new an effective robust adaptive sliding mode control for … · 2020. 5. 20. · control, robust...

International Journal of Engineering & Technology IJET-IJENS Vol:20 No:02 23

203602-4545-IJET-IJENS © April 2020 IJENS I J E N S

An Effective Robust Adaptive Sliding Mode Control

for Quadcopter UAVs Ha Le Nhu Ngoc Thanh1, Choong Hyun Lee 2, Sung Kyung Hong 2

1Faculty of Intelligent Mechatronics Engineering, Sejong University, Seoul, South Korea

2Faculty of Mechanical and Aerospace Engineering, Sejong University, Seoul, South Korea

Corresponding Author: Sung Kyung Hong

Abstract-- An effective robust adaptive sliding mode

controller to reject time-varying disturbances and

uncertainties for improving the tracking performance of

attitude and altitude control of quadcopter Unmanned

Aerial Vehicles (UAVs) is presented in this paper. The

flight controller provides a strict robustness, fast response,

and rapid adaptation for the vehicle under the undesirable

effect of perturbations. The proposed controller design is

based on the super-twisting algorithm for excellently

eliminating the chattering effect. In addition, an efficient

adaptive law is achieved from the Lyapunov stability to

ensure that the controller gains can be automatically

adjusted to compensate for any effect of perturbations.

Thus, the proposed algorithm greatly guarantees the

robust control for the vehicles even without knowing the

upper bound of a time-varying disturbance. A numerical

simulation was executed and compared with recently

alternative methods. A superior stability and excellent

tracking performance of the attitude and altitude control

of a quadcopter UAV in the simulation results

demonstrated the effectiveness of the proposed method.

Index Term-- Adaptive super-twisting sliding mode

control, robust adaptive control, unmanned aerial vehicles,

disturbance rejection, robustness.

I. INTRODUCTION Quadcopter UAVs are being used in many practical

applications such as data collection [1], [2], crop spraying

pesticide in agriculture [3], [4], search and rescue tasks

[5], [6], payload carrying [7], [8], disaster management

[9], [10], and military operations [11],[12]. As the

technological development of computers, electronics,

mechanics, control theories and communication technologies, the UAVs will become more and more

maneuverable and smarter. Thus, its applications will be

expanded in the future. However, their usability still

encounters various challenges such as ensuring the

robustness, safety, and high precision in the extremely

complex operation environments. Furthermore, the

operational principle of the UAVs is strictly affected by

the aerodynamics of multi-rotors making them more

sensitive from the external influences or uncertainties,

especially wind gust. Therefore, disturbance elimination is

one of crucial criteria and important objectives for

designing a controller. Many advanced control techniques

to solve this problem have been suggested in recent years.

In general, the existent studies can be gathered by two

groups. First, varieties of control algorithms have been

introduced by using an adaptive control, robust control,

and nonlinear control techniques to attenuate the effects of

perturbations on quadcopter UAVs. Second, the unknown disturbances or uncertainties are estimated by a nonlinear

disturbance observer, afterward the approximated value is

integrated with advanced control algorithms to

compensate for the effect of disturbances on the system.

In the UAVs control technique, the robustness of

attitude and altitude control is vital demand to sustain a

stability and safety of the vehicles during the flight.

Furthermore, it is easy to understand that a robust attitude

controller not only provides the stability of the attitude

performance but also guarantees the precision of the

vehicle‟s position in the mission flight. Improving and

developing a robust adaptive algorithm for the attitude and

altitude control of UAVs makes the vehicle become more

adaptive and stable to any unexpected effect of the

working environment such as disturbances or

uncertainties.

A. RELATED WORKS In the previous studies, several classical linear control

methods were introduced for controlling the attitude and

altitude of the UAVs such as Proportional Integral

Derivative (PID) [13]-[15] or Linear Quadratic Regulator

(LQR) [16]-[17]. However, these methods exhibited some

disadvantages such as slow response and inadaptable in the occurrence of external disturbances or uncertainties

because these control techniques are usually designed for

linear system models, while the UAVs are strict

nonlinearities, as a result the vehicle‟s stability may not

guarantee. Therefore, numerous adaptive controls, robust

controls and advanced control methods were proposed for

nonlinear system models. Among of them, Sliding Mode

Control (SMC) is famous for the strong robustness of its

control method in various system dynamics and

complicated environments, caused by its essential ability

of eliminating perturbations. Nevertheless, this control

technique always endures from the so-called “chattering

effect” because of a discontinuous term of the switching

law. In order to overcome this drawback, various studies

on the advanced SMC have been introduced. References

[18]-[20] presented the robust controllers based on a

second order SMC, global fast dynamic terminal SMC and PID-dynamic SMC techniques for improving the

tracking performance of the attitude and position control

of quadcopters. In [21]-[25], the robust nonlinear

controllers have been presented by a combination of

backstepping and SMC to attenuate the effect of



disturbances and improve the tracking performance of

the attitude and position control of a quadcopter.

Although the backstepping technique has exhibited a great performance, it still suffers from the so-called

“explosion of term”. In order to solve this problem, the

dynamic sliding surface is suggested in [26]. However

this method is not easy to implement in onboard

computing.

Besides the backtepping technique, a combination of

advanced SMC with fuzzy logic is also developed in

[27]-[29] to guarantee the stability of the vehicle when

unexpected faults have been occurred in actuators, and in

the presence of mismatched uncertainties or external

disturbances. In [29], an adaptive fuzzy PID based

nonsingular fast terminal SMC is introduced to improve

the robustness of attitude control for a spacecraft. An

adaptive law is obtained by a fuzzy logic to estimate

uncertain parameters of the vehicle. Following, a PID

nonsingular fast terminal SMC is applied to compensate

for the effects of perturbations. Although this method can

be achieved a satisfactory control performance of the vehicles, the fuzzy algorithm greatly depends on the

understanding and experiences of a controller designer

about the vehicles, its application, working conditions,

working environments, and type of external disturbances

and uncertainties.

In [30], [31], a robust adaptive controller for a finite-

time stability of the quadcopter attitude is proposed by

using an adaptive law to estimate the unknown

parameters. The algorithm can also be extended to

guarantee a stability of the vehicles in the attendance of

unstructured disturbances. In [32], another robust adaptive

tracking algorithm is proposed for a quadcopter through

the immersion and invariance methodology. In [33], [34],

an adaptive controller and robust controller are developed

to compensate for the control input saturations due to

actuator constrains. These algorithms also greatly

attenuate the effect of disturbances, parametric uncertainties, and delays problem. In [35], [36], other

adaptive controllers are designed by using a model

reference for UAVs to enhance the tracking performance

in the presence of perturbations.

Recently, along with the development of computer

science, the machine leaning technique is being applied to

control the UAVs. In [37]–[41], advanced robust adaptive

controllers are suggested to ensure the stability of attitude

and position of the vehicle based on integrating advanced

SMC, backstepping, and neural network (NN) technique.

The unknown parameters or disturbances are

approximated online by using the neural network.

Obviously, this approach may obtain a better performance

than classical control methods due to online estimation

capability of NN. However, the procedure to achieve

parameters of these controllers is not a simple task.

Especially, it is difficult to determine the proper network structure. Furthermore, in order to use NN in the practical

applications, the powerful hardware platform is required

due to its intensive computation.

Besides previous mentioned methods, another

efficient approach is also presented by an integration of a

disturbance observer with an advanced controller [42]-[45]. Although this method can obtain a satisfactory

performance of the vehicle, the algorithm is more

complicated and cumbersome with respect to single

controllers. Furthermore, the control system performance

almost depends on the accuracy of a disturbance observer.

Consequently, it is necessary to suggest a more effective

robust controller for quadcopter UAVs.

B. MAIN CONTRIBUTIONS Although plenty of controllers have been introduced

for quadcopter in the previous studies, majority of them

are not adaptive enough to control UAVs under

unexpected effect of uncertainties, especially time-varying

disturbances. Several controllers are too complicated to

implement in practical application in both design

procedure and onboard computing. Motivated by these

concerns, we suggested a simple and effective robust

adaptive super-twisting sliding mode control (RASTSMC)

for a quadcopter in the simultaneous presence of time-

varying external disturbances on the attitude (roll, pitch,

and yaw) and altitude dynamics. The contributions of this

study are presented in three issues. First, a general design

procedure of an effective RASTSMC is described through

a second-order nonlinear system, the controller gains can be automatically adjusted to rapidly adapt to the time-

varying disturbances even without knowing the upper

bound of perturbations. The stability of system is achieved

by the Lyapunov theory. Second, the proposed algorithm

not only uses to control UAVs but also applies for many

different engineering systems. Third, the technological

advantages of RASTSMC were demonstrated through a

comparison of simulation result from the new controller

with other previous methods. It can be concisely

summarized as follows: (i) the chattering effect is stronger

elimination with respect to the normal adaptive sliding

mode control (NASMC )[46]; (ii) the tracking

performance of attitude and altitude control of a

quadcopter UAV is more robust and faster convergence to

its desired trajectories regardless of the upper bound of

time-varying perturbations in comparison with NASMC

[46], super-twisting sliding mode control (STSMC) [47], [48], modified super-twisting sliding mode control

(MSTSMC), and nonsingular terminal sliding mode

control (NTSMC) (see [49]-[50]); and (iii) the new

controller is more simple and practical compared to

previous approaches [51]-[53], especially under the

consideration in the real-time implementation on the

embedded system.

The rest of this article is organized as follows. Section

2 concisely presents the mathematical model of the

quadcopter UAVs. The robust adaptive controller design

for quadcopters is described in Section 3. The

effectiveness of the proposed method is illustrated through

the simulation results and discussions provided in Section



4. Finally, the conclusions and future works are given in

Section 5.

II. MATHEMATICAL MODEL OF A QUADCOPTER The mathematical model of a quadcopter UAV is

presented in previous studies [54]-[58]. The essential

frames of the vehicle consist of an Earth frame, {E}, and

a body-fixed frame, {B}, as shown in Figure 1. The

attitude of the vehicle is described by Euler angle. Let

, , , 2 2 , 2 2 ,

represent the roll, pitch and yaw angles,

respectively; , ,x y z are the vehicle‟s position in the

Earth frame. The mathematical model of the vehicle

consisting of disturbances in attitude and altitude

dynamics can be described in Eq.(1) [54].

2

3

4

1

1

1

1

cos cos

cos sin cos sin sin

cos sin sin sin cos

yy zz r

xx xx xx

zz xx r

yy yy yy

xx yy

zz zz

I I J lU

I I II I J l

UI I I

I Iz U

I Ixy g U m

U m

U m

r

r

00

h

t

t

t

t

(1)

where , 1,2,3,4iU i denote the control inputs of

the quadcopter UAVs determined by Eq.(2) [54]:

2 2 2 21 2 3 4

2 212 4

22 2

3 1 3

2 2 2 241 2 3 4

1 2 3 4

bU

bUU bU

d

r

(2)

r is the total residual speed of rotors, and the gravitational acceleration is denoted by g = 9.81m/s

2;

, , , h are time-varying external disturbances

affecting on roll, pitch, yaw, and altitude dynamics of

quadcopter UAVs, respectively. The rest of parameters

are listed in Table 1.

Table I.

System parameters of a quadcopter UAV. Parameters Description

Ixx, Iyy, Izz (kgm2) Moments of inertia along x, y, and z axes in

the Earth frame, respectively.

m (kg) Total mass of a quadcopter UAV

l (m) Arm length of the vehicle frame

b (Ns2) Thrust coefficient

d (Nms2) Drag coefficient

Jr (kgm2) Moment of inertia of a rotor

Let 12 1X be a state vector which is defined as:

1 2 12

1 2 12

, ,..., , , , , , , , , , , ,

, ,..., , , , , , , , , , , ,

TT

TT

X x x x z z x x y y

X x x x z z x x y y

where 1x , 2 1x x , 3x , 4 3x x , 5x ,

6 5x x , 7x z , 8 7x x z , 9x x , 10 9x x x ,

11x y , 12 11x x y .

Eq.(1) can be described in state space as follows:

( ), ( )i iX f X t U t t (3)

where

2

4 6 1 4 2 1 2

4

2 6 3 2 4 2 3

6

2 4 5 3 4

8

1 3 1

10

1

12

1

( ), ( )

cos cos

i

x

y

xx x a x a bU

xx x a x a b U

xx x a b U

f X t U t xg x x U m

xu U m

xu U m

r

r

(4)

and ( ), ( 1,2,...,12)i t i are the external disturbances as

shown in Eq.(5); ,x yu u denote the position control

inputs described by Eq.(6).

0, ,0, ,0, ,0, ,0,0,0,0T

i ht t t t t (5)

1 3 5 1 5

1 3 5 1 5

cos sin cos sin sincos sin sin sin cos

x

y

u x x x x xu x x x x x

(6)

1 yy zz xxa I I I , 2 r xxa J I , 3 zz xx yya I I I ,

4 r yya J I , 5 xx yy zza I I I , 1 xxb l I ,

2 yyb l I ,

3 1 zzb I .

1F4F

3F

2F

Bz

Bx

BBy

l

z

Ey

x

1

3

2

4

Fig. 1. Configuration of a quadcopter, the thrust forces Fi (i=1,…,4)

generated by propellers‟ angular speeds (1,...,4) of four rotors.

III. CONTROLLER DESIGN FOR QUADCOPTERS A. ROBUST ADAPTIVE SUPER-TWISTING SLIDING MODE

CONTROLLER DESIGN

As previous mentioned, the chattering effect often provides a potential damage of the electromechanical

systems. In this Section, the RASTSMC for attitude and

altitude of a quadcopter UAV is designed by using the

super-twisting algorithm, which is well-known for its

excellent anti-chattering. The proposed controller is



described through a general procedure of designing a

controller for a second-order uncertain nonlinear system

model given as follows:

1 2

2 ( ) ( ) ( )f g u t

(7)

where 1 2,T

, 1 and 2 represent the states of a

system dynamics, u represents the controller input,

( )f and ( )g are the smooth functions in term of ,

and ( )t denotes an external disturbance.

The primary objective is to design a robust adaptive

controller that guarantee a good adaptation and rapid

convergence of the state 1 to a desired trajectory 1d .

A tracking error and its first derivative are computed as

1 1de and 1 1

de . The second derivative, e ,

can be obtained by Equation (7) as follows:

1 2

1 ( ) ( ) ( )

d

de

f g u t

(8)

Because ( )t is uncertainties or external

disturbances. Thus, the sign of ( )t is not known in

advanced. As a result, the Eq.(8) can be re-written as

follows:

1 ( ) ( ) ( )de f g u t (9)

Consider a sliding surface in space of error as follows:

( ) ( ) ( )s t e t e t (10)

where is a constant. We can see that once the

value of ( ) 0s t , meaning that the system

( ) ( ) 0 e t e t is asymptotically stable. Thus, the

tracking error, e(t), will converge to zero, lim ( ) 0

t

e t .

The derivative, ( )s t , is computed by Eq.(10):

( ) ( ) ( )s t e t e t (11)

From Eqs.(9) and (11), the derivative of sliding surface,

( )s t , can be described by:

1( ) ( ) ( ) ( ) ( )ds t e t f g u t (12)

The super-twisting sliding mode controller is obtained

by theorem 1.

Theorem 1:

Let us assume that 1 2

: ( ) ( )t s t , the sliding

surface, ( )s t , converges to zero if a super-twisting

sliding mode controller ( )u t is chosen by:

11/2

1 20

( ) ( )1( )

( ) sgn ( ) sgn ( )( )

d

t

e t fu t

k s t s t k s t dtg

(13)

where 1 2,k k are positive values and satisfy:

1

12 1

1

2

5 4

2 2

k

kk k

k

(14)

Proof:


( )s t , can be obtained as follows:

1 2

1 2 0( ) ( ) sgn ( ) sgn ( ) ( )

ts t k s t s t k s t dt t

(15)

Let a new state vector 1 2,T

be defined by:

1

2 2 0sgn ( )

ts

k s t dt

(16)

The derivative of is obtained from Eqs.(15) and (16)

as follows:

1 2

1 1 1 1 2

2 2 1

sgn ( )

sgn

k t

k

(17)

Now, let us refer to the Moreno‟s study of a

Lyapunov approach for the super-twisting algorithm in

the perturbed system [59],[60]. The Lyapunov function

candidate is considered as:

2

1 221 2 1 2 1 1 1 2

1 12 sgn( )

2 2T

V k k

P

(18)

where 21 21 2 1

1 1 21

1 4sgn( ) ,22

T k k kPk

The derivative of 1V is achieved from Eq.(18) as

follows:

1 1 21 2 1 21 1

1 T TV Q Q (19)

where

221 1 11 2 1

1 2 2

1

2, 2

2 2 21

Tk k kk k kQ Q k

k

Using the upper bound condition 1 2

( ) ( )t s t for

the disturbance as given in [59], the derivative, 1V , in

Eq.(19) satisfies the following equation:

1 1 21

1 TV Q (20)

where

2 21 1 2 1 1

1

1

42 2

22 1

kk k k k k

Q k

k

Obviously, 1V is negative definite if 0Q , it means

that the controller gains 1 2,k k must satisfy Eq.(14).

From the super-twisting controller in Eq.(13), it can

be seen that in order to exactly obtain the controller gains

1k and 2k , the value of must be given in advance such

that the Eq.(14) is always satisfied. However, in the

complex working environment, the disturbance is an

arbitrary signal. Thus, the value of is not easy to

determine. Furthermore, the uncertainty or external

disturbance can be a time-varying oscillation magnitude



and frequency, while the controller gains 1k and 2k are

fixed constants. As a result the stability and tracking

performance of the control system may not be guaranteed

during the working process. Thus, it is necessary to

propose a controller which is able to automatically adjust

the controller gains to ensure that the vehicle always

rapidly adapts to any unexpected changes of perturbations.

The proposed robust adaptive controller based on a super-

twisting sliding mode algorithm can be obtained by

theorem 2.

Theorem 2:

Let us assume that 1 2,d d , are constants and

always satisfy the following expression:

1 2

2 10

1sgn ( ) sgn ( ) ( )

2

td ds t s t dt t s t (21)

The sliding surface, ( )s t , converges to a neighborhood

of zero if the robust adaptive super-twisting sliding mode

controller ( )u t is chosen by:

11/2

1 20

( ) ( )1( ) ˆ ˆ( ) sgn ( ) sgn ( )( )

d

t

e t fu t

s t s t s t dtg

(22)

where 1 2ˆ ˆ, are the estimate of 1

d and 2d ,

respectively. The adaptive laws are given as follows:

1 2

1

20

1ˆ sgn

1ˆ sgnt

s s s

s s dt

(23)

where , are constants.

Proof:


( )s t , can be obtained as follows:

1 2

1 2 0

ˆ ˆ( ) ( ) sgn ( ) sgn ( ) ( )t

s t s t s t s t dt t (24)

Let 1 2,T

represents the adaptive errors formulated

by:

1 1 1

2 2 2

ˆ

ˆ

d

d

(25)

Consider a Lyapunov function as follows:

2 22

2 1 2

1 1 1

2 2 2V s (26)

From Eqs.(24), (25) and (26), the derivative, 2V , can be

obtained by:

2 1 1 2 21 2

1 2 0

1 1 1 2 2 2

ˆ ˆ( ) sgn ( ) sgn ( ) ( )

ˆ ˆ ˆ ˆ

t

d d

V ss

s s t s t s t dt t

(27)

From the adaptive law 1 2ˆ ˆ, in Eq.(23), the value of

2V in Eq.(27) can be achieved as follows:

1 2

2 1 2 0

1 2

1 2 0

( ) ( ) sgn ( ) sgn ( ) ( )

( ) sgn ( ) sgn ( )( )

( )sgn ( )

td d

td d

V s t s t s t s t dt t

s t s t s t dts t

t s t

From the condition (21), it can be seen that:

1 2

1 2 02

1 2

1

2

( ) sgn ( ) sgn ( )( )

( )

( ) ( ) 2 ( )

0

td d

d

s t s t s t dtV s t

t

s t s t t

V

Thus, the state 1 converges to the neighborhood of

desired trajectory 1d , and the adaptive errors converge

to the neighborhood of zero, 1 2, 0,0T T

,

meaning that 1d and 2

d are desired controller gains

estimated by 1 2ˆ ˆ, through an adaptive law in Eq.(23).

Moreover, the value of 1ˆ and 2

ˆ are automatically

adjusted through and , regardless of the upper bound of time-varying disturbances.

B. RASTSMC FOR A QUADCOPTER UAV

A quadcopter is strict nonlinear under-actuated

system. Therefore, in order to control this vehicle, the

controller of roll, pitch and yaw and altitude must be

simultaneously considered.

(i) Controller design for roll subsystem.

The dynamic of roll subsystem is re-written from

Eqs.(3), (4), and (5) as follows:

1 2

2 4 6 1 4 2 1 2 ( )x xx x x a x a bU t r

(28)

Let us considers 4 6 1 4 2 1( ) , ( )rf x x x a x a g x b ,

and d be a desired state of roll control. The tracking

error of roll angle is computed by 1de x , and a

sliding surface of controller is chosen as

( ) ( ) ( )s t e t e t . As per theorem 2, the robust

adaptive controller for roll subsystem can be described

as:

1/22

1 20

( ) ( )1ˆ ˆsgn sgn( )

dt

e t f xU

s s s dtg x

(29)

where the adaptive law 1 2ˆ ˆ, are obtained by:

1 2

1

20

1ˆ sgn

1ˆ sgnt

s s s

s s dt

(30)

and , , are positive constant values.

(ii) Controller design for pitch subsystem.



The dynamic of pitch subsystem is re-written from


3 4

4 2 6 3 2 4 2 3 ( )x xx x x a x a b U t r

(31)

Let us considers 2 6 3 2 4 2( ) , ( )f x x x a x a g x b r ,

and d be a desired state of pitch control. The tracking

error of pitch angle is computed by 3de x , and a

sliding surface of controller is given by

( ) ( ) ( )s t e t e t . The pitch controller is achieved

from theorem 2 as follows:

1/23

1 20

( ) ( )1ˆ ˆsgn sgn( )

dt

e t f xU

s s s dtg x

(32)

where the adaptive law 1 2ˆ ˆ, are given by:

1 2

1

20

1ˆ sgn

1ˆ sgnt

s s s

s s dt

(33)


Fee

db

ack

RASTSMCAltitude control

PID

x-position

PID

y-position

Conversion blockTo convert controller u

to attitude set-point

RASTSMCRoll control

RASTSMCPitch control

RASTSMCYaw control

Dynamics of Quadcopter UAVs

,

,

,

,,,

h hx xy y

Exte

rnal

D

istu

rban

ces

h

dhdx dy

xuyu

d

1U2U 3U 4U

d d

Fig. 2. Control structure of the quadcopter UAVs, the proposed

controller is used for attitude and altitude.

(iii) Controller design for yaw subsystem.

The dynamic of yaw subsystem is re-written from


5 6

6 2 4 5 3 4 ( )x xx x x a b U t

(34)

Let us considers 2 4 5 3( ) , ( )f x x x a g x b , and d

be a desired state of yaw control. The tracking error and

sliding surface of yaw controller are given by

5de x , and ( ) ( ) ( )s t e t e t . The

controller for yaw subsystem is achieved from theorem 2

as follows:

1/24

1 20

( ) ( )1ˆ ˆsgn sgn( )

dt

e t f xU

s s s dtg x

(35)

where the adaptive law 1 2ˆ ˆ, are chosen by:

1 2

1

20

1ˆ sgn

1ˆ sgnt

s s s

s s dt

(36)


(iv) Controller design of altitude subsystem.

The dynamic of altitude subsystem is obtained from


7 8

8 1 3 1cos cos ( )h

x xx g x x U m t

(37)

Let us considers

1 3( ) , ( ) 1 cos cosh hf x g g x m x x , and dh be a

desired state of altitude control. The tracking error of

altitude and the sliding surface of controller are given by

7h de h x , and ( ) ( ) ( )h h h hs t e t e t . The controller

for altitude dynamics is achieved from theorem 2 as

follows:

1/21

1 20

( ) ( )1ˆ ˆsgn sgn( )

h h d ht

h h h h hh

e t h f xU

s s s dtg x

(38)

where the adaptive law 1 2ˆ ˆ,h h are given by:

1 2

1

20

1ˆ sgn

1ˆ sgn

h h h h

ht

h h h

h

s s s

s s dt

(39)

and , ,h h h are positive constant values.

IV. SIMULATION RESULTS AND DISCUSSIONS

The numerical simulation is executed to

demonstrate the efficiency of the proposed RASTSMC

method presented in this Section. In order to highlight

the contributions and advantages of the proposed controller, we also simulated four other popular control

methods such as NASMC [46], STSMC [47], [48],

MSTSMC and NTSMC (see [49], [50]), with the same

flight conditions.

A. SIMULATION ASSUMPTIONS

The numerical simulation is carried out through

several assumptions: (i) the primary parameters of a quadcopter UAV, initial conditions, controller gains, and

desired trajectories are given in Table 2 and 3; (ii) The

external disturbances are the time-varying oscillation

amplitude and constant frequency. These perturbations

simultaneously affect on the dynamics of attitude and

altitude as following the situation (Figure 3).



Attitude: when 0,6 10,14t s , there are no

disturbances present, i.e. 0 . When

6,10t s , the vehicle is influenced by the time-

varying external disturbances described in

Eq.(40).

0.5

0.5

1.5 0.2 cos 2

4 0.08 cos 2.5

t

t

t t e t

t e t

(40)

Altitude: when 0,7t s , the quadcopter also

works in the absence of perturbations, i.e. 0h .

When 7t s , the vehicle is influenced by

external disturbances described in Eq.(41):

15 5cos 5h t t (41)

(iii) the desired trajectories of roll, pitch, yaw and

altitude are given by Eqs.(42).

0.8

0.8

( ) ( ) 10 180 cos 1.5

( ) 8 180 cos 2( ) 5

td d

td

d

t t e t

t e th t

(42)

Table II.

Parameters of quadcopter UAVs and initial values

Parameters Value Unit

m 1.12 kg

Ix 0.0119 kg.m2

Iy 0.0119

kg.m2

Iz 0.0223 kg.m2

b 7.73213 (10-6

) Ns2

d 1.27513 (10-7

) Nms2

Jr 8.5 (10-4

) kg.m2

l 0.23 m

(0, 0 , 0) (0,0,0) degree

h0 0 m

Table III.

Controller parameters of the proposed RASTSMC

Parameters Roll () Pitch () Yaw () Altitude (h)

0.0032 0.0032 0.001 0.8

0.0002 0.0002 0.0001 0.02

7.5 7.5 3.5 2.0

A. SIMULATION RESULTS

The simulation results of the proposed RASTSMC

and four other approaches in the same flight conditions

are presented in this Section. The results are shown from

Figures 4 to 7. Obviously, it can be seen that during the

first flight stage of the vehicle ( 0,6t s for attitude,

0,7t s for altitude), in the absence of perturbations,

the performance of attitude control (i.e. roll, pitch and

yaw angles) and altitude control of the five controllers

(i.e. MSTSMC, NTSMC, STSMC, NASMC and the

proposed RASTSMC) are almost identical in both time

response and tracking performance. The output

feedbacks of ( )t , ( )t , and ( )t rapidly converge to

the desired trajectories without any oscillations and

steady state errors as shown in Figure 4.

However, we can see that the performance of all

five controllers is no longer similar in the presence of

external disturbances stage ( 6,10t s for attitude and

7t s for altitude). Once the external disturbances, as

given in equations (41) and (42), simultaneously

influence on the attitude and altitude dynamics of a

quadcopter UAV, the proposed RASTSMC algorithm

exhibited a significantly superior performance compared

to other methods in both tracking performance and

chattering elimination. It can be clearly discussed as

follows:

Fig. 3. External disturbances simultaneously influence on the attitude

and altitude dynamics of a quadcopter UAV.

For attitude control performance: After a short time

from the first effect of external disturbances on the

vehicle ( 6 8.5s t s ), the performance of roll, pitch

and yaw angles of MSTSMC, STSMC, NTSMC and

NASMC exhibited the small oscillations (Figure 4a).

However, once the amplitude of time-varying

disturbances is larger ( 8.5 10s t s ) (as Figure 3),

these controllers cannot control the convergence of

attitude response to a desired trajectory, resulting in the

large oscillations of roll, pitch, and yaw angles (Figures

4a,b,c). Furthermore, a larger chattering effect is

generated by NASMC (Figure 5b,c,d). Conversely, the

proposed RASTSMC exhibited a much better

performance than the other approaches in both tracking

control and chattering elimination. We can see that when

disturbances are absent (t<6s), the controller gains 1ˆ

and 2ˆ of the proposed algorithm are automatically

adjusted until achieving the most suitable constant values ensuring the convergence of attitude response to a

desired trajectory (Figures 6a,b,c). Right after the vehicle

is influenced by the time-varying external disturbances

( 6 10s t s ), the adaptive gains 1ˆ and 2

ˆ of the

attitude controller are also immediately adjusted (Figures

6a,b,c) for guaranteeing a fast adaptation to the

unexpected changes of these disturbances. Moreover,

these adaptive gains ( 1ˆ and 2

ˆ ) are always sufficiently

adjusted until obtaining the most suitable constants



without increasing continuously. Therefore, the output

feedbacks of roll, pitch, and yaw angles rapidly converge

to a desired state as shown in Figures 4a,b,c, and the

chattering effect is minimized as shown in Figures

5b,c,d.

Fig. 4. Comparison of attitude and altitude control performance between the proposed RASTSMC with previous methods.

Fig. 5. Comparison of control signal U1, U2, U3, U4 between the proposed RASTSMC with previous methods.



Fig. 6. Adaptive law of the proposed RASTSMC

Fig. 7. Comparison of sliding surface between the proposed RASTSMC with previous methods.

When the effect of external disturbances on the

vehicle is suddenly dropped off to zero ( 10t s ) (Figure

3), the proposed RASTSMC algorithm is also exhibited

a superior performance, the output response of attitude

rapidly converges to its reference while the other

methods (MSTSMC, STSMC, NTSMC) take a longer

for returning to a steady state (Figures 4a, b, c).

For altitude control performance: Before any effect

of disturbances ( 7t s ), the response of altitude in all

five controllers is excellent in both time response and



tracking performance as shown in Figure 4d. However,

once the vehicle is influenced by external disturbances

( 7t s ), the proposed RASTSMC presented a fast

convergence of the output feedback to a desired

trajectory (Figure 4d), and strong elimination of the

chattering effect without using any anti-chattering

technique because the adaptive controller gains 1ˆ

h and

2ˆ

h are automatically adjusted until achieving the most

suitable constant values (Figure 6d) to compensate for

the effect of external disturbance. Conversely, the

performance of MSTSMC, NTSMC, and STSMC

exhibited the large oscillations and great delay before

returning to a steady state while the NASMC presented a

small oscillation but the output feedback never

converges to a reference for long times and exists a

steady state error (Figure 4d). Furthermore, the NASMC

also provides a very large chattering phenomenon

(Figure 5a). In Figure 7, the performance of sliding

surfaces in all five controllers is also presented. The

sliding surface of the proposed RASTSMC rapidly

converges to zero with respect to the other methods.

In summary, through the numerical simulation

results of the five considered controllers, although their

output feedbacks are almost equivalent in the absence of

disturbances (i.e., they presented both a fast response

and good tracking performance), the greatly different

performance appears once the vehicle is influenced by

time-varying disturbances. The previous approaches

such as MSTSMC, NTSMC, STSMC and NASMC did

not exhibit an excellent control performance because

these controllers cannot adapt to great unexpected

changes of perturbation. Thus, relatively large

oscillations are occurred in the performance of attitude

and altitude control. Conversely, the proposed controller

exhibited a superior performance: rapid adaptation, fast

response and excellent convergence of the output

feedbacks to its references with strong elimination of the

chattering phenomenon.

V. CONCLUSIONS In this study, we proposed a simple and effective

robust adaptive super-twisting sliding mode control

algorithm for greatly improving the tracking

performance of attitude and altitude control of the

quadcopter UAVs. This adaptive controller presents a

meaningful advancement of the existing state-of-the-art.

Moreover, the scope of working conditions and working

environments of the vehicle can be extended by using

this algorithm because its controller gains can be

automatically adjusted to guarantee a rapid adaptation to

any unexpected changes of environment even without

knowing the upper bound of perturbations. The

numerical simulations were executed and compared to the previous approaches. A superior stability and

excellent tracking performance of the attitude and

altitude control of a quadcopter UAV in the simulation

results demonstrated the robustness and effectiveness of

the proposed controller. However, the perturbation from

faults of actuator is not covered in this study. Thus, this

subject will be studied in the future.

VI. ACKNOWLEDGEMENT

This research was supported by Korea Institute for

Advancement of Technology (KIAT) grant funded by

the Korea Government (MOTIE) (N0002431, The

Competency Development Program for Industry

Specialist)

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