1. The Journal of Grey System 1 (2011 ) 35-46 35
Multi-objective and Multi-constrained UAV Path
Plan Optimum Selection Based on GRA*
Hu Zhong-hua'*, Zhao Min', Yao Min', Zhang Ke^
1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics,
Nanjing, 210016, China
2. College of Economics and Management, Nanjing University ofAeronautics and Astronautics,
Nanjing, 210016, China
'Corresponding Author: E-mail: chzhualong@yahoo.com.cn
Received January 2010
Abstract — To solve two-dimensional route planning problems of the Unmanned Aerial
Vehicle (UAV), optimal decision-making system of UAV flight path is established. And optimal
mathematical model of UAV flight path is also constructed. Grey relational analysis method is
applied to deal with the gray relational information among the various indicators and to solve
the model. Finally, the optimal model is used to plan optimum seeking for flight path planning
problem with seventeen radar threat nodes, five missile threat nodes, ten artillery threat nodes
and two climate threat nodes. The flight path with the best overall performance and minimum
comprehensive cost was obtained and the research provides a theoretical basis for further study
of the three-dimensional UAV flight path optimization.
Keywords: Grey Relational Analysis (GRA); UAV; Path planning; Radar; Missile; Artillery.
Introduction
Unmanned Aerial Vehicle (UAV) path plan optimum selection is the design of a flight
path from take-off area to the target area, and it must consider the fuel consumption
and avoid threats of radars, missiles, artilleries and climate, and then minimize the
overall comprehensive cost. Flight path planning is the most important part of UAV
•This research is supported by the foundation of Aviation Science Fund (Project no: 2009ZC52041)
and National Natural Science Foundation (Project no: 60974104).

2. 36 Hu Zhong-hua et al
combat mission planning. Mission is carried out by flight path, so reasonable path
planning can enable UAV to avoid threats effectively and improve survival probability
and operational efficiency [l].When getting a flight path, we must consider flve cost
indicators, they are fuel cost, radar threat cost, missile threat cost, artillery threat cost
and climate threat cost. In addition, we must also consider the constraints of them, just
like the greatest impact distance and the effective distance of radar threat and missiles
threats. There are certain correlations between these threats (e.g. radars detection
results can play an important role for guide the missiles to attack the UAV). Therefore,
the UAV flight path planning is a multi-constrained and multi-objective optimization
decision-making system, and is also an organic whole, all these factors are interrelated
and affect the system features jointly, and the impact is difficult to determine. That is to
say, it is a gray information system. The gray information often contains the correlation
between each indicator, so it is an overall information system, and in the design of path
plan optimization decision-making process, these information should be made full use.
The traditional approach of direct weighted sum often does not reflect the gray
information of these indicators, so this paper introduces grey relation analysis (GRA)
[2,3] and experience evaluation method to build UAV path plan optimum selection
model. And the optimal model is used to plan optimum seeking for flight path planning
problem with seventeen radar threat nodes, flve missile threat nodes, ten artillery threat
nodes and two climate threat nodes.
Flight Path Plan Problem Description
Flight path space representation
Because UAV usually maintain the level and speed unchanged in the cruise phase, and
the enemy's defensive zone is also in the flat region, there is no need to consider threats
avoidance by the use of terrain. Flight path planning issues can be simplified to a
two-dimensional space and it is a multi-object and multi-constrained optimum
selection problem. Survivability probability and effectiveness of UAV in the process of
implementation combat missions must also be considered, so it is one kind of special
optimum selection problem [4]. The flying space is divided by rectemgular grid. Fight
path is constituted by a group in the node vector, from the current node to the next
adjacent node. Therefore, the data structure of it is a Lo Shu Square with the current
node as the center and has eight adjacent nodes. Figure 1 is the adjacent node map of
nodes / .The adjacent nodes in path must be also adjacent in space. The size of grid

3. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 37
must be set according to the actual scale of the space and the distribution of the threats
nodes.
Adjacent Tlodes
Fig. 1. Net-construction for node.
The indicator of UAV fíght path
The indicator of UAV fight path consists mainly of fuel cost and the threats cost. And
the threats cost includes radars, missiles, artilleries and climate threat, as shown in
formula (1). The goal of path plan optimum selection is to make the overall
comprehensive cost minimum. And there are some constraints such as the greatest
impact distance and the effective distance for radars, missiles, artilleries and climate
models; therefore, the issue is a multi-objective and multi-constrained plan optimum
selection problem [4].
(1)
In formula (1): s is the UAV fiight path, s' is the optimum plan; WD(S) is radar threat
cost of s, and Wu(.s) stands for missile threat cost, WA(S) stands for artillery threat cost
and Wc(,s) stimds for climate threat cost. Wois) is cost of fuel consumption. Fuel cost is
a function of the voyage, and other threats cost is relative with detection range of
radars and the radius of destruction of missiles, artilleries and climate. It can be
specifically calculated as follows.
Establishment of threats models
Radars, missiles, artilleries, and climate threat model, respectively, are defined as
follows [5]:
Radars detection probability for UAV can be described as:
(2)

4. 38 Hu Zhong-hua et al.
In formula (2), P/^did is the probability of radars threats, dR stands for the distance
between the UAV and the radars, ¿/smax stands for the radius of maximum detection of
radars. When exceeding the distance, the return signal is so weak, and will be drowned
in the noise. <R i is a radius of effective detection of radars. Within this range, the
Í mn
detection probability is one. P!^dR)=l indicates that the detection probability of UAV is
1, so the radar threat is infinity. Pa(dii)=O indicates that the detection probability is 0
and then the cost of radar threat is zero, and as between the two, the probability is
Destruction probability of missiles, artilleries and climate for UAV can be described
as follows:
In formula (3), (4) and formula (5), the destruction probability of missiles, artilleries
and climate are described respectively, just like in formula (2). But there are two
differences. One is that subscripts have different means. M means missiles, A means
artilleries and C means climate. And another difference is that it is 1 / d^ in formula
(2),while l/d,m formula (3)^ (4) and formula (5).
After the indicator functions of optimization selection are defined, the indicator cost
can be calculated for a given path respectively.
GRA for Plan Set of UAV Flight Path
UAV fiight path plan set is composed by the n plans '*'. Each plan has m indicators set.
In this paper, UAV plan has five indicators, namely, the cost of radars threats, missiles
threats, artilleries threats, climate threat and the cost of fuel consumption.
Path / with m indicators in gray system can be expressed as a vector x,.
x,=ixn, xa, ••• , Xi„) , / = l,2,---,n, j = l,2,---,m (6)
And then gray system with n Paths and m indicators for each path can be expressed as

5. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 39
a matrix X„y„ as follows:
Xy
y =
'•2«
(7)
In order to facilitate grey relation analysis, each evaluation indicator values for all
alternative UAV flight path plans are treated as non-dimensional standardized
indicators. The treatment methods are shown as follows:
Path plan indicators in this article are the cost indicators, therefore, smaller for
comprehensive cost, better for overall performance, and the standardized formula is
shown as follows [7]:
-Xy +max
(8)
In formula(8), / = l,2,---,n,y = 1,2,---,OT.
After normalized treatment, matrix A^^m becomes series r, and is shown as follows:
rrinura, •••,r,„), / = 1,2,••-,«. (9)
UAV path plan optimum Selection for n plans has a relative comparison with each
other. That is to the say the relative importance of m evaluation indicators must be
considered during optimum selection for the gray system, therefore an ideal reference
plan is determined, denoted as follows:
/2 ' • • • ' / ; . • • • ' / « ] (10)
In formula {0), f° =m2x{rj, r2j,---,r„j,),j = l,2,---.,m. That is to say m evaluation
indicators of f* are the maximum of the corresponding evaluation indicator for all n
alternative paths, and it is considered as the ideal path plan (the ideal solution) and as
the standard. The ideal path plan is a reference sequence and al! these n a!temative
paths are comparative sequences which are compared with reference sequence
respectively [8] .The approach degree between reference sequence and comparative
sequence is usually measured by grey incidence coefficient, (^¡j is the grey incidence
coefficient between indicator rjj of sequence comparison /-,, and f° of reference
sequence

6. 40 Hu Zhong-hua et al
mm mm / f - r-' I + p max max
i
in
•' j i -'¡'-
J i = ,2,--,m. (11)
In formula (11), p€[0,], generally take p=0.5. And then grey incidence coefficient
matrix for the plans set of UAV fiight path scheme can be shown as follows:
9 21
(12)
7n2
Solution for optimum selection model
Evaluation system of UAV path plan selection includes fiiel cost and the cost of radars
threats, missiles threats, artillery threat and climate threat. Assume that there are
«paths, expressed respectively as: Si^2,-",Sj,---,s„. Among them, the composition of
indicators for path J can be expressed by Xj vector :
And n paths constitute a set of alternative plans: X(x, JC2,---,x„). After quantifying the
various performance indicators, a reference indicator set is determined and it is
constituted by choosing the best indicator of the value of UAV fiight path plans [9-10].
Reference indicator set describes a reference design of UAV flight path, and it is the
ideal solution. And then grey relational coefficient matrix for n kinds of design options
can be obtained and they are relative to reference design. Grey relational coefficient
matrix is described as H :
R2 Ml A2 C2 02
(13)
In formula (13), ^ is the grey relational coefficient of evaluation indicators relative to
the reference indicators.
The weight of fuel consumption and the cost of radars threats, missiles threats,
artillery threat and climate threat are calculated by using experience evaluation method

7. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 41
and shown as:So, SR, SM, SA and SQ,. The grey relational degree R{ru r2,-",r„)^of each
plan's decision-makers can be calculated as follows:
»fil Wfl Í4I ici 501
R can be sorted according to size, and the best one is the largest one and its plan is the
optimum plan s*, its corresponding indicators is are the optimum indicators: x*.
Examples of Path Plan Optimum Selection
In this paper. The UAV flight path parameters include UAV takeoff location
coordinates, the destination location coordinates and the coordinates for seventeen
missiles threats, ten artilleries threats, flve air missile threats and two climate threats,
as shown in Table 1 [11]. The entire flight path maps were drawn by using Matlab.
Tablel. Menace nodes, start node and destination node.
Start node (10,20) Destination node (40, 50)
No. (x,y) No. (x,y) No. (x,y)
1# (17,60) 7# (22,28) 3# (26,55)
2# (32,66.5) 8# (45,30) 14# (47,49)
Radar threat 3# (50,62) 9# (32,22) 15# (24,42)
nodes 4# (57,45) 10# (36,32) 16# (33,54)
5# (51.5,31) 11# (12,36) 17# (37,55)
6# (35,26) 12# (11,48)
1# (17,22) 4# (46, 54)
Missile threat
nodes 2# (40,62) 5# (56, 38)
3# (26, 30)
1# (14,46) 5# (26,22) 9# (20,30)
Artillery 2# (37,47) 6# (35,37) I0# (32,34)
threat nodes 3# (10,30) 7# (30,35)
4# (34,50) 8# (30,50)
Climate 1# (16,40)
threat nodes 2# (24,48)
The threats model parameters of radars, missiles, artilleries and atmospheric were set
to: dRmiB=4, £/ßmax=80, í4/min=4, <4/max=60, í/xinin=3, £(<niax=15, £¿Cmin=2, öti«ax=8. Their
weights were calculated by experience evaluation method.

8. 42 Hu Zhong-hua et al
Alternative plan set of UAV flight path
On the bases of meeting the threats constraints of radars, artilleries, missiles and
climate, thirty UAV fiight paths are determined as alternatives by identifying regions of
random search algorithm. Figures from Fig.2 - Fig.31 describe the thirty alternative
fiight path maps respectively.
o ••• "'
o o O O
o o o o o « o o
. " , Î »
'¿.3 Plan 2 llight path J.4 Plan 3 flighlpath
I
i ig.5 Plan 4 tlight path i- ig.ojr^lan :J iiigni patn r íg.b t'lan i iiignt pam
o o
o o o o
Fig. 8 Plan 7 flight paüi_^ Fig.9 Plan 8 flight path Fig. 10 Plan 9 flight path
o o o o o o
o o » o
itp- • ' . ^ <^
Fig 11 Plan 10 flight path Fig. 12 Plan 11 flight path Fig. 13 Plan 12 flight piith
o o
o « o o
;3nightpath í ig. 15 Plan 14 flight path Fig.lóPlanl;
o o o o o o
« o o o
3 ^
<» ..T^^*.
Fig. 17 Plan 16 flight path Fig. 18 Plan 17 flight path Fig. 19 Plan 18 flight path

9. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 43
o o o o
o o o
-' r o -»o
Sí—*— « .Ï
Fig.2O Plan 19 flight path Fig.21 Plaii 20 flight patl. Ig 22 pían iîfïiglu paili
O o
I-I
:Mg.24 Plan 23 flight path F i g . 2 5 r ,,;¡i . , i¡, , ;•;
O O ~ o o
o o
íí ¡e Ís-"-i'¿ P.
;.26 Plan 25 flight path Fig.27 Plan 26 flight patli Fig.28 Pian 27 íligiu JÚU
o"' i "a 1 (5 - —x._
o »
Hg.2y Plan 2Ü llight path Fig.3O Plan 29 tliglit paüi Fig.31 Plan 30 fliglit path
In these figures, square stands for the path starting node, and five-pointed stands for
the track nodes, and solid circle stands for the target nodes, and diamond stands for
radar threat nodes, and the triangle stands for the anti-aircraft artillery threat nodes, and
hexagonal stands for climate threat nodes, and hollow-point circle stands for the
missile threat nodes.
UAV flight path optimum selection
According to the formula (2-5), the thirty UAV flight path of the cost indicators are
calculated through the Matlab programs, the results are shown in Table 2. According to
the formula (8) and combining with the data of Table 2, the normalized values of each
indicator and reference normalized values are calculated. According to the formula (11)
the grey incidence coefficient for each indicator of every path plan are calculated. The
results are shown in Table 3. And they are compared with the ideal solution. These
indicators include the fuel consumption cost, the radar threat cost, artillery threat cost,
missile threat cost and climate threat cost.

10. 44 Hu Zhong-hua et al
Table 2.The initial value of each indicator.
Plan No. Fuel cost Radars cost Artilleries cost Missiles cost climate cost
1 26 0.0228 16.0365 7.4902 0.3349
2 23 0.0206 13.9257 6.7246 0.4355
3 23 0.0208 14.2023 6.8994 0.2968
4 22 0.0213 13.2377 6.3322 0.3349
5 22 0.0212 13.4521 6.3458 0.2968
6 22 0.0179 13.3222 6.3981 0.2968
7 23 0.0243 14.3953 6.7374 0.5936
8 22 0.0228 13.8197 6.2994 0.4930
9 23 0.0207 14.4064 6.8695 0.3349
10 22 0.0169 13.3413 6.5782 0.3349
11 21 0.0176 13.1398 6.3902 0.3349
12 21 0.0159 13.2410 6.2513 0.3349
13 21 0.0162 13.1843 6.1421 0.2968
14 25 0.0198 15.3825 7.5423 0.2968
15 22 0.0173 13.7966 6.5797 0.2968
16 21 0.0170 13.0862 6.2336 0.2968
17 21 0.0163 13.4766 6.2487 0.2968
18 26 0.0234 15.4881 7.3499 0.4736
19 23 0.0226 13.9014 6.8143 0.4355
20 23 0.0180 14.4413 6.8487 0.4355
21 23 0.0217 14.6534 6.8650 0.3349
22 21 0.0201 13.3901 6.2239 0.4355
23 22 0.0168 13.6482 6.6807 0.3349
24 24 0.0225 14.5526 6.9668 0.4355
25 22 0.0221 13.7040 6.5854 0.2968
26 22 0.0221 13.7040 6.5854 0.2968
27 27 0.0212 16.5315 7.9282 0.3349
28 23 0.0215 14.4353 6.9410 0.2968
29 21 0.0215 13.1435 6.1094 0.4930
30 22 0.0198 14.1398 6.6100 0.2968
First, the weight of each indicator was obtained by using experience evaluation method,
the result was respectively <5b=0.2, 4=0.1, ^^=0.3, 5^f^Çi.2 and ^ 0 . 2 . Then,
according to formula (14), the grey incidence degree R was calculated, and it was the
comprehensive performance of the each plan relative to the reference indicator of
reference plan (the ideal solution).And the result was i?=(0.4620, 0.5916, 0.6552,
0.7893, 0.8004, 0.8334, 0.5088, 0.6498, 0.6046, 0.7832, 0.8741, 0.9074, 0.9702,
0.5438, 0.7692, 0.9552, 0.9093, 0.4121, 0.5782, 0.5683, 0.5875, 0.7860, 0.7405,
0.5073, 0.7425, 0.7425, 0.4367, 0.6356, 0.8193, 0.7170). Where, the value of
comprehensive evaluation for the 13th plan was the maximum and the value was
0.9702. Therefore, the best solution for each path plan was the 13th plan. That was to

11. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 45
say, the optimal solution for the UAV path plan optimum seeking model was x*=(2I,
0.0162, 13.1843, 6.1421, 0.2968), so the optimal path s* was the 13th path (Fig.l4) and
its overall cost was minimal.
Table 3. Grey relational coefficient of each indicator.
Plan No. Fuel cost Radars cost Artilleries cost Missiles cost climate cost
1 0.3750 0.3784 0.3686 0.3971 0.7957
2 0.6000 0.4719 0.6723 0.5965 0.5169
3 0.6000 0.4615 0.6068 0.5351 1.0000
4 0.7500 0.4375 0.9192 0.8032 0.7957
5 0.7500 0.4421 0.8248 0.7937 1.0000
6 0.7500 0.6774 0.8795 0.7590 1.0000
7 0.6000 0.3333 0.5682 0.5915 0.3333
8 0.7500 0.3784 0.7014 0.8272 0.4306
9 0.6000 0.4667 0.5661 0.5447 0.7957
10 0.7500 0.8077 0.8710 0.6598 0.7957
11 1.0000 0.7119 0.9698 0.7641 0.7957
12 1.0000 1.0000 0.9175 0.8650 0.7957
13 1.0000 0.9333 0.9461 0.9653 1.0000
14 0.4286 0.5185 0.4286 0.3883 1.0000
15 0.7500 0.7500 0.7080 0.6591 1.0000
16 1.0000 0.7925 1.0000 0.8798 1.0000
17 1.0000 0.9130 0.8152 0.8672 1.0000
18 0.3750 0.3590 0.4177 0.4230 0.4563
19 0.6000 0.3853 0.6788 0.5633 0.5169
20 0.6000 0.6667 0.5597 0.5516 0.5169
21 0.6000 0.4200 0.5236 0.5462 0.7957
22 1.0000 0.5000 0.8500 0.8882 0.5169
23 0.7500 0.8235 0.7540 0.6142 0.7957
24 0.5000 0.3889 0.5402 0.5147 0.5169
25 0.7500 0.4038 0.7360 0.6564 1.0000
26 0.7500 0.4038 0.7360 0.6564 1.0000
27 0.3333 0.4421 0.3333 0.3333 0.7957
28 0.6000 0.4286 0.5608 0.5223 1.0000
29 1.0000 0.4286 0.9678 1.0000 0.4306
30 0.7500 0.5185 0.6205 0.6450 1.0000
Conclusions
To solve the problem of multi-objective and multi-constrained UAV path plan optimum
seeking, this paper established the threats model of UAV fiight path plan and its goal
system of optimizing and decision-making, including five decision-making objectives,
such as fuel cost and the cost of radars threats, missiles threats, artillery threat and

12. 46 Hu Zhong-hua et al.
climate threat. The constraints for the greatest impact distance and the effective
distance of these threats models are introduced into cost function. And by this way,
path optimization selection mathematical model of UAV flight path is established.
Then GRA method is used to solve the model. At last, the optimization model is
applied to real path optimization selection problem with seventeen radar threat nodes,
flve missile threat nodes, ten artillery threat nodes and two climate threat nodes. The
path with best comprehensive performance (minimum comprehensive cost) is sought
by the method. The method can avoid the subjectivity and randomness of traditional
selection and provide a theoretical basis for further study three-dimensional
multi-objective and multi-constrained UAV path plan optimum selection.
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