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1

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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2

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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15

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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16

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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19

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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20

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.

6. References

[1] Yang, L., Wen, R., Qin, J., et al.: 'A robotic system for overlapping radiofrequency ablation in large tumour

treatment', IEEE/ASME Trans. Mechatronics, 2010, 15, (6), pp. 887-897

[2] Guo, Y., Zhang, S., Ritter, A., et al.: 'A case study on a capsule robot in the gastrointestinal tract to teach robot

programming and navigation', IEEE Trans. Edu., 2014, 57, (2), pp. 112- 121

[3] Lukic, M., Barnawi, A., Stojmenovic, I.: 'Robot coordination for energy-balanced matching and sequence

dispatch of robots to events', IEEE Trans. Comput, 2015, 64, (5), pp. 1416-1428

of 22




21

[4] Mohan, R. E., Wijesoma, W. S., Calderon, C. A., et al.: 'False alarm metrics for human-robot interactions in

service robots', Adv Robot., 2010, 24, (13), pp. 1841-1859

[5] Craig. J. J.: 'Adaptive control of mechanical manipulators' (Addison-Wesley, 1988)

[6] Yang, Z. J., Fukushima, Y., Qin, P.: 'Decentralized adaptive robust control of robot manipulators using

disturbance observers', IEEE Trans. Control Syst. Technol., 2012, 20, (5), pp. 1357-1365

[7] Plestan, F., Shtessel, Y., Bregeault, Y., et al.: 'New methodologies for adaptive sliding mode control', Int. J.

Control, 2010, 83, (9), pp. 1907-1919

[8] Wai, R. J., Muthusamy, R.: 'Design of fuzzy-neural-network-inherited back-stepping control for robot

manipulator including actuator dynamics', IEEE Trans. Fuzzy Syst., 2014, 22, (4), pp. 709-722 [9] Krstic. M., Kanellakopoulos. I., Kokotovic. P.: 'Nonlinear and adaptive control design' (Wiley, 1995)

[10] Incremona, G. P., Felici, G. D., Ferrara, A., et al.: 'A supervisory sliding mode control approach for cooperative

robotic systems of systems', IEEE Syst. J., 2015, 9, (1), pp. 263-272

[11] Soltanpour, M. R., Otadolajam, P., Khooban, M. H.: 'Robust control strategy for electrically driven robot

manipulators: Adaptive fuzzy sliding mode', IET Sci. Meas. Technol., 2015, 9, (3), pp. 322-334

[12] Yu, S., Yu, X., Shirinzadeh, B., et al.: 'Continuous finite-time control for robotic manipulators with terminal

sliding mode control', Automatica, 2005, 41, (11), pp. 1957-1964

[13] Mondal, S., Mahanta, C.: 'Adaptive second order terminal sliding mode controller for robotic manipulators', J.

Franklin Inst., 2014, 351, (4), pp. 2356-2377 [14] Pan, Y., Kumar, K. D., Liu, G., et al.: 'Design of variable structure control system with nonlinear time-varying

sliding sector', IEEE Trans. Autom. Control, 2009, 54, (8), pp. 1981-1986

[15] Huang, Y. J., Kuo, T. C., Chang, S.H.: 'Adaptive sliding mode control for nonlinear systems with uncertain parameters', IEEE Trans. Syst., Man, Cybern, 2008, 38, (2), pp. 534-539

[16] Aghababa, M. P.: 'Design of hierarchical terminal sliding mode control scheme for fractional-order systems',

IET Sci. Meas. Technol., 2014, 9, (1), pp. 122-133

[17] Pisano, A., Rapaić, M. R., Jeličić, Z. D., et al.: 'Sliding mode control approaches to the robust regulation of

linear multivariable fractional‐order dynamics', Int. J. Robust Nonlinear Control, 2010, 20, (18), pp. 2045-2056

[18] Ferrara, A., Incremona, G. P.: 'Design of an integral suboptimal second order sliding mode controller for the

robust motion control of robot manipulators', IEEE Trans. Control Syst. Technol., 2015, 23, (6), pp. 2316-2325

[19] Mobayen. S.: 'Fast terminal sliding mode tracking of non-holonomic systems with exponential decay rate', IET

Control Theory Appl., 2015, 9, (8), pp. 1294-1301 [20] Feng, Y., Yu, X., Man, Z.: 'Non-singular terminal sliding mode control of rigid manipulators', Automatica,

2002, 38, (12), pp. 2159-2167

[21] Lin, C. K.: 'Nonsingular terminal sliding mode control of robot manipulators using fuzzy wavelet network', IEEE Trans. Fuzzy Syst., 2006, 14, (6), pp. 849-859

[22] Yang. C. C.: 'Robust adaptive terminal sliding mode synchronized control for a class of non-autonomous

chaotic systems', Asian J. Control, 2013, 15, (6), pp. 1677-1685 [23] Hilfer. R.: 'Applications of fractional calculous in physics' (World Scientific, 2001)

[24] Tavazoei, M. S., Haeri, M., Jafari, S., et al.: 'Some applications of fractional calculous in suppression of chaotic

oscillations', IEEE Trans. Ind. Electron., 2008, 55, (11), pp. 4094-4101

[25] Aghababa, M. P.: 'A fractional-order controller for vibration suppression of uncertain structures', ISA Trans.,

2013, 52, (6), pp. 881-887

[26] Zhong, J., Li, L.: 'Tuning fractional-order PI Dλ µ

controllers for a solid-core magnetic nearing system', IEEE

Trans. Control Syst., 2015, 23, (4), pp. 1648-1656

[27] Padula, F., Visioli, A.: 'Optimal tuning rules for proportional-integral-derivative and fractional-order

proportional-integral-derivative controllers for integral and unstable processes', IET Control Theory Appl., 2012, 6,

(6), pp. 776-786

[28] Charef, A., Assaba, M., Ladaci, S., et al.: 'Fractional order adaptive controller for stabilized systems via high-

gain feedback ', IET Control Theory Appl., 2013, 7, (6), pp. 822-828

[29] Yin, C., Chen, Y. Q., Zhong, S. M.: 'Fractional-order sliding mode based extremum seeking control of a class

of nonlinear systems', Automatica, 2014, 50, (12), pp. 3173-3181 [30] Yin, C., Stark, B., Chen, Y. Q., et al.: 'Fractional-order adaptive minimum energy cognitive lightening control

strategy for the hybrid lightening system', Energy Build., 2015, 87, pp. 176-184, 2015

of 22




22

[31] Pisano, A., Rapaić, M., Usai, E.: 'Second-order sliding mode approaches to control and estimation for fractional

order dynamics', (Sliding Modes after the First Decade of the 21st Century. Springer Berlin Heidelberg, 2012), pp.

169-197

[32] Aghababa, M. P.: 'Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic

(hyperchaotic) systems using fractional nonsingular terminal sliding mode technique', Nonlinear Dyn., 2012, 69, (1),

pp. 247-261

[33] Aghababa, M. P.: 'Robust finite-timestabilization of fractional-order chaotic systems based on fractional

Lyapunov stability theory', ASME. J. Comput. Nonlinear Dynam., 2012, 7, (2), pp. 021010

[34] Yin, C., Cheng, Y., Chen, Y. Q., et al.: 'Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems', Nonlinear Dyn., 2015, 82, (1), pp. 39-52

[35] Dadras, S., Momeni, H. R.: 'Fractional terminal sliding mode control design for a class of dynamical systems

with uncertainty', Commun. Nonlinear Sci. Numer. Simul., 2012, 17, (1), p. 367-377

[36] Aghababa, M. P.: 'Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order

dynamical systems', Int. J. Control, 2013, 86, (10), pp. 1744-1756

[37] Aghababa, M. P.: 'Control of non-linear Non-integer-order systems using variable structure control theory',

Trans. Inst. Measurement Control, 2013, 36, pp. 425-432

[38] Wang, Y., Luo, G., Gu, L., et al.: 'Fractional-order nonsingular terminal sliding mode control of hydraulic

manipulators using time delay estimation', J. Vib. Control, 2015, doi:10.1177/1077546315569518 [39] Podlubny. I.: 'Fractional differential equations' (Academic Press, 1999)

[40] Li, Y., Chen, Y. Q., Podlubny, I.: 'Mittag-Leffler stability of fractional order nonlinear dynamic systems',

Automatica, 2009, 45, (8), pp. 1965-1969 [41] Li, Y., Chen, Y. Q., Podlubny, I.: 'Stability of fractional-order nonlinear dynamic systems: Lyapunov direct

method and generalized Mittag-Leffler stability', Comput. Math. Appl., 59, (5), pp. 1810-1821

[42] Barambones, O., Etxebarria, V.: 'Energy-based approach to sliding composite adaptive control for rigid robots

with finite error convergence time', Int. J. Control, 2002, 75, (2), pp. 352-359

[43] Spong. M. W., Vidyasagar. M.: 'Robot dynamics and control' (Wiley, 1989)

[44] Yu, X., Man, Z.: 'Fast terminal sliding-mode control design for nonlinear dynamical systems', IEEE Trans.

Circuits Syst. I, Fundam. Theory Appl. (1993-2003), 2002, 49, (2), pp. 261-264

[45] Man, Z., Yu, X.: 'Terminal sliding mode control of MIMO linear systems', IEEE Trans. Circuits Syst. I,

Fundam. Theory Appl. (1993-2003), 1997, 44, (11), pp. 1065-1070 [46] Wu, Y., Yu, X., Man, Z.: 'Terminal sliding mode control design for uncertain dynamic systems', Syst. Control.

Lett., 1998, 34, pp. 281-288

[47] Zhang, F. R., Li. C. P.: 'Stability analysis of fractional differential systems with order lying in (1,2)', Adv. Difference Equ., 2011, Article ID 213485

[48] Utkin, V. I.: 'Sliding modes in control and optimization' (Springer, 1992)

[49] Bartolini, G., Pisano, A., Punta, E., et al.: 'A survey of applications of second-order sliding mode control to mechanical systems', Int. J. Control, 2003, 76, (9/10), pp. 875-892

[50] Zhao, D., Li, S., Gao, F.: 'A new terminal sliding mode control for robotic manipulators', Int. J. Control, 2009,

82, (10), pp. 1804-1813

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