Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile

Abstract

This paper proposes a synthesis of new guidance law to generate an evasive maneuver against enemy’s missile interception while considering its impact angle, acceleration, and field-of-view constraints. The first component of the synthesis is a new function of repulsive Artificial Potential Field to generate the evasive maneuver as a real-time dynamic obstacle avoidance. The terminal impact angle and terminal acceleration constraints compliance are based on Time-to-Go Polynomial Guidance as the second component. The last component is the Logarithmic Barrier Function to satisfy the field-of-view limitation constraint by compensating the excessive total acceleration command. These three components are synthesized into a new guidance law, which involves three design parameter gains. Parameter study and numerical simulations are delivered to demonstrate the performance of the proposed repulsive function and guidance law. Finally, the guidance law simulations effectively achieve the zero terminal miss distance, while satisfying an evasive maneuver against intercept missile, considering impact angle, acceleration, and field-of-view limitation constraints simultaneously.

Full text

Copyright ⓒ The Korean Society for Aeronautical & Space Sciences
Received: May 12,2017 Revised: September 7, 2017 Accepted: September 11, 2017
719 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480
Paper
Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)
DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719
Impact Angle Control Guidance Synthesis for Evasive Maneuver against
Intercept Missile
Y. H. Yogaswara*
Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
Department of Research and Development of Indonesian Air Force, Bandung 40174, Indonesia
Seong-Min Hong** and Min-Jea Tahk***
Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
Hyo-Sang Shin****
Cranfield University, Bedford MK43 0AL, United Kingdom
Abstract
is paper proposes a synthesis of new guidance law to generate an evasive maneuver against enemy’s missile interception
while considering its impact angle, acceleration, and eld-of-view constraints. e rst component of the synthesis is a new
function of repulsive Articial Potential Field to generate the evasive maneuver as a real-time dynamic obstacle avoidance.
e terminal impact angle and terminal acceleration constraints compliance are based on Time-to-Go Polynomial Guidance
as the second component. The last component is the Logarithmic Barrier Function to satisfy the field-of-view limitation
constraint by compensating the excessive total acceleration command. ese three components are synthesized into a new
guidance law, which involves three design parameter gains. Parameter study and numerical simulations are delivered to
demonstrate the performance of the proposed repulsive function and guidance law. Finally, the guidance law simulations
eectively achieve the zero terminal miss distance, while satisfying an evasive maneuver against intercept missile, considering
impact angle, acceleration, and eld-of-view limitation constraints simultaneously.
Key words: Missile guidance, Evasive maneuver, Impact angle control, Articial potential eld
1. Introduction
Since an Integrated Air Defense Systems (IADS) [1] have
been sophistically developed, a countermeasure action to
counteract the IADS becomes a signicant consideration in
a missile design. Surface attack missiles, which are launched
from air or surface platforms to attack designated surface
targets also need advanced solutions to respond the threats
of IADS. Guidance system design for surface attack missile in
this high threat environment is challenging since the attacking
missile must be delivered to its target, while maximizing
the survivability from the IADS. A proper guidance laws to
generate an evasive maneuver against intercept missiles
are rarely published in open literature. On the contrary,
the evasive maneuver of manned aircraft against intercept
missile has been studied extensively. ose evasive strategies
are based on continuously changing maneuver direction
such as barrel roll and the Vertical-S maneuver [2, p. 120] or
weaving maneuver [2, Ch. 27]. e strategies are not adequate
to be implemented into homing missiles or unmanned aerial
vehicles (UAVs) due to several reasons, i.e.: the movement can
be easily predicted, the maneuver might exceed seeker’s eld-
of-view (FOV) limitation, and the missile fails to satisfy zero
terminal miss distance.
This is an Open Access article distributed under the terms of the Creative Com-
mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-
nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc-
tion in any medium, provided the original work is properly cited.
*
Ph. D Student, Research Officer
**
Ph. D Student
***
Professor, Corresponding author: mjtahk@kaist.ac.kr
****
Associate Professor
(719~728)2017-97.indd 719 2017-12-29 오전 6:03:17

DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719
720
Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)
An optimal evasive maneuver for subsonic Anti-Ship
Missiles (ASM) has been studied against Close-in Weapon
System (CIWS) [3]. e CIWS is a common short-range IADS
for naval ships that are usually equipped with cannon as
the defensive weapon. Since the ASM moves with constant
acceleration at any moment, the CIWS is able to aim at the
predicted intercept position. In [3], the authors have dened
a time-varying weighted sum of the inverse of the aiming
errors as a cost function which to be minimized. Inspired
by the study above, a direct optimization technique using
Co-Evolutionary Augmented Lagrangian Method (CEALM)
was applied in [4], which the capturability analysis is proven
using Lyapunov-like approach. Both studies in [3] and [4]
generate a type of barrel roll evasive maneuver, however, a
large barrel roll maneuver could increase the miss distance
at the terminal time.
As studied in [5], the typical CIWS’s cannon system has
limitations due to single-threat-engagement capability and
re at a predicted intercept point of the threat’s trajectory.
is limitation motivates a new guidance law by using salvo
attack where multiple missiles are launched and guided to
arrive at a stationary target at the same desired time. e
guidance law adopting this idea has been rstly proposed by
Jeon et al. in [6] and called as Impact Time Control Guidance
(ITCG). To control the impact time, the ITCG suggests an
additional loop in a closed form solution based on the
linear formulation of traditional proportional navigation
guidance (PNG) law. e ITCG has been also proposed in
[7] by introducing a virtual leader approach. Sliding mode
control (SMC) method for ITCG has been adopted in [8]. A
Lyapunov-based law is the latest method proposed in [9] for
ITCG and simultaneous arrival.
Even though the ITCG has been broadly studied, the
concept of salvo attack for simultaneous impact time by
exploiting the limitation of CIWS is less eective. In fact,
if we may be allowed for a little exaggeration, it is not
applicable anymore. is knowledge is exposed since
CIWS or IADS introduces intercept missiles to enhance the
previous cannon system. e intercept missile has advanced
capability of multi-target engagement, longer intercept
range, and proximity warhead detonation for a higher
probability of kill. Regarding those new developments,
this paper proposes a new guidance law, which maximizes
survivability of an attack missile with respect to the threat of
advanced CIWS or IADS. In order to neutralize the threat of
intercept missile of enemy’s IADS, our attack missile must
have an evasive maneuver capability to counteract the
interception. In addition, the attack missile must satisfy an
impact angle constraint and zero terminal acceleration for
maximizing warhead detonation eect on the target. FOV
limitation of the attack missile is also considered to ensure
the seeker locks on the target.
Due to its elegant concept and simple computation, the
evasive maneuver of the attack missile in this paper is based
on the concept of Articial Potential Field (APF). e APF
was introduced by Krogh [10] and Khatib [11] for mobile
robots by dening functions of goal attractive potentials and
obstacle repulsive potentials. Inherent problems of APF were
systematically identied by Koren and Borenstein in [12]
due to a trap situations caused by local minima, no passage
between close space obstacle, and oscillation. In addition,
Goal Non-reachable with Obstacle Nearby (GNRON)
problem was also recognized and solved by Ge and Cui in
[13]. Modications of APF for moving obstacles have been
also proposed in [14]–[16]. Nevertheless, those modications
cannot be implemented to the missile problem since the
modied potential functions require a deceleration variable
to reduce velocity when facing the obstacle. Eventually,
inspired by Chen et al. in [17], this paper proposes a new
repulsive function that eective for the missile problem
and overcomes the GNRON problem at the same time.
e primary limitation of APF on trap situation due to
local minima will not be an issue in this evasive maneuver
scenario since the problem is one-on-one engagement with
relatively free dead-end trajectory, e.g. U-shaped obstacle.
Regarding the concept of APF, instead of the conventional
attractive potential function, this paper implements an
impact angle control guidance (IACG) method as the
attractive goal. e IACG attacks the weak spot of a target
to increase warhead detonation eect and maximize
its probability of kill towards the target. A generalized
formulation of energy minimization had been proposed in
[18] to achieve IACG. A PNG-based was proposed in [19]
for capturing all possible terminal impact angle. Lyapunov-
based pursuit guidance was introduced in [20] to reduce
the angle between the velocity and the distance vector. A
combination of dierential game and sliding mode control
were also proposed in [21]. e Time-to-go Polynomial
Guidance (TPG) as proposed by Lee et al. in [22] which
treat the guidance command as a function of time-to-go is
the most suitable to be adopted in this paper. In addition to
the impact angle constraint, the TPG also satises the zero
terminal acceleration to minimize the terminal angle-of-
attack (AOA) for eective impact angle, and zero terminal
lateral velocity to minimize zero eort miss.
In order to improve the practicality on the physical
concern of proposed guidance law, the limitation on
actuator’s acceleration command is considered in this paper.
Moreover, the seeker look angle is also limited to be inside the
FOV in order to ensure the missile seeker not losing its target.
(719~728)2017-97.indd 720 2017-12-29 오전 6:03:17

721
Y. H. Yogaswara Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile
http://ijass.org
Switching logic was implemented in [23] by switching the
guidance law from ITCG of [6] to the second law of constant
look angle guidance command, when the seeker look angle
exceeds the FOV limit. A rule of the cosine of a weighted
leading angle in the biased term was used in [24] to ensure
the ITCG was satised without violating the FOV constraint
during an engagement. Switching logic to constant seeker
look angle was also investigated in [25] for optimal IACG
of a missile with a strap-down seeker. Without applying the
switching logic, SMC was used in [26] to satisfy IACG by
implementing a control Integral Barrier Lyapunov Function
(iBLF) to design the reaching law. e implementation of
iBLF inspires this paper to apply a simpler internal penalty
function known as Logarithmic Barrier Function (LBF) to
limit the look angle.
is paper is organized as follows. Section 2 presents the
new proposed repulsive potential functions of APF with
its properties. In Section 3, the equations of motion, APF,
TPG, LBF and all corresponding problem formulation are
synthesized to generate a single acceleration command of
proposed guidance law. Guidance characteristics, analysis,
and solutions are presented in Section 4 by numerical
simulation at a particular missile engagement scenario. e
simulations successfully demonstrate the eectiveness of
the potential function and guidance law. Finally, concluding
remarks and future works are given in Section 5.
2. Proposed Repulsive Potential Function
e repulsive potential function that is implemented on
mobile robots, robot manipulators, and UAVs are broadly
based on the Khatib [11] or Ge and Cui [13]. However,
those repulsive potential functions are not suitable to be
used in missile engagement problem due to its potential
characteristics. e main properties of those functions is
a steep slope of repulsive potential, which the potential
values ascent sharply approaching the obstacle position.
is property is acceptable for mobile robots since it has the
capability of deceleration when approaching the obstacle,
stop the movement, and reroute its path. Conversely, the
missiles engagement scenario does not recognize those
capabilities.
e repulsive potential function for missile engagement
scenario needs a gradual ascent of potential approaching the
obstacle to anticipate the intercept in advance. e gradual
and gentle ascent of repulsive potential has been introduced
for UAV path planning by Chen et al. in [17]. Unfortunately,
this function does not consider the GNRON problem, which
drives a failure to achieve the goal position. e failure is
generated by shifting out the global minimum from the goal
position when the goal is within the inuence distance of the
obstacle.
A new repulsive potential function of APF is constructed
and proposed to generate a gradual ascent of potential
approaching the obstacle and solve the GNRON problem
at the same time. Assuming a generic problem of a point
masses vehicle, goal, and obstacle, which the vehicle moves
in a two-dimensional (2-D) space. e vehicle position in
the workspace is denoted by
5
problem formulation are synthesized to generate a single acceleration command of proposed guidance law.
Guidance characteristics, analysis,and solutions are presented in Section 4by numerical simulation at a
particular missile engagement scenario. The simulations successfully demonstrate the effectiveness of the
potential function and guidance law. Finally, concluding remarks and future works are given in Section 5.
2. Proposed Repulsive Potential Function
The repulsive potential function that is implemented on mobile robots, robot manipulators, and UAV s
are broadly based on the Khatib [11] or Ge and Cui [13]. However, those repulsive potential functionsare
not suitable to be used in missile engagement problem due to its potential characteristics. The main
properties of those functionsis a steep slope of repulsive potential,which the potential values ascent sharply
approaching the obstacle position. This property is acceptable for mobile robots since it has the capability of
deceleration when approaching the obstacle,stop the movement,and reroute its path. Conversely, the
missiles engagement scenario does not recognize those capabilities.
The repulsive potential function for missile engagement scenario needs a gradual ascent of potential
approaching the obstacle to anticipate the intercept in advance.The gradual and gentle ascent of repulsive
potential has been introduced for UAV path planning by Chen et al. in [17]. Unfortunately, this function does
not consider the GNRON problem, which drives a failure to achieve the goal position. The failure is
generated by shifting out the global minimum from the goal position when the goal is within the influence
distance of the obstacle.
A new repulsive potential function of APF is constructed and proposed to generate a gradual ascent of
potential approaching the obstacle and solve the GNRON problem at the same time. Assuming a generic
problem of a point masses vehicle, goal, and obstacle, which the vehicle movesin a two-dimensional (2-D)
space. The vehicle position in the workspace is denoted by
[ ]
T
AA
xy
=q
.The repulsive potential
( )
rep
Uq
at each vehicle position is defined by considering the relative distance of goal position into Chen’s equation
as follow
goal
, if
() 0, if
obst
d
obst o
rep
obst o
ed d d
Udd
ζ
e
−
≤

=>


q
(1)
. e repulsive
potential
5
problem formulation are synthesized to generate a single acceleration command of proposed guidance law.
Guidance characteristics, analysis,and solutions are presented in Section 4by numerical simulation at a
particular missile engagement scenario. The simulations successfully demonstrate the effectiveness of the
potential function and guidance law. Finally, concluding remarks and future works are given in Section 5.
2. Proposed Repulsive Potential Function
The repulsive potential function that is implemented on mobile robots, robot manipulators, and UAV s
are broadly based on the Khatib [11] or Ge and Cui [13]. However, those repulsive potential functionsare
not suitable to be used in missile engagement problem due to its potential characteristics. The main
properties of those functionsis a steep slope of repulsive potential,which the potential values ascent sharply
approaching the obstacle position. This property is acceptable for mobile robots since it has the capability of
deceleration when approaching the obstacle,stop the movement,and reroute its path. Conversely, the
missiles engagement scenario does not recognize those capabilities.
The repulsive potential function for missile engagement scenario needs a gradual ascent of potential
approaching the obstacle to anticipate the intercept in advance.The gradual and gentle ascent of repulsive
potential has been introduced for UAV path planning by Chen et al. in [17]. Unfortunately, this function does
not consider the GNRON problem, which drives a failure to achieve the goal position. The failure is
generated by shifting out the global minimum from the goal position when the goal is within the influence
distance of the obstacle.
A new repulsive potential function of APF is constructed and proposed to generate a gradual ascent of
potential approaching the obstacle and solve the GNRON problem at the same time. Assuming a generic
problem of a point masses vehicle, goal, and obstacle, which the vehicle movesin a two-dimensional (2-D)
space. The vehicle position in the workspace is denoted by
[ ]
T
AA
xy=q
.The repulsive potential
( )
rep
U
q
at each vehicle position is defined by considering the relative distance of goal position into Chen’s equation
as follow
goal
, if
() 0, if
obst
d
obst o
rep
obst o
ed d d
Udd
ζ
e
−
≤

=>


q
(1)
at each vehicle position is dened by
considering the relative distance of goal position into Chen’s
equation as follow
5
problem formulation are synthesized to generate a single acceleration command of proposed guidance law.
Guidance characteristics, analysis,and solutions are presented in Section 4by numerical simulation at a
particular missile engagement scenario. The simulations successfully demonstrate the effectiveness of the
potential function and guidance law. Finally, concluding remarks and future works are given in Section 5.
2. Proposed Repulsive Potential Function
The repulsive potential function that is implemented on mobile robots, robot manipulators, and UAV s
are broadly based on the Khatib [11] or Ge and Cui [13]. However, those repulsive potential functionsare
not suitable to be used in missile engagement problem due to its potential characteristics. The main
properties of those functionsis a steep slope of repulsive potential,which the potential values ascent sharply
approaching the obstacle position. This property is acceptable for mobile robots since it has the capability of
deceleration when approaching the obstacle,stop the movement,and reroute its path. Conversely, the
missiles engagement scenario does not recognize those capabilities.
The repulsive potential function for missile engagement scenario needs a gradual ascent of potential
approaching the obstacle to anticipate the intercept in advance.The gradual and gentle ascent of repulsive
potential has been introduced for UAV path planning by Chen et al. in [17]. Unfortunately, this function does
not consider the GNRON problem, which drives a failure to achieve the goal position. The failure is
generated by shifting out the global minimum from the goal position when the goal is within the influence
distance of the obstacle.
A new repulsive potential function of APF is constructed and proposed to generate a gradual ascent of
potential approaching the obstacle and solve the GNRON problem at the same time. Assuming a generic
problem of a point masses vehicle, goal, and obstacle, which the vehicle movesin a two-dimensional (2-D)
space. The vehicle position in the workspace is denoted by
[ ]
T
AA
xy=q
.The repulsive potential
( )
rep
Uq
at each vehicle position is defined by considering the relative distance of goal position into Chen’s equation
as follow
goal
, if
() 0, if
obst
d
obst o
rep
obst o
ed d d
U
dd
ζ
e
−
≤

=>


q
(1)
,
(1)
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x=q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 0246
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Position, x
Repulsive Pot ential, U
rep
e
=3,
ζ
=0.4 (Chen)
e
=2,
ζ
=0.4
e
=1,
ζ
=0.4
e
=1,
ζ
=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
,
(2)
where dobst, dgoal, d0, ε, and ζ are the minimal distance
between the vehicle and the obstacle, the distance between
the vehicle and the goal, the distance of inuence of the
obstacle, and both are positive design parameter gains,
respectively. is proposed function ensures the repulsive
potential approaches zero as the vehicle approaches the goal
and nally the goal position will be the global minimum of
total potential.
e eectiveness of the proposed repulsive potential
function is demonstrated in a case on one-dimensional
(1-D) space as shown in Fig. 1. e vehicle
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x
=
q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 02 4 6
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Position, x
Repulsive Pot ential, U
rep
e
=3,
ζ
=0.4 (Chen)
e
=2,
ζ
=0.4
e
=1,
ζ
=0.4
e
=1,
ζ
=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
is
moving along x-axis toward the goal
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x=q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 0246
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Position, x
Repulsive Pot ential, U
rep
e
=3,
ζ
=0.4 (Chen)
e
=2,
ζ
=0.4
e
=1,
ζ
=0.4
e
=1,
ζ
=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
while
avoiding the obstacle
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x=q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 02 4 6
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Position, x
Repulsive Pot ential, U
rep
e
=3,
ζ
=0.4 (Chen)
e
=2,
ζ
=0.4
e
=1,
ζ
=0.4
e
=1,
ζ
=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
. Assuming the distance
of inuence of the obstacle d0=6, the GNRON problem of the
predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the rst plot series. Since the goal position
near the obstacle, the generated repulsive potential is large
enough to create the non-reachable goal. is problem takes
place since the goal position is aected by the obstacle and
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x=q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 0 2 46
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Positi on, x
Repulsive Pot ential, U
rep
e=3, ζ=0.4 (Chen)
e=2, ζ=0.4
e=1, ζ=0.4
e=1, ζ=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
Fig. 1. Repulsive potential function in a 1-D space
(719~728)2017-97.indd 721 2017-12-29 오전 6:03:18

DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719
722
Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)
drive non-zero potential at the goal. Moreover, the potentials
are evenly distributed to the right and the left side of the
obstacle neglecting the goal. In the same case assumption,
the new proposed function shows signicant improvement
to handle the GNRON problem. e plot of three dierent
combinations of scaling gains maintains the minimum of
the potential at the goal position and the maximum of the
potential at the obstacle position. Furthermore, the scaling
gains show the freedom to control the properties of repulsive
potential. e higher value of ε, the higher peak value of
the potential. e higher value of ζ, the steeper ascent of
potential approaching the obstacle.
e corresponding repulsive force is given by the negative
gradient of the repulsive potential. According to Eq.(1), when
the vehicle is not at the goal, i.e.,
6
goal goal
obst obst
d
d
= −
= −
qq
qq
(2)
where
obst
d
,
goal
d
,
0
d
,
e
, and
ζ
are the minimal distance between the vehicle and the obstacle, the
distance between the vehicle and the goal,the distance of influence of the obstacle, and both are positive
design parameter gains, respectively. This proposed function ensures the repulsive potential approaches zero
as the vehicle approaches the goal and finally the goal position will be the global minimum of total potential.
The effectiveness of the proposed repulsive potential function is demonstrated in a case on one-
dimensional (1-D) space as shown in Fig. 1. The vehicle
[ ]
0
T
A
x=q
is moving along x-axis toward the
goal
[ ]
40
T
goal
=q
while avoiding the obstacle
[ ]
00
T
obst
=q
.Assuming the distance of influence of the
obstacle
0
6d=
, the GNRON problem of the predecessor function as mentioned by Chen et al. in [17] is
demonstrated in the first plot series.Since the goal position near the obstacle, the generated repulsive
potential is large enough to create the non-reachable goal.This problem takesplace since the goal position is
affected by the obstacle and drive non-zero potential at the goal. Moreover,the potentials are evenly
distributed to the right and the left side of the obstacle neglecting the goal. In the same case assumption, the
new proposed function shows significant improvement to handle the GNRON problem. The plot of three
different combinationsof scaling gains maintainsthe minimum of the potential at the goal position and the
maximum of the potential at the obstacle position. Furthermore, the scaling gains show the freedom to
control the properties of repulsive potential. The higher value of
e
, the higher peak value of the potential.The
higher value of
ζ
, the steeper ascent of potential approaching the obstacle.
-6 -4 -2 024 6
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Position, x
Repulsive Pot ential, U
rep
e
=3,
ζ
=0.4 (Chen)
e
=2,
ζ
=0.4
e
=1,
ζ
=0.4
e
=1,
ζ
=0.3
Obstacle
Goal
Fig. 1. Repulsive potential function in a 1-D space
The corresponding repulsive force is given by the negative gradient of the repulsive potential. According
to Eq.(1), when the vehicle is not at the goal, i.e.,
goal
≠qq
, the repulsive force is given by
the repulsive force
is given by
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
F
dd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
,
(3)
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
,(4)
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
,(5)
where
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
and
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
are unit vectors
pointing from the obstacle to the vehicle and from the
vehicle to the goal, respectively. ose unit vectors play an
important role since the
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
repulses the vehicle away from
the obstacle and the
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal
goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
attracts the vehicle towards the
goal.
To elaborate the properties of the force eld, the case on
Fig. 1 is developed into 2-D space, which the scaling gains
are dened as ε=20 and ζ=0.3. e repulsive potential eld
and repulsive force eld of the vehicle at every position in
a 2-D space are depicted in Fig. 2. e repulsive potential
eld keeps the goal as the global minima and the potential
peak at the obstacle. e repulsive potential force represents
the potential as a positive divergent vector eld outward the
obstacle and a negative divergent vector eld inward the
goal. Intuitively, the vehicle that aected by this vector eld
will be repulsed by the obstacle and attracted to the goal.
Regarding the implementation of the new repulsive
function into missile evasive maneuver, some nomenclatures
are adjusted. e vehicle of interest, the obstacle and the
goal are dened as the attack missile, the intercept missile,
and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario
as shown in the left illustration of Fig. 3. e friendly attack
missile has a constant velocity VA heading to enemy’s
stationary target while avoiding enemy’s intercept missile,
1
1. Affiliation’s postcode (우편번호) of 1st author:
Department of Research and Development of Indonesian Air Force, Bandung 40174, Indonesia
2. Affiliation’s postcode (우편번호) of 4th author:
Cranfield University, Bedford, MK43 0AL, United Kingdom
3. Change right figure of Fig. 3 (simplified vector):
Fig. 3. Guidance geometry (left) and guidance synthesis of acceleration command vector (right)
Fig. 3. Guidance geometry (left) and guidance synthesis of acceleration command vector (right)
7
( )
() ()
, if
() 0, if
rep rep
repObst obst repGoal goal obst o
rep
obst o
FU
F F dd
Fdd
= −∇
+≤

=>

qq
nn
q
(3)
obst
d
repObst goal
F de
ζ
eζ
−
=
(4)
obst
d
repGoal
Fe
ζ
e
−
=
(5)
where
obst obst
d= ∇n
and
goal goal
d= −∇n
are unit vectors pointing from the obstacle to the vehicle and from
the vehicle to the goal, respectively. Those unit vectors play an important role since the
obst
n
repulses the
vehicle away from the obstacle and the
goal
n
attracts the vehicle towards the goal.
To elaborate the properties of the force field, the case on Fig. 1 is developed into 2-D space, which the
scaling gains are defined as
20
e
=
and
0.3
ζ
=
. The repulsive potential field and repulsive force field of
the vehicle at every position in a 2-D space are depicted in Fig. 2. The repulsive potential field keeps the goal
as the global minima and the potential peak at the obstacle.The repulsive potential force represents the
potential as apositive divergent vector field outward the obstacle and a negative divergent vector field
inward the goal. Intuitively, the vehicle that affected by this vector field will be repulsed by the obstacle and
attracted to the goal.
-6 -4 -2 0246-10
0
10
0
10
20
30
40
50
60
70
80
90
Y
X
Repulsive Pot ential, U
rep
Obstacle
Goal
-4 -3 -2 -1 0 1 2 3 4 5
-4
-3
-2
-1
0
1
2
3
4
X
Y
Repulsive Force
Obstacle
Goal
Fig. 2. Repulsive potential field (left) and repulsive force field (right) in a 2-D space
Regarding the implementation of the new repulsive function into missile evasive maneuver, some
nomenclatures are adjusted. The vehicle of interest, the obstacle and the goal are defined as the attack missile,
the intercept missile, and the target, respectively.
3. Guidance Synthesis
Consider a two-dimensional homing guidance scenario as shown in the left illustration of Fig. 3. The
Fig. 2. Repulsive potential eld (left) and repulsive force eld (right) in a 2-D space
(719~728)2017-97.indd 722 2017-12-29 오전 6:03:25

723
Y. H. Yogaswara Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile
http://ijass.org
which has a constant velocity VI. Acceleration command a
is perpendicular to velocity vector to change the ight path
angle γ of each missile. e position of the attack missile, the
intercept missile, and the target are denoted as (xA, yA), (xI, yI),
and (xT, yT), respectively. eir relationships are denoted as
follows; relative range R(.), relative velocity V(.), line-of-sight
(LOS) angle σ(.), and seeker look angle λ(.). e subscripts 0,
f, A, I, TA, IA denote the initial time, terminal time, attack,
intercept, relationship of the attack missile regarding the
target, and the attack missile regarding the interceptor,
respectively.
e equation of motion in this homing problem for both
attack and intercept missile in inertial frame are generally
given by
8
friendly attack missile has a constant velocity
A
V
heading to enemy’s stationary target while avoiding
enemy’s intercept missile, which has a constant velocity
I
V
. Acceleration command
a
is perpendicular to
velocity vector to change the flight path angle
γ
of each missile. The position of the attack missile, the
intercept missile, and the target are denoted as
( )
,
AA
xy
,
( )
,
II
xy
, and
( )
,
TT
xy
, respectively. Their
relationships are denoted as follows; relative range
( )
R⋅
, relative velocity
( )
V
⋅
, line-of-sight (LOS) angle
( )
σ
⋅
,
and seeker look angle
( )
λ
⋅
.The subscripts
0, , , , ,f A I TA IA
denote the initial time, terminal time, attack,
intercept, relationship of the attack missile regarding the target, and the attack missile regarding the
interceptor, respectively.
Fig. 3. Guidance geometry (left) and guidance synthesis of acceleration command vector (right)
The equation of motion in this homing problem for both attack and intercept missile in inertial frame are
generally given by
cos
sin
dx
V
dt
dy V
dt
da
dt V
γ
γ
γ
=
=
=
(6)
With its boundary conditions are defined as follows:
( ) ( ) ( )
( ) ( ) ( )
00 0 0 00
ff f f ff
xt x yt y t
xt x yt y t
γγ
γγ
= = =
= = =
The guidance law is derived by using small angle approximation of AOA,that the velocity vector and body
orientation nearly have the same value. Hence, the seeker look angle of attack missile toward the target can
,
(6)
With its boundary conditions are dened as follows:
8
friendly attack missile has a constant velocity
A
V
heading to enemy’s stationary target while avoiding
enemy’s intercept missile, which has a constant velocity
I
V
. Acceleration command
a
is perpendicular to
velocity vector to change the flight path angle
γ
of each missile. The position of the attack missile, the
intercept missile, and the target are denoted as
( )
,
AA
xy
,
( )
,
II
xy
, and
( )
,
TT
xy
, respectively. Their
relationships are denoted as follows; relative range
( )
R⋅
, relative velocity
( )
V
⋅
, line-of-sight (LOS) angle
( )
σ
⋅
,
and seeker look angle
( )
λ
⋅
.The subscripts
0, , , , ,f A I TA IA
denote the initial time, terminal time, attack,
intercept, relationship of the attack missile regarding the target, and the attack missile regarding the
interceptor, respectively.
Fig. 3. Guidance geometry (left) and guidance synthesis of acceleration command vector (right)
The equation of motion in this homing problem for both attack and intercept missile in inertial frame are
generally given by
cos
sin
dx V
dt
dy V
dt
da
dt V
γ
γ
γ
=
=
=
(6)
With its boundary conditions are defined as follows:
( ) ( ) ( )
( ) ( ) ( )
00 0 0 00
ff f f ff
xt x yt y t
xt x yt y t
γγ
γγ
= = =
= = =
The guidance law is derived by using small angle approximation of AOA,that the velocity vector and body
orientation nearly have the same value. Hence, the seeker look angle of attack missile toward the target can
.
e guidance law is derived by using small angle
approximation of AOA, that the velocity vector and body
orientation nearly have the same value. Hence, the seeker
look angle of attack missile toward the target can be
approximated as subtraction of ight path and LOS angle
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ σ γ

= + −− − + − − −


(10)
.(7)
Recalling the right illustration of Fig. 3, the total
acceleration command of the guidance law
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ σ γ

= + −− − + − − −


(10)
,
which a synthesis of three components is simply
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ σ γ

= + −− − + − − −


(10)
.(8)
is formulation is described as follow. Principally,
the guidance synthesis implements the APF concept
by dening the attractive potential to be achieved and
repulsive potential to be avoided. Rather than implementing
classical attractive force function of APF, the acceleration
command of TPG aTPG is preferred to achieve zero terminal
miss distance at designated terminal impact angle and
zero terminal acceleration. e new proposed repulsive
force function of APF performs as the second component
to generate the evasive maneuver. Recalling Eq. (3), the
acceleration command of APF is taken from the proposed
repulsive potential force aAPF=Frep(q). Once the intercept
missile approaching, the repulsive force is generated as
aAPF and producing a new resultant vector of (aTPG+aAFP)
avoiding the interceptor. Finally, acceleration command of
LBF aLBF is also generated when the seeker look angle close
to its FOV limit by compensating the exceeding acceleration
command. rough this model, the synthesized guidance
law propose a responsive approach to achieve an eective
evasive maneuver while satisfying zero miss distance,
terminal impact angle, zero terminal acceleration, and FOV
limitation.
3.1 Time-to-Go Polynomial Guidance
e TPG demonstrates an eective IACG not only its
ability to satises the terminal impact angle, but also satises
the zero terminal acceleration to minimize the terminal AOA
for precise impact angle, and zero terminal lateral velocity
to minimize zero eort miss. Recalling the TPG on [22], the
missile acceleration command aTPG and the estimation of
time-to-go tgo for the curved path of the attack missile can be
formulated as follows:
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( )
( )( )
2 2 3 11
A
TPG A
f
go
V
at m n tmn tm n
t
σ
γγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ σ γ

= + −− − + − − −


(10)
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A
f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ σ γ

= + −− − + − − −


(10)
,
(9)
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
2
22
11
1 23
22
1
go
Af TAf Af TAf
A
R
tP P P
V
γγ σ
γ γγ σ γ

= + −− − + − − −


(10)
9
be approximated as subtraction of flight path and LOS angle
λγσ
= −
(7)
Recalling the right illustration of Fig. 3, the total acceleration command of the guidance law
A SYN
aa=
, which a synthesis of three components is simply
LBF
SYN TPG AFP
aaaa=++
(8)
This formulation is described as follow. Principally, the guidance synthesis implementsthe APF concept by
defining the attractive potential to be achieved and repulsive potential to be avoided. Rather than
implementing classical attractive force function of APF, the acceleration command of TPG
TPG
a
is
preferred to achieve zero terminal miss distance at designated terminal impact angle and zero terminal
acceleration. The new proposedrepulsive force function of APF performs as the second component to
generate the evasive maneuver.Recalling Eq. (3), the acceleration command of APF is taken from the
proposed repulsive potential force
( )
APF rep
aF=q
.Once the intercept missile approaching, the repulsive
force is generated as
APF
a
and producing a new resultant vector of
( )
TPG AFP
aa+
avoiding the interceptor.
Finally, acceleration command of LBF
LBF
a
is also generated when the seeker look angle close to its FOV
limit by compensating the exceeding acceleration command.Through this model, the synthesized guidance
law propose a responsive approach to achieve an effective evasive maneuver while satisfying zero miss
distance, terminal impact angle, zero terminal acceleration, and FOV limitation.
3.1 Time-to-Go Polynomial Guidance
The TPG demonstrates an effective IACG not only its ability to satisfies the terminalimpact angle, but
also satisfies the zero terminal acceleration to minimize the terminal AOA for precise impact angle, and zero
terminal lateral velocity to minimize zero effort miss. Recalling the TPG on [22], the missile acceleration
command
TPG
a
and the estimation of time-to-go
go
t
for the curved path of the attack missile can be
formulated as follows:
( ) ( )( ) ( ) ( ) ( ) ( )( )
2 2 3 11
A
TPG A f
go
V
at m n tmn tm n
t
σγγ

=− −+ + +++ ++ +

(9)
( ) ( )
( )
( ) ( )
{ }
222
11
1 23
22
1
go Af TAf Af TAf
A
R
tP P P
V
γγ σγ γγ
σ γ

= + −− − + − − −


(10)
,
(10)
10
( )( )( )
( )( )
1
2
3
1
2 32 3 3
22
33
22
Pm n mn
Pm n
Pm n
=+ + ++
=++
  
=++
  
  
(11)
where mand ndenote the guidance gains which are chosen to be any positive real values following
0nm>>
for zero terminal acceleration.If
1n=
and
0m=
, the performance of applied TPG results are
identical to the optimal guidance laws with terminal impact constraints, but without zero terminal
acceleration as studied in [18]. Higher values of mand ngains will not only satisfy desired impact angle but
also produce the zero terminal acceleration to avoid saturating commands and sufficiently small terminal
AOA to increase the lethality of the warhead.
On the other hand, the enemy’s intercept missile applies a Pure Proportional Navigation (PPN) in order
to intercept the attack missile. The PPN is chosen due to its natural characteristics in apractical sense as
concluded by Shukla and Mahapatra in [27].Referring to the literature, the acceleration command for the
intercept missile is formulated as
II IA
AI AI
IA
AI AI
a NV
RV
RR
σ
σ
= −
×
=


�
(12)
where the navigation constant is defined as N = 3.
3.2 Logarithmic Barrier Function
In areal application, the target should be located inside the FOV of the attack missile, and it is important
to keep the seeker look angle from exceeding the limitation. Regarding the look angle on conventional PNG
that decreases to zero as the missile approaches its target, the proposed guidance law is intended to achieve
an additional capability. This capability generates an uncommon trajectory that increases the look angle up to
exceeds the FO V. When the missile fails to lock on the target, it leads the missile into ahuge miss distance
and unsatisfied constraints at the terminal phase. Introducing the FOV limit as the barrier band the barrier
parameter
µ
, the final component in Eq. (8) can be easily derived by using LBF. Implementing the
characteristics of LBF into a compensated acceleration command ensures the command increases to
actuator’s limitation
max
a
as the current look angle approaching the barrier and keep the seeker look angle
,
(11)
where m and n denote the guidance gains which are chosen
to be any positive real values following n>m>0 for zero
terminal acceleration. If n=1 and m=0, the performance of
applied TPG results are identical to the optimal guidance
laws with terminal impact constraints, but without zero
terminal acceleration as studied in [18]. Higher values of m
and n gains will not only satisfy desired impact angle but also
produce the zero terminal acceleration to avoid saturating
commands and suciently small terminal AOA to increase
the lethality of the warhead.
On the other hand, the enemy’s intercept missile applies
a Pure Proportional Navigation (PPN) in order to intercept
the attack missile. e PPN is chosen due to its natural
characteristics in a practical sense as concluded by Shukla
and Mahapatra in [27]. Referring to the literature, the
acceleration command for the intercept missile is formulated
as
(719~728)2017-97.indd 723 2017-12-29 오전 6:03:26

DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719
724
Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)
2
4. Image file of equation 12 and 13:
5. Change right figure of Fig. 5 (change plotted variable):
Fig. 5. Seeker look angle (left), flight path angle (center), and acceleration command (right)
010 20 30 40 50 60
-40
-20
0
20
40
60
Time t, sec
Look Angle

, deg
Seeker Look Angle (

o
=

f
= -30
o
)
Non-Evasive

=90,

=3e
-4
,

= 0

=90,

=3e
-4
,

= 5

=90,

=3e
-4
,

=10
FOV =

lim
=

40
o
010 20 30 40 50 60 70
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path Angle

, deg
Flight Path Angle (

o
=

f
= -30
o
)
Non-Evasive

=90,

=3e
-4
,

= 0

=90,

=3e
-4
,

= 5

=90,

=3e
-4
,

=10

o
=

f
= -30
o
010 20 30 40 50 60
-80
-60
-40
-20
0
20
40
60
80
100
X: 58.14
Y: 0.06908
Time, sec
a
com
, m/s
2
Acceleration Command of Guidance Sy nthesis (

=90,

=3e
-4
,

=10 )
a
TPG
a
APF
a
LBF
a
SYN
a
lim
=

30 m/s
2
,
(12)
where the navigation constant is dened as N = 3.
3.2 Logarithmic Barrier Function
In a real application, the target should be located inside
the FOV of the attack missile, and it is important to keep the
seeker look angle from exceeding the limitation. Regarding
the look angle on conventional PNG that decreases to zero
as the missile approaches its target, the proposed guidance
law is intended to achieve an additional capability. is
capability generates an uncommon trajectory that increases
the look angle up to exceeds the FOV. When the missile fails
to lock on the target, it leads the missile into a huge miss
distance and unsatised constraints at the terminal phase.
Introducing the FOV limit as the barrier b and the barrier
parameter μ, the nal component in Eq. (8) can be easily
derived by using LBF. Implementing the characteristics of
LBF into a compensated acceleration command ensures the
command increases to actuator’s limitation alim as the current
look angle approaching the barrier and keep the seeker look
angle inside the FOV in a simple way. Acceleration command
in component of LBF can be formulated as
2
4. Image file of equation 12 and 13:
5. Change right figure of Fig. 5 (change plotted variable):
Fig. 5. Seeker look angle (left), flight path angle (center), and acceleration command (right)
010 20 30 40 50 60
-40
-20
0
20
40
60
Time t, sec
Look Angle

, deg
Seeker Look Angle (

o
=

f
= -30
o
)
Non-Evasive

=90,

=3e
-4
,

= 0

=90,

=3e
-4
,

= 5

=90,

=3e
-4
,

=10
FOV =

lim
=

40
o
010 20 30 40 50 60 70
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path Angle

, deg
Flight Path Angle (

o
=

f
= -30
o
)
Non-Evasive

=90,

=3e
-4
,

= 0

=90,

=3e
-4
,

= 5

=90,

=3e
-4
,

=10

o
=

f
= -30
o
010 20 30 40 50 60
-80
-60
-40
-20
0
20
40
60
80
100
X: 58.14
Y: 0.06908
Time, sec
a
com
, m/s
2
Acceleration Command of Guidance S ynthesis (

=90,

=3e
-4
,

=10 )
a
TPG
a
APF
a
LBF
a
SYN
a
lim
=

30 m/s
2
.
(13)
Finally, by substituting Eqs. (3), (9), and (13) into (8), the
total acceleration command of attack missile is synthesized
as a new proposed guidance law.
3.3 Survivability of Attack Missile
e survivability or Probability of Survival Ps of the
attack missile is important to be quantied to measure the
eectiveness of the proposed repulsive potential function and
guidance law. As introduced in [28, Ch. 1], the survivability
is the complement of the killability or Probability of Kill Pk
which quanties the probability of the aircraft being killed.
us, the relationship can be formulated as
11
inside the FOV in a simple way.Acceleration command in component of LBF can be formulated as
( )
( )
( )
max
o
o
if
log if 5
0if 5
LBF
ab
a b bb
b
λ
µλ λ
λ
≥


=− − >≥ −

<−


. (13)
Finally, by substituting Eqs. (3), (9), and (13) into (8), the total acceleration command of attack missile is
synthesized as a new proposed guidance law.
3.3 Survivability of Attack Missile
The survivability or Probability of Survival
S
P
of the attack missile is important to be quantified to
measure the effectiveness of the proposed repulsive potential function and guidance law. As introduced in
[28, Ch. 1], the survivability is the complement of the killability or Probability of Kill
K
P
which quantifies
the probability of the aircraft being killed. Thus, the relationship can be formulated as below.
1
K
S
PP= −
(14)
From apractical point of view, the killability
K
P
is a complex function,which is explained in detail in the
reference [29, Ch. 6.3]. The intercept missilestypically use a fragmentation warhead,which at a particular
distance to its target,detonates the explosive charge and breaks up the warhead case into smaller accelerating
fragments. The killability for this fragmentation warhead is defined as a function of five parameters, i.e.: the
presented target area, the vulnerable target area,the total number of fragments, spray angle of fragments, and
miss distance at detonation.Recalling the reference, to achieve
0.9
K
P=
, the number of hits on a target
with a vulnerable area is minimum 10% of the presented area, which needs aminimum of 22 hits from the
total fragmentation at the target.
Since all parameters in the reference are given for a rocket baseline against a drone aircraft, those
assumptions can be partially adopted into this paper. The first assumption is that all the five parameters are
known to achieve
0.9
K
P=
. Secondly, the function of the killability will only depend on miss distance at
detonation
det
d
.Finally, following the typical relation trends of
K
P
and
det
d
, a simple relation is
introduced as
.(14)
From a practical point of view, the killability Pk is a
complex function, which is explained in detail in the
reference [29, Ch. 6.3]. e intercept missiles typically use
a fragmentation warhead, which at a particular distance
to its target, detonates the explosive charge and breaks up
the warhead case into smaller accelerating fragments. e
killability for this fragmentation warhead is dened as a
function of ve parameters, i.e.: the presented target area,
the vulnerable target area, the total number of fragments,
spray angle of fragments, and miss distance at detonation.
Recalling the reference, to achieve Pk=0.9, the number of hits
on a target with a vulnerable area is minimum 10% of the
presented area, which needs a minimum of 22 hits from the
total fragmentation at the target.
Since all parameters in the reference are given for a rocket
baseline against a drone aircraft, those assumptions can
be partially adopted into this paper. e rst assumption
is that all the ve parameters are known to achieve Pk=0.9.
Secondly, the function of the killability will only depend on
miss distance at detonation ddet. Finally, following the typical
relation trends of Pk and ddet, a simple relation is introduced
as
12
0
2
det
exp
K
d
d
PR



= −





(15)
where
0
d
R
is defined as the reference distance. In order to achieve the
0.9
K
P=
at
det
6md=
,
0
18.5 m
d
R=
is chosen to fit the curve.
4. Numerical Analysis
4.1 Performance of Proposed Guidance Law
Let us consider a typical Suppression of Enemy Air Defenses (SEAD) mission ofour attack missile with
constant velocity. The attack missile is in its terminal phase maintaining the constant velocity and terminal
impact angle towards the stationary target. The defending enemy’s intercept missile is planted in front of the
target facing the attack missile. The intercept missile will be launched at arange where the attack missile can
be intercepted and neutralized in constant and 10% faster velocity than the attack missile.The intercept
missile has adesignated time of flight due to propellant burn time limitation and,if the time exceeds, it is
considered that the intercept missile fails to neutralize the attack missile.To implement the proposed
guidance law, the position of intercept missile and target are assumed to be clearly detected by the attack
missile as a result of active radar or passive seeker detection. Aparameter study is carried out to elaborate the
performance of evasive maneuver due to design parametersgains
[ ]
,,
eζµ
. The quantities of each
parameter for the SEAD engagement scenario are listed in Table 1.
,
(15)
where
12
0
2
det
exp
K
d
d
PR



= −




(15)
where
0
d
R
is defined as the reference distance. In order to achieve the
0.9
K
P=
at
det
6md=
,
0
18.5 m
d
R=
is chosen to fit the curve.
4. Numerical Analysis
4.1 Performance of Proposed Guidance Law
Let us consider a typical Suppression of Enemy Air Defenses (SEAD) mission ofour attack missile with
constant velocity. The attack missile is in its terminal phase maintaining the constant velocity and terminal
impact angle towards the stationary target. The defending enemy’s intercept missile is planted in front of the
target facing the attack missile. The intercept missile will be launched at arange where the attack missile can
be intercepted and neutralized in constant and 10% faster velocity than the attack missile.The intercept
missile has adesignated time of flight due to propellant burn time limitation and,if the time exceeds, it is
considered that the intercept missile fails to neutralize the attack missile.To implement the proposed
guidance law, the position of intercept missile and target are assumed to be clearly detected by the attack
missile as a result of active radar or passive seeker detection. Aparameter study is carried out to elaborate the
performance of evasive maneuver due to design parametersgains
[ ]
,,
eζµ
. The quantities of each
parameter for the SEAD engagement scenario are listed in Table 1.
is dened as the reference distance. In order to
achieve the Pk=0.9 at ddet=6m,
12
0
2
det
exp
K
d
d
PR



= −




(15)
where
0
d
R
is defined as the reference distance. In order to achieve the
0.9
K
P=
at
det
6md=
,
0
18.5 m
d
R=
is chosen to fit the curve.
4. Numerical Analysis
4.1 Performance of Proposed Guidance Law
Let us consider a typical Suppression of Enemy Air Defenses (SEAD) mission ofour attack missile with
constant velocity. The attack missile is in its terminal phase maintaining the constant velocity and terminal
impact angle towards the stationary target. The defending enemy’s intercept missile is planted in front of the
target facing the attack missile. The intercept missile will be launched at arange where the attack missile can
be intercepted and neutralized in constant and 10% faster velocity than the attack missile.The intercept
missile has adesignated time of flight due to propellant burn time limitation and,if the time exceeds, it is
considered that the intercept missile fails to neutralize the attack missile.To implement the proposed
guidance law, the position of intercept missile and target are assumed to be clearly detected by the attack
missile as a result of active radar or passive seeker detection. Aparameter study is carried out to elaborate the
performance of evasive maneuver due to design parametersgains
[ ]
,,
eζµ
. The quantities of each
parameter for the SEAD engagement scenario are listed in Table 1.
=18.5m is chosen to t the
curve.
4. Numerical Analysis
4.1 Performance of Proposed Guidance Law
Let us consider a typical Suppression of Enemy Air
Defenses (SEAD) mission of our attack missile with
constant velocity. e attack missile is in its terminal phase
maintaining the constant velocity and terminal impact
angle towards the stationary target. e defending enemy’s
intercept missile is planted in front of the target facing the
attack missile. e intercept missile will be launched at
a range where the attack missile can be intercepted and
neutralized in constant and 10% faster velocity than the
attack missile. e intercept missile has a designated time
of ight due to propellant burn time limitation and, if the
time exceeds, it is considered that the intercept missile fails
to neutralize the attack missile. To implement the proposed
guidance law, the position of intercept missile and target
are assumed to be clearly detected by the attack missile
as a result of active radar or passive seeker detection. A
parameter study is carried out to elaborate the performance
of evasive maneuver due to design parameters gains [ε, ζ, μ].
e quantities of each parameter for the SEAD engagement
scenario are listed in Table 1.
(719~728)2017-97.indd 724 2017-12-29 오전 6:03:27

725
Y. H. Yogaswara Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile
http://ijass.org
As seen in Fig. 4, the performance of evasive maneuver
is highly sensitive to the gains [ε, ζ, μ]. By dening LBF gain
μ=0, the performances of APF gains [ε, ζ] are evaluated on
the left and center of the Fig. 4. Recalling the properties of
the repulsive potential eld, the steepness of the potential is
veried by an early evasive maneuver when the attack missile
enters the distance of inuence radius d0. With respect to
the non-evasive maneuver, the evasive deviation width is
determined by the value of the APF gains [ε, ζ]. Starting from
ε≥70 with constant ζ, the attack missile is able to escape the
interception and gives wider evasive trajectories for higher
values. At a constant ε=90, wider evasive trajectories are also
demonstrated for higher values of ζ. Wide evasive trajectory
maximizes the escape of the attack missile, but intuitively
generates a huge ight path angle and exceeding FOV. For
the values of ζ ≥3e-4, the guidance law needs a compensation
command to suppress the ight path angle to ensure its
seeker look angle inside FOV limit. Right graph of Fig. 4
illustrates the eect for dierent values of LBF gain μ which
higher value of μ suppresses the guidance of attack missile
for narrower path trajectory. Regarding this parameter study,
gain combinations of ε=[90,100], ζ=3e-4, μ=[5,10] are picked for
further analysis since it complies with the initial requirement
to achieve zero miss distance and escaping an interception.
Elaboration of look angle λ suppression, ight path angle
γ, and acceleration command aSYN constraints are time
series plotted in Fig. 5. Once the look angle approaching
the barrier value λlim=±40o, the LBF gains μ=[5,10] generate a
suppression command to keep the look angle inside the FOV.
e higher value of gain, the more suppression applied to
the look angle. However, gain values μ≤2 in this scenario are
not adequate to maintain the look angle from violating the
limit. From the parametric study, the value μ=10 is a rational
choice to satisfy both constraints and will be used for further
examination. Depressed look angle when approaching
FOV limit aects the ight path angle correspondingly. By
implementing the IACG of TPG, the impact angle constraint
is satised even though its basic guidance law is combined
with other components by guidance synthesis. All impact
angle histories are relaxed into impact angle constraint γ0=-
14
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-1
0
1
2
3
4
5
6
7
8
Crossrange, km
Downrange, km
Trajectory of Evasive Maneuver (Parameter Study of Gain e )
Non-Evasive
e= 70, ζ=1e
-4
,µ=0
e= 90, ζ=1e
-4
,µ=0
e=110, ζ=1e
-4
,µ=0
Intercept Miss ile
Intercept Plat form
Stationary Target
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-1
0
1
2
3
4
5
6
7
8
Crossrange, km
Downrange, km
Trajectory of Evasive Maneuver (Parameter Study of Gain ζ )
Non-Evasive
e=90, ζ=1e-4,µ=0
e=90, ζ=2e-4,µ=0
e=90, ζ=3e-4,µ=0
e=90, ζ=4e-4,µ=0
Intercept Miss ile
Intercept Plat form
Stationary Target
-6 -5 -4 -3 -2 -1 012345
-1
0
1
2
3
4
5
6
7
8
Crossrange, km
Downrange, km
Trajectory of Evasive Maneuver (Parameter Study of Gain
µ
)
Non-Evasive
e
=90,
ζ
=3e
-4
,
µ
=2
e
=90,
ζ
=3e
-4
,
µ
=5
e
=90,
ζ
=3e
-4
,
µ
=10
e
=90,
ζ
=3e
-4
,
µ
=15
Intercept Miss ile
Intercept Plat form
Stationary Target
Fig. 4. Parameter study of APF gain
e
(left), APF gain
ζ
(center), and LBF gain
µ
(right)
Elaboration of look angle
λ
suppression, flight path angle
γ
, and acceleration command
SYN
a
constraints
are time series plotted in Fig. 5. Once the look angle approaching the barrier value
o
lim
40
λ
= ±
, the LBF
gains
[ ]
5, 10
µ
=
generate a suppression command to keep the look angle inside the FOV. The higher value
of gain, the more suppression applied to the look angle. However, gain values
2
µ
≤
in this scenario are
not adequate to maintain the look angle from violating the limit. From the parametric study, the value
10
µ
=
is a rational choice to satisfy both constraints and will be used for further examination. Depressed
look angle when approaching FOV limit affects the flight path angle correspondingly. By implementing the
IACG of TPG, the impact angle constraint is satisfied even though its basic guidance law is combined with
other components by guidance synthesis. All impact angle histories are relaxed into impact angle constraint
o
30
o
γ
= −
, and the look angle converges to zero. Recalling Fig. 5, components of acceleration command
express the synthesis of the new guidance laws for the gain setting
4
90, 3 , 10e
eζ µ
−
= = =
.The synthesized
total attack command
SYN
a
effectively satisfies the terminal zero acceleration of TPG and keeps inside
actuator’s acceleration limit
2
max
30a ms
−
= ±
.
010 20 30 40 50 60
-40
-20
0
20
40
60
Time t, sec
Look Angle
λ
, deg
Seeker Look Angle (
γ
o
=
γ
f
= -30
o
)
Non-Evasive
e
=90,
ζ
=3e
-4
,
µ
= 0
e
=90,
ζ
=3e
-4
,
µ
= 5
e
=90,
ζ
=3e
-4
,
µ
=10
FOV =
λ
lim
=
±
40
o
010 20 30 40 50 60 70
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path A ngle
γ
, deg
Flight Path A ngle (
γ
o
=
γ
f
= -30
o
)
Non-Evasive
e
=90,
ζ
=3e
-4
,
µ
= 0
e
=90,
ζ
=3e
-4
,
µ
= 5
e
=90,
ζ
=3e
-4
,
µ
=10
γ
o
=
γ
f
= -30
o
010 20 30 40 50 60
-80
-60
-40
-20
0
20
40
Time, sec
a
com
, m/s
2
Accelerati on Command of Guidance Synthes is (
e
=90,
ζ
=3e
-4
,
µ
=10 )
a
TPG
a
APF
a
LBF
a
SYN
a
max
=
±
30 m/s
2
Fig. 5. Seeker look angle (left), flight path angle (center), and acceleration command (right)
Fig. 4. Parameter study of APF gain ε (left), APF gain ζ (center), and LBF gain μ (right)
2
4. Image file of equation 12 and 13:
5. Change right figure of Fig. 5 (change plotted variable):
Fig. 5. Seeker look angle (left), flight path angle (center), and acceleration command (right)
010 20 30 40 50 60
-40
-20
0
20
40
60
Time t, sec
Look Angle

, deg
Seeker Look Angle (

o
=

f
= -30
o
)
Non-Evasive

=90,

=3e
-4
,

= 0

=90,

=3e
-4
,

= 5

=90,

=3e
-4
,

=10
FOV =

lim
=

40
o
010 20 30 40 50 60 70
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path A ngle

, deg
Flight Path A ngle (

o =

f = -30o )
Non-Evasive

=90,

=3e-4,

= 0

=90,

=3e-4,

= 5

=90,

=3e-4,

=10

o =

f = -30o
010 20 30 40 50 60
-80
-60
-40
-20
0
20
40
60
80
100
X: 58.14
Y: 0.06908
Time, sec
a
com
, m/s
2
Accelerati on Command of Guidance Synthes is (

=90,

=3e
-4
,

=10 )
a
TPG
a
APF
a
LBF
a
SYN
a
lim
=

30 m/s
2
Fig. 5. Seeker look angle (left), ight path angle (center), and acceleration command (right)
Table 1. Parameters of SEAD engagement scenario
13
Table 1. Parameters of SEAD engagement scenario
Parameter Symbol Va lue
Target,stationary position
( )
,
TT
xy
(4, 0) km
Attack missile,initial position
( )
00
,
AA
xy
(-5, 6) km
Intercept missile,initial position
( )
00
,
II
xy
(0, 0) km
Attack missile,velocity
A
V
200 m/s
Intercept missile,velocity
I
V
220 m/s
Attack missile,initial flight path angle
0
γ
-30 deg
Attack missile,terminal impact angle
f
γ
-30 deg
Intercept missile,time of flight - 20 s
Gain set of TPG
( )
,
mn
(2, 3)
Distance of Influence of Obstacle
0
d
6 km
As seen in Fig. 4, the performance of evasive maneuver is highly sensitive to the gains
[ ]
,,
eζµ
. By
defining LBF gain
0
µ
=
, the performancesof APF gains
[ ]
,
e ζ
are evaluated on the left and center of the
Fig. 4. Recalling the properties of the repulsive potential field, the steepness of the potential is verified by an
early evasive maneuver when the attack missile enters the distance of influence radius
0
d
.With respect to
the non-evasive maneuver, the evasive deviation width is determined by the value of the APF gains
[ ]
,
e ζ
.
Starting from
70
e
≥
with constant
ζ
, the attack missile is able to escape the interception and gives wider
evasive trajectories for higher values.At a constant
90
e
=
,wider evasive trajectories are also demonstrated
for higher values of
ζ
.Wide evasive trajectorymaximizes the escape of the attack missile, but intuitively
generates a huge flight path angle and exceeding FOV.For the values of
4
3e
ζ
−
≥
,the guidance law needs a
compensation command to suppress the flight path angle to ensure its seeker look angle inside FOV limit.
Right graph of Fig. 4illustratesthe effect for different values of APF gain
µ
which higher value of
µ
suppresses the guidance of attack missile for narrower path trajectory.Regarding this parameter study, gain
combinations of
[ ] [ ]
4
90, 100 , 3 , 5, 10e
e ζµ
−
== =
are picked for further analysis since it complies with the
initial requirement to achieve zero miss distance and escaping an interception.
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DOI: http://dx.doi.org/10.5139/IJASS.2017.18.4.719
726
Int’l J. of Aeronautical & Space Sci. 18(4), 719–728 (2017)
30o, and the look angle converges to zero. Recalling Fig. 5,
components of acceleration command express the synthesis
of the new guidance laws for the gain setting ε=90, ζ=3e-4,
μ=10. e synthesized total attack command aSYN eectively
satises the terminal zero acceleration of TPG and keeps
inside actuator’s acceleration limit alim=±30ms-2.
4.2 Interception Survivability
Applying the same engagement scenario as studied
above, the attack missile is simulated at its terminal phase
maintaining the prescribed impact angle, constant velocity,
and launched at all possible region above target. e initial
ight path angle of attack missile is determined as the
initial LOS angle to the target and dened as the terminal
impact angle to be achieved(γf =γ0=σ0). Fig. 6 compares the
survivability map of the attack missile with evasive maneuver
regarding the survivability map without evasive maneuver.
e higher survivability, towards Ps=1, represents that
the attack missile has more probability to survive from
interception. In contrast, the lower survivability towards
Ps=0 represents the attack missile has more probability to
be intercepted and neutralized before hitting the target. By
only implementing a single set of gain ε=90, ζ=3e-4, μ=10,
the guidance law demonstrates its eectiveness to increase
the survivability of the attack missile on the SEAD mission
scenario. e proposed guidance law extends the area of
initial position for the attack missile, which has the high
survivability to accomplish the mission.
4.3 Implementation in Operational Scenario
Based on the parametric study and considering the
survivability map, the trajectory plot in Fig. 7 elaborates
three operational scenarios of broader initial position
inside the initial distance of inuence. ose scenarios
represent a practical implementation of the guidance law
based on a single gain set ε=90, ζ=3e-4, μ=10. Dierent initial
positions, i.e.: [-4.9, 3.7] km, [-1.5, 4.8] km, and [1.5, 5.2] km
are dened as Scenario 1, 2, and 3 respectively to represent
a general engagement of SEAD scenario. In each scenario,
three trajectories of simulation result are displayed, i.e.:
3
6. Change both figure of Fig. 6 (high resolution):
Fig. 6. Survivability map as a function of initial position
Fig. 6. Survivability map as a function of initial position without evasive maneuver (left) and with evasive maneuver (right)
16
scenario, three trajectories of simulation result are displayed, i.e.: the trajectory of attack missile with and
without evasive maneuvers, and the trajectory of the intercept missile.Since the engagement scenario
implementing the proposed guidance law, the trajectories clearly validate the effectiveness of the guidance
law to escape from interception by generating an evasive maneuver for different and wider range of initial
position.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-1
0
1
2
3
4
5
6
7
8
Crossrange, km
Downrange, km
Trajectory of Evasive Maneuver against Intercept M issile (
e
=90,
ζ
=3e
-4
,
µ
=10)
Scenario 1
Scenario 2
Scenario 3
Intercept Plat form
Stationary Target
Fig. 7. Trajectory of operational scenario
Furthermore, as seen in Fig. 8, the constraints of terminal impact angle, look angle limitation, and zero
terminal acceleration are all satisfied in the three operational scenarios. Since the impact angles are defined
as the initial flight path
( )
0f
γγ
=
respectively, the terminal impact angles are all smoothly achieved in each
scenario. The look angles are also suppressed to keep inside the limitation of FOV and converge to zero look
angle. Finally, in addition to limit the acceleration command, the terminal accelerations are also convergedto
zero. All results verify that the synthesis of new guidance law based on the new proposed repulsive potential
function is effective and reliable to be implemented as a new solution for evasive maneuver against intercept
missiles.
0 5 10 15 20 25 30 35 40 45 50 55
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path A ngle
γ
, deg
Flight Path A ngle (
γ
o
=
γ
f
) - Operational Scenario
Scenario 1
Scenario 2
Scenario 3
0 5 10 15 20 25 30 35 40 45 50 55
-50
-40
-30
-20
-10
0
10
20
30
40
50
Time t, sec
Look Angle
λ
, deg
Seeker Look Angle (
γ
o
=
γ
f
) - Operational Scenario
Scenario 1
Scenario 2
Scenario 3
FOV =
λ
lim
=
±
40
o
0 5 10 15 20 25 30 35 40 45 50 55
-40
-30
-20
-10
0
10
20
30
40
Time, sec
a
com
, m/s
2
Accelerati on Command of Guidance Synthes is - Operational Scenari o
Scenario 1
Scenario 2
Scenario 3
a
lim
=
±
30 m/s
2
Fig. 8. Flight path angle (left), look angle (center), and acceleration command (right)
of operational scenarios
Fig. 8. Flight path angle (left), look angle (center), and acceleration command (right) of operational scenarios
16
scenario, three trajectories of simulation result are displayed, i.e.: the trajectory of attack missile with and
without evasive maneuvers, and the trajectory of the intercept missile.Since the engagement scenario
implementing the proposed guidance law, the trajectories clearly validate the effectiveness of the guidance
law to escape from interception by generating an evasive maneuver for different and wider range of initial
position.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-1
0
1
2
3
4
5
6
7
8
Crossrange, km
Downrange, km
Trajectory of Evasive Maneuver against Intercept M issile (
e
=90,
ζ
=3e
-4
,
µ
=10)
Scenario 1
Scenario 2
Scenario 3
Intercept Plat form
Stationary Target
Fig. 7. Trajectory of operational scenario
Furthermore, as seen in Fig. 8, the constraints of terminal impact angle, look angle limitation, and zero
terminal acceleration are all satisfied in the three operational scenarios. Since the impact angles are defined
as the initial flight path
( )
0f
γγ
=
respectively, the terminal impact angles are all smoothly achieved in each
scenario. The look angles are also suppressed to keep inside the limitation of FOV and converge to zero look
angle. Finally, in addition to limit the acceleration command, the terminal accelerations are also convergedto
zero. All results verify that the synthesis of new guidance law based on the new proposed repulsive potential
function is effective and reliable to be implemented as a new solution for evasive maneuver against intercept
missiles.
0 5 10 15 20 25 30 35 40 45 50 55
-100
-80
-60
-40
-20
0
20
Time t, sec
Flight Path A ngle
γ
, deg
Flight Path A ngle (
γ
o
=
γ
f
) - Operational Scenario
Scenario 1
Scenario 2
Scenario 3
0 5 10 15 20 25 30 35 40 45 50 55
-50
-40
-30
-20
-10
0
10
20
30
40
50
Time t, sec
Look Angle
λ
, deg
Seeker Look Angle (
γ
o
=
γ
f
) - Operational Scenario
Scenario 1
Scenario 2
Scenario 3
FOV =
λ
lim
=
±
40
o
0 5 10 15 20 25 30 35 40 45 50 55
-40
-30
-20
-10
0
10
20
30
40
Time, sec
a
com
, m/s
2
Accelerati on Command of Guidance Synthes is - Operational Scenari o
Scenario 1
Scenario 2
Scenario 3
a
lim
=
±
30 m/s
2
Fig. 8. Flight path angle (left), look angle (center), and acceleration command (right)
of operational scenarios
Fig. 7. Trajectory of operational scenario
(719~728)2017-97.indd 726 2017-12-29 오전 6:03:28

727
Y. H. Yogaswara Impact Angle Control Guidance Synthesis for Evasive Maneuver against Intercept Missile
http://ijass.org
the trajectory of attack missile with and without evasive
maneuvers, and the trajectory of the intercept missile.
Since the engagement scenario implementing the
proposed guidance law, the trajectories clearly validate the
eectiveness of the guidance law to escape from interception
by generating an evasive maneuver for dierent and wider
range of initial position.
Furthermore, as seen in Fig. 8, the constraints of terminal
impact angle, look angle limitation, and zero terminal
acceleration are all satised in the three operational
scenarios. Since the impact angles are dened as the initial
ight path (γ0=γf) respectively, the terminal impact angles
are all smoothly achieved in each scenario. e look angles
are also suppressed to keep inside the limitation of FOV and
converge to zero look angle. Finally, in addition to limit the
acceleration command, the terminal accelerations are also
converged to zero. All results verify that the synthesis of new
guidance law based on the new proposed repulsive potential
function is eective and reliable to be implemented as a new
solution for evasive maneuver against intercept missiles.
5. Conclusion
is paper proposes a synthesis of new guidance law
to generate an evasive maneuver against enemy’s missile
interception while considering its impact angle, acceleration,
and FOV constraints. e guidance law introduces a simple
approach but an eective result for a real-time avoidance
against dynamic obstacles. e new guidance law synthesizes
three components, starting with the new repulsive potential
function of APF to generate the evasive maneuver. e
zero terminal miss distance is satised by TPG as well as to
satisfy the impact angle and zero terminal acceleration. e
guidance law is nally synthesized with the LBF to guarantee
the look angle inside the FOV. e parametric study on the
gains of [ε, ζ, μ] are carried to generate a reliable gain set.
SEAD engagement scenarios of attack missile headed for
a stationary target that is defended by an intercept missile
are performed on numerical simulation. e gain set ε=90,
ζ=3e-4, μ=10 demonstrates the expected performance of
guidance law. e commanded acceleration is proven to
generate the evasive maneuver of attack missile avoiding
the enemy’s missile interception. e avoidance trajectory
of attack missile is adequate to escape the interception
while managing the constraints of impact angle, zero impact
acceleration, and FOV limit. Without the evasive maneuver,
the attack missile cannot survive the interception. However,
the survivability of the attack missiles is enhanced when the
guidance law is applied. e enhancement of survivability
is clearly compared in survivability maps for the scenarios
with and without evasive maneuver. Briey, the proposed
guidance law and new repulsive potential function perform
a simple, reliable and eective approach to generate evasive
maneuver against missile intercept while satisfying the
constraints.
Study on the optimization of gain values can be carried out
as further works. Optimized scaling gains will improve the
guidance performance and also enhance the survivability in a
broader area of initial position. An expansion of probability map
using optimized gains will ensure the success of engagement
mission over a high threat environment. Implementation of
the proposed guidance law for multiple missiles and another
engagement geometry such as air-to-air or surface-to-air
scenarios are also considered for future works.
Acknowledgement
Y. H. Yogaswara expresses his appreciation to Indonesia
Endowment Fund for Education (LPDP RI), for providing
scholarship on a doctoral program at the Korea Advanced
Institute of Science and Technology, Republic of Korea.
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