The necessary of the thesis
In these years, there is a growing recognition that mobile robots have the capability to operate
in a wide area and further the ability to manipulate in an automatic and smart way without any actions
taken by human. Hence, this project concentrated on researching and developing some control laws
for wheeled mobile robots.
The researching problems of this thesis
The author concentrated on radical control methods in order to deal with wheel slipping
whenever there exist slippage, model uncertainties, and external disturbances.
Object of study
So as to easily demonstrate the validity and performance of the proposed control methods,
the object of study was selected to be one threewheel mobile robot. To be specific, this robot consists
of two differential driving wheels and one caster wheel used to make gravity balance.
The purpose of researching
Proposing a number of radical control approaches so as to cope with the negative effects of
model uncertainties, external disturbances, and above all slippage.
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MINISTRY OF EDUCATION
AND TRAINING
VIETNAM ACADEMY OF
SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN VAN TINH
RESEARCHING AND DEVELOPING SOME
CONTROL LAWS FOR A WHEELED MOBILE
ROBOT IN THE PRESENCE OF SLIPPAGE
ENGINEERING DOCTORAL DISSERTATION
Major: Control and Automation Technology
Code: 9.52.02.16
SUMMARY OF ENGINEERING DOCTORAL
DISSERTATION
Ha Noi, 2018
This work is completed at:
Graduate University of Science and Technology
Vietnam Academy of Science and Technology
Supervisor 1: Dr. Pham Minh Tuan
Reviewer 1:
Reviewer 2:
Reviewer 3:
This Dissertation will be officially presented in front of the Doctoral
Dissertation Grading Committee, meeting at:
Graduate University of Science and Technology
Vietnam Academy of Science and Technology
At . hrs . day . month. year .
This Dissertation is available at:
1. Library of Graduate University of Science and Technology
2. National Library of Vietnam
ABSTRACT
The necessary of the thesis
In these years, there is a growing recognition that mobile robots have the capability to operate
in a wide area and further the ability to manipulate in an automatic and smart way without any actions
taken by human. Hence, this project concentrated on researching and developing some control laws
for wheeled mobile robots.
The researching problems of this thesis
The author concentrated on radical control methods in order to deal with wheel slipping
whenever there exist slippage, model uncertainties, and external disturbances.
Object of study
So as to easily demonstrate the validity and performance of the proposed control methods,
the object of study was selected to be one threewheel mobile robot. To be specific, this robot consists
of two differential driving wheels and one caster wheel used to make gravity balance.
The purpose of researching
Proposing a number of radical control approaches so as to cope with the negative effects of
model uncertainties, external disturbances, and above all slippage.
Approaches of study
The approach of study is illustrated as the following order:
Analyzing and building the kinematic and dynamic model of the mobile robot with
the occurrence of model uncertainties, external disturbances, and above all slippage.
Researching, analyzing stateoftheart control methods which were designed both
domestic and foreign for this topic. After that, some radical control methods were
proposed.
Proving the correctness and efficiency of the proposed control approaches via
Lyapunov standard and Barbalat lemma.
Demonstrating the abovementioned control methods through Matlab/Simulink tool.
Scientific and practical benefits of this project
Scientific benefits: Building novel control approaches for a wheeled mobile robot with the
purpose of compensating for the negative effects of model uncertainties, external disturbances, and
above all slippage.
Practical benefits: the proposed control methods in this project could be applied for wheeled
mobile robots operating in warehouses with the slippery floor and/or in orchards with wet land.
Structure of the thesis
Chapter 1: Overviewing domestic and foreign studies in recent years, and then showing a
process by which the kinematic and dynamic model of a wheeled mobile robot are established in the
presence of model uncertainties, external disturbances, and slippage.
Chapter 2: Designing an adaptive control law based a threelayer neural network.
Chapter 3: Designing a robust adaptive backstepping control law based a Gaussian wavelet
network.
Chapter 4: Designing a backtepping control law ensuring finitetime convergence at
dynamic level.
CHAPTER 1. OVERVIEWING AND MATHEMATIC MODELS
1.1. Problem statement
Motion control problem is fairly important in the field of mobile robot because the
performance of control laws affects the efficiency of the application of mobile robots in production
and life. Thus, this problem is chosen as the goal of this project.
These days, motion control problem for wheeled mobile robots has attracted the
consideration of researchers all over the world. Needless to say, a wheeled mobile robot is one of the
system subjected to nonholonomic constraint [1]. Furthermore, it is one multi input – multi output
nonlinear system [2]. It is thanks to the recent advances in control theory as well as engineering,
there were a large number of different control methods applied such as sliding mode control [34],
robust control [5], These control laws were under the assumption of pure rolling and no slippage.
Notwithstanding, in application practice, the violation of the above assumption can still
happen. That is to say, there exists slippage [1213].
Slippage is one of the key factors making the visible degradation of control performance.
Therefore, in such circumstances, so as to heighten control performance, a controller must be capable
of compensating slippage.
1.2. Domestic study
In Vietnam, until now, there have been reports researching autonomous vehicles such as the
group of the authors from University of Transport and Communications in Hanoi studying swarm
robots [1415]. One group of authors at Hanoi University of Science and Technology researched on
building a mathematical model for one fourwheel electrical car considering the interaction between
wheels and road [18]. Nevertheless, there have been still not many the studying results of addressing
slippage for wheeled mobile robots to be published.
1.3. Foreign study
There have been reports researching on control problems compensating slippage for wheeled
mobile robots. It is due to slippage that the performance of closed system deteriorates and even the
state of the system is unstable. Frequently, so as to cope with slippage, the values of friction
parameter and sideslip angle must be always measured in realtime accurately. Specifically, the
authors in [12] addressed slippage through compensating slipratios of wheels. Gyros and
accelerometers were utilized in [13] so as to compensate slippage in realtime. The study in [19]
reported a robust controller by which both slipkinematic and slipdynamic models were taken care
thanks to the framework of differential flatness.
1.4. Kinematic model
Let us consider one wheeled mobile robot under a nonholonomic constraint as Fig. 1.1.
Without slippage, the linear and angular velocities are calculated as follows [21]:
R L
R L
2
2
r
r
b
(1.1)
Where R L, are angular coordinates of the right and left wheel respectively.
Thereby, showing the kinematic model as follows [4]:
M
M
cos
sin
x
y
(1.2)
The nonholonomic constraint always assures the two following factors:
The direction of the linear motion is always perpendicular to the wheel shaft.
Both the linear and angular motion of this robot fully depend on the pure rolling of
the differential driving wheels.
Specifically, the mathematical model of this constraint is shown as follows [32]:
R M M0 cos sinr x y b (1.3)
L M M0 cos sinr x y b (1.4)
M M0 sin cosx y
(1.5)
By stark contrast, in the presence of slippage, the linear velocity along the longitudinal axis
is computed as follows:
R L
2
(1.6)
with R L, being the longitudinal slip coordinates of the right and left wheels respectively. Next,
the actual yaw rate is computed as follows:
R L
2b
(1.7)
Let us define as the lateral slip coordinate along the wheel shaft (see Fig. 2.1). In this
circumstance, the kinematic model of this object is illustrated as follows [30]:
F
2
F
1
F
3
Left wheel
Right
wheel
Caster
wheel
a
2b
F
4
L
R
Wheel
Shaft
G
M
Platform
θ
Figure 1.1. One wheeled mobile robot and slippage.
MM
cos sin
sin cos
x
y
(1.8)
Due to slippage, the nonholonomic constraint is represented as follows [32]:
R R M Mcos sinr x y b (1.9)
L L M Mcos sinr x y b (1.10)
M Msin cosx y (1.11)
1.5. Dynamic model
This dynamic model subjected to slippage, model uncertainties, and external disturbances
is expressed as follows:
d Mv Bv Bv Qγ C G τ τ (1.23)
Property 1: M has the invertible feature, is positive definition, and satisfies the following
inequality
2 2T
1 2M M x x Mx x
with M1 and M2 being upper and lower bound of M and satisfying 2 1M M 0 .
Property 2: matrix 2 M B v is skewsymmetric, that is to say
T 2 0 x M B v x with
2 1 x R .
1.6. Conclusion for Chapter 1
The attention and attempt of researchers all over the world for compensating slippage has
increasingly become more prevalent than ever before. However, most the studies were conducted
under the assumption that the sideslip angle and friction parameter always are measure exactly in
real time.
It goes without saying that accelerates and velocities are always directly measured via
affordable and feasible sensors. Yet, it is difficult and expensive to measure the sideslip angle and
friction parameter [40].
Taking into account all the factors mentioned above, this project is going to offer radical
control approaches so as to compensate slippage for a wheeled mobile robot without measuring the
sideslip angle and friction parameter. In stark contrast, the negative effects of slippage are going to
deal with in an indirect way via the proposed controllers here.
Moreover, the kinematic and dynamic model of the wheeled mobile robot subjected to
slippage, model uncertainties, and external disturbances were established successfully. These model
are going to be used for designing control laws in next chapters. This researching result was published
in the number 3 published material.
CHAPTER 2. DESIGNING AN ADAPTIVE CONTROL LAW BASED ON A THREE
LAYER NEURAL NETWORK
2.1. Problem statement
Due to the fact that the control law in Chapter 3 was designed under such an ideal, the
applicability of that control method is very limited. Therefore, in this chapter 4, one radical control
method is proposed under a more practical condition in order to heighten the applicability in
comparison to the method in Chapter 3.
To be specific, such a more practical condition involve the following factors:
There exist model uncertainties and external disturbances.
The velocities and accelerates of slippage are not measured.
Let D(xD,yD) be a target which is moving in a known desired trajectory (see Figure 3.1).
Without loss generality, the motion equation of D can be supposed as follows:
D 0
D 0
. cos( . )
. sin( . )
D
D
x T t R t x
y T t R t y
(2.1)
, TD, R, , x0, y0 are constant parameters, and time t varies from zero to infinity.
We assume that the tool location is at point P. So, the requirement of the position tracking control
problem is to control the WMR so that P has to track D with the position tracking errors being
uniformly ultimately bounded.
Remark 2.1: In Figure 2.1, we denote (xP, yP) as the position of P. Let (xP, yP, ) be the actual
posture of the WMR, and (xPd, yPd, d) be the desired one of the WMR. The presence of both the
longitudinal and lateral slips makes it impossible to control the WMR in the way that the actual
posture (xP, yP, ) tracks the desired one (xPd, yPd, d) with an arbitrarily good tracking performance
[32]. Instead of this, it is fully possible to control the WMR with the purpose of making the actual
position (xP, yP) track the desired one (xPd, yPd) with an arbitrarily good tracking performance [32].
2.2. Structure of the threelayer neural network (NN)
Admittedly, artificial neural networks have the ability of approximating nonlinear and
sufficiently smooth functions with arbitrary accuracy. In this subsection, a threelayer NN is
introduced briefly [8]. As illustrated in Figure 2.2, the output of the NN can be computed as
3
T
1 2, , ,..., Ny y y y W V
T T W σ V x where
1
T
1 21, , ,..., Nx x x x is the input vector, and
1
2
xD
yD
axis OY
C
xM
D (target)
P
M
axis OX
yM
x
P
O
axis
MY
axis
MX
Figure 2.1. Illustrating the target in the body coordinate system MXY.
ijw W and ijv
V are the NN weight matrices. (𝐳) = [𝟏,(𝒛𝟏),(𝒛𝟐), ]
𝐓 with 𝐳 =
[𝒛𝟏, 𝒛𝟐, ]
𝐓 . Next, () is the activation function of the NN. In this paper, the activation function
is chosen to be the sigmoid kind as (𝒛) = 𝟏/(𝟏 + 𝐞𝐱𝐩 (−𝒛)).
Let 𝐟(𝐱): 𝐑𝐍𝟏 → 𝐑𝐍𝟑 be a smooth function. There exist optimal weight matrices W and V
so that:
𝐟(𝐱) = 𝐖𝐓(𝐕𝐓𝐱) + , (2.3)
where
is the vector of optimal errors.
Assumption 2.1: is bounded. Especially, ‖‖ ≤ 𝒃 where 𝒃 expresses an upper bound of .
Let 𝐟(𝐱, �̂�, �̂�) = �̂�(𝐱, �̂�, �̂�) = �̂�(�̂�𝐓𝐱) denote an estimation of f(x), where �̂�, �̂� are estimation
matrices of 𝐖 and 𝐕, respectively, and they are provided by an online weight tuning algorithm to be
revealed subsequently.
2.3. Expressing the vector filtered tracking errors (FTE)
Let OXY be the global coordinate system, MXY be the body coordinate system which is
attached to the platform of the WMR (see Figure 2). The coordinate of the target is represented in
MXY as follows:
1 D M
2 D M
cos sin
sin cos
x y
y y
ζ (2.6)
Taking the first order derivative with respect to time of (2.6) yields
D
D
cos sin
sin cos
x
y
ζ hv χ (2.7)
where R
L
v ,
2 2
1 1
1 1
1 1
2 2
2 2
r r
b b
r r
b b
h ,
2
1
2
2
R L
R L
b
χ .
Taking the second order derivative with respect to time of (2.6) yields
1 2 ζ hv Ψ Ψ
(2.8)
where, D D D D
1
D D D D
cos sin sin cos
sin cos cos sin
x y x y
x y x y
Ψ hv
D D
2
D D
sin cos
cos sin
x y
x y
Ψ .
Remark 2.2: If 1 ≠ 0, then h is an invertible matrix.
Let us define the position tracking error vector as
T
1 2 de e e = ζ  ζ
(2.9)
where dζ is the desired coordinate vector of the target in MXY. According to the requirement of
the position tracking control problem mentioned above and Fig. 3.2, one can easily set
T
d ,0Cζ .
In order to tackle this problem via the novel proposed control method, first of all, the scheme
of entire closed loop system is proposed as Figure 2.3.
The vector FTE is defined as follows:
φ = e+Λe (2.10)
where Λ is one diagonal, positivedefinition, and constant matrix. It can be chosen arbitrarily.
2.4. Structure of the controller
2Ψ in (2.2) depends on the velocities and accelerates of slippage directly, so it is uncertain.
Thus, on auxiliary variable is proposed as follows:
1 d 1 κ = h ζ Λe Ψ (2.14)
On the other hand, (2.23) can be rewritten as follows:
d Mv τ Bv d τ
(2.15)
where d Qγ C G Bv .
Next, one control law is chosen via the computingtorque method as follows:
1 ˆˆ ˆ ˆ, , τ Mh Kφ f x W V
(2.19)
where K is a 2 × 2 diagonal, constant, positive definite matrix and is chosen arbitrarily. 𝐟(𝐱, �̂�, �̂�) is
the output of the NN in order to approximate 𝐟(𝐱).
In this work, let us propose the online weight tuning algorithm for the NN weights as follows:
T T T1ˆ ˆ ˆˆ W H σφ σV xφ φ W (2.24)
T T2ˆ ˆ ˆ V H xφ W σ φ V (2.25)
where 𝐇𝟏 is an (𝑵𝟐 + 𝟏) × 𝑵𝟑 positive definition constant matrix. 𝐇𝟐 is an (𝑵𝟏 + 𝟏) × 𝑵𝟐
positive definition constant matrix. is positive constants.
2.5. Stability
Theorem 1. For the WMR subject to wheel slip as in Eq. (1.8), let the control input be given
by Eq. (2.19) and the online weight tuning algorithm be provided by Eqs. (2.24) and (2.25). Then,
according to Lyapunov theory and LaSalle extension, the stability of the closedloop system is
assured to achieve the desired tracking performance where as well as the vector of the weight
errors are uniformly ultimately bounded [8] and can be kept arbitrarily small.
Figure 2.2. structure of the threelayer neural network.
∑
∑
Input layer Hidden layer Output layer
x1
x2
y1
y2
1
yN3 wN2N3
v1
v2
vN2
w1
wN3
x
N1
v
11
Figure 2.3. Scheme of the whole closed loop control system.
2.6. Simulation results
Example 2.1: target D moves in a straight line as follows:
D
D
2 3cos 0,2
0,5 3sin 0,2
x t
y t
(2.36)
Obviously, in Figures 2.5, 2.6, and 2.7, we can easily see that when the accelerations and
velocities of the unknown wheel slips were not measured and model uncertainties and unknown
bounded disturbances existed, the control approach in [8] could not compensate the undesired effects
while the proposed control method effectively dealt with the undesired effects.
Figure 2.4. the timelines of slip velocities.
2.7. Conclusion for chapter 2
All in all, in this chapter, an adaptive tracking controller based on a threelayer NN with the
online weight updating algorithm was developed to let the WMR track a desired trajectory with one
desired tracking performance. It has been clear that the convergence of both the position tracking
errors and the NN weight errors to an arbitrarily small neighborhood of the origin was ensured by
the standard Lyapunov criteria and LaSalle extension. The results of the Matlab simulations
illustrated the validity and efficiency of the proposed control method.
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
time (s)
v
e
lo
c
it
ie
s
o
f
w
h
e
e
l
s
lip
(
m
/s
)
velocities of wheel slip
longitudinal slip of the right wheel
longitudinal slip of the left wheel
lateral slip
controller
WMR
subject
to
slippag
e
Target
(xD, yD)
e
+

Eq. (3.2)
Threelayer
neural network
v
Eq. (4.6)
Bảng 2.1. Các tham số của rô bốt di động [21].
Variable Meaning value
r Radius of each wheel 0,065 (m)
b Haft of the wheel shaft 0,375 (m)
IG
The inertia of moment of the platform about the vertical axis
through G.
15,625 (kg.m2)
IW The inertia of moment of the wheel about the rotational axis . 0,0025 (kg.m2)
ID The inertia of moment of the wheel about the diameter axis . 0,005 (kg.m2)
mG The mass of the platform 30 kg
mW The mass of each wheel 1 kg
C The distance between M and P 0,5 m
a The distance between M and G 0,3 m
Figure 2.5. control performance comparison between two control methods in example 2.1.
Figure 2.6. Comparison of tracking errors between two control methods in example 2.1.
Figure 2.7. the torques of the proposed method in example 2.1.
CHAPTER 3. DESIGNING A ROBUST ADAPTIVE BACKSTEPPING CONTROL LAW
BASED A GAUSSIAN WAVELET NETWORK
3.1. Problem statement
Even though the control method in chapter 2 illustrated the efficiency to cope with model
uncertainties and external disturbances, the control accuracy, namely the tracking error vector e, still
not small enough in compared to the expectation of tasks demanding highaccuracy. The reason may
be:
There was the classification in a clear way for particular tasks. Especially, what control
terms are used to deal with the negative effects of slippage at the kine