International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 6
Number 1
March 2012
35
1 SAFE SHIP CONTROL
1.1 Structure of control system
The challenge in research for effective methods to
prevent collisions has become important with the in-
creasing size, speed and number of ships participat-
ing in sea carriage. An obvious contribution in in-
creasing safety of shipping has been application of
the ARPA (Automatic Radar Plotting Aids) anti-
collision system (Fig. 1).
Figure 1. The structure of safe ship control system.
1.2 Information of the state process
The ARPA system enables to track automatically at
least 20 encountered j objects as is shown on Figure
2, determination of their movement parameters
(speed V
j
, course ψ
j
) and elements of approach to
the own ship (
j
j
DCPAD =
min
- Distance of the
Closest Point of Approach,
j
j
TCPAT =
min
- Time to
the Closest Point of Approach) and also the assess-
ment of the collision risk r
j
(Bist 2000, Bole 2006).
Figure 2. Navigational situation passing of the own ship with j
met ship moving with V
j
speed and ψ
j
course.
The risk value is defined by referring the current
situation of approach, described by parameters
j
D
min
and
j
T
min
, to the assumed evaluation of the
situation as safe, determined by a safe distance of
approach D
s
and a safe time T
s
– which are necessary
to execute a collision avoiding manoeuvre with con-
sideration of distance D
j
to j-th met ship (Cahill
2002).
The Sensitivity of Safe Ship Control in
Restricted Visibility at Sea
J. Lisowski
Gdynia Maritime University, Electrical Engineering Faculty, Department of Ship Automation,
Gdynia, Poland
ABSTRACT: The structure of safe ship control in collision situations and computer support programmes on
base information from the ARPA anti-collision radar system has been presented. The paper describes the sen-
sitivity of safe ship control to inaccurate data from the ARPA system and to process control parameters altera-
tions. Sensitivity characteristics of the multi-stage positional non-cooperative and cooperative game and kin-
ematics optimization control algorithms on an examples of a navigational situations in restricted visibility at
sea are determined.
36
The functional scope of a standard ARPA system
ends with the trial manoeuvre altering the course
ψ∆±
or the ship's speed
V∆±
selected by the nav-
igator (Cockcroft & Lameijer 2006, Gluver & Olsen
1998).
1.3 Computer support of navigator
The problem of selecting such a manoeuvre is very
difficult as the process of control is very complex
since it is dynamic, non-linear, multi-dimensional,
non-stationary and game making in its nature. In
practice, methods of selecting a manoeuvre assume a
form of appropriate steering algorithms supporting
navigator decision in a collision situation. Algo-
rithms are programmed into the memory of a Pro-
grammable Logic Controller PLC (Fig. 3) (Lisowski
2008).
Figure 3. The system structure of computer support of naviga-
tor manoeuvring decision in collision situation.
2 COMPUTER PROGRAMMES OF
NAVIGATOR SUPPORT
2.1 Base model of process
The most general description of the own ship pass-
ing the j number of other encountered ships is the
model of a differential game of j number of moving
control objects (Fig. 4).
Figure 4. Block diagram of the basic differential game model
of safe ship control process.
The properties of control process are described by
the state equation:
( ) ( )
[ ]
tuuxxfx
mm
mmii
,...,,,...,,
,,0,,0
00
ννϑϑ
=
(1)
where:
( )
tx
0
,0 ϑ
- ϑ
0
dimensional
vector of process state of
own ship determined in time
],[
0 k
ttt ∈
,
( )
tx
j
j ϑ,
- ϑ
j
dimensional vector of the process state
for j-th ship,
( )
tu
0
,0 ν
- ν
0
dimensional control vector of own ship,
)(
,
tu
j
j ν
- ν
j
dimensional control vector of j-th ship
(Isaacs 1965, Lisowski 2010, Engwerda 2005).
The constraints of the control and the state of the
process are connected with the basic condition for
the safe passing of the ships at a safe distance D
s
in
compliance with COLREG Rules, generally in the
following form:
( )
0),(
min,,
≤−=
j
sjjj
DDuxg
jj
νϑ
(2)
For the class of non-coalition games, often used
in the control techniques, the most beneficial con-
duct of the own ship as a player with j-th ship is the
minimization of her goal function in the form of the
payments – the integral payment and the final one:
min)()()]([
0
0
2
,0,0
→++=
∫
k
t
t
kjj
tdtrdttxI
k
ϑ
(3)
The integral payment represents loss of way by
the ship while passing the encountered ships and the
final payment determines the final risk of collision
r
j
(t
k
) relative to the j-th ship and the final deflection
of the ship d(t
k
) from the reference trajectory (Fig. 5)
(Modares 2006, Nisan et al. 2007).
2.2 Programme of multi-stage positional non-
cooperative game MSPNCG
The optimal steering of the own ship
)(
0
tu
∗
, equiva-
lented for the current position p(t) to the optimal po-
sitional steering
)(
0
pu
∗
. The sets of acceptable strat-
egies
( )
[ ]
kj
tpU
0
are determined for the encountered
ships relative to the own ship and initial sets
( )
[ ]
k
jw
tpU
0
of acceptable strategies of the own ship
relative to each one of the encountered ship. The
pair of vectors
m
j
u
and
j
u
0
relative to each j-th ship
is determined and then the optimal positional strate-
gy for the own ship
)(
0
pu
∗
from the condition (4).
37
Figure 5. The final risk of collision r
j
(t
k
) relative and the final
deflection d(t
k
) from the reference trajectory in situation pass-
ing of three met ships.
)L,x(S
∫
dt)t(u
min
max
min
I
k00
t
t
0
U∈u
U∈u
U∈u
0
k
L
0
j
0
j
0
j
m
j
m
1j
j
0
0
∗∗
==
=
(4)
The function S
0
refers to the continuous function
of the manoeuvring goal of the own ship, character-
ising the distance of the ship at the initial moment t
0
to the nearest turning point L
k
on the reference p
r
(t
k
)
route of the voyage (Millington & Funge 2009, Os-
borne 2004).
The optimal control of the own ship is calculated
at each discrete stage of the ship’s movement by ap-
plying the Simplex method to solve the problem of
the triple linear programming, assuming the relation-
ship (4) as the goal function and the control con-
straints (2).
Using the function of lp – linear programming
from the Optimization Toolbox Matlab, the posi-
tional multi-stage game non-cooperative manoeu-
vring MSPNCG program has been designed for the
determination of the own ship safe trajectory in a
collision situation (Lisowski 2010).
2.3 Programme of multi-stage positional
cooperative game MSPCG
The quality index of control (4) for cooperative
game has the form:
)L,x(S
∫
dt)t(u
minminmin
I
k00
t
t
0
U∈u
U∈u
U∈u
0
k
L
0
j
0
j
0
j
m
j
m
1j
j
0
0
∗∗
==
=
(5)
2.4 Programme of non-game kinematic
optimization NGKO
Goal function (4) for kinematics optimization has
the form:
)L,x(Sdt)t(u
min
I
k00
t
t
0
Uu
0
k
L
0
m
1j
j
0
0
∗
∈
∗
=
∫
=
=
(6)
3 THE SENSITIVITY OF SAFE SHIP CONTROL
3.1 Definition of safe control sensitivity
The investigation of sensitivity of game control fetch
for sensitivity analysis of the game final payment (3)
measured with the relative final deviation of d(t
k
)=d
k
safe game trajectory from the reference trajectory, as
sensitivity of the quality first-order.
Taking into consideration the practical applica-
tion of the game control algorithm for the own ship
in a collision situation it is recommended to perform
the analysis of sensitivity of a safe control with re-
gard to the accuracy degree of the information re-
ceived from the anti-collision ARPA radar system
on the current approach situation, from one side and
also with regard to the changes in kinematical and
dynamic parameters of the control process (Lisowski
2009, Straffin 2001).
Admissible average errors, that can be contribut-
ed by sensors of anti-collision system can have fol-
lowing values for:
− radar,
− bearing: ±0,22
o
,
− form of cluster: ±0,05
o
,
− form of impulse: ±20 m,
− margin of antenna drive: ±0,5
o
,
− sampling of bearing: ±0,01
o
,
− sampling of distance: ±0,01 nm,
− gyrocompas: ±0,5
o
,
− log: ±0,5 kn,
− GPS: ±15 m.
The algebraic sum of all errors, influent on pictur-
ing of the navigational situation, cannot exceed for
absolute values ±5% or for angular values ±3
o
.
3.2 The sensitivity of safe ship control to
inaccuracy of information from ARPA system
Let X
0,j
represent such a set of state process control
information on the navigational situation that:
},,,,,{
,0 jjjjj
NDVVX ψψ=
(7)
Let then
ARPA
j,0
X
represent a set of information from
ARPA system containing extreme errors of meas-
urement and processing parameters:
38
},
,,,,{
,0
jjjj
jjjj
ARPA
j
NNDD
VVVVX
δ±δ±
δψ±ψδ±δψ±ψδ±=
(8)
Relative measure of sensitivity of the final pay-
ment in the game s
inf
as a final deviation of the ship's
safe trajectory d
k
from the reference trajectory will
be:
)(
)(
),(
,0
,
,0,0inf
jk
ARPA
jo
ARPA
k
j
ARPA
j
Xd
Xd
XXs ==
(9)
},,,,,{
inf
jjjj
NDV
V
sssssss
ψ
ψ
=
(10)
3.3 Sensitivity of safe ship control to process
parameters alterations
Let X
param
represents a set of parameters of the state
process control:
},,,{ VtDtX
ksmparam
∆∆=
(11)
Let then
'
param
X
represents a set of information
containing extreme errors of measurement and pro-
cessing parameters:
},,,{
'
VVttDDttX
kkssmmparam
∆δ±∆δ±δ±δ±=
(12)
Relative measure of sensitivity of the final pay-
ment in the game as a final deflection of the ship's
safe trajectory d
k
from the assumed trajectory will
be:
)(
)(
),(
''
'
paramk
paramk
paramparamdyn
Xd
Xd
XXs ==
(13)
},,,{
V
t
Dt
dyn
ssss
s
ksm
∆
=
(14)
where:
t
m
- advance time of the manoeuvre with respect to
the dynamic properties of the own ship,
k
t
- duration of one stage of the ship's trajectory,
D
s
– safe distance,
∆
V - reduction of the own ship's speed for a deflec-
tion from the course greater than 30
o
(Baba &
Jain 2001).
4 SENSITIVITY CHARACTERISTICS OF SAFE
SHIP CONTROL IN RESTRICTED
VISIBILITY AT SEA
Computer simulation of MSPNCG, MSPCG and
NGKO algorithms, as a computer software support-
ing the navigator manoeuvring decision, were car-
ried out on an example of a real navigational situa-
tions of passing j=3, 12 and 20 encountered ships.
The situations were registered in Kattegat Strait on
board r/v HORYZONT II, a research and training
vessel of the Gdynia Maritime University, on the ra-
dar screen of the ARPA anti-collision system Ray-
theon (Figs 6-7).
Figure 6. The place of identification of navigational situations
in Kattegat Strait.
Figure 7. The research-training ship of Gdynia Maritime Uni-
versity r/v HORYZONT II.
4.1 Navigational situation for j=3 met ships
Computer simulation of MSPNCG, MSPCG and
NGKO programmes was carried out in
Matlab/Simulink software on an example of the real
navigational situation of passing j=3 encountered
ships in Kattegat Strait in restricted visibility when
D
s
=2 nm and were determined sensitivity character-
39
istics for the alterations of the values X
0,j
and X
param
within ±6% or ±3
o
(Figs 8-14).
Figure 8. The 12 minute speed vectors of own ship and 3 en-
countered ships in situation in Kattegat Strait.
Figure 9. The safe trajectory of own ship for MSPNCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=3 encountered ships, r(t
k
)=0, d(t
k
)=7.60 nm.
Figure 10. Sensitivity characteristics of safe ship control ac-
cording to MSPNCG programme on an example of the naviga-
tional situation j=3 in the Kattegat Strait.
40
Figure 11. The safe trajectory of own ship for MSPCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=3 encountered ships, r(t
k
)=0, d(t
k
)=4.71 nm.
Figure 12. Sensitivity characteristics of safe ship control ac-
cording to MSPCG programme on an example of the naviga-
tional situation j=3 in the Kattegat Strait.
Figure 13. The safe trajectory of own ship for NGKO algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=3 encountered ships, r(t
k
)=0, d(t
k
)=3.70 nm.
Figure 14. Sensitivity characteristics of safe ship control ac-
cording to NGKO programme on an example of the naviga-
tional situation j=3 in the Kattegat Strait.
41
4.2 Navigational situation for j=12 met ships
Computer simulation of MSPNCG, MSPCG and
NGKO programmes was carried out in
Matlab/Simulink software on an example of the real
navigational situation of passing j=12 encountered
ships in Kattegat Strait in restricted visibility when
D
s
=2 nm and were determined sensitivity character-
istics for the alterations of the values X
0,j
and X
param
within ±6% or ±3
o
(Figs 15-21).
Figure 15. The 12 minute speed vectors of own ship and 12 en-
countered ships in situation in Kattegat Strait.
Figure 16. The safe trajectory of own ship for MSPNCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=12 encountered ships, r(t
k
)=0, d(t
k
)=3.20 nm.
Figure 17. Sensitivity characteristics of safe ship control ac-
cording to MSPNCG programme on an example of the naviga-
tional situation j=12 in the Kattegat Strait.
Figure 18. The safe trajectory of own ship for MSPCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=12 encountered ships, r(t
k
)=0, d(t
k
)=1.40 nm.
42
Figure 19. Sensitivity characteristics of safe ship control ac-
cording to MSPCG programme on an example of the naviga-
tional situation j=12 in the Kattegat Strait.
Figure 20. The safe trajectory of own ship for NGKO algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=12 encountered ships, r(t
k
)=0, d(t
k
)=1.23 nm.
Figure 21. Sensitivity characteristics of safe ship control ac-
cording to NGKO programme on an example of the naviga-
tional situation j=12 in the Kattegat Strait.
4.3 Navigational situation for j=20 met ships
Computer simulation of MSPNCG, MSPCG and
NGKO programmes was carried out in
MATLAB/SIMULINK software on an example of
the real navigational situation of passing j=20 en-
countered ships in Kattegat Strait in restricted visi-
bility when D
s
=2 nm and were determined sensitivi-
ty characteristics for the alterations of the values X
0,j
and X
param
within ±6% or ±3
o
(Figs 22-28).
43
Figure 22. The 12 minute speed vectors of own ship and 20 en-
countered ships in situation in Kattegat Strait.
Figure 23. The safe trajectory of own ship for MSPNCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=20 encountered ships, r(t
k
)=0, d(t
k
)=8.06 nm.
Figure 24. Sensitivity characteristics of safe ship control ac-
cording to MSPNCG programme on an example of the naviga-
tional situation j=20 in the Kattegat Strait.
Figure 25. The safe trajectory of own ship for MSPCG algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=20 encountered ships, r(t
k
)=0, d(t
k
)=6.64 nm.
44
Figure 26. Sensitivity characteristics of safe ship control ac-
cording to MSPCG programme on an example of the naviga-
tional situation j=20 in the Kattegat Strait.
Figure 27. The safe trajectory of own ship for NGKO algo-
rithm in restricted visibility D
s
=2 nm in situation of passing
j=20 encountered ships, r(t
k
)=0, d(t
k
)=6.94 nm.
Figure 28. Sensitivity characteristics of safe ship control ac-
cording to NGKO programme on an example of the naviga-
tional situation j=20 in the Kattegat Strait.
5 CONCLUSIONS
The sensitivity of the final game payment:
− is least relative to the sampling period of the tra-
jectory and advance time manoeuvre,
− most is relative to changes of the own and met
ships speed and course,
− it grows with the degree of playing character of
the control process and with the quantity of ad-
missible strategies.
The less sensitivity of safe ship control in collision
situations is represented by NGKO algorithm and
highest by MSPNCG algorithm.
The considered control algorithms are, in a cer-
tain sense, formal models of the thinking process of
navigator steering of the ship motion and making
manoeuvring decisions.
45
Therefore they may be applied in the construction
new model of ARPA system containing the comput-
er supporting of navigator decisions.
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