Research Article A Case Study on Air Combat Decision Using Approximated Dynamic Programming

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1 Mathematical Poblems in Engineeing, Aticle ID 84, pages Reseach Aticle A Case Study on Ai Combat Decision Using Appoximated Dynamic Pogamming Yaofei Ma, Xiaole Ma, and Xiao Song School of Automation Science and Electical Engineeing, Beijing Univesity of Aeonautics and Astonautics, Beijing 9, China Coespondence should be addessed to Yaofei Ma; mayaofeibuaa@6.com and Xiao Song; songxiao@buaa.edu.cn Received 6 June 4; Accepted August 4; Published 5 Septembe 4 Academic Edito: Minui Fei Copyight 4 Yaofei Ma et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. As a continuous state space poblem, ai combat is difficult to be esolved by taditional dynamic pogamming (DP) with discetized state space. The appoximated dynamic pogamming (ADP) appoach is studied in this pape to build a high pefomance decision model fo ai combat in vesus scenaio, in which the iteative pocess fo policy impovement is eplaced by mass sampling fom histoy tajectoies and utility function appoximating, leading to high efficiency on policy impovement eventually. A continuous ewad function is also constucted to bette guide the plane to find its way to winne state fom any initial situation. Accoding to ou expeiments, the plane is moe offensive when following policy deived fom ADP appoach othe than the baseline Min-Max policy, in which the time to win is educed geatly but the cumulated pobability of being killed by enemy is highe. The eason is analyzed in this pape.. Intoduction Unmanned aeial vehicle (UAV) plays an impotant ole in moden battlefield. Fo the past decades, UAV has made significant advancement on both hadwae and softwae and achieved mission capabilities including simple ones like intelligence, suveillance, and econnaissance (ISR), and complex ones like electonic attack, gound tagets stike, suppession o destuction of enemy ai defense (SEAD/DEAD), and othes. Accoding to the development oadmap [] poposed by UAV technical leading counties, even the mission of ai combat, which has been believed to be the dominated domain of human pilots due to the dynamic and complexity on tactical decisions, is possible to be caied out by autonomy UAV in the nea futue. Howeve, the decision technologies suppoting automatic ai combat ae fa fom matuity. They ae still on the way fo bette obustness, intelligence, autonomy, team coopeation, and adaption to complex envionment. The methods applied in this domain include game theoy [ 4], knowledge-based decision [5 9], gaphic based methods like influence diagam [], and othes. Dynamic pogamming (DP) [] isoneof themostpowefulmethodsfoitsadaptationtodynamic envionment and the capability to impove policy constantly by leaning []. Howeve, taditional DP appoach is not suitable to esolve continuous state space poblem like ai combat, in which the computation complex becomes intactable because of the cuse of dimensionality. In this pape, a tactical decision famewok employing 5 appoximated dynamic pogamming (ADP) method [ 5] ispoposedfoaicombatmission.thetaitofadpmethod is that the utility function is leaned fom mass sampled states in poblem space athe than fom scatch, which lead to high efficiency in policy conveging. As the esult, ADP canbeusedasasecondstagetooltoimpovethepolicy deived fom othe decision systems (denoted by the fist stage tool fo decision hee), fo example, a knowledge-based system. If we teat the combat taces poduced in the fist stage tool as the sampled states, then ADP algoithms can lean its utility function and policy fom these states diectly. Consideing the optimizing capability of ADP inheited fom DP appoach, the leaned policy can be impoved constantly to achieve bette decision pefomance. Thus, the meits of diffeent decision system ae combined togethe. The content of this pape would be aanged as follows. In Section, the vesus ai combat poblem is fomulated

2 Mathematical Poblems in Engineeing ±π R π/ π/ Antenna tain angle (+ATA) LOS of ada R Aspect angle (AA) Figue : The geometical measues of the fiing position. with DP fomation. In Section, the ADP method is biefly eviewed. Section 4 discusses the ewad function fo ai combat, which is designed to guide the UAVs to ente into goal states smoothly. Some featues ae also specified to gain an insight of engagement situation. In Section5, the key algoithms of ADP decision famewok ae poposed. The followed compaative expeiments (Section 6) validate the effectivenessofthepoposedfamewok.. Poblem Fomulation A vesus ai combat scenaio involves two opponent planes (denoted by ed and blue, whee the ed is supposed to be my side). Omitting vetical movement, the kinematic equations of the plane ae ψ= f n V x=v cos (ψ) y=v sin (ψ), whee V isthescalavalueofvelocity,whichisassumedto be const duing the combat. ψ [ π, π] is yaw angle and is defined as the deviation of velocity fom noth (the y axis). ψ is contolled by f n. f n is plane s nomal oveload, which always points ight fom the gavity cente of the plane and is othogonal to velocity. In ou contol schema, f n can take a value fom thee options once a time: {,, }. Theplane will tun counteclockwise, tun clockwise, and keep cuent velocity diection, espectively, with these values. The goal of the planes is to occupy advantage position by tactical decision and gain the fie oppotunities at its ival. The state space of ai combat can be descibed with vecto () x={x,y,ψ,x b,y b,ψ b }, () whee subscipt and b efe to ed and blue, espectively. Any state s is an instance of x.with(), the state tansition in combat space can be epesented as a function s =f(s, a =f n, a b =f n ) () which means the cuent state s will tansfe into a new state s afte pefoming a and a b. The goal state is eached when one plane gains oppotunities to fie at its opponent. The fiing position is defined by thee geometical measues: (a) Aspect angle <π/. Aspect angle (AA) is a elative angle between the longitudinal symmety axis (to the tail diection) of the taget plane and the connecting line fom taget plane s tail to attacking plane s nose. AA <π/efes to aea whee the killing pobability is high when attacking fom ea consideing most close-combat ai missiles ae infaed guidance; (b) Antenna tain angle < π/6. Antennatainangle (ATA) is the angle between attacking plane s longitudinal symmety axis and its ada s line of sight (LOS), as Figue shows. This citeion defines an aea fom which the taget plane is difficult to escape with ada locking. (c) Relative ange (R) between two planes: this citeion makessuethatthetagetplaneiswithintheattacking ange of ai-to-ai weapon.. ADP Method Review DP defines adaptive leaning pocess and its mathematic model is Makov decision pocess (MDP). In DP fomulation, the ai combat can be descibed as a discete time decision poblem with five-tuples: {S,A,P,R,U}: () S = {s} is the poblem space defined with state vaiable x; s is the instance of x; () A(s) is the finite action set available in state s, fom which the plane selects one to execute at each decision

3 Mathematical Poblems in Engineeing inteval. In ou poblem, A(s) is same fo each state s and thus can be simply denoted as A; () P(s s,a)isthepobability of tansition fom state s to s ; (4) R(s) is the ewad of state s. Ifs is visited multiple times duing the combat, the ewads ae discounted cumulatedtofomutilityvalueofthatstate; (5) U(s) is the utility of state s.itsvalueisthecumulated ewads of multiple visiting. If evey state is visited adequate times, the utility distibution will convege to the optimal one, by which the optimal policy is deived. The decision pocess stats fom an initial state s and then selects action to pefom. The action inteacts the envionment and leads to a new state, and so on. Then, the utility of the stating state is the expectation of discounted cumulated ewads on all states following the stat one: U (s) =E{R(s )+γ R(s )+γ R(s )+ s=s,π(s)}, (4) whee γ (, ) is the discounted coefficient, making sue U(s) conveges eventually. Policy π(s) a is a mapping fom state space to action space. Fo a fixed policy π, the utility satisfies Bellman equation U (s) = s P(s,π(s),s ) [R(s s,π(s))+γ U(s )]. The optimal utility U is the value function that simultaneously maximizes the expected cumulative ewad in all states s S.BellmanpovedthatU istheuniquesolutionof (5): U (s) = max { P (s, a, s ) [R(s s,a)+γ U (s )]}. a s (6) Actually, U can be obtained though iteations on Bellman equation: U k+ (s) =TU k (s) (5) = max { P (s, a, s ) [R(s s,a)+γ U k (s )]}. a s (7) T is denoted by Bellman opeato, epesenting the iteative impovement on U by tavesing states thoughout the space eventually. Duing this pocess, P(s, a, s ) would also convege to its tue distibution. Then, the optimal policy can be deived: π (s) = agmax { P (s, a, s ) [R(s s,a)+γ U (s )]}. a s (8) As we can see fom (4) (8), taditional DP method needs to tavese discete states iteatively, esulting in tabula utility function. This appoach is not suitable to esolve continuous state space poblem. Discetization on state space leads to two defects: (i) the uneasonable assuming that utility function is const in each discete state cell and (ii) the cuse of dimensionality. ADP method mitigates these poblems with two opeations: (i) sampling mass states effectively fom poblem space, thus educing the consumed time on space exploation; (ii) appoximating state utility U using sampled states, with which the nea-optimal policy, athe than the optimal one, is employed to detemine actions. Denoting the sampled states as a set S={s,s,...,s m },wehave U k+ =T U k, (9) whee U k is the cuent appoximation of utility; U k+ is one Bellman iteation fom U k.then, U k+ canbeappoximated based on U k+. Thee ae multiple options fo appoximation opeation [6, 7]; the least squaes appoximation is used hee: β k+ =(S T S) U k+, () U k+ =S β k+, whee β is appoximation coefficients vecto and S is sampled states set. Nomally, a set of featues need to be defined to gain an insight on chaacteistics of the studied poblem. The appoximated utility function will convege moe quickly and be moe pecise since these featues come fom pilots combat expeiences in eal wold. We have Φ (S) = {φ (S),...,φ c (S)}, β k+ =(Φ T Φ) U k+, U k+ =Φ β k+. () Φ is featue vecto. As a conclusion, the steps of ADP method can be biefly listed as follows: (a) to sample states set S in poblem space; (b) to get the one iteation impovement utility U k+ fom cuent utility U k ;theinitialvalueof U k can be set as theewadofinitialstates;thatis, U = R(S ),whee S is initial state set; (c) to update the next value of appoximated utility U k+ following ()and(); (d) if the policy still needs to be impoved, go back to (a).

4 4 Mathematical Poblems in Engineeing ADP Lean() Input vaiables: () S:sampledstatesset; () N: the numbe of leaning ound. () π mm (S): blue plane s policy deived fom Min-Max appoach. Output vaiables: () U k : utility function appoximated. Local vaiables: () A b : action vecto of blue plane deived fom Min-Max policy; () A : action vecto of ed plane deived fom cuent utility; (4) U k : one step impoved utility function by Bellman iteation; (5) Φ: featues vecto used to compute appoximation coefficients; (6) β k : vecto of appoximation coefficients. Code: () U k= = R(S); () FOR k=:n,do: () A b =π mm (S); (4) {A, U k }= (5) ag max {λ U k (f(s,a,a b )) + R (f (S, A, A b ))}; A (6) Φ(S) = {φ (S),φ (S),...} (7) β k =(Φ T Φ) Φ T U k (8) U k =Φ β k (9) END () RETURN U k=n ; Algoithm : Utility function appoximating based on sampled states. Rollout ADP Policy() Input vaiables: () s:thecuentstate; () N oll : the numbe of ollout steps; () U k=n : utility function appoximated in ADP Lean algoithm; (4) π appx (s): ed plane s policy deived fom U k=n,see(8). Output vaiables: () a best : the best action espect to cuent state s. Local vaiables: () U best : to cache the maximum utility esponding to diffeent actions; () s : the cache the next state computed by system equation. Code: () a best =NULL; () U best =NULL; () FOR a ={,,},DO: (4) s =f(s,a,π mm (s)); (5) FOR i=:n oll,do: (6) s f(s,π appx (s ), π mm (s )); (7) END (8) IF {γ U k=n (s ) + R(s )} > U best,then: (9) U best =γ U k=n (s ) + R(s ); () a best =a ; () END () END () RETURN a best ; Algoithm : Rollout decision pocedue using appoximated utility function.

5 Mathematical Poblems in Engineeing 5 Table : Featues evaluating combat situation. I III II IV Figue : Fou basic initial situations in ai combat. 4. Rewad Function and Combat Featues Befoe giving ADP algoithm, the ewad function R(s) needs to be discussed fistly since it is a necessay pat in ADP steps. As (4)shows,theutilityisactuallythediscountedcumulated R(s) alongstatetace.thus,apopelydefinedr(s) can bette guide plane appoach to goal state fom any stating state. The computation of R(s) is domain elated. As fo ou scenaio, the attacking plane in its goal state gets ewad +, and the taget plane in the same state gets aspunishment. The ewad in othe states is. Intuitively, with these discete ewads, the planes will spend moe time on space exploation to find tace fom stating state to goal state. To bette guide the plane, a ewad function is defined as R = [ ( AA /π) + ( ATA /π) ]e (( R RD)/(π k)), () whee R D is the expected attacking ange of weapons, R is theelativedistancebetweenplanes,andk is an coefficient adjusting the influence of R in total ewad. With (), the plane occupying fiing position (AA =, ATA =,andr=r D )getsewadr = ;theplane unde attack (AA = π,ata = π,andr=r D )gets ewad R =. In othe states, the ewad will incease continuously and monotonically fom the wost state to the most advantage state. To emphasize the punishment in bad state (the punishment will guide planes to avoid these states), a simple linea tansfomation is applied to R to get the final ewad function: R (s) = R. () To constuct the utility function, some geometic featues [8] ae specially defined to descibe combat situation, as Table shows. These featue ae optional fo utility appoximation in ADP steps because we can use sampled states instead, as () shows. Howeve, they ae moe staightfowad to captue the tue utility of states, and that is why human pilots also use them to judge thei situations in ealwold combat. In othe wods, these well-defined featues ae moe epesentative to appoximate utility function. 5. Method Inthecombatscenaio,theedplaneismakedas my side, and the blue one is maked as enemy. To descibe Featues (Φ) Desciption R Relative distance between ed and blue planes. AA Aspect angle. Refeence is Figue. AA The changing atio of AA. AA Absolute value of AA. ATA Antenna Tain angle. ATA Absolute value of ATA. ATA The changing atio of ATA. HCA The eo on yaw angle of both planes. HCA Absolute value of HCA. ADP appoach, a efeence decision algoithm, the Min- Max seach algoithm [9] is employed hee. The Min- Max algoithm looks into futue fo n steps, using domain knowledge to detemine the acting consequent befoe giving final decision. ADPappoachinvolvestwoalgoithms:(i)theleaning algoithm (ADP Lean()), in which the utility function is appoximated, and (ii) the decision algoithm (Rollout ADP Policy()), in which the final action is detemined based on ADP policy deived fom leaned utility. The ADP Lean() algoithm is displayed in Algoithm. In ADP Lean(), theutilityfunctionisappoximated with sampled states, which is expected to be sampled fom fequently visited space, to fully captue the changes on utility values in these aeas. An option is to use the tajectoies poduced in eal-wold combat o othe authoitative decision tools fo ai combat, since the tajectoies themselves indicate the high pobability of being visited in combat. In this pape, a scenaio is built to get combat tajectoies, whee two ival planes all take Min-Max policies, and thei tajectoies ae ecoded as S. The initial value of U k is assigned as the ewad of S (line in Code section)andthenisimpovedn ounds. In each ound, fistly, the blue plane s action A b is detemined by Min-Max policy (line ). Secondly, the ed plane s action A is selected by applying one step Bellman opeato. The changed utility U k is also ecoded (lines 4-5) fo futhe use. Thidly, the featue vecto is updated (line 6), with which the least squaes appoximation is pefomed to appoximate U k accoding to U k (lines 7-8). The appoximated utility function U k=n etuned by ADP Lean() aleady can be used to give decisions, as (8) shows. Howeve, a ollout pocedue is employed hee to futhe impove the quality of final decisions, as Algoithm shows. Assuming the ed plane is making decision in Rollout ADP Policy(). ItwillnotfollowADPpolicydiectly.On the contay, it ties each possible action (line in Code section). Fo each possible action, the ed plane s futue state

6 6 Mathematical Poblems in Engineeing s s s 6 s 6 s 9 s s s s 5 s 5 s 8 s s s 7 4 Figue : Baseline expeiment: combat stat fom Setup 4. Both ed and blue follow Min-Max policies. The plane icons appea at the stat position in each subfigue; the x, y positions (denoted as and ) ae scaled down fo bette epesentation. The ed plane wins at last and TTW = s. (s )isolledoutfon oll steps(lines5 7).Theedplane follows ADP policy duing this pocess and the blue one follows Min-Max policy. The sum of ewad and utility of s is compaed with the histoical best value U best :ifthe fome is bigge than U best,thenupdateu best and ecod the coesponding best action a best (lines 8 ). Having tied all possible actions, the ed plane gets the best action a best in state s.

7 Mathematical Poblems in Engineeing 7 Table : Fou initial setups of ai combat. Initial situation Desciption x b y b ψ b x y ψ Setup Offensive.5 Setup Neutal.75 π/ Setup Defensive Setup 4 Confonting π 5 s 4.5 s 4 s.5 s Figue 4: Engagement pocess stats fom Setup with ed plane following ADP policy π k=4 and blue plane following π mm b. TTW = 4.5 s. 4 Figue 5: Engagement pocess stats fom Setup with ed plane following ADP policy π k=4 and blue plane following π mm b. TTW =.5 s. 6. Simulation and Analysis The initial state of vesus ai combat can be classified into 4 basic situations (fom ed plane s pespective): offensive, neutal, defensive, and confonting, as Figue shows. In ou expeiments, all fou initial situations ae configued (Table ) to compae the pefomance of ADP policy and Min-Max policy. The expeiments ae aanged as follows. Fistly, a baseline expeiment is conducted in which both ed and blue planewouldtakemin-maxpolicy(denotedasπ mm ). The decision pefomance of π mm is teated as the baseline to compae ADP policy. Secondly, the leaned ADP policy is applied with same initial situations. ADP policy is denoted as π k=n,wheen is the leaning ounds. Fo example, π k=4 meansthispolicy is appoximated afte 4 ounds. The decision pefomance is measued with metics: (a) the aveage time to win (TTW); the winning states have been defined in Section ;the attackingplaneneedstoholdthatstatefoatleastseconds to win; (b) the accumulated pobability of being killed (APK); this indicates the total isks of one plane duing the combat, in which the pobability of being killed by enemy would be cumulated. A good policy would esult in both small TTW and APK. To speed up the expeiment, a bigge f n is assigned to ed plane which means it can change diection moe quickly. This measuement would avoid long time standoff when both planes follow the same policy. This pefomance advantage will not influence policy compaison since both set expeiments use the same configued planes. The baseline expeiments ae conducted fistly. Only the esult of Setup 4 (confont) is displayed hee consideing the pape space limitation, as Figue shows. Figues 4, 5, 6,and7 show the esult of each initial setup whee ed plane follows ADP policy and blue plane follows Min-Max policy. Compaing Figues 7 and, wecanseethe pefomance of ed plane is impoved geatly by taking ADP policy, in which the TTW is educed fom s to.5 s. The compaison on decision pefomance is displayed in Table. As we can see, the TTW of π k=4 is educed in all setups compaed to π mm,especiallyinsetup. This means π k=4 can guide the ed plane to get id of the chasing quickly and find its way to occupy the fiing position. On the othe hand, the APK is slightly highe with ADP policy. Theseesultsshowthataplaneismoeoffensivewhen following ADP policy. The plane is likely to occupy fiing position isk at the isk of being killed. This phenomenon can be explained fom the woking mechanism of two decision

8 8 Mathematical Poblems in Engineeing s.5 s.5 s 5 s 5 s 7.5 s 7.5 s s 7 s.5 s 4 7 Figue 6: Engagement pocess stats fom Setup with ed plane following π k=4 and blue plane following π mm b. TTW =.5 s. appoaches. In Min-Max algoithm, the decision is finally made consideing each possible eaction fom the opponent. This leads to a consevative style in decision making. By contast, the ADP appoach uses utility to guide the plane tomakebestpofitbyactingpopelyandavoidpunishment atthesametime.thissomehowmakestheplaneabandon consevative choice fo high ewad in the futue. This esult alsopovedthatadpappoachiseffectiveandofhigh pefomance to esolve ai combat poblem 7. Conclusion This pape studied the vesus ai combat decision poblem and employed ADP appoach to esolve it quickly and

9 Mathematical Poblems in Engineeing 9 s.5 s.5 s 5 s s 7.5 s s 9.5 s Figue 7: An engagement stats fom Setup 4 with ed plane following ADP policy π k=4 and blue plane following π mm b. TTW = 9.5 s. Initial setup Table : Compaison on pefomance of diffeent policies. TTW (s) APK 4 4 π mm π k= effectively. The ADP appoach involves two opeations: (i) leaning utility function fom mass sampled states athe than fom scatch; (ii) making final decision by evaluating the futue incoming of any possible action. Compaative expeiments show that the policy initially poduced by Min- Max algoithm is impoved geatly afte ADP pocess. In the futue wok, we plan to build a hybid-decision famewok fo UAV in which the decision functionality is divided into two pats. The fist pat is esponsible fo making initial acting policy fo specific task, which is a knowledge-based decision module, and can handle lage scale, complex task envionment. The second pat is the ADP module poposed in this pape. This ADP module gets state samples fom the fist pat, with which the utility function is appoximated and the acting policy is impoved. This pocess combines the advantages of diffeent decision famewoks togethe. Conflict of Inteests The authos declae that thee is no conflict of inteests egading the publication of this pape. Acknowledgment This eseach is suppoted by the Amament Reseach Foundation (94A4HK4). Refeences [] Defense Technical Infomation Cente, Unmanned aicaft systems oadmap 8 [R/OL],, [] R. Isaacs, Games of pusuit, 95. []L.A.Petosjan,Diffeential Games of Pusuit, vol.,wold Scientific, 99.

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11 Advances in Opeations Reseach Advances in Decision Sciences Jounal of Applied Mathematics Algeba Jounal of Pobability and Statistics The Scientific Wold Jounal Intenational Jounal of Diffeential Equations Submit you manuscipts at Intenational Jounal of Advances in Combinatoics Mathematical Physics Jounal of Complex Analysis Intenational Jounal of Mathematics and Mathematical Sciences Mathematical Poblems in Engineeing Jounal of Mathematics Discete Mathematics Jounal of Discete Dynamics in Natue and Society Jounal of Function Spaces Abstact and Applied Analysis Intenational Jounal of Jounal of Stochastic Analysis Optimization