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THE JOURNAL OF NAVIGATION, Page 1 of 21. c
The Royal Institute of Navigation 2018
doi:10.1017/S0373463318000796
COLREGS-Constrained Real-time Path
Planning for Autonomous Ships Using
Modified Artificial Potential Fields
Hongguang Lyu1,2 and Yong Yin2
1(Collaborative Innovation Research Institute of Autonomous Ship @ Dalian Maritime
University, Dalian Maritime University, Dalian 116026, China)
2(Navigation College, Dalian Maritime University, Dalian 116026, China)
(E-mail: lhg@dlmu.edu.cn)
This paper presents a real-time and deterministic path planning method for autonomous ships
or Unmanned Surface Vehicles (USV) in complex and dynamic navigation environments. A
modified Artificial Potential Field (APF), which contains a new modified repulsion potential
field function and the corresponding virtual forces, is developed to address the issue of Colli-
sion Avoidance (CA) with dynamic targets and static obstacles, including emergency situations.
Appropriate functional and safety requirements are added in the corresponding virtual forces
to ensure International Regulations for Preventing Collisions at Sea (COLREGS)-constrained
behaviour for the own ship’s CA actions. Simulations show that the method is fast, effective
and deterministic for path planning in complex situations with multiple moving target ships and
stationary obstacles and can account for the unpredictable strategies of other ships. The authors
believe that automatic navigation systems operated without human interaction could benefit
from the development of path planning algorithms.
KEYWORDS
1. Autonomous. 2. COLREGs. 3. Route Planning. 4. Ship Collision.
Submitted: 4 March 2018. Accepted: 15 September 2018.
1. INTRODUCTION. In June 2017, during its 98th session (MSC98), the Maritime
Safety Committee (MSC) of the International Maritime Organization (IMO) (2017)
approved a new work plan for Maritime Autonomous Surface Ships (MASS). In recent
years, MASS has become a popular research topic.
A key problem in researching MASS is planning a safe path at sea, in which automatic
Collision Avoidance (CA) of all moving and stationary obstacles (for example, other ships,
shallow waters, reefs, etc) is of vital importance. The complexities of CA for MASS lie in
the following aspects: necessity to comply with the International Regulations for Prevent-
ing Collisions at Sea (COLREGS) (IMO, 1972), complex situations with multiple vessels,
the existence of uncoordinated or uncertain CA actions for other Target Ships (TSs), and the
real-time CA decision making of the Own Ship (OS) affected by a changing environment.
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2HONGGUANG LYU AND YONG YIN
Early studies mainly addressed path planning or CA without considering COLREGS.
Some examples include an extension of an Evolutionary Planner/Navigator (EP/N) named
θEP/N++ that computes a safe and optimum path of a ship in given static and dynamic
environments (Smierzchalski and Michalewicz, 2000); a Line of Sight Counteraction Nav-
igation (LOSCAN) algorithm for CA in a two-ship encounter (Wilson et al., 2003); and
a genetic algorithm to compute a safe navigation path in an open sea (Zeng, 2003).
These methods are inherently unavailable for application in human-present ocean-going
CA applications where COLREGS prevail.
Many path planning techniques with consideration of COLREGS have been developed
in recent years, such as Artificial Potential Fields (APF) (Lee et al., 2004; Xue et al.,
2011; Lyu and Yin, 2017), a Velocity Obstacle (VO) (Kuwata et al., 2014), a Trajectory-
Based Algorithm (TBA) (Lazarowska, 2017), the Ant Colony Optimisation (ACO) method
(Lazarowska, 2015), Evolutionary Algorithms (EA) (Tam and Bucknall, 2010), Fuzzy
Logic (FL) (Brcko and Svetak, 2013), and neural networks and hybrid intelligent systems
(Perera et al., 2012). However, some studies claim to be “COLREGS compliant” while
allowing explicit violations of the rules in certain scenarios, such as making frequent and
minor course alterations that are difficult to detect by other ships (Tam and Bucknall, 2010)
or turning to port when a collision risk exists (Perera et al., 2012; Shah et al., 2014) even in a
head-on situation (Tam and Bucknall, 2010). Furthermore, some studies assume that all TSs
maintain their course and speed (Lee et al., 2004; Perera et al., 2012; Lazarowska, 2015;
Szlapczynski, 2015; Lazarowska, 2017; Lyu and Yin, 2017) or take COLREGS-compliant
manoeuvres (Tam and Bucknall, 2010) during problem solving. Once TSs perform changes
of speed and/or course, even with a non-protocol action, the original solution based on these
algorithms might be invalid. Again, some of them only simulated a single ship-to-ship
encounter situation (Lee et al., 2004; Brcko and Svetak, 2013; Szlapczynski, 2015).
A Cooperative Path Planning (CPP) algorithm is a new contribution to path planning for
MASS in multi-contact encounters. The main idea of this method is to compute a collision-
free path for all ships involved (Xue et al., 2011; Szlapczynski, 2011; Tam and Bucknall,
2013). This method supposes that all ships comply with COLREGS. Therefore, CA actions
of these ships are predictable. In fact, encountered ships at sea may include both unmanned
ships and human-operated ships, while non-COLREGS manoeuvres may exist and be hard
to predict. Adopting a CPP strategy for all encountered ships relies on a means of wireless
communication or coordination between them, such as the Automatic Identification System
(AIS) (Woerner and Novitzky, 2017), Vessel Traffic Services (VTS) (Szlapczynski, 2011)
or a future more advanced technique.
In addition, for planning the path approaches of ships, one noticeable limitation is the
computational time (at least a few seconds, for example, the CPP method and TBA, but
over 200 s for an EA-based approach), which is significantly increased with the number
of obstacles, making it difficult to achieve real-time navigation. One outstanding method
based on simplified Model Predictive Control (MPC) can be used to manage complex
scenarios with multiple dynamic obstacles with random motion (Johansen et al., 2016).
Unfortunately, this method does not consider static obstacles and might lead to higher
time consumption. However, an expected feasible path for the OS considering real nav-
igational constraints needs to be updated immediately due to the highly dynamic obstacles
and environmental disturbances.
The APF method in particular (Lee et al., 2004; Xue et al., 2011) has the ability to
handle static and/or moving obstacles and considers some key rules from the COLREGS.
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PATH PLANNING FOR AUTONOMOUS SHIPS 3
Additionally, an artificial repulsive potential was included in a Non-linear Model Predictive
Control (NMPC) method for controlling multiple Unmanned Surface Vehicles (USVs) in
arbitrary formations (Fahimi, 2007). However, this is a non-COLREGS potential field and
only used for static obstacle avoidance.
Lee’s method is capable of guiding a marine vehicle away from danger and return-
ing to the pre-determined path by combining two heading angles in ‘track-keeping’ mode
and ‘collision avoidance’ mode. However, the system incorporates over 200 fuzzy rules to
determine the heading angle, which presents a challenge in computation time. Additionally,
course alteration is fixed to a right-hand turn regardless of the obstacle’s mobility, and only
ship-to-ship encounter situations are presented in Lee’s paper.
Xue’s algorithm can address a complicated multi-ship encounter situation in the pres-
ence of some static obstacles. However, local minima limitations do exist in this method.
The OS is obstructed by the obstacle and cannot reach the destination when the OS, the des-
tination, and the obstacle lie on the same line with the obstacle in the middle. Additionally,
this method assumes that all TSs exhibit COLREGS-compliant behaviour. No emergency
scenarios or unforeseen circumstances are considered.
Table 1 shows a comparison of some typical approaches mentioned above in chrono-
logical order. These methods were evaluated according to the fulfilment of nine require-
ments/features: COLREGS compliance (a degree of fulfilment is specified as no, low,
medium, or high), consideration of static and dynamic obstacles (quantities and shapes),
motion features of the TSs (change of speed or/and course), consideration of dynamic
properties, speed change in CA actions, and emergency CA actions for the OS, as well
as computational time and repeatability of a solution. Some requirements are evaluated
based upon fulfilment (yes in the table) or failure of fulfilment (no in the table) of a defined
criterion. The computational time has its own scale of evaluation, where the time can be
very low (milliseconds), low (seconds), medium (several or tens of seconds) or high (hun-
dreds of seconds). In the last column this paper’s method is evaluated for comparison with
existing approaches.
The rest of the paper is organised as follows: Section 2describes the real-time path
planning problem to be studied, including some assumptions, environment representations
and requirements under COLREGS. Section 3presents the path planning strategy based on
a modified APF. Section 4includes a model of the ship’s motion. In Section 5, the results
of the simulation are given. The last section summarises the achievements and limitations.
2. PATH PLANNING PROBLEM. This paper is concerned with the process of path
planning for power-driven vessels in a complex environment modelled by static constraints
and dynamic TSs.
2.1. Environment. The environment data can be collected by navigational aids such
as a shipborne AIS, radar and Differential Global Positioning System (DGPS). Additional
information can be provided from an Electronic Chart Display and Information System
(ECDIS). The data input should include information about static obstacles and the real-time
position, speed, and course of moving obstacles (TSs).
The collision shapes of static obstacles are simplified as circles and modelled as TSs
with zero speed. This is not as precise as considering irregular shapes (Smierzchalski and
Michalewicz, 2000). However, the circle collision shapes are simple and easy to expand
into other shapes. For example, Xue et al. (2011) constructed coastlines and polygon land
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4HONGGUANG LYU AND YONG YIN
Table 1. Comparison between different selected path planning and CA methods and the author’s method for ships.
Methods θEP/N++ LOSCAN EA FL CPP VO ACO MPC TBA APF
Authors Smierzchalski and Wilson et al., Tam and Perera et al., Tam and Kuwata et al., Lazarowska, Johansen Lazarowska, Lee et al., Fahimi, Xue et al., Lyu and Yin
year Michalewicz, 2000 2003 Bucknall, 2010 2012 Bucknall, 2013 2014 2015 et al., 2016 2017 2004 2007 2011 (this paper)
COLREGS
compliance
no no low med①med high high high high med no med high
static obstacles (TSs) irregular no a circle no no a buoy polygons no a convex a circle two coastlines, a multiple
shapes polygon circles polygon land circles
dynamic obstacles
(TSs)
multiple 1 4 3 1∼444∼7 multiple 3∼51no1∼5 multiple
change of
speed/course
(TSs)
speed no course③no yes③random no yes no no n/a course③random
motion course
dynamic properties
(OS)
no no yes no yes yes partly④yes partly④yes yes yes yes
speed change (OS) yes no no⑤yes no⑤yes no yes no no n/a no no
emergency CA
actions (OS)
no no no no no no no yes no no n/a no yes
computational time med②? high ? low low med high? low high low ? very low
repeatability no yes no yes yes yes yes yes yes yes ? ? yes
Notes: symbol “?” means that this subject is not specified or presented clearly in corresponding papers, “med” is the abbreviation of “medium”;
①the magnitude in the course change is standardised to one value of 30◦;
②the running time is approximately linear with respect to the number of obstacles;
③a predictable change of speed or/and course as required by the COLREGS;
④dynamic properties of an OS are taken into account by considering the time needed for a ship to execute the calculated manoeuvre – course change;
⑤the turning manoeuvre directly affects the speed of the OS, not the speed change for CA action.
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PATH PLANNING FOR AUTONOMOUS SHIPS 5
Table 2. Encounters and Actions for OS and TSs.
Rules Encounter OS TSs Action by give-way vessel
R14 Head-on Give-way Give-way Alter course to starboard
R15/16/ 17 Crossing: A Give-way Stand-on Avoid crossing ahead of the other ship if
the circumstances of the case admitCrossing: B Stand-on Give-way
R13/16/17 Overtaking Stand-on (if slower) Give-way Keep out of the way of the vessel being
overtaken, and not relieved of the duty
until finally past and clear
Overtaking Give way (if faster) Stand-on
using discrete obstacle points, such as circle shapes, and convex and concave polygons in
Lazarowska (2017) are replaceable by the corresponding outer circles of these polygons.
Additionally, the collision shapes of moving TSs are approximated as circles, as adopted in
some recent studies (Tam and Bucknall, 2013; Johansen et al., 2016). Other domain shapes,
such as hexagons and ellipses, which are used in Lazarowska’s (2017) CA algorithm by
avoiding intersections with the boundaries of these shapes, are not suitable for use in the
APF method at the moment.
2.2. Requirements. The path planning for ships has to fulfil some key requirements
as follows:
•ability to avoid all the moving and stationary obstacles,
•COLREGS-compliant behaviours and the practice of good seamanship,
•nearly real-time operation for CA or path planning online,
•consideration of course changes for TSs, even with their uncoordinated behaviours,
•consideration of dynamic properties of the OS in a solved solution.
As compliance with COLREGS encompasses many different aspects, the focus in this
study is about general vessel encounters, including conduct of vessels in sight of one
another and operating under rules 13–17 for power-driven vessels. Categories of scope
for COLREGS can be found in Woerner et al. (2016) and Woerner (2016). Here, three
primary encounter situations are considered: crossing, head-on, and overtaking. It is also
worth considering Rule 8, which addresses general collision avoidance.
The CA action of the two ships potentially involved in a collision is defined as shown
in Table 2, with the corresponding five regions illustrated in Figure 1. These regions are
divided according to the relative bearings of TSs, such as 5◦, 112·5◦, etc. It should be
specified that a head-on situation is somewhat vaguely defined by Rule 14 as “. . . meeting
on reciprocal or nearly reciprocal courses...”, so 5
◦of the OS/TS heading line on either
side is configured by the author for a tolerance of head-on, although it is not specified in
the COLREGS. As required by Rule 8, the give-way vessel shall, as far as possible, take
early and substantial action to keep well clear.
Many open-ocean scenarios involve speed changes (sometimes exclusively). This
includes merchant ships slowing to a near stop rather than deviating course. However,
such actions only occur when appropriate preventive action is not taken in good time.
Under normal circumstances, mariners prefer to change heading rather than speed to
avoid a collision. Therefore, course change is assumed as the primary strategy adopted
for avoiding a collision in this article, which is suitable for the manoeuvres of the OS and
all TSs.
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6HONGGUANG LYU AND YONG YIN
Figure 1. An illustration of the encounter situations.
In previous studies, the path planning solution is fixed and determined in advance based
on ideal assumptions, such as all TSs maintaining their course and speed, or acting under
the COLREGS. However, in reality, TSs may take unexpected actions. Even as the stand-
on vessel, the OS has to take actions to avoid a collision due to absence or negligence
of actions from a give-way TS at a short range. The reactive avoidance routine of the
OS in an emergency collision situation should be considered to ensure safety (Campbell
et al., 2012). This is one major aspect of concern in this study that differs from previous
works.
3. THE PROPOSED STRATEGY.
3.1. Collision risk assessment. The collision risk is assessed by two factors:
•Checking Criterion of Collision Risk (CC), which is used to determine if there is a
collision risk with any TS when the current course for the OS is maintained;
•Checking Range of Collision Risk (CR), which is the distance the OS begins to
scrutinise the risk of collision (Xue et al., 2011).
The position and velocity of the OS and TS at one time step are denoted by pos,vos ,pts
and vts, respectively (see Figure 2). The safe boundary of the TS is expanded according to
the domain radius of the OS and TS (that is, Ros and Rts, respectively) and the allowable
safe distance (dsafe) is determined by the sea room, position sensor errors, and relative speed
of the OS and TS in one time step. Therefore, the expanded circle’s radius can be written
as:
dm=Ros +dsafe +Rts (1)
CR = dm+ρo(2)
where ρois the influence range of the TS set by operators and is larger under conditions of
low visibility, open waters, etc. The algorithm ignores obstacles outside the circle with a
radius of CR. A large CR means an early CA action for the OS.
Additionally, in Figure 2, two tangent lines are drawn from the point pos to the expanded
circle with a radius of dm.θmis defined as the angle between any tangent line and the
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PATH PLANNING FOR AUTONOMOUS SHIPS 7
Figure 2. The vectors and CA strategy for ship-to-ship encounters.
relative position vector pot (i.e., pts −pos), as in Equation (3).
θm=arctandm
ρ2(pos,pts )−d2
m(3)
θis the angle between pot and the relative speed vector vto (that is, vos −vts). A risk of
collision therefore exists when the extension line of vto crosses the circle with radius dm
around the TS. Therefore, the CC can be written as follows:
θ<θ
m(4)
3.2. Decision making with a modified APF. Original path planning to the goal is
achieved through an attractive potential field. The attractive potential Uatt(p) is defined
as a function of the relative distance between the OS and the goal:
Uatt(p)= 1
2ερ 2(pos,pg)(5)
where εis a positive scaling factor, and ρ(pos,pg) is the range between the OS and the goal.
The virtual attractive force is defined as the negative gradient of the attractive potential
in terms of position:
Fatt(p)=−∇ [Uatt(p)] = εp(p05,pg)−→
nog (6)
where −→
nog is a unit vector pointing to the goal position from the OS position.
If no risk is detected by the collision risk assessment module, the OS would proceed
to the goal under the virtual attractive force. Otherwise, the CA manoeuvre subroutine is
activated and computed through the repulsive potential field. Here, four subdivided zones
around the OS (see Figure 3) are defined to determine the corresponding virtual repul-
sive potential. The distances between the OS and any TSs are indicated by d=ρ(pos,pts ).
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8HONGGUANG LYU AND YONG YIN
Figure 3. Subdivided zones around the OS.
The τis a small radius of an artificial safety margin of the OS, which guarantees that at the
surface of the obstacles, the repulsive potential is sufficiently large but bounded (Ge and
Cui, 2002).
Based upon the above, a new subdivided repulsive potential field function is proposed,
which considers the CA for stationary and moving TSs giving sufficient sea-room, and
the emergency CA at close range. It is noted that there are some zones where the resultant
repulsive potential field is weakened or zero, even in the vicinity of an obstacle for the tradi-
tional APF model. A close-range emergency repulsion potential field function is included,
which can produce a larger repulsion potential field than usual so that the OS could take an
appropriate course alteration action immediately to keep well clear of all TSs. In addition
to the conditions of no collision risk, sometimes the repulsive potential is also not defined,
since there is no feasible solution for avoiding a collision with TSs in the case where TSs
intentionally move towards the OS. Finally, the modified repulsion potential function is
shown in Equation (7):
Urep (p,v)
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ηdRts(eθm−θ−1) 1
d−dm
−1
ρo2
d2
g,if vts =0, dm<d≤CR and θ<θ
m
1
2ηsRts 1
d−τ−1
ρo2
d2
g,if vts =0,d
m<d≤CR and θ<θ
m
ηeRts 1
d−τ−1
ρ22
+(Vto cos θ)2d2
g,if d ≤dm
not defined, others
(7)
where ηdand ηsare the positive scaling factors for the dynamic TSs and static obstacles,
respectively, at long distance, ηeis the positive scaling factor for emergency CA action for
the close-range obstacles, and dg=ρ(pos,pg).
The new corresponding repulsive force is defined as the negative gradient of the
repulsive potential in terms of the positions and velocities:
Frep (p,v)=−∇Urep(p,v)
=−∇pUrep(p,v)−∇
vUrep(p,v)
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PATH PLANNING FOR AUTONOMOUS SHIPS 9
=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Frd1+Frd2+Frd3,if vts =0, dm<d≤CR and θ<θ
m
Frs1+Frs3,if vts =0,d
m<d≤CR and θ<θ
m
Fre1+Fre2+Fre3,if d ≤dm
not defined, others
(8)
where:
Frd1=−ηdRtsd2
g1
d−dm
−1
ρ0eθm−θdm
dd2−d2
m
+sin θ
vto+eθm−θ−1
(d−dm)2−Fto0−→
not
(9)
Fto0=1
d−dm
−1
ρ0dm
dd2−d2
m
+sin θm
vto(10)
Frd2=±ηdRtsd2
g
1
d−dm
−1
ρ0eθm−θ1
pot+cos θ
vto+vot⊥(eθm−θ−1)
d(d−dm)2−Fto⊥−→
not⊥
(11)
Fto⊥=1
d−dm
−1
ρ0 1
pot+cos θm
vto(12)
Frd3=ηdRtsdg1
d−dm
−1
ρ0(eθm−θ−1)−→
nog (13)
Frs1=−ηsRts 1
d−τ−1
ρ0d2
g
d2−→
not (14)
Frs3=ηsRtsdg1
d−τ−1
ρ02−→
nog (15)
Fre1=−2ηeRts 1
d−τ−1
dmd2
g
(d−τ)2
−→
not (16)
Fre2=2ηeRts
dg
dVto2(cos θsin θ)−→
not⊥(17)
Fre3=2ηeRtsdg1
d−τ−1
dm2
+Vto2cos2θ−→
nog (18)
In the NZ, the repulsive force determines that the passing side is to port or starboard for
static obstacles yet only to starboard for dynamic TSs in order to meet the requirements of
COLREGS. Figure 4 shows an illustration of a new repulsive force for a dynamic TS, so
the total repulsive force Frd drives the OS to starboard.
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10 HONGGUANG LYU AND YONG YIN
Figure 4. Modified repulsive force for a dynamic TS.
For an emergency CA action, that is, TSs in the EZ, the forces of turning left or turning
right are chosen according to the minimum time to avoid collision. Frd1,Frs1and Fre1are
related to the relative positions of the OS and TSs, and they will keep the OS away from the
TS in the corresponding conditions. −→
not is the unit vector pointing to any TS from the OS,
and −→
not⊥is a unit vector that is perpendicular to −→
not (see Figure 2). Fre2will automatically
make the OS avoid danger from the right or left side of Pot depending on which side of the
Pot line the vector Vto is in. Frd2always makes the OS alter course to starboard at a range
of CR based on seafarers’ practice, as the appropriate passing side for each encounter is
determined by the COLREGS. Frd3,Frs3and Fre3force the OS to move towards the goal.
Based on the calculation of the attractive and repulsive forces, the total virtual force is
obtained by:
Ftotal =Fatt +Frep (19)
As there are multiple TSs, the repulsive force is given by:
Frep =
n
s=1
Freps(20)
where Frepsis the repulsive force generated by the s-th TS, and n is the number of encoun-
tered TSs. Therefore, the OS will take CA actions under the total resultant Ftotal for different
conditions and reach the goal, although the goal is near a stationary obstacle. A flow chart
of the modified APF method is given in Figure 5.
4. MODEL OF THE SHIP’S MOTION. A model of the ship’s motion is simulated on
the horizontal plane. All TSs can be set to change their course at any moment. Moreover,
the process of CA at sea is described with the use of a kinematic model of ship motion,
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PATH PLANNING FOR AUTONOMOUS SHIPS 11
Figure 5. Flow chart of the modified APF method.
where state variables and control values are represented in the following forms:
Pos =[pij ]m×2
˙pos
i1=Vos ×cos ψos(t)
˙pos
i2=Vos ×sin ψos(t)
(21)
Pts =[pts
ij ]k×2
Ptsi =⎡
⎢
⎣
pts
[n(i−1)+1]1 pts
[n(i−1)+1]2
.
.
..
.
.
pts
(n×i)1 pts
(n×i)2
⎤
⎥
⎦
Vts =[vij ]n×2
˙
Ptsi =Vts
(22)
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12 HONGGUANG LYU AND YONG YIN
where i=1,2,..., m is the i-th time step, k is the product of m and n, Pos and Pts are
coordinates of the OS and TS position, respectively, and Vos and Vts are velocities of the
OS and TS, respectively.
The course changes for TSs can be activated at some designated time step, which is
calculated by the following formulae:
a= sgn(rand(1) −0·5) ×ψts (23)
A=cos asin a
−sin acos a(24)
Vts−new =Vts ×A(25)
where ψts is the maximum value of course alteration of the TS at one time step, αis used
to determine the random magnitude and sign of the course alteration, Ais the transformation
matrix for changes of course and Vts−new is the new course of the TS at the designated time
step. The CA action of the TS is assumed to maintain course and speed (it does not give
way when it should), or it alters its course into a worse situation. Certainly, a coordinated
CA action can also be chosen in simulations.
The OS maximum turning angle to port or starboard at one time step is [-maxturn,
maxturn]. To approach this issue, let ψfbe the angle of the total force Ftotal, and ψos be the
heading of OS at the (i-1)-th time step. Then, the preliminary steering angle ψpis given
by:
ψp=ψf−ψos (26)
The final steering angle ψos (see Figure 5) can be determined by:
ψp=ψp−2πif ψp>π
ψp+2πif ψp<−π(27)
ψos =min(maxturn,ψp)ifψp≥0
max(−maxturn,ψp)ifψp<0(28)
The desired heading of the OS ψos at the i−th time step is given by:
ψos(i)=ψos (i−1) + ψos (29)
The dynamic properties of the OS are taken into account by considering the time needed
for a ship to execute the calculated manoeuvre, that is, course change (Lazarowska, 2017).
The course change should be large enough in a shorter period of time to ensure the avoid-
ance manoeuvre is obvious to other parties. Therefore, the configuration of the parameters
of maxturn and the time step are used to reflect the dynamic properties of the OS. This
mainly depends on the available rudder angle (≤35◦), the speed vos, the loading condition
and the delay of turning for a specific OS. For some large vessels, if 5◦is the maxturn
and 15 s is the time step, the total turning angle for 2 minutes will reach 40◦(satisfying
Rule 8b). However, a slightly bigger maxturn with a shorter time step could be selected for
more manoeuvrable vessels, which is also compliant with the “readily apparent manoeu-
vre” clause. It is therefore desirable to achieve a smooth path by incorporating the dynamic
properties, including the turning abilities of the OS.
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PATH PLANNING FOR AUTONOMOUS SHIPS 13
(a) (b) (c)
(d) (e) (f)
Figure 6. Single-TS Overtaking encounter simulation.
Meanwhile, this path contains the desired heading at any given point in near real-time.
This is convenient for modelling the tracks and designing an automatic pilot. To control
the course of the ships, slide mode controllers and a Proportion Integration Differentia-
tion (PID) control system may be used. Complex modelling for track keeping and course
controlling are beyond the scope of this paper.
5. RESULTS AND ANALYSIS. Extensive simulation cases were implemented using
MATLAB software. The simulations considered COLREGS compliance and a wide range
of cases, from a single TS avoidance to multi-ship avoidance, and from single predictable
TS trajectories to complex and random TS trajectories, as well as stationary obstacle
avoidance.
5.1. Experimental conditions and explanations. The calculations were conducted
using a PC with an Intel Core i5 3230 m 2·6 GHz processor, 6 GB RAM, and a 64-bit
Windows 7 Professional operating system. The following values for the parameters were
used in the presented simulation tests: ε= 3,000, ηd= 2,000, ηs= 300,000, ηe= 2,000,
τ=0·3 nautical mile (nm), Ros =0·5nm, dsafe =0·5∼1 nm (1 nm is selected for open
water), ρo=3∼5nm,maxturn =5
◦, and time step = 15 seconds.
The simulation results are illustrated in multiple figures, representing snapshots of sit-
uations. A two-dimensional Cartesian coordinate system presents the distance in nautical
miles (nm); the vertical axis in the positive direction shows North 000◦, and the horizontal
axis in the positive direction is 090◦. The following symbols and colour codes are applied:
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14 HONGGUANG LYU AND YONG YIN
(a) (b) (c)
(d) (e) (f)
Figure 7. Single-TS Head-on encounter simulation.
•The black curve is the OS trajectory, and its starting point and goal point are indi-
cated by the red square and blue triangle, respectively. In addition, the OS heading
data at every time step is plotted, to easily find the course alteration points.
•The paths of multiple TSs, including stationary obstacles, are identified by corre-
sponding TS numbers and different colours.
•The relative bearing of each TS and the range (scope 0∼6 nm) between the OS and
any TS at each time step is chosen to display with various colours.
•All the paths are marked at regular intervals, for example, 60 time steps.
5.2. Collision avoidance with one TS. Overtaking, Head-on, and Crossing encounters
are presented in this paper, see Figures 6–8(a),8(b) and 8(c), and the reactive avoidance
actions of the OS replying to the uncoordinated behaviours of the TS are illustrated in
Figures 6–8(d),8(e) and 8(f). The initial configuration of these simulations is described
in Table 3, which contains the start positions (pts), velocities (vts), and radii (Rts )ofthe
TSs. For example, (−8, 8) for vts means that the velocities on the x-axis negative direction
and y-axis positive direction are 8 kn and 8 kn, respectively. The OS’s start point (0, 0)
and destination (10, 10), as well as initial course 045◦and speed 12 kn are designated. The
simulation results, including the course changes and distances from the TS at the action
time for the OS, the final Distance of Closest Point of Approach (DCPA), and the total run
time of each simulation are also described in Table 3.
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PATH PLANNING FOR AUTONOMOUS SHIPS 15
(a) (b) (c)
(d) (e) (f)
Figure 8. Single-TS Crossing encounter simulation.
Table 3. Initial inputs and simulation results for three typical encounters.
vts Rts OS turn Action range DCPA Run time
Situation pts [kn] [nm] [◦] to TS [nm] [nm] [ms]
Overtaking (5, 2·6) (0, 2) 0·3 33 right 4·81·8 119
Head-on (7·3, 7·3) (−8, −8) 0·3 33 right >61·9 106
Crossing (9, 0) (−8, 8) 0·3 40 right >61·9 113
Overtaking Same initial inputs as above, only
lists the reactive CA actions and
total run time
35 left 3·41·0 100
Head-on 30 right 3 1·2 115
Crossing 50 right 3 2·0 130
It can be seen clearly that the OS’s behaviour complies with COLREGS Rules 13–17,
and the computation time is very low (no more than 130 milliseconds) to satisfy the real-
time requirement for CA or path planning online.
5.3. Collision avoidance with multiple obstacles. Two cases were chosen for this
scenario.
Case 1 presents an encounter between the OS and four moving TSs and two static
obstacles. The initial input for OS is no change. The initial configurations for the TSs are
described in Table 4 and an uncoordinated course alteration (30◦) of TSs are considered.
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16 HONGGUANG LYU AND YONG YIN
Table 4. The initial inputs for TSs at case 1.
No. pts vts [kn] Rts [nm]
1(7·8, 2·2) (0, 0) 0·4
2(6·8, 4·9) (0, 0) 0·2
3(7·0, 0) (−8, 8) 0·3
4(7·5, 7) (−8, −8) 0·2
5 (6, 8) (0, −3) 0·2
6 (4, 8) (2·8, −1·6) 0·15
Figures 9(a)–(h) indicate the temporary positions at the selected step and the trajectory his-
tory of all TSs. Figures 9(h1)–(h3) illustrate the distance between the OS and any TS, the
relative bearings of the TSs, and the heading of the OS at each time step.
In Figure 9(a), initially the OS moves towards the goal along the nominal path. It can
be observed that a TS4 head-on situation and a TS3 crossing situation from starboard are
developing. However, the distance between the OS and TS3 and even TS4 are so large
(more than 6 nm) that CA algorithms are considered to not apply according to the threshold
CR. When the distance between them reaches the CR (at step = 12 in Figure 9(h)) first for
TS3, the OS makes a course change 30◦to starboard to avoid the two risks.
In Figure 9(b), TS3 and TS4 change their courses to port to intercept the OS trajectory,
which is deemed as an uncoordinated CA action. Therefore, the OS makes a further course
change of 35◦to starboard (at step = 60) to avoid a collision with TS3, although it will
lead to a larger deviation from the planned trajectory. This shows that the method is able
to successfully handle and quickly react to the presence of obstacles with an unpredictable
course change.
In Figure 9(c), after passing TS3 at safe range 1·4 nm, the OS returns to the planned
route. Note that TS4 is arriving from the port side after turning left, and the OS has the
right to stand on. This stand-on behaviour means that it takes the OS far away from the
planned trajectory. However, this is compliant with the COLREGS and is safe.
In Figure 9(d), after passing at a safe distance (1·2 nm according to dm) in front of TS4
(according to CC: θ<θ
m), the OS makes a sharp turn 51◦(at step = 115) to port to avoid
collision with the static obstacle TS1. Finally, TS1 is passed at 1·4 nm (see Figure 9(h1))
on the OS’s starboard side.
In Figure 9(e), TS2 is ahead of the OS. Meanwhile, TS5 is arriving from the port side
such that it is a give-way ship. In this scenario TS5 does not respect its responsibility to
give way and the OS makes a course change (approximately 31◦at step = 155) to starboard
to avoid collision with the TS5 and to avoid a hazardous situation if TS5 alters course to
starboard in compliance with its responsibility to keep clear according to COLREGS. In
addition to the stationary obstacle, TS2 can be passed at a safe distance (approximately
1·3nm,seeFigure 9(h1), blue line) on the OS’s port side.
In Figure 9(f), after clearing TS2 and TS5, the OS turns left 30◦(at step = 190) towards
the goal, taking into account the influence of TS6. Since TS6 is approaching from the port
side and the DCPA (1·7nm, see Figure 9(h1), brown line) is enough to pass clear, the
OS stands on and moves to the goal without any collision risk, until reaching the goal at
step = 315.
Case 2 contains a more complex situation in which 11 TSs have random changes
in course and five TSs are stationary obstacles, as shown in Figure 10. The initial
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PATH PLANNING FOR AUTONOMOUS SHIPS 17
(a) (b) (c)
(d)
(g) (h1)
(h2)
(h3)
(e) (f)
Figure 9. Graphical solution of case 1. Video files are available as Supplementary Material.
configuration is shown in Table 5. The OS’s start point is set to (1, 1) and the other
parameters remain the same.
In Figures 10(a) and 10(b) the OS alters course 50◦to starboard to pass abaft TS8,
meanwhile avoiding the close-quarters situation with TS1. In Figures 10(c) and 10(d),
after passing clear of TSs #12, 7, 3 and 8 at a safe distance of 1 nm, the OS makes a large
starboard turn of 90◦to manage the uncoordinated behaviours of TS2 and passes clear at
0·9nm. In Figures 10(e) and 10(f), the OS avoids the static obstacle of TS #16, and 11,
then reaches the goal, although TS11 is near the goal.
Simulations show that the OS can make a collision-free path with multiple TSs making
unpredictable course changes and manage the reactive CA problem. The computational
time is also very short (140∼200 milliseconds, that is, ms at 50 repetitions of calculations)
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18 HONGGUANG LYU AND YONG YIN
(a) (b) (c)
(d) (e) (f)
(g)
Figure 10. Graphical solution of case 2. Video files are available as Supplementary Material.
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PATH PLANNING FOR AUTONOMOUS SHIPS 19
(h1)
(h2)
Figure 10. Continued.
Table 5. The initial inputs for TSs of case 2.
No. Start Vts[kn] Rts [nm]
1 (3, 5) (0, −13) 0·15
2 (11, 6) (−5, −5) 0·1
3 (10, 3) (−7, −3) 0·2
4 (0, 5) (0, 6) 0·2
5(6,0·5) (0, 0) 0·2
6(8,0·5) (0, 0) 0·4
7 (2, 5) (11, 0) 0·12
8(3·5, 1·0) (2·5, 14) 0·1
9(5,8·5) (−8, −4) 0·3
10 (9, 7·5) (−8, −0·6) 0·15
11 (9·3, 9·3) (0, 0) 0·3
12 (6·0, 8·0) (0·5, −9·5) 0·15
13 (9, 5) (−6, 0) 0·2
14 (7, 8) (0, 0) 0·2
15 (10, 1) (−5·5, 5·5) 0·2
16 (9, 6) (0, 0) 0·5
and is invariant in almost every simulation. Further, it does not obviously increase with an
increasing number of TSs or their random motion. Meanwhile, the solution for the same
situation is also deterministic for every run of the calculations. As the algorithm is based
on a simple deterministic structure with regard to the field and gradient theory, it is algo-
rithmically complete in such a way that it will generate the exact solution (if it exists). This
is because no random variables are involved in the computation and the outputs from the
algorithm are entirely consistent and predictable for the same known conditions. Addition-
ally, there is no feasible solution in the simulations for some extreme situation where the
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20 HONGGUANG LYU AND YONG YIN
TS suddenly alters course to a worse condition and rushes at the OS at a critical close range.
In this scenario a collision may be inevitable even in a real situation due to the limitation
of the manoeuvrability and reaction time of the OS.
Duetothemaxturn limit, the heading plot includes some small fluctuations, that is, the
small course changes. A smooth heading line is re-plotted using one-dimensional MAT-
LAB median filtering (see the magenta solid line in Figures 9(h3) and 10(h2)). Compared
with the original heading data (blue dotted line), the new heading line preserves the sub-
stantial course changes and is smoother than before so that it is feasible for the ship. By
inputting the new heading data, the new planned path can also be obtained (see the magenta
line in Figures 9(g) and 10(g)), which almost coincides with the original planned path (the
black line) and does not influence the collision avoidance effect. Other small course changes
are eliminated by adjusting the parameter dsafe, with a larger dsafe ensuring that the course
changes are readily apparent.
6. CONCLUSIONS. A real-time path planning method based on a new modified APF
model is presented in this paper. Simulations illustrate that this method is well-suited for
automatic path planning and collision avoidance in a complex environment, where static
and dynamic obstacles occur. Additionally, it is compliant with Rules 13–17 of COLREGS
for power-driven vessels and shows the potential to address the extreme or emergency
multi-ship collision situations mentioned by Xue et al. (2011) and Perera et al. (2011)in
their proposed future work. The proposed method is robust in the uncertainty of TS strate-
gies, such as unpredictable changes of course in the solution finding process. In addition, a
smoother path is achieved by considering the dynamics of the OS.
The strength of the approach appears in the capability to plan a collision-free path in
a very short time (millisecond level) for the OS. Additionally, the computational time is
barely affected by the number of obstacles and the random course changes of the TSs.
The presented simulation results prove the advantages and effectiveness for real time
path planning using the proposed method. Due to the deterministic and acceptable solution
derived for the complex collision avoidance task, a very low and almost constant com-
putational time for every scenario, repetition and robustness for the uncertainty of other
ship’s behaviours, it can be applied to an on board anti-collision decision-making system
and promotes the automation level of a USV or an autonomous ship.
The method can be further refined by considering speed reduction behaviours and more
accurate ship dynamics, as well as the uncertainty of environmental disturbances and area-
based obstructions.
ACKNOWLEDEGMENTS
This work was financially supported by Traffic Youth Science and Technology Talent Project
(Grant No. 36260401), Science and Technology Program of Yunnan Communication Depart-
ment (Grant No. YunJiaoKe2013(A)01), Natural Science Foundation of Liaoning Province of
China (Grant No.201602081), and the Fundamental Research Funds for the Central Universities
[No.3132018306].
Supplementary material
The supplementary material for this article can be found at https://doi.org/10.1017/S0373463318
000796.
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PATH PLANNING FOR AUTONOMOUS SHIPS 21
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