Three-Dimensional Impact Time and Angle Control Guidance

Research ArticleThree-Dimensional Impact Time and Angle Control GuidanceBased on MPSP

Yang Li , Hao Zhou , and Wanchun Chen

School of Astronautics, Beihang University, Beijing 100191, China

Correspondence should be addressed to Wanchun Chen; [email protected]

Received 16 June 2018; Revised 2 January 2019; Accepted 14 January 2019; Published 24 February 2019

Academic Editor: Christopher J. Damaren

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

A new nonlinear guidance law for air-to-ground missile cooperation attacks is proposed in this paper. This guidance law enablesmissiles with different initial conditions to attack targets simultaneously, and it can also precisely satisfy the terminal impactangle conditions in both flight-path angle and heading angle. The guidance law is devised using the model predictive staticprogramming (MPSP) method, and the control saturation constraint is incorporated in the MPSP algorithm. Thefirst-order-lag acceleration of the missile is taken as the state variable to realize the convergence of the terminal accelerationto zero. Moreover, a collision avoidance strategy for three-dimensional missile cooperative flight is proposed. The simulationresults show that the guidance law can make the missiles hit the target accurately at the same time with the ideal impactangles and can realize the control saturation constraints of the missiles. This can increase the attack effects and is significantfor collaborative attacks.

1. Introduction

When a missile strikes a target, its damage to the target canincrease at a particular attack angle. For example, in theapplications of anti-tank missiles, it is best to use the topattack, because the top of the tank is usually the weakest.Similarly, when anti-ship missiles attack ships, the rightattack angle can increase the attack effects. In these cases,the missile’s terminal flight-path angle and heading angle willusually be specified. The impact time is another importantaspect of improving the missile effectiveness, especially forcooperation attacks, where missiles should hit the targetsimultaneously. There are many self-defense measuresagainst anti-ship missiles in modern warfare ships, such assurface-to-air missiles, ECM systems, and CIWS (close-inweapon system). It is difficult to hit a ship through thesedefense systems. However, CIWS has one-by-one engage-ment characteristics, so salvo attack is an effective way tosolve this problem, in which several missiles strike the targetsimultaneously. In conclusion, it is necessary to control theimpact time and angle to make this cooperation attackmission successful.

There are many published articles on impact anglecontrol. In [1], Ryoo et al. studied the closed solution ofoptimal guidance laws to control the impact angle and termi-nal instability and presented a generalized form of optimalimpact-angle control guidance laws. To realize the impactangle constraint on the still target, Jeong et al. presented anew biased proportional navigation guidance law in [2]. Luet al. established the closed-loop nonlinear adaptive controllaw to realize the terminal impact angle control and appliedit to the terminal guidance of the hypersonic glider in [3].In [4], Xu et al. presented a missile integrated guidancescheme with impact angle constraint based on the idea ofreverse propulsion and VSC theory. In [5], Pamadi andOhlmeyer proposed two different guidance laws to achieveimpact angle control. The first one is a trajectory shapingcontrol law GENEX, and the second one, FITS, is a predictiveguidance method. In [6], Sang et al. proposed a guidance lawthat minimizes miss distance and satisfies impact angleconstraint. This method is based on Lyapunov stabilitytheory and applied parameter optimization. In [7], usingmodel predictive static programming (MPSP), Oza andPadhi proposed a guidance law that can satisfy terminal

HindawiInternational Journal of Aerospace EngineeringVolume 2019, Article ID 5631723, 16 pageshttps://doi.org/10.1155/2019/5631723

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https://doi.org/10.1155/2019/5631723

impact angle constraints accurately. In [8], Zhang et al.introduced a two-phased look angle control guidance schemewith the terminal impact angle constraints.

In recent years, there have been more and more articleson impact time control. In [9], Jeon et al. combined theproportional navigation guidance law and the feedback ofthe impact time error to achieve the impact time control. In[10], Shiyu and Rui proposed a collaborative guidancemethod combining coordination algorithm and local guid-ance law, and the impact-time-control guidance law obtainedin [9] was used. In [11], Zhang et al. designed a steadyimpact-time-control guidance law for multi-missiles. Aguidance law based on sliding mode control, which enablesa missile to hit a stationary target at a given impact time,was proposed by Kumar and Ghose in [12]. Besides, thereare other articles on impact time control, [13, 14].

However, only a few papers have been written on theguidance laws for controlling impact time and impactangle simultaneously. In [15], Zhang et al. propose aguidance law controlling both impact angle and impacttime. This guidance law is a combination of biased pro-portional navigation guidance (BPNG) and an additionalcommand for the impact time. A new impact time andangle control guidance law based on sliding mode wasproposed by Harl and Balakrishnan in [16]. In [17], toget a state feedback command with terminal constraintsof impact time and angle, Kim et al. introduced a polyno-mial function, which has three unknown coefficients. In[18], to make full use of the vulnerability of warships,Kang and Kim proposed a linear quadratic differentialgame guidance law to make the impact angle and timeof the missile controllable. A collaborative attack guidancelaw which can control impact time and angle based onsliding mode control was proposed by Yang et al. in[19]. In [20], Harrison proposed a new method to derivethe optimal missile guidance law and proposed a three-dimensional hybrid guidance law that constrains theimpact angle and time. Kumar and Ghose proposed aguidance strategy based on impact angle control to meetthe impact time constraint in [21].

Nevertheless, most of the existing articles consider thetwo-dimensional case. There are few articles that considerboth impact time control and terminal impact angle con-straints at the three-dimensional level.

In this paper, we have proposed a three-dimensionalimpact time and impact angle control guidance law tomake the cooperation attack mission successful, basedon MPSP. The model predictive static programming(MPSP) technique was proposed by Padhi and Kothariin [22]. MPSP is suitable for two-point boundary valueproblems with terminal constraints. Applications of thetechnology on different issues can be seen in recent arti-cles [22–24]. The following are the main contributions ofthis paper:

(1) The control saturation constraint is considered in theMPSP algorithm, the specific derivation process isgiven, and the effectiveness of the algorithm is veri-fied by simulation

(2) A collision avoidance strategy for three-dimensionalmissile cooperative flight is derived, and its effective-ness is verified

(3) The proposed guidance law achieves three-dimensional impact time and angle control simulta-neously, and it is an effective means of cooperationattack. The utility of MPSP in impact time and anglecontrol problem can be confirmed

(4) By treating the first-order-lag acceleration commandsas state variables, the terminal acceleration commandconverges to zero, which is beneficial to reduce themiss distance caused by random interference

This paper is arranged as follows. The mathematicalmodel and presentation of the problem is described inSection 2. The MPSP theory is described in Section 3. Theimplementation of the algorithm is described in Section 4,and the numerical simulations are provided in Section 5.Finally, Section 6 gives the conclusion.

2. Problem Statement

As shown in Figure 1, consider a scenario of a three-dimensional multi-missile coordinated attack. Several air-to-ground missiles are launched from different directionsand positions, and the target can be considered stationary.The missiles need to attack the target simultaneously at t ftime, and each missile needs to reach its required impactangle γmf i and ψmf i.

The dynamic equations of the system can be writtenas follows:

V = T −Dm

− g sin γ,

γ = −az − g cos γV

,

ψ =ay

V cos γ ,

x =V cos γ cos ψ,

y =V cos γ sin ψ,

z =V sin γ,

1

where x, y, and z are the positions of the missile inthree-dimensional space, γ is the flight path angle ofthe missile, and ψ is the heading angle of the missile.The value of the drag coefficient is given in [25]. In thispaper, the first-order autopilot model is adopted, and theaccelerations az t and ay t can be expressed in thefollowing formulas:

az t = azc 1 − e−t/τ ,

ay t = ayc 1 − e−t/τ ,2

2 International Journal of Aerospace Engineering

x

y

z�훾m1�휓m1

az

ayMissile 1

Missile 2Missile 4

Missile 3

Vm1

Vm2

Vm3

Vm4

Target

�훾mf1

�휓mf1

Figure 1: 3D engagement geometry of cooperation attack.

where azc and ayc are the acceleration commands in zand y directions, respectively, and τ is the time constant ofthe first-order system. Considering the lag of the system,the acceleration commands are replaced by az and ay,respectively. The acceleration vector and the velocity vectorare perpendicular to each other. To make the terminalacceleration command converge to zero, we consider azcand ayc as control variables and treat az and ay as statevariables; the added differential equations can be written as

az =azc − az

τ,

ay =ayc − ay

τ

3

Next, to ensure numerical stability, the state variables andcontrol variables are normalized:

Vn =VV∗ ,

γn =γ

γ∗,

ψn =ψ

ψ∗ ,

xn =xx∗

,

yn =yy∗

,

zn =zz∗

,

azn =azaz∗

,

ayn =ayay∗

,

azcn =azcaz∗

,

aycn =aycay∗

, 4

where the subscript n and superscript “∗” of statevariables and control variables, respectively, represent thedimensionless values and dimensionless constants; thedimensionless constants selected are referred to in Section 5(see Table 1 for details). Combining Eq. (1) and Eq. (3), thedimensionless form of the system dynamics equations canthen be written as

Vn =T −DmV∗ −

g sin γγ∗

V∗ ,

γn =−azng − g cos γγ∗

VnV∗γ∗

,

3International Journal of Aerospace Engineering

ψn =ayng

VnV∗ cos γγ∗ ψ∗ ,

xn =VnV

∗ cos γγ∗ cos ψψ∗

x∗,

yn =VnV

∗ cos γγ∗ sin ψψ∗

y∗,

zn =VnV

∗ sin γγ∗

z∗,

azn =azcn − azn

τ,

ayn =aycn − ayn

τ5

The mission objective is that all missiles hit the target at asame terminal time t f . Besides, missiles should have a specificterminal impact angle. Here, the control variable is definedas Un = azcn aycn T . The terminal constraints can be

written as Yn t f = γT ψT xT yT zT azT ayT T =γT ψT xT yT zT 0 0 T .

3. Model Predictive Static Programming(MPSP) with Control Saturation Constraints

In this section, we introduce the formula derivation of themodel prediction static programming (MPSP) [22] algo-rithm, which can solve the two-point boundary valueproblem. In addition, the control saturation constraints,namely, input inequality constraints, are considered in thealgorithm.

Considering the general nonlinear system, the discreteform of system state and output state is given by

Xk+1 = Fk Xk,Uk , 6

Yk = h Xk , 7

where X ∈ Rn,U ∈ Rm, Y ∈ Rp, and k = 1, 2,… ,N are the timesteps. The control history Uk, k = 1, 2,… ,N − 1, needs to be

determined to get the output at the last moment YN to adesired value YN

∗, i.e., YN ⟶ YN∗.

The MPSP algorithm starts with initial controlguesses. In general, initial control guesses cannot satisfyterminal constraints. Therefore, the control history canbe updated with a dynamic error. Iteration is kept untilYN ⟶ YN

∗.We define the error in the output as dYN ≜ YN − YN

∗.Then, using Taylor series expansion expanding YN aboutYN

∗ and ignoring the higher-order terms,

ΔYN ≅ dYN = ∂YN

∂XNdXN 8

According to Eq. (6), the state error at time step (k + 1)can be expressed as

dXk+1 =∂Fk

∂XkdXk +

∂Fk

∂UkdUk, 9

where dXk and dUk are the state and control errors at timestep k.

Expanding dXN in Eq. (9) and substituting it into Eq. (8),we get

dYN = ∂YN

∂XN

∂FN−1∂XN−1

dXN−1 +∂FN−1∂UN−1

dUN−1 10

Similarly, dXN−1 can be expanded and substituted intoEq. (8):

dYN = ∂YN

∂XN

∂FN−1∂XN−1

∂FN−2∂XN−2

dXN−2 +∂FN−2∂UN−2

dUN−2

+ ∂YN

∂XN

∂FN−1∂UN−1

dUN−1

11

Next, continuing until k = 1, we get

dYN = AdX1 + B1dU1 + B2dU2+⋯+BN−1dUN−1, 12

where

A ≜∂YN

∂XN

∂FN−1∂XN−1

⋯∂F1∂X1

,

Bk ≜∂YN

∂XN

∂FN−1∂XN−1

⋯∂Fk+1∂Xk+1

∂Fk

∂Uk

13

As the initial condition is given, the first term has noerror. Then, Eq. (12) is changed into

dYN = B1dU1 + B2dU2+⋯+BN−1dUN−1 = 〠N−1

k=1BkdUk 14

Table 1: Simulation parameters.

Parameters

PN guidance constant, N 4

Normalizing velocity 600m/s

Normalizing angle, γ∗, ψ∗ (50 deg, 50 deg)

Normalizing coordinates, x∗, y∗, z∗ (5 km, 5 km, 5 km)

Normalizing acceleration g = 9 81m/s2

Autopilot first-order lag, τ 0.3 s

Mass of missile 150 kg

Thrust force, T 0N

Surface area, sm 0.0324m2


The recursive calculation matrix Bk, k = 1,… , N − 1 ,can greatly simplify the calculation complexity of the MPSPalgorithm. First, we define

B0N−1 =

∂YN

∂XN15

Next, B0k, k = N − 2 , N − 3 ,… , 1, is computed as

B0k = B0

k+1∂Fk+1∂Xk+1

16

Finally, Bk, k = N − 2 , N − 3 ,… , 1, is computed as

Bk = B0k∂Fk

∂Uk17

Equation (14) is an incomplete constraint system ofequations. Using this feature and minimizing the followingperformance index,

J = 12 〠N−1

k=1U0

k − dUkTRk U0

k − dUk , 18

where U0k, k = 1,… , N − 1 , is the control history and

dUk is the corresponding error. Rk > 0 is a positive definiteweighting matrix, which should be chosen carefully. Theperformance functional in Eq. (18) needs to be minimizedunder the restriction of Eq. (14).

We consider the control saturation limits as

Uimin ≤Ui ≤Ui

max, i = 1,… ,m, 19

where Ui is the i component of the control U , and Uimax

and Uimin are its upper and lower bounds. Eq. (19) can

be written as

Uimax − CiU ≥ 0,

CiU −Uimin ≥ 0,

20

where C = Im×m and Ci is the line i of C.The Courant penalty function is used to solve this

constrained optimization problem. Using the optimal

control theory [26] and Courant penalty function theory[27, 28], the augmented cost function can be written as

J = 12 〠N−1

k=1U0

k − dUkTRk U0

k − dUk

+ λT dYN − 〠N−1

k=1BkdUk

+ 12 〠N−1

k=1〠m

i=1σki U

imax − Ci U

0k − dUk

2

+ 12 〠N−1

k=1〠m

i=1μki Ci U

0k − dUk −Ui

min2 ,

21

where λ is a Lagrange multiplier and

σki =ρ, Uki >Ui

max,0, Uki ≤Ui

max,

μki =ρ, Uki <Ui

min,0, Uki ≥Ui

min

22

In theory, ρ⟶ +∞; here we choose ρ = 107. Then,we have the following equations

∂J∂dUk

= −Rk U0k − dUk − BT

k λ

+ 〠m

i=1σkiC

Ti Ui

max − Ci U0k − dUk

− 〠m

i=1μkiC

Ti Ci U

0k − dUk −Ui

min = 0,

23

∂J∂λ

= dYN − 〠N−1

k=1BkdUk = 0 24

Solving for dUk from Eq. (23),

dUk = Rk + 〠m

i=1σkiC

Ti Ci + 〠

m

i=1μkiC

Ti Ci

−1

⋅ RkU0k + BT

k λ − 〠m

i=1σkiC

Ti Ui

max − CiU0k

+ 〠m

i=1μkiC

Ti CiU

0k −Ui

min

= R−1k RkU

0k + BT

k λ − 〠m

i=1σkiC

Ti Ui

max − CiU0k

+ 〠m

i=1μkiC

Ti CiU

0k −Ui

min ,

25

where Rk = Rk +∑mi=1σkiC

Ti Ci +∑m

i=1μkiCTi Ci.


Substituting for dUk from Eq. (25) into Eq. (24), we get

Aλλ + bλ = dYN , 26

where

Aλ ≜ 〠N−1

k=1BkR

−1k BT

k ,

bλ ≜ 〠N−1

k=1BkR

−1k RkU

0k − 〠

m

i=1σkiC

Ti Ui

max − CiU0k

+ 〠m

i=1μkiC

Ti CiU

0k −Ui

min

27

Aλ is a P × Pmatrix and bλ is a P × 1 vector. Assuming Aλto be nonsingular, λ is solved by Eq. (26):

λ = A−1λ dYN − bλ 28

Substituting Eq. (28) into Eq. (25) leads to

dUk = R−1k RkU

0k + BT

k A−1λ dYN − bλ

− 〠m

i=1σkiC

Ti Ui

max − CiU0k + 〠

m

i=1μkiC

Ti CiU

0k −Ui

min

29

Therefore, the updated control in step k = 1,… , N − 1is as follows:

Uk =U0k − dUk

=U0k − R

−1k RkU

0k + BT

k A−1λ dYN − bλ

− 〠m

i=1σkiC

Ti Ui

max − CiU0k + 〠

m

i=1μkiC

Ti CiU

0k −Ui

min

30

If the control saturation limit is not considered, σki andμki are zero; the updated control is as follows:

Uk =U0k − dUk = −R−1

k BTk A

−1λ dYN − bλ 31

The new control variables can be calculated by Eq. (30)and Eq. (31). We research the impact time and angle controlproblem based on this technology.

4. Implementation of the Guidance Law

To implement the MPSP algorithm, the Euler method [29]was used to discretize the dynamic equations.

Xnk+1= Fk Xnk

,Unk= Xnk

+ Δt f Xnk,Unk

, 32

where f Xnk,Unk

is the right side of Eq. (5).

The MPSP algorithm can constrain the terminal statethrough a specific output vector, so it is necessary to selectoutput vectors carefully. In this research, we have severaldifferent output vectors. To satisfy the impact time and angleconstrained, we need a same terminal impact time t f for allmissiles. We use PN guidance law [25] to calculate theterminal impact time of all missiles then choose the maxi-mum terminal time as the common t f of missiles. However,if there are many desired hard constraints, it is necessary tohave a good initial control guess history; otherwise, the algo-rithm may not converge. We first select the output vector asYn = xT yT zT azT ayT T ; all missiles need to satisfythe constraint. Then, the result of the calculation can be aguess history of the impact time and angle constraint case.The output vectors with impact time and angle constraintscan be written as Yn = γT xT yT zT azT ayT T and

Yn = γT ψT xT yT zT azT ayT T . The outputvectors vary with missiles, and different missiles have differ-ent terminal impact angle constraints. The output vector,Yk, is the value of Y at the time step k. DifferentiatingEq. (32) with respect to Xk, we get

∂Xnk+1

∂Xnk

= I8×8 + Δt ∂f k∂Xnk

, 33

where

∂f k∂Xnk

=∂f k∂Vnk

∂f k∂γnk

∂f k∂ψnk

∂f k∂xnk

∂f k∂ynk

∂f k∂znk

∂f k∂aznk

∂f k∂aynk 8×8

34

Then, we get the control derivative equation

∂Fk

∂Unk

= Δt ∂f k∂azcnk

∂f k∂aycnk 8×2

35

Finally, for Yn = xT yT zT azT ayT T ,

∂YnN

∂XnN 5×8

= O5×3⋮I5×5 5×8, 36

for Yn = γT xT yT zT azT ayT T ,

∂YnN

∂XnN 6×8

= O6×1⋮I1×1O5×1

⋮O6×1⋮O1×5

I5×5 6×8

, 37

for Yn = γT ψT xT yT zT azT ayT T ,

∂YnN

∂XnN 7×8

= O7×1⋮I7×7 7×8 38


4.1. Terminal Impact Time and Guess History Selection. Toachieve the cooperation attack, we need to choose a sameimpact time of missiles. We use the PN guidance law tocalculate each missile’s impact time then choose the max-imum as the impact time for all missiles. The PN guidancelaw is briefly introduced here.

σ represents the 3D line-of-sight angle between themissile and the target, and σ represents the line-of-sight rate;we get

σ = r × d/dt rr ⋅ r

= 1r2

ryrz − rzry

rzrx − rxrz

rxry − ryrx

=σx

σy

σz

, 39

where r is the missile-target relative range, and rx, ry, and rzare its three components, respectively. σx, σy, and σz are thethree components of the line-of-sight rate. Next, we convertthese rates into the velocity coordinate system

σpitch = − sin ψ σx + cos ψ σy,

σyaw = sin γ cos ψ σx + sin ψ σy + cos γ σz ,40

where σpitch and σyaw are the line-of-sight rates on the pitchand yaw planes, respectively. The closing velocity is asfollows:

Vc = −r =rxrx + ryry + rzrz

r41

Finally, the pitch and yaw acceleration commands aregiven by

azc =NeVcσpitch + g cos γ ,

ayc =NeVcσyaw,42

where azc and ayc are acceleration commands.Now, we have the common impact time t f . First, let all

missiles attack the target at impact time t f without impactangle constraint, and select zero as the guess history. Then,the result without impact angle constraint can be regardedas the guess history with impact angle constraint.

Then, we analyze the feasibility of the selected finalimpact time. According to the simulation conditions inSection 5, missile 1 is taken as an example for analysis.Guided by the PN guidance law, missile 1 flies 37.98 s to hitthe target, which has the longest flight time of the fourmissiles. The lateral acceleration of the missile is the control,which can make the trajectory straighter or more curved. Themore curved the trajectory is, the longer the flight distance ofthe missile will be and the longer the flight time will be.Similarly, the straighter the trajectory, the shorter the flighttime. However, the trajectory can only be straightened limit-edly, but theoretically it can be bent indefinitely. In fact, thereis aerodynamic drag, so the flight time can only be increasedin a limited range. Even so, the missile’s ability to increaseflight time is significantly stronger than that to reduce flighttime. Figure 2 shows the trajectories of missile 1 with flighttimes of 37 s, 37.98 s, 40 s, 45 s, and 50 s, respectively, fromwhich we can draw the same conclusion. For missile 1, it isobviously reasonable to choose a final time of 37.98 s, and

tf = 37 stf =37.98 stf = 40 s

tf = 45 stf = 50 s

0−2000

y (m)

−4000−6000

0−15000

2000

4000z (

m)

6000

8000

−10000

x (m)

−5000

50000

Figure 2: Missile 1 trajectories with different flight times.


for missiles 2, 3, and 4, 37.98 s is a larger final time,which is also reasonable. In conclusion, it is feasible toselect the maximum impact time as the final impact timeof all missiles.

4.2. Collision Avoidance Strategy. To avoid the collision ofmissiles in flight, we deduce a collision avoidance strategy.The following is a detailed derivation process.

The missiles can communicate with each other. Takingtwo missiles for example, the space positions and velocityvectors of missile 1 and missile 2 are known:

P1 = x1, y1, z1 ,

P2 = x2, y2, z2 ,

V1 = vx1, vy1, vz1 ,

V2 = vx2, vy2, vz2

43

When the distance between missiles 1 and 2 r12 is lessthan the threshold value Rm, collision risk is considered:

r12 = x1 − x22 + y1 − y2

2 + z1 − z22 1/2 < Rm 44

In this case, it is assumed that missile 1 and missile 2 willcollide. Then, the normal vectors of the plane determined byvelocity vectors V1 and V2 can be obtained

n ′ = vy1vz2 − vy2vz1, vz1vx2 − vz2vx1, vx1vy2 − vx2vy1 45

Unitizing the normal vector,

n = n ′n ′

, 46

where denotes the vector modulus length

n ′ = vy1vz2 − vy2vz12 + vz1vx2 − vz2vx1

2 + vx1vy2 − vx2vy12 1/2

47

Now give missile 1 a command acceleration in the ndirection, and give missile 2 a command acceleration inthe − n direction. The command accelerations need tobe decomposed into az and ay directions, as shown inFigure 3. According to the definition of az and ay inthe manuscript, the direction vectors of az and ay canbe obtained:

a z = sin γ cos ψ, sin γ sin ψ,− cos γ ,

a y = −sin ψ, cos ψ, 048

The unit command accelerations of missile 1 andmissile 2 can be obtained:

az1 =n ⋅ a z1a z1

,

ay1 =n ⋅ a y1a y1

,

az2 =− n ⋅ a z2

a z2,

ay2 =− n ⋅ a y2

a y2,

49

where a z1, a z2, a y1, and a y2 are determined by (48).Finally, the unit command accelerations are increased toan appropriate multiple. When the approaching speed ofthe two missiles is less than zero, the avoidance is endedand the attack path is replanned.

To verify the effectiveness of the collision avoidancestrategy, we design a collision situation and then use thecollision avoidance strategy to simulate it. The results areshown in Figure 4.

In Figure 4, the above two pictures are the situationswithout collision avoidance, while the following two picturesare the situations with collision avoidance. It can be seen thatthe proposed collision avoidance strategy is effective.

When there is a risk of collision between multiple mis-siles, the direction vector of the acceleration of each missile’santi-collision command is obtained firstly, which is thevector sum of the acceleration vectors of the anti-collisioncommand produced by this missile with all other missiles.Finally, the acceleration vector is decomposed into az anday directions of the missile. Thus, the collision avoidancecommand acceleration of each missile is obtained.

It should be noted that the missile will eventually hit thetarget at the same time; that is to say, the distance betweenthe missiles in the final stage will certainly be less than theset threshold Rm. Therefore, collision avoidance judgmentis not carried out in the whole flight process. When the

x

y

z

v1

v2n

−n

az1

ay1

Figure 3: Schematic diagram of command acceleration.


distance between the missile and the target rmt is less than acertain preset value Rt Rt > Rm , collision avoidance judg-ment is no longer carried out. In this study, we chose Rm =200(m) and Rt = 1000(m). In the final flight stage withoutcollision avoidance judgment, we set different terminalangles of each missile to ensure that the missiles would notcollide before hitting the target.

4.3. Guidance Law Implementation. The MPSP algorithmstarts with initial control guesses and iterates control untilit converges. The pseudocode for this algorithm is givenas below.

(1) Use the PN guidance law to obtain impact time t f ofmissiles

(2) Guess the initial control history. It is zero withoutimpact angle constraint. In the case of with impact

angle constraint, the guess history is the result ofwithout impact angle constraint

(3) Use the RK4method to simulate system dynamics forthe given guess history, and the collision avoidancestrategy is added to the simulation

(4) Compute Bk, k =N − 1,… , 1, matrices using Eq. (13)

0−100

50

z (m

)

100

x (m) y (m)

00

100 −100

100

M1M2

(a)

0

50

z (m

)

100

0 50−50 100y (m)

M1M2

(b)

0−100

50

z (m

)

100

00

x (m) y (m)

100 −100

100

M1M2

(c)

0

50

z (m

)

100

0 50−50 100

y (m)

M1M2

(d)

Figure 4: Simulation results of collision avoidance strategy.

Table 2: Initial conditions of the missile engagement.

V (m/s) γ (°) ψ (°) x, y, z (km) az , ay (m/s2)

Missile 1 578 0 10 (-12, -6, 7) (0, 0)

Missile 2 578 -5 -20 (-11.5, 2, 7.5) (0, 0)

Missile 3 578 0 -170 (12, 5, 6) (0, 0)

Missile 4 578 -10 -150 (12, 3, 8) (0, 0)

Target 0 \ \ (0, 0, 0) \


(5) Compute Aλ and Bλ, then compute new control Unusing Eq. (30) or (31)

(6) Go to step 3 until the algorithm converges

5. Numerical Simulations

In this section, simulation results are given to show theeffectiveness of the proposed guidance law in cooperationattacks. We first present the simulation results of the PNguidance law. Then we consider two cases, only impact timecontrol and impact time and angle control, without consider-ing the control saturation. Finally, the simulation resultsconsidering the control saturation are given. Table 1 liststhe other simulation parameters and the parameters usedby the PN guidance law (drag parameters; see Imado et al.[25]). In this paper, we present a simulation study of fourmissiles attacking the stationary target. The initial conditionsof the simulation are shown in Table 2.

5.1. Result of the PN Guidance Law and the Choice of ImpactTime. In this subsection, the PN guidance law is used forsimulation and the results are presented. The time stepΔt = 0 01 s, and the initial conditions are shown in Table 2.Simulation results are shown in Figures 5 and 6. Figure 5shows the cooperation attack trajectories of the four missilesunder different initial conditions, from which it can be seenthat the attack time of each missile is quite different. The mis-siles hit the target at 37.98 s, 32.62 s, 35.16 s, and 36.38 s,respectively. We chose the maximum impact time 37.98 s asthe terminal impact time t f for all missiles, which is used inthe MPSP algorithm. Figure 6 shows the relative distancesbetween missiles and the target. We can see that the PNguidance law does not have the ability to make missiles hitthe target at the same time.

5.2. Case with Impact Time Constraint. This subsectiongives the simulation results for impact time control.The initial conditions are shown in Table 2, and the ter-minal constraints are shown in Table 3. We choose thetime step Δt = 0 01 s, and the weighting matrix Rk = 1, k =1, 2,…N − 1. The convergence criterion for this simula-tion study is a miss distance of 1m, and acceleration tol-erance error is 1m/s2.

Figure 7 shows the missile trajectories with impact timecontrol by MPSP guidance. Figure 8 shows the relativedistances between missiles and the target. From Figures 7and 8, it can be observed that although the initial conditionsof the missiles are different, the missiles can hit the target at

0−2

2000

M2: 32.62 s

−1 5000

4000z (

m)

6000

x (m)

×104 0 0

M3: 36.38 s

y (m)

M1: 37.98 s8000

1 −5000

M4: 35.16 s

2 −10000

M1M2

M3M4

Figure 5: 3D trajectories by the PN guidance law.

0 5 10 15 20 25 30 35 40t (s)

0

2000

4000

6000

8000

10000

12000

14000

16000

R (m

)

M1M2

M3M4

Figure 6: Relative distances between missiles and the target by PN.


the specified impact time 37.98 s simultaneously. Figure 9presents the histories for the flight path angles γ and headingangles ψ, only with impact time constraint. The terminalangle is not controlled because there is terminal impactangle constraint. Figure 10 shows the histories of missiles’first-order-lag lateral acceleration. We can see that thelateral acceleration converges to zero at the terminal time,which can increase the anti-interference ability of missilesand is beneficial to reduce the miss distance caused byrandom interference. In addition, the control historiesare used as guess histories in the case of with impact timeand angle constraints.

5.3. Case with Impact Time and Angle Constraint. In thissubsection, results for impact time and angle control arepresented. As we mention in Section 1, good impact angleswill enhance the effect of a cooperation attack. Now, we letthe missiles attack the stationary target and constrain theimpact time and angle; terminal constraints and initial condi-tions are shown in Tables 4 and 2, respectively. The target isassumed to face -40 degrees. In this condition, one missilestrikes vertically, one strikes from the front at a -60-degreeflight path angle, and the other two can strike horizontally

0−2

2000

−1 5000

4000

z (m

)

6000

×104

x (m)

0 0

y (m)

8000

1 −50002 −10000

M1 by PNM2 by PNM3 by PNM4 by PN

M1 by MPSPM2 by MPSPM3 by MPSPM4 by MPSP

Figure 7: 3D trajectories with impact time control by MPSP.

0 5 10 15 20 25 30 35 40t (s)

0

2000

4000

6000

8000

10000

12000

14000

16000

R (m

)

M1M2

M3M4

Figure 8: Relative distances between missiles and the target withimpact time control.

Table 3: Terminal constraints with impact time control.

t f (s) V (m/s) γ (°) ψ (°) x, y, z (km) az , ay (m/s2)

Missile 1 37.98 / / / (0, 0, 0) (0, 0)

Missile 2 37.98 / / / (0, 0, 0) (0, 0)

Missile 3 37.98 / / / (0, 0, 0) (0, 0)

Missile 4 37.98 / / / (0, 0, 0) (0, 0)


0 5 10 15 20 25 30 35 40t (s)

−100

−80

−60

−40

−20

0

M1M2

M3M4

�훾 (º

)

(a)

M1M2

M3M4

0 5 10 15 20 25 30 35 40t (s)

−200

−300

−100

0

100

�휓 (º

)

(b)

Figure 9: Curve of angles with impact time control.

−20

−10

0

10

20

30

M1M2

M3M4

0 5 10 15 20 25 30 35 40t (s)

a z (m

/s2 )

(a)

M1M2

M3M4

0 5 10 15 20 25 30 35 40t (s)

−20

−10

0

10

20

a y (m

/s2 )

(b)

Figure 10: Missile acceleration with impact time control.


from both sides. When the flight path angle is -90 deg, itmakes no sense to constrain the heading angle, which iswhy the heading angle is not restricted under such circum-stances. The time step Δt = 0 01 s. We want to have a loweracceleration in the later stage; therefore, we chose the controlweightingmatrix asRk = 1 + 40e−0 03 t f −k×dt , k = 1, 2,…N − 1.The convergence criterion for this simulation study is a missdistance of 1m, 10−3 deg allowable error in angle, and 1m/s2

allowable error in acceleration.Figure 11 shows the missile trajectories with impact time

and angle control by MPSP guidance; Mi-T i = 1, 2, 3, 4means missile trajectories with impact time control, andMi-T-A i = 1, 2, 3, 4 means missile trajectories with impacttime and angle control. Figure 12 shows the relative distancesbetween missiles and the target. Figure 13 presents thehistories for the flight path angles γ and heading angles ψwith impact time and angle control. We can observe thatthe impact angles converge to the desired values at theexpected impact time 37.98 s, which verifies the validity ofthe proposed guidance law. Because both the initial and finalflight path angles of missiles 1 and 3 are zero, the missileacceleration az is larger at the beginning and end than atthe middle, as shown in Figure 14. But we can still get thefinal acceleration to zero by constraining the terminalacceleration, which is beneficial to reduce the miss distancecaused by random interference.

During an attack, the missile’s acceleration is generallylimited; hence, we need to verify the effectiveness of thealgorithm when the acceleration exceeds the boundary. Ofcourse, the acceleration is over the boundary for a period oftime, not all of the time. We chose the acceleration limitas 35m/s2, which may be greater in a real situation. Thehistories of missiles’ lateral acceleration are shown inFigure 15, and the algorithm still converges. The missilecan still strike the target accurately at the required impacttime and angle. Comparing Figure 15 with Figure 14, wecan find that when the acceleration exceeds the limit, theentire acceleration is adjusted to accommodate the newsituation. We can see that the algorithm is effective whenthe acceleration exceeds the boundary.

6. Conclusions

In this paper, a new guidance law for air-to-ground mis-sile cooperation attacks is presented based on the MPSPtechnique, which can simultaneously constrain the impacttime and angle. The main feature of this guidance is theability to guide missiles with different initial conditions toattack the target at the same time at a specific flight pathangle and heading angle. Control saturation restrictionsare considered in the MPSP algorithm, which widensthe application scope of the MPSP algorithm. Moreover,

Table 4: Terminal constraints with impact time and angle control.

t f (s) V (m/s) γ (°) ψ (°) x, y, z (km) az , ay (m/s2)

Missile 1 37.98 / 0 50 (0, 0, 0) (0, 0)

Missile 2 37.98 / -60 -40 (0, 0, 0) (0, 0)

Missile 3 37.98 / 0 -130 (0, 0, 0) (0, 0)

Missile 4 37.98 / -90 / (0, 0, 0) (0, 0)

0−2

2000

−1 5000

4000

z (m

)

6000

x (m)

×104 0 0

y (m)

8000

1 −50002 −10000

M1-TM2-TM3-TM4-T

M1-T-AM2-T-AM3-T-AM4-T-A

Figure 11: 3D trajectories with impact time and angle control by MPSP.


by taking the first-order-lag acceleration as a state vari-able, the missile acceleration converges to zero at theend, which can increase the anti-interference ability ofmissiles and is beneficial to reduce the miss distancecaused by random interference. In this paper, a collision

avoidance strategy applied to three-dimensional missilecooperative guidance flight is proposed, which can suc-cessfully avoid missile collision. It can be seen from thesimulation results that the guidance law can successfullycomplete the cooperation attack.

0 5 10 15 20 25 30 35 40t (s)

−100

−50

50

0

M1M2

M3M4

�훾 (º

)

(a)

0 5 10 15 20 25 30 35 40t (s)

−200

−100

0

100

M1M2

M3M4

�휓 (º

)

(b)

Figure 13: Curve of angles with impact time and angle control.

0 5 10 15 20 25 30 35 40t (s)

0

2000

4000

6000

8000

10000

12000

14000

16000

R (m

)

M1M2

M3M4

Figure 12: Relative distances between missiles and the target with impact time and angle control.


0 5 10 15 20 25 30 35 40t (s)

−40

−20

0

20

40

a z (m

/s2 )

M1M2

M3M4

(a)

0 5 10 15 20 25 30 35 40t (s)

−20

−10

0

10

20

a y (m

/s2 )

M1M2

M3M4

(b)

Figure 15: Limited missile acceleration with impact time and angle control.

0 5 10 15 20 25 30 35 40t (s)

−40

−20

0

20

40

60

M1M2

M3M4

a z (m

/s2 )

(a)

0 5 10 15 20 25 30 35 40t (s)

−20

−10

0

10

20

30

M1M2

M3M4

a y (m

/s2 )

(b)

Figure 14: Missile acceleration with impact time and angle control.


Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by Aeronautical Science Foundationof China (no. 20150151002).

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Three-Dimensional Impact Time and Angle Control Guidance