This document describes a master's thesis that designs an adaptive controller and state observer based on neural networks for a 4 degree-of-freedom parallel robot (4DOFPR). The thesis includes modeling of the 4DOFPR's kinematics and dynamics. It then designs three different controllers - a backstepping controller aggregated with sliding mode control, a radial basis function neural network-based adaptive controller, and a high-gain state observer to estimate states for the adaptive controller. Simulation results evaluating the performance of the controllers and observer are also presented.

1. HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
MASTER THESIS
Design an adaptive controller and a state
observer based on neural network for the
4DOF parallel robot
NGUYEN MANH CUONG
Control Engineering and Automation
Supervisor: Assoc. Prof. Nguyen Tung Lam
School: School of Electrical and Electronic Engineering
HA NOI, 2022

2. HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
MASTER THESIS
Design an adaptive controller and a state
observer based on neural network for the
4DOF parallel robot
NGUYEN MANH CUONG
Control Engineering and Automation
Supervisor: Assoc. Prof. Nguyen Tung Lam
School: School of Electrical and Electronic Engineering
HA NOI, 2022
Supervisor’s Signature

3. C NG HÒA XÃ H I CH T NAM
Ộ Ộ Ủ NGHĨA VIỆ
Độ ậ – ự – ạ
c l p T do H nh phúc
B N XÁC NH NH S A
Ả ẬN CHỈ Ử LUẬN VĂN THẠC SĨ
H và tên tác gi
ọ ả luận văn : Nguyễ ạnh Cườ
n M ng
tài lu
Đề ận văn: Thiết k b
ế ộ điều khi n thích nghi và b quan sát tr ng thái
ể ộ ạ
d a trên m
ự ạng nơ ron cho robot song song bốn b c t do
ậ ự (Design an adaptive
controller and a state observer based on neural network for the 4DOF
parallel robot)
Chuyên ngành: K thu u khi n và T ng hóa
ỹ ậ ề
t Đi ể ự độ
Mã số SV: 20202916M
Tác giả, Người hướng d n khoa h c và H
ẫ ọ ội đồng ch m lu
ấ ận văn xác nhận
tác gi a ch a, b sung lu n h p H ng ngày
ả đã sử ữ ổ ận văn theo biên bả ọ ội đồ
04/05/2022 v i các n i dung sau:
ớ ộ
- nh s i chính t m c
Chỉ ửa lỗ ả và đánh đề ụ
- nh s a các ký hi u và b sung danh m vi t
Chỉ ử ệ ổ ục từ ết tắ
Ngày 07 tháng 05 năm 2022
Giáo viên hướ ẫ ả ận văn
ng d n Tác gi lu
CH CH H NG
Ủ Ị
T Ộ Ồ
I Đ

4. 1
TABLE OF CONTENT
CHAPTER 1. OVERVIEW................................................................................. 1
1.1 .................. 1
The four degrees of freedom parallel robot (4DOFPR) model
1.2 ................................... 2
Trajectory tracking controllers and state observers
1.2.1 controllers.................................................... 2
Trajectory tracking
1.2.2 State Observers ........................................................................... 5
1.3 Conclusion.................................................................................................. 7
CHAPTER 2. DESIGN AN ADAPTIVE CONTROLLER AND A STATE
OBSERVER FOR THE FOUR DEGREES OF FREEDOM PARALLEL
ROBOT ................................................................................................................. 8
2.1 4DOFPR......................................................... 8
Mathematical model of the
2.1.1 ......................................................................... 9
Kinematic model
2.1.2 ......................................................................... 10
Dynamic model
2.2 ................................................................ 10
Controller design for 4DOFPR
2.2.1 Backstepping aggregated with SMC (BASMC)....................... 10
2.2.2 -based (RBFNNB) adaptive controller........................ 13
RBFNN
2.2.3 -gain observer for the adaptive controller......................... 17
High
2.3 Conclusion................................................................................................ 20
CHAPTER 3. SIMULATION RESULTS........................................................ 22
3.1 Results of the RBFNN based adaptive controller (RBFNNB) ................ 22
3.2 Simulation results of the adaptive controller using the high-gain state
observer................................................................................................................ 27
3.3 Conclusion................................................................................................ 35
CHAPTER 4. CONCLUSION AND FUTURE WORK................................. 36
4.1 Results of the thesis.................................................................................. 36
4.2 Future work .............................................................................................. 37
REFERENCES................................................................................................... 38

5. 2
LIST OF TABLES
Table 3.1 Reference trajectory parameters .......................................................... 22
Table 3.2 Control parameters............................................................................... 23
Table 3.3 Trajectory reference parameters .......................................................... 28

6. 3
LIST OF FIGURES
Figure 1.1 Parallel robot applied in the car motion simulator ............................... 2
Figure 1.2 Parallel robot applied in rehabilitation system [4] ............................... 2
Figure 2.1 (a) Robot coordinate; (b) Vector diagram of 4DOFP........................... 8
Figure 2.2 Structure of BASMC controller.......................................................... 12
Figure 2.3 RBFNN structure................................................................................ 14
Figure 2.4 Structure of the adaptive controller .................................................... 15
Figure 3.1 ...................................................................................... 23
External force
Figure 3.2 Motion trajectory of p ....................................................................... 24
Figure 3.3 Tracking error of p ............................................................................ 25
Figure 3.4 Approximated values.......................................................................... 26
Figure 3.5 Motion trajectory of q ........................................................................ 27
Figure 3.6 Tracking error of q ............................................................................. 27
Figure 3.7 Uncertain parts in the robot model ..................................................... 29
Figure 3.8 Observed values of q ......................................................................... 30
Figure 3.9 Observed values of q ......................................................................... 30
Figure 3.10 Observational error of q ................................................................... 31
Figure 3.11 Estimated values from RBFNN........................................................ 32
Figure 3.12 Robot’s trajectory ............................................................................. 33
Figure 3.13 Tracking error ................................................................................... 33
Figure 3.14 Observed position with different values of ob
 ................................ 34
Figure 3.15 Observed velocity with different values of ob
 ................................ 34

7. 4
LIST OF ABBREVIATIONS
Abbreviation Definition
4DOFPR Four Degrees of Freedom Parallel Robot
DOF Degrees of Freedom
SMC Sliding Mode Control
DSC Dynamic Surface Control
RBFNN Radius Basis Function Neural Network
BASMC
Backstepping aggregated with Sliding Mode
Control
RBFNNB
Radius Basis Function Neural Network-
based
DSCNN Dynamic Surface Control Neural Network

8. 1
CHAPTER 1. OVERVIEW
1.1 four degrees of freedom parallel robot PR) model
The (4DOF
Nowadays, robotic systems are being increasingly rapidly developed and
applied in several economic and social life fields because they are designed for
particularly complex and dangerous tasks or repetitive jobs and require high
accuracy. Moreover, apart from being almost precise and consistent, with their
flexible operating ability, robots are capable of working in hazardous
environments. In addition, the robot can perform tasks with heavy loads and toxic
substances and can adapt to particular environmental conditions. Thus, these
advantages have significantly contributed to productivity and quality
improvement, preventing accidents and saving labor costs.
In state-of- -art technology, parallel robots are increasingly prevalent in the
the
industry, military, medical, and entertainment. Various numbers of parallel
structures in [1], [2], [3], [4], and [5] have been taken into account, including the
six degrees of freedoms (DOF) robot in [1], which is capable of applied in medical
surgery, as well as rehabilitation in [4], and some other structures applied into
flight and automobile simulation. Most of these models have been implemented
based on the advantages of parallel structure, namely low inertia moment, high
load, and smooth transmission capacity [6]. From reality-based car models, to
assist trainees and drivers have an alternative approach to getting familiar with the
automobile’s movements, it is necessary to construct a driving simulation model
based on a class of parallel architectures and motion platforms developed recently
[3]. Moreover, car driving simulation models are also constructed with the purpose
of mitigating unexpected forces impacting drivers in practical and virtual reality
environments in relation to health care and rehabilitation [4], [5].
In order to describe the movement of the robot system, the demand for robot
modeling is imperative. Several studies [6], [7] showed the geometrical analysis
of a six DOF constrained parallel robot. Regarding the construction of the
mathematical model, a forward and inverse kinematics model of Quanser’s
Hexapod robot has been illustrated in [8]. In addition, the six DOF parallel robots
have a positive advantage of high accuracy movements. However, the complexity
of six actuators interaction and coordination gives the rising complexity in
’
designing trajectory tracking controllers of parallel robots, especially in the
presence of massive uncertainties. Therefore, the configuration with fewer joints
and DOF is able to mitigate the inevitable hysteresis and redundancy of actuators
shown in [9], [10], and [11]; , it would be more convenient in particular
thereby
practical applications and controller design considered uncertain elements. In
addition, in the attempt to reduce computation complexity and redundant
constraints, the group of authors has constructed the four DOF platform,
comprising the movements of rotating and translating along the vertical axis OZ,
rotating about the OX and OY axis.

9. 2
Figure 1.1 Parallel robot applied in the car motion simulator
Figure 1.2 Parallel robot applied in rehabilitation system [4]
From the reference and analysis of the above scientific works, moreover,
intending to reduce the computational complexity and redundant constraints while
still ensuring the necessary motion, the thesis puts focus on the four degrees of
freedom parallel robot platform with the movements of rotational and translational
movements along the OZ axis, rotation in the OX axis and the OY axis.
1.2 Trajectory tracking controllers and state observers
1.2.1 s
Trajectory tracking controller
In robot control, especially in orbital tracking control problems, modern
methods specially put focus on designing control algorithms capable of handling
problems related to uncertainties, perturbations, and unknown structural
components in the system model while still ensuring stability and tracking quality.
The 4DOFPR parallel robot model is considered to be a model being commonly
affected by nonlinear uncertain elements in practical applications, especially
external forces acting in different directions on the system.
The parallel structures are considered a nonlinear model in the control design
field; therefore, a control issue has attracted significant attention in the scientific
community. One of these designed methodologies for nonlinear control systems

10. 3
that the
have been interested in is Backstepping technique as in [12], [13], [14],
[15], and [16] in order to ensure the quality of trajectory tracking control. However,
when uncertainties or unmodeled components exist in the system model, the
“explosion of terms” phenomena adversely affects the control quality. Another
prominent control method is sliding mode control (SMC) which has been widely
used because of its robust characteristic as in [17], [18], and [19] when considering
the existence of unknown elements. However, the chattering phenomenon
generated by SMC controller is likely to demolish the system [20], as well as
the
the computational burden with the high order systems. Combining the two
aforementioned controllers is an approach to improving control performance
because it takes advantage of them. Then, the robustness characteristic is
enhanced, and the computational cost is reduced as in [20], [21], [22], and [23].
Nevertheless, the combined controller cannot cope with the chattering and
“explosion of terms” phenomena.
On the other hand, by taking advantage of the multiple sliding surface
controller and Backstepping technique, dynamic surface control (DSC) has been
proposed to address the problem “explosion of terms” in [24] and [25] by using a
low-pass filter for each computation step. However, the errors of the low-pass filter
in the DSC controller are a dilemma, majorly depending on a filter time constant
and being proven by complex mathematical conditions in [24], which may
correlate with the frequency of experimental devices. Alternatively, a more
efficient method in this paper handling mathematics difficulty is utilizing a neural
network to approximate virtual signals and alleviate the chattering phenomena.
In control theory, noise components are commonly considered to be an
inevitable part of the whole system, and analyzing noise is the key to finding a way
that assists the 4DOFPR system to be more stable and accurate. To be more
specific, stochastic disturbances are problematic, 4DOFPR system.
impacting the
In terms of non-Gaussian noises, the modified extended Masreliez Martin filter
–
constructed in [26] is an efficient approach to handle nonlinear systems when
environmental disturbances influence the whole system. Besides, stochastic
parameters have been taken into consideration in [27] by estimating stochastic
nonlinear systems. By taking into cautious consideration published in [28] and
[29], it is assumed that some stochastic disturbances as to an unknown varying
force from the input system act on actuators of the 4DOFPR system along the
vertical direction because of body weight and moment disturbance as well as
unknown parts. However, there have been several kinds of noises in external and
internal stochastic disturbances because of all range elements [30], from frictions,
vibrations, and changes of sudden forces to the shift in environmental conditions,
which are considered uncertainties. In this thesis, we assume that the 4DOFPR is
the model prone to the impact of stochastic uncertainty elements.
As mentioned above, for many conventional nonlinear controllers such as
SMC or Backstepping, there have been drawbacks in improving control
performances when it is challenging to identify the accurate model because of the

11. 4
existence of the two kinds of unknown elements, including matched and
unmatched uncertainties. In addition, in order to achieve high control quality,
almost conventional methodologies in control designs require the identification of
mathematical models that have uncertainty elements in advance. Thus, adaptive
controllers, in which the updated law is designed based on mathematical
characteristics to adapt unknown components, were considered as in [31], [32],
and [33]. However, the convergence of these controllers majorly depends on the
system model characteristics and control parameters, and thus the wide range of
uncertain elements can not be taken into account. Therefore, these controllers are
not adaptable to the change of unknown uncertainties and the diversity of the
model specifications in the 4DOFPR model. Hence, another approach using
approximate the uncertainties by using neural networks is investigated in this
paper.
A more modern and efficient approach approximating uncertainties by neural
networks was appreciated as a promising approach [34], [35], [36], [37], and [38].
In addition, this methodology has increasingly become a novel method for the
control design of the complex system affected by unknown uncertainties and
disturbances [34]. In [36], a radius basis function neural network ( ) has
RBFNN
been constructed to approximate the value of torques in web transport system.
a
According to [37], wheel slip phenomena and external disturbances have been
compensated by using the adaptive neural network to guarantee the tracking
quality for mobile robots. Meanwhile, an adaptive controller based on RBFNN as
in [38] has solved the problem of the dual-arm robot containing the noises.
Moreover, the idea to use the notable technique in approximating the
nonlinear function of the Radial Basis Function neural network was conducted in
[39] and [40]. By online tuning network parameters, the updated law of the
RBFNN shown in [40], [41], and [42 derived based on the Lyapunov theory,
] was
guaranteeing the minimization of the stochastic disturbance in terms of the overall
system. Besides using RBFNN, the fuzzy logic in which the weight value based
is
on the adaptive law has been designed [43] to effectively deal with the uncertain
dynamics problem by estimating the appropriate values for the controller.
By analyzing the above studies, solutions, and approaches, the thesis
proposes an adaptive con -SMC using RBFNN to
troller based on Backstepping
guarantee the tracking performance of the four degrees of freedoms parallel robot,
which contributes:
- method focuses on the problem of controller design for the uncertain
The
model of the 4DOFPR. Dissimilar to the paper [31], [32], and [33], the approach
based on RBFNN does not majorly depend on the mathematical characteristics of
parallel robots. Thus, the developed controller can cope with the range of
larger
uncertain elements and bounded disturbances because of its outstanding
characteristic in approximating nonlinear parts and compensating for the external
noises.

12. 5
- The proposed adaptive controller is designed based on the Backstepping
technique aggregated with SMC. Therefore, it takes advantage of these controllers
in eliminating nonlinear parts, as well as enhancing the robustness of the system.
Moreover, the adverse impacts of the two controllers, including “explosion of
terms” and chattering, are remarkably alleviated because the unknown elements,
which are the main factors causing the phenomena, are approximated and
compensated by the neural network. Thus, the developed controller not only
overcomes the disadvantage of the above-mentioned techniques but also increases
the robustness and adaptive behavior of the overall system.
- The RBFNN is designed to estimate values for all unknown components of
the dynamic system and compensate for the disturbances of the input torque and
external forces. Thus the controller can guarantee the control quality even when
only the inaccurate information about the system is known. Different from [37],
[38], [39], [40], and [43], neural network outputs shown in comparison to
are
nominal uncertain values in order to verify the approximating performance, as well
as the influence of the neural network on the system performance.
The purpose of the controller design is to construct an adaptive controller
based on RBFNN for the 4DOFPR to approximate uncertain components and
compensate for the disturbances in the model when these factors exist in the robot
model. In this study, the approach focus on constructing an adaptive control
es
algorithm for the 4DOFPR, which is affected by unknown external forces and the
uncertain components in the mathematical model. The BASMC supported by the
RBFNN demonstrates well-being tracking performance and closed-loop stability.
Given the self-learning capacity of the neural network, such considerable
information associated with complicated driving car modeling is not required for
neural controllers. Thus, neural-based controllers are able to cope with a more
extensive scope of uncertainty. are
The controller design and its performance
illustrated in Chapter 2 and Chapter 3.
1.2.2 State Observers
In robotic systems, system state estimation plays a vital role in controller
calculation or simply collecting information for visualization and monitoring. In
addition, system signals, including position and velocity, apparently need to be
accurately measured in the feedback control system architecture to ensure control
quality. In fact, improving the accuracy of the state measurement and information
about the system is continuously focused on research. A conventional way to solve
this problem is to use measurement sensors to obtain the necessary information,
including both direct and indirect data about the required quantities. Then, those
values are used to design observation algorithms to process this information to
generate reliable information about the entire system states. However, these
algorithms can only exist if the sensors can obtain enough information to determine
the system s state. In practice, the number and quality of sensors used are often
’

13. 6
limited by cost and physical constraints, so observers play an essential role in many
applications.
Commonly, linear observers and sliding mode observers are constructed for
linear systems. However, the fact that the system contains nonlinear components,
even uncertain components, it is challenging for these observers to guarantee the
quality of observation and control of the system. Therefore, nonlinear observers
have been researched and designed with specific structures and advantages [44],
[46], [47], [48], [49], and [50]. The most prominent and prevalent observers for
the class of nonlinear systems are the high-gain observers [44], [52], the sliding
mode observers [46], [47], [48], and extended observers. The four-degree-of-
freedom parallel robot model is an object that contains highly nonlinear
components and is also affected by noise and unknown components. Besides,
although it is robust to noise, the sliding observer has the well-known chattering
phenomenon. Therefore, extended or adapted observers or techniques were studied
in [46], [47], [48], and [49] to improve the quality and handle this phenomenon.
However, this also increases the complexity of the computation and
implementation of these observables. Therefore, due to its ability to minimize the
impact of model variation and uncertainty components along with a simple design
structure and ease of implementation in practice, the high-gain observer is chosen
to design for the robot object in the thesis.
In [44], [45], and [51], an overview of high-gain observers was introduced
for nonlinear systems and feedback control systems. In addition, the observer
design structure for the multi-input, multi-output nonlinear systems was also
constructed in [51] in two approaches, including a general method for the observed
nonlinear system and the other which structure is decomposable into linear and
nonlinear parts. It can be seen that the high-gain observer has the advantage of
observing systems with high-order nonlinear components as well as with
disturbances [52], [53], and [54]. Typically with the Quadrotor drone [52], the
high-gain nonlinear observer was used to monitor, analyze, and compensate for
the actuator s deviation and the measurement errors. In addition, the controller
’
designed in [52] reduced the use of sensors and actuators when using an observer.
When using a high-gain observer with feedback controllers for a nonlinear system,
especially a sliding mode controller, the system’s overall stability was ensured
even if the system contains uncertain components [48]. Besides, the study [54]
proposed a structure of the high-gain observer combined with control law for an
electro-hydraulic servo steering system because of its complex structure and
variable load. The automatic steering system plays a decisive role in the stability
of self-driving car systems. The flexible combination of the observer and the
controller not only reduces the complexity of using the sensors but also ensures
the stability of the system. In addition, adaptive controller using a high-gain
an
observer was also designed for the mobile robot to solve the problem of orbital
tracking [55].

14. 7
From the reference, analysis, and evaluation, the high-gain observer will be
constructed for the neural network-based adaptive controller for the uncertain
model of 4DOFP affected by exogenous noise. Thanks to the ability to quickly
converge as well as handle mathematical model errors, the high-gain observer can
ensure observation quality to guarantee the quality of the trajectory tracking
control problem. The observer design and its use for the adaptive controller will
be presented in Chapter 2, along with the results accompanied by the analysis and
evaluation of the whole system quality in Chapter 3.
1.3 Conclusion
In Chapter 1, an overview of the four-degree-of-freedom parallel robot was
presented along with various applications in several fields. From the study and
analysis of recent studies on nonlinear controllers in solving the trajectory tracking
problem, an adaptive controller based on a radial neural network is proposed to
solve the problem of robot model uncertainty as well as the disadvantages of
model-based controllers. Prominent representatives of state-observer sets were
also reviewed. From the tracking control problem for a parallel robot model with
four degrees of freedom combined with the analysis of related scientific
publications, the scope and research objectives of the thesis have been proposed
and presented.

15. 8
CHAPTER 2. DESIGN AN ADAPTIVE CONTROLLER AND A STATE
OBSERVER FOR THE FOUR DEGREES OF FREEDOM
PARALLEL ROBOT
This chapter presents the design of an adaptive controller and a state observer
for a four-degree-of-freedom parallel robot. First, the kinematics and dynamics
equations are described to serve as a premise for controller and observer design.
Then, an adaptive controller will be designed using the combination of the
Backstepping controller and the sliding mode control technique to take advantage
of the two controllers for nonlinear systems. Next, the radial radius neural network
and the update rule designed based on the Lyapunov stability criterion are
constructed to approximate the uncertain components to overcome the
disadvantages of the above controllers and enhance the control quality for the robot
system. Finally, a high-gain state observer is used to reduce the complexity of
using sensors to measure the motion velocity quantities of the robot.
2.1 Mathematical model of the 4DOFPR
(a) (b)
Figure 2.1 (a) Robot coordinate; (b) Vector diagram of 4DOFPR.
This section describes the robot’s kinematic and dynamic models followed by
a global problem in the control design procedure. Figure 2.1 (a) depicts the overall
system and system coordinates. The global coordinate is chosen as in Figure 2.1
(b). System states are rotational angular about Ox, Oy, Oz axes, and translational
position along the vertical axis of the upper platform’s panel. Robot movements
are performed by simultaneous vertical motions of three pistons combined with
the rotation of a rotating shaft attached below the base platform. The kinematic
model demonstrates a transformation between the piston’s movement and P ’s
position, shown in the following subsection.
Model-based control of robots requires a computationally efficient
formulation of motion equations achieved by expressing the dynamic model in
terms of a set of independent joint coordinates, referred to as minimal coordinates.
Minimal coordinates formulations have been widely used for nonlinear motion
control of robot manipulators. To this end, the joint coordinates of a robot

16. 9
manipulator that is subjected to kinematic loop constraints are split into a set of
dependent and independent coordinates
2.1.1 Kinematic model
The system state vector is denoted as  
, , ,
T
z
p   
=
p , where z
p is
vertical position and  ,  , and  are rotational angular about and
OX, OY, OZ
axes, respectively. Besides, vector  
1 2 3
T
l l l 
=
q describes the piston’s
lengths and OZ rotational angular with i
l is the piston length i ( 1,2,3)
i = . c is the
distance between origin O and base platform’s A
O , while a and b are the radius
of upper and base platforms, respectively cording to the vector diagram, a
. Ac
vector equation of 4DOF is computed as
PR :
,
i i i A i A
A B =OP+PB -O A -OO (2.1)
where 1,2,3
i = . The coordinates of A1, A2, A
and 3 in the local coordinate 1 1 1
PX YZ
associated with the base panel’s center are 1 sin( ) cos( ) 0 ,
6 6
T
base
A a a
 
 
=  
 
 
2 0 0 ,
T
base
A a
= − and 3 sin( ) cos( ) 0
6 6
T
base
A a a
 
 
= −
 
 
.
Assuming that a b
= , the coordinates of B1, B2, B
and 3 in the local coordinate
A
O XYZ associated with the upper panel’ center are
1 sin( ) cos( ) 0 ,
6 6
T
upper
B a a
 
 
=  
 
 
2 0 0 ,
T
upper
B a
= − and
1 sin( ) cos( ) 0
6 6
T
upper
B a a
 
 
= −
 
 
. Besides, the cent points of the base and upper
er
panel are  
0 0
T
A
O c
= and  
0 0
T
de z
p p
= in the global axis, respectively.
Then, we obtain fixed coordinates of i
A and i
B are .
i A ibase A
A T A O
= + , and
.
i P ibase de
B T B p
= + with
cos sin 0
sin cos 0
0 0 1
A
T
 
 
−
 
 
= 
 
 
and
cos cos cos sin sin cos sin sin sin cos cos sin
cos sin cos cos sin sin sin cos sin sin cos sin
sin cos sin cos cos
P
T
           
           
    
− +
 
 
= + −
 
 
−
 
.
Finally, the length of the robot’s legs is computed as
2
,
i
l = i i
A B (2.2)
where i i
B A
= −
i i
A B . From (2), q can be computed from p.

17. 10
2.1.2 Dynamic model
Commonly, to achieve more precise movement, the controller takes the
dynamic model, which additionally considers the kinetics energy of the system,
into account. Therefore, the elements containing forces, acceleration, mass, inertia,
and other system parameters are put effort to accurately identified. It is evident that
with the controller designed based on a dynamic model, the more reliable the
system is, the better control quality can obtain. Nevertheless, it is challenging to
identify these parameters exactly, and thus, there has been errors between the
actual and mathematical model. Therefore, assuming that the dynamic robot model
is a nominal model with bounded uncertain elements and noises.
According to [25], the dynamic model of the robot based on Euler-
the Lagrange
form is described by
,
M(q,q F (2.3)
in which
2
2
2
2
2
0
9 9 9
0
9 9 9
0
9 9 9
0 0 0
4
12
p py p p
dc
p p px p
dc
p p p
dc
m I m m
m m
m m I m
m m
m m m
a
m m
I
a
 
 
 
 
 
 
 
 
+ + +
 
 
 
 
+ + +
=  
 
 
+ +

 
  







M with
2 2
1 2
( ), ( ), ( ), ( ), ( . ), ( . ),
dc p px py
m kg m kg m kg m kg I kg m I kg m and 2
( . )
I kg m
 are the mass
of the covering of pistons, the mass of pistons, the mass of motors on the cylinders,
the mass of the mobile panels, and the inertia of the mobile panels about , ,
Ox Oy
Furthermore,
1
4
2
4
15
4
1
0 0 0
5
0 0 0
2
0 0 0
0 0 0
0
0
py
px
I l
I
a
a
l
 
 
 
 
=  
 
 
 
 
C and  
2 1 1 1 0
3
T
p
m
m
g
 
=  
 
+
D with
2
9.8( / )
g m s
 is the gravity acceleration. 1 2 3
T
F F F 

 
= 
F is defined as a
control signal vector.
2.2 Controller design for 4DOFPR
2.2.1 Backstepping aggregated with SMC (BASMC)
The controller designed by combining the two controllers can ensure that the
robot tends to track the desired trajectory with the identified mathematical model.
By combining Backstepping technique and SMC method, the controller takes
the

18. 11
advantage of the two control algorithms, which means it can eliminate the system’s
nonlinear elements and increase robust behavior to external noises
To facilitate the control design procedure, initially, assuming that , ,
M C and
D are accurately known, which means all the information concerning the robot
system is clearly identified. Therefore, the controller is designed under ideal
conditions with all information available. Subsequently, a reference for
trajectory
the dynamic model  
1 2 3
( ) ( ) ( ) ( ) ( )
T
d d d d
t l t l t l t t

=
d
q is computed from the
robot’s reference trajectory is defined as
and  
( ) ( ) ( ) ( ) ( )
T
zd d d d
t p t t t t
  
=
d
p
using the forward kinematic equations. To clearly describe the tracking process, in
the beginning, the difference between q and d
q in each control period is denoted
by
.
= −
1 d
ξ q q (2.4)
With the tracking error vector 1
ξ , the control objective is now equivalent to force
1
ξ tend to vicinity of zero. A virtual control which is chosen as the derivative
the
of q, which satisfies the following Lyapunov candidate function
1
1
.
2
V = T
1 1
ξ ξ (2.5)
From a first-order derivative of 1
V , it is straightforward to show that
1
V −
1 1 1 d (2.6)
the equation of q has a form of
= −
d 1 1
α q (2.7)
Next, the the
vector depicting between
error q and α is defined as
= −
2
ξ q (2.8)
Alternatively, we can write ,
= +
2
q and 1
V turns into
1 ,
V − T
1 2 1 1 1
ξ k ξ (2.9)
where
11
12
13
14
0 0 0
0 0 0
0 0 0
0 0 0
k
k
k
k
 
 
 
=
 
 
 
1
k is a diagonal positive definite matrix. In addition,
2
ξ is also considered as a sliding manifold to compute the system control input.

19. 12
Figure 2.2 Structure of BASMC controller
Figure 2.2 illustrates the structure diagram of the BASCM controller. The
tracking error is considered as an input for the Backstepping controller, and the
virtual control signal is calculated. In fact, the physical value of this signal is the
ideal velocity of the robot’s lengths and the rotational about Oz axis. After the
desired control signal is derived, the virtual signal error is fed into the sliding mode
control to calculate the control signal for the system.
Taking derivative of 2
ξ results in
= − + −
2
ξ (2.10)
Based on the Backstepping technique’s design steps, we propose the second
Lyapunov candidate function to calculate the system input. The system’s global
stability is taken into account according to this function
2 1
1
.
2
V V
= + T
2 2
ξ ξ (2.11)
Taking derivative of 2
V leads to
2 ( (
V − + − + −
T T -1 -1
1 2 2 1 2 1 1 1 2
ξ k ξ ξ M F M Cq (2.12)
The control signal F compris two parts: an equivalent control signal and
es
a switching control signal having the form of the SMC controller. The equivalent
control eq
F is chosen as follows
= + − −
eq 1
F Cq (2.13)
which holds the system state on the defined sliding manifold. Besides, the switch
control sw
F drives the system state to the sliding surface
( ( ) ),
sign
=− +
sw 2 2 3 2
F M k ξ k ξ (2.14)
where 21 22 23 24
( , , , )
diag k k k k
=
2
k and 31 32 33 34
( , , , )
diag k k k k
=
3
k are diagonal positive
definite matrices. One of the main factors causing the chattering phenomena is the
presence of the signum Commonly, to decrease the impact of this issue,
function.
the satlins function is , and the switching control signal (2.14)
selected is
transformed into
( ( ) ),
sat
=− +
sw 2 2 3 2
F M k ξ k ξ (2.15)
Eventually, the control signal is obtained by

20. 13
= +
eq sw
F F F . (2.16)
However, there will be a trade-off between the robustness of the system and
the smoothness of the control signal. In addition, this phenomenon can be
eliminated only in a certain range using this method. Theoretically, the chattering
phenomenon is majorly caused by the existence of unmodeled dynamic elements
in the systems. The RBFNN can estimate appropriate values for unmodeled
elements, thus reducing impacts of the chattering phenomenon.
the
Substituting (2. ) (2.12),
16 into 2
V turns into
2 ( ) .
V sat
− −
T T
1 1 1 2 2 2 2 3 2
ξ ξ k ξ ξ k ξ (2.17)
where 1
k , 2
k , and 3
k are chosen as above, 2
V satisfied the Lyapunov
which
stability standard.
Remark 1. BASMC controller is constructed based on the Backstepping
technique is
; therefore, the phenomenon called “explosion of terms” caused by
iteratively and incrementally taking derivative of the virtual control signal (2.7).
Especially when the system contains uncertain elements such as discontinuous
functions, this calculation can have an adverse effect on system performance.
Additionally, even though the satlins function in (2.15) has ability to reduce
the
the chattering’s impact, there will be a trade off in the system’s robust
-
characteristic since this model-based controller cannot capture all the dynamics in
the 4DOFPR model, which is often subjected to uncertainties, noises, and external
forces.
2.2.2 -based (RBFNNB) adaptive controller
RBFNN
In practice, it is difficult to precisely identify , ,
M C D, and external
disturbances d
τ . Therefore, with -based
a model controller such as BASMC, the
control law only uses nominal values of these matrices. Hence, BASMC may not
meet the tracking quality because of the “explosion of terms” phenomena
accompanied by the chattering problem. This section proposes a controller which
can solve the problems stemming from uncertain elements and external
disturbances.
, ,
M C and D are considered as the values of parameter
actual system
matrices. Besides, , ,
M C and D are their nominal values. Then, ΔM , ΔC , and
ΔD are denoted as the differences between ideal and nominal values of these
uncertainties. Without loss of generality, it can be shown that
, ,
= + = +
M M ΔM C C ΔC and = +
D D ΔD, the robot system (2.3) is transformed
into
+ + = + − + +
d
M (2.18)
To facilitate the control design steps, (2. ) is rewritten
18 as
= − + + − + +
d (2.19)

21. 14
Finally, system dynamic equations turn into
= − + + (2.20)
where 4 1
( ( )) .
R 
= − + + 
-1
d
δ M τ ΔM ΔD
Conducting the same procedure in the previous section, with the defined
as
uncertainties, 2
ξ becomes
= − + + −
2
ξ (2.21)
The iterative calculation of α causes “explosion of terms”, therefore, to
overcome this problem, propos an RBFNN and
ing to estimate the uncertainties
provide the appropriate value for the derivative of α. Moreover, estimated values
generated by RBFNN for the un parts can significantly reduce
certain dynamical
chattering impacts. The RBFNN has the ability of online learning to approximate
highly-nonlinear functions, and thus, the controller does not require full or even
any prior accurate knowledge about the system. Assuming that α is unknown and
considering 1 2 3
T

   
 
= = −
 
σ δ α
= − + +
2
ξ (2.22)
The control signal = +
eq sw
F F F with eq
F and sw
F s
are rewritten a
ˆ,
= + − −
eq 1
F Cq ξ Mσ (2.23)
( ( ) ).
sign
= − +
sw 2 2 3 2
F M k ξ k ξ (2.24)
respectively. In (2. ),
23 4 1
ˆ R 

σ is an out vector of the neural network. In addition,
σ is denoted as the idea value for σ̂ , and thus, RBFNN is designed to guarantee .
σ̂ . approaches σ with arbitrarily small error or estimate an appropriate value
an
for σ .
to ensure the system’s stability
Figure 2.3 RBFNN structure
Figure 2.3 shows the RBFNN’s structure with three layers: an input layer, a hidden
layer, and an output layer. The neural structure is constructed with
,
= +
T
σ W h ε (2.25)
ˆ
ˆ ,
T
=
σ W h (2.26)

22. 15
where ε is approximation error and 

ε with  is an arbitrarily small positive
constant. W and Ŵare ideal weight and updated weight matrices, respectively.
Additionally, ˆ
W W is considered as the error weight matrices. RBF function
is the
employed to calculate output of the hidden layer as follows
2
2
exp ,
2
 
−
= − 
 
 
x
h





(2.27)
where the system state q is chosen as an input vector, 



 is a center point, and 
presents the width value of RBF function
the h . Substitute the control signal in
(2.23) (2.24) into (2.22) we obtain
and
( )
sign
− +
2 2 2 3 2
ξ k ξ k ξ (2.28)
Adaptive law for RBFNN’s weight is chosen as
ˆ
( )

= −
T
2 2
W Γ hξ ξ W , (2.29)
where n n
R 

Γ positive definite matrix with
is a n being node number of the
hidden layer, a learning rate parameter is chosen in (0,1) .
Figure 2.4 Structure of the adaptive controller
The structure of the proposed controller based on RBFNN is shown in
Figure 2.4. In comparison to BASMC structure, the RBFNN with the input of
’s
robot states is constituted to estimate the appropriate values for σ̂ with online
weight updated law the adaptive controller.
in
The second Lyapunov candidate function is reselected as
2 1
1 1
(
2 2
V V tr
= + +
T
2 2
ξ ξ W (2.30)
and 2
V becomes
2
V 1 2 2 (2.31)
Substituting (2. ) (2.31) y lds
28 into ie

23. 16
2
( )
( ) (
V
sign
sign tr
= − − − + +
= − − − + +
1 2 1 2 2 3 2
T T T T T
1 1 1 2 2 2 2 3 2 2 2
T T T T
1 1 1 2 2 2 2 3 2 2 2
ξ k ξ ξ k ξ ξ k ξ ξ ε ξ W
ξ k ξ ξ k ξ ξ k ξ ξ ε W
(2.32)
With the updated law (2.29), 2
V is rendered as
2
V 1 1 1 2 2 2 2 3 2 2 2 (2. )
33
Applying the Cauchy-Schwarz inequality results in
2
( F
F F
tr W (2.34)
It can be shown that
2
2
2 2
2min 3min
1 1
(
4 2
N F
F F
N F F F
V
k k   
 − − − + + − −
1 1 1 2 2 2 3 2 2
T
1 1 1 2 2 2 2 2
ξ k ξ ξ ξ ξ ξ W ξ W W
(2.35)
where 2min
k and 3min
k are the minimum values of 2
k and 3
k . If the bounded
condition in the following inequality is satisfied
2
2min
3min
1
4
N F
k
k
 
+ −

2
W
ξ (2.36)
then
2
1
2 F F
V 1 1 1 2 (2.37)
and the system is global asymptotical stability. Furthermore, the updated law of
the network is designed and proven based on Lyapunov standard, and the
the
convergence of the method is ensured.
Remark 2. With the uncertain model of the system, the RBFNN plays a
key role in estimating the appropriate values for the uncertainties to ensure control
quality. ˆ (0)
W is an initial value of a weight matrix W , and it is chosen based on
the scope of the input data, especially the system’s states. Moreover, the updated
law is to calculate an appropriate value for W, which is proven based on the
Lyapunov standard to ensure convergence. In the radial basis function activation
h of the hidden layer: 



 is the center point, and η lue,
represents the width va and
they should be moderately chosen based on knowledge concerning the system and
input and output data. Center point
The 



 ha impact on the sensitivity of
s an the
RBF function to , and it opts range of the system
the input vector’s scope in the
state vector q. Besides, η expresses the covering scope of the network input. If
these parameters are chosen inappropriately, the RBF will not be effectively
mapped, and RBFNN will not take effect.

24. 17
2.2.3 -gain observer for the adaptive controller
High
Typically, a combination of measurement and data processing from sensors
is synthesized to obtain robot states position and velocity values . However, in
’
many circumstances, the construction of the sensor system and the calculation of
the necessary values are significantly complicated and costly to satisfy the
acceptable accuracy due to the influence of environmental conditions, noises, and
sensor quality. Therefore, the observer is constructed to estimate the system states,
namely the velocity values of the parallel robot system. On the other hand, a four-
degree-of-freedom parallel robot system is fast responsive; hence, the observed
velocity signal requires a fast convergence rate with the actual velocity value.
Thus, the high-gain observer is selected due to its advantages of convergence
speed, as well as resistance to model errors [44], [51], and .
[55]
To observer design process, the variables are defined as
facilitate the auxiliary
follows
   
 
1 2 3 1 2 3 4
1 2 3 1 2 3 4
,
T T
T T
l l l
l
    
   
 = =



= =
  

ω





(2.38)
Then the system states are denoted by    
T T
x q q  
= = , the ideal dynamic
model of the robot is rewritten as
( ) ( )
( ) ( )
, ,
   
= =
 
− −  
 
q q
x
M q q q q
(2.39)
with assuming that the control signal F .
is able to ensure the system’s stability
From that, the ideal values for the observers are generated by
( ) 1,2,3,4
i i
i i


=



=


(2.40)
with , ( 1,2,3,4)
i i i
  = are the ideal values of , 3,4)
i i
l l , respectively.
Next, defining ˆ
ˆ , ( 1,2,3,4)
i i i
  = as the high-gain observer outputs. The equations
of the high-gain observer are described by
( ) ( )
1
1
2 2
ˆ
ˆ, , , , , 1,2,3,4
i
i i i i i i
ob
i i i i i i
ob
k
j e e
k
f q q F j e f q q F e i
  




= + = +



 = =


(2.41)
with 1 2
ˆ , , 0,0
i i i i i ob
e k k
  
= −   The parameters 1 2
,
i i
k k to satisfy
are chosen
that 2
1 2
i i
s k s k
+ + polynomial.
is the Hurwitz
In (2.41), the ideal dynamic model is taken into account to update the
robot’s
obser . However, uncertain components exist evermore in the system
ver’s outputs
model. A considerable advantage of a high-gain observer is that it can minimize

25. 18
the influence of variable parameters and uncertainty components in the model.
Assuming that the dynamic .
matrices are nominal values containing uncertainties
Then (2.41) is rewritten as
( ) ( )
1
1
2 2
ˆ
ˆ, , , , , 1,2,3,4
i
i i i ii i i
ob
i i i ii i i
ob
k
j e e
k
f q q F j e f q q F e i
  




= + = +



 =


(2.42)
in which ( 1,2,3,4)
i
f i = are the nominal values experienced the effect of uncertain
parts and model errors of i
f . Additionally, assuming that the nominal model
1 2 3 4
T
f f f f
 
=  
f s
sati fies
ˆ
L M
−  − +
f f x x (2.43)
with L and M n-negative numbers.
are no
Then, let  
T
=
x ω 



 and ˆ
ˆ
ˆ
T
 
=  
x ω 



 . The error between x and x̂ are defined by
2
ˆ
( , , , )
i i
i ob
i i
i i i
ob
k
k
f e

  


 
 
 −
 


  − −
 
 
x
q q q q F
(2.44)
From (2. ), (2.41), and (2.42), the equation (2.44) is transformed into
40
ob
x (2.45)
where
 
=  
 
12 12
21 22
A A
A
A A
with
11
11
11
11
0 0 0
0 0 0
0 0 0
0 0 0
ob
ob
ob
ob
k
k
k
k




 
−
 
 
 
−
 
 
=
 
−
 
 
 
−
 
 
11
A ,
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
 
 
 
=
 
 
 
12
A ,
12
2
12
2
12
2
12
2
0 0 0
0 0 0
0 0 0
0 0 0
ob
ob
ob
ob
k
k
k
k




 
−
 
 
 
−
 
 
=
 
−
 
 
 
−
 
 
21
A ,
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
 
 
 
=
 
 
 
22
A ,

26. 19
Matrice
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
 
 
 
 
 
 
=
 
 
 
 
 
 
 
B , and ˆ
( , )
= −
ob
δ f q q .
To more explicitly present an ability to cope with uncertain model of the high-
the
gain observer, s observer error by a vector
caling the
3
1 2 4
5 6 7 8
T
ob ob ob ob
x
x
   
 
= 
 
ob
γ yields
( )
( )
( )
( )
5 1
6 2
1
3
2
3
4
5 12
1 1 1
6
22
7
2 2 2
8
32
3 3 3
42
4 4 4
ob
ob
ob
ob
ob
ob
ob
ob
ob
ob
ob
ob
ob
k
x
k
x
x k
x
x
x k
x
x
x
x f
x
x x f
x
k
x f
k
x f









 







−



−


 
−

 
 
−


= =
 
− + −
 


  − + −
 

− + −
− + −

ob
γ








 
 
 
  = +
 
 
 
 
 
 
 
 
 
 
 
 

ob ob ob
Dγ ε Bδ
(2.46)
with
11
21
31
41
12
22
32
42
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0








−
 
 
−
 
 
−
 
−
 
=
 
−
 
−
 
 
−
 
−
 
 
D . Moreover, with ob
 and the
condition (2.43), the Lipschitz condition is converted into
ob L M
  +
ob
γ (2.47)

27. 20
The Lyapunov candidate function of the high-gain observer (2.42) for 4DOFPR is
chosen as
= T
ob ob ob ob
V γ P γ (2.48)
where 0
= 
T
ob ob
P P is the solution of + = −
T
ob ob
P D D P I.
Taking derivative of ob
V
2
ob ob ob
V
 
+
T T
ob ob ob ob ob
γ γ P Bδ (2.49)
Using the condition (2.47), (2.49) turns into
2
2
2
2 2 2
1 1
2 ,
2 4
1
, 8
4
ob ob ob ob ob
ob ob ob
ob
V L M
M L
M
   
  

+  − + +
−
 +  
−
  
T T T
ob ob ob ob ob ob ob ob ob ob ob
ob ob ob
ob ob ob
γ γ P Bδ γ γ P B γ P B γ
γ P B P B
γ γ P B
(2.50)
According to Theorem 4.5 [45],
in it can be concluded that
( )
 
/
max e 0 , ,
nt
ob ob
m k M


−

ob ob
γ γ , , 0
ob
m n k
  (2.51)
This indicates that the observer s deviation from the actual value is a
’
bounded value. The smaller the value ob
 , the faster this error decreases. Therefore,
with a reasonably small value chosen of ob
 , the observer can guarantee the
convergence of the output value with the ideal value. However, from (2.51) it can
be seen that the peaking phenomenon [44] causes the output value at the initial
time to jump and reach large values when ˆ
(0) (0)

x x , and this phenomenon can
make the system becomes unstable. One of the methods to alleviate the impact of
this phenomenon on the system is to saturate the controller output. Then, values
beyond the response threshold of the system will be saturated; however, the control
quality of the system will be affected.
Remark 3. more explicitly demonstrate the
To ability to cope with noises
and uncertainties of the high-gain observer, the observer error x is scaled by ob
γ .
From (2.46), it can be seen that with the parameter 1
ob
  ,
is moderately chosen
the impact of ob
δ on the system performance is reduced. However, if ob
 is chosen
with positive number, it can result in the unstable of the system.
an infinitesimal
Therefore, this parameter needs to be tuned based on the knowledge about the
robot system input output data.
and and
2.3 Conclusion
In this chapter, a neural network-based adaptive controller and a high-gain
state observer have been proposed and designed for the trajectory tracking control
problem for the uncertain model of the four-degree-of-freedom parallel robot. The
adaptive controller can ensure system stability even when the model contains

28. 21
parameter variation, uncertainties, and noise components. Furthermore, the
proposed controller can minimize the effects of explosion of terms and
“ ”
chattering phenomena on the control quality of the system. In addition, the state
observer used estimate the system states even when the model errors exist.
can To
solve the control problem
First, design the Backstepping algorithm combined with sliding mode
+
control to solve the problem of tracking control for parallel robots with four
degrees of freedom with the assumption of an ideal robot model. Conduct
simulation to verify, evaluate, and analyze the advantages and disadvantages of
this method. Based on the analysis, propose an adaptive controller to improve the
control quality.
Propose a neural network-based adaptive control law for a four-degree-
+
of-freedom parallel robot model. The algorithm is constituted on the basis of the
BASMC controller and the radius basis function neural network. The RBFNN is
capable of approximating the appropriate values for uncertain components while
compensating for the adverse effects stemming from the phenomena of
conventional nonlinear controllers and noises. Then, the control quality of the
system is expected to be significantly improved. Furthermore, a high-gain state
observer is also used to estimate the system states, minimizing the complexity of
using and calculating these values from the sensor system.

29. 22
CHAPTER 3. SIMULATION RESULTS
To verify the correctness as well as efficiency and quality of the proposed
adaptive controller combined with the state observer, the robot simulation system
is constructed in Matlab/Simulink environment. The simulation scenario is divided
into two parts: when using the proposed adaptive controller and when using the
adaptive controller using the high-gain state observer. The control quality is
analyzed, evaluated, and compared with other recently published studies that have
been performed on the four-degree-of-freedom parallel robot model to
demonstrate the efficiency as well as the superiority of the method proposed in the
thesis.
3.1 Results of the RBFNN based adaptive controller (RBFNNB)
This section presents the simulation results of 4DOFPR using the RBFNNB
controller. Assuming that, the robot experiences certain elements
the model’s un
and external noises.
The following ternary function describes reference trajectories
2 3
( ) ,
t t t t
= + + +
d o 1 2 3
p a a a a (3.1)
in which, the coefficients 1 2
, , ,
o
a a a and 3
a from
are computed
2 3
0
2
1
2 3
2
2
3
1
0 1 2 3
1
0 1 2 3
o
o o
o o
f f f
f f
a
t t t
a
t t
a
t t t
a
t t
    
    
    
=
    
    
 
 
 
 
o
o
f
f
p
v
p
v
(3.2)
where o
p , o
v , and o
t are the initial position, velocity and time, while f
p , f
v , and
f
t those of the end. The desired trajectory ( )
t
d
p is constructed using the parameters
in 3.1.
Table
Table 3.1 Reference trajectory parameters
Initial position parameters End position parameters
0.05( / )
0.2( / )
0; ;
0.15( / )
0.1( / )
0.55( )
( )
5
.
( )
8
( )
6
o
m s
rad s
t
rad s
rad s
m
rad
rad
rad



 
 
 
= =
 
 
 
 
 
−
 
 
 
= −
 
 
 
 
 
0
0
v
p
0.0( / )
0.0( / )
5; ;
0.0( / )
0.0( / )
0.85( )
( )
5
.
( )
8
( )
3
f
m s
rad s
t
rad s
rad s
m
rad
rad
rad



 
 
 
= =
 
 
 
 
 
 
 
 
=
 
 
 
 
 
f
f
v
p

30. 23
Table 3.2 Control parameters
Control gains
1 2 3
8; 10; 10.
k k k
= = =
RBFNN parameters
Neural number: 50
n = ;
0.1
 = ; ˆ (0) 0
=
W ;
1
2
3
1
2
3
1 1
2 2
3 3
15
15
;
(
( , , )
( , , )
;
( , , )
, , )
15
5
l
l
l
linspace r r n
linspace r r n
linspace r r n
linspace r r
r
r
n
r
r
 
 
 =

 =

 =
= −


 = −


 = −

 =
 =




−
1 2 3
; 40.
T
l l l  
 
=     =
 





(2.5,2.5,....,2.5)
n n diag
 =
Γ
Figure 3.1 External force
Besides, system specifications include: a =b=0.25 (m) ; c=0.05 (m); ( )
z
a =0.050 m
, and nominal values of robot’s dynamic coefficients are p
m =15(kg) ; 1
m =0.5(kg)
; 2
m =3(kg) ; dc
m =3(kg) . Additionally, controller gains, as well as RBFNN
parameters, are given in 3.2.
Table
The simulation scenario is conducted with nominal values of matrices , ,
M C and
D; therefore, the system parameters are also nominal values. Moreover, the system
is influenced by an external force in the vertical direction, whose value is described
in Figure 3.1.
The results are demonstrated by comparing the three controllers, including
BASMC, Dynamic Surface Control Neural Network (DSCNN) in [25], and the
proposed controller RBFNNB. With the external force acting on the system as well
as inaccurate information of the system parameters, model-based controllers like
BASMC cannot achieve acceptable control quality despite robust behavior
their

31. 24
inherited from SMC, as in Figure 3.2. Moreover, -state errors exist in the
steady
system’s tracking process when CDS is affected by an external force. Specifically,
the system is not likely to compensate for the deviations caused by an unexpected
impact because of the lack of information in the system’s mathematical model
On the other hand, with a self-learning capacity, the neural network-based
controllers do not require any information related to the system. Therefore, these
controllers can meet the control requirement although the mathematical is
inaccurate or the system is affected by bounded s. As a
external impact result,
DSCNN and RBFNNB significantly perform than BASMC. To clarify this
statement, the tracking errors of the system state p are shown in Figure 3.3, with
maximum values belonging to BASMC and minimum ones of RBFNNB. Besides,
in [25], DSCNN is demonstrated with the neural network ε and low-pass filter
errors are ignored, and thus a learning rate is not considered in an adaptive law.
Therefore, DSCNN ‘s convergence speed is slower than RBFNNB, which leads to
D RBFNNB.
SCNN ‘s tracking error being bigger than
(a) Motion trajectory of z
p (b) Motion trajectory of 
(c) Motion trajectory of  (d) Motion trajectory of 
Figure 3.2 Motion trajectory of p

32. 25
(a) Tracking error of z
p (b) Tracking error of 
(c) Tracking error of  (d) Tracking error of 
Figure 3.3 Tracking error of p
Initially, RBFNN takes time to estimate unknown information concerning
the system. Hence, there is a fluctuation in the early period in responses of DSCNN
and RBFNN when the estimated value considerably affects the control system’s
quality, as in Figure 3.2 Figure 3.3. However, with the proper design,
and
RBFNNB takes advantage of RBFNN to simultaneously estimate appropriate
values for uncertain parts and compensate for the disadvantages of model-based
controllers, in this case, Backstepping and SMC. As a result, the RBFNNB’s
fluctuation is not trivial. Moreover, when the system experiences abrupt changes
of external forces at time instants 1s, 2s, and 4s, the system with RBFNNB
an
controller can promptly drive the system back to the stability states because
RBFNN has almost immediately estimated the proper values (approximately
0.08s) as in Figure 3.4.
It can be seen from Figure 3.4 that the solid lines depict the idea values
which can guarantee the system’s stability and is unknown to the controllers.
RBFNN is designed to estimate these values in the proposed controller, whose
output is the dashed line. At the initial step, the neural network commences
estimating the value for  with a convergence speed depending on the network
structure and the chosen learning rate. By self-learning ability, RBFNN can

33. 26
generate unknown values that track the idea values even when sudden system
information changes.
(a) Approximated value of 1
 (b) Approximated value of 2

(c) Approximated value of 3
 (d) Approximated value of 

Figure 3.4 Approximated values
(a) Motion trajectory of 1
l (b) Motion trajectory of 2
l

34. 27
(c) Motion trajectory of 3
l
Figure 3.5 Motion trajectory of q
(a) Tracking error of 1
l (b) Tracking error of 2
l
(c) Tracking error of 3
l
Figure 3.6 Tracking error of q
Finally, the tracking performance of q is shown in Figure 3.5 and Figure 3.6,
and their trend tends to be similar to p . In general, the proposed controller
RBFNNB gives the best performance compared to BASMC and DSCNN.
3.2 Simulation results of the adaptive controller using the high-gain state
observer
The high-gain state observer is to generate the input state signal for the
adaptive controller designed in the previous chapter. Both the controller and the

35. 28
neural network demand the available values of the position and velocity of the
system in order to calculate the control signal and appropriate estimated values to
guarantee the system s stability. These values will be derived from the state
’
observer, so the control performance majorly depends on the observer s output
’
quality. Therefore, a simulation scenario is built to analyze and evaluate the
observer quality and its influence on the control system, especially in the presence
of uncertainties, model errors, and noises.
For this scenario, the reference trajectory’s parameters are given in Table
3.3. The initial position for 4DOFPR is reselected as 01 02
=0.55( ), =0.45(m),
l m l
03 0.58( ),
l m
= and 0 0( )
rad
 =
Table 3.3 Trajectory reference parameters
Initial position End position
0.( / )
0( / )
0; ;
0( / )
0( / )
0.55( )
( )
12
.
( )
8
( )
12
o
m s
rad s
t
rad s
rad s
m
rad
rad
rad



 
 
 
= =
 
 
 
 
 
−
 
 
 
= −
 
 
 
 
 
0
0
v
p
0.0( / )
0.0( / )
5; ;
0.0( / )
0.0( / )
0.65( )
( )
8
.
( )
9
( )
3
f
m s
rad s
t
rad s
rad s
m
rad
rad
rad



 
 
 
= =
 
 
 
 
 
 
 
 
=
 
 
 
 
 
f
f
v
p

36. 29
Figure 3.7 Uncertain parts in the robot model
Besides, to verify the ability of insensitivity model errors and
to
uncertainties, assuming that the ideal model parameters p
m =15(kg) ; 1
m =0.5(kg);
2
m =3(kg); dc
m =3(kg) and those of nominal models are p
m =25(kg) ; 1
m =1(kg) ;
2
m =10(kg); dc
m =2(kg) . Furthermore, the system is assumed to be experienced
uncertain parts, as in Figure 3.7. The uncertain elements, as well as the model
variation, have adverse impacts on the quality of both controller and observer.
However, as above-mentioned in the previous section, the RBFNNB adaptive
controller appropriate
can estimate the values for the controller.

37. 30
Figure 3.8 Observed values of q
Figure 3.9 Observed values of q

38. 31
Figure 3.10 Observational error of q
The control parameters are chosen in Table 3.2, and the observer parameters
are selected 1 2
0.01; 1.0; 2.0( 1,2,3,4)
ob i i
k k i
 = = = = . Because the objective is to
estimate the robot’s velocity values and the robot position is still necessary for the
observer, assuming that the initial position of the observer coincides with that of
the robot.
The observer outputs are given in Figure 3.8 and Figure 3.9. In Figure 3.8,
it can be seen that the observed position tracks the actual values with minimal
errors (approximate 10-4
) with
(m the robot’s legs and 10-4
(rad) of the angle  )
within the assumption is not different in the initial stage. However, this
that it
quantity is possibly measured to be used to calculate the observer outputs.
Therefore, the comparison of position values is to demonstrate the high-gain
performance.
In addition, the observed velocity values in Figure 3.9 show that the observer
can track the actual velocity. At the initial time, the velocities rapidly increase
because the is different from that of the desired trajectory
robot’s initial position .
Thus, the controller generates the appropriate value to quickly steer the system to
the reference. Moreover, it is evident that, although affected by the uncertain
components and model variation, the velocity values of the robot legs and the
angular velocity around OZ are still able to follow actual values with minor
deviations within the acceptable range. Specifically, the observed velocity value
errors are shown in Figure 3.10. In the beginning, the required observer will need
time to calculate the appropriate value, hence the internal oscillation as shown in

39. 32
Figure 3.10. However, this time interval is also minimal (about 0.03s), and
the convergence speed is fast, which is also one of the outstanding advantages of
the high-gain observer. At the steady-state, the maximum deviation of the legs’
rate is only approximately 6.10-3
and that of the angular velocity is roughly
(m/s),
5.10-4
(rad/s). In addition, it can be seen that there is a steady error phenomenon in
the deviation values of the velocity observed because, according to (2.46), with a
sufficiently small value of ob
 , the observer is able to minimize the adverse effect
of the mathematical model error. However, because of 0 ob

 , this
characteristic cannot eliminate errors entirely, but the parameter of the observer
will be selected so that the deviation is kept in an acceptable range. From the afore
analysis, it can be seen that the high-gain observer can ensure fast convergence
speed and simultaneously minimize the impact of uncertainty components and
model noise.
With the ability to approximate the wide ranges of uncertainties, combined
with the high-gain outputs, RBFNN still ensures that it can generate suitable values
tracking ideal values for unknown parts to guarantee the system stability, as in
Figure 3.11. Therefore, the control quality, as well as tracking performance, are
ensured in Figure 3.12. Thus, the robot states can track their references with minor
tracking errors (about 1,2.10-3
(m)) in Figure 3.13. Besides, there is also a steady
error phenomenon occurring in the observer as .
evaluations mentioned above
However, with the 4DOFPR, the errors are acceptable.
Figure 3.11 Estimated values from RBFNN

40. 33
Figure 3.12 Robot’s trajectory
Figure 3.13 Tracking error
To more extensively demonstrate the ability to alleviate the influence of
model errors, different values of ob
 are chosen. It can be seen that the minor ob

is, the better performance the system can achieve. However, if it is determined to
be extremely small, peaking phenomena will affect the system, leading to
[44]
instability.

41. 34
Figure 3.14 Observed position with different values of ob

Figure 3.15 Observed velocity with different values of ob

In Figure 3.14 Figure 3.15, with
and 0.05
ob
 = and 0.1
ob
 = the observed
quality is not as good as 0.01
ob
 = . The value 0.01
ob
 = ensures superior quality
in comparison to that of the others. On the contrary, with 0.1
ob
 = , the observed

42. 35
error is the st significant deviation. Specifically, the steady error exists in both
mo
position and velocity values. Thus it can have a significant deteriorative impact on
the system performance.
3.3 Conclusion
This chapter illustrated the simulation results of the proposed adaptive
controller based on the RBF neural network compared to existing methods in the
literature. Moreover, the controller performance supported by the high-gain
observer is also presented and analyzed.
In general, the proposed approach demonstrated superior performance to
other methods because of the ability to cope with uncertain parts, external noises,
and problems of conventional controllers. In detail, the tracking errors of
RBFNNB are significantly more minor than those of the others, even in the
presence of the assumed external force and unknown model parameters for both
simulation scenarios. Moreover, a standard high-gain observer can ensure the
acceptable quality of the observed states. Therefore, with the input generated from
the observer, the control performance is still guaranteed. However, both controller
and observer parameters have to be moderately chosen. Thus an appropriate
method to tune them can remarkably improve the system performance. The future
work section in the following chapter will discuss this problem.

43. 36
CHAPTER 4. CONCLUSION AND FUTURE WORK
Parallel robots are increasingly widespread in various fields, including those
in industry or essential operations. Therefore, problems of tracking control, dealing
with uncertain components, model variation, or the impact of exogenous noise are
the content of interest. Moreover, the design of intelligent controllers based on
neural networks to improve control quality has also been a potential research field
for this robot model. In addition, the combined use of the state observer provides
the basis for reducing reliance on complex sensor systems and machining costs.
4.1 Results of the thesis
The thesis has achieved the objectives of researching and proposing a new
adaptive control algorithm based on a radius basis function neural network for the
trajectory tracking problem of a four-degrees of freedom parallel robot. The model
is considered to experience the uncertain elements and the effects of exogenous
noise components as well as model variability. The controller has the ability to
ensure system stability simultaneously and minimize the adverse impact on the
control quality of conventional nonlinear controllers, even when combined with a
high-gain state observer. The main contributions of the thesis:
- thesis focuses on the problem of controller design for the uncertain
The
model of the 4DOFPR The approach based on RBFNN does not majorly depend
.
on the mathematical characteristics of parallel robots. Thus, the developed
controller can cope with the larger range of uncertain elements and bounded
disturbances because of its outstanding characteristic in approximating nonlinear
parts and compensating for the external noises. The neural network outputs are
presented in comparison to nominal uncertain values in order to verify the
efficiency of approximating uncertainties as well as the influence of the neural
network on the system performance (Publication 1)
- The proposed adaptive controller is designed based on the Backstepping
technique aggregated with SMC. Therefore, it takes advantage of these controllers
in eliminating nonlinear parts, as well as enhancing the robustness of the system.
Moreover, the adverse impacts of the two controllers, including “explosion of
terms” and chattering, are remarkably alleviated because the unknown elements,
which are the main factors causing the phenomena, are approximated and
compensated by the neural network. Thus, the developed controller not only
overcomes the disadvantage of the above-mentioned techniques but also increases
the robustness and adaptive behavior of the overall system. (Publication 1)
- The adaptive controller is designed using a high-gain state observer to
estimate robot states to reduce the complexity of using sensor systems. The high-
gain observer ensured observation quality, and thus it can guarantee the control
quality and system stability even under the influence of model errors. The
simulation results show the availability and correctness of the theoretical analysis
and the efficiency of the proposed controller. (Publication 16)

44. 37
In addition, in the process of completing the thesis, research on control
algorithms as well as adaptive control algorithms for nonlinear objects and
parallel robots have also been published at scientific conferences and in
prestigious national and international journals (Publications 1, 2, 3, 9, 12, 14,
15, and 16)
4.2 Future work
Parallel robots are not only affected by uncertainties, model variations, and
exogenous noises, but there are many constraints, both in terms of robot state and
motion of actuator structures. Therefore, considering and using these constraints
for the control law calculation is necessary to improve the control quality and
calculate the reasonable control signal for the system.
The high-gain state observer has an observation quality that is majorly
dependent on the appropriate choice of the parameters, especially when a wide
range of model errors exists. For each stage in the operational process, observer
parameters have different appropriate values, and thus parameter tuning can
improve the system’s quality. Fuzzy logic rules can be designed to provide suitable
values for them based on knowledge about the system as well as the input and
output data sets of the robot model. Moreover, it can also tune control parameters
and those of the neural network to enhance the control performance.

45. 38
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