Design of Adaptive Fuzzy Sliding Mode Controller for Mobile Robot - IJMERR
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International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
Design of Adaptive Fuzzy Sliding Mode
Controller for Mobile Robot
Phan Thanh Phuc, Tuong Phuoc Tho, Nguyen Dao Xuan Hai, and Nguyen Truong Thinh
Department of Mechatronics, HCMC University of Technology and Education, Vietnam
Email: phanthanhphuc@gmail.com, thotp@hcmute.edu.vn, ndxuanhai@gmail.com, thinhnt@hcmute.edu.vn
Abstract—A Wheeled Mobile Robot (WMR) system is one of approach to control problems mimics how a person
the well-known non-holonomic systems. In this paper, an would make a decision. The main advantages of a fuzzy
Adaptive Fuzzy Sliding Mode Controller (AFSMC) is navigation strategy lie in the ability to extract heuristic
proposed for trajectory tracking control of a non-holonomic rules from human experience. The hallmark [6]
system, which the centroid doesn’t coincide to the
investigates the application of an adaptive neuro-fuzzy
connection center of driving wheels. First, a Sliding Mode
Controller (SMC) is proposed to the convergence of WMR inference system (ANFIS) to path generation and obstacle
on the desired position, velocity and orientation trajectories. avoidance for an autonomous mobile robot in a real-
However, the SMC is still fluctuation around trajectory world environment [7]. A control structure that makes
tracking also the system response time is slow. So the second possible the integration of a kinematic controller and an
fuzzy logic controller (FLC) is combined with SMC to adaptive fuzzy controller for trajectory tracking is
improve quality of control WMR for quick response time. developed for non-holonomic mobile robots. Sliding
The results of Matlab/Simulink demonstrated the efficiency mode control is widely utilized to control robotics system
of the AFSMC proposed good working. also SMC has been considered recently to improve the
performance of the nonlinear controller [8]. However, the
Index Terms—sliding mode control, adaptive control,
trajectory tracking, wheeled mobile robots, autonomous
chattering phenomenon caused by the Sign function leads
to fluctuation in object high – frequency dynamics.
I. INTRODUCTION Moreover, selecting a large value of the switching gain in
the sliding mode control to ensure the effectiveness of the
Mobile robots have been used in many applications control system could cause severe solicitation in the
and areas such as industrial, transportation, inspection control inputs and increase the chattering phenomenon.
and other fields. WMRs are considered as the most In this paper, we proposed FLC combine with SMC to
widely used class of mobile robots. Mobile robots are improve quality control of WMR which created error
complex and combine many technologies such as sensors, trajectory tracking by SMC. The paper is organized as
controller design, electronic components. In the process follows: the modeling of WMB is shown in Section II.
of design and development of a mobile robot, the Section III presents SMC and AFSMC. The simulation
controller is an important role because of the working results are indicated in Section IV. The conclusion is
ability of a mobile robot based on the controller. The address in Section V.
possible motion tasks can be classified as follows: point
to point motion, path following motion but stability and II. STRUCTURE AND MODELING OF THE MOBILE ROBOT
trajectory tracking control is the key issue in the control
also interested in many research. A. Kinematic Model
Many studies are conducted to develop an adaptive The considered mobile robot is composed of two
controller to control the motion of a mobile robot with similar driving wheels mounted on a bar and
disturbances and uncertainties [1], [2]. The global independently controlled by two actuators, as indicated in
trajectory tracking problem has been discussed based on
back-stepping [3]-[5], proposed method to controller Fig. 1. Where R is the wheel radius (m), vR and vL are
WMR by kinematic controller is designed first so that right and left drive wheel velocities respectively (m/s), (x,
tracking error between a real robot and a reference robot y) present mobile robot that is defined to mid-point A, on
converges to zero, and secondly a torque controller is the axis between the wheels, center of axis of wheels in
designed by using back-stepping so that the velocities of world Cartesian coordinates in (m) and 2L is axle length
a mobile robot converge to desired velocities which are between the drive wheel (m). The orientation of the
given by the kinematic controller designed at first step. mobile robot is given by the angle (rad) between the
The fuzzy logic controller has been found to be the most instant linear velocities of the mobile robot body. The
attractive. The theory of fuzzy logic systems is inspired center of mass C of the robot is assumed to be on the axis
by the remarkable human capability to operate on and of symmetry at a distance d from the origin A. The
reason with perception-based information. Fuzzy logic position vector state of the robot is defined as q ( x, y, ) .
The linear and the angular velocities of the mobile robot
Manuscript received August 5, 2020; revised January 25, 2021.
© 2021 Int. J. Mech. Eng. Rob. Res 54
doi: 10.18178/ijmerr.10.2.54-59International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
are expressed by (1) and the kinematic model equation of 0 md cos 0 0 0
the mobile robot is given by (2) as follows [9].
0 md sin 0 0 0
v vL ( L ) V ( q, q ) 0 0 0 0 0
v R R R
2 2 0 0 0 0 0
vR vL ( R L ) 0 0
0 0 0
R
2L 2 (1)
0 0
0 0
x cos 0
v
q y sin 0 B(q) 0 0
0 1 1 0
(2) 0 1 ,
y sin cos cos 1
cos
sin sin 2
xr T ( q ) 0 L L x 3
yr l
R
0 4
0
d
C 0 0 R 5
r
ya Next, the system described by (4) is transformed into
A an alternative form which is more convenient for the
purpose of control and simulation. The main aim is to
2L
2R
eliminate the constraint term T(q) in equation (4) since
the Lagrange multipliers i are unknown. This is done
x first by defining the reduce vector:
xa
(5)
Figure 1. Structure of WMR R
L
B. Dynamic Modeling of the WMR Next, by expressing the generalized coordinates
Consider the following non-holonomic mobile robot velocities using the forward kinematic model we have
that is subject to m constraints [10]
M ( q ) q V ( q, q ) q F ( q ) G ( q ) d R cos R cos
(3)
B(q) T (q) x R sin R sin
y
where q Rn is generalized coordinates, Rn is the
1
R R R
input vector, Rm is the vector of constraint forces,
M(q) Rnxn is asymmetric and positive-definite inertia 2 L L L (6)
R
matrix, V(q, q’) Rnxn is the centripetal and Coriolis 2 0
matrix, F(q’) Rn is the surface friction matrix, G(q) Rn L
is the gravitational vector, B(q) Rrxn is the input
transformation matrix, d Rr is the vector of bounded 0 2
unknown disturbances including unstructured un-
modeled dynamics, and (q) Rmxn is the matrix This can be written in the form:
associated with the constraints.[9] the equation (3) can be
q S (q) (7)
represented as follows:
M (q)q V (q, q)q B(q) T (q) (4) It can be verified that the transformation matrix S(q) is
in the null space of the constraint matrix (q). Therefore
where: we have:
m 0 md sin 0 0
S T (q)T (q) 0 (8)
0 m md cos 0 0
M (q ) md sin md cos I 0 0 The derivative of (7) we have:
0 0 0 0 0 q S (q) S (q) (9)
0 0 0 0 I w
Substituting equation (7) and (9) in the main equation
(4) we obtain:
© 2021 Int. J. Mech. Eng. Rob. Res 55International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
M (q)[S(q) S (q) ] V (q, q)[S(q) ] (10) are defined as qr = [Xr, Yr, θr] the reference velocity and
B(q) (q)T acceleration vectors are derived from qr as zr = [vr, ωr,]T
and z’r = [v’r, θ”r,]T respectively. The real world robotic
Rearranging the equation and multiplying both sides systems have inherent system disturbances such as
by lead to: parameter uncertainties, friction, etc.
Therefore, the real dynamic equation of the WMR is
S T (q)M (q)S (q) S T (q)[M (q)S (q) V (q, q)S (q)] (11) considered as:
S T (q) B(q) S T (q)T (q)
Mz V (q, z ) d (14)
Now the new matrix is defined as follows:
where, the disturbance vectors, τd = [τd1, τd2]T is defined to
M (q) S ( q) M (q) S (q)
T
include the disturbance effects in the dynamical equations.
It is assumed that τd is bounded and satisfies the
V S T (q)M (q)S (q) S T (q)V (q, q)S(q) uncertainty matching condition as:
B S T ( q ) B( q ) d M p
The dynamics equations are reduced form:
p [ p1 , p2 ]T , p1 p1m , p2 p2m
M (q) V (q, q) B(q) (12)
p1m and p2m are upper bounds of the perturbations.
where: 2) Controller design
First, the position and orientation errors are considered
R2 R2 as:
I w 2
(mL2 I ) 2
(mL2 I )
4 L 4L
M (q) Xe Xc Xr
R2 R2
I w 4 L2 (mL I ) I w 2 (mL I ) Ye Yc Yr
2 2
4L
e c r
R 2
0 mc d , To stabilize the tracking errors, the sliding surfaces are
2L
V ( q, q ) defined as [11]:
R2
2 L mc d 0
s1 X e K1 X e (15)
1 0
B (q)
1
s2 Ye K2Ye (16)
0
Equation (12) shows that dynamics problem of WMR where K1 and K 2 are positive constant parameters. If s1
is express only as a function of angular velocities (φ’R, is asymptotically stable, Xe and X’e converge to zero
φ’L) or the right and left wheels, the robot angular asymptotically. Because if s1 = 0 then X’e +K1Xe =0.
velocity θ’ and the driving motor torques (τR, τL). The Therefore, if X’e ≤ 0 then Xe ≥ 0 and if X’e ≥ 0 then Xe≤ 0.
equations of (12) can be also transformed into an Therefore, the equilibrium state Xe X’e is asymptotically
alternative form which is represented by linear and stable. Similarly, if s2 is asymptotically stable, Ye and Y’e
angular velocities (v, w) of WMR. Using kinematic asymptotically converge to zero. Thus, if s1 and s2 are
model equation (1), it showed that equations (12) can be stabilized, the convergence of WMR to the reference
rearranged as follows: trajectory is guaranteed.
As a feedback linearization of the system, the control
2I w 1
m R 2 v mc d R ( R L )
2
inputs are defined by the computed-torque method as
(13) follows [12].
I 2 L2
I L
mc d v ( R L )
w
R 2
R Mzr V (q, z) Musmc (17)
Here usmc = [u1, u2]T is the control law which
determines error dynamics. Applying the control input
III. DESIGN OF CONTROLLER
(17) in the dynamic equation of WRM (14), the feedback-
linearized dynamic equation is given as:
A. Sliding Mode Controller
1) Problem statement z p zr usmc (18)
The purpose of the SMC is the design of control inputs
τ = [τR, τL]T. which make the WMR track a feasible Thus from (18), we have
trajectory with bounded posture errors. The three- vc p1m vr u1 (19)
dimensional posture variables of the reference trajectory
© 2021 Int. J. Mech. Eng. Rob. Res 56International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
c p2m r u2 (20) By combining FLC with SMC we have FSMC law can
be developed as:
The control law u1 and u2 which stabilizes the sliding
surface s1 and s2 are proposed as follows. u fsmc usmc ur (26)
u1 D1sign( s1 ) K1 X e X e vc vr E1 X e sign(s1 X e ) The membership functions corresponding to the input
u2 D2 sign( s2 ) K 2Ye Ye c r E2Ye sign(s2Ye ) and output fuzzy sets of e, e’ and uf are present in Fig. 3.
where D1, D2 are bigger than p1m, p2m respectively and K1,
K2, E1, E2 are real positive constant values. To prove the
stability of s1 and s2 when u1 and u2 are applied, the
Lyapunov direct method is used. Substituting u1 and u2
respectively in (19) and (20), X”e and Y”e are derived as:
X e D1sign(s1 ) p1m K1 X e E1 X e sign(s1 X e ) (21)
Ye D2 sign(s2 ) p2m K2Ye E2Ye sign(s2Ye ) (22) Figure 3. Membership function of inputs e, e’and output uf.
According to Lyapunov’s direct method, the following
In this figure, P, N, and Z stand for positive, negative
Lyapunov function is applied [11]:
and zero, respectively. Corresponding to the three
1 membership functions for each input variable, 9 if-then
V ( s12 s22 ) (23) rules of Table I are obtained using the expert engineering
2
knowledge and experiences in the field of WMR.
The time derivative of V along the trajectory is given
TABLE I. RULE BASE OF FSMC
by:
e
V s1s1 s2 s s1 ( X e K1 X e ) s2 (Ye K2Ye ) (24) N Z P
e
Replacing X”e and Y”e from (21) and (22) in (24) N P P Z
result in the following negative definite function : Z P Z P
V ( D1 s1 p1m s1 E1 s1 X e ) (D2 s2 p2m s2 E2 s2Ye ) P Z P P
Therefore s1 and s2 are asymptotically stable and the IV. SIMULATION RESULTS
WMR converges to the reference trajectory. However, the
WMR has slowly response time also fluctuation around The effects of proposed methods to improve the
orientation which causes by sign function. convergence of WMR to reference position and
orientation trajectories are calculated by Matlab/Simulink
B. Fuzzy Sliding Mode Controller to implement the control structure shown in Fig. 2 using
As address above the SMC has slow response time, so the control law given by equations (18) and (26). In all
we propose Sugeno type fuzzy to adapt to change errors simulations, the robot starts at position (0.2, 0.1, 0) (m)
and derivate errors of WMR, the control following form: and should follow a circular trajectory of reference. The
center of the reference circle is at x= 0.0 (m) and y = 0.0
ur K f u f (25) (m) and follows a circle having a radius of 0.15 (m) and τd
disturbance is 0.
where Kf is the normalizing factor. The output fuzzy
variables uf are continuously adjusted using an if-then
rule base with respect to both e and e’. The configuration
of AFSMC is shown in Fig. 2.
Figure 2. Structure controller of AFSMC for control WMR
Figure 4. Reference trajectory for SMC and AFSMC
© 2021 Int. J. Mech. Eng. Rob. Res 57International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
Fig. 4 indicates a reference trajectory for controller V. CONCLUSION
AFSMC and SMC. Fig. 5 and Fig. 6 showed the response
In this paper, we proposed AFSMC based on robust
time of AFSMC quickly and smooth than SMC follows a
SMC for control trajectory tracking WMR by adapting
circle reference at approximate time 0–11s. Similar, the
changing e, e’. The simulation results showed AFSMC
Fig. 7 and Fig. 8 demonstrated AFSMC response time
for quick response time also stability.
better SMC for along X and Y axis.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
AUTHOR CONTRIBUTIONS
Start
Nguyen Truong Thinh, Nguyen Dao Xuan Hai,
Point contributed to the analysis and implementation of the
research, to the analysis of the results and to the writing
of the manuscript. All authors discussed the results and
contributed to the final manuscript. Besides, Nguyen
Truong Thinh conceived the study and were in charge of
overall direction and planning. Nguyen Truong Thinh is a
corresponding author.
Figure 5. The actual trajectory for SMC
ACKNOWLEDGMENT
The authors wish to thank Ho Chi Minh City
Start Point
University of Technology and Education, Viet Nam. This
work was supported in part by a grant from Vietnam
Ministry of Education and Training.
REFERENCES
[1] B. BeloboMevo, M. R. Saad, and R. Fareh, "Adaptive sliding
mode control of wheeled mobile robot with nonlinear model and
uncertainties," in Proc. 2018 IEEE Canadian Conference on
Electrical & Computer Engineering (CCECE), 2018, pp. 1-5.
[2] N. Hu and C. Wang, "Adaptive tracking control of an uncertain
nonholonomic robot," Intelligent Control and Automation, vol. 2,
p. 396, 2011.
[3] R. Fierro and F. L. Lewis, "Control of a nonholomic mobile robot:
Figure 6. The actual trajectory for AFSMC Backstepping kinematics into dynamics," Journal of robotic
Systems, vol. 14, pp. 149-163, 1997.
[4] W. G. Wu, H. T. Chen, and Y. J. Wang, "Global trajectory
tracking control of mobile robots," Acta Automatica Sinica, vol.
27, pp. 326-331, 2001.
[5] G. Zidani, S. Drid, L. Chrifi-Alaoui, A. Benmakhlouf, and S.
Chaouch, "Backstepping controller for a wheeled mobile robot," in
Proc. 2015 4th International Conference on Systems and Control
(ICSC), 2015, pp. 443-448.
[6] P. K. Mohanty and D. R. Parhi, "Navigation of autonomous
mobile robot using adaptive network based fuzzy inference
system," Journal of Mechanical Science and Technology, vol. 28,
pp. 2861-2868, 2014.
[7] T. Das and I. N. Kar, "Design and implementation of an adaptive
fuzzy logic-based controller for wheeled mobile robots," IEEE
Figure 7. Trajectory tracking of WMR along the X axis Transactions on Control Systems Technology, vol. 14, pp. 501-510,
2006.
[8] C. C. Tsai, M. B. Cheng, and S. C. Lin, "Robust tracking control
for a wheeled mobile manipulator with dual arms using hybrid
sliding‐mode neural network," Asian Journal of Control, vol. 9,
pp. 377-389, 2007.
[9] R. Dhaouadi and A. A. Hatab, "Dynamic modelling of
differential-drive mobile robots using lagrange and newton-euler
methodologies: A unified framework," Advances in Robotics &
Automation, vol. 2, pp. 1-7, 2013.
[10] T. Fukao, H. Nakagawa, and N. Adachi, "Adaptive tracking
control of a nonholonomic mobile robot," IEEE transactions on
Robotics and Automation, vol. 16, pp. 609-615, 2000.
[11] V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electro-
mechanical Systems, CRC press, 2009.
[12] J. M. Yang and J. H. Kim, "Sliding mode control for trajectory
Figure 8. Trajectory tracking of WMR along the Y axis tracking of nonholonomic wheeled mobile robots," IEEE
© 2021 Int. J. Mech. Eng. Rob. Res 58International Journal of Mechanical Engineering and Robotics Research Vol. 10, No. 2, February 2021
Transactions on Robotics and Automation, vol. 15, pp. 578-587, Tuong Phuoc Tho is a Lecturer of Faculty of
1999. Mechanical Engineering at Ho Chi Minh City
University of Technology and Education, Viet
Copyright © 2021 by the authors. This is an open access article Nam. His work focuses on Robotics and
distributed under the Creative Commons Attribution License (CC BY- Mechatronic system.
NC-ND 4.0), which permits use, distribution and reproduction in any
medium, provided that the article is properly cited, the use is non-
commercial and no modifications or adaptations are made.
Phan Thanh Phuc received the B.S degree in
Mechatronic Engineering from Industrial
University of Ho Chi Minh, Ho Chi Minh, Viet Nguyen Dao Xuan Hai hold a B.E. in Machine
Nam, in 2015 and the M.E degree in Manufacturing Engineering in 2020. As head of
Mechatronic Engineering from Ho Chi Minh a Mechatronics Laboratory in Ho Chi Minh City
University of Technology and Education, Ho University of Technology and Education - Viet
Chi Minh, Vietnam, in 2018 Nam, he supervises the work of the
Currently, he is working at Central interdisciplinary group that support researching
Transportation 6 College, Ho Chi Minh city, in theory and experiment.
Vietnam
Nguyen Truong Thinh is Dean of Faculty of
Mechanical Engineering at Ho Chi Minh City
University of Technology and Education, Viet
Nam. He is also Associate Professor of
Mechatronics. He obtained his PhD. In 2010 in
Mechanical Engineering from Chonnam
National University. His work focuses on
Robotics and Mechatronic system. Projects
include: Service robots, Industrial Robots,
Mechatronic system, AI applying to robot and
machines, Agriculture smart machines...
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