Precise tip-positioning control of a single-link flexible arm using a fractional-order sliding mode controller

Abstract

This article presents an efficient scheme based on fractional order sliding mode control approach for precision tip position control of a single link flexible robot arm. The proposed control strategy is robust against the system parameters variations such as payload and viscous friction variations in the presence of the sinusoidal disturbance and the unknown Coulomb friction disturbances. The aim of controller design is reduction of the deviation caused by the link flexibility and then the precision tip positioning control of the single-link flexible arm. In this regard, sliding mode control strategy is performed in two stages. In the first stage, the difference between the motor angle (load angle) and the tip angle of the flexible link is reduced by applying the proposed fractional order sliding mode controller and then, in the second step, the precision tracking of the tip position of the link is done by adding another sliding mode control scheme. The feasibility and effectiveness of the proposed control scheme is demonstrated via numerical simulation results.

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Article
Journal of Vibration and Control
2020, Vol. 0(0) 1–14
© The Author(s) 2020
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DOI: 10.1177/1077546320902548
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Precise tip-positioning control of
a single-link flexible arm using
a fractional-order sliding mode controller
Farzaneh Hamzeh Nejad
1
, Ali Fayazi
1
,
Hossein Ghayoumi Zadeh
1
, Hassan Fatehi Marj
1
and
S Hassan HosseinNia
2
Abstract
This article presents an efficient scheme based on fractional order sliding mode control approach for precision tip position
control of a single link flexible robot arm. The proposed control strategy is robust against the system parameters variations
such as payload and viscous friction variations in the presence of the sinusoidal disturbance and the unknown Coulomb
friction disturbances. The aim of controller design is reduction of the deviation caused by the link flexibility and then the
precision tip positioning control of the single-link flexible arm. In this regard, sliding mode control strategy is performed in
two stages. In the first stage, the difference between the motor angle (load angle) and the tip angle of the flexible link is
reduced by applying the proposed fractional order sliding mode controller and then, in the second step, the precision
tracking of the tip position of the link is done by adding another sliding mode control scheme. The feasibility and ef-
fectiveness of the proposed control scheme is demonstrated via numerical simulation results.
Keywords
Precision tip positioning control, a single-link flexible arm, fractional order control, sliding mode control
1. Introduction
In recent decades, a range of applications of robotics in
various fields of science and engineering have led to the
development of research into the control of flexible arms
(Wang and Gao, 2003). Among these applications can refer
to the industrial applications that tend to use the lightweight
materials in the construction of flexible robotic arms to
improve the performance of the industrial robots, which are
currently heavy and bulky; motion control of large struc-
tures, such as boom cranes, and fire rescue turntable ladders,
which are treated as flexible link robots (Sawodny et al.,
2002); motion control of a sensing antenna that slides on
a surface and detects an object (Feliu-Talegon and Feliu-
Batlle, 2017a,2017b;Feliu-Batlle et al., 2017). There are
several research works in the field of dynamic modeling of
flexible multibody systems (Bauchau, 2011;Benosman and
Le Vey, 2004;Boscariol et al., 2017;Shabana, 1997). Mod-
eling, control and some of the most important applications
of the flexible manipulators have also been reviewed in
Benosman and Vey (2004),Dwivedy and Eberhard (2006),
Lochan et al. (2016),Castillo-Berrio et al. (2017),Kiang
et al. (2015),Fayazi et al. (2019),Khorrami et al. (1992),
Korayem et al. (2012),Nagaraj et al. (1997),Korayem and
Shafei (2015),Ettefagh et al. (2018) and Khodaei et al.
(2018). Over the past decades, flexible robots have attracted
much attention from both engineers and researchers because
of their advantages over rigid robots (Castillo-Berrio et al.,
2017;Dwivedy and Eberhard, 2006;Kiang et al., 2015;
Lochan et al., 2016). Some of these significant advantages
include faster movements and therefore higher maneuver-
ability, easier transportability, relatively smaller actuators,
and thus lower energy consumption, and higher payload-to-
arm weight ratio. Despite these useful features, the flexibility
of the flexible link lead to more nonlinearity of the dynamic
equations and oscillatory behavior at the tip of the link.
Therefore, precise positioning of the tip of the link is difficult
1
Department of Electrical Engineering, Vali-e-Asr University of Rafsanjan,
Iran
2
Department of Precision and Microsystem Engineering, Delft University of
Technology, Netherlands
Received: 25 October 2019; accepted: 28 December 2019
Corresponding author:
Ali Fayazi, Department of Electrical Engineering, Vali-e-Asr University of
Rafsanjan, Imam Khomeini Square, 7718897111 Rafsanjan, Kerman, Iran.
Email: a.fayazi@vru.ac.ir
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and a very effective control scheme is required. The con-
troller applied to this type of flexible link robot has to ac-
complish two different tasks: the elimination of the deviation
caused by the flexibility of the link and the precision tip
position control of the flexible link. To achieve these control
objectives, many control methods have been applied to
flexible link robots such as fractional order position control
(Monje et al., 2007), the fuzzy logic control method com-
bined with genetic algorithm (GA) optimization (Alam and
Tokhi, 2008), the conventional proportional–integral–derivative
(PID) control scheme (Castillo-Berrio and Feliu-Batlle, 2015),
the slide mode control strategy (Mamani et al., 2012), the
adaptive control method (Shaheed and Tokhi, 2013), inverse
dynamics based control technique (Feliu et al., 2003), the pole
placement based control method (San-Millan et al., 2015), the
neural network control scheme (Su and Khorasani, 2001), and
the reinforcement learning control method (Ouyang et al.,
2017). In Feliu-Talegon and Feliu-Batlle (2017a,2017b),
a control strategy, consisting of two inner and outer loops, is
proposed to realize the position control of a two degrees of
freedom flexible link. An inner loop is proposed to control the
position of two motors by a PID controller and an outer loop is
provided to eliminate flexible link vibrations by applying
a state-input feedback linearization strategy. In Pereira et al.
(2009), two sliding mode nested control loops have been
proposed to improve the performance of the position control of
the flexible single link robot, which is robust against variations
in the payload and viscous friction of the motor. In Castillo-
Berrio and Feliu-Batlle (2015), fractional order controllers are
presented for the vibration damping and precise tip-positioning
of a single link-flexible arm. In Pereira et al. (2009) and (2011),
a hybrid control scheme includes two nested control loops
based on the oscillatory integral controller (IRC) scheme are
provided for vibration damping and precise tip-positioning
control of a single-link flexible which are robust against un-
modeled components of the friction, parameter uncertainties,
and unmodeled dynamics. In Morales et al. (2012),twonested
control loops based on a generalized proportional integral
controller (GPI) have been proposed to realize the position
control of the flexible link that is robust to a wide range of
payload changes. In Yang and Tan (2018), sliding mode
control method has been presented to robust position control of
flexible-link manipulators based on adaptive neural networks.
To improve the system robustness, the system uncertainties
include the model uncertainty, and external disturbances have
been estimated by an adaptive neural network. In Ta hir et a l.
(2019), an intelligent hybrid control is proposed for vibrations
and intelligent tracking control of a single link flexible ma-
nipulator. The tip link deflection was eliminated by an output
based filter (OBF). Moreover, the hybrid control includes the
OBF, the linear quadratic regulator control, and the fuzzy
logic has beenaccomplished to track the desired trajectory. In
Feliu-Talegon and Feliu-Batlle (2018), a passivity-based
control is provided for vibrations and position control of
asinglelinkflexible arm using fractional controllers. The
control scheme includes two nested control loops. An inner
loop based on a PID controller is proposed for the position
control of the motor and an outer loop based on a fractional
PI controller is provided for the vibration damping of the
flexible link. In Garcia-Perez et al. (2019), a control scheme
consisting of two parts is proposed for trajectory tracking
and vibration control of flexible link robot manipulators. In
the first part, a sliding mode controller (SMC) based on PID
control strategy is implemented for the position control of
the flexible link and in the second part; an active control
scheme is then accomplished for surpassing the system
vibration and damping the vibration of the flexible link.
In most articles which are related to the position control
of a single-link flexible arm, to simplify the flexible link
model, the angular position of the motor is considered to be
equal to the tip angular position of the link and then different
control strategies is proposed based on this simplification.
The main contribution of this article is that in the dynamics
modeling part of the single-link flexible arm, to achieve
a more accurate dynamics model of the single-link flexible
arm, the angular position of the motor is not considered to
be equal to the tip angular position of the link. Therefore, in
the first step, a control strategy based on fractional-order
sliding mode control is proposed to eliminate the deflection
caused by the link flexibility. Then, in the second step,
another sliding mode control strategy is added to the first
control strategy to realize the precise tip-positioning control
of the single-link flexible arm. As far as the authors of this
article are aware, the precise tip-positioning control of
a single-link flexible arm based on an accurate dynamics
model has been rarely addressed in the literature. Moreover,
fractional-order control (FOC) plays an important role in the
field of mechatronics and biological systems (Fayazi et al.,
2018;Feliu-Talegon et al., 2019;Krijnen et al., 2018;
Tejado et al., 2014). Therefore, in this study, we have taken
the advantage of the combination of the FOC and the SMC
is to enhance the performance of the controller.
This article is structured as follows. Section 2 describes
the modeling of a single-link flexible arm. The definition of
the fractional order operator is given in Section 3.Section 4
introduces the proposed position control strategy based on
fractional-order sliding mode controller (FOSMC) and
presents the stability analysis of closed-loop control system.
Section 5 presents the definitions of the performance cri-
teria. Section 6 provides the simulation results to depict the
effectiveness of our proposed control scheme. Finally, our
conclusions are summarized in Section 7.
2. Dynamic modeling of a single-link
flexible arm
In this section, a dynamic model for a single-link flexible
arm is presented. Dynamic modeling of a single link
flexible manipulator consists of two parts: the first part of
the dynamic is related to the flexible-link actuator, which
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includes a direct currrent (DC) mini servo motor, a re-
duction gear n= 50, and power supply. The other part of the
dynamic is related to the single-link flexible arm. In the
following, the dynamics of the DC motor and the single-link
flexible arm are described, respectively.
2.1. Motor dynamics
As aforementioned, the flexible-link actuator includes a DC
motor, a reduction gear and a current servo-amplifier. The
servo amplifier controls the input current to the DC motor,
which is proportional to the voltage applied to the servo by
the controller. Figure 1 shows a block diagram of the electric
actuator (servo amplifier + motor + gear). The dynamic
model of the DC motor with a reduction gear of ratio 1:n
and a current servo-amplifier is given as follows
Motor: Γm¼kmi¼J€
θmþν_
θmþΓcoul þΓcoup (1)
Servo-amplifier: i¼kaV(2)
Gear: θm¼nθl;Γcoup ¼Γl
n(3)
where k
m
is the electromechanical constant of the motor, iis
the current supplied to the motor by the servo amplifier, vis
the viscous friction of the motor, Jis the motor inertia, Γ
coul
is the unknown Coulomb friction torque, Γ
coup
is the
coupling torque between the flexible slewing link and the
motor shaft, Vis the voltage supplied to the servo-amplifier
generated by the computer, k
a
is the servo-amplifier gain, n
is the reduction ratio of the gear, θ
m
is the motor angle, and
θ
l
is the load angle (or the motor angle in the gear outlet).
The complete actuator system dynamics is obtained by
combining the above equations (1)–(3), as follows
KnV ¼Jn2€
θlþνn2_
θlþΓcoul þΓl(4)
where K=k
m
k
a
is the motor constant and Γ
coul
represents
the Coulomb friction perturbation which affects the system
dynamics. This perturbation depends on the sign of the
motor angular velocity, for which its model is given by the
following equation
Γcoul ¼Γcsign_
θm_
θm≠0
signðVÞminðkjnV j;ΓcÞ_
θm¼0(5)
where Γ
c
is an unknown constant value which is different
for each motor.
2.2. Link dynamics
There are three main methods to model a single-link flexible
arm, i.e. the lumped parameter model Bayo (1987), the
lumped-mass model Feliu et al. (1992), and the truncation
of a distributed parameter model Cannon and Schmitz
(1984). In this article, the lumped-mass model is used to
obtain the reduced flexible link dynamics with tip mass to
achieve a simple and convenient design control (Payo et al.,
2009). Figure 2 depicts a single-link flexible arm. In the
following, the assumptions are considered as described in
Morales et al. (2012):
1. Assumption 1: There are no rebounds.
2. Assumption 2: All the mass concentrated at the tip
position.
3. Assumption 3: The tip mass is considered to freely
rotate, and therefore, no rotational inertia from it
affects the link dynamics.
4. Assumption 4: The single-link flexible manipulator
rotates in a horizontal plane (the zaxis perpendicular
to the plane of the figure).
5. Assumption 5: The stiffness parameter EI is constant
throughout the beam.
6. Assumption 6: The deformations in the structure is
very small, which allows assumption of geometrical
linearity, i.e. sin(x)x, tan(x)x.
7. Assumption 7: The structure oscillates with the fun-
damental mode of vibration without the higher modal
densities being excited because it is assumed that the
mass of the load is much larger than the arm’s weight.
Figure 1. Block diagram of the actuator.
Hamzeh Nejad et al. 3
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Assumption 6 allows us to disregard the other modes
of vibration to obtain a simple model for the controller
design. The linear dynamics model for the tip angle can
be described by the following equations (Payo et al.,
2009)
ml2€
θt¼cðθlθtÞ(6)
Γl¼cðθlθtÞ(7)
where mis the tip mass, lis the length of the arm, Γ
l
is the
coupling torque, θ
m
is the angular position of motor, θ
t
is the
angular position of the tip mass (load angle), and c¼3EI =l
is the rotational stiffness coefficient of the arm, which is
considered constant throughout the whole flexible structure
as occurs in Payo et al. (2009). Let us we choose the state
vector as follows
xðtÞ¼ θlðtÞ_
θlðtÞθtðtÞ_
θtðtÞT(8)
By considering the equations (4), (6), and (7) and se-
lecting the state variables according to the equation (8), the
linear model of the single-link flexible arm in the state-space
representation is obtained as follows
_
x1ðtÞ¼x2ðtÞ
_
x2ðtÞ¼
ν
Jx2ðtÞ c
Jn2ðx1ðtÞx3ðtÞÞ  Γcoul
Jn2þK
Jn V
_
x3ðtÞ¼x4ðtÞ
_
x4ðtÞ¼ c
ml2ðx1ðtÞx3ðtÞÞ
yðtÞ¼"1000
0010
#xðtÞ
(9)
3. The definition of fractional operator
Fractional-order integral-derivative differential operator is
defined by aDq
tas follows (Caldern et al., 2006)
aDq
t¼
dq
dtqq>0
1q¼0
Zt
a
ðdτÞqq<0
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(10)
where qis the fractional-order and ais a constant that
corresponds to the initial condition. This operator is
a symbol of the fractional-order derivative and integral
operator. Thus, the aDq
trepresents the derivative operator
and the integral operator for the positive and the negative
values of q, respectively. The definitions that are commonly
used to the fractional-order derivative include Grunwald–
Letnikov, Riemann–Liouville, and Caputo as follows:
Definition 1. The Riemann–Liouville fractional derivative
operator is defined by the following equation (Miller and
Ross, 1993)
RL
aDq
tfðtÞ¼ 1
ΓðmqÞ
dm
dtmZt
a
fðτÞ
ðtτÞ1ðmqÞdτ(11)
where m1<q<mand mis the first integer greater than q.
The Γ() is a well-known gamma function. The operator
DðqÞ
tfðtÞ¼dqfðtÞ=dtqis the fractional derivative of f(t)of
the order q>0.
Definition 2. The Grunwald–Letnikov fractional derivative
operator is given by the following equation (Podlubny,
1998)
GL
aDq
tfðtÞ≡lim
N→∞N
tq
X
N
j¼1ΓðjqÞ
ΓðqÞΓðjþ1Þ
fðNjÞt
N
(12)
Definition 3. The Caputo fractional derivative operator is
given by the following equation (Caputo, 1967)
8
>
>
>
<
>
>
>
:
c
aDq
tfðtÞ¼ 1
ΓðmqÞZt
a
fðmÞðτÞ
ðtτÞ1ðmqÞdτm1<q<m
dm
dtmfðtÞq¼m
(13)
where mis the first integer greater than q> 0. The Laplace
transform of Caputo fractional derivative can be expressed
as follows
Figure 2. Single-link flexible arm.
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LdqfðtÞ
dtq¼sqLffðtÞg  X
m1
k¼0
sq1kfðkÞð0Þ;
m1<q≤m2N
(14)
By considering zero initial conditions we have
LdqfðtÞ
dtq¼sqLffðtÞg (15)
In the following, the D
q
x(t) is introduced as a symbol of
the Caputo fractional derivative operator of order q>
0 according to the definition of the fractional-order de-
rivative operator in equation (10).
4. Controller design
4.1. SMC
SMC is a nonlinear control technique that is robust against
uncertainties and external disturbances. In general, the
design of sliding mode control consists of two steps: the first
step involves the selection of a suitable sliding surface to
restrict the dynamics of the system in the sliding manifold.
The second stage involves the design of a suitable control
law that forces the trajectory of the system to the sliding
surface and maintain it there.
4.2. SMC design
In this section, an SMC is designed to precise tip posi-
tioning control of a very lightweight flexible link. Figure 3
shows the block diagram of proposed closed loop control
system.
As can be seen from Figure 3, a two-stage controller is
proposed for precise tip positioning control of the single-
link flexible robot arm. The proposed control strategy is
based on two SMCs. In the first stage, the FOSMC is
designed to eliminate the deflection of the tip link caused by
the flexibility. In the second step, another SMC is then
designed with respect to disregard the angle difference
between the angular position of the tip beam and the angular
position of the motor. Thus, the second control law is added
to the first control law to get perfect tracking of the desired
reference trajectory. Next, we define a new state variable
based on the angle difference between the angular position
of the motor and the angular position of the tip beam as
follows
z1ðtÞ¼x1ðtÞx3ðtÞ(16)
Therefore, by defining the new state variable in accor-
dance with the above equation, the equation (10) can be
rewritten as follows
_z1ðtÞ¼z2ðtÞ
_z2ðtÞ¼
c
Jn2þc
ml2z1ðtÞþgðx2ðtÞÞ þ K
Jn V
(17)
where g(x
2
(t)) which is given by the following equation
gðx2ðtÞÞ ¼ ν
Jx2ðtÞþΓcoul
Jn2(18)
Next, in the first step, to design a fractional-order SMC
to stabilize the system in accordance with the equation (17),
the sliding surface is first defined as follows
s1ðtÞ¼kp1z1ðtÞþkd2Dqz1ðtÞ(19)
where k
p1
,k
d2
, and qare constant control parameters that are
chosen by the designer. Equation (19) can be rewritten as
follows
s1ðtÞ¼kp1z1ðtÞþkd1Dq1_z1ðtÞ(20)
Taking the derivative of both sides of the equation (20)
and substituting the equation (17) obtains
_
s1ðtÞ¼kp1z2ðtÞþkd1Dq1c
Jn2þc
ml2z1ðtÞ
þgðx2ðtÞÞ þ K
Jn V(21)
where the input voltage, i.e. Vis defined as a control signal,
i.e. u
1
in accordance with the following equation
Figure 3. Closed-loop control system scheme.
Hamzeh Nejad et al. 5
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u1¼ueq1þus1(22)
The component ueq1is used to compensate the known
terms and the discontinuous sentence us1with its smooth
approximation is used to reduce the chattering phenomenon
at the reaching phase in the sliding mode control. The
equivalent control law ðueq1Þis obtained by equating the
equation (21) to zero (˙
s
1
(t) = 0) as follows
ueq1¼Jn
Kc
Jn2þc
ml2z1ðtÞgðx2ðtÞÞ
kp1
kd1
D1qz2ðtÞks1
kd1
s1ðtÞ (23)
The switching control law ðus1Þcan also be obtained as
follows
us1¼
kst1
kd
satðs1ðtÞÞ (24)
where kst1is switching gain in the feedback control and the
saturation function of s
1
(t), i.e. satðs1ðtÞÞ is given by the
following equation
satðs1ðtÞÞ ¼ sgn s1ðtÞif js1j≥φ
s1ðtÞ=φif js1j<φ(25)
ks1,kst1, and φare constant control parameters that are
chosen by the designer. By substituting the equations (22)
and (23) into equation (21), yields
_
s1ðtÞ¼ks1s1ðtÞkst1satðs1ðtÞÞ (26)
As can be seen from the equation (26), by selecting
appropriate control parameters, i.e. ks1> 0 and kst1> 0, the
sliding surface dynamics will be stable. Therefore, by ap-
plying the control law, the z
1
(t) tends to zero after a lapse of
τ(s). Thus, the angle difference between the angular po-
sition of the motor and the angular position of the tip beam
can be ignored. The aim of the next step is to design an SMC
to perfect tracking of the desired reference trajectory.
Therefore, by considering θ
t
=θ
l
and the equations (4), (6),
and (7), the dynamics of actuator system is obtained as
follows
Γm¼kmi¼J€
θmþν_
θmþΓcoul (27)
By defining the state vector as xðtÞ¼θmðtÞ_
θmðtÞT,
the linear dynamics model of a single-link flexible arm in
the state-space representation (equation (9)) can be re-
written as follows
_
x1ðtÞ¼x2ðtÞ
_
x2ðtÞ¼
ν
Jx2ðtÞΓcoul
Jn2þK
Jn V
(28)
Let us define the error between the angular position of
motor and the desired reference trajectory as follows
eðtÞ¼x1ðtÞxdðtÞ(29)
Next, the SMC is design to realize the control objective
in the second step, which is perfect for tracking the desired
reference trajectory. Therefore, a suitable sliding surface is
first selected as follows
s2ðtÞ¼kp2eðtÞþkd2DqeðtÞ(30)
where k
p2
,k
d2
,andqare constant control parameters that are
chosen by the designer. Equation (30) can be rewritten as
follows
s2ðtÞ¼kp2eðtÞþkd2Dq1_
eðtÞ(31)
Taking derivative from both sides of the equation (31)
and substituting in (28) yields
_
s2ðtÞ¼kp2_
eðtÞþkd2Dq1€
eðtÞ
¼kp2ðx2ðtÞ _
xdðtÞÞ þ kd2Dq1ð_
x2ðtÞ€
xdðtÞÞ
¼kp2ðx2ðtÞ _
xdðtÞÞ þ kd2Dq1
ν
Jx2ðtÞΓcoul
Jn2þK
Jn V€
xdðtÞ
(32)
where the input voltage, i.e. Vis defined as a control signal,
i.e. u
2
in accordance with the following equation
u2¼ueq2þus2(33)
As aforementioned in the first stage, the component ueq2
is used to compensate the known terms and the discon-
tinuous sentence us2with its smooth approximation used to
reduce the chattering phenomenon at the reaching phase in
the sliding mode control. The equivalent control law ðueq2Þ
is obtained by equating the equation (32) to zero (˙
s
2
(t)=0)
as follows
ueq2¼Jn
Kν
Jx2ðtÞþΓcoul
Jn2þ€
xdðtÞ
kp2
kd2
D1qðx2ðtÞ _
xdðtÞÞ  ks2
kd2
s2ðtÞ (34)
The switching control law ðus2Þis derived from similar
equation in accordance with the equations (24) and (25). By
substituting the equations (33) and (34) into (32), we can
obtain
_
s2ðtÞ¼ks2s2ðtÞkst2satðs2ðtÞÞ (35)
As can be seen from the equation (35), by choosing
suitable control parameters, i.e. ks2> 0 and kst2> 0, the
sliding surface dynamics will be stable. The control law
obtained at this stage (u
2
) will be added to the first stage
control law (u
1
) after a period of τ(s) to achieve the precise
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tip-positioning control of the single-link flexible arm. Thus,
the control input law is obtained as follows
uðtÞ¼u1ðtÞt<τ
au1ðtÞþbu2ðtτÞt≥τ(36)
where aand bare the control parameters which are de-
termined by the designer to achieve perfect tracking and to
ensure the stability of the closed loop control system.
4.3. Stability analysis of closed-loop control system
In this section, the stability conditions of closed loop control
system are investigated. For this purpose, the Lyapunov
function is defined as follows
Vðs1;s2Þ¼1
2s2
1þs2
2(37)
Taking the derivative of both sides of the equation (37)
and using the equations (26) and (35) yields
_
Vðs1;s2Þ¼s1ðks1s1kst1satðs1ÞÞþ s2ðks2s2kst2satðs2ÞÞ
≤ks1s2
1ks2s2
2kst1satðs1Þs1kst2satðs2Þs2
(38)
The dynamics of sliding surfaces, i.e. s
1
and s
2
are as-
ymptotically stable. Therefore, the asymptotic stability of
the closed loop control system can be achieved by the
suitable selection of control parameters.
5. Performance criteria
In this section, the following performance criteria are
defined to evaluate the performance of the proposed
controller:
1. Root mean squared error (RMSE) for the trajectory
tracking.
RMSE ¼RMSE1þRMSE2
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
tf
t¼τ
kθdðtÞθtðtÞk2T
tf
v
u
u
t
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
tf
t¼0
kθlðtÞθtðtÞk2T
tf
v
u
u
t
(39)
where RMSE
1
represents the RMSE value for
tracking the desired reference trajectory and RMSE
2
represents the RMSE value for evaluating the per-
formance of controller in eliminating the deflection
caused by the link flexibility.
2. RMSEyfor the tip position trajectory tracking in the y
direction.
RMSEy¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
tf
t¼0
kydðtÞylðtÞk2T
tf
v
u
u
t(40)
where RMSE
y
represents the RMSE value for the tip
position trajectory tracking in the ydirection.
3. Mean absolute error (MAE) for the trajectory
tracking.
MAE ¼MAE1þMAE2
¼X
tf
t¼τ
kθdðtÞθtðtÞk T
tf
þX
tf
t¼0
kθlðtÞθtðtÞk T
tf
(41)
where MAE
1
is the MAE value for tracking the de-
sired reference trajectory and MAE
2
represents the
MAE value for evaluating the controller performance
in eliminating the deflection.
4. MAE
y
for the tip position trajectory tracking in the y
direction.
MAEy¼X
tf
t¼0
kydðtÞylðtÞk T
tf
(42)
where MAE
y
represents the RMSE value for the tip
position trajectory tracking in the ydirection.
5. Root mean square value (RMSV) of the control input
voltage.
RMSV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
tf
t¼0
kUðtÞk2T
tf
v
u
u
t(43)
The above performance criteria include RMSE, RMSV,
and MAE, which are used as objective numerical measures
of tracking performance of an entire error curve, where Tis
the sample time, t
f
represents the total running time of the
simulations. In our simulation, these parameters are chosen
as T=1×10
3
s, t
f
= 10 s. The energy consumption is
determined by the RMSV criterion.
6. Computer simulation results
System numerical simulations are performed using
SIMULINK/MATLAB R2016a. The CRONE Toolbox and
RungeKutta solver with a fixed step size of 0.001 are used to
solve the sets of fractional-order differential equations re-
lated to the FOSMC. The physical parameters of the system
are depicted in Table 1 which includes the flexible-link
parameters and the motor parameters. Most of these
physical parameters of the system were taken from Payo
et al. (2009).Table 2 shows the control parameters of
Hamzeh Nejad et al. 7
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system. The control parameters have been selected by trial
and error which made the closed-loop control system stable.
However, this parameter can be optimized by optimization
methods such as GAs or particle swarm optimization.
In the following, the simulation has been carried out
in the presence of the unknown Coulomb friction dis-
turbances. The desired reference trajectory and the desired
tip position trajectory have been defined as the following
equations
θdðtÞ¼
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
5ðt1Þ21≤t≤2
10ðt1Þ52<t≤4
10 4ðt1Þðt1Þ2
2!50 4 < t≤5
30 t>5
(44)
ydðtÞ¼lsinðθdðtÞÞ (45)
Next, the simulation results are provided for evaluating
the performance of the proposed FOSMC controller.
6.1. Simulation results of the FOSMC controller
In this section, the simulation has been carried out for
evaluating the efficiency of the proposed FOSMC con-
troller. Figure 4 illustrates the simulation results. Figure 4(a)
shows the desired trajectory tracking (tip-positioning
control). Figure 4(b) shows the coupling torque applied
to the single-link flexible arm. Figure 4(c) shows the control
input voltage to the DC motor.
As can be seen in Figure 4(a), the angle difference
between the angular position of the motor and the angular
position of the tip link tends to zero by applying the first
control input signal, i.e. u
1
, within a period of 1 s, and after
1 s, the desired reference trajectory is perfectly followed by
the tip of the link with null steady-state error by adding the
second control input signal, i.e. u
2
, to the first control input
signal (u
1
). As shown in Figure 4(c), the control effort is
very small and never saturates the amplifier which supplies
the motor at 10 and 10 V, providing a proper trajectory
tracking without overheating the electrical system.
6.2. Simulation results of the FOSMC controller in
the presence of disturbance and noise
In this section, we added a sinusoidal disturbance (d(t)=
sin(20πt)) to the control input signal at (t= 1 s) to dem-
onstrate the effectiveness of the proposed control scheme in
dealing with disturbance during the trajectory. We also
added a white noise signal with zero mean and finite var-
iance to the output signal (tip angular position). Figure 5
Table 1. Physical parameters of the system.
Parameter Description Value
Data of the flexible arm
EYoung modulus 122 × 10
9
Pa
ICross section inertia 3.017 × 10
12
m
4
pDensity 1800 kg/m
3
lLength 0.98 m
dDiameter 2.8 × 10
3
mm
mTip mass 43.71 × 10
3
kg
Data of the motor-gear set
JTotal motor inertia + reduction gear 6.87 × 10
5
kg m
2
vViscous friction 1.041 × 10
3
kg m
2
/s
nReduction ratio of the motor gear 50
KMotor constant 2:1×10
1Nm=V
V
sat
Saturation voltage of the servo amplifier ±10 V
Table 2. Control parameters of the closed loop control system.
Parameter Description Value
qFractional derivative order 0.95
k
p
Control parameter 5.4
k
d
Control parameter 5.4
ks1Control parameter in the feedback control 10
kst1Switching gain in the feedback control 0.5
βControl parameter 1.25
ks2Control parameter in the feedback control 8
kst2Switching gain in the feedback control 0.5
aControl parameter 0.6
bControl parameter 0.4
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Figure 4. System transient response. (a) Tracking of the prescribed desired trajectory (tip-positioning control), (b) coupling torque, and
(c) control input voltage.
Hamzeh Nejad et al. 9
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shows the simulation results in the presence of the dis-
turbance and the white noise during the trajectory tracking.
As it can be observed from Figure 5, the proposed control
strategy has a good performance in compensating the si-
nusoidal disturbance during trajectory tracking.
6.3. Effects of possible variations in link weight and
viscous friction of the motor upon the system
response
In this section, effects caused in the system response
because of variations in link weight (m) and viscous
friction of the motor (v) have been evaluated. These
payload changes can happen, for example because of the
use of various tools placed at the end-effector of the
manipulator. A range of possible payload values (m
n
±
0.25m
n
,m
n
= 0.044 kg being the nominal value) have been
chosen so that the arm could support under normal op-
erating condition. The control system robustness has also
been investigated under variations in the viscous friction
parameter (v
n
±0.1v
n
,v
n
=1.041×10
3
kg m
2
/s being the
nominal value). The other remaining parameters of the
system have been assumed unchanged. Therefore, there is
no uncertainty in these parameters. Figure 6 shows the
simulation results when both parameters vary simulta-
neouslyinthedefined range. As it can be observed from
Figure 6, the maximum deviation of the system response
with respect to the nominal case (m
n
,v
n
)isalmost0.5°,
when mis equal to m
max
=m
n
+0.25m
n
and vis equal to
v
min
=v
n
0.1v
n
.
Figure 5. Transient response of the system for q= 0.95 in the presence of disturbance and noise during the trajectory tracking. (a)
Tracking of the prescribed desired trajectory (tip-positioning control) and (b) control input voltage.
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Figure 6. Transient response of the system for q= 0.95 when payload and viscous friction vary simultaneously in the defined range
(a) tracking of the prescribed desired trajectory (tip-positioning control), (b) trajectory tracking error, and (c) control input voltage.
Hamzeh Nejad et al. 11
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6.4. Comparison between the proposed FOSMC
and the conventional SMC and PID controllers
In this section, to investigate the capability of the proposed
FOSMC in comparison with the conventional SMC and
PID controllers, several simulations are carried out. The
performance results of the FOSMC, the SMC, and the PID
controller based on the evaluation criteria without distur-
bance presence and in the presence of disturbance are
compared in Tables 3 and 4, respectively. From Tables 3 and
4, it can be seen that the proposed FOSMC controller has the
smaller error and better trajectory tracking as compared with
the conventional SMC and PID controllers.
7. Conclusion
In this research, an FOSMC method is proposed to precise
tip-positioning control of a single-flexible robot arm. The
purpose of this controller design is first to reduce the de-
flection due to the link flexibility by eliminating the angle
difference between the angular position of the motor and the
angular position of the tip link and then perfect tracking
of the desired reference trajectory by the angular position
of the tip link. Real-time simulation was performed in
MATLAB/Simulink to show the robustness and effective-
ness of the proposed controller. The simulation results
indicate the effectiveness of the proposed control scheme in
eliminating the deflection caused by the flexibility of the
link and the precise tip-positioning control of the single-link
flexible arm, such that the proposed FOSMC control scheme
has 56% and 48% improvement over the conventional PID
control strategy according to the MAE criterion under
without disturbance and in the presence of the disturbance,
respectively. Besides, the proposed control system depicts
robustness against the variations in the weight of the link,
such that with 25% variation in the link weight and with 10%
variation in viscous friction of the motor and the maximum
deviation from the nominal case is less than 0.5°. Further-
more, the disturbance rejection is realized by the proposed
controller. Moreover, the practical implementation of the
proposed control approach may be feasible because of the
control input voltage features which never exceed the pos-
itive or negative saturation voltages. Future work will focus on
the experimental implementation of the controller, the exper-
imental verification of the proposed control scheme, and ex-
tension of the methodology to multiple-input, multiple-output
(MIMO) systems. Some useful experimental implementation
canbefoundintheworksbyKhorrami et al. (1992),Korayem
et al. (2012),Nagarajetal.(1997),Korayem and Shafei (2015),
Ettefagh et al. (2018),andKhodaei et al. (2018).
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with re-
spect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, au-
thorship, and/or publication of this article.
ORCID iDs
Ali Fayazi https://orcid.org/0000-0003-3237-1060
Hossein Ghayoumi Zadeh https://orcid.org/0000-0002-5390-
3938
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