International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
347
Adaptive Neuro-Fuzzy Formation Control for Leader-Follower Mobile
Robots
Hung-Wei Lin, Wei-Shou Chan, Chia-Wen Chang, Cheng-Yuan Yang, and Yeong-Hwa Chang
Abstract1
reliant on heuristic experience. Therefore, in recent year,
the neural fuzzy control that combining with the
This paper aims to investigate the formation capability of fuzzy reasoning to handle uncertain
control of multi-robot systems, where the kinematic nonlinear information and the capability of artificial
model of a differentially driven wheeled mobile robot neural networks to learn from processes has been
is considered. Based on the graph-theoretic concepts popularly addressed in robot systems. A number of
and locally distributed information, an adaptive neuro-fuzzy logic approaches have been implemented
neural fuzzy formation controller is designed with the for mobile robot systems. In [1], a neuro-fuzzy reasoning
capability of on-line learning. The learning rules of algorithm, having the advantage of greatly reducing the
controller parameters can be derived from the number of fuzzy rules, was proposed to fulfill the
analyzing of Lyapunov stability. In addition to navigation task of mobile robots. In [2], a
simulations, the proposed techniques are applied to behavior-based neuro-fuzzy controller for mobile robot
an
experimental
multi-robot
platform
for system was addressed for the navigation problem. In this
performance validations. From simulation and work, a neuro-fuzzy method was applied to implement
experimental results, the proposed adaptive neural the behavioral function. A neuro-fuzzy algorithm was
fuzzy protocol can provide better formation presented and implemented in an 8-bit microcontroller to
responses compared to conventional consensus control the mobile robot for multiple tasks [3]. The
algorithms.
authors Alavander and Nigam proposed a new approach
to deal with the problem of inverse kinematics in robot
Keywords: Multi-robot systems, formation control, control by adaptive neuro-fuzzy method [4]. In the work
neural fuzzy control, Lyapunov stability.
of [5], a torque controller was presented to manipulate a
PUMA-600 robotic arm using an adaptive neuro-fuzzy
1. Introduction
inference. In [6], a neural integrated fuzzy controller was
successfully implemented for the navigation task of a
Over the last decade, robot control has attracted great multi-robot system under the information exchange in a
attention for its applications in service robot, rescue complete topology. However, the stability of multi-robot
robot, and unmanned autonomous vehicle, to name but a system associated with the communication topology has
few. In designing robot system, the challenges how to not been well considered.
derive the nonlinear model and how to deal with the
Distributed multi-agent coordination has attracted
uncertainty are repeatedly encountered. It is well known much attention in many fields, such as autonomous
that classical control strategies are usually relied on the vehicles [7], wireless sensor networks [8], smart
availability of precise models. To overcome this problem, buildings [9], group decision making [10], multi-agent
fuzzy logic control is suitable for controlling a robot architecture with cooperative fuzzy control [11], and
system due to the tolerance to noise and disturbance power systems [12], where only the information
even under uncertainty. However, the design of fuzzy available locally is required for each agent. In general,
rules leaks systematic methodology and it is frequently information transmitted among agents is based on a
communication topology that could be static or dynamic.
Corresponding Author: Y.-H. Chang are with the Department of Recently, graph theory has been used to describe
Electrical Engineering, Chang Gung University, Taiwan.
network topologies in the study of consensus protocols
E-mail: [email protected]
[13]-[16]. A general consensus problem solving is to
H.-W. Lin is with the Department of Electrical Engineering,
find a distributed control strategy such that the states of
Lee-Ming Institute of Technology, Taiwan.
W.-S. Chan and C.-Y. Yang are with the Department of Electrical agents converge to a common value. In [13],
Engineering, Chang Gung University, Taiwan.
average-consensus problem was investigated for
C.-W. Chang is with the Department of Information and distributed networks, in which the analysis of consensus
Telecommunication Engineering, Ming Chuan University, Taiwan.
Manuscript received 12 Dec. 2012; revised 12 April 2013; accepted protocols was based on the graph theory, matrix theory,
and control theory. A distributed impulsive control
30 July 2013.
© 2013 TFSA
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
348
protocol was presented for multi-agent linear dynamic
systems [17]. In [18], leader-following consensus
problem was discussed for agents with a general n -th
order linear model. Qin et. al. [19] discussed the
condition of communication delays to achieve consensus
for second-order agents under switching topology. In
[20], consensus problems were addressed for a group of
high-order dynamic agents with switching topology and
time-varying communication delays. Moreover, linear
consensus protocol and saturated consensus protocol
were presented for a heterogeneous multi-agent system
[21]. In most of the existing results, first- and
second-order linear models are commonly considered to
address the consensus stability of multi-agent systems.
However, aforementioned the cooperative problem of
multi-agent system associated with parameter
uncertainty or external disturbance has not been well
considered.
This paper aims to investigate the formation control of
multi-robot systems with parameter uncertainty or
external disturbance, where the kinematic model of a
differential wheeled robot is considered. The graph
theory is used to model the communication topology
between robots. To improve the control performance, a
novel formation algorithm, adaptive neural fuzzy
formation control, is proposed for multi-robot systems in
directed graphs. The proposed formation controller has
the capability of on-line learning, and the adaptive
learning rules can be derived using the Lyapunov
stability analysis. An experimental platform is also
applied to validate the formation performance.
This paper is organized as follows. In Sec. 2, some
graph-theoretic concepts and a differential wheeled robot
with kinematics are introduced. In Sec. 3, the framework
of an adaptive neural fuzzy formation control is
presented. In Sec. 4, the stability analysis of
leader-follower mobile robots are investigated. In Sec. 5,
simulation and experimental results are provided for
performance validations. Some concluding remarks are
given in Sec. 6.
2. Preliminaries
A. Graph theory
In this section, some fundamental concepts in
algebraic graph theory used for multi-agent systems will
be introduced. Considering a multi-agent system of n
agents, let G  ( ,  ) be a directed graph (digraph),
consisting of a vertex set   { 1 , 2 , , n } and an edge
set     . The vertexes  i and  j represent the
i th and j th agents, respectively. In digraphs, an edge of
G is an ordered pair of distinct nodes ( j , i )   , in
which  i and  j are the head and tail of the edge,
respectively [14]. The weighted adjacency matrix of a
digraph G is denoted as
 a11 a12  a1n 
a
a22  a2 n 
21
(1)

   n n ,
A

   


 an1 an 2  ann 
where aij is the weight of link ( j , i ) ,
 aij  1, ( j , i )  

aij  0, ( j , i )  
 a  0.
 ii
The degree matrix of G is a diagonal matrix,
D  diag{d1 , d 2 , , d n }   nn , where the in-degree of node
 i is defined as
n
d i   aij , i  1, 2,  , n.
(2)
j 1
Then the Laplacian matrix associated with the digraph
G is defined as L  D  A   n n . In this paper, a
leader-follower problem will be dealt with, where the
multi-agent system consists of n agents, one leader and
n  1 followers. In notations, the agents indexed by
1, 2,  , n  1 are followers and the item n is the leader.
Assume that the leader agent has only transmitting
capability, i.e. the leader acquire no information from
followers, anj  0 , j  1, , n . In this case, let the
topology relationship of follower agents be denoted as
G , a subgraph of G . Then, the associated adjacency
matrix of G is represented as
a12

a1( n 1) 
 a11
 a
a22
 a2( n 1) 
21
   ( n 1)( n 1) .
A
(3)
 






 a( n 1)1 a( n 1) 2  a( n 1)( n 1) 
Similarly, let D  diag{d1 , d 2 , , d n 1} be the degree
matrix of digraph G , where d i   nj 11aij , i  1, 2,  , n  1 .
The Laplacian matrix of the digraph G can be
analogously defined as L  D  A . Then the following
condition holds
L 1n 1  0,
(4)
T
n 1
where
. Consequently, the
1n 1  [111]  
connection relationship between the leader and followers
can be described as B  diag{b1 , b2 , , bn 1} , where
bi  ain , i  1, 2,  , n  1 .
B. Kinematic model of wheeled mobile robot
Considering a group of n identical wheeled mobile
robots, the robot structure is shown in Fig. 1. The
Hung-Wei Lin et al.: Adaptive Neuro-Fuzzy Formation Control for Leader-Follower Mobile Robots
kinematic characteristics of the i th robot are described
by
 xci (t )  vi (t ) cos( i (t )),

(5)
 y ci (t )  vi (t ) sin( i (t )),
  (t )   (t ),
 i
i
i
riL
vi
 xhi , yhi 
lh
i
y
 xci , yci 
riR
2lw
r
x
Figure 1. Configuration diagram of wheeled mobile robots.
where ( xci (t ), yci (t )) is the center position, vi (t ) is the
linear velocity, i (t ) is the angular velocity, i (t ) is
the rotation angle, i  1, 2,  , n . It is denoted
( xhi (t ), yhi (t )) as the head position of the i th robot, such
that
 xhi (t )   xci (t ) 
 cos( i (t )) 
(6)
 y (t )    y (t )   lh  sin( (t ))  ,
 hi   ci 


i
in which lh is the length between the center position
and the head position. Taking the derivative of (6), the
governing equations of the head position for the i th
robot can be determined as follows
 xhi (t )   cos( i (t )) lh sin( i (t ))   vi (t ) 
(7)
 y (t )    sin( (t )) l cos( (t ))   (t )  .
 hi  
 i 
i
h
i
According to the configuration in Fig. 1, the motion
equations of vi (t ) and i (t ) corresponding to the leftand right-wheel angular velocities, iL (t ) and iR (t ) ,
can be obtained as
0.5r  iL (t ) 
 vi (t )   0.5r
(8)
 (t )    0.5rl 1 0.5rl 1   (t )  ,
 i  

w
w   iR
where r is the radius of wheel, and lw is the distance
between the center of robot and the wheel. Substituting
(8) into (7) and defining l  0.5lh / lw , it leads to the
following equivalent model of a wheeled robot
 xhi (t ) 
T11 (t ) T12 (t )  iL (t ) 
 y (t )   T (t ) T (t )   (t )  ,
(9)

 hi 
21
22

  iR 
Ti ( t )
where
T11 (t )  0.5r cos(i (t ))  rl sin(i (t )),
T12 (t )  0.5r cos(i (t ))  rl sin(i (t )),
349
T21 (t )  0.5r sin(i (t ))  rl cos(i (t )),
T22 (t )  0.5r sin(i (t ))  rl cos(i (t )).
From (9), the perturbed model of a wheeled robot can
be represented as [22]
 xhi (t ) 
iL (t ) 
(10)
 y (t )   (Ti (t )  Ti (t ))  (t )  ,
 hi 
 iR 
in which Ti (t ) is the perturbation term of augmented
uncertainties.
Let
 u xi (t ) 
iL (t )   xi (t ) 
iL (t ) 
u (t )   Ti (t )  (t )  ,  (t )   Ti (t )  (t )  ,
 iR   yi 
 iR 
 yi 
where u xi (t ) and u yi (t ) are considered as control
actions to the head position, and  xi (t ) and  yi (t ) are
described as the uncertainty terms of the x -axis and
y -axis, respectively. Therefore, the kinematic equations
of a wheeled robot with uncertainties can be described in
the following compact form,
 xhi (t )   u xi (t )   xi (t ) 
(11)
 y (t )   u (t )    (t )  .
 hi   yi   yi 
It is noted that the kinematic equations of a wheeled
robot in (11) can be decoupled into two one-dimensional
subsystems. To derive consensus controllers, it suffices
to consider only the subsystem in the following.
3. Adaptive Neural Fuzzy Formation Control
In this section, an adaptive neural fuzzy formation
control (ANFFC) is proposed to deal with the
leader-following formation problem, where the perturbed
kinematic model of (11) is considered.
First, let the x -axis error functions be defined as
exi (t ) 
n 1

aij [( xhi (t )  pxi )  ( xhj (t )  pxj )]
j 1, j  i
(12)
 bi [( xhi (t )  pxi )  ( xhn (t )  pxn )],
where pxi indicates the coordinate position regarding to
a desired formation pattern in x -axis, i  1, 2,  , n . It is
noticed that aij  1 means that the j th robot can send
position information to the i th robot. In addition, bi  1
implies that the leader robot can send position
information to the i th robot. The controller input of the
i th robot is designated as follows
S xi  c1exi  c2  exi dt ,
t
0
(13)
in which c1 and c2 are positive constants.
The network structure of neural fuzzy control (NFC)
is shown in Fig. 2. The fuzzy rules are given in Table 1,
where the input and output spaces are fuzzily partitioned
into six fuzzy sets, Negative Big (NB), Negative
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
350
Medium (NM), Negative Small (NS), Positive Small
(PS), Positive Medium (PM) and Positive Big (PB). The
input and output membership functions are depicted in
Fig. 3.
using the centroid defuzzification technique, the NFC
output can be calculated as follows
6
u xi , NFC ( S xi ,  xi ,  xi ) 
Table 1. Fuzzy rule base.
S xi ( S yi )
NB NM
NS
PS
PM PB
PB
PS
NS
NM NB
u xi , NFC (u yi , NFC )
PM

xik
 xik
k 1
6
  xik
  xiT  xi ,
(16)
k 1
where  xik are the values of the corresponding output
fuzzy singletons,  xi  [ xi1 xi 2  xi 6 ]T and
6
 xi  [ xi1 /  xi  xi 2 /  xi  xi 6 /  xi ]T ,  xi    xik .
u xi , NFC (u yi , NFC )
k 1
Since the robot kinematics is transformed to a
collection of decoupled subsystems as (11), it suffices to
design an ANFFC for one subsystem. Consider that
u xi , NFC  u xi* , NFC ( S xi ,  xi* ,  xi* )  uˆ xi , NFC ( S xi , ˆxi , ˆxi )
(17)
  xi*T  xi*  ˆxiT ˆxi ,
where u xi* , NFC and uˆxi , NFC are the optimal and estimated
 xi1  xi 2  xi 3  xi 4  xi 5  xi 6
( yi 2 ) ( yi 3 ) ( yi 4 ) ( yi 5 ) ( yi 6 )
( yi1 )
 xi1
( yi1 )
 xi 2
( yi 2 )
 xi 3
( yi 3 )
 xi 6
 xi 5
( yi 5 ) ( yi 6 )
 xi 4
( yi 4 )
NFC control actions, respectively,
 xi*  [ xi* 1  xi* 2   xi* 6 ]T ,
ˆxi  [ˆxi1 ˆxi 2  ˆxi 6 ]T ,
 xi*  [ xi* 1  xi* 2   xi* 6 ]T ,
ˆxi  [ˆxi1 ˆxi 2  ˆxi 6 ]T .
S xi ( S yi )
It is noticed that  xi* ,  xi* , and u xi* , NFC are unknown
Figure 2. Structure of NFC.
optimal parameters and control output. In addition, ˆxi
and ˆxi are the estimated parameters corresponding to
 xi* and  xi* , respectively. Let the estimation errors of
c xi1 c xi 2 c xi 3 c xi 4 c xi 5 c xi 6
(c yi1 ) (c yi 2 ) (c yi 3 ) (c yi 4 ) ( c yi 5 ) (c yi 6 )
S xi
( S yi )
 x6i  x5i  x 4i  x3i  x2i  x1i
( y 6i ) ( y 5i ) ( y 4i ) ( y 3i ) ( y 2i ) ( y1i )
uxi , NFC
(u yi ,NFC )
(a)
(b)
Figure 3. Membership functions of NFC: (a) input
membership functions, (b) output membership functions.
The corresponding if-then fuzzy rules for the i th
robot are expressed as
Rik : IF S xi is M ik THEN uxi , NFC is Gik
(14)
where M ik and Gik are the fuzzy sets of antecedent and
consequence parts, respectively, i  1, 2,  , n  1 ,
k  1, 2,  , 6 .
In Fig. 3, the k th nodes of the membership layer are
Gaussian functions represented as
 1  S  c  2 
 xik ( S xi ,  xi , C xi )  exp   xi xik   ,
(15)
 2   xik  
where  xi  [ xi1 xi 2  xi 6 ]T , Cxi  [cxi1cxi 2  cxi 6 ]T ,
i  1, 2,  , n  1 . In (15), cxik and  xik are means and
standard deviations, respectively, k  1, 2,  , 6 . By
controller parameters be defined as xi   xi*  ˆxi and
xi   xi*  ˆxi .
The vector of Gaussian functions is linearized using
Taylor expansion, xi can be written as [23]
xi  Gxi C xi  H xi xi  h.o.t.,
(18)
in which
  *
Gxi   xi*1
 C xi
 
 
H xi  
*
xi1
*
xi
 xi* 2
Cxi*


*
xi 2
*
xi
T
 xi* 6 


C xi*  C* Cˆ
xi

xi
T
 

  * ˆ
xi
xi
*
xi 6
*
xi
and H xi are matrices of  66 , C xi  C xi*  Cˆ xi ,
 xi   xi*  ˆ xi , and
Gxi
*
 xik
*
*
  xik
 xik


*

 xik
,
*
*
* 
 cxi1 cxi 2 cxi 6 
*
*
  *  xik
 xik
 * 
 xik
  xik*
,
*
*
 xi   xi1  xi 2  xi* 6 
C xi*
Hung-Wei Lin et al.: Adaptive Neuro-Fuzzy Formation Control for Leader-Follower Mobile Robots
i  1, 2,  , n  1 , k  1, 2,  , 6 . In (18), the term h.o.t .
stands for the high-order terms associated with the Talor
expansion. From (17) and (18), it can be obtained that
uˆ xi , NFC  u *xi , NFC  u xi , NFC
 (xi  ˆxi )T (xi  ˆxi )  xiT xi
 ˆxiT (Gxi C xi  H xi xi  h.o.t.)  (ˆxiT +xiT )ˆxi .
(20)
where ucxi (t ) is a compensation controller. Substituting
(19) into (20), the control law of (20) can be rewritten as
c1
xi
n 1

j 1, j  i
xi
xi
xi
cxi
2 xi
(21)
From (11) and (21), the derivative of (13) can be
represented as
S xi  ˆxiT (Gxi C xi  H xi xi )  xiT ˆxi  xi  ucxi (t )
 n 1

(22)
 j 1

Then (22) can be equivalently written as
S xi  ˆxiT (Gxi C xi  H xi xi )  xiT ˆxi  ucxi   xi ,
(23)
where
 n 1

 xi  xi  c1   aij ( xi   xj )  bi ( xi   xn )  ,
 j 1

is
an
uncertain
term.
Suppose
is
bounded
such
 xi
 xi
that |  xi | Exi for a constant Exi . However, E xi is
unknown a priori, and then it is rather to be estimated.
Denoted Eˆ xi as the estimation of Exi , the estimation
error is defined as
E xi  E xi  Eˆ xi .
(24)
The adaptive laws and compensation control ucxi (t )
for the dynamic equation of (11) are designed as follows


Eˆ xi  r1 | S xi |, Cˆ xi  r2 S xi GxiT ˆxi , ˆ xi  r3 S xi H xiT ˆxi ,

ˆxi  r4 S xiˆxi , ucxi   Eˆ xi sgn( S xi ),
obtained as

Vxi  C xiT ( S xi GxiT ˆxi  r21Cˆ xi )   xiT ( S xi H xiT ˆxi  r31ˆ xi )


 xiT ( S xiˆxi  r41ˆxi )  S xi (ucxi   xi )  r11 E xi Eˆ xi .
(27)
1

Vxi  S xi (ucxi   xi )  E xi Eˆ xi .
r1
(28)

Vxi  S xi (ucxi   xi )  r11 E xi Eˆ xi

 S xi (  Eˆ xi sgn( S xi )   xi )  r11 ( E xi  Eˆ xi ) Eˆ xi
(29)
| S xi | (|  xi |  Exi ).
aij u xj  c1bi u xn  .
 c1   aij ( xi   xj )  bi ( xi   xn )  .
2r4
  Eˆ xi | S xi |  S xi xi  Exi | S xi |  Eˆ xi | S xi |
1
xi
2r3
Furthermore, substituting (25) into (28), it leads to
j 1
xi
2r2
Substituting (25) into (27), it gives that
 c2 exi  c1  aij u xj  c1bi u xn ]
n 1
 

u xi , ANFFC (t )  c1  bi   aij   xiT ˆxi
j 1, j  i
 
 
T
ˆ (G C  H  )    u (t )  c e
2r1
i  1, 2,  , n  1 . From (23), the derivative of (26) can be
1
n 1
networks designed as (20) and (25), the x -axis
formation stability of the i th robot is guaranteed.
Proof: A Lyapunov function candidate is chosen as
E 2 C T C
 T 
 T 
1
Vxi  S xi2  xi  xi xi  xi xi  xi xi ,
(26)
2
(19)
Let xi  ˆxiT ( h.o.t.)  ˆxiT ˆxi . The proposed ANFFC for
the x -axis dynamic equation of (11) is designed as
n 1
 

u xi , ANFFC (t )   c1  bi   aij   [uˆ xi , NFC  ucxi
j 1

 
351
(25)
where ri  0 , i  1, 2, 3, 4 .
Theorem 1: Consider the x -axis dynamic equation of
(11). With the ANFFC and adaptive laws of neural fuzzy
From the assumption that |  xi | Exi , it can be concluded
that Vxi is negative semi-definite. From the negative
semi-definiteness of Vxi , it implies that S xi , E xi , C xi ,
 xi , and xi are all bounded. Furthermore, it can be
obtained that Vxi is also bounded. Then using the
Barbalat's lemma [24], it can be concluded that Vxi  0
as t   , i.e. S xi  0 and exi  0 as t   . Hence,
it can be concluded that the x -axis formation stability
of the i th robot is guaranteed. ■
Remark 1: The proposed ANFFC for the y -axis
dynamic equation of (11) is designed as
 
n 1
 
j 1

1
u yi , ANFFC (t )  c1  bi   aij   [uˆ yi , NFC  ucyi

n 1
(30)
 c2 eyi  c1  aij u yj  c1bi u yn ].
j 1
The adaptive laws and compensation control ucyi (t )
for the y -axis dynamic equation of (11) are designed as
follows

Eˆ yi  r1 | S yi |,

Cˆ yi  r2 S yi G yiT ˆyi , ˆ yi  r3 S yi H yiT ˆyi ,
(31)

ˆyi  r4 S yiˆyi , ucyi   Eˆ yi sgn( S yi ).
Similarly, the notations E yi , C yi ,  yi , yi , and Vyi
can be reciprocally defined for the y -axis dynamic
equation of (11). Consequently, the convergent
properties, S yi  0 and e yi  0 , can be also obtained.
As a result, the guaranteed stability of the i th robot
implies that the required formation goal for the i th
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
352
robot can be asymptotically achieved by employing the
proposed ANFFC approach.
4. Stability Analysis of Leader-Follower Mobile
Robots
The proposed ANFFC protocol can be extended to
multi-robot systems with one leader and n  1 followers.
From (20) and (30), the output of ANFFC in a
multi-robot system can be grouped together as
U  [c1 ( D  B )  I 2 ]1[c2 (( L  B )  I 2 (  ))
 c2 (( B 1n 1  I 2 )( Z n  Pn ))  c1 ( A  I 2 )U
(32)
 Uˆ NFC  U c  c1 ( B 1n 1  I 2 )U n ].
where the symbol  stands for the Kronecker product,
I 2  diag{1,1} , and
Uˆ NFC  [Uˆ 1,T NFCUˆ 2,T NFC Uˆ nT1, NFC ]T   2( n 1) ,
U c  [U cT1 U cT2 U cT( n 1) ]T   2( n 1) ,
  [ Z1T Z 2T  Z nT1 ]T   2( n 1) ,
  [ P1T P2T  PnT1 ]T   2( n 1) ,
U  [U 1T U 2T U nT1 ]T   2( n 1) ,
and Uˆ i , NFC  [uˆ xi , NFC uˆ yi , NFC ]T , U ci  [ucxi ucyi ]T ,
Z i  [ xhi yhi ]T , Pi  [ pxi p yi ]T , U i  [u xi , ANFFC u yi , ANFFC ]T .
From (32), it is noticed that the output of ANFFC is
related to the communication graph of a underlying
multi-robot system. In other words, ANFFC can be
derived while the communication graph of a multi-robot
system is provided.
From (32), it can be equivalently obtained that
{I 2( n 1)  [c1 ( D  B )  I 2 ]1[c1 ( A  I 2 )]}U
 2( n 1)
ˆ  [ˆ1T ˆnT1 ]T ,
Cˆ  [Cˆ1T  Cˆ nT1 ]T ,
ˆ  [ˆ1T ˆ nT1 ]T , and ˆ  [ˆ1T  ˆnT1 ]T are vectors of
12( n 1) ; Eˆ  [ Eˆ Eˆ ]T and S  [ S S ]T are vectors of
of
i
;
xi
yi
i
xi
yi
12
ˆi  [ˆxi ˆyi ]T , i  1, 2,  , n  1 , are vectors of  .
 c2 (( B 1n 1  I 2 )( Z n  Pn ))  U c  c1 ( B 1n 1  I 2 )U n ]
(33)
Furthermore, the term on the left-hand side of (33) can
be rewritten as
1
{I 2( n 1)  [c1 ( D  B )  I 2 ] [c1 ( A  I 2 )]}U
 [c1 ( D  B )  I 2 ] [c1 ( D  B  A )  I 2 )]U
where Eˆ  [ Eˆ1T  Eˆ nT1 ]T and S  [ S1T  S nT1 ]T are vectors
 2 ; Cˆ i  [Cˆ xiT Cˆ yiT ]T , ˆi  [ˆxiT ˆyiT ]T , ˆ i  [ˆ xiT ˆ Tyi ]T , and
 [c1 ( D  B )  I 2 ]1[Uˆ NFC  c2 (( L  B )  I 2 (  ))
1
only if L has a simple zero eigenvalue [14]. In a
leader-follower system, L can be rewritten as
 a12
  a1( n 1)
 a1n 
 d1
 a
d2
  a2( n 1)
 a2 n 
21


L 



 


  a( n 1)1  a( n 1) 2  d ( n 1) a( n 1) n 
 0

0
0
0 
(36)
b1 

 LB

b2



 .


bn 1 

 0 0  0
0 
It means that Rank ( L )  Rank ( L  B )  n  1 and
( L  B ) are invertible. Thus, the ANFFC of (35) is
computable. The adaptive laws in a multi-robot system
can be grouped together as
 Eˆ  r1 | S |

  r1 | S x1 | | S y1 |  | S x ( n 1) | | S y ( n 1) | T ,

(37)
Cˆ  r FG T ˆ,
 2 T
ˆ  r3 FH ˆ,
 ˆ
  r4 Fˆ,
(34)
Moreover,
 F1 0
0 F
2
F
 

0 




1
 [c1 ( D  B )  I 2 ] [c1 ( L  B )  I 2 )]U .
Substituting (34) into (33), the control actions to a
leader-following multi-robot system can be obtained as
U  [c1 ( L  B )  I 2 ]1[Uˆ NFC  c2 (( L  B )  I 2 (  ))
 c2 (( B 1n 1  I 2 )( Z n  Pn ))  U c  c1 ( B 1n 1  I 2 )U n ].
(35)
It is known that if graph G has a spanning tree, then
a right eigenvector of L associated with the zero
eigenvalue is 1n . In other words, Rank ( L)  n  1 if and
H
0 
G1 0
0 G
2
G



0 
 
,
0 

Fn 1 
 H1
0



0

0 

 

0 
0 
0

H2

 


0 


H n 1 
,

Fi  diag{S xi , S yi , S xi , S yi ,  , S xi , S yi },
 ˆxi1 ˆyi1
,
 cˆxi1 cˆ yi1
, ,

 Gn 1 
and
Gi  diag 
,
ˆxi 6 ˆyi 6 

,

cˆxi 6 cˆ yi 6 

Hung-Wei Lin et al.: Adaptive Neuro-Fuzzy Formation Control for Leader-Follower Mobile Robots

ˆyi 6 
ˆ
 ˆxi1 ˆyi1
,
, , xi 6 ,
.
ˆ xi 6 ˆ yi 6 

 ˆ xi1 ˆ yi1
H i  diag 
It is noticed that Fi , Gi , and H i are matrices of 1212 ,
i  1, 2,  , n  1 .
The compensation control U c (t ) in a multi-robot
system can be grouped together as
ˆ
 ucx1    Ex1sgn( S x1 ) 


 u 
ˆ sgn( S ) 

E

cy
1
y
1
y
1





Uc      
(38)
.


ˆ


u
 Ex ( n 1) sgn( S x ( n 1) )
 cx ( n 1) 

ucy ( n 1)   ˆ

   E y ( n 1) sgn( S y ( n 1) ) 
In matrix forms, E xi , E yi , C xi , C yi ,  xi ,  yi , xi , and
yi in multi-robot systems can be represented as
E
C


  *  ˆ ,
  *  ˆ,

E  Eˆ ,
C *  Cˆ ,

C  [C  C ] , and   [
*
*T
1
12( n 1) ;
*T T
n 1
*
Ei  [ Exi E yi ]T
is
*T
1
 *  [1*T  n*T1 ]T
;

*T T
n 1
a
vector
]
are vectors of
of
2 ;
i  1, 2,  , n  1 , are vectors of 12 .
Theorem 2: Considering the dynamic equation of (11), it
is assumed that the communication graph of a
multi-robot system has a directed spanning tree. With the
ANFFC and adaptive law designed as (35) and (37)-(38),
the stability of leader-follower formation control is
guaranteed.
Proof: To verify the overall stability of an
ANFFC-controlled multi-robot system, a Lyapunov
function candidate is chosen as
S T S E T E C T C  T   T 




.
V
(40)
2r1
2r2
  [( L  B )  I 2 ](  )  ( B 1n 1  I 2 )( Z n  Pn )
 0 2( n 1) ,
where
02( n 1)  [0 0]T   2( n 1)
,
(43)
  [1T   Tn 1 ]T
,
 i  [exi eyi ]T , and i  1, 2,  , n  1 . From (4), it can be
obtained that
[( L  B )  I 2 ](   )  [( L  B )  I 2 ](1n 1  I 2 )( Z n  Pn ). (44)
It is noticed that ( L  B ) is invertible. From (44), it can
be equivalently obtained that
  (1n 1  I 2 ) Z n  [  (1n 1  I 2 ) Pn ].
(45)
It concludes that formation control in the multi-robot
systems is guaranteed. Hence the stability of the
multi-robot systems is guaranteed, and the
leader-following formation goal can be asymptotically
achieved using the proposed ANFFC protocol. ■
(39)
Ci*  [C xi*T C yi*T ]T ,  i*  [ xi*T  *yiT ]T , and  i*  [ xi*T  yi*T ]T ,
2
as t   , i.e. S xi  0 , S yi  0 , exi  0 , and e yi  0 ,
as t   , i  1, 2,  , n  1 . From (12), the error function
in a multi-robot system can be grouped together as
5. Simulation and Experimental Results
E  [ E1T  EnT1 ]T   2( n 1)
where
353
2r3
2r4
From (35), the derivative of (40) can be obtained as

V  C T ( FG T ˆ  r21Cˆ )   T ( FH T ˆ  r31ˆ )


 T ( F ˆ  r41ˆ)  S T (U c   )  r11 E T Eˆ ,
(41)
where   [ x1  y1  x 2  y 2  x ( n 1)  y ( n 1) ]T .
Furthermore, substituting (37)-(38) into (41), it leads to
V | S T | (|  |  E )  0.
(42)
From (42), implies that S , E , C ,  , and  are all
bounded, i  1, 2,  , n  1 . Furthermore, it can be
obtained that V is also bounded. Then using the
Barbalat's lemma [24], it can be concluded that V  0 ,
A. Simulation results
The case of four robots, one leader and three followers,
is considered, where r  0.029 , lw  0.034 , and lh  0.04
( m) . The communication topology is shown in Fig. 4,
where the circles labeled 1 to 3 denote the follower
robots and the circle 4 represents the leader robot.
Figure 4. Communication topology of multi-robot systems.
It is clear that the information exchange graph forms a
spanning tree, where the adjacency matrix and Laplacian
matrix can be determined as follows
 2 1 0 1
0 1 0 1 
 1 1 0 0 
1 0 0 0 
 , L
.
A
0 0 0 1 
 0 0 1 1




0 0 0 0 
 0 0 0 0
Consequently, it can be obtained that
0 1 0
1 0 0 
 1 1 0 




A  1 0 0 , B  0 0 0 , L   1
1 0 .






 0 0 0 
 0 0 0 
 0 0 1 
The follower robots are initially placed on (0,1.1) ,
(0, 0.8) , and (0, 0.3)( m) , respectively, and the initial
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
position of leader is (0, 0.6)(m) . The formation pattern
are designated as
( p x1 , p y1 )  ( 0.2, 0.2) , ( p x 2 , p y 2 )  ( 0.4, 0) ,
1.4
1.2
Y-axis(m)
354
1
0.8
( p x 3 , p y 3 )  ( 0.2, 0.2) , ( p x 4 , p y 4 )  (0, 0)( m ) .
0.4
The parameters for the ANFFC are initially chosen as
c1  3 , c2  2 , Eˆ xi (0)  Eˆ yi (0)  0 ,
0.5
0
0.2
0.4
X-axis(m)
Y-axis Error(m)
X-axis Error(m)
0.2
-0.2
r1  0.03 , r2  0.19 , r3  0.19 , r4  0.19 ,
ˆ
C (0)  Cˆ (0)  [ 1  0.67  0.33 0.33 0.67 1]T
xi
F1
F2
F3
Leader
Leader
Followers
0.6
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
1.2
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
(a)
1.4
yi
0.4
0.2
-0.2
Accordingly, the simulation results are shown in Figs.
5-6 corresponding to the cases C1 and C 2 ,
respectively. In each subplot, both the robot trajectories
and formation errors are presented, where unique
distributed strategies, CA [25], NFC, and ANFFC, are
considered. The performance comparisons for the x and y -axis error functions are summarized in Table 2,
where ITSE is the integral time square error, ITAE is the
integral time absolute error,


3
2
ITSE   t   (exi  eyi )  dt ,
0



i 1
3
ITAE   t |  (exi  eyi ) | dt.

0
i 1
Table 2. Performance comparison (simulation).
Performance
index
CA
NFC
ANFFC
C1
C2
C1
C2
C1
C2
ITSE
47.4
50.9 10.4
11.4
1.2
1.6
ITAE
256.7 418.0 113.6 188.6 18.3 37.8
0
0.2
0.4
X-axis(m)
Y-axis Error(m)
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
1.2
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
(b)
1.4
and  yi (t ) in (11) are given as 0.03*rand(), where
Y-axis(m)
1.2
1
0.8
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
0.2
-0.2
X-axis Error(m)
rand()[-1,1] is a uniformly distributed random
number.
C 2 . The uncertainties addressed in the case C1 are
taken. Moreover, a time-varying leader velocity,
(u x 4 , u y 4 )  (0.06, 0.08 cos(0.2 t ))( m / s ) , is considered.
F1
F2
F3
Leader
Leader
Followers
0
0.2
0.4
0.5
X-axis(m)
Y-axis Error(m)
In the following, three cases are respectively
investigated:
C1 . A constant velocity command is assigned to the
leader, (u x 4 , u y 4 )  (0.1, 0.04)( m / s ) . In addition,  xi (t )
1
0.8
0.6
X-axis Error(m)
and
ˆ xi (0)  ˆ yi (0)  [0.5 0.5 0.5 0.5 0.5 0.5]T , i  1, 2, 3 .
Y-axis(m)
1.2
ˆxi (0)  ˆyi (0)  [ 1  0.67  0.33 0.33 0.67 1]T
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
1.2
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
(c)
Figure 5. Simulation results of leader-follower formation
control (case C1): (a) CA [25], (b) NFC, (c) ANFFC.
In Fig. 5, it is illustrated that the desired formation can
not be achieved using conventional consensus algorithm
[25]. The error divergence is improved with the NFC
control, however, there exists some steady-state
formation errors in the x -axis. Moreover, the formation
errors are significantly reduced using the proposed
ANFFC subject to uncertainties. Formation results
associated with time-varying leader velocity are shown
in Fig. 6. It is noticed that formation outcomes of CA
and NFC become much worse. However, the formation
errors of robots asymptotically converge and the desired
formation pattern can be achieved using the proposed
ANFFC. In addition, from Table 2, it can be observed
that the proposed ANFFC has the advantages of
formation responses in all performance indexes.
B. Experimental results
In this paper, the e-puck education robot is used as the
test platform for the formation control of multi-robot
systems, in which there are three followers and one
leader. It is noticed that the e-puck robot is a modular,
robust, and desktop-sized robot that is designed by
Francesco Mondada and Michael Bonani from Ecole
Hung-Wei Lin et al.: Adaptive Neuro-Fuzzy Formation Control for Leader-Follower Mobile Robots
Polytechnique Federale de Lausanne (EPFL) for research
and educational purposes [22], [26]. The e-puck robot is
powered by a dsPIC processor and is equipped with
sensors, and its mobility is ensured by a differential
driven configuration. In addition, the e-puck hardware
and software are fully open source, providing low-level
access to every electronic device and offering extension
possibilities. The e-puck robot can communicate with a
computer or with other devices by Bluetooth
communication. The control scheme of the multi-robot
system is shown in Fig. 7. The control scheme of the
multi-robot system consists of one leader, three
followers, Bluetooth communication module, a computer,
and a webcam. The webcam is utilized to obtain the
position of each robot by the techniques of image
processing. One robot can receive the position
information from its neighbors, and the control
algorithm is implemented in the dsPIC processor to
achieve the purpose of formation control. Two main
tasks are executed in the PC, namely, transmitting the
position information to each robot through the Bluetooth
and recording the responses of all robots.
1.2
Y-axis(m)
1
0.8
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
-0.2
0
0.2
0.4
X-axis(m)
0.5
Y-axis Error(m)
X-axis Error(m)
0.2
0
-0.5
0
5
time(sec)
10
15
0.6
0.8
1
1.2
10
15
0.5
0
-0.5
0
5
time(sec)
(a)
1.2
Y-axis(m)
1
0.8
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
-0.2
0
0.2
0.4
X-axis(m)
0.5
Y-axis Error(m)
X-axis Error(m)
0.2
0
-0.5
0
5
time(sec)
10
15
0.6
0.8
1
1.2
10
15
0.5
0
-0.5
0
5
time(sec)
(b)
1.2
Y-axis(m)
1
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
0
0.2
0.4
X-axis(m)
Y-axis Error(m)
X-axis Error(m)
-0.2
0.5
0
-0.5
0
Figure 7. Control scheme of the multi-robot system.
The initial positions and desired formation pattern of
robots are the same as simulation settings. The sampling
time of experiments is selected to be 100 ms. The
communication topology is the same digraph shown in
Fig. 4. In experimental validations, two scenarios are
considered, constant and time-varying leader velocities.
The assigned velocities are the ones given in simulation
cases. Since the position of each robot is determined
through certain image processing, the associated position
deviations can be viewed as uncertainties.
Experimental results are shown in Figs. 8-9, where
three formation strategies, CA, NFC, and ANFFC, are
considered. In Fig. 8, there exhibit significant formation
errors in both axes using the conventional CA, i.e. the
designated formation pattern of this four-robot system
cannot be achieved. Alternatively, either NFC or
ANFFC can effectively improve the formation responses.
Furthermore, the associated formation results are getting
worse using CA and NFC under the circumstance of
time-varying leader velocity, shown in Fig. 9. In
particular, comparing with the formation results of NFC
and ANFFC, it can be seen that the steady-state
responses of ANFFC is much better than that of NFC.
Ultimately, the proposed parameter tuning rules can
further improve the formation responses as expected.
The comparison of experiment results are shown in
Table 3, it can be seen that the ANFFC has better
performance than other methods.
Table 3. Performance comparison (experimentation).
0.8
0.2
355
5
time(sec)
10
15
0.6
0.8
1
1.2
10
15
0.5
0
-0.5
0
5
time(sec)
(c)
Figure 6. Simulation results of leader-follower formation
control (case C2): (a) CA [25], (b) NFC, (c) ANFFC.
CA
Performance
index
C1 C 2
NFC
C1
C2
ANFFC
C1
C2
ITSE
408.9 50.1 231.6 4.5 191.2 3.4
ITAE
326.4 182.3 246.6 73.7 220.9 73.4
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
1.4
1.4
1.2
1.2
1
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
0.2
0.4
X-axis(m)
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
0
-0.5
0
0.4
0.2
-0.2
1.2
0.5
2
4
time(sec)
6
8
F1
F2
F3
Leader
Leader
Followers
0.6
X-axis Error(m)
0
0.5
Y-axis Error(m)
X-axis Error(m)
0.2
-0.2
1
0.8
10
0
0.2
0.4
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
1.2
1.2
1
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
0.5
X-axis(m)
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
0.5
0
-0.5
0
0.2
-0.2
2
4
time(sec)
6
8
0
0.2
0.4
0.5
10
2
4
time(sec)
6
8
10
F1
F2
F3
Leader
Leader
Followers
0.6
0.4
2
4
time(sec)
6
8
10
6
8
10
0.6
0.8
1
1.2
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
0
-0.5
0
2
4
time(sec)
6
8
F1
F2
F3
Leader
Leader
Followers
0.4
0.2
-0.2
1.2
0.5
1
0.6
X-axis Error(m)
X-axis(m)
Y-axis Error(m)
X-axis Error(m)
-0.5
0
time(sec)
0.8
10
0
0.2
0.4
0.5
X-axis(m)
Y-axis Error(m)
0.8
Y-axis(m)
Y-axis(m)
1.2
0
4
(b)
1
0.5
X-axis(m)
0
-0.5
0
1.2
0.4
2
0.4
1.2
1.4
0.2
0
-0.5
0
F1
F2
F3
Leader
Leader
Followers
(b)
0
1.2
1
1.4
0.2
-0.2
1
0.6
X-axis Error(m)
0.4
Y-axis Error(m)
X-axis Error(m)
0.2
0.8
0.8
Y-axis Error(m)
0.8
0
0.6
0.5
(a)
1.4
Y-axis(m)
Y-axis(m)
(a)
1.4
0.2
-0.2
X-axis(m)
Y-axis Error(m)
0.8
Y-axis(m)
Y-axis(m)
356
0
-0.5
0
2
4
time(sec)
6
8
10
0.6
0.8
1
1.2
0.5
0
-0.5
0
2
4
time(sec)
6
8
10
(c)
(c)
Figure 8. Experimental results of leader-follower formation
control (constant leader velocity): (a) CA [25], (b) NFC, (c)
ANFFC.
Figure 9. Experimental results of leader-follower formation
control (time-varying leader velocity): (a) CA [25], (b) NFC,
(c) ANFFC.
6. Conclusion
Acknowledgment
This paper presents an adaptive neural fuzzy
formation control for multi-robot systems. A graph
theoretic digraph is used to describe the communication
topology between agents. The kinematic model of
wheeled robots is addressed with the consideration of
uncertainties. With the proposed ANFFC, the desired
leader-following formation of multiple robots can be
asymptotically achieved. The stability of the distributed
control system is guaranteed by the Lyapunov theorem.
Moreover, updating rules for ANFFC parameters are
derived. In addition to simulations, the effectiveness of
the designed ANFFC for multi-robot systems is validated
by experimental results. Compared to the conventional
consensus algorithm, the proposed ANFFC is superior in
performance and robustness.
This work is supported by the National Science
Council under Grant NSC 101-2221-E-182-041-MY2
and the High Speed Intelligent Communication Research
Center, Chang Gung University, Taiwan.
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Hung-Wei Lin received the B.S. degree
in electrical engineering from Feng Chia
University, Taichung, Taiwan, in 1997,
and the M.S. and Ph.D. degrees in
electrical engineering from National
Defense University, Chung Cheng
Institute of Technology, Tao-Yuan,
Taiwan, in 2000 and 2004, respectively.
He is currently an Assistant Professor
with the Department of Electrical Engineering, Lee-Ming
Institute of Technology, New Taipei City, Taiwan. His
research interests include digital signal processors and
358
International Journal of Fuzzy Systems, Vol. 15, No. 3, September 2013
embedded systems.
Wei-Shou Chan received the B.S.
degree in electrical engineering from
Lunghwa University of Science and
Technology, Tao-Yuan, Taiwan and the
M.S. degree in electrical engineering
from Chang Gung University, Tao-Yuan,
Taiwan, in 2006 and 2008, respectively.
He is working toward the Ph.D. degree in
the Department of Electrical Engineering
at Chang Gung University, Tao-Yuan, Taiwan, R.O.C. His
current research interests include intelligent systems, dynamic
surface control, and multi-robot systems.
Chia-Wen Chang received the B.S.
degree in electrical engineering from
National I-Lan Institute of Technology,
I-Lan, Taiwan, R.O.C., in 2003, and the
M.S. and Ph.D. degrees in electrical
engineering
from
Chang
Gung
University, Tao-Yuan, Taiwan, R.O.C.,
in 2005 and 2011, respectively. He is
currently an Assistant Professor with the
Department of Information and Telecommunication
Engineering, Ming Chuan University, Tao-Yuan, Taiwan,
R.O.C. His research interests include fuzzy sliding-mode
control, intelligent computation, and multi-robot systems.
Cheng-Yuan Yang received the B.S.
de- gree in electrical engineering from
National I-Lan Institute of Technology,
I-Lan, Taiwan, in 2001, and the M.S.
degrees in electrical engineering from
Chung Hua University, Hsin-Chu,
Taiwan, in 2007. He is working toward
the Ph.D. degree in the Department of
Electrical Engineering at Chang Gung
University, Tao-Yuan, Taiwan. His current research interests
are on the intelligent control systems including fuzzy
sliding-mode control systems and multi-robot systems.
Yeong-Hwa Chang received the B.S.
degree in electrical engineering from
Chung Cheng Institute of Technology,
Tao-Yuan, Taiwan, the M.S. degree in
control engineering from National Chiao
Tung University, Hsin-Chu, Taiwan, and
the Ph.D. degree in electrical
engineering from the University of
Texas at Austin, U.S.A., in 1982, 1987,
and 1995, respectively. He is currently a Professor with the
Department of Electrical Engineering, Chang Gung University,
Tao-Yuan, Taiwan. His research interests include intelligent
systems, multi-robot systems, haptic control systems, and
embedded control systems.