Suboptimal Midcourse Guidance with Terminal-Angle Constraint for Hypersonic Target Interception

Home , Missile guidance, Pursuit guidance

Hindawi International Journal of Aerospace Engineering Volume 2019, Article ID 6161032, 13 pages https://doi.org/10.1155/2019/6161032

Research Article Suboptimal Midcourse Guidance with Terminal-Angle Constraint for Hypersonic Target Interception

Shizheng Wan , Xiaofei Chang, Quancheng Li, and Jie Yan

School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Shizheng Wan; [email protected]

Received 17 November 2018; Revised 28 December 2018; Accepted 16 January 2019; Published 7 April 2019

Academic Editor: Seid H. Pourtakdoust

Copyright © 2019 Shizheng Wan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For the problem of hypersonic target interception, a novel midcourse guidance method with terminal-angle constraint is proposed. Referring to the air-breathing and the boost-gliding hypersonic targets, flight characteristics and difficulties of interception are analyzed, respectively. The requirements of midcourse guidance for interceptors are provided additionally. The kinematics model of adversaries is established concerning line-of-sight coupling in longitudinal and lateral planes. Suboptimal guidance law with terminal-angle constraint, specifically the final line-of-sight angle or impact angle, is presented by means of model predictive static programming. The trajectory is optimized and the load factor would finally converge after penalizing control sequence and output deviations. The realization of terminal angle is firstly verified with a constant speed target. A full interception scenario is further simulated focusing on a typical boost-gliding target, which flies along a skipping trajectory. Results show the success of providing handover conditions for intercepting hypersonic targets.

1. Introduction to increase the interception rate. Midcourse guidances for intercepting conventional aviation targets or ballistic missiles The hypersonic vehicle, typically traveling greater than Mach are relatively mature. For example, concerning air-to-air 5 in the near space, has developed rapidly in recent years, and missile, singular perturbation theory was applied to the deri- their weaponization has accelerated. Compared with conven- vation of near-optimal midcourse guidance in articles [5–7], tional missiles, hypersonic weapons show great advantages but they did not consider the angle alignment constraints. of diverse launching platforms, fast flight speed, and strong Focusing on the optimal guidance problem with angular con- penetration capability. Aiming at time-sensitive and high- straint, Indig et al. [8] proposed an analytic guidance law value targets, they pose severe challenges to the traditional with a linear dynamic model, which would cause control sat- antiaircraft or antimissile defense system. Hypersonic vehi- uration at the time of interception. Liu et al. [9] designed cles are going through intensive flight testing, for instance, a biased proportional navigation guidance method with atti- the “Hypersonic Air-breathing Weapon Concept” and “Tac- tude constraint and line-of-sight angle rate control, but it tical Boost Glide” [1], the Russian “Zircon” hypersonic cruise mainly aimed at stationary ground target, not to mention missile [2], and the Chinese “DF-ZF” hypersonic glide vehi- the capability of energy management. Taking constant accel- cle [3, 4]. The defense system should be upgraded and pre- eration into consideration, Seo and Tahk [10] provided a pared to meet the requirements of the future battlefield. closed-loop midcourse guidance law, but the terminal con- In order to eliminate the target in time, modern intercep- straint can be fulfilled only under the assumption that the tors generally require the ability of beyond-visual-range air impact point was precisely predicted. To maximize the kill combat. Midcourse guidance is vital to the entire interception probability, Phillips and Drake [11] designed a two-tier process. The purpose is to lead the interceptor to a desired approach for surface-to-air midcourse trajectory with an position, where not only the target can be acquired but also auxiliary constraint on an intersection approach angle. Three a good initial angle is obtained for the terminal phase, thus classes of midcourse guidance laws, namely, optimal fuel, 2 International Journal of Aerospace Engineering optimal energy, and closed-loop explicit strategy, are adopts the line-of-sight coupled interception model and pro- proposed by Jalali-Naini and Pourtakdoust [12] with a poses a novel midcourse guidance method with the MPSP zero-effort miss differential equation. Selecting different technique according to the characteristics of a specific target, objectives and constraint functions, Reza and Abolghasem which can optimize the interceptor’s trajectory and also meet [13] solved an optimal interception problem with receding the handover constraints. horizon control; commands are generated online by means The rest paper is organized as follows. In Section 2, the of a differential flatness concept and nonlinear programming. characteristics and technical requirements of midcourse Characterized by ultrafast speed and large-scale maneuver- guidance for hypersonic target interception are analyzed, ability, the requirements for hypersonic target interception and a three-dimensional interception model is established. become extremely strict, thus failing most of the existing In Section 3, suboptimal guidance law with constrained ter- midcourse guidance methods. New approaches must be minal line-of-sight or impact angle is derived utilizing the found for these special targets, which can not only optimize MPSP approach. The guidance scheme is validated in Sec- trajectory online but also satisfy terminal-angle constraint. tion 4, considering a constant speed target and a typical To solve the real-time suboptimal problem with terminal boost-gliding target. Finally, the text is briefly summarized constraints, Dwivedi et al. [14, 15] and Padhi and Kothari in Section 5. [16] put forward the model predictive static programming (MPSP) technique, which combines the “model predictive control” and “approximate dynamic programming” philoso- 2. Midcourse Guidance Problem phies. With the qualities of accurate convergence and low computational cost, the MPSP technique has attracted exten- 2.1. Consideration for Hypersonic Interceptor. Referring to the hypersonic target, many aspects must be taken into consider- sive attention. Applying the MPSP algorithm, Oza and Padhi ’ [17] presented a guidance law for air-to-ground missiles, ation while designing the interceptor s midcourse guidance. which satisfies the constraint of terminal impact angle; supe- (1) Different from most ballistic missiles, hypersonic tar- riority was demonstrated in comparison with augmented get can perform a wide range of maneuver strategies proportional navigation guidance and explicit linear optimal without affecting the final attack performance during guidance. Halbe et al. [18] presented a robust suboptimal the strike mission, which raises the demand of accu- reentry guidance scheme for a reusable launch vehicle with racy and timeliness for target tracking and prediction the help of the MPSP algorithm. Apart from hard constraints of terminal conditions, soft constraints on path and control (2) Limited by the working condition of structure and were also introduced through an innovative cost function, detection system, the interceptor’s speed is gener- which consists of several components with various weighting ally slower than that of the target’s. Therefore, factors. To solve the state variable inequality constraint, Bhi- head-on or head pursuit guidance [26, 27] remains tre and Padhi [19] converted the MPSP problem into an the only choice unconstrained but higher dimension one by means of a slack variable technique. Afterwards, the state constrained MPSP (3) In view of the high closing speed and the capability of formulation was tested on an air-to-air engagement scenario. penetration, the velocity heading error must be small Considering the soft landing problem of a lunar craft during enough at the beginning of the terminal phase. Head- fi the powered descent phase, Sachan and Padhi [20] proposed on condition is bene cial for a successful interception a fuel optimal guidance law under the help of G-MPSP, (4) As the hypersonic target flies in a large airspace and fi where both terminal position and velocity were satis ed by midcourse guidance takes a relatively long time, the adjusting the magnitude and angle of thrust. Bringing in slid- abilities of rapid trajectory correction and flight energy ing mode control, S. Li and X. Li [21] solved the divergence management become essential for the interceptor problem of the common MPSP algorithm caused by model inaccuracy; better robustness was proved when applied to Hypersonic vehicles are mainly classified into two cate- maneuvering target interception. Moreover, Maity et al. [22, gories, the air-breathing type equipped with scramjet engine 23], Kumar et al. [24], and Guo et al. [25] also promoted like the X-51 and the boost-gliding type powered by rockets the MPSP theory and applied it to the guidance design. like the HTV-2 [28]. Significant differences can be found in These aforementioned MPSP guidance laws in [15, 17] their maneuvering strategy and capability. The air-breathing only lead the vehicle to a desired position defined by Y X hypersonic target is critically restricted by the operation and Z X . However, when referring to a defense scenario, conditions of the scramjet. It generally adopts horizontal the optimal handover point changes with the maneuvering flight in the longitudinal plane during cruise segment, while target, which requires additional prediction in every guid- bank-to-turn control is applied in the lateral plane to achieve ance cycle. In the process of midcourse guidance design for evasive maneuvers. Being absent of thrust after reentry, three-dimensional interception, the line-of-sight is not con- the boost-gliding target however employs skip ballistic to sidered, which is vital to the flight trajectory and the startup increase flight distance; bank angle is adjusted to ensure the of detection system. In addition, state-of-the-art guidance heading direction. Target identification can be done with methods for intercepting hypersonic speed and large-scale the help of early warning systems. Midcourse guidance is maneuverable target are rarely researched to the best of the then designed, respectively, according to the differences authors’ knowledge. To solve these problems, this paper between air-breathing and boost-gliding hypersonic targets. International Journal of Aerospace Engineering 3

plane, the axis MY1 is normal to MX1 and points upward. Y The axis MZ1 is defined by the right-hand rule and is normal Y1 aT T to MX1 and MY1. AXYZ is the inertial reference coordinate X1 system, and for clarity of illustration, its origin A is translated R to coincide with the origin M. Hereafter, two Euler angles are introduced [30]. qε stands for the elevation line-of-sight, qε namely, the angle between the line-of-sight and horizontal M (A) T′ plane ZAX. qβ is the azimuth line-of-sight, which acts on q� the horizontal plane with respect to reference direction AX. aM R represents the relative distance between the interceptor Z X and target. Z1 The unit vectors of the inertial reference coordinate AX YZ and the line-of-sight coordinate MX1Y1Z1 are i, j, k and i1, j1, k1 , respectively. Transformation from MX1Y1 Figure 1: Interceptor and target relationship. Z1 to AXYZ can be done within two steps: clockwise rotate qε angle with respect to axis MZ1 and then clockwise rotate Based on the above analyses, the instructions of the mid- qβ angle with respect to axis MY1. In this way, we get course guidance design for the hypersonic target interceptor are as follows: i i1 “ ff ” (1) To make use of the zero-e ort miss concept, the j = Lqε, qβ j1 , 1 line-of-sight rate had better approach zero at the k k handover of the midcourse phase and terminal phase. 1 So long as the target does not maneuver, the interceptor will strike on the target without extra control. In where the transformation matrix is expressed as other words, the load factor of the interceptor tends cos 0 sin cos −sin 0 to zero, which is beneficial to the separation from qβ qβ qε qε the booster and the overload reservation for compen- Lq, q = 010 sin cos 0 sating target perturbations ε β qε qε −sin 0 cos (2) To satisfy the capture conditions of the on-board qβ qβ 001 fi seeker and improve the success rate of nal inter- cos qβ cos qε −cos qβ sin qε sin qβ ception, it is important to meet the limit of the final line-of-sight angle or impact angle. The optimal = sin qε cos qε 0 interception geometry will be obtained if an approximate head-on scenario is realized −sin qβ cos qε sin qβ sin qε cos qβ (3) Considering the trajectory smoothness and energy 2 constraint, the control command for the interceptor should be optimized After then, the rotation angular velocity ω of the line-of- sight coordinate relative to the inertial reference coordinate is (4) For the target implementing a large maneuver, its obtained by flight path angle or velocity azimuth angle changes frequently; midcourse guidance with velocity angle ω = qβ j + qεk1 = qβ sin qεi1 + qβ cos qε j1 + qεk1 3 alignment for the interceptor becomes overreact- ing and guidance performance will suffer. In such The range vector in the line-of-sight coordinate is R circumstances, line-of-sight constrained midcourse ,0,0T guidance is preferred to realize the optimal aspect R . Applying the vector derivative method, the relative angle. Otherwise, guidance laws with a velocity angle velocity between the interceptor and the target is given as constraint will be better if the target employs a small V = Ri1 + ω × Ri1 = Ri1 + Rqε j1 − Rqβ cos qεk1 4 maneuver. In this way, the optimized impact angle can be achieved [29], thus reducing the overload demand for the terminal phase Considering nonlinear point mass dynamics, the acceleration vectors of the interceptor and target on the line-of-sight T coordinate system are denoted as a = a , a ε, a β and 2.2. Coupled Line-of-Sight Dynamics. The kinematic relation- M MR M M a = , , T ship between the pursuer and evader in a three-dimensional T aTR aTε aTβ , respectively. Applying the vector deriv- space is shown in Figure 1, where M and T are centroids of ative method once again to relative velocity gives the interceptor and target, respectively. MX1Y1Z1 is the ∂V line-of-sight coordinate system. The axis MX1 is aligned with a − a = ω × V + 5 the line-of-sight and points to target T. Staying in a vertical T M ∂t 4 International Journal of Aerospace Engineering

Substituting (3) and (4) to (5) and rearranging equations, changes of its acceleration at all times. It is not reliable the line-of-sight coupled 3D interception model is obtained: to predict the future maneuvers of the hypersonic target. Therefore, the acceleration is neglected briefly 2 2 2 a − a = R − Rq − Rq cos q , TR MR ε β ε (2) Applying the quasisteady assumption, the closing 2 velocity of the adversaries is regarded as a constant, a ε − a ε = Rqε +2Rqε + Rqβ sin qε cos qε, T M = − namely, R Vc. Guidance law acts periodically, − = − cos − 2 cos +2 sin aTβ aMβ Rqβ qε Rqβ qε Rqεqβ qε the closing velocity is updated in every guidance 6 cycle, and the deviations would be corrected Discretizing the plant dynamics (7) with Euler’s integra- Generally, the thrust magnitude of the interceptor is tion formula gives designed previously according to the velocity characteristics, and the drag on the vehicle is closely related to its dynamic X = F X , U = X + X , U , 8 ffi k+1 k k k k f k k T pressure. As a consequence, the term aMR is di cult to be controlled during flight. In fact, like proportional navigation where X ∈ ℜ4, U ∈ ℜ2, subscript =1,2,… , is the time (PN) and most other guidance laws do, the control of aMR is k N not necessary as long as R <0. Thereby, the equation along index, and T is the step size. the MX1 direction can be ignored. Assuming that the interceptor can receive immediate , , , , Defining the state variables x1 = qε, x2 = qε, x3 = qβ, and position xT yT zT and velocity VxT VyT VzT of the target from remote data-link during the midcourse, together x4 = qβ, the state transition equation for the line of sight with its position x , y , z and velocity V , V , V and its rate can be written as M M M xM yM zM from an on-board IMU system, the relative position and

x1 = x2, velocity can be obtained by

2 T R 2 aMε aTε r = , , x2 = − x2 − x4 sin x1 cos x1 − + , xMT yMT zMT R R R 7 = − , − , − T, = , xT xM yT yM zT zM x3 x4 9 V = , , T 2 a β a β VxMT VyMT VzMT = − R +2 tan + M − T x4 x4 x2x4 x1 cos cos = − , − , − T R R x1 R x1 VxT VxM VyT VyM VzT VzM

The objective in the midcourse guidance design is to opti- Therefore, the line-of-sight information is derived as mize the control history of aMε and aMβ, meanwhile satisfying the constraints in the guidance process and at the handover. yMT qε = arctan , 2 + 2 3. Angle Constrained MPSP Guidance xMT zMT − In this section, the suboptimal midcourse guidance law with xMTVyMT yMTVxMT q = , a terminal-angle constraint is derived by means of the MPSP ε 2 + 2 + 2 xMT yMT zMT 10 algorithm, in particular, the line-of-sight constrained MPSP zMT guidance and the impact angle constrained MPSP guidance. qβ =arctan − , xMT 3.1. Line-of-Sight Constrained Guidance. A brief principle of z V − x V MPSP can be described as follows. Firstly, expand the q = MT xMT MT zMT fi β 2 + 2 + 2 observed output with rst-order Taylor series at the expected xMT yMT zMT terminal point. Then, the deviation between the predicted output and the desired value can be formulated by an initial T 4 Now, the measured output Y = qε, qε, qβ, qβ ∈ ℜ is state error and control error history. Applying an appropriate selected in terms of specific guidance requirements, right in cost function, the corrected control is derived from iterative Y = X calculation, when the final output value approximates the the state of the system, and we get k k. Marking N as the prediction horizon of the MPSP method, the missile- expected one. = − R / − 1 Because of the guidance iteration, flight trajectory is target range step is expressed as Rstep Rinit end N , corrected in each cycle to get rid of external disturbance. where Rinit indicates the missile-target distance at every guid- The following assumptions are acceptable while processing ance epoch and Rend means the handover distance. Then, the = / the guidance design: increment of time is obtained by T Rstep Vc. Through appropriate control modification dU, MPSP ff Y (1) Di erent from the terminal phase, midcourse guid- drives the predicted terminal output N to the expected value Y∗ fi ance lasts longer. The hypersonic target is most likely N . In consideration of the nal line-of-sight constraints and to adopt various maneuver strategies, causing the parallel approach method, the desired output is select as International Journal of Aerospace Engineering 5

T ∂Y ∂F ∂F Y∗ = ∗ ,0, ∗ ,0 11 Y = N N−1 X + N−1 U 17 N qεf qβf d N ∂X ∂X d N−1 ∂U d N−1 N N−1 N−1 Y Applying the Taylor series expansion to N at the point In this way, the output deviation is expressed as a Y∗ N and ignoring higher-order terms, the output deviation function of the initial state error and control error is written as sequences. ∂Y Y = N X , 12 dY = AdX1 + B1dU1+⋯+B −1dU −1, 18 d N ∂X d N N N N N A B wherein the matrix and the sensitivity matrices k are where deﬁned as 1000 ∂Y N−1 ∂F A = N m , ∂Y 0100 ∂X ∂X N = 13 N m=1 m ∂X 19 N 0010 ∂Y N−1 ∂F ∂F B = N m k 0001 k ∂X ∂X ∂U N m=k+1 m k The transfer function of the state error can be derived from (8): Given the initial condition of the missile-target con- frontation, no error exists in the ﬁrst state, i.e., dX1 =0, ∂F ∂F equation (18) yields X = k X + k U 14 d k+1 ∂X d k ∂U d k k k N−1 Y = B U +⋯+B U = 〠 B U 20 d N 1d 1 N−1d N−1 kd k X U =1 In the formula, d k and d k represent the state error k and control error at the k step, respectively; the partial B derivatives of the state variables and the control variables The sensitivity matrices k can be solved recursively in are expressed as the following manner:

1 00 ∂Y T B0 = N , N−1 ∂X ∂F a21 a22 0 a23 N k = , ∂X ∂F k 001T B0 = B0 k+1 , 21 k k+1 ∂X +1 a41 a42 0 a44 k 15 ∂F 00 B = B0 k k k ∂U k ∂F b21 0 k = , ∂U k 00 To optimize the trajectory of the interceptor and fl fi 0 b42 maintain a smooth ight, de ne the quadratic objective function as where 1 N−1 = 〠 U0 − U T R U0 − U , 22 = − 2 cos 2 , J 2 k d k k k d k a21 x4k x1k T k=1 =1+2 / , a22 VcT Rk where R is the positive definite weighting matrix and U0 = − sin 2 , k k a24 x4k x1k T is the previous control solution. Note that the weight matrix 2 a β tan x1 is alternatively chosen to adjust the magnitude of the con- = x2kx4k + M k k , a41 2 T trol history at different time steps. cos x1 R cos x1 16 k k k For the constrained optimal problem defined in equa- =2 tan a42 x4k x1k T tions (20) and (22), static optimization theory is applied, =1+2 / +2 tan , which gives the augmented cost function a44 VcT Rk x2k x1k T = − / , b21 T Rk 1 N−1 ′ = 〠 U0 − U T R U0 − U J k d k k k d k b42 = T/ R cos x1 2 k k k=1 23 N−1 = − 1 + λT Y − 〠 B U Taking the step k N in (14) and substituting it to d N kd k (12), we obtain k=1 6 International Journal of Aerospace Engineering

60 Y 40 � imp 20 � VT Tf

( ° ) 0 f

V q M T −20 �Mf −40 qf M −60 O X 0 100 200 300

�imp (°) v = 1.5 v = 2.5 v = 2.0 v = 3.0 θ (a) Impact triangle (b) imp and qf relationship

Figure 2: Interception angle in a 2D plane.

The necessary conditions for minimal J′ are given by process is done in the longitudinal plane and the lateral plane is similar. ′ The impact angle is defined as the angle between velocity ∂J 0 k = −R U − dU − BT λ =0, vectors of the adversaries; particularly, we have ∂dU k k k k k 24 −1 ∂ ′ N θimp = θ − θ , 27 Jk = Y − 〠 B U =0 T M ∂λ d N kd k k=1 where θimp is the impact angle in the longitudinal plane and θ θ fl Solving for the corrected control U , we get T and M are the ight path angles of the target and inter- k ceptor, respectively. A perfect head-on interception scenario θ = π 0 −1 T −1 is formed by restricting imp . U = U − dU = R B A dY − bλ , 25 k k k k k λ N When the midcourse guidance comes to the end, the line- of-sight rate q has been driven to zero. The interceptor and where target stay on an impact triangle geometry, as shown in Figure 2(a). Under this circumstance, the zero-effort miss N−1 A = − 〠 B R−1BT , can be guaranteed if both the interceptor and the target do λ k k k k=1 not execute any further maneuvers from the instant time 26 onward. Then, the relationship between the terminal line- N−1 b = 〠 B U0 of-sight angle and impact angle is given by λ k k k=1 sin θ − − sin θ − =0, 28 VT Tf qf VM Mf qf In conclusion, the line-of-sight constrained MPSP guidance is numerically solved by the control (25), considering where V and V are the velocities of the hypersonic target the discrete guidance problem (8) with the required output T M and interceptor, respectively, q stands for the line-of-sight (11) and cost function (22). Note that the control sequence in a 2D plane, and subscript f represents the final value in U k =1,2,… , N − 1 is solved in each guidance period, k midcourse guidance. but only U1 is employed as the current guidance command. Noting the target-to-missile speed ratio v = V /V , rear- The suboptimal midcourse guidance law for target intercep- T M ranging (28) yields tion not only optimizes the flight control but also satisfies the terminal constraints of the line-of-sight angle and its rate. sin θ = θ − arctan imp − π 29 qf Tf n 3.2. Impact Angle Constrained Guidance. As already men- cos θimp − v tioned in Section 2.1, the midcourse guidance with velocity angle alignment exhibits better handover properties if the ff target’s maneuver is somehow weak. However, only the line Di erent from the situation in [31], the interceptor no >1 of sight and line-of-sight rate are involved while modeling longer has velocity advantage, namely, v ; the singular the interception scenario. A solution can be found in [31], problem is structurally avoided. One-to-one condition of qf where the desired impact angle is able to be transferred into and θimp is further analyzed, in which a monotonous zone the final line-of-sight angle. For simplicity, the following including head-on condition can be found. Considering an International Journal of Aerospace Engineering 7

θ = π Table incoming target with Tf , the relationship between qf 1: Convergence results of LOS constrained guidance. θ and imp changes according to the speed ratio v, as shown ° rad/ Rm qf q s nyf g in Figure 2(b). Since qf is continuous, extreme points can / θ =0 cos = −5° Default 60000 -5 0 be obtained by driving dqf d imp , which gives v qf -0.149 Real 59960 -5.000 −5 908e − 5 θimp =1. In conclusion, for any desired impact angle θimp ∈ ° Default 60000 0 0 arccos 1/v ,2π − arccos 1/v , a corresponding q can be q =0 f f −8 377 − 6 −1 800 − 5 -0.052 calculated by (29). Real 59972 e e Applying (29), the aforementioned line-of-sight con- =5° Default 60000 5 0 qf -0.057 strained MPSP guidance is transferred into the impact angle Real 59940 5.000 −2 21e − 6 constrained MPSP guidance, which actually constrains the final flight path angle for the interceptor in the midcourse earth model are applied, where the Coriolis and centrifugal phase. Note that the prediction of the target’s terminal flight inertial force are technically ignored. path angle is required. = sin θ, 4. Simulation r V V cos θ sin ψ ff ff λ = , Three di erent scenarios are presented to validate the e ec- r cos ϕ tiveness of the constrained MPSP midcourse guidance law. cos θ cos ψ The first two scenarios refer to the cruising hypersonic target ϕ = V , with a constant velocity in the longitudinal plane. Line-of- r sight constrained and impact angle constrained midcourse D 31 V = − − g sin θ, guidance are applied, respectively; The last scenario con- m siders the gliding target with a skipping trajectory in a L cos σ V2 cos θ three-dimensional space, and a full guidance scheme includ- θ = + − g , ing midcourse is adopted. mV r V sin σ 4.1. Problem Setup ψ = L + V cos θ sin ψ tan ϕ, mV cos θ r 4.1.1. Interceptor and Target Model. The interceptor is modeled using three-degree-of-freedom (3DOF) particle where r is the radial distance from the earth’s center to the dynamics [15]. gliding vehicle; λ and ϕ are the longitude and latitude of the location; and V is the vehicle speed. The flight path angle x = V cos θ cos ψ, and velocity azimuth angle are marked as θ and ψ; L and D indicate the aerodynamic lift and drag; and σ is the controlled = sin θ, y V bank angle. = cos θ sin ψ, z V 4.1.2. Overall Guidance Scheme. In order to improve mobility − = T D − sin θ, and timeliness of the interceptor platform, meanwhile V g expanding the defensive area, the air-launched interceptor m 30 1 is adopted. The ability of attack beyond visual range is vital θ = a − g cos θ , to a multiple-layer intercept. Taking limited energy and load V y factor into account, the interceptor prefers a parabola trajec- ψ = az , tory for its boost guidance. In this phase, the interceptor sep- cos θ V arates from the air platform and climbs to the required T altitude. The drag can be significantly reduced in a high alti- m = , Isp tude and its operational range is extended thereafter. The trigonometric guidance command [33] is applied for the smooth shifting from separation; thus, the overload would where x, y, and z are the positions of the interceptor, V is increase gently as expected. The flight path angle command the vehicle speed, and m stands for mass. The relative and velocity azimuth angle command are given by flight path angle and velocity azimuth angle are θ and ψ, respectively; a and a represent the acceleration commands π π y z θ = θ sin t , in the longitudinal and lateral planes; T and D donate the M opt 2t0 2 t0 thrust and drag on the interceptor; and Isp is the specific 32 π π impulse of the rocket or reaction control system (RCS). ψ = ψ sin t , M 2 opt 2 Taking conventional gliding vehicle interception as an t0 t0 example, the target adopts bank angle constrained predictor- corrector reentry guidance [32]. The particle dynamics of a where t represents the guidance time after separation, common aero vehicle (CAV) over a nonrotating spherical while t0 is the total time preset for the initial phase; θopt 8 International Journal of Aerospace Engineering

×104 5 6

4 4 2

3 0 q ( º ) y (m) −2 2 −4

1 −6 012340 20 40 60 80 100 x (m) ×105 t (s) qf = −5° qf = 5° qf = −5º qf = 0° Target qf = 0º qf = 5º

(a) Interception trajectory (b) Line-of-sight angle ×10−3 2 6

0 2

(g) 0 y q (rad/s) n −2 −2

−4

−4 −6 0 20406080100 0 20406080100 t (s) t (s) qf = −5º qf = −5º qf = 0º qf = 0º qf = 5º qf = 5º

Figure 3: Simulation results of LOS constrained guidance.

Table 2: Convergence results of impact angle constrained guidance. and ψopt are the expected flight path angle and velocity azimuth angle depending on the predicted impact point and rad/ θ ° Rm q s imp nyf g maximum altitude of the parabola trajectory. ° Default 60000 0 170 The proposed terminal-angle constrained MPSP guid- θimp = 170 5 614 − 6 0.015 ance law is applied for interceptor’s midcourse flight. This Real 59981 e 170.016 approach requires the line-of-sight and line-of-sight rate ° Default 60000 0 180 θimp = 180 -0.051 information in real time. However, on-board detection Real 59959 −1 465e − 5 179.958 system will not be activated until the capture conditions θ = 190° Default 60000 0 190 are satisfied. Assuming that the interceptor can receive tar- imp -0.104 −2 948e − 5 get’s information provided by ground radar or space-based Real 59945 189.915 infrared system, additional filter and prediction are necessary for signal extraction and denoising. For those air-breathing nonlinear Markov acceleration model [34] exhibits better hypersonic targets, the interacting multiple model algorithm performance for boost-gliding targets. with commonly “CA” and “current” statistical models is pre- In the terminal phase, the goal of guidance is trying to ferred accounting for its complex maneuverability. However, minimize the miss distance. However, the aerodynamic force it is reported in the literature that filter based on the in the near space is not enough. The reaction control system International Journal of Aerospace Engineering 9

×104 4.5 190

4 180

3.5 170

3 ( ° ) 160 imp y (m) � 2.5 150

2 140

1.5 130 012340 20406080100 x (m) ×105 t (s)

�imp = 170º �imp = 190º �imp = 170º �imp = 180º Target �imp = 180º �imp = 190º

(a) Interception trajectory (b) Impact angle ×10−3 0.5 6

0 4

−0.5 2 -1 (g) y n q (rad/s) 0 −1.5

−2 −2

−2.5 −4 0 20 40 60 80 100 0 20406080100 t (s) t (s)

�imp = 170º �imp = 170º �imp = 180º �imp = 180º �imp = 190º �imp = 190º

Figure 4: Simulation results of impact angle constrained guidance.

is designed to accelerate the control response and increase where η, η1 >0, δ, and δ1 are small positive scalars and the available overload. In addition, the on-board detection R1 = R cos qε . system is activated and target information is acquired without delay. Therefore, the guidance and control period 4.2. Line-of-Sight Constrained Guidance. Interception con- has shortened obviously. For the above considerations, an frontation scenario is presented to verify the proposed analytic guidance called adaptive sliding mode guidance line-of-sight constrained MPSP midcourse guidance. The (ASMG) [35] is applied for terminal engagement, and the following conditions are set assuming the adversaries are in sign function is replaced by a saturation function to eliminate the longitudinal plane. chattering. The acceleration command is expressed as (1) The initial position of the interceptor 0,17 km, the ° = +1 + η qε , average velocity at 1200 m/s, 0 initial ﬂight path aMε k R qε qε + δ angle, and 5 g maximum overload 33 qβ (2) The initial position of hypersonic target 400,35 km, a = k +1 R1 q + η ° MB β 1 the average velocity at 2400 m/s, 180.286 initial ﬂight qβ + δ1 path angle, and without maneuver 10 International Journal of Aerospace Engineering

×104 1400 5

1200 4 1000 3

y (m) 800 z (m/s) 2 600

1 400 4 2 6 5 4 200 ×10 0 0 2 x (m) z (m) ×105 0 50 100 150 200 t (s) Missile CAV target

(a) Interception trajectory (b) Velocity ×104 ×105 5 6

4 4

3 y (m) z (m) 2 2

1 0 0123 0123 x (m) ×105 x (m) ×105

Missile Missile CAV target CAV target

2 5 (g) 0 (g) y z n n

0 −2

−4 −5 0 50 100 150 200 0 50 100 150 200 t (s) t (s)

(e) Longitudinal load factor (f) Lateral load factor

Figure 5: Continued. International Journal of Aerospace Engineering 11

20 0

−20 0

−40 � ( º ) � ( º )

−20 −60

−40 −80 050100 150 200 0 50 100 150 200 t (s) t (s)

(g) Flight path angle (h) Velocity azimuth angle

Figure 5: Simulation results of CAV interception.

(3) Terminal handover distance is 60 km, simulation step (2) The interceptor’s rocket works in two levels, the first of 20 ms, and the scheduled final line-of-sights are level lasts 2 s and produces 30kN thrust, while the ° ° ° -5 ,0, and 5 , respectively second level lasts 22 s and produces 8 kN thrust (4) Initial control guess utilizes the three-dimensional (3) The hypersonic target maintains pullup and pull- proportional guidance law down gliding trajectory with an initial altitude of 60 km and speed of 5100 m/s The results are shown in Table 1 and Figure 3. From Figure 3(a), we can see that the interceptor is able Assuming that the interceptor is released once the to reach the desired position under the command of the missile-target range meets the desired distance, simulation line-of-sight constrained MPSP guidance algorithm, right in results are shown in Figure 5. From that, we can see the the front of the hypersonic target. Figures 3(a) and 3(b) indi- progress of the interceptor. After launching from an air- cate that the terminal constraints of both line-of-sight and borne platform, the interceptor climbs to upper air under line-of-sight rate have been satisfied with high accuracy. In trigonometric guidance. Until 15.00 s later, it turns to mid- Figure 3(d), the overload command can gradually converge course phase and keeps accelerating. A maximum speed of to zero, thus providing a favorable initial condition for the 1389.92 m/s is achieved when the rocket engine burns out endgame interception. at 24.00 s. The trajectory reaches a maximum altitude of 42.67 km after 113.36 s. Separation from booster rocket 4.3. Impact Angle Constrained Guidance. To validate the appears at 166.33 s when the handover range is reached and fi impact angle constrained MPSP midcourse guidance, the terminal guidance takes over. The nal miss distance is fl same simulation conditions are set as in Section 4.2, except 2.47 m after 186.44 s ight, which demonstrates the correct- ° ° ° that the required impact angles are 170 , 180 , and 190 , ness of the proposed guidance strategy. ° where 180 represents a typical head-on collision. The simulation results are shown in Table 2 and Figure 4. 5. Conclusions Similar conclusions can be made in contrast to the previous line-of-sight constrained MPSP guidance method. As This paper proposes a novel midcourse guidance strategy for shown in Figure 4(a), the interceptor is able to reach the interceptors against near-space hypersonic targets. The char- desired position under the command of the impact angle acteristics of hypersonic weapon defense and the require- constrained MPSP guidance algorithm. Figures 4(b) and ments of midcourse guidance design for interceptors are 4(c) imply that the terminal constraints of both impact angle analyzed. The line-of-sight coupled interception model is and line-of-sight rate have been satisfied. In Figure 4(d), established in a three dimensional space. Under the assump- the overload command gradually converges to zero and tion that the closing velocity remains constant and the target therefore provides a favorable initial condition for the end- maneuver is briefly negligible, a MPSP method-based subop- game interception. timal midcourse guidance law is derived for two different purposes that either satisfies the terminal line-of-sight angle 4.4. Full Trajectory Interception. To verify the effectiveness of constraint or satisfies the terminal impact angle constraint. full trajectory guidance scheme, parameters are set as follows: Results indicate that the designed angle can be realized with ° maximum error less than 0 . In addition, the line-of-sight (1) Initial position of interceptor is 0,12,0 km, initial rate approaches zero at handover time, which implies that velocity is 360 m/s, and initial flight path angle and the final overload command can converge to a small value. ° velocity azimuth angle are both 0 In the interception of a hypersonic gliding target, the final 12 International Journal of Aerospace Engineering miss distance turns out to be 2.47 m, which proves the effec- Navigation, and Control Conference and Exhibit, pp. 1–17, tiveness of this guidance law. San Francisco, CA, USA, 2005. [13] R. Jamilnia and A. Naghash, “Optimal guidance based on Data Availability receding horizon control and online trajectory optimization,” Journal of Aerospace Engineering, vol. 26, no. 4, pp. 786–793, The data used to support the findings of this study are avail- 2013. able from the corresponding author upon request. [14] P. N. Dwivedi, A. Bhattacharyya, and R. Padhi, “Computation- ally efficient suboptimal mid course guidance using model pre- Conflicts of Interest dictive static programming (MPSP),” IFAC Proceedings Volumes, vol. 41, no. 2, pp. 3550–3555, 2008. The authors declare that they have no conflicts of interest. [15] P. N. Dwivedi, A. Bhattacharya, and R. Padhi, “Suboptimal midcourse guidance of interceptors for high-speed targets with ” Acknowledgments alignment angle constraint, Journal of Guidance, Control, and Dynamics, vol. 34, no. 3, pp. 860–877, 2011. This research was supported by the National Natural Science [16] R. Padhi and M. Kothari, “Model predictive static program- Foundation of China (Grant no. 61503301) and the National ming: a computationally efficient technique for suboptimal ” Safety Academic Fund (Grant no. U1630127). control design, International Journal of Innovative Comput- ing, Information and Control, vol. 5, no. 2, pp. 399–411, 2009. “ References [17] H. B. Oza and R. 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