895
1 INTRODUCTION
Over 80% of global transportation goods are
conducted by ship. [1]The prosperity and
globalization of the world trade economy depend on
the rapid development of the maritime shipping
industry. However, the larger, and faster ships bring
increased risk to the individual, societies, and
environment, in terms of dire consequences such as
loss of life, the economy lost, and environmental
pollution. Assessing the risk of ship collision is
therefore of great importance as it provides a cost-
effective and practical way to mitigate risk.
To analyze the risk of ship collision, various
research has been carried out from various
perspectives. Among all the studies of quantitative
risk analysis, indicator-based models and safety
boundary models are quite popular, see [2]. Ship
indicators are utilized in the modeling of indicator‐
based approaches, such as the Closest Point of
Approach (CPA), etc. Many efforts have been taken to
integrate multiple indicators into one, for a better and
more accurate characterization of collision risk. CRI
index [3]was put forward which integrates the TCPA
and DCPA into one indicator. To combine more
indicators such as relative speeds and bearing, the
VCRO index is proposed[4]. Huang et al. [5] and Chen
et al. [6]analyzed collision risk by projecting pair
ships’ distance into velocity space, which is another
way to integrate distance and time into one indicator.
Most collision risk analysis methods are limited to
the encounters of pair ships by now. However, in
navigation practice, multi-ship encounters are quite
common and often more dangerous. Therefore, it is
surely necessary to analyze the collision risk of multi-
ship encounters. To analyze that, a two-stage MC
simulation algorithm was presented[7], and CPA was
improved to estimate collision risk in multi-ship
Multi-ship Encounter Situation Analysis with the
Integration of Elliptical Ship Domains and Velocity
Obstacles
Z
. Cheng
1,2
, P. Chen
1,2
, J. Mou
1,2
& L. Chen
1,2
1
School of Navigation, Wuhan University of Technology, Wuhan, China
2
Hubei Key Laboratory of Inland Shipping Technology, Wuhan, China
ABSTRACT: With economic globalization, ships tend to be larger and faster, and the volume of maritime traffic
is increasing. Ships sailing in waters with dense traffic flow are easy to fall into complicated multi-ship
encounter situations and have a high risk of collision. Thus, it is crucial to conduct risk analysis in such
situations. In this paper, a modified collision analysis method for detecting dangerous multi-ship encounters in
ports and waterways is proposed. The velocity obstacle algorithm is utilized to detect encounters. The model of
the elliptic ship domain was integrated into the algorithm as the criteria. The Boolean operation was also used
in the multi-ship encounter. A case study is conducted to illustrate the efficacy of the improved model, and a
comparison between the existing method and the formal model is also performed. The results indicate that with
the integration of the ship domain, the proposed method can effectively detect the encounters of multiple ships
which are dangerous to collide.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 17
Number 4
December 2023
DOI: 10.12716/1001.17.04.
16
896
encounters. In[8], a negative exponential function was
used to characterize the collision risk for each cluster
of encounter vessels with DCPA and TCPA. The value
of multi-ship collision risk was obtained by using
fuzzy logic theory and Analytic Hierarchy Process
(AHP) to incorporate the impact factors of DCPA,
TCPA, etc., see[9]. Although several methods have
been proposed to get the collision risk of multi-ship
encounters, they are all depended on multiple
separate metrics, like DCPA and TCPA. Chen et
al.[10] applied a velocity obstacle-based risk
measurement to measure the risk of collision between
multiple ships from the velocity perspective, which
provided an interpretable method that incorporated
multi-ship encounters into the linear algorithm. In
[11], the modified TD‐NLVO algorithm with the
integration of an elliptical ship domain was applied to
detect the pair-ship encounter. Based on this, this
paper aims to improve the collision risk detection
model of multi-ship encounter, incorporate multi-ship
encounter into the model of time-discrete and
nonlinear velocity obstacles, and combine it with the
elliptical ship domain to make it more practical.
In this paper, an improved Time Discrete Non-
Linear Velocity Obstacle algorithm is used to combine
the multi-ship encounter model with the ship domain
model, to further improve the accuracy of the results.
Firstly, the Non‐ linear velocity obstacle algorithm
is introduced as the basic tool to detect multi-ship
encounter situations from the perspective of the
process; Then, the elliptical ship domain model is
integrated into the algorithm to detect the candidate
ships. A case study using actual AIS (Automatic
Information System) data is conducted, together with
compassion between the old and new algorithms. The
arrangement of the article is as follows: Section 2
illustrates the methodology of this paper, followed by
the design of the algorithm in Section 3. A case study
is performed in section 4 to show the results of the
algorithm and the comparison. Section 5 makes a
conclusion.
2 METHODOLOGY
Chen et al.[10] proposed a TD-NLVO algorithm
integrated with Boolean operation on the individual
NLVO, which provided an effective tool to detect
multiple ship encounter situations using historical AIS
data. Therefore, in this paper, the objective is to
improve the criteria of the multi-ship encounter
detection algorithm from circle to elliptical ship
domain, which is a common domain model in
maritime practice. The improved algorithm is then
applied to determine the dangerous multi-ship
encounters according to the violation of the own
ship’s domain through the process of the encounter.
To do so, the TD‐NLVO algorithm is adopted as the
basic framework for collision analysis, and the
elliptical ship domain model is integrated as the
criteria.
3 DETECTION OF MULTI-SHIP ENCOUNTER
3.1 TD
‐
NLVO algorithm
The velocity obstacle algorithm is a series of
algorithms that can represent the potential of collision
as it determines the range of velocities that will result
in a collision with those obstacles based on
geometrical calculations of the velocities of obstacles
relative to an object. The velocity obstacle (VO)
algorithm was originally used as a method for
obstacle avoidance in robotics and autonomous
systems [12]. It has been extensively studied and
improved upon in the literature, with applications in
various fields such as robotic navigation, ship
collision avoidance, and autonomous systems.
Assessing the risk of ship collision using the VO
algorithm is a great idea. In maritime shipping area,
this idea was first applied by Degre and Lefevre [13]
in 1981 and further developed by Lenart [14] in 1983
into the Collision Threat Parameter Area (CTPA), also
known as Linear Velocity Obstacle (LVO) analysis.
Like the principle of radar ARPA, the principle of this
algorithm is relatively simple, with a small calculation
amount, and fast solution speed, and it can be
calculated in real-time. Moreover, the model
established based on collision maneuver is more in
line with the maritime navigation, it can effectively
reflect the collision risk of ships and visualize it.
Currently, the common velocity obstacle algorithms
used in collision avoidance models on safety support
system for marine traffic include Linear Velocity
Obstacle (LVO), Nonlinear Velocity Obstacle (NLVO)
[5, 6], Probabilistic Velocity Obstacle (PVO)[15],
Generalized Velocity Obstacle [16]et al. Those
methods respectively considered different degrees of
motion constraints, and the simulated ship encounter
process tends to the encounters in actual navigation
practice.
The Nonlinear Velocity Obstacle algorithm can
adopt nonlinear trajectories of the robot and obstacles
while maintaining real-time performance. Since the
NLVO algorithm can update the motion state of the
ship and the target ships, and the operation speed is
fast enough with few constraints, it is suitable for
detecting dangerous encounters during the process of
multi-ship encounters. Therefore, just as the non‐
linear velocity obstacle algorithm was applied in [6] as
the fundamental tool for collision candidate detection,
this work utilizes the algorithm to detect collision
candidates and analyze the collision risk level.
PA
PB
VA
VB
X
Y
a.in geographic space
Vx
Vy
b.in ship A's velocity space
Figure 1. The basic illustration of the Non‐linear Velocity
Obstacle algorithm
As shown in Fig. 1, the spatiotemporal relationship
between own ship A and target ship B was projected
into the velocity space of ship A. ConfP are all the
dangerous positions for ship A that may collide with
897
ship B, defined as a circular area with radius R. NLVO
is the set of all the dangerous velocity circles, as
shown in Fig. 1(b), means the set of the velocity of
ship A which may collide with ship B.
The basic theory of the NLVO algorithm is
expressed as Eq.1, according to [5].
(1)
where P
i(t) means the position of ship i at time t, VA
means the velocity of own ship A which may collide
with ship B, t
0 means the current detection time step, tf
means one time step in the time period of detection.
3.2 Elliptical ship domain
In the preceding section, the ConfP is defined as a
circular area with a fixed radius, which is similar to
the Collision Diameter defined by Fujii[17] and
Pedersen[18]. However, this definition has some
limitations in practice, as any violation of the area
results in physical contact due to the small size of the
circle. To overcome these limitations, this paper
expands the ConfP and introduces the static elliptical
ship domain model as a new criterion for detecting
collision candidates. This model is based on the
fundamental concept of the ship domain, which
represents an area around a ship that must be kept
clear of other vessels to avoid collisions. We
respectively set the parameters of semi-major and
semi-minor axes, according to the research by [19], at
8 and 4 times the own ship's length. Integration of this
domain into the TD-NLVO requires a mathematical
function, and variables can be obtained from own
ship's information: length and course over the
ground. This information can be acquired from
historical AIS data. Overall, this paper replaces the
circular ConfP with a static elliptical ship domain
model and adopts the following mathematical
function to integrate the model into the TD-NLVO.
X
Y
O
Ship(x
0
,y
0
)
Elliptic Ship Domain
V
θ
Figure 2. Illustration of elliptic ship domain
The parametric equations for oblique ellipse are
shown as Eq.2:
222
22 2 2
22
2 2 22
22
00
''' 0
sin ' cos '
2( )sin 'cos '
cos ' sin '
' ,' ,'
θθ
θθ
θθ
θ αθ
+++=
= +
= −
= +
= −
=−=−=−
Ax By Cy D
Aa b
B ab
Ca b
D ab
x xxy y y
(2)
where x
0, y0 means the horizontal and vertical
coordinates of the centre of the ellipse, a, b means the
length of semi-major and semi-minor axes,
θ
means
the angle between the major and horizontal axes of
the elliptical ship domain. Those equations were
incorporated into the algorithm described in the next
section.
3.3 Collision risk detection of multi-ship encounter
When ships go through busy waterways, such as
ports and important water channels, the encounter of
ships can be even more complicated because they may
encounter multiple ships and may pose a pressing
situation. The International Regulations for
Preventing Collisions at Sea (COLREGs) only provide
guidance on how to avoid collisions between two
ships, leaving the officers responsible for evaluating
and prioritizing the response to each target ship's
collision risk. This can be particularly difficult in ports
and waterways that are critical to commercial
activities or national security. This work has been
discussed in [10], and the TD-NLVO algorithm has
been improved for multiple encounter situations. The
principle is shown in Fig.3. Based on the TD-NLVO,
the technique of Boolean operation on polygons[20]
has been introduced into constructing the dangerous
velocity set combined with multiple target ships, as is
shown in Figure 3(b). Combine multiple VO of each
obstacle to form the own ship's reachable velocity
space (RVO), which defines the range of velocities
that the own ship can take while avoiding obstacles.
V
A
V
B
X
Y
O
V
C
P
A
P
B
P
C
a.in geographic space
O
Vx
Vy
VO
B
VO
C
b.in ship A's velocity space
Figure 3. Illustration of multiple encounter situation and
Boolean operation on polygons.
To detect the collision candidates from multi-ship
encounters, this paper used the TD-NLVO algorithm
integrated with improved criteria (as shown in Eq.3).
The dangerous encounters were determined by the
violation of the combined TD-NLVO of the target
ships, which criteria are determined by domain of
elliptic. The available range of speed was also
considered in the construction of the TD-NLVO.
898
|
00
ellipse
||
1
||
()
()
{|| ( ) ( ) || }
()
∞
=
= ⊕
−−
= −≤
=
=
Bt
f
f
Bt Bt
ff
Bt
f
jf
DOE
A ship
t
ff
DOE B A
n
A allship A ship
j
DOE A allship region A
Pt
ConfP
NLVO
tt tt
ConfP P t P t R
NLVO NLVO
NLVO NLVO V
(3)
where
|
Bt
f
A ship
NLVO
denotes the dangerous velocity set
of ship A calculated by the TD-NLVO algorithm
induced by the target ship B, and the elliptic ship
domain mentioned in Section 3.2 was integrated into
the algorithm replacing the ConfP mentioned in
Section 3.1 as
DOE
ConfP
.
|
Bt
f
A allship
NLVO
denotes the
union of all the velocity sets of ship A induced by all
target ships in the multi-ship encounter.
DOE
NLVO
means the polygon intersection of
|
Bt
f
A allship
NLVO
and
|
region A
V
which contains all the adoptable velocity range
of own ship A.
This research is discussing the development of an
improvement of the encounter detection method for
multiple ships, which integrates an improved TD‐
NLVO and elliptical ship domain model. This new
version method for multi-ship encounters focuses on
detecting through the entire process of the encounter,
rather than just at a certain time interval during the
encounter. To achieve this, the trajectory data of ships
in the area is reconstructed using their MMSI and then
divided into subsets to speed up computation. The
design of the new model is shown in Figure 4.
Select ship trajectory in
an encounter
Implement domain based
on TD-NLVO
Choose one encounter,
choose one ship as own ship
Any Violation?
No
Yes
choose another ship as target ship
Is the process finished?
Detect the encounter situation
based on NLVO
DOE
Ship AIS trajectory
database
No
Yes
Record information
Figure 4. Flow chart of the ship domain-based encounter
detection model
4 CASE STUDY
In this section, two case studies on implementing the
domain‐based TD‐NLVO were illustrated. Each
case is an encounter situation involving three ships.
Historical AIS data provided by the Wuhan
University of Technology was utilized as the test
datasets. The case study was performed to verify the
capability of the proposed method for detecting the
encounter process of multiple ships.
We introduced two sets of AIS data on May 20,
2019, in the East China Sea, at the estuarine waters of
the Yangtze River as the test dataset. To apply the
proposed algorithm according to the actual waterway
and traffic, the semi-major and semi-minor axes of the
safety region are respectively set as 1000m and 500m.
The parameter T
threshold is set to be 10s, Tscan is set to be
20mins. The own ship's available speed is set to be
within 20 m/s.
Case one is a multi-ship encounter situation for
5mins from “13:15:05” to “13:20:13” between ship
“413XXX250”, ship “413XXX080”, and ship
“413XXX480”. Ship “413XXX250” was chosen as the
research object (Own ship). Based on the proposed
method, a three-ship encounter between the own ship
“413XXX250” and target ship1“413XXX480”, target
ship2“413XXX080” was detected, which are shown
with their trajectories and encounter situation in
velocity space at a certain time step. Fig.5 shows their
trajectories and relative distance from “13:15:05” to
“13:20:13”, on May 20, 2019.
Figure 5. Trajectory and relative distance between ships-case
1
From the information of ship trajectories, we can
see that at the encounter process, own ship
“413XXX250” were in “head on” situation with target
ship1“413XXX480”, and in “crossing” situation with
target ship2“413XXX080” at last.
Taking the encounter situation at 10:15:05, and
10:16:15 AM as examples, the spatiotemporal
relationships between two target ships in the own
ship’s velocity space are represented. To be more
specific, a snapshot of the positions of ships,
individual NLVO, and combined NLVO are
illustrated in Fig. 6, 7.
From Fig. 6 (b), (c), one can see that the combined
NLVO is made by two different individual NLVOs,
and the violated NLVO induced by ship “413XXX480”
is part of the combined NLVO. The results indicate
that the own ship only has a violation with ship
“413XXX480”, which indicates that at “10:15:05” AM
own ship has a potential collision only with ship
“413XXX480” in 5mins if keeping the current velocity.
The NLVO of the other targets is “combined” into the
large NLVO during the Boolean operation “Union”
and the velocity o
f the own ship does not have a
violation with it. In Fig. 7 (b), (c), we can see that the
own ship’s velocity didn’t violate any NLVO, which
illustrates the encounter was safe if keeping the
899
current velocity. Over time, the position and
encounter situation of the two ships changed, and the
detection result of the encounter had changed from
dangerous to safe, which reflected the collision risk in
the future period.
With this design of algorithm, the encounter
process can be detected in the following: firstly,
determine if there is a violation of combined NLVO at
each step, and if so, determine which individual ship
is responsible for the violation with the own ship. In
addition, since the modified TD‐NLVO considered
course information in domain modelling, the coverage
of velocity obstacles during the multi-ship encounter
process varies as the courses of target ships change
constantly. This allows a detailed analysis of the
encounter.
Figure 6. Positions and Vos of ships at “10:15:05” –case 1.
Figure 7. Positions and Vos of ships at “10:16:15” –case 1.
Case two is another three-ship encounter for
8mins, ship “413XXX220”, ship “413XXX350” and ship
“477XXX900” are involved in the duration. Ship
“413XXX220” was chosen as the research object (Own
ship). The detection results of the three-ship
encounter between the own ship “413XXX220” and
target ship1“413XXX350”, target ship2“477XXX900”
were shown with their trajectories and encounter
situation in velocity space at a certain time step. Fig.8
shows their trajectories from “13:16:35” to “13:24:23”
and the relative distance between the two target ships
and the own ship, on May 20, 2019. We can see that at
the beginning of the encounter process, the own ship
“412XXX450” and two target ships respectively were
in a “crossing” situation.
Figure 8. Trajectory and relative distance between ships-case
2
Fig. 9, 10 illustrate the encounter situation
respectively at “10:18:15”, “10:19:05” AM by a
snapshot of the positions of ships, individual NLVO
and combined NLVO. The two individual NLVOs has
an intersection, and the area of the combined NLVO is
smaller than the sum of the two individual areas. In
Fig. 9 (a) we can see target ship “413XXX350” was
close to own ship at 10:18:15 AM, which may be
dangerous. From Fig. 9 (b) and (c), we can see that
there is no violation of any NLVO, which indicates
that own ship was safe but need pay attention if a
turn was needed. From Fig. 10 (b) and (c), we can see
that own ship’s velocity violated the individual
NLVO induced by the ship “413XXX350”, which
indicates that own ship may collide with the target
ship in 8 mins if keeping this velocity. The detection
results changed during this period, reflecting the
change of collision risk in this period.
Figure 9. Positions and Vos of ships at “10:18:15” –case 2.
900
Figure 10. Positions and Vos of ships at “10:19:05” –case 2.
5 DISCUSSION
In this section, a comparison between the original TD
‐ NLVO (M1) [10]and ship domain‐based TD‐
NLVO (M2) is conducted. The comparison has two
components: 1) comparison between results from M1
and M2 and 2) analysis of the detection results by two
different methods. The AIS dataset of encounters
utilized is the same as the two cases in the case study.
The parameter setting between the two methods is
shown in the Table 1.
Table 1. Parameter setting of the two methods
________________________________________________
Parameter M1 M2
________________________________________________
Tthreshold 60s 60s
T
scanning 20mins 20mins
Available V ±20m/s ±20m/s
Criteria Circular safety region Elliptical ship domain
Radius 800m Semi‐major: 1000m
Semi‐minor: 500m
________________________________________________
The two methods were applied to the same AIS
data, and the danger of the same multi-ship
encounters was detected at the same time step. In the
case at the same time steps, the detection results are
not always the same. For case one, at 10:15:05 AM, the
velocity of the own ship just violated the NLVO
induced by one target ship, which is the same in M1
and M2, as shown in Fig.11. At 10:16:15 AM, the
velocity of the own ship violated the individual
NLVO in M1, while in M2 there was no violation, as
shown in Fig.12. The detection results of two methods
at every time step of case one are shown in Appendix
Ⅰ.
Figure 11. Illustration of NLVO with M1 and M2 at
“10:15:05”-case 1
Figure 12. Illustration of NLVO with M1 and M2 at
“10:16:15”-case 1
For case two, at 10:18:15 AM the velocity of the
own ship just violated the NLVO induced by one
target ship in M1, but in M2 there was no violation,
see Fig.13. At 10:19:05 AM, the velocity of the own
ship violated individual NLVO in both M1 and M2,
see Fig.14. The detection results of the whole
encounter process are shown in Appendix
Ⅱ.
Figure 13. Illustration of NLVO with M1 and M2 at
“10:18:15”-case 2
Figure 14. Illustration of NLVO with M1 and M2 at
“10:19:05”-case 2
The difference can be explained by the difference
in criteria choices. From the comparison, we found
901
out that M2 was more sensitive to the course of target
ships. The NLVO area located longitudinally on the
target ship covers a larger area than in the lateral
direction. In a result, any change in the course angle
of the target ship will change the shape and area of
the NLVO region and may make a difference to the
detection results of the multi-ship encounter.
Compared with the criteria of the circular region,
integrating the elliptic region into TD‐NLVO is more
reasonable in this area, because it has the preference
of coverage on different directions around the ship
based on various aspects, e.g., the experience of the
officers on watch, ship maneuverability, etc. What’s
more, compared with the detection model for pair-
ship encounter in[11], we can find that the criteria of
the algorithm were adjusted according to the different
waterway and traffic situations. If more ships are
involved in an encounter, a larger ship domain for the
safe meeting may be required.
6 CONCLUSIONS
This paper proposes a modified non-linear velocity
obstacle algorithm for the multi-ship encounter by
integrating an elliptical ship domain. The algorithm
includes a case study that shows its effectiveness in
comparison to the existing VO algorithm. The
integration of an elliptical ship domain helps in
identifying collision candidates that consider the
course and length of ships, which is shown in the
discussion. The maritime transport system is the
backbone of the global transportation system. This
method presents a fresh perspective for considering
multi-ship encounter cases. Port authorities and
maritime safety administration can utilize this method
to appreciate the collision risk in the region and
facilitate the decision-making process of safety
measures. However, the choice of parameters, such as
the ship domain parameters, can have a significant
impact on the detection results of collision candidates.
Further efforts can be dedicated to determining the
criteria considering the traffic characteristics of the
region, such as the distribution of ship length. The
parameter of the ship domain used in different waters
needs to be adjusted according to the traffic data.
Another aspect that requires further work is how to
improve data quality to avoid underestimation of the
results, as data can have missing information about
ship length. In future studies, the method will be used
to quantify the collision risk of ships in encounters to
further quantify the risk and ensure the relative safety
of all ships within their domain.
ACKNOWLEDGMENT
This work is supported by the National Natural Science
Foundation of China under grants 52101402, 52271367, and
52271364. The historical AIS data is provided by the Wuhan
University of Technology.
APPENDIX I
DETECTION RESULTS OF BOTH MODELS
________________________________________________
Detection Detection Violation Violation
Case Time Numbers in M1 Numbers in M2
________________________________________________
Case one 10:15:05 1 1
10:15:15 0 1
10:15:25 0 1
10:15:35 0 1
10:15:45 0 1
10:15:55 0 1
10:16:05 0 1
10:16:15 0 1
10:16:25 0 1
10:16:35 0 1
10:16:45 0 1
10:16:55 0 1
10:17:05 0 1
10:17:15 0 1
10:17:25 0 1
10:17:35 0 1
10:17:45 0 0
10:17:55 0 0
10:18:05 0 0
10:18:15 0 0
10:18:25 0 0
10:18:35 0 0
10:18:45 0 0
10:18:55 0 0
10:19:05 0 0
10:19:15 0 0
Total 1 16
________________________________________________
“Violation numbers” means the numbers of violation of
Individual NLVO.
APPENDIX II
DETECTION RESULTS OF BOTH MODELS
________________________________________________
Detection Detection Violation Violation
Case Time Numbers in M1 Numbers in M2
________________________________________________
Case two 10:16:35 0 0
10:16:45 0 0
10:16:55 0 0
10:17:05 0 0
10:17:15 0 0
10:17:25 0 0
10:17:35 0 0
10:17:45 0 0
10:17:55 0 0
10:18:05 0 0
10:18:15 0 0
10:18:25 0 0
10:18:35 0 1
10:18:45 1 1
10:18:55 1 1
10:19:05 1 1
10:19:15 1 1
10:19:25 1 1
10:19:35 1 1
10:19:45 1 1
10:19:55 1 1
10:20:05 1 1
10:20:15 1 1
10:20:25 1 1
10:20:35 1 1
10:20:45 1 1
10:20:55 1 1
10:21:05 1 1
10:21:15 1 1
10:21:25 1 1
10:21:35 1 1
10:21:45 1 1
10:21:55 1 1
902
10:22:05 1 1
10:22:15 0 0
10:22:25 0 0
10:22:35 0 0
10:22:45 0 0
10:22:55 0 0
Total 21 22
________________________________________________
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