Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against
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14-02-2011, 05:59 PM


Guidance Synthesis for Evasive Maneuver of
Anti-Ship Missiles
Against Close-In Weapon Systems

Seminar Report

Presented By
Ms Sarika Raju
st
1 Semester, M.Tech, Roll No:10GNC10

Department of Electrical Engineering
College of Engineering, Trivandrum
Thiruvananthapuram-16
2010

Abstract
Evasive maneuvers of anti-ship missiles are complicated since the missile should home to
the target ship alive while avoiding the defensive weapons of the target ship. Recent devel-
opments of anti-air defense systems are threatening the survivability of anti-ship missiles.
Here a 3 dimensional evasive maneuvers of anti-ship missiles against CIWS are investigated.
By using a parameter optimization technique, trajectory optimization is performed using 2
methods. Based on the optimization results a 3 dimensional biased proportional navigation
guidance (BPNG) that generates a barrel roll maneuver that enhance the survivability of
anti-ship missile is also proposed.


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Contents
1 Introduction 1
2 Equations of Motion and Aiming Errors 3
2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Aiming Errors of CIWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Optimal Evasive Maneuver Against CIWS 6
3.1 Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Biased PNG for Barrel - Roll Maneuvers 12
5 Conclusions 15
iii
List of Figures
2.1 Definitions of coordinate systems and angles . . . . . . . . . . . . . . . . . . . 3
3.1 (a)Optimum trajectory (method 1). (b) Trajectory project and implimentationed on the Y-Z plane
(method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Acceleration command profile (method 1) . . . . . . . . . . . . . . . . . . . . 8
3.3 Aiming errors (method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 (a)Optimum trajectory (method 2) (b)Trajectory project and implimentationed on Y-Z plane
(method 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5 Acceleration command profile (method 2) . . . . . . . . . . . . . . . . . . . . 11
3.6 Aiming errors (method 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 Direction of uBR and uP P N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
iv
Chapter 1
Introduction

The capturability of the biased PNG is ensured from the Fighter aircraft needs evasive ma-
neuvers just to evade attacking missiles. For anti-ship missiles, however, evasive maneuvers
are complicated since the missile should home to the heavily defended target alive, while
avoiding the defensive weapons of the target ship. Recent developments of anti-air defense
systems are threatening the survivability of anti-ship missile. It is presumed that policies of
evasive maneuvers are empirically determined by anti-ship missiles versus defense systems
engagement simulation results, rather than theoretical analysis.
On the other hand, evasive maneuvers to enhance the survivability of fighter aircraft have
been found in many scholarly articles. Continuously changing the maneuver direction, such
as weaving or barrel-roll maneuvers, is believed to be effective to enhance the survivability
of fighter aircraft. However these are not adequate for the anti-ship missiles because they
cannot satisfy a zero terminal miss distance.
Some evasive maneuver policies for missiles against proportional navigation (PN)-guided
air defense missiles has been studied. The optimal evasive maneuver pattern has been char-
acterized by a deformed conical barrel-roll, which is obtained from a constrained trajectory
optimization problem that maximizes the time-varying weighted sum of instantaneous miss
distances. The results are supported by the fact that spiral motions of tactical ballistic mis-
siles, which are caused by mass or configurational asymmetries, impose difficulty on anti-air
defense systems .
A 3-dimensional optimal evasive maneuver problem for anti-ship missiles against close-in
weapon stytems (CIWS), which is the most common defensive weapon for naval ships, is
investigated in this paper. The acceleration command is assumed to be the sum of a con-
ventional pure proportional navigation guidance (PNG) term and a bias term, the former
to guarantee the homing capability and the latter to minimize the performance index. Sim-
ilarly the performance index is defined in terms of the time-to-go weighted aiming errors of
the CIWS. To simplify the numerical analysis, the bias term is constrained to be normal to
the usual guidance plane of PNG, which is a plane including the line of sight (LOS) vector
and the missile velocity vector. Again the optimal trajectory turns out to be the type of
barrel-roll.
Based on the numerical optimization results, a 3-dimensional biased PNG law to induce
a barrel-roll maneuver is also proposed. As in the numerical optimization problem, the pro-
posed guidance law consists of a conventional pure PNG term to guarantee homing to the
target and a time-varying bias term to produce evasive barrel-roll maneuvers. To define
a barrel-roll maneuver, the barrel-roll axis and the barrel-roll frequency are introduced as
follows. The barrel-roll axis is chosen as the LOS vector from the missile to the target, and
the barrel-roll frequency is a user-defined parameter that determines the angular velocity of
1
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
the barrel-roll.
The acceleration command for the barrel-roll maneuver is computed from the cross prod-
uct of the missile velocity vector and the desired angular velocity vector of the barrel-roll.
Since the barrel-roll command is always normal to the velocity vector, the proposed guidance
law can be easily implemented to a missile system by using the pure PNG law. Note that the
altitude of the anti-ship missile should be bounded in order to prevent the anti-ship missile
from crashing into the sea. An additional altitude controller may be required to control the
altitude, but such a method reduces the evasive performance of the proposed guidance law.
Note that the altitude of the anti-ship missile should be bounded in order to prevent
the anti-ship missile from crashing into the sea. An additional altitude controller may be
required to control the altitude, but such a method reduces the evasive performance of the
proposed guidance law. Fortunately the radius of the curvature of the barrel-roll is maximal
when the barrel-roll maneuver is initiated, and it monotonically decreases as the anti-ship
missile approaches the target. Hence by a proper choice of the direction of the barrel-roll ma-
neuver, we can guarantee that the anti-ship missile does not go down below a specified height.
In this paper the capturability of the proposed 3-dimensional biased PNG with barrel-
roll maneuver against a stationary target is proved by using the Lyapunov-like approach. In
our capturability analysis the nonlinear dynamics of a 3-dimensional engagement are taken
into accountfact that the barrel-roll command does not affect the capturability of PN if the
direction of the barrel-roll command is carefully chosen.
Department of Electrical Engineering, College of Engineering, Trivandrum 2
Chapter 2
Equations of Motion and Aiming

Errors
2.1 Equations of Motion
We consider the engagement geometry between the anti-ship missile and the target, as shown
in Fig1. To simplify the equations of motion of the missile, we assume that the anti-ship
missile is a point mass and that its speed is constant. We further assume that the autopilot
and the seeker dynamics of the missile are fast enough to be neglected. Gravity is also
assumed to be compensated for by autopilot. The target ship is modeled as being stationary
since the maneuverability and the speed of ships are not comparable with those of anti-ship
missiles.
Figure 2.1: Definitions of coordinate systems and angles
Three coordinate frames are used to describe the motion of the missile: the inertial
coordinate frame, the LOS coordinate frame, and the missile velocity coordinate frame,
denoted by the subscripts I, L, and M, respectively. The relationship between theI-frame
and the L-frame is obtained by the coordinate transformation given by
    
iL cθL cψL cθL sψL sθL iI
 jL  =  −sψL cψL 0   jI  (2.1)
−sθL cψL −sθL sψL cθL
kL kI
Similarly the relationship between the M-frame and the L-frame is as follows.
3
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
    
iM cθM cψM cθL sψM sθM iL
 jM  =  −sψM cψM 0   jL  (2.2)
−sθM cψM −sθL sψM
kM cθM kL
By considering the engagement geometry depicted in Fig.2.1, we derive the following
equations from the definitions of the coordinate frame and the well-known classical principles
of dynamics.
r = rT − rM = riL (2.3)
d
dr
= (riL ) = r. iL + ΩLL = −vM iM (2.4)
dt dt
AM = ay jM + az kM = (ΩL + ΩM ) × vM (2.5)
M
M
ΩL = ΨL sinθL iL + ΨL cosθL kL − θ ̇L
̇ ̇ (2.6)
ΩM = ψ ̇ sinθM iM − θM jM + ψ ̇ cosθM kM
̇ (2.7)
M M
By using the above equations and the aforementioned assumptions, we obtain the non-
linear equations of motion of the anti-ship missile given by
r = −vM cosθM cosψM
̇ (2.8)
̇
rλy = vM sinθM (2.9)
̇
rλz = −vM cosθM sinψM (2.10)
ay tanθL sinθM cosψM sinθM sinψM 1
̇ M
ψM = + (−vM cosθM sinψM + (vM sinθM ) + (vM cosθM sinψM )
vM cosθM rcosθM rcosθM r
(2.11)
aZ tanθL sinψM cosψM
̇
θM = M + (vM cosθM sinψM ) + (vM sinθM ) (2.12)
vM r r
Department of Electrical Engineering, College of Engineering, Trivandrum 4
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
2.2 Aiming Errors of CIWS
A CIWS is a naval shipboard defense weapon used to destroy incoming anti-ship missiles
or enemy aircrafts at a short range. The CIWS usually is equipped with an automatic gun
system of high firing rates and radar systems for tracking anti-ship missiles. To hit the target
traveling at a high speed, the CIWS has to fire the shells to the predicted impact point since
mid-course corrections of the shells are not possible. For the prediction of the impact point
of the missile and the shells fired from the CIWS, the CIWS relies on the estimated missiles
states of the current time.
In this paper it is assumed that the CIWS exactly measures the states of the anti-ship
missile, rM , VM , and AM . We also assume that the shells fired from the CIWS fly straight
and maintain the muzzle velocity vCIW S . Then we can predict the flight time of each shell
as
r(t)
τ (t) ≈ (2.13)
vM + vC IW S
Typically the fire control system of the CIWS calculates the intercept point at τt from
the current time t, assuming that the missile maintains the current acceleration. If this is the
case, any time-varying maneuver of the missile can produce an aiming error of the CIWS,
which is calculated as
t+τ (t) s2
1
AM (s1 )ds1 ds2 − AM (t)[τ (t)]2
(t + τ (t)) = (2.14)
2
t t
Equation 2.14 physically represents the miss distance of the anti-ship missile and the CIWS
shell at the moment of encounter. The aiming error of the CIWS, given in eqn 2.14, is used
to analyze the evasion performance of the guidance laws designed for anti-ship missiles.
Department of Electrical Engineering, College of Engineering, Trivandrum 5
Chapter 3
Optimal Evasive Maneuver Against
CIWS
To investigate the optimal evasive maneuver of anti-ship missiles against the CIWS, consider
the following optimal control problem. Find u, which minimizes
tf
τ 2 (t)
J= dt (3.1)
(t)
t0
subject to the constraints described by eqn 2.8 to eqn 2.12, the terminal inequality
constraint
|rM (tf ) − rT | ≤ D (3.2)
and the bound on control
|u| ≤ umax (3.3)
Here D is the admissible miss distance of the missile, and umax is the limit of the ac-
celeration command. In order treat more realistic problems, the missile is assumed to have
first-order dynamics:
1
A ̇M = (u − AM ) (3.4)
τM
where u is the acceleration command and where τM represents the time constant of
the autopilot. A performance index of the optimal control problem should be carefully
determined since the aiming error (t), given by eqn 2.14, goes to 0 as the missile approaches
to the target. For example a performance index to maximize the integral of the aiming error
could produce zero aiming errors even at long ranges. On the other hand a performance
index to maximize the minimum of the aiming error would not be much more meaningful
since the aiming error eventually converges to 0. For these reasons τ 2 (t) is included in the
performance index in order to put less emphasis on the aiming errors in the final phase.
The closed-form solution of this optimal control problem could not be derived due to the
nonlinearities included in the performance index and the inequality constraints. To find
the policy of a 3-dimensional evasive maneuver of the missile against the CIWS, we rely
on numerical optimization techniques. The optimal control problem is converted into a
parameter optimization problem which treats the discretized control and the total flight
time as an unknown parameter vector, as shown below.
6
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
3.1 Method 1
X = [upitch (i), uyaw (i), tf ]T, i = 0, ....., N − m.
Note that the number of unknown parameters of method 1 is 2(N-m+1)+1. Since the
anti-ship missile should hit the target within an admissible miss distance, we assume that
the acceleration command during the final m intervals is void of evasive maneuvers. Let us
consider the following acceleration command:
 
0 1 0 0 0
 0 0 −1 1 0 
 
 0 0 −1/τM 0
A = 0  (3.5)

−1/τT 1/τT
 0 0 0 
0 0 0 0 0
= u(i) = uP P N f ori = N − m − 1..., N
Figure 3.1: (a)Optimum trajectory (method 1). (b) Trajectory project and implimentationed on the Y-Z plane
(method 1)
Finding u(i) without uP P N may be a natural approach. However we cannot obtain a con-
sistent solution from this approach. Optimizing the 2-dimensional maneuver of the missile
with the intercept constraint seems to have many local minima. To circumvent this difficulty
Department of Electrical Engineering, College of Engineering, Trivandrum 7
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
we propose method 1, which assumes a biased PNG for homing to the target and optimizes
the bias term for effective evasion. This approach reduces the optimal control problem to
a 1-dimensional maneuver problem without the terminal constraint, which is a lot simpler
than the original 2-dimensional maneuver problem.
Figure 3.2: Acceleration command profile (method 1)
For this optimization problem a coevolutionary augmented Lagrangian method (CEALM)
is used. This method converts a constrained optimizationproblem into a minimax problem
using the augmentedLagrangian formulation, which is solved by the evolution of two pop-
ulation groups. The advantage of the CEALM is that it does not require the gradient
information of the cost and the constraints functions and that it is very robust to the initial
guess of the solution. Integration of the equations of motion to evaluate the values of the
performance index and the constraints is done by the Euler method.
Method 1 has not provided consistent optimization results. Nonetheless all the results
show sinusoidal changes in the lateral acceleration, which are characteristic of barrel-roll
maneuvers. Two sample trajectories, depicted in Fig.3.1, imply that the optimal maneuver
of the anti-ship missile against CIWS should be a barrel-roll type maneuver. The acceleration
profile shown in Fig 3.2 clearly supports this conclusion. Both of the two cases have sinusoidal
changes in the normal and the lateral accelerations, with a phase shift of 90◦ between the
two axes. It is also noted that the total acceleration is saturated to its maximum value for
most of the flight. Fig 3.3 shows the CIWS aiming errors caused by the sinusoidal change of
the acceleration commands. Note that the aiming errors of the two cases have very similar
time histories, although the trajectories are somewhat different. This may be the reason for
the difficulties in obtaining a unique optimal solution; the cost function in the solution area
may be fairly flat or may have numerous local minima. The costs of two cases are 124.526
and 129.820, respectively. This is a big improvement over the PNG, for which the cost turns
Department of Electrical Engineering, College of Engineering, Trivandrum 8
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
Figure 3.3: Aiming errors (method 1)
out to be 3038.471. To obtain more consistent solutions, the missiles evasive maneuver is
now constrained to the direction perpendicular to the plane composed by the LOS vector
and the missiles velocity vector. This plane is called the PN guidance plane in this paper
since the missile trajectory is confined to this plane if the PN guidance is applied. (The
gravity effect is ignored here.) The usual PN guidance command is given in the PN guidance
plane so that homing to the target is guaranteed. This approach, referred to as method 2,
significantly simplifies the optimization problem, and the number of unknown parameters is
reduced to (N-m+1)+1.
3.2 Method 2
X = [k(i), tf ]T, i = 0, ....., N − m
iLM
f ori = 0....., N − m − 1
u(i) = uP P N + k(i) (3.6)
|iLM |
u(i) = uP P N f ori = N − m..., N (3.7)
Two sample trajectories are depicted in Fig 4.1. The result of case 1 shows a barrel-roll
maneuver. In case 2, however, the rotating direction of the missile changes once in the mid-
dle. Consequently the acceleration command of case 2 has an abrupt change, as shown in
Fig 4.2, which produces large aiming errors of the CIWS observed in Fig 4.3. However these
two cases have very similar costs: 214.595 and 214.245 for case 1 and 2, respectively. We
have obtained various results showing the reversal of the rotating direction observed in case
2. This may be the reason for the difficulties in obtaining the unique optimal solution.
From the results of the numerical analysis on optimal evasive trajectory, we see that the
optimal evasive maneuver of the anti-ship missiles against the CIWS should have sinusoidal
acceleration commands to produce a barrel-roll type trajectory. Motivated by this observa-
tion we propose, in the next section, a guidance law, which is a biased PNG, to generate a
homing trajectory with a barrel-roll maneuver.
Department of Electrical Engineering, College of Engineering, Trivandrum 9
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
Figure 3.4: (a)Optimum trajectory (method 2) (b)Trajectory project and implimentationed on Y-Z plane
(method 2)
Department of Electrical Engineering, College of Engineering, Trivandrum 10
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
Figure 3.5: Acceleration command profile (method 2)
Figure 3.6: Aiming errors (method 2)
Department of Electrical Engineering, College of Engineering, Trivandrum 11
Chapter 4
Biased PNG for Barrel - Roll
Maneuvers
To obtain a trajectory of the barrel-roll type, consider the following 3-dimensional biased
PNG:
uBP N = uP P N + uBR (4.1)
where uP P N is the conventional pure PN(PPN) guidance command for homing and where
uBR is a bias term to generate barrel-roll maneuvers. Pure PNG: The conventional PPNG
law generates an acceleration command proportional to the LOS rate, which is expressed as
uP P N = NM ΩL × VM = NM vM (−λ ̇x sinθM cosψM − λy sinθM sinψM + λz cosθM )jM + (4.2)
̇ ̇
+NM vM (λ ̇x sinψM − λy cosψM )kM
̇
Since the on-board seeker cannot measure , the 3-dimensional PPNG law is practically
given by
̇ ̇ ̇
uP P N = NM vM (−λy sinθM sinψM + λz cosθM )jM + NM vM (−λy cosψM )kM (4.3)
Barrel-Roll Command : A barrel-roll maneuver can be defined by the barrel-roll axis and
the barrel-roll frequency. The command for a barrel-roll maneuver can be defined by
uBR = ωBR × VM (4.4)
If the barrel-roll axis is in the same direction as the velocity, the barrel-roll maneuver,
does not occur
ωBR //VM =⇒ uBR = 0 (4.5)
Therefore a forced maneuver is needed to generate a velocity component which is normal
to the barrel-roll axis. The command magnitude for a barrel-roll is calculated as
1 − cos2 θM cos2 ψM
|uBR | = ωBR vM (4.6)
As the missile velocity approaches the direction of the LOS vector, the command magnitude
decreases.
3-Dimensional Biased PNG: A 3-dimensional BPNG law for the barrel-roll maneuver is
given by
̇ ̇
uBP N = NM vM (−λY sinθM sinψM + λz cosθM − ωBR vM sinθM cosψM jM (4.7)
̇
+NM vM (−λy cosψM ) + ωBR vM sinψM
12
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
Figure 4.1: Direction of uBR and uP P N
For any anti-ship missile, the missile altitude should be lower-bounded properly in order
to prevent a crash into the sea. An altitude controller can be used for this purpose, but
it may deteriorate the evasive performance of the proposed guidance law by reducing the
magnitude of the barrel-roll command. In the following we investigate the time history of
the radius of the barrel-roll to check the possibility of a sea crash.
The barrel-roll maneuver occurs in the maneuver plane, and its instantaneous radius of
curvature is calculated as
LP
1 − cos2 θM cos2 ψM
VM (t) vM
ρ(t) = = (4.8)
ωBR ωBR
LP
where VM represents the project and implimentationion of the missile velocity on the maneuver plane. The
role of PPN is to align the velocity vector of the missile with the LOS vector. The PPN
LP
command project and implimentationed on the maneuver plane is always in the opposite direction of VM , as
shown in Fig 4.1. Hence, the time rate of the radius of curvature of barrel-roll is calculated
as
LP uLP
VM (t) /dt
= − PPN
ρ(t) =
̇ (4.9)
ωBR ωBR
where uLP N is the PPN command project and implimentationed onto the maneuver plane. From eqn 4.2 the
PP
magnitude of uLP N is calculated as
PP
2
N M vM
uLP N = 1 − cos2 θM cos2 ψM
cosθM cosψM (4.10)
PP
r
By substituting eqn 4.9 into eqn 4.10, we obtain an alternative expression of ρ· (t), givenby
It is noted that the radius of curvature decreases as long as the missile is heading toward
the target:
2
NM vM
ρ· (t) = 1 − cos2 θM cos2 ψM ≤ 0, f or − π/2 ≤ θM , ψM < π/2 (4.11)
cosθM cosψM
rωBR
Department of Electrical Engineering, College of Engineering, Trivandrum 13
Guidance Synthesis for Evasive Maneuver of Anti-Ship Missiles Against Close-In Weapon
Systems
In general the anti-ship missile attacks the target with sea skimming during the midcourse
guidance phase. Hence the initial flight path angle θM (0) can be assumed zero, and then the
shape of the barrel-roll is determined by ψM (0). The barrel-roll maneuver occurs counter-
clockwise in the maneuver plane. If ψM (0) > 0 and θM (0 = 0, then the missile arises
from the sea surface. Since the radius of the curvature of the barrel-roll is monotonously
decreasing, the altitude of the missile is always higher than the initial altitude. For ψM (0)<0
the direction of the barrel-roll axis should be reversed to avoid a crash into the sea.
For ψM (0) = 0, the barrel-roll maneuver does not occur so that a forced maneuver is
required to generate some velocity component in the maneuver plane.
Capturability of 3-D BPNG: To treat more general situations, we consider the case that
the speed of the anti-ship missile is time varying. In this case the acceleration of the missile,
given by (5), is modified as
AM = ax iM + ay jM + az kM = ax iM + (ΩL Ω) (4.12)
M M M
M
The other equations of the motion of the anti-ship missile do not need correction. For
capturability analysis of the proposed 3-dimensional BPNG law, it is assumed that the initial
missile heading satisfies
cosθM (0)cosψM (0) > 0 (4.13)
This condition requires that the anti-ship missile be launched toward the target. A missile
guided by the proposed BPNG law for the barrel-roll maneuver always captures a stationary
target within a finite time, provided that the navigation constant is larger than one.
Department of Electrical Engineering, College of Engineering, Trivandrum 14
Chapter 5
Conclusions

In this paper 3-dimensional evasive maneuvers of the anti-ship missile against the CIWS are
investigated. By using a direct parameter optimization technique, trajectory optimization
is performed with two methods for control parameterization: 1) the direction of the lateral
acceleration is free, and 2) it is constrained to be orthogonal to the plane generated by the
velocity vector and by the LOS vector. For the first method the optimal evasive maneuver
of the missile turns out to be sinusoidal acceleration commands in the pitch and the yaw
channel, but a consistent solution is not obtained. When the bias term of acceleration
command is constrained, the solutions look more like a barrel-roll, although we have not
been able to obtain this unique solution again. Based on the optimization results, we propose
a 3-dimensional BPNG that generates a barrel-roll maneuver to enhance the survivability
of anti-ship missiles. In the proposed guidance law, a barrel-roll maneuver is defined by
the barrel-roll frequency and the barrel-roll axis. The barrel-roll frequency determines the
radius of the curvature of the barrel-roll project and implimentationed on the maneuver plane so that it can be
used as a trajectory design parameter. The target capture of the BPNG is guaranteed even
for the case of time-varying missile speed. Furthermore a proper choice of the barrel-roll
direction enables the missile to avoid the possibility of a sea crash. By being compared
with the results of the numerical optimization, the proposed guidance law has an evasion
performance close to the case of the constrained maneuver treated in method 2, which is
asignificant improvement over the evasion performance of PPNG. Due to the time lag of
the missile dynamics, large barrel-roll maneuvers could increase the miss distance in reality.
Hence the barrel-roll frequency should be carefully selected, and the initial heading error
must not be too large.
15
References
[1] Yoon-Hwan ,K., Chang-Kyung ,R., and Min-Jea,T.,“Guidance Synthesis for Evasive
Maneuver of Anti-Ship Missiles Against Close-In Weapon Systems” IEEE Transactions
on Aerospace and Electronic Systems, Vol 46, No.3, 2010, pp.1376-1387
[2] Zarchan, P “Proportional navigation and weaving targets”. Journal of Guid-
ance,Control and Dynamics, 18, 5 (1995),969-974.
[3]Imado, F. and Uehara,“S.High-g barrel roll maneuvers against proportional
navigation from optimal control viewpoint”.Journal of Guidance, Control, and Dynam-
ics, 21, 6 (1998),876-881.
[4] Imado, F. and Miwa, S. “Missile guidance algorithm against high-g barrel
maneuvers”. Journal of Guidance, Control, and Dynamics, 17, 1 (1994), 123-128.
16
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