The Aeronautical JournalOctober 2015 Volume 119 No 1220
1271
3D UAV trajectory planning
using evolutionary algorithms:
A comparison study
M. Bagherian
[email protected]
Applied Math Department
Faculty of Mathematical Science
University of Guilan
Rasht
Iran
A. Alos
[email protected]
ABSTRACT
This paper focuses on the three dimensional flight path planning for an unmanned aerial vehicle
(UAV) on a low altitude terrain following\terrain avoidance mission. The UAV trajectory planning
problem is to compute an optimal or near-optimal trajectory for a UAV to do its mission objectives
in a surviving penetration through the hostile enemy environment, considering the shape of the
earth and the kinematics constraints of the UAV. Using the three dimensional terrain information,
three dimensional flight path from a starting point to an end point, minimising a cost function and
regarding the kinematics constraints of the UAV is calculated. The geographic information of the
earth shape and enemy locations is generated using digital terrain model (DTM) and geographic
information system (GIS), and is displayed in a 3D environment. Using 3D-maps containing
the geographic data accompanied by DTM, and GIS, the problem is modelled by deriving the
motion equations of the UAV. Two heuristic algorithms are proposed for this problem: genetic
and particle swarm algorithms. Genetic and particle swarm algorithms are general purposes
Paper No. 4356. Manuscript received 8 February 2015, accepted 7 May 2015.
1272
The Aeronautical JournalOctober 2015
algorithms, because they can solve a wide range of problems, so they have to be adjusted to solve
the trajectory planning problem. To test and compare the paths obtained from these algorithms,
a software program is built using GIS tools and the programming languages C# and MATLAB.
NOMENCLATURE
DTM
DEM
Height(x,y)
γ
ψ
Jdistance Jlength
Jheight
Jobstacle
DTED
∆v ∆γ
∆ψ
f j (x j, y j, z j)
digital terrain model
data elevation model
the value of any earth point elevation
climb angle
turn rate
distance cost
path length cost
height cost
obstacle cost
digital terrain elevation data
speed rate
climb angle rate
heading angle rate
fitness function
1.0 INTRODUCTION
It is very important to plan a trajectory for a UAV before sending it to do its mission; this process
has received considerable research attention, because following the planned trajectory maximises
the likelihood of mission succession of a flying vehicle, Chen(1). The objective of the UAV trajectory
planning is to find the optimum flight path that maximises survivability while satisfying appropriate
flight path constraints. Past approaches to this problem have usually formulated the problem
as a matter of numeric cost minimisation to be solved by methods like dynamic programming,
(Belman(2)), numerical procedures, (Betts and Huffman(3)) or geometric algorithms like A* search
(Rippel et al(4)). A*-based algorithms which are variants of the shortest path algorithm, have
difficulties with route constraints. Moreover these algorithms are strongly based on the cost map
which should be produced and stored; and the cost map production is a very time consuming task.
In dynamic programming a local cost is assigned to each link of the grid which spans the area of
operation. This assumes that the cost of flying over an area is independent of the path by which
the UAV reached the area, which is not always true. Both dynamic programming and A* search
techniques suffer from their relatively slow execution. Some researchers have formulated the
optimal path planning problem as an optimal control problem which is equivalent to a two-point
boundary value problem, (Betts and Huffman(3)). As discussed in Betts and Huffman(5) this
approach is problematic, since the adjoin equations are nonlinear and obtaining them is difficult
for complex dynamics. Moreover, convergence of the corresponding iterative solution procedure
is often very sensitive to the accuracy of the initial guess of the adjoin variables; and there is no
physical interpretation of the adjoin variables to provide an intuition into what constitutes a good
initial guess. Many authors have investigated alternative approaches to the solution of the optimal
control problem: Enright and Conway(6), Hall(7) and Lu(8). However their approaches suffer from
long convergence time. The interested reader is refereed to Hwang and Ahuja(9) and Latombe(10)
Bagherian and Alos
3D UAV trajectory planning using evolutionary algorithms...
1273
for more discussion about different route planning approaches. Chen et al (1) presented a good
approach of the GA algorithm for two dimensional trajectory planning with a variable-length
chromosome; however it was only suitable for 2D areas, and did not handle terrain following (TF)/
threat avoidance (TA) missions. So, in this paper, this algorithm is extended to be applicable for
three dimensional trajectories and accept more different changes in genes parameters values like
(speed rate, climbing angle rate, and heading angle rate). In addition by collecting the information
about some important DTM concepts and motion equations a model is constructed for the problem.
Then two evolutionary algorithms are proposed to solve the problem: genetic algorithm (GA),
and particle swarm optimisation (PSO) algorithm which are general purpose algorithms. Since
they can be used to solve a large domain of problems, they have to be adjusted to be applicable
to trajectory planning problem.
Without loss of generality, the path planning problem considered in this paper can be described
as follows.
Given a start point at (x0, y0, z0); a target point at (xT, yT, zT); and a set of obstacles located at (x i0,
i
y 0, z i0 ), where i = {1, 2, ….N}, respectively, find a near-optimal UAV trajectory (x j, y j, z j) = {(xk,
yk, zk)} defined at times {tk, k = 0, 1 … N}, which advances toward the target point, considering
the UAV dynamics. So the path should be generated as a sequence of speed rate, climb angle rate,
and heading angle rate at discrete times {tk, k = 0, 1 … N}, respectively, while taking into account
the physical constraints of the climb angle, γ and turn rate, ψ. The rest of the paper is organised
as follows. Section 2 describes the terrain representation. Section 3 models the UAV. Section 4
describes the UAV path planning. Section 5 and 6 dedicates to GA and PSO algorithms respectively.
Computational results are presented in Section 7 and Section 8 ends the paper with conclusion.
2.0 TERRAIN REPRESENTATION
A topographic† 3D-map can be defined as a cartographic‡ representation of a landscape section
in a perspective view, combined with topographic information that is defined in a legend. The
topographic information of a topographical 3D-map includes all terrain characteristics (height,
exposition, slope) and other topographic elements such as lakes, rivers, settlement areas as well as
other infrastructures, road and railway networks, land use and vegetation patterns, (Haeberling(11)).
Basically 3D-maps are topographic information plus DTM. The digital terrain model (DTM) is
simply a statistical representation of the continuous surface of the ground by a large number of
selected points with known x, y, z co-ordinates in an arbitrary co-ordinate system. In computer
language the earth surface or earth shape will be digital terrain model (DTM), or data elevation
model (DEM).
This DEM or DTM should be accurate and real, and geo-referenced. In this paper, WGS84 is
used for the geographical co-ordinate system (longitude, latitude), and UTM for the projected
co-ordinate system. The DEM will be a geographic TIFF raster image file representing the area
as a rectangular grid of cells according to a suitable resolution, each cell will have a height value
and its location is defined by two values (x, y), the resolution of the DEM is defined by the cell
size. In this paper the resolution of some TIFF images is represented by:
†
‡
cellsize = 85.9890611, 85.9890611 metres.
Topographic: is the description of earth’s surface shape and features, especially their depiction on the map
Cartographic: is the study and practice of making maps
�������� � 8
85.9890611, 85.9890611Meters.
1274 2-1 sshows a graphhical representtation
TheofAthe
eronautical JournalOctober 2015
Figure
D
DEM.
Figure 1. Terrain representation.
Figure 2-11- Terrain reprresentation
The DEM iis a grey scalee raster, the vaalue of each ppixel is an integer bounded
d from 0 to 2555. The value of any earth
point elevaation is calculated using minimum
m
heigght value and
d maximum height
h
value by the follow
wing simple
Figure 1 shows a graphical representation of the DEM.
The DEM is a grey scale raster, the value of each pixel is an integer bounded from 0 to 255.
The value of any earth point elevation is calculated using minimum height value and maximum
height value by the following simple equation:
equation:
1
2
Topographicc: is the descriptioon of earth’s surfaace shape and feaatures, especially their depiction on
n the map
Cartographicc: is the study andd practice of makiing maps.
Height(x,y) = (cellvalue(x,y)*(h
3 max – hmin))/255 + hmin
. . . (1)
Here x, y are the cell column order, and row order in the raster grid respectively.
3.0 UAV MODEL
In this paper the flight model equations and UAV physical constraints are similar to Zardashti and
Bagherian (12). Cost function components are like Chen(1), Anargyros(13) and Ioannis and Kimon(14).
Based on Call(15) an algorithm is proposed to avoid obstacles and threats.
A simplified kinematics model of a UAV flying in a three-dimensional airspace is of interest.
So the UAV is considered as a point in a 3D space, and its translational and angular states at time
t are defined as:
X(t) = (x(t), y(t), z(t), γ(t), ψ(t))T G
G = R3 × (γmin, γmax) × (–π, +π)
The co-ordinates x, y and z are taken with respect to an inertial reference frame. The flight speed
is V(t). The state equations could be written as:
.
X (t) = V(t)Cos γ(t)Cos ψ(t)
. . . (2)
.
Y (t) = V(t)Cos γ(t)Sin ψ(t)
. . . (3)
.
Z (t) = V(t)Sin γ(t)
. . . (4)
Bagherian and Alos
3D UAV trajectory planning using evolutionary algorithms...
1275
Also it will be useful to add some physical constraints on the climb angle and turn rate γ, ψ,
respectively; for example:
γmin ≤ γ ≤ γmax
. .
ψ ≤ ψ max
. . . (5)
. . . (6)
4.0 UAV PATH PLANNING
The low altitude flight and penetration mission can be divided into two main steps in mission
execution. The first step is the mission preparation, in which the task is to collect information
about the mission area, prepare the airborne electronic terrain map, and plan an optimal reference
flight route so as to improve the mission survivability of the UAV. The second step is the mission
execution, where the airborne computer calculates the optimal flight trajectory based on the
planned flight route and local information of the terrain and threats, then follows the trajectory
to finish the flight mission. In this step, because the information detected and processed by the
airborne equipment is limited, a planned flight route obtained before the mission is necessary
for the trajectory optimisation so as to ensure the optimality of the mission. Therefore the flight
route planning is very important for improvement of mission survivability and successability of
the UAV Li Qing et al(16).
As mentioned, the path planning problem considered in this paper can be described as follows.
Suppose we have a start point at (x0, y0, z0); a target point at (xT, yT, zT); a set of obstacles located at
(x i0, y i0, z i0), where i = {1, 2, … N} respectively, and we have to find a near-optimal UAV trajectory
(x j, y j, z j) = {(xk, yk, zk)} defined at times k = {1, 2, …, N}, which advances toward the target point.
However, the previous model cannot be used directly while considering the UAV dynamics, and
the path should be generated from a sequence of speed rate, climb angle rate, and heading angle
rate at discrete times {tk, k = 0, 1 … N} respectively, taking into account the physical constraints
.
of the climb angle and turn rate (γ, (ψ)), as mentioned above.
The path changes should be decoded to generate a corresponding UAV trajectory (x j, y j, z j)
for the flight. Using Equations (2), (3), and (4), and considering discrete times, the motion of the
UAV is described by the following equations:
v(k + 1) = v(k) + ∆v
. . . (7)
γ(k + 1) = γ(k) + ∆γ
. . . (8)
ψ(k + 1) = ψ(k) + ∆ψ
. . . (9)
x(k + 1) = x(k) + v(k +1)Cos γ(k+ 1)Cos ψ(k + 1)
. . . (10)
y(k + 1) = y(k)+v(k +1)Cos γ(k + 1)Sin ψ(k + 1) . . . (11)
z(k + 1) = z(k) + v(k + 1)Sin γ(k + 1)
. . . (12)
Where v is the UAV speed, γ is the climb angle, ψ is the heading angle, and x, y, z are the inertial
co-ordinates of the UAV position.
1276
The Aeronautical JournalOctober 2015
Table 1
Path changes
Row Angle rates
v
0
2
10
6
∆γ
0
–2
0
2
∆ψ
–2
0
4
0
3
0
2
5
1
1
3
0
0
1
–1
3
0
0
–1
where v, ∆γ, ∆ψ are the speed, climb and heading angle rates, respectively
This discrete UAV model is based on the assumption that there exist inner and outer loop
navigation control laws, which enable the UAV to track an acceptable and safe trajectory as long
as speed, climb angle and heading angle rates satisfy the physical constraints.
The process of finding the desired path must minimise a cost function J(x j, y j, z j), Chen(1).
5.0 GA APPROACH
Briefly, the GA algorithm is initiated with an initial set of randomly generated chromosomes called
the initial population. The genes of each chromosome are a sequence of numerical values. In this
paper each chromosome or individual represents a UAV trajectory constrained by UAV dynamics.
The population will be changed repeatedly in each generation by genetic operations (crossover,
mutation, selection, insertion, deletion); the changed elements (chromosomes) of the population
will be selected according to some fitness function. The target of this process is minimising the
fitness function as far as required, or finding the chromosome which has the near-minimal fitness
value finally. This chromosome is called the near-optimal solution.
A detailed explanation of GA algorithm could be found in references Back(17), Goldberg(18) and
Melanie(19).
As mentioned reference Chen(1) presented a good approach of the GA algorithm for two dimensional problems with a variable-length chromosome; however it was only suitable for 2D areas,
and did not handle TF/TA problems. So in this paper this algorithm is extended to be applicable
for three dimensional trajectories, and accept more different changes in genes parameters values
like (speed rate, climbing angle rate, and heading angle rate). Finally a cost function component
is used to handle the TF/TA problem, considering the UAV physical constraints like maximum
speed, maximum climbing angle … etc.
Table 1 shows an example of the speed, climb angle and turn angle rates sequenced in a random
length chromosome.
5.1 Cost function
Usually, the cost function J(x j, y j, z j) is a weighted positive scalar function, which must reflect all
the forces that conspire to derail the intensions of the UAV, and consists of several components
including distance cost, path length cost, obstacle and the height costs.
5.1.1 Distance cost Jdistance
The distance cost is the distance between the terminal point on a trajectory, and the target point.
Suppose the termination time of the flight is tN, which is a free parameter, because in this paper a
variable-length sequence is used to present a flight path. So Jdistance can be computed as:
Obstacle coost���������
Whereand
�, ��,
�� are the
climb and heading
respectively.
Bagherian
Alos
3D speed,
UAV trajectory
planningangle
usingrates
evolutionary
algorithms...
1277
The UAV is consideredd as a moving
g ball of radiius,���� mov
ving along a particular patth segment frrom time ��
Where �, ��, �� are the speed, climb and heading angle rates respectively.
to���� , as sshown in figurre 5-1.
5-1.
Cost Function
5-1.
Cost Function
Usually, the cost function ��� � , � � , � � � is a weighted positive scalar function, which must reflect all the forc
� �
Usually,tothe
costthe
function
��� �of
, � the
, � UAV,
� is a weighted
positive
scalarcomponents
function, which
mustdistance
reflect all
thepath
forc
conspire
derail
intensions
and consists
of several
including
cost,
conspire
to derail
theheight
intensions
cost,
obstacle
and the
costs.of the UAV, and consists of several components including distance cost, path
cost, obstacle and the height costs.
Distance cost ���������
Distance cost ���������
The distance cost is the distance between the terminal point on a trajectory, and the target point. Suppo
The distance
is the
distance
terminal
point because
on a trajectory,
and the
target point. Suppo
termination
timecost
of the
flight
is� , between
which is the
a free
parameter,
in this paper
a variable-length
seque
�
termination
of the
flight
which
acomputed
free
parameter,
because in this paper a variable-length sequ
�2., Obstacle
used
to presenttime
a flight
path.
So is�
���������
can isbecost
as:
Figure
calculating.
Figure
5-1 obstacle
calculating
cost
used to present a flight path. So ��������� can be computed as:
� ��� �, �������
This compoonent ������������������
is calculate
using the foollowing
expreessions:
�d����������
(5-1)
��������� � �������������� �, �������
(5-1)
. . . (13)
∑���� ����
������
� ∑���
(5-3)
����value
�� ���
Minimizing this
cost
insures
nearing
the target point.
Minimizing this cost value insures nearing the target point.
Minimising
this cost value insures
the�����
target
point.
�:����nearing
����������
��
��
(5-4)
��������
Path Length ����
cost��
0:���
�������
Path
Length
cost
�
5.1.2 Path length cost������
J
length
Where c is the obstacles count, N is Trrajectory poinnts count, D iss a suitable constant value, �
������ is the ��� obstacle
is used
to find
shorter
paths.
It can
expressedasas
This This
cost cost
is used
to find
shorter
paths.
It can
bebe
expressed
area.This cost is used to find shorter paths. It can be expressed as
�
�
������� � ∑���
��� ���������� ����� � , � ��� ��
In this papeer the obstaclee areas
con
nsidered
as a rrandomly
� �� shapped
∑��� ����������
�� �� orr concave polyygon.
� , ��convex
� are �
������
���
���
�
. . . (5-2)
(14)
(5-2)
Where
N isthhe
thepolygon
of the
sample
intervals.
Where
N
isnumber
the number
of the
sample
Consider
reppresented
in fi
igure 5-2intervals.
Minimising
cost
valueofinsures
to near
to the shortest flying path.
Where Nthis
is the
number
the sample
intervals.
Minimizing this cost value insures to near to the shortest flying path.
5.1.3 Height
cost
Minimizing
thisJcost
heightvalue insures to near to the shortest flying path.
This Height
cost is used
avoid hitting the ground, and assures that the UAV flies in low altitudes. To
cost �to
������
Height
cost ������� f3, we need the vertical distance hi between any point in the trajectory and
calculate
this component
the surface
of the
x, y, zthe
areground,
the co-ordinates
of the
iththe
point
pi offlies
the trajectory
then: To calcula
This cost
is Earth.
used toAssuming
avoid hitting
and assures
that
UAV
in low altitudes.
This cost is used to avoid hitting the ground, and assures that the UAV flies in low altitudes. To calcula
componentf� , we need the vertical=distance
h� between any point in the trajectory and the surface of the
h
pdistance
(z) - terrain(x,y)
i
i
componentf , we need the vertical
h7between any point in the trajectory and the surface of the
�
�
Assuming
x, y, z must
are thereeach
coordinates
of
ith point
the trajectory
then:
Every poinnt in
the trajectory
a vertica
al the
distance,
whose
w p� ofvalue
iss bounded
beetween the miinimum and
Assuming
x, y,
z are the must
coordinates
of
the ith point
p� of whose
the trajectory
then:
Every
point
in
the
trajectory
reach
a
vertical
distance,
value
is
bounded
between the
maximum ssafe height. Thhus the cost ff� is calculatedd using the folllowing equations:
minimum and maximum safe height. Thus thehcost
f
is
calculated
using
the
following
equations:
�z�
�
p
�
t����i��x,
y�
�
3�
�z� � t����i��x, y�
� h� � p�
J������� � � cost� ���
cost � � �
6
6
C:if��h� � �i�imu
umheight�or �h� � ���im
mumheight�
0: otherwisee
Here C isHere
a suitable
tant value.
C is aconst
suitable
constant value.
5.1.4 Obstacle cost Jobstacle
The UAV is considered as a moving ball of radius, R_UAV moving along a particular path segment
, as shown in Fig. 2.
k
(k+1)
Obstaclefrom
coost�
��������
time
t to t
The UAV is consideredd as a moving
g ball of radiius,���� mov
ving along a particular patth segment frrom time ��
to���� , as sshown in figurre 5-1.
Obstacle coost�
1278 Obstacle coost�����������������The Aeronautical JournalOctober 2015
The UAV is consideredd as a moving
g ball of radiius,���� mov
ving along a particular patth segment frro
The UAV is consideredd as a moving
g ball of radiius,���� mov
ving along a particular patth segment frrom time
to���� , as sshown in figurre 5-1.
to���� , as sshown in figurre 5-1.
Figure 3. Calculating if the point is inside the polygon
Figure 5-2 calculatinng if the point is inside the polygon
p
Figure 5-1 obstacle cost calculating
Figure 5-1 obstacle cost calculating
Let point A be a pointt included in the trajectorry, and the obstacle
o
polyg
gon is constrructed in a way
w that the
This compo
onentis�
calculate
using the foollowing
expreessions:
�������� is using
ThisThis
component
Jobstacle
calculated
thedthe
following
expressions:
compo
onent
�
is
calculate
d
using
fo
ollowing
expre
essions:
��
its vertices.
points� � � � � � … � are ������
It
I is needed too know wheth
her the point A is inside thee obstacle areea or not. To
�
�
�
�
���
�
������
� ∑���� ∑fr
ts�� � �� ���� : �: ��� � �������
solve this pproblem; the sum
s
of the
ang
gles
from
the��point
�������
c (5-3)
if
����
. . . (15)is calculated,
��� ����
��� formed
���������� � ∑��� ∑��� ������ the sum equuals 360° then the point A iss inside the obbstacle area, otherwise
o
not.
�:���� ����������
������ �
�����:���
� � � � ����������
��
������������
��
0:° ���
������
� �� ⟺∑ �
� � ��
������������
� 360 � �� ���
0:
�������
(5-3)
...
(5-5)
(5-4)
(16)
(5-4)
N is points
Trrajectory
Where
c is the
obstacles
poinnts
D iss aconstant
suitable co
nstantEspot
value, �
������ is the ��
Where
the
obstacles
N is count,
trajectory
count,
D iscount,
������
��c�is�
�������
� � �� �
��� � count,
i � is the ��� obst
count,
N
is Trrajectory
Where
c is
the
obstacles
poin
nts count,
Daisssuitable
a suitable
constantvalue,
value, �
�����
area. area.
is the ith obstacle
Figure 5-2
calculatin
ng if the
inside
the polygon
p the polygon
Figure
5-2 calculatin
ngpoint
if theispoint
is inside
p
area.
In
this
paper
the obstacle
considered
5-2.
Proposed
GA
A forareas
UAVare
pa
ath
planningas a randomly shaped convex or concave polygon.
Consider the
polygon
in Fig.
In this
papeer represented
the obstaclee areas
are3.con
nsidered as a rrandomly shap
ped convex orr concave polyygon.
Let next
point
A point
be the
a point
inconthe
trajector
ry,
andtrrajectory
the
obstacle
o the
gon
is
constr
ructed
in a inway
wa w
Let
Ade
bet aincluded
t the
included
the
and
obstacle
ois polyg
polyg
gon
constr
ructed
In this
obstacle
e point
areas
are
nsidered
as atrajector
rrandomly
shap
ped
convex
orrincludin
concave
poly
ygon.
The
suub-sections
escribe
theinpro
oposed
GAinallgorithm
for ry,
plan
nning;
ng
rep
presentation,
Let point
Apape
be er
a point
included
trajectory,
and
the
obstacle
polygon
constructed
ingenetic
ais
way
its
vertices.
points�
�points�
��Consider
� �p�1,� …
It
I isGA
needed
to
o know
theher
point
A
is inside
thee obstacle
areea ar
o
that
the points
p�2� ,�
p(i�functio
are
ItIigure
is
to
know
whether
the point
A
its
vertices.
�pth
�are
are its
is needed
needed
too wheth
knowher
wheth
point
A is
is inside
inside
thee obstacle
polygon
presented
in op
fi
5-2
�me
�he
chromosom
decoding,
f,�…
n,vertices.
and
perators.
�
�� …
3fitness
)� rep
Consider
thheorpolygon
rep
presented
in
fiigure 5-2
the
obstacle
area
not.
To
solve
this
problem;
the
sum
of
the
angles
formed
from
the
pointsp
,
i
���:���������
solve thissolve
pproblem;
the sum
s
of the
fr
from the
point
ts�
����
: ��: ���
�������
is calc
c is
this pproblem;
the
s ang
sum
ofgles
theformed
ang
gles formed
fr
from
the
point
ts�
: ��� � �������
�������
� � ��
� � ��
A,
p(i + 1):i:(1..polygon
order)
is calculated,
if the sum equals 360° then the point A is inside the
5-2-1.
Geneti
ic
representa
tion
°uals 360° then the point A is
the
sum
equ
inside
the obbstacle
o not. not.
the sumarea,
equ
uals
360
then
the point A iss inside sthe
obbstacle
area, otherwise
oarea, otherwise
obstacle
otherwise
not.
mosome consiists of three rows
r
of differrent integers, representing the speed raate, climb ang
gle rate, and
Each chrom
∑� �°�� � 360° � � ��
������������
⟺360
� � ��
������������
⟺ ∑� ��� �
. . . (17) (5-5)
(5-5)
heading anggle rate, at diiscrete times ��
� � � � � 0� � … �� respectiv
vely, taking in
nto account thhe physical co
onstraints of
�the
�
�������
�
� �� ��
� � �� ���� �
maximum
sspeed,
mentioned prev
viously. Tablee 5-2 shows ann example of the changes
�������� �������
�
�������
�mb
where
� clim
��� �and turn rate �� � as m
contained inn a random chhromosome with
w random lenngth, Pellazarr and Miles [20].
5-2.
GA
A for
UAV
pa
ath planning
5.2
GAProposed
for
path
planning
5-2.Proposed
Proposed
GA
AUAV
for UAV
pa
ath
planning
7
7
Table 5-2
2 Genetic
reprresentation
The next sub-sections
describe the
proposed
GA
algorithm
for trajectory
planning;
including
The next suub-sections
deescribe
the pro
oposed
GA allgorithm
for trrajectory
plan
nning;
includinng genetic re
The next
suub-sectionschromosome
deescribe the decoding,
pro
oposed GA
allgorithm
trrajectory
plan
nning; includinng genetic rep
pres
genetic
representation,
fitness
function,for
and
GA operators.
row fitness
repres
entation
chromosom
me decoding,
f
function,Genetic
and GA
opperators.
chromosom
me decoding, fitness
f
function, and GA opperators.
5.2.1 Genetic representation
-3
2
0
1
2
-3
-3
0
∆� -5
5-2-1.
Geneti
ic
representa
tion
Each
chromosome consists
three rowstion
of different integers, representing the speed rate, climb
5-2-1.
Genetiic of
representa
0 discrete
-2
0times2 {t , k0 = 0,11 …0N} respectively,
-1
0
angle rate, and heading angle ∆�
rate, at
taking into
k
rent
integers,
mosome
consi
ists
of
three
rows
r
of
differ
representing
the speed raate, climb an
Each
chrom
account the physical constraints of maximum speed, the climb and turn rate γ, ψ, as mentioned
rent2 integers,
mosome consiists ∆�
of three
rows
r 0 of
differ
representing
the speed raate, climb ang
gle
Each chrom
-2
4
3
-1
0
1
0
previously.heading
Table 2ang
shows
an example
the changes
a random
with thhe physical c
gle rate,
at diiscreteoftimes
��
� � � � �contained
0� � … �� in
respectiv
vely, chromosome
taking in
nto account
(20)
headinglength,
anggle Pellazar
rate, at di
iscrete
times
� � � � � 0� � … �� respectiv
vely, taking in
nto account thhe physical co
onst
random
and
Miles
. ��
maximum sspeed, the clim
mb and turn rate �� �� as m
mentioned prev
viously. Tablee 5-2 shows ann example of
maximum sspeed, the clim
mb and turn rate �� �� as m
mentioned prev
viously. Tablee 5-2 shows ann example of the
contained inn a random chhromosome with
w random lenngth, Pellazarr and Miles [20].
Where
∆�� ∆
∆��
are theech
speed
rate, climb
c with
raate, and
headin
ng angle
rateMiles
respectively.
r
contained
inn a∆�
random
hromosome
w angle
random
lenngth,
Pellazar
r and
[20].
5-2-2.
Chrom
mosome decod
ding
Table 5-22 Genetic reprresentation
Table 5-22 Genetic reprresentation
Bagherian and Alos
3D UAV trajectory planning using evolutionary algorithms...
1279
Table 2
Genetic representation
RowGenetic representation
∆v –5 –3
2
0
1
2
∆γ
0
–2
0
2
0
1
∆ψ –2
0
4
0
2
1
–3
0
0
–3
–1
3
0
0
–1
where ∆v, ∆γ, ∆ψ are the speed rate, climb angle rate, and heading angle rate, respectively
Now
the genes
of each should
chromosome
shouldtobe
decodeda corresponding
to generate a corresponding
UAV
�� � trajecto
Now the genes
of each
chromosome
be decoded
generate
UAV trajectory
� � � � �� �
it is usedthe
tofitness
calculate
the fitness
andresult
showing
the resultEquations
graphically.
Equation
the flight; as itthe
is flight;
used toascalculate
function,
and function,
showing the
graphically.
(4-1),
(4-2)
5.2.2 Chromosome decoding
3), (4-4), (4-5), (4-6) are used to decode and encode the chromosomes.
3), (4-4),
(4-5), of
(4-6)
arechromosome
used to decode
and be
encode
the chromosomes.
Now
the genes
each
should
decoded
to generate a corresponding UAV trajectory
j
j
j
(x , y , z ) for the flight; as it is used to calculate the fitness function, and showing the result
5-2-3.
Fitness Function
5-2-3.
Fitness
graphically.
Equations
(7)Function
to (12) are used to decode and encode the chromosomes.
Thefunction
fitness
function
goodness
of each chromosome.
It must
accurately
measureoft
The fitness
function
evaluates
the evaluates
goodness the
of each
chromosome.
It must accurately
measure
the quality
5.2.3
Fitness
chromosomes
in the
eachgoodness
generation,
and chromosome.
in itourshould
problem
should
the
of a giv
chromosomes
in each
generation,
and
in ourofproblem
calculate
the calculate
total cost
of atotal
given
The
fitness function
evaluates
each
It it
must
accurately
measure
thecosttrajectory,
quality
of the
chromosomes
in
generation,
and
in Qinget.
our
problem
it This
should
calculate
totalfunction
et. al.[13],
Ioannis
et. al.[14],
al. [16]
and
Chen[1].
Thisthe
fitness
need
Anargyros
et. Anargyros
al.[13],
Ioannis
et. each
al.[14],
Li Qinget.
al. Li
[16]
and
Chen[1].
fitness
function
needs
to be optimi
(13)
(14)
(16)
(1)
cost of a giventotrajectory,
see
Anargyros
et
al
,
Ioannis
et
al
,
Li
Qing
et
al
and
Chen
.
This
�
� � � � �is� �used
here is
of sum
the
an optimal
trajectory.
Thus
the fitness
� � �� �here
� � � � � � �used
the inverse
ofthe
theinverse
weighted
to get an optimalget
trajectory.
Thus
the fitness
function
� � ��function
fitness function needs to be optimised, to get an optimal trajectory. Thus the fitness function f j (x j,
the is
four
costs:
j four
previous
costs:
ythe
, z j) used
here
theprevious
inverse of
the weighted sum of the four previous costs:
�
�
�
� ��
� � � �� � �� � �� � �
� � �� � � � � � � � ������ � � � � �� �
� ∑
�
where
where
∑��� �� �� �� �� �� ���� � �
where
�
�
� ∑�
�
J �� � � � � � � � � � �
��
��
��������
������
������
� �� ��������� � �������
∑��� J� �� � � � ����
� �������
� ����������
(w� � arecorresponding
the weights corresponding
totoeach
In which
which (w
(w�i)�In
arewhich
the weights
weights
corresponding
eachcost.
cost.to each cost.
.(5-6)
. . (18)
� ����������(5-7)
. . . (19)
(5-6)
(5-7)
5-2-4.
Genetic operators
5.2.4
operators
5-2-4. Genetic
Genetic operators
Selection: This operator gives the high-quality chromosome a better chance to reproduce and get
Selection:
This
gives the
high-quality
awheel
better
chance and
to which
reproduce
Selection:
This
operator
givesoperator
the
high-quality
achromosome
better
chance
to reproduce
get copiedand
to get
the
copied
to the
next
generation.
In this
paper
wechromosome
use the weighted
roulette
selection,
generation.
In
this
paper
we
use
the
weighted
roulette
wheel
selection,
which
is
based
on
the
indi
is
based
on
the
individual’s
selection
probability
(relative
fitness).
Thus,
high-fitness
individuals
generation. In this paper we use the weighted roulette wheel selection, which is based on the individual’s selec
(1)
(20)
are likely but probability
not guaranteed
to
be
reproduced
(Chen
and
Pellazar
).
(relative
Thus,individuals
high-fitnessareindividuals
but not
to be(Chen
repr
probability (relative fitness).
Thus,fitness).
high-fitness
likely but are
not likely
guaranteed
to guaranteed
be reproduced
Crossover: The crossover operation allows two individuals to exchange information by swapping
and Pellazar [20]).
and Pellazar
some
part of[20]).
their representation with other individuals. This creates new individuals that may or
may not perform better than the parent individuals. The choice of which individuals to crossover
Crossover:
crossover
allows
two
individuals
to
by swapping
(20)
Crossover:
crossover
operation
allows
two
to
exchange
byinformation
swapping
part of t
and
at what The
point
is doneThe
randomly,
andoperation
this
is individuals
what
gives
GA
much ofinformation
itsexchange
power, Pellazar
. Insome
representation
with
other
individuals.
This
creates
new
individuals
that
may
or
may
not
this
paper
we
used
one-point
crossover.
Figure
4
shows
an
example
of
the
crossover
procedure,
representation with other individuals. This creates new individuals that may or may not perform betterperfor
than
(20)
Chen(1) and Pellazar
.
parent
individuals.
The
choice
of
which
individuals
to
crossover
and
at
what
point
is
done
randoml
parent individuals. The choice of which individuals to crossover and at what point is done randomly, and this is w
Mutation: The random mutation operator is used to prevent the loss of information that occurs
gives
GApower,
much Pellazar
of its power,
[20]. we
In this
we used
one-point crossover.
Figure
s
gives GA
of its
[20]. Pellazar
In thisthat
paper
usedpaper
one-point
crossover.
5-3 shows
an 5-3
exam
when
theremuch
are many
nearly-optimal
individuals
are missing
an
important
value inFigure
their genes.
ofadds
the
crossover
procedure,
Chen[1]
and Pellazar
[20].without affecting the rest of the
This
a random
factorand
to the
individual
members,
of theoperator
crossover
procedure,
Chen[1]
Pellazar
[20].
population. The individuals can ‘jump’ out of local minima and come closer to the optimal value.
Figure 5-3Example
of the
crossover procedure
Figure 5-3- Example
of the
crossover
procedure
parent individuals. The choice of which individuals to crossover and at what point is done randomly, and this is what
gives GA much
of its power, Pellazar [20]. In
paper we used
one-point
crossover. Figure
5-3 shows an example
1280
Thethis
Aeronautical
Journal
O
ctober 2015
of the crossover procedure, Chen[1] and Pellazar [20].
into the seleected chromossome, while th
he deletion opperator deletess a gene from the candidatee chromosomee. Figure 5-
shows two examples of these
t
operatorFigure
s. 4. Example of the crossover procedure.
Figure 5-3- Example of the crossover procedure
Mutation: The random mutation operator is used to prevent the loss of information that occurs when there are many
nearly-optimal individuals that are missing an important value in their genes. This operator adds a random factor to
he individual members, without affecting the rest of the population. The individuals can “jump” out of local minima
nd come closer to the optimal value. Figure
Mutation
must
be used
carefully,
since it often causes near-optimal individuals
5. The
insertion
and deletion
procedures.
Figu
ure 5-4 the inssertion and deeletion proced
dures
o perform worse. The mutation rate is sometimes decreased linearly during the simulation, Pellazar [20].
Mutation must be used carefully, since it often causes near-optimal individuals to perform worse.
6.
PSO ap
pproach
The mutation rate is sometimes decreased linearly during the simulation, Pellazar(20).
nsertion and Deletion: these operators are necessary to implement variable-length chromosomes. They are added
Insertion and Deletion: these operators are necessary to implement variable-length chromosomes.
swaarm
optimizat
tion
wa
as originally
ddesigned
and
introduced
i to some
and
doperator
Kennedy
[21]. The
PSO
Eberhart
oParticle
the mutation
function,
where
apply
to somewhere
of the
candidate
parents.
The
insertion
a gene
They
are added
to (PSO)
thethey
mutation
function,
they
apply
ofby
the
candidate
parents.
Theinserts
insertion operator inserts a gene into the selected chromosome, while the deletion operator deletes
is a populattion based search algorithm
m based on thhe simulation of the social behavior of bbirds, bees orr a school o
a gene from the candidate chromosome. Figure 9
5 shows two examples of these operators.
fishes. Eveery bird recallls its flying experience w
which is the shortest disttance betweenn food and itself. Bird
communicaate their flyingg experience with
w each otheer, and then th
hese behaviorss lead all birdss into the locattion of food
6.0 PSO APPROACH
so it is calleed Swarm Inttelligence, (Leeandro [22], C
Chun-Yao and
d Yi-Xing [23] and Shi [24]]). Each indiv
vidual withi
Particle swarm optimisation (PSO) was originally designed and introduced by Eberhart and
(21)
the swarm is
represented
d by
a vector
r in multidim
mensional
searcch space.
is simulation
vector hasofalso
one assiigned vecto
Kennedy
. The
PSO
is a population
based
search algorithm
based onThi
the
the social
behaviour of birds, bees or a school of fishes. Every bird recalls its flying experience which is the
nt of the parti
ticle and is caalled the velo
ocity vector. E
Each particlee updates it
which deteermines the next movemen
shortest distance between food and itself. Birds communicate their flying experience with each other,
velocity lead
an
nd all
thebirds
bestinto
poosition
it hasofexplored
far;
f and
alsointelligence,
based on thee global bes
velocity baased
and on
thencurren
thesentbehaviours
the location
food, so itso
is called
swarm
(Leandro(22), Chun-Yao and Yi-Xing(23), and Shi(24)). Each individual within the swarm is represented by
position expplored by swaarm. The PSO
O process thenn is iterated a fixed numbeer of times or until a minim
mum error i
a vector in multidimensional search space. This vector has also one assigned vector which determines
achieved baased
on amovement
desired performan
nce index
zinica
the next
of the particle
and isLaz
called
the[25].
velocity vector. Each particle updates its velocity
based on current velocity and the best position it has explored so far; and also based on the global
best position
explored
by swarm.
PSOcprocess
then is
a fixed
number
of times
until a population o
Particle sw
warm
optimizaation
is simila
ar to The
a genetic
algorithm
in
niterated
that the
systtem
is initiali
izedorwith
a minimum error is achieved based on a desired performance index Lazinica(25).
random sollutions.
It swarm
is unlike
a gen
netic
algorithm
however,
in that
each
potential
sollution with
is also
o assigned
Particle
optimisation
is similar
to m,
a genetic
algorithm
in that
the system
is initialized
population
of
random
solutions.
It is unlike
algorithm,
in that each
potential
particles,
are thenhowever,
flow
randomizedd avelocity,
and
d the
potential
l solutions,
callleda genetic
wn through
hypperspace.
solution is also assigned a randomised velocity, and the potential solutions, called particles, are
then flown through hyperspace.
References [23],
[24] andd23,
[25]
th
heused
PSOthe
algori
ithm
to solvetothe
trajectory
planning
probblem
without considerin
References
24 used
and 25
PSO
algorithm
solve
the trajectory
planning
problem
(26)
without
considering
kinematic
constraints
of the
Roberge
et
al algorithm
used GA
PSO e UAV pat
constraints
s of thethe
vehic
le. Roberge
eet. al.
[26]vehicle.
used GA
SO
ms and
for real-tim
the kinemattic
and PS
algorithms for real-time UAV path planning and obtained quasi-optimal trajectories for fixed
fixed
win
[27] in
prop
an imp
planning annd
obtained
q
quasi-optimal
trajectories
fforPSO
ng UAVs.
Qiang
proved PSO
(27)
wing
UAVs. Qiang
proposed
an improved
algorithm
for 3D path
planning
a posed
concentric
spherical
co-ordinate
trajectory
will have
the form
a variable-length
algorithm ffor
3D path
pllanning system.
in a concentric
c The generated
sphherical
coordin
nate
system.
The of
generate
ed trajectory will
w have th
line that advances slowly towards the target point, avoiding obstacles and enemy spots.
form of a vaariable-length
h lineresembles
that advaances
slowly
ttowards
the ittaarget
voidingoperations
obstacclessuch
and as
enemy
y spots.
PSO algorithm
genetic
algorithm,
except
does point,
not do av
genetic
crossover and mutation. The particles move randomly till they find the target, but through their
movement,
the
on theexc
particles
least
values.
PSO algoriithm
resembl
esalgorithm
Genetic focuses
Algorithm,
A
cept it with
doesthenot
docost
Gene
etic operationns such as cro
ossover an
mutation. T
The particles move
m
randomly till they findd the target, but through theeir movement,, the algorithm
m focuses o
the particless with the leasst cost values.
6-1.
Repressenting particles
Bagherian and Alos
3D UAV trajectory planning using evolutionary algorithms...
1281
6.1 Representing particles
Each particle will be the UAV mass centre itself in one of its states, its location and orientation is
defined by six parameters: x, y, z, v, γ, ψ and all the particles start from the point (x0, y0, z0, v0, γ0,
ψ0), and fly in the virtual space according to motion Equations (7) to (12). In every time step (∆t)
PSO makes each particle move ahead one step, stores the state of the particle (location, orientation)
in an array like (x, y, z, v, γ, ψ), and calculates the cost value of the current location. Eventually
PSO will store a set of arrays of locations, each array will be a potential UAV trajectory, but PSO
will focus only on the best trajectory with the least cost value. PSO will stop whenever one of
the trajectories reaches a location relatively acceptable to be near the target point and its cost is
the least value.
6-2.
Cost Value
6-2.
CostThe
Value
6.2 The
cost The
value
Thecost
costThe
value
calculated
showto
the
goodness
theparticle’s
particle’s
trajectory
eachstep.
The
cost
is calculated
show
the goodness
of the particle’s
trajectory
atstep.
each
step.
Cost
The
value
isisvalue
calculated
totoshow
the
goodness
ofofthe
trajectory
at ateach
The
CostThe
function
j
j
j j
Cost function
� � f� (x
� , y , z ) can be considered as:
�x , ybe
�can be considered
, z considered
as:
f � �x � , y � , zf� �can
as:
�
�
�
� J� �x � , y � , z � �
f�� �x � , y ��, z � � � ∑�� w
f �x , y , z � � ∑��� w� J� �x���
, y , z� ��
where
where
where:
�
�∑� �
J ��� � , � � , � � � � �
��
��
� ����������(6-2)
∑��� J� �� , �
������� � �������
���, �� � � ��������� ���������
������ � �������
���������
and
wweights
are thecorresponding
weights
corresponding
to cost.
each cost.
and
the
to each
cost.
� weights
andww� are
are
the
corresponding
to each
(6-1) . . . (6-1)
(20)
(6-2)
. . .(21)
i
7.
7. Experimental
Experimental
results results
7.0 EXPERIMENTAL RESULTS
Using
GIS
tools, MATLAB
environment
and C# programming
the performance
of
Using GIS
tools,
MATLAB
environment
(m files),(m
andfiles),
C# programming
language,language,
the performance
of the two
Using GIS tools, MATLAB environment (m files), and C# programming language, the performance
algorithms
are compared.
algorithms
compared.
of the twoare
algorithms
are compared.
The DEM over which the trajectory is planned, is represented by a TIFF raster projected using
The
DEM
over
the trajectory
is is
planned, is represented
a TIFF
raster projected
using
(UTM
-zone
The
DEM
over
which
thewhich
trajectory
planned,
by a nearly(450,407
TIFFbyraster
projected
using (UTM
-zone
38N)
and3
2
(UTM
-zone
38N)
and
the
geoid isWGS84
andrepresented
covers an area
× 541,342)m
. The
�
�
. The elevation
ranges
between
-74 a
geoid and
WGS84
and
covers
an area
nearly�450407
� 54��42�m
. The
elevation
ranges
between
and 4788
the
geoidthe
WGS84
covers
area4,788
nearly�450407
� 54��42�m
elevation
ranges
between
74anand
metres
above
the sea
level.
The
maximum
speed
of the-74
UAV isabove
300 metres/seconds.
The
climb
ranges
between45°,
andmeters/seconds.
45°.
The
initialThe
speed
isThe
0angle
meters
the sea
level.
The angle
maximum
of the
is
300
meters/seconds.
climb
angle
meters
theabove
sea level.
The
maximum
speed
of speed
the
UAV
is UAV
300
climb
ranges
metres/seconds. The initial climb angle is 45°. Assuming that the 0°angle direction points towards
the East; the initial direction angle will be 270°. The step time is 30 seconds, i.e. every 30 seconds
°
° be270° . The step time is 30 sec
0° angleplanning
direction
points the
towards
the
the
initial
direction
angle
will
0the
angle
direction
points towards
East;
the East;
initial
direction
angle will
be270
. flight
The step
timeofisthe
30 seconds i.e.
trajectory
algorithm
registers
and adds
a trajectory
point.
The
height
every
30
seconds
the
trajectory
planning
algorithm
registers
and
adds
a
trajectory
point.
The
flightofheigh
trajectory
point
should
range
between
100
and
600
metres.
every 30 seconds the trajectory planning algorithm registers and adds a trajectory point. The flight height
the
Figure
6
shows
four
obstacle
zones
–
the
polygon
areas
filled
with
the
red
colour
–
that
the
point
should
range 100
between
100meters.
and 600 meters.
trajectorytrajectory
point should
range
between
and 600
UAV has to avoid flying near.
The distance
between
the
and
target
is nearly
471,731
metres.
shows
fourstart
obstacle
zones
–points
the areas
polygon
areas
with
the red
that the
has flying
to avo
Figure 7-1Figure
shows7-1
four
obstacle
zones
– the
polygon
filled
withfilled
the red
colorthatcolorthe UAV
hasUAV
to avoid
°
°
°
°
between�45
. The
initial
is 0 meters/seconds.
The
initial
climb
angle
is45° . Assuming
between�45
, and45°,. and45
The initial
speed
is speed
0 meters/seconds.
The initial
climb
angle
is45
. Assuming
that the
near.
near.
7.1 Testing
GA
The initial population is chosen to be 20 genes, and the maximum number of the population is
30 genes; increasing these values is possible but it will cause the algorithm to run very slowly,
especially if the distance between the start and the target point is very large.
The result of ten executions of the genetic algorithm is shown in Table 3.
trajectory point should range between 100 and 600 meters.
Figure 7-1 shows four obstacle zones – the polygon areas filled with the red color- that the UAV has to avoid flying
1282
near.
The Aeronautical JournalOctober 2015
Figure 6. Obstacle areas.
Figure 7-1- obstacle areas
Table
3 meters.
The distance between the start and target points is nearly
471731
Testing
11 GA
Name Points Count
Length
Calc Time Time to reach target
Cost
(m)(Sec)
(Sec)
GA
GA1
GA2
GA3
GA4
GA5
GA6
GA7
GA8
59
58
61
57
57
58
58
56
56
487,950
490,530
509,730
482,370
475,710
487,530
486,450
474,600
478,140
63.8
63
65.9
62.8
61.6
62.6
62.9
60.8
61.7
1,740
1,710
1,800
1,680
1,680
1,710
1,710
1,650
1,650
928,141.5
992,703,401.7
121,180,590.8
70,864,453.9
59,163,507.7
99,093,020.4
131,297,755.7
94,164,936
79,444,360.8
It can be seen that in every execution a totally new trajectory is created, this is acceptable because
GA depends on stochastic operations. The best trajectory is GA which has the least cost, even
though it does not reach the target in the shortest period of time. Since the cost function does not
depend only on the shortest path, but it is calculated using other parameters such as (How much
the trajectory points are far from the obstacle areas, or how much their heights are in the range
between the minimum and the maximum value of height …), for example trajectory GA7 has the
shortest length, but not the least cost, see Fig. 7.
Note that trajectory GA7 is too near to the obstacle zone 1, this way the cost will be larger.
Figure 8 shows a 3D representation of the two trajectories:
7.2 Testing PSO
By using the same start and target points, and the maximum particle count to be 20 particles, the
PSO algorithm is tested, and the results are shown in Table 4.
ers such
s
as (How much the traj
ajectory pointss are far from the obstacle areas, or how
w much their
nge between
b
the minimum
m
and tthe maximum
m value of heig
ght…), for exaample trajecto
ory GA7 has
but not the least co
ost,
see figure
7-2.
Bagherian
and
Alos
3D UAV trajectory planning using evolutionary algorithms...
1283
Figuree 7-2- GA trajeectories
Note that trrajectory GA77 is too near to
o the obstacle zone 1, this way
w the cost will
w be larger.
Figure 7-3 sshows a 3D reepresentation of the two traj
ajectories:
Figure 8. 3D representation of GA trajectories.
Figure 7. GA trajectories.
Figu
Table
4ure 7-3- 3D reepresentation of GA trajectoories
The results ofTestin
the gexecution
of the PSO algorithm
7-2.
PSO
12
maximum
parrticle countCost
same start and Calc
the to
m
to
o be 20 particcles, the PSO
using thhe Length
target time
points, and
Name
PointsBycount
Time
reach target
(m)
(Sec)
(Sec)
tested, and the results aree shown in tab
ble 7-2.
PSO
PSO 1
PSO 2
PSO 3
PSO 4
PSO 5
PSO 6
PSO 7
PSO 8
PSO 9
63
57
58
59
57
57
59
60
57
57
503,280
475,710naame
490,140
492,420
PSO
480,330
PSO 1
478,530
PSO 2
489,120
PSO 3
490,530
PSO 4
468,660
PSO 5
471,660
.5
Table
the executio
on of the
PSO algorithm
567-2- The results of1,860
217,132,086
51.5
Points
Length
52.5 (m)
Count
53.5
51.5
57 .
51 3
58
53.1
59
53.4
57
51
57
51.5
475710
51.55
490140
52.55
492420
53.55
480330
51.55
63
503280
1,680
124,785,824.1
1,740
1860
1,680
1680
1,680
1710
1,740
1740
1,770
1680
1,680
1680
1,680
120,488,710
2171320866.5
91,051,760
124785824
4.1
63,021,741.1
1102495944
121,135,090.7
1204887100
156,138,674
91051760
123,907,850.7
63021741..1
134,386,337.2
Callc. Time Time to reach target(Sec) Cost
110,249,594
(Se c) 1,710
56
478530
51.33
59
489120
53. 1
PSO 7
60
490530
53.44
1770
1561386744
PSO 8
57
468660
51
1680
1239078500.7
PSO 6
1740
1211350900.7
57 cost471660
51.55 .1. This
1680
1343863377.2
The best trajectory is PSO5 which hasPSO
the9 least
63,021,741
trajectory is represented
in Fig. 9. Figure 10 shows a 3D representation of the best PSO trajectory:
Each algorithm has generated a near-optimal trajectory. Table 5 compares the two best trajectories generated by GA
and
PSO
algorithms.
The
best
trajectory
iss PSO5 which
h has the least Cost 6302174
41.1. This trajjectory is reprresented in fig
g
GA algorithm generated the better trajectory during the longest period of calculation time but
PSO generated a trajectory which is not very different from the trajectory generated by GA using
an acceptable period of calculation time (51.3 seconds).
These results and values are changeable according to the area and constraints, but in general the
PSO algorithm is faster than GA, but the trajectory generated by the genetic algorithm is better.
1284
The Aeronautical JournalOctober 2015
Figure 7-4- PSO trajectories
Figure 7-5 sshows a 3D reepresentation of the best PS
SO trajectory:
Figure 9. PSO trajectories.
Figure 10. 3D representation of PSO trajectories.
Figure 7-4- PSO trajectories
Figu
ure 7-5- 3D rep
epresentation of
o PSO trajecttories
Table 5
Each algoriithm has
geneerated a near-optimal trajecctory. Table 7-3
7 compares the two best trajectories
Comparing
results
a 3D reepresentation of the best PS
SO trajectory:
GA and PSO algorithms..
Name Points count Length
Calc time
Time to reach target
Cost
7-3- Comparing
g results (Sec)
(m) Table7
(Sec)
.5
nam
me
C
Calc. Time Tim
me to reach
targe
et(Sec)
GA
59
487,950
63.8Points Couunt Length 1,740
928,141
((Sec)
(m)
PSO 5
57
478,530
51.3
1,680
63,021,741.1
Cost
GA
A
59
487950
663.8
17
740
9281411.5
PSO
O5
57
478530
51.3
16
680
630217741.1
c
tim
me, but PSO
GA algorithhm generatedd the better trrajectory durinng the longesst period of calculation
8.0 CONCLUSION
trajectory w
which is not very
v
different from the trajeectory generatted by GA usiing an acceptaable period
This paper presented time
two (51.3
evolutionary
sseconds). approaches to solve the trajectory planning problem: GA
and PSO algorithms. The optimal trajectory was planned considering three dimensional terrain
environment with static and moving obstacles which must be avoided.14
Simulation results show
that
each
approach
finds
a
near-optimal
solution
in
a
very
effective
and
easy manner. GA does
Figu
ure 7-5- 3D rep
epresentation of
o PSO trajecttories
more calculations than PSO, but the solution obtained by GA is better, while PSO algorithm needs
as geneerated a near--optimal trajecctory. Table 7-3
7 compares the two best trajectories generated
g
by
less calculations, the solution it finds is worse. For further research in this area, it is suggested to
rithms..
combine genetic and fuzzy to make genetic-fuzzy algorithm, or PSO and fuzzy (PSO-fuzzy) to
find more improved algorithms, this may be done by using GA, or PSO to construct fuzzy rules,
Table77-3- Comparing
g results
and then build
a fuzzy system
that uses these rules to solve the trajectory planning problem, this
way
GA
or
PSO
will
generate
the
besttarge
route
least cost route) using a very fast fuzzy system,
nam
me
Points Cou
unt Length
C
Calc. Time Tim
me to reach
et(Sec)(the
Cost
((Sec)
(m) genetic-fuzzy
this is called
algorithm or PSO-fuzzy.
GA
A
59
487950
663.8
17
740
9281411.5
PSO
O5
57
478530
51.3
16
680
630217741.1
ACKNOWLEDGEMENTS
This work is supported by a grant from University of Guilan, Rasht, Iran. The authors appreciate
neratedd the better
trrajectory comments
durinng the longes
st period
of calculation
canonymous
tim
me,
but PSO generated a
the valuable
provided
by the
referees.
s not very
v
different from the trajeectory generatted by GA usiing an acceptaable period off calculation
s).
14
Bagherian and Alos
3D UAV trajectory planning using evolutionary algorithms...
1285
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