Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 2, pp. 279-295, Warsaw 2010

DYNAMICS OF A CONTROLLED ANTI-AIRCRAFT MISSILE

LAUNCHER MOUNTED ON A MOVEABLE BASE

Zbigniew Koruba

Zbigniew Dziopa

Izabela Krzysztofik

Kielce University of Technology, Faculty of Mechatronics and Machine Building, Kielce, Poland

e-mail: ksmzko@tu.kielce.pl; zdziopa@tu.kielce.pl; pssik@tu.kielce.pl

In the work, dynamics of a controlled anti-aircraft missile launcher mo-
untedonamoveablebase (wheeledvehicle) is analysed.Pre-programmed
controls were used for basic motion of the launcher during target inter-
ception, and stabilizing controlswere applied to counteract disturbances
resulting fromthe operationof the anti-aircraft system.This type of con-
trol allows setting the longitudinal axis of themissile with respect to the
target line of sight irrespective of the carrier vehiclemotion. The system
vibrations are due to road-induced excitations and they act directly on
the vehicle, themissile launch and the control of launcher basic motion.
Selected results of computer simulation were presented graphically.

Key words: dynamics, control, launcher, missile

1. Introduction

Modern self-propelled anti-aircraft missile systems with short-range self-
guidedmissiles should be able to detect, identify and track aerial targets when
the launcher-carrier vehicles are in motion. To improve the system accuracy,
it is necessary to apply pre-programmed controls for target detection and cor-
rective controls for target tracking (Dziopa, 2004, 2005, 2006a; Koruba, 2001;
Koruba and Osiecki, 1999).
Synthesis of the launcher control involves determination the impact that

road-induced kinematic excitations may have on the launcher performance. It
is desirable that any transitional process accompanying terrain unevenness (a
bump) be optimally attenuated.
In this paper, motion of a launcher-missile system was considered in a

three-dimensional Euclidean space andEarth’s gravitational field. There were



280 Z. Koruba et al.

six degrees of freedom in the discrete model of the system described by ana-
lytical relations in the form of equations of motion, kinematic relations and
three equations of equilibrium (Dziopa, 2006b,c).

In the general case, a launcher-missile system is not symmetric about the
longitudinal vertical plane going through the centre of the systemmass (Dzio-
pa, 2008; Mishin, 1990; Mitschke, 1972; Svietlitskiy, 1963). The symmetry
refers to selected geometrical dimensions and properties of flexible elements.
In the general case, the inertial characteristic departs from this symmetry.
The launcher turret can rotate with respect to the carrier together with the
guide rail and the missile. The turret rotates in accordance with the azimuth
angle ψpv, which is the turret yaw angle. The turret and the guide rail mo-
unted on it constitute a rotary kinematic pair. The guide rail can rotate with
respect to the turret in accordance with the elevation angle ϑpv, which is the
guide rail pitch angle. This leads to an asymmetric distribution of masses.
The system is reduced to a structural discrete model describing the pheno-
mena that are mechanical excitations in character. The basic motion of the
turret is reduced to basic motion of the carrier. The turret is an object whose
inertial characteristic is dependent on the position of the target with respect
to the anti-aircraft system. The mass of the turret remains constant, but its
moments of inertia andmoments of deviation change.

The launcher wasmodelled as two basic masses, four deformable elements
and a control system, as shown schematically in Fig.1.

Fig. 1. Physical model of the controlled launcher



Dynamics of a controlled anti-aircraft missile... 281

The launcher is a perfectly stiff body with mass mv, moments of inertia
Ivx, Ivy and Ivz and a moment of deviation Ivxz. The launcher is mounted
on the vehicle using four passive elastic-attenuating elements with linear pa-
rameters kv11 and cv11, kv12 and cv12, kv13 and cv13 aswell as kv14 and cv14,
respectively.

The turret is a perfectly stiff body with mass mw and main central mo-
ments of inertia Iwξ′

v
, Iwη′

v
, Iwζ′

v
. The guide rail is also a perfectly stiff body

with mass mpr andmain central moments of inertia Iprξpv, Iprηpv, Iprζpv.

If the basic motion of the launcher is not disturbed, then the 0vxvyvzv,
Svxvyvzv and Svξvηvζv coordinate systemscoincideat anymoment.Theturret
modelas anelementof the spatially vibrating systemperformscomplexmotion
with respect to the 0vxvyvzv reference systemconsisting of straightlinemotion
of themass centre Sv, in accordancewith a change in the yv coordinate, rotary
motion about the Svzv axis in accordance with a change in the pitch angle ϑv
and rotarymotion about the Svxv axis in accordance with a change in the tilt
angle ϕv.

Prior to the launch, the missile is rigidly connected with the guide rail.
Themounting prevents themissile frommoving along the guide rail. Once the
missile motor is activated, the missile moves along the rail.

The guide rail-missile system used in the analysis ensures collinearity of
points on the missile and the guide rail.

It is assumed that the longitudinal axis of the missile coincides with the
longitudinal axis of the guide rail at anymoment and that themissile is a stiff
bodywith an unchangeable characteristic of inertia. Themissile is a perfectly
stiff bodywithmass mp andmain central moments of inertia Ipxp, Ipyp, Ipzp.
The geometric characteristic of the guide rail-missile system shown in Fig.2a
and Fig.2b can be used to analyze the dynamics of the controlled system.

The main view of the turret-missile system model is presented in Fig.2a.
It includes an instantaneous position of the missile while it moves along the
guide rail. Figure 2b shows the top view of the turret-missile system model.
It includes the instantaneous position of the missile while it moves along the
guide rail.

The missile model performs straightline motion with respect to the
Svξpvηpvζpv reference system according to a change in the ξpv coordinate.

Themodel of the launcher-missile systemhas six degrees of freedom,which
results from the structure of the formulated model. The positions of the sys-
tem were determined at any moment assuming four independent generalized
coordinates:



282 Z. Koruba et al.

yv – vertical displacement of the turret mass centre Sv,

ϕv – angle of rotation of the turret about the Svxv axis,

ϑv – angle of rotation of the turret about the Svzv axis,

ξpv – straightline displacement of the missile mass centre Sp along the Svξpv
axis.

Fig. 2. (a) Side view of the turret-and-missile model; (b) top view of the
turret-and-missile model

2. Model of motion of the launcher on a moveable base

Themathematical model of the systemwas developed basing on the physical
model. Six independentgeneralized coordinateswere selected to determine the
kinetic and potential energy of the model and the distribution of generalized
forces basing on the considerations for the physicalmodel.Byusing the second
order Lagrange equations, it was possible to derive equations of motion of the
analysed system.The systemwas reduced toa structural discretemodel,which
required applying differential equations with ordinary derivatives.

Six independent coordinates were used to determine motion of the
launcher-missile system model:

• turret vibrations: yv, ϑv, ϕv,

• basic motion of the turret and the guide rail ψpv and ϑpv, respectively,

• missile motion with respect to the guide rail ξpv.



Dynamics of a controlled anti-aircraft missile... 283

Motion of the controlled launcher mounted on a moveable base can be
illustrated bymeans of the following equations

ϑ̈v =
1

b7
[z2− (b2ÿv + b8ϕ̈v + b9ξ̈pv + b10ψ̈pv + b11ϑ̈pv)]+Mϑ

ϕ̈v =
1

b12
[z3− (b3ÿv + b8ϑ̈v + b13ξ̈pv +b14ψ̈pv + b15ϑ̈pv)]+Mϕ

ÿv =
1

b1
[z1− (b2ϑ̈v + b3ϕ̈v + b4ξ̈pv + b5ψ̈pv + b6ϑ̈pv)]+Fy

(2.1)

ψ̈pv =
1

b19
[z5− (b5ÿv +b10ϑ̈v + b14ϕ̈v + b17ξ̈pv + b20ϑ̈pv)]+M

p
ψ

ϑ̈pv =
1

b21
[z6− (b6ÿv + b11ϑ̈v +b15ϕ̈v + b18ξ̈pv + b20ψ̈pv)]+M

p
ϑ

ξ̈pv =
1

b16
[z4− (b4ÿv + b9ϑ̈v + b13ϕ̈v +b17ψ̈pv + b18ϑ̈pv)]

where

yv – vertical displacement of the turret mass centre Sv,

ϕv,ϑv – angle of rotation of the turret about the Svxv and Svzv axis, respec-
tively,

ξpv – straightline displacement of the missile mass centre Sp along the Svξpv
axis,

ψpv – turret yaw angle,

ϑpv – guide rail pitch angle,

M
p
ϑ
,M

p
ψ
– preprogrammed moment of control of the turret pitch angle and

the guide rail yaw angle, respectively,

Fy – force stabilizing vertical progressive motion of the launcher,

Mϑ,Mϕ –moments stabilizingangularmotionsof the launcher about the Svzv
and Svxv axes,

bi (where i =1, . . . ,21) – coordinate functions yv, ϑv, ϕv, ψpv, ϑpv, ξpv,

zi (where i = 1, . . . ,6) – coordinate functions yv, ϑv, ϕv, ψpv, ϑpv, ξpv and
their derivatives with respect to time ẏv, ϑ̇v, ϕ̇v, ψ̇pv, ϑ̇pv, ξ̇pv.

Explicit forms of the functions bi(yv,ϑv,ϕv,ψpv,ϑpv,ξpv) and
zi(yv,ϑv,vpv,ψpv,ϑpv,ξpv, ẏv, ϑ̇v, ϕ̇v, ψ̇pv, ϑ̇pv, ξ̇pv) are long mathematical
expressions, which developed analytically are available at the authors of the
article.



284 Z. Koruba et al.

3. Control of motion of the launcher on a moveable base

The launcher under consideration is mounted on a moveable base (wheeled
vehicle). It canbeput into rotarymotionbyangle ψpv about the Svyv axis and
rotarymotion by angle ϑpv about theSvzv axis.When themissile longitudinal
axis Svxp is set, for instance, to coincide with the target line of sight, it
is necessary to apply the following pre-programmed control moments of the
launcher

M
p
ψ
= kψpv(ψpv −ψ

p
pv)+ c

ψ
pv(ψ̇pv − ψ̇

p
pv)

(3.1)
M

p
ϑ
= kϑpv(ϑpv −ϑ

p
pv)+ c

ϑ
pv(ϑ̇pv − ϑ̇

p
pv)

where

kψpv,k
ϑ
pv – gain coefficients of the control system,

cψpv,c
ϑ
pv – attenuation coefficients of the control system.

The principle of operation of the launcher control system is presented in a
schematic diagram in Fig.3.

Fig. 3. Diagram of the system for automatic control of the launcher on the
moveable base



Dynamics of a controlled anti-aircraft missile... 285

The quantities ψppv and ϑ
p
pv present in the above formulas are pre-

programmed angles changing in time according to the following laws

ψppv = ω
ψ
pvt ϑ

p
pv = ϑpv0 sin(ω

ϑ
pvt+ϕpv0)

ψ̇ppv = ω
ψ
pv ϑ̇

p
pv = ϑpv0ω

ϑ
pv cos(ω

ϑ
pvt+ϕpv0)

(3.2)

where

ωψpv =
2π

Tψ
ωϑpv =

2π

Tϑ

Once the launcher is in thepre-determinedfinal state, the longitudinal axis
of the guide rail needs to coincide with the target line of sight irrespective of
motions of the base or external disturbances.
The stabilizing (corrective) controls used for the automatic control of the

launcher need to have the following form

Mϑ = kϑ(ϑv −ϑ
p
v)+ cϑ(ϑ̇v − ϑ̇

p
v) Mϕ = kϕ(ϕν −ϕ

p
v)+ cϕ(ϕ̇ν − ϕ̇

p
v)
(3.3)

Fy = ky(yν −y
p
v)+ cϕ(ẏν − ẏ

p
v)

where

ϑpv,ϕ
p
v,y

p
v – quantities defining the pre-determined position of the launcher in
space during angular motions of the base,

kϑ,kϕ,ky – gain coefficients of the regulator,

cϑ,cϕ,cy – attenuation coefficients of the regulator.

4. Results

4.1. System parameters

Motion of the hypothetical controlled anti-aircraft missile launcher was
describedusing the systemof equations denoted by (2.1). The launcher system
was assumed to have the following parameters:

• Parameters of the launcher (turret and guide rail)

mv = mw +mpr Ivη′
pv
=7kgm2

Ivx =(Iwξ′
v
+Iprξpv cos

2ϑpv + Iprηpv sin
2ϑpv)cos

2ψpv +

+(Iwζ′
v
+ Iprζpv)sin

2ψpv

Ivy = Ivη′
pv
+ Iprξpv cos

2ϑpv + Iprηpv sin
2ϑpv



286 Z. Koruba et al.

Ivz =(Iwξ′
v
+ Iprξpv cos

2ϑpv + Iprηpv sin
2ϑpv)sin

2ψpv +

+(Iwζ′
v
+ Iprζpv)cos

2ψpv

Ivxz =(Iwξ′
v
+ Iprξpv cos

2ϑpv + Iprηpv sin
2ϑpv − Iwζ′

v
− Iprζpv) ·

·cosψpv sinψpv

mw =50kg Iwξ′
v
=10kgm2

Iwη′
v
=7kgm2 Iwζ′

v
=12kgm2

mpr =30kg Iprξpv =0.6kgm
2

Iprηpv =4kgm
2 Iprζpv =3.5kgm

2

• Launcher suspension parameters

kv11 =30000N/m cv11 =150Ns/m

kv12 =30000N/m cv12 =150Ns/m

kv13 =30000N/m cv13 =150Ns/m

kv14 =30000N/m cv14 =150Ns/m

• Missile parameters

mp =12kg Ipxp =0.01kgm
2

Ipyp =2kgm
2 Ipzp =2kgm

2

• Geometric characteristics

l1 =0.3m l2 =0.3m

d1 =0.2m d2 =0.2m

lp =1.6m lsp =0.8m

• Thrust and operation time of the missile starter motor

Pss =4000N tss =0.07s

4.2. Kinematic excitations

The terrain unevenness, i.e. a bump, was assumed to be a kinematic exci-
tation. The basic motion of the vehicle carrying the launcher was defined as

sn = Vn(t− tgb)

where



Dynamics of a controlled anti-aircraft missile... 287

Vn =8.3m/s – velocity of the vehicle with the launcher,

tgb =0.5s – time in which the vehicle carrying the launcher goes over the
bumpwith the front wheels.

In the case considered here, all the vehicle wheels climb a single bump.

The excitations have the following form

y01 = y0 sin
2(ω0sn) ẏ01 = y0ω0Vn sin(2ω0sn)

y02 = y0 sin
2(ω0sn) ẏ02 = y0ω0Vn sin(2ω0sn)

y03 = y0 sin
2[ω0(sn − lwn)] ẏ03 = y0ω0Vn sin[2ω0(sn − lwn)]

y04 = y0 sin
2[ω0(sn − lwn)] ẏ04 = y0ω0Vn sin[2ω0(sn − lwn)]

where y0 =0.05m, l0 =0.35m, ω0 = π/l0, lwn = l1+ l2.

• Control parameters

Tψ =1s Tϑ =1s ϑpv0 =
π

2
ϕpv0 =

π

4
or ϕpv0 =0

kϑ =50000 kϕ =20000 ky =50000

cϑ =5000 cϕ =2000 cy =5000

— low values of the coefficients of the pre-programmed controls

kψpv =2000 k
ϑ
pv =2000

cψpv =200 c
ϑ
pv =200

— high values of the coefficients of the pre-programmed controls

kψpv =10000 k
ϑ
pv =10000

cψpv =2000 c
ϑ
pv =2000

Figures 4-21 show selected results of computer simulation conducted for
the hypothetical controlled missile launcher.

Figures 5-13 illustrate the performance of the launcher with no or some
correction applied to the automatic control system (Figures a and b, respec-
tively) during motion of the vehicle over the uneven terrain (a single road
bump). By analogy, the effects of the pre-programmed controls are presented
in Figs. 14-21. The low and high coefficients of the control system are shown
in Figs. 17-21 a and b, respectively.



288 Z. Koruba et al.

Fig. 4. (a) Time-dependent profile of the terrain unevenness (bump);
(b) time-dependent translatory displacements of the rocket with regard to the guide

rail

Fig. 5. Time-dependent vertical displacement y of the launcher without (a) and
with (b) correction

Fig. 6. Time-dependent angular displacement ϑv of the launcher turret without (a)
and with (b) correction



Dynamics of a controlled anti-aircraft missile... 289

Fig. 7. Time-dependent angular displacements of the launcher without (a) and
with (b) correction, ϑpv and ψpv, respectively

Fig. 8. Time-dependent vertical velocity of the launcher without (a) and with (b)
correction, ẏ

Fig. 9. Time-dependent angular velocity of the launcher turret without (a) and
with (b) correction, ϑ̇v



290 Z. Koruba et al.

Fig. 10. 11a. Time-dependent angular velocities of the launcher without (a) and
with (b) correction, ϑ̇pv and ψ̇pv, respectively

Fig. 11. Time-dependent vertical acceleration of the launcher without (a) and
with (b) correction, ÿ

Fig. 12. Time-dependent angular acceleration of the launcher turret without (a) and
with (b) correction, ϑ̈v



Dynamics of a controlled anti-aircraft missile... 291

Fig. 13. Time-dependent angular accelerations of the launcher without (a) and
with (b) correction, ϑ̈pv and ψ̈pv, respectively

Fig. 14. Time-dependent angular displacements of the launcher turret during
pre-programmed control without (a) and with (b) correction, ϑv and ϕv,

respectively

Fig. 15. Time-dependent angular velocities of the launcher turret during
pre-programmed control without (a) and with (b) correction, ϑ̇v and ϕ̇v,

respectively



292 Z. Koruba et al.

Fig. 16. Time-dependent angular velocities of the launcher turret during
pre-programmed control without (a) and with (b) correction, ϑ̈v and ϕ̈v,

respectively

Fig. 17. Time-dependent pre-programmed control moments M
p
ϑ
and M

p
ψ
at low (a)

and high (b) gain coefficients

Fig. 18. 19a. Time-dependent pre-determined and real angular pitches of the
launcher at low (a) and high (b) gain coefficients, ϑzpv and ϑpv, respectively



Dynamics of a controlled anti-aircraft missile... 293

Fig. 19. Time-dependent pre-determined and real angular velocities of the launcher
at low (a) and high (b) gain coefficients, ϑ̇zpv and ϑ̇pv, respectively

Fig. 20. Time-dependent pre-determined and real angular displacements of the
launcher at low (a) and high (b) gain coefficients, ψzpv and ψpv, respectively

Fig. 21. Time-dependent pre-determined and real angular velocities of the launcher
at low (a) and high (b) gain coefficients, ψ̇zpv and ψ̇pv, respectively



294 Z. Koruba et al.

5. Conclusions

The following conclusions were drawn from the theoretical considerations and
simulation tests:

• corrective controlsmust to be applied to prevent the occurrence of trans-
itional processes resulting from terrain unevenness conditions (bumps);

• pre-programmed controls of the launcher result in negative vibrations of
all its elements; the vibrations can be removed effectively by optimal se-
lection of the coefficients of gain and attenuation in the launcher control
system;

• the launcher control systemallows immediate positioning of the launcher
in space so that the guide-rail longitudinal axis coincides with the target
line of sight.

A feasibility study should to be conducted for the proposed control sys-
tem. Particular attention has to be paid to values of the pre-programmed and
corrective control moments.

References

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3. Dziopa Z., 2006a, An anti-aircraft self-propelled system as a system determi-
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Dynamics of a controlled anti-aircraft missile... 295

6. Dziopa Z., 2008,The Modelling and Investigation of the Dynamic Properties
of the Sel-Propelled Anti-Aircraft System, Published by Kielce University of
Technology, Kielce

7. Koruba Z., 2001,Dynamics and Control of a Gyroscope on Board of an Fly-
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8. Koruba Z., Dziopa Z., Krzysztofik I., 2009, Dynamics and control of a
gyroscope-stabilized platform in a self-propelled anti-aircraft system, Journal
of Theoretical and Applied Mechanics, 48, 1, 5-26

9. Koruba Z., Osiecki J., 1999, Construction, Dynamics and Navigation of
Close-Range Missiles – Part 1, University Course Book No. 348, Kielce Uni-
versity of Technology Publishing House, PL ISSN 0239-6386 [in Polish]

10. Koruba Z., Tuśnio J., 2009, A gyroscope-based system for locating a point
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Applied Mechanics, 47, 2, 343-362

11. Mishin B.P., 1990,Dinamika raket, Mashinostroyenie,Moskva

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13. Svietlitskiy V.A., 1963, Dinamika starta letatelnykh apparatov, Nauka,
Moskva

Dynamika sterowanej przeciwlotniczej wyrzutni rakietowej umieszczonej

na ruchomej platformie

Streszczenie

W pracy dokonano analizy dynamiki sterowanej przeciwlotniczej wyrzutni poci-
sków rakietowych umieszczonej na ruchomej podstawie (pojeździe samochodowym).
Zastosowanosterowanieprogramowedla ruchupodstawowegowyrzutni realizowanego
wprocesie przechwytywania celu oraz sterowanie stabilizujące zaburzenia wynikające
zdziałania zestawuprzeciwlotniczego.Tego rodzaju sterowaniepozwalanaustawienie
osi podłużnej pocisku rakietowegowzględem linii obserwacji celu niezależnie od ruchu
pojazdu, na którym znajduje się wyrzutnia. Drgania spowodowane są wymuszeniami
działającymi bezpośrednio na pojazd samochodowy od strony drogi, startem pocisku
rakietowego oraz sterowanego ruchu podstawowego samej wyrzutni. Niektóre wyniki
badań symulacji komputerowej przedstawione są w graficznej postaci.

Manuscript received May 14, 2009; accepted for print June 19, 2009