The Journal of Grey System 1 (2011 ) 35-46                                                         35  Multi-objective and...
36                                 Hu Zhong-hua et alcombat mission planning. Mission is carried out by flight path, so re...
Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 37must be set according to the actual s...
38                               Hu Zhong-hua et al.In formula (2), P/^did is the probability of radars threats, dR stands...
Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 39 a matrix X„y„ as follows:           ...
40                                Hu Zhong-hua et al          mm mm  / f - r- I + p max max              i               i...
Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 41 and shown as:So, SR, SM, SA and SQ,....
42                                             Hu Zhong-hua et alAlternative plan set of UAV flight pathOn the bases of me...
Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 43                                     ...
44                                    Hu Zhong-hua et alTable 2.The initial value of each indicator.Plan No. Fuel cost Rad...
Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 45say, the optimal solution for the UAV...
46                                    Hu Zhong-hua et al.climate threat. The constraints for the greatest impact distance ...
Copyright of Journal of Grey System is the property of Research Information Ltd. and its content may not becopied or email...
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Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA*

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

  1. 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.IntroductionUnmanned Aerial Vehicle (UAV) path plan optimum selection is the design of a flightpath from take-off area to the target area, and it must consider the fuel consumptionand avoid threats of radars, missiles, artilleries and climate, and then minimize theoverall 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. 2. 36 Hu Zhong-hua et alcombat mission planning. Mission is carried out by flight path, so reasonable pathplanning can enable UAV to avoid threats effectively and improve survival probabilityand operational efficiency [l].When getting a flight path, we must consider flve costindicators, they are fuel cost, radar threat cost, missile threat cost, artillery threat costand climate threat cost. In addition, we must also consider the constraints of them, justlike the greatest impact distance and the effective distance of radar threat and missilesthreats. There are certain correlations between these threats (e.g. radars detectionresults 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 optimizationdecision-making system, and is also an organic whole, all these factors are interrelatedand affect the system features jointly, and the impact is difficult to determine. That is tosay, it is a gray information system. The gray information often contains the correlationbetween each indicator, so it is an overall information system, and in the design of pathplan optimization decision-making process, these information should be made full use.The traditional approach of direct weighted sum often does not reflect the grayinformation of these indicators, so this paper introduces grey relation analysis (GRA)[2,3] and experience evaluation method to build UAV path plan optimum selectionmodel. And the optimal model is used to plan optimum seeking for flight path planningproblem with seventeen radar threat nodes, flve missile threat nodes, ten artillery threatnodes and two climate threat nodes.Flight Path Plan Problem DescriptionFlight path space representationBecause UAV usually maintain the level and speed unchanged in the cruise phase, andthe enemys defensive zone is also in the flat region, there is no need to consider threatsavoidance by the use of terrain. Flight path planning issues can be simplified to atwo-dimensional space and it is a multi-object and multi-constrained optimumselection problem. Survivability probability and effectiveness of UAV in the process ofimplementation combat missions must also be considered, so it is one kind of specialoptimum selection problem [4]. The flying space is divided by rectemgular grid. Fightpath is constituted by a group in the node vector, from the current node to the nextadjacent node. Therefore, the data structure of it is a Lo Shu Square with the currentnode as the center and has eight adjacent nodes. Figure 1 is the adjacent node map ofnodes / .The adjacent nodes in path must be also adjacent in space. The size of grid
  3. 3. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 37must be set according to the actual scale of the space and the distribution of the threatsnodes. Adjacent Tlodes Fig. 1. Net-construction for node.The indicator of UAV fíght pathThe indicator of UAV fight path consists mainly of fuel cost and the threats cost. Andthe threats cost includes radars, missiles, artilleries and climate threat, as shown informula (1). The goal of path plan optimum selection is to make the overallcomprehensive cost minimum. And there are some constraints such as the greatestimpact distance and the effective distance for radars, missiles, artilleries and climatemodels; therefore, the issue is a multi-objective and multi-constrained plan optimumselection problem [4]. (1)In formula (1): s is the UAV fiight path, s is the optimum plan; WD(S) is radar threatcost of s, and Wu(.s) stands for missile threat cost, WA(S) stands for artillery threat costand Wc(,s) stimds for climate threat cost. Wois) is cost of fuel consumption. Fuel cost isa function of the voyage, and other threats cost is relative with detection range ofradars and the radius of destruction of missiles, artilleries and climate. It can bespecifically calculated as follows.Establishment of threats modelsRadars, missiles, artilleries, and climate threat model, respectively, are defined asfollows [5]: Radars detection probability for UAV can be described as: (2)
  4. 4. 38 Hu Zhong-hua et al.In formula (2), P/^did is the probability of radars threats, dR stands for the distancebetween the UAV and the radars, ¿/smax stands for the radius of maximum detection ofradars. When exceeding the distance, the return signal is so weak, and will be drownedin the noise. <R i is a radius of effective detection of radars. Within this range, the Í mndetection probability is one. P!^dR)=l indicates that the detection probability of UAV is1, so the radar threat is infinity. Pa(dii)=O indicates that the detection probability is 0and 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 describedas follows:In formula (3), (4) and formula (5), the destruction probability of missiles, artilleriesand climate are described respectively, just like in formula (2). But there are twodifferences. One is that subscripts have different means. M means missiles, A meansartilleries 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 costcan be calculated for a given path respectively.GRA for Plan Set of UAV Flight PathUAV 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, missilesthreats, 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. 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 allalternative UAV flight path plans are treated as non-dimensional standardizedindicators. The treatment methods are shown as follows: Path plan indicators in this article are the cost indicators, therefore, smaller forcomprehensive cost, better for overall performance, and the standardized formula isshown 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 eachother. That is to the say the relative importance of m evaluation indicators must beconsidered during optimum selection for the gray system, therefore an ideal referenceplan 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 evaluationindicators of f* are the maximum of the corresponding evaluation indicator for all nalternative paths, and it is considered as the ideal path plan (the ideal solution) and asthe standard. The ideal path plan is a reference sequence and al! these n a!temativepaths are comparative sequences which are compared with reference sequencerespectively [8] .The approach degree between reference sequence and comparativesequence is usually measured by grey incidence coefficient, (^¡j is the grey incidencecoefficient between indicator rjj of sequence comparison /-,, and f° of referencesequence
  6. 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 coefficientmatrix for the plans set of UAV fiight path scheme can be shown as follows: 9 21 (12) 7n2Solution for optimum selection modelEvaluation system of UAV path plan selection includes fiiel cost and the cost of radarsthreats, missiles threats, artillery threat and climate threat. Assume that there are «paths, expressed respectively as: Si^2,-",Sj,---,s„. Among them, the composition ofindicators for path J can be expressed by Xj vector :And n paths constitute a set of alternative plans: X(x, JC2,---,x„). After quantifying thevarious performance indicators, a reference indicator set is determined and it isconstituted 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 theideal solution. And then grey relational coefficient matrix for n kinds of design optionscan be obtained and they are relative to reference design. Grey relational coefficientmatrix is described as H : R2 Ml A2 C2 02 (13)In formula (13), ^ is the grey relational coefficient of evaluation indicators relative tothe 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. 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 plans decision-makers can be calculated as follows: »fil Wfl Í4I ici 501R can be sorted according to size, and the best one is the largest one and its plan is theoptimum plan s*, its corresponding indicators is are the optimum indicators: x*.Examples of Path Plan Optimum SelectionIn this paper. The UAV flight path parameters include UAV takeoff locationcoordinates, the destination location coordinates and the coordinates for seventeenmissiles 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 setto: dRmiB=4, £/ßmax=80, í4/min=4, <4/max=60, í/xinin=3, £(<niax=15, £¿Cmin=2, öti«ax=8. Theirweights were calculated by experience evaluation method.
  8. 8. 42 Hu Zhong-hua et alAlternative plan set of UAV flight pathOn the bases of meeting the threats constraints of radars, artilleries, missiles andclimate, thirty UAV fiight paths are determined as alternatives by identifying regions ofrandom search algorithm. Figures from Fig.2 - Fig.31 describe the thirty alternativefiight 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 tlan 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. 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 forthe track nodes, and solid circle stands for the target nodes, and diamond stands forradar threat nodes, and the triangle stands for the anti-aircraft artillery threat nodes, andhexagonal stands for climate threat nodes, and hollow-point circle stands for themissile threat nodes.UAV flight path optimum selectionAccording to the formula (2-5), the thirty UAV flight path of the cost indicators arecalculated through the Matlab programs, the results are shown in Table 2. According tothe formula (8) and combining with the data of Table 2, the normalized values of eachindicator and reference normalized values are calculated. According to the formula (11)the grey incidence coefficient for each indicator of every path plan are calculated. Theresults are shown in Table 3. And they are compared with the ideal solution. Theseindicators include the fuel consumption cost, the radar threat cost, artillery threat cost,missile threat cost and climate threat cost.
  10. 10. 44 Hu Zhong-hua et alTable 2.The initial value of each indicator.Plan No. Fuel cost Radars cost Artilleries cost Missiles cost climate cost1 26 0.0228 16.0365 7.4902 0.33492 23 0.0206 13.9257 6.7246 0.43553 23 0.0208 14.2023 6.8994 0.29684 22 0.0213 13.2377 6.3322 0.33495 22 0.0212 13.4521 6.3458 0.29686 22 0.0179 13.3222 6.3981 0.29687 23 0.0243 14.3953 6.7374 0.59368 22 0.0228 13.8197 6.2994 0.49309 23 0.0207 14.4064 6.8695 0.334910 22 0.0169 13.3413 6.5782 0.334911 21 0.0176 13.1398 6.3902 0.334912 21 0.0159 13.2410 6.2513 0.334913 21 0.0162 13.1843 6.1421 0.296814 25 0.0198 15.3825 7.5423 0.296815 22 0.0173 13.7966 6.5797 0.296816 21 0.0170 13.0862 6.2336 0.296817 21 0.0163 13.4766 6.2487 0.296818 26 0.0234 15.4881 7.3499 0.473619 23 0.0226 13.9014 6.8143 0.435520 23 0.0180 14.4413 6.8487 0.435521 23 0.0217 14.6534 6.8650 0.334922 21 0.0201 13.3901 6.2239 0.435523 22 0.0168 13.6482 6.6807 0.334924 24 0.0225 14.5526 6.9668 0.435525 22 0.0221 13.7040 6.5854 0.296826 22 0.0221 13.7040 6.5854 0.296827 27 0.0212 16.5315 7.9282 0.334928 23 0.0215 14.4353 6.9410 0.296829 21 0.0215 13.1435 6.1094 0.493030 22 0.0198 14.1398 6.6100 0.2968First, 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 thecomprehensive performance of the each plan relative to the reference indicator ofreference 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 ofcomprehensive evaluation for the 13th plan was the maximum and the value was0.9702. Therefore, the best solution for each path plan was the 13th plan. That was to
  11. 11. Multi-objective and Multi-constrained UAV Path Plan Optimum Selection Based on GRA 45say, 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) andits overall cost was minimal.Table 3. Grey relational coefficient of each indicator.Plan No. Fuel cost Radars cost Artilleries cost Missiles cost climate cost1 0.3750 0.3784 0.3686 0.3971 0.79572 0.6000 0.4719 0.6723 0.5965 0.51693 0.6000 0.4615 0.6068 0.5351 1.00004 0.7500 0.4375 0.9192 0.8032 0.79575 0.7500 0.4421 0.8248 0.7937 1.00006 0.7500 0.6774 0.8795 0.7590 1.00007 0.6000 0.3333 0.5682 0.5915 0.33338 0.7500 0.3784 0.7014 0.8272 0.43069 0.6000 0.4667 0.5661 0.5447 0.795710 0.7500 0.8077 0.8710 0.6598 0.795711 1.0000 0.7119 0.9698 0.7641 0.795712 1.0000 1.0000 0.9175 0.8650 0.795713 1.0000 0.9333 0.9461 0.9653 1.000014 0.4286 0.5185 0.4286 0.3883 1.000015 0.7500 0.7500 0.7080 0.6591 1.000016 1.0000 0.7925 1.0000 0.8798 1.000017 1.0000 0.9130 0.8152 0.8672 1.000018 0.3750 0.3590 0.4177 0.4230 0.456319 0.6000 0.3853 0.6788 0.5633 0.516920 0.6000 0.6667 0.5597 0.5516 0.516921 0.6000 0.4200 0.5236 0.5462 0.795722 1.0000 0.5000 0.8500 0.8882 0.516923 0.7500 0.8235 0.7540 0.6142 0.795724 0.5000 0.3889 0.5402 0.5147 0.516925 0.7500 0.4038 0.7360 0.6564 1.000026 0.7500 0.4038 0.7360 0.6564 1.000027 0.3333 0.4421 0.3333 0.3333 0.795728 0.6000 0.4286 0.5608 0.5223 1.000029 1.0000 0.4286 0.9678 1.0000 0.430630 0.7500 0.5185 0.6205 0.6450 1.0000ConclusionsTo solve the problem of multi-objective and multi-constrained UAV path plan optimumseeking, this paper established the threats model of UAV fiight path plan and its goalsystem 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. 12. 46 Hu Zhong-hua et al.climate threat. The constraints for the greatest impact distance and the effectivedistance 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 isapplied to real path optimization selection problem with seventeen radar threat nodes,flve missile threat nodes, ten artillery threat nodes and two climate threat nodes. Thepath with best comprehensive performance (minimum comprehensive cost) is soughtby the method. The method can avoid the subjectivity and randomness of traditionalselection and provide a theoretical basis for further study three-dimensionalmulti-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 route planning based on the sparse A* algorithm. Journal of Huazhong University of Science and Technology (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 Its Realization 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 for Unmanned Combat Air Vehicles, Ada Aeronáutica ET Astronáutica Sinica, 24(5):435-438. (in Chinese).[6] Dang Yao-guo, Liu Si-feng, Liu Bin et al. (2004). Study on Incidence Decision Making Model of 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 Relational Analysis, 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 training program 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 Antibiotic Selection, The Journal ofGgrey System, 20(04): 311-318.[10] Ren Shiyan,Zou Ningxin, Dong Jiahong, et al. (2008). Grey relational analysis of value of CA19-9 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).
  13. 13. Copyright of Journal of Grey System is the property of Research Information Ltd. and its content may not becopied or emailed to multiple sites or posted to a listserv without the copyright holders express writtenpermission. However, users may print, download, or email articles for individual use.

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