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
Modeling, Design and Control of a Ship Carried 3 DOF Stabilized Platform
Leghmizi Said, Liu Sheng, Naeim Farouk and Boumediene Latifa
College 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 are
discussed in this study. Relevant mathematics model of these components are established. Thus, the
dynamic model of the system is deduced including the effects of friction, inertia and torque motors. Using
Solid 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 the
concept 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 were
established using Simulink of Matlab. The obtained results have demonstrated the feasibility and
effectiveness 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 can
isolate motion of the vehicle and can measure the
change of platform’s motion and position incessantly. It
can make the equipment which is fixed on the platform
aim at and track object fastly and exactly. In the
stabilized platform systems, the basic requirements are
to maintain stable operation even when there are
changes in the system dynamics and to have very good
disturbance rejection capability.
Since they began to be utilized about 100 years
ago, stabilized platforms have been used on every type
of moving vehicle, from satellites to submarines and are
even used on some handheld and ground-mounted
devices (Hilkert, 2008; Debruin, 2008). Its application
is quite abroad and it becomes investigative hotspot in
most countries all the time.
The considered platform is a class of multivariable
servomechanisms with multiple axes. The control of
such multivariable servomechanisms is, in general, not
a simple problem, as there exist cross-couplings, or
interactions, between the different channels. In
addition, this system is required to maintain stable
operation even when there are changes in the system
dynamics. In the stabilized platform systems, the basic
requirement is to have very good disturbance rejection
capability. Presence of inherent nonlinearities such as
striction, 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 each
subsystem is controlled independently based on local
information. However, the decentralized approaches are
restricted to stabilization and the dynamics of each
subsystem 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 has
been 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 the
dynamics model and the 3D design of the ship carried
stabilized platform. Then, apply the PID and Fuzzy
controllers for stabilizing the platform. By using the
decentralized control concept, we developed a control
structure composed of three separate control channels,
each of which is associated with a single variable
controller. The simulation results in applying the two
proposed controller to a 3-DOF stabilization system are
presented which demonstrates the effectiveness of the
proposed fuzzy controller comparing to the PID
controller.
Corresponding Author: Leghmizi Said, College of Automation, Harbin Engineering University, Harbin, Heilongjiang
150001, China
3843
Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
 (Yaw)
 (Pitch)
 (Roll)
Fig. 1: The 3-DOF platform structure
= Relative angular rate between the case and the
outer gimbal, measured about the outer gimbal X
axis (Xo)
SYSTEM DYNAMICS
System description: The considered system in this
study is composed of the platform, inner gimbal, outer
gimbal 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):
θ = 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)
ψ = 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)
At each member of the gimbaled system we
associate an orthogonal coordinate system (Fig. 2)
platform (Xp, Yp, Zp), inner gimbal (XI, YI, ZI), outer
gimbal (XO, YO, ZO) and case (XC, YC, ZC).
The considered system platform is fixed on the
ship. Generally the satellite dish antenna is based on the
back part of the ship presented in Fig. 3.
Dynamics model: The mathematical modeling was
established using Euler theory. The Euler’s moment
equations are:
M  i H
(1)
The net torque M consists of driving torque applied
by the adjacent outer member and reaction torque
applied by the adjacent inner member:
i H 
i H
3844 dH
 m H  m  H
dt
: Inertial derivative of the vector
(2)
H
Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
Fig. 2: The system topology
Fig. 3: The case coordinates in the body-fixed frames
mH : Derivative of H calculated in a rotating frame of
reference
of second-order differential equations in the state
variables. Solving this system of equations we obtain:
m : Absolute rotational rate of the moving reference
frame
H : Inertial angular momentum
M : External torque applied to the body
By applying Eq. (2) on the different parts of the
platform system, the system may be expressed as a set
3845  
C i B o  C o Bi
Ai Bo  Ao Bi
(3)
 
C o Ai  C i Ao
Ai Bo  Ao Bi
(4)
Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
 
Cp
Bp

Ap
Bp
*
C i B o  C o Bi
Ai Bo  Ao Bi
(5)
where,
A p  sin 
Bp  1
Cp 
M
Fig. 4: The base unit of the platform
*
Ipy
 MPY
I py
 I px  I pz 
Ai  cos cos sin  

 I iz 
I px
I px 

 cos 2 
Bi  1  sin 2 

I iz
I iz 

M *  MIZ
Ci  oiz
I iz
 I ix  I px cos 2   I pz sin 2  
 I iy 
2
Ao  1  cos 2  
  sin   
I ox
 I ox 


Fig. 5: The outer gimbal
 I px  I pz 
Bo  cos sin  cos 

 I ox 
Co 
*
M cox
 MCX
I ox
Detailed equations computation is presented in the
study (Leghmizi et al., 2011).
ESTABLISHING THREE DIMENSIONAL
MODEL OF THE STABILIZED PLATFORM
Fig. 6: The inner gimbal
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 cause
inaccurate 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
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 3DOF 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 final
actuator, shown in Fig. 8c, is attached the left side of
the inner gimbal.
The platform gimbal showed in Fig. 7 is
responsible for the platform’s 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.
3846 Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
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 noting
that 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:
Fig. 7: The platform gimbal
 
Dco
1
 
I px  I ix  I ox
I px  I ix  I ox
Fco (sgn) 
 
I pz  I py  I px
I px  I ix  I ox
 Too
1
Doi
 
I pz  Iiz
I pz  Iiz
I I I
Foi (sgn )  py px pz   Tmm
I pz  Iiz
Fig. 8: The machine boxes used in the platform
 
Fig. 9: Final assembly of the platform
Table 1: Platform parameters
Platform
Inner gimbal
Ipx
0.119
Iix
0.406
Ipy
0.119
Iiy
0.845
Ipz
0.237
Iiz
1.020
0.110
Doi
0.260
Dip
Outer gimbal
Iox
Ioy
Ioz
Dco
1.05
1.05
1.10
0.32
Using this software we can also calculate the
principal 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 in
Table 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
(6)
Dip  1
I  I pz  I py 

Fip (sgn )  px
  TII
I py
I py
I py
(7)
(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 platform
separately 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 can’t handle. For this an introduction of a non
linear controller is necessary.
3847 Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
Fig. 10: PID control for the 3-DOF platform
Ki
1.8
5
Kd
11.3
Gimbals position (rad)
1.5
1.0
0.5
0
0
0.2
0.4
0.6
0.8 1.0 1.2
Time (s)
1.4
1.6 1.8
0.5
0
-0.5
0
0
0.5
1.0
1.5
2.0
2.5
Time (s)
2
3
4
5
Time (s)
6
7
8
9
1.0
0.5
0
0
1
1.5
Platform
Inner gimbal
Outer gimbal
1.0
1
Gimbals position (rad)
Gimbals position (rad)
1.5
2
Fig. 13: Position tracking response using PID controller
2.0
Fig. 11: Step response of the platform using PID controller
3
-1
Platform
Inner gimbal
Outer gimbal
Platform
Inner gimbal
Outer gimbal
Reference
4
Gimbals position (rad)
Table 2: PID parameters
Kp
8.8
3.0
3.5
4.0
Fig. 12: Impulse response of the platform using PID controller
Platform
Inner gimbal
Outer gimbal
0
0.5
1.0
1.5
2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time (s)
Fig. 14: Disturbance rejection of the 3DOF platform using
PID controller
3848 Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012



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 as
shown in Fig. 3 (Chuen, 1990).
The general TS fuzzy systems in this study use 2
input variables. eθ, eψ, e and , , 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 θ, ψ, , , 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
NB
NM
NS
ZR
PS
e\
θ
PB
PM
PS
ZR
NS
NM
NB
ZR
NS
NM
NB
NB
NB
NB
PS
ZR
NS
NM
NB
NB
NB
PM
PS
ZR
NS
NM
NB
NB
PB
PM
PS
ZR
NS
NM
NB
PB
PB
PM
PS
ZR
NS
NM
PM
PB
PB
PB
PB
PM
PS
ZR
NS
PB
PB
PB
PB
PM
PS
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 eθ is Ai and is Bi then:
c i j  p 0  p1e  p 2
where, Ai and Bi are linguistic terms which in this
study 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
3849 Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
1.2
1.0
0.8
0.6
0.4
0.2
0
0
0.2
0.4
PID
TS Fuzzy
Inner gimbal position (rad)
Platform position (rad)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0
PID
TS Fuzzy
1.4
0.6 0.8 1.0 1.2
Time (s)
1.4 1.6 1.8 2.0
0.2
0.4
0.6 0.8 1.0 1.2
Time (s)
(a)
1.4 1.6 1.8 2.0
(b)
Outer gimbal position (rad)
1.5
PID
TS Fuzzy
1.0
0.5
0
0
0.2
0.4
0.6 0.8 1.0 1.2
Time (s)
1.4 1.6 1.8 2.0
(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:
Tl 
c
i 1
N
j
i
i

i 1
(9)
i
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
Platform position (rad)
N
Platform response
Desired trajectory
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
0
0.5
1.0
1.5
2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time (s)
Fig. 17: Position tracking response of the platform
(dashed lines) shows step responses of the stabilized
platform system when controlled by three separated
PID controllers. As shown in Fig. 16 the responses
(solid lines) were significantly improved with smaller
overshoot, shorter rising time.
Figure 17 illustrates the position tracking responses
using TS fuzzy PD controller. It can be seen that this
controller present good tracking performance with
minor rise time.
3850 Res. J. Appl. Sci. Eng. Technol., 4(19): 3843-3851, 2012
CONCLUSION
Our research focused on the common coordinate,
kinematics, dynamics, control system and software
design for ship carried stabilized platform. For that in
this study we developed a dynamics modeling of the
platform and a 3D model of the platform using
Simulink and SolidWorks. Then, we considered the
problem
of
controlling
this
multivariable
servomechanism where there exist cross-couplings
between the channels. A fuzzy PD control strategy
using a Takagi-Sugeno fuzzy model has been proposed
and by comparing it with PID controller, it has been
shown in the study that uniformly stable operation is
achieved together with asymptotic tracking of the
reference command signals.
In the study, simulation results in applying the
proposed TS fuzzy PD controller to a 3-Dof
stabilization system have been presented which
demonstrates the effectiveness of the fuzzy controller.
Future study is directed to the optimization of the
scaling factors of the fuzzy system and the intelligent
method to generate an effective rule base.
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