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Adaptive Fractional-order Non-singular Fast Terminal Sliding Mode Control for Robot Manipulators
Donya Nojavanzadeh, Mohammad Ali Badamchizadeh*
Department of Control Engineering, Faculty of Electrical and Computer Engineering,
University of Tabriz, Tabriz, Iran *[email protected]
Abstract: In this paper, an adaptive fractional-order terminal sliding mode controller (FO-TSMC) is
proposed for controlling robot manipulators with uncertainties and external disturbances. An adaptive
tuning method is utilized to deal with uncertainties which upper bounds are unknown in practical cases.
Fast convergence is achieved by using non-singular fast terminal sliding mode control. Also, fractional-
order controller is used to improve tracking performance of controller. After proposing a new stable
fractional-order non-singular and nonlinear switching manifold, a sliding mode control law is designed.
The stability of the closed-loop system is proved by Lyapunov stability theorem. Simulation results
demonstrate the effectiveness and high-precision tracking performance of this controller in comparison
with integer-order terminal sliding mode controllers.
1. Introduction
During recent years, there have been serious efforts for controlling robot manipulators due to their
importance in various fields [1-4], like industrial automation, space exploration and medical domains
requiring high precision in trajectory tracking. Robot manipulators entail inherent nonlinearities and
uncertainties produced by model inaccuracies, payload variations, external disturbances and so on, which
makes their control complex. In order to deal with these problems, advanced control methods are
implemented; like adaptive control and advance decentralized control [5-7], back-stepping design
techniques, nonlinear feedback control [8, 9] and sliding mode control (SMC) [10, 11].
Amongst these methods, sliding mode control is a well-known and well-improved control method
which is insensitive to parametric uncertainties. SMC is useful for controlling a wide range of linear and
nonlinear uncertain systems especially in controlling robots [12, 13]. Conventional sliding mode control
uses a control law with large control gains that yields undesired chattering when the system is on sliding
mode. Chattering phenomenon is undesired because it excites high frequency dynamics and leads to
instability. There are a number of methods for dealing with this problem. To attenuate chattering, in [14]
instead of sliding mode, a sliding sector has been proposed. Also, methods like boundary layer technique
[15], second-order or higher-order sliding mode is introduced in [16-18]. Second-order controllers are
really successful in eliminating chattering.
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In conventional sliding mode control, a linear manifold is considered as a sliding surface, which can
guarantee asymptotic stability of the system. Besides chattering, another disadvantage of conventional
SMC is that the system states cannot reach the equilibrium point in finite time, although the parameters of
linear sliding mode can be adjusted to make the convergence faster. In high precision control, like motion
control in robotics, faster convergence is a high priority which can be achieved only at the expense of large
control input. By utilizing nonlinear sliding surface, the terminal sliding mode controller (TSMC) can
reach the fast convergence of the states, without spending large control efforts. TSMC in comparison with
conventional SMC has numerous superiorities like faster, finite time convergence and higher control
precision [19]. However, there are two disadvantage for terminal sliding mode control. The first is
singularity problem and the second is the requirement of prior knowledge of upper bounds of uncertainties
that is not viable in practice. The first problem can be tackled by non-singular terminal sliding mode
control [19, 20], and the second problem can be solved by methods like fuzzy wavelet networks to
estimate unknown functions [21], and adaptive control [22]. In proposed controller, adaptive non-singular
terminal sliding mode (NTSM) control is utilized to solve these problems.
Fractional calculus is a mathematical analysis that dates to more than 300 years ago and is a
generalization of integer-order integration and differentiation to non-integer order ones [23]. In recent
years, the application of fractional order in science and engineering makes it an attractive subject between
researchers. Designing fractional-order (FO) controllers is one of these applications [24, 25]. Many
different FO controllers such as fractional PID controllers, FO optimal controllers [26, 27] and FO
adaptive controllers [22, 28] are introduced. In [29, 30], FO optimal control schemes are introduced for
minimizing energy usage for a class of nonlinear systems. Recently, fractional calculus has been combined
with sliding mode control in the controller design for fractional-order and integer-order systems. Second-
order sliding mode control proposed in [31] is very useful in reducing chattering phenomenon for a class
of fractional-order systems. It has been demonstrated that many fractional-order systems exhibit chaotic
behaviour, and the study of stabilizing chaotic systems have attracted considerable interests. In [32],
fractional-order non-singular terminal sliding mode control is used to study the finite-time
stabilization/synchronization problem of fractional-order autonomous/nonautonomous
chaotic/hyperchaotic systems, and in [33], the fractional Lyapunov stability theory is used for finite-time
stabilization of fractional-order chaotic systems. Fractional-order controllers have some superiority like
faster tracking performance and higher control accuracy in comparison with integer-order ones. Some
recent studies were focused on reachability conditions and used to show the superiorities of fractional-
order controllers over integer-order ones by comparing the reaching times [34]. Besides FO controllers,
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terminal SMC is useful in finite-time stabilization. In [35-38], FO-SMC and FO-NTSM controllers are
designed for control of integer-order or fractional-order systems. In this paper, to enhance the performance
of TSM, FO-NTSM for control of robot manipulators has been proposed.
To the authors' best knowledge adaptive fractional-order non-singular fast terminal sliding mode
control has not been used for controlling robot manipulators, so far. Consequently, this study proposed a
sliding surface by utilizing a nonlinear sliding surface (TSM) to achieve the goal of tracking in finite time.
Also, adaptive control is used to estimate the upper bounds of uncertainties. Utilizing fractional-order
control with appropriate fractional-order, leads to high precision and fast tracking performance. After
proposing a new stable fractional-order switching manifold, a fractional fast terminal sliding mode control
law is designed. Then, the stability of closed-loop system is proved by Lyapunov stability theorem. The
simulation results have demonstrated the effectiveness and robustness of the proposed method. Besides,
the proposed controller is compared with TSMC in [20], which results in superior performance of the
proposed controller.
The rest of this paper is organized as follows. In section 2, the preliminaries and definitions of
fractional calculus are provided. In section 3, the design procedure of adaptive FO-TSMC for robot
manipulators is explained. In section 4, numerical simulations are performed. Finally, concluding remarks
are included in section 5.
2. Preliminaries of Fractional Calculus
Definition 1: [39], The α th-order Reimann-Liouville fractional integration of function ( )f t with
respect to t is given by the following:
0
01
1 ( )( )
( ) ( )
t
t tt
fI f t d
t
αα
ττ
α τ −=Γ −∫ (1)
where ( )αΓ is the Gamma function and 0t is the initial time.
Definition 2: [39], The Reimann-Liouville fractional derivative of α th-order of function ( )f t is
defined as follows:
0
01
( ) 1 ( )( )
( ) ( )
mt
t t m mt
d f t d fD f t d
dt m dt t
αα
α α
ττ
α τ − += =
Γ − −∫ (2)
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where 1m mα− < ≤ , m N∈ . In this paper, the notation Dα indicates the Reimann-Liouville fractional
derivative.
Property 1: [39], The Reimann-Liouville fractional derivative operator 0t tDα
commutes with
n nd dt ,i.e.,
0 0 0
( )( ( )) ( ) ( )
n nn
t t t t t tn
d d f tD f t D D f t
dt dt
α α αα
+= = (3)
Theorem 1: [40], Let 0x = be an equilibrium point for the non-autonomous fractional-order system
( ) ( , )D x f x tα = (4)
where ( , )f x t satisfied the Lipschitz condition with Lipchitz constant 0l > and (0,1)α ∈ . Assume that
there exist a Lyapunov function ( , ( ))V t x t satisfying
1 2( , )a
x V t x xα α≤ ≤ (5)
3( , )V t x xα≤& (6)
where 1α , 2α , 3α and a are positive constants. Then the equilibrium point of the system (4) is Mittag-
Leffler stable.
Remark 1: Mittag-Leffler stability implies asymptotically stability [41].
Theorem 2: [40], Let 0x = be an equilibrium point for the non-autonomous fractional order system
( ) ( , )D x f x tα = (7)
Assume that there exist a Lyapunov function ( , ( ))V t x t and class-K function ( 1, 2,3)i iα = satisfying
1 2( ) ( , ) ( )x V t x xα α≤ ≤ (8)
3( , ) ( )D V t x xβ α≤ − (9)
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where (0,1)β ∈ . Then the equilibrium point of system (7) is asymptotically stable.
Lemma 1: [42] Assume that a continuous positive-definite function ( )V t satisfies the differential
inequality
0 0( ) ( ) , , ( ) 0V t V t t t V tηα≤ − ∀ ≥ ≥& (10)
where 0α > , 0 1η< < are constant. Then, for any given 0t , ( )V t satisfies the inequality
1 1
0 0 0 1( ) ( ) (1 )( ) ,V t V t t t t t tη η α η− −≤ − − − ≤ ≤ (11)
and
1( ) 0,V t t t= ∀ ≥ (12)
with 1t given by
1
01 0
( )
(1 )
V tt t
η
α η
−
= +−
(13)
3. Design Approach of Adaptive FO-NTSM Controller for Trajectory Tracking of Robot Manipulators
The dynamic of n-link robot manipulator can be derived in joint space as follows [43]:
0 0
0
( ( ) ( )) ( ( , ) ( , ))
( ( ) ) d
M q M q q C q q C q q q
G q G τ τ
+ ∆ + +∆
+ +∆ = +
&& & & &
(14)
where q , q& , nq R∈&& are position, velocity and acceleration vectors of the joint respectively.
0( ) ( ) ( )M q M q M q= + ∆ is inertia matrix, 0( , ) ( , ) ( , )C q q C q q C q q= + ∆& & & is centripetal Coriolis matrix,
0( ) ( ) ( )G q G q G q= + ∆ is gravitational matrix, ( )n
t Rτ ∈ is the joints' torque vector and ( ) n
d t Rτ ∈ is
disturbance torque vector. ( )M q∆ , ( , )C q q∆ & , ( ) n nG q R ×∆ ∈ are parameter uncertainties of the robot
manipulator and 0 ( )M q , 0 ( , )C q q& , 0 ( )G q are the nominal terms [20].
Assumption 1. The dynamic model of the robot manipulator can be written as:
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0 0 0( ) ( , ) ( ) ( , , )dM q q C q q q G q F q q qτ τ+ + = + +&& & & & && (15)
where ( , , ) ( ) ( , ) ( )n
F q q q M q q C q q q G q R= −∆ −∆ −∆ ∈& && && & & is the lumped uncertainty of the system, which is
bounded by the following function [20]:
0 1( , , )F q q q B B q< +& && (16)
where 0B and
1B are the given positive constants.
The control aim is to lead robot manipulator described by Eq. (14) to tracking a reference trajectory.
Assume q and dq as actual and desired position vectors respectively, therefore tracking error and
derivatives are as de q q= − , de q q= −& & & and de q q= −&& && && . The system (15) can be written in the following
form:
1
0 0 0( ( , ) ( ) ( , , )) de M C q q q G q F q q q qτ−= − − + + −&& & & & && && (17)
where ( , , ) ( , , )dF q q q F q q qτ= +& && & && such that
0 1( , , )F q q q B B q< +& && (18)
here 0B and 1B are positive constants.
The adaptive FO-NTSM control design procedure entails two main steps to achieve faster
convergence speed with high precision tracking performance in comparison with integer-order terminal
sliding mode controller in [20]. The first step is to design a suitable fractional terminal sliding surface. In
this paper, a novel fractional non-singular fast terminal sliding surface is defined as:
1( )pqS t D e Ae eα λ+= + + (19)
where 1 2( ) [ , ,..., ]T n
nS t s s s R= ∈ . The second and third term are defined as reported fast TSM control in
[44], and A is a constant diagonal matrix whose diagonal elements are ia R∈ , ( 1, 2,..., )i n= , and λ is a
design positive constant, p and q are positive odd integers who satisfied 2p q p< < . For having the
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advantages of TSM control and fractional-order control, the first term is define as shown in (19) with
(0,1)α ∈ which is a real number. However, TSM controller has a singularity problem, that is, in some
areas of the state space, the TSM control may require to be infinitely large in order to maintain the ideal
TSM motion. Substantial research has been done to overcome the singularity problem in TSM control
systems. In [45, 46] indirect approaches are adopted to avoid the singularity problem, and in [20] simple
non-singular TSM controls are proposed, which are able to avoid this problem completely. In this paper,
the proposed method in [20] has been selected. Therefore, by satisfying 1 2q p< < , the proposed TSM
controller does not have singularity problem.
Remark 2: [47] The fractional sliding mode system (19), with Riemann-Liouville derivative is stable,
if arg( ) 0A ≠ , arg( ( )) 2spec A απ> .
Taking time derivative of Eq. (19) and using property 1 yields:
( )
( ) ( ) ( ) ( ) ( ) ( )
p q
qpS t D e t Ae t e t e t
q
α λ−
= + +& && & & o (20)
where o operator denotes element-wise multiplication between two vectors. After stablishing the
appropriate manifold, the next step is to design an adaptive robust control law for robot manipulators in
order to achieve the control aims, that is, fast convergence in finite time and robustness against
uncertainties with unknown upper bounds. Therefore, the control law is introduced as follows:
0 0 0
0
0
( )
0 0
1 0
( ) ( , ) ( ) ( )
( ( ) ( )
( ) ( ) s ( )
( ) ( ) ( ) ( ) ( )
( )s ( ))
d
p q
q
t C q q q G q M q q
D M q S t
M q S t ignS t
pM q Ae t M e t e t
q
B q B ignS t
α
µ
τ
λ
−
−
= + +
− Θ
+ Θ
+ +
+ +
& & &&
o
& & o
(21)
where 0 ( )M q , 0 ( , )C q q& , 0( )n n
G q R×∈ are nominal mass, centripetal Coriolis and gravity matrix of robot
manipulator respectively, 0Θ > is a n n× diagonal matrix whose elements are equal to θ , θ is positive
constant, (0,1)µ ∈ is a constant for reducing chattering phenomenon, 1 2( ) [ , ,..., ]T n
nS t s s s R= ∈ , (0,1)α ∈
is the fractional order of the sliding surface (19), 1 2( ) [ , ,..., ]T
nS t s s sµ µ µ µ= ,
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1 2( ) [ ( ), ( ),..., ( )]TnsignS t sign s sign s sign s= and 1B , 0B are supposed to be known in Eq. (21). It is obvious
the implementation of Eq. (21) depends on the amount of 0B and 1B . In practice, this is so restrictive,
because the exact model of robot manipulator and consequently aforementioned amounts are not obtained
easily. This issue brings in mind the proposal of adaptive control that estimates the bounds of 0B and 1B .
The following adaptive fractional-order non-singular fast terminal sliding mode controller has the
advantage of adaptive control and also ensures smooth chattering in control signal.
Statement 1. To stabilize the uncertain nonlinear system, namely, two degree of freedom robot
manipulator in presence of external disturbances, an adaptive FO-FTSM control law is suggested as
follows:
0 0 0
0
0
( )
0 0
1 0
( ) ( , ) ( ) ( )
( ( ) ( )
( ) ( ) s ( )
( ) ( ) ( ) ( ) ( )
ˆ ˆ( )s ( ))
d
p q
q
t C q q q G q M q q
D M q S t
M q S t ignS t
pM q Ae t M e t e t
q
B q B ignS t
α
µ
τ
λ
−
−
= + +
− Θ
+ Θ
+ +
+ +
& & &&
o
& & o
(22)
By defining the adaptation error as 1 1 1
ˆB B B= −%
and 0 0 0
ˆB B B= −%
, the adaptive laws are defined as:
1
0 0
ˆ ( )B S tυ −=&
(23)
1
1 1
ˆ ( )B S t qυ −=&
(24)
where 0υ and 1υ are positive tuning parameters.
Proof: The stability of the system is proved by choosing the Lyapunov function as
2 2
0 0 1 1
1 1 1( ) ( ) ( )
2 2 2
TV t S t S t B Bµ µ= + +% % (25)
Differentiating ( )V t with respect to time yields
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0 0 0 1 1 1
ˆ ˆ( ) ( ) ( )TV t S t S t B B B Bµ µ= + +
& &% %&& (26)
Using Eq. (20) gives
( )
0 0 0 0 1 1 1 1
( ) ( )( ( ) )
ˆ ˆ ˆ ˆ( ) ( )
p q
T qpV t S t D e Ae e e
q
B B B B B B
α λ
µ µ
−
= + +
+ − + −
& && & & o
& &
(27)
Inserting Eq. (17), (23) and (24) to the right hand side of Eq. (27) yields
1
0 0 0
( )
1
0 0 0 0
1
1 1 1 1
( ) ( )( ( ( ( , ) ( )
( , , )) )
( ) )
ˆ( ) ( )
ˆ( ) ( )
T
d
p q
q
V t S t D M C q q q G q
F q q q q
pAe e e
q
B B S t
B B S t q
α
τ
λ
µ υ
µυ
−
−
−
−
= − −
+ + −
+ +
+ −
+ −
& & &
& && &&
& & o (28)
Based on control law (22)
1
0 0 0
0 0
0
0
0
0
( )
0
1 0
( )
0
( ) ( )( ( ( ( , ) ( )
( , ) ( )
( )
( ( ) ( )
( ) ( ) s ( )
( ) ( )
( ) ( ) ( )
ˆ ˆ( ) s ( ))
( , , )) )
( ) )
T
d
p q
q
d
p q
q
V t S t D M C q q q G q
C q q q G q
M q q
D M q S t
M q S t ignS t
M q Ae t
pM e t e t
q
B q B ignS t
F q q q q
pAe e e
q
α
α
µ
λ
λ
µ υ
−
−
−
−
= − −
+ +
+
− Θ
+ Θ
+
+
+ +
+ −
+ +
+
& & &
& &
&&
o
&
& o
& && &&
& & o
1
0 0 0
1
1 1 1 1
ˆ( ) ( )
ˆ( ) ( )
B B S t
B B S t qµυ
−
−
−
+ − (29)
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According to the facts2TS S S= , ( )TS sign S S= and ( ( )) ( )TS sign S sign S S
µ µ=o , one can
obtain
2
1
0 1 0
1
0
1
0 0 0 0
1
1 1 1 1
( ) ( ) ( )
ˆ ˆ( ) ( )
( , , ) ( )
ˆ( ) ( )
ˆ( ) ( )
V t S t S t
B q B S t
F q q q S t
B B S t
B B S t q
µθ θ
µ υ
µυ
−
−
−
−
≤ − −
− Μ +
+ Μ
+ −
+ −
&
& && (30)
And based on this fact that 2 2
( ( ) ( ) ) ( )K S t S t S tµθ θ≤ − + ≤ −
2
1
0 1 0
1
0
1
0 0 0 0
1
1 1 1 1
( ) ( )
ˆ ˆ( ) ( )
( , , ) ( )
ˆ( ) ( )
ˆ( ) ( )
V t S t
B q B S t
F q q q S t
B B S t
B B S t q
θ
µ υ
µυ
−
−
−
−
≤ −
− Μ +
+ Μ
+ −
+ −
&
& && (31)
Now, by adding and subtracting the term1
0 1 0( ) ( )B q B S t−Μ + , one can obtain
2
1
0 1 0
1
0
1
0 0 0 0
1
1 1 1 1
1
0 1 0
1
0 1 0
( ) ( )
ˆ ˆ( ) ( )
( , , ) ( )
ˆ( ) ( )
ˆ( ) ( )
( ) ( )
( ) ( )
V t S t
B q B S t
F q q q S t
B B S t
B B S t q
B q B S t
B q B S t
θ
µ υ
µυ
−
−
−
−
−
−
≤ −
− Μ +
+ Μ
+ −
+ −
+ Μ +
− Μ +
&
& &&
(32)
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and
1
0 1 0
1
0
1 1
0 0 0 0 0
1 1
0 1 1 1 1
( ) ( ( ) ( )
( , , ) ) ( )
ˆ( ( ) ( ) ) ( )
ˆ( ( ) ( ) ) ( )
V t S t B q B
F q q q S t
S t S t B B
S t S t q B B
θ
µ υ
µυ
−
−
− −
− −
≤ − + Μ +
− Μ
− Μ − −
− Μ − −
&
& &&
(33)
00
0
11
1
( ) 2( ) 2
22
2
2
S tV t B
B
σ γ
δ
µβ β
µ
µβ
µ
≤ − −
−
%&
%
(34)
where 1 1
0 1 0 0( ) ( ) ( , , )S t B q B F q q qσβ θ − −= + Μ + − Μ & && , 1 1
0 0 0( ) ( )S t S tγβ µ υ− −= Μ − and
1 1
0 1 1( ( ) ( ) )S t S t qδβ µυ− −= Μ − .
so
0 1
0 10 1
2 2( ) min{ 2 , , }
( ).( )
2 22
V t
S tB B
σ γ δβ β βµ µ
µ µ
≤ −
+ +
&
% %
(35)
From inequality (35), the following inequality can be derived
1 2( ) ( )c
V t V tβ≤ −&
(36)
where 0 1
2 2min{ 2 , , }c σ γ δβ β β β
µ µ= and 0cβ > . The inequality (36) holds if σβ , γβ , 0δβ > [7].
Remark 3: According to lemma 1, system states can reach the fast TSM as 1t t≥ , 1
01 0
( )( )
(1 )
V tt t
η
α η
−
= +−
and 0t is the initial time. Based on inequality (36), 1 2, cη α β= = and if 0 0t =
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1 2
1
(0)
(1 2)c
Vt
β= (37)
Hence, the system (14), which is controlled with the law of (22), is finite time stable and the sliding
manifolds converge to the origin in a known finite time 1t . Besides, if sliding manifolds converge to zero,
the tracking errors of the robotic manipulator de q q= − will also reach zero.□
4. Simulation Results
In this section, the proposed adaptive fractional-order non-singular fast terminal sliding mode
controller is applied for trajectory tracking of robot manipulator shown in Fig.1.
Fig. 1. Two-link robot manipulator
The mass, centripetal Coriolis and gravity matrix of a two-link robot manipulator presented in (14)
are given as follows:
1 2
3 4
( )M M
M qM M
=
,
2
2 1 2 2 1 2 1 2 2 1 2
2
2 1 2 2 2
2( , )
m l l s q m l l s q qC q q q
m l l s q
− −=
& & && &
& ,
1 2 1 2 2 2 1 2
2 2 1 2
( ) cos( )( )
cos( )
m m l gc m l g q qG q
m l g q q
+ + + = +
where 2 2
1 1 2 1 2 2 2 1 2 2 1( ) 2M m m l m l m l l c J= + + + + , 2
2 2 2 2 1 2 2M m l m l l c= + , 2
3 2 2 2 1 2 2M m l m l l c= + , 2
4 2 2 2M m l J= +
1 1cos( )c q= , 2 2cos( )c q= , 1 1sin( )s q= , 2 2sin( )s q= . Here 1 2[ , ]Tq q q= , 1 2[ , ]Tq q q=& & & are the angular
position and velocity vector respectively. 1 2[ , ]Tτ τ τ= is the input torque and 1 2[ , ]Td d dτ τ τ= is the external
disturbance as follows:
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1
2
2sin( ) 0.5sin(200 )
cos(2 ) 0.5sin(200 )
d
d
t t
t t
τ π
τ π
= +
= + (38)
Parameters of two-link robot manipulator are assigned in table 1.
Table 1 Parameters of two-link robot manipulator [20]
parameters values
1l (Length of the first link) 1m
2l (Length of the second link) 0.85m
1m (Mass of the first link) 0.5kg
2m (Mass of the second link) 0.5kg
1m̂ (Nominal mass of the first link) 0.4kg
2m̂ (Nominal mass of the second link) 1.2kg
1J (Moment of inertia of the first DC motor) 5kgm
2J (Moment of inertia of the second DC motor) 5kgm
g (Gravitational constant) 29.81 /m s
The desired trajectories are assumed as
4
1
4
2
1.25 7 5 7 20
1.25 1 4
t t
d
t t
d
q e e
q e e
− −
− −
= − +
= + − (39)
The controller parameters are chosen as (10,10)A diag= , the order of derivatives in applied
controller is 0.1α = , and (2, 2)diagλ = .
Remark 4: The value of θ is a trade-off between the reaching time and control input. Based on Eq.
(22), one can see that the control effort is proportional to the value of parameter θ , while according to the
following statements, the reaching time is proportional to the inverse value of parameter θ .
Selecting the Lyapunov function as 2 1( ) ( )V t S t= and taking its time derivative, one has
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2 ( ) ( ) ( ( ))TV t S t sign S t= && (40)
using Eq. (17), (20) and (21) yields
( )
2
1
0 0 0
( )
( ) ( ( ) ) ( ( ))
( ( ( ( , ) ( )
( , , )) )
( ) ) ( ( ))
p q
Tq
d
p q
Tq
pV t D e Ae e e sign S t
q
D M C q q q G q
F q q q q
pAe e e sign S t
q
α
α
λ
τ
λ
−
−
−
= + +
= − −
+ + −
+ +
& && & & o
& &
& && &&
& & o
(41)
According to Eq. (21), (18) and based on the fact that 1
( )TS sign S S= ,
1( ( )) ( )TS sign S sign S S
µ µ=o and after some simplifications, one can see
12 1 1
( )( ) ( ( ) ( ) )
d S tV t S t S t
dt
µθ= ≤ − +& (42)
after some simple calculations and using a similar approach in [32, 33], one can obtain
1 1 1
1
1 1 1
1
1
1
1
( ) ( ) ( )
( ( ) ( ) ) ( ( ) 1)
( )1
(1 ) ( ( ) 1)
d S t S t d S tdt
S t S t S t
d S t
S t
µ
µ µ
µ
µ
θ θ
θ µ
−
−
−
−
≤ − = −+ +
= −− +
(43)
taking integral of both sides of Eq. (43), from 0 to the reaching time rt , and setting ( ) 0rS t = , yields
1( )
1
1(0)
1
1 ( )
(0)1
1
1
( )1
(1 ) ( ( ) 1)
1ln( ( ) 1) |
(1 )
1ln( (0) 1)
(1 )
r
r
S t
rS
S t
S
d S tt
S t
S t
S
µ
µ
µ
µ
θ µ
θ µ
θ µ
−
−
−
−
≤ −− +
= − +−
= − +−
∫
(44)
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Thus, achieving smaller reaching times is possible by tuning θ . In applied controller for robot
manipulator, it is assumed as 5θ = .
The constant 1µ ≈ is chosen to alleviate undesirable chattering in control inputs, and the fractional
power of sliding manifold is 0.9p q = . The constant values of adaptive laws are 0 10.05υ υ= = . The
initial conditions are selected as 1(0) 1q = , 2 (0) 1.5q = , 1 2(0) (0) 0q q= =& & and 1 0
ˆ ˆ(0) (0) 5B B= = .
The simulation results by implementing the proposed controller are shown in Fig. 2-5, which clearly
demonstrate the asymptotical convergence of the controller. In Fig. 2 and Fig. 3, the position and velocity
tracking performances of joints are presented respectively. Fig. 4 illustrates the control inputs of joints
without any chattering. In Fig. 5, the parameter estimations are depicted. Fig. 6 shows the designed sliding
surfaces that confirms the fast convergence of system.
Fig. 2. Output position tracking performances of joint 1 and joint 2 using the proposed controller
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Fig. 3. Output velocity tracking performances of joint 1 and joint 2 using the proposed controller
Fig. 4. Control inputs of joint 1 and joint 2 using the proposed controller
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Fig. 5. Parameter estimations using the proposed controller
Fig. 6. Sliding surfaces using the proposed controller
Remark 5: The appropriate amount for α in proposed controller is 0.1 and is chosen by trial and
error. Position tracking performances of proposed controller with different fractional orders are presented
in Fig. 7.
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Fig. 7. Tracking performance with different fractional orders
4.1. Comparison between FO-TSMC, IO_TSMC and SO-SMC
Second-order sliding mode control (SO-SMC) is very successful in eliminating chattering
phenomenon, as it is introduced for a class of linear uncertain multivariable FO dynamic systems in [31].
Extending integer-order derivations and integrals to fractional-order items has a solid foundation. The
main advantage of fractional-order controllers is the greater flexibility in improving the robustness and
control performance. In [34], a FO switching-type control law is designed to show the superiority of FO-
SMC over IO-SMC. The sign function used in the control law is fractional-order and control law is
chatter-free. Also, the smaller reaching time in comparison with IO-SMC is proved mathematically.
In this paper, according to control law, Eq. (22), one can see the sign function is of fractional-order
and can attenuate chattering, like [34]. FO-SMC approaches show faster tracking and higher control
accuracy in comparison with IO-SMC, as it is shown in Fig. 8 and Fig. 9. High precision in tracking
performance and smaller reaching times is very important in control of robot manipulators. In proposed
controller, this goal is achieved by taking the advantages of terminal sliding mode control and FO-SMC
and tuningθ . To demonstrate the exactness and effectiveness of the proposed controller, a comparison of
the control method suggested in this study, with IO-TSMC method in [20], and similar second-order
sliding mode control scheme used in [49], is presented.
The control law of the controller in [20] is given as:
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0 1
0 0 0 0
1 1 2
0
1 0 1 2
( , ) ( ) ( )
( ) ( )
( )( ( ) )
dC q q q G q M q q
M q sig e
M q K s K sig s
γ
ρ
τ τ τ
τ
β γ
τ
− − −
= +
= + +
−
= − +
& & &&
&
% %
(45)
where ( )s e sig eγβ= +% & is sliding manifold, 1 2γ< < and 1
K , 2K are positive definite diagonal feedback
control gain matrices. In Fig. 8 and Fig. 9, the position and velocity tracking of robot manipulator by using
the proposed controller in this paper, SO-SMC and TSMC in [20] are presented respectively. By
comparing the results, one can see that the proposed controller, namely, FO-TSMC is more robust in
presence of disturbances and yields high exactness in convergence, in comparison with IO-TSMC in [20].
Besides, the reaching time in proposed controller is less than 1 second, whereas it is near 3 seconds in IO-
TSM controller and more than 6 seconds in SO-SMC. Finally, proposed controller has little reaching time
in comparison with SO-SMC and IO-TSMC, and shows higher precision in comparison with IO-TSMC.
Another controller is suggested in [50] for control of robot manipulators, which does not need the
model of robot manipulator, however it needs the upper bounds of uncertainties. Comparing the tracking
results of this controller for supposed robot manipulator with proposed controller in this study shows the
tracking performance of both control approaches are similar. The control forces of two control methods
are chatter-free, and the comparison also shows the effectiveness of the proposed new FO-FTSM control.
Fig. 8. Output position tracking performances of joint 1 and joint 2 using proposed controller in [20]
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Fig. 9. Output velocity tracking performances of joint 1 and joint 2 using proposed controller in [20]
5. Conclusion
In this paper, a new adaptive fractional-order non-singular fast terminal sliding mode controller is
proposed for trajectory tracking of robot manipulators. Uncertainties and external disturbances are taken in
to account. Also, by using the adaptive control the requirement of upper bounds of uncertainties are not
necessary in this proposed controller. The fast convergence is obtained by terminal sliding mode control.
Fractional-order controllers have some superiority like faster tracking performance and higher control
accuracy in comparison with integer-order ones. In this study, utilizing fractional-order control with
appropriate fractional-order, leads to high precision in tracking performance. To illustrate the effectiveness
of proposed controller the simulation results are compared with IO-TSM and SO-SMC control. The
simulation results reveal the exactness and effectiveness of proposed fractional-order non-singular fast
TSM controller for trajectory tracking of robot manipulators.
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