Chapter 3 is the main contribution of the thesis. In this chapter, the DSC algorithm,
and new adaptive algorithms such as AFDSC, AFNNDSC are proposed to solve the
trajectory tracking control problem for FWOMR. The steps to solve this problem is as
follows:
+ Applying the DSC algorithm to solve the trajectory tracking control problem for
FWOMR. Simulating the DSC control system for FWOMR, analyzing and evaluating the
advantage and disadvantage to propose AFDSC to improve the control quality FWOMR.
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e Backstepping feedback method is a viable solution to
solve affined models [58] and [59]. However, with high order nonlinear systems, the
computational volume is large, complicated, and takes too many computation time due to the
need to calculate the derivative in each iteration step.
4
Sliding mode control (SMC) has also been used [60], [61], [62] and [63 for its superior
properties in the case of the system affected by noise. However, the limitation of the SMC
algorithm is chattering, and reducing this phenomenon requires an accurate object model. It
goes against the properties of the robot model, which is the parameter uncertainty.
In order to improve the quality of control as well as to limit some of the disadvantages
of the Backstepping and SMC controllers, a dynamic surface control (DSC) technique is
introduced in [64] and [65]. The design steps are similar to those of the Backstepping, but to
avoid derivative steps for the DSC virtual control signal, a low pass filter is added, just to get
information about the lead. medium function to filter the high-frequency internal noises
occurring in the control object [65].
For OMR, it is challenging to build an accurate mathematical model because factors
such as friction, load change, and environmental conditions are not known. Therefore, the
effective modern design methods, in this case, are to use adaptive algorithms to tune
controller parameters using Fuzzy logic or approximate the uncertainties using neural
networks. This adaptive controller significantly improves the quality of the nonlinear
dynamics [60], [61], [62], [66], [68], [69], [70], [71] and [72].
With the above reference and analysis, a new adaptive control structure based on a radial
basis function neural network (RBFNN) and fuzzy logic system for the trajectory tracking
controller is researched and developed based on the Dynamic Surface Control (DSC)
technique. A novel adaptive controller with RBFNN for the approximation of the nonlinear
uncertain parameters of the FWOMR and fuzzy logic to tune the controller's parameter is
proposed in the thesis.
1.4. Conclusion
Chapter 1 presented an overview of mobile robot classification and autonomous mobile
robots, which focuses on an autonomous mobile four-wheeled robot (FWOMR) being the
main research object of the thesis. Chapter 1 also focused on a research overview of domestic
and international research on OMR modeling and trajectory tracking control algorithms for
OMR published and analyzed the advantages and disadvantages of these methods from which
to draw appropriate research directions for the thesis.
2. MODELING AND TRAJECTORY CONTROL ALGORITHMS FOR THE
FOUR-WHEELED OMNIDIRECTIONAL MOBILE ROBOT
Building the system of kinematic and dynamic equations for OMR is the very first
problem needed for the synthesis of the trajectory tracking control. In the thesis, the
research object considered is an autonomous four-wheeled robot using Omni-type wheels,
which moves on the plane is affected by friction.
2.1. Building the kinematic and dynamic models of the four-wheeled omnidirectional
mobile robot
2.1.1. The Omni wheel
Omni wheels are arranged perpendicular to the axis of the motor, the wheels are
spaced 3600 / n apart. Omni wheels are widely used in autonomous robots because it allows
the robot to move immediately to a position on a plane without having to rotate before.
Furthermore, the translational movement along a straight trajectory can be combined with
5
rotational movement that causes the robot to move to the desired position with the accuracy
orientational angle.
2.1.2. Kinematic model of the four-wheeled omnidirectional mobile robot [41], [42]
An equation presenting a relationship between the two coordinates is also the robot
kinematic model.
cosθ sinθ
sinθ cosθ
- 0
0
0 0 1
q Hv v (2.1)
where:
cosθ sinθ
sinθ cosθ
- 0
= 0
0 0 1
H is a transition matrix.
From (2.1), we calculate an equation presenting a relationship between the robot’s
position and the velocity of wheels:
1
2
3
4
( )
x
y g
với 2( )g HH
(2.4)
2.1.3. Dynamic model of the four-wheeled omnidirectional mobile robot [41], [42]
The kinematic and dynamic models of FWOMR are constructed based on a robot
model accompanied by the Omni wheels which are positioned 450 apart from the dynamic
coordinate and 900 apart from the beside ones.
From which, the robot’s dynamic equation has the following formula
( ) sgn( ) M q v Cv G v τ Bτ d (2.8)
with: [ ]Tx yv v v is a velocity vector
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
r r r r
r r r r
d d d d
r r r r
B is a control parameter matrix.
0 0
( ) 0 0
0 0
m
q m
J
M
is a matrix with m is the robot mass and J is the inertia moment.
0 0
0 0
0 0
x
y
B
B
B
C
and
0 0
0 0
0 0
x
y
C
C
C
G are viscous friction parameter matrix and
Coulomb friction matrix, respectively.
6
2.2. Several existing trajectory tracking control algorithms for the four-wheeled
omnidirectional mobile robot.
2.2.1. PID controller for FWOMR
The PID controller for FWOMR is proposed in [43] and [44]. These studies have
designed the PID controller based on the kinematic model of OMR. Hence, the effects of
external forces on the system in the robot's dynamic equation were not taken into account.
1
2
1
3
4
( )
( ) ( )
( )
d d
d d
d d
x t x x
y t y g y
t
e
(2.11)
We need to find the angular velocity vector of the wheels for the closed-loop controller
to be stable.
01
2 1
3 0
4
0
( )( ( ) ( ))
t
e
e t
T T
P e I e
e t
e
x d
x
g g g K y K y d
d
(2.12)
with PK and IK are diagonal positive definite matrices.
2.2.2. Sliding mode control for FWOMR
SMC in [60], [61], [62] and [63] is commonly used for robot systems in general and
FWOMR in particular because of its robust characteristic with external noises.
From (2.1) and (2.8), let 1
2
x q
x v
, we have state equations:
1 2
2 2 2sgn( ) d
x Hx
Mx Cx G x τ Bτ
(2.19)
with dτ is uncertain and not accurately measured, and thus, this component will not
exist during the calculation of SMC, MSSC controllers.
Define the sliding surface with conditions and assumptions.
Define the errors 1 1 1d
2 2 2d
e x x
e x x
with 1dx is a reference trajectory and
1
2d 1d
x H x is
reference velocity.
Choose the sliding surface
1 1 S e e (2.20)
with >0 is a sliding surface coefficient.
Take the sliding surface’s derivative
1 1
2 2 1 2 2 2 2(M ( sgn( )) ( ) )d
S He He e H Bτ Cx G x x H H e (2.21)
Choose a Lyapunov candidate function
7
21
2
V S (2.22)
Take its derivative, we obtain
1
2 2 2 2( ( sgn( )) ( ) )dV
-1SS SH M Bτ Cx G x x H H e (2.23)
With the control signal is chosen as follows
1 1
2 2 2 2 1( ) ( (( ) ) sgn( ) sgn( ))
T T
d K
τ B BB M H H e x Cx G x S (2.24)
1sgn( ) 0V K S S
, which satisfies the Lyapunov standard.
The sliding controller (2.24) is designed for stability and durability when the system
exists with model deviation and impact interference. The function V in the formula (2.22)
with control law (2.24) for the FWOMR system is the Lyapunov function of the closed
system.
2.2.3. Multiple sliding surface control for FWOMR
- Consider the robot’s state equations
1 2
2 2 2sgn( ) d
x Hx
Mx Cx G x τ Bτ
(2.36)
with 1
x
y
x và 2
x
y
v
v
x
- Consider the sliding surface
11
1 12 1 1
13
d
S
S
S
S x x (2.37)
- Take derivative of 1S and use (2.37), we obtain
1 1 1 2 1 d d S x x Hx x
(2.38)
Choose a virtual control signal
1
2 1 1 1( )d dK
x H S x (2.39)
- Choose the first Lyapunov candidate function
1 1 1
1
2
TV S S
(2.40)
- Take derivative of 1V , and use (2.38) and (2.39)
1 1 2 1 1 1
T TV K S S S S (2.41)
- Consider 2S as the second sliding surface
2 2 2 )( d S x xH (2.42)
Taking derivative of 2S
2 2 2 2 2
1
2 2 2 2 2
( ) ( )
( ( sgn( )) ) ( )
d d
d d
S H x x H x x
H M Bτ Cx G x x H x x
(2.43)
Combine (2.39), (2.40), (2.43), and (2.44), we obtain:
8
1 2 1 1H K S S S
(2.44)
- Choose th control signal as follows:
1 1
2 2 2 2 2 2 2( ) ( ( ( ) ) ) sgn( ) )
T T
d d K
τ B BB M H H x x x Cx G x S (2.45)
We have:
2 2 2K S S
(2.46)
Choose the second Lyapunov candidate function
2 1 1 2 2
1 1
2 2
T TV S S S S
(2.47)
Take derivative of 2V and combine with (2.45), (2.46), (2.47), and (2.48)
2 22 1 1 1 1 1 12 2 2 2
T T TV K K S S S S S SS S S S (2.48)
We have
2
2 2 2
2
2 1 1 1V K K SS S S
(2.49)
- Choose 21
1
2
KK K with 0K , and we obtain:
2 2
2 1 1 1
2
1 1
2 2
2 2 2
2 2
2 2
1 1
2
1
(
2
2
)
V K K
K K
S SS S S
S S
S
S S
- Thus, 2V
is the Lyapunov function of the close-loop system.
One disadvantage of this approach is that it is necessary to compute the derivative of
the virtual control signal 2dx because this input depends on the slip surface and state
variables of the system (2.43). That is also the difficulty when using the MSSC method.
2.3. Conclusion
In chapter 2, the thesis has obtained the following results:
Model a four-wheeled omnidirectional mobile robot with the selected structure,
construct the kinetic and dynamics equations and analyze the dynamic of FWOMR based on
numerical simulation.
Research some typical trajectory tracking control algorithms applied to FWOMR,
survey and evaluate the advantages and disadvantages of these control methods by Matlab /
Simulink software such as:
- PID
- Sliding mode control
- Multiple sliding surface control
Based on theoretical analysis and simulation results, the sliding multi-surface control
method (MSSC) will be further researched and developed in the following chapter.
3. DESIGN AN ADAPTIVE TRAJECTORY TRACKING CONTROLLER FOR
THE FOUR-WHEELED OMNIDIRECTIONAL ROBOT
In Chapter 3, a novel control algorithm is proposed for the FWOMR. The control
algorithm is designed based on the basis of the DSC technique which is developed on
9
MSSC combined with the Backstepping technique. An adaptive DSC is constructed using a
fuzzy rule and a neural network for FWOMR to overcome the disadvantages of the DSC
and expand the application field for FWOMR which has uncertain nonlinear elements and is
influenced by noises. The adaptive DSC controller is simulated and evaluated by Matlab-
Simulink software.
The studies in Chapter 3 propose new adaptive algorithms, including AFDSC and
AFNNDSC, to solve the trajectory tracking control problem for FWOMR, in the case of
uncertain components in the robot model, as well as the effects of noises.
3.1. Dynamic Surface Control
Design a trajectory tracking controller using the dynamic surface control for
FWOMR
To simplify the calculation and demonstration of the control system stability, system
state variables are set as follows:
1
2
[ ]
[ ]
T
T
x y
x y
v v
x q
x v
(3.1)
and we obtain the system state equations as follows
2
1 2
2 2 sgn d
x Hx
Mx Cx G τ Bτx
(3.2)
With the assumption that an accuracy model is identified and dτ is considered as the
unknown external noises, the FWOMR model with the existence of disturbances has the
formula as follows
1 2
22 2 sgn
x Hx
Mx Cx G Bτx
(3.3)
First, define 1 1 1d e x x as a tracking error vector, where 1
T
d d d dx y x is
the desired trajectory vector. The control target is to ensure that 1x approach 1dx or 1e
tends to 0.
Take derivative of 1e
1 1 1 2 1d d e x x Hx x (3.4)
Assuming that fα is a virtual control signal in the design of DSC controller. α is an
input of the first-order lowpass filter
1 1 1 1dc
H e xα (3.5)
with
1
1 1
1
0 0
0 0
0 0
x
y
c
c c
c
is a appropriate diagnonal matrix containing positive
elements.
The first-order lowpass filter has a formula
T f fα α α (3.6)
With T is chosen small enough not to increase the calculation time of the DSC. A
10
Lyapunov candidate function is proposed
1 1 1
1
2
TV e e (3.7)
Take derivative of 1V
1 1 1 1 2 1 1 1 1 1 1 1 2 1
T T T T
d dV c c e e e Hx x e e e e Hx x
(3.8)
It can be seen that from (3.8) with the virtual control signal (3.5), 1 1 1 1 0
TV c e e and
that leads to the condition 1 1 1 1 0
TV c e e is satisfied.
Define the virtual signal error
2 2 f e x α (3.9)
Choose the sliding surface
1 2 S e He (3.10)
where is a coefficient.
Take derivative of S
11 2 2 1 2 22 sgn f S e He He e He H M C G Bτ αxx (3.11)
The second Lyapunove candidate function is chosen as
2
1
2
TV S S (3.12)
The control signal includes the two elements eqτ và swτ
eqτ keeps the system states on the sliding surface. eqτ is calculated from solving
0S .
11 2 2 21 2( ) sgndT Teq G e He x Cτ B BB xM xH (3.13)
The equation of swτ is chosen as follows:
1 3w 1 2( ) sgnT Ts c c τ B BB M SH S (3.14)
with
2
2 2
2
0 0
0 0
0 0
x
y
c
c c
c
and
3
3 3
3
0 0
0 0
0 0
x
y
c
c c
c
are the diagonal positive definite
matrixes. Finally, the control signal is the sum of eqτ and swτ :
eq sw τ τ τ (3.15)
Theorem 3.1: Consider the FWOMR model is described by (2.3), the controller (3.15)
with eqτ in (3.13) and swτ in (3.14) guarantees that the close-loop system is stable and the
tracking error tends to 0.
Proof
Taking derivative of 2V
2
TV SS (3.16)
From (3.11), 2V
becomes
11
12 1 2 2 2sgn TV fS e He H M Cx G Bτx α (3.17)
With the control signal (3.13), the derivative of 2V can be rewritten as
2 2 3
T TV c sgn c S SS S (3.18)
By choosing appropriate values for 2c and 3c , we obtain
2 2 3 0
T TV c sgn c S S SS (3.19)
That satisfies the Lyapunov standard, and the Theorem 1 is proven!
3.2. An adaptive fuzzy dynamic surface control for trajectory tracking control for
FWOMR
3.2.1. An adaptive fuzzy dynamic surface control.
The outstanding point of the DSC controller is its stability with variable system
parameters (uncertainties vary in the limited range). However, this strength is only available
when the system state is on the sliding surface or the vicinity of the sliding surface. The
schematic diagram of a fuzzy DSC system is shown in Figure 3.7.
Figure 3.7. The structure of the adaptive fuzzy dynamic surface control system for
FWOMR
Based on the DSC simulation results for FWOMR, we found that the quality of the
system significantly depends the determination of the DSC parameters 1 2 3( , , )c c c . 1c is a
parameter directly affecting the tracking quality of the robot, while 2c and 3c take impact on
the speed of approaching the sliding surface of the system states, as well as the ability to
keep the system states on the sliding surface. In each state, if the right set of parameters is
selected, the system will achieve high-quality performance, especially when the system is
affected by noise. Thus, in this chapter, an adaptive fuzzy DSC is proposed for FWOMR.
The fuzzy inputs are the tracking error 1e and its derivative 1e . Fuzzy sets for
linguistic variables are described in Figure 3.8 and Figure 3.9.
-10 -5 -0.01 0 0.01 5 10
NB NS Z PS PB
-25 -12 -0.06 0 0.06 12 25
NB NS Z PS PB
Figure 3.8. Fuzzy set for 1e Figure 3.9. Fuzzy set for 1e
12
With the input and output data obtained when simulating the DSC controller, the
fuzzy sets of the input language variable, as well as the output values and the constituent
rules for the fuzzy tuner, are built based on the Sugeno fuzzy model. The fuzzy sets for the
input linguistic variables 1e và 1e are triangle forms, while 1 2 3, ,c c c are chosen through
experiment. Fuzzy linguistic variables and their meanings are shown in Table 3.1. The fuzzy
output values are shown in Table 3.2.
Table 3.1. Fuzzy sets of the input linguistic
variable
Linguistic
1e
Linguistic
1e
Meaning
NB NB Negative big
NS NS Negative small
Z Z Zero
PS PS Positive small
PB PB Positive big
Bảng 3.2. Output values
Output
variabl
e
Meanin
g
Output
value of
1c
Output
values of 2c
and 3c
VS Very
small
1.5 20
S Small 4.25 25
M Medium 6.5 30
B Big 8 35
VB Very
big
10 40
Bảng 3.3. Fuzzy rule of 1c
1e 1e
NB NS Z PS PB
N M S VS S M
NS B M S M B
Z VS B M B VS
PS B M S M B
PB M S VS S M
Bảng 3.4. Fuzzy rule of 2c ( 3c )
1e
1e
NB NS Z PS PB
NB M B VB B M
NS S M B M S
Z VB S M S VB
PS S M B M S
PB M B VB M M
3.2.2. Simulation
The external disturbance has the form in Figure 3.10.
Figure 3.10. The external disturbance
The reference trajectory is described by:
0
0
cos( )
sin( )
r
r
r
x r t
y r t
The paramters of the FWOMR model and the controller are chosen as in Table 3.5
13
Table 3.5. System parameters and control parameters
Dynamic parameters 210 ; J=0.56 kgm ; 0.3 ; 0.06 m kg d m r m
Trajectory parameters 00 15, 10t r m
Control parameters (10,10,10); 25diag b
Figure 3.11. x-axis motion
Figure 3.12. y-axis motion
Figure 3.13. angular motion
It can be seen that the controllers ensured the tracking quality, but the AFDSC showed
the most considerable performance.
The parameters 1 2 3( , , )c c c of the AFDSC are in the Figures 3.14, 3.15, 3.16.
Figure 3.17 describes the motion of FWOMR in the Oxy coordinate. It can be seen
that the efficiency of the proposed algorithm when the robot’s trajectory tracks remarkably
close to the reference.
Figure 3.17. Motion of FWOMR
Figure 3.15. 2c Figure 3.16. 3c Figure 3.14. 1c
14
3.3. Adaptive fuzzy neural network dynamic surface control for FWOMR
AFDSC has been a suitable recommendation to improve the tracking quality for
FWOMR in the case of model deviation and noise with a small amplitude. But in the case of
large model deviation, the control quality is no longer guaranteed. Therefore, the estimation
of model deviation and compensation in controller components will ensure to improve the
quality of this controller.
Figure 3.18. The structure of AFNNDSC
3.3.1. Approximation of the uncertainty in FWOMR model using the radial basis
function neural network.
The FWOMR model contains the uncertainties described by dτ in (2.8). Therefore,
the calculated control signal τ in the previous chapter may not reach the good quality in
many cases. Besides, other uncertainties make the AFDSC difficult to perform. The thesis
proposes an estimator using RBF neural network for uncertain components in the AFDSC
controller.
The uncertain elements are described by an equation:
1 22 sgn d Θ x G xM C τ (3.20)
which is a (3x1) vector containing the uncertainties of FWOMR. The equations
describing FWOMR is rewritten as:
1 2
1
2
x Hx
x Θ M Bτ
(3.21)
Conduct the calculation steps which are the same as previous chapter for control
design, the sliding surface’s derivative has the form
11 2 2 1 2 2d S e He He e He H Θ M Bτ x (3.22)
The system control signal is
eq sw τ τ τ (3.23)
with
1 1 1 2 2( ) ˆT Teq d τ B BB M H e He x Θ (3.24)
15
1 1 2 3( ) sgnT Tsw c c τ B BB M SH S (3.25)
where Θˆ is trained online to approximate the system. The radial basis function neural
network contains three layers, including input layer, hidden layer, and output layer.
Figure 3.19. Radial basis function neural network
Lựa chọn các giá trị để tính toán luật thích nghi cho Θˆ
T Θ γ εR (3.26
ˆ ˆ TΘ γR (3.27)
with Θ is idea value of the uncertainty. While, Θˆ is the neural network output and it
is also the value used for the controller.
Next, ˆ R R R is defined as the weight error. The hidden layer output is calculated
by a radial basis function
2 2
1 1 2 2
2
exp i ii
i
x x
γ (3.28)
where 1x and 2x are input vectors of the RBFNN. 1i and 2i are the center vectorsn
nơ-ron, and i indicates the standard deviation. With the designed neural network
structure, the updated law is chosen as
ˆ ˆT γS H SR R (3.29)
where is a square positive definite matrix with n dimension, in which n is a neural
number. is a learning rate, which is chosen in the range (0,1) .
Theorem 3.2: Consider the FWOMR model (3.2), with the controller (3.23) and the
adaptive law (3.29), if the bounded condition
2
3
4
F
N
minc
S
R
(3.30)
is satisfied, then the system stability is validated according to the Lyapunov standard.
Proof
Consider a Lyapunov candidate function:
12
1 1
2 2
T TV tr S S R R (3.31)
Take derivative of 2V
16
12 ˆT TV tr S S )R R (3.32)
Combine (3.22) with the control signal (3.23), the derivative of 2V becomes
2 2 3 1ˆ ˆs ( )gnT T T TV c c tr S HS S S S Θ Θ R R (3.33)
Use (3.22), (3.24), and (3.25), we obtain
12 2 3 ˆsgn T T TT T TV c c tr S S Hε HΘ γS S S S R R (3.34)
After some calculation steps, the derivative of 2V becomes
12 2 3 1 ˆsgnT T T T TV c c tr S S HS S S γS HM R R (3.35)
With the adaptive law (3.29), 2V
is rewritten as
2 2 3sgn T T T TV c c tr S SS S S Hε S R R R (3.36)
Apply the Cauchy-Schwarz inequality
2T
FF F
tr R R R R R R (3.37)
We obtain
22 2 3sgnT T N FF FV c c S SS S S S R R R (3.38)
With the bouned condition (3.30), 2V
becomes
2
2 2
1
sgn
2
T
FF
V c
S SS R R (3.39)
2 0V
and Theorem 2 is proven!
3.3.2. Construct the fuzzy law for AFNNDSC
The fuzzy law is described in the section 3.21. The fuzzy inputs are a vector
containing 1 1 1, ,x y e e e and its derivative, which are shown in Table 3.6. The fuzzy sets for
the input linguistic variables are described in Figure 3.20. In addtion, The fuzzy law outputs
are given in Table bẳng 3.7.
The first output is the tuned value for 1 , ,ic i x y . The other is the value for
2 , ,ic i x y và 3 , ,ic i x y . To simplify the paramters choosing for AFNNDSC, 2ic is
chosen so that it is equal 3ic . The fuzzy rule is given in Table 3.6.
Figure 3.20. The fuzzy sets for the input
17
Figure 3.6. Fuzzy rules for 1 2( )i ic c
1e 1e
NB NS Z PS PB
NB M(M) S(B) VS(VB) S(B) M(M)
NS B(S) M(M) S(B) M(M) B(S)
Z VS(VB) B(S) M(M) B(S) VS(VB)
PS B(S) M(M) S(B) M(M) B(S)
PB M(M) S(B) VS(VB) S(B) M(M)
Figure 3.7. Output values of 1 2( )i ic c
VS 3.0 (5)
S 4.15 (10)
M (20)
B 7.5 (25)
VB 12 (30)
3.3.3. Simualtion results.
In this section, the simulations are performed in Matlab/Simulink environment.
3.3.3.1. Robot model affected by the external disturbances
The reference trajectory for FWOMR is a circular trajectory described by a equation
with the radius 0 5r m ; 0 10t . In addition, the robot parameters are: m=10kg,
J=0.56kgm2, d=0.3m, r=0.06m. The initial position of the robot is chosen as ( ; ) (2;2)x y
In this case, the system is simulated and evaluted in the condition, which is directly
influenced by the Gaussian-type moment noise affecting the motors. Moreover, the impact
of friction force is ignored. The sliding surface coefficient is chosen as: 10,10,10diag
Figure 3.21. Moment noise (Nm)
The simulation results are shown as the following figures. The Figure 3.22, 3.23, and
3.24 compare the tracking error of FWOMR when using the thr
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