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MISSILE AUTOPILOT AERODYNAMICS

1. The motion of a guided missile in flight can be resolved into two types:-

(a) The translational motion of centre of gravity of the missile and

(b) The rotational motion about the centre of gravity of the missile.

2. The forces acting on a missile in flight are generally not applied at the centre of gravity (c.g.) of the missile. However, each force can be decomposed into two parts:-

(a) The forces that cause translational motion of c.g. and

(b) The moments that cause rotary motion with respect to c.g.

Forces Acting on a Missile

3. The c.g. moves in flight due to three main forces namely

(a) Engine thrust,

(b) Resultant aerodynamic forces i.e., lift and drag and

(c) Earth’s gravitational force.

4. The forces acting on a missile are pictorially depicted in Fig.1 below:

V is the velocity vector.

T is the engine thrust.

D is the drag force (aerodynamic force component acting opposite to velocity vector).

L is the lift (aerodynamic force component directed perpendicular to velocity vector).

G is the gravitational force acting vertically downwards .

Θ is the trajectory angle or angle between velocity vector and horizontal.

ν is the pitch angle or angle between thrust vector and horizontal. This determines the position of the missile longitudinal axis with respect to horizontal.

R is the resultant of the aerodynamic forces L and D.

5. Now the forces are resolved into two components namely

(a) The components acting in the direction of flight, FT and

(b) The components acting perpendicular to the direction of flight, FN.

6. The sum of all the force components acting in the direction of flight is given by the equation

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FT=−G∗sin θ−D+T∗cosα ____________________(1)

While the sum of all the force components acting perpendicular to the direction of flight is given by

FN=−G∗cosθ+L+T∗sinα ____________________(2)

The force FT changes the value of the velocity vector while the force FN changes the direction of motion of the missile. Hence in order to control the direction of motion of the missile, it is required to control FN.

7. Since the value of angle of attack during the missile flight is comparatively small, it can be assumed that sin α=α. Thus equation (2) can be written as

FN=−G∗cosθ+L+Tα ______________________(3)

8. For small angles of attack, the lift of the guided missile is given by

L=lα α _______________________________________(4)

Where lα is a constant of proportionality between lift and angle of attack.

9. Substituting for L, eqn.(3) becomes

FN=−G∗cosθ+(l¿¿α+T )α ¿ ______________________(5)

10. The gravity force component –G*cos θ cannot be used as a control force. Thus the direction of motion of the missile can be controlled by changing the component (l¿¿α+T )α ¿ which consists of the lift and thrust components. For a constant thrust, T, this force FN will be proportional to the angle of attack, α. Hence it is required to create an angle of attack in order to guide the missile in flight.

11. The angle of attack can be changed by providing canard or tail control. In case of canard control, the nose part of the missile is provided with ruddervators which control the translational (pitch and yaw) (and ailerons for rotary (roll)) motion of the missile with respect to c.g. by changing the angle of attack.

Moments Acting on a Missile in Flight

12. The moments that cause rotational motion of the missile in flight can be due to:

(a) Control moments of ruddervators and ailerons and

(b) Disturbing and stabilising moments.

13. Control moments of ruddervators and ailerons. The guided missile aerodynamics consists of two pairs of ruddervators located towards the nose part which control the pitch and yaw motion of the missile and hence its trajectory of flight. The two ailerons located on the trailing edges of the wings ensure roll angle stabilisation. If the aileron deviates through an angle δ from its equilibrium position, this creates an angle of attack due to which lift is generated by the aileron. This is shown in Fig. 2.

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(a) Considering that the lift, La, is generated along the Y axis, the aileron generates two moments, one along the X axis and the other along the Z axis. The moment along X axis is given by the product of the lift, La, and the distance between the centre of pressure (c.p.) of aileron and centre of gravity (c.g.) of the missile along the X axis. The moment along Z axis is given by the product of the lift, La, and the distance between the c.p. of aileron and c.g. of missile along Z axis.

(b) The ailerons 1 and 2 are interconnected. Hence any deflection of one of them creates a deflection exactly opposite on the other, i.e., if one of them deflects upward, the other deflects downward. Hence when the sum of moments of lift force of both the ailerons acting on the missile are considered, the moments along the lateral axis (Z) are opposite to each other and will cancel out. The moments along the longitudinal axis (X) will add up and equal to 2* La *lx.

(c) The ruddervators 3 and 4, as well as 5 and 6 are connected in pairs and turn together but in opposite directions. In a similar fashion to the aileron, it can be shown that the moments created by any deflection to the ruddervators 3 and 4 creates a resultant moment which acts only along the Z-axis and similarly, the deflection to the ruddervators 5 and 6 creates a resultant moment which acts along Y-axis.

(d) The control moments of the ruddervators, Mrud, and that of the ailerons, M xd, are

proportional to the angles of deflection of the respective ruddervators and ailerons. Hence,

M rud=M δ δ kgf-m _____________________(6)

and

M xd=M x

δ δail kgf-m ___________________(7)

where δ and δail are the deflection angles of ruddervators and ailerons. The maximum angle is

limited to 30 degrees. Mδ and M xδ are proportionality constants given by

M δ=mδ qSw b kgf-m/deg _______________(8)

and

M xδ=mx

δ qSw l kgf-m/deg _________________(9)

where mδ and mδx are efficiency factors of the ruddervators and ailerons,

q is the ram pressure in kgf/m2,

Sw is the wing area of the missile in m,

b is the mean aerodynamic chord in m and

l is the wing span in m.

Factors mδ and m xδ depend on the geometric sizes of the ruddervators and their position

with respect to c.g. of the missile as well as Mach number (M=v/a, where v=missile velocity; a=velocity of sound).

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(e) Signs and directions of ruddervator deflection.

Direction of turn of missile

Direction of deflection of trailing edge Sign of ruddervator deflection

For ruddervators 3 & 4

For ruddervators 5 & 6

For right aileron

Nose up Down Down - + (plus)Nose down Up Up - - (minus)Clockwise turn - - Up - (minus)

14. Disturbing and Stabilising Moments. The missile is subjected to external disturbances like gust of wind etc. , as also aerodynamic moments due to uncertainty in design parameters. Aerodynamic coefficients are seen to vary due to changes in density, height/altitude etc., as the missile traverses different levels of atmosphere. These disturbing moments tend to turn the missile with respect to its c.g. which is undesired. Stabilising moments are those moments which counteract the disturbing moments thus preventing the turning of the missile and helping it to retain its initial position. This is generally called roll stabilisation.

15. Stabilising moments can be categorised into the following three types:-

(a) Static stability moments

(b) Damping moments and

(c) Inertia moments.

16. Static Stability Moments. Static stability means that when an external force is applied the missile automatically comes back to the equilibrium position at which the sum of moments equals to zero. The missile can be designed to be statically stable by ensuring that the centre of pressure (c.p.) from where lift and drag act is positioned aft of c.g. i.e., closer to the tail. Thus if the external force makes the missile nose to turn upward, the lift force will create a moment which will force the missile nose downwards and back to its initial or equilibrium position. Thus the sign of the stabilising moments is opposite to that of the external disturbing moments.

(a) Since lift is proportional to angle of attack, the static balancing moment, Ms, will also be proportional to the angle of attack. Thus

M s=−M ααkgf-m ________________________(10)

Where M α is the proportionality factor given by

M α=mα qSw b kgf-m/rad __________________(11)

(b) Since the missile body is symmetrical about the longitudinal or X-axis (axial symmetry), longitudinal stability does not exist under the consideration that the missile is neutral relative to X-axis.

17. Damping Moments. Damping moments are due to the interaction of air flowing about the guided missile, particularly when the missile is turning through an angular rate, ω. The resultant angular rate vector, ω⃗, can be resolved along the pitch (ωp), yaw (ωy) and roll (ωx) axes. The

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damping moments which counteract these components are given by M pd , M y

d and M xd. Thus the

damping moments can be expressed by the equations

M pd=M y

d=−Mωω _______________________(12)

With regard to the lateral axis (since missile is symmetrical) and

M xd=−M x

ωωx kgf-m ______________________(13)

With regard to the longitudinal axis. Mω and M xω are coefficients of proportionality given by

Mω=mωq Sw b kgf.m/sec __________________(14)

M xω=m x

ωq Sw l kgf.m/sec __________________(15)

The factors mω and m xω depend on geometrical characteristics of missile and Mach number.

18. Inertia Moments. Inertia moments are similar to damping moments and are caused due to interaction of air flowing about the guided missile. The difference is that they are caused due

to the angular acceleration, dωdt

, unlike the damping moments which are due to the angular rate, ω.

Thus inertia moments are proportional to the angular acceleration component at any given instant.

(a) In case of lateral axis, the inertia moment is given as

M i=−J dωd t

kgf-m ______________________(16)

(b) In case of longitudinal axis, the inertia moment is given by the equation

M ix=−J xd ωx

dt kgf-m ________________(17)

Where J and J x are the guided missile inertia moments.

In short, inertia moments counterbalance the external disturbance caused by interaction of air due to angular acceleration caused during the rotary motion of the missile.

19. The tendency of the missile to rotate or turn, is thus due the sum of the control and stabilising moments. The above equations can now be combined together to give a general expression for the moments acting on the guided missile at any given instant. Thus,

M rud=M s+M d+M i ____________________(18)

and

M ail=M xd+M ix _________________________(19)

From equations (6) and (7),

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M rud=M δ δ kgf-m _____________________(6)

and

M xd=M x

δ δail kgf-m ___________________(7)

where

M δ=mδ qSw b kgf-m/deg _______________(8)

and

M xδ=mx

δ qSw l kgf-m/deg _________________(9)

where q is the ram pressure in kgf/m2,

Sw is the wing area of the missile in m,

b is the mean aerodynamic chord in m and

l is the wing span in m.

20. Substituting for M rud , M s, etc., in eqns. (14) and (15), we can rewrite the equations as

M δ δ=M α α+Mωω+J dωdt

______________________(20)

M xδ δ ail=M x

ωωx+J xdωx

dt ________________________(21)

21. From Fig.1, through the geometry of the figure,

ν=θ+α ____________________________________(22)

Also,

ω=dνdt

dωdt

=d2 νdt2

ωx=d ν xdt

d ωx

dt=d2 νxdt 2

Thus the full differential equations for the moments acting on the guided missile along the longitudinal and lateral axes are given by

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M δ δ=M α α+Mωdνdt

+J d2 νdt 2

______________________(23)

M xδ δ ail=M x

ω d νxdt

+J xd2 ν xdt2

________________________(24)

22. Hinge Moments of Ruddervators and Ailerons. The lift forces, L, generated by the ruddervators and ailerons create the control moments and also the moments with respect to the rotational axes of the ruddervators and ailerons, which are called hinge moments.

(a) The values of the hinge moments depend on the distance, lp, from the c.p. to the hinge axis and on the value of lift force, L. The lift and the c.p. change during the flight since they depend on ram pressure, q, Mach number, M, the ruddervator turning angle, δ, the aileron turning angle, δail, and the angle of attack, α.

(b) The ruddervator hinge moments can be expressed as the sum of two components as

M h=M hα α+M h

δ δ _________________________(25)

where M hα is the hinge moment component due to angle of attack, α and

M hδ is the hinge moment component due to missile turning angle, δ.

(c) The aileron hinge moment is proportional to the turning angle of the ailerons and given by

M h=M hδail δail _______________________________(26)

where M hδail is hinge moment factor with respect to aileron deflection angle.

(d) The centre of pressures of all the ruddervators are positioned aft of their hinge axes (closer to the tail). Due to this fact, the hinge moments tend to set the ruddervators in the zero position (feathering effect). Hence, in order to deflect the ruddervator through the required angle during flight, the control surface actuator of the autopilot must overcome the respective hinge moment of the ruddervator. The moment developed by actuator hence is made higher than the value of hinge moment so that the travel, σ, of the actuator rod depends only on the control signal and not on the hinge moment.

(e) Since the hinge moments change with flying conditions, they can be used for adjusting the control moments by using spring mechanisms arranged between the rods of the control surface actuators and the corresponding ruddervators. Thus as the control moment acting on the missile increases with ram pressure, this moment is automatically compensated by reducing the ruddervator deflection angle which is caused by compression of the spring mechanism under the action of the hinge moment. This ensures a constant relation between the control signal and the rate of turn of the guided missile under all conditions of flight of the guided missile.

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SYSTEMS OF EQUATIONS FOR RADIO CONTROLLED FLIGHT OF GUIDED MISSILE ON KINEMATIC TRAJECTORY WITHOUT STABILISATION

23. The motion of the guided missile with respect to centre of gravity and on the kinematic trajectory can be shown to be determined by a system of three differential equations as shown below:-

(a) Equation of Moments

M δ δ=M α α+Mωdνdt

+J d2 νdt 2

______________________(23)

(b) Force Equation

MVdθdt

=( y¿¿α+T )α ¿__________________________(24)

(c) Geometry (Angles Equation)

ν=θ+α ____________________________________(22)

24. Thus the dependent variables are θ, ν and α or the angular co-ordinates, while the independent variable is the ruddervator control moment M δ δ . From eqn.(23), it is seen that θ, ν and α not only determine the position of the missile but also its velocity vector at any given instant.

25. The relation between the ruddervator deflection angle and the radio command control signal is called the control law. This is given by the equation

δ=K AP KSPR R_________________________________(25)

Where KAP is the autopilot coefficient, KSPR is the spring coefficient which depends on hinge moments and R is the radio command control signal.

26. As was previously seen, to ensure stability of the missile, the centre of pressure is positioned aft of c.g. i.e., closer to tail. Thus the stabilising moment sign was shown to be opposite to the sign of the external moment applied, i.e., static balancing moment,

M st=−M αα kgf-m___________________________(26)

27. Thus any change in the control input signal, R, will result in a corresponding change in δ. This will result in change in θ, ν and α. The change in α will be countered by a static balancing moment of opposite sign and will act till the time the missile comes back to its original α.

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28. Thus if the radio controlled flight involves high manoeuvrability i.e., change in δ frequently, this will result in a highly oscillating motion of the missile with respect to c.g. if there is no other stabilisation applied to the missile. Therefore, if the missile is designed to be highly maneuverable, then it requires to be stabilised by employing an autopilot.

Note: M α α is made zero by the stabilising moment. Thus eqn.(23) will have no constant term thus resulting in a pole at origin. This explains the oscillatory motion of the missile mathematically.

SYSTEMS OF EQUATIONS FOR RADIO CONTROLLED FLIGHT OF GUIDED MISSILE ON KINEMATIC TRAJECTORY WITH STABILISATION

29. The oscillatory motion of the missile when it is not stabilised will cause the following problems:-

(a) Flying speed is reduced due to oscillatory motion

(b) Guidance process becomes difficult.

(c) The guidance accuracy worsens.

30. Thus there is a requirement for stabilisation or reducing the oscillatory motion of the missile. Stabilisation can be achieved using an autopilot. The general principle of working of the autopilot is as given below:-

(a) The autopilot consists of sensing elements which are generally angular rate sensors. The missile angular rates are quite high during oscillations.

(b) The control surface actuators of the autopilot receive the control signal from the angular rate sensors. This causes deflection of the ruddervators at an angle proportional to the angular rate.

(c) Thus the moments of the ruddervators act in a direction opposite to the direction of angular rate.

(d) This results in an induced damping moment which prevents the oscillation of the missile. The higher the autopilot coefficient, the higher is the angular rate.

(e) The dependence of ruddervator deflection angle on the angular rate is called the “stabilisation law” and is given by the equation

δ (s)=−K DG KSPR s γ (s) __________________________________(27)

Where KDG is the autopilot coefficient due to the damping gyro (angular rate sensor).

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31. The principle and construction of an autopilot in case of a command guided missile is as given below:-

(a) The input to the autopilot is the radio control command signal from the ground radar and the information from the sensing elements of the autopilot itself.

(b) This control signal is applied to the actuator which converts the electrical signal into angular displacement of the ruddervator.

(c) Spring mechanisms are provided at the hinges of ruddervator which provide automatic correction of the ruddervator angle with respect to hinge moment thus resulting in preservation of the relation between the resultant control signal and motion of the ruddervator at any given operating condition.

STABILISATION OF GUIDED MISSILE IN LAUNCH PHASE

32. During the launch phase, the missile cannot be radio controlled. This is because the guided missile is influenced by moments due to the force of wind as well as due to forces of its aerodynamic asymmetry and disturbances caused as the missile leaves the launcher. In this case, the autopilot stabilises the missile trajectory. The autopilot basically ensures that the guided missile flies along a straight trajectory at a constant angle θ. This stabilisation along a straight trajectory is required in order to reduce any transient errors introduced during the launch phase of the missile and ensure that the missile enters the antenna beam of the radar for its radio control after lock-on.

33. The trajectory angle, θ, is computed in the earth-based system of co-ordinate axis. This cannot be measured in flight inside the guided missile. Hence the pitch angle, γ, is used. Pitch angle can be obtained by integrating the angular turn rate which is measured by the angular rate sensors. Thus the disturbing moments are counterbalanced by the stabilisation system which uses angular rate signal and the integral of the angular rate of the missile as it turns with respect to Y and Z body axes.

34. When the guided missile is launched, the integrating circuit of the angular rate signal is switched on. The zero position of the ruddervators will correspond to the pitch angle of the guided missile on the launcher at the instant of launch. Thus if there is any deviation from the launch direction after launch, the ruddervator will deflect proportionally with the change of pitch angle and remain in that deflected position until the longitudinal axis of the missile returns to its initial position. Thus the guided missile trajectory is stabilised during launch phase.

RADIO CONTROLLED FLIGHT

35. During the radio controlled flight, the control signal is composed of the difference between the radio command control signal and the stabilisation signal from the angular rate sensor. The stabilisation signal ensures that the transient processes are rapidly damped and the guided missile follows the trajectory dictated by the radio control signal quickly.

STABILISATION OF ROLL ANGLE

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36. Since command guidance method is used, the missile is required to be roll-stabilised in angles. The roll angle is measured between the body axis system and the conditionally stationary system of co-ordinate axes in the lateral Z-axis plane, using a roll sensor. This can be a free gyro in the autopilot. Any roll signal causes the displacement of the rod connected to the missile ailerons. Thus the ailerons are deflected from zero position. The moments caused due to this deflection turns the missile along its body X-axis until the roll angle reduced to minimum permissible value.

37. However roll angle stabilisation cannot be achieved only with the application of the free gyro signal. At times small values of damping moments, say, due to external disturbance, along X-axis may result in divergent oscillations which may cause the missile to turn unnecessarily resulting in loss of control. Hence to ensure steady stabilisation, angular rate sensor signal is also added to the roll angle signal. The resulting signal causes the ailerons to deflect through an additional angle which creates a moment for artificial damping of the oscillations.

38. The roll autopilot ensures that a constant roll angle is maintained with respect to the conditionally stationary system of co-ordinate axes. Thus the system of co-ordinate axes of the missile in flight is correlated with the system of axes of the ground based missile guidance system.

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REFERENCES

1. SIOURIS, G.M.,”TACTICAL MISSILE GUIDANCE AND CONTROL”

2. CHIN, ”MISSILE CONFIGURATION DESIGN”

3. GARNELL, P., “GUIDED MISSILE CONTROL SYSTEM”

4. BLAKELOCK, J.H., “AUTOMATIC CONTROL OF AIRCRAFT AND MISSILES”