Adaptive Fuzzy Control of a Four-Wheeled Mobile Robot Subject to
Wheel Slip
ZENON HENDZEL1, MACIEJ TROJNACKI2
1Rzeszow University of Technology,
al. Powstancow Warszawy 12, 35-959 Rzeszow,
POLAND
2Eduroco, ul. Lakowa 3/5, 90-562, Lodz,
POLAND
Abstract: - In this paper, the adaptive fuzzy tracking control of a four-wheeled mobile robot subject to wheels
slip is considered. We proposed an adaptive scheme in that fuzzy logic approximators are used to approximate
the unknown system functions in designing the adaptive tracking control of a mobile robot. Fuzzy systems are
expressed as a series expansion of basis functions, to adaptively compensate for the mobile robot nonlinearities.
The proposed control system works online, parameter adaptation is realized in every discrete step of the control
process, and a preliminary learning phase of fuzzy system parameters is not required. The stability of the
algorithm is established in the Lyapunov sense, with tracking errors converging to a neighborhood of zero.
Simulation results illustrate the effectiveness of the approach.
Key-Words: - Fuzzy system, Lyapunov stability, mobile robot, tracking control, wheels slip.
Received: September 17, 2022. Revised: May 2, 2023. Accepted: May 19, 2023. Published: June 12, 2023.
1 Introduction
Application of modern methods of realization of
motion of wheeled mobile robots, in which a
fundamental role is played by artificial intelligence
methods, belongs to priority research direction in
the field of modern technologies of autonomous
robots. Despite significant advances in the field of
autonomous robotics, still, many problems remain
unsolved. Most difficulties are associated with a
description of the natural work environment of an
autonomous robot. Usually, the knowledge about
the environment is, in general, incomplete,
uncertain, and approximate. To this field belong, for
example, the problems concerning the inclusion of
the phenomena of mobile robot wheel slips into
control algorithms. Recently, a lot of attention is
devoted to the problems of modeling and control of
wheeled mobile robots taking into account wheel
slips [3], [5], [6], [7], [8], [9], [11], [12], [13], [18],
[23], [24], [29], which follows from possibility of
using those objects in practical applications,
characterized, for instance, by irregular surfaces and
various parameters of wheels contact with the
ground. In the conventional control theory, most of
the control problems are usually solved by
mathematical tools based on the system models.
Fuzzy controllers are assumed to work in situations
where the plant parameters and structures have
some uncertainties or unknown variations. As we
know, based on the universal approximation
theorem, [26], [27], where fuzzy logic systems have
been shown to be capable of uniformly
approximating any well-defined nonlinear function
to any degree of accuracy, many important adaptive
fuzzy control schemes have been developed to
directly incorporate the expert information
systematically and various stable performance
criteria are guaranteed by theoretical analyses, [20],
[21], [22], [28]. Based on the established fuzzy
system properties, various adaptive fuzzy control
schemes have been systematically developed, by
which the stability of the closed-loop system can be
guaranteed by theoretical analyses, [22], [27].
Among these approaches, the adaptive tracking
control method with a radial basis function fuzzy
system, [17], is proposed for nonlinear systems to
adaptively compensate the nonlinearities of the
systems, [4]. The indirect and direct adaptive
control schemes using fuzzy systems for nonlinear
systems have also been shown in [19], to provide
design algorithms for stable controllers. In addition,
control systems based on a fuzzy control scheme are
augmented with variable structure control, [27],
[29], to ensure global stability and robustness to
disturbances.
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In this paper, the intelligent stable adaptive fuzzy
control system for the position and heading of a
four-wheeled mobile robot with the inclusion of
longitudinal and lateral slips is proposed, in which
fuzzy systems are used for compensation of
nonlinearities and variable operating conditions of a
mobile robot.
The structure of the paper is as follows. In section
2 basic kinematic relationships are discussed, and
generalized velocities required for realization of the
desired robot motion, understood as kinematic
controller, are determined using the backstepping
method. Dynamic equations of motion of a four-
wheeled mobile robot taking into account wheel
slips are given in section 3. Section 4 concerns the
description of the adopted structure of an adaptive
fuzzy system for compensation of robot
nonlinearities. In section 5 synthesis of tracking
control of mobile robots is conducted and stability
analysis of the control algorithm is carried out based
on Lyapunov’s theory. In section 6 obtained results
of simulations of the introduced solution are
presented. Conclusions are given in section 7.
2 Kinematic Controller for WMR
The object analyzed in the present article is a four-
wheeled mobile robot. A diagram of its kinematic
structure is shown in Fig. 1, [24], [25].
Fig. 1: Model of the analyzed robot
In the model, the following basic robot
assemblies can be distinguished: 0 – mobile
platform (body with additional control and
measurement frame attached to it), 1-4 – wheels, 5-6
– toothed belts (caterpillars). In the analyzed robot,
the front wheels are coupled with the back wheels
by means of the toothed belts. The following
symbols are adopted for i-th wheel: Ai – geometric
centre, ri – radius, θi – wheel spin angle. Mobile
platform spin angle is denoted
oz
o
. It is assumed
that the motion of the mobile robot occurs in the Oxy
plane (as shown in Fig. 1). Position and orientation of
the mobile platform are described by generalized
coordinates vector:
T
oz
o
R
o
R
oo yx
,,q
(1)
where: oxR, oyR – coordinates of the point R of the
mobile platform, φz = oφoz – the spin angle of the
mobile platform with respect to z-axis of stationary
coordinate system {O}. Generalized velocities
vector
q
can be determined based on the value of
the velocity of motion of the point R of the robot
along the direction of the x-axis of the {R} system
connected with the robot, that is vR, and angular
velocity of spin of the mobile platform, that is
,
based on the kinematic equations of motion in the
form:
.
10
0sin
0cos
R
z
z
z
R
Rv
y
x
q
(2)
The above equation is valid if the robot moves on
horizontal ground. In the control of the position and
heading of the robot, one assumes that the motion of
the robot is realized based on the desired vector of
its position and heading, which has the form:
T
dRdRddxx
,,q
, (3)
where: xRd, yRd – desired coordinates of the
characteristic point R of the robot in the {O}
coordinate system in (m), φd = oφozd – the desired
spin angle of the mobile platform with respect to z-
axis of {O} coordinate system in (rad). To define
the problem of tracking control, based on the
relationship (2) let us define desired parameters of
motion of the point R in the form of the equation:
d
Rd
d
d
d
Rd
Rd
d
v
y
x
10
0)sin(
0)cos(
q
(4)
where: vRd, ωd – respectively desired linear velocity
of the characteristic point R of the robot in (m/s) and
desired angular velocity of its mobile platform in
(rad/s), in the stationary coordinate system {O}. In
the problem of tracking control, one should
determine the vector of control of position and
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heading of the robot us = [vs, ωs]T, such that q → qd
for
t
. The errors of the robot’s position and
heading in the coordinate system associated with the
robot {R} and in the stationary system {O} can be
determined from the relationship:
zzd
RRd
RRd
z
R
z
zz
O
L
F
eyy
xx
e
e
e
100
0cossin
0sincos
q
zzd
RRd
RRd
y
x
yy
xx
e
e
e
e
(5)
where
OLF eee ,,
are respectively longitudinal
position error in (m), lateral position error in (m),
and heading error in [rad]. Generalized velocities
required for the desired motion of the robot can be
determined using various methods. A popular
method used for this purpose is the so-called
backstepping method, [1], [2], [7], [15]. According
to it, the vector of desired generalized velocities of
motion of the robot’s mobile platform expressed in
the robot’s coordinate system {R} can be
determined based on the following relationship:
oRdoRdLLd
oRdFF
s
s
devkvek
evek
v
sin
cos
u
(6)
where:
ss
v
,
– desired velocities of robot motion
expressed in the coordinate system {R}, that is, the
linear velocity of characteristic point R in (m/s) and
angular velocity of the mobile platform in (rad/s), kF
(s–1), kL (rad/m2), ko (rad/m) – chosen positive
parameters.
3 Dynamic Model of a WMR Subject
to Wheel Slip
In Fig. 2 a schematic diagram of the analyzed robot
with marked reaction forces acting on the robot in
the wheel-ground plane of contact is presented, [24].
Fig. 2: Diagram of reaction forces acting on the
robot in the wheel-ground contact plane
In the description of the motion of the four-
wheeled robot, it is assumed that the tire-ground
coefficient of adhesion changes according to the
Kiencke model and values of longitudinal slip ratios
3
and
4
depend respectively on angular
velocities of driven wheels
3
and
4
. Additionally,
equality of driving torques for passive and active
wheels is assumed, that is,
31
and
42
. After
taking into account the above assumptions, dynamic
equations of motion for the hybrid chassis system,
i.e. with wheels and toothed belts, are written as
[24]:
,
sgn2
/4sgn2
sgn2
/4sgn2
20
02
4
3
4876
543
2
4
2
442
3876
543
2
3
2
332
4
3
1
1
RyRx
RyRxpp
RyRx
RyRxpp
aaaaa
aaaaaa
aaaaa
aaaaaa
a
a
(7)
where
RyRxpaa ,,
are respectively: a constant
associated with a model of wheel-ground adhesion,
projections of acceleration of characteristic point R
of the robot in the coordinate system associated with
the robot {R}. In turn, constants ai that occur in
equation (7) result from geometry, masses, and
distribution of masses of the analyzed robot and
were determined in the work, [24]. From the
kinematic relationships of the analyzed model of the
mobile robot, one can determine angular velocities
of driven wheels as functions of control signals that
realize the desired trajectory of the robot’s motion,
according to the following relationship:
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,
2/1
2/1
1
4
3R
W
W
ru
(8)
where the control signals’ vector has the form:
.
T
RR v
u
(9)
After introducing equation (8) into dynamic
equations of motion of a mobile robot (7), one
obtains:
,ττuFuM zRRR
Waa
Waa
r11
11
2
2
1
M
, (10)
where: M is a constant positive-definite inertia
matrix,
12
)( x
RR RuF
is a vector describing robot
nonlinearities,
12x
zRτ
denotes bounded unknown
disturbances which include, for example, motion
phenomena not taken into account in the
description,
12x
Rτ
is control signals’ vector
identical with torques of robot driving wheels 3 and
4. The dimensionality of equation (10) results from
the assumption of active torques of wheels 3 and 4
and passive torques of wheels 1 and 2 being
respectively equal.
4 Fuzzy Systems, Fuzzy Basis
Function Expansion and Function
Approximation
Problems of control of wheeled mobile robots with
the inclusion of wheels’ slips are complex and their
solution requires the application of complex
methods. Because of the lack of a systematic
approach to analysis and synthesis of control of
nonlinear systems so far, the adaptive fuzzy systems
became an attractive tool used in the theory of
nonlinear systems. Fig. 3 shows an adaptive fuzzy
system. An adaptive fuzzy system is defined as a
fuzzy system equipped with a learning algorithm,
where the fuzzy system is constructed from a set of
fuzzy IF-THEN rules using fuzzy logic principles
and the learning algorithm adjusts the parameters of
the fuzzy system based on the training information,
[14], [16], [17], [20], [26], [28]. Adaptive fuzzy
systems can be viewed as fuzzy logic systems
whose rules are automatically generated through a
training process. In this section, we will give the
mathematical formulas of fuzzy systems and fuzzy
basis functions.
Fuzzifier Defuzzifier
Fuzzy Rule Base
Fuzzy Inference Engine
Adaptation/
Learning Signal
xy
Fig. 3: Adaptive fuzzy system
Without loss of generality, we assume that fuzzy
systems are MISO systems
VU:f n
,
where
n
n21 U...UUU
is the input space
and
RV
is the output space. Consider a fuzzy
logic system (FLS) with rules in the following form
j
isy THEN
j
n
A is
n
x...and and
j
1
A is
1
x IF :
j
R
,
N,...,2,1j
(11)
Where
j
i
A
are fuzzy sets defined by their respective
membership functions
n1,2...,i ,
xA i
j
i
and
j
are singleton rule consequents. When a
product and operator and a product implication
method are used together with the center of gravity
defuzzification method, this leads to a fuzzy logic
system with the following form
N1j i
x
n1i A
N1j ji
x
n1i A
)(fy
j
i
j
i
x
. (12)
Which coincides with the Takagi-Sugeno model,
[22]. When all the parameters of the FLS in (12) are
considered free, methods such as back-propagation
learning can be applied. The idea introduced in [14],
[20], [26], is to fix the premise parameters of the
FLS such that the resulting fuzzy system is
equivalent to a linear combination of nonlinear
functions called fuzzy basis functions.
Definition 4.1, [26], defines fuzzy basis functions
(FBF) as
N1j i
x
n1i A
i
x
n1i A
j
p
j
i
j
i
, j=1,2,…,N.
(13)
Now the fuzzy system (12) is equivalent to a linear
combination of an FBFs
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j
N
1j j
pf
xx
. (14)
In order to develop learning algorithms for these
fuzzy systems, we need to specify the functional
form of the fuzzy membership function for a fuzzy
set
j
i
A
. The membership function can be any
continuous bounded function, e.g., the Gaussian
membership function
2
w
2
cx
exp),c;x(A
, (15)
and Fig. 4 shows an example of FBFs in one-
dimensional premise space.
Fig. 4:. Membership functions
It has been shown, [10], [21], [27], that FLS possess
the universal approximate property. That is, for any
given continuous function
)(g x
on a compact set U
and any given real number
0
, there exists a
fuzzy system
xf
in the form (14) such that
xx
Ux fgsup
. (16)
Therefore, the fuzzy system (14) is qualified to
estimate the unknown non-linear function
)(g x
. In
fact, there exist, ideal control representatives
P
,
centroids c and widths
w
, so that the non-linear
function can be represented as
xPΓxT
g
, (17)
with the estimation error bounded by
m
.
Then an estimate of
)(g x
can be given by
xPΓxT
ˆ
g
ˆ
, (18)
where
Γ
ˆ
is an estimate of ideal values provided by a
learning algorithm. It is shown in [26], [27], that
Gaussian basis functions do have the best
approximation property. This is the main reason we
choose the Gaussian function as the membership
function. In this work, an FBF can be generated
based on a numerical input-output pair.
5 Adaptive fuzzy Control Algorithm
and Stability
In the present section, the synthesis of control of
position and heading of a wheeled mobile robot
using the control structure of nonlinear systems will
be conducted, which takes into account
compensation for robot nonlinearities realized by
means of the FBFs linear with respect to parameters
described in section 4. The task of this control will
be the reduction of the actual control vector (9) to
the control vector resulting from the analysis of
kinematics (6). To this end, let us define the velocity
tracking error:
Rd uus
. (19)
After differentiating relationship (19) and inserting
it into (7), one obtains dynamic equations of motion
written as a function of the velocity error:
ττxfsM z
)(
, (20)
where
z
represents bounded disturbances so that
Z
z
and nonlinear function has the form:
)()( Rd uFuMxf
. (21)
Vector x allowing determination of the value of the
nonlinear function can be defined as:
T
T
R
T
duux ,
, (22)
and it should be available for measurement. The
function f(x) involves all parameters of the analyzed
wheeled mobile robot such as masses, mass
moments of inertia, coefficients of motion
resistance, and description of the slip phenomenon.
Quantities of this kind usually can be described only
in an approximate way. Because the function f(x) is
described approximately, if one adopts the law of
control with the inclusion of this approximation in
the form:
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δs
p
kxfτ )(
ˆ
, (23)
where
)(
ˆxf
is an output of a fuzzy system,
p
k
is a
positive-definite diagonal matrix, and
δ
is a control
signal robust to non-modeled phenomena and other
disturbances, then the description of a closed system
one may express as:
δτxfs
p
ksM z
)(
~
, (24)
where velocity tracking error s in a significant way
will depend on the correct approximation of robot
nonlinearities. Approximation of the control
compensating for nonlinearities f(x) is often applied
in practice. For the approximation, a fuzzy system
may be used. It is convenient to use a fuzzy system
linear with respect to the parameters, described in
section 4. Then, the nonlinear function
approximated by the fuzzy system one can write in
the form:
εxΓxf P
T
, (25)
where
ε
is approximation error satisfying condition
m
εε
,
0constεm
.The estimate of the f(x)
function can be written as:
xΓxf P
T
ˆ
ˆ
, (26)
where
Γ
ˆ
is the matrix of estimated parameters of an
ideal fuzzy system. After using (26) in the control
law with the robot’s nonlinearities compensation,
the control law in the following form is obtained:
δskxPΓτ p
T
ˆ
. (27)
Substitution of (25) and (26) into (24) yields:
δτxfsksM p z
~
, (28)
where
xf
~
is an error of approximation of f(x)
function, equal to:
εxΓεxΓxΓxfxfxf P
T
~
P
T
ˆ
P
T
ˆ
~
,
(29)
where
ΓΓΓ ˆ
~
is an error in the estimation of
weights of the neural network. After using
relationship (29), equation (28) is written as:
δτεxPΓsksM p z
T
~
.(30)
The structure of the system for adaptive fuzzy
control of robot generalized velocities is shown in
Fig. 5. For a derivation of an algorithm of
ˆ
weights learning, the theory of Lyapunov stability is
used.
f(x)
^
---
sMobile
Robot
Kinematic
controller kp
Robust
Term
ud
udud
.uR
uR
x
y
R
R
z
z
x
y
R
R
zd
d
d
e
F u z z if ie r D e f u zz ifie r
F u zzy R ule B as e
F u zzy I nfe re n ce E n gin e
Adaptation/
Learning Signal
Fig. 5: Adaptive fuzzy feedback control scheme.
Let us take a scalar positive-definite function:
,ecos1vk2
eek
~~
tr
2
1
2
1
V
oRdo
2
L
2
FF
1TT
ΓFΓMss
(31)
where
0
T FF
is a design matrix. A derivative of
the V function with respect to time, one can write as:
ooRdo
LLFFF
1TT
esinevk2
eeeek2
~~
trV
ΓFΓsMs
.
(32)
After inserting the expression
sM
from equation
(30), one obtains:
.esinevk2
eeeek2
)(
~~
trkV
ooRdo
LLFFFz
T
T1T
p
T
δτεs
sxPΓFΓss
(33)
After choosing the law of adaptation of weights as:
T
~sxFPΓ
, (34)
and after introducing the robust control signal:
s
s
Zεδ M
, (35)
Relationship (33) is transformed into the form:
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s
s
Zs
esinevk2eeeek2
s
s)x(P
~
(
~
trkV
m
T
ooRdoLLFFF
z
T
T1T
p
T
Fss
(36)
After writing in expanded form the error of desired
velocities (15) as:
22
11
2
1
Rd
Rd
uu
uu
s
s
, (37)
and after determining a derivative of error (15), and
putting
doLF kkk
, one gets:
2
minp
2
2oRdo
2
1FF
2
L
2
Rd
2
L
o
22
Rd
2
o
2
F
2
F
sk
sesinvk
sekevk
esinvkekV
(38)
Since V is positive definite for
0s
and
V
is
negative semidefinite, both s,
~
are bounded
according to Lyapunov’s theorem. Such a synthesis
of the adaptive fuzzy control permits proper
operation of the control system with a proportional
controller until the fuzzy system starts adapting.
6 Simulation Results
This section shows some simulation results of the
fuzzy logic system using a four-wheeled robot
subject to wheel slip whose objective is to follow
the given reference trajectory. An adaptive fuzzy
system with adaptive learning rules (34) was used in
the simulation. The three Gaussian membership
functions were selected along each input dimension,
therefore 9 fuzzy IF-THEN rules can be generated.
The initial and final membership function shapes are
shown in Fig. 6.
Remark: We omitted the signal
d
u
in learning the
conclusion of the rules because nothing brings on in
the process of learning as numerous simulations
showed, but the dimensionality of the problem
grows considerably.
a) b)
Fig. 6: Membership functions along a)
R
v
and b)
dimension
For use in simulation investigations, one assumes
the following robot parameters:
geometric dimensions (A1A3 = A2A4 = L, A1A2 =
A3A4 = W – see Fig. 1), L = 0.35 m, W = 0.386 m,
ri = 0.0965 m, i = {1, …, 4},
masses of particular bodies: m0 = 15.02 kg, mi =
0.66 kg, m5 = m6 = 0.17 kg,
rolling resistance coefficient fr = 0.03,
whereas the constants ai occurring in equation (7)
were determined using the methodology described
in works, [3], [4]. The following values of gains for
the controller were assumed: kL = 15, kF = 10, ko = 5,
kp = diag(20, 20). Desired motion parameters of the
robot’s wheels, kinematic parameters of point R,
and motion path of point R are shown in Fig. 7. In
simulation three phases of motion are assumed:
acceleration, motion with constant velocity of the
point R (
m/s3.0
R
v
), and braking. For an
approximation of nonlinearities and variable robot
operating conditions, the fuzzy system described in
section 4 is used with Gaussian functions describing
fuzzy sets, assuming each element of the f vector is
approximated with 6 rules. In the simulation,
parametric disturbance occurring
st 12
is assumed
in the form of an increase in the rolling resistance
coefficient
03.0 r
f
, when the characteristic point
R of the robot moves along a curvilinear path.
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a) b)
c) d)
Fig. 7: Desired kinematic quantities used in
simulation: a) kinematic parameters of the point R,
b) desired velocities: linear of the point R, and
angular of the robot’s body, c) angular velocities of
driven wheels, d) desired motion path of the point R.
Trajectory tracking performances for two cases were
considered, firstly with action generation fuzzy
compensation of the nonlinearity of the robot
according to the scheme shown in Fig. 5 (case 1),
and secondly without the compensation of the
nonlinearity of the robot and robust term (case 2).
Case 1.
In Fig. 8 are shown obtained control signals, and in
Fig. 9, errors of neural control of position and
heading of the robot. The obtained control signals
43,
(i.e., desired torques for driven wheels) that
realize desired trajectory of motion of the point R of
the mobile robot are shown in Fig .8a. Values of
torques are the largest during motion of the mobile
robot along a circular trajectory, their value is
constant until the occurrence of a parametric
disturbance. This corresponds to robot motion with
constant velocity. At the moment of occurrence of
the parametric disturbance, the value of the
3
torque increases whereas the value of the
4
torque decreases, which results from an increase in
the adopted motion resistance. For time
12t
s
values of torques decrease, which corresponds to the
phase of braking and finishing motion along the
rectilinear path.
a) b)
c) d)
Fig. 8: Control signals according to a relationship
(23)
The discussed total control signals are generated
based on control signals compensating for robot
nonlinearities shown in Fig. 8b, signals generated by
a P-type regulator (Fig. 8c), and robust control
signals (Fig. 8d). The fuzzy compensation control
has the largest influence on the total control signal,
as far as level and character are concerned. In turn,
the stabilizing P control and robust
δ
control have
the largest values during periods of occurrence of
disturbances associated with wheels’ slips or
resulting from the character of desired velocity,
desired motion path, or the occurring parametric
disturbances. It follows from the fact that in those
motion states, the fuzzy compensation adapts to
changing operating conditions of the robot, and only
after the adaptation period the fuzzy system
generates dominant control signals. This fact of the
significance of the influence of fuzzy compensating
control on the overall quality of control is confirmed
by results shown in Fig. 9a-c, in which errors of
neural control of position and heading of the robot
are presented. Error-values are the largest during the
period of motion along a circular trajectory, and
then as the process of fuzzy adaptation progresses,
they decrease. The occurring parametric disturbance
as well as changing robot operating conditions,
excite the proposed control structure, which as a
result generates control signals that make the control
errors
eee yRxR ,,
bounded, which confirms the
theoretical considerations. In Fig. 9d are shown the
desired and actual paths realized with small errors,
marked as ‘trajd’ and ‘traj’, respectively.
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a) b)
c) d)
Fig. 9: Errors of fuzzy control of robot’s position
and heading
For quantitative evaluation of the generated control
signals and realized tracking motion, the following
quality indices are introduced:
maximum values of the errors exRmax, eyRmax in (m)
and
max
e
in (rad),
))(abs( (.)(.) kee max
, k =1,2,…n,
the square root of the mean squared error
(RMSE) of motion realization
n
kRRdxR k-xkx
n
e
1
2
)()(
1
(m),
n
kRRdyR k-yky
n
e
1
2
)()(
1
(m),
n
kdk-k
n
e
1
2
)()(
1
(rad),
where k is the ordinal number of a discrete value
and n = 23 000 is the total number of discrete
values. Values of all quality indices of realization of
tracking motion are given in Table 1.
Table 1. Values of the introduced quality indices
exR
eyR
eφ
exRmax
eyRmax
eφmax
0.0101
0.005403
0.01001
0.02004
0.01345
0.01787
Case 2.
To gauge the effectiveness of the fuzzy
compensation of the nonlinearity of the robot it is
useful to compare the performance of the closed-
loop system without the output of action generating
compensation of the nonlinearity of the robot and
without robust term. The output tracking
performance, in this case, is shown in Fig. 10. The
output tracks corresponds to the desired trajectory
with the bigger values of the introduced quality
indices, a) b)
c) d)
Fig. 10: Errors of fuzzy control of the robot’s
position and heading
which are given in Table 2, because of the non-
linearity of the robot dynamics. There exist state
errors resulting from the nonlinear appearance of the
system.
Table 2. Values of the introduced quality indices
exR
eyR
eφ
exRmax
eyRmax
eφmax
0.01324
0.009465
0.01001
0.02278
0.02314
0.04633
7 Conclusions
In the article, a stable algorithm of control of
position and heading in tracking the motion of a
four-wheeled mobile robot is designed. In the
algorithm, the fuzzy system linear with respect to
estimated parameters is used. The algorithm does
not require prior knowledge of the dynamic
properties of the controlled object and is robust to
occurring longitudinal and lateral slips of wheels as
well as to parametric disturbances. After the fuzzy
logic system has compensated partially for the non-
linearity of the controlled system through adaptive
learning, the output tracking of the plant follows the
reference trajectory quite satisfactorily. The same
controller works even if the behavior or structure of
the system has changed. Results of conducted
simulation investigations lead to the conclusion that
intelligent control with a correctly designed
kinematic controller significantly increases the
accuracy of the realization of tracking motion.
Additionally, the proposed fuzzy control algorithm
operates online and does not require initial learning
of fuzzy parameters.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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