Research ArticleDynamic Modeling and Coupling Characteristic Analysis ofTwo-Axis Rate Gyro Seeker

Shixiang Liu ,1 Tianyu Lu ,2 Teng Shang,2 and Qunli Xia 3

1School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China2Beijing Aerospace Automatic Control Institute, Beijing 100854, China3School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Shixiang Liu; liushi-xiang@163.com

Received 5 November 2017; Revised 20 July 2018; Accepted 5 September 2018; Published 22 October 2018

Academic Editor: Giovanni Palmerini

Copyright © 2018 Shixiang Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A dynamic model of a two-axis rate gyro-stabilized platform-based seeker with cross-coupling, mass imbalance, and disturbancetorque is developed on the basis of the working principle of seeker two-loop steady tracking theory; coordinate transformationsare used to analyze the effects of seeker servo control mode on missile guidance and control systems. Frequency domain is usedto identify the servo motor transfer function. Furthermore, a block diagram of the two-gimbal-coupled system is developed, andthe coupling characteristics of gimbal angle are analyzed with different missile body inputs. Simulation results show that theanalysis conforms with the actual movement rule of seeker gimbal and optical axis, and cross-coupling exists between the twogimbals. The lag compensation network can increase the open loop gain and increase the capacity for disturbance rate rejection.Simulations validate the theory and technology support for developing the seeker servo control model in engineering.

1. Introduction

The radar seeker has many advantages, such as high-range,all-weather detection capability, compared with traditionalphotoelectric seeker [1]. Therefore, a radar seeker can be amore effective choice in developing military power orupgrading missile systems than traditional photoelectricseeker [2]. Radar seeker servo system is a key part of a pre-cise, stable target tracking system. The core of the radarseeker is a precise electromechanical servo device that com-pensates for the missile body disturbance to the beam axis ofthe seeker [3, 4] and maintains the beam-pointing stabilityin space, thus tracking the line of sight quickly. Two majortypes of stable servo seeker in history [5], namely, dynamicgyro and rate gyro stabilization modes, are used to meet dif-ferent needs. In [6], the dynamic modeling and simulation ofthe dynamic gyro-type seeker were conducted, and the influ-ence of the missile disturbance on the servo control systemof the seeker was analyzed; however, the disadvantages ofthe dynamic gyro seeker include low load capacity, large for-ward torque, and delayed response. In [7], the stability

tracking principle of rate gyroscope-stabilized laser seekerwas investigated; the stability of the seeker load is high,and the reaction speed is fast.

When a radar system for tracking and measuring target isaccommodated on a moving carrier, a line of sight stabiliza-tion system is the basis for high-precision tracking. In orderto eliminate the influence of the carrier, a stable platform isusually available to achieve the stability of the line of sight;according to different stable objects and stability require-ments, there can be various ways of stability [8–10]. In orderto precisely control the line of sight of the detection systemand make it track the target motion accurately while isolatingthe missile body disturbance, some scholars adopted a varietyof stable platform schemes and control methods [11–13].Unlike the airborne or ground-based tracking radar case,the radar seeker will output line of sight (LOS) angle ratefor terminal guidance of homing missile using proportionalnavigation [14]. In [15], a stable platform control loop of aseeker was designed and the effect of disturbance momenton LOS tracking was analyzed. The guidance accuracy ofthe homing missile will be affected by the LOS rate output

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 8513684, 14 pageshttps://doi.org/10.1155/2018/8513684

accuracy because the output error of the LOS rate caused byservo platform disturbance torque would induce the disturb-ing rate rejection, thereby resulting in its parasitical loop. In[16–18], the effect of disturbing rate rejection caused byspring and damping torques on the stability of the guidancesystem was analyzed. Any deviation in the course of theseeker control loop will make the body attitude angle coupleinto the LOS rate output, because the missile will be subject toairflow, disturbance torque in the flight process, therebyaffecting the stability and tracking of the radar seeker beamaxis and reducing the missile guidance accuracy [19–22].Therefore, the two-channel rate gyro servo platform-typeseeker-coupled model must be studied by using the platformkinematic [23, 24] and dynamic models [25].

The current study is aimed at building a dynamicmodel of two-axis rate gyro radar seeker and analyze thecoupling characteristics between the two channels. The restof this paper is organized as follows: Section 2 introduces therate gyro stabilization platform tracking principle. Section 3builds a dynamic model of two-axis platform radar seeker,including mass imbalance, disturbance torque, and cross-coupling. Section 4 introduces a frequency domain servomotor transfer function identification method, which pro-vides reasonable design parameters for the seeker controlsystem. Section 5 designs the lag compensation networkparameters for tracking loop and stability loop, respectively,analyzes the characteristic in frequency domain and at thesame time analyzes the influence of the missile motion onthe angle of the gimbal, disturbance rate rejection character-istic and coupling effect between the two gimbals by simula-tion. Section 6 concludes the study.

2. Rate Gyro Stabilization PlatformTracking Principle

The radar seeker servo device consists of a gyro assembly, DCtorque motor, angular position sensor, radar signal process-ing system, and yaw and pitch gimbals [26]. Figure 1 illus-trates the schematic of two-axis rate gyro stability platform.The outer gimbal is the pitch gimbal, the inner gimbal isthe yaw gimbal, and they are connected by the bearing; thus,

the ring gimbal can rotate around each other. The outergimbal shaft is mounted on two brackets through the bearinggimbal, and the bracket and the base are fixed together.The torque motor is mounted on one end of the shaft toenable the rotation of the shaft. The potentiometer gauge ismounted on the other end of the shaft to measure the rota-tion angle of the gimbal relative to the base. The inner gimbalpasses through the bearing gimbal on the outer gimbal. Thetorque motor is also mounted on one end of the inner gimbalshaft, and the potentiometer is mounted on the other end.The two rate gyros are mounted on the inner gimbal to mea-sure the rotation angle rate of the inner gimbal relative to theinertial space. The measured value is equal to the true LOSangle rate and can be used to guide the missile. The radarantenna is also mounted on the inner gimbal.

The radar antenna and inertial device (rate gyro) aremounted on the inner gimbal of the stable platform. Anymovement of the platform in inertial space will be sensedby the inertial devices. The real antenna beam off-centeredangle signals are sent to the corresponding electric motor tocontrol the platform back to the initial reference orientationin the inertial space, which is a function of the seeker stabilityloop. The radar seeker antenna measures the angle errorbetween the beam direction and the true LOS to track the tar-get, thus forming negative angle feedback to achieve closedloop tracking control by angle tracking loop.

3. Dynamic Model of Two-Axis PlatformRadar Seeker

3.1. Coordinate System Definition and Transformation.Figure 2 depicts the geometric schematic between the seekergimbal and missile body coordinates. The coordinate systemsare defined as follows.

(1) Inertial coordinate system Oxiyizi: the Oxi-axispoints to the earth plane flat, and the Oyi-axis pointsalong the vertical direction. The Ozi-axis directioncan be obtained by the right-hand rule. The subscripti denotes the inertial

Pitch motor Pitch rate gyro Yaw rate gyro

Yaw motor

Angular sensor

Angular sensor

Stabilized platform(inner gimble)

Yaw axis

Pitch axis

Flexible cable

Radar antenna

Figure 1: Two-axis rate gyro-stabilized platform-based radar seeker.

2 International Journal of Aerospace Engineering

(2) Missile body coordinate system Axb′yb′zb′: the Axb′-axiscoincides with the longitudinal axis of the missile andpoints to the front of the missile. The Ayb′-axis islocated within the missile longitudinal symmetryplane and is perpendicular to the Axb′-axis pointingup. The Azb′-axis direction is obtained by the right-hand rule. A point is the missile center of mass; Opoint is the gimbal rotating center, which coincideswith the center of the antenna beam. Oxbybzb is thenew coordinate system obtained by translating themissile body coordinate system Axb′yb′zb′ from A pointto O point. The subscript b denotes the missile body

(3) Outer gimbal coordinate system Oxwywzw: the Oxw-axis is perpendicular to the outer gimbal plane andpoints to the target. The Ozw-axis coincides withthe Ozb-axis of the missile body coordinate system.Oyw-axis is located within the Oxbyb plane and canbe obtained by the right-hand rule. The subscript wdenotes the outer gimbal

(4) Inner gimbal coordinate system Oxnynzn: the Oxn-axis points to the target and coincides with the beamdirection. The Oyn-axis coincides with the Oyw-axisof the outer gimbal coordinate system. Ozn-axis islocated within the Oxwzw plane and can be obtainedby the right-hand rule. The subscript n denotes theinner gimbal

(5) LOS coordinate system Oxsyszs: the Oxs-axis pointsto the target. The Oys-axis is located within the verti-cal plane that contains the Oxs-axis and is perpendic-ular to theOxs-axis pointing up. The Ozs-axisdirection can be obtained by the right-hand rule.The subscript s denotes the LOS

According to the coordinate systems above, from Oxbybzb toOxwywzw,Ozb-axis is the axis of rotation; φw is the angle

of rotation. From Oxwywzw to Oxnynzn, Oyw-axis is the axisof rotation; φn is the angle of rotation. From Oxnynzn toOxsyszs, first, the axis of rotation is Oyn-axis and the angleof rotation is εy, and then, the axis of rotation is Ozn′-axis(transition axis), and the axis of rotation is εp.

Figure 3 displays the coordinate transformationrelationship.

According to the coordinate systems above, from Oxiyizito Oxbybzb, ψ, ϑ, and γ denote the missile body yaw, pitch,and roll angles; the axis of rotation is Oyi-axis, Ozi′-axis (tran-sition axis), and Oxi′-axis (transition axis), respectively. FromOxiyizi to Oxsyszs, qy and qp are the LOS angles in the yaw

and pitch directions; the axis of rotation is Oyi-axis, and Ozi′-axis (transition axis), respectively. From Oxbybzb to Oxnynzn, φw and φn are the gimbal angles in the pitch and yawdirections; the axis of rotation is Ozb-axis and Oyw-axis,respectively. From Oxnynzn to Oxsyszs, εy and εp are theseeker antenna beam off-centered angles in the yaw and pitchdirections; the axis of rotation is Oyn-axis and Ozn′-axis(transition axis), respectively.

The transformation matrix of the missile body coordinatesystem to the outer gimbal coordinate system can be writtenas follows:

Lb→w φw =

cos φw sin φw 0

−sin φw cos φw 0

0 0 1

1

The transformation matrix of the outer gimbal coordi-nate system to the missile body coordinate system can bewritten as follows:

L−1b→w φw =

cos φw −sin φw 0

sin φw cos φw 0

0 0 1

, 2

y′b

x′b

z′b

z′nOA

xb

yb

.

.

yn (yw)

�휑w

�휑w

�휑w

xn

zn

Beam axis

xw

zw (zb)

�휀p

Targetxs

�휑n

�휑n

�휑n

�휀y

�휀y

Figure 2: Coordinate systems of seeker and missile body.

Lw→n (�휑n)-axis isthe axis ofrotation

Oyw

Ozbthe axis ofrotation

Oxbybzb Oxwywzw

Oxiyizi

Oxsyszs

Oxnynzn

The first transformation methodThe second transformation method

Li→s (qy qp)

Li→b (�휗 �휓 �훾)

L−1i→s (qy qp) L−1

n→s (�휀y �휀p)

Lb→w (�휑w)

Ln→s (�휀y �휀p)

is-axis

Figure 3: Coordinate system transformation relationship.

3International Journal of Aerospace Engineering

where Lb→w denotes the transformation matrix fromOxbybzbto Oxwywzw, L

−1b→w denotes the transformation matrix from

Oxwywzw to Oxbybzb, L−1b→w is the inverse of the matrix

Lb→w, and the transformation angle is φw. The other coordi-nate transformation relationships are similar.

3.2. Model of the Antenna Beam Off-Centered Angle. Figure 4presents the geometric relationship for beam off-centeredangle calculation. The antenna boresight unit vector in Oxs

yszs is 1 0 0 T when the unit vector a is along with the lineof sight, which can be projected to Oxnynzn directly, whichis denoted as xn1 yn1 zn1

T in Oxnynzn.

xn1 yn1 zn1T = L−1n→s a , 3

where L−1n→s denotes the transformation matrix from Oxsyszsto Oxnynzn; the transformation angles are εy and εp. The sub-

script n1 denotes the unit vector a which is transformed tothe n frame with the first transformation method.

At the same time, the unit vector a can be projectedto Oxiyizi, then be projected to Oxbybzb, and finally be pro-jected to Oxnynzn, which is denoted as xn2 yn2 zn2

T inOxnynzn

xn2 yn2 zn2T = Lb→nLi→bL

−1i→s a , 4

where L−1i→s denotes the transformation matrix from Oxsyszsto Oxiyizi, Li→b denotes the transformational matrix fromOxiyizi to Oxbybzb, and Lb→n denotes the transformationalmatrix from Oxbybzb to Oxnynzn. The subscript n2 denotes

the unit vector a which is transformed to the n frame withthe second transformation method.

Due to xn1 yn1 zn1T = xn2 yn2 zn2

T, theantenna beam off-centered angle calculation formula isexpressed as follows:

εy = arctan −zn2xn2

,

εp = arctanyn2xn2

6

3.3. Dynamic Derivation of the Seeker Platform. The projec-tion of the missile angular rate with respect to the inertialspace in the missile body frame is

ωb/ib = ωbx ωby ωbz

T, 7

where the superscript i denotes inertial coordination system,b/i denotes missile body with respect to the inertial space, andthe subscript b denotes the projection in the missile system.

zb

zn

ynyb

xn�휀p�휀y

xb�휑w

�휑n

�휑w

�휑n

Targetxs

Figure 4: Geometric relationship for practical antenna beam off-centered angle calculation.

L−1n→s =

cos εp cos εy sin εp −sin εy cos εp−sin εp cos εy cos εp sin εp sin εy

sin εy 0 cos εy

,

L−1i→s =

cos qy cos qp −cos qy sin qp sin qy

sin qp cos qp 0

−sin qy cos qp sin qy sin qp cos qy

,

Li→b =

cos ϑ cos ψ sin ϑ −cos ϑ sin ψ

−sin ϑ cos ψ cos γ + sin ψ sin γ cos ϑ cos γ sin ϑ sin ψ cos γ + cos ψ sin γ

sin ϑ cos ψ sin γ + sin ψ cos γ −cos ϑ sin γ −sin ϑ sin ψ sin γ + cos ψ cos γ

,

Lb→n =

cos φn cos φw sin φw −sin φn cos φw

−cos φn sin φw cos φw sin φn sin φw

sin φn 0 cos φn

5

4 International Journal of Aerospace Engineering

The rotation relationship of the missile coordinate sys-tem to the inner gimbal coordinate system can be expressedas follows:

Ω = φ n + φ w, 8

where Ω denotes the inner gimbal angular rate with respect

to the missile body, φ n coincides with the Oyw-axis, and

φ w coincides with the Ozb-axis.The inner gimbal angular rate with respect to the missile

body in the inner gimbal coordinate system can be expressedas follows:

ωn/Bn = −φw sin φn φn φw cos φn

T 9

The missile body angular rate with respect to the inertialspace in the inner gimbal coordinate system can be expressedas follows:

ωb/in = Lw→n φn Lb→w φw ωb/i

b

=

ωbx cos φw + ωby sin φw cos φn − ωbz sin φn

−ωbx sin φw + ωby cos φw

ωbx cos φw + ωby sin φw sin φn + ωbz cos φn

10

The inner gimbal angular rate with respect to the inertialspace in the inner gimbal coordinate system can be obtainedby (9) and (10).

ωn/in = ωn/b

n + ωb/in = ωn/i

nx ωn/iy ωn/i

nz

T, 11

where

ωn/inx = ωbx cos φw + ωby sin φw cos φn − ωbz + φw sin φn,

ωn/iny = −ωbx sin φw + ωby cos φw + φn,

ωn/inz = ωbx cos φw + ωby sin φw sin φn + ωbz + φw cos φn

12

The missile body angular rate with respect to the inertialspace in the outer gimbal coordinate system can be expressedas follows:

ωb/iw =

ωbx cos φw + ωby sin φw

−ωbx sin φw + ωby cos φw

ωbz

13

The outer gimbal angular rate with respect to the inertialspace in the outer gimbal coordinate system can be expressedas follows:

ωw/iw =

ωw/iwx

ωw/iwy

ωw/iwz

= ωw/bw + ωb/i

w

=

ωbx cos φw + ωby sin φw

−ωbx sin φw + ωby cos φw

ωbz + φw

14

We obtain (15) by substituting (14) into (12).

ωn/inx

ωn/iny

ωn/inz

=

ωw/iwx cos φn − ωw/i

wz sin φn

ωw/iwy + φn

ωw/iwx sin φn + ωw/i

wz cos φn

15

A derivative of (15) is as follows:

ωn/inx

ωn/inz

=ωn/inx

∗ − ωw/iwz sin φn

ωn/inz

∗ + ωw/iwz cos φn

, 16

where

ωn/inx

∗ = ωw/iwx − ωw/i

wzφn cos φn − ωw/iwxφn sin φn,

ωn/inz

∗ = ωw/iwx − ωw/i

wz φn sin φn + ωw/iwxφn cos φn

17

Applying Newton’s laws to the inner gimbal, the dynamicequation of the inner gimbal according to the theorem of themomentum of the rigid body rotating about a moving point ocan be expressed as follows:

ΣMn = Mnx Mny MnzT

=dHn

dt+ ω

n/in ×Hn + ρn × anmn,

Hn =

Hnx

Hny

Hnz

=

Jnxx 0 0

0 Jnyy 0

0 0 Jnzz

ωn/inx

ωn/iny

ωn/inz

,

18

where moving point o is the intersection of the inner gimbal

and outer gimbal (rotating center of the gimbal), Hn denotesthe angular momentum of the inner gimbal about the pointo, Jn denotes the inertia terms of the inner gimbal about thepoint o, ρn × anmn is the cross product of the moment armfrom the point o to the center of mass of the inner gimbalassembly and the mass of the inner gimbal times the specific

5International Journal of Aerospace Engineering

force acceleration of the inner gimbal at the point o, ρndenotes the moment arm from the point o to the center ofmass of the inner gimbal, an denotes the specific force accel-eration of the inner gimbal at the point o, and mn denotesthe inner gimbal system mass.

Since the inner gimbal only has rotational freedom aboutthe Oyn, the second component Mny of (11) is taken. Theexternal torque Mny is equal to the applied commanded tor-

que from the inner gimbal controller torqueMCTny , minus the

spring or cable torque KSnφn, minus the viscous torque KVnφn, and minus the Coulomb friction torque Kcouln φn/ φn .

Mny =MCTny − KSnφn − KVnφn − Kcouln

φn

φn, 19

where KSn, KVn, and Kcouln denote the spring, viscous, andCoulomb friction torque factors, respectively. The distur-bance torque between the missile body and the seeker willaffect the stability of the seeker. Therefore, the influence ofthe disturbance torque must be suppressed by the design ofthe seeker.

Now applying Newton’s laws to the outer gimbal, thedynamic equation can be expressed as follows:

ΣMw = Mwx Mwy MwzT

=dHw

dt+ ω

w/iw ×Hw + ρw × awmw,

Hw=

Hwx

Hwy

Hwz

=

Jwxx 0 0

0 Jwyy 0

0 0 Jwzz

ωw/iwx

ωw/iwy

ωw/iwz

,

20

where Hw denotes the angular momentum of the outergimbal about the point o, Jw denotes the inertia terms ofthe outer gimbal about the point o, ρw × awmw is thecross product of the moment arm from the point o tothe center of mass of the outer gimbal assembly and themass of the outer gimbal times the specific force accelera-tion of the outer gimbal at the point o, ρw denotes themoment arm from the point o to the center of mass ofthe outer gimbal, aw denotes the specific force accelerationof the outer gimbal at the point o, and mw denotes theouter gimbal system mass.

Since the outer gimbal only has rotational freedom aboutthe Ozw, the second component Mwz of (13) is taken. Theexternal torque Mwz is equal to the applied commanded tor-que from the outer gimbal torque MCT

wz , minus the spring orcable torque KSwφw, minus the viscous torque KVwφw, minusthe Coulomb friction torque Kcoulw φw/ φw , and minus thereaction torque of the inner gimbal on the outer gimbal.

Mwz =MCTwz − KSwφw − KVwφw − Kcoulw

φw

φw

− −Mnx sin φn +Mnz cos φn ,21

where

Mcc =MCTwz − KSwφw − KVwφw − Kcoulw

φw

φw22

We obtain (23) by substituting Mnx and Mnz into (21).

Mcc + Jnxxωn/inx

∗ − Jnxxωw/iwz sin φn + ωn/i

ny Hnz

− ωn/inz Hny + ρn2an3 − ρn3an2 mn sin φn

− Jnzzωn/inz

∗ − Jnzzωw/iwz cos φn + ωn/i

nxHny

− ωn/iny Hnx + ρn1an2 − ρn2an1 mn cos φn

= Jwzzωw/iwz + ωw/i

wxHwy − ωw/iwy Hwx

+ ρw1aw2 − ρw2aw1 mw

23

Note that (23) and Mny contain the common terms ωn/iny

and ωw/iwz . It is these terms that need to be solved for using

these two equations. These two equations can be put intothe form of a matric equation as shown below.

Jnyy 0

0 Jwzz + Jnxx sin2φn + Jnzz cos2φn

ωn/iny

ωw/iwz

=F1 +Mny

F2 +Mcc

24

The solutions for ωn/iny and ωw/i

wz are found by inverting thematrix and multiplying by the vector containing the F1 +Mny and F2 +Mcc terms.

ωn/iny

ωw/iwz

=

Jwzz + Jnxx sin2φn + 2Jnxy cos φn sin φn + Jnzz cos2φn 0

0 JnyyJnyy Jwzz + Jnxx sin2φn + 2Jnxy cos φn sin φn + Jnzz cos2φn

F1 +Mny

F2 +Mcc

25

6 International Journal of Aerospace Engineering

According to (25), ωn/iny and ω

w/iwz can be written as follows:

ωn/iny =

J11F1 + J12F2 + J11Mny + J12Mcc

Jeq,

ωw/iwz =

J12F1 + JnyyF2 + J12Mny + JnyyMcc

Jeq

26

If the inertia product and the mass imbalance momentare zero, then the parameters J11, J12, F1, F2, and Jeq can besimplified as follows:

J11 = Jwzz + Jnxx sin2φn + Jnzz cos2φn,

J12 = 0,

F1 = −ωn/in zHnx + ωn/i

n xHnz ,

F2 = Jnxxωn/inx

∗ + ωn/iny Hnz − ωn/i

nz Hny sin φn

− Jnzzωn/inz

∗ + ωn/inxHny − ωn/i

ny Hnx cos φn

− ωw/iwxHwy + ωw/i

wy Hwx ,

Jeq = Jnyy Jwzz + Jnxx sin2φn + Jnzz cos2φn ,

27

correspondingly.

Figure 5 demonstrates the LOS rate extraction model forthe two-gimbal-coupled system. The model starts from theantenna beam off-centered angles computed by (6) and thecorrective network using the lag compensation network toincrease open loop gain with lag compensation coefficient βand time constant T ; L is inductance and R is resistance inthe motor model. First, through the antenna beam off-centered angle model, the εy and εp can be obtained, and thenthe off-centered angles can be amplified by the gain K1 or K3to be LOS angle rate signal; the LOS angle rate error (qc − qs)can be amplified by the gain K2 or K4 to be voltage signal.The lag compensation network in the tracking loop (whichis an angle position loop; its main function is to receive theoff-centered angle signal, control the movement of the seekerplatform, and realize the closed loop tracking to the target)can reduce the tracking error. The lag compensation networkin the stability loop (which is an angle rate loop; its mainfunction is to isolate the missile body motion and keep thebeam stable in inertial space) can increase the open loop gain.The voltage signal through the motor model, moment coeffi-cients KT1 or KT2, and load moment of inertia Jeq can outputgimbal angular acceleration signal; gimbal angular rate canbe obtained by integrating the gimbal angular accelerationsignal. Finally, the gimbal angular rate feedback to the inputpoint of the stability loops through the rate gyro model. Theobjective of the controller in Figure 5 is to align the antenna

F1

F2

�휀p

KSn

KVn

Kcouln

�휑n

�휑n

�휔wy

�휑w

KVw

Kcoulw

Ksw

cos �휑w

sin �휑n

w/i

.

�휔nyn/i

�휔wz

�휔Bz

�휔Bx

�휔By

.�휑w

w/i

�휔nzn/i

�훽1 (T1s + 1)�훽1T1s + 1)

�훽3 (T3 s+ 1)�훽3T3s + 1)

�훽4 (T4 s+ 1)�훽4T4s + 1)

qyc�훽2 (T2s + 1)�훽2T2s + 1

Beam off-centered anglecalculation model

qyt

�휓

�휗

�훾

qpt

�휀y

qpc

J11

J11

J12

J12

J12

J12

Jnyy

1Jeq

1Jeq

1s

1s

+

++

++

+

++

− +−+−

++

++

++

++++

++

++

−

+−

++

++

++

+

1s

cos �휑n

q̂ys

1s

sin �휑w

q̂ps

−

Jnyy

+− +−1L1s + R1

L2s + R2

KT1

1KT2

KE1

KE2

K1K2

K3K4

.

.

Inner gimbal(yaw)

Outer gimbal(pitch)

Motormodule

Rate gyromodel

Counter EMF loop

Disturbancetorque

Motormodule

Disturbancetorque

Counter EMF loop

Rate gyromodel

Stability loop

Tracking loop

Stability loop

Lag compensationnetwork

Lag compensationnetwork

Lag compensationnetwork

Lag compensationnetwork

Tracking loop

Figure 5: Two-gimbal-coupled system block diagram.

7International Journal of Aerospace Engineering

boresight with the LOS and eliminate the εy and εp promptlyand in a stable way.

4. Dynamic Model Identification of the SeekerControl Loop

The analysis of the transfer function of the seeker platformand the inertial device is necessary before designing theseeker control system. The mathematical model of thetransfer function is typically obtained through the systemparameter identification method. The transfer function infrequency domain can be obtained by measuring the fre-quency characteristics of the inertial device to identify thesystem. In a sufficiently wide frequency range, the frequencyof the input signal of the inertial device is continuously chan-

ged, the ratio of the output characteristic of the inertialdevice to the input characteristic is measured, and the decibelnumber for identifying the logarithmic amplitude-frequencycharacteristics is obtained.

In this study, in order to validate the method, weassume that the transfer function of the torque motor isa second-order model, which contains the motor model1/ Ls + R , moment coefficient KT , back EMF coefficientKE, and moment of inertia Jeq. The torque motor modelis shown in Figure 6.

The transfer function can be expressed as follows:

qs sua s

=1/KE

LJ/KTKE s2 + RJ/KTKE s + 1, 28

where ua s denotes the voltage signal, qs denotes the LOSangle rate, back KE denotes EMF coefficient, L is inductanceand R is resistance in the motor model, and KT denotesmoment coefficient.

Setting ωm = KTKE / LJeq , km = 1/KE, and ξm =R Jeq / 2 KTKEL , the stability loop can be shownin Figure 7.

In Figure 7, the closed loop transfer function can beexpressed as follows:

where qs s is LOS rate input signal, qc s is LOS rate outputsignal, K f is amplifier gain, ωm is motor transfer function fre-quency, ξm is damping ratio, and G s denotes the closedloop transfer function.

Then,

Kf kgkm1 + Kf kgkm

= kn,

1/ω2m

1 + Kf kgkm= 1ω2n,

2 1/ωm ξm1 + K f kgkm

= 21ωn

ξn

30

The relationship between the resonance peak (Mr) anddamping (ξn) under the sinusoidal input signal can beexpressed as follows:

Mr =1

2ξn 1 − ξn2

31

Simultaneously, the relation for resonant frequency wr ,inverse of the natural frequency tn, and damping ξn can beexpressed as follows:

wr =1 − 2ξn2

tn32

Thus, by measuring the resonant frequencywr and calcu-lating ξn by using (31), the value of tn can be determined by

tn =1 − 2ξn2

wr33

Data fitting results are obtained for the experimental dataat different input frequencies. Thus, we can measure theangle rate ratio qs ω /qc ω . We can identify the value ofthe system parameters kn, tn, and ξn by using (31)–(33),and the values of the torque motor parameters km, tm, andξm can be calculated by (30).

The input signal is the angular velocity signal obtained bydifferentiating the angular position signal, where the ampli-tude is 0.2°, and the frequency is 1.25–15.7 rad/s, with theinner gimbal as the measured object. The closed loop

1

Ls + R+−

KT

1

Jeqs

KE

qs.

TMua

Ea

Figure 6: The torque motor model.

G s =qs sqc s

=K f kgkm / 1 + K f kgkm

1/ω2m / 1 + Kf kgkm s2 + 2ξm/ωm / 1 + Kf kgkm s + 1

=kn

s2/ω2n + 2 ξn/ωn s + 1

, 29

8 International Journal of Aerospace Engineering

frequency characteristic of the seeker can be obtained byaccessing the data processing. Figure 8 presents the frequencycharacteristic curve, where the dotted line is the measureddata, and the solid line is the magnitude-frequency character-istics of the second order.

The identification transfer function can be expressedas follows:

G s =1

0 00648s2 + 0 0507s + 134

Then, we can obtain ωn = 12 42 rad/s (1.99Hz), ξn =0 315; the closed loop bandwidth is 17.9 rad/s (2.8Hz).

Using the parameters Kf = 1 6, kg = 1, and Jeq =0 0025995 in the stability loop, we can obtain the torquemotor parameters km = 1 87, ωm = 6 2 rad/s, and ξm = 0 63by using (30); they are matched with the true motor model;the frequency of the stability loop is nearly 2 times that ofthe motor model.

5. Simulation Analysis of the Seeker Two-Axis-Coupled Dynamic Model

Equations (19), (21), and (32) and Figure 5 reflect thecomplexity of the two-gimbal platform seeker system.The dynamic model includes not only the cross-couplingand mass imbalance but also the disturbance torque fromthe spring, viscous, and Coulomb friction torque factors.

Moreover, the body attitude angle will couple into the seekerLOS rate output. We can simulate the tracking performancesand coupling characteristics of the seeker according to thetwo-gimbal-coupled system block diagram illustrated inFigure 5. We assume that the inertia product and the massimbalance moment are zero to verify the accuracy and avail-ability of the seeker dynamic model. According to the testmodel of the seeker in laboratory, the dynamic parametersare summarized in Table 1.

In order to increase the gain of the lower frequencies bynearly 15 times in tracking loop and reduce the tracking errorof the LOS, the lag compensation network in tracking loop isdesigned as follows:

β1 T1s + 1β1T1s + 1

=β3 T3s + 1β3T3s + 1

=15 s/2 5 + 115 s/2 5 + 1

35

Figure 9 shows the open loop Bode diagram of trackingloop with and without the lag compensation.

In order to increase the gain by more than 2 times in thefrequency of the missile body oscillation and the phase lag

qc qsqs. .. ua

k�Kf 2+

km

�휉ms2 s + 1ω2m ωm

Motor transfer functionGain

+−

Angular rategyro

Stability loop

Figure 7: The single-axis stability loop contains a torque motor.

100 101 102

w(rad/s)

−200−150−100

−500

50

Phas

e (de

g)

100 101 102−40−30−20−10

010

Mag

nitu

de (d

B)

Figure 8: Frequency characteristic curve (amplitude is 0.2°).

Table 1: Value of the parameter.

Parameter Value

Operational amplifier K1 = K3 = 12

Power amplifier K2 = K4 = 12

Lag compensation network

T1 = T3 = 0 4T2 = T4 = 0 04

β1 = β2 = β3 = β4 = 15

Inductance L2 = L4 = 0 0035H

Resistance R1 = R2 = 8ΩTorque coefficient KT1 = KT2 = 0 2344N ×m/A

Back EMF coefficient KE1 = KE1 = 0 5344V/rad/s

Inner gimbalmoment of inertia

Jnxx = 0 002N ×m × s2

Jnyy = 0 02N ×m × s2

Jnzz = 0 004N ×m × s2

Outer gimbalmoment of inertia

Jwxx = 0 004N ×m × s2

Jwyy = 0 005N ×m × s2

Jwzz = 0 045N ×m × s2

Angular rate gyro H s = 1/ 0 000004s2 + 0 0028s + 1

9International Journal of Aerospace Engineering

within 15 deg., the lag compensation network in stability loopis designed as follows:

β2 T2s + 1β2T2s + 1

=β4 T4s + 1β4T4s + 1

=15 s/25 + 115 s/25 + 1

36

Figure 10 shows the open loop Bode diagram of stabilityloop with and without the lag compensation.

If the missile and the target are stationary and the relativedistance is fixed, then the yaw angle is a constant value of theLOS angle input, and the missile body yaw angle is the posi-tive rotation signal input. Figure 11 displays the simulationcurve with body yaw direction input. The yaw gimbal willswing in the opposite direction when the missile body isyaw swinging to ensure the stability of the LOS because theLOS angle input has the constant value of 0. Therefore, the

Frequency (rad/s)10−2 10−1 100 101 102 103 104 105

−270−225−180−135

−90−45

System: sysclosePhase margin (deg): 90.5Delay margin (sec): 0.129At frequency (rad/s): 12.3Closed loop stable? Yes

Phas

e (de

g)

Tracking loop with lag compensationTracking loop without lag compensation

Bode diagram

−200−150−100

−500

50100

System: syscloseGain margin (db): 31.1At frequency (rad/s): 347Closed loop stable? Yes

System: syscloseGain margin (db): 31At frequency (rad/s): 345Closed loop stable? Yes

Mag

nitu

de (d

b)

System: sysclosePhase margin (deg): 80Delay margin (sec): 0.111At frequency (rad/s): 12.5Closed loop stable? Yes

Figure 9: The open loop Bode diagram of tracking loop.

Frequency (rad/s)10−2 10−1 100 101 102 103 104 105

−360

−270

−180

−90

0

System: stability loop without lag compensationPhase margin (deg): 59.4Delay margin (sec): 0.00622At frequency (rad/s): 167Closed loop stable? YesPh

ase (

deg)

Stability loop with lag compensationStability loop without lag compensation

System: stability loop with lag compensationPhase margin (deg): 51.2Delay margin (sec): 0.0053At frequency (rad/s): 168Closed loop stable? Yes

Bode diagram

−200−150−100

−500

50100

Mag

nitu

de (d

b)

System: stability loop withoutLag compensationGain margin (db): 10.5At frequency (rad/s): 441Closed loop stable? Yes

System: stability loop with lag compensationGain margin (db): 9.96At frequency (rad/s): 426Closed loop stable? Yes

Figure 10: The open loop Bode diagram of stability loop.

0.0 0.5 1.0 1.5 2.0−6

−4

−2

0

2

4

6Only the missile body yaw motion

Body yaw attitude angle Inner gimbal angleInput yaw LOS angle Output yaw LOS angle

Yaw

dire

ctio

n an

gle (

deg)

Time (s)

Figure 11: Simulation curve with body yaw direction.

10−1 100 101 102 103−225−180−135

−90−45

0

Phas

e (de

g)Bode diagram

Frequency (rad/s)

−50

−40

−30

−20

−10

0M

agni

tude

(db)

System: syscloseFrequency (rad/s): 12.3Magnitude (db): -3

Figure 12: The closed loop Bode diagram of tracking loop.

−100−80−60−40−20

020

Mag

nitu

de (d

b)

101 102 103 104 105−180

−135

−90

−45

0

Phas

e (de

g)

Bode diagram

Frequency (rad/sec)

Figure 13: The closed loop Bode diagram of stability loop.

10 International Journal of Aerospace Engineering

LOS angle output is near zero in the case of missile yaw dis-turbances, and the swing is not large. The lag compensationnetwork can increase the open loop gain 15 times. The closedloop Bode diagram of tracking loop is shown in Figure 12,and the closed loop bandwidth is 12.37 rad/s (1.97Hz), whichis close to the frequency of oscillation (2Hz). The open loopBode diagram is shown in Figure 10; we can find that the lagcompensation increases the open loop gain. The closed loopBode diagram about stability loop is shown in Figure 13:the closed loop bandwidth is 370 rad/s (59Hz), which islarger than the tracking loop and the frequency of oscillationand can improve the disturbance rejection capability.

Figure 14 demonstrates the coupling swing curve of thepitch gimbal caused by the missile yaw movement. The mis-sile yaw movement causes not only the yaw gimbal swing butalso the pitch gimbal coupling swing.

0.0 0.5 1.0 1.5 2.0−6

−4

−2

0

2

4

6

8

10A

ngle

(deg

)

Time (s)

Inner gimbal angleOuter gimbal angle

(a)

0.5 1.0 1.5

9.9

10.0

10.1

Outer gimbal angle

Ang

le (d

eg)

Time (s)

Partially enlarged detail

(b)

Figure 14: Coupled curve of the inner and outer gimbals.

0.0 0.5 1.0 1.5 2.0−6

−4

−2

0

2

4

6Only the missile body pitch motion

Body pitch attitude angle Outer gimbal angleInput pitch LOS angle Output pitch LOS angle

Pitc

h di

rect

ion

angl

e (de

g)

Time (s)

Figure 15: Simulation curve with body pitch direction.

0.0 0.5 1.0 1.5 2.0−6

−4

−2

0

2

4

6Together the missile body pitch and yaw motion

Body pitch attitude angle Output pitch LOS angleOuter gimbal angle Input pitch LOS angle

Pitc

h di

rect

ion

angl

e (de

g)

Time (s)

Figure 16: Simulation curve with body pitch and yaw direction.

0.3590 0.3595 0.3600

4.858

4.860

4.862

4.864

Only body pitch motionouter gimbal angle

Ang

le (d

eg)

Time (s)

Body pitch and yawmotion outer gimbal angle

Figure 17: Outer gimbal contrast curve with missile double andpitch directions.

11International Journal of Aerospace Engineering

Figure 15 depicts the simulation curve with body pitchattitude angle input signal. The pitch gimbal will swing inthe opposite direction when the body is swinging toensure the stability of the LOS. The output pitch LOSangle is close to the input pitch LOS angle (which is zero).The closed loop bandwidth is greater than the frequencyof oscillation. The frequency characteristic is the same as ofthe yaw gimbal.

Figure 16 depicts the simulation curve with body pitchangle and yaw angle input. The pitch gimbal will swing inthe opposite direction of the body pitch motion; the yaw gim-bal will swing in the opposite direction of the body yawmotion, when the missile body pitch and yaw is swinging toensure the stability of the LOS. Figure 17 presents the outer

gimbal simulation curve with body pitch and yaw angle inputor only the pitch angle input. We find that the outer gimbalangle swing amplitude is higher with body pitch and yawinput than only the pitch angle input given the couplingbetween the gimbal and the missile body and the couplingeffect between the inner and outer gimbals.

Figure 18 depicts the schematic of the single-axis trackingcontrol system, where GD s denotes the disturbance torquetransfer function. Spring and viscous torque factors are alsoapplied to the system, and the disturbance torque transferfunction can be expressed as GD s = Ks/s and GD s = Kv ,respectively; Δqf is the angular velocity output due to bodydisturbance. The disturbing rate rejection transfer function[10] can be expressed as follows:

1s

qt �휀 1

Ls + RKT

1

Jeqs

KE

qs qs.

.

.

�휑r .�휗

TD

TM

Ea

GD(s)

u

qfStability loop

Tracking loopGyro model

K1�훽1 (T1s + 1) �훽2 (T2s + 1)�훽1T1s + 1 �훽2T2s + 1

K2+− +−−+ −

−

+

+

H (s)

Figure 18: Schematic of the single-axis seeker tracking control system.

1%3%

5%7%

Bode diagramThe magnitude is highest

Serious rejection area

100 101 10210−1

Frequency (Hz)

−80

−60

−40

−20

Mag

nitu

de (d

b)

−450

−360

−270

−180

−90

Phas

e (de

g)

Figure 19: Function Bode diagram of viscous torque disturbing raterejection transfer.

1%3%

5%7%

Bode diagramThe magnitude is highest

Serious rejection area

100 101 10210−1

Frequency (Hz)

−45

0

45

90

135

180Ph

ase (

deg)

−80

−60

−40

−20

0

Mag

nitu

de (d

b)

Figure 20: Function Bode diagram of spring torque disturbing raterejection transfer.

Δqf sϑ s spring

=sKEKTH s + KsH s Ls + R

s Js Ls + R + KEKT +G2 s KTH s + Ks Ls + R +G1 s G2 s KT,

Δqf sϑ s vis

=s KEKTH s + KvH s Ls + R

s Js Ls + R + Kv Ls + R + KEKT +G2 s KTH s +G1 s G2 s KT

37

12 International Journal of Aerospace Engineering

Figures 19 and 20 illustrate the disturbing rate rejectiontransfer function in frequency domain caused by viscousand spring torques, respectively. The disturbing rate rejectionamplitudes are 1%, 3%, 5%, and 7% when the frequency is2Hz. In Figure 19, we find that the maximum value of theamplitude of the disturbing rate rejection transfer functionis located near 2Hz, where the operating frequency of theseeker is maximized and has a significant influence on theparasitic loop. In Figure 20, we find the serious rejection areaunder the action of spring torque which is located in the low-frequency band and presents the integral link characteristic.Spring and viscous torques will affect the tracking accuracyof the seeker.

6. Conclusions

The two-axis rate gyro-stabilized platform dynamic model-based seeker with cross-coupling, mass unbalance, and dis-turbance torque is built according to the two-axis rate gyrostabilization platform tracking principle, applying the torquemotor drive and seeker servo control theory and combin-ing the platform framework kinematics theory and spatialcoordinate transformation method. Simultaneously, two-gimbal-coupled system block diagram is built. The frequencydomain method is used to identify the seeker sensor transferfunction. This study comprehensively considers the condi-tions of the disturbance moment of the platform, couplingcharacteristics, and gimbal angle changing curve analyzedin different missile body inputs. Furthermore, the disturbingrate rejection caused by spring and viscous torque factors isanalyzed. On the basis of the analysis, the coupling effect ofthe missile body to the gimbal is strong given the presenceof disturbance torque. The missile motion is coupled to theseeker output, but the seeker servo stable platform can reducethe influence of the missile disturbance by swinging in theopposite direction. The coupling effect between the innerand outer gimbals is evident. The serious segregation area islocated in the middle- and low-frequency band by analyzingthe function frequency characteristics of disturbing raterejection transfer caused by spring and viscous torque fac-tors. Spring and viscous torque factors will affect the trackingaccuracy of the seeker.

The results of this study can support the equivalencedemonstration of the seeker servo control system in theclosed loop simulation test and actual work conditions andthus uphold the study of the seeker servo control systemand semiphysical simulation.

Conflicts of Interest

The authors declare that there is no conflict of interestsregarding the publication of this article.

Acknowledgments

This work is supported in part by research grants from theAviation Science Foundation, China (20150172001).

References

[1] A. J. Fenn, D. H. Temme, W. P. Delaney, and W. E. Courtney,“The development of phased-array radar technology,” LincolnLaboratory Journal, vol. 12, no. 2, pp. 321–340, 2000.

[2] C. R. Janice, H. M. James, P. B. Joel, and T. Hudson, “The past,present, and future of electronically-steerable phased arrays indefense application,” in 2008 IEEE Aerospace Conference,pp. 1–7, Big Sky, MT, USA, March 2008.

[3] M. K. Masten and J. M. Hilker, “Electromechanical systemconfigurations for pointing, tracking and stabilization applica-tions,” in Proceedings of SPIE 0779, Electromechanical SystemInteraction with Optical Design, pp. 75–87, Orlando, FL,USA, October 1987.

[4] L. A. Stockum, J. Burge, and G. Plunk, “Electro-mechanicaldesign for precision pointing and tracking systems,” inProceedings of SPIE, Electromechanical System Interactionwith Optical Design, pp. 66–74, Orlando, FL, USA, October1987.

[5] P. Garnell, Z. K. Qi, and Q. L. Xia, Guided Weapon ControlSystem. Second Revision, Beijing Institute of Technology, 2004.

[6] H. Li, J. H. Wu, X. Hu, H.-P. Zhao, and R.-G. Yin, “Dynamicmodeling and simulation of dynamic-gyro laser hominghead,” Laser & Infrared, vol. 41, no. 7, pp. 758–762, 2011.

[7] H. Li, J. H. Wu, Z. Zhu, R. G. Yin, and W. Dai, “Stabilizationtracking principle and simulation of rate gyroscope stabilizedlaser seeker,” Infrared and Laser Engineering, p. 4, 2011.

[8] R. T. Rudin, “Strapdown stabilization for imaging seekers,”in Annual Interceptor Technology Conference, pp. 1–10,Albuquerque, NM, USA, June 1993.

[9] R. D. Wetherington and J. L. Birchfield, Techniques forGenerating Guidance Signals from Body-Fixed Sensors, GeorgiaInstitute of Technology, 1970.

[10] A. S. Larry, B. James, and P. Gerry, “Electro-mechanical designfor precision pointing and tracking systems,” in Proceedings ofSPIE 0779, Electromechanical System Interaction with OpticalDesign, pp. 66–74, Orlando, FL, USA, October 1987.

[11] A. K. Rue, “Precision stabilization systems,” IEEE TransactionsAerospace and Electronic Systems, vol. AES-10, no. 1, pp. 34–42, 1974.

[12] J. J. Rodden, “Mirror line of sight on a moving base,” inGuidance and control 1989: Proceedings of the Annual RockyMountain Guidance and Control Conference, pp. 255–272,Keystone, CO, USA, February 1989.

[13] W. J. Bigley and F. Schupan, “Wide-band base motion isola-tion control for a mobile platform,” in 1987 American ControlConference, pp. 1483–1490, Minneapolis, MN, USA, June1987.

[14] P. Zarchan, Tactical and Strategic Missile Guidance, AmericanInstitute of Aeronautics and Astronautics, Virginia, USA,2004.

[15] C. J. Zhang, “Analysis on design of control loop for seeker sta-bilized platform,” Journal of Projectiles, Rockets, Missiles andGuidance, vol. 29, no. 5, pp. 46–48, 2009.

[16] T. Song, D. F. Lin, and J.Wang, “Influence of seeker disturbancerejection rate on missile guidance system,” Journal of HarbinEngineering University, vol. 34, no. 10, pp. 1234–1241, 2013.

[17] F. G. Li, Q. L. Xia, and Z. K. Qi, “Effect of parasitic loop onstrap-down seeker and compensated with identificationmethod,” Systems Engineering and Electronics, vol. 35, no. 8,pp. 1717–1722, 2013.

13International Journal of Aerospace Engineering

[18] X. Du and Q. L. Xia, “The research of guidance performance ofthe phased array seeker with platform for air-to-air missile,”Optik, vol. 127, no. 22, pp. 10322–10334, 2016.

[19] P. J. Kennedy and R. L. Kennedy, “Direct versus indirect line ofsight (LOS) stabilization,” IEEE Transactions of Control Sys-tems Technology, vol. 11, no. 1, pp. 3–15, 2003.

[20] J. X. Wang, D. F. Lin, Z. K. Qi, and Z. X. Cheng, “Study on dis-turbance rejection rate parasitical loop of strapdown phasedarray radar seeker,” Transactions of Beijing Institute of Tech-nology, vol. 33, no. 11, pp. 1124–1129, 2013.

[21] J. M. Song, G. H. Cai, L. X. Kong, and J. Fan, “The guidancesystem design of the semi-strapdown homing guided missile,”Proceedings of the Institution of Mechanical Engineers, Part I:Journal of Systems and Control Engineering, vol. 226, no. 6,pp. 761–774, 2012.

[22] R. N. Bhattacharya, T. V. Tao, and S. Sadhu, “Control struc-tures and properties of missile seekers,” Journal of the Institu-tion of Engineers: Electrical Engineering Division, vol. 82,pp. 253–261, 2002.

[23] B. Ekstrand, “Equations of motion for a two-axes gimbal sys-tem,” IEEE Transactions on Aerospace and Electronic Systems,vol. 37, no. 3, pp. 1083–1091, 2001.

[24] J. K. Peter and R. L. Kennedy, “Direct versus indirect line-of-sight (LOS) stabilization,” IEEE Transaction on ControlSystems Technology, vol. 11, no. 1, pp. 3–15, 2003.

[25] J. Royalty, “Development of kinematics for gimbal mirror sys-tems,” in Proceedings of SPIE 1304, Acquisition, Tracking, andPointing IV, pp. 262–274, Orlando, FL, USA, September 1990.

[26] V. Saini and H. B. Hablani, “Air-to-air tracking of a maneuver-ing target with gimbaled radar,” Journal of Guidance Control,and Dynamics, vol. 39, no. 2, pp. 262–274, 2016.

14 International Journal of Aerospace Engineering

International Journal of

AerospaceEngineeringHindawiwww.hindawi.com Volume 2018

RoboticsJournal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Shock and Vibration

Hindawiwww.hindawi.com Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwww.hindawi.com

Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwww.hindawi.com Volume 2018

International Journal of

RotatingMachinery

Hindawiwww.hindawi.com Volume 2018

Modelling &Simulationin EngineeringHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Navigation and Observation

International Journal of

Hindawi

www.hindawi.com Volume 2018

Advances in

Multimedia

Submit your manuscripts atwww.hindawi.com