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 misalignment of the missile from the line of sight (LOS). The effectiveness of a guided missile weapon system, in terms of accuracy and probability 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 depends 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 gimbaled motors, flexible nozzles (ball and socket), interference methods (spoilers/vanes), secondary fluid or gas injection and vernier engines (external or extra engines). Aerodynamic control methods are generally used for tactical missiles.
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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
11
Euler’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
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
2.6
3 Control Surface Configurations 3.1
3.2
3.3
25
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.1
Cartesian Control . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.2
Polar Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
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
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
3.5
3.6
Comparison of Forward and Tail Control Missiles . . . . . . . . . . . . .
32
3.4.1
Advantages of Forward Control . . . . . . . . . . . . . . . . . . .
32
3.4.2
Drawbacks of Forward Control . . . . . . . . . . . . . . . . . . . .
32
Control Surface Configuration for Cartesian Missiles . . . . . . . . . . . .
33
3.5.1
Plus Type Cruciform Configuration . . . . . . . . . . . . . . . . .
33
3.5.2
Cross Type Cruciform Configuration . . . . . . . . . . . . . . . .
36
Sign Convention for Moments . . . . . . . . . . . . . . . . . . . . . . . .
38
References
39
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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 consistent ”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 (unamplified) 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 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 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: -
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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 K1 K2 K3 m/s2 /m or K1 K2 K3 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.
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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 21 ρu2 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 especially 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.
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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.
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(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.
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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
R (e) Moments of inertia Ix = (y 2 + z 2 ) dm = Σ(yi2 + zi2 )mi (f) Products of inertia Iyz =
2.2
R
yz dM (when body not symmetrical).
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 d(mU ) dt d(mV ) ΣFy = dt d(mW ) ΣFz = dt
(2.1)
ΣFx =
(b) Summation of Moments d(hx ) dt d(hy ) ΣMy = dt d(hz ) ΣMz = dt
ΣMx =
(2.2)
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 = m
dV dt
(2.4)
ˆ where V = uˆi + vˆj + 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 ˆi, ˆj and kˆ are considered as not varying with time, then Newton’s law will no longer be valid since ˆi, ˆj and kˆ 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]
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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 illustrates the angular rates of the Euler angles.
Figure 2.4: Euler Angles and Angular Rates [2]
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2.3
Inertial Form of Force Equation in terms of Eulerian Axes
Since we now consider ˆi, ˆj and kˆ 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 dV )I = ( )B + ω X V dt dt
(2.5)
Substituting for V in ( dV ) and since ˆi, ˆj and kˆ are considered constant in this body dt B axes form, we get (
dV du dv dw )B = ˆi + ˆj + kˆ dt dt dt dt
(2.6)
The cross-product ω X V can now be given as ˆ X (uˆi + vˆj + z k) ˆ ω X V = (pˆi + qˆj + rk)
(2.7)
or
ˆi ˆj kˆ
ω X V = det p q r u v w
(2.8)
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Expanding the determinant gives ˆ ω X V = ˆi(qw − rv) + ˆj(ru − pw) + k(pv − qu)
(2.9)
Substituting equations (2.6) and (2.9) in (2.5) gives (
du dv dw ˆ dV )I = ˆi + ˆi(qw − rv) + ˆj + ˆj(ru − pw) + kˆ + k(pv − qu) dt dt dt dt
(2.10)
Hence the Force equation (2.4) can be written/resolved in terms of X, Y and Z components acting along x,y and z axes respectively as: du + (qw − rv)) dt dv Y = m( + (ru − pw)) dt dw Z = m( + (pv − qu)) dt
X = m(
2.4
(2.11)
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 )I dt
(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 vector cross product of angular velocity (ω ) and distance from c.g.(r) (Coriolis Equation). That is vˆ = ω ˆ X rˆ Linear Momentum=dm ∗ vˆ=dm ∗ (ω X r) Angular Momentum (dH)=ˆ r X Linear Momentum=ˆ r X dm ∗ (ˆ ω X rˆ) Hence Z H=
(ˆ r X (ˆ ω X rˆ))dm
(2.13) 14
ˆ their cross product is given by Considering ω ˆ = pˆi + qˆj + rkˆ and rˆ = xˆi + yˆj + z k, ˆi ˆj kˆ ω ˆ X rˆ = det p q r (2.14) x y z Expanding the determinant gives ˆ ω ˆ X rˆ = ˆi(qz − ry) + ˆj(rx − pz) + k(py − qx) The vector cross product rˆ X (ˆ ω X rˆ) is now given as ˆi ˆj kˆ rˆ X (ˆ ω X rˆ) = det x y z
(2.15)
(2.16)
(qz − ry) (rx − pz) (py − qx) Expanding the above determinant gives rˆ X (ˆ ω X rˆ) = [p(y 2 +z 2 )−qxy−rxz]ˆi+[q(x2 +z 2 )−ryz−pxy]ˆj+[r(x2 +y 2 )−pxz−qyz]kˆ (2.17) Hence the total angular momentum is given by Z ˆ H = ([p(y 2 +z 2 )−qxy−rxz]dmˆi+[q(x2 +z 2 )−ryz−pxy]dmˆj+[r(x2 +y 2 )−pxz−qyz]dmk) (2.18) Defining the moment of inertia along the x,y and z axes respectively as Z
(y 2 + z 2 ) dm
Ix = Z Iy = Z Iz =
(2.19)
(x2 + z 2 ) dm (x2 + y 2 ) dm
and similarly Z Ixy =
(xy) dm
(2.20)
Z Ixz =
(xz) dm Z
Iyz =
(yz) dm 15
the equation for H can be rewritten as H = [pIx − qIxy − rIxz ]ˆi + [qIy − rIyz − pIxy ]ˆj + [rIz − pIxz − qIyz ]kˆ
(2.21)
Thus the moment acting on the body dH )I M =( dt can also given by dH M =( )B + (ω X H) dt The term ( dH ) is given by dt B
(2.22)
(2.23)
dH d d d )B = [pIx − qIxy − rIxz ]ˆi + [qIy − rIyz − pIxy ]ˆj + [rIz − pIxz − qIyz ]kˆ (2.24) dt dt dt dt and the term ω X H is given by ˆ X [[pIx −qIxy −rIxz ]ˆi+[qIy −rIyz −pIxy ]ˆj+[rIz −pIxz −qIyz ]k] ˆ (2.25) ω X H = [pˆi+qˆj+rk] which can be given by
ˆi
ˆj
kˆ
ω X H=
p
q
r
(pIx − qIxy − rIxz ) (qIy − rIyz − pIxy ) (rIz − pIxz − qIyz )
(2.26)
Expanding the determinant we get ω X H = ˆi[qrIz − qpIxz − q 2 Iyz − rqIy + r2 Iyz + rpIxy ]
(2.27)
+ˆj[rpIx − rqIxy − r2 Ixz − rpIz + p2 Ixz + pqIyz ] 2 2 ˆ +k[pqI y − prIyz − p Ixy − pqIx + q Ixy + 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 dH )B + (ω X H) M =( dt and
(2.28)
M = Lˆi + M ˆj + N kˆ
(2.29)
as: ˙ − rI ˙ ] + [qrIz − qpIxz − q 2 Iyz − rqIy + r2 Iyz + rpIxy(2.30) L = [pI ˙ x + pI˙x − qI ˙ xy − q Ixy ˙ xz − rIxz ] ˙ − pI ˙ ] + [rpIx − rqIxy − r2 Ixz − rpIz + p2 Ixz + pqIyz ] M = [qI ˙ y + q I˙y − rI ˙ yz − rIyz ˙ xy − pIxy ˙ − qI ˙ ] + [pqIy − prIyz − p2 Ixy − pqIx + q 2 Ixy + qrIxz ] N = [rI ˙ z + rI˙z − pI ˙ xz − pIxz ˙ yz − q Iyz 16
2.5
Mathematical Modeling for Missile Lateral Autopilots
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.,I˙x , I˙y , I˙z , I˙xy , I˙xz , I˙yz are zero. Hence moment equations will simplify as
L = [pI ˙ x − qI ˙ xy − rI ˙ xz ] + [qrIz − qpIxz − q 2 Iyz − rqIy + r2 Iyz + rpIxy ](2.31) M = [qI ˙ y − rI ˙ yz − pI ˙ xy ] + [rpIx − rqIxy − r2 Ixz − rpIz + p2 Ixz + pqIyz ] N = [rI ˙ z − pI ˙ xz − qI ˙ yz ] + [pqIy − prIyz − p2 Ixy − pqIx + q 2 Ixy + 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 = [pI ˙ x − rI ˙ xz ] + [qrIz − qpIxz − rqIy ] M = [qI ˙ y ] + [rpIx − r2 Ixz − rpIz + p2 Ixz ] N = [rI ˙ z − pI ˙ xz ] + [pqIy − pqIx + qrIxz ] 17
(2.32)
This can be simplified as L = pI ˙ x − qr(Iy − Iz ) − (pq + r)I ˙ xz
(2.33)
M = qI ˙ y ] − pr(Iz − Ix ) + (p2 − r2 )Izx N = rI ˙ z − pq(Ix − Iy ) + (qr − p)I ˙ xz (f) Missile is symmetrical on xy plane; then Ixz = 0 (in case of cruciform configuration). This will not be true for aircraft and cruise missiles. Thus the moment equations will further simplify as :L = pI ˙ x − qr(Iy − Iz )
(2.34)
M = qI ˙ y ] − pr(Iz − Ix ) N = rI ˙ z − 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 = pI ˙ x
(2.35)
M = qI ˙ y ] − pr(Iz − Ix ) N = rI ˙ z − 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 = pI ˙ x
(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)
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M = qI ˙ y
(2.37)
N = rI ˙ z
(2.38)
and
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: du + (qw − rv)) dt dv Y = m( + (ru − pw)) dt dw Z = m( + (pv − qu)) dt
X = m(
(2.39)
(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.
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(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:du + (qw − rv)) dt dv Y = m( + ru) dt dw Z = m( − qu) dt
(2.40)
X = m(
(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 dU + (qw − rv)) dt dv Y = m( + rU ) dt dw Z = m( − qU ) dt
(2.41)
X = m(
(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., dv + rU ) dt dw Z = m( − qU ) dt
(2.42)
Y = m(
are linear.
2.6
Translational and Rotational Dynamics of Missile Autopilot
The final simplified equations for forces and moments acting on the missile which represent the translational and rotational dynamics of the missile respectively are: dv + rU ) dt dw Z = m( − qU ) dt L = pI ˙ x Y = m(
(2.43)
M = qI ˙ y N = rI ˙ z
2.6.1
Dynamics of Yaw Autopilot
It can be seen that the equations Y = m(
dv + rU ) dt N = rI ˙ z
(2.44)
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 − qU ) dt M = qI ˙ y
(2.45)
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 = pI ˙ x
2.7
(2.46)
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
˙ φ˙ and ψ˙ in terms of the Euler angles roll The equations for the angular velocities i.e., θ, (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 φ˙ = p + (q sin φ + r cos φ)tan θ ψ˙ = (q sin φ + r cos φ)sec θ
(2.47)
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]: x˙ cos θ cos ψ sin φ sin θ cos ψ − cos φ sin ψ cos φ sin θ cos ψ + sin φ sin y˙ = cos θ sin ψ sin φ sin θ sin ψ + cos φ cos ψ cos φ sin θ sin ψ − sin φ sin z˙ −sin θ sin φ cos θ cos φ cos θ
u ψ v w
ψ
(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 leftright 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 L0 . L0 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 Methods
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. 25
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 symmetrically 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 direction. 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
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This is the most commonly used form of missile control.It is used in long-range air-toair 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 shortrange 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.
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Figure 3.2: Missile Aerodynamic Design Characteristics [1]
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3.4
Comparison of Forward and Tail Control Missiles
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 Missiles
The cruciform configuration of missile will be considered for those using Cartesian Control. 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 1 η = (ξ1 − ξ3) 2
(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 1 η = (ξ2 − ξ4) 2
(3.2)
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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 1 η = (ξ1 + ξ2 + ξ3 + ξ4) 2
(3.3)
The figure below shows the control surface orientation for roll left movement of the missile.
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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 N c = 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.
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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.
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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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