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Head Pursuit Variable Structure Guidance Law for Three-Dimensional Space Interception

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Head Pursuit Variable Structure Guidance Law for Three-Dimensional Space Interception Chinese Journal of Aeronautics Chinese Journal of Aeronautics 21(2008) 247-251 www.elsevier.com/locate/cja Head Pursuit Variable Structure Guidance Law for Three-dimensional Space Interception Ge Lianzheng*, Shen Yi, Gao Yunfeng, Zhao Lijun Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China Received 21 September 2007; accepted 25 December 2007 Abstract This article aims to develop a head pursuit (HP) guidance law for three-dimensional hypervelocity interception, so that the effect of the perturbation induced by seeker detection can be reduced. On the basis of a novel HP three-dimensional guidance model, a nonlinear variable structure guidance law is presented by using Lyapunov stability theory. The guidance law positions the interceptor ahead of the target on its flight trajectory, and the speed of the interceptor is required to be lower than that of the target. A numerical example of maneuvering ballistic target interception verifies the rightness of the guidance model and the effectiveness of the proposed method. Keywords: head pursuit; three-dimensional guidance model; nonlinear variable structure; Lyapunov stability theory; guidance law 1 Introduction1 channels. Based upon the planar model, a HP variable structure guidance law is then developed. In tactical ballistic missile interception, many However, as the actual missile interception occurs in interceptors employ an infrared seeker to detect the three-dimensional space, a three-dimensional HP target. However, the detection precision is often guidance method is more useful in practical degraded by aerodynamic heating[1]. To solve the applications. aerodynamic ablation problem, a head pursuit (HP) Various classic guidance methods have been guidance law, whichpositions the interceptor mi- examined for implementation of three-dimensional ssile ahead of the target on its flight trajectory to guidance interception since the origination of the destroy the target, has been developed recently[2]. three-dimensional pure proportional navigation gui- Using this guidance law, the interceptor can fly in dance law proposed by Adler[5]. Refs.[6-11] have the same direction with the target at a lower speed developed the three-dimensional guidance model than that of the target. Compared to a head-on and given a guidance law based on Lyapunov sta- engagement, the low closing speed is achieved with bility theory. These guidance laws are only suitable reduced energy requirements. The HP guidance for head-on interception, their interception styles method is further improved in Refs.[3-4], where the and kinematics models are different from the HP relative motion model is separated into two guidance method. As an intuitive and robust control perpendicular channels and the guidance problem technique, the sliding mode variable structure can be treated as a planar problem in each of those [12-15] control has been utilized in various guidance *Corresponding author. Tel.: +86-451-86418285. applications to address highly nonlinear systems E-mail address: [email protected] Ge Lianzheng et al. / Chinese Journal of Aeronautics 21(2008) 247-251 · 248 · with large modeling errors and uncertainties. the relative motion, as follows (details are omitted). Different from the above-mentioned studies, dL VV vvr i i iu: L (1) the authors propose to design a guidance law here, dt mtmmttl l based on the three-dimensional HP engagement rv mmmttcosTM cos v cos TM cos t (2) model, using the sliding mode theory. The guidance azt vm Mt sinTMTttl cos tan < law is verified via the digital simulation of a guided vrttcosTT cos t missile against a maneuvering target. vt vm (cosTMmm sin cos TM t sin t ) < vrm 2 HP Kinematics Model v v (cosTM sint cos TM sin ) m < mmvr ttcosT Considering a typical three-dimensional missile mt v homing guidance problem, the authors assume that: sinTMT sin (sin t sin T ) (3) ttmv t ķ the missile and the target are point masses; ĸ the m a dynamical characteristics of the missile are yt vm TMTTTttlmm sin tan (cos sin neglected; Ĺ the maneuvering accelerations of the vrt missile and the target can only change the speed vvttvm cosTTt sinttmt ) cos MT (sin sin T ) (4) direction. The assumptions are reasonable and often vrvmm used in the design of a guidance law for aero- avzmm Mmmml sinTMT cos tan < dynamically controlled missiles. vrmmcosTT cos m The missile-target endgame geometry of HP vt vm (c o sTMmm s i n c o s TM t s i n t ) < three-dimensional interception is depicted in Fig.1. vrm In contrast to the conventional configuration, target vt (c o sTMmm s i n c o s TM t s i n t ) T is located behind the slower interceptor M. The vm v v distance between the target and interceptor is r. L is m t sinTMTmmm sin (sin sin T t ) (5) rvcos Tt m the line of sight (LOS) vector. Txiyizi, Txlylzl, Mx y z ,andTx y z are the inertia coordinate a m m m t t t ym vm TMTTMmmlmm sin tan (cos sin system, the LOS coordinate system, the interceptor vrm body coordinate system, and the target body coor- vvttvm cosTTtt sin ) cos M m (sin T m sin T t ) (6) dinate system, respectively.Ml, Tl, Mm, Tm, Mt, and Tt vrvmm are the Euler angles.il, jl, kl, im, jm, km, it, jt, and kt where Vm and Vt are the velocity vectors of the are the unit vectors corresponding to the coordinate interceptor and the target. :l is the angular velocity systems of Txlylzl, Mxmymzm, and Txtytzt, respectively. vector of LOS. ayt and azt are the pitch maneuvering The velocities of target and interceptor are denoted acceleration and the yaw maneuvering acceleration by vt and vm, respectively. of the target, respectively. aym and azm are the pitch maneuvering acceleration and the yaw maneuvering acceleration of the interceptor, respectively. HP guidance law requires that the speed of the interceptor is lower than that of the target, so the speed ratio is defined as Nvv mt/1 (7) To hit the target, it is not only required that Fig.1 Missile-target engagement geometry. r 0 at the interception point, but also required With the help of the classical mechanics, the that both the target and the interceptorfly in the authors can derive the HP kinematical equations of same direction, hence Ge Lianzheng et al. / Chinese Journal of Aeronautics 21(2008) 247-251 · 249 · limTTmt 0, lim 0 (8) system without disturbances, U the proportion rroo00 s item, Un the control item for the uncertain factor. limMMmt 0, lim 0 (9) rroo00 And The objective of the HP guidance law is to 1 UCgXCfXeq [(,)][(,)] 1 tt (15) bring the interceptor to the point, which is confined 1 by Eqs.(8)-(9). Hence, the interceptor’s lead angles UCgXKSs1 [(,)]t (16) 1 Tm and Mm are required to be proportional to the UCgXCgXSSn1 [(,)](,)ttb 2 (17) target’s lead angles relative to LOS pp where K R u is a nonnegative definite matrix. MM nn, TT (10) m1tm2t Lemma Using the control law of Eq.(14), where n1 and n2 are the guidance constants. Thus, Eq.(11) is asymptotically stable. the relations mentioned earlier can guarantee that Tm Proof Considering the Lyapunov function T vanishes with Tt, and Mm vanishes with Mt. It is then VSS /2 and differentiating Eq.(13), the follow- necessary to find out the relation between the ing equation is achieved: angular condition defined by Eq.(10) and the inter- S KS Cg22(,) Xttb V Cg (,) X S S (18) ceptor acceleration. then TTT 3 HP Variable Structure Guidance Law VSSSKSSCgXV 2 (,)t btCg(,) X S ST KS 0 3.1 Variable structure control law 2 (19) Considering the nonlinear multiple-input mul- It can be seen that VV!00 . It is easily tiple-output (MIMO) uncertain system checked that VX 00 . Hence, the system is asymptotically stable, and the conditions of the XfXgXU (,)ttttt12 (,)() gXW (,)() (11) sliding mode variable structure control theory are n p where X R is the state variable, U()t R the satisfied, thus Lemma1 holds. control variable, and fX(,)t the uncertain 3.2 The design of nonlinear guidance law nonlinear item. gX1(,)t and gX2 (,)t are vector functions, which have suitable dimensions. W ()t It follows that to realize the control law one Rs is the acceleration disturbance of the target and only needs to know the scope of acceleration of the limited by 0()W tb, where bis the upper target, and it can be applied in the course of limited of target acceleration, and it is a positive intercepting the unknown acceleration target. constant. The aim of the control method is to design Usually no control is taken in the LOS direction, as the control signal U(t) such that the system app- long as other parameters are kept sliding on the sliding surface. When the target catches the roaches the equilibrium point X s 0 asymptoti- cally, that is, interceptor, the guidance control is completed. The limX (t ) 0 (12) other coupling parameters can be treated as tto s disturbances, so one can design the guidance law by According to the variable structure control using the design method[16] of single channel. theory, the sliding surface is defined as The authors have design the nonlinear variable S CX (13) control guidance law by using the yaw channel as an where C R pnu is the constant satisfying example. The aim is to bring the system into the Cg1(,)0 X t z , after which the control law is sliding surface and keep the dynamical charac- designed as teristics of the system. According to Eqs.(11), (13)- UU eq U s U n (14) (14), the variables are defined as: ut() azm is the control variable.
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