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Novel Formulation for Line of Sight Guidance and Obstacle Avoidance
for Under-actuated Ships
NASSIM KHALED1, RAMI ALKHATIB2
1Mechanical Engineering Department, Prince Mohammad Bin Fahd University, nkhaled@pmu.edu.sa,
Al Khobar, Saudi Arabia
2Mechanical and Mechatronics Engineering Department, Rafik Hariri University,
khatibrh@rhu.edu.lb, Mechref, Damour, Lebanon
Abstract: - Automatic control of under-actuated ships is a challenging task due to the external factors and limited
actuators onboard a ship. It is even more so when the controller needs to seamlessly integrate with a guidance
system and obstacle avoidance for the purpose of autonimity. In this paper, line of sight guidance system for
marine surface vessels is augmented to include obstacle avoidance. The process of directing the ship movement
to avoid a stationary and moving obstacles is tackled by introducing an iterative mathematical formulation for
the circle of avoidance algorithm. Unlike learning based guidance system, the proposed formulation has an
explicit solution that is updated at each instant in time. Three simulations are conducted to assess the performance
of the overall guidance and avoidance system. The developed algorithm is validated through simulation results
of a 6-degree of freedom model of a ship. The simulation results prove the effectiveness of the developed
technique to converge the ship to the desired trajectory autonomously while avoiding obstacles along the path.
Key-Words: - Ship guidance system, trajectory planning, Line-of-Sight (LOS), Circle-of-Avoidance (COA) and
obstacle avoidance.
Received: January 15, 2021. Revised: February 25, 2021. Accepted: March 8, 2021. Published: March 16, 2021.
1 Introduction
Marine vehicle utilization is under high demand due
to their importance in transportation for both goods
and passengers, oceanography and research, fishing,
and defense. Their control systems are widely carried
out by researchers to enhance their functionality.
Developing heading and speed control frameworks is
considered the main step for a vessel to follow a
predefined trajectory with minimal position error.
The challenging problem in vessel’s path following
become more complicated in the presence of an
obstacle under the environmental uncertainties and
the nonlinear nature of the large dynamic model.
Nonetheless, collision avoidance and maneuvering
around an obstacle is a leading problematic fact that
obstructs the employment of completely autonomous
vehicles including ships.
1.1 Previous Work
Marine surface vessel maneuvering along a
predefined path impressively covered in the
literature. Signal data collected from the radar
combined by visual inspection allow safe ship
navigation. However, visual observations can lead to
misinterpretation of the actual scenario that may lead
to ship collision. Oceanic autonomous surface
vessels are divided into autonomous sailboats and
autonomous vessels based on the type of vessels. In
both cases, navigation and path planning form the
basis for marine transportation safety and oceanic
data collection [1]. Breivik et al. developed a
guidance algorithm based on estimating and
nonlinearly controlling the velocity vector that
converges to an anticipated geometrical track [2].
The performance of path following has improved
through combining model predictive control (MPC)
and line-of-sight (LOS). the MPC is linearized along
the LOS via quadratic programming [3]. In another
work, adaptive feedback control is combined with a
modified LOS guidance law to be less susceptible to
environmental perturbations [4]. Moreover, the
rudder angle is mainly used to address the path
following based on feedback dominance-nonlinear
controller. The simplification of the controller is done
through additional parameters used in Lyapunov
function [5]. Furthermore, an adaptive integral LOS
guidance law is introduced on another work to
compensate for uncertainties and input saturation.
The author used the backstepping technique
improved by a robust adaptive radial basis
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function neural network within the adaptive robust
control system [6]. B. Martinsen et al. employed the
policy search algorithm in deep reinforcement
learning to solve the straight-path following for
under-actuated marine vessels without any previous
knowledge of the controlled system. Results indicate
improved results compared to LOS guidance law [7].
Zhang et al. proposed adaptive obstacle avoidance
algorithm made up of two parts: local avoidance
module and adaptive learning module. The algorithm
is based on Sarsa on-policy reinforcement learning
which is tested in complicated marine environments
[8]. Moreover, deep reinforcement learning, through
deep deterministic policy gradient and historical data
provided by ship automatic identification system, is
used to generate intelligent path planning of
unmanned ships, particularly in unknown
environments [9]. Artificial potential field is used by
Petres et al. to create virtual gravitational field to help
the vessel in avoiding obstacles through complex
navigation environment [10]. On another work, a
way-point structure based on a vector field algorithm
yields good performance on the path following
controller. The parameter identification is based on
Lyapunov stability and support vector machine [11].
Shen et al. used deep Q-learning for automatic
collision avoidance of multiple ships in highly
complicated situations [12]. Pedersen et al.
employed the potential flower solver on which the
marine vessel follows a streamline and not the
gradient of potential. This method yield an acceptable
performance through regulating the target point and
vessel steering control through rudder. However,
stagnation line may cause obstacle collision and the
length of the path generated is not guaranteed to be
the shortest [13]. Relative value iterative gradient
algorithm implemented on autonomous ships and
simulated on Unity3D game engine software by
Yang et al. The algorithm performed well in
simulated navigation environment [14].
It is more challenging for an autonomous surface
vessel to avoid both static and dynamic obstacles
while following a specified path. A deterministic path
planning algorithm which is COLREGS [15], Coast
Guard Collision Regulations defined by the
International Maritime Organization, compliant
(collision prevention regulation especially in
presence of more than one vessels) has been
developed by Tam et al. to avoid obstacles [16].
Recently, new methods for ship obstacle avoidance
rules has been introduced. Namely, distance to the
point of approach and time at the point of approach
[17]. As those two methods can roughly estimate the
ship target, the degree of domain violation and the
time to domain violation are introduced by
Szlapczynski et al. [18]. Furthermore, Li et al.
realized that multi-objective optimization algorithm
(NSGA-II), that takes into consideration both
security and economic aspects, as a major parameter
in collision avoidance. NSGA-II and ship domain are
then combined together to calculate the ship collision
avoidance risk [19]. Xie et al. employed an improved
beetle antenna search algorithm to improve the
predictive collision for surface ships. Although this
method minimized economic cost and improved
safety, it is computationally expensive making it
adaptable for offline path planning rather than on
real-time application [20]. In another work, the
evidential reasoning theory is used to assess the
potential of collision risks in unmanned surface
vehicles. Maneuvering is implemented with the help
of optimal reciprocal collision avoidance algorithm
[21]. Such modern maneuvering quality has been
included in computer based and electronics ship
guidance system as in ECDIS [22]. Naeem et al.
developed a reactive path-planning algorithm for a
manned ship that helps in safely steering the craft.
The approach combines both LOS waypoint
guidance and manual biasing scheme [23]. Abdelaal
et al. have used the nonlinear disturbance observer in
the prediction model that approximate External
environmental forces. This has improved the
trajectory tracking controller robustness and
especially by embedding the collision avoidance as a
time-varying nonlinear constraint [24]. In another
work, energy planner is used for autonomous marine
vehicles rather than using temporal planning.
Accordingly, the vehicle dynamics and Monte Carlo
simulation are used to estimate the energy
consumption during attaining a certain mission. This
helps the vehicle to withstand faults and yield
advance reasoning without the operator assistance
[25].
1.2 Main Contribution
Marine surface vessels travel predictably along a
trajectory defined with relative to the surface of the
earth. The author has reported a unique vessel path
planning based on an integrated guidance and control
system despite external disturbances and modelling
imprecisions [26]. In another work, the propeller and
rudder were used to take actions by the controller on
a six degree of freedom ship model, which developed
for marine tasks [27]. The controller takes into
account takes into consideration the surge, sway and
yaw motions. A robust output feedback disturbance
rejection has been achieved in the presence of wind
and sea-current resistive loads, retardation forces,
wave excitation and nonlinear restoring forces [28].
Accordingly, calculations, under different
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environmental perturbations, are fed to the ship’s
autopilot to ensure a safe movement towards the
destination. However, the problem of collision
avoidance in ships is not well tackled. This paper
introduces a novel algorithm for obstacle avoidance
for ships based on the intersection between the circle
of avoidance, LOS concept and the acceptance circle
around the waypoints. The paper contains the
physical interpretation of proposed algorithm along
with its compatibility to its control framework. Thus,
the guidance system along with the obstacle
avoidance system highly contribute to having a fully
automated ship with minimal intervention of the
crew.
1.3 Problem Statement
The primary objective behind path following in
marine surface vessels is to keep the position of the
vehicle within a certain threshold from the trajectory.
Under different environmental disturbances, the
dynamical behavior behind ships become
tremendously significant for control systems to keep
the ship on the path. The second objective is to enable
the ship to avoid obstacles to prevent collisions. This
obstacle, which predefined by its shape and motion
descriptor, avoidance task enforces a specific route
planning procedure. In this paper, the obstacle is
assumed to be stationary and the ship needs to plan
the path and maneuver properly through a collision
course.
2 Guidance System
The ship considered in this paper under-actuated. The
two actuators are the propeller thrust and rudder
torque. is used to provide forward speed
control. is used to deliver the desired
rudder angle of attack which changes the heading of
the ship. The heading control problem has to
simultaneously control the sway displacement and
yaw angle of the ship [29, 30]. This is traditionally
accomplished by coupling the guidance system with
the heading controller as in Fig.1.
Line-of-sight (LOS) is a common technique for
the ship guidance system. The desired heading angle
of the ship is assigned by the guidance system. The
goal is to point the ship towards a fictitious point on
the desired trajectory. This way the distance of the
ship to the desired trajectory (also referred to as
cross-track error) decreases to zero when the ship
converges to the path. The path is defined by a series
of way-points connected by straight lines shown in
Fig.2. The ship location with respect to a global
coordinate frame is given by . and
are the coordinates of two consecutive
way-points on the desired trajectory.
Figure 2: Illustration of the LOS circle and way path
Consider a circle centered at the center of the ship
with a radius, . This circle moves with the ship
intersecting the desired trajectory at two points. The
forward-looking intersection point provides the
guidance system with the reference where the ship
should be headed towards in order to converge to the
path as indicated in Fig.2.
Figure 1: Guidance and control block diagram for the ship
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The radius is usually chosen to be bigger than the
length of the ship, . Having a small radius means
that the intersection point with the trajectory is too
close to the ship center. This will cause oscillations
around the desired trajectory. When the vessel is in
the vicinity of the desired trajectory, the circle will
intersect the line passing through and
at two points, and . Note that
corresponds to the closest intersection point to the
forward waypoint . The line joining the
center of the ship OShip to is called the line-of-sight
(LOS). The angle between the LOS and the reference
X-axis is given by . This
angle is the setpoint for the heading controller of the
ship, . If the ship moves along the direction of
LOS it will be heading towards the desired trajectory.
If the radius of the circle of LOS is constant and
the distance from the path is greater than then
the LOS scheme will fail as there will be intersection
between the circle of LOS and the desired path. This
is why there are many techniques to vary in order
to maintain bigger than . For example, in [30],
authors presented a guidance scheme which varies
linearly as a function of cross-track error. By
choosing to be (where L is the length of the
ship), the guidance system will always yield an
appropriate value for that will guide the ship to
the desired trajectory irrespective of the magnitude of
the cross-track error [30]. In the author’s previous
work, an improvement over the linear LOS concept
was implemented [26]. The radius was varied
exponentially using lambertw [32] function. This
allowed the ship to have smaller values or at same
distance which produced faster convergence to the
desired path.
In this work, we improve further upon the
guidance system proposed in [30, 26] by accounting
for obstacles in the path planning for the guidance
system. We proposed a methodology to modify
setpoint for the heading controller of the ship, to
avoid the obstacle while at the same time minimize
cross-tracking error. The algorithm relies on creating
a circle (circle of avoidance) centered around the
obstacle. The radius of the circle is comparable to the
size of the ship.
When there is no intersection between the circle
of LOS and circle of avoidance, the guidance and
avoidance system is identical to that in [26] which is
given in Fig.2.
The guidance and avoidance logic saves the
intersection points E and F as in Fig.3. The point with
closer distance is identified as the starting point of the
new path and is denoted with point “E” in Fig.3.
Figure 3: Illustration of the LOS circle, avoidance circle
and the waypath
Furthermore, the algorithm determines whether
the ship will avoid the obstacle by maneuvering
around it clockwise or anti-clock wise. This is done
by finding the sign of the angle
. The
maneuvering will follow same orientation as the
. We call the new path that the ship will track
as the avoidance arc illustrated in Fig.4. The first
instance the circle of LOS and circle of avoidance
intersect, the LOS is defined as the line connecting
center of the ship and E (or F depending of which one
is closer to the original path). The next step time, the
intersection between the circle of LOS and the arc of
avoidance is found. The intersection represents the
new endpoint of LOS. Once the ship reaches the
vicinity of point D, the waypath is switched back to
the original path. Fig.5. shows the flowchart for
executing the proposed guidance and avoidance
system.
Figure 4: Illustration of the avoidance arc
3 Dynamic Model of the Ship
The ship used in the simulations has a length of
100m. It has six degrees of freedom, namely, surge,
sway, heave, roll, pitch and yaw. Two coordinate
systems have been used. Following the work the
author did in [26], the equations used in modeling the
ship translational motion are:
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(1)
FX, FY and FZ are the components of the total
forces acting on the ship.
The equations used to model the roll, pitch and
yaw are:
(2)
where
,
and
are the components of the
total moments applied on the ship. Both
F
and
Externally applied forces and moments include wave
excitations, retardation forces, wind and current
loads, linear viscous damping terms, nonlinear
restoring forces in addition to the control the
propeller and the rudder forces.
4 Simulation Results
4.1 Case A: Stationary Obstacle:
To test the guidance and avoidance system, the
obstacle is arbitrarily placed in the path of the ship.
The radius of the circle of avoidance is chosen to be
200m. The choice of the radius depends on the size
of the ship, size of the obstacle and the turning
dynamics of the ship. Fig.6 shows the guidance,
avoidance system and control diagram of the ship
used in the simulation.
Figure 5: Flowchart of the guidance and avoidance system
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To show the impact of the new logic, two
simulations were conducted. The first one had the
avoidance feature turned off. Then it was turned on
in the second simulation. In both cases, the ship is
navigating from A to B, then to C.
Fig.7 shows the position of the center of the ship
in relation with the desired trajectory in the lower
section.
The upper part of Fig.7 shows the way the
guidance and avoidance algorithm react in the
presence of an obstacle. The objective is to minimize
the distance from the path, while minimizing the risk
of impact to the obstacle.
A fictious circle is drawn around the obstacle of
radius 200m. This represents a dangerous proximity
to the obstacle. If the ship is within this zone, then
Figure 6: Guidance, avoidance and control block diagram for the ship
Figure 7: Guidance, avoidance and control block diagram for the ship
Figure 8: Heading and rudder angles of the ship during tracking
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there is a great risk of collision, depending on the size
of the obstacle.
From the right plot of Fig. 8, one can notice how
the ship successfully follows the top arc for the circle
when it gets into the 200m proximity of the obstacle.
At the point (600,200), the ship non-minimum phase
behavior is clearly demonstrated. The ship is existing
the avoidance mode. The guidance system resumes
navigating under the assumption that the obstacle has
been avoided.
Fig.8 shows that at 40 seconds, the guidance and
avoidance system detects an obstacle in the path of
the ship to the right of the figure. The rudder is turned
to its maximum value in the negative direction to
embrace a circular maneuver. Then at around 80
seconds, the guidance and avoidance system decides
to switch back to follow the straight line trajectory.
In contract to the right part of Fig.7, the rudder
was almost zero in the times between 40 to 80
seconds in Fig.7 left part where the avoidance logic
was turned off.
It is worthwhile mentioning that in both plots of
Fig. 8, there is a small offset between the ship
position and the desired trajectory. This is a resultant
of the external wave, wind and current forces.
Despite the fact that the ship is headed (tilted)
towards the path, the external forces prohibit the ship
from converging to the path. The guidance system
logic by default doesn’t account for such offset.
4.2 Case B: Moving Obstacle:
To test the effectiveness of the guidance system
against a moving obstacle, an object is positioned at
coordinates of (600,0) at time zero. This point is
denoted as point M in figure 8. The obstacle moves
to coordinates of (600,200) after 60 seconds. The
final location of the obstacle is denoted as point N in
Fig.9. In the left plot of the figure, the avoidance logic
was turned off. The ship and the obstacle cross paths
around the same time.
This means that there is a high risk of collision.
Where as in the right plot of figure 8, the avoidance
system pushes the ship above the path (from x=0 m
till x=250 m) since the obstacle was underneath the
path AB and the avoidance system determined that
the best avoidance path is to push the ship above AB.
But when the obstacle crossed above AB, the
avoidance logic determined that the ship needs to
navigate from the bottom part of the path AB. The
avoidance path is updated at every second. This leads
to bigger adaptability to the position of the obstacle.
5 Summary
In this paper we presented an enhanced guidance
system for the ship to account for a stationary
obstacle. The logic was validated using nonlinear
simulation model for a 100m navigating against
waves, wind and current. The two simulation
scenarios included a stationary and moving obstacle.
The results demonstrated the effectiveness of the
disclosed combined guidance and avoidance model.
Simulations presented prove that the guidance and
avoidance system is able to minimize the risk of
collision in the presence of an obstacle. In future
work, we will optimize the radius of avoidance and
include the drift correction logic disclosure achieved
in previous work [31]. Furthermore, we will test the
logic in presence of multiple moving obstacles in the
pathway of the ship.
Figure 9: Navigation Results in Presence of a Moving Obstacle
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Author Contributions:
Nassim Khaled carried out the simulation and testing
of the proposed algorithms
Rami Alkharib carried the derivation and verification
of the mathematical equations and experimentations.
Acknowledgement:
The authors wish to acknowledge the support from
Prince Mohammad Bin Fahd University.
Creative Commons Attribution License 4.0
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WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2021.20.1
Nassim Khaled, Rami Alkhatib
E-ISSN: 2224-266X
9
Volume 20, 2021
