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

Authors: Shizheng Wan; Xiaofei Chang; Quancheng Li; Jie Yan
Journal: International Journal of Aerospace Engineering
 (2019)

---

## Abstract

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.


---

## Body

## 1. Introduction

The hypersonic vehicle, typically traveling greater than Mach 5 in the near space, has developed rapidly in recent years, and their weaponization has accelerated. Compared with conventional missiles, hypersonic weapons show great advantages of diverse launching platforms, fast flight speed, and strong penetration capability. Aiming at time-sensitive and high-value targets, they pose severe challenges to the traditional antiaircraft or antimissile defense system. Hypersonic vehicles are going through intensive flight testing, for instance, the “Hypersonic Air-breathing Weapon Concept” and “Tactical Boost Glide” [1], the Russian “Zircon” hypersonic cruise missile [2], and the Chinese “DF-ZF” hypersonic glide vehicle [3, 4]. The defense system should be upgraded and prepared to meet the requirements of the future battlefield.In order to eliminate the target in time, modern interceptors generally require the ability of beyond-visual-range air combat. Midcourse guidance is vital to the entire interception process. The purpose is to lead the interceptor to a desired position, where not only the target can be acquired but also a good initial angle is obtained for the terminal phase, thus to increase the interception rate. Midcourse guidances for intercepting conventional aviation targets or ballistic missiles are relatively mature. For example, concerning air-to-air missile, singular perturbation theory was applied to the derivation of near-optimal midcourse guidance in articles [5–7], but they did not consider the angle alignment constraints. Focusing on the optimal guidance problem with angular constraint, Indig et al. [8] proposed an analytic guidance law with a linear dynamic model, which would cause control saturation at the time of interception. Liu et al. [9] designed a biased proportional navigation guidance method with attitude constraint and line-of-sight angle rate control, but it mainly aimed at stationary ground target, not to mention the capability of energy management. Taking constant acceleration into consideration, Seo and Tahk [10] provided a closed-loop midcourse guidance law, but the terminal constraint can be fulfilled only under the assumption that the impact point was precisely predicted. To maximize the kill probability, Phillips and Drake [11] designed a two-tier approach for surface-to-air midcourse trajectory with an auxiliary constraint on an intersection approach angle. Three classes of midcourse guidance laws, namely, optimal fuel, optimal energy, and closed-loop explicit strategy, are proposed by Jalali-Naini and Pourtakdoust [12] with a zero-effort miss differential equation. Selecting different objectives and constraint functions, Reza and Abolghasem [13] solved an optimal interception problem with receding horizon control; commands are generated online by means of a differential flatness concept and nonlinear programming. Characterized by ultrafast speed and large-scale maneuverability, the requirements for hypersonic target interception become extremely strict, thus failing most of the existing midcourse guidance methods. New approaches must be found for these special targets, which can not only optimize trajectory online but also satisfy terminal-angle constraint.To solve the real-time suboptimal problem with terminal constraints, Dwivedi et al. [14, 15] and Padhi and Kothari [16] put forward the model predictive static programming (MPSP) technique, which combines the “model predictive control” and “approximate dynamic programming” philosophies. With the qualities of accurate convergence and low computational cost, the MPSP technique has attracted extensive attention. Applying the MPSP algorithm, Oza and Padhi [17] presented a guidance law for air-to-ground missiles, which satisfies the constraint of terminal impact angle; superiority was demonstrated in comparison with augmented proportional navigation guidance and explicit linear optimal guidance. Halbe et al. [18] presented a robust suboptimal reentry guidance scheme for a reusable launch vehicle with the help of the MPSP algorithm. Apart from hard constraints of terminal conditions, soft constraints on path and control were also introduced through an innovative cost function, which consists of several components with various weighting factors. To solve the state variable inequality constraint, Bhitre and Padhi [19] converted the MPSP problem into an unconstrained but higher dimension one by means of a slack variable technique. Afterwards, the state constrained MPSP formulation was tested on an air-to-air engagement scenario. Considering the soft landing problem of a lunar craft during the powered descent phase, Sachan and Padhi [20] proposed a fuel optimal guidance law under the help of G-MPSP, where both terminal position and velocity were satisfied by adjusting the magnitude and angle of thrust. Bringing in sliding mode control, S. Li and X. Li [21] solved the divergence problem of the common MPSP algorithm caused by model inaccuracy; better robustness was proved when applied to maneuvering target interception. Moreover, Maity et al. [22, 23], Kumar et al. [24], and Guo et al. [25] also promoted the MPSP theory and applied it to the guidance design.These aforementioned MPSP guidance laws in [15, 17] only lead the vehicle to a desired position defined by YX and ZX. However, when referring to a defense scenario, the optimal handover point changes with the maneuvering target, which requires additional prediction in every guidance cycle. In the process of midcourse guidance design for three-dimensional interception, the line-of-sight is not considered, which is vital to the flight trajectory and the startup of detection system. In addition, state-of-the-art guidance methods for intercepting hypersonic speed and large-scale maneuverable target are rarely researched to the best of the authors’ knowledge. To solve these problems, this paper adopts the line-of-sight coupled interception model and proposes a novel midcourse guidance method with the MPSP technique according to the characteristics of a specific target, which can optimize the interceptor’s trajectory and also meet the handover constraints.The rest paper is organized as follows. In Section2, the characteristics and technical requirements of midcourse guidance for hypersonic target interception are analyzed, and a three-dimensional interception model is established. In Section 3, suboptimal guidance law with constrained terminal line-of-sight or impact angle is derived utilizing the MPSP approach. The guidance scheme is validated in Section 4, considering a constant speed target and a typical boost-gliding target. Finally, the text is briefly summarized in Section 5.


## 2. Midcourse Guidance Problem

### 2.1. Consideration for Hypersonic Interceptor

Referring to the hypersonic target, many aspects must be taken into consideration while designing the interceptor’s midcourse guidance.(1)
Different from most ballistic missiles, hypersonic target can perform a wide range of maneuver strategies without affecting the final attack performance during the strike mission, which raises the demand of accuracy and timeliness for target tracking and prediction
(2)
Limited by the working condition of structure and detection system, the interceptor’s speed is generally slower than that of the target’s. Therefore, head-on or head pursuit guidance [26, 27] remains the only choice
(3)
In view of the high closing speed and the capability of penetration, the velocity heading error must be small enough at the beginning of the terminal phase. Head-on condition is beneficial for a successful interception
(4)
As the hypersonic target flies in a large airspace and midcourse guidance takes a relatively long time, the abilities of rapid trajectory correction and flight energy management become essential for the interceptorHypersonic vehicles are mainly classified into two categories, the air-breathing type equipped with scramjet engine like the X-51 and the boost-gliding type powered by rockets like the HTV-2 [28]. Significant differences can be found in their maneuvering strategy and capability. The air-breathing hypersonic target is critically restricted by the operation conditions of the scramjet. It generally adopts horizontal flight in the longitudinal plane during cruise segment, while bank-to-turn control is applied in the lateral plane to achieve evasive maneuvers. Being absent of thrust after reentry, the boost-gliding target however employs skip ballistic to increase flight distance; bank angle is adjusted to ensure the heading direction. Target identification can be done with the help of early warning systems. Midcourse guidance is then designed, respectively, according to the differences between air-breathing and boost-gliding hypersonic targets.Based on the above analyses, the instructions of the midcourse guidance design for the hypersonic target interceptor are as follows:(1)
To make use of the “zero-effort miss” concept, the line-of-sight rate had better approach zero at the handover of the midcourse phase and terminal phase. So long as the target does not maneuver, the interceptor will strike on the target without extra control. In other words, the load factor of the interceptor tends to zero, which is beneficial to the separation from the booster and the overload reservation for compensating target perturbations
(2)
To satisfy the capture conditions of the on-board seeker and improve the success rate of final interception, it is important to meet the limit of the final line-of-sight angle or impact angle. The optimal interception geometry will be obtained if an approximate head-on scenario is realized
(3)
Considering the trajectory smoothness and energy constraint, the control command for the interceptor should be optimized
(4)
For the target implementing a large maneuver, its flight path angle or velocity azimuth angle changes frequently; midcourse guidance with velocity angle alignment for the interceptor becomes overreacting and guidance performance will suffer. In such circumstances, line-of-sight constrained midcourse guidance is preferred to realize the optimal aspect angle. Otherwise, guidance laws with a velocity angle constraint will be better if the target employs a small maneuver. In this way, the optimized impact angle can be achieved [29], thus reducing the overload demand for the terminal phase


### 2.2. Coupled Line-of-Sight Dynamics

The kinematic relationship between the pursuer and evader in a three-dimensional space is shown in Figure1, where M and T are centroids of the interceptor and target, respectively. MX1Y1Z1 is the line-of-sight coordinate system. The axis MX1 is aligned with the line-of-sight and points to target T. Staying in a vertical plane, the axis MY1 is normal to MX1 and points upward. The axis MZ1 is defined by the right-hand rule and is normal to MX1 and MY1. AXYZ is the inertial reference coordinate system, and for clarity of illustration, its origin A is translated to coincide with the origin M. Hereafter, two Euler angles are introduced [30]. qε stands for the elevation line-of-sight, namely, the angle between the line-of-sight and horizontal plane ZAX. qβ is the azimuth line-of-sight, which acts on the horizontal plane with respect to reference direction AX. R represents the relative distance between the interceptor and target.Figure 1
Interceptor and target relationship.The unit vectors of the inertial reference coordinateAXYZ and the line-of-sight coordinate MX1Y1Z1 are i,j,k and i1,j1,k1, respectively. Transformation from MX1Y1Z1 to AXYZ can be done within two steps: clockwise rotate qε angle with respect to axis MZ1 and then clockwise rotate qβ angle with respect to axis MY1. In this way, we get
(1)ijk=Lqε,qβi1j1k1,where the transformation matrix is expressed as
(2)Lqε,qβ=cosqβ0sinqβ010−sinqβ0cosqβcosqε−sinqε0sinqεcosqε0001=cosqβcosqε−cosqβsinqεsinqβsinqεcosqε0−sinqβcosqεsinqβsinqεcosqβ.After then, the rotation angular velocityω of the line-of-sight coordinate relative to the inertial reference coordinate is obtained by
(3)ω=q̇βj+q̇εk1=q̇βsinqεi1+q̇βcosqεj1+q̇εk1.The range vector in the line-of-sight coordinate isRR,0,0T. Applying the vector derivative method, the relative velocity between the interceptor and the target is given as
(4)V=Ṙi1+ω×Ri1=Ṙi1+Rq̇εj1−Rq̇βcosqεk1.Considering nonlinear point mass dynamics, the acceleration vectors of the interceptor and target on the line-of-sight coordinate system are denoted asaM=aMR,aMε,aMβT and aT=aTR,aTε,aTβT, respectively. Applying the vector derivative method once again to relative velocity gives
(5)aT−aM=ω×V+∂V∂t.Substituting (3) and (4) to (5) and rearranging equations, the line-of-sight coupled 3D interception model is obtained:
(6)aTR−aMR=R¨−Rq̇ε2−Rq̇β2cos2qε,aTε−aMε=Rq¨ε+2Ṙq̇ε+Rq̇β2sinqεcosqε,aTβ−aMβ=−Rq¨βcosqε−2Ṙq̇βcosqε+2Rq̇εq̇βsinqε.Generally, the thrust magnitude of the interceptor is designed previously according to the velocity characteristics, and the drag on the vehicle is closely related to its dynamic pressure. As a consequence, the termaMR is difficult to be controlled during flight. In fact, like proportional navigation (PN) and most other guidance laws do, the control of aMR is not necessary as long as Ṙ<0. Thereby, the equation along the MX1 direction can be ignored.Defining the state variablesx1=qε, x2=q̇ε, x3=qβ, and x4=q̇β, the state transition equation for the line of sight and its rate can be written as
(7)ẋ1=x2,ẋ2=−2ṘRx2−x42sinx1cosx1−aMεR+aTεR,ẋ3=x4,ẋ4=−2ṘRx4+2x2x4tanx1+aMβRcosx1−aTβRcosx1.The objective in the midcourse guidance design is to optimize the control history ofaMε and aMβ, meanwhile satisfying the constraints in the guidance process and at the handover.


## 2.1. Consideration for Hypersonic Interceptor

Referring to the hypersonic target, many aspects must be taken into consideration while designing the interceptor’s midcourse guidance.(1)
Different from most ballistic missiles, hypersonic target can perform a wide range of maneuver strategies without affecting the final attack performance during the strike mission, which raises the demand of accuracy and timeliness for target tracking and prediction
(2)
Limited by the working condition of structure and detection system, the interceptor’s speed is generally slower than that of the target’s. Therefore, head-on or head pursuit guidance [26, 27] remains the only choice
(3)
In view of the high closing speed and the capability of penetration, the velocity heading error must be small enough at the beginning of the terminal phase. Head-on condition is beneficial for a successful interception
(4)
As the hypersonic target flies in a large airspace and midcourse guidance takes a relatively long time, the abilities of rapid trajectory correction and flight energy management become essential for the interceptorHypersonic vehicles are mainly classified into two categories, the air-breathing type equipped with scramjet engine like the X-51 and the boost-gliding type powered by rockets like the HTV-2 [28]. Significant differences can be found in their maneuvering strategy and capability. The air-breathing hypersonic target is critically restricted by the operation conditions of the scramjet. It generally adopts horizontal flight in the longitudinal plane during cruise segment, while bank-to-turn control is applied in the lateral plane to achieve evasive maneuvers. Being absent of thrust after reentry, the boost-gliding target however employs skip ballistic to increase flight distance; bank angle is adjusted to ensure the heading direction. Target identification can be done with the help of early warning systems. Midcourse guidance is then designed, respectively, according to the differences between air-breathing and boost-gliding hypersonic targets.Based on the above analyses, the instructions of the midcourse guidance design for the hypersonic target interceptor are as follows:(1)
To make use of the “zero-effort miss” concept, the line-of-sight rate had better approach zero at the handover of the midcourse phase and terminal phase. So long as the target does not maneuver, the interceptor will strike on the target without extra control. In other words, the load factor of the interceptor tends to zero, which is beneficial to the separation from the booster and the overload reservation for compensating target perturbations
(2)
To satisfy the capture conditions of the on-board seeker and improve the success rate of final interception, it is important to meet the limit of the final line-of-sight angle or impact angle. The optimal interception geometry will be obtained if an approximate head-on scenario is realized
(3)
Considering the trajectory smoothness and energy constraint, the control command for the interceptor should be optimized
(4)
For the target implementing a large maneuver, its flight path angle or velocity azimuth angle changes frequently; midcourse guidance with velocity angle alignment for the interceptor becomes overreacting and guidance performance will suffer. In such circumstances, line-of-sight constrained midcourse guidance is preferred to realize the optimal aspect angle. Otherwise, guidance laws with a velocity angle constraint will be better if the target employs a small maneuver. In this way, the optimized impact angle can be achieved [29], thus reducing the overload demand for the terminal phase


## 2.2. Coupled Line-of-Sight Dynamics

The kinematic relationship between the pursuer and evader in a three-dimensional space is shown in Figure1, where M and T are centroids of the interceptor and target, respectively. MX1Y1Z1 is the line-of-sight coordinate system. The axis MX1 is aligned with the line-of-sight and points to target T. Staying in a vertical plane, the axis MY1 is normal to MX1 and points upward. The axis MZ1 is defined by the right-hand rule and is normal to MX1 and MY1. AXYZ is the inertial reference coordinate system, and for clarity of illustration, its origin A is translated to coincide with the origin M. Hereafter, two Euler angles are introduced [30]. qε stands for the elevation line-of-sight, namely, the angle between the line-of-sight and horizontal plane ZAX. qβ is the azimuth line-of-sight, which acts on the horizontal plane with respect to reference direction AX. R represents the relative distance between the interceptor and target.Figure 1
Interceptor and target relationship.The unit vectors of the inertial reference coordinateAXYZ and the line-of-sight coordinate MX1Y1Z1 are i,j,k and i1,j1,k1, respectively. Transformation from MX1Y1Z1 to AXYZ can be done within two steps: clockwise rotate qε angle with respect to axis MZ1 and then clockwise rotate qβ angle with respect to axis MY1. In this way, we get
(1)ijk=Lqε,qβi1j1k1,where the transformation matrix is expressed as
(2)Lqε,qβ=cosqβ0sinqβ010−sinqβ0cosqβcosqε−sinqε0sinqεcosqε0001=cosqβcosqε−cosqβsinqεsinqβsinqεcosqε0−sinqβcosqεsinqβsinqεcosqβ.After then, the rotation angular velocityω of the line-of-sight coordinate relative to the inertial reference coordinate is obtained by
(3)ω=q̇βj+q̇εk1=q̇βsinqεi1+q̇βcosqεj1+q̇εk1.The range vector in the line-of-sight coordinate isRR,0,0T. Applying the vector derivative method, the relative velocity between the interceptor and the target is given as
(4)V=Ṙi1+ω×Ri1=Ṙi1+Rq̇εj1−Rq̇βcosqεk1.Considering nonlinear point mass dynamics, the acceleration vectors of the interceptor and target on the line-of-sight coordinate system are denoted asaM=aMR,aMε,aMβT and aT=aTR,aTε,aTβT, respectively. Applying the vector derivative method once again to relative velocity gives
(5)aT−aM=ω×V+∂V∂t.Substituting (3) and (4) to (5) and rearranging equations, the line-of-sight coupled 3D interception model is obtained:
(6)aTR−aMR=R¨−Rq̇ε2−Rq̇β2cos2qε,aTε−aMε=Rq¨ε+2Ṙq̇ε+Rq̇β2sinqεcosqε,aTβ−aMβ=−Rq¨βcosqε−2Ṙq̇βcosqε+2Rq̇εq̇βsinqε.Generally, the thrust magnitude of the interceptor is designed previously according to the velocity characteristics, and the drag on the vehicle is closely related to its dynamic pressure. As a consequence, the termaMR is difficult to be controlled during flight. In fact, like proportional navigation (PN) and most other guidance laws do, the control of aMR is not necessary as long as Ṙ<0. Thereby, the equation along the MX1 direction can be ignored.Defining the state variablesx1=qε, x2=q̇ε, x3=qβ, and x4=q̇β, the state transition equation for the line of sight and its rate can be written as
(7)ẋ1=x2,ẋ2=−2ṘRx2−x42sinx1cosx1−aMεR+aTεR,ẋ3=x4,ẋ4=−2ṘRx4+2x2x4tanx1+aMβRcosx1−aTβRcosx1.The objective in the midcourse guidance design is to optimize the control history ofaMε and aMβ, meanwhile satisfying the constraints in the guidance process and at the handover.


## 3. Angle Constrained MPSP Guidance

In this section, the suboptimal midcourse guidance law with a terminal-angle constraint is derived by means of the MPSP algorithm, in particular, the line-of-sight constrained MPSP guidance and the impact angle constrained MPSP guidance.


### 3.1. Line-of-Sight Constrained Guidance

A brief principle of MPSP can be described as follows. Firstly, expand the observed output with first-order Taylor series at the expected terminal point. Then, the deviation between the predicted output and the desired value can be formulated by an initial state error and control error history. Applying an appropriate cost function, the corrected control is derived from iterative calculation, when the final output value approximates the expected one.Because of the guidance iteration, flight trajectory is corrected in each cycle to get rid of external disturbance. The following assumptions are acceptable while processing the guidance design:(1)
Different from the terminal phase, midcourse guidance lasts longer. The hypersonic target is most likely to adopt various maneuver strategies, causing the changes of its acceleration at all times. It is not reliable to predict the future maneuvers of the hypersonic target. Therefore, the acceleration is neglected briefly
(2)
Applying the quasisteady assumption, the closing velocity of the adversaries is regarded as a constant, namely,Ṙ=−Vc. Guidance law acts periodically, the closing velocity is updated in every guidance cycle, and the deviations would be correctedDiscretizing the plant dynamics (7) with Euler’s integration formula gives
(8)Xk+1=FkXk,Uk=Xk+fXk,UkT,where X∈ℜ4, U∈ℜ2, subscript k=1,2,…,N is the time index, and T is the step size.Assuming that the interceptor can receive immediate positionxT,yT,zT and velocity VxT,VyT,VzT of the target from remote data-link during the midcourse, together with its position xM,yM,zM and velocity VxM,VyM,VzM from an on-board IMU system, the relative position and velocity can be obtained by
(9)r=xMT,yMT,zMTT=xT−xM,yT−yM,zT−zMT,V=VxMT,VyMT,VzMTT=VxT−VxM,VyT−VyM,VzT−VzMT.Therefore, the line-of-sight information is derived as(10)qε=arctanyMTxMT2+zMT2,q̇ε=xMTVyMT−yMTVxMTxMT2+yMT2+zMT2,qβ=arctan−zMTxMT,q̇β=zMTVxMT−xMTVzMTxMT2+yMT2+zMT2.Now, the measured outputY=qε,q̇ε,qβ,q̇βT∈ℜ4 is selected in terms of specific guidance requirements, right in the state of the system, and we get Yk=Xk. Marking N as the prediction horizon of the MPSP method, the missile-target range step is expressed as Rstep=Rinit−Rend/N−1, where Rinit indicates the missile-target distance at every guidance epoch and Rend means the handover distance. Then, the increment of time is obtained by T=Rstep/Vc.Through appropriate control modificationdU, MPSP drives the predicted terminal output YN to the expected value YN∗. In consideration of the final line-of-sight constraints and parallel approach method, the desired output is select as
(11)YN∗=qεf∗,0,qβf∗,0T.Applying the Taylor series expansion toYN at the point YN∗ and ignoring higher-order terms, the output deviation is written as
(12)dYN=∂YN∂XNdXN,where
(13)∂YN∂XN=1000010000100001.The transfer function of the state error can be derived from (8):
(14)dXk+1=∂Fk∂XkdXk+∂Fk∂UkdUk.In the formula,dXk and dUk represent the state error and control error at the k step, respectively; the partial derivatives of the state variables and the control variables are expressed as
(15)∂Fk∂Xk=1T00a21a220a23001Ta41a420a44,∂Fk∂Uk=00b210000b42,where
(16)a21=−x4k2cos2x1kT,a22=1+2VcT/Rk,a24=−x4ksin2x1kT,a41=2x2kx4kcos2x1k+aMβktanx1kRkcosx1kT,a42=2x4ktanx1kTa44=1+2VcT/Rk+2x2ktanx1kT,b21=−T/Rk,b42=T/Rkcosx1k.Taking the stepk=N−1 in (14) and substituting it to (12), we obtain
(17)dYN=∂YN∂XN∂FN−1∂XN−1dXN−1+∂FN−1∂UN−1dUN−1.In this way, the output deviation is expressed as a function of the initial state error and control error sequences.(18)dYN=AdX1+B1dU1+⋯+BN−1dUN−1,wherein the matrix A and the sensitivity matrices Bk are defined as
(19)A=∂YN∂XN∏m=1N−1∂Fm∂Xm,Bk=∂YN∂XN∏m=k+1N−1∂Fm∂Xm∂Fk∂Uk.Given the initial condition of the missile-target confrontation, no error exists in the first state, i.e.,dX1=0, equation (18) yields
(20)dYN=B1dU1+⋯+BN−1dUN−1=∑k=1N−1BkdUk.The sensitivity matricesBk can be solved recursively in the following manner:
(21)BN−10=∂YN∂XN,Bk0=Bk+10∂Fk+1∂Xk+1,Bk=Bk0∂Fk∂Uk.To optimize the trajectory of the interceptor and maintain a smooth flight, define the quadratic objective function as(22)J=12∑k=1N−1Uk0−dUkTRkUk0−dUk,where Rk is the positive definite weighting matrix and Uk0 is the previous control solution. Note that the weight matrix is alternatively chosen to adjust the magnitude of the control history at different time steps.For the constrained optimal problem defined in equations (20) and (22), static optimization theory is applied, which gives the augmented cost function
(23)J′=12∑k=1N−1Uk0−dUkTRkUk0−dUk+λTdYN−∑k=1N−1BkdUk.The necessary conditions for minimalJ′ are given by
(24)∂Jk′∂dUk=−RkUk0−dUk−BkTλ=0,∂Jk′∂λ=dYN−∑k=1N−1BkdUk=0.Solving for the corrected controlUk, we get
(25)Uk=Uk0−dUk=Rk−1BkTAλ−1dYN−bλ,where
(26)Aλ=−∑k=1N−1BkRk−1BkT,bλ=∑k=1N−1BkUk0.In conclusion, the line-of-sight constrained MPSP guidance is numerically solved by the control (25), considering the discrete guidance problem (8) with the required output (11) and cost function (22). Note that the control sequence Ukk=1,2,…,N−1 is solved in each guidance period, but only U1 is employed as the current guidance command. The suboptimal midcourse guidance law for target interception not only optimizes the flight control but also satisfies the terminal constraints of the line-of-sight angle and its rate.


### 3.2. Impact Angle Constrained Guidance

As already mentioned in Section2.1, the midcourse guidance with velocity angle alignment exhibits better handover properties if the target’s maneuver is somehow weak. However, only the line of sight and line-of-sight rate are involved while modeling the interception scenario. A solution can be found in [31], where the desired impact angle is able to be transferred into the final line-of-sight angle. For simplicity, the following process is done in the longitudinal plane and the lateral plane is similar.The impact angle is defined as the angle between velocity vectors of the adversaries; particularly, we have(27)θimp=θT−θM,where θimp is the impact angle in the longitudinal plane and θT and θM are the flight path angles of the target and interceptor, respectively. A perfect head-on interception scenario is formed by restricting θimp=π.When the midcourse guidance comes to the end, the line-of-sight rateq̇ has been driven to zero. The interceptor and target stay on an impact triangle geometry, as shown in Figure 2(a). Under this circumstance, the zero-effort miss can be guaranteed if both the interceptor and the target do not execute any further maneuvers from the instant time onward. Then, the relationship between the terminal line-of-sight angle and impact angle is given by
(28)VTsinθTf−qf−VMsinθMf−qf=0,where VT and VM are the velocities of the hypersonic target and interceptor, respectively, q stands for the line-of-sight in a 2D plane, and subscript f represents the final value in midcourse guidance.Figure 2
Interception angle in a 2D plane.
(a)
Impact triangle
(b)
θimp and qf relationshipNoting the target-to-missile speed ratiov=VT/VM, rearranging (28) yields
(29)qf=θTf−arctansinθimpcosθimp−v−nπ.Different from the situation in [31], the interceptor no longer has velocity advantage, namely, v>1; the singular problem is structurally avoided. One-to-one condition of qf and θimp is further analyzed, in which a monotonous zone including head-on condition can be found. Considering an incoming target with θTf=π, the relationship between qf and θimp changes according to the speed ratio v, as shown in Figure 2(b). Since qf is continuous, extreme points can be obtained by driving dqf/dθimp=0, which gives vcosθimp=1. In conclusion, for any desired impact angle θimp∈arccos1/v,2π−arccos1/v, a corresponding qf can be calculated by (29).Applying (29), the aforementioned line-of-sight constrained MPSP guidance is transferred into the impact angle constrained MPSP guidance, which actually constrains the final flight path angle for the interceptor in the midcourse phase. Note that the prediction of the target’s terminal flight path angle is required.


## 3.1. Line-of-Sight Constrained Guidance

A brief principle of MPSP can be described as follows. Firstly, expand the observed output with first-order Taylor series at the expected terminal point. Then, the deviation between the predicted output and the desired value can be formulated by an initial state error and control error history. Applying an appropriate cost function, the corrected control is derived from iterative calculation, when the final output value approximates the expected one.Because of the guidance iteration, flight trajectory is corrected in each cycle to get rid of external disturbance. The following assumptions are acceptable while processing the guidance design:(1)
Different from the terminal phase, midcourse guidance lasts longer. The hypersonic target is most likely to adopt various maneuver strategies, causing the changes of its acceleration at all times. It is not reliable to predict the future maneuvers of the hypersonic target. Therefore, the acceleration is neglected briefly
(2)
Applying the quasisteady assumption, the closing velocity of the adversaries is regarded as a constant, namely,Ṙ=−Vc. Guidance law acts periodically, the closing velocity is updated in every guidance cycle, and the deviations would be correctedDiscretizing the plant dynamics (7) with Euler’s integration formula gives
(8)Xk+1=FkXk,Uk=Xk+fXk,UkT,where X∈ℜ4, U∈ℜ2, subscript k=1,2,…,N is the time index, and T is the step size.Assuming that the interceptor can receive immediate positionxT,yT,zT and velocity VxT,VyT,VzT of the target from remote data-link during the midcourse, together with its position xM,yM,zM and velocity VxM,VyM,VzM from an on-board IMU system, the relative position and velocity can be obtained by
(9)r=xMT,yMT,zMTT=xT−xM,yT−yM,zT−zMT,V=VxMT,VyMT,VzMTT=VxT−VxM,VyT−VyM,VzT−VzMT.Therefore, the line-of-sight information is derived as(10)qε=arctanyMTxMT2+zMT2,q̇ε=xMTVyMT−yMTVxMTxMT2+yMT2+zMT2,qβ=arctan−zMTxMT,q̇β=zMTVxMT−xMTVzMTxMT2+yMT2+zMT2.Now, the measured outputY=qε,q̇ε,qβ,q̇βT∈ℜ4 is selected in terms of specific guidance requirements, right in the state of the system, and we get Yk=Xk. Marking N as the prediction horizon of the MPSP method, the missile-target range step is expressed as Rstep=Rinit−Rend/N−1, where Rinit indicates the missile-target distance at every guidance epoch and Rend means the handover distance. Then, the increment of time is obtained by T=Rstep/Vc.Through appropriate control modificationdU, MPSP drives the predicted terminal output YN to the expected value YN∗. In consideration of the final line-of-sight constraints and parallel approach method, the desired output is select as
(11)YN∗=qεf∗,0,qβf∗,0T.Applying the Taylor series expansion toYN at the point YN∗ and ignoring higher-order terms, the output deviation is written as
(12)dYN=∂YN∂XNdXN,where
(13)∂YN∂XN=1000010000100001.The transfer function of the state error can be derived from (8):
(14)dXk+1=∂Fk∂XkdXk+∂Fk∂UkdUk.In the formula,dXk and dUk represent the state error and control error at the k step, respectively; the partial derivatives of the state variables and the control variables are expressed as
(15)∂Fk∂Xk=1T00a21a220a23001Ta41a420a44,∂Fk∂Uk=00b210000b42,where
(16)a21=−x4k2cos2x1kT,a22=1+2VcT/Rk,a24=−x4ksin2x1kT,a41=2x2kx4kcos2x1k+aMβktanx1kRkcosx1kT,a42=2x4ktanx1kTa44=1+2VcT/Rk+2x2ktanx1kT,b21=−T/Rk,b42=T/Rkcosx1k.Taking the stepk=N−1 in (14) and substituting it to (12), we obtain
(17)dYN=∂YN∂XN∂FN−1∂XN−1dXN−1+∂FN−1∂UN−1dUN−1.In this way, the output deviation is expressed as a function of the initial state error and control error sequences.(18)dYN=AdX1+B1dU1+⋯+BN−1dUN−1,wherein the matrix A and the sensitivity matrices Bk are defined as
(19)A=∂YN∂XN∏m=1N−1∂Fm∂Xm,Bk=∂YN∂XN∏m=k+1N−1∂Fm∂Xm∂Fk∂Uk.Given the initial condition of the missile-target confrontation, no error exists in the first state, i.e.,dX1=0, equation (18) yields
(20)dYN=B1dU1+⋯+BN−1dUN−1=∑k=1N−1BkdUk.The sensitivity matricesBk can be solved recursively in the following manner:
(21)BN−10=∂YN∂XN,Bk0=Bk+10∂Fk+1∂Xk+1,Bk=Bk0∂Fk∂Uk.To optimize the trajectory of the interceptor and maintain a smooth flight, define the quadratic objective function as(22)J=12∑k=1N−1Uk0−dUkTRkUk0−dUk,where Rk is the positive definite weighting matrix and Uk0 is the previous control solution. Note that the weight matrix is alternatively chosen to adjust the magnitude of the control history at different time steps.For the constrained optimal problem defined in equations (20) and (22), static optimization theory is applied, which gives the augmented cost function
(23)J′=12∑k=1N−1Uk0−dUkTRkUk0−dUk+λTdYN−∑k=1N−1BkdUk.The necessary conditions for minimalJ′ are given by
(24)∂Jk′∂dUk=−RkUk0−dUk−BkTλ=0,∂Jk′∂λ=dYN−∑k=1N−1BkdUk=0.Solving for the corrected controlUk, we get
(25)Uk=Uk0−dUk=Rk−1BkTAλ−1dYN−bλ,where
(26)Aλ=−∑k=1N−1BkRk−1BkT,bλ=∑k=1N−1BkUk0.In conclusion, the line-of-sight constrained MPSP guidance is numerically solved by the control (25), considering the discrete guidance problem (8) with the required output (11) and cost function (22). Note that the control sequence Ukk=1,2,…,N−1 is solved in each guidance period, but only U1 is employed as the current guidance command. The suboptimal midcourse guidance law for target interception not only optimizes the flight control but also satisfies the terminal constraints of the line-of-sight angle and its rate.


## 3.2. Impact Angle Constrained Guidance

As already mentioned in Section2.1, the midcourse guidance with velocity angle alignment exhibits better handover properties if the target’s maneuver is somehow weak. However, only the line of sight and line-of-sight rate are involved while modeling the interception scenario. A solution can be found in [31], where the desired impact angle is able to be transferred into the final line-of-sight angle. For simplicity, the following process is done in the longitudinal plane and the lateral plane is similar.The impact angle is defined as the angle between velocity vectors of the adversaries; particularly, we have(27)θimp=θT−θM,where θimp is the impact angle in the longitudinal plane and θT and θM are the flight path angles of the target and interceptor, respectively. A perfect head-on interception scenario is formed by restricting θimp=π.When the midcourse guidance comes to the end, the line-of-sight rateq̇ has been driven to zero. The interceptor and target stay on an impact triangle geometry, as shown in Figure 2(a). Under this circumstance, the zero-effort miss can be guaranteed if both the interceptor and the target do not execute any further maneuvers from the instant time onward. Then, the relationship between the terminal line-of-sight angle and impact angle is given by
(28)VTsinθTf−qf−VMsinθMf−qf=0,where VT and VM are the velocities of the hypersonic target and interceptor, respectively, q stands for the line-of-sight in a 2D plane, and subscript f represents the final value in midcourse guidance.Figure 2
Interception angle in a 2D plane.
(a)
Impact triangle
(b)
θimp and qf relationshipNoting the target-to-missile speed ratiov=VT/VM, rearranging (28) yields
(29)qf=θTf−arctansinθimpcosθimp−v−nπ.Different from the situation in [31], the interceptor no longer has velocity advantage, namely, v>1; the singular problem is structurally avoided. One-to-one condition of qf and θimp is further analyzed, in which a monotonous zone including head-on condition can be found. Considering an incoming target with θTf=π, the relationship between qf and θimp changes according to the speed ratio v, as shown in Figure 2(b). Since qf is continuous, extreme points can be obtained by driving dqf/dθimp=0, which gives vcosθimp=1. In conclusion, for any desired impact angle θimp∈arccos1/v,2π−arccos1/v, a corresponding qf can be calculated by (29).Applying (29), the aforementioned line-of-sight constrained MPSP guidance is transferred into the impact angle constrained MPSP guidance, which actually constrains the final flight path angle for the interceptor in the midcourse phase. Note that the prediction of the target’s terminal flight path angle is required.


## 4. Simulation

Three different scenarios are presented to validate the effectiveness of the constrained MPSP midcourse guidance law. The first two scenarios refer to the cruising hypersonic target with a constant velocity in the longitudinal plane. Line-of-sight constrained and impact angle constrained midcourse guidance are applied, respectively; The last scenario considers the gliding target with a skipping trajectory in a three-dimensional space, and a full guidance scheme including midcourse is adopted.


### 4.1. Problem Setup

#### 4.1.1. Interceptor and Target Model

The interceptor is modeled using three-degree-of-freedom (3DOF) particle dynamics [15].
(30)ẋ=Vcosθcosψ,ẏ=Vsinθ,ż=Vcosθsinψ,V̇=T−Dm−gsinθ,θ̇=1Vay−gcosθ,ψ̇=azVcosθ,ṁ=TIsp,where x, y, and z are the positions of the interceptor, V is the vehicle speed, and m stands for mass. The relative flight path angle and velocity azimuth angle are θ and ψ, respectively; ay and az represent the acceleration commands in the longitudinal and lateral planes; T and D donate the thrust and drag on the interceptor; and Isp is the specific impulse of the rocket or reaction control system (RCS).Taking conventional gliding vehicle interception as an example, the target adopts bank angle constrained predictor-corrector reentry guidance [32]. The particle dynamics of a common aero vehicle (CAV) over a nonrotating spherical earth model are applied, where the Coriolis and centrifugal inertial force are technically ignored.
(31)ṙ=Vsinθ,λ̇=Vcosθsinψrcosϕ,ϕ̇=Vcosθcosψr,V̇=−Dm−gsinθ,θ̇=LcosσmV+V2r−gcosθV,ψ̇=LsinσmVcosθ+Vrcosθsinψtanϕ,where r is the radial distance from the earth’s center to the gliding vehicle; λ and ϕ are the longitude and latitude of the location; and V is the vehicle speed. The flight path angle and velocity azimuth angle are marked as θ and ψ; L and D indicate the aerodynamic lift and drag; and σ is the controlled bank angle.


#### 4.1.2. Overall Guidance Scheme

In order to improve mobility and timeliness of the interceptor platform, meanwhile expanding the defensive area, the air-launched interceptor is adopted. The ability of attack beyond visual range is vital to a multiple-layer intercept. Taking limited energy and load factor into account, the interceptor prefers a parabola trajectory for its boost guidance. In this phase, the interceptor separates from the air platform and climbs to the required altitude. The drag can be significantly reduced in a high altitude and its operational range is extended thereafter. The trigonometric guidance command [33] is applied for the smooth shifting from separation; thus, the overload would increase gently as expected. The flight path angle command and velocity azimuth angle command are given by
(32)θ̇M=π2t0θoptsinπ2tt0,ψ̇M=π2t0ψoptsinπ2tt0,where t represents the guidance time after separation, while t0 is the total time preset for the initial phase; θopt and ψopt are the expected flight path angle and velocity azimuth angle depending on the predicted impact point and maximum altitude of the parabola trajectory.The proposed terminal-angle constrained MPSP guidance law is applied for interceptor’s midcourse flight. This approach requires the line-of-sight and line-of-sight rate information in real time. However, on-board detection system will not be activated until the capture conditions are satisfied. Assuming that the interceptor can receive target’s information provided by ground radar or space-based infrared system, additional filter and prediction are necessary for signal extraction and denoising. For those air-breathing hypersonic targets, the interacting multiple model algorithm with commonly “CA” and “current” statistical models is preferred accounting for its complex maneuverability. However, it is reported in the literature that filter based on the nonlinear Markov acceleration model [34] exhibits better performance for boost-gliding targets.In the terminal phase, the goal of guidance is trying to minimize the miss distance. However, the aerodynamic force in the near space is not enough. The reaction control system is designed to accelerate the control response and increase the available overload. In addition, the on-board detection system is activated and target information is acquired without delay. Therefore, the guidance and control period has shortened obviously. For the above considerations, an analytic guidance called adaptive sliding mode guidance (ASMG) [35] is applied for terminal engagement, and the sign function is replaced by a saturation function to eliminate chattering. The acceleration command is expressed as
(33)aMε=k+1Ṙq̇ε+ηq̇εq̇ε+δ,aMB=k+1Ṙ1q̇β+η1q̇βq̇β+δ1.where η, η1>0, δ, and δ1 are small positive scalars and R1=Rcosqε.


### 4.2. Line-of-Sight Constrained Guidance

Interception confrontation scenario is presented to verify the proposed line-of-sight constrained MPSP midcourse guidance. The following conditions are set assuming the adversaries are in the longitudinal plane.(1)
The initial position of the interceptor 0,17 km, the average velocity at 1200 m/s, 0° initial flight path angle, and 5 g maximum overload
(2)
The initial position of hypersonic target 400,35 km, the average velocity at 2400 m/s, 180.286° initial flight path angle, and without maneuver
(3)
Terminal handover distance is 60 km, simulation step of 20 ms, and the scheduled final line-of-sights are -5°, 0°, and 5°, respectively
(4)
Initial control guess utilizes the three-dimensional proportional guidance lawThe results are shown in Table1 and Figure 3.Table 1
Convergence results of LOS constrained guidance.
Rm
qf°
q̇rad/s
nyfg
qf=−5°
Default
60000
-5
0
-0.149
Real
59960
-5.000
−5.908e−5
qf=0°
Default
60000
0
0
-0.052
Real
59972
−8.377e−6
−1.800e−5
qf=5°
Default
60000
5
0
-0.057
Real
59940
5.000
−2.21e−6Figure 3
Simulation results of LOS constrained guidance.
(a)
Interception trajectory
(b)
Line-of-sight angle
(c)
Line-of-sight rate
(d)
Normal commandFrom Figure3(a), we can see that the interceptor is able to reach the desired position under the command of the line-of-sight constrained MPSP guidance algorithm, right in the front of the hypersonic target. Figures 3(a) and 3(b) indicate that the terminal constraints of both line-of-sight and line-of-sight rate have been satisfied with high accuracy. In Figure 3(d), the overload command can gradually converge to zero, thus providing a favorable initial condition for the endgame interception.


### 4.3. Impact Angle Constrained Guidance

To validate the impact angle constrained MPSP midcourse guidance, the same simulation conditions are set as in Section 4.2, except that the required impact angles are 170°, 180°, and 190°, where 180° represents a typical head-on collision. The simulation results are shown in Table2 and Figure 4.Table 2
Convergence results of impact angle constrained guidance.
Rm
q̇rad/s
θimp°
nyfg
θimp=170°
Default
60000
0
170
0.015
Real
59981
5.614e−6
170.016
θimp=180°
Default
60000
0
180
-0.051
Real
59959
−1.465e−5
179.958
θimp=190°
Default
60000
0
190
-0.104
Real
59945
−2.948e−5
189.915Figure 4
Simulation results of impact angle constrained guidance.
(a)
Interception trajectory
(b)
Impact angle
(c)
Line-of-sight rate
(d)
Normal commandSimilar conclusions can be made in contrast to the previous line-of-sight constrained MPSP guidance method. As shown in Figure4(a), the interceptor is able to reach the desired position under the command of the impact angle constrained MPSP guidance algorithm. Figures 4(b) and 4(c) imply that the terminal constraints of both impact angle and line-of-sight rate have been satisfied. In Figure 4(d), the overload command gradually converges to zero and therefore provides a favorable initial condition for the endgame interception.


### 4.4. Full Trajectory Interception

To verify the effectiveness of full trajectory guidance scheme, parameters are set as follows:(1)
Initial position of interceptor is 0,12,0 km, initial velocity is 360 m/s, and initial flight path angle and velocity azimuth angle are both 0°
(2)
The interceptor’s rocket works in two levels, the first level lasts 2 s and produces 30kN thrust, while the second level lasts 22 s and produces 8 kN thrust
(3)
The hypersonic target maintains pullup and pulldown gliding trajectory with an initial altitude of 60 km and speed of 5100 m/sAssuming that the interceptor is released once the missile-target range meets the desired distance, simulation results are shown in Figure5. From that, we can see the progress of the interceptor. After launching from an airborne platform, the interceptor climbs to upper air under trigonometric guidance. Until 15.00 s later, it turns to midcourse phase and keeps accelerating. A maximum speed of 1389.92 m/s is achieved when the rocket engine burns out at 24.00 s. The trajectory reaches a maximum altitude of 42.67 km after 113.36 s. Separation from booster rocket appears at 166.33 s when the handover range is reached and terminal guidance takes over. The final miss distance is 2.47 m after 186.44 s flight, which demonstrates the correctness of the proposed guidance strategy.Figure 5
Simulation results of CAV interception.
(a)
Interception trajectory
(b)
Velocity
(c)
Longitudinal trajectory
(d)
Lateral trajectory
(e)
Longitudinal load factor
(f)
Lateral load factor
(g)
Flight path angle
(h)
Velocity azimuth angle


## 4.1. Problem Setup

### 4.1.1. Interceptor and Target Model

The interceptor is modeled using three-degree-of-freedom (3DOF) particle dynamics [15].
(30)ẋ=Vcosθcosψ,ẏ=Vsinθ,ż=Vcosθsinψ,V̇=T−Dm−gsinθ,θ̇=1Vay−gcosθ,ψ̇=azVcosθ,ṁ=TIsp,where x, y, and z are the positions of the interceptor, V is the vehicle speed, and m stands for mass. The relative flight path angle and velocity azimuth angle are θ and ψ, respectively; ay and az represent the acceleration commands in the longitudinal and lateral planes; T and D donate the thrust and drag on the interceptor; and Isp is the specific impulse of the rocket or reaction control system (RCS).Taking conventional gliding vehicle interception as an example, the target adopts bank angle constrained predictor-corrector reentry guidance [32]. The particle dynamics of a common aero vehicle (CAV) over a nonrotating spherical earth model are applied, where the Coriolis and centrifugal inertial force are technically ignored.
(31)ṙ=Vsinθ,λ̇=Vcosθsinψrcosϕ,ϕ̇=Vcosθcosψr,V̇=−Dm−gsinθ,θ̇=LcosσmV+V2r−gcosθV,ψ̇=LsinσmVcosθ+Vrcosθsinψtanϕ,where r is the radial distance from the earth’s center to the gliding vehicle; λ and ϕ are the longitude and latitude of the location; and V is the vehicle speed. The flight path angle and velocity azimuth angle are marked as θ and ψ; L and D indicate the aerodynamic lift and drag; and σ is the controlled bank angle.


### 4.1.2. Overall Guidance Scheme

In order to improve mobility and timeliness of the interceptor platform, meanwhile expanding the defensive area, the air-launched interceptor is adopted. The ability of attack beyond visual range is vital to a multiple-layer intercept. Taking limited energy and load factor into account, the interceptor prefers a parabola trajectory for its boost guidance. In this phase, the interceptor separates from the air platform and climbs to the required altitude. The drag can be significantly reduced in a high altitude and its operational range is extended thereafter. The trigonometric guidance command [33] is applied for the smooth shifting from separation; thus, the overload would increase gently as expected. The flight path angle command and velocity azimuth angle command are given by
(32)θ̇M=π2t0θoptsinπ2tt0,ψ̇M=π2t0ψoptsinπ2tt0,where t represents the guidance time after separation, while t0 is the total time preset for the initial phase; θopt and ψopt are the expected flight path angle and velocity azimuth angle depending on the predicted impact point and maximum altitude of the parabola trajectory.The proposed terminal-angle constrained MPSP guidance law is applied for interceptor’s midcourse flight. This approach requires the line-of-sight and line-of-sight rate information in real time. However, on-board detection system will not be activated until the capture conditions are satisfied. Assuming that the interceptor can receive target’s information provided by ground radar or space-based infrared system, additional filter and prediction are necessary for signal extraction and denoising. For those air-breathing hypersonic targets, the interacting multiple model algorithm with commonly “CA” and “current” statistical models is preferred accounting for its complex maneuverability. However, it is reported in the literature that filter based on the nonlinear Markov acceleration model [34] exhibits better performance for boost-gliding targets.In the terminal phase, the goal of guidance is trying to minimize the miss distance. However, the aerodynamic force in the near space is not enough. The reaction control system is designed to accelerate the control response and increase the available overload. In addition, the on-board detection system is activated and target information is acquired without delay. Therefore, the guidance and control period has shortened obviously. For the above considerations, an analytic guidance called adaptive sliding mode guidance (ASMG) [35] is applied for terminal engagement, and the sign function is replaced by a saturation function to eliminate chattering. The acceleration command is expressed as
(33)aMε=k+1Ṙq̇ε+ηq̇εq̇ε+δ,aMB=k+1Ṙ1q̇β+η1q̇βq̇β+δ1.where η, η1>0, δ, and δ1 are small positive scalars and R1=Rcosqε.


## 4.1.1. Interceptor and Target Model

The interceptor is modeled using three-degree-of-freedom (3DOF) particle dynamics [15].
(30)ẋ=Vcosθcosψ,ẏ=Vsinθ,ż=Vcosθsinψ,V̇=T−Dm−gsinθ,θ̇=1Vay−gcosθ,ψ̇=azVcosθ,ṁ=TIsp,where x, y, and z are the positions of the interceptor, V is the vehicle speed, and m stands for mass. The relative flight path angle and velocity azimuth angle are θ and ψ, respectively; ay and az represent the acceleration commands in the longitudinal and lateral planes; T and D donate the thrust and drag on the interceptor; and Isp is the specific impulse of the rocket or reaction control system (RCS).Taking conventional gliding vehicle interception as an example, the target adopts bank angle constrained predictor-corrector reentry guidance [32]. The particle dynamics of a common aero vehicle (CAV) over a nonrotating spherical earth model are applied, where the Coriolis and centrifugal inertial force are technically ignored.
(31)ṙ=Vsinθ,λ̇=Vcosθsinψrcosϕ,ϕ̇=Vcosθcosψr,V̇=−Dm−gsinθ,θ̇=LcosσmV+V2r−gcosθV,ψ̇=LsinσmVcosθ+Vrcosθsinψtanϕ,where r is the radial distance from the earth’s center to the gliding vehicle; λ and ϕ are the longitude and latitude of the location; and V is the vehicle speed. The flight path angle and velocity azimuth angle are marked as θ and ψ; L and D indicate the aerodynamic lift and drag; and σ is the controlled bank angle.


## 4.1.2. Overall Guidance Scheme

In order to improve mobility and timeliness of the interceptor platform, meanwhile expanding the defensive area, the air-launched interceptor is adopted. The ability of attack beyond visual range is vital to a multiple-layer intercept. Taking limited energy and load factor into account, the interceptor prefers a parabola trajectory for its boost guidance. In this phase, the interceptor separates from the air platform and climbs to the required altitude. The drag can be significantly reduced in a high altitude and its operational range is extended thereafter. The trigonometric guidance command [33] is applied for the smooth shifting from separation; thus, the overload would increase gently as expected. The flight path angle command and velocity azimuth angle command are given by
(32)θ̇M=π2t0θoptsinπ2tt0,ψ̇M=π2t0ψoptsinπ2tt0,where t represents the guidance time after separation, while t0 is the total time preset for the initial phase; θopt and ψopt are the expected flight path angle and velocity azimuth angle depending on the predicted impact point and maximum altitude of the parabola trajectory.The proposed terminal-angle constrained MPSP guidance law is applied for interceptor’s midcourse flight. This approach requires the line-of-sight and line-of-sight rate information in real time. However, on-board detection system will not be activated until the capture conditions are satisfied. Assuming that the interceptor can receive target’s information provided by ground radar or space-based infrared system, additional filter and prediction are necessary for signal extraction and denoising. For those air-breathing hypersonic targets, the interacting multiple model algorithm with commonly “CA” and “current” statistical models is preferred accounting for its complex maneuverability. However, it is reported in the literature that filter based on the nonlinear Markov acceleration model [34] exhibits better performance for boost-gliding targets.In the terminal phase, the goal of guidance is trying to minimize the miss distance. However, the aerodynamic force in the near space is not enough. The reaction control system is designed to accelerate the control response and increase the available overload. In addition, the on-board detection system is activated and target information is acquired without delay. Therefore, the guidance and control period has shortened obviously. For the above considerations, an analytic guidance called adaptive sliding mode guidance (ASMG) [35] is applied for terminal engagement, and the sign function is replaced by a saturation function to eliminate chattering. The acceleration command is expressed as
(33)aMε=k+1Ṙq̇ε+ηq̇εq̇ε+δ,aMB=k+1Ṙ1q̇β+η1q̇βq̇β+δ1.where η, η1>0, δ, and δ1 are small positive scalars and R1=Rcosqε.


## 4.2. Line-of-Sight Constrained Guidance

Interception confrontation scenario is presented to verify the proposed line-of-sight constrained MPSP midcourse guidance. The following conditions are set assuming the adversaries are in the longitudinal plane.(1)
The initial position of the interceptor 0,17 km, the average velocity at 1200 m/s, 0° initial flight path angle, and 5 g maximum overload
(2)
The initial position of hypersonic target 400,35 km, the average velocity at 2400 m/s, 180.286° initial flight path angle, and without maneuver
(3)
Terminal handover distance is 60 km, simulation step of 20 ms, and the scheduled final line-of-sights are -5°, 0°, and 5°, respectively
(4)
Initial control guess utilizes the three-dimensional proportional guidance lawThe results are shown in Table1 and Figure 3.Table 1
Convergence results of LOS constrained guidance.
Rm
qf°
q̇rad/s
nyfg
qf=−5°
Default
60000
-5
0
-0.149
Real
59960
-5.000
−5.908e−5
qf=0°
Default
60000
0
0
-0.052
Real
59972
−8.377e−6
−1.800e−5
qf=5°
Default
60000
5
0
-0.057
Real
59940
5.000
−2.21e−6Figure 3
Simulation results of LOS constrained guidance.
(a)
Interception trajectory
(b)
Line-of-sight angle
(c)
Line-of-sight rate
(d)
Normal commandFrom Figure3(a), we can see that the interceptor is able to reach the desired position under the command of the line-of-sight constrained MPSP guidance algorithm, right in the front of the hypersonic target. Figures 3(a) and 3(b) indicate that the terminal constraints of both line-of-sight and line-of-sight rate have been satisfied with high accuracy. In Figure 3(d), the overload command can gradually converge to zero, thus providing a favorable initial condition for the endgame interception.


## 4.3. Impact Angle Constrained Guidance

To validate the impact angle constrained MPSP midcourse guidance, the same simulation conditions are set as in Section 4.2, except that the required impact angles are 170°, 180°, and 190°, where 180° represents a typical head-on collision. The simulation results are shown in Table2 and Figure 4.Table 2
Convergence results of impact angle constrained guidance.
Rm
q̇rad/s
θimp°
nyfg
θimp=170°
Default
60000
0
170
0.015
Real
59981
5.614e−6
170.016
θimp=180°
Default
60000
0
180
-0.051
Real
59959
−1.465e−5
179.958
θimp=190°
Default
60000
0
190
-0.104
Real
59945
−2.948e−5
189.915Figure 4
Simulation results of impact angle constrained guidance.
(a)
Interception trajectory
(b)
Impact angle
(c)
Line-of-sight rate
(d)
Normal commandSimilar conclusions can be made in contrast to the previous line-of-sight constrained MPSP guidance method. As shown in Figure4(a), the interceptor is able to reach the desired position under the command of the impact angle constrained MPSP guidance algorithm. Figures 4(b) and 4(c) imply that the terminal constraints of both impact angle and line-of-sight rate have been satisfied. In Figure 4(d), the overload command gradually converges to zero and therefore provides a favorable initial condition for the endgame interception.


## 4.4. Full Trajectory Interception

To verify the effectiveness of full trajectory guidance scheme, parameters are set as follows:(1)
Initial position of interceptor is 0,12,0 km, initial velocity is 360 m/s, and initial flight path angle and velocity azimuth angle are both 0°
(2)
The interceptor’s rocket works in two levels, the first level lasts 2 s and produces 30kN thrust, while the second level lasts 22 s and produces 8 kN thrust
(3)
The hypersonic target maintains pullup and pulldown gliding trajectory with an initial altitude of 60 km and speed of 5100 m/sAssuming that the interceptor is released once the missile-target range meets the desired distance, simulation results are shown in Figure5. From that, we can see the progress of the interceptor. After launching from an airborne platform, the interceptor climbs to upper air under trigonometric guidance. Until 15.00 s later, it turns to midcourse phase and keeps accelerating. A maximum speed of 1389.92 m/s is achieved when the rocket engine burns out at 24.00 s. The trajectory reaches a maximum altitude of 42.67 km after 113.36 s. Separation from booster rocket appears at 166.33 s when the handover range is reached and terminal guidance takes over. The final miss distance is 2.47 m after 186.44 s flight, which demonstrates the correctness of the proposed guidance strategy.Figure 5
Simulation results of CAV interception.
(a)
Interception trajectory
(b)
Velocity
(c)
Longitudinal trajectory
(d)
Lateral trajectory
(e)
Longitudinal load factor
(f)
Lateral load factor
(g)
Flight path angle
(h)
Velocity azimuth angle


## 5. Conclusions

This paper proposes a novel midcourse guidance strategy for interceptors against near-space hypersonic targets. The characteristics of hypersonic weapon defense and the requirements of midcourse guidance design for interceptors are analyzed. The line-of-sight coupled interception model is established in a three dimensional space. Under the assumption that the closing velocity remains constant and the target maneuver is briefly negligible, a MPSP method-based suboptimal midcourse guidance law is derived for two different purposes that either satisfies the terminal line-of-sight angle constraint or satisfies the terminal impact angle constraint. Results indicate that the designed angle can be realized with maximum error less than 0°. In addition, the line-of-sight rate approaches zero at handover time, which implies that the final overload command can converge to a small value. In the interception of a hypersonic gliding target, the final miss distance turns out to be 2.47 m, which proves the effectiveness of this guidance law.



---

*Source: 6161032-2019-04-07.xml*