TERMINAL IMPACT ANGLE AND ANGLE-OF-ATTACK CONTROL GUIDANCE FOR SURFACE-TO-AIR MISSILE USING TVC

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1 ERMINAL IMPAC ANGLE AND ANGLE-OF-AACK CONROL GUIDANCE FOR SURFACE-O-AIR MISSILE USING VC Seong-Min Hong*, Min-Guk Seo*, Min-Jea ahk* * KAIS Kewords: wo-staged surace-to-air missile, Impact angle control, Optimal guidance law Abstract his paper deals with impact angle and angleo-attack control guidance or a two-staged surace-to-air missile. he equation o motion is simpliied as pith planar motion. he vehicle can be controlled during the propellant burning phase, terminal impact angle constraint should be satisied within the limited time. o conduct a stable stage separation, urthermore, the light path angle and pitch attitude angle should be aligned. Namel, the angle o attack o the missile should be zero. In this paper, the guidance law to satis the constraints described above is obtained rom an optimization tool, GPOPS. Also, the optimal path and control input results are depicted. Introduction Generall, impact angle guidance laws have been developed or anti-tank missiles or anti-ship missiles to maximize their warhead eect and penetrate the most vulnerable point o the target [-4]. Impact angle guidance laws are also used or surace-to-air missiles to accomplish the eective engagement geometr [6]. I the missile knows the states o the target, an impact angle guidance law could be used. For anti-ballistic missiles, multi-staged missile coniguration is used such as erminal High Altitude Area Deense (HAAD), Standard Missile-3 (SM-3) and Aster series. Especiall, HAAD does not have tail ins or canard or the control and onl use hrust Vector Control (VC) sstem. he missile can be controlled during the boosting phase and the impact angle constraint should be satisied within the limited time. Ater the burn-out moment, the missile and the warhead are separated. During the separation, a small angle o attack causes the aerodnamic unstable so that the warhead collides with the separated boost rocket motor. hus, the angle o attack o the missile should be zero at the burn-out time. In order to satis the impact angle constraint, man impact angle guidance laws have been developed based on the optimal control theor [-5]. Roo et al. suggested impact angle control guidance law considering lag-ree sstem and st order lag sstem []. In [3], the derived the optimal guidance law with the energ cost unction weighted b a power o the time-to-go. Cho [5] investigated the relationship between Proportional Navigation Guidance (PNG) laws and optimal guidance laws weighted b time-to-go. Not onl the impact angle, but the impact time is also adjusted [4]. Lee et al. had derived a guidance law considering impact angle and time simultaneousl,. For the surace-to-air missiles using VC, the missile should be delivered to a predicted handover point (PHP) during the boost phase. Miss distance based on the engagement geometr between the missile and the target should be compensated b the kill vehicle, however, the magnitude o lateral acceleration o a kill vehicle is generall smaller than that o VC sstems. Also, the speed o the missile varies due to the thrust and the mass variation o the missile. In this paper, an optimal problem or the missiles considering the constraints described above is deined. hen, the optimal guidance command is calculated b the optimization tool.

2 SEONG-MIN HONG, MIN-GUK SEO, MIN-JEA AHK Y u x M, M M V zt PHP P x, P Generall, the missile velocit is constant and a small angle approximation is used. Equation () can be linearized as z V v v a V (3) erminal constraints or the problem are given below. Fig.. Missile Engagement Geometr his paper is composed o as ollows. Section deals with the problem ormulation. wo ormulations, general case and the missiles using VC case are compared. Optimization results using a commercial optimization tool, GPOPS, are depicted and the conclusion o this paper is ollowed. Problem Formulation or Optimal Impact Angle Guidance Laws. General Impact Angle Control Guidance Law In [-4], optimal guidance laws considered that the missile is a point mass and the engagement kinematics is deined on the impact angle rame xp, P. he missile engagement geometr is depicted in Fig., where xm, M is missile position, V is velocit, u is control input, is light path angle, is terminal impact angle constraint and z t X is lateral position with respect to the impact angle rame. hrust and angle o attack is described in the next section. () he missile kinematics are described as ollowing equation. z V sin t V a () vt z t (4) he cost unction is determined to minimize the control input a. where x t min J S u tdt u (5) t is state vector, ut is control input and S is weighting matrix. he optimal control or this problem is given as 6 4 u v t t (6). Impact Angle Guidance using VC to In the previous section, we assume that the missile is a point mass and the velocit does not change. his paper deals with the missile using VC, we make ive basic assumptions as ollows: A. he target (PHP) is predetermined, thus, stationar. A. he engagement occurs in a horizontal plane. A3. he thrust vector is aligned with the bod axial direction. A4. he missile is controlled b using the missile angle o attack. A5. he mass variation, aerodnamic orces and the gravit are neglected. Under the assumptions and or a stationar target, we can obtain the engagement kinematics as ollows: go

3 ERMINAL IMPAC ANGLE AND ANGLE OF AACK CONROL GUIDANCE FOR SURFACE-O-AIR MISSILE USING VC x V cos V sin cos V m sin mv (7) where is thrust, is angle o attack and m is mass o the missile. As the missile approaches to the PHP, the downrange is monotonicall decreases. Now we can replace the independent variable time with another monotone variable, the downrange. hen, equation (7) can be rewritten as d V tan dv cos mv cos d sin mv cos (8) Following relationships are obtained b using small angle approximation. sin cos tan cos (9) he equation (8) is simpliied b substituting equation (9) into (8). d dv mv d mv () In the assumption A5, the mass variation is neglected, so the velocit o missile can be integrated as a unction o downrange. V V x x () m Finall, we can obtain the equation o - directional position and light path angle b substituting equation () into (). where ' V m below. t subscript d d m V x x ' () is missile initial velocit and. he state boundaries are given x t t x t t (3) equals to the burn-out moment and means the terminal constraints o the states because the missile should be located at the PHP when the propellant burns out. he independent variable is changed as downrange, equation (3) should be changed using ollowing terms. x x x x x (4) Now, we can deine the optimal control problem. As we mentioned above, the angle o attack should be zero as the propellant burn-out moment nears. he cost unction is deined considering angle o attack constraint. min J x (5) x x x he denominator decreases as the missile approaches to the target, magnitude o the 3

4 SEONG-MIN HONG, MIN-GUK SEO, MIN-JEA AHK control input should decrease to minimize the cost unction (5). o ind the optimal control input, deined the Hamiltonian. H where ' x x mv x x and are co-states. (6) he equations o co-states can be obtained rom the equation (6). d d H H (7) We can get the co-state equations b integrating equation (7) rom o. x x x (8) he optimalit condition is obtained b partiall dierentiating the Hamiltonian b control input. hen, substitutes equation (8). m V x x x x ' m V x x xx ' 3 (9) he optimization scenario is deined as ollows: the missile is launched rom the origin o the two-dimensional Euclidean plane. Initial and terminal conditions are given below. xt t V t t m s 85deg t t 3km 6km 35deg () In the optimization scenario, equation (7) is used but mass variation o the missile should be considered. he parameters associated with VC are given as ollows: kn m 35kg m kg sec t burn sec () he cost unction is set as equation (5). he optimal trajector, velocit proile, light path angle, mass variation and the optimal control input are depicted rom Fig. -6. In Fig. and Fig. 4, the missile approaches to the target with terminal light path angle constraint. he mass varies rom initial mass 35kg to 5kg with the propellant consumption rate -kg/sec and the burn time seconds Optimization Results o ind the undetermined coeicients in equation (9) is tedious procedure. Previousl, the optimal control and trajector are obtained b a commercial optimization tool, GPOPS. his tool transcribes optimal control problem into parameter optimization problems b using pseudo-spectral method. Altitude (km) Downrange (km) Fig.. Missile rajector 4

5 ERMINAL IMPAC ANGLE AND ANGLE OF AACK CONROL GUIDANCE FOR SURFACE-O-AIR MISSILE USING VC Velocit (m/s) ime (sec) Fig. 3. Velocit Proile Angle o Attack (deg) ime (sec) Fig. 6. Angle o attack Proile Flight Path Angle (Deg) ime (sec) Fig. 4. Flight Path Angle Proile 4 Conclusion In this paper, the optimal guidance or the missiles using VC is dealt. Unlike the general impact angle guidance law based on the optimal control theor, the missile kinematics is deined considering the missile velocit variation and the control input is deined as angle o attack. he optimization is conducted b using the commercial optimization tool instead deriving rigorous analtic solution which is tedious procedure. he optimization results are satising the terminal constraints. Also, the optimal control input can be calculated Contact Author Address mailto:smhong@dcl.kaist.ac.kr Mass (kg) ime (sec) Fig. 5. Mass Variation he angle o attack in Fig. 6 also satisies the terminal condition. At the inal time, angle o attack becomes zero. here exist jump point near 8 sec, however, it is caused b the tool. Acknowledgements his work was conducted at High-Speed Vehicle Research Center o KAIS with the support o Deense Acquisition Program Administration and Agenc or Deense Development Reerences [] M. Kim, and K. Y. Grider. erminal guidance or impact attitude angle constrained light trajectories. IEEE ransactions on Aerospace and Electronics Sstems, Vol. 9, No. 6, pp ,

6 SEONG-MIN HONG, MIN-GUK SEO, MIN-JEA AHK [] C. K. Roo, H. Cho, and M. J. ahk. Optimal guidance law with terminal impact angle constraint. Journal o Guidance, Control and Dnamics. Vol. 8, No. 4, pp74-73, 5. [3] C. K. Roo, H. Cho, and M.J. ahk. ime-to-go weighted optimal guidance with impact angle constraints. IEEE ransactions on control Aerospace and Electronics Sstems, Vol. 4, No. 3, pp , 6. [4] J. I. Lee, I.S. Jeon, and M. J. ahk, Guidance law to control impact time and angle. IEEE ransactions on Aerospace and Electronic Sstems, Vol. 4, No., pp 6-66, 6. [5] H, Cho. Navigation constants in PNG law and the associated optimal control problems. Proceedings o the 99 Korean Automatic Control Conerence, Seoul, Korea, 99. [6] S.W. Shim, S.M. Hong, M.G. Seo, and M.J. ahk, Preliminar Engagement Eect Analsis o Isotropic Kinetic Energ Warhead, Journal o he Korean Societ or Aeronautical and Space Science (Korean), Vol. 45, No. 5, pp , 5. Copright Statement he authors conirm that the, and/or their compan or organization, hold copright on all o the original material included in this paper. he authors also conirm that the have obtained permission, rom the copright holder o an third part material included in this paper, to publish it as part o their paper. he authors conirm that the give permission, or have obtained permission rom the copright holder o this paper, or the publication and distribution o this paper as part o the ICAS proceedings or as individual o-prints rom the proceedings. 6