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DDCLS.2017.8068053

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2017 IEEE 6th Data Driven Control and Learning Systems Conference
May 26-27, 2017, Chongqing, China
Fuzzy Neural Network Based Adaptive Iterative Learning Control
Scheme for Velocity Tracking of Wheeled Mobile Robots
Xiaochun Lu1,2, Juntao Fei1, Jiao Huang1
1. College of IOT Engineering, Hohai University, Changzhou, 213022, China
2. Jiangsu Key Laboratory of Power Transmission & Distribution Equipment Technology, Changzhou, 213022, China
E-mail: Luxc@hhu.edu.cn
Abstract: The velocity tracking problem of wheeled mobile robots (WMRs) which work with repeatable trajectories and
different initial errors is discussed in the paper. Three mathematical models of WMR, namely, kinematic model, dynamic model
and DC motor driven model, are deduced and the stratagem of fuzzy neural network based adaptive iterative learning control
(FNN-AILC), which includes the components of fuzzy neural network, approximation errors and feedback, is presented. The
proposed scheme can deal with MIMO system, which is distinguished from previous research work. The simulation is presented
and the result verifies the effectiveness of the controller.
Key Words: Fuzzy Neural Network Based Adaptive Iterative Learning Control, Wheeled Mobile Robot, Velocity Tracking
Control
1
proposed[15-18]. In [15], Lipschitz condition was relaxed,
and Lyapunov-like approach was adopted. An adaptive
iterative learning controller, which contains two adaption
laws in both time-domain and iteration-domain and does not
use any priori knowledge of robot parameters, was designed
in [16]. An AILC algorithm including a parametric
estimation and an iterative controller is designed to improve
the tracking performance of MEMS gyroscope in [17].
Usually, fuzzy technique and neural network are effective
approaches for the adaptive control of nonlinear systems, so
a fuzzy AILC (FAILC) was proposed in [18].
In some applications, especially in industrial fields,
WMRs usually have the repeated tasks and move along with
relatively fixed trajectory, which is the typical application
area of ILC. Hence, taking advantage of the feature of
iterative learn, ILC can be used to solve the problem of
WMR trajectory tracking, which means that the promotion
of tracking performance is realized by iteration in stationary
time intervals. The combination of WMR trajectory tracking
and ILC has received attention. In [19], the discrete dynamic
kinematical model of WMR was presented, and an ILC
strategy was designed to realize trajectory tracking for WMR.
An ILC algorithm based on the kinematical model of farm
mobile robots was developed in [20], which also analyze the
convergence of the controller. An ILC law with predictive,
current and past learning items to solve the high-precision
trajectory tracking issue of WMR was derived in [21], and
the convergence of the algorithm was proofed rigorously. In
[22], a FAILC strategy is proposed to implement the velocity
tracking of WMRs, and the Lyapunov-like method was used
to proof the convergence of the controller.
In terms of literature review, few research reports have
covered the combination of ILC and trajectory tracking of
WMR, especially based on WMR dynamic model. However,
dynamic model of WMRs is more suitable to describe a real
robots in mathematics than kinetic model. The paper, which
introduce a new FNN-AILC algorithm, is an extension of
[22].The proposed algorithm can work with the dynamic
model of WMRs. The main features of FNN-AILC with
Introduction
Due to the features of efficiency and flexibility, WMRs
have been widely applied in the fields of military,
transportation, and industry, and also drawn much attention
of researchers. There are three main researching problems in
the motion control of WMRs, i.e. path following [1],
trajectory tracking [2], and point stabilization [3]. The
requirement in common of path following and trajectory
tracking is that WMRs must follow a line or a curve.
However, the difference between path following and
trajectory tracking is specific velocity requirement at
specific location, which is needed in trajectory tracking and
not needed in path following. Therefore, solving the
trajectory tracking with unknown model parameters has
profound significance. Over the past decade, many
meaningful research results in this field have emerged. A lot
of control methods, such as feedback linearization [4],
backstepping control [5], sliding mode control [6], adaptive
control [7], neural network technique [8], fuzzy system [9]
and model predictive method [10, 11], were proposed based
on kinematical model and dynamic model.
ILC algorithm, which has the advantage of dealing with
nonlinear systems, has been made great progress in
mathematical description and convergence analyses. As a
branch of intelligent control, ILC can work in simple way,
especially in low cost of computing [12-14]. The iterative
learning stratage, robustness, convergence and application
are four essential problems in research of ILC. Initially,
D-type, PD-type and PID-type ILC based on contraction
map theory were proposed[12]. Traditional ILC schemes are
simple and easy to be realized, but these methods need strict
Lipschitz condition[13]. In order to remove some strict
conditions for technical analysis, adaptive ILC (AILC)
which can handle the issue of resetting condition, input noise
and iteration-varying desired trajectory, has been
*
This work is supported by Open Foundation of Jiangsu Key Laboratory
of Power Transmission & Distribution Equipment Technology under grant
No. 2015JSSPD06.
978-1-5090-5461-9/17/$31.00 ©2017 IEEE
106
DDCLS'17
ª x º ªcos T 0 º
« y » «sin T 0 » ªX º
(3)
« » «
» «Z »
¬
¼
«¬T »¼ «¬0
1 »¼
For the sake of simplicity, Eq. (3) can be rewritten as:
(4)
q S (q)V
application in the velocity tracking of WMRs are highlighted
as follows:
1) The algorithm of FNN-AILC which works on the WMR
dynamic model, can improve the velocity tracking
performance of WMR.
2) The FNN-AILC controller, which contains a FNN
approximation component, an approximation error
compensator and a feedback controller, has the advantages
of deal with MIMO systems and random initial errors.
This paper is organized as follows. In section 2, the
kinematical model of WMR, the dynamic model and the DC
motor driven model are deduced. The FNN-AILC strategy is
derived in section 3. Then the simulation results are
manifested in section 4. Finally, the section of Conclusion
summarizes the whole thesis.
2
where q [ x, y,T ]T is the generalized velocity vector and
V
By using of the Euler-Lagrangian formulation, the
dynamic model of WMR is formulated as:
M (q)q C(q, q)q G(q) E(q)W A(q)O
where q [ x, y,T ] is generalized acceleration vector;
M (q)  3u3 , C (q, q)q  3 and G(q)  3 denote the
symmetric positive definite inertia matrix, the centripetal and
Coriolis force vector, the gravitational force vector
respectively; E (q)  3u2 is input transformation matrix;
The diagram of typical two-wheeled mobile robot is
shown in Fig. 1. This WMR, which has two independent
driving wheels on the same axis, can move in the coordinate
system for the inertial frame OXY . The center of mass of the
robot locates at point C , which is also the center of driving
axle.The location of the robot is at point C with
coordinates ( x, y) and the heading angle is T . The length of
driving axle and the radius of driving wheels are 2R and r ,
respectively. The vector of generalized coordinate is defined
as q [ x, y,T ]T .
W
[W L ,W R ]T denotes the input torque vector; A(q)O  3
denotes the constraint forces vector, and O is the Lagrange
multiplier.
Assume that WMR move on a plane, so its potential
energy is constant. Considering that the WMR mass is m
and the moment of inertia of WMR is I , and ignoring the
moment inertia of wheels, the total kinetic energy is
1
1
m( x 2 y 2 ) IT 2 , and the other parameters are
2
2
ªm 0 0º
M (q) «« 0 m 0 »» , C (q, q) 0 , G(q) 0 ,
«¬ 0 0 I »¼
Y
X
Y
E (q )
C
O
x
X
cos T º
sin T »» . So, Eq. (5) can be rewritten as:
R »¼
q
Fig.1 The diagram of WMR
S (q)V S (q)V
S(
(7)
T
Substitute Eq. (7) into Eq. (6), and multiply by S on
both sides of Eq. (6), then we get
S T MSV
SV S T MS
MSV
V S T EW S T AO
(8)
ªm 0º
1 ª 1 1º
S 0 , ST E
where S T MS «
, S T MS
,
»
r «¬ R R »¼
¬0 I¼
Assuming that the contact of driving wheels and ground
satisfies the condition of "pure rolling without sliding", the
model of two-wheeled mobile robot can be simplified as
unicycle-type mobile robot. The nonholonomic constrains of
the WMR, which implies that the robot cannot move
sideways, is expressed as:
(1)
x sin T y cosT 0
Eq. (1) can be written in the form of vector:
ª xº
T
[ i T , cos T , 0] «« y »»
A (q)q [sin
¬«T »¼
ªcos T
1«
sin T
r«
«¬ R
(6)
M (q)q E(q)W A(q)O
With the purpose of omitting the Lagrange multiplier O ,
the derivative of Eq. (4) can be written as
2r
2R
(5)
T
WMR model
y
[X , w]T is the input vector of WMR kinematical model.
0
ST A 0 .
Define M S T MS , B S T E , and substitute the
definition into Eq.(8), then we get the simplified
formulation:
MV
V BW
(9)
The wheels of WMR are driven by DC motors with
mechanical gears, and electrical equation of motor armature
is formulated as:
(2)
The linear velocity of WMR at point C is defined as X ,
and angular velocity at the same point is defined as Z . Then,
the kinematical model of WMRs can be formulated as:
u
107
L
di
Ra i KbZm
dt
(10)
DDCLS'17
where u is input voltage, L is armature inductance, Ra is
clear that I j (t ) is a monotonic decreasing function in the
iterative period [0, T ] .
Define the error function as:
ª S Ij1(t ) º
j
i
j
i
(16)
(
t
)
SI
« j » e (t ) S (t ) ˜ I (t )
(
t
)
¬S I 2 ¼
j
ª
º
e1 (t )
0
« Sat ( j )
»
(
)
t
I
1
» , and sat (˜) is
where S j (t ) «
j
«
»
(
)
t
e
0
Sat ( 2j ) »
«
I 2(t ) »¼
«¬
saturation function which is defined as
j
j
­
1 ,
e i (t ) !I i (t )
°
j
j
°
e (t )
e i (t )
j
j
Sat ( ij ) ®
,
(i 1, 2)
e i (t ) d I i (t )
I i (t ) °
I ij (t )
j
j
°
1 ,
e i (t ) I i (t )
¯
armature resistance, i is armature current, Zm is angular
velocity of motor and K b is back electromotive force
constant. With the ignorance of armature inductance,
considering relations between angle velocity and torque
before gear and after gears ( Zw Zm / n , W n ˜W m )and the
relation between armature current and motor torque
( W M KW ˜ i ), we get the following expression:
where K1
ªW l º
ª ul º
ªZl º
«W » K1 «u » K 2 «Z »
¬ r¼
¬ r¼
¬ r¼
nKW / Ra , K2 nKb K1 .
(11)
According to the relation of velocity vector V and
angle velocity of wheels Zw , we have the formulation of
1 º ªZl º
rª 1
« » , which can be rewritten as:
«
2 ¬ 1/ R 1/ R »¼ ¬Zr ¼
ªZl º 1 ª1 R º ªX º
(12)
«Z »
«
»« »
¬ r ¼ r ¬1 R ¼ ¬Z ¼
Substitute Eq. (12) into Eq. (11) , then we have
ªW l º
ª ul º K 2 ª1 R º ªX º
(13)
«W » K1 «u » «
»« »
¬ r¼
¬ r ¼ r ¬1 R ¼ ¬Z ¼
ªX º
«Z »
¬ ¼
.
By use of the definition of saturation function, the error
function can be rewritten as
­ e ij (t ) I ij (t ), e ij (t ) !I ij (t )
°°
j
j
. When S Iji (t ) 0 ,
S I i (t ) ® 0ˈ e ij (t ) d I i (t )
° j
j
j
j
°̄e i (t ) I i (t )ˈ e i (t ) I i (t )
Substitute Eq. (13) into Eq. (9) , and multiply by M -1 at the
both sides of Eq. (9), then we get the dynamic model of
WMR combining the DC motor model, which is formulated
as
V
3
t  0, T @ .
Because of the feature of the monotonic deceasing
function I ij (t ) which will converge to zero rapidly in the
0 º
2 K 2 ª1/ m
K ª 1/ m 1/ m º
2K
V 1«
u
»
2
2 «
R / I¼
r ¬ R / I R / I »¼ (14)
r ¬ 0
f (V ) Bu
condition of large value of k , e ij (t ) can converge to zero in
the time period 0,T @ . Now, the time derivative of
T
j
j
S I (t ) ˜ S I (t ) can be deduced as :
d
( S Ij (t )T ˜ S Ij (t )) 2 S Ij (t )T ˜ S Ij (t )
dt
­ 2 S Ij (t )T ˜ (e j (t ) I j (t ))
°
0
®
° j T
j
j
¯2S I (t ) ˜ (e (t ) I (t ))
ª sgn(e1j (t ))
º j
ªsgn
0
2S Ij (t )T (e j (t ) «
» ˜ I (t ))
j
sgn(e 2(t )) ¼
0
¬
FNN-AILC controller design
In the view of ILC system, the dynamic model of WMR
can be rewritten as:
(15)
V j (t ) f (V j (t )) Bu j (t )
where j  is the times of iteration, and t  [0, T ] .
Define the desired velocities as Vd
>Xd
Zd @ . What
T
this paper concerns is the tracking problem of WMR.
Therefore, the control objective is to design an appropriate
control strategy to make WMR tracking the desired
velocities Vd asymptotically when the iteration times trends
to infinity. The velocity tracking errors can be defined as
ª X j Xd º j
ªe j º
2u1
, then the
e j « 1j » V j Vd « j
», e 
Z
Z
e
d¼
¬ 2¼
¬
derivative of velocity tracking errors are e j V j Vd ,
j
2S Ij (t )T [k e j (t ) sgn(e j (t )T )I (t )]
2S Ij (t )T (V j Vd k e j (t ))
j
2k S Ij (t )T S Ij (t ) 2k S Ij (t )T ˜ I j (t ) 2 S Ij (t )T I (t )
2 S Ij (t )T [ Bu (V d f (V j ) k e j (t ))]
2k S Ij (t )T S Ij (t ) 2 S Ij (t )T [ Bu (V d f (V j ) k e j (t ))]
where e j  2u1 .
In order to handle the problem of random initial errors,
design the time-varying boundary layer function as
I j (t ) ª¬I 1j (t ) I 2j (t ) º¼
T
If Bu =Vd f (V j ) ke j (t ) ,
d
( S Ij (t )T ˜ S Ij (t )) 2k S Ij (t )T S Ij (t ) d 0 , which implies that
dt
the S Ij (t ) is decreasing function, and S Ij (t ) 0 in the time
H j e kt , where
T
period > 0,T @ since S Ij (0) 0 . However, f (V j ) in
dynamic model of WMR is unknown.
Usually, fuzzy logic and neural networks can be used to
approximate unknown function. Each of them has special
ª¬H 1j H 2j º¼ is the absolute value of initial
errors in No. j iteration, H 1j ! 0 , H 2j ! 0 , H j  2u1 . It is
Hj
e j (0)
(17)
108
DDCLS'17
properties that make them suited for particular problems and
not for others. So, some intelligent hybrid methods which
combine two or more techniques, such as fuzzy neural
networks, are created to overcome the limitation of
individual technique. In this paper, we use FNN technique to
approximate f (V j ) . The structure diagram of FNN is
illustrated in Fig. 2. The FNN has four layers, including input
layer, premise layer, rule layer and output layer.
Vd ke j (t )
UC
(21)
where J ! 0 ˈ T (t ) 
denotes approximation errorˈ
U A is the component of FNN adaptive iterative learning
2u1
j
controller, U B is the component of approximation error
compensator, and U C is the component of feedback
controller.
Fig.3 Block diagram of FNN-AILC for WMR
For the purpose of ensuring the controller’s convergence
along with both time axis and iteration axis, define the
parameter errors as
Fig. 2 The architecture of FNN
j
W (t )
T
ª¬ x1j x2j º¼ [X j Z j ]T  2u1 as input
signal in the input layer of FNN. In premise layer, we choose
( x m )2
Gaussian function exp{ i 2 il } .
Define X j
V
j
j
j
T (t ) ª«T 1 (t ) T 2 (t ) º»
¬
T*
il
ª¬ L(3)1 ( X j ) L(3)2 ( X j )
L(3)
( ) M ( X j ) º¼  M u1
is the fuzzy basis function vector in rule layer. The FNN has
two output signals
L(3)( X j )
ª¬ L(4)1 ( X j ,W j ) L(4)2 ( X j ,W j ) º¼  2u1 ,
which can be further written as:
ªW jT L(3)( X j ) º
L(4)( X j ,W j ) « 1 jT
W jT L(3)( X j ) (18)
j »
¬W2 L(3)( X ) ¼
W j (t )T L(3)( X j (t )) T j (t ) U B
j
j
j
¬ªW1 W2 ¼º
ª¬W W º¼ , and then we have
f (V j (t )) L(4)( X j (t ),W * ) H ( X j(t )) DŽ
Define virtual input as
U Bu Vd ke
k j (t ) U A U B UC , U 
W
1
J
(t ) W j (t ) J SIj (t )T L(3)( X j (t ))
j 1
(t ) T j (t ) J SIj (t )
W j 1 (t )
proj (W
ˈand we
(t ))
ª proj (T 1 j 1 (t )) º
«
»
« proj (T 2 j 1 (t )) »
¬
¼
(25)
(26)
where proj (˜) is mapping function and defined as:
­c
if c(t ) t c
°°
proj (c(t )) ®c
if c(t ) t c ˈ c! 0 .
° c(t ) otherwise
°̄
Now, the result is presented as follows.
(19)
SIi (t )( L(3)( X j )T L(3)( X j )) SIi (t )
2
2
j 1
ª proj (W 11 (t )) ... proj (W 1n j 1 (t )) º
«
»
« proj (W j 1 (t )) ... proj (W j 1 (t )) »
21
2
n
¬
¼
B (U Vd ke (t )) . U A , U B
J
(24)
j 1
j
W j (t )T L(3)( X j ) T j (t )
(23)
and
and U C are designed as˖
UA
j 1
T j 1 (t )
2u1
(22)
j
T
*
2
can get the real input u
UB
U A U B L(4)( X i (t ),W * j (t )) H ( X i (t ))
W (t )T L(3)( X j (t )) T (t ) U B
Design the adaptive laws of parameters along with
iterative axis as:
H 2 ( X j (t ) d H , and optional weights are defined as
W
T
j
*
2
*
1
T j (t ) T * , where
d W (t )T L(3)( X j (t )) T * T j (t ) U B
T
*
¼
W j (t ) W * ,
W *T L(3)( X j (t )) H ( X i (t ))
w11MM º
ª w11 w12
M u2
«w
» 
w
w
¬ 21
2M
22
2
M ¼
are the adaptive parameter vectors in output layer. The
optional approximation error is defined as
ª H ( X j (t ) º
H ( X j (t )) « 1 j » , where H1 ( X j (t ) d H1* ,
¬H 2 ( X (t ) ¼
where W j
T
T
ª¬H1* H 2* º¼ .
U f (V i ) U C
T
L(4)( X j ,W j )
ªW 1 j (t ) W 2 j (t ) º
«¬
»¼
(20)
109
DDCLS'17
Theorem 1. If the WMR dynamic model (Eq. 15) satisfies
the assumption of known and invertible input gain matrix B ,
and the FNN-AILC controllers (Eq. 19, Eq. 20 and Eq. 21)
and the parameter adaption laws(Eq. 23-26) are adopted, the
following conclusions can be obtained.
(1) The parameters of fuzzy system ( W j (t ) , T j (t ) ) are
bounded;
(2) Error function S Ij (t ) will converge to zero when
iteration times trends to infinite, i.e. lim S Ij (t )
j of
0;
(3) Tracking errors eij (t ) will converge to zero rapidly
after first time if k is large enough.
4
Fig. 4 Linear velocities in 5 times of iterations
Simulations
For the purpose of demonstrating the effectiveness of the
FNN-AILC method for WMR velocity tracking,
Matlab/Simulink is utilized for the numerical simulation.
Parameters of WMR are listed as: m 15kg , I 10kgm2 ,
2R 0.3m , r 0.1m , K1 5.2 , K2 2.3 . In this example
of simulation, the desired velocity trajectories can be
described as follow:
ªX (t ) º ª1.2 1.1sin(t S / 2) º
Vd (t ) « d » «
» and t  >0, 20@ .
0.3sin(0.4t )
¼
¬ wd (t ) ¼ ¬
In the FNN system, The Gaussian functions are chosen as
( x m )2
exp{ i 2 il } , which are also called membership
V
Fig.5 Angular velocity in 5 times of iterations
il
functions and have five rules for xi . The centers of Gaussian
functions are set as [m11
[m21
m25
2 ]
m15
1 ]
Five times of velocity tracking curves in time
interval [0, 20s] are illustrated in Fig. 4 and Fig. 5. It is
obviously that the velocity tracking cures converge
asymptotically to the desired velocity curves(black) in the
presence of different initial states when the iteration times
keep increasing. In the No. 5 iterative learning control, actual
velocity curves(red) can not converge to the desired
curves(black) in the first two seconds because of different
initial states, and then the velocity curves almost overlap the
desired velocity curves.
>0,0.6,1.2,1.8, 2.4@ and
>1, 0.5,0,0.5,1@ , and the variances of
Gaussian functions are V1l 6 and V 2l 4 . The initial
value of FNN-AILC parameters are set as
T
W 1 (t )
T 1 (t )
ª0.02 ... 0.02 º
1
«0.02 ... 0.02 » and W (t ) 
¬
¼
>0.02
25u2
,
0.02@ . The upper bounder c of project
T
function is 1. The random initial errors e j (0) step from the
5
random initial states V j (0) . In 5 times of iterations,
Conclusions
In this paper, the models of two-wheeled mobile robots
are deduced in detail, which include the kinematical model,
dynamic model and DC motor model. Then, in order to deal
with the random initial errors of velocity and the unknown
parameters in mathematic model, the FNN-AILC strategy is
proposed. The controller for velocity tracking, which
contains three components (FNN approximation component,
approximation error compensator and feedback control
component) and two adaptive laws in iterative direction, can
work on MIMO dynamic model. The effectiveness and
convergence of the control method is verified by the
simulation results.
V j (0) are set as
ª0.3 0.2 0.15 0.25 0.22 º
«0.1 0.1 0.09 0.15 0.05» .
¬
¼
When k 0.9 and J 0.9 , the velocity tracking curves are
shown in Figs.4-5 which show the good performance in
velocity tracking.
ª¬V 1 (0) V 5 (0)
( )ºº¼
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DDCLS'17
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