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Research Journal of Applied Sciences, Engineering and Technology 4(19): 3843-3851, 2012
ISSN: 2040-7467
Maxwell Scientific Organization, 2012
Submitted: May 08, 2012 Accepted: May 29, 2012 Published: October 01, 2012
Corresponding Author: Leghmizi Said, College of Automation, Harbin Engineering University, Harbin, Heilongjiang
150001, China
3843
Modeling, Design and Control of a Ship Carried 3 DOF Stabilized Platform
Leghmizi Said, Liu Sheng, Naeim Farouk and Boumediene LatifaCollege of Automation, Harbin Engineering University, Harbin, Heilongjiang 150001, China
Abstract: The system for stabilizing platform of a ship carried antenna a nd its core component arediscussed in this study. Relevant mathematics model of these components are established. Thus, thedynamic model of the system is deduced including the effects of friction, inertia and torque motors. UsingSolid Works, we built the mechanical structure, including the servo machine of each part of the system.The system under consideration is a system with strong interactions between three channels. By using theconcept of decentralized control, a control structure is developed that is composed of three control loops,each of which is associated with a single-variable controller. First, PID controller was applied; then,Takagi-Sugeno (TS) fuzzy controller was used for controlling the platform. Simulation tests wereestablished using Simulink of Matlab. The obtained results have demonstrated the feasibility andeffectiveness of the proposed fuzzy approach comparing to the PID controller. Simulation results are
represented in this study.
Keywords: Decentralized control, dynamic model, fuzzy controller, PID contoller, simulink, solidworks,Takagi-Sugeno (TS)
INTRODUCTION
The stabilized platform is the object which canisolate motion of the vehicle and can measure thechange of platforms motion and position incessantly. Itcan make the equipment which is fixed on the platformaim at and track object fastly and exactly. In thestabilized platform systems, the basic requirements areto maintain stable operation even when there arechanges in the system dynamics and to have very gooddisturbance rejection capability.
Since they began to be utilized about 100 yearsago, stabilized platforms have been used on every typeof moving vehicle, from satellites to submarines and areeven used on some handheld and ground-mounteddevices (Hilkert, 2008; Debruin, 2008). Its applicationis quite abroad and it becomes investigative hotspot inmost countries all the time.
The considered platform is a class of multivariableservomechanisms with multiple axes. The control ofsuch multivariable servomechanisms is, in general, nota simple problem, as there exist cross-couplings, orinteractions, between the different channels. Inaddition, this system is required to maintain stableoperation even when there are changes in the systemdynamics. In the stabilized platform systems, the basicrequirement is to have very good disturbance rejectioncapability. Presence of inherent nonlinearities such asstriction, friction, saturation of actuators, etc., also must
be taken into account.Many approaches have been proposed to control
such a complex interconnected system for example the
decomposition-coordination approach, the aggregation
approach, the multitime-scale approach and the
decentralized control approach (Linkens and Nyongesa,1998; Lee et al., 1995) Since the decentralized control
approach is reliable and practical in view of the
implementation, it is the most popular method that
attempts to design control schemes, where eachsubsystem is controlled independently based on local
information. However, the decentralized approaches arerestricted to stabilization and the dynamics of eachsubsystem and the interconnection terms are assumed to
be known (Nie, 1997; Shi and Singh, 1992). In practice,
the model of the considered platform contains vast
unknown uncertainties. Since fuzzy logic control hasbeen considered as an alternative to traditional control
schemes to deal with system dynamics uncertainty and
obtain the best performance of the system. Fuzzy
control is adopted as the subject of this study (Yeh,1999).
The objective of this study is to develop thedynamics model and the 3D design of the ship carriedstabilized platform. Then, apply the PID and Fuzzy
controllers for stabilizing the platform. By using thedecentralized control concept, we developed a controlstructure composed of three separate control channels,each of which is associated with a single variablecontroller. The simulation results in applying the two
proposed controller to a 3-DOF stabilization system arepresented which demonstrates the effectiveness of theproposed fuzzy controller comparing to the PIDcontroller.
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(Pitch)
(Roll)
(Yaw)
Fig. 1: The 3-DOF platform structure
SYSTEM DYNAMICS
System description: The considered system in this
study is composed of the platform, inner gimbal, outergimbal and the case (Fig. 1); each member is assumed
to be rigid and has one degree of freedom (Leghmizi
and Liu, 2010).
According to the Euler definition of the rotation
angles, we define the angles and rates relating the four
members of the gimbaled system as following (Barnes,
1971):
= Relative angle between the inner gimbal and the
platform, measured about the platform Y axis (Yp)
= Relative angular rate between the inner gimbal and
the platform, measured about the platform Y axis
(Yp) = Relative angle between the outer and inner
gimbals, measured about the inner gimbal Z axis
(ZI)
= Relative angular rate between the outer and inner
gimbals, measured about the inner gimbal Z axis
(ZI)
= Relative angle between the case and the outer
gimbals, measured about the outer gimbal X axis
(Xo)
= Relative angular rate between the case and theouter gimbal, measured about the outer gimbal Xaxis (Xo)
At each member of the gimbaled system weassociate an orthogonal coordinate system (Fig. 2)
platform (Xp, Yp, Zp), inner gimbal (XI, YI, ZI), outergimbal (XO, YO, ZO) and case (XC, YC, ZC).
The considered system platform is fixed on theship. Generally the satellite dish antenna is based on the
back part of the ship presented in Fig. 3.
Dynamics model: The mathematical modeling wasestablished using Euler theory. The Eulers momentequations are:
HiM (1)
The net torque M consists of driving torque appliedby the adjacent outer member and reaction torqueapplied by the adjacent inner member:
HHmdt
dHHi m
(2)
Hi : Inertial derivative of the vectorH
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Fig. 2: The system topology
Fig. 3: The case coordinates in the body-fixed frames
Hm : Derivative of H calculated in a rotating frame ofreference
m : Absolute rotational rate of the moving reference
frameH : Inertial angular momentum
M : External torque applied to the bodyBy applying Eq. (2) on the different parts of the
platform system, the system may be expressed as a set
of second-order differential equations in the statevariables. Solving this system of equations we obtain:
iooi
iooi
BABA
BCBC
(3)
iooi
oiio
BABA
ACAC
(4)
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iooi
iooi
p
p
p
p
BABA
BCBC
B
A
B
C
* (5)
where,
sinpA
1pB
py
Ipy
pI
MPYMC
*
iz
pzpx
iI
IIA sincoscos
iz
px
iz
pxi
II
IIB 22 cossin1
iz
oiz
iI
MIZMC
*
ox
iy
ox
pzpxix
oI
I
I
IIIA
2
22
2sin
sincoscos1
ox
pzpx
oI
IIB cossincos
ox
coxo
I
MCXMC
*
Detailed equations computation is presented in the
study (Leghmizi et al., 2011).
ESTABLISHING THREE DIMENSIONAL
MODEL OF THE STABILIZED PLATFORM
The design of a three-axis platform requires a total
of four bodies. These bodies include the base (case), the
inner, outer and platform gimbals.
The base, shown in Fig. 4 is designed to provide a
solid foundation to the three gimbals that will be
rotating around their respected axis. A rigid base is
responsible for preventing vibrations that will causeinaccurate movement and give unpredictable dynamic
responses. In the right section of the base is attached a
servo machine boxe, shown in Fig. 8a, responsible of
the rotation of the outer gimbal.
The outer gimbal, shown in Fig. 5, controls the
system platform roll. The outer gimbal will be directly
attached to the actuator mounted to the base. When the
actuator is activated it will allow the outer gimbal to
Fig. 4: The base unit of the platform
Fig. 5: The outer gimbal
Fig. 6: The inner gimbal
rotate 180. The second actuator boxe, shown in Fig.
8b, is contained inside the outer gimbal. The actuator is
firmly attached inside and is enclosed by a cap.
The inner gimbal, shown in Fig. 6, controls the 3-
DOF platform yaw. The middle gimbal is rotated by the
actuator housed in the outer gimbal. The middle gimbal
allows the platform to rotate 90. The third and finalactuator, shown in Fig. 8c, is attached the left side of
the inner gimbal.
The platform gimbal showed in Fig. 7 is
responsible for the platforms pitch. The platform
gimbal does not include housing for an actuator as there
are no additional gimbals to rotate.
The platform system assembly is represented in
Fig. 9.
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Fig. 7: The platform gimbal
Fig. 8: The machine boxes used in the platform
Fig. 9: Final assembly of the platform
Table 1: Platform parameters
Platform Inner gimbal Outer gimbal
Ipx 0.119 Iix 0.406 Iox 1.05
Ipy 0.119 Iiy 0.845 Ioy 1.05
Ipz 0.237 Iiz 1.020 Ioz 1.10
Dip 0.110 Doi 0.260 Dco 0.32
Using this software we can also calculate theprincipal moments of inertia taken at the center of mass
of each part of the platform system. These moments of
inertia will be used in the following simulation section.
The values found for these parameters are given inTable 1.
SIMULATION RESULTS AND DISCUSSION
The complete nonlinear dynamics for the stabilized
platform system was developed in the previous section.
Here, it suffices to note that designing a simulation for
the system based on these complete nonlinear dynamics
is extremely difficult. It is thus necessary to reduce the
complexity of the problem by considering the linearized
dynamics (Lee et al., 1996). This can be done by notingthat the gimbal angles variation are effectively
negligible and that the ship velocities effect is
insignificant.Applying the above assumptions to the nonlinear
dynamics, the following equations are obtained:
oo
oxixpx
pxpypz
co
oxixpxoxixpx
co
TIII
IIIF
IIIIII
D
)(sgn
1
(6)
mm
izpz
pzpxpy
oi
izpzizpz
oi
TII
IIIF
IIII
D
)(sgn
1
(7)
II
py
pypzpx
ip
pypy
ipT
I
IIIF
II
D
)(sgn
1 (8)
PID controller simulation: In the 3-D of platform we
will apply a decentralized PID which consiste of three
PID controllers applied to each part of the platformseparately as shown in Fig. 10 the PID parameters are
calculated for each part using the Ziegler-Nichols
method (Ziegler and Nichols, 1942). The obtained
parameters are listed in Table 2.
In order to see the outcome of the designed
controller, we performed a simulation in closed-loop
mode. This simulation was particularly useful for the
recognition of the effect of each PID coefficient to the
response of the system.
Simulation results will be presented to illustrate the
gimbals behavior to different PID parameters. They are
presented in Fig. 11, which contain the step response of
the platform system using the PID controller and in Fig.
12, which contain the impulsion response of the
platform response using the PID controller.
As shown in Fig. 11 and 12, the responses are
significantly acceptable but the response characteristics
still not well improved.
Figure 13 illustrates the position tracking responses
using PID controller. It can be seen that this controller
present bad tracking performance with big rising time.
This is due to the nonlinearities of the system that the
controller cant handle. For this an introduction of a non
linear controller is necessary.
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Fig. 10: PID control for the 3-DOF platform
Table 2: PID parameters
Kp Ki Kd
8.8 1.8 11.3
Fig. 11: Step response of the platform using PID controller
Fig. 12: Impulse response of the platform using PID controller
Fig. 13: Position tracking response using PID controller
Fig. 14: Disturbance rejection of the 3DOF platform using
PID controller
0
0.5
1.0
1.5
Gimbalsposition(rad)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Time (s)
Platform
Inner gimbal
Outer gimbal
0
0.5
1.0
1.5
Gi
balsposition(rad)
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time (s)
-0.5
Platform
Inner gimbal
Outer gimbal
0Gimbalsposition(rad)
0 1 2 3 5
Time (s)
-1
Platform
Inner gimbal
Outer gimbal
4 6 7 98
1
2
3
4
5
Reference
0
0.5
1.0
1.5
Gimbalspositio
n(rad)
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time (s)
Platform
Inner gimbal
Outer gimbal
4.5 5.0
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Fig. 15: TS fuzzy controller for the 3-DOF platform
Also, in order to see the disturbance rejection
aptitude of the PID controller, small disturbance was introduced. The injected disturbance was a pulse of
0.1rad amplitude and adds to the system input at
time instant 3s.
The disturbance rejection capability of each part of
the stabilized platform using PID controller is plotted in
Fig. 14 They show that the controller is also unable of
dealing with this situation.
Fuzzy controller simulation: In the 3-D of platform
we will apply a decentralized Fuzzy controller which
consiste of three TS-Fuzzy controllers applied to each
part of the platform separately as shown in Fig. 15.
The structure of a complete fuzzy control system is
composed from the following blocs: Fuzzification,
Knowledge base, Inference engine, Defuzzification asshown in Fig. 3 (Chuen, 1990).
The general TS fuzzy systems in this study use 2
input variables. e, e, eand , , are selected as
input variables of each subsystem respectively and
defined as two variables representing the situation. cji is
selected as output of the jth subsystem and defined as a
variable representing the action. Notice that variables
for, , , , and assume linguistic terms as their
values such as positive-big, negative-small and zero,
etc.
Table 3: Rule base of the fuzzy logic controller
e\ NB NM NS ZR PS PM PB
PB ZR PS PM PB PB PB PB
PM NS ZR PS PM PB PB PB
PS NM NS ZR PS PM PB PB
ZR NB NM NS ZR PS PM PB
NS NB NB NM NS ZR PS PM
NM NB NB NB NM NS ZR PS
NB NB NB NB NB NM NS ZR
Using the Takagi-Sugeno model (Takagi and
Sugeno, 1985), the fuzzy system is characterized by a
set of p If-Then rules stored in a rule-base and
expressed as Ri: IF e is Ai and is Bi then:
210 peppcj
i
where, Ai and Bi are linguistic terms which in thisstudy can be NL, NM, NS, ZR, PS, PL and PB.
The rule base of this controller is summarized in
Table 3 for simplicity; the same universe of discourse
and the same fuzzy set are adopted for fuzzy input
variables. The membership functions of isosceles
triangles are used as the fuzzification function.
The Sugeno type fuzzy controller employ linear
functions of input variables as rule consequent, so the
steps of aggregation and defuzzification of fuzzy rules
are simultaneously and the final output of the system
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(a) (b)
(c)
Fig. 16: Step response of the platform system using PID and TS Fuzzy controller, (a) platform, (b) inner gimbal, (c) outer gimbal
is the weighted average of all rule outputs, computed
as:
N
i
i
N
i
i
j
i
l
c
T
1
1
(9)
The zero-order Sugeno model is applied, the output
level z is a constant (p1 = p2 = 0). The value of p0
depends on the linguistic term of the output. For
example if the output is NB (according to the rule base)
so p0 = -1.To observe the performance of the designed fuzzy
controller a comparison between the step response of
the platform using PID controller and the step response
of the platform using the fuzzy controller was done.
The results of the simulation investigating
positioning performance comparison of platform
system are shown in Fig. 16.
Figure 16 (Solid lines) shows step responses of the
stabilized platform system when controlled by three
separated order-0 TS fuzzy controllers. Figure 16
Fig. 17: Position tracking response of the platform
(dashed lines) shows step responses of the stabilizedplatform system when controlled by three separatedPID controllers. As shown in Fig. 16 the responses(solid lines) were significantly improved with smallerovershoot, shorter rising time.
Figure 17 illustrates the position tracking responsesusing TS fuzzy PD controller. It can be seen that thiscontroller present good tracking performance withminor rise time.
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Time (s)
PID
TS Fuzzy
Platformp
osition(rad)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Time (s)
PID
TS Fuzzy
Innergimbalposition(rad)
0
0.5
1.0
1.5
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Time (s)
PID
Outergi
balposition(rad)
TS Fuzzy
-0.2
0.6
1.0
1.2
Platform
position(rad)
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time (s)
Platform response
Desired trajectory
4.5 5.0
0.8
0.0
0.4
0.2
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CONCLUSION
Our research focused on the common coordinate,kinematics, dynamics, control system and softwaredesign for ship carried stabilized platform. For that inthis study we developed a dynamics modeling of the
platform and a 3D model of the platform usingSimulink and SolidWorks. Then, we considered the
problem of controlling this multivariableservomechanism where there exist cross-couplings
between the channels. A fuzzy PD control strategyusing a Takagi-Sugeno fuzzy model has been proposedand by comparing it with PID controller, it has beenshown in the study that uniformly stable operation isachieved together with asymptotic tracking of thereference command signals.
In the study, simulation results in applying theproposed TS fuzzy PD controller to a 3-Dofstabilization system have been presented whichdemonstrates the effectiveness of the fuzzy controller.Future study is directed to the optimization of thescaling factors of the fuzzy system and the intelligentmethod to generate an effective rule base.
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