Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 3, pp. 629-639, Warsaw 2014

ANALYSIS OF THE DYNAMICS AND CONTROL OF THE MODIFIED

OPTICAL TARGET SEEKER USED IN ANTI-AIRCRAFT ROCKET

MISSILES

Daniel Gapiński, Izabela Krzysztofik, Zbigniew Koruba

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

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

The paper presents the concept of programmed control of the designed optical target seeker
in the phase of searching the air space. The controlling of the programmed movements
of the seeker axis has been developed with simultaneous consideration of the process of
scanning of the air space by an optoelectronic system of the device. Numerical analysis of
the dynamics of the proposed optical scanning seeker as well as the analysis of selection of
velocity and suitable trajectories of the displacement of its axis were conducted. The results
were presented in a graphical form.

Keywords: self-guidance, dynamics and control, target seeker, rocketmissile

1. Introduction

Optical scanning heads of self-guided rocket missiles are devices requiring high accuracy and
precision of making (Awrejcewicz andKoruba, 2012; Moir and Seabridge, 2006; Siouris, 2004).
Structural solutions of those devices are constantly improved because detection and deter-

mination of the location of faster and faster moving air targets in real conditions is additionally
hindered by various thermal disruptions (Yanushevsky, 2008). In connection with the above,
optical systems and structural solutions of those devices are still improved what is proven by
the most recent European and American patents published in the years 2010-2012 (Barenz et
al., 2012; Kröner, 2012; Rueger and Zoz, 2012; Shaffer, 2011; Wellman et al., 2010, Wild and
Leavy, 2012). The possibility of using innovative structural solutions is considered, and the opti-
mization of the process of scanning (searching) of the air space by the designed scanning seeker
is presented in the paper. Innovation and the principle of operation have been presented in the
patent description (Gapiński, 2005).
The design of the developed scanning seeker (Gapiński, 2005), including themost significant

constituents, is shown in Fig. 1.

• The gyroscope rotor consists of the following elements presented in Fig. 1:

2 – primarymirror, 3 – secondarymirror guard, 4 – adjustable secondarymirror, 5 – secon-
darymirror roller, 6 – pattern board with electromagnet, 8 – retaining ring, 9 – fastening
lid, 10 – body of mechanical gyroscope, 11 – corrective lens system, 12 – permanent ma-
gnet of driving motor, 14 – additional movable mirror, 18 – gear transmission, 19 – snap
ring, 20 – bearing, 21 – two spacer bushings.

• The complete inner housing consists of:

1 – coil of driving motor, 15 – electric motor controlling inner housing, 13 – two in-
frared radiation receivers, 17 – pattern board with electromagnet, 24 – fastening board,
25 – two fibre-optic gyroscopes, 26 – inner housing, 29 – bearing fastening, 31 – moving
axis, 32 – bearing.



630 D. Gapiński et al.

Fig. 1. Structure of the modified scanning seeker

• The complete outer housing consists of:

15 – electric motor controlling outer housing, 27 – outer housing, 29 – bearing fastening,
31 – moving axis, 32 – bearing.

2. Model of the seeker movement

Figure 2 shows a diagram of the seeker, including the adopted systems of coordinates as well as
markings of individual angles of rotations of the respective systems in relation one to another.

Fig. 2. The diagram of the seeker, including the adopted systems of coordinates

A gyroscope movement can take place under the influence of moments of external forces
MZ and MW generated by controlling motors (15) as well as angular movements of the rocket
missile deck determined with the use of angular velocities ωxP , ωyP , ωzP causing its rotation
around individual axes of the system OXPYPZP with the respective angles αxP , αyP and αzP .
The following systems of coordinates (Awrejcewicz and Koruba, 2012; Baranowski, 2013;

Krzysztofik, 2012) were introduced:

OXKYKZK – system of coordinates connected with the direction set in space,

OXPYPZP –moving coordinate system connected with themissile,



Analysis of the dynamics and control of the modified optical target seeker... 631

OXCZYCZZCZ –moving coordinate system connected with the outer housing,

OXCWYCWZCW – moving coordinate system connected with the inner housing,

OXRYRZR –moving coordinate system connected with the rotor.

The followingmarkings of angles of rotation and the order ofmeasuring themwere adopted:

ψ – angle of rotation OXCZYCZZCZ in relation to OXPYPZP around axis ZCZ,

ϑ – angle of rotation OXCWYCWZCW in relation to OXCZYCZZCZ around axis XCW ,

φ – angle of rotation OXRYRZR in relation to OXCWYCWZCW around axis YR.

Hence, the location of gyroscope rotor in relation to the system OXPYPZP is determined
with the use of three angles: ψ, ϑ, φ.
As given quantities the following were adopted:

1. JxCZ,JyCZ ,JzCZ – moments of complete inertia of the outer housing,

2. JxCW ,JyCW ,JzCW –moments of complete inertia of the inner housing,

3. JxR,JyR,JzR – moments of inertia of the rotor,

4. ωP(ωxP ,ωyP ,ωzP) – missile angular velocity;

5. MZ – moment of missile forces interacting with the outer housing,

6. MW – moment of forces of the outer housing interacting with the inner housing,

7. MR –moment of forces of the inner housing interacting with the rotor,

8. MTR – moment of friction forces in rotor bearings and aerodynamic resistance,

9. MTW ,MTZ – moments of friction forces in the bearings of the inner and outer housing,
respectively.

The equations of seeker (gyroscope) motion have been introduced with the use of Lagrange
equations of the II-nd kind (Awrejcewicz and Koruba, 2012; Baranowski, 2013; Krzysztofik,
2012)

JzCZ
d

dt
ωzCZ +JyCW

d

dt
(ωyCW sinϑ)+JzCW

d

dt
(ωzCW cosϑ)+JyR

d

dt
(ωyR sinϑ)

+JzR
d

dt
(ωzCW cosϑ)− (JxCZ −JyCZ)ωxCZωyCZ − (JxCW +JxR)ωxCWωyCZ

+JyCW ωyCWωxCZ cosϑ− (JzCW +JzR)ωzCWωxCZ sinϑ+JyRωyRωxCZ cosϑ

= MZ −MTZ

JxCW
d

dt
ωxCW +JxR

d

dt
ωxCW − (JyCW −JzCW −JzR)ωyCWωzCW −JyRωyRωzCW

= MW −MTW

JyR
d

dt
(ωyCW + φ̇)= MR−MTR

(2.1)

where

ωxCZ = ωxP cosψ+ωyP sinψ ωyCZ =−ωxP sinψ+ωyP cosψ

ωzCZ = ψ̇+ωzP ωxCW = ωxCZ + ϑ̇

ωyCW = ωyCZ cosϑ+ωzCZ sinϑ ωzCW =−ωyCZ sinϑ+ωzCZ cosϑ

MTW = cwϑ̇ MTZ = czψ̇

and cw is the friction coefficient in the bearing of the inner housing, cz – friction coefficient in
the bearing of the outer housing.
We assume that MR = MTR, then ωyR = ωR = n = const (n is the angular velocity of the

rotor) andmotion of the seeker axis is governed by equations (2.1)1,2.



632 D. Gapiński et al.

3. Controlling motion of the seeker axis

On contemporary battlefields, the precise aiming of a missile at a moving air target is undo-
ubtedly a difficult thing to accomplish. The scanning seeker proposed in the paper will allow
one to aim the missile only in the direction of the foreseen location of the target. The optical
axis of the seeker performs programmed movements (e.g. in a circle) and, simultaneously, the
system ofmirrors scans the surface on an n-leaved rosette with the so called big scanning angle.
It increases the area of searching and, with a suitable selection of velocity of the programmed
movement of the seeker axis, gives satisfactorily dense scanning of space. At the moment of in-
tercepting the target by the scanning seeker, the angles by which the seeker axis is to bemoved
so that it overlapped with the line of sight (LOS) are determined. It is still the programmed
control which can be carried out, e.g. in a straight line. During that control phase, the angles
ϑZ, ψZ determine the set (desired) programmed movement which should be made by the axis
of the scanning seeker so that it overlapped with LOS. After intercepting the target and the
programmed presentation of the seeker axis on LOS, there is the second phase consisting in
the tracking of the target. During that phase, the rocket missile is launched, while the angles
ϑZ, ψZ are determined systematically by the target seeker optical system. At the moment of
finishing the operation by the missile start motor (stabilisation of the trajectory of the missile
flight), there appears a decrease in the angles δW and δD what causes the narrowing down of
the area of the scanned space. The angles δW and δD are inclination angles of the primary
and secondary mirror respectively in relation to the seeker axis. They are shown in Fig. 1. The
selection of those angles is presented in the paper by Gapiński (2013).
Thepassingof the seeker fromthe searchingmodeto the trackingmode,andthe simultaneous

interaction of the rocket missile deck adversely affect the long maintaining with a sufficient
precision the programmed movement and the tracking through its axis. We eliminate by that
the suitably chosen correction system.
Thecontrol law for theaxis of the scanning seeker (Awrejcewicz andKoruba,2012;Blakelock,

1991; Ładyżyńska-Kozdraś, 2009) can be written in the following way

MW =Π(to, tw)M
p
W
(t)+Π(ts, tk)M

s
W MZ =Π(to, tw)M

p
Z
(t)+Π(ts, tk)M

s
Z (3.1)

where M
p
W , M

p
Z are program controls for the seeker axis, M

s
W , M

s
Z – tracking controls for the

seeker axis, Π(to, tw), Π(ts, tk) – functions of rectangular impulse, t0 –moment of the beginning
of the scanning of space, tw –moment of detecting the target, ts –moment of the beginning of
the tracking of the target (ts = tw), tk –moment of the finishing of the process of self-guidance.

The program controls M
p
W , M

p
Z can be determined from the dependence (Koruba et al.,

2010)

M
p
W =−kw(ϑ−ϑZ)+kz(ψ−ψZ)−hz(ϑ̇− ϑ̇Z)

M
p
Z =−kz(ϑ−ϑZ)−kw(ψ−ψZ)−hz(ψ̇− ψ̇Z)

(3.2)

where ϑ, ψ are real angles determining the location of the seeker axis in space, ϑZ, ψZ – set
angles determining the location of the seeker axis in space, kw, kz, hz – controller coefficients.

The tracking controls MsW , M
s
Z can be determined from the dependence (Koruba et al.,

2010)

MsW =−kw(ϑ−ε)+kz(ψ−σ)−hz(ϑ̇− ε̇)

MsZ =−kz(ϑ−ε)−kw(ψ−σ)−hz(ψ̇− σ̇)
(3.3)

The angles ε, σ are angles determining the current location of LOS in space, determined
systematically through the optoelectronic system of the scanning seeker.



Analysis of the dynamics and control of the modified optical target seeker... 633

4. Results

Numerical research was conducted for the designed scanning seeker intended for the close-range
surface-to-air rocket missiles.

4.1. Parameters of the seeker

Themoments of inertia of individual elements of the seeker have been calculated in relation
to respective axes of the adopted systems of coordinates (shown in Fig. 2). The beginnings of
all systems of coordinates overlap and are at the intersection of the axis of rotation of the outer
housing with the axis of rotation of the inner housing of the seeker. The maximum torques of
individual motors controlling the outer and inner housing have been determined based on the
previous analysis of the designed seeker dynamics. The selection of a suitable gear ratio (18)
results from the analysis of the cooperation of individual seeker mirrors.

• Moments of inertia of the rotor are:

JxR =0.00114143kgm
2, JyR =0.00157911kgm

2, JzR =0.00158234kgm
2

• Moments of complete inertia of the inner housing:

JxCW =0.0016663kgm
2, JyCW =0.0011666kgm

2, JzCW =0.0011463kgm
2

• Moments of complete inertia of the outer housing:

JxCZ =0.0003383kgm
2, JyCZ =0.0002213kgm

2, JzCZ =0.0002583kgm
2

• Angular velocity of the rotor:

n =600rad/s

• Friction coefficients in the bearing of the inner and outer housing:

cw =0.05Nms, cz =0.05Nms

• Maximum torque of the controlling motors:

Mmax =0.8Nm

Figure 3a presents the scope of the air space scanned by the optoelectronic system of the
seeker with a large top angle of scanning βw amounting to 3.84 degrees, while Fig. 3b presents
the scope of the scanned space with a small angle of scanning βw amounting to 0.56 degrees,
however βx, βy are the constituents of angle β = βw/2.

Fig. 3. Scanning of space by the optoelectronic systemwith a large scanning angle (a) and
a small scanning angle (b)



634 D. Gapiński et al.

Figure 4 shows the direct detection of the air target by the optoelectronic system moving
with the velocity of 200m/s, which is 3000m away from the firing position, where XC, YC –
coordinates of target location, XS, YS – coordinates of the line of scanning.

Fig. 4. Direct detection of a target

The time of detecting the target amounted to 0.168s. The set angular coordinates of the
location of the detected target in relation to the seeker axis are as follows: σ = 0.435deg,
ε =1.5063deg.

In the case when the target is outside the area scanned by the optoelectronic system of the
device, the optical axis of the seeker is additionally set into programmed motion. It increases
the area of searching and, with a suitable selection of velocity of the programmedmovement of
the seeker axis, gives satisfactorily dense scanning of space.

Controlling theaxis of the scanning seekerwas checkedbysetting itsmovement on the surface
of a circular cone and on the surface of the unwinding coil. Theparameters of the controller have
been chosen according to (Awrejcewicz and Koruba, 2012) and assumed the following values

kw =100 kz =
1

2

√

2+4kw hz =
√

2+4kw

Figures 5-9 present the results of computer simulation of the control of the scanning seeker
axis setting itsmovement on the surface of a circular cone at the same time taking into conside-
ration the scanning of the air space by the optoelectronic system of the device. The target was
moving with the velocity of 250m/s and was 3000m away from the firing position. Figure 5a
presents the diagram of the set angular velocity ωO for the seeker axis moving on the surface
of a circular cone. In the initial phase of control, the scanning seeker axis accelerates till the
moment of achieving the set angular velocity amounting to 360deg/s. In the further phase of
control, the angular velocity of the axis is maintained at the constant level of 360deg/s.

Figures 10-14 present the results of computer simulation of the control of the scanning seeker
axis setting its movement on the surface of an unwinding spiral at the same time taking into
consideration the scanning of the air space by the optoelectronic systemof the device.The target
was moving with the velocity of 70m/s and was 3000m away from the firing position.

Figure 10 presents the diagram of the set angular velocity ωO for the seeker axis moving on
the surface of an unwinding spiral. In the initial phase of control, the axis accelerates till the
moment of achieving the set angular velocity amounting to 180deg/s.

In the further phase of control, the angular velocity of the axis ismaintained at the constant
level of 180deg/s.

Figures 12 and 14 present the searching of the air space by the seeker and the detection of
the target. In the first of the presented simulations, the time of detecting the target amounted



Analysis of the dynamics and control of the modified optical target seeker... 635

Fig. 5. (a) Set angular velocity of circling around a circular cone by the seeker axis; (b) desired and
actual rotation of the outer housing around the axis ZCZ

Fig. 6. (a) Desired and actual rotation of the inner housing around the axis XCW ; (a) trajectory of the
set TZ and actual TR movement of the seeker axis

Fig. 7. Moments controlling the outer and inner housing of the seeker

Fig. 8. Controllingmoments in transitional periods



636 D. Gapiński et al.

Fig. 9. Searching of the air space by setting the seeker axis into programmedmovement on the surface
of the circular cone and detecting the air target

Fig. 10. (a) Set angular velocity of the seeker axis in its movement on the surface of the unwinding
spiral; (b) desired and actual rotation of the outer housing around the axis ZCZ

Fig. 11. (a) Desired and actual rotation of the inner housing around the axis XCW ; (b) trajectory of
the set TZ and actual TR movement of the seeker axis

to 6.085s, while the set angular coordinates of the location of the detected target amounted to:
σ =3.93deg, ε = 10.192deg. In the second case, the time of detecting the target amounted to
5.397s, while the set angular coordinates of the location of the detected target amounted to:
σ =−10.523deg, ε =−1.88deg.

Figures 15 and 16 show the results of computer simulation of controlling the seeker axis by
setting it in the first phase inmovement on the surface of a circular cone.When the seeker does
not detect a target, it passes to the second phase of control on the surface of the unwinding
spiral. At the same time, the scanning of the air space by the optoelectronic systemof the device



Analysis of the dynamics and control of the modified optical target seeker... 637

Fig. 12. Searching of the air space by setting the seeker axis into programmedmovement on the surface
of the unwinding spiral and detecting the air target

Fig. 13. (a)Moments controlling the outer and inner housing of the seeker; (b) controllingmoments in
transitional periods

Fig. 14. Searching of the air space by setting the seeker axis into programmedmovement on the surface
of the unwinding spiral and detecting the air target

was taken into consideration. The target wasmovingwith the velocity of 70m/s andwas 3000m
away from the firing position.

Figure 15a presents the diagram of the set angular velocity ωO for the seeker axis moving
in the first and second phase of control. In the initial phase of control, the scanning seeker axis
accelerates till it reaches the set angular velocity amounting to 360deg/s, which is maintained
at a constant level of 360deg/s until a full circle has been made, after which the second phase
of control follows. In the second phase of control, the angular velocity of moving in a spiral
decreases till it reaches the set angular velocity amounting to 180deg/s.



638 D. Gapiński et al.

Fig. 15. (a) Set angular velocity of the seeker axis in its movement on the surface of the circular cone
and the unwinding spiral; (b) trajectory of the set TZ and actual TR movement of the seeker axis

Fig. 16. Searching of the air space by setting the seeker axis into programmedmovement and detecting
the air target

Figure 16presents the searching of the air space by the seeker and thedetection of the target.
The time of detecting the target amounted to 5.9s. The set angular coordinates of the location
of the detected target are as follows: σ =−1.707deg, ε =15.137deg.

5. Conclusions and final remarks

From the conducted analysis of the dynamics and control of the designed target seeker, it
appears that the programmed control of its axis in the phase of searching the air space is done
with the precision sufficient for detecting both slowly moving air targets such as helicopters,
transport aeroplanes, as well as for detecting targets moving with higher velocities amounting
to 1000km/h. The selection of a suitable trajectory as well as angular velocities of movement
of the seeker axis depends, among others, on the type of the air assault means. The searching
of the air space in the way presented in Fig. 11b is characterized by a large area of searching
the radius of which can exceed even 800m for the scanned plane that is 3000m away from the
firing position. However, with such a large scanned area, it is possible to successfully detect the
target when its velocity does not exceed 300km/h. Whereas, for the way presented in Fig. 6b,
the area of searching is smaller, its radius amounts to 300m for the scanned plane that is also
3000m away, however the time of scanning is much shorter due to which it is possible to detect
targets moving with higher velocities.



Analysis of the dynamics and control of the modified optical target seeker... 639

A rocket missile can of course be equipped with a switch enabling one to choose the type of
searching of the air space, however on contemporary battlefields, because of quick response time
required, the automation of such type of processes is simply unavoidable. The way of searching
shown in Fig. 15b, in which the control of the seeker axis is done on the surface of a circular
cone, and then when the seeker does not detect any target, it automatically switches to the
second phase of control on the surface of the unwinding spiral, can be a suggestion.

Acknowledgements

Thework reported hereinwas undertaken as a part of the researchproject supported by theNational

Centre for Research and Development over the period 2011-2014.

References

1. Awrejcewicz J., Koruba Z., 2012,Classical Mechanics. Applied Mechanics and Mechatronics,
Vol. 30, Springer, NewYork

2. BaranowskiL., 2013,Equationsofmotionofa spin-stabilizedprojectile forflight stability testing,
Journal of Theoretical and Applied Mechanics, 51, 1, 235-246

3. Barenz J., Kordulla H., Eckhardt R., Kuppel H., Tholl H. D., Baumann R., 2012,
Verfahren zum Steuern eines Lenkflugkörpers und Suchkopf für einen Lenkflugkörper, European
Patent EP 2466247A1

4. Blakelock J.H., 1991, Automatic Control of Aircraft and Missiles, John Wiley & Sons, New
York

5. Gapiński D., 2005, Optical scanning seeker, Patent PL 199721 B1

6. Gapiński D., 2013, Analysis of the optoelectronic system of modified target seeker (in Polish),
Proceedings of 14th Conference ASMOR 2013, Jastrzębia Góra, Poland, 79-87

7. Koruba Z., Krzysztofik I., Dziopa Z., 2010, An analysis of the gyroscope dynamics of an
anti-aircraft missile launched from a mobile platform, Bulletin of the Polish Academy of Sciences
– Technical Sciences, 58, 4, 651-656

8. Krzysztofik I., 2012, The dynamics of the controlled observation and tracking head located on
amoving vehicle, Solid State Phenomena, 180, 313-322

9. Kröner Ch., 2012, Suchkopf fr einen zielverfolgenden Flugkörper, European Patent EP 2533002
A1

10. Ładyżyńska-Kozdraś E., 2009, The control laws having a form of kinematic relations between
deviations in the automatic controlof aflyingobject,Journal ofTheoretical andAppliedMechanics,
47, 2, 363-381

11. Moir I., Seabridge A., 2006,Military Avionics Systems, JohnWiley & Sons Ltd, Chichester

12. Rueger R., Zoz J., 2012, Infrared seeker head, United States Patent US 2012/0248238A1

13. Shaffer J., 2011, Scanning array for obstacle detection and collision avoidance, United States
Patent US 7982662B2

14. Siouris G. M., 2004,Missile Guidance and Control Systems, Springer, NewYork

15. Wellman W., Borchard J. F., Anderson D., 2010, Optical system for wide field of view
staring infrared sensor having improved optical symmetry, European Patent EP 1618358B1

16. Wild N. R., Leavy P. M., 2012,Optical detection system,United States PatentUSRE43681E

17. YanushevskyR., 2008,ModernMissile Guidance, CRCPressTaylor&FrancisGroup,NewYork

Manuscript received September 19, 2013; accepted for print January 21, 2014