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Flight control system for guided rolling-airframe missile
Conference Paper · March 2016
DOI: 10.1109/AERO.2016.7500499
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Flight Control System for Guided Rolling-airframe Missile
Saeb AmirAhmadi Chomachar
Aerospace Eng.,
AmirKabir University of Technology
424 Hafez Ave., Tehran, Iran.
+98-939-469-7639
amirahmadi@phd.guilan.ac.ir
Alireza Mohammadi Fard
Aerospace Eng.,
AmirKabir University of Technology
+98-912-647-2027
alireza.mfard@aut.ac.ir
is such a tactical short-range missile that has very worthwhile
characteristics. Various types of RAM could be used for
airborne, naval and ground engagements. As the name
implies, guided RAM has a rolling airframe, it means it has a
revolving bank angle (for some reasons with a constant rate).
The rolling bank angle brings with itself a challenge; the
tricyclic motion. The tricyclic motion does not happen for
symmetric missiles, whereas, guided RAM is not symmetric
because the need for cambered actuating fins obliges the
missile to be slightly asymmetric. The theory of the linear
motion of a ‘slightly asymmetric’ missile was first developed
by Nicolaides [1]. The aerodynamic characteristic of a
slightly asymmetric missile is not very different from a
symmetric missile; however, in the case of asymmetric
missile, a constant amplitude force and moment are produced
whose orientations are fixed relative to the missile. This is the
reason for a trim angle-of-attack that rolls with the missile
(tricyclic motion). For a constant roll-rate, the amplitude of
this trim angle-of-attack is in direct relation with the roll-rate,
while the maximum angle occurs when the roll-rate equals
the natural frequency of pitch (resonance) [2]. In the current
study, it is assumed that roll equation is decoupled from pitch
equation, hence Magnus-moments and their accompanied
induced tricyclic motion are not present. Moreover, the
constant roll-rate of the missile is obtained at launch stage
while it is as large as to be invariant during the engagement.
Besides attenuating the tricyclic motion, the constant roll-rate
as well provides an opportunity to apply a simple two-point
guidance strategy by sampled excitation of actuating fins.
Abstract—A sampled-data system associated with the guided
rolling-airframe missile (RAM) is digitally controlled. The
digital control system is open-loop and missile dynamics (pitchrate-to-elevator) is assumed to be of second-order type. The
square-wave input, corresponding to the elevator deflections,
stabilizes system online output that is the rate of the line-of-sight
(LOS) angle. The output stabilization results in two-point
guidance-law to be actively realized, hence the missile
approaches the target until a hit. The guidance strategy is openloop (it doesn’t require active homing), whereas the missile can
hit dynamic targets moving uniformly on a linear pathway.
Moreover, only the initial triggering of the missile is targetoriented and requires active target data, to be provided by a
visual device at the launch stage where the missile is to be
directly pointed towards the target before being fired. During
the engagement, the missile is assumed to have constant roll-rate
(obtained at launch stage) and also constant forward velocity.
The missile is also slightly asymmetric due to actuating fins
geometry. Moreover, it could be assumed almost symmetric
hence the linear motion theory analysis is valid. Meanwhile, as
long as the roll-rate is not close to the pitch natural frequency,
the unwanted tricyclic motion is avoided and this is preferred.
In the current study, and through the numerical simulations, the
roll-rate is so much higher than the pitch-rate, hence normally,
the Magnus-moment and its accompanied tricyclic motion are
not present. Novel ideas for technology development of surfaceto-air RAM (SARAM) and seaborne RAM (SEARAM) are
presented. The simulations is performed in Matlab Simulink
software environment with discrete-time blocks. The missdistance is almost zero and the simulations outcome is
satisfactory.
TABLE OF CONTENTS
The guided RAM technology has not been reported in the
existing literature. In this research, a theoretical study
regarding technological development of flight control system
for guided RAM is presented. The sampled-data system
associated with the RAM is digitally controlled by an openloop strategy. The input to the missile system is square-wave;
corresponding to the elevator deflections (on-off actuation).
The response of the second order system (missile) to the
square wave input is stable, hence an open-loop guidance
strategy is available to nullify the system online output that is
the rate of the line-of-sight (LOS) angle. Based on this output
stabilization, two-point guidance-law is actively realized,
therefore, by keeping the rate of the LOS angle at zero, the
missile approaches the target until a hit. Missile passive
dynamics is assumed to be highly damped. The aerodynamic
turning-rate time-constant is assumed zero. It means the
missile swiftly responds to actuating fins deflection while the
response is highly damped. Meanwhile, the velocity vector is
assumed to be tightly aligned with the missile axis of
1. INTRODUCTION ....................................................... 1
2. PITCH-RATE-TO-ELEVATOR DYNAMICS.............. 2
3. DESIGN OF GUIDANCE LAW.................................... 2
4. INITIAL TRIGGERING .............................................. 5
5. TECHNOLOGY CHALLENGES .................................. 6
6. SIMULATIONS ....................................................... 6
7. SUMMARY ............................................................... 8
NOMENCLATURE ..................................................... 8
REFERENCES............................................................... 8
BIOGRAPHY ................................................................ 9
1. INTRODUCTION
Guided missiles are of major interest in warfare industry.
Short-range tactical guided missiles play a critical role in
military capabilities. Guided rolling-airframe missile (RAM)
978-1-4673-7676-1/16/$31.00 ©2016 IEEE
1
the axis of symmetry (longitudinal axis); hence, the angle-ofattack (α) could be assumed negligible. Meanwhile, the
symmetry and this happens when the missile flies at high
speed. Taking all these into account then the angle-of-attack
is assumed almost negligible. During the engagement, the
missile is assumed to have constant roll-rate (obtained at
launch stage) and also constant forward velocity, while the
target is at uniform motion on a linear pathway. The target is
typically unaware of missile attack hence is supposed to
move unswerving (on a linear pathway) during a possible
engagement scenario. In this study surface-to-air RAM
(SARAM), sea-borne RAM (SEARAM) and surface-tosurface RAM (SSRAM) which operates similar to SEARAM,
are theoretically studied.
turning-rate
JM
is related to the angle-of-attack by Eq. (2);
see [5]:
JM
D
(2)
TD
Based on these assumptions, if the missile is aerodynamically
designed to be highly maneuverable ( J
M is comparatively
large) and also flies at high speed with small angle-of-attack
variation, then based on Eq. (2), Tα could be ignored and this
was assumed in the simulation.
In the following, at first, the assumptions for the missile
dynamics (transfer function) is given. Thereafter, guidancelaw together with the whole missile system for simulation is
portrayed. After describing the initial triggering process of
the missile, technology challenges are discussed and then
computer simulation is presented.
You should be notified that although the transfer function Eq.
(1) is in Laplace domain, however for numerical simulation
we used a couple of discrete-time integrators to model missile
dynamics.
2. PITCH-RATE-TO-ELEVATOR DYNAMICS
3. DESIGN OF GUIDANCE LAW
For guidance-law to be implemented, the missile dynamics
should be already known. There are several mathematical
models available for the inner-loop of a typical guidance
system. In this study, the dynamics of the missile is modeled
by pitch-rate-to-elevator transfer function. The open-loop
response of the model to square-wave input (corresponding
to elevator deflections) is stable. This gives a free-hand to
nullify system online output that is the rate of the line-of-sight
(LOS) angle. Missile pitch-rate-to-elevator transfer function
is usually assumed to be of second order type [3, 4]:
Proportional navigation (PN) is a well-established and highly
reliable guidance-law which has had a long-standing history
of successful implementation. By PN, the line-of-sight (LOS)
angle is kept constant. By keeping the LOS angle at constant
(or by nullifying its rate), the missile and target move on a
collision course and finally hit one another. The PN is utilized
here to formulate a 3-D engagement, while the missile and
target are normally on a common plane, hence the
engagement, although is 3-D, could be considered co-planar
as illustrated in Fig. 1. As discussed in the preceding section,
angle-of-attack (α) is assumed negligible. An example for an
engagement based on PN is illustrated in Fig. 2 where the
dashed lines are the LOS, while its orientation is kept
constant. The stair-case shows missile flight-path and it
should be noticed that, in a real-world engagement, the staircase is not necessarily observable and the missile path is
continuous, however if the missile path is zoomed in, then the
stair-case trend could be possibly observed. This stair-case
trend is the result of sampled excitation of the actuating fins.
At each sample-time, the on-off actuation gives a small pitchangle jump. The idea says if the missile is directly pointed
towards target before being fired, and an open-loop strategy
inspired by PN is implemented, then the goal of guidance that
means hitting the target is achieved.
T
G
b (1
( T D s )
2] s
s
AF 1
2
2
ZAF
(1)
ZAF
where Z AF is the airframe natural frequency and ζAF is the
airframe damping constant. Additionally, Tα is the
aerodynamic-turning-rate time constant. In this study, it is
assumed that missile dynamics is highly damped (ζAF≈1)
and this is the goal of missile off-line aerodynamic design. If
during the operation, the missile flies at a constant and
noticeably high speed (at sea-level atmosphere), then the
velocity vector could be assumed to be tightly aligned with
2
Figure 1. Kinematics of missile-target engagement (two-point guidance) and the related parameters
Figure 2. An example for a missile-target engagement based on proportional navigation (PN)
be kept at zero, and the PN guidance-law is realized. It could
be also viewed as a classical guided missile autopilot which
acts open-loop and in the reverse direction. Regarding the
current study, it should be noticed that in the real-world
engagement, the missile acts open-loop and there is no
feedback loop or online controller, whereas for the simulation
purpose, to verify and validate the proposed guidance
strategy we need active target data to be provided by a
numerical model. This numerical model is provided by a
Matlab SimulinkTM model built with discrete-time blocks.
The missile should be aerodynamically designed to have a
desired passive response. Since there is no controller in the
loop, for hitting certain class of targets, the passive
aerodynamic design of the missile could be case-sensitive
and is a technology challenge.
PN has been the guidance-law utilized in this study, however
it is implemented by an open-loop strategy not a feedback
method. In contrast to classical guidance laws where the
inner-loop (guidance-loop) drives the outer-loop (bodyloop), in this study and through the numerical simulation
model, the outer-loop drives the inner-loop. It means, at the
block-diagram of the simulation model the body-loop is
before the guidance-loop (the guidance strategy is openloop), while the whole idea is illustrated in Fig. 3. In Fig. 3
the missile system which was assumed to be of second-order
type, was modelled by two discrete-time integrators. As was
reported in the existing literature, sampled excitation of a
second-order discrete model (the missile system in Fig. 3) by
square-wave, outs a stable response. Subsequently, by
sampled excitation of the missile actuating fins, the online
output of the system that is the rate of the LOS angle, could
3
Figure 3. The digital control system associated with the guided RAM (simulation model)
SEARAM which might have a mono-wing actuation setup,
if߶଴ ൌ Ͳ, the actuation happens as (+1, -1,…) at ߶ ൌ
݇ߨǡ ݇ ൌ Ͳǡ ͳǡ ʹǡ ǥ. Also notice that for SARAM the actuation
గ
happens as (+1, +1, -1, -1, …) at ߶ ൌ ݇ ǡ ݇ ൌ Ͳǡ ͳǡ ʹǡ ǥ.
ଶ
Meanwhile, for the purpose of simulation, based on the
discussions, all fin strokes (whether positive or negative) give
a pitch moment perpendicular to the engagement plane and at
the same direction, hence the fin stroke signal could be
modelled by the sampled-data input illustrated by Fig. 4. This
is true if the roll dynamics is taken into account. The cross
fins are assumed to be similarly cambered such that missile's
symmetry is respected and only slightly changed.
Fig. 4 illustrates the strategy which results in PN guidance.
The guidance strategy is implemented by sampled actuation
of fins. It means after the first fin stroke which is event-based
(described at the next section), fins are actuated at a sampletime of Tact
గ
=థሶ. This is illustrated by Fig. 4 where fins
stroke anytime they can produce a pitching moment
perpendicular to the engagement plane. It means if ߶ ൌ
గ
గ
߶଴ ൅ ‫ ; ݐ݌‬then p=
(for two actuating fins) and p=
(for
୘ୟୡ୲
ଶ୘ୟୡ୲
four actuating fins). For example if ߶଴ ൌ Ͳ then fin actuation
happens at any ߶ ൌ ݇ߨ (for mono-wing setup) and any ߶ ൌ
గ
݇ (for cross-wing setup) where k={1, 2, 3, …}. For
ଶ
Figure 4. Sampled actuation of fins
4
4. INITIAL TRIGGERING
plane (RHP).
Here, the initial setup of the RAM is illustrated. Although it
was already stated that RAM operates open-loop, however
the initial triggering is target-oriented or event-based. This
mainly depends on the launch camera that has two duties. The
first is to directly point the missile towards target before the
firing of the missile. The second is to trigger the missile if a
certain event is spotted. The initial fin stroke (at trigger stage)
should give a missile pitch-moment perpendicular to the
engagement plane. This is what was graphically illustrated in
Fig. 5. Initial triggering is event-based, it means the missile
launch computer should be programmable to trigger if a
desired event is spotted by the launcher camera. The trigger
events are:
0 ≤ ϕ ≤ π (sin ϕ ≥0), OT perpendicular to AB, T in upper halfplane (UHP)
π <ϕ <2π (sin ϕ <0), OT perpendicular to AB, T in lower halfplane (LWHP)
0 ≤ ϕ ≤ π (sin ϕ >0), OT perpendicular to CD, T in left halfplane (LHP)
π < ϕ <2π (sin ϕ <0), OT perpendicular to CD, T in right half-
In all the events, target is assumed to be in the inner circle of
the camera (see Fig. 5); it means the missile is almost directly
pointed towards the target before being fired. This happens
as long as target is at the cross-hairs. Since the missile
operates open-loop; ߶଴ (the roll angle at which the missile
triggers), is not required to be processed, however it varies
continuously for various surface-to-air engagements.
Additionally, for SEARAM, ߶଴ could be set to an offline
constant value (e.g. zero), if the launcher is integrated with
the vessel which has a known maneuver (also could be at
stop) during a conflict scenario; see Fig. 6. Moreover, a
multiple RAM launcher, could fire at high rounds-per-minute
(RPM) towards targets aggregation location and destroy them
in bulk; see Fig. 6.
Figure 5. Missile active camera and the parameters for initial triggering
Figure 6. A possible conflict scenario for seaborne engagement (missile and target move at the same level)
5
5. TECHNOLOGY CHALLENGES
For technological development of guided RAM, there are
several issues to be considered. For sea-borne RAM
(SEARAM), the engagement happens at the sea surface (the
missile and target are both on the same level). This is what
makes the initial triggering an easy job in comparison to
surface-to-air RAM (SARAM). SEARAM launcher could be
integrated with the vessel, based on a presumed off-line
conflict scenario. It means some ideas could be weighed ‘how
to initially trigger the SEARAM without active target
homing’ and only with pointing the missile towards the target
before firing it.
specific challenges which is a technology problem.
In addition to all was discussed, to avoid the tricyclic motion
mode in the RAM, the missile could be only slightly
asymmetric to be governed by the linear motion theory. The
need for actuating fins, poses a challenge for the missile's
symmetry, however it could be possibly coped with. All these
challenges together, give way to a daunting design job while
if the missile is a rigid body, then the design is reduced from
an aeroservoelastic problem to an aeroservo problem.
6. SIMULATIONS
For surface-to-air RAM (SARAM) the situation differs. Due
to three-dimensional engagement, the initial triggering
depends on active target data that is to be provided by a visual
device, it means the initial triggering is certainly event-based.
The main challenge for SARAM is the question, ‘could it be
shoulder-fired (as Stinger) or it requires large and probably
weighty launcher setup?’ At the first sight it comes to mind
that for the SARAM to obtain a relatively large constant rollrate, there should be a heavy high-power still launcher,
however as the technology advances, there should be some
ideas for a portable or shoulder-fired SARAM to be
developed. Hence, for a portable SARAM to be developed,
the main challenge is the weight and power source, however
technological development of its dormant type could be an
easy job. Since sea targets are not very agile, hence multiple
fin actuation for high-rate sampling is not necessarily needed,
therefore SEARAM could have only two actuating fins
(monowing setup). Moreover, for SARAM to hit a dynamic
aerial target (which is typically faster than sea targets), highrate actuation is required and this is accomplished by four
actuating fins installed at cross. Surface-to-surface RAM
(SSRAM) operates as SEARAM and does not differ widely
from that. It means as sea-borne engagement, the missile and
target are almost at the same level (sea surface or earth
surface) during the engagement. Hence, the idea of
SEARAM could be easily generalized to SSRAM. The other
challenge is the actuating fin frequency (strokes per second)
and the feasible roll-rate. It means the higher is the fin
actuation frequency and the higher is the roll-rate, the higher
is the accuracy of the RAM. However, there is a trade-off for
these high-rate actuation and high roll-rate and this yields a
design problem. Regarding the structural geometry of the
RAM, low-damping and high-maneuverability requires
slender-body missile, however for a stable response, high
rigidity is of importance. It means the missile should have
high agility together with high rigidity. The aerodynamic (or
aeroelastic) design of slender-body missile is of its own
In this section, simulation results are illustrated. Regarding
missile transfer function Eq. (1), the numerical values for
simulation are given as b=0.1, ωAF =1, ζAF =0.9, VM=0.1,
VT=0.01ξ͵ . Target inertial velocity components are:
VTx=0.01, VTy=0.01 and VTz=0.01 where ‘M’ denotes
Missile and ‘T’ denotes target. Notice that for a successful
mission, missile's velocity should be sufficiently larger than
target's velocity. Moreover, x, y and z are inertial frame
coordinates. The sample-time of the Simulink model used for
simulations is Ts=0.02. The initial distance from missile to
target is r0=0.5. Because an experimental prototype was not
in access, the numerical values are set as non-dimensional.
The simulation of missile-target real-world engagement is
illustrated in Fig. 7 where missile’s flight-path trajectory is
shown in black and target’s is shown in blue. The rate of LOS
angle is portrayed in Fig. 8. Notice that at the hit stage the
lateral acceleration becomes extremely large and this is a
known characteristic of the PN guidance strategy. In Fig. 9,
the distance from missile to target is plotted over time. As
you see, missile distance from target monotonically decreases
to zero. Fin deflections are portrayed in Fig. 10 as a constant
signal of a value equal to one. Be warned that although the
fin deflection signal seems continuous, however it is of
sampled-data type and is typically an on-off actuation. Also
notice that a sampled constant signal is normally shown as a
continuous signal in computer simulations. By Fig. 11 we
arrive at the stable pitch-rate dynamics during the
accomplished engagement. In Fig. 12 you see the plot of rollangle versus time. It also implies constant roll-rate while
గ
ϕ0=0. As illustrated by Fig. 4, Ԅሶ ൌ
, henceԄሶ ൌp4 =750
ଶ்ೌ೎೟
rpm (revolution per minute). Fig. 12 illustrates the time signal
of roll-angle associated with a RAM with four actuating fins,
each one strokes 25 times per second (frequency=25 Hz).
6
0.5
0.45
0.06
0.05
0.35
distance; r
0.04
z
0.4
Target flight-path
Missile flight-path
0.03
0.02
0.01
0.3
0.25
0.2
0.15
0
0.06
0.1
0.8
0.04
0.6
0.05
0.4
0.02
0.2
0
y
0
0
0
x
2
3
time (seconds)
4
5
6
Figure 9. Missile distance from target is strictly
monotone decreasing to zero
Figure 7. Missile intercept of the target (simulation of
real-world engagement)
2
9
1.8
8
1.6
7
1.4
fin deflection
10
6
dO /dt
1
5
4
1.2
1
0.8
0.6
3
0.4
2
0.2
1
0
0
1
2
3
time (seconds)
4
5
0
0
6
1
2
3
time (seconds)
4
5
Figure 10. Actuating fin deflections over time (the
thick line is sampled-data and is discrete); on-off
actuation
Figure 8. Rate of LOS angle versus time
7
6
0.1
NOMENCLATURE
0.09
a M = Missile's lateral acceleration (latax)
λ = Line-of-sight (LOS) angle
Δz = deviation from LOS
T= Target
M= Missile
V M = Missile's velocity vector
V T = Target's velocity vector
V C = closing velocity vector
Θ = pitch angle
r = distance from missile to target
X b = Missile's body x-orientation
α = angle-of-attack
δ = elevator deflection
ωAF =air-frame frequency
ζAF =air-frame damping
s = Laplace domain variable
Tα = the aerodynamic-turning-rate time constant
ߛሶெ = Missile's turning-rate
p = roll rate
φ= roll angle
Tact = actuation sample time
z = Z-domain variable
dT/dt (pitch rate)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
time (seconds)
4
5
6
Figure 11. Missile stable pitch-rate dynamics during
the accomplished engagement
450
400
350
REFERENCES
I (roll angle)
300
[1] J. D. Nicolaides, “On the Free Flight Motion of Missiles
Having Slight Configurational Asymmetries”. Ballistic
Research Laboratories Report No. 858, AD 26405, June
1953; also Institute of the Aeronautical Sciences Preprint
395, January 1953
250
200
150
[2] Charles H. Murphy, June 1972. “Generalized
Subharmonic Response of a Missile with Slight
Configurational Asymmetry”. BRL Report No. 1591.
100
50
0
0
1
2
3
4
5
6
[3] John H. Blakelock, “Automatic Control of Aircraft and
Missiles”. Second Edition, John Wiley & Sons, Inc,
February 1991, p. 49.
time (seconds)
Figure 12. Roll-angle versus time (constant roll-rate);
ϕ0=0
[4] Paul Zarchan, “Tactical and Strategic Missile Guidance”.
Third Edition, American Institute of Aeronautics and
Astronautics, Volume 176, Progress in Astronautics and
Aeronautics, A Volume in the AIAA Tactical Missile Series,
p. 474.
7. SUMMARY
A theoretical study regarding design of flight control
system for guided rolling-airframe missile (RAM) was
presented. It was shown that by sampled actuation of the
control surface it was possible to realize a two-point
guidance law hence missile and target moved on a collision
course. Several challenges for technological development
of the missile were underlined. Simulation outcomes
proved the feasibility of the theoretical design plan.
[5] N. A. Shneydor, “Missile Guidance and Pursuit;
Kinematics, Dynamics and Control”. Horwood Publishing
Chichester, 1998, p. 73 & pp. 232-233.
8
BIOGRAPHY
Saeb AmirAhmadi C. was born
on 4th July 1984 in Rasht,
Guilan, Iran. He holds an
M.Sc. in aerospace engineering
(flight mechanics) from the
AmirKabir
University
of
Technology
(Tehran
Polytechnic). He has served as
a reviewer for the IEEE
(Aerospace and Electronic
Systems, Transactions on),
AIAA (Journal of Guidance, Control and Dynamics) and
ASME-IMECE-2012. His research interests are flight
dynamics and controls in general and particularly the
aeroservoelasticity. He is also an active researcher on
dynamics (fluids and solids), stabilization, vibration, and
optimization.
Alireza Mohammadi Fard was
born in 1986 in Tehran, Iran, and
is currently a Ph. D. student in
aerospace engineering (flight
mechanics) at the Center of
Excellence in Flight Dynamics
and Controls; the Aerospace
Engineering Department of the
AmirKabir
University
of
Technology (AUT), Tehran, Iran,
where he studied towards receiving the M.Sc. degree since
2008 until 2011. As well, he holds a B.Sc. in aerospace
engineering from the AUT. He has been involved in both
theoretical and practical aspects of his course. His areas
of interest are: control and guidance of autonomous
vehicles, modeling and simulation, flight trajectory design
and optimization and optimal Control.
9
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