Neural network guidance based on pursuit-evasion games with enhanced performanceIntroductionTwo-dimensional pursuit-evasion gameStructure of neural network guidance lawSynthesis of neural network guidanceNeural network trainingVerification of neural network approximationPerformance enhancement of the guidance lawAdditional network trainingGame solutions along the fictitious trajectoriesGeneral geometries for shorter-range engagementsPerformance comparisonHybrid guidanceComparison with proportional navigationConclusionsAcknowledgementsReferencesControl Engineering Practice 14 (2006) 735–742Neural network guidance based on pursuit-evasion games withenhanced performance$Han-Lim Choi, Min-Jea Tahk, Hyochoong BangDivision of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701,Republic of KoreaReceived 9 August 2003; accepted 11 March 2005Available online 3 May 2005AbstractThis paper addresses a neural network guidance based on pursuit-evasion games, and performance enhancing methods for it.Two-dimensional pursuit-evasion games solved by the gradient-based method are considered. The neural network guidance lawemploys the range, range rate, line-of-sight rate, and heading error as its input variables. Additional pattern selection methods and ahybrid guidance method are proposed for the sake of the interception performance enhancement. Numerical simulations areaccompanied for the verification of the neural network approximation, and of the improved interception performance by theproposed methods. Moreover, all proposed guidance laws are compared with proportional navigation.r 2005 Elsevier Ltd. All rights reserved.Keywords: Missile; Guidance system; Differential games; Neural networks; Feedback control1. IntroductionThis study deals with missile guidance based onpursuit-evasion games. Pursuit-evasion game, which wasintroduced by Isaacs (1967) in the first place, hasbecome an attractive concept in missile guidance, as theneed for a guidance law guaranteeing good interceptionperformance against a smart target increased. (Ehtamo& Raivio, 2001; Faber & Shinar, 1980; Shima & Shinar,2002) Since pursuit-evasion game considers the worst-case design, it is expected to warrant acceptableinterception performance even when a target aircraftmaneuvers in a very intelligent way. Pursuit-evasiongame considers a minimax optimization problembetween the missile and the target. In other words, themissile makes an effort to minimize a specified payofffunction, while the target maximizes it. Interceptiontime and miss distance are frequently chosen as thepayoff of the game. (Breitner, Pesch, & Grimm, 1993;Shima & Shinar, 2002 ; Tahk, Ryu, & Kim, 1998;Ehtamo & Raivio, 2001) Intercept time has beenpreferred as the payoff if the model dynamics arecomplicated, since it entails easier mathematicalformulation.It is needed to obtain a feedback guidance law forreal-time implementation of pursuit-evasion game. Noone ca n expect good interception performance whenusing pre-programmed open-loop guidance, since realengagement situations are not exactly same as those oneconsidered before. Unfortunately, many solvers forpursuit-evasion game merely give open-loop solutions.Some researches have been conducted for obtainingfeedback type game solutions. (Ben-Asher, 1996; Faber& Shinar, 1980; Menon, 1989) However, all these worksled to co mplicated problem formulations, which inevi-tably limited the extension of each idea to very simplecases. A neural network can be a good help, since itis a universal approximator (Hornik, Stinchcombe, &ARTICLE IN PRESSwww.elsevier.com/locate/conengprac0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.conengprac.2005.03.001$This is the revised version of the paper with Editorial ReferenceNumber CD 2379.Corresponding author. Tel.: +82 42 869 3718;fax: +82 42 869 3710.E-mail address: [email protected] (M.-J. Tahk).White, 1989). It can yield an approximate functionalrelation between the state variables and the game-optimal control inputs. In addition, Song and Tahk(1998, 1999, 2001, 2002) have substantiated the feasi-bility of this concept in missile midcourse guidance,although they considered one-sided optimal problemsrather than game-optimal problems. For this reason, inthis work, a neural network is employed to synthesize afeedback guidance law from open-loop solutions.The authors have studied neural network guidancelaw based on the pursuit-evasion game solution sobtained by using the gradient-based method for bothtwo-dimensional and three-dimensional situations.(Choi, Park, Lee, & Tahk, 2001; Choi, Tahk, Bang, &Lee, 2001; Lee, Choi, Tahk, & Bang, 2001). Whileinvestigating the outcomes, however, it is observed thatto select neural network input variable plays a key rolein determining the performance of the guidance law, andit is also observed that the performance of the neuralnetwork guidance law degrades too much, when thetarget does not maneuver along the game-optimaltrajectory obtained in advance. Based on these twoobservations, this paper focuses on the selection of thenetwork input variables and on the ways of overcomingthe undesirable feature above.This paper derives a neural network guidance lawfrom two-dimensional pursuit-evasion games . Thisstudy focuses on only two-dimensional situations, sincethey are more appropriate for elucidating the qualitativefeatures of the pursuit-evasion game and the neuralnetwork guidance. Four variables, i.e. the range, rangerate, heading error, and line-of-sight (LOS) rate areselected as neural network input variables. Two methodsare also proposed for the sake of improving theinterception performance against not game-optimallymaneuvering targets: additional pattern scenario selec-tion, and hybrid guidance. In addition, perfor mance ofthe neural network guidance laws is compared withproportional navigation.2. Two-dimensional pursuit-evasion gameTwo-dimensional pursuit-evasion situation is consid-ered as described in the Fig. 1. The equations of motionof the missile and the target are express ed as follows._xi¼ vicos gi,_yi¼ visin gi,_gi¼viRiui¼1viv2iRiui¼aivi; juijp1,_vi¼v2iRiðaiþ biu2iÞ,ði ¼ M; TÞ, ð1Þwhere x, y are the missile’s or the target’s position, v isthe speed and g is the flight path angle, respectively. u isthe normalized control input, and R is the minimumturn radius . In addition, a is the