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Neural Control of Bimanual Robots With Guaranteed Global Stability and Motion Precision

TL;DR: In order to extend the semiglobal stability achieved by conventional neural control to global stability, a switching mechanism is integrated into the control design and effectiveness of the proposed control design has been shown through experiments carried out on the Baxter Robot.
Abstract: Robots with coordinated dual arms are able to perform more complicated tasks that a single manipulator could hardly achieve. However, more rigorous motion precision is required to guarantee effective cooperation between the dual arms, especially when they grasp a common object. In this case, the internal forces applied on the object must also be considered in addition to the external forces. Therefore, a prescribed tracking performance at both transient and steady states is first specified, and then, a controller is synthesized to rigorously guarantee the specified motion performance. In the presence of unknown dynamics of both the robot arms and the manipulated object, the neural network approximation technique is employed to compensate for uncertainties. In order to extend the semiglobal stability achieved by conventional neural control to global stability, a switching mechanism is integrated into the control design. Effectiveness of the proposed control design has been shown through experiments carried out on the Baxter Robot.

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IEEE Transactions on Industrial Informatics
Cronfa URL for this paper:
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Paper:
Yang, C., Jiang, Y., Li, Z., He, W. & Su, C. (2017). Neural Control of Bimanual Robots With Guaranteed Global
Stability and Motion Precision. IEEE Transactions on Industrial Informatics, 13(3), 1162-1171.
http://dx.doi.org/10.1109/TII.2016.2612646
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1
Neural Control of Bimanual Robots with
Guaranteed Global Stability and Motion Precision
Chenguang Yang, Senior Member, IEEE, Yiming Jiang, Zhijun Li, Senior Member, IEEE, Wei He, Senior
Member, IEEE, and Chun-Yi Su, Senior Member, IEEE,
Abstract—Robots with coordinated dual arms are able to
perform more complicated tasks that a single manipulator could
hardly achieve. However, more rigorous motion precision is
required to guarantee effective cooperation between the dual
arms, especially when they grasp a common object. In this case,
the internal forces applied on the object must also be considered
in addition to the external forces. Therefore, a prescribed tracking
performance at both transient and steady states is first specified,
and then a controller is synthesized to rigorously guarantee
the specified motion performance. In the presence of unknown
dynamics of both the robot arms and the manipulated object,
the neural networks approximation technique is employed to
compensate for uncertainties. In order to extend the semiglob-
al stability achieved by conventional neural control to global
stability, a switching mechanism is integrated into the control
design. Effectiveness of the proposed control design has been
shown through experiments carried out on the Baxter Robot.
Index Terms—Neural networks; Bimanual robots; Tailored
tracking performance; Global uniformly ultimately boundedness
(GUUB)
I. INTRODUCTION
With bimanual cooperation, our humans are able to perform
delicate and complicated manipulations. There has been a
pronounced tendency in the robotics and automation com-
munity to shift focus of studies from single manipulators to
coordinated dual-arm robots [1]–[6]. In comparison to a single
arm robot, a dual-arm robot has prominent advantages in the
handling capability, loading capability as well as manipulative
skills. For example, in tool using tasks such as carving
or screwing, distribution of motions and forces required by
the tasks between the two robot arms greatly reduces the
complexity and energy cost of manipulation, compared with
that of a single robot arm. Therefore, the topics of dual
arms robot control have attracted much research attention over
Manuscript received June 25, 2016; revised August 26, 2016; and
accepted September 15, 2016. This work was partially supported by
National Nature Science Foundation (NSFC) under Grants 61473120,
61573147 and 91520201, Guangdong Provincial Natural Science Founda-
tion 2014A030313266 and International Science and Technology Collabo-
ration Grant 2015A050502017, Science and Technology Planning Project
of Guangzhou 201607010006 and the Fundamental Research Funds for the
Central Universities under Grant 2015ZM065.
C. Yang, Y. Jiang. Z. Li and C.-Y. Su are with Key Lab of Au-
tonomous Systems and Networked Control, Ministry of Education, South
China University of Technology, Guangzhou, China (e-mail: cyang@ieee.org;
ym.jiang@qq.com; zjli@ieee.org; cysu@alcor.concordia.ca). C. Yang is also
with Zienkiewicz Centre for Computational Engineering, Swansea University,
UK. C.-Y. Su is on leave from Concordia University, Canada.
W. He is with School of Automation and Electrical Engineering, U-
niversity of Science and Technology Beijing, Beijing, China (e-mail:
hewei.ac@ustb.edu.cn).
the past decades [7]–[10]. The early studies of coordinative
control schemes of two robotic arms were reported in [11]
and [12], where the position tracking and force control were
addressed. To deal with the unknown output hysteresis in the
control of coordinate robot, an adaptive neural control was
presented with computational efficiency [6]. In [7], a dual
NN has been used to resolve the distribution problem of
redundant coordination robot systems by using a multicriteria
to minimize the global kinetic energy.
It should be emphasized that the motion precision is of great
importance in the robot operation, especially for the dual arm
manipulation [13]. A precise coordination of both arms can
ensure that no excessive internal force would occur, and also
reduce possible variation of the internal forces. In this regards,
the rigorous requirement of motion precision implies that the
transient performance in the operation must also be taken into
account. Therefore, much effort in the control community has
been made to achieve a desired transient performance [14]–
[17]. For this purpose, an effective tracking algorithm was
proposed to control a five-bar closed-chain robot based on
transformation of tracking errors in [16]. In [17], a constraint
on output was considered for control of a class of multi-
input-multi-output (MIMO) systems. The above mentioned
control approaches rely on purposely built transformations
with appropriate inverses which increase the complexity of
the control design.
In practice, usually the kinematics information of robots
can be accurately known from the manufacturer, but there exist
inevitable uncertainties of the dynamics of the robot [18]–[22].
Nevertheless, we can always access the input-output data of
an robot system, thus it is desirable to use available input-
output data to approximate the unknown robot dynamics, in
order to design a controller with satisfactory performance.
One of the most successful control approaches is the neural
network (NN) based intelligent controller, which utilizes the
powerful universal approximation ability of NN to compensate
for unknown dynamics [23]–[34]. In [35], the NN was used
to approximate the hypersonic flight vehicle dynamics in the
tracking control of strict-feedback systems. In [15], the NN
was used to compensate for the complicated nonlinearity in
the closed-loop robot dynamics.
It should be noted that the above mentioned NN control
methods only ensure stability in the sense of semiglobally
uniformly ultimately boundedness (SGUUB) of the closed-
loop signals, because the NN’s approximation only holds over
a certain compact set, so called NN’s approximation domain.
Therefore, the range of state variable must be within this ap-

2
proximation domain during operation. However, such compact
set is impossible to be identified precisely beforehand, espe-
cially for highly nonlinear complicated systems with multiple
inputs and multiple outputs (MIMO). Therefore, it is important
to develop an NN controller with guaranteed global stability.
In [36], a robust adaptive neural controllers was developed
to achieve global uniformly ultimately boundedness (GUUB)
stability. An adaptive NN control for hypersonic flight vehicle
systems was proposed to ensure GUUB stability in [35].
However, only single-input-single-output (SISO) systems were
reported in most existing works, and few of them consider
transient performance at the same time.
In this paper, we aim to achieve both tailored transient per-
formance and guaranteed global stability at the same time, by
exploiting the barrier Lyapunov functions (BLFs). The BLFs
were originally developed in the nonlinear control community
to deal with the state and output constrains [37]–[40]. A BLF-
based controller was developed to control a robot manipulator
with joint space constraints in [37]. In [40], an asymmetric
time-varying BLF was presented for nonlinear systems in
strict-feedback form.
It is noted that by posing constraints to the behavior of the
states or outputs, tracking errors can be indirectly constrained
using the technique of BLFs. Motivated by this, in this paper
the BLFs technique was exploited to achieve the tailored
tracking performance at both transient and steady states.
Comparing with the regulation of steady state responses, the
shaping of the transient control is much more difficult. By
constructing a prescribed tracking performance requirement
function, a proper BLF is proposed for controller synthesis
of a dual-arm robot, such that both transient and steady state
tracking performance can be ensured. Meanwhile, a switching
mechanism is introduced into the NN controller design to
ensure global stability. In comparison to the conventional
NN controllers which only ensure the stability of SGUUB,
our proposed NN controller guarantees global stability of the
closed-loop system. This is practically much more useful as
the requirement of the NN inputs is greatly relaxed.
II. PROBLEM FORMULATION AND MODELLING
PROCEDURE
A. Problem Formulation
Consider a bimanual robot grasping a common object, our
objective is to design a robot controller such that the ma-
nipulated object could track a desired trajectory x
d
specified
in the task space, as shown in Fig. 1, while simultaneously
guarantee (i) the tracking errors fall into the predefined bounds
to achieve tailored tracking performances; (ii) all the signals in
the close-loop bimanual robot system remain GUUB; and (iii)
the internal forces between the end-effectors and the object
converge to a small neighborhood of specified values.
B. Modeling of the Bimanual Robot
The position and orientation of the manipulated object could
be defined by a vector x R
N
0
, where N
0
is the object’s
degree of freedom (DOF). Assume that both arms grasp the
object rigidly so that there is no relative motion in between
X
Y
O
Z
O
Z
O
X
O
O
ARM 1
ARM 2
O
X
2
e
X
2
e
Y
2
e
Z
2
e
Y
2
e
Z
O
Z
Reference
Trajectory
d
x
o
F
1
oe
F
1
e
X
O
Y
2
oe
F
Fig. 1. An overwive of the dual arm robot manipulated a common object
the object and the end-effectors. Then, based on the forward
kinematics of robot manipulator, the relations between task
space and robot joint space can be calculated in the following
manner:
x = p
i
(q
i
), ˙x = ˙p
i
(q
i
) = J
i
(q
i
) ˙q
i
(1)
where q
i
R
N
i
and ˙q
i
R
N
i
are vectors of joint variable
and joint velocity of the ith robotic arm, respectively, and N
i
is the DOF of the ith robotic arm. p
i
is a continues function,
and J
i
(q
i
) is the Jacobian matrix. The following assumptions
are considered to facilitate the modeling procedure of the
bimanual robot system:
Assumption 1: The dynamics of the robot manipulators are
uncertain, while the kinematics is accurately available. The
robotic arms are operating away from any singular configura-
tions during the motion.
Assumption 2: The rigid object would not be deformed by
the exerted forces.
Then, the dynamics of each robot arm are described in the
following Lagrangian form:
M
i
(q
i
)¨q
i
+ C
i
(q
i
, ˙q
i
) ˙q
i
+ G
i
(q
i
) = τ
i
+ J
T
e
i
(q
i
)F
e
i
(2)
where M
i
(q
i
) R
N
i
×N
i
, C
i
(q
i
, ˙q
i
) R
N
i
×N
i
, G
i
(q
i
) R
N
i
are the inertial matrix, Coriolis and centrifugal matrix and
gravity vector, respectively. J
T
e
i
(q
i
) represents the robotic
arm’s Jacobian matrix, while τ
i
R
N
i
is the joint torque,
F
e
i
R
N
0
is the force vector exerted at end-effector. The
dynamics of the object’s motion can be described as:
M
o
(x)¨x + C
o
(x, ˙x) ˙x + G
o
(x) = F
o
(3)
where M
o
(x), C
o
(x, ˙x) and G
o
(x) denote the inertial, Coriolis
and centrifugal matrix, and the gravitational vector of manip-
ulated object, respectively, while F
o
R
N
0
is the resulting
force given as follows
F
o
= F
oe
1
F
oe
2
, F
oe
i
= f
i
+ f
o
i
(4)
where F
oe
i
is the interaction force applied on the end-effector
of ith robotic arm. F
oe
i
are decomposed into an external force
f
o
i
and an internal force f
i
, where the external forces f
oi
derive the motion of the object, and the internal forces f
i
cancel with each other and satisfy the constraint f
1
+f
2
= 0
[n]
.
Combination of equation (3) and (4) yields
f
i
= F
oe
i
D
i
(t)(f
o
1
+ f
o
2
) (5)

3
Robotic Arm 1
Common
Object
Dual arm robot
System
1
2
Virtual
Controller
Global NN
Control
Transient
Control
Adaptive NN
Learning
Robotic Arm 1
Bimanual Robot
Neural Controller
,
ab

21
z
1
1
q
d
x
e
x
d
x
di
f
q
Fig. 2. The framework for the bimanual robot controller
where D
i
(t) R
N
0
×N
0
is the object load distribution matrix
satisfying D
1
(t) + D
2
(t) = I
N
0
, where I
N
0
R
N
0
×N
0
is an
identity matrix.
Combination of (2), (3), (4), (5) and the kinematic equation
(1) yields a compact form below:
τ
i
= M
i
(q
i
)¨q
i
+ C
i
(q
i
, ˙q
i
) ˙q
i
+ G
i
(q
i
) J
T
i
(q
i
)f
i
(6)
where M
i
= M
i
+ D
i
M
o
, M
o
= J
T
i
M
o
J
i
, C
i
= C
i
+
D
i
(M
D
+ C
o
), C
o
= J
T
i
C
o
J
i
, M
D
= J
T
i
M
o
˙
J
i
, G
i
= G
i
+
D
i
G
o
, G
o
= J
T
i
G
o
. To be self-contained, the fundamental
properties of robot manipulator dynamics, which will be used
later for control design and analysis, are described below:
Property 1: [10] The skew-symmetric matrix 2C
i
(q
i
, ˙q
i
)
[
˙
M
i
(q
i
)
˙
D
i
(t)M
o
(q
i
, ˙q
i
)] satisfies that:
T
n
2C
i
(q
i
, ˙q
i
)
˙
M
i
(q
i
)
˙
D
i
(t)M
o
(q
i
, ˙q
i
)
o
= 0,
Property 2: [10] The matrix
˙
D
i
(t)M
0
(q
i
) is bounded and
uniformly continuous while satisfies the following inequality:
k
˙
D
i
(t)M
0
(q
i
)k 2, t 0 (7)
where is a positive constant.
III. CONTROL DESIGN
Before proceeding to control design, let us introduce the
following tracking error signals:
e = x x
d
, z
i
= ˙q
i
α
i
i = 1, 2 (8)
where e = [e
1
, e
2
, · · · , e
N
0
] R
N
0
stands for the po-
sition tracking error of the manipulated object, z
i
=
[z
i1
, z
i2
, · · · , z
iN
i
] R
N
i
stand for the velocity tracking error
of each robotic arm in joint space, and α
i
is a virtual controller
to be specified in (19), x
d
is the reference trajectory of the
manipulated object. Our control strategy is illustrated in Fig.
2.
A. Specification on Requirement for Tracking Performance
To specify tracking performance, especially transient per-
formance (e.g., overshoot, undershoot and coverage rate), we
construct a series of smoothly decreasing functions φ(t) = [φ
1
,
φ
2
, · · · , φ
N
0
] to shape the motion of the object as
φ
k
(t) = (ρ
0k
ρ
k
)e
a
k
t
+ ρ
k
(9)
where ρ
0k
, ρ
k
and a
k
(k = 1, 2, · · · , N
0
) are properly
chosen positive constants. Let us define ϕ
a,k
(t) = β
1k
φ
k
(t)
and ϕ
b,k
(t) = β
2k
φ
k
(t), with positive constants β
1k
and β
2k
to be specified by the designer.
Remark 1: The functions ϕ
a,k
(t) and ϕ
b,k
(t) specify the
tracking transient response, i.e., the exponential term a
k
regu-
lates the required convergence rate of tracking errors, β
1k
ρ
0k
,
β
2k
ρ
0k
define the maximum overshoot and undershoot,
and β
1k
ρ
k
, β
2k
ρ
k
regulates the bounds of the steady
errors, as shown in Fig. 3. This implies that we are able
to regulate both transient and steady-state performance by
properly choosing parameters β
1k
, β
2k
, ρ
0k
, ρ
k
and a
k
.
The following coordinate transformation of tracking errors
will be used in the later design.
ξ
a
=
e
1
ϕ
a,1
, · · · ,
e
N
0
ϕ
a,N
0
T
ξ
b
=
e
1
ϕ
b,1
, · · · ,
e
N
0
ϕ
b,N
0
T
ξ
k
= h
k
(e
k
)ξ
b,k
+ (1 h
k
(e
k
))ξ
a,k
(10)
where ξ
a,k
, ξ
b,k
are the kth element of the vectors ξ
a
, ξ
b
,
respectively, and h
k
(e
k
) is defined as
h
k
(e
k
) =
1 e
k
0
0 otherwise
(11)
B. Controller Design Using BLF and Backstepping
Inspired by the work [40], an asymmetric time-varying
barrier function is constructed for the ith robotic arm as
V
i1
=
N
0
X
k=1
h
k
2
ln
1
1 ξ
2
b,k
+
1 h
k
2
ln
1
1 ξ
2
a,k
!
(12)
The differentiation of (12) with respect to time gives us
˙
V
i1
=
N
0
X
k=1
h
k
1 ξ
2
b,k
ξ
b,k
˙
ξ
b,k
+
1 h
k
1 ξ
2
a,k
ξ
a,k
˙
ξ
a,k
!
(13)
According to definitions of ξ
a,k
, ξ
b,k
, and substituting (8) into
(13) we have
˙
V
i1
=
N
0
X
k=1
ξ
2
k
(1 ξ
2
k
)e
k
˙e
k
+
N
0
X
k=1
(1 h
k
)ξ
2
a,k
(1 ξ
2
a,k
)
˙ϕ
a,k
ϕ
a,k
+
h
k
ξ
2
b,k
(1 ξ
2
b,k
)
˙ϕ
b,k
ϕ
b,k
)
!
(14)
Then, by defining a transient control vector
P = [
ξ
2
1
(1 ξ
2
1
)e
1
,
ξ
2
2
(1 ξ
2
2
)e
2
, · · · ,
ξ
2
N
0
(1 ξ
2
N
0
)e
N
0
]
T
(15)
and substituting it into (14), we rewrite V
i1
as below:
˙
V
i1
= P
T
˙e +
N
0
X
k=1
(1 h
k
)ξ
2
a,k
(1 ξ
2
a,k
)
˙ϕ
a,k
ϕ
a,k
+
h
k
ξ
2
b,k
(1 ξ
2
b,k
)
˙ϕ
b,k
ϕ
b,k
)
!
(16)
Note that the relation between ˙x and ˙q
i
as specified in (1)
always hold. According to the definitions of e and z
i
in (8),
we have
˙e = J
i
(q)(z
i
+ α
i
) ˙x
d
i = 1, 2 (17)

4
Fig. 3. Relationship between the
tracking error e
k
(t) and the perfor-
mance function
Fig. 4. Global tracking performance
Substituting (17) into (16) yields
˙
V
i1
= P
T
(J
i
(q)(z
i
+ α
i
) ˙x
d
)
+
N
0
X
k=1
(1 h
k
)ξ
2
a,k
(1 ξ
2
a,k
)
˙ϕ
a,k
ϕ
a,k
+
h
k
ξ
2
b,k
(1 ξ
2
b,k
)
˙ϕ
b,k
ϕ
b,k
)
!
(18)
Then, let us design a virtual controller α
i
as
α
i
= J
+
i
(q) ( ˙x
d
K
1
e σ(t)e) (19)
where J
+
i
(q
i
) is the Moore-Penrose inverse of J
i
(q
i
), K
1
=
diag{k
11
, k
12
, · · · , k
1N
0
} with k
1
k
being positive constants.
And σ(t) = diag{σ
1
(t), σ
2
(t), · · · , σ
N
0
(t)} with σ
k
(t) =
q
(
˙ϕ
a,k
ϕ
a,k
)
2
+ (
˙ϕ
b,k
ϕ
b,k
)
2
+ k
a
, where k
a
selected as a positive
parameter that ensures the boundedness of ˙α
i
when ˙ϕ
a,k
(t),
˙ϕ
b,k
(t) are zero. Substituting (19) into (18) yields
˙
V
i1
= P
T
J
i
(q)z
i
P
T
(K
1
e + σ(t)e)
+
N
0
X
k=1
h
k
ξ
2
b,k
(1 ξ
2
b,k
)
(
˙ϕ
b,k
ϕ
b,k
) +
(1 h
k
)ξ
2
a,k
(1 ξ
2
a,k
)
(
˙ϕ
a,k
ϕ
a,k
)
!
(20)
Note that the following inequality holds
σ
k
(t) h
k
˙ϕ
a,k
ϕ
a,k
(1 h
k
)
˙ϕ
b,k
ϕ
b,k
0 (21)
Using the definition of P in (15) and in terms of (21),
equation (20) can be rewritten as
˙
V
i1
N
0
X
k=1
k
1
k
ξ
2
k
(1 ξ
2
k
)
+ P
T
J
i
(q)z
i
(22)
C. Global Adaptive NN (GANN) Control
1) Radial basis function neural network (RBFNN) [41]:
In this paper, the following RBFNNs are used to
approximate a continuous vector function F (Z) =
[f
1
(Z), f
2
(Z), · · · , f
n
(Z)]
T
R
n
,
ˆ
F (Z) =
ˆ
W
T
S(Z) (23)
where
ˆ
F (Z) R
n
is the estimate of F (Z), Z
Z
R
q
is
NN inputs vector, and q denotes the demonstration of the input;
ˆ
W = [
ˆ
W
1
,
ˆ
W
2
, · · · ,
ˆ
W
n
] R
n×l
is the estimation of NN
optimal weight matrix W
, and l is the number of NN nodes.
S(Z) = [s
1
(Z), s
2
(Z), · · · , s
l
(Z)]
T
R
l
is the regressor
vector with s
i
(·) being a radial basis function. In general,
the most commonly used Gaussian radial basis functions are
employed as follows:
s
i
(kZ µ
i
k) = exp
(Z µ
i
)
T
(Z µ
i
)
ϑ
2
i
(24)
where µ
i
(i = 1, · · · , l) are distinct points in state space,
µ
i
= [µ
i1
, µ
i2
, · · · , µ
iq
]
T
is the center of the neural and ϑ
i
is the Gaussian function’s width. It has been established that,
with sufficiently large node number, an arbitrary continuous
function F (Z) can be approximated by the RBFNN (23) over
a compact set
Z
as
F (Z) = W
T
S (Z) + ε(Z), Z
Z
(25)
where W
is an ideal constant weight vector, and ε(Z) R
n
is the approximation error. There exist ideal weight vector W
such that |ε(Z)| < ε
with constant ε
> 0 for all Z
Z
.
2) Global NN control design: Let us define a positive
Lyapunov function as,
V
i2
= V
i1
+
1
2
z
T
i
M
i
z
i
(26)
Substituting (6) and (8) into its derivative, and considering
Properties 1 and 2, we can derive from (26) that
˙
V
i2
˙
V
i1
+
i
z
T
i
z
i
+ z
T
i
(τ
i
M ˙α
i
C
i
α
i
G
i
+ J
T
i
(q
i
)f
i
)
(27)
where M
i
, G
i
and C
i
are abbreviations of M
i
(q), G
i
(q) and
C
i
(q, ˙q), respectively, and
i
is a positive constant specified in
(7).
Considering the dynamics of robot in (6), we reformulate it
by using a function vector F
i
(Z
i
) R
N
i
as
F
i
(Z
i
) = (M ˙α
i
+ C
i
α
i
+ G
i
) (28)
where F
i
(Z
i
) = [f
i,1
(Z
i
), f
i,2
(Z
i
), · · · , f
i,N
i
(Z
i
)]
T
, Z
i
=
[q
T
i
, ˙q
T
i
, α
T
i
, ˙α
T
i
]
T
R
ν
i
, with ν
i
= 4N
i
. It should be noted
that, for the functions f
i,j
(Z
i
) R, j = 1, 2, · · · , N
i
, there
exist known bounded nonnegative smooth functions f
U
i,j
(Z
i
)
such that |f
i,j
(Z
i
)| f
U
i,j
(Z
i
), Z R
ν
i
.
Applying RBFNN described in Section III.C, we see that
over a compact set
i1
,
ˆ
F
i
(Z
i
) =
ˆ
W
T
i
S
i
(Z
i
) + ε
i
(29)
where
ˆ
W
i
= [
ˆ
W
i,1
,
ˆ
W
i,2
, · · · ,
ˆ
W
i,N
i
]
T
R
l
i
×N
i
is the
estimation of optimal neural weight matrix W
i
, and
ˆ
W
i,j
=
[ˆω
i,j1
, ˆω
i,j1
, · · · , ˆω
i,jl
i
] R
l
i
, S
i
(Z
i
) R
l
i
is the basis vector
function with l
i
being the NN nodes number, and ε
i
is the NN
construction error satisfying |ε
i
| < ¯ε
i
.
Prior to proceed to control design, let us introduce a set of
smooth switching functions Q
i
(Z
i
) R
N
i
×N
i
as
Q
i
(Z
i
) = diag
M
i1
(Z
i
), M
i2
(Z
i
), · · · , M
iN
i
(Z
i
)
(30)
where M
ij
(Z
i
) =
ν
i
Q
c=1
m(z
ic
), and m(z
ic
) is designed as
m(z
ic
) =
1 |z
ic
| < d
1,ic
d
2
2,ic
z
2
ic
d
2
2,ic
d
2
1,ic
e
z
2
ic
d
2
1,ic
ω
i
(d
2
2,ic
d
2
1,ic
)
2
otherwise
0 |z
ic
| > d
2,ic
(31)

Citations
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Journal ArticleDOI
TL;DR: With the proposed control, the stability of the closed-loop system is achieved via Lyapunov’s stability theory, and the tracking performance is guaranteed under the condition of state constraints and uncertainty.
Abstract: This paper investigates adaptive fuzzy neural network (NN) control using impedance learning for a constrained robot, subject to unknown system dynamics, the effect of state constraints, and the uncertain compliant environment with which the robot comes into contact. A fuzzy NN learning algorithm is developed to identify the uncertain plant model. The prominent feature of the fuzzy NN is that there is no need to get the prior knowledge about the uncertainty and a sufficient amount of observed data. Also, impedance learning is introduced to tackle the interaction between the robot and its environment, so that the robot follows a desired destination generated by impedance learning. A barrier Lyapunov function is used to address the effect of state constraints. With the proposed control, the stability of the closed-loop system is achieved via Lyapunov’s stability theory, and the tracking performance is guaranteed under the condition of state constraints and uncertainty. Some simulation studies are carried out to illustrate the effectiveness of the proposed scheme.

498 citations


Cites background from "Neural Control of Bimanual Robots W..."

  • ...However, while the robotic end-effector comes in contact with the environment, it is inevitable that an interaction force will develop between the robot and its environment [40]–[42]....

    [...]

Journal ArticleDOI
TL;DR: A robot control/identification scheme to identify the unknown robot kinematic and dynamic parameters with enhanced convergence rate was developed, and the information of parameter estimation error was properly integrated into the proposed identification algorithm, such that enhanced estimation performance was achieved.
Abstract: For parameter identifications of robot systems, most existing works have focused on the estimation veracity, but few works of literature are concerned with the convergence speed. In this paper, we developed a robot control/identification scheme to identify the unknown robot kinematic and dynamic parameters with enhanced convergence rate. Superior to the traditional methods, the information of parameter estimation error was properly integrated into the proposed identification algorithm, such that enhanced estimation performance was achieved. Besides, the Newton–Euler (NE) method was used to build the robot dynamic model, where a singular value decomposition-based model reduction method was designed to remedy the potential singularity problems of the NE regressor. Moreover, an interval excitation condition was employed to relax the requirement of persistent excitation condition for the kinematic estimation. By using the Lyapunov synthesis, explicit analysis of the convergence rate of the tracking errors and the estimated parameters were performed. Simulation studies were conducted to show the accurate and fast convergence of the proposed finite-time (FT) identification algorithm based on a 7-DOF arm of Baxter robot.

321 citations

Journal ArticleDOI
TL;DR: A novel control scheme is developed for a teleoperation system, combining the radial basis function (RBF) neural networks (NNs) and wave variable technique to simultaneously compensate for the effects caused by communication delays and dynamics uncertainties.
Abstract: In this paper, a novel control scheme is developed for a teleoperation system, combining the radial basis function (RBF) neural networks (NNs) and wave variable technique to simultaneously compensate for the effects caused by communication delays and dynamics uncertainties. The teleoperation system is set up with a TouchX joystick as the master device and a simulated Baxter robot arm as the slave robot. The haptic feedback is provided to the human operator to sense the interaction force between the slave robot and the environment when manipulating the stylus of the joystick. To utilize the workspace of the telerobot as much as possible, a matching process is carried out between the master and the slave based on their kinematics models. The closed loop inverse kinematics (CLIK) method and RBF NN approximation technique are seamlessly integrated in the control design. To overcome the potential instability problem in the presence of delayed communication channels, wave variables and their corrections are effectively embedded into the control system, and Lyapunov-based analysis is performed to theoretically establish the closed-loop stability. Comparative experiments have been conducted for a trajectory tracking task, under the different conditions of various communication delays. Experimental results show that in terms of tracking performance and force reflection, the proposed control approach shows superior performance over the conventional methods.

297 citations

Journal ArticleDOI
TL;DR: A decentralized adaptive formation controller is designed that ensures uniformly ultimate boundedness of the closed-loop system with prescribed performance and avoids collision between each vehicle and its leader.
Abstract: This paper addresses a decentralized leader–follower formation control problem for a group of fully actuated unmanned surface vehicles with prescribed performance and collision avoidance. The vehicles are subject to time-varying external disturbances, and the vehicle dynamics include both parametric uncertainties and uncertain nonlinear functions. The control objective is to make each vehicle follow its reference trajectory and avoid collision between each vehicle and its leader. We consider prescribed performance constraints, including transient and steady-state performance constraints, on formation tracking errors. In the kinematic design, we introduce the dynamic surface control technique to avoid the use of vehicle's acceleration. To compensate for the uncertainties and disturbances, we apply an adaptive control technique to estimate the uncertain parameters including the upper bounds of the disturbances and present neural network approximators to estimate uncertain nonlinear dynamics. Consequently, we design a decentralized adaptive formation controller that ensures uniformly ultimate boundedness of the closed-loop system with prescribed performance and avoids collision between each vehicle and its leader. Simulation results illustrate the effectiveness of the decentralized formation controller.

273 citations


Cites background from "Neural Control of Bimanual Robots W..."

  • ..., robot manipulators [23]–[25] and marine vehicles [26], where the prescribed performance controllers were designed in a centralized manner....

    [...]

Journal ArticleDOI
TL;DR: A Lyapunov function is proposed to prove the closed-loop system stability and the semi-global uniform ultimate boundedness of all state variables and a series of simulation results indicate that proposed controllers can track desired trajectories well via selecting appropriate control gains.
Abstract: The research of this paper works out the attitude and position control of the flapping wing micro aerial vehicle (FWMAV). Neural network control with full state and output feedback are designed to deal with uncertainties in this complex nonlinear FWMAV dynamic system and enhance the system robustness. Meanwhile, we design disturbance observers which are exerted into the FWMAV system via feedforward loops to counteract the bad influence of disturbances. Then, a Lyapunov function is proposed to prove the closed-loop system stability and the semi-global uniform ultimate boundedness of all state variables. Finally, a series of simulation results indicate that proposed controllers can track desired trajectories well via selecting appropriate control gains. And the designed controllers possess potential applications in FWMAVs.

269 citations


Cites methods from "Neural Control of Bimanual Robots W..."

  • ...[38], [39] used NN approximation techniques to compensate the unknown dynamics of both the robot arms and manipulated objects by the Baxter robot....

    [...]

References
More filters
Journal ArticleDOI
TL;DR: Adaptive neural network control for the robotic system with full-state constraints is designed, and the adaptive NNs are adopted to handle system uncertainties and disturbances.
Abstract: This paper studies the tracking control problem for an uncertain ${n}$ -link robot with full-state constraints The rigid robotic manipulator is described as a multiinput and multioutput system Adaptive neural network (NN) control for the robotic system with full-state constraints is designed In the control design, the adaptive NNs are adopted to handle system uncertainties and disturbances The Moore–Penrose inverse term is employed in order to prevent the violation of the full-state constraints A barrier Lyapunov function is used to guarantee the uniform ultimate boundedness of the closed-loop system The control performance of the closed-loop system is guaranteed by appropriately choosing the design parameters Simulation studies are performed to illustrate the effectiveness of the proposed control

1,021 citations

Journal ArticleDOI
TL;DR: A barrier Lyapunov function (BLF) is introduced to address two open and challenging problems in the neuro-control area: for any initial compact set, how to determine a priori the compact superset on which NN approximation is valid; and how to ensure that the arguments of the unknown functions remain within the specified compact supersets.
Abstract: In this brief, adaptive neural control is presented for a class of output feedback nonlinear systems in the presence of unknown functions. The unknown functions are handled via on-line neural network (NN) control using only output measurements. A barrier Lyapunov function (BLF) is introduced to address two open and challenging problems in the neuro-control area: 1) for any initial compact set, how to determine a priori the compact superset, on which NN approximation is valid; and 2) how to ensure that the arguments of the unknown functions remain within the specified compact superset. By ensuring boundedness of the BLF, we actively constrain the argument of the unknown functions to remain within a compact superset such that the NN approximation conditions hold. The semiglobal boundedness of all closed-loop signals is ensured, and the tracking error converges to a neighborhood of zero. Simulation results demonstrate the effectiveness of the proposed approach.

818 citations

Journal ArticleDOI
TL;DR: It is shown that asymptotic output tracking is achieved without violation of the time-varying constraint, and that all closed loop signals remain bounded.

688 citations


Additional excerpts

  • ...The BLFs were originally developed in the nonlinear control community to deal with the state and output constraints [37]–[40]....

    [...]

  • ...Inspired by the work [40], an asymmetric time-varying barrier function is constructed for the ith robotic arm as...

    [...]

  • ...In [40], an asymmetric time-varying BLF was presented for nonlinear systems in a strict-feedback form....

    [...]

Journal ArticleDOI
01 Mar 2016
TL;DR: In this article, an adaptive impedance controller for a robotic manipulator with input saturation was developed by employing neural networks. But the adaptive impedance control was not considered in the tracking control design, and the input saturation is handled by designing an auxiliary system.
Abstract: In this paper, adaptive impedance control is developed for an ${n}$ -link robotic manipulator with input saturation by employing neural networks. Both uncertainties and input saturation are considered in the tracking control design. In order to approximate the system uncertainties, we introduce a radial basis function neural network controller, and the input saturation is handled by designing an auxiliary system. By using Lyapunov’s method, we design adaptive neural impedance controllers. Both state and output feedbacks are constructed. To verify the proposed control, extensive simulations are conducted.

685 citations


Additional excerpts

  • ...In order to well approximate the robot dynamics and considering both the accuracy and the computational efficiency, we divide the inputs of RBFNN into two groups, with one group containing [qTi , α̇ T i ] T ∈ R6 and another [qTi , q̇Ti , αTi ]T ∈ R9 , and employ three centers for each input dimension of the NNs, and ended up with totally l1 = 20 412 NN nodes for each NN....

    [...]

  • ...1) Radial Basis Function Neural Network (RBFNN) [42]: In this paper, the following RBFNNs are used to approximate a continuous vector function F (Z) = [f1(Z), f2(Z), . . . , fn (Z)]T ∈ Rn , F̂ (Z) = Ŵ T S(Z) (23) where F̂ (Z) ∈ Rn is the estimate of F (Z), Z ∈ ΩZ ⊂ Rq is NN inputs vector, and q denotes the demonstration of the input; Ŵ = [Ŵ1 , Ŵ2 , . . . , Ŵn ] ∈ Rn × l is the estimation of NN optimal weight matrix W ∗, and l is the number of NN nodes....

    [...]

  • ...Applying RBFNN described in Section III-C, we see that over a compact set Ωi1 F̂i(Zi) = Ŵ Ti Si(Zi) + εi (29) where Ŵi = [Ŵi,1 , Ŵi,2 , . . . , Ŵi,Ni ] T ∈ Rli ×Ni is the estimation of optimal neural weight matrix W ∗i , and Ŵi,j = [ω̂i,j1 , ω̂i,j1 , . . . , ω̂i,j li ] ∈ Rli , Si(Zi) ∈ Rli is the basis vector function, with li being the NN node number, and εi is the NN construction error satisfying |εi | ε̄i ....

    [...]

  • ...1) Radial Basis Function Neural Network (RBFNN) [42]: In this paper, the following RBFNNs are used to approximate a continuous vector function F (Z) = [f1(Z), f2(Z), ....

    [...]

  • ...The transient responses such as overshoot, settling time, and final tracking RBFNNs are employed to approximate the unknown dynamics of both the robot arms and the manipulated object....

    [...]

Journal ArticleDOI
TL;DR: It is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed-loop system.
Abstract: In this paper, robust adaptive neural network (NN) control is investigated for a general class of uncertain multiple-input-multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities. For nonsymmetric input nonlinearities of saturation and deadzone, variable structure control (VSC) in combination with backstepping and Lyapunov synthesis is proposed for adaptive NN control design with guaranteed stability. In the proposed adaptive NN control, the usual assumption on nonsingularity of NN approximation for unknown control coefficient matrices and boundary assumption between NN approximation error and control input have been eliminated. Command filters are presented to implement physical constraints on the virtual control laws, then the tedious analytic computations of time derivatives of virtual control laws are canceled. It is proved that the proposed robust backstepping control is able to guarantee semiglobal uniform ultimate boundedness of all signals in the closed-loop system. Finally, simulation results are presented to illustrate the effectiveness of the proposed adaptive NN control.

670 citations