This document proposes a new guidance strategy for intercepting stationary and moving targets using a virtual target approach, focusing on achieving specified impact time and angle while considering nonlinear kinematics and acceleration limits. The guidance method is divided into two stages, with a terminal sliding-mode law for the virtual target and a proportional navigation law for the actual target, optimizing control energy and simplifying user effort. Numerical simulations confirm the effectiveness of this strategy in various engagement scenarios.

1. New Impact Time and Angle Guidance Strategy via
Virtual Target Approach
Qinglei Hu∗
and Tuo Han†
Beihang University, 100191 Beijing, People’s Republic of China
and
Ming Xin‡
University of Missouri, Columbia, Missouri 65211-2200
DOI: 10.2514/1.G003436
In this paper, a new guidance strategy is proposed for intercepting stationary and constant-velocity moving targets
with desired impact time and angle via a virtual target approach, subject to nonlinear engagement kinematics and
lateral-acceleration limits. The guidance procedure is divided into two stages by introducing a virtual target.
Specifically, for the first stage, the nonsingular terminal sliding-mode guidance law is employed to intercept the
virtual target with a specified impact angle in finite time. Furthermore, to achieve the desired impact angle on the real
target, the proportionalnavigation guidancelaw is used in the second stage to keep the missile traveling with invariant
flight-path angle. The impact time and angle on the real target are simultaneously controlled by selecting a proper
guidance parameter whose optimal value is obtained via an optimization routine with a small number of iterations.
This technique is user/designer-friendly in that it does not involve the time-to-go estimation procedure. In addition,
less control energy is required in the overall guidance process. Numerical simulations with comparisons are
conducted to validate the effectiveness and feasibility of the proposed guidance strategy in different engagement
scenarios.
I. Introduction
ADVANCED guidance laws aim to achieve zero miss distance
and meet different terminal constraints for raising the kill
probability and improving survivability during the homing phase. To
achieve these goals, terminal constraints such as desired impact time
and impact angle are critical for homing missiles to intercept modern
warships, tanks, and ballistic missiles. The impact angle constraint is
required not only to increase lethality of the missile’s warhead but to
escape the limited defense zone of the target. The impact time
requirement is desired for antiship missiles to facilitate a salvo attack
against the advanced close-in weapon systems (CIWS) [1–3]. The
effectiveness of a salvo attack can be greatly enhanced by controlling
the impact time and impact angle simultaneously, which motivates
the guidance law design that can achieve both objectives.
Plenty of literature has been published on the impact angle or
impact time control problems. One of the earliest impact angle
guidance (IAG) laws was proposed in [4] (dating back to the 1970s),
where the impact angle constraint is addressed by posing a linear
quadratic control problem with time-varying gains. In [5–9], the
desired impact angle can be achieved through the optimal control,
sliding-mode control, and other advanced control methods. More
recently, the impact angle control problem was solved by converting
the optimal control problem into a second-order cone programming
problem [10]. In [11], the estimated terminal flight-path angle was
used to derive the impact angle guidance law for various missile and
target motions. The feedback linearization was used in [12] to control
the impact angle with varying guidance gains. In the meanwhile, the
study on guidance laws that can achieve the desired impact time has
attracted more attention in recent years. For instance, the impact time
guidance (ITG) law was first designed in [13], in which the desired
impact time is achieved by combining the proportional navigation
guidance (PNG) law with an additional feedback term. Thereafter,
the approach of [13] was improved in [14,15]. Besides, the
sliding-mode control [16], Lyapunov-based control [17], feedback
linearization [18], and some other advanced guidance strategies
[19–23] have been successfully employed for satisfying the impact
time constraint.
Although the impact angle or impact time control problem has
attracted a great deal of attention, there has been only a few works on
controlling both simultaneously. The existing impact time and angle
guidance (ITAG) approaches can be classified into two main
categories, nonswitching ITAG law and switching ITAG strategy. For
the former category, the ITAG laws are derived without a switching
logic through the optimal control theory [24,25], polynomial shaping
approach [26,27], and time-varying sliding-mode control [28].
However, the target is assumed to be stationary in [24,25,27], and the
kinematics equations in [24,26–28] are required to be transformed
with the downrange variable as the independent variable. For the
second category, several ITAG strategies with the switching logic
between different guidance laws were presented in [29–33]. One
early research on the design of the switching ITAG strategy was given
in [29], where different guidance laws are integrated in two stages
under the generation of desired waypoints. Therein, an IAG law and
an ITG law are combined in the first stage, whereas the well-known
PNG law is employed in the second stage. A hybrid ITAG guidance
scheme was proposed in [30] by switching three different guidance
laws (the first mode of the guidance logic enforces impact angle
constraint and the other guidance modes are used to control the
impact time), where the switching logic is determined by the impact
time error and the range rate. A switching logic was employed
between a sliding-mode impact time guidance law and an optimal
impact angle guidance law in [31] to meet the impact time and angle
constraints. However, the downside of [31] is that the missile’s flight
trajectory needs to be quickly curved as soon as the missile is
launched, which may lead to saturation of the missile’s lateral
acceleration (latax). Besides, an ITAG strategy with three phases was
presented in [32] by considering the look angle and acceleration
constraints under the nonlinear optimal control framework. More
recently, the guidance performance of [31] has been improved in [33]
Received 6 November 2017; revision received 10 January 2018; accepted
for publication 11 January 2018; published online 21 February 2018.
Copyright © 2018 by the American Institute of Aeronautics and Astronautics,
Inc. All rights reserved. All requests for copying and permission to reprint
should be submitted to CCC at www.copyright.com; employ the ISSN
0731-5090 (print) or 1533-3884 (online) to initiate your request. See also
AIAA Rights and Permissions www.aiaa.org/randp.
*Professor, School of Automation Science and Electrical Engineering;
huql_buaa@buaa.edu.cn. Senior Member AIAA (Corresponding Author).
†
Ph.D. Candidate, School of Automation Science and Electrical
Engineering; hantuo@buaa.edu.cn. Student Member AIAA.
‡
Associate Professor, Department of Mechanical and Aerospace
Engineering; xin@missouri.edu. Associate Fellow AIAA.
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2. via selecting the impact time error as the switching surface. The
guidance switching logic given in [33] is identical to that of [31].
Unlike all of the aforementioned ITAG laws, which consider
stationary targets only [24,25,27,29,30,32], employ a linearized
kinematic model [24,27], demand error-prone estimates of the time-
to-go information [24,28–31,33], convert the differential equations
with the downrange variable as the independent variable [24,26–28],
create large control effort, or involve frequent guidance switching
[31,33], the proposed ITAG strategy attempts to cope with these
drawbacks concurrently with simple design procedures. It is worth
noting that the design of an ITAG strategy against moving targets
encountered in practice is a very challenging and complex task
considering the nonlinear engagement kinematics with physical
limits on achievable accelerations and without the information of
time to go. The guidance strategy to be proposed in this paper differs
fromwhat is available inthe literature in that theproblem of achieving
both desired impact time and impact angle against moving targets
with the nonlinear kinematics is addressed by employing a novel
virtual target approach. The virtual target approach has been applied
to lead the missile to a desired interception point [34], guide an
autonomous glider to a desired destination with an expected impact
angle [35], guarantee the missile to hit the target with no-fly zone
constraints [36], and develop a guidance logic for unmanned aerial
vehicle path following [37]. Note that none of these virtual target
methods addresses the impact time and impact angle constraints
together. To the best of the authors’ knowledge, the new virtual target
approach proposed in this paper can control the impact time and
impact angle simultaneously against a stationary or moving target for
the first time.
The main contributions of the new ITAG approach are presented as
follows.
1) The guidance procedure is divided into two stages by
introducing a virtual target, the location of which can be adjusted to
ensure the desired impact time on the real target with only one tuning
parameter.
2) The IAG law based on the nonsingular terminal sliding
mode and PNG law are employed in the two stages, respectively,
with a simple switching logic, to guarantee that the missile can
meet the desired impact angle on the virtual target as well as the
real target.
3) The desired impact time and angle on the real target can
be achieved simultaneously by selecting a proper guidance
parameter whose optimal value is obtained via an optimization
routine.
4) Less control energy is generated during the whole
guidance process because less control effort is needed in the
second stage.
The other keyfeature of the proposed ITAG strategy is that missiles
can achieve a salvo attack of a moving target with desired impact
angle and without the information of time to go.
The rest of this paper is organized as follows. In Sec. II, the
engagement geometry and nonlinear kinematics against a moving
target are presented. Section III represents the virtual target approach
against stationary and constant-velocity moving targets. The ITAG
strategy is presented in Sec. IV. The performance of the ITAG is
validated by simulations with comparisons in Sec. V. Finally,
conclusion remarks are given in Sec. VI.
II. Problem Formulation
Consider a planar homing engagement scenario of a missile M
against a moving target T, as shown Fig. 1. The missile is assumed to
travel with a constant velocity VM, which is perpendicular to the
missile latax AM. The angles ϕM, γ, and λ represent the lead angle,
flight-path angle, and line of sight (LOS) angle of the missile,
respectively. The relative range between the missile and target is
denoted as r. The target moves at a constant speed VT, and the flight-
path angle of the target is denoted as γT. All the angles are defined as
positive in the counterclockwise direction. As described in Fig. 1, the
positions of the missile (xM, yM) and target (xT, yT) are defined in an
inertial x–o–y frame. Suppose that the missile is cruising with a
constant flight height, which is equal to the altitude of the intercept
point. Then, the adopted x–o–y frame is horizontal.
According to the missile–target geometry relationship, the
nonlinear engagement kinematics for this problem can be
given by
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
_
xM VM cos γ
_
yM VM sin γ
_
γ AM∕VM
_
xT VT cos γT
_
yT VT sin γT
_
γT 0
_
r VT cosγT − λ − VM cosγ − λ
(1)
r_
λ VT sinγT − λ − VM sinγ − λ (2)
Differentiating Eq. (2) with respect to time, the LOS
dynamics can be given by
λ −
2_
r _
λ
r
−
AM cos ϕM
r
(3)
where ϕM γ − λ is the lead angle.
According to the result in [6], the relation between the desired
impact angle (denoted as γimp) and final LOS angle (denoted as λF)
can be expressed as λF γTf − tan−1sin γimp∕cos γimp − ρ,
where ρ VT∕VM 1 denotes the target-to-missile velocity ratio,
and γTf represents the target’s flight-path angle on the collision
course. Obviously, there exists a one-to-one correspondence between
the impact angle and the LOS angle on the collision. Thus, the
guidance problem with the impact angle constraint can be converted
to control of the terminal LOS angle. Although it can be seen from
Eq. (3) that the LOS angle cannot be controlled if jϕMj π∕2, it has
been checked in [6,38] that ϕM will not stay on the condition
jϕMj π∕2 because _
ϕM ≠ 0 when jϕMj π∕2. Indeed, such a
special scenario would rarely appear and can be easily avoided by
appropriately selecting the relative positions of the missile and target.
Therefore, the guidance mission with impact time and angle
constraints studied in this work can be described as the following
objectives:
8
:
rf → 0;
timp → Td;
λTd → λF
(4)
where rf denotes the miss distance, timp is the final impact time, and
Td represents the desired impact time.
M
T
λ
γ M
VM
VT
AM
y
x
Trajectory
r
o
γ T
φ
Fig. 1 Homing guidance geometry of missile–target engagement.
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3. III. Virtual Target Approach for Impact Time and
Angle Constraints
In this section, a novel virtual target approach is proposed for
achieving the guidance objectives presented in Eq. (4) (i.e., the
impact time and angle constraints as well as the zero miss distance).
The key feature of the proposed virtual target approach is that the
guidance procedure is divided into two stages by a virtual target
located in the plane of the missile and the real target. Thevirtual target
approach aiming to meet the impact time and angle constraints is
developed in the following subsections to intercept a stationary target
and a constant-velocity moving target, respectively.
A. Virtual Target Approach Against a Stationary Target
In this subsection, the virtual target approach is developed for
achieving the desired impact time and angle against a stationary
target. More specifically, the virtual target is assumed as a stationary
point, which is located in the plane of the missile and the real target
and to be captured by the missile ahead of the real collision on the real
target, as shown in Fig. 2. The virtual target is denoted by T 0. γMV and
γM represent the flight-path angles of the missile on the collision
point with respect to the virtual target and the real target, respectively.
λFV and λF denote the desired LOS angles for thevirtual target andthe
real target, respectively. Note that the desired impact angle can be
expressed as λF −γimp and γM −γimp for a stationary target,
which implies that the missile’s flight-path angle is equal to the LOS
angle on the collision point.
To achieve the desired impact angle and impact time, the guidance
procedure is divided into two stages by the virtual target T 0. In Fig. 2,
the IAG law is employed for the first stage (depicted as stage 1). In
this stage, the mission of the IAG law is to capture the virtual target at
a specified impact angle, which is selected to be equal to the
desired impact angle on the real target (which means that
γMV γM λFV λF). For the second stage (depicted as stage 2),
the PNG law is applied to attack the real target as soon as the virtual
target is captured with the desired impact angle. The “switching
point” in Fig. 2 describes the point where the guidance command
switches from IAG law to PNG law, which occurs at the moment that
the guidance mission in stage 1 is accomplished. It is important to
emphasize that the desired impact angle on the real target can always
be satisfied by employing the PNG law because the LOS rate turns
into zero in stage 2, which leads to AM 0 and _
γ 0; then, the
missile will keep traveling in a straight line with an invariant flight-
path angle in stage 2.
In the meanwhile, it should be noted that the total flight time of the
missile varies with different positions of the virtual target, which
means that there exists a desired impact time if the virtual target is
located at a desired position. Therefore, choosing a proper position of
the virtual target is the key to meet the desired impact time. To attain
this goal, the position of the virtual target is computed as follows
(taking the scenario in Fig. 2 as an example):
(
xT 0 xT − bTd • VM cosλF
yT 0 yT − bTd • VM sinλF
(5)
where (xT 0 , yT 0 ) denotes the position of the virtual target, and
b ∈ 0; 1 represents the parameter to be selected.
It can be seen from Eq. (5) that the position of the virtual target can
be adjusted by tuning the single parameter b. The flight time of the
missile in stage 2 is bTd, and the flight time in stage 1 is 1 − bTd.
Thus, the desired impact time and angle on the real target can be
easily achieved because there always exists a coefficient b such that
the IAG law in stage 1 can capture the virtual target with the desired
impact angle at the impact time 1 − bTd.
B. Virtual Target Approach Against a Constant-Velocity Target
The virtual target approach against a stationary target in the
preceding subsection can be naturally extended to the case of
intercepting a constant-velocity (nonmaneuvering) target because the
IAG law and PNG law are capable of intercepting a moving target.
Consider the planar engagement geometry for intercepting a moving
target as shown in Fig. 3. Note that the IAG law employed in the
previous subsection is to capture the virtual target that is assumed to
be a stationary point. In this subsection, the virtual target is
considered moving with a constant velocity because the real target is
moving.
In Fig. 3, T and T 0 denote the positions of the real target and the
virtual target at the initial moment, respectively. VT 0 and γT 0 denote
the velocity and flight-path angle of the virtual target, respectively.
The initial position of the virtual target is located in the plane of the
missile and target, and the flight trajectory of the virtual target is
parallel to that of the real target (which means γT 0 γT). R1 is the
distance from the initial position of the virtual target to the virtual
collision point. R2 is the distance from the initial position of the real
target to the real collision point. In addition, the condition λFV λF
still holds for this case.
The initial position of the virtual target is calculated as
(
xT 00 xT0 − bTd • VM cosλF
yT 00 yT0 − bTd • VM sinλF
(6)
where λF can be computed from the relationship
λF γTf − tan−1sin γimp∕cos γimp − ρ. (xT 00, yT 00) and (xT0,
yT0) denote the initial positions of the virtual target and the real target,
respectively. b ∈ 0; 1 is the same as the parameter defined in Eq. (5).
According to the geometry in Fig. 3, it is known that the flight time
of the missile in stage 2 is still equal to bTd, and the flight time in
stage 1 should be 1 − bTd if the desired impact time is Td.
Considering that the flight trajectories of thevirtual target and the real
targetare parallel to each other, the following conditions are obtained:
M
T
VM
AM
y
x
o
Virtual target
Switching point
Stage 2 : IAG
Stage 1 : PNG
λFV
λF
T’
MV
M
VM
VM
γ
γ
Fig. 2 Illustration of the virtual target approach against a stationary
target.
M
T
VM
AM
y
x
o
Stage 1: IAG
Stage 2 : PNG
T’
VT’
VT
λFV
λF
γT
γT’
Virtual collision point
Real collision point
λF
λFV
R1
R2
Fig. 3 Illustration of the virtual target approach against a constant-
velocity target.
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4. 8
:
R1 VT 0 • 1 − bTd
R2 VT • Td
R1 R2
(7)
From Eq. (7), it can be obtained that VT 0 VT∕1 − b. In
addition to the speedof thevirtual target,the kinematics for thevirtual
target and the relative equations of motion for the missile and virtual
target can be given as
8
:
_
xT 0 VT 0 cos γT 0
_
yT 0 VT 0 sin γT 0
_
γT 0 0
rV
_
λV VT 0 sinγT 0 − λV − VM sinγ − λV
_
rV VT 0 cosγT 0 − λV − VM cosγ − λV
(8)
where xT 0 and yT 0 represent the position of the virtual target along the
x axis and y axis, respectively. rV and λV represent the relative range
and LOS angle between the missile and the virtual target,
respectively.
The IAG law is applied in stage 1 to capture the constant-velocity
virtual target with the desired impact angle. The PNG law is
employed in stage 2 to intercept the real moving target as soon as the
virtual collision occurs. Note that the desired impact angle on the real
target can always be satisfied by employing the PNG law because the
conditions AM 0 and _
γ 0 still hold in stage 2. Then, the missile
will keep traveling in a straight line with the invariant flight-path
angle before the real target is captured.
Similarly, the impact time and angle constraints on the real moving
target can be easily satisfied because there always exists a b with
which the IAG law in stage 1 can capture the constant-velocity virtual
target with the desired impact angle at the specified impact
time 1 − bTd.
IV. Impact Time and Angle Guidance Strategy
In this section, a new guidance strategy including two guidance
laws is developed to achieve both the impact time and impact angle
requirements against stationary targets as well as moving targets via
the virtual target approach. The proposed guidance strategy consists
of two stages. For the first stage, a nonsingular sliding-mode impact
angle guidance law is employed to send the missile to meet the virtual
target with a specified impact angle, which equals the desired impact
angle on the real target. Once the virtual target is captured with the
desired impact angle, the PNG law is employed to attack the real
target in the second stage. With this switching logic, the impact time
requirement can be achieved by selecting the initial position of the
virtual target properly.
A. Impact Angle Guidance Law for the Virtual Target
In view of the proposed virtual target approach, it is clear that the
primary task for achieving the desired impact time and angle is to
design an IAG law that can capture the desired virtual target with a
specified impact angle. To this end, a nonsingular terminal sliding-
mode impact angle guidance law is developed for intercepting a
stationary or moving target. With the engagement kinematics of the
virtual target and the missile in Eq. (8), the LOS dynamics in Eq. (3) is
recalled here to represent the LOS dynamics for the missile and the
virtual target:
λV −
2_
rV
_
λV
rV
−
AM cos ϕMV
rV
(9)
where ϕMV γ − λV.
As mentioned previously, control of the impact angle can be
converted to control of the LOS angle on the collision point. Thus, in
this work, the objective of the IAG law is to control the terminal LOS
angle under the LOS dynamics in Eq. (9). Define a tracking error as
e λV − λFV. Then, the nonsingular terminal sliding surface is
chosen as
s e kj_
ejα
sign_
e (10)
where k 0, 1 α 2.
To guarantee the sliding mode on s 0, the missile latax AM is
designed by combining an equivalent controller and a discontinuous
controller (denoted as Aeq
M and Adisc
M , respectively) in the form of
AM Aeq
M Adisc
M (11)
It is known that Aeq
M can be obtained by taking the time derivative of
Eq. (10), which leads to
_
s _
e kαj_
ejα−1
λV (12)
Substituting Eq. (9) into Eq. (12) yields
_
s _
e kαj_
ejα−1
−
2_
rV
_
λV
rV
−
AM cos ϕMV
rV
(13)
Let _
s 0. Aeq
M can be given as
Aeq
M
rV
cos ϕMV
j_
ej2−α
kα
sign_
e −
2_
rV
_
λV
rV
(14)
and the discontinuous controller is chosen as
Adisc
M
k1
cos ϕMV
signs (15)
where k1 is a positive parameter to be designed.
Then, substituting Eqs. (14) and (15) into Eq. (11), the IAG law
against the virtual target is obtained as
AM
rV
cos ϕMV
j_
ej2−α
kα
sign_
e −
2_
rV
_
λV
rV
k1
rV
signs
(16)
Recall that the lead angle cannot stay on π∕2 permanently; thus,
ϕMV is not an attractor (which means that ϕMV π∕2 can only occur
momentarily). Therefore, the guidance command in Eq. (16) will not
be hindered to control the LOS angle.
To prove that the LOS tracking error can converge to zero in finite
time under the IAG law in Eq. (16), a Lyapunovfunction is selected as
V
1
2
s2
(17)
Taking the time derivative of Eq. (17) results in _
V s_
s, and
substituting Eq. (16) into Eq. (13), _
V becomes
_
V −kk1α∕rVj_
ejα−1
jsj (18)
It can be seen from Eq. (18) that _
V is negative-definite for all
_
e ≠ 0 when s ≠ 0. Under this condition, the sliding mode can
always occur. However, _
e may equal zero when s ≠ 0, which would
hinder the convergence of the sliding mode. Thus, it is necessary to
discuss whether _
e can stay on zero or not. Taking the time derivative
of _
e leads to
e −
2_
rV
_
λV
rV
−
AM cos ϕMV
rV
(19)
Substituting Eq. (16) into Eq. (19) leads to
e −
j_
ej2−α
kα
sign_
e −
k1
rV
signs (20)
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5. Then, for _
e 0 and s ≠ 0,
e −k1∕rVsigns (21)
It can be seen from Eq. (21) that
e −k1∕rV for s 0, and
e k1∕rV for s 0. Consequently, the condition _
e 0 is not an
attractor (which means that _
e cannot stay on zero permanently,
and _
e 0 can only occur momentarily), and the sliding mode
s 0 must occur. Next, it is important to claim that the LOS
tracking error can converge to zero in finite time after the sliding
mode occurs. It is observed from Eq. (10) that, once the sliding
surface s 0 is guaranteed, e −kj_
ejαsign_
e. It has been
shown in [39] that the condition e 0 is guaranteed in finite
time. For brevity, the proof of the convergence for e 0 is
omitted here.
B. Achievement of the Impact Angle on the Real Target
With the understanding of the virtual target approach, the
impact angle constraint on the real target can be achieved via a
switching logic between two guidance laws (i.e., the IAG law in
Eq. (16) and the PNG law). Specifically, the impact angle
guidance law against the real target with a switching logic is
proposed as follows:
AM
8
:
rV
cos ϕMV
j_
ej2−α
kα sign_
e − 2_
rV
_
λV
rV
k1
rV
signs
; if jej ζ
NVc
_
λ; if jej ≤ ζ
(22)
where e is the LOS tracking error defined in the previous
subsection for stage 1, and ζ 10−3
is selected as the minimal
tolerance on the impact angle error for the virtual target. N
denotes the effective navigation gain (usually in the range of
3–5), and Vc is the missile–target closing velocity.
Note that the nonsingular sliding-mode IAG law has
guaranteed e → 0 in finite time. Thus, the minimal tolerance ζ
can always be met and is introduced in Eq. (22) just for
switching the guidance command from the IAG law to the PNG
law. In addition, it can be seen from Eq. (22) that the guidance
command always keeps the form of Eq. (16) in stage 1 and then
employs the PNG law in the subsequent stage 2. That is to say,
the switching occurs only once during the whole guidance
process. Therefore, the proposed switching logic will not result
in frequent switches between different guidance laws, compared
to the switching logics in [31,33].
Up to now, it is worth noting that the guidance law in Eq. (22) has
achieved two guidance objectives presented in Eq. (4) (i.e., λTd →
λF and rf → 0). The remaining guidance objective is to meet the
desired impact time such that timp → Td, which is achieved in the
following subsection.
C. Parameter Selection of b: Meeting the Desired Impact Time
To meet the desired impact time on the real target, it is important to
select a proper value of b such that timpv 1 − bTd, where timpv
denotes the actual impact time on the virtual target. Define the final
impact time error in stage 1 as ε timpv − 1 − bTd. Tuning of b
properly for minimizing ε requires the relationship between b and ε.
To investigate this relationship, a set of schematic flight trajectories of
the missile against a stationary target under various values of b with
the same impact angle are analyzed. Recall the calculation of the
virtual target’s position in Eq. (5); it is evident that the virtual target is
farther from the real target with a larger b. Thus, three flight
trajectories (denoted as P0, P1, P2) of the missile using the concept of
virtual target with b 0 and b b1, b2 (b2 b1) are presented in
Fig. 4. Therein, the flight times corresponding to P0, P1, P2 are
denoted as T0, T1, T2, respectively.
From Fig. 4, it can be observed that the flight times of the three
trajectories satisfy the condition T2 T1 T0 because the speed of
the missile is assumed constant, which means that a longer desired
impact time requires a larger value of b. In other words, if the actual
impact time is longer or shorter than the desired impact time, thevalue
of b should be decreased or increased. This reveals the relationship
between b and ε; if ε 0 or ε 0, then the value of b should be
decreased or increased to meet the desired impact time. Note that,
regardless of the value of λF, this relationship is valid and
independent of the impact angle constraint for both stationary and
moving targets because the desired impact angle can always be
achieved by the IAG law.
Although tuning the parameter b by trial and error is not a difficult
task with the relationship described previously, it is still time-
consuming. In this work, instead of tuning b by hand, an optimization
routine “fminsearch” (a Nelder–Mead simplex direct search
algorithm in MATLAB) is employed to minimize the impact time
error with a specified performance index given by [26,28]
J jεj jtimpv − 1 − bTdj (23)
During the optimization, the termination tolerances on the
performance index and the searching variable b are set to 0.01 s and
0.01, respectively. The initial value is set to b0 0.2. Taking an
example of the simulation case of intercepting a stationary target
(see the simulation conditions in Sec. V.A) with Td 50 s and
λF −30 deg, the convergence of the performance index is
presented in Fig. 5. Note that only nine iterations are taken for
M
T
y
x
o
Virtual target: b=b1
Virtual target: b=b2b1
λF
λF
λF
b=0
P0
P1
P2
Fig. 4 Schematic flight trajectories under various values of b with the
same desired impact angle.
0 2 4 6 8 10
Iteration
0
0.2
0.4
0.6
0.8
1
1.2
Performance
index
Stop Pause
Fig. 5 Performance index in each iteration against a stationary target
with Td 50 s and λF −30 deg.
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6. meeting the minimal performance index. Actually, the performance
index can converge to a small value (0.005 s) after only seven
iterations, which means that the optimal value of b is easy to find.
Moreover, for the various simulation cases in Sec. V, an average of
eight iterations are needed while running the optimization algorithm.
Thus, with such a small number of iterations, it is feasible to
determine the optimal value of b by the onboard computer with an
online optimization solver.
With the optimal value of b, the condition timpv 1 − bTd is
achieved. Thus, the total flight time of the missile can be obtained as
timp 1 − bTd bTd Td, which meets the guidance objective
timp → Td presented in Eq. (4).
D. Implementation of the Impact Time and Angle Guidance Strategy
In the previous subsections, the switching logic presented in
Eq. (22) guarantees the desired impact angle and zero miss distance
on the real target, and the optimal value of b obtained via the
optimization routine ensures the desired impact time on the real
target. In this subsection, the complete design of the ITAG strategy
using the virtual target approach against the real target is presented in
a block diagram, as shown in Fig. 6.
Therein, the IAG law shown in the region with a dashed
boundary is employed in stage 1 for intercepting the virtual target
with a desired impact angle at the specified impact time 1 − bTd
via an optimal b determined by the simplex optimization
algorithm. The PNG law presented in the region with a solid
boundary is used in stage 2 for intercepting the real target with a
desired impact angle at the expected impact time Td via the
switching logic in Eq. (22). Hence, the desired impact time and
angle on the real target can be achieved simultaneously with the
proposed ITAG strategy shown in Fig. 6. To further illustrate
the implementation procedure during the practical operation, a
pseudocode is presented in Algorithm 1.
With the ITAG strategy presented in Fig. 6 and implemented with
Algorithm 1, the missile is capable of intercepting a stationary target
as well as a moving target with the desired impact angle at a specified
impact time. Note that the error-prone estimate of time to go is
necessary for designing the ITAG law against a moving target in
[28,30,31,33]. However, the ITAG strategy proposed in this paper
does not rely on such information in spite of intercepting a moving
target.
E. Practical Modification of the Guidance Law
While operating the guidance law in practice, the time lag due to
the missile autopilot dynamics should be considered, which can be
modeled as a first-order transfer function. Mathematically,
Aa
M
1
1 TmS
AM (24)
where Aa
M denotes the achieved accelerations in practice,
and Tm 0.3 s is selected as the time constant of the
autopilot.
In addition to the autopilot dynamics, an important approximation
of the discontinuous function sign (s) in Eq. (16) to reduce chattering
should be considered while operating theguidance law in practice. As
shown in [40], the generated guidance command will be continuous
and smooth by replacing sign (s) with the following sigmoid
function:
sgmfs
s
jsj a
; a 0 (25)
where a 0.1 is selected in this work.
Furthermore, it can be noticed from Eq. (16) that AM → ∞ if
jϕMVj → π∕2. But recall that jϕMVj π∕2 is not a stable equilibrium
during the guidance procedure. Thus, the proposed guidance law can
be restricted by the following saturation function to avoid the infinite
guidance command and to satisfy the physical limit of the missile
acceleration as well:
AM
(
AM maxsignAM; if jAMj AM max
AM; if jAMj ≤ AM max
(26)
where AM max is the missile’s maximum allowable lateral
acceleration.
V. Numerical Simulations
In accordance with the previous sections, the performance of the
proposed ITAG strategy is validated through a series of numerical
simulations for the cases of intercepting a stationary target and a
constant-velocity target. In all the cases, the missile travels with a
speed of 250 m∕s while curving its trajectory toward the target
located 11 km away from the point of launch. The initial positions of
the missile and target are selected as (0, 0.5) and (11, 0) km,
respectively. The initial flight-path angle of missile is chosen as
30 deg. In addition, the upper bound of the missile latax is limited by
AM max 100 m∕s2
. To ensure the minimal miss distance, the final
Missile
Missile
Virtual
Target
Virtual
Target
Real
Target
Real
Target
Impact
Time Error
Impact
Time Error
Switching
Logic
Switching
Logic
IAG
Law
IAG
Law
PNG
Law
PNG
Law
Impact
Angle Error
Impact
Angle Error
Optimal
Value of b
Optimal
Value of b
Performance
Index
Performance
Index
Simplex
Algorithm
Simplex
Algorithm ζ
ε
J
e e
Eq. (16)
Eq. (22)
Stage I Stage II
Fig. 6 ITAG strategy via the virtual target approach.
Algorithm 1 Pseudocode for the proposed ITAG strategy
Input: The initial conditions; the terminal constraints: Td and λF; the guidance parameters: k, k1, α, N
Output: The miss distance, impact time, and impact angle on the real target
Step 0: Assign ζ 10−3 // The minimal tolerance on the impact angle error for the virtual target
Assign b ∈ 0; 1; // The tuning parameter for meeting the impact time constraint
Step 1: Calculate the guidance command from Eq. (16); // IAG law in stage 1
Step 2: while jεj ≥ 0.01 // The iteration will be terminated if this condition is not satisfied
Step 3: Find the optimal value of b via minimizing the performance index in Eq. (23);
Step 4: end while
Step 5: Store the optimal b, and update the position of the virtual target by Eq. (5) or Eq. (6);
Step 6: Employ the guidance command from Eq. (16); // IAG law in stage 1
Step 7: if jej ζ // The impact angle error in Stage I
Switch the guidance command to PNG law; // PNG law in stage 2
end if
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7. range of the missile and target is set to be less than 1 m. For all the
cases, the design parameters are set to k 1, α 1.2, k1 300,
N 3, while the values of b are given in the following subsections
under different simulation cases.
A. Various Impact Times and Impact Angles Against a Stationary
Target
The first set of simulations is conducted for the scenario when the
target is stationary. To verify the effectiveness of the proposed ITAG
strategy in this case, various desired impact times and impact angles
are considered as terminal constraints. The simulation results are
presented in Fig. 7 for selected impact times (50 and 55 s) and impact
angles (30, 45, 60, and 75 deg). The values of the guidance parameter
b and the guidance errors resulting from the autopilot lag under
various terminal constraints are presented in Table 1.
It can be observedfrom Fig. 7a that thevirtual target is farther away
from the real target if a smaller impact angle is required with the same
impact time. Meanwhile, it is clearly shown in Fig. 7b that larger
missile latax is generated with the growth of the desired impact time
and impact angle. Moreover, all the missile latax commands are
within the limits of the missile acceleration. By inspecting Fig. 7c, the
flight-path angles can meet the desired values at various desired
impact times. Note that the missile’s flight-path angle keeps invariant
after the virtual target (switching point) is met. It is shown in Fig. 7d
that the relative range can converge to zero at the desired impact
times. Additionally, it can be observed from Table 1 that guidance
errors on the impact time and angle due to the autopilot lag are very
small. They are all within an acceptable range, which shows the
robustness of the proposed ITAG law.
Recall that the relation between thevalueof b and the desired impact
time has been analytically discussed in Sec. IV.C. To further validate
such a relationship in the simulation, thevalues of b under a desired set
of impact times and angles are shown in Fig. 8 for the current
engagement geometry via the optimization routine presented
a) Flight trajectories b) Missile latex trends
c) Flight-path angle variations d) Relative range histories
Fig. 7 Simulation results for various impact times and impact angles against a stationary target.
Table 1 Values of b and guidance errors against a stationary target
Desired
impact time, s
Desired impact
angle, deg b
Impact time
error, s
Impact angle
error, deg
50 30 0.295 0.007 −0.23
50 45 0.065 0.013 −0.16
55 60 0.133 0.011 −0.57
55 75 0.062 0.024 −0.28
Fig. 8 Variations of b with various impact time (in seconds) and angle
(in degrees) contours.
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8. previously. Therein, the trends of b under the same impact time
(described as impact time counters) with various impact angles are
presented in solid lines, and the changing histories of b under the same
impact angle (described as impact angle counters) with various impact
times are plotted in dashed lines. The selected impact angles are
distributed between 15 and 75 deg, whereas the impact times vary
between 50 and 70 s. The values of b are shown from 0 to 0.6 for better
visibility. As expected, the higher the desired impact time is, the larger
the parameter b is, which validatesthe analysis inSec. IV.C. Moreover,
it is observed that b increases as the impact angle decreases. Another
observation is that the impact time and impact angle counters become
denser if the desired impact angle and time are smaller, respectively. It
shouldbe noted that the impactanglecounterwith20deg only covers a
small range of impact times (up to 56.7 s), which is reasonable because
the missile’s acceleration limit will be violated severely with too large
desired impact times but small impact angles.
B. Various Impact Times and Impact Angles Against a
Constant-Velocity Target
In this subsection, the proposed ITAG strategy is employed to
attack a constant-velocity moving target at various impact times and
impact angles. The velocity of the target is assumed as 10 m∕s
(corresponding to the typical speed of a fast moving modern warship,
roughly equal to 20 kt), and the path angle of the target is assumed as
60 deg. The values of b and the guidance errors resulting from the
autopilot lag under various terminal constraints for this case are
presented in Table 2.
The simulation results are presented in Fig. 9. Note that the
missile’s flight-path angle on the collision point is γM γTf − γimp;
then, the final flight-path angles are −30, −45, −70, and −85 deg,
respectively, according to the impact angle constraints listed in
Table 2. From Fig. 9a, it can be seen that the missile can intercept the
moving target with various flight trajectories to satisfy various impact
time and angle constraints via the proposed virtual target approach. In
Fig. 9b, it shows that larger missile acceleration is required with the
increase of the desired impact time and impact angle. By inspecting
Fig. 9c, the flight-path angles can meet the desired final values at the
desired impact times. Note that the missile’s flight-path angle keeps
invariant after the virtual target (switching point) is met for
intercepting a moving target. It is also shown in Fig. 9d that the
relative range can converge to zero at the desired impact times.
Similarly, it can be observed from Table 2 that the impact time error
and impact angle error due to the autopilot lag are still within an
acceptable small range, which has little impact on the guidance
performance.
C. Comparison Study
To further validate the proposed guidance strategy, it is compared
with the technique given in [31], which employs a switching logic
between an impact time guidance law and an impact angle guidance
law. Furthermore, the energy-optimal solution with desired terminal
constraints is used as a benchmark through minimizing the total
Table 2 Values of b and guidance errors against a constant-velocity
target
Desired
impact time, s
Desired impact
angle, deg b
Impact time
error, s
Impact angle
error, deg
50 90 0.166 0.008 −0.17
55 105 0.196 0.015 −0.24
60 130 0.134 0.019 −0.31
65 145 0.137 0.022 −0.35
a) Flight trajectories b) Missile latex trends
c) Flight-path angle variations d) Relative range histories
Fig. 9 Simulation results for various impact times and impact angles against a constant-velocity target.
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9. control effort (defined as 1∕2∫ A2
Mt dt). The optimal flight
trajectories under minimal energy consumption are obtained via the
MATLAB numerical optimization function fmincon, which is the
solver for constrained nonlinear optimal control problems. The
desired impact time is chosen as 50 s, and the expected impact angle
on the collision course is selected as 45 deg. Simulation results of the
proposed ITAG, the ITAG in [31], and the optimal energy solution
against a stationary target are presented in Fig. 10. The other
simulation conditions are chosen to be the same as the previous
settings in Sec. V.A.
It can be seen from Fig. 10 that all three guidance strategies can
intercept the stationary target at the desired impact time and impact
angle. In Fig. 10a, the guidance strategy in [31] starts to curve the
missile’s flight trajectory as soon as the missile is launched, whereas
the curves generated by the other two guidance laws change more
gradually. In Fig. 10b, larger missile acceleration is required under
the guidance strategy in [31] during the initial phase than the other
two guidance laws do. It can be also observed from Fig. 10b that the
missile spends no control effort under the proposed ITAG after the
virtual target is captured, whereas the other two still generate nonzero
guidance commands to meet the terminal constraints. Additionally, it
can be observed from Fig. 10c that the missile’s flight-path angle is
immediately curved to a large value after launch using the guidance
strategy in [31], whereas the proposed guidance strategy and the
optimal solution can both meet the terminal constraints without
creating large flight-path angles during the initial period. Besides, the
results generated by the proposed approach are closer to the optimal
solution.
To further investigate the control effort generated by the three
methods, the control energy corresponding to the ITAG strategy in
[31], the optimal solution, and the proposed approach are calculated
as 15,794, 1626, and 2388 m2
∕s3
, respectively. Thus, the proposed
ITAG strategy leads to much less control effort, which is closer to
the one generated from the optimal solution. Comparison studies
(omitted here for space limitation) are also conducted for the case of
intercepting a moving (nonmaneuvering) target with the desired
impact time and angle, and the same results can be observed, which
show that the proposed ITAG strategy performs better with lower
energy consumption.
Note that the preceding optimal solution is merely used as a
benchmark for energy comparison. It is not suitable for real-time
guidance law design because the nonlinear constrained optimization
requires intensive computations to find the optimal and feasible
solution.
D. Salvo Attack
The modern warship is always equipped with CIWS, which could
detect and even destroy the incoming missiles. To compromise the
CIWS, one possible way is to facilitate the salvo attack strategy with
impact angle constraints. Specifically, several missiles are launched
to attack the target simultaneously from different positions and
directions with specified impact angles. To this end, four missiles
a) Flight trajectories b) Missile latex trends
c) Flight-path angle variations d) LOS angle histories
Fig. 10 Simulation results for comparison study against a stationary target with γimp 45 deg, Td 50 s.
Table 3 Initial positions, terminal constraints, and values of b for
salvo attack
Missile
Initial
position, km
Desired impact
time, s
Desired impact
angle, deg b
1 (0, −10) 65 30 0.325
2 (0, 10) 65 90 0.730
3 (0, −5) 65 15 0.675
4 (0, 5) 65 105 0.608
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10. with the same initial flight-path angle and the same speed (selected as
0 deg and 250 m∕s, respectively) are chosen to intercept a constant-
velocity moving target (with the speed and path angle set to 10 m∕s
and 60 deg, respectively) simultaneously from different positions
with desired impact angles. The initial positions, terminal constraints,
and values of b for these missiles are given in Table 3.
The simulation results for salvo attack with impact angle constraints
against a moving target are given in Fig. 11. The flight trajectories of
the missiles are presented in Fig. 11a. It can be seen that all the missiles
can attack the target simultaneously with the desired impact angles
from different positions and directions. According to the impact angle
constraints listed in Table 3, the final flight-path angles for those
missiles are 30, −30, 45, and −45 deg, respectively. As shown in
Fig. 11b, the flight-path angles of all the missiles are curved in various
shapes to satisfy the impact time and angle constraints, whereas they
keep invariant after the virtual target is captured.
To sum up, the numerical results with comparison studies show
that the proposed ITAG strategy is effective and feasible for
intercepting the stationary target as well as the moving target with
desired impact times and angles. The IAG law employed in stage 1
enables the capture of the virtual target at the desired impact angle,
and the PNG law applied in stage 2 guarantees the desired impact
angle and zero-miss distance. The desired impact time is achieved by
selecting the initial location of the virtual target properly through
determining an optimal b with the optimization routine. The key
features of the proposed ITAG strategy are guaranteed satisfaction of
the terminal guidance constraints and less control effort via the
concept of virtual target and the switching logic of the two guidance
laws, without estimating the time-to-go information.
VI. Conclusions
In this work, the guidance problem with terminal impact time and
impact angle constraints against a stationary target and a moving
target is solved via a novel virtual target approach. The guidance
process is divided into two stages to address this problem by
introducing a virtual target located in the plane of the missile and
target. The switching logic between the nonsingular sliding-mode
impact angle guidance law and the proportional navigation guidance
law is employed to guarantee the desired impact angle on the virtual
target as well as the real target. The proper initial position of the
virtual target can be determined by an optimization routine to achieve
the desired impact time and angle on the real target simultaneously.
Compared with other impact time and angle guidance (ITAG) laws,
the proposed ITAG strategy not only demands less control effort but
also requires no information about the time to go. Extensive
simulation studies with comparisons show the superiority and
feasibility of the proposed ITAG strategy.
Acknowledgment
This work was supported partially by the National Natural Science
Foundation of China (project numbers 61522301 and 61633003); the
authors greatly appreciate the financial support. The authors would
also like to thank the associate editor and reviewers for their valuable
comments and constructive suggestions that helped to improve the
paper significantly.
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DOI:
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