Analytical Modeling of Beyond Visual Range AirCombatDarin Corman, [email protected] (A project report writtenunder the guidance of Prof. Raj Jain)DownloadAbstractThe paper reviews fundamental beyond-visual-range air-combat analytical models and enhances thesemodels. In particular, the fundamental models reviewed are sequential missile exchanges. A one-versus-onemodel is developed to handle the case when both sides are destroyed in a single exchange. The paper thandevelops an analytical model that takes into account engagement geometry, detection range, aircraft speed,missile speed, and missile seeker type for a single missile exchange. This type of model can be used to predictthe order of missile exchanges.Keywords: Air Combat, Beyond Visual Range, Analytical ModelingTable of Contents1. Introduction2. Fundamental Analytical Models 2.1 Undefended Target - One-Sided Engagements 2.2 One Versus One Sequential Engagements 2.3 Many Versus Many One-Sided Engagements3. Extentions To Fundamental Analytical Models 3.1 One Versus One Sequential Engagement - Mutual Kills 3.2 Including Engagement Geometry, Detection, Speed, And Different Weapon Technologies4. Conclusion5. Acronyms6. References1. IntroductionThere is an extensive history of applying mathematics to military science. During World War II the professionoperations research was created to use mathematics to address military problems. During the same timeperiod, warfare was changing rapidly due to new technology such as aircraft. A new branch of the UnitedStates Military was founded, the Air Force, to best exploit this new domain of warfare. Since its inception, theAir Force has relied on science and mathematics to determine investments in technology and form itsdoctrine. The quintessential model of combat was developed by Lanchester while contemplating the impactaircraft would have on warfare (1). During the cold war simple analytical models were developed and used tounderstand air-to-air combat in the jet-age. The Rand Corporation played a key role in the development ofthese models.Analytical Modeling of Beyond Visual Range Air Combat1 of 13Today, simulation is the main tool used to study complex military problems. Performance analysis of variousmilitary systems and systems-of-systems is often the subject. There are many reasons for the predominance ofsimulation in the study of the performance of military systems. The reasons are not unique to the study ofmilitary systems, for example Dr Jain's book on computer systems performance analysis outlines the strengthsof computer simulation (2). Among them are moderate fidelity, moderate abstraction, and the ability to studysystems that do not yet exist. All of these reasons make it more saleable to decision makers. Nevertheless,simulation can be a time intensive and still requires verification from other methods such as measurement,analytical models, or expert opinion. Analytical models, while lacking detail, are simple and fast. They areuseful as screening tools and understanding more complex simulations. They can also be used in largecomputer simulations when fast run times are necessary or when a simulation requires modeling aphenomenon but it is not the subject of the study. In order to be accepted by decision makers, theassumptions that are made in analytical models must not impair the purpose of the analysis.Historic analytical models to study air combat for beyond-visual-range (BVR) missile exchanges, whilehaving there place, do not take into account factors important to current-day aircraft. At the very least auseful model should include the geometry of the engagement, detections, speeds, and types of missiles used.The purpose of this paper is two-fold. The first objective is to survey several analytical models of air-to-airBVR missile exchanges. The second objective is to expand on these models to create a useful model toanalyze current and future aircraft performance.2. Fundamental Analytical Models2.1 Undefended Target - One-Sided EngagementsBasic missile exchange models start with the probability of a single missile destroying a target (3). Thisquantity is known as the single shot probability of kill and is denoted by PKSS. Sequential exchanges ofmissiles between forces can be modeled to determine either the expected number of survivors or kills. Forexample, for a one-sided exchange between a fighter aircraft carrying air-to-air missiles (AAM) and a bombernot carrying AAM, the equations are particularly simple. The probability of the bomber surviving the fighterattack is determined by equation 1 and the probability of the fighter destroying the target is given by equation2.(1)(2)k - number of shotsp - probability of a single shot destroying the targetS(k) - probability of target surviving salvo of shotsP(k) - probability of target destroyed by salvo of shotsFigure 1 illustrates equation 1 and 2 for various values of p and k. Although, this model is extremely simple, itis appropriate for modeling the attack of a slow-moving undefended target.Analytical Modeling of Beyond Visual Range Air Combat2 of 13Figure 12.2 One Versus One Sequential EngagementsA similar model can be developed for an exchange of missiles between two aircraft (3). This is known as a1v1 engagement where v stands for versus. As is convention, one side will be known as blue and the otherred. For example, we will assume blue fires first. Blue fires first and the probability of red being destroyed ispB and the probability that red survives is 1-pB. Red responds and the probability of blue being destroyed isnow pR(1-pB) and probability of blue surviving is 1-pR(1-pB). The sequential exchange continues until bothsides are out of weapons. When blue shoots first, equation 3 and 4 are used to represent the probability of redbeing destroyed by blue after n blue weapons are fired and the probability of blue being destroyed by n redweapons, respectively. Conversely if red fires first, equation 5 and 6 are used. Figure 2a illustrates equations 3and 4 for various values of pB and pR, and figure 2b illustrates equation 5 and 6.(3)(4)(5)(6)PBR(n) - when blue shoots first, probability red is destroyedPBB(n) - when blue shoots first, probability blue is destroyedPRR(n) - when red shoots first, probability red is destroyedAnalytical Modeling of Beyond Visual Range Air Combat3 of 13PRB(n) - when red shoots first, probability blue is destroyedpB - blue probability of single shot destroying redpR - red
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