Cooperative Guidance Law against Highly Maneuvering Target with Dynamic Surrounding Attack

Abstract

In this paper, a new dynamic surrounding attack cooperative guidance law against highly maneuvering target based on decoupled model is proposed. First, a new dynamic surrounding guidance strategy is proposed, and virtual targets are introduced to establish the cooperative guidance model for dynamic surrounding attack. Second, a dynamic inverse method is used to decouple the cooperative guidance model, and extended state observers (ESOs) are introduced to estimate the disturbances caused by target maneuver. Then, the impact time and dynamic surrounding guidance (ITDSG) law against highly maneuvering target is designed based on a prescribed-time stable method and the decoupled model. Finally, numerical simulations are performed to illustrate the superiority and effectiveness of the proposed ITDSG.

Full text

Research Article
Cooperative Guidance Law against Highly Maneuvering
Target with Dynamic Surrounding Attack
Zhikai Wang,
1
Wenxing Fu ,
2
Yangwang Fang,
2
Zihao Wu,
1
and Mingang Wang
1
1
School of Astronautics, Northwestern Polytechnical University, Xi’an, China
2
Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an, China
Correspondence should be addressed to Wenxing Fu; wenxingfu@nwpu.edu.cn
Received 9 December 2020; Revised 15 March 2021; Accepted 29 March 2021; Published 19 April 2021
Academic Editor: Paolo Castaldi
Copyright © 2021 Zhikai Wang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a new dynamic surrounding attack cooperative guidance law against highly maneuvering target based on decoupled
model is proposed. First, a new dynamic surrounding guidance strategy is proposed, and virtual targets are introduced to establish
the cooperative guidance model for dynamic surrounding attack. Second, a dynamic inverse method is used to decouple the
cooperative guidance model, and extended state observers (ESOs) are introduced to estimate the disturbances caused by target
maneuver. Then, the impact time and dynamic surrounding guidance (ITDSG) law against highly maneuvering target is
designed based on a prescribed-time stable method and the decoupled model. Finally, numerical simulations are performed to
illustrate the superiority and effectiveness of the proposed ITDSG.
1. Introduction
Cooperative guidance laws for multimissiles have captured
the interest of many researchers since the seminal work by
I.S. Jeon first appeared [1]. Cooperative guidance law against
a static or low-speed target has been extensively studied over
the past decades [2, 3]. The early cooperative guidance laws
mainly focused on the cooperative proportional navigation
(CPN) proposed by Tahk et.al. [4, 5], which were used to
solve the problems of antiship missiles attacking the ship
simultaneously. In fact, the CPN is not a true sense of coop-
erative guidance law, because it does not consider the infor-
mation interaction between the missiles. Soon afterwards,
the true sense of cooperative guidance laws including central-
ized [6] or distributed [7–9] cooperative guidance laws were
studied.
Considering the communications between missiles, some
researches have proposed cooperative guidance laws against
static or constant velocity target. Zhou and Yang [10] used
the range-to-go as a covariable to avoid the estimation of
times-to-go and designed a cooperative guidance law against
a static target with an undirected communication topology.
Zhao et.al. [7, 11] designed a multimissile cooperative guid-
ance law against single stationary target based on model-
predictive-control (MPC) and CPN.
Unlike the above scenarios, some researchers have
focused on the research of cooperatively intercepting maneu-
vering target with multimissiles in recent years [2, 3]. As is
known to all, a missile can intercept a low maneuvering tar-
get (such as early warning aircraft) easily [12–15], but it is
difficult for a missile to intercept a target with highly maneu-
verability (such as unmanned aerial vehicle). Thus, it is nec-
essary to develop cooperative guidance law for multiple
missiles against the highly maneuvering target. Some
scholars have studied this problem in recent years. In [16],
an ESO was introduced to estimate the unknown disturbance
caused by the acceleration of target; furthermore, the cooper-
ative guidance law was designed with the estimated distur-
bances and finite time consensus theory [17, 18]. Based on
the optimal control method, Nikusokhan and Nobahari
[19] proposed a novel approach to derive a cooperative guid-
ance law for two pursuers against one zero-lag evader with a
random step maneuver. However, the shortage of the
approach is that the maneuverability of the missile is sup-
posed to be unlimited, which is impracticable in engineer
application. Because of the limitation of the missiles’
Hindawi
International Journal of Aerospace Engineering
Volume 2021, Article ID 6623561, 16 pages
https://doi.org/10.1155/2021/6623561

overload, by setting the virtual target to make each missile’s
joint reachable range cover different subintervals of target
maneuvering range, the cooperative guidance law was
designed in [20] based on BPN to ensure that at least one
missile hits the target. For the three-dimensional (3D) termi-
nal cooperative guidance law against the maneuvering target,
Song et.al. [21, 22] proposed two-direction cooperative guid-
ance laws including acceleration commands in the LOS and
normal LOS directions. The acceleration command in the
LOS direction was developed to ensure all the missiles to
hit the target simultaneously in a finite time; the normal
acceleration command was designed to guarantee that LOS
angular rate and LOS angle converge to the desired values.
Unfortunately, the aforementioned methods did not consider
the couplings between the LOS and normal LOS direction.
Furthermore, the existing guidance strategies for surround-
ing attack require to set the desired LOS angles in advance,
in fact this is a static strategy. However, in the actual combat
environment, it is difficult to set the suitable LOS angles in
advance.
In order to improve the effort of attacking a target, sur-
rounding attack strategy must be considered. In the exist-
ing literatures, in order to achieve the surrounding attack,
the term LOS angle constraint is added to the basic guid-
ance law, so that the designed guidance law can ensure
the missile to approach the target with a desired LOS angle.
The guidance laws for single missile attacking single target
with a LOS angle constraint have been extensively studied
[23–25]. For the guidance problem of multimissile attack
a target with time and LOS angle constraints, some
researchers have studied them [26–29]. However, in the
actual application, the relative relationship between the
missile and the maneuvering target like the above assump-
tions cannot be predicted, so it is difficult to set a suitable
desired LOS angle in advance.
Motivated by the aforementioned papers, in order to
achieve the dynamic surrounding attack of multiple missiles
against a highly maneuvering target and further consider the
couplings between the LOS and normal LOS directions in the
cooperative guidance model, we proposed the ITDSG in this
paper. The main contributions of this paper can be summa-
rized as follows:
(1) We proposed the strategy of dynamic surrounding
attack and established the cooperative guidance
model between multiple missiles and virtual targets.
Due to the traditional cooperative guidance laws with
angle constrain need a preset LOS angle [26–29], it is
difficult to set the appropriate angle in advance due to
the unknown maneuver of the target. But the
dynamic surrounding attack strategy proposed in this
paper can avoid this
(2) By using the dynamic inverse method, the coupled
cooperative guidance model is decoupled, and the
ITDSG is designed based a prescribed-time stable
method subject to the decoupled model. This is
opposed to Ref [21, 22, 30], where the guidance laws
are designed without considered the coupling
(3) The proposed cooperative guidance in this paper
requires less maximum overload and energy con-
sumption compared with Ref [21], which is more
conducive to engineering realization
The remainder of this paper is organized as follows. Sec-
tion 2 presents some necessary preliminaries and model
description. Section 3 presents the dynamic surrounding
attack strategy and the detailed design processes of proposed
cooperative guidance law. Finally, the effectiveness of
dynamic surrounding attack strategy and cooperative guid-
ance law are verified through simulations in Section 4; con-
clusions are drawn in Section 5.
2. Problem Formulation
2.1. Model Description. In this section, the model description
and the basic knowledge of consensus protocol are
introduced.
The relative motion geometry of single missile and single
target can be formulated as follows:
_
r=Vtcos q−θt
ðÞ
−Vmcos q−θm
ðÞ
,ð1Þ
r_
q=−Vtsin q−θt
ðÞ
+Vmsin q−θm
ðÞ
:ð2Þ
Differentiating Eqs. (1) and (2) with respect to time t, the
equations are obtained as follows:
€
r=wr−ur+r_
q2,ð3Þ
€
q=wq−2uq−2_
r_
q
r:ð4Þ
In Eqs. (3) and (4), urand wrdenote the components of
the acceleration of the missile and target in the LOS direction,
respectively; uqand wqdenote components of the accelera-
tion of the missile and target in the normal LOS direction,
respectively.
In order to ensure that missiles hit the target simulta-
neously, a variable tgo called time-to-go is introduced in this
paper, which can be estimated as
tgo =−r
_
r:ð5Þ
Compute the derivation of tgo with respect time tas
follows:
_
tgo =−1+ r2_
q2
_
r2−r
_
r2ur+r
_
r2wr:ð6Þ
Combining Eqs. (1) and (2) with Eqs. (4) and (6), the
state equations of the ith missile can be described as follows:
2 International Journal of Aerospace Engineering

_
x1i=r2
i
_
r2
i
x2
2i−
_
T∗
fi −ri
_
ri2uri +dri,
_
x2i=−2_
ri
ri
x2i−1
ri
uqi +dqi,
8
>
>
>
<
>
>
>
:
ð7Þ
where x1i=tgoi +t−T∗
fi and x2i=_
qiare the state variables
and dri =ðr/_
r2Þwrand dqi =ðwq/rÞare the disturbances
caused by target maneuver.
Considering the problem of nmissiles intercept a maneu-
vering target simultaneously, a dynamic surrounding attack
strategy is proposed in this paper. By this new strategy, the
problem is transformed to that multiple missiles attack multi-
ple virtual, where the virtual targets approach the real target
with missiles approaching the real target. Figure 1 shows the
guidance geometry of Missiles-Target-Virtual Targets in
two-dimensional plane, where Miand Tiv i=ð1, 2,⋯,nÞrep-
resent the ith missile and the ith virtual target; Tdenotes the
real target; Vmi and θmi i=ð1, 2,⋯,nÞdenote the velocity
and the flight path angle of ith missile respectively; aT is the
normal acceleration of target; Vt and θtdenote the velocity
and the flight path angle of target, respectively; aT is the nor-
mal acceleration of target; and qiand rii=ð1, 2,⋯,nÞdenote
the LOS angle and distance between ith missile and ith virtual
target, respectively. The subscripts m,t,andvdenote the state
variables of missiles, target, and virtual targets, respectively.
For multimissiles against single target system, the subscript i
denotes the ith missile’sorvirtualtarget’s state variable.
Assumption 1. The disturbance dri and dqi are bounded, that
is ∣dri ∣<εri and ∣dqi ∣<εqi, where εri and εqi are the given pos-
itive constants.
2.2. Graph. Suppose that communication network graph
between agents can be expressed as G=fV,E,Cg, where V
=f1, 2,⋯,ngdenotes the set of vertices, E⊆V×Vdenotes
the edge set of graph, the subscripts idenotes the ith agent,
eij is the edge of graph G, and eij ⊆Eindicates that the agent
iand jcan receive message from each other in an undirected
graph. If there is a connection between any two agents in the
graph, then the graph Gis connected. In the directed graph,
eij ⊆Emeans that the agent ican receive message from agent
j. In addition, C=ðcijÞ∈Rn×nis the adjacency matrix of
graph G.Ifeij ⊆E, then cij >0; otherwise, cij =0. It is worth
noting that cij =cji when the communication topology is an
undirected graph. The Laplace matrix of the graph Gis
defined as L=ðLijÞ∈Rn×n, which can be expressed as:
Lij =
−cij i≠j,
〠
n
j=1,j≠i
cij i=j:
8
>
>
<
>
>
:
ð8Þ
In this paper, the communication network graph of mul-
tiple missiles is assumed to be undirected and connected.
2.3. Multiagent Consensus. The first-order multiagent system
with nagents is described as
_
xi=ui,ð9Þ
where xidenotes the state of ith agent and uiis the consensus
protocol to be designed with the information of the ith agent
and its neighbors.
Lemma 2 [31]. For a first-order multiagent system as Eq. (9),
if the graph of communication topology is undirected and con-
nected, the state variables xiði=1,⋯,NÞof nagents can be
convergent in a prescribed-time Tby the following consensus
protocol.
ui=−k+c
_
μt
ðÞ
μt
ðÞ

〠
j∈Ni
aij xi−xj

,ð10Þ
Y
OX
Vm1
M1
M2
Mn
Vmn
rnVt
rnv
qn
qnv
𝜃m1
𝜃m2
𝜃mn
𝜃t
Tnv
at
𝜃t
𝜃t
Vt
T2v
r2v
Vm2
𝜃t
at
T
q1v
q2v q2
q1
r1v
V1
at
T
1v
atV1
r1
r2
Figure 1: Guidance geometry on Missiles-Target (virtual targets) engagement.
3International Journal of Aerospace Engineering

2000
1500
1000
500
0
–1000
–500
–1500
–2000
R
R
0 500 1000 1500 2000 2500
(a) Target escape area in Case
Figure 2: Continued.
4 International Journal of Aerospace Engineering

where i=1,⋯,N;k>0and c>1are design parameters; and
μðtÞis a time-varying scaling function as
μt
ðÞ
=
Th
T+t0−t
ðÞ
h,t∈t0,T
½Þ
1,t∈T,∞
½Þ
8
>
<
>
:
,ð11Þ
where h>2is a real number and T>Ts>0with the mini-
mum communication interval Ts.
Lemma 3 [32]. Consider a nonlinear system defined as
_
xt
ðÞ
=ft,xt
ðÞðÞ
+dt
ðÞ
+u,t∈R+:ð12Þ
Select a continue and differentiable Lyapunov candidate
function Vðt,xðtÞÞ >0and Vðt,0Þ=0. The state vector xis
prescribed-time stable with a given time Tin Eq. (11), if the
differentiation of Lyapunov function satisfies
_
V≤−bV −kð_
μ
/μÞ,ont∈½t0,∞Þ. In addition, the Lyapunov candidate func-
tion holds
Vt
ðÞ
≤μ−kexp−bt−t0
ðÞ
Vt
0
ðÞ
,t∈t0,T
½Þ
Vt
ðÞ
≡0,t∈T,∞
½Þ
(:ð13Þ
Lemma 4 [16, 33]. For a nonlinear system with an unknown
and bounded disturbance dðtÞas Eq. (12). Suppose that the
state vector xand the control input ucan be measured. By
a second order ESO Eq. (14), there exist observer gain γ11 ,
γ12,μ1, and δ1such that the estimated states Z11 and Z12
converge into a neighborhood actual states xand dðtÞ,
respectively.
E11 =Z11 −x,
_
Z11 =Z12 +ft,xt
ðÞðÞ
+u−γ11E11 ,
_
Z12 =−γ12fal E11 ,μ1,δ1
ðÞ
,
8
>
>
<
>
>
:
ð14Þ
where E11 is the observed error of the state x;γ11,γ12 ,μ1,
and δ1are the parameters of ESO; and fal is defined as
2000
L
L
R
R
1500
1000
500
0
–1000
–500
–1500
–2000
0 500 1000 1500 2000 2500
(b) Target escape area in Case 2
Figure 2: Target escape area.
5International Journal of Aerospace Engineering

fal E11,μ1,δ1
ðÞ
=E11i
jj
μ1sgn E11
ðÞ
,E11
jj
>δ1
E11/δ1
1−μ1,E11
jj
≤δ1
(:ð15Þ
3. Main Result
Firstly, a novel dynamic surrounding attack strategy is pro-
posed in this section by introducing the virtual targets. Then,
we propose the calculation formulas of the position of virtual
targets using measurable real target information. Finally,
considering the couplings between the LOS and normal
LOS directions of the guidance model in Eq. (7), an adaptive
dynamic inverse method is proposed to decouple them. And
an impact time and dynamic surrounding cooperative guid-
ance law (ITDSG) is designed subject to the decoupled
models. Besides, the accelerations of target in LOS and nor-
mal LOS directions are regarded as disturbances and esti-
mated by ESOs.
Remark 5. The difference between dynamic and static sur-
rounding attack is whether a preset LOS angle constraint is
required. The static surrounding attack is achieved by preset-
ting different LOS angle constraint for each missile to achieve
multiple missiles attacking the target from different direc-
tions. This attack strategy is determined at the beginning of
the missile terminal guidance, which will not change with
the maneuvering of the target. By introducing multiple vir-
tual targets, the dynamic surrounding attack strategy without
preset LOS angle constraint is achieved. The virtual targets
are calculated according to the movement of real target,
and multiple virtual targets gradually approach the real target
from different directions.
3.1. The Strategy of Dynamic Surrounding Attacking. In the
scenario of single missile intercepting a target, at the terminal
phase of intercepting, the target usually maneuvers to avoid
being hit by a missile. Figure 2(a) shows the target escape
area, where target moves with a fixed overload from −
nTmax to nTmax in a finite time. Rdenotes the minimum
turning radius of the target. In a real combat environment,
the purpose of the target turning left or right is to obtain a
maximum lateral distance perpendicular to the line of sight.
When the target is maneuvering with the abovementioned
strategy, escape area of the target is shown in Figure 2(b). R
denotes the minimum turning radius of the target; Ldenotes
the distance of the target flying along a straight line after a 90
°
turning.
Figure 3 shows the novel guidance strategy ITDSG. In the
scenario that nmissiles intercept a maneuvering target,
where T0and Miði=1,2,⋯,nÞrepresent the initial positions
of target and missiles, respectively. T1represents the upper
boundary point of the target perpendicular to the initial
velocity direction in the forward maximum positive overload
maneuver. Similarly, Tirepresents the lower boundary point
when the target maneuvers with negative maximum over-
load. Tpis the middle boundary point of the target with a
zero overload. The escape area is determined by the velocity
and the maximum overload of the target. In this paper, the
escape area of the target is divided into nsubescape areas,
where nis the number of missiles; the center point of the
subescape area is set as the virtual target which denotes as
Tiv ði=1,2,⋯,nÞ. The goal of cooperative guidance is to
make the reachable sets of the nmissiles cover these n
subescape areas, respectively. When the distances between
missiles and the virtual targets become smaller, their
times-to-go gradually approach zero, and multiple virtual
targets gradually approach the real target from different
Mp
Mpv
M1
M1v
Mi
Miv
Ti
Tiv
T
1
T
pv
T
1v
T
1
T
p
L/n
L/n
L/n
L/n
L
T0
–L
L/n
L/n
Figure 3: Guidance geometry on Missiles-Targets engagement.
6 International Journal of Aerospace Engineering

directions as well as the escape areas of multiple virtual
targets coincide gradually. Since each missile aiming at a
virtual target, thus the dynamic surrounding attack is
achieved.
3.2. Dynamic Models of Missiles and Virtual Targets. The
dynamic models of the missiles and virtual targets can be
established by the three steps. Firstly, calculate the coordinate
of boundary points subject to the real target position and
velocity information. Then, the coordinates of nvirtual tar-
gets are calculated based on the information of the boundary
points. Finally, by using the information of virtual targets and
the existing dynamic model of single missile and target, the
dynamic models of multiple missiles and multiple virtual tar-
gets are established.
Step 1. Calculate the boundary point.
Figure 4 shows the geometry between the target bound-
ary point with the target initial position and velocity. The
time of the target flying a quarter of a circle with maximum
overload is defined as Tqc.T0denotes the initial point of
the target; Tu1and Tu2represent the lateral boundary point
when the target is maneuvering with a positive overload
and the flight time is less and greater than Tqc, respectively;
Td1and Td2represent the lateral boundary point when the
target is maneuvering with a negative overload and the flight
time is less and greater than Tqc , respectively; Tgrepresents
the boundary point when the target is moving in a straight
line without any maneuvering.
However, the seeker can only obtain the relative informa-
tion of the missile and the target, such as r,q,_
r, and _
q; the
information of the missile such as the position (xmi,ymi )
and velocity (Vmi,θmi ) can be obtained through its own
inertial navigation device; the velocity of the target (Vt,
θT) can be calculated with Eqs. (1) and (2); the target’s
coordinates can be calculated as xT=xmi +rcosðqiÞand
yT=ymi +rsinðqiÞ. Suppose that at least one missile can
obtain the position and velocity of the target which are
transmitted to other missiles through the communication
network. Then, each missile can calculate boundary points
and virtual target points based on the real target
information.
The boundary points can be calculated as follows. When
Tp≤Tqc, then
xu1=−RT1−cos α
ðÞ½
sin θt
ðÞ
+RTsin α
ðÞ
cos θt
ðÞ
+xT0,
yu1=RT1−cos α
ðÞ½
cos θt
ðÞ
+RTsin α
ðÞ
sin θt
ðÞ
+yT0,
(
xd1=RT1−cos α
ðÞ½
sin θt
ðÞ
+RTsin α
ðÞ
cos θt
ðÞ
+xT0,
yd1=−RT1−cos α
ðÞ½
cos θt
ðÞ
+RTsin α
ðÞ
sin θt
ðÞ
+yT0,
(
ð16Þ
when Tp>Tqc, then
xu2=−RT+L0
ðÞ
sin θt
ðÞ
+RTcos θt
ðÞ
+xT0,
yu2=RT+L0
ðÞ
cos θt
ðÞ
+RTsin θt
ðÞ
+yT0,
(
xd2=RT+L0
ðÞ
sin θt
ðÞ
+RTcos θt
ðÞ
+xT0,
yd2=−RT+L0
ðÞ
cos θt
ðÞ
+RTsin θt
ðÞ
+yT0:
(ð17Þ
What is more, the coordinates (xg,yg) of the bounded
point can be calculated as:
xg=VtTpcos θt
ðÞ
+xT0,
yg=VtTpsin θt
ðÞ
+yT0,
(ð18Þ
where RT=Vt2/atmax denotes the minimum turning radius
of the target, θtis the flight path angle of the target, T0is
the initial position of the target denoted as (xT0,yT0), and
the other variables can be calculated asTqc =πRT/2VT,α=
πTp/2Tqc, and L0=VTðTp−Tqc Þ.
Step 2. Calculate the coordinates of virtual targets.
Considering the scenario that nmissiles cooperatively
intercept a highly maneuvering target, based on the ITDSG
interception strategy, the missile’s attack range covers the
escape area of the target by introducing nvirtual targets.
The coordinates (xiv,yiv ) of the virtual targets Tiv ði=1,
2,⋯,nÞcan be calculated as follows. When Tp>Tqc, then
xiv =−n−2i+1
nRT+L0
ðÞ
sin θt
ðÞ
+RTcos θt
ðÞ
+xT0,
yiv =n−2i+1
nRT+L0
ðÞ
cos θt
ðÞ
+RTsin θt
ðÞ
+yT0:
8
>
>
<
>
>
:ð19Þ
y
Ou
L0
L0
T0
𝜃TRT
Ox
Od
𝛼
𝛼
Tu2(x u2,yu2)
Tu1(xu1,yu1)
T
g(xg,yg)
Td1(xd1,yd1)
Td2(xd2,yd2)
RT
Figure 4: Relative geometrical between target boundary point and
initial position of target.
7International Journal of Aerospace Engineering

When Tp≤Tqc, then
xiv =−n−2i+1
nRT1−cos α
ðÞ½
sin θt
ðÞ
+RTsin α
ðÞ
cos θt
ðÞ
+xT0,
yiv =n−2i+1
nRT1−cos α
ðÞ½
cos θt
ðÞ
+RTsin α
ðÞ
sin θt
ðÞ
+yT0,
8
>
>
<
>
>
:ð20Þ
where ðn−2i+1Þ/nði=1,2,⋯,nÞis designed to adjust the
position of ith virtual target in the normal direction of
target velocity. By setting the virtual targets, the reachable
sets of all missiles can cover the escape area of the target.
In Eqs. (19) and (20), Tpdetermines the values RTand
L0, which is defined as follows:
Tpt
ðÞ
=
Tpmax,t<Ts
Tgop −t−δt,Ts≤t≤Te
0, t>Te
8
>
>
<
>
>
:
,ð21Þ
where Ts=Tgop –δt−Tpmax,Te=Tgop −δt,Tgop =Tgoð0Þ,
Tpmax,andδtare the given positive constants and trepresents
the flight time.
As a result, by applying Eqs. (19)–(21), the position of
virtual targets (xiv,yiv)(i=1,2,⋯,n) can be obtained.
Step 3. Establish the dynamic model.
Similar to Eq. (7), the dynamic models of multiple mis-
siles attacking multiple virtual targets based on the informa-
tion of the virtual targets can be written as follow
_
x1iv =r2
iv
_
r2
iv
x2
2iv −
_
T∗
fiv −riv
_
riv2uri +driv ,
_
x2iv =−2_
riv
riv
x2iv −1
riv
uqi +dqiv,
8
>
>
>
<
>
>
>
:
ð22Þ
where x1iv =tgoiv +t−Tfiv =−ðriv /_
rivÞ+t−Tfiv;x2iv =_
qiv;
riv =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxiv −xmiÞ2+ðyiv −ymi Þ2
q;_
qiv =dq/dt;_
riv =dr/dt
;qiv = arctan ðyiv −ymi Þ/ðxiv −xmiÞ;xmi and ymi are the coor-
dinates of missile i, which are obtained from the missile’s
inertial navigation system, xiv and yiv cannot be measured,
which can be calculated by Step 1 and Step 2. Besides, driv
=ðriv/_
ri2Þwr+driϵand dqiv =ð1/rivÞwq+dqiϵare the distur-
bances, where wrand wqare the components of the real
target’s acceleration in LOS and norm LOS directions and
driϵand dqiϵare the virtual acceleration caused by the virtual
target approaching the real target.
In the initial stage of intercepting, the ith missile is aim-
ing at the ith virtual target. Then, with the time ttending to
Te, the virtual targets are approaching the real target. In the
final stage, the virtual targets and real target are coincident,
that is, all missiles aim at the real target.
3.3. The Guidance Law Design of ITDSG. In this section, an
adaptive dynamic inverse method is used to decouple and
linearize Eq. (22); then, the ITDSG is designed with a
prescribed-time stable control method. Equation (22) can
be rewritten as
_
xi=fixi
ðÞ
+gixi
ðÞ
ui+dit
ðÞ
,ð23Þ
where
fixi
ðÞ
=
r2
iv
_
riv2x2
3iv
−2_
riv
riv
x3iv
2
6
6
6
6
4
3
7
7
7
7
5
,ui=
uri
uqi
2
43
5,
gixi
ðÞ
=
−riv
_
riv20
0−1
riv
2
6
6
6
4
3
7
7
7
5
,dit
ðÞ
=
driv
dqiv
2
43
5:
ð24Þ
Applying Lemma 4, the disturbances driv and dqiv can be
estimated by
̂
driv and
̂
dqiv with two ESOs, and there exist
bounded real numbers η1i>0 and η2i>0 satisfying ∣driv −
̂
driv ∣≤η1i,∣dqiv −
̂
dqiv∣≤η2i. The inverse matrix of giðxiÞcan
be calculated as follows:
gixi
ðÞ
−1=−
_
r2
i
ri
0
0−ri
2
6
43
7
5
:ð25Þ
According to the dynamic inverse method, the controller
can be computed as follows:
uit
ðÞ
=gixi
ðÞ
−1Vit
ðÞ
−fixi
ðÞ
−dit
ðÞ½
:ð26Þ
Substituting the above controller uiðtÞinto Eq. (23), we
obtain
_
xit
ðÞ
=Vit
ðÞ
:ð27Þ
M2
M1
M3
Figure 5: Communication topology for three missiles.
Table 1: Simulation conditions for missiles.
Missile Initial
position (m) θmi0(
°
)ami (g)Vmi0(m/s) R0(km)
Missile 1 (8000, 0) 2 ±30 g 600 18
Missile 2 (9000, 1500) 6 ±30 g 600 19
Missile 3 (7000, -800) -6 ±30 g 600 17
8 International Journal of Aerospace Engineering

The above equation is the expected linearization one;
ViðtÞis the desired dynamic equation of xi, considering
the estimation errors of the disturbances and applying
prescribed-time control method; the desired dynamic
equation of xican be designed as:
Vit
ðÞ
=−Kit
ðÞ
xi,ð28Þ
where
Kit
ðÞ
= diag k11i+k12i
μt
ðÞ
:
μt
ðÞ+η1isign x1iv
ðÞ
,k21i+k22i
μt
ðÞ
:
μt
ðÞ+η2isign x1iv
ðÞ
"#
,
ð29Þ
is a diagonal matrix to be designed, where k11i>0,k12i>0,
k21i>0,k22i>0,η1i, and η2iare the parameters of
controller.
Finally, combining with the Eqs. (26)–(28), the controller
is designed as Eq. (30).
Based on the above design method of the decoupled guid-
ance model, a nonlinear and strong coupled guidance prob-
lem can be transformed into linear control stabilization
problem. By designing the controller, all state variables can
converge to 0. x1iv =0means tgo f i ði=1,2,⋯,nÞof all nmis-
siles tend to a same value. When x2iv =0is satisfied, each mis-
sile can be guaranteed to hit the target.
uri
uqi
"#
=
_
riv2
riv
−k11i+k12i
μt
ðÞ
:
μt
ðÞ+η1isign x1iv
ðÞ
x1iv −riv
_
riv
2x2
2iv +uinom −
̂
driv
"#
−riv k21i+k22i
μt
ðÞ
:
μt
ðÞ+η2isign x2iv
ðÞ
!!−2ri
:x2iv −
̂
dqi
2
6
6
6
6
6
4
3
7
7
7
7
7
5
:
ð30Þ
Remark 6. It is easy to obtain that, as ttends to T,μðtÞand
_
μðtÞ/μðtÞboth approach to infinity, which may cause the
control input to be infinity. However, the state values x1i
and x2iwill approach zero at the prescribed-time T, such that
the control input uri and uqi are both bounded, which has
been proved in Ref. [31]. In addition, due to the use of a
prescribed-time control method, the convergence rate of
the state variables is slower in the initial phase, when the time
tapproaches T, the convergence rate gradually increases.
This will avoid large control inputs in the initial stage, which
is more conducive to engineering realization.
Theorem 7. For the multiple-missile and multiple-virtual tar-
get system Eq. (23), suppose that the disturbances dri and dqi
can be estimated by ESOs designed as Eq. (14). The proposed
guidance Eq. (30) can guarantee that all the tgo f i converge to
a same value (all the missiles hit the target simultaneously),
and q˙
iconverges to 0 (each missile can hit the target) in a
prescribed-time T, respectively.
Proof. Define the following Lyapunov candidate function:
V1=1
2xiTxi:ð31Þ
The derivative of Eq. (31) is given as
_
V1=xiT_
xi:ð32Þ
By applying Eqs. (30) and (23) to Eq. (32) and rearran-
ging, the following inequality can be obtained:
3000
2000
1000
0
–1000
–2000
–8000–10000 –4000 –2000–6000
Y (m)
T
Tup
Tdown
Tgo
Figure 6: Virtual targets with constant velocity target.
9International Journal of Aerospace Engineering

Target
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
5000
4000
3000
2000
1000
Y(m)
0
–1000
–2000
–3000
–1 –0.5 0
X(m)
0.5 1
×104
(a) Trajectories for missiles and target
2.5
2
1.5
1
0.5
0
0510
Time (s)
Relative range (m)
15 20
×104
M1 (ITDSG)
M2 (ITDSG)
M3 (ITDSG)
M1 (SMCG)
M2 (SMCG)
M3 (SMCG)
(b) Distance between missiles and target
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
LOS angle (º)
40
30
20
10
0
–10
–20
–30
0510
Time (s)
15 20
(c) LOS angles for missiles
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
Time-to-go (s)
0
0
5
10
15
20
510
Time (s)
15 20
(d) Time-to-go for missiles
Figure 7: Continued.
10 International Journal of Aerospace Engineering

_
V1=x1iv
_
x1iv +x2iv
_
x2iv =x1iv
r2
iv
_
riv2x2
3iv +
̂
driv −riv
_
r2
iv
uri

+x2iv −2riv
_
riv
x3iv +
̂
dqiv −1
riv
uqi

=x1iv ψ1t
ðÞ
+η1isign x1iv
ðÞ
−
̂
driv +driv

x1iv
+x2iv ψ2t
ðÞ
+η2isign x2iv
ðÞ
−
̂
dqiv +dqiv

x2iv,
ð33Þ
where ψ1ðtÞ=−ðk11i+k12iðμðtÞ/μðtÞÞÞ$and $ψ2ðtÞ=−ðk21i
+k22iðμðtÞ/μðtÞÞÞ.
Since ∣−
̂
driv +driv ∣≤η1i,∣−
̂
dqiv +dqiv∣≤η2i; then, we
have
_
V1≤ψ1t
ðÞ
x2
1iv −ψ2t
ðÞ
x2
2iv≤− λmin 1 +λmin 2
_
μt
ðÞ
μt
ðÞ

V1,
ð34Þ
where λmin1 = min fk11i,k21igand λmin2 = min fk12i,k22ig.
Then, according to Lemma 3, we can obtain that the state
variables x1iv and x2iv gradually converge to zero in a
prescribed-time T. According to the definition of x1iv and
x2iv, it is clear to see that all the tgo f i can converge to T∗
gof ,
and all the _
qiconverge to zero in a prescribed-time T, respec-
tively. Besides, the T∗
gof can converge to a same value in a
prescribed-time Tby applying consensus protocol Eq. (10),
which has been proved in Lemma 2.
Remark 8. The convergence speed of the system states is con-
trollable with the proposed guidance law in this paper. From
Eq. (21), it can be seen that the time for the virtual target to
approach the real target is determined by Te. By applying
the prescribed-time convergence protocol, the desired flight
time Tgof i of the missiles converge to a same value at time
T. In addition, by introducing a time-varying scaling func-
tion, the convergence time of the LOS angular rates _
qiand
the times-to-go Tgoi of the missiles are also determined by
the prescribed-time T. In other words, by setting different
values of Teand T, the convergence speed of multimissile
cooperative attack a highly maneuvering target can be set
arbitrarily.
4. Simulation
To illustrate the effectiveness and superiority of the proposed
ITDSG, we design two groups of simulations in the scenario
of three missiles cooperative attacking a target with constant
velocity and highly maneuvering target, respectively. Simula-
tions with the slide mode cooperative guidance law (SMCG)
in [21] are carried out for comparison.
4.1. Case 1: Constant Velocity Target. Let us consider the sce-
nario of three missiles attacking single target from different
directions; the first group simulations are performed to verify
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
LOS acceleration (g)
–25
–20
–15
–10
–5
0
5
10
15
0510
Time (s)
15 20
(e) Accelerations on the LOS direction
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
100
80
40
60
20
0
–20
–40
0 5 10 15 20
Time (s)
Normal LOS acceleration (g)
(f) Accelerations on the normal LOS direction
Figure 7: Simulation curves for a constant velocity target.
Table 2: Energy consumption.
JuMissile Case 1 Case 2
ITDSG SMCG ITDSG SMCG
Jur (m/s
2
)
Missile 1 254.4 220.4 314.5 1025.4
Missile 2 133.9 86.5 217.9 167.8
Missile 3 222.5 565.9 504.6 548.4
Juq (m/s
2
)
Missile 1 553.9 1024.7 1021.5 1858.7
Missile 2 53.7 41.8 944.9 911.1
Missile 3 647.3 4605.3 1076.7 1740.4
11International Journal of Aerospace Engineering

the performances of the proposed ITDSG compared with
SMCG.
Table 1 shows the simulation initial conditions of three
missiles. Suppose that the target has a constant velocity,
and the initial position of the target is (−10000 m, 0 m), the
initial flight path angle of the target is θT0=0
°.
Assume that the communication network graph of three
missiles is connected and undirected, Figure 5 shows the
communication typology between the three missiles.
The ESO parameters in LOS and normal LOS direction
are the same; they are set as γ11 = 200,γ12 = 300,δ=0:01,
and μ=0:5. The parameters of ITDSG are set as k11i=1,
k12i=1,k21i=0:5,k22i=1,η1i=0:1, and η2i=0:05; besides,
the parameters of μðtÞare set as h=2 and T=19; what is
more, there are two parameters for calculation of virtual tar-
gets Tpmax =8and δt=1:5. In order to compare SMCG with
ITDSG under the same conditions, we set the desired LOS
angles of SMCG as the same with the terminal LOS angles
qfi (i=1,2,3) of ITDSG.
Note 1. Since the energy of the missile is limited, so the
smaller energy needs to provide by the guidance law is, the
more favorable application the guidance law is. Therefore,
an energy consumption evaluation indicators are defined as
Eq. (35).
Jur =ðt
0
urt
ðÞ
jj
dt
ðÞ
,
Juq =ðt
0
uqt
ðÞ
dt
ðÞ
:
8
>
>
>
<
>
>
>
:
ð35Þ
Figure 6 shows the trajectories of the real target and vir-
tual targets, where Tdenotes the real target and Tup,Tdown ,
and Tgo denote the virtual target from different directions.
Figure 7 shows the comparative simulation results of three
missiles against a target with SMCG and ITDSG methods,
in the scenario of three missiles cooperatively hitting a con-
stant velocity target. Further, Figure 6 shows the trajectories
of real target and virtual targets; in ITDSG, all missiles aim
the virtual targets. It is clear to see that all the three virtual
targets gradually approach the real target.
The effectiveness of the proposed ITDSG against a con-
stant velocity target can be verified as follows: Figures 7(a)–
7(c) show that all of the missiles can hit the target simulta-
neously, which indicates that the proposed ITDSG subject
to the decoupled model is effective for the time consistency.
Figure 7(c) shows that all missiles with ITDSG approach
the real target from different directions, which shows that
the proposed dynamic surrounding attack strategy and the
cooperative guidance law are effective.
In terms of missile trajectories, ITDSG has a better per-
formance than SMCG in Figure 7(a). In the initial stage, the
trajectories of ITDSG are smoother than SMCG; then, in
the final stage, the trajectories are similar. The reason is that
when applying SMCG, there are bigger errors between the
initial LOS angles and desired angles of each missile; in order
to decrease the errors, larger lateral accelerations must be
generated to change the missile’sflight path to meet the con-
straint of LOS angles; but when applying the proposed
ITDSG strategy, all the missiles aim at the virtual targets,
and there are no LOS angle constrains; thus, the missile’s tra-
jectories are relatively smooth. With the same reason,
Figure 7(f) shows that in the initial stage the control inputs
uqin SMCG are bigger than in ITDSG. In the whole guidance
process, the energy consumption in ITDSG is also smaller
than in SMCG that is listed in Table 2.
In terms of the impact time, the SMCG and ITDSG have
the same performance. From the simulation curves in
–2000–4000–6000–8000
X(m)
–10000
–3000
–2000
–1000
0
1000
2000
3000
Y(m)
T
Tup
Tdown
Tgo
Figure 8: Real and virtual targets with high maneuvering.
12 International Journal of Aerospace Engineering

Target
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
6000
4000
2000
–2000
–4000
–1 –0.5 0 0.5 1
X(m)
Y(m)
0
×104
(a) Trajectories for missiles and target
M1 (ITDSG)
M2 (ITDSG)
M3 (ITDSG)
M1 (SMCG)
M2 (SMCG)
M3 (SMCG)
0
0
0.5
1
1.5
2
2.5
Relative range (m)
51015
Time (s)
20
×104
(b) Distance between missiles and target
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
40
30
20
10
0
–10
–20
–30
LOS angle (º)
0 5 10 15
Time (s)
20
(c) LOS angles for missiles
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1 (SMCG)
M2 (SMCG)
M3 (SMCG)
0
0
5
10
15
20
Time-to-go (s)
51015
Time (s)
20
(d) Time-to-go for missiles
Figure 9: Continued.
13International Journal of Aerospace Engineering

Figure 7(d), it is clear that even though the missile initial
values of tgoi are different, they can converge to the same
value under the action of uri, which indicates that the SMCG
and the proposed ITDSG law subject to the decoupled model
with ITDSG strategy are both effectiveness in impact time.
Besides, in Figure 7(e), bigger accelerations in the LOS direc-
tion are generated because of the bigger errors between the
times-to-go and their desired values of different missiles.
What is more, Figures 7(e) and 7(f) show clearly that there
are small fluctuations of acceleration curves in LOS and nor-
mal LOS directions with ITDSG after 9 seconds, which is
caused by the maneuver of virtual target approach the real
target.
As a result, in the scenario of intercepting a constant
velocity target, ITDSG and SMCG both show good perfor-
mance. Because ITDSG uses a dynamic surrounding attack
strategy without LOS angle constraint, the trajectories by
ITDSG are smoother than that by SMCG; the accelerations
in normal LOS directions are also smaller.
4.2. Case 2: Highly Maneuvering Target. In this case, the per-
formance of the proposed ITDSG is investigated in the sce-
nario of three missiles attacking single highly maneuvering
target from different directions.
In order to further illustrate the superiority of ITDSG
over SMCG, simulations are conducted with the same coop-
erative guidance laws and the same missile parameters as
those in Case 1, the only difference is that the target maneu-
ver with a bigger overload nt=8gsinððπ/4ÞtÞ, where g=9:8
m/s
2
. Furthermore, the selection principle of qfi (i=1,2,3)
in SMCG is the same as in Case 1. Figure 8 shows the curves
of virtual targets with highly maneuvering target subject to
the real target motion.
Similar to Case 1, Figure 8 shows the trajectories of the
real target and virtual targets, where Tdenotes the real target
and Tup,Tdown , and Tgo denote the virtual target from differ-
ent directions; Figure 9 shows the effectiveness of the pro-
posed ITDSG strategy against a highly maneuvering target.
Figures 9(a) and 9(c) show that the proposed dynamic
surrounding attack strategy is effective; three missiles can
intercept the target from different directions; Figures 7(a),
7(d), and 7(b) show that the proposed ITDSG is also effective
for the time consistency.
Similar to the analysis in Case 1, Figure 9(a) shows that
trajectories of ITDSG are smoother than SMCG; Figure 9(f )
and Table 2 show that the maximum overloads in LOS and
normal LOS directions of ITDSG are smaller than SMCG,
and the total control energy of ITDSG is also smaller. The
time constraint performance of the ITDSG is the same as
SMCG, which shows in Figures 9(b) and 9(d). What is more,
the difference between Case 1 and Case 2 is that the acceler-
ation curves of all missiles in the two directions can both con-
verge but not zero due to the target maneuver.
As a result, the proposed guidance law ITDSG can be
suitable for constant velocity speed as well as highly maneu-
vering target. Besides, by introducing virtual targets instead
of impact LOS angle to achieve dynamic surrounding attacks,
ITDSG has better performance in trajectories and energy
consumption than SMCG.
5. Conclusion
In this paper, we developed a new dynamic surrounding
attack cooperative guidance law against highly maneuvering
target without a preset LOS angle constraint. Firstly, we pro-
posed the strategy of dynamic surrounding attack by intro-
ducing virtual targets and then established the cooperative
guidance models between multiple missiles and multiple vir-
tual targets. Finally, by using the dynamic inverse method to
decouple the coupled cooperative guidance model, and the
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
–10
–5
0
5
10
15
20
25
30
LOS acceleration (g)
0 5 10 15
Time (s)
20
(e) Accelerations on the LOS direction
M1(ITDSG)
M2(ITDSG)
M3(ITDSG)
M1(SMCG)
M2(SMCG)
M3(SMCG)
100
80
60
40
20
0
–20
–40
Normal LOS acceleration (g)
0 5 10 15
Time (s)
20
(f) Accelerations on the normal LOS direction
Figure 9: Simulation curves for a highly maneuvering target.
14 International Journal of Aerospace Engineering

ITDSG subject to the decoupled model is designed based a
prescribed-time stable method. Additionally, ESOs are intro-
duced to estimate the disturbances in the LOS and normal
LOS directions. In order to demonstrate the effectiveness
and superiority of the proposed ITDSG, two groups of com-
parison simulations are carried out with SMCG in the scenar-
ios of three missiles cooperative attack a constant velocity or
highly maneuvering target.
Compared with the existing works, the advantages of the
proposed ITDSG are that it does not require to set desired
LOS angles in advance, and the couples in the cooperative
guidance model are also considered. The limitation of the
proposed guidance law in this paper lies in the need for pre-
cise target information, such as q,_
q,r, and _
r. In the future
work, we would develop a novel cooperative guidance law
with dynamic surrounding attack when there are seeker mea-
surement errors.
Data Availability
The data used to support the findings of this study are
included within the article.
Conflicts of Interest
The authors declared that they have no conflicts of interest to
this work.
Acknowledgments
Thanks are due to the financial support provided by the
National Natural Science Foundation of China (Grant No.
61973253), the Aviation Science Foundation of China
(20180153001), and the Foundation of National Defense Sci-
ence and Technology Key Laboratory (6142219180202).
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