Adaptive fuzzy control of unmanned underwater vehicles
S. A. Salman1, Sreenatha A. Anavatti2 & T. Asokan3
1Mechanical Engineering Department, Assiut University, 71515 Assiut, Arab Republic of Egypt
2School of Engineering and Information Technology, UNSW@ADFA, Canberra Act 2600, Australia.
3Department of Engineering Design, Indian Institute of Technology Madras,Chennai-600 036, India [E-mail: s_saalman@yahoo.com1, a.sreenatha@adfa.edu.au2, asok@iitm.ac.in3]
Received 23 March 2011; revised 28 April 2011
Unmanned Underwater Vehicles (UUVs) have been playing an increasingly important role in military and civilian operations and been widely used in various applications. The main issue associated with the development and design of UUV’s is the control system design. These vehicles have nonlinear dynamics and coupling, and tend to exhibit time varying characteristics. In addition they are subject to different environmental disturbances. Successful completion of the UUV missions depends on the control provided by the autopilot unit mounted on board. These controllers need to be tuned and analysed before implementing them in real environment. In the present work, the first objective is to demonstrate the capability of adaptive network fuzzy inference system, namely ANFIS for modelling of UUVs and the second objective is to design a fuzzy controller using the ANFIS model. The input output data from the UUV are used for the ANFIS modelling.
This model is used in the design and validation of the fuzzy controller and the results are compared with a conventional PID controller.
[Keywords: Dynamics, Seabed, Fuzzy system, Inertia matrix, Navigate, Autonomous]
Introduction
Autonomous Underwater Vehicles (AUVs) have gained importance over the years as specialized tools for performing various underwater missions1. This has lead to an increase in the research for these vehicles considerably. Underwater robotics technology can be employed for the investigation of the ocean and sea environmental issues2, the study of coastal dynamics and the protection of the seas and ocean resources from pollutants3,4 . Furthermore, they can be deployed in mapping and exploration of marine resources such as fisheries, mining, oil and gas drilling activities.
They have also been used in the exploration of the deep water with hazardous and unstructured undersea and seabed environments5. Thus due to the variety of missions, they operate in uncertain environments exposing themselves to unpredictable external disturbances such as sea currents and drag effects.
Hence, the development of fully autonomous underwater vehicles is an extremely challenging issue and there is a necessity for accurate and robust controllers6. A good model of the UUV helps in reducing the tuning time and design for the controllers.
Identification plays a fundamental role in the design of guidance and control systems for AUVs7,8. Identification is defined as a tool or an algorithm that builds a mathematical model of a dynamical system from measured data. In the literature, different system identification techniques have been proposed. The complexity and difficulties of the traditional identification methods is in the model structure selection which requires an engineering intuition combined with a prior knowledge of the system behaviour. The alternative method, the intelligent techniques, only needs to be trained to learn the actual non-linear relationships between inputs and outputs of the system under study. The only information required for training the intelligent system is the input and output data. It means that there is no need for any prior knowledge of the physical relationship inside the system and it offers a ‘black box’ modelling tool.
Intelligent and adaptive systems such as fuzzy and neural networks provide the appropriate solutions for such systems. Fuzzy systems are known for their capabilities to approximate any non-linear dynamic system9. On the other hand, neural networks system brings learning capabilities to fuzzy inference
learning algorithm that combines least square method with back propagation is used to adjust the premise and consequent parameters of fuzzy system. In the present paper the ANFIS is considered for the modelling of AUV dynamics using input and output data.
Due to the non-linearity and uncertainty associated with the UUVs, it is a better idea to use an intelligent robust control technique for tuning and implementing the PID controller. One such technique is the Fuzzy PD controller. Fuzzy controllers are shown to handle non-linear models in a natural way and provide robustness against parameter variations13,14. The present work provides the design and validation of a fuzzy PD controller for the UUVs. Numerical results are compared with the conventional PID controller.
Materials and Methods
Underwater Vehicles
Figure 1 shows a typical underwater vehicle model.
The main specifications of the model used in this study are given in Table 1. Two electrical thrusters power the UUV for forward motion as well as manoeuvring in the horizontal plane. Two control planes on the sides help the UUV to navigate in the vertical plane. The inner box is used for carrying the sensors, battery and the electronic accessories. A camera pan tilt mechanism is provided at the front of the UUV for video/still camera mounting. A transparent, hemispherical cover is provided at the
w represent the forward, lateral and vertical speeds along ox, oy and oz axes respectively. Similarly, the hydrodynamic moments on UUV will be denoted by
, and
L M Nacting around ox oy, and oz axis respectively. The angular rates will be described by the p q, and rcomponents.
The equations of motion for the UUV are derived from Newton’s second law of motion. A detailed description of the model is given in [15]. The general dynamic model for underwater vehicle can be presented as;
… (1) where represents the body inertia matrix, and added mass , denotes the centrifugal and Coriolis matrix which includes both the rigid body and the added mass terms, is damping matrix which represents the hydrodynamic damping and the lift force, and is restoring matrix due to gravity and buoyancy. is the vector of linear and angular velocities of the UUV with respect to the vehicle's body attached frame. The vector is the vector of generalized forces on the UUV which is supplied by the thrusters. is the generalized coordinate denoting the position and orientation of the UUV with respect to the inertial frame.
The rigid body inertia matrix, can be represented as
Fig. 1—Underwater vehicle coordinate system.
Table 1—Main specifications of the vehicle
Total Length = 1.065m
Hull Diameter = 0.250m
Vehicle speed = 1m/s
Coefficient of Axial Drag = 0.3
Coefficient of Cross flow Drag = 1.1 Frontal Area (in the YZ plane) = 0.0491m2 Frontal Area (in the XZ plane) = 0.204m2
Stern plane Area = 0.1125
Density of fresh water = 1000 kg/m3
Mass of the Vehicle = 45.1kg
Weight of the Vehicle = 453N
Buoyancy of the Vehicle = 453N
Moment of Inertia in X axis = 0.34 kgm2 Moment of Inertia in X axis = 3.7 kgm2 Moment of Inertia in X axis = 3.7 kgm2
Centre of Gravity = [0 0 0] m
Centre of Buoyancy = [0 0 0] m
… (1-a) The added mass matrix, can be denoted as
… (1-b) The rigid body Coriolis and centripetal matrix, is given by
… (1-c) The added mass Coriolis and centripetal matrix, is given by
… (1-d) where,
The damping of an underwater vehicle moving in 6 degree of freedom can be separated into two different terms, a linear and quadratic term. The linear term can be given as
… (1-e) whereas the quadratic term can be represented as
… (1-f) The restoring force and moment vector in the body fixed coordinate system is denoted as
… (1-g) where and represents the weight and buoyancy of the vehicle respectively.
The dynamic model of thrusters and control surfaces has been included in the present study.
Table 2 represents the main calculated parameters for the model. The AUV model is simulated by a mathematical model based on physical laws and design data as shown in Fig. 2. Each block in Fig. 2 represents a specified term in equation 1. In the following sections PID and fuzzy controllers have been designed for the above model.
ANFIS Modeling
UUV dynamic model in body coordinates can be represented as
… (2)
where .
Thrust can be represented as and where there are two engines in both sides of the AUV body.
The two control surface deflections are and . In the present study, there are six ANFIS models. The first system is the forward velocity, which can be represented in the general non-linear discrete form as
… (3) Similarly, suitable models are written for the other five degrees of freedom by taking into account suitable couplings between different degrees of freedom.
To obtain the fuzzy model for the above non-linear dynamic models, a T-S fuzzy model has been used to represent the dynamic models as
: ij( ) ij m( 1) j Rule j If input k is M Then x k+ is B
where Mij(i=1, 2)are fuzzy membership sets for
ij( )
input k , while Bjdescribes the fuzzy memberships for the outputs .
Use the Gaussian membership function for all inputs, as
,
,
( ) 2
( ) exp 0.5 i j
i j
i input
M ij
input
input k c input
µ σ
−
= − … (4)
where
,
inputi j
c and
,
inputi j
σ represent the Gaussian membership function parameters.
Using the product inference rule and the singleton fuzzifier with the centre of area defuzzifier method for the forward velocity fuzzy system, one obtains [9]
,
,
,
,
2 7
1 1
2 7
1 1
( ) exp 0.5
( 1)
exp 0.5 ( )
σ
σ
= =
= =
−
−
∑ ∏ + =
−
−
∑∏
i j
i j
i j
i j
l i input
j
j i input
m
l i input
j i input
input k c B
u k
input k c
… (5)
Fig. 2—Underwater vehicle block diagram.
where
Similarly one can write the expressions for the other five fuzzy systems. As shown above, each ANFIS is considered as a seven input/single-output neuro-fuzzy network, in which, the inputs consist of the linear velocities, angular rates and control inputs. The input variables are directly sent to the ANFIS identifier.
Each universe of discourse is partitioned into two primary fuzzy sets.
The above fuzzy systems can be represented as five multilayer networks as shown in Fig. 3. input ki( ) passes through layer 1 and the output of layer 1 is the grade of membership for each input. It consists of i’s nodes. Since Gaussian membership has been used for
Xv|v -0.737 Kg/m Xv -0.31 Kg/sec
Xw|w -0.737 Kg/m Xw -0.31 Kg/sec
Xq|q -1.065 Kg.m/rad Xq -0.51 Kg.m/sec
Xr|r -1.065 Kg.m/rad Xr -0.51 Kg.m/sec
Yv|v -112.2 Kg/m Yv -62.45 Kg/sec
Yr|r 0.250 Kg.m/rad Yr 0.12 Kg.m/sec
Zw|w -112.2 Kg/m Zw -62.45 Kg/sec
Zq|q -0.250 Kg.m/rad Zq 0.12 Kg.m/sec
Kp|p -0.5975 Kg.m2/rad2 Kp -0.3125 Kg.m2/sec
Mw|w 2.244 Kg.m/rad Mw 1.2 Kg.m/sec
Mq|q -119.5 Kg.m2/rad2 Mq -59.75 Kg.m2/sec
Nv|v -2.244 Kg.m/rad Nv 1.2 Kg.m/sec
Nr|r -59.75 Kg.m2/rad2 Nr -31.25 Kg.m2/sec
X'ْu (Xudot) -1.17 Kg K'p (Kpdot) 0 Kg.m/rad
Y'v (Yvdot) -34.834 Kg M'w (Mwdot) -1.042 Kg.m/rad
Y'r (Yrdot) 1.042 Kg.m/rad M'q (Mqdot) -2.659 Kg.m/rad
Z'w (Zwdot) -34.834 Kg N'v (Nvdot) -1.042 Kg.m/rad
Z'q (Zqdot) -1.042 Kg.m/rad N'r (Nrdot) -2.659 Kg.m/rad
all inputs, the output of node i for this layer can be represented as
( )
1,i Mi
O =µ input
… (6) where
( )
( ) 2
exp 0.5
i
input M
input
input k c input
µ σ
−
= −
Parameters cinput and σinputin this layer are referred to as premise parameters.
Layer 2 is called the product layer. Every node in this layer is marked as П. It has j nodes. The output of each node can be represented as
,
,
2 7
2, 1
( ) exp 0.5
σ
=
−
=∏ −
i j
i j
i input
j
i input
input k c O
… (7) The third layer is called the normalized layer. The output of this layer can be denoted as
,
,
,
,
2 7
1
3, 2
7
1 1
( ) exp 0.5
( ) exp 0.5
σ
σ
=
= =
−
∏ −
= ∑∏ − −
i j
i j
i j
i j
i input
i input
i
l i input
j i input
input k c
O
input k c
… (8) Layer 4 is the defuzzification layer and the output can be obtained by
,
,
,
,
2 7
1
4, 2
7
1 1
( ) exp 0.5
( ) exp 0.5
σ
σ
=
= =
−
∏ −
= ∑∏ − −
i j
i j
i j
i j
i input
j
i input
i
l i input
j i input
input k c B
O
input k c
… (9) Parameter Bj in this layer is referred to as consequent parameters.
The last layer is a single node layer which computes the overall output as the summation of incoming signals.
The above description represents the ANFIS architecture. In the present work a hybrid learning algorithm has been used to update the premise and consequent parameters. The premise parameters are
updated by gradient method while the consequent parameters are identified by least squares method8 .
Fuzzy Controller
Figure 4 shows the basic configuration of a fuzzy system, where it consists of four main components;
the fuzzifier, the fuzzy rule base, the fuzzy inference and the defuzzifier. The fuzzifier transfers the measured input data into corresponding fuzzy sets which can be understood by the fuzzy inference system. The fuzzy rule base describes the correlation between input and output fuzzy sets in the forms of If- Then rules. It is the core of the whole system. The fuzzy inference engine uses techniques in approximate reasoning to determine a mapping from the fuzzy sets in the input space U⊂Rn to the fuzzy sets in the output space V⊂Rm. The defuzzifier transfers fuzzy sets in the output space into numerical data in the output space.
The fuzzy controller is shown in Figure 5, where the inputs to the fuzzy controller are the error and error difference. The fuzzy controller output can be represented as
Fig. 3—ANFIS Architecture.
Fig. 4—Basic configuration of fuzzy system.
error difference. The fuzzy controller rule base is represented by,
: e( ) is and e( ) is e THEN ( ) is
FC e u
j j j j
R If t A ∆ t A∆ u t B
where Aej and Aj∆e are the fuzzy sets for error and change of error respectively and Bujis the rule consequent parameter for fuzzy singletons.
Using the fuzzy inference based upon product sum gravity at given input ( ( ),e t ∆e t( )) and the Gaussian membership functions for all fuzzy sets, the final output of the fuzzy controller is given by
2 2
1
2 2
( ) ( )
exp 0.5 exp 0.5
( )
( ) ( )
exp 0.5 exp 0.5
h j j
j
j j
j
j j
j j
e t ce e t c e
B e e
u t
e t ce e t c e
e e
σ σ
σ σ
=
− ∆ − ∆
− −
∑ ∆
= 1 − − − ∆ ∆− ∆ h
j=
∑
… (11) cej,σej and c e∆ jand σ∆ejrepresent the centre and width of Gaussian memberships for the error e t( ) and error difference ∆e t( ) respectively for the rule j and
h are the number of rules of fuzzy controller.
Results and Discussion
Simulation Results
A. ANFIS modelling results:
Figures 6 illustrate the results of the identification process where the red lines are the measured values, and the blue lines are outputs of the ANFIS identifiers. Training begins at t=0 s and ends at t=250 s. As can be seen, the ANFIS identifiers identify the dynamics of the UUV successfully. At the conclusion of training, the trained ANFIS identifiers are tested after 250 seconds of training. The results with the other fuzzy systems are also satisfactory and suggest that the ANFIS identifiers can be very effective in UUV applications.
the depth control and pitch angle control for UUV are considered. Generally the depth of UUV can be maintained by controlling the pitch angle where the pitch angle can be obtain as
… (12) The pitch angle is controlled by deflecting the control surfaces. The inputs to fuzzy controller are the pitch angle error and its error difference. The outputs of the fuzzy controller are the control voltages to the control surface actuator. The memberships functions for e t( ) and ∆e t( )are shown in Fig. 7. The total number of rules used for fuzzy controller is 9.
Fig. 5—Fuzzy controller.
Fig. 6—Forward velocity model and its inputs memberships.
Fig. 7—Memberships functions for the error and the derivative of the error.
Fig. 8—Depth Control.
Fig. 9—Pitch angle control.
A PID controller has been designed based on the ANFIS model of the UUV. The gains for the PID controller have been tuned by trial and error to achieve a comparative performance. Due to the nature of trial and error process, the time consumed in the development of the PID controller has been significantly higher compared to the Fuzzy controller.
The controller output with the PID controller is given by;
0
1 ( )
( ) ( ( ) ( ) )
t
p d
it
u t K e t e t dt T de t
T ∫ dt
= + +
… (13) Figure 8 shows the depth control for UUV using PID and fuzzy control technique whereas the pitch angle control is shown in Fig. 9. The hydrodynamic coefficients vary during the simulation depending
upon the speed and other parameters like the effect of added mass.
It is seen that the Fuzzy controller does a better job compared to the conventional PID in terms of accuracy as well as the speed. The PID controller is seen to have higher transients as well as steady state error in pitch angle. The results with varying hydrodynamic coefficients and noise indicate similar trends with the fuzzy PID doing a better job than the conventional PID.
Conclusion
The paper presents the numerical simulation results for a typical under water vehicle using PID and Fuzzy Controllers. ANFIS modeling is used to identify the model of the UUV using input-output data. The controllers are designed using this model. Since the Fuzzy Controllers incorporate robustness in the design, they are better suited for non-linear and time- varying systems. The problems associated with tuning the PID gains for non-linear systems do not exist here.
Further simulations to validate the controller under noisy conditions and varied environments are currently undertaken along with real time implementation.
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