the manoeuvre and the adoption of a non-linear mathematical description of the ship's dynamic features in the rudder control system according to Nomoto and the linear one in the control system of the rotational speed of the propeller (Fossen, 1994):
x
= V sin'!'
y
= V cos'!'
Xl.k+1 = xl.k + Xu . ~tk+1 . sin X~. k X2 . k+1 = xa + Xu . ~tk+1 . cos Xu XU+I
= Xu + Xu . ~tk+1
XU-t-I
= Xu + ~ (-x u
-al ' XuI X4.kI+ (2)
I
k,
= Xu + x n.k . ~tk+1
Xn,k+1
= x n.k + T T
I
[-(T2 + T1)x n.k - X",k +
2 1
k2 .
X7,k+1
nm . U 2.k ]~tkT l
= Xu + ~tk-t-I
where:xl=x, X2=y, X3='I', '4='f'
Xs=V,
2.3 Constraints of the control and process state
(I)
• 2
The generated constraints are a result of the necessity to consider physical values characterising the process:
380
Moreover, a criterium of optimisation is taken into consideration in the form of smallest possible way loss for safe passing of the objects, which, at a constant speed of the own ship, leads to the timeoptimal control :
to ensure safe shipping having regard, at the same time, to the recommendations of the regulations on the priority way law:
t ~
I
(5) together with consideration to real navigational constraints::
=
f Vdt
Ik
=V
f dt
(11 )
2.6 Generation of the ship's domains by an artificial neural network NEUROCONSTR
(6)
The area of a collision risk - the ship's domain is defined by the operator of the anti-collision ARPA system on the basis of the information expressed in numbers supplied by the navigational devices and measuring systems installed on board a ship and also according to the operator's subjective impressions concerning the degree of the collision risk in the analysed navigational situation. The adopted ship's domain is represented in the form of a circle, parable, ellipse or hexagon the dimensions of which depends on the relative speed of the object being passed and these are modified on the basis of the answer from an appropriately prepared neural network which assesses the degree of the collision risk. Let's consider a network which has five inputs and one output which aim is to ascribe one among the acceptable values of the responses, with the smallest error possible, to particular input vectors (Hunt, et aI., 1998; Leondes, 1998):
2.4 Initial and final conditions
In practice the process state is controlled on a current basis by the arrangements of the ARPA anti-collision system (radar, gyrocompass, ship's log, angular and rotational speed indicators).The initial conditions of the process are determined by a variety e", described during the initiation of the automatic monitoring of the encountered objects by the ARPA system:
e
Having regard to the final conditions of k two types of control tasks have been adopted: steering with a determined final point of the trajectory (DPP)
y
=r[Wx]
(8)
(12) (13)
which corresponds to the situation of the ship's control in restricted waters, steering with a determined final course in the open waters (OPC)
y = [0 I]
(14)
the following is to be found: (15)
(9) where: Yk - network response, Yek - expected network activation functions of neural network response, layers, Pj - position of "j" object, YJ - speed of the "j" object, Y - speed of own ship, "'wj - relative course of the "j"object, IVw) - relative speed, k index of the time moment. The values of the elements of the Xk vector are provided from the ARPA system, and the Yk values determine the degree of the collision risk through the dimension of domain imputed to "j" object. The one way network has three layers of neurons. The nonlinear activation functions in the first and second layers represent a tangent nature and the output layer represent the sigmoidal nature. The network was modelled with the use of Neural Network Toolbox from the MATLAB package (Fig. 3).
r-
2.5 Index of the control quality
The basic criterium for the ship's control is to ensure safe passing of the vessels, which is considered in the state constraints: (10)
The dependence (10) is determined by the area of the collision hazard the so-called ship's domain and which assumes the form of a circle, parable, ellipse, hexagon, etc. (Colley, et aI., 1983; Oavie, et aI., 1980). The ships' domains may have a permanent or variable shapes generated, for example, by an artificial neural network (Lisowski, et aI., 2(00).
381
generate the optimal strategies from the point of the state resulting from the first decision . It results from this that the calculations using this method are usually initiated from the final stage and then the process goes toward the first one (Bell man and Dreyfus, 1957).
1,
1.
,- -, ~-""--I
<
r r--'---
1,
3. J Bel/man's fun ctional equation
1.
In the work mentioned in (Lisowski, 1981) it has been demonstrated that the process of the collision prevention fulfils the duality conditions, therefore the optimal trajectory of the ship under a collision situation is determined using the optimisation principle and is commenced from the calculation of the first stage and then it is directed toward the final stage. The optimal time for the ship to go through k stages is as follows :
Fig. 3. The structure of the neural network generating the ships' domains. The learning process used the algorithm of the backward propagation of the error with adaptively matched learning step and the momentum.
2.7 The International Regulations for Preventing Collisions at Sea COLREG rules
+ ~k[XI. k' X 2.k ' XI.k+I(XI.k' x:u (x:U _ 1' XU_I
The incorporation of the maritime navigation COLREG rules into the mathematical model is effected in the following manner: - in good visibility conditions the selection of the alternate manoeuvre depends on the angle ~j (Fig. I). At the value of
o $;
~
j
$;
180
(X U - 2 ' U I. k- 2 '
X6,k-1 , (X6.k-2' U U - 2 '~k-2 )~k-I» ' XU+I (x u
' x1,k (X:\k_I' X4.k_1 (XU
~k-2'~k-I )x'i,k (XU
(18)
_ 2 ' Ul.k-2'
_1' X6.k-I(XIl.k -2 ' U U - 2 '
~k-2 )'~k-I » ,X".k (x\k-1' X6.k-1 (XIl.k-2'}
(16)
U2.k-2 · ~k-2 )'~k-I)]
the own ship encounters the object from starboard, whilst for - 180 "
~k-2 ) '~k-I)' X,.k (X,.k_I'
$; ~
1< 0
k=3,4, ...,K The optimal time for the ship to go through the k stages is a function of the system state at the end of the k-\ stage and control (u L k- 2 ' U 2.k-2 ) at the k-2
( 17)
there happens an encounter with the object from port side. According to the COLREG rules the objects which are approaching from starboard must give way to our ship. The conditions (16) and (17) are being implemented into the procedure NEUROCONSTR by adequate formulation of state variables constraints (5). For angle ~j satisfying condition (17) a circle constraint is imposed. For objects appearing on the starboard (16) constraint in the form of a parable, ellipse or hexagon seems the most adequate. - in restricted visibility conditions the starboard rule giving way to the object approaching from starboard does not apply. but the own ship must keep a greater safe approach distance. Ds= 2-3 nm.
stage. The time of going through the k-th stage be determined as follows : L1t k =
~S k (x
where: .6.S k
-
I.k ' X 2.k ' X I. k + I ' X 2. k + I )
~
t k may
( 19)
a distance of the route from point
(x Lk' X 2.k) to point (x I.k +I ' X 2.k+I)· By going from the first stage to the last one the formula (18) determines the Bellman's functional equation for the process of the ship's control by the alteration of the angle of the rudder angle and the rotational speed of the propeller. The constraints for the state variables and the control values generate the NEUROCONSTR procedure in the algorithm for the determination of the safe ship's trajectory. The consideration of the constraints resulting from maintaining safe approaching distance and the recommendations of the way priority law is performed by checking whether the state variables have not exceeded constraints in each of the
3. ALGORITHM OF SAFE CONTROL OF SHIP Determination of the optimal control of the ship in terms of an adopted index of the control quality may be performed by applying Bellman's principle of optimisation. The principle describes the basis feature of the optimal strategy - whatever the initial state and decisions are the remaining decisions must
382
intersections intersections discovered.
considered in which
To2 - time of delay equal approximately to the time constant for the propulsion system main engine - propeller shaft - propeller. Therefore the control at the k-I stage has the following form :
and by rejecting the the excess has been
3.2 Simulation model of the ship's dynamics The dynamic features of the ship during the course alteration by an angle .1.'1' is described in a simplified manner with the use of transfer function : To
G I (s)
k(
'fI. - 'fI._ 1
_
(20)
~t._ 1 -
To' + a
~t._1 -
To'
(23)
s
T.
approximately equal to the time constant of the ship as a course control object, k (
U 2k -
.1.'1'
k,e-T. ,s
-
To2
+ V. _I (24)
I
4. SIMULATION OF THE SAFE SHIP'S TRAJECTORY BY PROGNEURAL PROGRAMME
tm=T o' +-.(21) 'I' Differential equation of the second order describing the ship's behaviour during the change of the speed by Ll Y is approximated with the use of the first order inertion with a delay: = ---"..---
V.-V._I dt k-I
= 1,2, .. ., K - I
results from the static characteristics of the rudder steering. The course manoeuvre delay time is as follows:
dyeS)
To'
UI.k._1 -
where: To' == T, - manoeuvre delay time which is
G 2 (s) =
'fI. - 'fI._ 1 1'fI. - 'fI. 11
~t._1 -
The results of learning by the network were used during the simulation of encountering own ship with other ships. The ships were assigned the domains which aims was to provide data on the areas dangerous for the anti-collision system of own ship. All ships moved along rectilinear courses. At every step of calculations the values of the domains were updated (Fig. 4).
(22)
dn(s) 1+ Tks where: Tk - the time constant of the ship's hull and the mass of the accompanying water,
383
(1)
(2a)
mls ZENIT La Manche Range 12 nm, !!J=2', 60 objects
20
15
10
".
.
".
...
5
1000
2000
)000
5000
1000
7000
1000
lOaD
(2b)
....•• ,.
~V-
0
~
-5
..• ,.
~ ;
-5
0
5
.,. 10
Fig. 4 . Continued over page.
384
1000
2000
SOOO
.000
5000
.000
7000
1000
'000
The correctness of the assessment of the safety of the passing vessels with the use of the networks depends, to a decisive degree, on the correctness of the data used in the process of the network learning. The use of the experience of a few experienced navigators during the learning by the network may lead to the situation when the network acquires their averaged knowledge. The introduction of elements of the artificial intelligence represented by a properly prepared network, to the systems of determining the ship's domain, and as a consequence - her safe trajectory in a collision situation may help less experienced navigator supervising the anti-collision system assist the navigational situation, increase the safety of the anti-collision manoeuvre and accelerate the process of selecting a manoeuvre avoiding the collision.
REFERENCES
·10
, .0
0
1000
2000
SOOO
~OOO
5000
5
.000
7000
(3b)
'00
,
.0
.'
20
'000
0000
8000
8000
,
'.
,
00 '0
10
:
~
120
,
. ~
0 '0
o
·5
Bellman, R.E., and S .E. Dreyfus (1957) . Dynamic programming. Princeton University Press, New York. Colley, BA, Curtis, R.G., and c.r. Stockel (1983). Manoeuvring times, domains and arenas. Journal of Navigation. 36, 324-328. Cross, S.l. (1994). Objective assessment of maritime simulator training. Proc. of the Int. Con! the Development and Implementation of International Maritime Training Standards. Malmo. Davie, P.V., Dove, MJ., and C.T. Stockel (1980). A computer simulation of marine traffic using domains and arenas. Journal of Navigation. 33, 215-222. Fossen, T.1. (1994). Guidance and control of ocean vehicles. 10hn Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapore. Hunt, KJ., Irwin, G.R., and K. Warwick (1998). Neural network engineering in dynamic control systems. Advances in Industrial Control. Springer. Leondes, C.T. (1998) . Control and dynamic systems. Neural network systems techniques and applications, 7, Academic Press. Lisowski, J. (1981). Ship's anti-collision systems. Maritime Publishing, Gdansk (in polish). Lisowski, J. (1998). The application of modern control theory in synthesis of computer simulation programmes of ship's safe manoeuvring for effective training of marine engineers, Global Congr. on Eng. Education. Cracow. Lisowski, J., Rak, A. and W. Czechowicz (2000). Neural network classifier for ship domain assessment. Journal of Mathematics and Computers in Simulation. Elsevier Science B.V., 51, 399-406.
1000
2 000
3000
"000
15000
' 0 00
7000
Fig. 4. Simulation of the safe own ship's trajectory in the situation 60 encountered objects: 1- radar picture of the situation in the La Manche region, 2a- safe trajectory for con-ro 20 000 DWT own ship in good visibility, 3a- safe trajectory for this ship in poor visibility, 2b and 3b- reference rotational speed, 2c and 3c- reference rudder angle.
5. CONCLUSION The neural networks presented in this paper may be used as the elements of the systems for the assessment of the safety of the ships' passing by the introduction of the possibility to make a current correction of the sizes ships' domains. They are able to represent the heuristic knowledge similar to that which is represented by an experienced navigator.
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