Tactical Missiles

Autopilot Design

Aerodynamic Control

— D Viswanath

Acknowledgment

I am most grateful to my Dr. S. E. Talole, for introducing me to this subject. His

teachings have been my source of motivation throughout this work.

(D Viswanath)

Feb 2011

1

Synopsis

Broadly speaking autopilots either control the motion in the pitch and yaw planes, in

which they are called lateral autopilots, or they control the motion about the fore and

aft axis in which case they are called roll autopilots. Lateral ”g” autopilots are designed

to enable a missile to achieve a high and consistent ”g” response to a command. They

are particularly relevant to SAMs and AAMs. There are normally two lateral autopilots,

one to control the pitch or up-down motion and another to control the yaw or left-right

motion.

The requirements of a good lateral autopilot are very nearly the same for command

and homing systems but it is more helpful initially to consider those associated with

command systems where guidance receiver produces signals proportional to the mis-

alignment of the missile from the line of sight (LOS).

The effectiveness of a guided missile weapon system, in terms of accuracy and prob-

ability of kill, depends greatly on the response characteristics of the complete guidance,

control, and airframe loop. Since the accuracy or effectiveness of a guided missile de-

pends greatly on the dynamics of the missile, particularly during the terminal phase of

its flight, it is often desirable to predict its flight dynamics in the early preliminary-design

phase to assure that a reasonably satisfactory missile configuration is realized.

The missile control methods can be broadly classified under aerodynamic control and

thrust vector control. Aerodynamic control can be further classified into Cartesian and

polar control methods while thrust vector control can be further classified under gim-

baled motors, flexible nozzles (ball and socket), interference methods (spoilers/vanes),

secondary fluid or gas injection and vernier engines (external or extra engines). Aero-

dynamic control methods are generally used for tactical missiles.

2

Contents

Acknowledgment 1

Synopsis 2

Contents 3

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Lateral Autopilot Design Objectives . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Maintenance of near-constant steady state aerodynamic gain . . . 4

1.2.2 Increase weathercock frequency . . . . . . . . . . . . . . . . . . . 5

1.2.3 Increase weathercock damping . . . . . . . . . . . . . . . . . . . . 5

1.2.4 Reduce cross coupling between pitch and yaw motion . . . . . . . 6

1.2.5 Assistance in gathering . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Mathematical Modelling :

Rigid Body Flight Dynamics 7

2.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Euler’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3

2.3 Inertial Form of Force Equation in terms of Eulerian Axes . . . . . . . . 13

2.4 Inertial Form of Moment Equation in terms of Eulerian Axes . . . . . . . 14

2.5 Mathematical Modeling for Missile Lateral Autopilots . . . . . . . . . . . 17

2.5.1 Linearising Moment Equations . . . . . . . . . . . . . . . . . . . . 18

2.5.2 Linearising Force Equations . . . . . . . . . . . . . . . . . . . . . 20

2.6 Translational and Rotational Dynamics of Missile Autopilot . . . . . . . 22

2.6.1 Dynamics of Yaw Autopilot . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 Dynamics of Pitch Autopilot . . . . . . . . . . . . . . . . . . . . . 22

2.6.3 Dynamics of Roll Autopilot . . . . . . . . . . . . . . . . . . . . . 23

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Kinematics of the Missile . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Control Surface Configurations 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Cartesian Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2 Polar Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Comparison of Polar and Cartesian Control Methods . . . . . . . . . . . 26

3.2.1 Advantages of Polar Control over Cartesian control . . . . . . . . 26

3.2.2 Disadvantages of Polar control . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Advantages of Cartesian control . . . . . . . . . . . . . . . . . . . 28

3.3 Classification of Aerodynamic control . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Tail Controlled Missiles . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.2 Canard Controlled Missiles . . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Wing Controlled Missiles . . . . . . . . . . . . . . . . . . . . . . . 30

4

3.4 Comparison of Forward and Tail Control Missiles . . . . . . . . . . . . . 32

3.4.1 Advantages of Forward Control . . . . . . . . . . . . . . . . . . . 32

3.4.2 Drawbacks of Forward Control . . . . . . . . . . . . . . . . . . . . 32

3.5 Control Surface Configuration for Cartesian Missiles . . . . . . . . . . . . 33

3.5.1 Plus Type Cruciform Configuration . . . . . . . . . . . . . . . . . 33

3.5.2 Cross Type Cruciform Configuration . . . . . . . . . . . . . . . . 36

3.6 Sign Convention for Moments . . . . . . . . . . . . . . . . . . . . . . . . 38

References 39

5

Chapter 1

Introduction

Broadly speaking autopilots either control the motion in the pitch and yaw planes, in

which they are called lateral autopilots, or they control the motion about the fore and

aft axis in which case they are called roll autopilots.

(a) Lateral ”g” autopilots are designed to enable a missile to achieve a high and con-

sistent ”g” response to a command.

(b) They are particularly relevant to SAMs and AAMs.

(c) There are normally two lateral autopilots, one to control the pitch or up-down

motion and another to control the yaw or left-right motion.

(d) They are usually identical and hence a yaw autopilot is explained here.

(e) An accelerometer is placed in the yaw plane of the missile, to sense the sideways

acceleration of the missile. This accelerometer produces a voltage proportional to

the linear acceleration.

(f) This measured acceleration is compared with the ’demanded’ acceleration.

(g) The error is then fed to the fin servos, which actuate the rudders to move the

missile in the desired direction.

(h) This closed loop system does not have an amplifier, to amplify the error. This is

because of the small static margin in the missiles and even a small error (unam-

plified) provides large airframe movement.

1

Figure 1.1: Lateral Autopilot[1]

1.1 Overview

The requirements of a good lateral autopilot are very nearly the same for com-

mand and homing systems but it is more helpful initially to consider those associated

with command systems where guidance receiver produces signals proportional to the

misalignment of the missile from the line of sight (LOS). A simplified closed-loop block

diagram for a vertical or horizontal plane guidance loop without an autopilot is as shown

below: -

2

Figure 1.2: Basic Guidance and Control System [1]

(a) The target tracker determines the target direction θt.

(b) Let the guidance receiver gain be K1 volts/rad (misalignment). The guidance

signals are then invariably phase advanced to ensure closed loop stability.

(c) In order to maintain constant sensitivity to missile linear displacement from the

LOS, the signals are multiplied by the measured or assumed missile range Rm

before being passed to the missile servos. This means that the effective d.c. gain

of the guidance error detector is K1 volts/m.

(d) If the missile servo gain is K2 rad/volt and the control surfaces and airframe

produce a steady state lateral acceleration of K3 m/s2/rad then the guidance loop

has a steady state open loop gain of K1K2K3 m/s2/m or K1K2K3 s

−2.

(e) The loop is closed by two inherent integrations from lateral acceleration to lateral

position. Since the error angle is always very small, one can say that the change in

angle is this lateral displacement divided by the instantaneous missile range Rm.

(f) The guidance loop has a gain which is normally kept constant and consists of the

product of the error detector gain, the servo gain and the aerodynamic gain.

3

Consider now the possible variation in the value of aerodynamic gain K3 due to change

in static margin. The c.g. can change due to propellant consumption and manufacturing

tolerances while changes in c.p. can be due to changes in incidence, missile speed and

manufacturing tolerances. The value of K3 can change by a factor of 5 to 1 for changes

in static margin (say 2cm to 10 cm in a 2m long missile). If, in addition, there can be

large variations in the dynamic pressure 12ρu2 due to changes in height and speed, then

the overall variation in aerodynamic gain could easily exceed 100 to 1.

1.2 Lateral Autopilot Design Objectives

The main objectives of a lateral autopilot are as listed below: -

(a) Maintenance of near-constant steady state aerodynamic gain.

(b) Increase weathercock frequency.

(c) Increase weathercock damping.

(d) Reduce cross-coupling between pitch and yaw motion and

(e) Assistance in gathering.

1.2.1 Maintenance of near-constant steady state aerodynamic

gain

A general conclusion can be drawn that an open-loop missile control system is not

acceptable for highly maneuverable missiles, which have very small static margins espe-

cially those which do not operate at a constant height and speed. In homing system,

the performance is seriously degraded if the ”kinematic gain” varies by more than about

+/ − 30 per cent of an ideal value. Since the kinematic gain depends on the control

system gain, the homing head gain and the missile-target relative velocity, and the latter

may not be known very accurately, it is not expected that the missile control designer

will be allowed a tolerance of more than +/− 20 per cent.

4

1.2.2 Increase weathercock frequency

A high weathercock frequency is essential for the stability of the guidance loop.

(a) Consider an open loop system. Since the rest of the loop consists essentially of

two integrations and a d.c. gain, it follows that if there are no dynamic lags in the

loop whatsoever we have 180 deg phase lag at all frequencies open loop.

(b) To obtain stability, the guidance error signal can be passed through phase advance

networks. If one requires more than about 60 degrees phase advance one has to use

several phase advance networks in series and the deterioration in signal-to-noise

ratio is inevitable and catastrophic.

(c) Hence normally designers tend to limit the amount of phase advance to about 60

deg. This means that if one is going to design a guidance loop with a minimum

of 45 deg phase margin, the total phase lag permissible from the missile servo and

the aerodynamics at guidance loop unity gain cross-over frequency will be 15 deg.

(d) Hence the servo must be very much faster and likewise the weathercock frequency

should be much faster (say by a factor of five or more) than the guidance loop

undamped natural frequency i.e., the open-loop unity gain cross-over frequency.

(e) This may not be practicable for an open-loop system especially at the lower end

of the missile speed range and with a small static margin. Hence the requirement

of closed loop system with lateral autopilot arises.

1.2.3 Increase weathercock damping

The weathercock mode is very under-damped, especially with a large static margin and

at high altitudes. This may result in following: -

(a) A badly damped oscillatory mode results in a large r.m.s. output to broadband

noise. The r.m.s. incidence is unnecessarily large and this results in a significant

reduction in range due to induced drag. The accuracy of the missile will also be

degraded.

5

(b) A sudden increase in signal which could occur after a temporary signal fade will

result in a large overshoot both in incidence and in achieved lateral g. This might

cause stalling. Hence the airframe would have to be stressed to stand nearly twice

the maximum designed steady state g.

1.2.4 Reduce cross coupling between pitch and yaw motion

If the missile has two axes of symmetry and there is no roll rate there should be no cross

coupling between the pitch and yaw motion. However many missiles are allowed to roll

freely. Roll rate and incidence in yaw will produce acceleration along z axis. Similarly

roll rate and angular motion induce moments in pitch or yaw axis. These cross coupling

effects can be regarded as disturbances and any closed-loop system will be considerably

less sensitive to any disturbance than an open-loop one.

1.2.5 Assistance in gathering

In a command system, the missile is usually launched some distance off the line of sight.

At the same time, to improve guidance accuracy, the systems engineer will want the

narrowest guidance beam possible. Thrust misalignment, biases and cross winds all

contribute to dispersion of the missile resulting in its loss. A closed-loop missile control

system (i.e., an autopilot) will be able to reasonably resist the above disturbances and

help in proper gathering.

6

Chapter 2

Mathematical Modelling :

Rigid Body Flight Dynamics

2.1 Notations and Conventions

The reference axis system standardized in the guided weapons industry is centred

on the c.g. and fixed in the body as shown in the figure below:

Figure 2.1: Missile Axes [2]

(a) x axis, called the roll axis, forwards, along the axis of symmetry if one exists, but

7

in any case in the plane of symmetry.

(b) y axis called the pitch axis, outwards and to the right if viewing the missile from

behind

(c) z axis, called the yaw axis, downwards in the plane of symmetry to form a right

handed orthogonal system with the other two.

Thus the missile has six degrees of freedom which consists of three translations and

three rotations along the three body axes as shown in figure below:-

Figure 2.2: Missile Axes Depicting Six Degrees of Freedom[2]

Table given below defines the forces and moments acting on the missile, the linear

and angular velocities, and the moments of inertia.

8

*Missile velocity along x-axis U is denoted by a capital letter to emphasise that it is

a large positive quantity changing at most only a few percent per second

(a) Linear velocity ν or V = ui + vj + zk

(b) Rotational velocity ω = pi + qj + rk

(c) Force F = Xi + Yj + Zk

(d) Moments M = Li + Mj + Nk

9

(e) Moments of inertia Ix =∫

(y2 + z2) dm = Σ(y2i + z2

i )mi

(f) Products of inertia Iyz =∫yz dM (when body not symmetrical).

2.2 Equations of Motion

The equations of motion of a missile with controls fixed may be derived from Newton’s

second law of motion, which states that the rate of change of momentum of a body is

proportional to the summation of forces applied to the body and that the rate of change

of the moment of momentum is proportional to the summation of moments applied to

the body. Mathematically, this law of motion may be written as (Reference axis can be

taken as the inertial axis (fixed) x,y,z): -

(a) Summation of Forces

ΣFx =d(mU)

dt(2.1)

ΣFy =d(mV )

dt

ΣFz =d(mW )

dt

(b) Summation of Moments

ΣMx =d(hx)

dt(2.2)

ΣMy =d(hy)

dt

ΣMz =d(hz)

dt

where hx, hy, hz are moments of momentum about x, y and z and may be written

in terms of moments of inertia and products of inertia and angular velocities p,q

and r of the missile as follows: -

hx = pIx − qIxy − rIxz (2.3)

hy = qIy − rIyz − pIxy

hz = rIz − pIxz − qIyz

10

For designing an autopilot, we can consider a particular point in space instead of

considering the complete trajectory (system parameters will not be the same at different

points of the trajectory). In that case, mass can be assumed as constant. Hence the

force equations can be rewritten as

ΣF = mdV

dt(2.4)

where V = ui+ vj + wk.

2.2.1 Euler’s Equations

The equations of motion as per Newton’s laws of motion for translational system are

written about an inertial or fixed axis. They are extremely cumbersome and must be

modified before the motion of the missile can be conveniently analysed. In eqn (1), if

i, j and k are considered as not varying with time, then Newton’s law will no longer be

valid since i, j and k with respect to missile body frame change with time. Hence there

is a requirement for expressing the orientation of the fixed axis co-ordinate system with

respect to another moving axis co-ordinate system i.e., co-ordinate transformations need

to be applied. The moving-axis system called the Eulerian axes or Body axis (for

rotational system) is commonly used. This axis system is a right-handed system of

orthogonal coordinate axes whose origin is at the center of gravity of the missile and

whose orientation is fixed with respect to the missile. The Euler angles are designated

as roll (φ), pitch (θ) and yaw (ψ) and are as shown in the figure below:-

Figure 2.3: Euler Angles [3]

11

The two main reasons for the use of the Eulerian axes in the dynamic analysis of the

airframe are: -

(a) The velocities along these axes are identical to those measured by instruments

mounted in the missile and

(b) The moments and products of inertia are independent of time.

The disadvantage of Eulerian axis system is the mathematical singularity singularity

that exists when the pitch angle approaches 90 degrees.

Another view of the Eulerian angles is as shown in the figure below which also illus-

trates the angular rates of the Euler angles.

Figure 2.4: Euler Angles and Angular Rates [2]

12

2.3 Inertial Form of Force Equation in terms of Eu-

lerian Axes

Since we now consider i, j and k also as variables, the derivative of linear velocity, V, in

the force equation has to cater for the velocity of the center of mass of the missile as well

as the velocity of the element relative to the center of mass. If ω is the angular velocity

of the missile and r is the position of the mass element measured from the c.g. as shown

in figure below, the force equation is given by teh well-known Coriolis equation as

(dV

dt)I = (

dV

dt)B + ω X V (2.5)

Substituting for V in (dVdt

)B and since i, j and k are considered constant in this body

axes form, we get

(dV

dt)B = i

du

dt+ j

dv

dt+ k

dw

dt(2.6)

The cross-product ω X V can now be given as

ω X V = (pi+ qj + rk) X (ui+ vj + zk) (2.7)

or

ω X V = det

i j k

p q r

u v w

(2.8)

13

Expanding the determinant gives

ω X V = i(qw − rv) + j(ru− pw) + k(pv − qu) (2.9)

Substituting equations (2.6) and (2.9) in (2.5) gives

(dV

dt)I = i

du

dt+ i(qw − rv) + j

dv

dt+ j(ru− pw) + k

dw

dt+ k(pv − qu) (2.10)

Hence the Force equation (2.4) can be written/resolved in terms of X, Y and Z compo-

nents acting along x,y and z axes respectively as: -

X = m(du

dt+ (qw − rv)) (2.11)

Y = m(dv

dt+ (ru− pw))

Z = m(dw

dt+ (pv − qu))

2.4 Inertial Form of Moment Equation in terms of

Eulerian Axes

The moments acting on a body are equal to the rate of change of angular momentum

that is given by

M = (dH

dt)I (2.12)

Angular momentum is equal to the moment of linear momentum whereas the linear

momentum is product of mass and velocity where velocity for a rotating mass is the vec-

tor cross product of angular velocity (ω ) and distance from c.g.(r) (Coriolis Equation).

That is

v = ω X r

Linear Momentum=dm ∗ v=dm ∗ (ω X r)

Angular Momentum (dH)=r X Linear Momentum=r X dm ∗ (ω X r)

Hence

H =

∫(r X (ω X r))dm (2.13)

14

Considering ω = pi+ qj + rk and r = xi+ yj + zk, their cross product is given by

ω X r = det

i j k

p q r

x y z

(2.14)

Expanding the determinant gives

ω X r = i(qz − ry) + j(rx− pz) + k(py − qx) (2.15)

The vector cross product r X (ω X r) is now given as

r X (ω X r) = det

i j k

x y z

(qz − ry) (rx− pz) (py − qx)

(2.16)

Expanding the above determinant gives

r X (ω X r) = [p(y2+z2)−qxy−rxz ]i+[q(x2+z2)−ryz−pxy]j+[r(x2+y2)−pxz−qyz]k

(2.17)

Hence the total angular momentum is given by

H =

∫([p(y2+z2)−qxy−rxz]dmi+[q(x2+z2)−ryz−pxy]dmj+[r(x2+y2)−pxz−qyz]dmk)

(2.18)

Defining the moment of inertia along the x,y and z axes respectively as

Ix =

∫(y2 + z2) dm (2.19)

Iy =

∫(x2 + z2) dm

Iz =

∫(x2 + y2) dm

and similarly

Ixy =

∫(xy) dm (2.20)

Ixz =

∫(xz) dm

Iyz =

∫(yz) dm

15

the equation for H can be rewritten as

H = [pIx − qIxy − rIxz ]i+ [qIy − rIyz − pIxy]j + [rIz − pIxz − qIyz]k (2.21)

Thus the moment acting on the body

M = (dH

dt)I (2.22)

can also given by

M = (dH

dt)B + (ω X H) (2.23)

The term (dHdt

)B is given by

dH

dt)B =

d

dt[pIx− qIxy− rIxz ]i+

d

dt[qIy− rIyz−pIxy]j+

d

dt[rIz−pIxz− qIyz]k (2.24)

and the term ω X H is given by

ω X H = [pi+qj+rk]X [[pIx−qIxy−rIxz ]i+[qIy−rIyz−pIxy]j+[rIz−pIxz−qIyz]k] (2.25)

which can be given by

ω X H =

i j k

p q r

(pIx − qIxy − rIxz) (qIy − rIyz − pIxy) (rIz − pIxz − qIyz)

(2.26)

Expanding the determinant we get

ω X H = i[qrIz − qpIxz − q2Iyz − rqIy + r2Iyz + rpIxy] (2.27)

+j[rpIx − rqIxy − r2Ixz − rpIz + p2Ixz + pqIyz]

+k[pqIy − prIyz − p2Ixy − pqIx + q2Ixy + qrIxz]

Hence the Moment equation can be resolved in terms of L, M and N components acting

along x,y and z axes respectively using

M = (dH

dt)B + (ω X H) (2.28)

and

M = Li+Mj +Nk (2.29)

as: -

L = [pIx + pIx − qIxy − q ˙Ixy − rIxz − r ˙Ixz] + [qrIz − qpIxz − q2Iyz − rqIy + r2Iyz + rpIxy](2.30)

M = [qIy + qIy − rIyz − r ˙Iyz − pIxy − p ˙Ixy] + [rpIx − rqIxy − r2Ixz − rpIz + p2Ixz + pqIyz]

N = [rIz + rIz − pIxz − p ˙Ixz − qIyz − q ˙Iyz] + [pqIy − prIyz − p2Ixy − pqIx + q2Ixy + qrIxz]

16

2.5 Mathematical Modeling for Missile Lateral Au-

topilots

It is found from the above equations for force and moments, that these are simultaneous

non-linear coupled first order equations that are difficult to solve. Since we are concerned

with the design of an autopilot for a missile i.e., math-modelling, we try to linearise these

equations by considering certain basic assumptions.

2.5.1 Linearising Moment Equations

The moment equations are linearised based on the following assumptions:-

(a) Mass is constant. (This has already been considered).

(b) Missile and control surfaces are rigid bodies i.e., they are non-elastic. This is not

always true for control surfaces/wings.(This has been already considered).

(c) C.G. and center of body frame are coincident. This is not true since c.g. keeps

changing as propellant burns and msl moves in angles.(This is already considered).

(d) Rate of change of moment inertia is approximately zero i.e.,Ix, Iy, Iz, Ixy, Ixz, Iyz

are zero. Hence moment equations will simplify as

L = [pIx − qIxy − rIxz] + [qrIz − qpIxz − q2Iyz − rqIy + r2Iyz + rpIxy](2.31)

M = [qIy − rIyz − pIxy] + [rpIx − rqIxy − r2Ixz − rpIz + p2Ixz + pqIyz]

N = [rIz − pIxz − qIyz] + [pqIy − prIyz − p2Ixy − pqIx + q2Ixy + qrIxz]

(e) Missile is symmetrical about xz plane. This is true for aircraft and missiles with

mono-wing configuration (cruise or polar coordinate missiles). In this case, Ixy =

Iyz = 0. Thus moment equations will further simplify as:-

L = [pIx − rIxz] + [qrIz − qpIxz − rqIy] (2.32)

M = [qIy] + [rpIx − r2Ixz − rpIz + p2Ixz]

N = [rIz − pIxz] + [pqIy − pqIx + qrIxz]

17

This can be simplified as

L = pIx − qr(Iy − Iz)− (pq + r)Ixz (2.33)

M = qIy]− pr(Iz − Ix) + (p2 − r2)Izx

N = rIz − pq(Ix − Iy) + (qr − p)Ixz

(f) Missile is symmetrical on xy plane; then Ixz = 0 (in case of cruciform configu-

ration). This will not be true for aircraft and cruise missiles. Thus the moment

equations will further simplify as :-

L = pIx − qr(Iy − Iz) (2.34)

M = qIy]− pr(Iz − Ix)

N = rIz − pq(Ix − Iy)

(g) Consider missile to be a solid cylinder. Then the moment of inertia about y and

z axes will be the same i.e., Iz = Iy. Hence equations will further reduce to:-

L = pIx (2.35)

M = qIy]− pr(Iz − Ix)

N = rIz − pq(Ix − Iy)

(h) Missiles are roll-stabilised i.e., roll rate is made zero (p=angular velocity about x

axis = 0). Hence the above equations are reduced to

L = pIx (2.36)

(Note:-p can be zero does not necessarily mean that dp/dt is zero since as shown in

figure below p can be zero at a certain point of time only and have values varying

with time at all other times)

18

M = qIy (2.37)

and

N = rIz (2.38)

2.5.2 Linearising Force Equations

The force equations can be linearised based on the following assumptions:-

(a) The Force equation (2.4) resolved in terms of X, Y and Z components acting along

x,y and z axes respectively was derived as: -

X = m(du

dt+ (qw − rv)) (2.39)

Y = m(dv

dt+ (ru− pw))

Z = m(dw

dt+ (pv − qu))

(i) The term mpw in Y is saying that there is a force in the y direction due to

incidence in pitch ( = w/U) and roll motion i.e., there is an acceleration along

y axis due to to roll rate and incidence in pitch. In other words the pitching

motion of the missile is coupled to the yawing motion on account of roll rate.

19

(ii) The term mpv in Z is also saying that yawing motion induces forces in the

pitch plane if rolling motion is present i.e., acceleration along z axis due to

roll rate and incidence in yaw.

(iii) The presence of the above two terms is most undesirable since we require the

pitch and yaw channels to be completely uncoupled. Cross-coupling between

the planes must contribute to system inaccuracy. To reduce these undesirable

effects the designer tries to keep roll rates as low as possible and in simplified

analysis p is considered zero.

(b) Thus the force equations can be simplified as given below under the assumption

that p is zero:-

X = m(du

dt+ (qw − rv)) (2.40)

Y = m(dv

dt+ ru)

Z = m(dw

dt− qu)

(c) The component of velocity in x direction i.e., u also has thrust along its direction

that is of a larger magnitude. Also, this component of velocity will only add to

the thrust in a small way. Hence u is normally written in capital letters to denote

as a constant quantity. Thus the force equations can be written as

X = m(dU

dt+ (qw − rv)) (2.41)

Y = m(dv

dt+ rU)

Z = m(dw

dt− qU)

(d) Thus it is found that the equation for X is of not much use in control system

since the force (thrust) in the x direction does not affect any maneuver; we are

interested in the acceleration perpendicular to the velocity vector as this will result

in a change in the velocity direction. In any case in order to determine the change

in the forward speed we need to know the magnitude of the propulsive and drag

forces.

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(e) The forces in y and z direction are responsible for yaw and pitch maneuvers. From

the final equations, it can be seen that the Y and Z equations are linear i.e.,

Y = m(dv

dt+ rU) (2.42)

Z = m(dw

dt− qU)

are linear.

2.6 Translational and Rotational Dynamics of Mis-

sile Autopilot

The final simplified equations for forces and moments acting on the missile which rep-

resent the translational and rotational dynamics of the missile respectively are: -

Y = m(dv

dt+ rU) (2.43)

Z = m(dw

dt− qU)

L = pIx

M = qIy

N = rIz

2.6.1 Dynamics of Yaw Autopilot

It can be seen that the equations

Y = m(dv

dt+ rU) (2.44)

N = rIz

are coupled and produce moments about z axis or torque about z axis or the yaw

movement and are used for design of yaw autopilot.

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2.6.2 Dynamics of Pitch Autopilot

Similarly the eqns

Z = m(dw

dt− qU) (2.45)

M = qIy

are for pitching dynamics and are used for design of pitch autopilot.

2.6.3 Dynamics of Roll Autopilot

The roll autopilot dynamics is represented by the equation

L = pIx (2.46)

2.7 Summary

Thus pitch, yaw and roll dynamics have been decoupled. In other words, a multivariable

system has been decomposed into single variable three sets of equations. This is possible

only in missiles. Design of autopilot for aircraft is much more difficult since this kind of

decoupling is not possible.

2.8 Kinematics of the Missile

The equations for the angular velocities i.e., θ, φ and ψ in terms of the Euler angles roll

(θ), pitch (φ) and (ψ) and the rates p, q and r which are the roll rate, pitch rate and

yaw rate respectively can be resloved from the figure given below:-

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Figure 2.5: Euler Angles and Angular Rates [2]

θ = q cos φ− r sin φ (2.47)

φ = p+ (q sin φ+ r cos φ)tan θ

ψ = (q sin φ+ r cos φ)sec θ

Thus if x,y and z are the position variables in the fixed frame, they can be related to

u, v and w in the moving frame by the relations as shown below [2]:-xyz

=

cos θ cos ψ sin φ sin θ cos ψ − cos φ sin ψ cos φ sin θ cos ψ + sin φ sin ψ

cos θ sin ψ sin φ sin θ sin ψ + cos φ cos ψ cos φ sin θ sin ψ − sin φ sin ψ−sin θ sin φ cos θ cos φ cos θ

uvw

(2.48)

As can be seen from the above equation, the above transformation can result in

ambiguities or singularities if θ, φ and ψ → 90 degrees. This can be avoided by limiting

the ranges of the Euler angles accordingly.

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Chapter 3

Control Surface Configurations

3.1 Introduction

The missile has a total of six degrees of freedom of movement. Out of these, three

are translational or linear about the three axis viz x,y and z; while the other degrees

are rotational about the three axes and termed as pitch, yaw and roll. Pitch is the

turn of missile when it climbs up or down. Yaw is its turn to left or right. The roll is

when the missile rotates about its longitudinal axis i.e., one running from nose to tail.

Aerodynamic control can be further classified into Cartesian and polar control methods.

3.1.1 Cartesian Control

In a Cartesian system, the guidance angular error detector produces two signals, a left-

right signal and an up-down signal, which are transmitted to the missile. The method

is also called skid-to-turn (STT) method. Here there are two pairs of control surfaces.

Hence there will be two lift forces acting in perpendicular directions simultaneously and

independently say Fx and Fy. The resultant of Fx and Fy, say F , will help missile to

move towards target. This is as shown in Fig (a).

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3.1.2 Polar Control

The same information could be expressed in polar coordinates i.e., R and φ. The usual

method is to regard the signal as a command to roll through an angle φ measured from

the vertical and then to maneuver outwards by means of the missile’s elevators. The

method is called polar control or twist and steer method or bank-to-turn (BTT) method.

Here there is just one pair of control surface as in aircraft. Hence lift is produced in only

one plane. To achieve the aim of reaching the target, the missile will roll by and angle

φ and so lift changes from say from L to L′. L′ is proportional to the range R of the

target.This is as shown in Fig (b).

Figure 3.1: Missile Configurations [1]

3.2 Comparison of Polar and Cartesian Control Meth-

ods

3.2.1 Advantages of Polar Control over Cartesian control

(a) Weight is less due to elimination of one pair of control surfaces.

(b) Drag also is less due to only one pair of control surfaces.

(c) Due to the above reasons, payload can be increased.

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3.2.2 Disadvantages of Polar control

(a) Controlled maneuvers are not precise since roll, yaw and pitch are simultaneously

coupled or roll, pitch and yaw are cross-coupled. Due to this, the system becomes

a multivariable system with three inputs and three outputs. Hence calculation of

output is not very accurate and the maneuver will not be precise.

(b) Let us consider how course is altered in a glider or aeroplane that follow polar

control.

(i) First the ailerons are used to bank i.e., roll by an angle φ .

(ii) Simultaneously, elevators are used to slightly increase the lift force so that

the vertical component of lift (which will be less than original lift L) equals

the weight. Thus the horizontal component of lift will equal to the total lift

times sin . This causes the flight path to change whereas the heading of the

aircraft (attitude) will remain the same (sideslip).

(iii) To avoid this sideslip and bring the aircraft to the required heading, a small

amount of rudder is applied in an attempt to make the general airflow directly

along the fore and aft axis of the aircraft and in the plane of the wings. Thus

there will be no net side force.

(iv) This is the preferred method of maneuvering since lifting forces are most

efficiently generated perpendicular to the wings: the lift-to-drag ratio is a

maximum in this condition. Also from the passenger’s point of view this is

comfortable maneuver since the total force he experiences is always symmet-

rically through the seat of his pants.

(v) Thus it is found that the polar control is a slow process since the full maneuver

cannot take place until the full bank angle is achieved. This might not be

acceptable with some systems designed to hit fast moving targets.

(vi) Also it is not a very precise method of maneuvering since if the elevators are

moved at the same time as the ailerons, there will be some movement in the

plane perpendicular to the desired one. If one waits until banking is complete,

there is an additional delay.

(c) Hence majority of missiles use Cartesian control method.

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3.2.3 Advantages of Cartesian control

(a) Pitch and yaw channels can be considered as independent two-dimensional problem

with roll being zero unless purposefully introduced. Hence system is more accurate.

(b) Thus Cartesian method is a quicker method of moving laterally in any one direc-

tion. Also analysis of the performance of Cartesian system is simpler.

(c) Polar control cannot be used where roll stabilization is required (in case of homers

where high roll rate may disturb the homing head).

(d) With a Cartesian control system, the pitch control system is made identical to the

yaw control system. Hence we need to discuss one channel only and with missiles,

lateral movement usually means up-down or left-right. With polar control one rolls

and elvates and lateral control equivalent will apply for elevation channel only.

3.3 Classification of Aerodynamic control

Aerodynamic control can be further classified as: -

(a) Rear control surfaces (Tail Control)

(b) Forward control (Canards)

(c) Wing control

3.3.1 Tail Controlled Missiles

27

This is the most commonly used form of missile control.It is used in long-range air-to-

air missile (AMRAAM) and surface-to-air missiles like PATRIOT and ROLAND. Tail

control provides excellent manoeuvrability at high angles of attack, to intercept a highly

manoeuvrable aircraft. These missiles have non-movable wings to provide additional lift

and improve range. Tail control provides lot of convenience in placing the propellant,

warhead and guidance system in the missile. In subsonic missiles, the controls are used

as ”flaps” immediately behind the wings. In supersonic missiles, the tail control surfaces

are placed as far as possible at the rear, to exert maximum moment on the missile.

3.3.2 Canard Controlled Missiles

Canard control surfaces are also called as ”Forward Control” surfaces, where the

control surfaces are kept at the front of the missile. They are commonly used in short-

range air-to-air missiles like AIM - 9L Sidewinder. The primary advantage of canard

control is better manoeuvrability at low angles of attack. The initial control force due

to canard deflection is in the same direction that of the total normal force. However to

retain the stability in the canard configuration, the missiles have large fixed tails kept at

the rear, as far as possible. Hence canard controlled missiles do not have conventional

wings. Canard controls are not generally used for homing missiles; since the homing

head occupies more space in the front, leaving little room for the canard servos. Canard

controlled missiles have difficulties in performing roll control. If the missile has to be

rolled to a certain angle (Polar control), the roll effect produced by the canards is nullified

by the fixed tail surfaces.

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3.3.3 Wing Controlled Missiles

It is the earliest form of missile control, but less common in today’s design. The

primary advantage of wing control is that the deflections of the wings produce a very fast

response with little motion of the body. This feature results in small seeker tracking error

and allows the missile to remain locked on to the target even during large manoeuvres.

Long-range missiles like Sparrow, Sea Skua etc use this type of control. The major

disadvantage is that the wings must usually be quite large in order to generate both

sufficient lift and control effectiveness, which makes the missiles rather large overall.

Further bigger wings require bigger servos for actuation and hence the power requirement

to operate them is also high. Wing control also generates strong vortices that may

adversely interact with the tails causing the missile to roll, known as induced roll. If the

induced roll is strong enough, a separate control system is required to compensate it.

29

Figure 3.2: Missile Aerodynamic Design Characteristics [1]

30

3.4 Comparison of Forward and Tail Control Mis-

siles

In tail control, c.p. is behind c.g. and N and Nc are oriented opposite to each other

(N in upward direction and Nc in downward direction). Whereas, in forward or canard

control, c.p. is ahead of c.g. and N and Nc are in the same directions (upwards). Thus

the net force NT in case of tail control will be N − Nc and in case of forward control

N +Nc.

3.4.1 Advantages of Forward Control

(a) Maneuverability of forward control missile is higher than that with tail control

since NT produced (N+Nc) is greater than that of tail control(N-Nc).

(b) The speed of response is higher with forward control and missile starts nose-up

immediately since control surfaces move in same direction. Whereas with tail

control since control surfaces move in opposite directions, there is a lag and hence

speed of response is low.

(c) Forward control is a minimum phase transfer function model that is easy to design;

whereas tail control is a non-minimum phase transfer function model for which

design of compensator is difficult.

3.4.2 Drawbacks of Forward Control

But most missiles have tail control due to following drawbacks in forward control: -

(a) Downwash and wake effects. Local Mach number is proportional to square of

velocity of air and hence reduces near the wings due to turbulence (downwash)

created by the presence of forward control surfaces. Since wings produce maximum

lift, this will affect lift produced and hence affect maneuverability. Also this may

cause roll reversal when the torque generated by wing becomes greater than that

produced by control surfaces. (Cross-configuration reduces this downwash effect).

Hence canards are not used for roll control.

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(b) Location of two pairs of control surfaces difficult due to homing head being placed

in the same region thus creating problem for placement of actuating mechanism.

3.5 Control Surface Configuration for Cartesian Mis-

siles

The cruciform configuration of missile will be considered for those using Cartesian Con-

trol. This can be of two types: the plus(+)-type and the cross(X)-type.

3.5.1 Plus Type Cruciform Configuration

In case of plus type, the control surfaces are designated clockwise from right as 1, 2, 3 and

4. Here, 1 and 3 will act as rudders or yaw controls and when deflected will maneuver

the missile in a flat turn, right or left.. 2 and 4, likewise, will act as elevators or pitch

controls and when deflected will maneuver the missile vertically, up or down.

Figure 3.3: Plus Type Cruciform Configuration [1]

In the plus type cruciform missile, the control surfaces are designated clockwise from

right as 1, 2, 3 and 4 and their deflections are designated ξ1,ξ2,ξ3 and ξ4. The sign

convention to be followed is that when right hand is holding the respective control

32

surface of the missile with thumb pointing away from the body of the missile, then if

the deflection is in the direction towards which the other fingers are pointing then it is

designated positive and if it is in the opposite direction of the fingers it is negative.

Elevator Deflection

The elevator deflection required for pitch up or down movement is achieved through the

movement of the control surfaces ξ1 and ξ3 and is given by the equation

η =1

2(ξ1− ξ3) (3.1)

The figure below shows the control surface orientation for pitch up movement of a canard

controlled missile. (Note: For tail control, the action will be reversed since the c.g. acts

as fulcrum).

Figure 3.4: Control Surface Movement for Pitch Up [1]

Rudder Deflection

The rudder deflection required for yaw right or left movement is achieved through the

movement of the control surfaces ξ2 and ξ4 and is given by the equation

η =1

2(ξ2− ξ4) (3.2)

33

The figure below shows the control surface orientation for yaw left movement of the

missile.

Figure 3.5: Control Surface Movement for Yaw Left [1]

Aileron Deflection

The aileron deflection required for rolling the missile clockwise (right) or anticlockwise

(left) is achieved through the movement of all four control surfaces and is given by the

equation

η =1

2(ξ1 + ξ2 + ξ3 + ξ4) (3.3)

The figure below shows the control surface orientation for roll left movement of the

missile.

34

Figure 3.6: Control Surface Movement for Roll Left [1]

Mathematical Analysis

Thus we have three equations for elevator, rudder and aileron deflection. Since there

are three equations and four unknown variables in them, there is no unique solution for

these three set of equations i.e., there is an infinite set of solutions. Hence we can only

choose the best (or unique) solution from this set of solutions depending on a particular

requirement (e.g., choose ξ1,ξ2,ξ3 and ξ4 such that drag is to be kept minimum). This

is called optimization.

3.5.2 Cross Type Cruciform Configuration

In case of cross type, there is no such demarcation of control surface pair for a particular

maneuver. On the other hand all four surfaces act together for any kind of maneuver i.e.,

pitch or yaw. The control surfaces are in pairs (same hinge for cross surfaces). Hence if

both the sets of control surfaces are deflected by the same attitude, then this will result

35

in pitching action. If both sets are deflected in opposite directions then this will result

in yawing motion.

Figure 3.7: Cross Type Cruciform Configuration [1]

Advantages of Cross Type Configuration

If the control force Nc = x newtons in + configuration of a particular missile and

the same missile is configured into x type, then Nc will be√

2 times x; hence control

effectiveness has been increased by√

2 times without changing the control surfaces. Also,

the magnitude of each set of control deflections is approximately 0.7 of that required for

the case of plus type.

Drawbacks

Here there is a need to deflect all four control surfaces; hence power consumption from

the servos and the drag forces due to the control surfaces will be higher than for plus

type. Also for roll control ailerons will have to be incorporated.

36

3.6 Sign Convention for Moments

For standardisation of sign convention for moments, consider the right fist to be holding

either x, y or z-axis with the thumb pointing towards positive direction of respective

axes. Then, the direction in which the other fingers are pointing or facing decides

positive value of moments.

37

References

[1] P. Garnell, Guided Weapon Control Systems. London: Brassey’s Defence Publishers,

1980.

[2] G. M. Siouris, Missile Guidance and Control Systems. New York: Springer, 2003.

[3] R. Yanushevsky, Modern Missile Guidance. Boca Raton: CRC Press, Taylor Francis

Group, 2008.

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