Australian Journal of Basic and Applied Sciences
ISSN:1991-8178Journal home page: www.ajbasweb.com
Corresponding Author: Z.M. Zain, Universiti Malaysia Pahang, Robotics and Unmanned Research Group (RUS), Instrument & Control Engineering (ICE) Cluster, Faculty of Electrical and Electronics Engineering, 26600 Pekan, Pahang, Malaysia.
E-mail: [email protected]. Ph: +609-4246123
Stabilization Control for an X4-AUV using Backstepping Control Method
1
Z.M. Zain, 2D. Pebrianti, 3N. Harun
123Universiti Malaysia Pahang, Robotics and Unmanned Research Group (RUS), Instrument & Control Engineering (ICE) Cluster, Faculty
of Electrical and Electronics Engineering, 26600 Pekan, Pahang, Malaysia.
A R T I C L E I N F O A B S T R A C T Article history:
Received 15 September 2014 Accepted 5 October 2014 Available online 25 October 2014
Keywords:
Underactuated system, X4-AUV, Backstepping control.
Background: A nonlinear control method is considered for stabilizing all attitudes and positions of an underactuated X4-AUV with four thrusters and six degrees-of-freedom (DOF). Objective: The objective of this paper is to stabilize all the attitudes and positions for an X4-AUV system using backstepping control method. A control system composed of translational and rotational subsystems. A controller for the translational subsystem stabilizesthe positions, whereas controllers for the rotational subsystems generate the desired roll, pitch and yaw angles. Results:Thus, the rotational controllers stabilize all the attitudes of the X4-AUV at a desired position of the vehicle. Some numerical simulations are conducted to demonstrate the effectiveness of the proposed controllers. Conclusion: From the simulation, write the main conclusion for your paper. From the simulation and exprimental results, when using the FLC controller and the MRAC in coordination in which the control law is used to cope with parameter variation and load disturbances, the controlled system can be robustly stabilized all the time.
© 2014 AENSI Publisher All rights reserved. ToCite ThisArticle:Z. M. Zain, D. Pebriantiand N. Harun., Stabilization Control for an X4-AUV Using Backstepping Control Method. Aust. J. Basic & Appl. Sci., 8(15): 140-146, 2014
INTRODUCTION
Recently, the control problems of underactuated vehicles motivate the development of new design methodologies in nonlinear control. The control system for such vehicles has fewer number of control inputs than the number of generalized state variables to be controlled. Hence, the dynamical equations of the vehicle exhibit so-called second-order nonholonomic constraints, i.e., non-integrable conditions imposed on the acceleration in one or more DOFs directions due to the lacks of capability to command instantanenous accelerations in these directions of the configuration space (Kolmanovsky, 1995). As pointed in a celebrated paper of Brockett in 1983 (Brockett, 1983), such nonholonomic systems cannot be stabilized by usual smooth, time-invariant and state feedback control algorithms. Underactuation arises out of the need to reduce the actuators cost and weight or to increase the reliability of the system in case of actuator failure. The typical applications include space robotics, mobile robots, surface vehicles and underwater vehicles. Many researchers proposed the nonlinear feedback control among others for the underactuated underwater systems control. The nonholonomic property of any mechanical system can be seen in the underactuated systems, even though the connection between the nonholonomic control systems and the underactuated systems is not completely understood. Many control approaches have been studied to solve the nonholonomic systems control, e.g., Kolmanovsky and McClamroch presented a vast literature and overview on the nonholonomic control in their work (Kolmanovsky, 1995). An X4-AUV with an ellipsoid hull shape was studied by Zain (Zain et al., 2010), in which it makes only use of four thrusters to control the vehicle without using any steering rudders, falls into the class of underactuated AUVs and has nonholonomic features. The consideration of nonholomic systems is an interesting study from a theoretical standpoint, because as pointed out in the earlier works of Brockett, they cannot be asymptotically stabilized to a fixed point in the configuration space using continuously differentiable, time-invariant and state feedback control laws (Brockett, 1983).
BacksteppingControl of an X4-AUV system:
The dynamic equations of motion for an X4-AUV is given by:
4 3 2 1 3 1 2 1 1 ) ( ) ( ) ( sin sin cos cos cos lu J I I I lu J I I I u I I I u z m u y m u x m t y x z t x z y z y x (1)
where the derivation of the X4-AUV dynamic model can be seen in (Zain et al., 2010)and (Zain et al., 2013).
The model (1), can be rewritten in a state-space form X f(X,U)by introducing 𝑋 = 𝑥1⋯ 𝑥12 𝑇𝜖 ℜ12 as state vector of the system as follows:
𝑥1= 𝑥 𝑥7= 𝜙
𝑥2= 𝑥 1= 𝑥 𝑥8= 𝑥 7= 𝜙
𝑥3= 𝑦 𝑥9= 𝜃
𝑥4= 𝑥 3= 𝑦 𝑥10 = 𝑥 9= 𝜃
𝑥5= 𝑧 𝑥11= 𝜓
𝑥6= 𝑥 5= 𝑧 𝑥12 = 𝑥 11= 𝜓 (2)
where the inputs 𝑈 = 𝑢1⋯ 𝑢2 𝑇 𝜖 ℜ4. From (1) and (2), we obtain:
𝑓 𝑋, 𝑈 =
𝑥2
cos 𝜃 cos 𝜓 1 𝑚1 𝑢1 𝑥4 𝑢𝑦 1 𝑚2 𝑢1 𝑥6
− 𝑢𝑧
1 m3
𝑢1
𝑥8
𝑥10𝑥12 𝑎1+ 𝑏1𝑢2
𝑥10
𝑥8𝑥12 𝑎2− 𝑎3𝑥12Ω+ 𝑏2𝑢3
𝑥12
𝑥8𝑥10 𝑎5+ 𝑎4𝑥10Ω+ 𝑏3𝑢4
(3)
with:
𝑎1= 𝐼𝑦 − 𝐼𝑧 /𝐼𝑥
𝑎2= 𝐼𝑧 − 𝐼𝑥 /𝐼𝑦
𝑎3= 𝐽𝑡/𝐼𝑦
𝑎4= 𝐽𝑡/𝐼𝑧
𝑎5= 𝐼𝑥 − 𝐼𝑦 /𝐼𝑧
𝑏1= 1/𝐼𝑥
𝑏2= 𝑙/𝐼𝑦
𝑏3= 𝑙/𝐼𝑧
(4) 𝑢𝑦= cos 𝑥9sin 𝑥11(5)
Fig. 1: Connection of rotational and translational subsystems
It is worthwhile to note in the latter system that the angles and their time derivatives do not depend on translation components. On the other hand, the translations depend on the angles. We can ideally imagine the overall system described by (3) as constituted of two subsystems, the angular rotations and the linear translations, see Fig. 1.
Backstepping Control of the Rotations Subsystem:
Using the backstepping approach, one can synthesize the control law forcing the system to follow the desired trajectory. For the first step we consider the tracking-error:
𝑧1= 𝑥7𝑑− 𝑥7 (6) And we use the Lyapunov theorem by considering the Lyapunov function 𝓏1 positive definite and it’s time derivative negative semi-definite:
𝑉 𝑧1 = 1
2𝑧1
2 (7)
𝑉 𝑧1 = 𝑧1 𝑥 7𝑑 − 𝑥8 (8)
The stabilization of 𝑧1 can be obtained by introducing a virtual control input 𝑥8:
𝑥8= 𝑥 7𝑑+ 𝛼1𝑧1with:𝛼1> 0 (9)
The equation (8) is then:
V 𝑧1 = −α1𝑧12 (10) Let us proceed to a variable change by making:
𝑧2= 𝑥8− 𝑥 7𝑑 −α1𝑧1 (11)
For the second step we consider the augmented Lyapunov function:
V 𝑧1, 𝑧2 = 1
2𝑧1
2 +1
2𝑧2
2 (12)
And it’s time derivative is then:
V 𝑧1, 𝑧2 = 𝑧2 𝑎1𝑥10𝑥12+ 𝑏1𝑢2 − 𝑧2 𝑥 7d−α1 𝑧2+α1𝑧1 − 𝑧1𝑧2−α1𝑧12 (13)
The control input 𝑢2 is then extracted 𝑥 1,2,3d = 0 , satisfying V 𝑧1𝑧2 < 0:
𝑢2= 1
𝑏1 𝑧1− 𝑎1𝑥10𝑥12−α1 𝑧2+α1𝑧1 –α2𝑧2 (14)
The term α2𝑧2 with α2> 0 is added to stabilize 𝑧1. The same steps are followed to extract 𝑢3and 𝑢4.
𝑢3= 1
𝑏2( 𝑧3− 𝑎2𝑥8𝑥12− 𝒶3𝑥12Ω) −α3 𝑧4+α3𝑧3 –α4𝑧4 (15) 𝑢4=
1
with:
𝑧3= 𝑥9d − 𝑥9
𝑧4= 𝑥10 − 𝑥 9d–α3𝑧3
𝑧5= 𝑥11d − 𝑥11
𝑧6= 𝑥12− 𝑥 11d−α5𝑧5
(17)
Backstepping Control of the Linear Subsystem: 1) Altitude Control:
The altitude control u1is obtained using the same approach described in the backstepping control of the
rotational subsystem.
𝑢1=
𝑚1
cos 𝓍9cos 𝓍11 𝑧7− 𝛼7 𝑧8+ 𝛼7𝑧7 − 𝛼8𝑧8 (18)
with:
𝑧7= 𝑥1d − 𝑥1
𝑧8= 𝑥2 − 𝑥 1d–α7𝑧7
(19)
2) Linear y and z Motion Control:
From the model(1) one can see that the motion through the axes yandzdepends on u1. In fact u1is the total
thrust vectororiented to obtain the desired linear motion. If we consideruyand uzthe orientations of u1responsible
for the motionthrough yand zaxis respectively, we can then extractfrom (5) the roll and pitch angles necessary to computethe controls uyand uzsatisfying V 𝑧1𝑧2 < 0. The yawcontrol is then given as a desired angle.
𝑢𝑦=
𝑚2
𝑢1 𝑧9−α9 𝑧10+α9𝑧9 –α10𝑧10 (20)
𝑢𝑧= −
𝑚3
𝑢1 𝑧11−α9 𝑧12+α11𝑧11 –α12𝑧12 (21)
RESULTS AND DISCUSSION
A nonlinear control strategy is implemented to stabilize the AUV. The position and angles of the X4-AUV are stabilized by using control input u1, u2, u3, and u4 respectively. Backstepping controllers were
introduced for controlling each orientation angle and the positions of the X4-AUV system. The angles and their time derivatives of rotational subsystem do not depend on translation components, whereas the translations depend on the angles. Ideally, it can be imagined as two subsystems: the angular rotations and the linear translations. Due to its complete independence from the other subsystem, the angular rotation-related subsystem is tuned first. The rotational control keeps a 3D orientation of the X4-AUV to the desired state and the translational control moves the vehicle to the desired position.
The controllers have been implemented on MATLAB and the simulation results for stabilizing the X4-AUV in x-positions are shown in Fig. 2-7. The system started with an initial state
T
X ,0)
4 , 0 , 4 , 0 , 4 , 0 , 0 , 0 , 0 , 0 , 0 ( 0
and we wanted the final x-positions, at 3 m with all zero orientation
angles. As shown inFig. 2,it is seen that all orientation angles, and x-positions converge to the targets, where α1
= 1, α1 = 1, α2 = 2, α3 = 3, α1 = 1, α4 = 1, α5 = 1, α6 = 1,α7 = 1.6,α8 = 1.4,k1 = 1.0, k2 = 2.0. The physical
parameters for X4-AUV that has been used for simulating the dynamic model is shown in Table 1. Note that this technique also used for a Quadrotor studied in (Bouabdallah, 2005).
Table 1: Physical parameters for X4-AUV
Parameter Description Value Unit
mb Mass 21.43 kg
ρ Fluid density 1023.0 kg/m3
l Distance 0.1 m
r Radius 0.1 m
b Thrust factor 0.068 Ns2
d Drag factor 3.617e4 Nms2
Jbx Roll inertia 0.0857 kgm2
Jby Pitch inertia 1.1143 kgm2
Jbz Yaw inertia 1.1143 kgm2
Fig. 2: Attitude control
Fig. 3: Attitude rate control
Fig. 4: A Position control
0 5 10 15
-0.2 0 0.2 0.4 0.6 0.8
Time t [s]
A
tt
it
ude
[ra
d]
Attitude , and
0 5 10 15
-0.2 0 0.2 0.4 0.6 0.8
Time t [s]
A
tt
it
ude
[ra
d]
Attitude , and
0 5 10 15
0 1 2 3 4
Time t [s]
Pos
it
ion
x
[m
]
Controlled result of position x
Fig. 5: A Position rate control
Fig. 6: A control inputs
Fig. 7: A control inputsin rotation
Conclusion:
In this paper, a nonholonomic control method has been presented for stabilizing all attitudes and positions of an underactuated X4-AUV with four thrusters and 6-DOFs. The controller design was separated into two parts: the rotational and translational dynamics-related control designs. The stabilization strategy is based on backstepping control method. The backstepping controller proves the ability to stabilize all attitudes and positions of the system.
0 5 10 15
-0.5 0 0.5 1 1.5 2
Time t [s]
Pos
it
ion ra
te
of
x
[m
]
Controlled result of position rate of x
Rate of x
0 5 10 15
-100 0 100 200 300 400 500
Time t [s]
Cont
rol
input
u i
Control inputs u1,..., u4
u
1
u
2
u
3
u4
0 5 10 15
0 10 20 30 40
Time t [s]
Cont
rol
input
i
[ra
d/
s]
Control inputs in rotation 1,..., 4
1 2
REFERENCES
Bouabdallah, S. and R. Siegwart, 2005. Backstepping and sliding-mode techniques applied to an indoor micro quadrotor, International Conference on Robotics and Automation, pp: 2259-2264.
Brockett, R.W., 1983. Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (R. W. Brockett, R. S. Millman and H. J. Sussmann, eds.), Birkhauser, Boston, pp: 181-191.
Kolmanovsky, I and N.H. McClamroch, 1995. Developments in nonholonomic control problems, IEEE Control Systems Magazine, 15(6): 20-36.
Zain, Z.M., K. Watanabe, T. Danjo, K. Izumi and I. Nagai, 2010. Stabilization control for an X4-AUV. Proc. of the 3rd International Conference on Underwater System Technology: Theory and Applications (USYS '10): 13-17.
Zain, Z.M., K. Watanabe, T. Danjo, K. Izumi and I. Nagai, 2013. Exponential stabilization of second-order nonholonomic chained systems, The 2013 International Conference on Intelligent Robotics and Applications
