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€ 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 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 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 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 r ψ 1 d d d2 ¼ 2 ð13Þ r= cos2 d dr v r cos 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 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 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 ðsÞ ¼ðrÞ þ r ðÞ ; 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€ 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 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 ri ¼ € velocities vT and v. i ri ri1 and i ¼ i i1. Defining Bi ¼ , we have 2 _i 3. Solutions for a Non-Maneuvering Target the homogeneous equation: d2r dr In the case of a non-maneuvering straight-line target, the i þ B i r ¼ 0 ð23Þ d 2 i d i target velocity v is assumed to be constant. It is easily i i T 1 shown that: The roots of the characteristic equation are B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 i 2 2 v sin Bi þ 4 , and we obtain the solution ria;b ¼ _ ¼ T ð18Þ ÀÁpffiffiffiffiffiffiffiffiffi 1 B B 24 x e 2 i i i . A general solution of Eq. (23) is: 2 3 ÀÁpffiffiffiffiffiffiffiffiffi 2vT sin cos 1 2 € Bi Bi þ4 i ¼ ð19Þ r ð Þ¼c e 2 x2 i i 1 ÀÁpffiffiffiffiffiffiffiffiffi 1 2 € _2 Biþ Bi þ4 i = ¼ 2 cot ð20Þ þ c2e 2 ð24Þ where, x is the constant horizontal distance of the target Hence, we yield ri0ði0Þ¼c1 þ c2 ¼ 0 by the first initial from the reference axis.6) condition. Differentiation gives: € pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁpffiffiffiffiffiffiffiffiffi 0 1 1 B B 2þ4 Substituting from Eq. (20), Eq. (9) can be rewritten as: r ð Þ¼ c B B 2 þ 4 e 2 i i i _2 i0 i0 2 1 i i ÀÁffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 1 2 d r dr 2 Biþ Bi þ4 i þ 2 cot r ¼ 0 ð21Þ c2 Bi þ Bi þ 4 e 2 ð25Þ d2 d 2 This is the case treated by Jalali-Naini and Esfahanian.4) Equation (16) leads to another type of initial condition. Letting The solution in this case is given by r sin ¼ A1ð 0Þ, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and one can easily derive the transverse acceleration as: dr ds 2 2 2 K ¼ ¼ r ; 2A1vt 3 d d a ¼ sin ð22Þ x we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which is also shown as Eq. (22) in Jalali-Naini and v 2 4) i 2 Esfahanian. Ki ¼ ri : ð26Þ _ 0 4. Solutions for Maneuvering Target Hence, ri ðiÞ¼Ki is the second initial condition. Together:

There are two techniques for deriving the CLOS trajecto- Ki c1i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ ry differential equation. By assuming an ideal case in which 2 Bi þ 4 the pursuer is always on the line between the reference Aug. 2007 C.-C. CHAO et al.: Analytical Study of CLOS Guidance Law against Head-on High-Speed Maneuvering Targets 91

Ki drT c2i ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð28Þ ¼ v cosð Þð39Þ 2 T T Bi þ 4 dt ÀÁpffiffiffiffiffiffiffiffiffi d Ki 1 2 Bi Bi þ4 i r ¼ v ð ÞðÞ riðiÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 T T sin T 40 2 dt Bi þ 4 ÀÁpffiffiffiffiffiffiffiffiffi Rewrite the differential Eq. (38) in the form of: Ki 1 þ 2þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Bi Bi 4 i e ð29Þ 2 2 þ dr d d Bi 4 þ r dt dt dt2 Overall, we have d d dv ¼ v cosð Þ þ sinð Þð41Þ r ¼ r1 þ r2 þ r3 þ ... ð30Þ dt dt dt To avoid divergence, we suggest the periodic ‘‘restarting’’ Substituting Eqs. (37) and (38) into Eq. (41), the preced- of operating Eqs. (29) and (30). From Eq. (16), we have: ing differential equation can be simplified as: 2 i ¼ i þ i d d r d 1 dv d sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 þ ð42Þ dt dt dr=dt dt2 v dt dt 2 1 ri ¼ sin ð31Þ 2 2 i 2 2 By taking the time derivative of Eq. (40), (d =dt ) can be ri þ Ki obtained as follows: Referring to Fig. 2, we have: d2 1 Z ¼ 2v 2 cosð Þ sinð Þ 2 2 T T T dt rT sz ¼ v cos dt ð32Þ Z dT dvT þ rT cosðT ÞvT þ rT sinðT Þ ð43Þ sx ¼ v sin dt ð33Þ dt dt By assuming that the missile has a constant velocity (nom- When inal atmospheric flight): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2 dT i 2 a ¼ v ð44Þ >_ 0; Ki ¼þ ri NT T _ dt

where, aNT represents lateral target acceleration. szn ¼ rn1 cos rn1 sin ð34aÞ d a ¼ v ð45Þ sxn ¼ rn1 sin þ rn1 cos ð34bÞ N dt When aN represents the lateral acceleration of the missile as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2 it follows the CLOS guidance trajectory. Substituting i 2 <_ 0; Ki ¼ ri Eqs. (42), (40) and (43) into Eq. (45), it can be written as: _ r vt sinðT Þ szn ¼ rn1 cos þ rn1 sin ð35aÞ aN ¼ rT cosð Þ s ¼ r r ð Þ xn n1 sin n1 cos 35b 2v cosð Þ 2v cosð Þ þ T T This procedure is repeated to obtain the overall trajectory: r rT sz ¼ sz1 þ sz2 þ sz3 þ ... ð36aÞ 1 dvT 1 dv cosðT Þ þ þ þ aNT ð46Þ sx ¼ sx1 þ sx2 þ sx3 þ ... ð36bÞ vT dt v dt cosð Þ

Method B: The trajectory approach with a lateral acceler- vN represents the lateral missile velocity and vs represents ation calculation. the tangent flight path velocity.Z Shourci15) showed that a closed-form solution derived from the two-point GPN concept exists for the three-point vN ¼ aNdt ð47Þ CLOS system. The basic two-point equation of motion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 for an ideal CLOS guidance system, which first appeared vs ¼ v vN ð48Þ in the previous section, is repeated to obtain the CLOS The missile trajectory is obtained as follows: Z course trajectory. It is: sz ¼ ðvs cos vN sin Þdt ð49Þ dr ¼ v cosð Þð37Þ Z dt s ¼ ðv sin þ v cos Þdt ð50Þ d x s N r ¼ v sinð Þð38Þ dt 92 Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 168

5. Quantitative Studies the proposed analytical methods (solution), A and B, are illustrated. The results from methods A and B are compared The CLOS guidance scheme is one in which the missile is with a numerical solution. guided on a LOS course in an attempt to remain on line, 6.1. Missile trajectory model joining the target and the point of control. To do so, the Here, the trajectory model of missile interception using missile requires a velocity component (vM) that is equal Tracking Via Missile (TVM) is applied. It uses the modified to the LOS velocity and perpendicular to it: CLOS in the terminal phase. The major difference compared with conventional CLOS is the observation (reference) v ¼ r_ ð51Þ M point, which is changed from the ground control station to In general, the missile flies a pursuit guidance course at ini- seeker lock-on point in the collision course. The recently tiation of the entry the beam and flies on an approximately developed modified CLOS guidance law with an advanced constant bearing course (CBC) near impact. This is ob- inertial measurement unit (IMU) system serves as a possible served from the velocity equation:13) approaches for solving TVM interception control prob- 19) r lems. There are continuing efforts to improve the per- vM ¼ vT ð52Þ formance of modified CLOS for intercepting maneuvering r T targets. More detailed models have been investigated to de- From performance analysis, a zero-lag guidance loop can rive modified CLOS. The inclusion of LOS angle measure- be developed to remove heading error using a fixed beam ment only lead to the introduction of Method A into guid- ( ¼ Const.) step by step. ¼ and _ ¼ _ _ ¼ _ ance. When the target acceleration is unknown, Method B give a guidance law between and corresponding to the is used to account for target maneuvering. The missile trajectory described by the differential equation given by acceleration demand has been dealt with in Method B. For Eq. (15). the missile, v ¼ 1500 m/s, at ‘lock-on’ point x ¼ 0 m and d2r z ¼ 0 m, ¼ 95 deg, ¼ 95 deg (Fig. 1). r2 þ r d2 6.2. Simplified ballistic target model 2 ¼ 0 ð53Þ Based on earlier assumptions,18) suppose that only drag dr 2 r2 þ and gravity act on the endoatmospheric ballistic target. d Let the target have velocity vT and initial reentry angle Equation (53) can be rewritten as: T. The downrange target is xT. The altitude is zT. Note that 2 the drag force F acts in a direction opposite the velocity d2r 2 dr drag r ¼ 0 ð54Þ vector, and gravity g always acts downward. Therefore, d2 r d if the drag effect is greater than that of gravity, the target By noting that _=_ ¼1, will decelerate. The target reentry angle can be computed ! d2r _ 1 dr 2 _ using the two inertial target velocity components: 1 þ r ¼ 0: ð55Þ d2 _ r d _ vTz ¼ tan1 ð57Þ T v When the proportional navigation (PN) law _=_ ¼ k is Tx used, where k is a constant.18) PN is known to effectively The simple ballistic target acceleration components in the engage an evasive target; ac ¼ Nv_ ¼ v_. PN is also a guid- inertial downrange and altitude directions can be expressed ance law in which the angular rate of the missile flight path in terms of the target’s weight Wt, reference area Atref , zero is directly proportional to the angular LOS rate of change; lift drag CtD0 and gravity g, or more simply, in terms of the _ ¼ _ þ _ ¼ N:_ ballistic coefficient according to the following equation: dv F So, it get _=_ ¼ k. Tx ¼ drag cosð Þ dt m T k ¼ 1, this is pursuit guidance. T In the CLOS case where k ¼1, which is Eq. (19) of Qg ¼ cosðTÞð58Þ Shourci,15) the cumulative velocity increment is: Z tF dvtz Fdrag V ¼ ja jdt ð56Þ ¼ sinðtÞg N dt mt 0 Qg where, t is the time of flight and t the time, which is related ¼ sinð Þg ð59Þ F t to the corresponding propellant mass required for effective intercept in exoatmospheric flight. where, mt is the target mass and

Wt 6. Illustrative Example ¼ : ð60Þ CtD0Atref In this section, two target trajectories are used as case The air density is measured in kg/m3 and is approximated studies. Through these examples, the superior property of as Aug. 2007 C.-C. CHAO et al.: Analytical Study of CLOS Guidance Law against Head-on High-Speed Maneuvering Targets 93

4 x 10 CLOS-ATBM trajectory The lateral acceleration of CLOS-ATBM and Target 2.5 16

Dashed:Target 14 Dashed:Target Solid:Missile 2 Solid:Missile 12

10 1.5 8

ZM(m) 6 1 G-value(G) 4

2 0.5

0

0 -2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 XM(m) 4 x 10 Time(sec)

Fig. 4. CLOS trajectory for a simplified ballistic target; vt ¼ 2700 m/s, Fig. 6. Engagement lateral acceleration profile for a simplified ballistic v ¼ 1500 m/s. target.

4 x 10 CLOS-ATBM trajectory Target and CLOS-ATBM velocity profiles 2.5 1500 Dashed:Target Solid: Missile 1000 Dashed:Target 2 Solid:Missile 500

0 1.5

-500 ZM(m) 1 -1000 Velocity(m/sec) -1500 0.5 -2000

-2500 0 2500 2000 1500 1000 500 0 -3000 0 5 10 15 XM(m) Time(sec)

Fig. 7. CLOS trajectory for a sinusoidal target, vt ¼ 1800 m/s, v ¼ Fig. 5. Engagement velocity profile for simplified ballistic target. 1500 m/s.

Zt ¼ 515ð0:0034Þe6705:6 for Zt > 9144 m: ð61Þ Target and CLOS-ATBM velocity profiles 1500

The dynamic pressure Q is defined as: 1000 2 Q ¼ 0:5vt ð62Þ 500 Dashed: Target Solid: Missile the total target velocity v is obtained from: 0 t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 vt ¼ vtx þ vtz ð63Þ -500 Velocity(m/sec) Because the acceleration equations are in a fixed or -1000 inertial frame, they can be integrated directly to yield the -1500 velocity and position. The initial condition is as follows. -2000 v ¼ ¼ ðx ; z Þ¼ 0 1 2 3 4 5 6 78 For the target, t 2700 m/s, t 95 deg (Fig. 1), t t Time(sec) 2 ð20; 25Þ km, reference area Atref ¼ 0:15 m , zero lift drag co- Fig. 8. Engagement velocity profile for a sinusoidal target. efficient CtD0 ¼ 1:25, and weight Wt ¼ 1000 kg. Figure 4 shows the simulated engagement trajectory for the missile guided by CLOS command against a simplified ballistic these trajectories were plotted in rectangular coordinates target. Note that these trajectories are plotted in rectangular with the reference point at the lock-on point. Figures 8 coordinates with the reference point at the origin. Figures 5 and 9 show the CLOS engagement velocity and lateral and 6 show the CLOS engagement velocity and lateral acceleration profile. acceleration profile. 6.4. Missile trajectory of CLOS course 6.3. Sinusoidal maneuvering target model To demonstrate how the analytical solution is presented Ballistic targets may sometimes have maneuvering in methods A and B, the solution was programmed on a dig- capabilities to avoid a counter-attack. This simulation ital computer and subsequently compared with a numerical used a simple target model. The sinusoidal target maneuver simulated trajectory. The discussion is divided into two satisfies vtz ¼1800 m/s, and vtx ¼ 50 sin 2 t m/s. Figure 7 parts: 1) simplified ballistic target, and 2) sinusoidal target. shows the CLOS trajectory for a sinusoidal target. Note that 1) Figure 10 shows the defensible trajectory against a bal- 94 Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 168

The lateral acceleration of CLOS-ATBM and Target CLOS-ATBM trajectory 25 10000 Solid: Simulation 20 9000 Dashed:Method A Dashdot: Method B 15 8000

10 7000 7700 5 6000 7600 0 5000 7500 ZM(m) G-value(G) -5 4000 7400 7300 -10 3000 7200 -15 Dashed: Target 2000 7100 Solid: Missile -20 1000 7000 1400 1300 1200 1100 -25 0 1 2 3 4 5 6 78 0 2500 2000 1500 1000 500 0 Time(sec) XM(m)

Fig. 9. Engagement lateral acceleration profile for sinusoidal target. Fig. 11. Trajectory compared with the simulation; methods A and B (sinusoidal target).

CLOS-ATBM trajectory 10000 Solid:Simulation The lateral acceleration of CLOS-ATBM trajectory 9000 Dashed:Method A 15 Dasheddot:Method B 8000 Solid:Simulation Dashed:Method B 7000

6000 5400 10 5000 ZM(m) 5200 4000

3000 5000 G-value(G) 5 2000 4800

1000 4600 7000 6500 6000 0 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 XM(m) 4 x 10 0 0 5 10 15 Time(sec) Fig. 10. Trajectory compared with the simulation; methods A and B (simple ballistic target). Fig. 12. Missile acceleration for a ballistic target. listic target with the missile guided using simulated CLOS not unexpected and, in fact, was manifested in all of the data. The trajectory is in good comparison with the trajecto- geometries characterized with a high LOS rate. From the ries from analytical methods A and B. The results show plot, the error made by Method B is lower than that by that the trajectories obtained using methods A and B were Method A. The Method B scheme is more complex than close to that obtained using the numerical simulated data. Method A because the missile and target variables are The Method A trajectory could be obtained on a digital included. computer using a stepwise accumulation algorithm such 6.5. Lateral acceleration in CLOS course as Eqs. (30) and (31). It is worth noting that Eqs. (30) The missile acceleration saturation problem is the most and (31) depend on Ki entirely. When _ is very close to zero, important guidance system nonlinearity in the performance a singular point problem occurs and Eqs. (30) and (31) comparison. Because the CLOS guide uses LOS, less lateral become anomalous. The error of Bi is also caused mainly acceleration is required than the PN case when a maneuver- by _ approaching zero, which is especially sensitive to ing target must be destroyed. In Figs. 6 and 9, where lateral Eq. (24). Consequently, each time a new engagement acceleration is plotted versus the time for the missile and tar- geometry is acquired, _ and € must be solved in advance. get, it is evident that the CLOS law requires a lower accel- The analytical solution accuracy using Method A is limited eration advantage than the PN law to complete the mission. by these two conditions. This is the major advantage of CLOS: it can use reasonable 2) Figure 11 shows the defensible trajectory against a lateral acceleration in the last phase when a maneuvering weaving target with the missile guided using simulated target must be destroyed. The lateral missile acceleration CLOS data. The target has the same velocity (1800 m/s) for the CLOS course is produced using Eq. (46). As shown but also has maneuvering capability. This trajectory is in in Fig. 12, the missile approaches the target at a lower speed good comparison with analytical methods A and B. The and velocity ratio. For a sinusoidal target, the lateral accel- results show that the trajectories obtained using methods eration required for the missile is the same as the target ac- A and B are close to that obtained using the numerical simu- celeration, as shown in Fig. 13. The maximum acceleration lated data. An examination of Fig. 11 reveals that methods during flight can be found using Eq. (46). The results A and B exhibit analytical solutions with characteristics presented here demonstrate that Eq. (46) is both a stable exactly like those derived using theory. Such behavior is and predictable estimation algorithm for the CLOS course. Aug. 2007 C.-C. CHAO et al.: Analytical Study of CLOS Guidance Law against Head-on High-Speed Maneuvering Targets 95

The lateral acceleration of CLOS-ATBM trajectory Energy Cost 25 600

Simulation 20 500

15 400 Method B

10 300 G-value(G)

5 200 Cumulative velocity increment (m/s) 0 100 Dashed:Method B Solid:Simulation -5 0 0 1 2 3 4 5 6 7 8 0 5 10 15 Time(sec) Time(sec)

Fig. 13. Missile acceleration for a sinusoidal target. Fig. 16. Energy cost for a simplified ballistic target.

Normalized Missile Acceleration Due to Target Maneuvering Energy Cost 0 600

-20 Method B 500

-40 Simulation 400

-60 300 N / Nt -80 Method B

200 -100 Cumulative velocity increment (m/s) 100 -120 Simulated

-140 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 67 t t / F Time(sec)

Fig. 14. Normalized missile acceleration for a simplified ballistic target. Fig. 17. Energy cost for a sinusoidal target.

6.7. Energy cost due to target maneuvering Normalized Missile Acceleration Due to Target Maneuvering 30 A high energy cost is generated with extensive target ma- neuvering. It is simply illustrated with a typical example as 25 described in this section. Thus, target maneuvering increas- 20 Simulated es the energy cost required to effectively intercept a target, Method B 15 as depicted in Figs. 16 and 17, respectively. The energy cost 10 is mainly affected by T and aNT. N / Nt

5 7. Conclusion 0

-5 A CLOS guidance scheme concept was proposed for -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 guiding a missile engaging an incoming target at very high t t / F speed. An analytic solution for the differential equations Fig. 15. Normalized missile acceleration for a sinusoidal target. describing the missile LOS trajectory for a head-on high-speed maneuvering target was derived, which ise more general and comprehensive than those obtained before. The 6.6. Normalized missile acceleration due to target CLOS trajectory problem was described and solutions maneuvering for methods A and B, and a simulated trajectory were The approximate solution for the missile acceleration compared. The general equations for the two methods and response due to two example target maneuvers are displayed derivations were given for comprehensive analysis and ap- in normalized form in Figs. 14 and 15, respectively. We can plication to the CLOS guidance issue. Analytical expres- see that the CLOS guidance scheme inhibits the acceleration sions were obtained for method A using only the LOS angle requirements at the end of the flight. From a system sizing measurement. Numerical results were given for two exam- point of view, the designer must ensure that the acceleration ples with a maneuvering target. Additionally, some signifi- capability of the missile is adequate at the end of the flight cant characteristics such as missile acceleration demand, so that saturation can be avoided and the missile can hit normalized acceleration and energy cost were investigated the target. and discussed in detail. 96 Trans. Japan Soc. Aero. Space Sci. Vol. 50, No. 168

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