Missile Lateral Autopilot
— 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) Jan 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).
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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 : Aerodynamic Derivatives and Transfer Functions
7
2.1
Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.1
Euler’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
10
2.3
Inertial Form of Force Equation in terms of Eulerian Axes . . . . . . . .
10
2.4
Inertial Form of Moment Equation in terms of Eulerian Axes . . . . . . .
11
2.5
Mathematical Modeling for Missile Lateral Autopilots . . . . . . . . . . .
14
2.5.1
Linearising Moment Equations . . . . . . . . . . . . . . . . . . . .
15
2.5.2
Linearising Force Equations . . . . . . . . . . . . . . . . . . . . .
17
Translational and Rotational Dynamics of Missile Autopilot . . . . . . .
19
2.6.1
Dynamics of Yaw Autopilot . . . . . . . . . . . . . . . . . . . . .
19
2.6.2
Dynamics of Pitch Autopilot . . . . . . . . . . . . . . . . . . . . .
19
2.6.3
Dynamics of Roll Autopilot . . . . . . . . . . . . . . . . . . . . .
20
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.6
2.7
References
21
4
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 : Aerodynamic Derivatives and Transfer Functions 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 follows: (a) x axis, called the roll axis, forwards, along the axis of symmetry if one exists, but 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. Table given below defines the forces and moments acting on the missile, the linear and angular velocities, and the moments of inertia.
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*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 νor V = ui + vj + zk (b) Rotational velocity ω = pi + qj + rk (c) Force F = Xi + Yj + Zk (d) Moments M = Li + Mj + Nk 8
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 ΣFx =
(2.1)
(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 9
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 a 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 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.
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 is given by (
dV dV )I = ( )B + ω X V dt dt
(2.5) 10
Substituting for V in ( dV ) and since ˆi, ˆj and kˆ are considered constant in this body dt B axes form, we get (
du dv dw dV )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
(2.8)
ω X V = det p q r u v w 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 (
dV du dv dw ˆ )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
(2.11)
X = m(
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 )I dt
(2.12) 11
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). 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)
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 (qz − ry) (rx − pz) (py − qx) 12
(2.15)
(2.16)
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
(2.19)
(x2 + z 2 ) dm
Iy = Z
(x2 + y 2 ) dm
Iz = and similarly Z Ixy =
(xy) dm
(2.20)
Z Ixz =
(xz) dm Z
Iyz =
(yz) dm
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 M =(
dH )I dt
(2.22)
can also given by M =(
dH )B + (ω X H) dt
(2.23) 13
The term ( dH ) is given by dt B 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 M =(
dH )B + (ω X H) dt
(2.28)
and 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
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 14
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 ]
(2.32)
M = [qI ˙ y ] + [rpIx − r2 Ixz − rpIz + p2 Ixz ] N = [rI ˙ z − pI ˙ xz ] + [pqIy − pqIx + qrIxz ] This can be simplified as L = pI ˙ x − qr(Iy − Iz ) − (pq + r)I ˙ xz M = qI ˙ y ] − pr(Iz − Ix ) + (p2 − r2 )Izx N = rI ˙ z − pq(Ix − Iy ) + (qr − p)I ˙ xz
15
(2.33)
(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)
M = qI ˙ y
(2.37) 16
and N = rI ˙ z
2.5.2
(2.38)
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. (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
X = m(
(2.40)
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(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
X = m(
(2.41)
(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. (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.
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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.
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.
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2.6.3
Dynamics of Roll Autopilot
The roll autopilot dynamics is represented by the equation L = pI ˙ x
2.7
(2.46)
Conclusion
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.
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References [1] P. Garnell, Guided Weapon Control Systems. London: Brassey’s Defence Publishers, 1980.
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