JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 25, No. 4, July– August 2002 Nonorthodox Guidance Law Development Approach for Intercepting Maneuvering Targets J. Shinar ¤ and T. Shima † Technion—Israel Institute of Technology, 32000 Haifa, Israel Two aspects of a new mindset are applied to interceptor guidance: 1) the mathematical formulation of an interception scenario against maneuverable targets and 2) the relationship between the estimation process and optimal guidance law design. The interception of a maneuverable antisurface missile is formulated as a zero- sum pursuit evasion game. The perfect information game solution, which guarantees a robust hit-to-kill homing accuracy, requires, among other things, the knowledge of the target maneuver. This variable cannot be directly measured and has to be estimated based on noisy measurements. The greatest error source in the estimation of time- varying target maneuvers is the inherent delay due to the convergence time of the process. The estimation process is modeled as a pure information delay. Such a delay is partially compensated by a new guidance law based on the solution of a delayed information pursuit– evasion game. The new approach represents a potential breakthrough in guidance law design predicting reduced miss distances and robustness even in stressing interception environments. The improved accuracy is conrmed by a set of Monte Carlo simulations in a ballistic missile interception scenario with noisy measurements. I. Introduction I N contrast to the impressive technologicalprogress achieved by the guided missile community, the concepts of interceptor guid- ance law development have remained (unfortunately) conservative. This conservative mindset can be attributed to that until recently the target of an interceptormissile was an inhabitedaircraft,against which the missile had substantial advantage in speed, maneuver- ability, and agility. Moreover, miss distances of the order of a few meters (compatible with the lethal radius of the missile warhead) were considered admissible. Thus, the effort to develop improved guidance laws may have seemed to be unnecessary. The 1991 Gulf War introduced a new type of target, namely, the tactical ballistic missile (TBM), able to carry nonconventional warheads. Successful interception of a TBM, much less vulnerable than an aircraft, requires a very small miss distance or even a di- rect hit, creatinga new challengefor the guided missile community. Several defense systems against ballistic missiles are currently in development to meet the threat. All of them have been designed against ballistic targets, ying predictable trajectories. The avail- ableadvancedtechnologyallowedthesedefensesystems(ARROW, PAC-3, THAAD) to demonstrate, in spite of using conventional guidance and estimation concepts, an excellent homing accuracy (sometimes even a kinetic hit to kill) against such nonmaneuvering targets. 1¡3 Although known TBMs were not designed to maneuver, due to their high reentry velocity they have a substantial maneuverabil- ity potential. Moreover, this great maneuverabilitypotential can be made applicable by a modest technical effort. The same is true for futurehigh-speedantishipor cruisemissiles.Paradoxically,thesuc- cessfulcurrentdevelopmentof antiballisticmissile defensesystems Presented as Paper 2000-4273 at the AIAA Guidance, Navigation, and Control Conference, Denver, CO, 14– 17 August 2000; received 12 April 2001; revision received 6 December 2001; accepted for publication 18 January 2002. Copyright c ° 2002 by J. Shinar and T. Shima. Published by the American Institute of Aeronautics and Astronautics, Inc., with per- mission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/02 $10.00 in correspondence with the CCC. ¤ Professor Emeritus, Faculty of Aerospace Engineering; aer4301@ aerodyne.technion.ac.il. Fellow AIAA. † Doctoral Student, Faculty of Aerospace Engineering; currently System Engineer, RAFAEL, Ministry of Defense, Department 35, P.O. 2250, 31021 Haifa, Israel; talsh@rafael.co.il. Member AIAA. can motivate the development of a new generation of highly ma- neuverable antisurface missiles. Against such threats, interceptor missiles will have only a marginal maneuverabilityadvantage. Re- cent simulation studies of anticipated antimissile defense scenarios clearly indicate that currently used guidance laws and estimation techniques are unable to guarantee an adequate homing accuracy for a hit to kill in the interception of the expected highly maneuver- ing targets. 4¡6 The great majority of advanced missile guidance laws in current use were developed by using a linearized kinematical model and solvingalinearquadraticoptimalcontrolproblem,wherethelimited maneuver potential of the interceptor was not explicitly taken into account. 7 As is well known, the optimalcontrolconceptrequiresin- formationon the currentandfuturetargetmaneuvers.In most cases, for the sake of simplicity, constant target maneuvers were assumed. There is a basicdeciencyin formulatingtheinterceptionof a ma- neuverable target as an optimal control problem. Target maneuvers are independently controlled; thus, future target maneuvers cannot be predicted. As a consequence, the optimal control formulation of such problems is not appropriate. The mathematical framework for analyzing conicts controlled by two independent agents is in the realm of dynamic games. Thus, the scenario of intercepting a maneuverable target has to be formulated as a zero-sum pursuit– evasion game. The game solution provides simultaneouslythe opti- mal pursuerstrategy(the missile’s guidancelaw), the optimal evader strategy (the worst target maneuver), and the value of the game (the miss distance guaranteed to the interceptor missile as well as to the maneuvering target, by using the respective optimal strategies). Although the concept of such a formulation dates back to the 1950s and was published in the seminal book by Isaacs in 1965, 8 the guided missile industry has not fully recognized the potential involved in it. Nevertheless, the idea has raised some academic in- terest, as is evident in the open literature.Whereas a linear quadratic game formulationwith an ideal dynamic model lead to proportional navigation 9 as an optimalguidancelaw, a more realisticanalysisrec- ognized that the controls are bounded and interceptor missile dy- namics should be represented at least by a rst-order transfer func- tion. Such an analysis, 10 limited to a planar scenario,was published in 1979 and was extended later to three dimensions. 11 It yielded a game optimal guidance law that explicitly accounts for the limited interceptormaneuverability,and the assumption of ideal target ma- neuver dynamics (the worst case for the interceptor) eliminated the need of knowing the actual target maneuver. This differential game guidance law (DGL/0) provides robustnesswith respect to the type of target maneuver but cannot guarantee a zero miss distance. If, in 658