Adaptive Fuzzy-Neural Control Utilizing Sliding Mode Based Learning Algorithm for Robot Manipulator

Thuy Van Tran. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 3, ( Part -4) March 2016, pp.131-137
www.ijera.com 131 | P a g e
Adaptive Fuzzy-Neural Control Utilizing Sliding Mode Based
Learning Algorithm for Robot Manipulator
Thuy Van Tran*, YaoNan Wang**
* (College of Electrical and Information Engineering, Hunan University, China;
Faculty of Electrical Engineering, Hanoi University of Industry, Vietnam;
Email: tranthuyvan.haui@gmail.com)
** (College of Electrical and Information Engineering, Hunan University, China;
Email: yaonan@hnu.edu.cn)
ABSTRACT
This paper introduces an adaptive fuzzy-neural control (AFNC) utilizing sliding mode-based learning algorithm
(SMBLA) for robot manipulator to track the desired trajectory. A traditional sliding mode controller is applied to
ensure the asymptotic stability of the system, and the fuzzy rule-based wavelet neural networks (FWNNs) are
employed as the feedback controllers. Additionally, a novel adaptation of the FWNNs parameters is derived
from the SMBLA in the Lyapunov stability theorem. Hence, the AFNC approximates parameter variation,
unmodeled dynamics, and unknown disturbances without the detailed knowledge of robot manipulator, while
resulting in an improved tracking performance. Lastly, in order to validate the effectiveness of the proposed
approach, the comparative simulation results of two-degrees of freedom robot manipulator are presented.
Keywords – traditional sliding mode control (TSMC), adaptive fuzzy neural control (AFNC), fuzzy rule-based
wavelet neural network (FWNN), sliding mode-based learning algorithm (SMBLA), degrees of freedom robot
manipulator (DOFRM)
I. INTRODUCTION
Generally, various uncertainties comprising
parameter variation, unmodeled dynamics, and
unknown disturbances influence the tracking
performances of robot manipulator [1, 2]. In the
designing of reference model based control system,
it is difficult for determining a mathematical model
correctly. Because the traditional controllers (i.e.,
robust controller [3], sliding mode controller [4]) are
time-invariant controllers, this term causes
nonlinearities and discontinuities which renders
traditional control invalid. So the requirement of the
intelligent control approaches (ICAs) is that
reducing the impact of the various uncertainties in
the design process. During the last decades, the ICAs
(i.e., neural network control (NNC) [5], and fuzzy
logic control (FLC) [6]) have been largely applied
for controlling the motion of robot manipulators [7,
8]. The topical trend of researches is that integrating
the traditional control methods with the ICAs for the
improvement in the performance of system [9-11].
Besides, based on the combination of the rule
reasoning of fuzzy systems and the learning
capability of neural networks without the prior
knowledge, the fuzzy-neural network control
(FNNC) methods are also designed to provide higher
robustness than both NNC and FLC [12-14].
In the training of artificial neural networks
(ANNs) and fuzzy-neural networks (FNNs),
different learning algorithms containing gradient
descent-based algorithm (GDBA) [15] and
evolutionary computation-based algorithm (ECBA)
[16, 17] have been utilized. However, the
convergence rate of GDBA is sluggish due to the
involvement of partial derivatives, specifically when
the solution space is complicated. For the ECBA, the
stability and optimal values are difficultly reached
by using stochastic operators, and the high
calculation is still a burden. It is well-known that
sliding mode control (SMC) is a method which can
ensure the stability and robustness in both the case
of uncertainties and computationally intelligent
systems [18]. By using the SMC strategy in the
online learning for ANNs and FNNs, sliding mode-
based learning algorithm (SMBLA) can guarantee
better convergence and more robust than
conventional learning approaches [19, 20]. It is
different from GDBA in feedback-error learning
[21], the network parameters are updated by
SMBLA in the way that the learning error is
enforcedly satisfied a stable equation.
In this paper, an adaptive fuzzy-neural
control (AFNC) using SMBLA is proposed for
tracking desired trajectory of robot manipulator. In
the proposed control method, the traditional sliding
mode controller (TSMC) is applied for guaranteeing
the asymptotic stability of the control system, and
the fuzzy rule-based wavelet neural networks
RESEARCH ARTICLE OPEN ACCESS

(FWNNs) are employed as feedback controllers to
approximate the uncertainties. Moreover, a novel
SMBLA strategy is suggested to train the FWNNs
using wavelet basis membership function (WBMF)
[22]. By using Lyapunov theorem to prove the
stability of the SMBLA, the fast convergence ability
of the FWNNs parameters is ensured, and an
adaptive updating law is achieved. Hence, the
proposed method approximates the uncertainties
without the detailed knowledge of robot
manipulator, while resulting in an improved
performance. Last of all, the comparative simulation
results of two-degrees of freedom (DOF) robot
manipulator are presented for validating the
effectiveness of the proposed AFNC system.
The remainder of the paper is organized as
follows: section 2 represents the preliminaries. In
section 3, the AFNC scheme and the SMBLA are
presented. Section 4 provides the comparative
simulation results of two-DOF robot manipulator.
Finally, the conclusion is shown in section 5.
II. PRELIMINARIES
1. Dynamic Model of Robot Manipulator
Consider an 𝑛-DOF robot manipulator, the
dynamics can be represented in Lagrange formation
[23]:
𝑴 𝑟 𝜽 𝜽 + 𝑽 𝑟 𝜽, 𝜽 𝜽 + 𝒈 𝑟 𝜽 + 𝜼 𝑒 = 𝒖 𝜏 (1)
where 𝑴 𝑟 (𝜽) ∈ 𝑅 𝑛×𝑛
is the inertial matrix,
𝑽 𝑟(𝜽, 𝜽) ∈ 𝑅 𝑛×𝑛
is the Coriolis-centripetal matrix,
𝒈 𝑟 (𝜽) ∈ 𝑅 𝑛
is the gravity vector, 𝜼 𝑒 ∈ 𝑅 𝑛
is the
vector of unknown disturbances, 𝒖 𝜏 ∈ 𝑅 𝑛
is the
vector of control torques, and 𝜽 𝑡 ∈ 𝑅 𝑛
, 𝜽 𝑡 , and
𝜽(𝑡) are the vectors of joint positions, corresponding
velocities, and corresponding accelerations,
respectively.
2. FWNN Structure and Fuzzy If-Then Rule
The structure of a five-layer FWNN, as
depicted in Figure 1, contains two input neurons,
𝑝 + 𝑞 membership neurons, 𝑝 × 𝑞 rule neurons,
𝑝 × 𝑞 normalization neurons, and one output
neuron.
Figure 1: Structure of FWNN
Consider a zeroth-order Takagi-Sugeno-
Kang model containing two input variables, the
fuzzy If-Then rules is described as follows:
𝑟𝑖𝑗 : 𝐼𝑓 𝑦1 𝑖𝑠 𝐴𝑖 𝑎𝑛𝑑 𝑦2 𝑖𝑠 𝐵𝑗 , 𝑇𝑕𝑒𝑛 𝜑𝑖𝑗 = 𝑑𝑖𝑗 (2)
where 𝑦1 and 𝑦2 are the input variables of FWNN,
𝜑𝑖𝑗 is a zeroth-order function in the consequent
element of the rule 𝑟𝑖𝑗 , and 𝐴𝑖 and 𝐵𝑗 denote the
fuzzy sets of 𝑦1 and 𝑦2, respectively.
Input Layer (Layer 1): Given a input vector of two
crisp variables 𝒚 = [𝑦1, 𝑦2] 𝑇
∈ 𝑅2
, their values are
transmitted to the next layer by the neurons in this
layer.
Membership Layer (Layer 2): By using WBMF,
the membership neurons map 𝑦1 and 𝑦2 into
fuzzified values. These membership neurons have
the WBMFs represented by:
𝜇 𝐴 𝑖
𝑦1 = 1 − 𝛿 𝐴 𝑖
𝑦1 − 𝛼 𝐴 𝑖
2
𝑒− 𝛿 𝐴 𝑖
𝑦1−𝛼 𝐴 𝑖
2
𝜇 𝐵 𝑗
𝑦2 = 1 − 𝛿 𝐵 𝑗
𝑦2 − 𝛼 𝐵 𝑗
2
𝑒
− 𝛿 𝐵 𝑗
𝑦2−𝛼 𝐵 𝑗
2
(3)
where 𝜇 𝐴 𝑖
𝑦1 and 𝜇 𝐵 𝑗
𝑦2 are the membership
values, 𝛼 𝐴 𝑖
and 𝛼 𝐵 𝑗
are the translation parameters,
and 𝛿 𝐴 𝑖
and 𝛿 𝐵 𝑗
are the dilation parameters of
WBMF for input variables 𝑦1 and 𝑦2, respectively.
𝑖 = 1,2, … , 𝑝 and 𝑗 = 1,2, … , 𝑞.
Rule Layer (Layer 3): The output of each rule
neuron expresses a firing strength 𝑤𝑖𝑗 of
corresponding rule, and it is calculated by
multiplying two incoming signals:
𝑤𝑖𝑗 = 𝜇 𝐴 𝑖
𝑦1 𝜇 𝐵 𝑗
𝑦2 (4)
Normalization Layer (Layer 4): In this layer, the
normalization of all of the firing strengths is
performed. Then, the normalized value of every
neuron can be denoted as:
𝑤𝑖𝑗 =
𝑤 𝑖𝑗
𝑤 𝑖𝑗
𝑞
𝑗=1
𝑝
𝑖=1
(5)
Output Layer (Layer 5): The defuzzification is
performed in this layer. The output linguistic
variable is computed according to the weighted sum
technique of all incoming signals:
𝑧 = 𝑤𝑖𝑗 𝜑𝑖𝑗
𝑞
𝑗=1
𝑝
𝑖=1 =
𝑤 𝑖𝑗 𝑑 𝑖𝑗
𝑞
𝑗=1
𝑝
𝑖=1
𝑤 𝑖𝑗
𝑞
𝑗=1
𝑝
𝑖=1
(6)
III. DESIGN OF AFNC USING SMBLA
1. AFNC Scheme
An AFNC scheme, as illustrated in Figure
2, presents a combination of the sliding mode
controller in parallel with the FWNNs.

Figure 2: Structure of AFNC system
The first, the sliding mode controller is
designed for ensuring the asymptotic stability of the
control system. A sliding surface 𝜷 𝜀 is specified by:
𝜷 𝜀 𝜺, 𝜺 = 𝜺 + 𝜦 𝛽 𝜺 = 𝜽 𝑑 − 𝜽 + 𝜦 𝛽 𝜽 𝑑 − 𝜽 =
𝛽𝜀
1
, … 𝛽𝜀
𝑘
, … 𝛽𝜀
𝑛 𝑇
(7)
where 𝜦 𝛽 is a diagonal and positive definite constant
matrix defining the sliding surface slope, 𝑘 =
1,2, … 𝑛, and the vectors of desired positions, desired
velocities, feedback position errors, and feedback
velocity errors are denoted by
𝜽 𝑑 = 𝜃 𝑑
1
, … 𝜃 𝑑
𝑘
, … 𝜃 𝑑
𝑛 𝑇
,
𝜽 𝑑 = 𝜃 𝑑
1
, … 𝜃 𝑑
𝑘
, … 𝜃 𝑑
𝑛 𝑇
,
𝜺 = 𝜀1
, … 𝜀 𝑘
, … 𝜀 𝑛 𝑇
, and
𝜺 = 𝜀1
, … 𝜀 𝑘
, … 𝜀 𝑛 𝑇
, respectively. Then, the
sliding control law is defined as follows:
𝒖 𝑆 = 𝑢 𝑆
1
, … 𝑢 𝑆
𝑘
, … 𝑢 𝑆
𝑛 𝑇
= 𝜦 𝑆 𝜷 𝜀 =
𝜆 𝑆
1
… 0
⋮ ⋱ ⋮
0 … 𝜆 𝑆
𝑛
𝛽𝜀
1
, … 𝛽𝜀
𝑘
, … 𝛽𝜀
𝑛 𝑇
(8)
where 𝜦 𝑆 is a diagonal and positive definite gain
matrix, with 𝜆 𝑆
𝑘
> 0.
The second, the FWNNs are used as the
feedback controllers to approximate the uncertainties
in the system. For the 𝑘 𝑡𝑕
FWNN, the two inputs 𝑦1
𝑘
and 𝑦2
𝑘
are considered as 𝜀 𝑘
and 𝜀 𝑘
, and the output
𝑧 𝑘
is applied as the output of 𝑘 𝑡𝑕
feedback
controller. Then, the output vector of feedback
controllers is obtained as
𝒖 𝐹 = 𝑢 𝐹
1
, … 𝑢 𝐹
𝑘
, … 𝑢 𝐹
𝑛 𝑇
, where 𝑢 𝐹
𝑘
= 𝑧 𝑘
=
𝑤𝑖𝑗
𝑘
𝑑𝑖𝑗
𝑘𝑞
𝑗=1
𝑝
𝑖=1 .
Thus, the control input vector of the joint
torques, 𝒖 𝜏, is determined by:
𝒖 𝜏 = 𝒖 𝑆 − 𝒖 𝐹 = 𝑢 𝜏
1
, … 𝑢 𝜏
𝑘
, … 𝑢 𝜏
𝑛 𝑇
(9)
2. Sliding Mode-Based Learning Algorithm
Assumption 1: Consider that all of the input signals
(i.e., 𝑦1
𝑘
and 𝑦2
𝑘
) and their time derivatives (i.e., 𝑦1
𝑘
and 𝑦2
𝑘
) are bounded by:
𝑦1
𝑘
𝑡 ≤ 𝑏 𝑦 ; 𝑦1 𝑡 ≤ 𝑏 𝑦 ; ∀𝑡
𝑦2
𝑘
𝑡 ≤ 𝑏 𝑦 ; 𝑦2 𝑡 ≤ 𝑏 𝑦 ; ∀𝑡
(10)
with 𝑏 𝑦 and 𝑏 𝑦 are known positive constants.
Assumption 2: Suppose that all of the control input
torques and their time derivatives are bounded by:
𝑢 𝜏
𝑘
𝑡 ≤ 𝑏 𝑢 ; 𝑢 𝜏
𝑘
𝑡 ≤ 𝑏 𝑢 ; ∀𝑡 (11)
with 𝑏 𝑢 and 𝑏 𝑢 are known positive constants.
Definition 1: By utilizing the SMC theory in [24],
𝒖 𝑆 can be defined as a time-varying sliding surface:
𝜷 𝑢 𝒖 𝜏, 𝒖 𝐹 = 𝒖 𝑆 𝑡 = 𝒖 𝐹 𝑡 + 𝒖 𝜏 𝑡 = 0 (12)
Definition 2: A sliding motion sustains on (12) after
finite time 𝑡 𝑢 , if the satisfaction of the inequality
𝜷 𝑢 𝑡 𝑇
𝜷 𝑢 𝑡 < 0 is achieved for all time 𝑡 in
some non-trivial semi-open sub-interval of a form as
𝑡, 𝑡 𝑢 ⊂ (−∞, 𝑡 𝑢 ).
Theorem 1: Based on the above Assumptions and
Definitions, given an initial value 𝒖 𝑆 0 , the
convergence of the learning error 𝒖 𝑆 𝑡 to zero
within 𝑡 𝑢 can be guaranteed, if the adaptive learning
laws for the parameters of FWNNs are designed as:
𝛼 𝐴 𝑖
𝑘
= 𝑦1
𝑘
+ 𝑐𝐴 𝑖
𝑘
𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
𝛼 𝐵 𝑗
𝑘
= 𝑦2
𝑘
+ 𝑐 𝐵 𝑗
𝑘
𝑘
𝛿 𝐴 𝑖
𝑘
= 𝛿 𝐴 𝑖
𝑘
+
𝑕 𝐴 𝑖
𝑘
𝛿 𝐴 𝑖
𝑘
𝑐 𝐴 𝑖
𝑘
2 𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
𝛿 𝐵 𝑗
𝑘
= 𝛿 𝐵 𝑗
𝑘
+
𝑕 𝐵 𝑗
𝑘
𝛿 𝐵 𝑗
𝑘 𝑐 𝐵 𝑗
𝑘
2 𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
𝜑𝑖𝑗
𝑘
= −
𝑤 𝑖𝑗
𝑘
𝒘 𝑘 𝑇
𝒘 𝑘
𝑘
(13)
where 𝑐 𝐴 𝑖
𝑘
= 𝑦1
𝑘
− 𝛼 𝐴 𝑖
𝑘
, 𝑐 𝐵 𝑗
𝑘
= 𝑦2
𝑘
− 𝛼 𝐵 𝑗
𝑘
, 𝑕 𝐴 𝑖
𝑘
=
1 − 𝛿 𝐴 𝑖
𝑘
𝑐 𝐴 𝑖
𝑘 2
2 − 𝛿 𝐴 𝑖
𝑘
𝑐 𝐴 𝑖
𝑘 2
, 𝑕 𝐵 𝑗
𝑘
=
1 − 𝛿 𝐵 𝑗
𝑘
𝑐 𝐵 𝑗
𝑘
2
2 − 𝛿 𝐵 𝑗
𝑘
𝑐 𝐵 𝑗
𝑘
2
, 𝒘 𝑘
=
𝑤11
𝑘
, … 𝑤𝑖𝑗
𝑘
, … 𝑤𝑝𝑞
𝑘 𝑇
, 𝑠𝑔𝑛 . is the sign
function, and the learning speed 𝜗 is a sufficiently
large positive constant which is designed for
satisfying the condition 𝑏 𝑢 < 𝜗.
Proof of Theorem 1:
From (3), the time derivatives of the
membership functions in the 𝑘 𝑡𝑕
FWNN are written
as follows:
𝜇 𝐴 𝑖
𝑘
𝑦1
𝑘
= −2
𝜎 𝐴 𝑖
𝑘
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 𝜇 𝐴 𝑖
𝑘
𝑦1
𝑘
𝜇 𝐵 𝑗
𝑘
𝑦2
𝑘
= −2
𝜎 𝐵 𝑗
𝑘
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘 𝜇 𝐵 𝑗
𝑘
𝑦2
𝑘
(14)
where:
𝜎𝐴 𝑖
𝑘
= 𝛿 𝐴 𝑖
𝑘
𝑐 𝐴 𝑖
𝑘
= 𝛿 𝐴 𝑖
𝑘
𝑦1
𝑘
− 𝛼 𝐴 𝑖
𝑘
𝜎 𝐵 𝑗
𝑘
= 𝛿 𝐵 𝑗
𝑘
𝑐 𝐵 𝑗
𝑘
= 𝛿 𝐵 𝑗
𝑘
𝑦2
𝑘
− 𝛼 𝐵 𝑗
𝑘
(15)
By differentiating (15), yields:
𝜎𝐴 𝑖
𝑘
= 𝛿 𝐴 𝑖
𝑘
𝑦1
𝑘
− 𝛼 𝐴 𝑖
𝑘
+ 𝛿 𝐴 𝑖
𝑘
𝑦1
𝑘
− 𝛼 𝐴 𝑖
𝑘
𝜎 𝐵 𝑗
𝑘
= 𝛿 𝐵 𝑗
𝑘
𝑦2
𝑘
− 𝛼 𝐵 𝑗
𝑘
+ 𝛿 𝐵 𝑗
𝑘
𝑦2
𝑘
− 𝛼 𝐵 𝑗
𝑘
(16)
The time derivative of 𝑤𝑖𝑗
𝑘
is expressed as:
𝑤𝑖𝑗
𝑘
= 𝜇 𝐴 𝑖
𝑘
𝜇 𝐵 𝑗
𝑘
+ 𝜇 𝐵 𝑗
𝑘
𝜇 𝐴 𝑖
𝑘
= −𝑠𝑖𝑗
𝑘
𝑤𝑖𝑗
𝑘
(17)
where:

𝑠𝑖𝑗
𝑘
= 2
𝜎 𝐴 𝑖
𝑘
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 +
𝜎 𝐵 𝑗
𝑘
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘 (18)
According to (17) and (18), 𝑤𝑖𝑗
𝑘
is determined as
𝑤𝑖𝑗
𝑘
= −𝑤𝑖𝑗
𝑘
𝑠𝑖𝑗
𝑘
+ 𝑤𝑖𝑗
𝑘
𝑤𝑖𝑗
𝑘
𝑠𝑖𝑗
𝑘𝑞
𝑗=1
𝑝
𝑖=1 (19)
From (13), (15) and (16), it can be obtained that
𝜎 𝐴 𝑖
𝑘
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 =
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 𝛿 𝐴 𝑖
𝑘
𝑐𝐴 𝑖
𝑘
+ 𝛿 𝐴 𝑖
𝑘
𝑦1
𝑘
− 𝛼 𝐴 𝑖
𝑘
= 𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
𝜎 𝐵 𝑗
𝑘
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘 =
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘 𝛿 𝐵 𝑗
𝑘
𝑐 𝐵 𝑗
𝑘
+ 𝛿 𝐵 𝑗
𝑘
𝑦2
𝑘
− 𝛼 𝐵 𝑗
𝑘
= 𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
(20)
Take a Lyapunov function as follows:
ℒ1 𝑡 =
1
2
𝜷 𝑢 𝑡 𝑇
𝜷 𝑢 𝑡 =
1
2
𝒖 𝑆
𝑇
𝒖 𝑆 =
1
2
𝑢 𝑆
𝑘 2𝑛
𝑘=1 (21)
By differentiating (21) with respect to time
yields:
ℒ1 𝑡 = 𝜷 𝑢 𝑡 𝑇
𝜷 𝑢 𝑡 = 𝑢 𝑆
𝑘
𝑢 𝑆
𝑘𝑛
𝑘=1 =
𝑢 𝐹
𝑘
+ 𝑢 𝜏
𝑘
𝑢 𝑆
𝑘𝑛
𝑘=1 (22)
By using (13), (18), (19), and (20), 𝑢 𝐹
𝑘
is
obtained as follows:
𝑢 𝐹
𝑘
= 𝑤𝑖𝑗
𝑘
𝜑𝑖𝑗
𝑘
+ 𝑤𝑖𝑗
𝑘
𝜑𝑖𝑗
𝑘𝑞
𝑗=1
𝑝
𝑖=1 =
−𝑤𝑖𝑗
𝑘 𝑤 𝑖𝑗
𝑘
𝒘 𝑘 𝑇
𝒘 𝑘
𝑘
+
𝑞
𝑗=1
𝑝
𝑖=1
−2𝑤𝑖𝑗
𝑘
𝜎 𝐴 𝑖
𝑘
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 +
𝜎 𝐵 𝑗
𝑘
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘 +
𝑤𝑖𝑗
𝑘
2𝑤𝑖𝑗
𝑘
𝜎 𝐴 𝑖
𝑘
𝜎 𝐴 𝑖
𝑘
𝑕 𝐴 𝑖
𝑘 +
𝜎 𝐵 𝑗
𝑘
𝜎 𝐵 𝑗
𝑘
𝑕 𝐵 𝑗
𝑘
𝑞
𝑗=1
𝑝
𝑖=1 𝜑𝑖𝑗
𝑘
=
−𝑤𝑖𝑗
𝑘 𝑤 𝑖𝑗
𝑘
𝒘 𝑘 𝑇
𝒘 𝑘
𝑘
+
𝑞
𝑗=1
𝑝
𝑖=1
−4𝑤𝑖𝑗
𝑘
𝑘
+
𝑤𝑖𝑗
𝑘
4𝑤𝑖𝑗
𝑘
𝑘𝑞
𝑗=1
𝑝
𝑖=1 𝜑𝑖𝑗
𝑘
=
𝑘
−𝑤𝑖𝑗
𝑘 𝑤 𝑖𝑗
𝑘
𝒘 𝑘 𝑇
𝒘 𝑘
+ −4𝑤𝑖𝑗
𝑘
+
𝑞
𝑗=1
𝑝
𝑖=1
4𝑤𝑖𝑗
𝑘
𝑤𝑖𝑗
𝑘𝑞
𝑗=1
𝑝
𝑖=1 𝜑𝑖𝑗
𝑘
=
𝑘
−𝑤𝑖𝑗
𝑘 𝑤 𝑖𝑗
𝑘
𝒘 𝑘 𝑇
𝒘 𝑘
𝑞
𝑗=1
𝑝
𝑖=1 = −𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
(23)
Based on (23), the Assumptions and the
condition 𝑏 𝑢 < 𝜗, it is concluded that ℒ 𝑡 must be
lower than zero for satisfying the stability of the
learning:
ℒ1 𝑡 = −𝜗𝑠𝑔𝑛 𝑢 𝑆
𝑘
+ 𝑢 𝜏
𝑘
𝑢 𝑆
𝑘𝑛
𝑘=1 ≤
−𝜗 𝑢 𝑆
𝑘
+ 𝑢 𝜏
𝑘
𝑢 𝑆
𝑘𝑛
𝑘=1 ≤
−𝜗 𝑢 𝑆
𝑘
+ 𝑏 𝑢 𝑢 𝑆
𝑘𝑛
𝑘=1 < 0, ∀𝑢 𝑆
𝑘
≠ 0 (24)
This completes the proof.
Assumption 3: Assume that the desired position
vector 𝜽 𝑑 𝑡 is uniformly continuous and
differentiable, and the vectors 𝜽 𝑑 𝑡 , 𝜽 𝑑 𝑡 and
𝜽 𝑑 𝑡 are bounded.
Theorem 2: Consider a dynamics system as (1),
under all of Assumptions and Definitions, if an
AFNC law is defined as (9), and an online
adaptation strategy for the parameters of FWNNs is
designed as (13), then the convergence of tracking
errors and the stability of proposed control system
can be ensured.
Proof of Theorem 2:
By using (8) and (12), a relation between 𝜷 𝜀 and
𝜷 𝑢 is presented as follows:
𝜷 𝜀 = 𝜦 𝑆
−1
𝜷 𝑢 = 𝜦 𝑆
−1
𝒖 𝑆 =
1
𝜆 𝑆
1 … 0
⋮ ⋱ ⋮
0 …
1
𝜆 𝑆
𝑛
𝑢 𝑆
1
, … 𝑢 𝑆
𝑘
, … 𝑢 𝑆
𝑛 𝑇
(25)
For analyzing the tracking performance of the
control system, a Lyapunov function is considered as
follows:
ℒ2 𝑡 =
1
2
𝜷 𝜀 𝑡 𝑇
𝜷 𝜀 𝑡 (26)
Based on (25) and Theorem 1, the negative-
definiteness of the time derivative of ℒ2 𝑡 can be
guaranteed:
ℒ2 𝑡 = 𝜷 𝜀 𝑡 𝑇
𝜷 𝜀 𝑡 =
1
𝜆 𝑆
𝑘 2 𝑢 𝑆
𝑘
𝑢 𝑆
𝑘𝑛
𝑘=1 ≤
1
𝜆 𝑆
𝑘 2 −𝜗 𝑢 𝑆
𝑘
+ 𝑏 𝑢 𝑢 𝑆
𝑘𝑛
𝑘=1 < 0, ∀𝑢 𝑆
𝑘
≠ 0
(27)
This completes the proof.
IV. COMPARATIVE SIMULATION
RESULTS
Consider a two-DOF robot manipulator
with the dynamics parameters as follows:
𝑴 𝑟 𝜽 =
𝑙1
2
𝑚1 + 𝑙1
2
+ 𝑙2
2
𝑚2 + 𝜉 𝑚 , 𝑙2
2
𝑚2 + 𝜉 𝑚
𝑙2
2
𝑚2 + 𝜉 𝑚 , 𝑙2
2
𝑚2
(28)
𝑽 𝑟 𝜽, 𝜽 =
−𝑙1 𝑙2 𝑚2 𝜃2 𝑠𝑖𝑛 𝜃2 , −𝑙1 𝑙2 𝑚2 𝜃1 + 𝜃2 𝑠𝑖𝑛 𝜃2
𝑙1 𝑙2 𝑚2 𝜃1 𝑠𝑖𝑛 𝜃2 , 0
(29)
𝒈 𝑟 𝜽 =
9.81
𝑙1 𝑚1 + 𝑚2 𝑐𝑜𝑠 𝜃2 + 𝑙2 𝑚2 𝑐𝑜𝑠 𝜃1 + 𝜃2
𝑙2 𝑚2 𝑐𝑜𝑠 𝜃1 + 𝜃2
(30)
where 𝜉 𝑚 = 2𝑙1 𝑙2 𝑚2 𝑐𝑜𝑠 𝜃2 .
In order to demonstrate the robustness and
the superior control performance of the proposed
AFNC, both the AFNC system and the proportional
differential control (PDC) system [2] are simulated
for comparison.
The PDC system is illustrated in Figure 3,
and the PDC law is defined as
𝒖 𝑝𝑑 = 𝑲 𝑝 𝜺 + 𝑲 𝑑 𝜺 (31)
where the gain matrices 𝑲 𝑝 and 𝑲 𝑑 are derived from
the tuning rules of Ziegler Nichols [25] by a
compromise between the control performance and
the stability:

𝑲 𝑝 =
1100 0
0 600
; 𝑲 𝑑 =
45 0
0 30
(32)
Figure 3: Structure of PDC system
In the AFNC system, all values of 𝛿 𝐴 𝑖
𝑘
, 𝛼 𝐴 𝑖
𝑘
,
𝛿 𝐵 𝑗
𝑘
, and 𝛼 𝐵 𝑗
𝑘
are randomly initialised in the range of
−0.1, 0.1 , and other detailed parameters are
given as follows:
𝑝 = 𝑞 = 5; 𝜗 = 0.01; 𝜦 𝛽 =
5 0
0 5
; 𝜦 𝑆 =
60 0
0 60
(33)
The nominal parameters of the robot system
are given as in Tables 1.
Table 1: The nominal parameters of the
robot system
DOF DOF 1 DOF 2
Mass (kg) 𝑚1 = 3 𝑚2 = 1.5
Length (m) 𝑙1 = 0.5 𝑙2 = 0.9
Initial position (rad) 𝜃1 0 = 0.8 𝜃2 0 = 0.8
Initial velocity (rad/s) 𝜃1 0 = 0 𝜃2 0 = 0
Desired trajectory
(rad)
𝜃 𝑑1
𝑡 = 𝑒−𝑡
𝜃 𝑑2
𝑡 = 𝑒−2𝑡
Herein, the simulation is implemented in
two cases as follows:
Case 1: Have no the parameter variation, and
consider the external disturbances term as:
𝜼 𝑒 = 4𝑒−0.6𝑡
, 6𝑒−0.4𝑡 𝑇
(34)
Case 2: 𝜼 𝑒 as in (34), and the parameter variation
(i.e., a tip load, 1 (kg), on DOF 2) is considered.
Besides, the root mean square error (RMSE)
method is utilized to record the individual
performance of control systems:
𝑅𝑀𝑆𝐸𝑘 =
1
𝑇 𝜔
𝜃 𝑑
𝑘
(𝜔) − 𝜃 𝑘 (𝜔) 2𝑇 𝜔
𝜔=1 (35)
where 𝜃 𝑑
𝑘
(𝜔) is the 𝜔 𝑡𝑕
element of 𝜃 𝑑
𝑘
, 𝜃 𝑘
(𝜔) is the
𝜔 𝑡𝑕
element of 𝜃 𝑘
, 𝑇𝜔 is the total sampling instants,
and 𝑘 = 1, 2.
The simulation results of the PDC system
and the AFNC system in two cases, which comprise
joint position, tracking error, and control torque, are
depicted in Figures 4-7, respectively. Moreover, the
values of RMSEs in both the PDC system and the
AFNC system are expressed in Table 2.
Figure 4: The simulation of PDC in Case 1
Figure 5: The simulation of PDC in Case 2
Figure 6: The simulation of AFNC in Case 1
Figure 7: The simulation of AFNC in Case 2

Table 2: RMSEs of PDC system and AFNC
system in two cases
RMSE
(rad)
Case 1 Case 2
PDC AFNC PDC AFNC
𝑅𝑀𝑆𝐸1 0.0517 0.0279 0.0609 0.0298
𝑅𝑀𝑆𝐸2 0.0478 0.0264 0.0578 0.0276
From Figures 4 and 5, the tracking
performances of the PDC system are good.
However, the convergences of tracking errors are
still slow. In Figures 6 and 7, the joint positions can
closely track the desired trajectories under the
existence of the uncertainties, and the tracking errors
are regularly reduced because of the learning ability
of FWNNs.
The simulation results in Figures 4-7 and
Table 2 show that the proposed AFNC system
reaches the control performance improvement than
that of the PDC system, while the convergence of
tracking errors as well as the RMSEs of the
proposed AFNC method is better than ones of the
PDC method.
V. CONCLUSION
This paper has successfully applied an
AFNC approach utilizing SMBLA for tracking the
desired trajectory of robot manipulator. The AFNC
scheme represents a parallel combination of FWNNs
and TSMC, which not only approximates the various
uncertainties but also guarantees the stability of the
whole system. Additionally, the parameters of
FWNNs are updated by a novel SMBLA that its
convergence is proven by employing Lyapunov
theorem. Hence, the proposed control system
resulting in a robust and improved tracking
performance without the detailed knowledge of
robot manipulator. The comparative simulation
results of two-DOF robot manipulator demonstrate
that the tracking errors of the proposed AFNC
method converge faster than ones of the PDC
method.
VI. ACKNOWLEDGEMENTS
The authors would like to thank the editor and
the reviewers for their valuable comments.
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