







































INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 2 (June), pp. 341-352

Robust Adaptive Neural-Fuzzy Network Tracking Control for
Robot Manipulator

T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

ThanhQuyen Ngo, YaoNan Wang, T. Long Mai,
M. Hung Nguyen, Jun Chen
College of Electrical and Information Engineering
Hunan University, Changsha, Hunan Province 410082, P.R.China
Faculty of Electrical Engineering
HCM City University of Industry, HCM City, Vietnam
thanhquyenngo2000@yahoo.com, yaonan@hnu.cn,
mailongtk@gmail.com, manhhung@yahoo.com, 2011junchen@gmail.com

Abstract: This paper presents a robust adaptive neural-fuzzy network con-
trol (RANFNC) system for an n-link robot manipulator to achieve the high-
precision position tracking. Initially, the model dynamic of an n-link robot
manipulator is introduced. However, it is difficult to design a conformable
model-based control scheme, for instance, external disturbances, friction forces
and parameter variations. In order to deal with this problem, the RANFNC
system is investigated to the joint position control of an n-link robot manipu-
lator. In this control scheme, a four-layer neural-fuzzy-network (NFN) is used
for the main role, and the adaptive tuning laws of network parameters are
derived in the sense of a projection algorithm and the Lyapunov stability the-
orem to ensure network convergence as well as stable control performance. The
merits of this model-free control scheme are that not only the stable position
tracking performance can be guaranteed but also unknown system information
and auxiliary control design are required in the control process. The simula-
tion results are provided to verify the effectiveness of the proposed RANFNC
methodology.
Keywords: Adaptive control, Neural-fuzzy network, robot manipulator.

1 Introduction

In the past decade, the applications of intelligent control techniques (fuzzy control or neural
control) to the motion control of robotic manipulator have received considerable attention [1]-[3].
In general, robotic manipulators have to face various uncertainties in their dynamics, such as
friction, and external disturbance. It is difficult to establish exactly mathematical model for
the design of a model-based control system. Thus, the general claim of these intelligent control
approaches is that they can reduce the effects of structured parametric uncertainty and the
unstructured disturbance by using their powerful learning ability without a detailed knowledge
of the controlled plant in the design processes.

Recently, the concept of incorporating fuzzy logic into a neural network has grown into a
popular research topic [5]-[8]. The integrated neural-fuzzy-network system possesses the merits
of both fuzzy system [9] (e.g., humanlike IF-THEN rule thinking and ease of incorporating expert
knowledge) and neural network [10] (e.g., learning and optimization abilities and connectionist
structure). In this way, one can bring the low-level learning and computational power of neural
network into fuzzy systems as well as the high-level humanlike IF-THEN rule thinking and
reasoning of fuzzy systems into neural network.

Copyright c⃝ 2006-2012 by CCC Publications



342 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

The main of this paper is to design an intelligent control system scheme for the position
control of an n-link robot manipulator by using neural-fuzzy-network controller to compensated
uncertainness dynamic model and external disturbance via capability self-learning of neural net-
work and human intuitive.

This paper is organized as follows: Section 2 described a dynamic model of an n-link robot
manipulator briefly [11]. Section 3 presents a structure of neural-fuzzy-network. The design
process of the RANFNC system is investigated to control an n-link robot manipulator for periodic
motion in section 4. The design procedures of the proposed RANFNC system are described
in detail. The adaptive learning laws in the RANFNC system are designed in the sense of
the Lyapunov stability theorem [12], [13] so that the network convergence and system tracking
stability can be guaranteed in the closed-loop control system. Numerical simulation results of
a two-link robot manipulator under the possible occurrence of uncertainties are provided to
demonstrate the tracking control performance of the proposed RANFNC system in section 5.
Conclusions are drawn in section 6.

2 System Description

2.1 Robotic Dynamic Model

In general, the dynamic of an n-link robot manipulator may be expressed in [11] as:

M(q)q̈ + C(q, q̇)q̇ + G(q) + F(q̇) + τd = τ (1)

Where q, q̇, q̈ ∈ ℜn are the joint position, velocity and acceleration vectors, M(q) ∈ ℜn×n
denotes the inertia matrix, C(q, q̇) ∈ ℜn×n expresses the matrix of centripetal and Coriolis forces,
G(q) ∈ ℜn is the gravity vector, and F(q̇) ∈ ℜn is the friction. Bounded unknown disturbances
are denoted by τd and the control input torque is τ(t) ∈ ℜn. In this paper a robot manipulator
is shown in Fig.1 which is utilized to verify dynamic properties are given in section 5.

Y

Y0

X0 X

q1

X1

Y1

lc1

m1

lc2

m2
g

p

X2

Y2

lt1

lt2

q2

Figure 1: Architecture of two-link
robot manipulator.

1

1

j

1

z

1

1

b

j

b

z

b

1

r

j

r

z

r

1
z bz rz

k

jb
w

1
l

k
l

1
y

2
y

n
y

ik
w

p
l

Input

Layer

Membership

Layer

Rule Layer

Output

Layer

Figure 2: Structure of four-layer NFN

Given a desired arm trajectory qd(t) ∈ ℜn the tracking error is:

e(t) = qd(t) − q(t) (2)

And the filtered tracking error (in standard use in robotics) is:

r(t) = ė + λe (3)



Robust Adaptive Neural-Fuzzy Network Tracking Control for Robot Manipulator 343

Where λ = λT > 0, differentiating r(t) and using (1), the arm dynamics may be written in
terms of the filtered tracking error as:

M(q)ṙ = −C(q, q̇)r − τ + M(q)(q̇d + λė) + C(q, q̇)(q̇ + λe) + F(q̇) + G(q) + τd (4)

Where the nonlinear robot function is:

f(x) = M(q)(q̈d + λė) + C(q, q̇)(q̇ + λe) + G(q) + F(q̇) (5)

Substituting (5) equation into (4) we have:

M(q)ṙ = −C(q, q̇)r − τ + f(x) + τd (6)

And, for instance x =
[
eT ėT qT q̇T q̈T

]T
2.2 Defined Control Law

Now, we define a control input torque as:

τ0 = f̂(x) + Kvr (7)

With f̂(x) is an estimate of f(x) and a gain matrix Kv = KTv the closed-loop system becomes:

M(q)ṙ = −(Kv + C(q, q̇))r + f̃(x) + τd ≡ −(Kv + C(q, q̇))r + ζ0, ζ0 = f̃(x) + τd (8)

Where functional estimate error is given by:

f̃(x) = f(x) − f̂(x) (9)

This is a system error wherein filtered tracking error is driven functional estimate error.
The control τ0 incorporates a proportional plus derivative (PD) term in Kvr = Kv(ė + λe).
In the remainder of the paper we shall use (8) to focus on selecting NFN weight tuning

algorithms that guarantee the stability of the filtered tracking error r(t). Then, since (3), with
the input considered as r(t) and the output as e(t) describes a stable system, standard techniques
[13], [15] guarantee that e(t) exhibits stable behaviour. In fact ∥e∥2 ≤ ∥r∥2/σmin, ∥ë∥2 ≤ ∥r∥2
with σmin(λ) the minimum singular value of λ. Generally λ is diagonal, so that σmin(λ) is the
smallest element of λ.

The following properties of the robot dynamics are required [15]. They hold for all revolute
rigid-link manipulator.

The inertial matrix M(q) is symmetric and positive definite. It is also bounded as a function of
q : m1I ≤ m2I. Ṁ(q)−2C(q, q̇) is a skew symmetric matrix, that is yT [Ṁ(q)−2C(q, q̇)]y, where
y is a n × 1 nonzero vector. The gravity vector G(q) is bounded as a function of q : G(q) ≤ gd.
The unknown disturbance satisfies ∥τd∥ ≤ db and this property is new, that is, the dynamic (8)
from ζ0(t) to r(t) are a state strict passive system.

3 Structure of NFN

In the RANFNC scheme, as shown in Fig.3, an NFN estimator is designed to tune the
nonlinear dynamic function vector, and then, the estimative vector is used to indirectly develop
a stable RANFNC law. An NFN controller is directly designed to imitate a predetermined
model-based stabilizing control law, and then, the stable control performance can be achieved
by using joint position and filtered error vector and information. In this paper, a four-layer NFN



344 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

d/dt
e

+

-q

iq

Neural-Fuzzy

Network

(NFN)

Adaptive

Law

Proposed Robust Adaptive Neural-Fuzzy-Network Control System

i

Robot Manipulator

e

e

)(tq
r Reference

Model

d
q

eer
+

+

v
K

r

Sgn(•) fK

-

)(ˆ xf

F )( xL

ŵ

d/dt

e

e

)(tq
d

Figure 3: Block diagram of RANFNC scheme

structure showns in Fig.2, which is composed of input, membership, rule, and output layers, is
adopted to implement the NFN estimate in RANFNC. The signal propagation and the basic
function in each layer of the NFN are introduced as follows.

1. Input layer transmits the input linguistic variables zb|1,2,··· ,r to the next layer.
2. Membership layer represents the input values with the following Gaussian membership

functions:

µ
j
b(zb) = exp

[
−
(zb − m

j
b)

2

(t
j
b)

2

]
(10)

Where exp[·] is the exponential function, mjb and t
j
b(b = 1,2, · · · ,r;j = 1,2, · · · ,z) are the

mean and the standard deviation of the Gaussian function in the jth term of the bth input
variable zb to the node of membership layer, respectively. It can be referred as the fuzzification
procedure.

3. The output of each node in the rule layer is determined by fuzzy And operation. Each
node in this rule layer is denoted by

∏
, which multiplies the input signals and output the result

of the product. The product operation is utilized to determine the firing strength. It can be
referred as the fuzzy inference mechanism. The output of this layer is given as:

lk =

r∏
b=1

wkjbµ
j
b(zb) (11)

Where lk|k = 1,2, · · · ,p represents the kth output of the rule layer, wkjb which represents the
weights between the membership layer and the rule layer, is assumed to be unity, and p is the
number of rules.

4. Final layer is the output layer, and nodes in this layer represent the output linguistic
variables. Each node in the output layer yi(1,2, · · · ,n) is labelled as

∑
, which computes the



Robust Adaptive Neural-Fuzzy Network Tracking Control for Robot Manipulator 345

overall output as the summation of all input signals, and be represented as:

yi =

p∑
k=1

wiklk (12)

The output node, together with the links connected to it, acts as a defuzzifier. It can be
referred as the normal defuzzification procedure. Moreover, it can be rewritten in the following
vector form:

y =
[
y1 y2 . . . yn

]T
= WL (13)

Where

W =



w11 w12 · · · w1p
w21 w22 · · · w2p
...

...
. . .

...
wn1 wn2 · · · wnp


 =

[
w1 w2 · · · wn

]
∈ ℜn×p,

L =
[
l1 l2 · · · ln

]T ∈ ℜp×1
In the RANFNC scheme, the NFN is used to estimate unmodeled nonlinear function, More-

over, the RANFNC law and adaptive tuning algorithms for NFN are introduced from the stability
analyses of the closed-loop system by using Lyapunov method. The input of the NFN estimator
are the elements in the filtered error vector and joint positions signal, the output of the NFN
estimator are the nonlinear dynamic function vectors in the local models.

Based on the powerful approximation ability [4], there exists an optimal NFN estimator to
approximate the nonlinear dynamic function in (5) such that

f(x) = W∗L(x) + ε(x) (14)

With W∗ the ideal weight matrix and the estimative error vector ε(x) ∈ ℜn×1 are assumed
to be given by

W̃ = arg min
Ŵ∈Mx

[
sup
x∈Mx

∥ f(x) − ŴL(x) ∥
]
, ∥ ε ∥≤ εN (15)

In which ∥ · ∥ is the Euclidean norm, Mx and Mw are the predefined compact sets of x and
Ŵ , and the positive constant εN can be reduced arbitrarily by increasing the number of rules.

4 RANFNC Design

Define the NFN functional estimate by

f̂(x) = ŴL(x) (16)

With Ŵ the current values of the NFN weight as provided by the tuning algorithm. With
the ideal weights required in (14) define the weight deviations or weight estimation errors as

W̃ = W∗ − Ŵ (17)

With τ0 defined to be (7), select the control input

τ = τ0 − v = ŴL(x) + Kvr − v (18)



346 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

v = (εN + db)sgn(r) = Kfsgn(r) (19)

With v(t) a function to be determined to provide robustness in the face of the net recon-
struction error ε. Then, the closed-loop filtered error dynamics become

Mṙ = −(Kv + C(q, q̇))r + W̃L(x) + (ε + τd) + v = −(Kv + C(q, q̇))r + ζ1 (20)

Theorem 1: Consider an n-link robot manipulator represented (1). If the RANFNC law is
designed as (18), (19) and the weight update law is designed as (21), then the stability of the
proposed RANFNC system can be ensured

+

-

)(tq

)(tq

)(tq

e

e

)(tq
d

)(tq
r

)(tq
d

d/dt
Reference

Model

d/dt

Robot Manipulator 

MCG)(tq

)(),(

)(

qGqqqC

qekrM
dcfl

)sgn()sgn(
1 T

c
erMk

fl

r
K

a
k

r

r

)sgn(rd/dt

Error

Function
Sgn(•)

+

+

Robust Feedback Linearization Control System

(c)

+

-

)(tq
d

)(tq

e

k k

pd )(tq
)()()( tektekt

pd

PD Control System

Robot Manipulator 

d/dt

e

(a)

+

-

)(tq

Computed Control System

)(tq

)(tq

e

e

)(tq
d

)(tq
r

)(tq
d

ct

)(),(

)(

qGqqC

ekekqM
badct

a
k bk

d/dt
Reference

Model

d/dt

Robot Manipulator 

MCG)(tq

(b)

Figure 4: (a) PD control system, (b) computed torque control (CTC) system, (c) robust feedback
linearization control (RFLC) system



Robust Adaptive Neural-Fuzzy Network Tracking Control for Robot Manipulator 347

˙̂
W = FL(x)rT (21)

Proof: Define a Lyapunov function candidate as

V (r(t),W̃) =
1

2
rT Mr +

1

2
tr(W̃T F−1W̃) (22)

Where tr(·) is a trace operator. By differentiating (22) with respect to time and using (19),
(20), (21), and using properties of the robot dynamics are introduced in section 2, one can obtain.

V̇ =
1

2
rT Mṙ +

1

2
rT Ṁr + tr(W̃T F−1W̃)

= −rT Kvr +
1

2
rT (Ṁ − 2C)r + trW̃(F−1 ˙̃W + LrT )

+ rT (ε + τd) − rT (εN + db)sgn(r)
= −rT Kvr + rT (ε + τd) − rT (εN + db)sgn(r)
= −rT Kvr + rT (ε + τd) − ∥r∥(εN + db)
≤ −rT Kvr ≤ 0

(23)

Since V̇ (r(t),W̃) ≤ 0, V̇ (r(t),W̃) is a negative semidefinite function, i.e. V (r(t),W̃) ≤
V (r(0),W̃). It implies that r(t) and W̃ is bounded functions. Let function h(t) ≡ rT Kvr ≤ −V̇
and integrate function h(t) with respect to time

τ∫
0

h(t)dτ ≤ V (r(0),W̃) − V (r(t),W̃) (24)

Because V (r(0),W̃) is a bounded function, and V (r(t),W̃) is a nonincreasing and bounded
function, the following result can be concluded:

lim
t→∞

τ∫
0

h(t)dτ < ∞ (25)

In addition, ḣ(t) is bounded; thus, by Barbalats lemma can be shown that lim
t→∞

h(t) = 0. It
can imply that r(t) will be converging to zero as time tends to infinite.

5 Numerical Simulation

A two-link robot manipulator as shown in Fig.1 is utilized in this paper to verify the effective-
ness of the proposed control scheme. The detailed system parameters of this robot manipulator
are given as: link mass m1,m2(kg), lengths l1, l2(m), angular positions q1,q2(rad).

The parameters for the equation of motion (1) are adopted in [11].

M(q) =

[
(m1 + m2l

2
1) m2l1l2(s1s2 + c1c2)

m2l1l2(s1s2 + c1c2) m2l
2
2

]

C(q, q̇) = m2l1l2(c1s2 + s1c2)

[
0 −q̇2

−q̇1 0

]
, G(q) =

[
−(m1 + m2)l1gs1

−m2l2gs2

]



348 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

Where q ∈ ℜ2 and the shorthand notations c1 = cos(q1), c2 = cos(q2), s1 = sin(q1) and
s2 = sin(q2) are used.

For the convenience of the simulation, the nominal parameters of the robotic system are
given as m1 = 4.6(kg), m2 = 2.3(kg), l1 = 0.5(m), l2 = 0.2(m), g = 9.8(m/s2) and the initial
conditions q1(0) = 0.5, q2(0) = 0.5, q̇1(0) = 0, q̇2(0) = 0. The desired reference trajectories are
qd1(t) = sin(2t), qd2(t) = cos(2t) respectively.

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(a)

P
os

it
io

n 
li

nk
 1

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(b)

P
os

it
io

n 
li

nk
 2

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−0.6

−0.4

−0.2

0

0.2

Time(s)
(c)

E
rr

or
 l

in
k 

1 
(r

ad
)

0 2 4 6 8 10
−1

−0.5

0

0.5

1

Time(s)
(d)

E
rr

or
 l

in
k 

2 
(r

ad
)

0 2 4 6 8 10
−50

0

50

100

Time(s)
(e)

C
on

tr
ol

 T
or

qu
e 

1 
(N

m
)

0 2 4 6 8 10
−10

0

10

20

Time(s)
(f)

C
on

tr
ol

 T
or

qu
e 

2 
(N

m
)

Figure 5: Simulated position responses, tracking errors, and control torques of the CTC control
system at joints 1 and 2

The most important parameters that effect the control performance of the robotic system are
the external disturbance and the friction term which are denoted tl(t) and f(q̇), in simulation,
parameter variation situation and disturbance situation occurring at 5s are considered. The pa-
rameter variation situation is that 1(kg) weight is added to the mass of link 2, i.e. m2 = 3.3(kg).
The disturbance situation is that external forces are injected into the robotic system, and their
shapes are expressed as follows: tl(t) =

[
5 sin(5t) 5 cos(5t)

]T
. In addition, friction forces are

also considered in this simulation and given as: f(q̇) =
[
2q̇1 + 0.8sgn(q̇1) 4q̇2 + 0.1sgn(q̇2)

]T
.

To this end, the simulation situations are adopted to demonstrate the robust property of the
proposed control scheme. In order to exhibit the superior control performance of the proposed
RANFNC scheme, three extra control systems including an RFLC system shows in Fig. 4(c),
a conventional computed torque control (CTC) and a proportional differential (PD) control are
examined in the mean time [13]. Moreover the conventional CTC system as shown in Fig.4(b)can
be expressed as

τcd = M(q̈d + kaė(t) + kbe(t)) + C(q, q̇) + G(q) (26)



Robust Adaptive Neural-Fuzzy Network Tracking Control for Robot Manipulator 349

The PD control system as shown in Fig. 4(a) can be expressed as

τpd = kαe(t) + kβė(t) (27)

The gain in these control system are given as

kα =

[
2500 0

0 1000

]
, kβ =

[
20 0

0 25

]
, ka =

[
6 0

0 6

]
, kb =

[
9 0

0 9

]
, Kv =

[
5 0

0 5

]
,

η = 20, Kr = db = 5. (28)

The gain matrices of ka, and kb, are determined so that the roots of the characteristic poly-
nomial of kaė + kbe lie strictly in the open left haft of the complex plane, i.e. lim

t→∞
e(t) = 0. It

means that the CTC system shown in (26) is globally asymptotically stable as the root dynamics
(1) without the consideration of system uncertainties. However, the stability of the closed-loop
control system may be destroyed if the system dynamics are perturbed by external disturbance.
However gains kα and kβ in the PD control are selected according to the Ziegler-Nichols tuning
rule. In addition, the selection of learning rates η is dependent on the significance of tuning
objects.

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(a)

P
os

it
io

n 
li

nk
 1

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(b)

P
os

it
io

n 
li

nk
 2

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−0.5

0

0.5

Time(s)
(c)

E
rr

or
 l

in
k 

1 
(r

ad
)

0 2 4 6 8 10
−0.5

0

0.5

Time(s)
(d)

E
rr

or
 l

in
k 

2 
(r

ad
)

0 2 4 6 8 10
−500

0

500

Time(s)
(e)

C
on

tr
ol

 T
or

qu
e 

1 
(N

m
)

0 2 4 6 8 10
−100

−50

0

50

100

Time(s)
(f)

C
on

tr
ol

 T
or

qu
e 

2 
(N

m
)

Figure 6: Simulated position responses, tracking errors, and control torques of the RFLC control
system at joints 1 and 2

The simulated results of CTC system, the responses of joint position, tracking error and
control torque are depicted Fig. 5(a-d), and (e-f), respectively. From the simulated results, in
interval from beginning to 5s, favourable tracking responses can only be obtained for the nominal
situation. However, since the control gains in (28) are determined without considering the joint



350 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

friction and external disturbance. So, poor tracking responses after 5s are resulted due to the
occurrence of joint friction and external disturbance. In the RFLC system are depicted in Fig.
6. The joint position responses, tracking error and control torque are depicted in Fig. 6(a-d)
and (e-f), respectively. The robust control performance of the RFLC system is obvious under
the occurrence of system uncertainties. However, the undesirable chattering phenomenon in the
control torque.

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(a)

P
os

it
io

n 
li

nk
 1

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(b)

P
os

it
io

n 
li

nk
 2

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−0.5

0

0.5

Time(s)
(c)

E
rr

or
 l

in
k 

1 
(r

ad
)

0 2 4 6 8 10
−0.2

0

0.2

0.4

0.6

Time(s)
(d)

E
rr

or
 l

in
k 

2 
(r

ad
)

0 2 4 6 8 10
−1000

−500

0

500

1000

Time(s)
(e)

C
on

tr
ol

 T
or

qu
e 

1 
(N

m
)

0 2 4 6 8 10
−400

−200

0

200

400

Time(s)
(f)

C
on

tr
ol

 T
or

qu
e 

2 
(N

m
)

Figure 7: Simulated position responses, tracking errors, and control torques of the PD control
system at joints 1 and 2

The PD control system based on model-free design is offered to apply comparable responses
to manifest the performance of the RANFNC system. The simulated responses of joint position,
tracking error, and control torque, are represented in Fig. 7(a-d) and (e-f). From simulated
results see that, the tracking responses are greatly improved and the chattering phenomenon are
much reduced.

Now, the proposed RANFNC system depicted in Fig. 3 is applied to control the robot
manipulator for comparison. The simulated results of joint position responses, tracking error
and control torque are depicted in Fig. 8(a-d) and (e-f), respectively. Because of, all parameters
in the NFN are roughly initialized, the tracking errors are gradually decreased though online
training process whether the uncertainties exist or not. Furthermore, robust control performance
of the RANFNC system, both in the condition of joint friction, parameter variation, and external
disturbance are obvious. Compared these results with the CTC, RFLC and PD control systems,
the control torque of proposed RANFNC system is not chattering phenomenon.



Robust Adaptive Neural-Fuzzy Network Tracking Control for Robot Manipulator 351

6 Conclusions

This paper has successfully implemented an RANFNC system to control the joint position
of a two-link robot manipulator for achieving desired position control. All the system dynamic
could be unknown and no strict constraints. The NFN is used to compensate the uncertainty of
the system. All adaptive learning laws in the RANFNC system were derived in the sense of a pro-
jection algorithm and Lyapunov theorem so that the network convergence and system-tracking
stability of the closed-loop control system can be ensured whether or not the uncertainties oc-
cur. Simulated results of a two link robot manipulator via various existing control frameworks
including CTC RFLC and PD control were also applied in this paper to compare and display
the manipulative performance of the proposed control system. According to the result as depict
in Figs. 5-8, the desired position tracking response of the RANFNC system can be controlled
closely follow specific reference trajectories under wide range of disturbance. The main of the
paper is to construct a simpler and more efficient intelligent control system without dynamic
knowledge of plant. While ensuring the convergence and tracking stability of the closed-loop
system. The proposed RANFNC system can also be applied to other systems, such as mobile
robotic, AC servo system and so on.

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(a)

P
os

it
io

n 
li

nk
 1

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−2

−1

0

1

2

Time(s)
(b)

P
os

it
io

n 
li

nk
 2

 (
ra

d)

 

 
Desired Position

0 2 4 6 8 10
−0.6

−0.4

−0.2

0

0.2

Time(s)
(c)

E
rr

or
 l

in
k 

1 
(r

ad
)

0 2 4 6 8 10
−0.2

0

0.2

0.4

0.6

Time(s)
(d)

E
rr

or
 l

in
k 

2 
(r

ad
)

0 2 4 6 8 10
−50

0

50

100

Time(s)
(e)

C
on

tr
ol

 T
or

qu
e 

1 
(N

m
)

0 2 4 6 8 10
−100

−50

0

50

100

Time(s)
(f)

C
on

tr
ol

 T
or

qu
e 

2 
(N

m
)

Figure 8: Simulated position responses, tracking errors, and control torques of the RANFNC
control system at joints 1 and 2

7 Acknowledgment

This work was supported by the National Natural Science Foundation of China (60775047;
60835004), the National High Technology Research and Development Program of China (863



352 T. Ngo, Y. Wang, T.L. Mai, M.H. Nguyen, J. Chen

Program) (2007AA04Z244; 2008AA04Z214). The authors would like to thank the associate
editor and the reviewers for their valuable comments.

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