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788 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
Trajectory Tracking Control of Omnidirectional
Wheeled Mobile Manipulators: Robust Neural
Network-Based Sliding Mode Approach
Dong Xu, Dongbin Zhao, Member, IEEE, Jianqiang Yi, Member, IEEE, and Xiangmin Tan
Abstract—This paper addresses the robust trajectory tracking
problem for a redundantly actuated omnidirectional mobile ma-
nipulator in the presence of uncertainties and disturbances. The
development of control algorithms is based on sliding mode control
(SMC) technique. First, a dynamic model is derived based on
the practical omnidirectional mobile manipulator system. Then,
a SMC scheme, based on the fixed large upper boundedness of
the system dynamics (FLUBSMC), is designed to ensure trajec-
tory tracking of the closed-loop system. However, the FLUBSMC
scheme has inherent deficiency, which needs computing the up-
per boundedness of the system dynamics, and may cause high
noise amplification and high control cost, particularly for the
complex dynamics of the omnidirectional mobile manipulator
system. Therefore, a robust neural network (NN)-based sliding
mode controller (NNSMC), which uses an NN to identify the
unstructured system dynamics directly, is further proposed to
overcome the disadvantages of FLUBSMC and reduce the online
computing burden of conventional NN adaptive controllers. Using
learning ability of NN, NNSMC can coordinately control the
omnidirectional mobile platform and the mounted manipulator
with different dynamics effectively. The stability of the closed-loop
system, the convergence of the NN weight-updating process, and
the boundedness of the NN weight estimation errors are all strictly
guaranteed. Then, in order to accelerate the NN learning effi-
ciency, a partitioned NN structure is applied. Finally, simula-
tion examples are given to demonstrate the proposed NNSMC
approach can guarantee the whole system’s convergence to the
desired manifold with prescribed performance.
Index Terms—Omnidirectional mobile manipulators, robust
neural network (NN), sliding mode control (SMC), trajectory
tracking control, uncertainties.
I. INTRODUCTION
AMOBILE manipulator is ordinarily regarded as a manipu-
lator mounted on a moving platform. Due to the mobility
Manuscript received January 9, 2008; revised August 14, 2008. This
work was supported in part by the NSFC Projects under Grants 60475030,
60575047, 60621001, and 60874043, by the National 863 Program under
Grant 2007AA04Z239, and by the Joint Laboratory of Intelligent Sciences
and Technology under Grant JL0605 China. This paper was recommended by
Associate Editor S. X. Yang.
D. Xu is with Key Laboratory of Complex Systems and Intelligence Science,
Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China,
and also with the IC Processing Equipment R&D Center, Beijing Sevenstar
Electronics Co., Ltd., Beijing 100016, China (e-mail: dong.xu@ia.ac.cn).
D. Zhao, J. Yi, and X. Tan are with Key Laboratory of Complex Systems and
Intelligence Science, Institute of Automation, Chinese Academy of Sciences,
Beijing 100190, China (e-mail: dongbin.zhao@ia.ac.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSMCB.2008.2009464
of the platform, this system with motion redundancy can per-
form manipulations in a much larger workspace than a fixed
manipulator. Among many types of the mobile manipulators,
the omnidirectional mobile platform, which is considered as
holonomic parallel system, has 3 degrees of freedom (DOF)
in the horizontal motion plane. Therefore, it has more advan-
tages than the nonholomic differential-driven platform [1], [2]
which is provided with only two independent driving inputs.
Thus, it can fully use the null space motions in cabined work
environment and improve overall dynamic properties of the
manipulator [3]–[5].
Generally, the omnidirectional mobile platform has closed-
chain mechanism with redundant actuation. The redundant
actuation provides an effective means for eliminating singulari-
ties, and improving the performance such as Cartesian stiffness
and homogeneous output forces. At the same time, a major dif-
ficulty, which prevents vast control strategies from applying in
redundant actuated systems, is the lack of an efficient dynamic
model and control method. In order to solve the aforementioned
problems, there is a stringent need to derive an efficient and
correct dynamic model. Without dynamic control, it is difficult
to perform coordinated motion of the mobile platform and the
dynamically controlled manipulator.
There are many researches about dynamic modeling and
control of omnidirectional mobile manipulators with redundant
or nonredundant actuation [6]–[9]. The decentralized control
structure for cooperative tasks of multiple mobile manipulators
was addressed in [6]. Because of the complex mechanism of
the platform and intense coupling effect between the platform
and manipulator, Tan and Xi [7] regarded the platform as a
simplified motion point and proposed a unified model approach
for planning and control. In [8], three lateral orthogonal wheel
modules were used in the omnidirectional mobile manipulator
for dynamic modeling and motion control. However, the re-
searches on robust tracking control considering uncertainties
and disturbances are scarce. In recent years, several results
about dynamic modeling and control of holonomic systems
were published [10]–[12], which provided effective means for
such problems of omnidirectional mobile manipulators, but the
robust control schemes for these uncertain system dynamics are
not fully explored.
Along with the development of classical adaptive control
algorithms, using regressor matrix to design adaptive control
schemes has become popular. In such cases, a nonlinear dy-
namic system with unknown (or uncertain) system parameters
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XU et al.: TRAJECTORY TRACKING CONTROL OF OMNIDIRECTIONAL WHEELED MOBILE MANIPULATORS 789
is expressed by a product of a regressor matrix and an unknown
parameter vector. Li et al. [13] applied robust adaptive tech-
nique to holonomic system with uncertainties and got satisfied
performances. Learning control was presented in [14], but it
was applicable only when the task was repetitive. An adaptive
fuzzy approach was presented in [15], but the asymptotic
convergence performance depended on a strong assumption that
the approximation error plus external disturbance belonged to
L2∩L∞, and the tracking accuracy depended on the choice of
the number of the fuzzy sets. Therefore, these works mainly
consider the simple case that the dynamic models are assumed
to be perfect, and update laws are then employed to estimate the
unknown parameters, which are assumed to be constant or vary
slowly.
It is well known that the main advantage of using slid-
ing mode control (SMC) is strong robustness with respect
to system uncertainties and external disturbances. SMC is a
special discontinuous control technique applicable to various
practical systems [16]. By designing switch functions of state
variables or output variables to form sliding surfaces, SMC
under matching condition can guarantee that when tracking
trajectories reach the sliding surfaces, the switch functions keep
the trajectories on the surfaces, thus yielding desired system
dynamics. Therefore, it is attractive for many highly nonlinear
uncertain systems, such as the holonomic and nonholonomic
constrained mechanical systems [17], [18].
Neural network (NN), one of the most popular intelligent
computation approaches, has an inherent learning ability and
can approximate a nonlinear continuous function to arbitrary
accuracy. This feature is crucial in the controller design for
complex model identifying and unstructured uncertainties com-
pensating. Thus, in the past two decades, the development of
intelligent control, particularly NN control (NNC) in robotic
fields has attracted considerable interest. Polycarpou [19] pre-
sented a systematic methodology to identify a nonlinear system
usinganNN.Jinet al. [20] developed a rigorous theoretical
basis to design a stable NN-based controller for simple ma-
nipulator system. Lewis et al. [21] proposed an NNC scheme
that guaranteed the closed-loop performance in terms of small
tracking errors and bounded controls. An NN-based control
methodology was proposed for the joint space position control
of a mobile manipulator in [22], which was composed by a
linear control term (classical PID) and an NN compensation
term. Hu and Woo [23] proposed a control scheme which con-
sisted of an SMC and an NNC, and their contribution proportion
was determined by a fuzzy supervisory controller. Adaptive
control scheme of robot manipulators using Chebyshev NN
under actuator constraints was developed in [24]. An adaptive
wavelet NNs control scheme of mobile robots with unmodeled
dynamics and disturbances was addressed in [25].
However, there is little research work on the dynamic model
and robust control of redundantly actuated holonomic con-
strained mechanical systems with uncertainties, such as the
redundantly actuated omnidirectional wheeled mobile manip-
ulator whose dynamic property is similar to serial-parallel
manipulator in essence.
Hence, in this paper, we focus on a redundantly actuated
omnidirectional mobile manipulator with holonomic kinematic
Fig. 1. Mobile manipulator assembly.
constraints. The dynamic properties of the mechanical system
are used to form a conservative upper boundedness of the
system dynamics. Then, an SMC algorithm with this fixed large
upper boundedness (FLUBSMC) is derived to guarantee that
given trajectories can be tracked in the presence of structured
and/or unstructured uncertainties. Although using fixed large
boundedness can guarantee good performance, this control
scheme is conservative in essence and cannot be applied in
practical systems directly.
Therefore, a robust control scheme using an NN combined
with an SMC (NNSMC) is further proposed to solve the
aforementioned problem. SMC is designed to be robust to
disturbance with a guarantee of the stability of the system. NN
approximates the system dynamics directly and overcomes the
structured uncertainty by learning. Therefore, the control can
coordinately track the trajectory of the mobile platform and the
manipulator with different dynamics effectively. Furthermore,
no preliminary learning stage is required for the NN weights.
The control scheme is capable of disturbance rejection in
the presence of unknown bounded disturbances. The tracking
stability analysis of the closed-loop system is proved by the
Lyapunov theory. The convergence of the NN learning process
and the boundedness of the NN weight estimation errors are
all rigorously proven. In comparison to the traditional NNC,
NNSMC improves the robustness of the system.
The rest of this paper is organized as follows: The dynamic
model of the redundantly actuated omnidirectional mobile
manipulator is derived, and some preliminaries are given in
Section II. Section III presents some remarks and the design
process of FLUBSMC. Then, in order to overcome the disad-
vantages of FLUBSMC, NNSMC is further proposed, and its
stability is strictly proved, and then the partitioned NN structure
is adopted to reduce the computing burden of the integrated NN
in Section IV. In Section V, an illustrative example is applied to
validate the dynamic model, the FLUBSMC and the NNSMC
algorithms. Finally, conclusions are given in Section VI.
II. SYSTEM DESCRIPTIONS
The mobile manipulator considered here moves on the hor-
izontal plane which is in the global frame OXY. It is cylinder-
shaped; three identical castor wheels are axed symmetrically
on the bottom of the platform, and a two-links manipulator
locates on the gravity center C(which coordinate is defined
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790 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
Fig. 2. Side view of wheel module i.
as [xyθ]T) of the platform, as shown in Fig. 1. The angle
between two neighboring wheels is 2π/3. The wheels 1, 2,
and 3 are assigned clockwise. l1and l2are the lengths of the
two links, and θ1and θ2are the angle displacements of those
two links, respectively.
The side view is shown in Fig. 2. The radius of each wheel
is r, and the horizontal distance between Cand the center Fi
of the vertical axis of each wheel is D. The offset dis the
horizontal distance between the center of each wheel Oiand
Fi. The angle displacements for wheel rolling and steering are
ϕiand ηi, respectively.
A. Kinematic Model
According to the aforementioned description, we first define
the following state variables for easy reference:
q=qT
1qT
2T
=[xyθθ
1θ2ϕ1η1ϕ2η2ϕ3η3]T
q1=[xyθθ
1θ2]T
q2=[ϕ1η1ϕ2η2ϕ3η3]T
ζ=[θ1θ2ϕ1η1ϕ2η2ϕ3η3]T
qGeneralized coordinates of the mobile
manipulator.
q1Pose of the mobile manipulator.
q2Drive variables of the platform.
ζDrive variables of the mobile
manipulator.
0m×nm×nzero matrix.
In×nn×nidentity matrix.
λmin(∗)and λmax (∗)Minimum and maximum eigenvalues
of (∗).
Thus, the whole mobile manipulator’s kinematic model can
be described as
˙
ζ=Jq1(β,η)˙q1(1)
Jq1(β,η)=⎡
⎢
⎣
02×3I2
J102×2
J202×2
J302×2
⎤
⎥
⎦(2)
where
Ji=−1
rcos βi−1
rsin βi−D
rsin ηi
1
dsin βi−1
dcos βi−D
dcos ηi−1
,(i=1,2,3).
Property 1: For Jq1(β, η),ifδ=(d/D)≤(1/2), rank
Jq1(β,η)=5holds.
Proof: Calculate det(JT
q1(β,η)Jq1(β,η)) as follows:
det JT
q1(β,η)Jq1(β,η)
=D6
r9d927 + 18δ(cos η1+cosη2+cosη3)+18δ2−6δ2
×cos η1−η2+2π
3+cosη2−η3+2π
3
+cosη3−η1+2π
3.
Only when the expressions (cos η1+cosη2+cosη3)=−3
and cos(η1−η2+(2π/3))+ cos(η2−η3+(2π/3))+ cos(η3−
η1+(2π/3)) = 3, the expression det(JT
q1(β,η)Jq1(β,η))
can achieve its minimum. Thus, det(JT
q1(β,η)Jq1(β,η)) ≥
(D6/r9d9)(9 −18δ)can be guaranteed. Therefore, rank
Jq1(β,η)=5can be always held, if δ≤1/2.
B. Dynamic Model
In most of the dynamic model researches about the omni-
directional mobile manipulator systems, the redundantly actu-
ated property of the platform was not emphasized. Moreover,
the integrated dynamic model of the omnidirectional mobile
manipulator which was derived from the driving wheels was
not explicitly addressed. According to Lagrange theory [26],
based on the kinematic model, the dynamic equations of the
mechanical system with external disturbances can be derived as
M(q)¨q+C(q, q)˙q+G(q)+d(t)=B(q)u+AT(q)λ(3)
where u∈Rnis the vector of generalized input torques,
M(q)∈Rn×nis a symmetric bounded positive definite inertia
matrix, C(q, ˙q)˙q∈Rnrepresents the vector of centripetal and
Coriolis torques, G(q)∈Rnis the gravitational torque vector,
d(t)∈Rndenotes the external disturbances, B(q)∈Rn×nis
a full rank input transformation matrix and is assumed to be
known because it is the function of fixed geometry of the
system, A(q)∈Rm×nis the kinematic constraints matrix, and
λ∈Rmis a constraint force vector. The details of the dynamic
model description are addressed in appendix.
The kinematic constraints are considered independent of
time and can be expressed as
A(q)˙q=0 (4)
where the effect of the kinematic constraints can be viewed as
restricting the dynamics on a constraint manifold as
Ω={(q, ˙q)∈Rn×Rn:A(q)˙q=0}.(5)
The presence of mkinematic constraints causes the system
to lose mDOF; hence, the system only has n−mDOF left.
Based on the kinematic model, we know q1(t)∈Rn−mis the
vector of the independent generalized coordinates of the mobile
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XU et al.: TRAJECTORY TRACKING CONTROL OF OMNIDIRECTIONAL WHEELED MOBILE MANIPULATORS 791
manipulator system, and q2(t)∈Rmis the vector of driving
variables of the platform.
Property 2: Define H=[I303×2]and R(q)=H
Jq1.
Based on Property 1, the matrix RT(q)B(q)is of full rank, so
the Moore-Penrose inverse matrix of R(q)always exists.
In order to reduce complexity of the whole system, applying
the projection of the dynamic model into the Null space of the
constraints, we can obtain
˙q=R(q)˙q1
¯
M(q)¨q1+¯
C(q, ˙q)˙q1+¯
G(q)+ ¯
d=¯
Eu (6)
where ¯
M(q)=RT(q)M(q)R(q),¯
C(q, ˙q)=RT(q)C1(q, ˙q),
C1(q, ˙q)=M(q)˙
R(q)+C(q, ˙q)R(q),¯
G(q)=RT(q)G(q),
¯
d=RT(q)d,¯
E(q)=RT(q)B(q). Equations (4) and (5) lead
to A(q)R(q)˙q1=0. In other words, A(q)R(q)=0, since
A(q)R(q)is of full columns and the configuration of the whole
system has n−mDOF. For (6), some fundamental properties
are addressed below.
Property 3: The matrix ¯
M(q)=RT(q)M(q)R(q)is sym-
metric and positive definite.
Property 4: ˙
¯
M(q)−2¯
C(q, ˙q)is skew symmetric, i.e.,
ξT(˙
¯
M(q)−2¯
C(q, ˙q))ξ=0,∀ξ∈Rn−m.
Property 5: There exist positive scalars βi(i=1,...,5)
such that ∀q∈Rn,∀˙q∈Rn:M(q)≤β1<∞,C(q, ˙q)≤
β2+β3˙q,G(q)≤β4and supt≥0d(t)≤β5.
It should be noted that the whole system (6) consists of a new
dynamic model together with a pure kinematic relationship.
This structure makes it possible to design the SMC law.
III. FL UB SM C DESIGN
This section considers the trajectory tracking problem of
the redundantly actuated mobile manipulator system discussed
above. Based on the fixed large upper boundedness of the
system dynamics, an SMC scheme, (FLUBSMC), is de-
signed to ensure trajectory tracking of the closed-loop system.
FLUBSMC consists of two components: fixed large upper
boundedness and SMC. In order to develop FLUBSMC, the
following assumptions are required throughout this section.
Assumption 1: The vectors q1,˙q1are bounded and uniformly
continuous, and have bounded and uniformly continuous deriv-
atives up to the second order. Moreover, the matrices R(q)
and ˙
R(q)are also bounded as R(q)≤β6,˙
R(q)≤β7˙q,
where βi(i=6,7) are positive constants.
Remark 1: Property 1 and the structure of R(q)can guaran-
tee all n−mDOF are independently actuated, which always
holds for redundantly actuated holonomic mechanical systems
such as parallel manipulator and so on.
Remark 2: According to Li et al.’s [13] claim, if q1(t)is
bounded, then R(q)is bounded. If ˙q1(t)is bounded, then ˙
R(q)
is bounded. The purpose of Assumption 1 is to try to make full
use of the available knowledge to reduce control gains.
A trajectory tracking control objective for perturbed mo-
bile manipulator system is formulated as: on the basis of the
vector q1and ˙q1, given a desired q1dand ˙q1d, develop a
controller such that for any (q(0),˙q(0)) ∈Ωas in (5), q1(t)
and ˙q1(t)can asymptotically converge to a manifold Ωd
specified as:
Ωd={(q, ˙q)|q1(t)= q1d(t),˙q1(t)= ˙q1d(t),q(t)= R(q)q1(t)}
(7)
where the vector q1(t)can be considered as n−m“output
equations” of the system.
Assumption 2: The desired reference trajectory q1dis as-
sumed to be bounded and uniformly continuous, and has
bounded and uniformly continuous derivatives up to the sec-
ond order. Therefore, there exists a constant q1B, such that
qT
1d˙qT
1d¨qT
1d≤q1Balways holds.
In the following, some variables are defined as:
e=q1−q1d(8)
˙q1r=˙q1d−Λe(9)
where eand q1rdenote the tracking error and a set of auxiliary
signals, respectively. Λis a positive definite matrix which
eigenvalues are strictly in the right-hand of complex plane.
Then, a sliding variable is defined as
s=˙q1−˙q1r=˙e+Λe. (10)
Remark 3: From Assumptions 1 and 2, there exist positive
scalars βi(i=8,9) such that ∀˙q1r∈Rn−m,∀¨q1r∈Rn−m:
˙q1r≤β8+Λe,¨q1r≤β9+Λ˙e.
When the sliding surface s=0, according to the theory of
SMC, the sliding mode is governed by the following differential
equation:
˙e=−Λe. (11)
Obviously, the behavior of the system on the sliding surface is
determined by the structure of the matrix Λ. In other words,
when s=0, the tracking error transient response is completely
governed by the aforementioned equation.
Based on (3) and the first equation of (6), the closed-loop
system in terms of the sliding variable scan be obtained by
M(q)R(q)˙s=B(q)u−(M(q)R(q)¨q1r+C1(q, ˙q)˙q1r
+G(q)+d(t)) −C1(q, ˙q)s+AT(q)λ. (12)
Multiplying left RT(q)and using Property 3 lead to
¯
M(q)˙s=¯
E(q)u−¯
H(¨q1r,˙q1r,˙q, q, t)−¯
C(q, ˙q)s
(13)
¯
H(¨q1r,˙q1r,˙q, q, t)=(¨q1r,˙q1r,˙q, q)+RT(q)d(t)(14)
H(¨q1r,˙q1r,˙q, q)=RT(q)(M(q)R(q)¨q1r
+C1(q, ˙q)˙q1r+G(q))
=RT(q)(M(q)R(q)(¨q1d−Λ˙e)
+C1(q, ˙q)( ˙q1d−Λe)+G(q))
=H(x)(15)
where x=[¨qT
1d˙qT
1dqT
1d˙qTqT]T∈R3×(n−m)+2×n.
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792 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
Fig. 3. FLUBSMC scheme.
Because the omnidirectional mobile manipulator moves in
the horizontal plane, RT(q)G(q)≡0.
Before we design control law, following Vicente et al.’s [27]
work, we have the result that the expression ¯
H(¨q1r,˙q1r,˙q, q, t)
in (13) is upper bounded; and the boundedness is a state-
dependent function. Therefore, according to Property 4, the
following can be reached:
¯
H(¨q1r,˙q1r,˙q, q, t)
≤¯
β1·¨q1r+¯
β2˙q1r
+¯
β3˙q1r·˙q+¯
β4(16)
where ¯
β1=β2
6β1,¯
β2=β2β2
6,¯
β3=β1β6β7+β2
6β3,¯
β4=β6β5.
Here, we denote ρ(t)= ¯
β1·¨q1r+¯
β2˙q1r+¯
β3˙q1r·
˙q+¯
β4and call ρ(t)as the upper bounded function, which
is obviously a function of the state variables.
Remark 4: From the aforementioned description, we know
the upper boundedness of the system dynamics is a positive
definite second-order polynomial. What we are concerned with
is only the boundedness of ¯
H(¨q1r,˙q1r,˙q, q, t), thus the mea-
surement of acceleration signals is not required. This forms the
foundation of FLUBSMC.
Assuming that the parameters ¯
βi(i=1,2,3,4) of the upper
boundedness of the system dynamics are known, we design the
FLUBSMC law according to [28]
¯
E(q)u=−Ks −Kssgn(s)−s·(ρ(t))2
s·ρ(t)+γ(t)(17)
where Kand Ksare (n−m)×(n−m)positive definite
gain matrixes determined by the designer, ρ(t)is the upper
boundedness of the system, γ(t)>0is a time varying function
and satisfies t
0γ(ν)dν =a<∞.
From (13) and (17), by selecting a large gain Ks, it is easy to
obtain:
˙sTs≤−σ|s|(18)
where σis a positive constant. The aforementioned expression
shows that any trajectories away from the sliding surface are
forced to reach the sliding surface in finite time.
The whole FLUBSMC scheme shown in Fig. 3 consists of
the controller design component and the actual mobile ma-
nipulator system component. The Tracking Loop provides the
feedback signals for the controller designing. Correspondingly,
the first two terms of (17) is a sliding mode controller, the third
term of (17) is the Fixed Large Upper Boundedness term.
Remark 5: From [28], consider the uncertain dynamic sys-
tem (6), the FLUBSMC law is chosen as (17). Then, for
any (q(0),˙q(0)) ∈Ω, the tracking error eand its derivative ˙e
converge to the sliding surface and are restricted to the surface
for all future time. Furthermore, when t→∞,eand ˙easymp-
totically converge to zero.
IV. NNSMC DESIGN
Using fixed large boundedness can guarantee good perfor-
mance, but it is not recommended in practice as large bounded-
ness implies high noise amplification and high control cost. In
addition, due to the different dynamic properties of the mobile
platform and the manipulator, and highly coupled dynamics
between them, the FLUBSMC law (17) which considers the
two different dynamic subsystems as a whole system will affect
the manipulator’s driving outputs obviously. Furthermore, this
boundedness cannot be easily obtained in practice. In order to
solve the aforementioned problems, a more effective control
method NNSMC is first proposed in this section to coordinately
control the mobile platform and the mounted manipulator with
different dynamics. Because NN has ability to approximate
a nonlinear continuous function to arbitrary accuracy, NN in
NNSMC is directly used to identify the uncertainty vector of
the system dynamics, not to compute the Kronecker product,
so the weight matrixes structures are simplified to accelerate
the tuning speed. Finally, in order to accelerate the NN learning
efficiency and reduce the computing burden, a partitioned NN
structure is applied.
During the development of NNSMC, we assume that M(q),
C1(q, ˙q), and G(q)are completely unknown, then an NN ap-
proximator is derived to approximate the integrated uncertain
term H(x)in (15).
A. Robust NNSMC Scheme Design
Fig. 4 shows the structure of three layers radial basis function
NN (RBFNN) adopted in NNSMC, which includes an input
layer, a hidden layer, and an output layer. The relationships
among these layers can be described as follows.
Let
W=⎡
⎢
⎢
⎣
w11,w
21,...,w
h1
w12,w
22,...,w
h2
.
.
..
.
..
.
.
w1k,w
2k,...,w
hk
⎤
⎥
⎥
⎦
,φ=⎡
⎢
⎢
⎣
φ1
φ2
.
.
.
φh
⎤
⎥
⎥
⎦
,Y =⎡
⎢
⎢
⎣
y1
y2
.
.
.
yk
⎤
⎥
⎥
⎦
.(19)
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XU et al.: TRAJECTORY TRACKING CONTROL OF OMNIDIRECTIONAL WHEELED MOBILE MANIPULATORS 793
Fig. 4. RBFNN.
In this way, the relationship of the output layer can be
written as
Y=WTφ(20)
where Wis the weight matrix of the RBFNN, φis the excitation
function vector.
For any given real continuous functions on a compact set U,
define a smooth function f(·):U→Rn. If we choose f(x)∈
Cr(U), where Cr(U)is the space of the continuous functions,
there exists an RBFNN in the form of (20) such that
f(x)=WTφ(x)+ε(21)
ε<ε
N(22)
where εNis a constant.
This property makes it possible for applying RBFNN to
solve almost any nonlinear modeling problems. Because of the
universal approximation ability of RBFNN, H(x)defined in
(15) can be identified using RBFNN with enough number of
hidden layer neurons such that
H(x)=WTφ(x)+ε(23)
where W∈Rh×kis assumed to be constant and bounded by
W≤WB(24)
where WBis one known positive constant. The basis func-
tion vector φ(x)is usually chosen as the Gaussian functions
defined in [24].
The estimate of the uncertain term H(x)is expressed as
ˆ
H(x)= ˆ
WTφ(x)(25)
where ˆ
Wis the tuning parameter matrix of the network and is
adjusted in the learning process.
Assumption 3: In the RBFNN, the input vector x[defined
in (15)] is used to identify the uncertain term H(x). There
exist positive constants c0and c1, such that the following ex-
pression holds:
x(t)≤qB+c0s(0)+c1s(t)(26)
where sis defined in (10). Assume that (23) holds, according
to the definitions in (10) and (11), we know that all xis in
the compact set Ωx≡{x|x<b
x}, where bx>q
B. Define
the compact set as Ωs≡{s|s<(bx−qB)/(c0+c1)}with
s(0) ∈Ωs. Thus, the RBFNN approximation property always
holds for all sin the compact set Ωs.
Consider the uncertain dynamic system (6) with plant un-
certainties and external disturbance. Using RBFNN to identify
the uncertain term H(x). For any (q(0),˙q(0)) ∈Ωand desired
trajectory q1d(t), let the NNSMC law be given by
¯
E(q)u=−Ks −Kssgn(s)+ ˆ
WTφ(x)(27)
˙
ˆ
W=−αφsT−μαsˆ
W(28)
where Kand Ksare the same as (6) and (17), αis a positive
constant representing the learning rate of the network, μis a
small positive design constant.
From Fig. 5, we can see that the NNSMC scheme is different
from the FLUBSMC scheme by adopting NN to identify the
uncertain term of system dynamics directly.
Theorem 1: For the uncertain dynamic system (6), there
exist appropriate parameters in the NNSMC law (27), and the
following performance can be achieved.
1) Because the matrix RT(q)B(q)is of full rank, the input
torque u(t)is bounded for all t>0.
2) With suitable positive gain constants Kand Ks,the
tracking error eof the uncertain dynamic system will be
uniformly ultimately bounded.
3) With sufficient large Ks, the tracking errors eand ˙ewill
asymptotically converge to zero.
Proof:
1) Since all the variables in the right side of (27) are
bounded, according to Property 2 and Assumption 1, it
is easy to conclude that u(t)is bounded for all t>0.
2) Consider a Lyapunov candidate function as [22]
V=1
2sT¯
Ms+1
2αtr{˜
WT˜
W}(29)
where ˜
W=W−ˆ
W. By substituting (27) and (23) into
(13), the time derivative of Vleads to
˙
V=1
2sT˙
¯
Ms+sT¯
M˙s+1
αtr{˜
WT˙
˜
W}
=1
2sT˙
¯
Ms+sT−Ks −Kssgn(s)+ ˆ
WTφ(x)−WTφ(x)
−ε(x)−RT(q)d(t)−¯
C(q, ˙q)s
+1
αtr{˜
WT˙
˜
W}
=sT−Ks −Kssgn(s)+ ˆ
WTφ(x)−WTφ(x)
−ε(x)−RT(q)d(t)+1
αtr{˜
WT˙
˜
W}
=sT−Ks −Kssgn(s)−˜
WTφ(x)−ε(x)
−RT(q)d(t)+1
αtr{˜
WT˙
˜
W}
=sT−Ks −Kssgn(s)−˜
WTφ(x)−ε(x)
−RT(q)d(t)−1
αtr{˜
WT˙
ˆ
W}
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794 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
Fig. 5. NNSMC scheme.
=sT−Ks −Kssgn(s)−˜
WTφ(x)−ε(x)
−RT(q)d(t)−1
αtr ˜
WT−αφsT−μαsˆ
W
=sT−Ks −Kssgn(s)−ε(x)−RT(q)d(t)
+μstr ˜
WTˆ
W).(30)
According to the matrix trace, inner product, and Frobenius
norm theory, the following property holds [21]:
tr{˜
WTˆ
W}=tr˜
WT(W−˜
W)
=˜
W,WF−˜
W2
F
≤˜
WFWF−˜
W2
F.(31)
Thus
˙
V≤sT−Ks −Kssgn(s)−ε(x)−RT(q)d(t)
+μs˜
WFWF−˜
W2
F
≤−λmin(K)s2−Kss+εNs+β6β5s
+μs˜
WFWF−˜
W2
F
≤−λmin(K)s2−Kss+εNs+β6β5s
+μs˜
WFWB−˜
W2
F
=−sλmin(K)s+Ks−εN−β6β5
+μ˜
W2
F−˜
WFWB
=−sλmin(K)s+Ks−εN−β6β5
+μ˜
WF−WB
22
−μW2
B
4.(32)
From the aforementioned proving process, in order to ensure
the approximation effect of RBFNN estimator, we know that if
the following expression (33) holds, the sliding variable should
be always constrained in the compact set Ωs:
s>εN+β6β5+μW2
B
4−Ks
λmin(K)≡bs.(33)
This can be achieved by choosing suitable positive constants
to construct the gain matrix Kto satisfy
λmin(K)>εN+β6β5+μW2
B
4−Ks(c0+c1)
bx−qB
.(34)
Therefore, the compact set defined by s>b
sis con-
strained by Ωs. As a result, the RBFNN approximation property
always holds.
Thus, ˙
Vis guaranteed to be negative outside a compact set.
Based on the standard Lyapunov theory and LaSalle theory, this
property ensures the uniform ultimate boundedness of the slid-
ing variable s, the NN weights ˜
W, and the weight estimates ˆ
W.
Moreover, the norm of the sliding variable scan be made
arbitrarily small by increasing the gain matrix K.
3) Furthermore, we choose a sufficient large Kssuch that
εN+β6β5+μW2
B
4−Ks
λmin(K)≤0.(35)
Since s≥0and λmin(K)>0, we know
˙
V≤−λmin(K)s2≤0.(36)
Define
VR=V(t)−
t
0˙
V(ν)+λmin(K)s(v)2dν. (37)
Its time derivatives up to the second order are
˙
VR=˙
V(t)−˙
V(t)−λmin(K)s2
=−λmin (K)s2(38)
¨
VR=−2λmin (K)sT˙s. (39)
We can conclude that |¨
VR|is bounded and ˙
VRis continuous.
We know that V(t)is bounded from (29) and the second
term of VRis a finite integral. Thus, VRis bounded. In ad-
dition, t
0˙
VR(ν)dν =VR(t)−VR(0) is also bounded. Then,
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XU et al.: TRAJECTORY TRACKING CONTROL OF OMNIDIRECTIONAL WHEELED MOBILE MANIPULATORS 795
according to Barbalat’s lemma, |˙
VR|=λmin(K)s2→0,
when t→∞.
According to the aforementioned proof, the sliding variable
s→0, when t→∞. Therefore, based on Remark 5, the track-
ing error eand ˙easymptotically converge to zero as t→∞.
Remark 6: The NNSMC law (27) consists of two com-
ponents. The first component [i.e., −Ks −Kssgn(s)]isthe
sliding mode controller which is used to guarantee the stability
of the system and achieve the uniformly ultimately bounded
performance, and the second component [i.e., ˆ
WTφ(x)]isthe
adaptive RBFNN to approximate the uncertain term. Hence,
in practice, the NNSMC law is a hybrid adaptive robust
controller.
Remark 7: Most of the conventional adaptive control
schemes consider the uncertain dynamics of the holonomic
mechanical systems as a product of a regressor matrix and a
basis vector. The conventional adaptive control schemes can
be employed only to deal with parameter uncertain systems in
which the unknown parameters must be assumed to be constant
or vary slowly. It is obvious that NNSMC resembles the con-
ventional adaptive method to a great extent. Hence, the analysis
and design developed in this paper can also be directly applied
to the holonomic constrained mechanical systems with linear
parameterization property. Therefore, in essence, the NNSMC
law (−Ks −Kssgn(s)+ ˆ
WTφ(x)) can be categorized into the
indirect adaptive NN controller because NNSMC uses NN to
identify the unknown dynamics of the model.
Remark 8: The proposed scheme is nonregressor based and
requires no information about the system dynamics. We use NN
to estimate the uncertain term H(x)and avoid using fixed large
boundedness like (17) to guarantee good performance, because
large boundedness implies high noise amplification and high
control cost.
Remark 9: For the NNSMC law (27), once the control
parameters such as K,Ksare determined, the input torque u(t)
is bounded for all t>0. If we choose a sufficient large Ks,
we can ensure the asymptotical stability, but u(t)may be
larger than the maximum output torque of driving motors. This
situation is prohibited in practice and need to be considered first
in parameters choosing procedure. In addition, large Ksalso
causes severe chattering. Therefore, when we design a suitable
controller, there is a critical tradeoff between the actual input
torque and the tracking performance.
B. Partitioned NNs
Given a holonomic constrained mechanical system, it is
desired to select a set of basis functions and determine the NN
reconstruction error bound so that (23) holds; that is, determine
a basis function vector φ(x), so that the uncertain term defined
in (15) can be expressed by (23). However, when we face a
complex mechanical system like the omnidirectional mobile
manipulator studied in this paper, the inputs and outputs di-
mensions of NN are large, the computing burden of the learning
process is big. These two problems above severely prohibit the
application of a single NN in practical system.
From [30], we know that the NN controller allows one
to design partitioned NNs to achieve the same performance
Fig. 6. Partitioned NN structure.
as a single NN. Furthermore, using partitioned structure has
following advantages:
1) simplifying the structure design and parameters choosing,
so the suitable basis functions for each individual parti-
tioned subsystem can be separately determined;
2) making the network weights tuning procedure faster,
so the parameters in each subsystem can be tuned
respectively.
For the omnidirectional mobile manipulator system, there is
a unique function ϑ:Rn−m→Rm, such that the holonomic
constraint depicted in Section III can always be expressed
explicitly as [29]
q2=ϑ(q1).(40)
Then, we can easily obtain that the uncertain H(x)defined
in (15) can be rewritten as the following form:
H(¯x)=RT(q1)M(q1)R(q1)¨q1r+RT(q1)C1(q1,˙q1)˙q1r(41)
where ¯x=[¨qT
1r˙qT
1r˙qT
1qT
1]T∈R4×(n−m).
From (41), there are two separated RT(q)M(q)R(q)¨q1rand
RT(q)C1(q, ˙q)˙q1rin the H(¯x). Therefore, we can apply two
separated NNs to identify them, respectively. Using the similar
analysis method as [30] and [31], we can derive that
ˆ
H(¯x)=ˆ
WT
Mˆ
WT
CφM
φC
=ˆ
WT
MφM+ˆ
WT
CφC.(42)
From the partitioned NN structure (42) shown in Fig. 6, it
is easy to know that the partitioned decoupled NNs can be
tuned, respectively. Then, the learning algorithm (28) can be
replaced by
˙
ˆ
WM=−αMφMsT−μMαMsˆ
WM(43)
˙
ˆ
WC=−αCφCsT−μCαCsˆ
WC.(44)
According to the theorems in [30], we can conclude that the
performance of the partitioned NN structure is equivalent to the
single NN.
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796 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
TAB L E I
PARAMETERS FOR THE MOBILE MANIPULATOR
Remark 10: In essence, the uncertain term H(x)that we
pay attention to is a vector, not a matrix. Using the partitioned
structure is to simplify the NN design process and accelerate
the tuning speed. The partitioned NNs in [30] and partitioned
fuzzy logic systems in [31] require Kronecker product to con-
struct new basis function for identifying the uncertain matrix
[RT(q)M(q)R(q)]. Compared to directly using NN to identify
the uncertain integrated vector (RT(q)M(q)R(q)¨q1r),employ-
ing Kronecker product to estimate the whole uncertain matrix
increases the computing burden of tuning weight matrixes
obviously, particularly for the complex mechanical systems.
Thus, their method only partitions the structure of the NN ap-
proximator, but does not simplify the tuning procedure. In this
paper, we employ two separated NNs to directly approximate
two decoupled RT(q)M(q)R(q)¨q1r, and RT(q)C1(q, ˙q)˙q1rof
the uncertain term H(x). This method does not need Kronecker
product; therefore, the weight matrixes structures are more
compact, and tuning speed is faster.
V. S IMULATION RESULTS
As an example to verify the proposed approach, this section
discusses the simulation of the dynamic model and trajectory
tracking controller. The physical parameters for the omnidirec-
tional mobile manipulator system are shown in Table I, where
m0,m1, and m2are the masses of the platform, link 1, and
link 2, respectively. I0,I1, and I2are the inertias of the
platform, link1, and link2, respectively. Iwb and Ibw are the
rolling and steering inertias for the wheel module.
Let the desired output trajectory q1dbe
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
xd=2cos(t/2)
yd=2sin(t/2)
θd=2sin(t/2)
θ1d=sin(t)
θ2d=cos(t).
(45)
Assume that the disturbances to the system is d=
[20 sin(t)20cos(t)20cos(t)3sin(t)2cos(t)0
1×6]T.
Then, an NNSMC is designed to track the desired trajectory
of the omnidirectional mobile manipulator system. Based on
the analysis of Section V, a partitioned NN structure is em-
ployed to learn the behavior of H(x).
First, we design an NN to identify the uncertain term
RT(q)M(q)R(q)¨q1r. This term relates to the exactly known
position measurement and the variable ¨q1r. Choose the basis
function defined in [26] as
φMj=exp!−xM−cj2
2σ2
j",(j=1,2,...,5) (46)
Fig. 7. Trajectory of the mobile platform.
Fig. 8. Position errors of Xand Y.
where input state vector xM=q1
¨q1r, the center vector is
[c1,c
2,...,c
5]T=[−3,−1.5,0,1.5,3]T, and the width vector
is [σ1,σ
2,...,σ
5]T=[3,3,3,3,3]T.ˆ
WMis a 5 ×11 matrix
and all elements are initialized to zero.
Second, we design another NN to approximate the uncertain
term RT(q)C1(q, ˙q)˙q1r. This term depends on the exactly
known position measurement, velocity measurement, and the
variable ˙q1r. The basis function is characterized by the follow-
ing parameters:
φCj=exp!−xC−cj2
2σ2
j",(j=1,2,...,7) (47)
where input state vector xC=⎡
⎣
q1
˙q1
˙q1r⎤
⎦, the center vector
is [c1,c
2,...,c
7]T=[−3,−2,−1,0,1,2,3]T, and the width
vector is [σ1,σ
2,...,σ
7]T=[2,2,2,2,2,2,2]T.ˆ
WCis a
7×11 matrix, and all elements are initialized to zero.
Now, the estimate of the uncertain term H(x)is concretely
described by (41). Then, the NNSMC law is modified as
¯
E(q)u=−Ks +ˆ
WT
MφM+ˆ
WT
CφC−Kssgn(s).(48)
For the convenience of simulations, choose the control
parameters as Λ = diag[2,2,2,2,2],αM= 100,μM=0.1,
αC= 200,μC=0.1,K= diag[300,300,300,20,20], and
Ks= diag[10,10,10,5,5],ρ(t)= 80,γ(t)= e−t, respectively.
The proposed NNSMC is applied to the omnidirectional
mobile manipulator system and the simulation results are
shown in Figs. 7–9. The actual computed trajectory of the
platform is shown in Fig. 7. Position tracking errors are shown
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XU et al.: TRAJECTORY TRACKING CONTROL OF OMNIDIRECTIONAL WHEELED MOBILE MANIPULATORS 797
Fig. 9. Position errors of θ,θ1,andθ2.
Fig. 10. Output torques of rolling motor.
Fig. 11. Output torques of steering motor.
in Figs. 8 and 9. It can be seen that all signals are bounded. From
the aforementioned explanations, it is clear that the system
converges to the desired trajectory quickly and achieves good
tracking performance. Therefore, the proposed control scheme
is robust to both the plant uncertainty and time varying external
disturbance.
By using the FLUBSMC law and the NNSMC law, the
torques exerted on the omnidirectional mobile manipulator
system are given in Figs. 10–13. In order to show the output
torques of the mobile platform clearly, we only choose the
outputs of the wheel module 1 in Figs. 10 and 11, because the
outputs of the other two wheel modules are similar to the wheel
module 1.
Remark 11: It is well known that the mobile manipula-
tor system has two subsystems: the mobile platform and the
mounted manipulator. Usually, these two subsystems have en-
tirely opposite dynamic properties: the mobile platform has
large mass and inertia, and responses slowly; the manipula-
tor has small mass and inertia, and responses quickly. The
FLUBSMC law considers these two subsystems as an inte-
grated entity and ignores the difference of the dynamic proper-
Fig. 12. Comparison of output torques with link 1 motor.
Fig. 13. Comparison of output torques with link 2 motor.
ties. On the contrary, the NNSMC law employs NN to identify
the uncertain term, i.e., identifying these two subsystems, re-
spectively. From Figs. 10–13, we can clearly see that the driving
torques of the mobile platform in FLUBSMC and NNSMC
are similar. However, the driving toques of the manipulator
in FLUBSMC are much larger than those in NNSMC. It is
not acceptable in practice as large boundedness introduces
high noise amplification and high control cost. In addition,
much large output driving torques cannot be obtained easily in
practical systems. These results adequately illuminate that the
FLUBSMC law not only increases the output torques, but also
exerts different influence on the mobile platform and the ma-
nipulator. Therefore, compared to NNSMC, it is conservative
in essence and cannot be applied in practical systems directly.
From Figs. 12 and 13, by using NNSMC scheme, the output
torques of the motor at joint 1 is reduced up to 80%, and the
output torques of the motor at joint 2 is reduced up to 90%.
Remark 12: The values of the center vector and the width
vector in the Gaussian functions heavily influence the smooth-
ness of the output surface of RBFNN. How to choose the values
is an optimization problem, which is beyond the scope of this
paper. By lots of experiments, for this problem, we conclude
that the larger the distance between the center elements and the
sharper the functions are, the less smooth the output surface
is. Therefore, during the NN structure design, these parameters
are selected to make the functions smooth so that our proposed
control output can also be smooth. In addition, because the
SMC in (27) can efficiently compensate the approximation
error ε, we can choose finite number basis functions to achieve
good tracking performance.
Remark 13: SMC guarantees its robustness in the presence
of large disturbance, but the chattering effect is the inherent
deficiency. This phenomenon is clearly shown in Figs. 12
and 13. Therefore, if we need smooth control output in practice
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798 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 3, JUNE 2009
application, and the tracking effect is satisfying, the NNSMC
law (27) can be modified as
¯
E(q)u=−Ks +ˆ
WTφ(x).(49)
This is a conventional NN adaptive controller. When the extern
disturbances are small or equal to zero, the trajectory tracking
performance can be ensured.
Remark 14: NNSMC can take advantage of the strong
robustness performance of SMC, and nonlinear mapping prop-
erties and the self-learning ability of NN to deal with the
unstructured uncertainties. Moreover, NN can alleviate the
chattering to some extent and the learning process is online.
The best NNSMC parameters, which depend significantly on
individual conditions, could be considerably different from
these adopted here. In this simulation, the number of hidden
layer neurons, the center vector, and the width vector are chosen
by trials to make the simulation more perfect. In this paper’s
research work, although the center vector and the width vector
are chosen as equal interval distance, and no optimization
method is applied to NN, the system dynamics identification
and trajectory tracking performance are effective. However, if
there are some standard optimization methods such as gradient
descent to be applied in the determination of the weights
and center vector of NNSMC, the controller output and the
simulation results will be more satisfying.
VI. CONCLUSION
In this paper, the problems of dynamic model and trajec-
tory tracking are addressed for a redundantly actuated omni-
directional mobile manipulator system with uncertainties and
external disturbances. First, FLUBSMC at the dynamic level
is developed to solve the trajectory tracking problem. Then,
a more effective NNSMC is studied to overcome the inherent
deficiency of FLUBSMC, achieving coordinately control the
manipulator system effectively. On the one hand, NN in
NNSMC is used to identify the system dynamics directly to
make the weights matrix structure compact and tuning speed
fast. On the other hand, a partitioned NN structure is applied to
reduce the computing burden further. The stability and conver-
gence of the whole control system are proved using Lyapunov
theory and related lemmas. From the discussion and simulation
results, the following conclusions can be reached.
1) NNSMC can take advantage of the strong robustness of
SMC and the self-learning and nonlinear mapping prop-
erties of NN to deal with both structured and unstructured
uncertainties.
2) Taking advantage of SMC and NNC, NNSMC can co-
ordinately control the mobile platform and the mounted
manipulator with different dynamics effectively.
3) NN is directly used to identify the uncertain vector of
system dynamics in the controller design. No preliminary
learning stage is required for the NN weights, the NN
learning process is performed online, and its uniform ulti-
mate boundedness is guaranteed by the stability analysis.
4) Compared to FLUBSMC, NNSMC requires no informa-
tion of the mathematical model and/or the parameteri-
zation of the mechanical dynamics, and can overcome
the defects of FLUBSMC, which may cause large output
torques, particularly for the manipulator.
5) Although the overall structure of NNSMC looks compli-
cated, it is very suitable for real-time application after
adopting the partitioned structure, because the partitioned
structure developed in this paper simplifies the NN design
process and accelerates the tuning speed.
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Dong Xu received the B.S. and M.S. degrees from
the Harbin Institute of Technology, Harbin, China,
in 2003 and 2005, respectively, and the Ph.D. degree
from the Institute of Automation, Chinese Academy
of Sciences, Beijing, China, in 2008.
He is currently a Research Engineer with the
IC Processing Equipment R&D Center, Beijing
Sevenstar Electronics Co., Ltd., Beijing. His research
interests lie in the area of robotics, industrial control,
fuzzy systems, and neural networks.
Dongbin Zhao (M’06) received the B.S., M.S., and
Ph.D. degrees in materials processing engineering/
welding robotics and automation from the State
Key Laboratory of Advanced Welding Production
Technology, Harbin Institute of Technology, Harbin,
China, in 1994, 1996, and 2000, respectively.
He was a Postdoctoral Fellow in humanoid robot
with the Department of Mechanical Engineering,
Tsinghua University, Beijing, China, from May 2000
to January 2002. He is currently an Associate Pro-
fessor with the Key Laboratory of Complex Systems
and Intelligence Science, Institute of Automation, Chinese Academy of Sci-
ences, Beijing. His current research interests are in the area of computational
intelligence, robotics, and intelligent transportation systems.
Dr. Zhao is an Associate Editor of IEEE Intelligent Transportation System
Magazine.
Jianqiang Yi (M’03) received the B.E. degree from
the Beijing Institute of Technology, Beijing, China,
in 1985 and the M.E. and Ph.D. degrees from the
Kyushu Institute of Technology, Kitakyushu, Japan,
in 1989 and 1992, respectively.
He worked as a Research Fellow with Computer
Software Design, Inc., Tokyo, Japan, from 1992 to
1994. From 1992 to 1994, he was with the Computer
Software Development Company, Tokyo. From 1994
to 2001, he was a Chief Researcher and Chief Engi-
neer with MYCOM, Inc., Kyoto, Japan. Currently,
he is a Full Professor with the Key Laboratory of Complex Systems and Intelli-
gence Science, Institute of Automation, Chinese Academy of Sciences, Beijing.
His research interests include theories and applications of intelligent control,
intelligent robotics, underactuated system control, sliding-mode control, flight
control, etc. His research interests lie in the area of intelligent control and
robotics.
Dr. Yi is an Associate Editor for the IEEE Computational Intelligence
Magazine,Journal of Advanced Computational Intelligence and Intelligent
Informatics,andJournal of Innovative Computing, Information and Control.
Xiangmin Tan is currently working toward the
Ph.D. degree in the Key Laboratory of Complex
Systems and Intelligence Science, Institute of Au-
tomation, Chinese Academy of Sciences, Beijing,
China.
His research interests lie in the area of robotics, in-
dustrial control, fuzzy systems, and neural networks.
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