multi-objective and multi-constrained uav path plan optimum selection based on gra*

13
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: [email protected] 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).

Upload: laurence-musgrove

Post on 16-May-2015

403 views

Category:

Technology


2 download

TRANSCRIPT

Page 1: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

The Journal of Grey System 1 (2011 ) 35-46 35

Multi-objective and Multi-constrained UAV PathPlan 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: [email protected]

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).

Page 2: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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

Page 3: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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)

Page 4: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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. <ÍRmin is a radius of effective detection of radars. Within this range, the

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

Page 5: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 39

a matrix X„y„ as follows:

y =

Xy

'•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{r\j, 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

Page 6: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

40 Hu Zhong-hua et al

mm mmJ i

in / f - r + p max maxi \ •' -' 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

7n2

(12)

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

Page 7: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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

Radar threat

nodes

Missile threatnodes

Artillerythreat nodes

Climatethreat nodes

(10,20)

No.

1#

2#3#4#5#6#

1#

2#3#1#

2#

3#4#

1#2#

(x,y)

(17,60)

(32,66.5)(50,62)(57,45)

(51.5,31)(35,26)

(17,22)

(40,62)(26, 30)

(14,46)(37,47)

(10,30)

(34,50)

(16,40)(24,48)

Destination node (40, 50)

No.

7#

8#9#10#11#12#4#

5#

5#

6#

7#

8#

(x,y)

(22,28)

(45,30)(32,22)(36,32)(12,36)(11,48)

(46, 54)

(56, 38)

(26,22)(35,37)

(30,35)

(30,50)

No.

\3#

14#15#16#17#

9#

I0#

(x,y)

(26,55)

(47,49)(24,42)(33,54)(37,55)

(20,30)(32,34)

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.

Page 8: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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 oo o o o

. " , Î »

O O

o « o o

\'¿.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

Fig. 8 Plan 7 flight paüi_^

o o

Fig.9 Plan 8 flight path Fig. 10 Plan 9 flight path

o o o o

itp- • ' . ^ <

o oo o » o

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;

3 ^

o o o o« o o o

o o

<» . . T ^ ^ * .

Fig. 17 Plan 16 flight path Fig. 18 Plan 17 flight path Fig. 19 Plan 18 flight path

Page 9: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 43

o oo o o

\-' r o -»o

o o

Sí—*— « .Ï

Fig.2O Plan 19 flight path Fig.21 Plaii 20 flight patl.

I-I

O o

Ig 22 pían iîfïiglu paili

:Mg.24 Plan 23 flight path~O O o o

o o

íí ¡e Ís-"-i'¿ P.

F i g . 2 5 r , , ; ¡ i . , i ¡ , , ;•;

;.26 Plan 25 flight path Fig.27 Plan 26 flight patlio"' i "a

Fig.28 Pian 27 íligiu JÚU1 (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.

Page 10: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

44

Table 2.

Plan No.

123456789101112131415161718192021222324252627282930

Hu Zhong-hua et al

The initial value of each indicator.

Fuel cost

262323222222232223222121212522212126232323212224222227232122

Radars cost

0.02280.02060.02080.02130.02120.01790.02430.02280.02070.01690.01760.01590.01620.01980.01730.01700.01630.02340.02260.01800.02170.02010.01680.02250.02210.02210.02120.02150.02150.0198

Artilleries cost

16.036513.925714.202313.237713.452113.322214.395313.819714.406413.341313.139813.241013.184315.382513.796613.086213.476615.488113.901414.441314.653413.390113.648214.552613.704013.704016.531514.435313.143514.1398

Missiles cost

7.49026.72466.89946.33226.34586.39816.73746.29946.86956.57826.39026.25136.14217.54236.57976.23366.24877.34996.81436.84876.86506.22396.68076.96686.58546.58547.92826.94106.10946.6100

climate cost

0.33490.43550.29680.33490.29680.29680.59360.49300.33490.33490.33490.33490.29680.29680.29680.29680.29680.47360.43550.43550.33490.43550.33490.43550.29680.29680.33490.29680.49300.2968

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

Page 11: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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.

123456789101112131415161718192021222324252627282930

Fuel cost

0.37500.60000.60000.75000.75000.75000.60000.75000.60000.75001.00001.00001.00000.42860.75001.00001.00000.37500.60000.60000.60001.00000.75000.50000.75000.75000.33330.60001.00000.7500

Radars cost0.37840.47190.46150.43750.44210.67740.33330.37840.46670.80770.71191.00000.93330.51850.75000.79250.91300.35900.38530.66670.42000.50000.82350.38890.40380.40380.44210.42860.42860.5185

Artilleries cost

0.36860.67230.60680.91920.82480.87950.56820.70140.56610.87100.96980.91750.94610.42860.70801.00000.81520.41770.67880.55970.52360.85000.75400.54020.73600.73600.33330.56080.96780.6205

Missiles cost

0.39710.59650.53510.80320.79370.75900.59150.82720.54470.65980.76410.86500.96530.38830.65910.87980.86720.42300.56330.55160.54620.88820.61420.51470.65640.65640.33330.52231.00000.6450

climate cost

0.79570.51691.00000.79571.00001.00000.33330.43060.79570.79570.79570.79571.00001.00001.00001.00001.00000.45630.51690.51690.79570.51690.79570.51691.00001.00000.79571.00000.43061.0000

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

Page 12: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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.

References

[I] Zhou Cheng-ping, Chen Qian-yang, Qin Xiao-wei (2005). Parallel algorithm of 3D routeplanning based on the sparse A* algorithm. Journal of Huazhong University of Science andTechnology (Nature Science), 33 (5): 42-45 (in Chinese).

[2] Deng Julong (2009). Binary Grey Relational Analysis (BGRA), The Journal of Grey System.21(03): 225-230.

[3] LIU Si-feng, LIN "Yl (2006). Grey Information: Theory and Practical Applications,Spinger-Verlag London, 9-130.

[4] ZHANG Qingjie, XU Ha (2007). HUO Desen .Path Planning of Reconnaissance UAV and ItsRealization Based on Improved ANT Algorithm, Operations Research and Management Science,16(3):97-112. (in Chinese).

[5] GAO Xiao-guang, YANG You-long (2003). Initial Path Planning Based-on Different Threats forUnmanned Combat Air Vehicles, Ada Aeronáutica ET Astronáutica Sinica, 24(5):435-438. (inChinese).

[6] Dang Yao-guo, Liu Si-feng, Liu Bin et al. (2004). Study on Incidence Decision Making Modelof Multi-Attribute Interval Number, JOURNAL OF NANJING UNIVERSITY OF AERONAUTICS& ASTRONAUTIC, 36(3):403-^06 (in Chinese).

[7] Rao C J, Peng J, Li C F, et al. (2009). Group Decision Making Model Based on Grey RelationalAnalysis, The Journal of Grey System, 21(01):15-24.

[8] Fan C K, Tsai H Y, Lee Y H (2008). The selection of life insurance sales representatives trainingprogram by using the AHP and GRA, The Journal of Grey System, 20(2):149-160.

[9] Zeng Qianglin,Xie Haiying,Dai Qihua. (2008). Grey Relational Analysis in Clinical AntibioticSelection, The Journal ofGgrey System, 20(04): 311-318.

[10] Ren Shiyan,Zou Ningxin, Dong Jiahong, et al. (2008). Grey relational analysis of value ofCA19-9 levels in predictability of respectability of pancreatic canner. The Journal of Grey System,20(03): 281-293.

II1] Liu Changan (2003). Study on Path Planning for UAV[D]. Northwestern Polytechnic University,(in Chinese).

Page 13: Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

Copyright of Journal of Grey System is the property of Research Information Ltd. and its content may not be

copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written

permission. However, users may print, download, or email articles for individual use.