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Analytical Study of CLOS Guidance Law Against Head-On High-Speed Maneuvering Targets

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Analytical Study of CLOS Guidance Law Against Head-On High-Speed Maneuvering Targets Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 168, pp. 88–96, 2007 Analytical Study of CLOS Guidance Law against Head-on High-Speed Maneuvering Targets By Chia-Chi CHAO,1Þ Sou-Chen LEE2Þ and Chen-Yaw SOONG3Þ 1ÞDepartment of Weapon System Engineering, Chung Cheng Institute of Technology, National Defense University, Taoyuan, Taiwan, R.O.C. 2ÞCollege of Humanities and Sciences, Lunghwa University of Science and Technology, Taoyuan, Taiwan, R.O.C. 3ÞDepartment of Electronics, China Institute of Technology, Taipei, Taiwan, R.O.C. (Received November 28th, 2005) A command to line-of-sight (CLOS) guidance law is developed against head-on high-speed maneuvering targets. Preliminary studies have shown that the aspect angle of the interceptor at lock-on near 180 deg is a fundamental require- ment for achieving small miss distance against a very high-speed incoming target. The solutions of missile trajectory obtained before under this guidance scheme seemed tedious and incomplete. Now, in this study, exact and complete solutions which are more general and comprehensive than those obtained before are derived for head-on high-speed maneuvering targets. Some related important characteristics such as lateral acceleration demand and normalized missile acceleration are investigated and discussed. Additionally, illustrated examples of target maneuvering are introduced to easily describe the trajectory of the target. The results obtained in this study are very significant and practical, and will be useful in actual application. Key Words: Guidance Law, CLOS, Analytical Solution 1. Introduction 2. Equations of Motion In a CLOS trajectory, the pursuer always lies on the line An important assumption for successful engagement is between the target tracker and target. If the pursuer is always that the missile intercepts the target at a near head-on geom- on the LOS, then the pursuer will surely hit the tar- etry. Based on this assumption, the LOS rotational rates are get.1–3,7,11–14,16,18) In other words, in CLOS guidance, the minimized and unsuitable command signals are avoided at pursuer maneuvers so as to be on the LOS between the target the terminal phase. The guidance scheme proposed here tracker and target.2,5,8,10) A closed-form solution of a CLOS possesses three phases: midcourse, shaping and terminal. trajectory for maneuvering target and a pursuer with con- Assume that an ideal midcourse guidance law navigates stant velocity is not available. In 1955, Locke8) presented the missile first. As the seeker ‘‘sees’’ the incoming target, a 10-term CLOS series solution with simplified assump- the missile then enters the terminal phase. The shaping tions. In addition, Macfadzean10) claimed that there is no phase is between the midcourse and terminal phases. The closed-form solution for LOS trajectory, even with simplify- geometric relationship between a defense missile and an ing assumptions such as those made by Locke. In 2000, incoming target is illustrated in Fig. 1.6) 4) Jalali-Naini and Esfahanian claimed a closed-form solu- In Fig. 1, ZM is the axis of the missile system along a tion for LOS trajectory for non-maneuvering targets. In theoretical missile inertial reference unit lock-on point. 15) 2001, Shoucri showed that some scattered published re- OM is defined as the missile forward direction (i.e., the di- sults in the guidance theory field were similar to the gener- rection the missile is required to go for a direct-hit). XM is alized proportional navigation (GPN) guidance law. In this the lateral axis of the missile system, which is perpendicular paper, a closed-form solution of CLOS trajectory for to ZM, and is positive in the missile travel direction. The maneuvering targets is proposed. Here, the total pursuer ac- planar missile-target engagement geometry motion is shown celeration is assumed to be equal to the required accelera- in Fig. 2, where M and T are the missile and target, respec- tion in the direction normal to LOS, whereas Locke assumed tively. Point O is fixed and represents the ‘‘lock point’’ or the pursuer velocity is constant, that is, the pursuer acceler- the reference axis at the lock-on point, where r is the vector ation is restricted in the direction normal to pursuer velocity. corresponding to the line-of-sight (LOS) distance, rM is the Shoucri’s solution seems tedious and imcomplete. Now, in missile distance and rT is the target distance from the origin. this paper, we try to derive an exact and complete closed- The equation of motion is depicted by point mass dynam- form solution for a maneuvering target. Some important ics. A particle p (M or T) in the polar coordinates ðr;Þ, ac- and significant characteristics related to the system perform- cording to Fig. 2, can be written in the following form: ance, such as lateral acceleration and normailized missile vp ¼ rre_ r þ r_e ð1Þ acceleration, are investigated and discussed in detail. 2 ap ¼ðr€ À r_ Þer þðr€ þ 2r__Þe ð2Þ Ó 2007 The Japan Society for Aeronautical and Space Sciences Aug. 2007 C.-C. CHAO et al.: Analytical Study of CLOS Guidance Law against Head-on High-Speed Maneuvering Targets 89 2 dr 2 d r T ZM r€ ¼ € þ _ ð8Þ d d2 Terminal Predicted Equation (5) can be rewritten in the following form: interception point d2r € dr X OM(Lock-on point), O’ þ À r ¼ 0 ð9Þ M ∆X Shaping d2 _2 d Midcourse By differentiating z ¼ r cos and x ¼ r sin , one can obtain z (z_ ¼ v cos ) and x components (x_ ¼ v sin ) for Zr(Altitude) the velocity vector r_. This can also be written as: Boosting v cos ¼ r_ cos À r_sin ð10Þ _ Xr(Range) Or(Launcher) v sin ¼ r_ sin þ r cos ð11Þ with Fig. 1. Engagement geometry. r_ sin þ r_cos tan ¼ ð12Þ r_ cos À r_sin where is the angle between the direction of velocity v and γ ZM T T the Z axis. By differentiating Eq. (12) with respect to , Reference M r one can obtain the following equation: vT T γ 2 dr d2r r2 þ 2 À r ψ 1 d d d2 ¼ 2 ð13Þ r= cos2 d dr v r cos À r sin M θ M d 1 Noting that tan2 þ 1 ¼ sec2 ¼ , we have OM cos2 XM dr 2 þ r2 Fig. 2. Scenario for a missile target interception. 1 d ¼ ; ð14Þ cos2 dr 2 e e cos À r sin where, r and are all unit vectors. If the missile is always d on the LOS, the missile will hit the target. Therefore, the basic guidance law is defined as: and finally derive at: d2r ¼ MðtÞ¼TðtÞð3Þ r2 þ r d d2 where, and are the LOS angles from the reference ¼ 2 À ð15Þ M T d dr 2 point to the missile and to the target, respectively. We r2 þ assume an ideal case in which the missile is always on the d line between the reference point and the target without The LOS trajectory geometry can also be shown using an- error. Based on this point, the required lateral acceleration other form in Fig. 3, where s is the arc length of trajectory. for the missile on LOS is: Referring to Fig. 3, we have, 2 2 2 2 aM ¼ rM€T þ 2r_M_T ð4Þ ðÁsÞ ¼ðÁrÞ þ r ðÁÞ ; Let r ¼ r, and using the theorem, a is perpendicular to and M M the LOS. Therefore, if the missile is initially aimed at the 2 2 dr 2 ds target and accelerated according to Eq. (4), it will always þr ¼ ; ð16Þ d d lie on the reference-target LOS line. We assume that the to- tal lateral acceleration is equal to the required acceleration, Now, using the sine function (found in Fig. 3), Eq. (16) can thus from Eq. (2), we have: be written as: r2 r€ À r_2 ¼ 0 ð5Þ sin2 ¼ ð17Þ dr 2 € _ r2 þ r þ 2r_ ¼ aM ð6Þ d Using the relationship, Equation (17) is the same as that of Eq. 15 in the report 15) dr by Shoucri, where ¼ À is the angle between v r_ ¼ _ ð7Þ and r. This development only utilizes the fundamental guid- d ance equation concept. Shoucri utilized the GPN concept 90 Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 168 T point and the target without any lateral error; if the pursuer is always on the LOS, the pursuer will surely hit the target. ZM Therefore, the basic guidance law for CLOS is ðtÞ¼TðtÞ vT and ψ ∆θ r ∆r dðtÞ d ðtÞ ¼ T : ∆S dt dt S Define two cases to examine the robustness of the trajec- r tory solution as depicted in methods A and B. Method A: The trajectory approach with LOS angle ∆θ θ measurement only. From the observation of and , Eq. (9) is employed. Because Eq. (9) is a homogeneous equation, no particular solution can be produced. We assume that the missile trajec- X OM M tory is simple and has a continuous first derivative r0ðsÞ, which is different from the zero vector for all s under Fig. 3. CLOS Geometry. consideration. Then missile trajectory is a smooth curve; that is, it has a unique tangent at each of its points whose treated by Yang et al.17) and the result derived by Lu.9) This direction varies continuously as we traverse the curve. is the major difference between the two approaches. Note We subdivide missile trajectory into n portions by the that, in the derivation of Eqs. (16) and (17), no restriction LOS measurement: let P0ð0; r0Þ; P1ð1; r1Þ; P2ð2; r2Þ; ...; is placed on the target and missile time variation in the Pnðn; rnÞ be the end points of these portions, and let Ári ¼ € velocities vT and v.
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