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Missile Aerodynamics and Air-to-Air Missile CodesVersion 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 page iv

List of Tables 1

1 Missile and Target 2

1.1 Introduction 2

1.2 Dimensional Governing Equations 2

1.3 Configuration 6

1.4 Fin Deflections 7

1.5 Fin Dynamics 10

1.6 Roll Rate Autopilot 13

1.7 Pitch Rate Autopilot 14

1.8 Yaw Rate Autopilot 16

1.9 Pitch Acceleration Autopilot 18

1.10 Yaw Acceleration Autopilot 20

1.11 Proportional Navigation 21

1.12 Image and Seeker Blur 22

1.13 Duty Cycle 22

1.14 Target 24

1.15 Examples 24

2 Missile Aerodynamics Code 26

2.1 Overview 26

2.2 Input File datain n 27

2.2.1 <initial> 27

2.2.2 <flight> 28

2.2.3 <reference> 29

2.2.4 <axisymmetric> 30

2.2.5 <inertia> 31

2.2.6 <finset> 31

ii

Contents iii

2.2.7 <autopilot> 33

2.3 Execution 35

3 Air-to-Air Missile Code 36

3.1 Overview 36

3.2 Input file datain 36

3.2.1 <reference> 37

3.2.2 <simulation> 37

3.3 Execution 37

Bibliography 38

List of Illustrations

1.1 Earth and Body Frames 31.2 AIM-7 Sparrow 71.3 Missile (view from tail) 71.4 Missile for δ1c < 0, δ2c < 0, δ3c > 0 and δ4c > 0 91.5 Missile for δ1c

> 0, δ2c> 0, δ3c

> 0 and δ4c> 0 10

1.6 Missile for δ1c< 0, δ2c

> 0, δ3c< 0 and δ4c

> 0 111.7 Sequence of deflection commands 121.8 Duty cycle 231.9 Effect of navigation constant on average miss distance 252.1 Definition of geometric parameters 302.2 Definition of geometric parameters 312.3 Airfoil section 34

iv

List of Tables

1.1 Variables 61.2 Control Combinations 81.3 Duty Cycle 242.1 missile aerodynamics Files 262.2 Categories 273.1 missile aerodynamics Files 363.2 Categories 37

1

1

Missile and Target

1.1 Introduction

This book is the user manual for the missile aerodynamics and air-

to-air missile codes. Together these codes simulate the interception of

a target by a missile. The missile aerodynamics code defines the aero-

dynamic, 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 ta-

bles of the missile. The missile aerodynamics code also defines the aero-

dynamic 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 yE−axis points

2


east, and the zE−axis forms a right-handed coordinate system and thus

points into the earth†. Thus, a positive altitude for the aircraft corresponds

to a negative zE.

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 yB−axis is perpendicular to the

xB−axis and also perpendicular to the vertical plane of symmetry of the

aircraft. The zB−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 yE−axis is alignedwith the local line of constant latitude.

4 Missile and Target

The moments of inertia of the missile are†

Ixx =

∫ (y2

B+ z2

B

)dm (1.1)

Iyy =

∫ (x2

B+ z2

B

)dm (1.2)

Izz =

∫ (x2

B+ y2

B

)dm (1.3)

Ixy =

∫xByB dm (1.4)

Ixz =

∫xBzB dm (1.5)

Iyz =

∫yBzB dm (1.6)

The six-degree-of-freedom equations for the motion of the missile are†

m (u+ qw − rv) = X −mg sin θ (1.7)

m (v + ru− pw) = Y +mg cos θ sinφ (1.8)

m (w + pv − qu) = Z +mg cos θ cosφ (1.9)

Ixxp− Iyz(q2 − r2

)− Ixz (r + pq)− Ixy (q − rp)− (Iyy − Izz) qr = L

(1.10)

Iyy q − Ixz(r2 − p2

)− Ixy (p+ rq)− Iyz (r − pq)− (Izz − Ixx) pr = M

(1.11)

Izz r − Ixy(p2 − q2

)− Iyz (q + pr)− Ixz (p− 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 ˙


relating the earth frame of reference to the body frame of reference, for

example

ωB = TφTθTψ ωE (1.13)

where the vector† ωE 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 asso-

ciated with Euler angles

q0 = −12 (pq1 + qq2 + rq3) (1.14)

q1 = 12 (pq0 + rq2 − qq3) (1.15)

q2 = 12 (qq0 − rq1 + pq3) (1.16)

q3 = 12 (rq0 + qq1 − pq2) (1.17)

and the Euler angles are obtained from the quaternions as

ψ = tan−1

[2 (q1q2 + q0q3)

q20 + q2

1 − q22 − q2

3

](1.18)

θ = sin−1 [2 (q0q2 − q1q3)] (1.19)

φ = tan−1

[2 (q0q1 + q2q3)

q20 + q2

3 − q21 − q2

2

](1.20)

The Euler angles and angular velocities are related by

φ = p+ (q sinφ+ r cosφ) tan θ (1.21)

θ = q cosφ− r sinφ (1.22)

ψ = (q sinφ+ r cosφ) 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., ωB = (p, q, r).


x = u cosψ cos θ + v (cosψ sin θ sinφ− sinψ cosφ) +

w (sinψ sinφ+ cosψ sin θ cosφ) (1.24)

y = u sinψ cos θ + v (cosψ cosφ+ sinψ sin θ sinφ) +

w (sinψ sin θ cosφ− cosψ sinφ) (1.25)

z = −u sin θ + v cos θ sinφ+ w cos θ cosφ (1.26)

Table 1.1. Variables

Variable Definition Representedin Frame

Dependent Variablesx, y, z Cartesian coordinates of CG Eu, v, w Velocity of CG with respect to E Bp, q, r Angular velocity of vehicle with respect to E Bψ, θ, φ Euler angles Bq0, q1, q2, q3 Quaternions B

Specified PropertiesIxx, . . . , Moments of inertia BX,Y, Z Aerodynamic forces on vehicle BL,M,N Aerodynamic moments on vehicle Bm mass of vehicleg gravitational constant

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 yB axis. The missile has tetragonal symmetry

and consequently Ixy = Ixz = Iyz = 0. A solid rocket motor propels the

missile.


Fig. 1.2. AIM-7 Sparrow

1.4 Fin Deflections

The rear missile fins are numbered beginning with the quadrant defined by

the positive yB axis and negative zB 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.

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yB

zB

+

++

+

1

23

4

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).


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 com-

mands as indicated in Table 1.2. Each combination of control commands

is associated with a deflection command to deflect a specific fin by a spe-

cific 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 CombinationsN o. δφc δθc δψc δic

1 + + + δ4c

2 + + - δ3c

3 - + + δ2c

4 - + - δ1c

According to Table 1.2 the deflection commands are

δ1c = −δφc + δθc − δψcδ2c = −δφc + δθc + δψc

δ3c = +δφc + δθc − δψcδ4c = +δφc + δθc + δψc (1.27)

Inotherwords, given the control commands δφc, δθc and δψc, then the de-

flection 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 controlsystem. There are a total of eight possible linear combinations of the three commands δφc, δθcand δψc. Note that the other four combinations are simply the negative of the combinationsshown in Table 1.2. Since the definition of positive deflection of the fin is arbitrary, theremaining four combinations simply represent the opposite definitions of positive deflection,and therefore are omitted.


δic . The above system of equations (1.27) may be inverted‡

δφc = 14 [−δ1c − δ2c + δ3c + δ4c ] (1.28)

δθc = 14 [+δ1c + δ2c + δ3c + δ4c ] (1.29)

δψc = 14 [−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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yB

zB

+

−

−+

14

23

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 zB axis (shown as dotted arrow)

generates a negative pitch moment about the yB 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 yB 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

δ1c + δ2c + δ3c + δ4c = 0

thus implying that there are only three linearly independent equations.


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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.....

.....

.................

yB

zB

+

+

4

3

+

+

1

2

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 yB axis (shown as dotted arrow) generates a negative

yaw moment about the zB 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 zB 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)

d2δ

dt2+ 2 ζ ω

dδ

dt+ ω2δ = ∆(t) (1.31)

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

∆(t) =

{0 t < 0

ω2δc t ≥ 0(1.32)

1.5 Fin Dynamics 11

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yB

zB

+

−4

3

+

2

1

−

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) = ω2δcH(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

dt

∣∣∣∣t=0

= δo (1.34)

The solution δ(t) to (1.31) for t > 0 subject to the initial conditions (1.34)

may be obtained using Laplace transforms as

δ(t) = δc

[H(t) +

1

(ν − µ)

(ν−1e−νωt − µ−1e−µωt

)]+

1

(ν − µ)

[δo(νe−µωt − µe−νωt

)+δoω

(e−µωt − e−νωt

)](1.35)

† The Heaviside function is

H(t) =

{0 t < 01 t > 0


where

µ = ζ −√ζ2 − 1 (1.36)

ν = ζ +√ζ2 − 1 (1.37)

The real part of (1.35) is assumed. For ζ > 1, µ and ν are real and δ → δcfor 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

∆(t) = ω2

δct1 t1 ≤ t ≤ t2δct2 t2 ≤ t ≤ t3. . .

(1.38)

where δcti is the deflection command issued by the control system at t = tias illustrated in Fig. 1.7.

........................

........................t

∆(t)ω2

t1 t2 t3 t4 t5 t6 t7

. . .

0

δct1

δct2

δct3

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= γ

dγ

dt= −2 ζ ω γ − ω2δ + ∆ (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

dt≤ dδ

dt

∣∣∣∣max

(1.40)

1.6 Roll Rate Autopilot

Assuming small perturbations about a uniform flight condition† the dimen-

sional 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 p = Lp p+ Lδ δφc (1.41)

φ = p (1.42)

where‡

Lp ≡∂L

∂p(1.43)

Lδ ≡∂L

∂ δφc(1.44)

In Eqs (1.41) and (1.42) the roll rate p and roll angle φ represent the per-

turbation 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 dat-

com§.

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

d2φ

dt2+ I−1

xx (KpLδ − Lp)dφ

dt+ I−1

xxKφLδ φ = I−1xxKφLδφc (1.46)

† A uniform flight condition assumes zero linear acceleration and zero angular rotation of themissile 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.


This is the equation for a damped harmonic oscillator

d2φ

dt2+ 2ζω

dφ

dt+ ω2φ = ∆ (1.47)

where ω is the natural frequency, ζ is the damping coefficient and ∆ = ω2φcis the forcing function. Thus

Kφ =ω2IxxLδ

(1.48)

and

Kp =2ζωIxx + Lp

Lδ(1.49)

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

f(s) =

∫ ∞o

f(t)e−stdt (1.50)

where f(s) is the Laplace transform of f(t). Taking the Laplace transform

of Eqs (1.41), (1.42) and (1.45) yields

Ixxs 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

φ(s)

φc(s)=

KφLδI−1xx

s2 + I−1xx (KpLδ − Lp) s+KφLδI

−1xx

(1.54)

The Laplace transform of the damped harmonic oscillator (1.47) yields

φ(s)

φc(s)=

ω2

s2 + 2ζωs+ ω2(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 dimen-

sional conservation of linear momentum (1.9) and angular momentum (1.11)

1.7 Pitch Rate Autopilot 15

may be simplified as (Zipfel 2007)

mV α = −(Nα α+ Nδ δθc

)+mV q (1.56)

Iyy q = Mα α+Mq q +Mδ δθc (1.57)

where†

Mα ≡ ∂M

∂α(1.58)

Mq ≡∂M

∂q(1.59)

Mδ ≡∂M

∂ δθc(1.60)

Nα ≡ −∂Z∂α

(1.61)

Nδ ≡ − ∂Z

∂ δθc(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, Nα and Nδ are obtained

from missile datcom.

The angle of attack α is defined by

α = tan−1 w

u(1.64)

and hence for small departures from a uniform flight condition

w = u tanα ≈ V α and thus w = V α (1.65)

where V =√u2 + v2 + 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 −Kqq (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)

δθc(s)=

c(s+ d)

s2 + as+ b(1.67)

† The notation N denotes the normal force which is the negative of the Z force. The quantityN is not to be confused with the yaw moment N .


where

a =1

mVNα −

1

IyyMq (1.68)

b = − 1

Iyy

(Mα +

1

mVMqNα

)(1.69)

c =1

IyyMδ (1.70)

d =1

mV

[Nα −

Mα

MδNδ

](1.71)

Substituting Eq (1.66) in (1.67) yields

q(s)

qc(s)=

c (s+ d)

s2 + (a+Kqc) s+ (b+Kqcd)(1.72)

Writing

q(s)

qc(s)=

c (s+ d)

s2 + 2ζωs+ ω2(1.73)

yields

Kq = −1

c

(a− 2ζ2d

)+

1

c

[(a− 2ζ2c

)2 − (a2 − 4ζ2b)]1

2(1.74)

1.8 Yaw Rate Autopilot

Assuming small perturbations about a uniform flight condition the dimen-



mV β = Yβ β + Yδ δψc −mV r (1.75)

Izz r = Nβ β +Nr r +Nδ δψc (1.76)

1.8 Yaw Rate Autopilot 17

where

Nβ =∂N

∂β(1.77)

Nr =∂N

∂r(1.78)

Nδ =∂N

∂δψc(1.79)

Yβ =∂Y

∂β(1.80)

Yδ =∂Y

∂δψc(1.81)

In Eqs (1.75) and (1.76) the yaw rate r and yaw angle β represent the per-

turbation 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 tanβ ≈ V β and thus v = V β (1.83)

where V =√u2 + v2 + 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)

δψc=

c (s+ d)

s2 + as+ b(1.85)


where

a = − 1

mVYβ −

1

IzzNr (1.86)

b =1

Izz

[Nβ +

1

mVYβNr

](1.87)

c =1

IzzNδ (1.88)

d =1

mV

[−∂Y∂β

+∂N

∂β

(∂N

∂δψc

)−1 ∂Y

∂ δψc

](1.89)

(1.90)

Substituting Eq (1.84) into (1.85) yields

r(s)

rc(s)=

c (s+ d)

s2 + (a+Krc) s+ (b+Krcd)(1.91)

Writing

r(s)

rc(s)=

c (s+ d)

s2 + 2ζωs+ ω2(1.92)

yields†

Kr = −1

c

(a− 2ζ2d

)+

1

c

[(a− 2ζ2c

)2 − (a2 − 4ζ2b)]1

2(1.93)

1.9 Pitch Acceleration Autopilot

Assuming small disturbances about a uniform flight condition the dimen-



mw = Zα α+ Zδ δθc (1.94)

Iyy q = 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 =∂Z

∂α(1.96)

Zδ =∂Z

∂δθc(1.97)

Mα =∂M

∂α(1.98)

Mq =∂M

∂q(1.99)

Mδ =∂M

∂δθc(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 = w (1.101)

and can be expressed as

a = V (q + α) (1.102)

Differentiating (1.94) and using (1.102) in (1.94) and (1.95) yields

ma = −Zαq + ZαV−1a (1.103)

Iyy q = Mq q +Mα

Zαma+

(Mδ −

ZδZα

Mα

)δθc (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

ac=

c

s2 + as+ b(1.106)


where

a = − 1

mVZα −

1

IyyMq (1.107)

b =1

Iyy

(1

mVMqZα +Mα

)+

1

mIyy(ZαMδ − ZδMα)Kθ (1.108)

c = − 1

mIyy(ZαMδ − ZδMα)Kθ (1.109)

Equating

a = 2ζω (1.110)

b = ω2 (1.111)

and thus

Kθ = mIyy (ZαMδ − ZδMα)−1 ·[1

4ζ2

(1

mVZα +

1

IyyMq

)2

− 1

mV IyyMqZα −

1

IyyMα

](1.112)

1.10 Yaw Acceleration Autopilot

Assuming small disturbances about a uniform flight condition the dimen-


may be simplified to (Zipfel 2007)

mv = Yβ β + Yδ δψc (1.113)

Izz r = Nβ β +Nr r +Nδ δψc (1.114)

1.11 Proportional Navigation 21

where

Yβ =∂Y

∂β(1.115)

Yδ =∂Y

∂δψc(1.116)

Nβ =∂N

∂β(1.117)

Nr =∂N

∂r(1.118)

Nδ =∂N

∂δψc(1.119)

The acceleration in the y−direction is defined as

a = v (1.120)

and may be expressed as

a = V(b− r

)(1.121)

Following a similar derivation as in the case of the pitch acceleration autopi-

lot yields

ma = Yβ r + YβV−1a (1.122)

Izz r = Nr r +Nβ

Yβma+

(Nδ −

YδYβNβ

)δψc (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 (YβNδ − YδNβ)−1 ·[1

4ζ2

(1

mVYβ +

1

IzzNr

)2

− 1

mV IzzNrYβ +

1

IzzNβ

](1.125)

1.11 Proportional Navigation

The Pure Proportional Navigation (ProNav) rule is (Shneydor 1998)

aMc = Nω × vM (1.126)


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

dt=dr

dter + ω × r (1.127)

with r = rT−rM and r = |r| and er is the instantaneous unit vector aligned

with r. Taking the vector cross product of (1.127) with r yields

ω =1

r2

(r × dr

dt

)=

1

r2[r × (vT − vM)] (1.128)

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 = rlos (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 = f−1d (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 = f−1p (1.131)

using Eq (1.45). Similarly, the pitch/yaw acceleration autopilot is opera-

tional 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 = f−1a (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 δψcautopilot commands are determined at each time interval

∆qr = f−1qr (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 autopi-

lots is clearly inconsistent. Additionally, the frequencies must satisfy

fp ≥ d−1p fd

fa ≥ d−1a fd

fqr ≥ d−1qr fd (1.135)

Therefore, at any instant of time there exists roll, pitch and yaw com-

mands according to Eq (1.27) which are repeated here

δ1c = −δφc + δθc − δψcδ2c = −δφc + δθc + δψc

δ3c = +δφc + δθc − δψcδ4c = +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).

........................

................................................

................................................ ........................ ........................

................................................

t

Pitch/YawRoll Rate

Pitch/YawAcceleration

Roll Rate

DutyCycle

Fig. 1.8. Duty cycle


Table 1.3. Duty Cycle

Symbol Definition

da fraction of duty cycle for pitch/yaw accelerationdp fraction of duty cycle for roll rate autopilotdqr fraction of duty cycle for pitch/yaw roll rate autopilotfd frequency of duty cycle (Hz)fa frequency of pitch/yaw acceleration update (Hz)fp frequency of roll rate update (Hz)fqr frequency of pitch/yaw rate update (Hz)∆d period of duty cycle (sec)∆a period of pitch/yaw acceleration update (sec)∆p period of roll rate update (sec)∆qr 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-to-

air 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 25

Fig. 1.9. Effect of navigation constant on average miss distance

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 E missile aerodynamics executable filemd.exe E missile datcom executable filedatain n I input file for agent ndataou n O output file for agent n (n = 0, . . .)agent n O output file for agent n (n = 0, . . .)for00m O output files for missile datcom (m = 3, . . .)

legendE executableI inputO output

26


2.2 Input File datain n

The datain n file is in ASCII format and utilizes a simplified XML nota-

tion. The file comprises several sections each beginning with <designator>

and ending with </designator> where <designator> is one of the cate-

gories 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> initial condition<flight> flight condition<reference> reference quantities<axisymmetric> missile body description<inertia> moments of inertia<finset> finset description<autopilot> autopilot desscription

2.2.1 <initial>

The <initial> section defines the initial state of the missile or target. The

data descriptors are

AgentType agent is either MISSILE or CONSTANT

Azimuth initial azimuth ψ (deg)

DynamicsRoll 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)

28 Missile Aerodynamics Code

SpecificImpulse specific impulse of motor (sec)

Speed initial speed (m/s)

XCoordinate initial value of xE (m)

YCoordinate initial value of yE (m)

ZCoordinate initial value of zE (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 extrapolate aerodynamic coefficients

when angle of attack exceeds range of

coefficient tables (YES) or terminate

simulation (NO)

ExtrapolateAngleOfYaw extrapolate aerodynamic coefficients

when angle of yaw exceeds range of


simulation (NO)

ExtrapolateDp extrapolate aerodynamics coefficients

when roll command exceeds range of


simulation (NO)

ExtrapolateDq extrapolate aerodynamics coefficients

when pitch command exceeds range of


simulation (NO)

ExtrapolateDr extrapolate aerodynamics coefficients

when yaw command exceeds range of


simulation (NO)

ExtrapolateMach extrapolate aerodynamics coefficients

when Mach number exceeds range of


simulation (NO)


MaximumAltitude maximum altitude (m)

MinimumAltitude minimum altitude (m)

MaximumAngleOfAttack maximum α for calculating

aerodynamic coefficient tables (deg)

MinimumAngleOfAttack minimum α for calculating


MaximumAngleOfYaw maximum β for calculating


MinimumAngleOfYaw minimum β for calculating


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

2.2.3 <reference>

The <reference> section defines reference parameters for missile datcom.

The data descriptors are

BoundaryLayerType Turbulent (TURB) or natural transition

(NATURAL)

LateralReferenceLength Lateral reference length (m)

LongitudinalPositionOfCG Longitudinal position of CG (m)

LongitudinalReferenceLength Longitudinal reference length (m)

ReferenceArea Reference area (m2)

RoughnessHeightRating Arithmetic average roughness height variation

(millionths of inch) See Table 2 in Blake (1998)

VehicleScaleFactor See Section 3.1.2, page 9 in Blake (1998)

VerticalPositionOfCG 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.


2.2.4 <axisymmetric>

The <axisymmetric> section defines missile body parameters for missile

datcom. The data descriptors are

AfterbodyDiameterAtBase diameter of afterbody at base (m)

AfterbodyLength length of afterbody (m)

AfterbodyShape conical (CONICAL) or tangent ogive (OGIVE)

CenterbodyDiameterAtBase diameter of centerbody at base (m)

CenterbodyLength length of centerbody (m)

LongitudinalCoordinateNoseTip value of xB at nose

NoseBluntnessRadius radius of nose (m)

NoseDiameterAtBase diameter of nose at base (m)

NoseLength length of nose (m)

TypeOfNoseShape 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


Fig. 2.2. Definition of geometric parameters

2.2.5 <inertia>

The <inertia> section defines missile moments of inertia. The data de-

scriptors are

Ixx Ixx (kg·m2) see Eq (1.1)

Ixy Ixx (kg·m2) see Eq (1.4)

Ixz Ixx (kg·m2) see Eq (1.5)

Iyy Ixx (kg·m2) see Eq (1.2)

Iyz Ixx (kg·m2) see Eq (1.6)

Izz Ixx (kg·m2) see Eq (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 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

NumberOfPanels Number of panels in each finset

NumberOfSemiSpanLocations Number of semi-span locations

AirfoilSection HEX


ActuatorDamping Value of ζ in Eq (1.31)

ActuatorFrequency Value of ω in Eq (1.31)

Chord Chord (m) at semi-span location no. 1

Chord Chord (m) at semi-span location no. 2

ChordStation 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)




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)


MaximumDeltaQ Maximum δθc for aerodynamic coefficients (deg)

MaximumDeltaR Maximum δψc for aerodynamic coefficients

MinimumDelta Minimum allowable fin deflection (deg)

MinimumDeltaP Minimum δφc for aerodynamic coefficients (deg)

Minimum DeltaQ Minimum δθc for aerodynamic coefficients (deg)

MinimumDeltaR Minimum δψc for aerodynamic coefficients (deg)

RollAngle Angle of fin no. 1




SemispanLocation Location of first semi-span section (m)

SemispanLocation Location of second semi-span section (m)

SweepAngle Sweep angle of semi-span no. 1 (deg)

SweepAngle Sweep angle of semi-span no. 2 (deg)

ThicknessToChordLowerSurface tl/c for semi-span no. 1 (Fig. 2.3)

ThicknessToChordLowerSurface tl/c for semi-span no. 2 (Fig. 2.3)

ThicknessToChordUpperSurface tu/c for semi-span no. 1 (Fig. 2.3)

ThicknessToChordUpperSurface 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 de-

scriptors are

AutoilotAccel perform Proportional Navigation (ON)

AutoPilotRoll engage roll autopilot (ON)

AutoPilotPitch engage pitch autopilot (ON)

AutoPilotYaw engage yaw autopilot (ON)

AutopilotAccelDamping ζ in (1.112) and (1.125)

AutopilotAccelDutyCycle Fraction of duty cycle for autopilot


........................................

........................................

........................................

........................................

........................................

............... ............................................................................................................................................................................................................................................................................................................................................................................................................................... ............................

............................

............................

............................

............................

............

.................... .................... .................... ....................

.................... .................... .................... ....................

.................... ....................

....................

....................

....................

....................

au bu

al bl

c

tu

tl

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 35

2.3 Execution

The missile aerodynamics code is executed using the command

ma.exe -na n

where n is the number of agents.

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 E air-to-air missile executable filedatain I input filedataou O output fileagent n I output file for agent n (n = 0, . . .)dataou n O output file for agent n (n = 0, . . .)trajectory n O trajectory file for agent n (n = 0, . . .)

legendE executableI inputO 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 end-

ing with </designator> where <designator> is one of the categories indi-

36

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> reference quantities<simulation> simulation quantities

3.2.1 <reference>

The <reference> section defines the reference quantities of the simultion.


altitude altitude (m) used to define reference density and velocity

gravity gravitational constant (m/s2)

length reference length (m)

machnumber reference Mach number

3.2.2 <simulation>

The <simulation> section defines the additional quantities of the simultion.


impact distance (m) defined as impact of missile and target

maxtime maximum duration of engagement (s)

timestep 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 VehiclesDirectorate, Wright-Patterson AFB, Ohio.

Shneydor, N. (1998) Missile Guidance and Pursuit - Kinematics, Dynamics andControl. 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) MISSILEDATCOM Volume I - Final Report. AFWAL-TR-86-3091. Flight DynamicsLaboratory, Air Force Wright Aeronautical Laboratories, Air Force SystemsCommand, Wright-Patterson AFB, Ohio.

Zipfel, P. (2007) Modeling and Simulation of Aerospace Vehicle Dynamics SecondEdition. American Institute of Aeronautics and Astronautics, Reston, VA.

38

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