Missile Aerodynamics and Air-to-Air Missile Codes

Missile Aerodynamics and Air-to-Air Missile Codes
Version 1.05
1 August 2013
Draft
Doyle D. Knight
Department of Mechanical and Aerospace Engineering
Rutgers − The State University of New Jersey
New Brunswick, New Jersey USA 08903
i
Contents
List of Illustrations
List of Tables
page iv
1
1
Missile and Target
1.1 Introduction
1.2 Dimensional Governing Equations
1.3 Configuration
1.4 Fin Deflections
1.5 Fin Dynamics
1.6 Roll Rate Autopilot
1.7 Pitch Rate Autopilot
1.8 Yaw Rate Autopilot
1.9 Pitch Acceleration Autopilot
1.10 Yaw Acceleration Autopilot
1.11 Proportional Navigation
1.12 Image and Seeker Blur
1.13 Duty Cycle
1.14 Target
1.15 Examples
2
2
2
6
7
10
13
14
16
18
20
21
22
22
24
24
2
Missile Aerodynamics Code
2.1 Overview
2.2 Input File datain n
2.2.1 <initial>
2.2.2 <flight>
2.2.3 <reference>
2.2.4 <axisymmetric>
2.2.5 <inertia>
2.2.6 <finset>
26
26
27
27
28
29
30
31
31
ii
Contents
2.3
3
2.2.7 <autopilot>
Execution
Air-to-Air Missile Code
3.1 Overview
3.2 Input file datain
3.2.1 <reference>
3.2.2 <simulation>
3.3 Execution
Bibliography
iii
33
35
36
36
36
37
37
37
38
List of Illustrations
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.1
2.2
2.3
Earth and Body Frames
AIM-7 Sparrow
Missile (view from tail)
Missile for δ1c < 0, δ2c < 0, δ3c > 0 and δ4c > 0
Missile for δ1c > 0, δ2c > 0, δ3c > 0 and δ4c > 0
Missile for δ1c < 0, δ2c > 0, δ3c < 0 and δ4c > 0
Sequence of deflection commands
Duty cycle
Effect of navigation constant on average miss distance
Definition of geometric parameters
Definition of geometric parameters
Airfoil section
iv
3
7
7
9
10
11
12
23
25
30
31
34
List of Tables
1.1
1.2
1.3
2.1
2.2
3.1
3.2
Variables
Control Combinations
Duty Cycle
missile aerodynamics Files
Categories
missile aerodynamics Files
Categories
6
8
24
26
27
36
37
1
1
Missile and Target
1.1 Introduction
This book is the user manual for the missile aerodynamics and airto-air missile codes. Together these codes simulate the interception of
a target by a missile. The missile aerodynamics code defines the aerodynamic, guidance and control properties of the missile. The open source
software missile datcom (Vukelich 1986, Blake 1998) is utilized by the
missile aerodynamics code to calculate the aerodynamic coefficient tables of the missile. The missile aerodynamics code also defines the aerodynamic properties of the target (e.g., maneuvering or non-maneuvering).
The air-to-air missile code simulates the six-degree-of-freedom motion
of the missile in the interception of the three-degree-of-motion target. The
missile aerodynamics and air-to-air missile codes are open source and
available on the author’s website (http://coewww.rutgers.edu/knight/).
1.2 Dimensional Governing Equations
The dynamical equations for a missile are based upon Newton’s laws and
Euler’s equations. The following simplifying assumptions are made:
• The earth is flat and is an inertial system
• The missile is a rigid body
• The missile is flying in a quiescent atmosphere
There are two separate frames of reference used to describe each vehicle.
The origin of the earth frame of reference E is affixed to an arbitrary point
on the the earth’s surface. The xE −axis points north, the y E −axis points
2
1.2 Dimensional Governing Equations
3
east, and the z E −axis forms a right-handed coordinate system and thus
points into the earth†. Thus, a positive altitude for the aircraft corresponds
to a negative z E .
The origin of the body frame of reference B is affixed to the center of
gravity of the aircraft. The xB −axis points forward and is aligned with the
vertical plane of symmetry of the aircraft. This requirement alone does not
uniquely specify the direction of the xB −axis, however, and therefore an
particular orientation of the xB − axis within the vertical plane of symmetry
needs to be specified by the user. The y B −axis is perpendicular to the
xB −axis and also perpendicular to the vertical plane of symmetry of the
aircraft. The z B −axis is defined by assuming a right-handed coordinate
system.
Fig. 1.1. Earth and Body Frames
† Thus, the xE −axis is aligned with the local line of constant longitude and the y E −axis is aligned
with the local line of constant latitude.
4
Missile and Target
The moments of inertia of the missile are†
Z
Ixx =
yB2 + zB2 dm
Z
Iyy =
x2B + zB2 dm
Z
x2B + yB2 dm
Izz =
Z
Ixy =
xB yB dm
Z
Ixz =
xB zB dm
Z
Iyz =
yB zB dm
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
The six-degree-of-freedom equations for the motion of the missile are†
Ixx ṗ − Iyz
m (u̇ + qw − rv) = X − mg sin θ
(1.7)
m (v̇ + ru − pw) = Y + mg cos θ sin φ
(1.8)
m (ẇ + pv − qu) = Z + mg cos θ cos φ
(1.9)
q 2 − r2 − Ixz (ṙ + pq) − Ixy (q̇ − rp) − (Iyy − Izz ) qr = L
(1.10)
2
2
2
2
Iyy q̇ − Ixz r − p
− Ixy (ṗ + rq) − Iyz (ṙ − pq) − (Izz − Ixx ) pr = M
(1.11)
Izz ṙ − Ixy p − q
− Iyz (q̇ + pr) − Ixz (ṗ − qr) − (Ixx − Iyy ) pq = N
(1.12)
Eqs (1.7) to (1.9) are the conservation of linear momentum and Eqs (1.10)
to (1.12) are the conservation of angular momentum. The variables are
summarized in Table 1.1. The components (u,v,w) of the velocity of the
center-of-gravity of the missile relative to the earth coordinate system origin
are represented in the body frame of reference B. The components (p,q,r)
of the angular velocity of the body frame of reference B with respect to the
earth frame of reference E are represented in the body frame of reference B.
The Euler angles (ψ,θ,φ) represent the three successive angular rotations
† A subscript B is used instead of superscript to avoid double superscripts.
† A derivative with respect to time is denoted by an overdot ˙
1.2 Dimensional Governing Equations
5
relating the earth frame of reference to the body frame of reference, for
example
ω B = Tφ Tθ Tψ ω E
(1.13)
where the vector† ω E is the rate of rotation of frame B with respect to the
inertial frame E and represented in frame E.
Quaternions are introduced to avoid the “gimbal lock” phenomenon associated with Euler angles
q̇0 = − 21 (pq1 + qq2 + rq3 )
q̇1 =
q̇2 =
q̇3 =
1
2
1
2
1
2
(1.14)
(pq0 + rq2 − qq3 )
(1.15)
(qq0 − rq1 + pq3 )
(1.16)
(rq0 + qq1 − pq2 )
(1.17)
and the Euler angles are obtained from the quaternions as
ψ = tan−1
2 (q1 q2 + q0 q3 )
q02 + q12 − q22 − q32
θ = sin−1 [2 (q0 q2 − q1 q3 )]
2 (q0 q1 + q2 q3 )
−1
φ = tan
q02 + q32 − q12 − q22
(1.18)
(1.19)
(1.20)
The Euler angles and angular velocities are related by
φ̇ = p + (q sin φ + r cos φ) tan θ
(1.21)
θ̇ = q cos φ − r sin φ
(1.22)
ψ̇ = (q sin φ + r cos φ) sec θ
(1.23)
The missile center-of-gravity position (x, y, z) is represented in the earth
frame of reference E and is defined by
† Vectors are denoted by boldface, e.g., ω B = (p, q, r).
6
Missile and Target
ẋ =
u cos ψ cos θ + v (cos ψ sin θ sin φ − sin ψ cos φ) +
w (sin ψ sin φ + cos ψ sin θ cos φ)
ẏ =
(1.24)
u sin ψ cos θ + v (cos ψ cos φ + sin ψ sin θ sin φ) +
w (sin ψ sin θ cos φ − cos ψ sin φ)
ż = −u sin θ + v cos θ sin φ + w cos θ cos φ
(1.25)
(1.26)
Table 1.1. Variables
Variable
Definition
Represented
in Frame
Dependent Variables
x, y, z
u, v, w
p, q, r
ψ, θ, φ
q0 , q1 , q2 , q3
Cartesian coordinates of CG
Velocity of CG with respect to E
Angular velocity of vehicle with respect to E
Euler angles
Quaternions
E
B
B
B
B
Specified Properties
Ixx , . . . ,
X, Y, Z
L, M, N
m
g
Moments of inertia
Aerodynamic forces on vehicle
Aerodynamic moments on vehicle
mass of vehicle
gravitational constant
B
B
B
1.3 Configuration
The missile is comprised of a cylindrical centerbody with a shaped nose and
truncated aftbody, and two sets of four fins each. An example is the AIM-7
Sparrow shown in Fig. 1.2. The forward fins (finset no. 1) are fixed with zero
deflection and the rear fins (finset no. 2) are movable. The fins are located
at 45◦ , 135◦ , 225◦ and 315◦ where the angle of the fin is measured in the
clockwise direction from the y B axis. The missile has tetragonal symmetry
and consequently Ixy = Ixz = Iyz = 0. A solid rocket motor propels the
missile.
1.4 Fin Deflections
7
Fig. 1.2. AIM-7 Sparrow
1.4 Fin Deflections
The rear missile fins are numbered beginning with the quadrant defined by
the positive y B axis and negative z B axis† as indicated in Fig. 1.3 where the
missile is viewed from the tail. A positive deflection of the fin (or fin flap)
is indicated. The deflection of fin i in degrees is denoted δi .
4
1
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3
2
Fig. 1.3. Missile (view from tail)
The missile fins are actuated in response to commands from the autopilot
and guidance systems. The function of the fins is to change the angles of
roll (φ), pitch (θ) and/or yaw (ψ) of the missile. It is assumed that there
† This convention is identical to Zipfel (2007).
8
Missile and Target
are three possible control commands δφc , δθc and δψc determined by the
autopilot and guidance system of the missile (Zipfel 2007). The units of
δφc , δθc and δψc are radians.
There are four possible linear combinations† of the three control commands as indicated in Table 1.2. Each combination of control commands
is associated with a deflection command to deflect a specific fin by a specific number of radians. For example, δ1c is the commanded deflection in
radians for fin no. 1. The identification of a given combination of control
commands in Table 1.2 with a specific fin deflection command will become
evident below.
Table 1.2. Control Combinations
N o.
δφc
δθc
δψc
δic
1
2
3
4
+
+
-
+
+
+
+
+
+
-
δ4c
δ3c
δ2c
δ1c
According to Table 1.2 the deflection commands are
δ1c
= −δφc + δθc − δψc
δ2c
= −δφc + δθc + δψc
δ3c
= +δφc + δθc − δψc
δ4c
= +δφc + δθc + δψc
(1.27)
Inotherwords, given the control commands δφc , δθc and δψc , then the deflection commands δ1c , δ2c , δ3c and δ4c are determined from Eqs (1.27).
We now consider the aerodynamic effect of the commanded deflections
† With no loss of generality, we may assume that the three commands are combined in the form
±δφc ± δθc ± δψc
with unit coefficients, since the magnitude of each command is determined by the control
system. There are a total of eight possible linear combinations of the three commands δφc , δθc
and δψc . Note that the other four combinations are simply the negative of the combinations
shown in Table 1.2. Since the definition of positive deflection of the fin is arbitrary, the
remaining four combinations simply represent the opposite definitions of positive deflection,
and therefore are omitted.
1.4 Fin Deflections
9
δic . The above system of equations (1.27) may be inverted‡
δφc =
δθc =
δψc =
1
4
1
4
1
4
[−δ1c − δ2c + δ3c + δ4c ]
(1.28)
[+δ1c + δ2c + δ3c + δ4c ]
(1.29)
[−δ1c + δ2c − δ3c + δ4c ]
(1.30)
Consider the roll command (1.28) and the fin deflections illustrated in Fig. 1.4
where δ1c < 0, δ2c < 0, δ3c > 0 and δ4c > 0. The lift force on each fin is
indicated by the arrow. The resultant set of lift forces generates a positive
roll moment about the xB axis and hence a net change in the roll angle φ.
Assuming the drag forces are the same for each fin, there is no net pitch or
yaw moment.
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4
1
−
+
y
−
+
z
3
2
Fig. 1.4. Missile for δ1c < 0, δ2c < 0, δ3c > 0 and δ4c > 0
Consider the pitch command (1.29) and the fin deflections illustrated in
Fig. 1.5 where δ1c > 0, δ2c > 0, δ3c > 0 and δ4c > 0. The component of
lift force on each fin anti-parallel to the z B axis (shown as dotted arrow)
generates a negative pitch moment about the y B axis since the center of
gravity is assumed ahead of the fins. The components of the lift force on
each fin anti-parallel (or parallel) to the y B axis cancel assuming the lift force
‡ At first glance, the system of equations (1.27) would appear to be overdetermined. However,
it is straightforward to show
δ 1c + δ 2c + δ 3c + δ 4c = 0
thus implying that there are only three linearly independent equations.
10
Missile and Target
on the each fin is the same. Assuming the drag forces are the same for each
fin, there is no net roll or yaw moment.
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1
4
+
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y
+
+
z
2
3
Fig. 1.5. Missile for δ1c > 0, δ2c > 0, δ3c > 0 and δ4c > 0
Consider the yaw command (1.30) and the fin deflections shown in Fig. 1.6
where δ1c < 0, δ2c > 0, δ3c < 0 and δ4c > 0. The component of lift force on
each fin parallel to the y B axis (shown as dotted arrow) generates a negative
yaw moment about the z B axis since the center of gravity is assumed ahead
of the fins. The components of the lift force on each fin anti-parallel (or
parallel) to the z B axis cancel assuming the lift force on the each fin is the
same. Assuming the drag forces are the same for each fin, there is no net
pitch or yaw moment.
1.5 Fin Dynamics
The missile fins do not respond instantly to a command due to their inherent
inertia. A simple second-order model of the response of a fin to a deflection
command is a damped harmonic oscillator (Zipfel 2007)
dδ
d2 δ
+ ω 2 δ = ∆(t)
(1.31)
+ 2ζ ω
2
dt
dt
where δ is the fin deflection (e.g., δ represents δi ), ω is the natural frequency
and ζ is the damping ratio. The forcing function is
0
t<0
∆(t) =
(1.32)
2
ω δc t ≥ 0
1.5 Fin Dynamics
11
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............... ......................
.....
.....
......
.....
.....
....
.... .....
.....
.....
.....
..... ........
.....
.
.
.
.
.
.
.
.
.
.....
..... .....
...
.
.
.....
.
.
.
.
.
.
..........
......
...
.....
.
.
.
.
.
.....
...
....
.
.
.
..... .....
.....
.
.
.
.
......
..... ......... .......................
....
.
.
.
.
.
.
.
.....
.....
....
.....
.....
.....
....
.....
..... ......
.....
.........
..... .........
.....
.....
....
.....
.......
.....
....
.
.....
.....
.
.....
.....
...................
.....
.....
B
.
..... ........
..... .....
.....
4
1
−
+
y
−
+
z
3
2
Fig. 1.6. Missile for δ1c < 0, δ2c > 0, δ3c < 0 and δ4c > 0
where δc is the deflection command from the control system (e.g., δc is δic ).
This may be expressed as
∆(t) = ω 2 δc H(t)
(1.33)
where H(t) is the Heaviside function†. The solution to (1.31) is subject to
the initial conditions
δ|t=0 = δo
dδ = δ̇o
dt (1.34)
t=0
The solution δ(t) to (1.31) for t > 0 subject to the initial conditions (1.34)
may be obtained using Laplace transforms as
1
−1 −νωt
−1 −µωt
δ(t) = δc H(t) +
+
ν e
−µ e
(ν − µ)
"
#
1
δ̇
o
δo νe−µωt − µe−νωt +
e−µωt − e−νωt (1.35)
(ν − µ)
ω
† The Heaviside function is
H(t) =
0
1
t<0
t>0
12
Missile and Target
where
p
ζ2 − 1
p
ν = ζ + ζ2 − 1
µ = ζ−
(1.36)
(1.37)
The real part of (1.35) is assumed. For ζ > 1, µ and ν are real and δ → δc
for t (ωµ)−1 and t (ων)−1 without oscillations. For ζ < 1, µ and ν are
complex, and the deflection executes a damped oscillation.
During flight the missile fins are subjected to a sequence of commands of
the form (1.33) and thus

 δct1
∆(t) = ω 2
δ
 ct2
...
t1 ≤ t ≤ t2
t2 ≤ t ≤ t3
(1.38)
where δcti is the deflection command issued by the control system at t = ti
as illustrated in Fig. 1.7.
...
....
∆(t)
ω2
δct2
δct1
...
δ c t3
0
t
t1
t2
t3
t4
t5
t6
.........
t7
Fig. 1.7. Sequence of deflection commands
The governing equation for each fin deflection can be represented as a
system of first order differential equations
dδ
dt
dγ
dt
= γ
= −2 ζ ω γ − ω 2 δ + ∆
(1.39)
which are solved using a Runge-Kutta algorithm.
The above equations for fin deflection are subject to the additional con-
1.6 Roll Rate Autopilot
13
straints
δ(t) ≤ δmax
dδ
dδ ≤
dt
dt max
(1.40)
1.6 Roll Rate Autopilot
Assuming small perturbations about a uniform flight condition† the dimensional conservation of angular momentum equation in roll (1.10) and relation
between the roll angle and angular velocity (1.21) may be simplified as
Ixx ṗ = Lp p + Lδ δφc
φ̇ = p
(1.41)
(1.42)
where‡
Lp ≡
Lδ ≡
∂L
∂p
∂L
∂ δφc
(1.43)
(1.44)
In Eqs (1.41) and (1.42) the roll rate p and roll angle φ represent the perturbation to the uniform state (p = 0, φ = 0) and δφc is the roll command
(Section 1.4). The derivatives Lp and Lδ are obtained from missile datcom§.
The closure of Eqs (1.41) and (1.42) requires the specification of δφc in
terms of p and φ. The linear roll autopilot model of Zipfel (2007) is
δφc = Kφ (φc − φ) − Kp p
(1.45)
where φc is the desired stable roll angle and Kφ and Kp are constants. Since
δφc > 0 yields a positive rolling moment (see Section 1.4), the form of the
model Eq (1.45) implies that Kφ and Kp are positive.
Substituting Eq (1.45) into Eqs (1.41) and (1.42) yields the following
second order equation for the roll angle
d2 φ
dφ
−1
−1
−1
+ Ixx
(Kp Lδ − Lp )
+ Ixx
Kφ Lδ φ = Ixx
Kφ Lδ φc
2
dt
dt
(1.46)
† A uniform flight condition assumes zero linear acceleration and zero angular rotation of the
missile with u v and u w.
‡ The equivalence symbol ≡ is used to introduce simplifying notation.
§ More precisely, the dimensionless coefficients proportional to the derivatives Lp and Lδ are
obtained from missile datcom.
14
Missile and Target
This is the equation for a damped harmonic oscillator
dφ
d2 φ
+ 2ζω
+ ω2φ = ∆
dt2
dt
(1.47)
where ω is the natural frequency, ζ is the damping coefficient and ∆ = ω 2 φc
is the forcing function. Thus
ω 2 Ixx
Lδ
(1.48)
2ζωIxx + Lp
Lδ
(1.49)
Kφ =
and
Kp =
A damped oscillation occurs for ζ < 1 with frequency ω. Thus, the roll
autopilot is defined by the selection of ω and ζ.
Alternately, Eqs (1.41) and (1.42) can be solved using Laplace transforms.
Defining
Z ∞
f (s) =
f (t)e−st dt
(1.50)
o
where f (s) is the Laplace transform of f (t). Taking the Laplace transform
of Eqs (1.41), (1.42) and (1.45) yields
Ixx s p(s) − Lp p(s) − Lδ δφc (s) = 0
(1.51)
s φ(s) − p(s) = 0
(1.52)
δφc (s) − Kφ (φc (s) − φ(s)) + Kp p(s) = 0
(1.53)
which can be solved to obtain
−1
Kφ Lδ Ixx
φ(s)
= 2
−1
−1
φc (s)
s + Ixx (Kp Lδ − Lp ) s + Kφ Lδ Ixx
(1.54)
The Laplace transform of the damped harmonic oscillator (1.47) yields
φ(s)
ω2
= 2
φc (s)
s + 2ζωs + ω 2
(1.55)
Equating terms in (1.54) and (1.55) yields (1.48) and (1.49).
1.7 Pitch Rate Autopilot
Assuming small perturbations about a uniform flight condition the dimensional conservation of linear momentum (1.9) and angular momentum (1.11)
1.7 Pitch Rate Autopilot
15
may be simplified as (Zipfel 2007)
mV α̇ = − Ñα α + Ñδ δθc + mV q
Iyy q̇ = Mα α + Mq q + Mδ δθc
(1.56)
(1.57)
where†
Mα ≡
Mq ≡
Mδ ≡
Ñα ≡
Ñδ ≡
∂M
∂α
∂M
∂q
∂M
∂ δθc
∂Z
−
∂α
∂Z
−
∂ δθc
(1.58)
(1.59)
(1.60)
(1.61)
(1.62)
(1.63)
In Eqs (1.56) and (1.57) the pitch rate q and angle of attack α represent
the pertubation to the uniform state (q = 0, α = 0) and δθc is the pitch
command (Section 1.4). The derivatives Mα , Mq , Ñα and Ñδ are obtained
from missile datcom.
The angle of attack α is defined by
w
u
and hence for small departures from a uniform flight condition
α = tan−1
w = u tan α ≈ V α
where V =
√
and thus
ẇ = V α̇
(1.64)
(1.65)
u2 + v 2 + w2 .
The closure of (1.56) and (1.57) requires a model of δθc in terms of α and
q. The linear pitch autopilot model of Zipfel (2007) is
δθc = qc − Kq q
(1.66)
where qc is the desired pitch rate. Substituting into Eqs (1.56) and (1.57)
and taking the Laplace transform yields
q(s)
c(s + d)
= 2
δθc (s)
s + as + b
(1.67)
† The notation Ñ denotes the normal force which is the negative of the Z force. The quantity
Ñ is not to be confused with the yaw moment N .
16
Missile and Target
where
1
1
Ñα −
Mq
mV
Iyy
1
1
b = −
Mα +
Mq Ñα
Iyy
mV
1
c =
Mδ
Iyy
Mα
1
Ñα −
Ñδ
d =
mV
Mδ
a =
(1.68)
(1.69)
(1.70)
(1.71)
Substituting Eq (1.66) in (1.67) yields
q(s)
c (s + d)
= 2
qc (s)
s + (a + Kq c) s + (b + Kq cd)
(1.72)
q(s)
c (s + d)
= 2
qc (s)
s + 2ζωs + ω 2
(1.73)
Writing
yields
Kq = −
1h
2
i 12
1
a − 2ζ 2 d +
a − 2ζ 2 c − a2 − 4ζ 2 b
c
c
(1.74)
1.8 Yaw Rate Autopilot
Assuming small perturbations about a uniform flight condition the dimensional conservation of linear momentum (1.8) and angular momentum (1.12)
may be simplified as (Zipfel 2007)
mV β̇ = Yβ β + Yδ δψc − mV r
(1.75)
Izz ṙ = Nβ β + Nr r + Nδ δψc
(1.76)
1.8 Yaw Rate Autopilot
17
where
Nβ =
Nr =
Nδ =
Yβ =
Yδ =
∂N
∂β
∂N
∂r
∂N
∂δψc
∂Y
∂β
∂Y
∂δψc
(1.77)
(1.78)
(1.79)
(1.80)
(1.81)
In Eqs (1.75) and (1.76) the yaw rate r and yaw angle β represent the perturbation to the uniform state (r = 0, β = 0) and δψc is the yaw command
(Section 1.4). The derivatives Nβ , Nr , Nδ , Yβ and Yδ are obtained from
missile datcom.
The yaw angle β is defined by
β = tan−1
v
u
(1.82)
and hence for small departures from a uniform flight condition
v = u tan β ≈ V β
where V =
√
and thus
v̇ = V β̇
(1.83)
u2 + v 2 + w2 .
The closure of Eqs (1.75) and (1.76) rquires a model of δψc in terms of β
and r. The linear yaw autopilot model of Zipfel (2007) is
δψc = rc − Kr r
(1.84)
where rc is the desired yaw rate. Substituting into Eqs (1.75) and (1.76)
and taking the Laplace transform yields
r(s)
c (s + d)
= 2
δψc
s + as + b
(1.85)
18
Missile and Target
where
1
1
Nr
Yβ −
mV
Izz
1
1
Nβ +
Yβ Nr
Izz
mV
1
Nδ
Izz
"
#
∂Y
∂N −1 ∂Y
1
∂N
−
+
mV
∂β
∂β ∂δψc
∂ δψc
a = −
(1.86)
b =
(1.87)
c =
d =
(1.88)
(1.89)
(1.90)
Substituting Eq (1.84) into (1.85) yields
r(s)
c (s + d)
= 2
rc (s)
s + (a + Kr c) s + (b + Kr cd)
(1.91)
r(s)
c (s + d)
= 2
rc (s)
s + 2ζωs + ω 2
(1.92)
Writing
yields†
1h
i 12
1
2
2 2
2
2
Kr = − a − 2ζ d +
a − 2ζ c − a − 4ζ b
c
c
(1.93)
1.9 Pitch Acceleration Autopilot
Assuming small disturbances about a uniform flight condition the dimensional conservation of linear momentum (1.9) and angular momentum (1.11)
may be simplified as (Zipfel 2007)
mẇ = Zα α + Zδ δθc
(1.94)
Iyy q̇ = Mα α + Mq q + Mδ δθc
(1.95)
† Note that the expressions for a, b,c and d are given by Eqs (1.86) to (1.89).
1.9 Pitch Acceleration Autopilot
19
where
Zα =
Zδ =
Mα =
Mq =
Mδ =
∂Z
∂α
∂Z
∂δθc
∂M
∂α
∂M
∂q
∂M
∂δθc
(1.96)
(1.97)
(1.98)
(1.99)
(1.100)
where the angle of attack α and pitch rate q represent the perturbation to
the uniform state (q = 0, α = 0) and δθc is the pitch command (Section 1.4).
The derivatives Zα , Zδ , Mα , Mq and Mδ are obtained from missile datcom.
The acceleration in the z−direction is defined as
a = ẇ
(1.101)
a = V (q + α̇)
(1.102)
and can be expressed as
Differentiating (1.94) and using (1.102) in (1.94) and (1.95) yields
mȧ = −Zα q + Zα V −1 a
Mα
Zδ
Iyy q̇ = Mq q +
ma + Mδ −
Mα δθc
Zα
Zα
(1.103)
(1.104)
A linear autopilot pitch acceleration law is
δθc = Kθ (ac + a)
(1.105)
where ac is the command pitch acceleration. Taking the Laplace transform
of (1.103) and (1.104) and using (1.105) yields
a
c
= 2
ac
s + as + b
(1.106)
20
Missile and Target
where
1
1
Mq
Zα −
mV
Iyy
1
1
b =
M q Z α + Mα
Iyy mV
1
+
(Zα Mδ − Zδ Mα ) Kθ
mIyy
1
c = −
(Zα Mδ − Zδ Mα ) Kθ
mIyy
a = −
(1.107)
(1.108)
(1.109)
Equating
a = 2ζω
(1.110)
b = ω2
(1.111)
and thus
Kθ = mIyy (Zα Mδ − Zδ Mα )−1 ·
"
#
2
1
1
1
1
1
Zα +
Mq −
Mq Z α −
Mα
4ζ 2 mV
Iyy
mV Iyy
Iyy
(1.112)
1.10 Yaw Acceleration Autopilot
Assuming small disturbances about a uniform flight condition the dimensional conservation of linear momentum (1.8) and angular momentum (1.12)
may be simplified to (Zipfel 2007)
mv̇ = Yβ β + Yδ δψc
Izz ṙ = Nβ β + Nr r + Nδ δψc
(1.113)
(1.114)
1.11 Proportional Navigation
21
where
Yβ =
Yδ =
Nβ =
Nr =
Nδ =
∂Y
∂β
∂Y
∂δψc
∂N
∂β
∂N
∂r
∂N
∂δψc
(1.115)
(1.116)
(1.117)
(1.118)
(1.119)
The acceleration in the y−direction is defined as
a = v̇
(1.120)
a = V ḃ − r
(1.121)
and may be expressed as
Following a similar derivation as in the case of the pitch acceleration autopilot yields
mȧ = Yβ r + Yβ V −1 a
Nβ
Yδ
Izz ṙ = Nr r +
ma + Nδ −
Nβ δψc
Yβ
Yβ
(1.122)
(1.123)
A linear autopilot yaw acceleration law is
δψc = Kψ (ac + a)
(1.124)
where ac is the command yaw acceleration. Taking the Laplace transform
and solving for Kψ in a manner similar to the pitch acceleration yields
Kψ = mIzz (Yβ Nδ − Yδ Nβ )−1 ·
"
#
2
1
1
1
1
1
Yβ +
Nr −
Nr Yβ +
Nβ
4ζ 2 mV
Izz
mV Izz
Izz
(1.125)
1.11 Proportional Navigation
The Pure Proportional Navigation (ProNav) rule is (Shneydor 1998)
aMc = N ω × v M
(1.126)
22
Missile and Target
where aMc is the acceleration command to the missile and ω is the rate of
rotation of the separation vector r as defined by
dr
dr
=
er + ω × r
(1.127)
dt
dt
with r = r T −r M and r = |r| and er is the instantaneous unit vector aligned
with r. Taking the vector cross product of (1.127) with r yields
1
dr
1
ω= 2 r×
= 2 [r × (v T − v M )]
(1.128)
r
dt
r
1.12 Image and Seeker Blur
A simple model of the effect of image and seeker blur is incorporated by
replacing r in (1.126) by
r = r los (1 + ε℘)
(1.129)
where ε is constant and ℘ is a uniformly distributed (white noise) random
variable between −1 and +1.
1.13 Duty Cycle
The contributions from the several autopilot functions described in Sections
1.6 to 1.10 are combined into the command deflections δic of the fins during
a repeated time interval denoted the duty cycle. The time period is
∆d = fd−1
(1.130)
where fd is the frequency (Hz) of the duty cycle.
The concept is illustrated in Fig. 1.8. The roll rate autopilot is operational
for a fraction dp of the duty cycle where dp ≤ 1. During this time interval,
the roll rate autopilot is updated at the frequency fp . Inotherwords, a new
roll autopilot command δφc is determined at each time interval
∆p = fp−1
(1.131)
using Eq (1.45). Similarly, the pitch/yaw acceleration autopilot is operational for a fraction da of the duty cycle where da ≤ 1. During this time
interval, the pitch/yaw acceleration autopilot is updated at the frequency
fa , i.e., new pitch δθc and yaw δψc autopilot commands are determined at
each time interval
∆a = fa−1
(1.132)
1.13 Duty Cycle
23
Finally, the pitch/yaw rate autopilot is operational for a fraction dqr of the
duty cycle where dqr ≤ 1. During this time interval, the pitch/yaw rate
autopilot is updated at the frequency fqr , i.e., new pitch δθc and yaw δψc
autopilot commands are determined at each time interval
−1
∆qr = fqr
(1.133)
Additionally, the pitch/yaw acceleration and pitch/yaw rate autopilot duty
cycles must satisfy
da + dqr ≤ 1
(1.134)
Note that the pitch/yaw acceleration autopilot and pitch/yaw rate autopilot
must be executed sequentially. Simultaneous operation of these two autopilots is clearly inconsistent. Additionally, the frequencies must satisfy
fp ≥ d−1
p fd
fa ≥ d−1
a fd
fqr ≥ d−1
qr fd
(1.135)
Therefore, at any instant of time there exists roll, pitch and yaw commands according to Eq (1.27) which are repeated here
δ1c
= −δφc + δθc − δψc
δ2c
= −δφc + δθc + δψc
δ3c
= +δφc + δθc − δψc
δ4c
= +δφc + δθc + δψc
(1.136)
which determine the forcing function ∆(t) in Eq (1.32) for solution of the
individual fin deflections according to Eq (1.31).
Duty
Cycle
.........
.........
.........
Roll Rate
Pitch/Yaw
Acceleration
..................
.........
.........
Pitch/Yaw
Roll Rate
.........
.........
Fig. 1.8. Duty cycle
t
24
Missile and Target
Table 1.3. Duty Cycle
Symbol
Definition
da
dp
dqr
fd
fa
fp
fqr
∆d
∆a
∆p
∆qr
fraction of duty cycle for pitch/yaw acceleration
fraction of duty cycle for roll rate autopilot
fraction of duty cycle for pitch/yaw roll rate autopilot
frequency of duty cycle (Hz)
frequency of pitch/yaw acceleration update (Hz)
frequency of roll rate update (Hz)
frequency of pitch/yaw rate update (Hz)
period of duty cycle (sec)
period of pitch/yaw acceleration update (sec)
period of roll rate update (sec)
period of pitch/yaw rate update (sec)
1.14 Target
The current versions of the missile aerodynamics and air-to-air missile
codes assume a constant velocity target. Future versions will include target
maneuvering.
1.15 Examples
An example of the execution of the missile aerodynamics and air-toair missile codes is presented in Fig. 1.9. The figure displays the average
miss distance for a planar (x − y) engagement of a AIM-7 missile with a
constant velocity target. A total of 55 different origins of the target are
assumed within a planar region extending ±5 km in the y−direction and
1 km to 5 cm in the x−direction from the initial location of the missile.
The simulations indicate that the average miss distance is insensitive to the
Navigation constant N in (1.126) for N ≥ 4.
1.15 Examples
Fig. 1.9. Effect of navigation constant on average miss distance
25
2
Missile Aerodynamics Code
2.1 Overview
The missile aerodynamics code utilizes the files listed in Table 2.1. The
executable files for the missile aerodynamics and missile datcom codes
are ma.exe and md.exe, respectively. For each agent there is a file datain n
where n is the agent number. The missile is n = 0 and the target is n =
1. The datain n file is read by ma.exe and written to dataou n. This
provides a direct check that the datain n file has been read correctly. The
missile aerodynamics code generates the agent n file for each agent. The
missile datcom code generates several files for00n. However, these files are
rewritten during every execution of missile datcom, and therefore contain
information for the last missile datcom execution upon completion of the
missile aerodynamics code.
Table 2.1. missile aerodynamics Files
File
Type
Description
ma.exe
md.exe
datain n
dataou n
agent n
for00m
E
E
I
O
O
O
missile aerodynamics executable file
missile datcom executable file
input file for agent n
output file for agent n (n = 0, . . .)
output file for agent n (n = 0, . . .)
output files for missile datcom (m = 3, . . .)
legend
E executable
I input
O output
26
2.2 Input File datain n
27
2.2 Input File datain n
The datain n file is in ASCII format and utilizes a simplified XML notation. The file comprises several sections each beginning with <designator>
and ending with </designator> where <designator> is one of the categories indicated in Table 2.2. The data within each category are written one
item per line and can be listed in any order. Data is in free format (i.e.,
white space is ignored); however, there must be at least one blank space
between the data descriptor and its value. Examples of complete datain 0
and datain 1 files for a missile and target are provided in the download
MissileAerodynamics.zip.
Table 2.2. Categories
Designator
Description
<initial>
<flight>
<reference>
<axisymmetric>
<inertia>
<finset>
<autopilot>
initial condition
flight condition
reference quantities
missile body description
moments of inertia
finset description
autopilot desscription
2.2.1 <initial>
The <initial> section defines the initial state of the missile or target. The
data descriptors are
agent is either MISSILE or CONSTANT
initial azimuth ψ (deg)
roll angular momentum Eq (1.10) is updated (YES)
or omitted (NO)
DynamicsPitch
pitch angular momentum Eq (1.11) is updated (YES)
or omitted (NO)
DynamicsYaw
yaw angular momentum Eq (1.12) is updated (YES)
or omitted (NO)
Elevation
initial elevation θ (deg)
FuelMassFraction fuel fraction of initial mass
Mass
initial mass (kg)
MotorEndTime
time at end of motor operation (sec)
MotorStartTime
time at start of motor operator (sec)
AgentType
Azimuth
DynamicsRoll
28
Missile Aerodynamics Code
SpecificImpulse
Speed
XCoordinate
YCoordinate
ZCoordinate
specific impulse of motor (sec)
initial speed (m/s)
initial value of xE (m)
initial value of y E (m)
initial value of z E (m)
Notes:
(i) ZCoordinate is negative
(ii) Elevation is positive downwards
2.2.2 <flight>
The <flight> section defines flight conditions for the missile or target. The
data descriptors are
ExtrapolateAngleOfAttack
ExtrapolateAngleOfYaw
ExtrapolateDp
ExtrapolateDq
ExtrapolateDr
ExtrapolateMach
extrapolate aerodynamic coefficients
when angle of attack exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
extrapolate aerodynamic coefficients
when angle of yaw exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
extrapolate aerodynamics coefficients
when roll command exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
extrapolate aerodynamics coefficients
when pitch command exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
extrapolate aerodynamics coefficients
when yaw command exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
extrapolate aerodynamics coefficients
when Mach number exceeds range of
coefficient tables (YES) or terminate
simulation (NO)
2.2 Input File datain n
29
maximum altitude (m)
minimum altitude (m)
maximum α for calculating
aerodynamic coefficient tables (deg)
MinimumAngleOfAttack
minimum α for calculating
aerodynamic coefficient tables (deg)
MaximumAngleOfYaw
maximum β for calculating
aerodynamic coefficient tables (deg)
MinimumAngleOfYaw
minimum β for calculating
aerodynamic coefficient tables (deg)
MaximumMachNumber
maximum Mach number for calculating
aerodynamic coefficient tables
MinimumMachNumber
minimum Mach number for calculating
aerodynamic coefficient tables
NumberOfTableValuesPerVariable number of values for each
independent variable in aerodynamic tables
SaveFOR004
Saves last for004 file from md.exe
SaveFOR006
Saves last for004 file from md.exe
MaximumAltitude
MinimumAltitude
MaximumAngleOfAttack
2.2.3 <reference>
The <reference> section defines reference parameters for missile datcom.
The data descriptors are
BoundaryLayerType
LateralReferenceLength
LongitudinalPositionOfCG
LongitudinalReferenceLength
ReferenceArea
RoughnessHeightRating
VehicleScaleFactor
VerticalPositionOfCG
Turbulent (TURB) or natural transition
(NATURAL)
Lateral reference length (m)
Longitudinal position of CG (m)
Longitudinal reference length (m)
Reference area (m2 )
Arithmetic average roughness height variation
(millionths of inch) See Table 2 in Blake (1998)
See Section 3.1.2, page 9 in Blake (1998)
Vertical position of CG (m)
Notes:
(i) The Longitudinal PositionOfCG is measured from the nose of the
missile.
(ii) LateralReferenceLength and LongitudinalReferenceLength must
be the same. This is an assumption of air-to-air missile code.
30
Missile Aerodynamics Code
2.2.4 <axisymmetric>
The <axisymmetric> section defines missile body parameters for missile
datcom. The data descriptors are
AfterbodyDiameterAtBase
AfterbodyLength
AfterbodyShape
CenterbodyDiameterAtBase
CenterbodyLength
LongitudinalCoordinateNoseTip
NoseBluntnessRadius
NoseDiameterAtBase
NoseLength
TypeOfNoseShape
diameter of afterbody at base (m)
length of afterbody (m)
conical (CONICAL) or tangent ogive (OGIVE)
diameter of centerbody at base (m)
length of centerbody (m)
value of xB at nose
radius of nose (m)
diameter of nose at base (m)
length of nose (m)
conical (CONICAL) or tangent ogive (OGIVE)
Notes:
(i) The geometric parameters are defined in Figs. 2.1 and 2.2
Fig. 2.1. Definition of geometric parameters
2.2 Input File datain n
31
Fig. 2.2. Definition of geometric parameters
2.2.5 <inertia>
The <inertia> section defines missile moments of inertia. The data descriptors are
Ixx
Ixy
Ixz
Iyy
Iyz
Izz
Ixx
Ixx
Ixx
Ixx
Ixx
Ixx
(kg·m2 )
(kg·m2 )
(kg·m2 )
(kg·m2 )
(kg·m2 )
(kg·m2 )
see
see
see
see
see
see
Eq
Eq
Eq
Eq
Eq
Eq
(1.1)
(1.4)
(1.5)
(1.2)
(1.6)
(1.3)
2.2.6 <finset>
The <finset> sections define the fin parameters for missile datcom. The
first <finset> record in datain n refers to the forward set of fins, and the
second <finset> record refers to the rear fins. Only the rear fins may be
deflected and therefore any parameters referring to deflection of the fins for
the forward finset are read but ignored. The data descriptors are
FinDynamics
NumberOfPanels
NumberOfSemiSpanLocations
AirfoilSection
SECONDORDER indicates that the fin
deflection is governed by the dynamics
in Section 1.5. NODYNAMICS indicates
that the fins deflect instantaneously in
response to the commands
Number of panels in each finset
Number of semi-span locations
HEX
32
Missile Aerodynamics Code
Value of ζ in Eq (1.31)
Value of ω in Eq (1.31)
Chord (m) at semi-span location no. 1
Chord (m) at semi-span location no. 2
Determines chord station for measuring
sweep for Chord No. 1 (see Notes)
ChordStation
Determines chord station for measuring
sweep for Chord No. 2 (see Notes)
Dihedral
Dihedral angle for panel no. 1 (deg)
Dihedral
Dihedral angle for panel no. 2 (deg)
Dihedral
Dihedral angle for panel no. 3 (deg)
Dihedral
Dihedral angle for panel no. 4 (deg)
FlapChordToFinChord
ratio of flap chord to fin chord
for chord no. 1
FlapChordToFinChord
ratio of flap chord to fin chord
for chord no. 2
FractionOfChordOfConstantThicknessLowerSurface
bl /c for semi-span no. 1 (Fig. 2.3)
FractionOfChordOfConstantThicknessLowerSurface
bl /c for semi-span no. 2 (Fig. 2.3)
FractionOfChordOfConstantThicknessUpperSurface
bu /c for semi-span no. 1 (Fig. 2.3)
FractionOfChordOfConstantThicknessUpperSurface
bu /c for semi-span no. 2 (Fig. 2.3)
FractionOfChordToMaxThicknessLowerSurface
al /c for semi-span no. 1 (Fig. 2.3)
FractionOfChordToMaxThicknessLowerSurface
al /c for semi-span no. 2 (Fig. 2.3)
FractionOfChordToMaxThicknessUpperSurface
au /c for semi-span no. 1 (Fig. 2.3)
FractionOfChordToMaxThicknessUpperSurface
au /c for semi-span no. 2 (Fig. 2.3)
HingeLine
Distance of hinge line from origin (m)
HingeLineSweepback
Angle of sweepback of hinge line (deg)
LeadingEdge
Distance of leading edge of root chord from origin
LeadingEdgeRadius
Leading edge radius for semispan location no. 1
LeadingEdgeRadius
Leading edge radius for semispan location no. 2
MaximumDelta
Maximum allowable fin deflection (deg)
MaximumDeltaP
Maximum δφc for aerodynamic coefficients (deg)
ActuatorDamping
ActuatorFrequency
Chord
Chord
ChordStation
2.2 Input File datain n
MaximumDeltaQ
MaximumDeltaR
MinimumDelta
MinimumDeltaP
Minimum DeltaQ
MinimumDeltaR
RollAngle
RollAngle
RollAngle
RollAngle
SemispanLocation
SemispanLocation
SweepAngle
SweepAngle
ThicknessToChordLowerSurface
ThicknessToChordLowerSurface
ThicknessToChordUpperSurface
ThicknessToChordUpperSurface
33
Maximum δθc for aerodynamic coefficients (deg)
Maximum δψc for aerodynamic coefficients
Minimum allowable fin deflection (deg)
Minimum δφc for aerodynamic coefficients (deg)
Minimum δθc for aerodynamic coefficients (deg)
Minimum δψc for aerodynamic coefficients (deg)
Angle of fin no. 1
Angle of fin no. 2
Angle of fin no. 3
Angle of fin no. 4
Location of first semi-span section (m)
Location of second semi-span section (m)
Sweep angle of semi-span no. 1 (deg)
Sweep angle of semi-span no. 2 (deg)
tl /c for semi-span no. 1 (Fig. 2.3)
tl /c for semi-span no. 2 (Fig. 2.3)
tu /c for semi-span no. 1 (Fig. 2.3)
tu /c for semi-span no. 2 (Fig. 2.3)
Notes:
(i) The number of entries for Chord, ChordStation, FlapChordToFinChord
is equal to the value of NumberOfSemiSpanLocations
(ii) Panel sweep is measured from leading edge (ChordStation set to 0)
or trailing edge (ChordStation set to 1)
(iii) Semispan locations are measured from centerline of missile. Thus,
the first SemispanLocation is the radius of the missile centerbody
at the location of the fin.
2.2.7 <autopilot>
The <autopilot> section defines the autopilot parameters. The data descriptors are
AutoilotAccel
AutoPilotRoll
AutoPilotPitch
AutoPilotYaw
AutopilotAccelDamping
AutopilotAccelDutyCycle
perform Proportional Navigation (ON)
engage roll autopilot (ON)
engage pitch autopilot (ON)
engage yaw autopilot (ON)
ζ in (1.112) and (1.125)
Fraction of duty cycle for autopilot
34
Missile Aerodynamics Code
au
.......
.....
.
tl
.
....
................
bu
........
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.....
.......
.......
.......
.
.
.
.
.
.
.....
.
.
.
.
.
.
....
.......
.......
.......
.......
.........
..........
..........
..........
..........
.
.
.
.
.
.
.
.
.
.........
..........
..........
..........
...............
.......
.......
.......
.......
.......
.......
.......
.......
.......
......
.......
.......
al
................
bl
c
.....
.
tu
......
........
........
Fig. 2.3. Airfoil section
AutopilotAccelFrequency
Not used
AutopilotAccelMaximum
maximum allowable value of a in (1.126)
AutopilotAccelUpdateFrequency
frequency for application of acceleration autopilot (Hz)
AutopilotDutyCycleFrequency
frequency for overall duty cycle (Hz)
AutopilotKpMaximum
maximum allowable value for Kp
AutopilotKphiMaximum
maximum allowable value for Kφ
AutopilotKqMaximum
maximum allowable value for Kq
AutopilotKrMaximum
maximum allowable value for Kr
AutopilotPitchDamping
ζ in (1.74)
AutopilotPitchFrequency
ω in (1.74)
AutopilotPitchYawRateUpdateFrequency
frequency for application of pitch and yaw rate autopilot (Hz)
AutopilotRollDamping
ζ in (1.49)
AutopilotRollFrequency
omega in (1.48)
AutopilotRollRateDutyCyclefraction of duty cycle for rate autopilot
AutopilotRollRateUpdateFrequency
frequency for application of roll autopilot (Hz)
AutopilotYawDamping
ζ in (1.93)
AutopilotYawFrequency
ω in (1.91)
LimitDpDqDr
limits δφc , δθc , δψc (YES)
LimitDeltaCommand
limit δic (YES)
LimitDelta
limit fin deflection (YES)
NavigationConstant
N in (1.126)
SeekerImageBlurAndPixelRandomError
ε in (1.129)
2.3 Execution
2.3 Execution
The missile aerodynamics code is executed using the command
ma.exe -na n
where n is the number of agents.
35
3
Air-to-Air Missile Code
3.1 Overview
The air-to-air missile code utilizes the files listed in Table 3.1. The
executable file for the air-to-air missile code is aam.exe. For each agent
there is a file agent n where n is the agent number. The missile is n = 0
and the target is n = 1. The datain and agent n files are read by aam.exe.
The output files are dataou, dataou n and trajectory n.
Table 3.1. missile aerodynamics Files
File
Type
Description
aam.exe
datain
dataou
agent n
dataou n
trajectory n
E
I
O
I
O
O
air-to-air missile executable file
input file
output file
output file for agent n (n = 0, . . .)
output file for agent n (n = 0, . . .)
trajectory file for agent n (n = 0, . . .)
legend
E executable
I input
O output
3.2 Input file datain
The datain file is in ASCII format and utilizes a simplified XML notation.
The file comprises two sections each beginning with <designator> and ending with </designator> where <designator> is one of the categories indi36
3.3 Execution
37
cated in Table 2.2. The data within each category are written one item per
line and can be listed in any order. Data is in free format (i.e., white space
is ignored); however, there must be at least one blank space between the
data descriptor and its value. An example of a complete datain file for a
missile and target are provided in the download Air-to-Air-Missile.zip.
Table 3.2. Categories
Designator
Description
<reference>
<simulation>
reference quantities
simulation quantities
3.2.1 <reference>
The <reference> section defines the reference quantities of the simultion.
The data descriptors are
altitude
gravity
length
machnumber
altitude (m) used to define reference density and velocity
gravitational constant (m/s2 )
reference length (m)
reference Mach number
3.2.2 <simulation>
The <simulation> section defines the additional quantities of the simultion.
The data descriptors are
impact
maxtime
timestep
distance (m) defined as impact of missile and target
maximum duration of engagement (s)
timestep (s)
3.3 Execution
The air-to-air missile code is executed using the command
aam.exe -na n
where n is the number of agents.
Bibliography
Blake, W. (1998) MISSILE DATCOM User’s Manual 1997 Fortran 90 Revision.
AFRL-VA-WP-TR-1998-3009. Air Force Research Laboratory, Air Vehicles
Directorate, Wright-Patterson AFB, Ohio.
Shneydor, N. (1998) Missile Guidance and Pursuit - Kinematics, Dynamics and
Control. Horwood Publishing, Chichester, West Sussex, England.
Stevens, B. and Lewis, F. (2003) Aircraft Control and Simulation Second Edition.
Dover, New York.
Vulkelich, S., Stoy, S., Burns, K., Castillo, J. and Moore, M. (1986) MISSILE
DATCOM Volume I - Final Report. AFWAL-TR-86-3091. Flight Dynamics
Laboratory, Air Force Wright Aeronautical Laboratories, Air Force Systems
Command, Wright-Patterson AFB, Ohio.
Zipfel, P. (2007) Modeling and Simulation of Aerospace Vehicle Dynamics Second
Edition. American Institute of Aeronautics and Astronautics, Reston, VA.
38