Adaptive Super-twisting Second-order Sliding Mode
for Attitude Control of Quadcopter UAVs
1

Van Truong Hoang1, Quang Hieu Pham2
University of Technology Sydney Broadway, New South Wales, Australia
2
Naval Academy, Nha Trang, Khanh Hoa, Vietnam
E-mail: vanTruong.hoang@uts.edu.au, hieu.phamquang@gmail.com

Abstract
This work addresses the modelling and control aspects
for quadcopter or drone unmanned aerial vehicles
(UAVs). First, the mathematical model of the drone is
derived by identifying significant parameters and the
negligible ones are treated as disturbances. The control
design begins with the switching surface selection,
then, an Adaptive Super Twisting Sliding Mode
(ASTSM) Control algorithm is applied to adjust
attitudes of the quadcopter under harsh conditions such
as nonlinear, strong coupling, high uncertainties and
disturbances. Simulation results show that the
proposed controller can achieve robust operation with
disturbance rejection, parametric variation adaptation
as well as chattering attenuation. Comparisons with
some commonly used and advanced controllers in a
quadcopter model show advantages of the proposed
control scheme.

Keywords
Quadcopter, drone, Second-order, Adaptive control,
Super-twisting, Sliding mode control, Second-order
sliding.

1. Introduction
Quadcopters, known as quadrotors or drones, belong to
a particular type of Vertical Take-Off and Landing
aircraft with four directed rotors upward. The electric
motors and their corresponding propellers are usually
placed in a square formation with an equal distance to
the centre of mass. Quadrotors are controlled by
adjusting angular velocities of the propellers. They
have been used in numerous real-world applications,
such as surveillance, search and rescue operations,
infrastructure inspections [1], emergencies management and product home delivery [2].
In research, the quadcopter is an exemplary design for
small unmanned aerial vehicles with six degrees of
freedom but only four independent inputs, thus, make
it critically underactuated. To gain the six degrees of
freedom, rotational and translational motions are
coupled. As a result, dynamics of this flying object are
highly nonlinear, particularly under the effect of the
aerodynamics. Besides that, quadrotor has microscopic
friction to prevent its movement, so it must yield its
own damping to block the move and remain in a steady
state. As a consequence, the design of controllers for
the quadcopter becomes extreme problematic tasks.
A vast volume of controllers has been developed for
quadrotors in literature, such as PID [3], H∞ [4], optimal

[5], SMC [7], [8] and potential field [6]. Among them,
SMC has been widely used because of its capability to
robustly control systems under uncertainties and
disturbances. Even though, chattering phenomenon
remains as a significant disadvantage of the method. To
eliminate chattering, high-order sliding modes (HOSM)
[10], [11], [22] have been offered as a most likely
preferable solution [12].
HOSM is a higher-order derivative of the conventional sliding mode for sliding function [10]. This
creates an attraction for researchers to continuously
develop related mathematical problems, accompanied
by brilliant solutions, i.e., [12], [13]. HOSM is capable
of removing the condition to have the relative degree
to be equivalent to one for the tradition SMC and
reduce the chattering effect. Another advantage of
HOSM is in the construction of an accurate, robust
differentiator with finite time convergence [14] or
fixed-time convergence [15].
The second order sliding mode (SOSM) controllers,
i.e., twisting, super twisting and accelerated twisting
[9], [15], [16], [21], quasi-continuous [17], sub-optimal
[18], and drift algorithm [19] have been extensively
developed during the last two decades. The main idea
of SOSM is not only to drive the sliding surface but
also its derivatives to zero. Among them, supertwisting sliding mode (STSM) is a unique continuous
sliding mode algorithm, which ensures all essential
properties of the first-order SMC together with
chattering rejection. However, the performance of
STSM depends on the knowledge of the bound of
perturbations. In practical scenarios, the drones are
affected by disturbances, uncertainties, modelling
errors and parameter variations that may downgrade
the control efficiency, but their boundaries are not
obvious. To address this concern, STSM controller
with an adaptive gain has been applied to drive the
switching variable and its derivative to zero in the
presence of both additive and multiplicative
disturbances [16].
In this paper, the ASTSM algorithm is proposed to
control the attitude of quadcopters, which subject to
nonlinear dynamics, strong coupling, high uncertainties and disturbances. The mathematical model of the
drone is derived by adopting possible vital variables
while some others are considered to be uncertainties.
The controller mentioned above is proposed to achieve
the robustness while rejecting disturbances and
parametric variations as well as decreasing affection of
the chattering phenomenon. This control performance

is demonstrated by extensive simulation and
comparison with the conventional ProportionalIntegral-Derivative (PID), the classical first-order
Sliding Mode (SMC) and the second-order Accelerated
Twisting Sliding Mode (ATSM) controllers to show its
advantageous feasibility.
The paper is organized as follow. The nonlinear
dynamic model of quadrotor is presented in Section 2
followed by an introduction of second-order super
twisting sliding mode with an adaptive scheme in
Section 3. Section 4 describes simulation results and
comparison strategies. The paper ends with a
conclusion and recommendation for the future study.

2. Dynamic Model
2.1 Kinematics
The quadcopter model is shown in the Fig.1. Its
dynamics is set up by two coordinate systems, namely
earth frame (inertial frame) and the body fixed frame
(body frame). The inertial frame ( xE , yE , zE ) is defined
by the ground, with zE pointing down to the earth
centre. The body frame ( xB , yB , zB ) is specified by the
orientation of the quadcopter, with the rotor axes
pointing downward and the arms pointing in xB and yB
directions.

1

ω =  0
0


0
cφ
− sφ

− sθ 
cθ sφ 
cθ cφ 

(1)

where sx = sin( x ) and cx = cos( x ) .
The below transformation matrix defines the relation
between the body frame to earth frame translational
velocities:
cψ cθ cψ sθ cφ − sψ cφ cψ sθ cφ + sψ sφ 


R =  sψ cθ sψ sθ sφ + cψ cφ sψ sθ cφ − cψ sφ 
(2)
 s

c
s
c
c
ψ θ
θ φ
 θ

2.2 Quadcopter Dynamics
Since the objective of this study is the attitude control,
only torque elements that are capable of varying the
quadcopter orientation are taken into account. They
include torques caused by thrust forces τ , body
gyroscopic effects τb , propeller gyroscopic effects τ p ,
and aerodynamic friction τ a .
The torque τ is produced by the quadcopter in the
body frame including roll, pitch and yaw components,
T
i.e., τ =  τφ ,τθ ,τψ  . They are performed as:
τφ = l( F2 − F4 )
(3)

τθ = l( −F1 + F3 )
τ ψ = c( −F1 + F2 − F3 + F4 )

(4)
(5)

where Fk , k = 1…4, is the thrust force generated by the
propeller k, l is the distance from a motor to the center
of mass and c is a force-to-torque scaling coefficient.
The body gyroscopic torque is modelled as:
τb = S( ω )I ω
(6)
where S( ω ) is a skew-symmetric matrix for the given
vector ω , and is expressed as follows:

Fig. 1 A schematic diagram of quadcopter

The equations representing the motion of the
quadcopter are basically those of a rotating rigid body
with six degrees of freedom, i.e., three translational and
three rotational motions. The translational movements
are defined in the earth frame, where the position is
presented in vector form as ξ =(x, y, z)T and the vector
ɺ (x,y,z)
ɺ ɺ ɺ T denotes its linear velocity. The drone
ξ=
attitude is defined by using the three Euler angles,
named roll, pitch, and yaw are determined in the body
frame as Θ=(ϕ,θ, ψ)T , their corresponding angular rates

ɺ = (ϕɺ ,θɺ ,ψɺ )T .
are performed as Θ
T

Let ω = ( p, q, r ) represents the angular rate vector in
the inertial frame. Then, the following rotational
kinematics is achieved to show the relation between the
earth angular velocity and the Euler angle rate vectors:

 0 −r

0
S( ω ) =  r

−q p


q 

− p

0 

(7)

The resultant of torques generated by propeller
gyroscopic effects τ p is determined as:
 I rΩr q 


τ p = − I r Ω r p 




0



where I r is

the

(8)
inertial

moment

of

rotor,

Ωr = −Ω1 + Ω2 −Ω3 + Ω4 is the residual angular
velocity of rotor in which Ωk denotes the angular
velocity of the propeller k. The aerodynamic friction
torque τa is given by:
τ a = ka ω 2

(9)

where ka is a positive definite matrix of aerodynamic
friction coefficients, ka = diag  kax ,kay ,kaz  .
Using the aforementioned torques, the overall attitude
dynamic model of the quadcopter is derived as:
ɺɺ = τ + τ + τ − τ
IΘ
(10)
b
p
a

where I is a diagonal positive definite matrix of
inertia tensors when the quadrotor is assumed to
be symmetrical, I = diag  I xx ,I yy ,I zz  .
In our study, the gyroscopic and aerodynamic torques
are considered as external disturbances, and they are
supposed to be removed by the advancement of the
proposed controller. Therefore, the control inputs
mainly depend on torque τ and from (3), (4) and (5),
they can be represented as:
uφ   τ φ   0
    
 u θ   τ θ   −l
 =  = 
 u   τ  − c
 ψ  ψ 
u   F   1
 z    

l

0

0

l

c

−c

1

1

− l   F1 
 
0   F2 
 
c   F3 
1   F4 

(11)

pitch and yaw torques, uZ represents the total thrust
acting on the four rotors and F denotes the UAV lift
k

produced by the four propellers, F = ∑ Fk . In this
i =1

paper, uZ is supposed to accommodate with the gravity
when we consider the rotational control only. In view
of the equations from (3) to (7), the second-order
nonlinear dynamics of quadcopters for attitude control
can be described by the following equations:
I yy − I zz
1
1
φɺɺ =
qr +
uφ +
dφ
I xx
I xx
I xx

(12)

I − I xx
1
1
θɺɺ = zz
pr +
uθ +
dθ
I yy
I yy
I yy

(13)

I xx − I yy
1
1
ψɺɺ =
pq +
uψ +
dψ
I zz
I zz
I zz

(14)

where dϕ ,dθ and dψ are the disturbances, including the
two terms τ a in (8), τ p in (9) to the system in real time.
Let us define the following state variables:
X1 = Θ
ɺ
X =Θ

(15)

2

Then, the dynamics of quadcopters can be represented
in the following form as:
 Xɺ = X
2
 1
(16)
ɺ
 X 2 = I −1 [ f ( X ) + u( t ) + d( t )]

T
where u = uφ ,uθ ,uψ  is the input vector, d is the
T

vector, d =  dφ ,dθ ,dψ  ,

3. Control Design
The control signals uφ ,uθ and uψ in (16) are used to
T

control the Euler angle Θ = [ϕ, θ, ψ ] to reach the
T

where uφ ,uθ and uψ respectively represent the roll,

disturbance

A.4 The velocity and the acceleration of the
quadcopter are bounded.
 π π
A.5. The orientation angles are limited to φ ∈ − ,  ,
 2 2 
 π π
θ ∈ − ,  and ψ ∈ [−π ,π ]
 2 2 
A.6 The rotational speeds of rotors are bounded.

and f ( X ) is

represented as:

 ( I yy − I zz )qr 


f ( X ) = −S( ω )I ω =  ( I zz − I xx ) pr 
(17)
( I − I )pq 
yy
 xx

In our system, the following assumptions are made:
A.1 The quadcopter structure is rigid and symmetric.
The propellers are rigid.
ɺ can be measured by onA.2 The signals Θ and Θ
board sensors.
A.3 The reference trajectories and their first and
second time derivatives are bounded.

reference value Θd = (ϕd ,θd , ψd ) .
The overall control law is presented as:
u( t ) = ueq ( t ) + uD ( t )

(18)

where ueq ( t ) is a continuous part defined by the
controlled variables and reference values, uD ( t ) is the
discontinuous part that contains a switching element.
3.1. Sliding Manifold
The sliding surface equation determines the dynamics
of the system, so it is presented as:
(19)
σ = eɺ + Λe
where Λ = diag( λϕ ,λθ ,λψ ) is a positive definite
matrix to be designed, and e is the control error:
e = X1 − X1d
where X1d is the vector of desired angles. Thus, the first
derivative of the error vector will be:
eɺ = Xɺ 1 − Xɺ 1d
3.2. Design ueq
The equation (19) can be rewritten for the quadcopter
attitude sliding surface as:
σ = ( Xɺ 1 − Xɺ 1d ) + Λ( X 1 − X1d )
(20)
Taking time derivative of σ we have:
σɺ = ( Xɺɺ1 − Xɺɺ1d ) + Λ( Xɺ 1 − Xɺ 1d )
(21)
or
σɺ = − Xɺɺ1d + Xɺ 2 + Λeɺ
(22)
When the system is in its nominal condition, i.e., d( t ) = 0
, we can substitute Xɺɺ from (16) into (22), which yields:
σɺ = − Xɺɺ + I −1 [ f ( X ) + u + d ] + Λeɺ
(23)
1d

During the time the system is in sliding phase, u can be
considered as the equivalent control ueq . By driving the
derivative of sliding surface to zero, we found the
control rule for the continuous part:
ueq = I( Xɺɺ1d −Λeɺ ) − f ( X )
(24)
3.3. Design uD and problem formulation
The second-order super twisting sliding mode controller is
given in [10], uD is redefined as follows:

u = −α σ sign( σ ) + ν
 D

(25)
νɺ = − β sign( σ )

2
Where α and β are definite positive diagonal matrices
of corresponding gains to be adjusted.
Thus, we have the accomplished control equation for
the quadrotor attitude:
u = I( Xɺɺ1d −Λeɺ ) − f ( X ) + uD
(26)
with f ( X ) = −S( ω )I ω , we can represent (26) in the
following form:
u = I( Xɺɺ − Λeɺ ) + S( ω )I ω + u
(27)
1d

D

The quadcopter uncertainties are subjected to variations, modelling errors, as well as some disturbances
such as aerodynamic frictions, propeller gyroscopic
effects and environmental affections, particularly wind
gusts while flying outdoor. Let I = I 0 + ∆I where I0
and ∆I represent the known nominal and unknown
uncertain parts of the inertia matrix, respectively. The
term Xɺ 2 in (16) becomes:
Xɺ 2 = ( I 0 + ∆I )−1 S( ω )I 0 ω + ( I 0 + ∆I )−1 u
(28)
+ ( I 0 + ∆I )−1 d + ( I 0 + ∆I )−1 S( ω )∆I ω
We have
( I 0 + ∆I )−1 = I 0−1 ( 1 + I 0−1∆I )−1
(29)
= AI 0−1 + B( 1 + I 0−1∆I )−1
With A and B are diagonal matrices of constants found
by breakdown analysis (29).
The sliding surface (23) will be derived as:
σɺ =−Xɺɺ1d +Λeɺ +(I0 +∆I )−1 S( ω )I0ω +( I0 +∆I )−1u
(30)
+( I0 +∆I )−1d( I0 +∆I )−1 S( ω )∆Iω
Substitute (29) to (30), we have:
σɺ =−Xɺɺ1d +Λeɺ + AS( ω )I0ω +( I0 +∆I )−1 S( ω )∆I ω
+ B(1+ I0−1∆I )−1 S( ω )I0ω +( I0 +∆I )−1 d
−1
0

−1
0

(31)

−1

+ [ AI + B(1+ I ∆I ) ]u
Let we rewrite σɺ in the following form:
σɺ = a( x,t ) + b( x,t )u

(32)

3

Where the function a( x,t ) ∈ ℝ is presented as:

a( x,t ) = a1 ( x,t ) + a2 ( x,t ) ,
a ( x,t ) = − Xɺɺ + Λeɺ + AS( ω )I ω
1

1d

0

a2 ( x,t ) = ( I 0 + ∆I )−1 S( ω )∆I ω
+ B( 1 + I 0−1∆I )−1 S( ω )I 0 ω + ( I 0 + ∆I )−1 d
We assume that a1 ( x,t ) and a2 ( x,t ) bounded, but their
boundaries are not yet clarified. Also, the function
b( x,t ) ∈ ℝ3 is uncertain and represented as:

b( x,t ) = b0 ( x,t ) +∆b( x,t ),
where b0 ( x,t ) = AI0−1 and ∆b( x,t ) = B(1+I0−1∆I )−1 are a
known function and a bounded perturbation,
respectively. An assumption for this case is:
∆b( x,t ) / b( x,t ) < γ( x,t ) < γ1 < 1,

with γ1 is an unknown boundary. Thus, it can be seen
clearly that the input-output dynamics (32) contains of
both additive and multiplicative perturbations.
The STSM controller (25) can robustly handle the given
problems while their boundaries are known. However, the
bound is not easy to evaluate in practice and besides, a
high value of sliding gain α and/or β will lead to high
chattering magnitude. Therefore, the problem is now to
drive the sliding variable σ and its derivative σɺ to zero
in finite time by means of super-twisting SMC without
exaggeration of the control gains.
3.4. Adaptive STSM Control Design
The adaptive gains for (25) is defined as:
α = α( σ ,σɺ ,t )

(33a)
β = β( σ ,σɺ ,t )
(33b)
The ASTSM control gains [16] are proposed to
decrease the chattering phenomenon and converge σ
and σɺ to zero in the presence of disturbances and
uncertainties. The gains are chosen as:

ϖ γ1 sign ( σ − µ) if α > α
m
(34)
αɺ = 
2

η
if α ≤ αm
(35)
β = 2εα
where ϖ > 0, γ1,µ ,η and ε are arbitrary positive scalars,

αm is the threshold of the adaptation. The significant
property of the adaptive scheme is non-exaggerating
the values of the gains to be adjusted. The global
Lyapunov function candidate is defined as follows:
1
1
V = V0 ( σ ,σɺ ) + ( α − α* )αɺ + ( β − β* )βɺ (36)
γ1
γ2
Where V0( σ,σɺ ) is a Lyapunov function for ( σ ,σɺ ), γ1
and γ2 are arbitrary positive numbers, α* and β * are
maximum possible values of α and β .
The derivative of the Lyapunov function (36) is
obtained as:
Vɺ ( σ ,α ,β ) ≤ −η0 V( σ ,α , β ) + ξ
(37)
where η0 is a positive constant, V( σ ,α ,β ) ≥ 0 is a
function of σ ,α and β , and
1
1
ω 
ω 


ξ = − εα  αɺ − 1  − εβ  βɺ − 2  (38)
 γ1
 γ 2
2γ1 
2γ 2 
where εα = α − α* ,εβ = β − β* , ω1 and ω2 are some
arbitrary positive constants.
It can be seen that the finite time convergence of the
system is guaranteed given (34) and (35) [8].

4. Simulation Results
This section presents extensive simulations to
demonstrate the performance of the ASTSM controller.
The quadcopter model is the 3DR Solo drone shown in
Fig. 2. Technical specifications and accessories of the
quadcopter are described in [20]. The physical

parameters L( ⋅ ),d( ⋅ ),r( ⋅ ) and h( ⋅ ) in the figure are
measured distances, which are used to calculate model
properties, as listed in Table I. The parameters of the
proposed controller, are shown in Table 2.
Numerical simulation results have been done in three
different conditions, i.e., responses of the system in
nominal conditions, under the appearance of
disturbances and parametric variations. The initial
conditions of the quadrotor are assumed to be in its
steady state, where all control angles and angular
velocities are zeros. The desired angles are changed in
the simulation as φ = −100 , θ = 100 and ψ = 450 at
time 0.5s, 1s and 2s, respectively.

(a)

(b)

Fig. 2 The 3DR Solo drone with body coordinate frame.

In disturbance scenario, a torque of 0.5N.m is separately added in each axis of the drone. Particularly, to
demonstrate the performance of the ASTSM controller
in dynamic variation conditions, simulation parameters
are varied to counteract some modelling errors, the
most capable payload 0.8 kg of the 3DR Solo, is added
to the model and the inertial matrix is varied with the
uncertainties as in Table 3.
Table 1. Parameters of the quadcopter model

Parameter
m
l
g
I xx

Value
1.50
0.205
9.81
8.85.10-3

Unit
kg
m
m/s2
kg.m2

I yy

1.55.10-3

kg.m2

I zz

23.09.10-3

kg.m2

(c)
Fig. 3 Responses of angles and their angular velocities in nominal
condition

Table 2. Control design parameters

Variable
λϕ

Value
3.89

Variable

λθ

3.89

λψ
η

4.36

γ1
ε

0.60

0.01

αm

0.01

ϖ

Value
200
6.60

Table 3. Uncertainties added to the inertia matrix

∆I
x
y
z

x

y

z

0.4825
0.0044
-0.0077

0.0044
0.2437
0.0115

-0.0077
0.0115
0.2437

Fig. 4 Yaw angle and angular velocity responses in the presence of
external disturbances

Results of simulation are additionally compared with
the ATSM [15], the conventional SMC, and the PID
controller that is practically implemented to the Solo
drone. Results of tracking behaviour in nominal
situations are shown in the Fig.3. The outputs of the
controllers shown in Fig.5, where the time scale is

zoomed in to observe the gain response to adapt to
various changes of the system, such as references and
coupling effects. The adaptive gain of the ASTSM
controller for yaw ( α3 ( t )) responses is shown in Fig.7.
The system response of the distinctive controllers
under disturbances are presented in Fig.4. Fig.6 shows
the simulation results in parametric variation in
comparison with the nominal conditions.

(b)
Fig. 6 Pitch (a) and yaw (b) angle and angular velocity responses
in the presence of parametric variations

(a)

Fig. 7 The adaptation of gain α3 ( t ) in three scenarios: Top:
Nominal condition; Middle: Occurrence of disturbances;
Bottom: Parametric variations.

(b)
Fig. 5 Roll control inputs ( u1 ) in: (a) Nominal condition; and
(b) Occurrence of disturbances

Fig.3 illustrates that the controllers smoothly drives the
angles to the required values with comparatively
unremarkable overshoot within two seconds for all
cases mentioned above. Robustness of the three sliding
controllers under the presence of disturbances is
presented in Fig.4. However, by the advantage of its
adaptation, the designed controller performs its faster
convergence.

(a)

The leading cause of chattering effect is modelling
errors in conjunction with high gain selected to
preserve the robustness of the system under
perturbations [12]. It results in high chattering
amplitude and frequency at the origin, as shown in case
of ATSM and SMC control inputs in Fig.5. By
comparing control inputs in the roll attitude, chattering
effect is attenuated greater by ASTSM when the
adaptive gain is adjusted to its threshold. Furthermore,
the smaller magnitude of the control inputs compared
to ATSM, SMC and PID illustrate that less energy is
required by the designed control scheme.
There exist strong coupling relations among the control
variables as pointed out in Eqs. (12-14) and the
response of PID controller in Fig.6a shows clearly this
phenomenon. However, the improvement of the
proposed controller is able to solve this issue by
managing the Euler angles to reach their desired values
and then track them with no relative perturbation.
Responses of the system with parametric variations in
Fig.6 show that ASTSM is capable of preserving its
robust properties better than ATSM, SMC and PID in
particular. It can be seen that the three sliding
controllers reach the reference value without causing
much overshoot or oscillation. However, the faster
responses of ASTSM in the two above cases, again,
illustrates its advancement.
Time histories of α3 ( t ) in Fig.7 show how the adaptive
gain adjusted to response with the sudden change of the

desired signals, coupling effect, disturbances (Fig.5a)
and variation (Fig.5b). The higher gain magnitudes are
observed in the two bottom subfigures imply more
energy is required to stabilise the system in dealing
with disturbances and uncertainties. This also suggests
the feasibility of the control scheme.

5. Conclusion and Future work
The paper deals with the design of the adaptive second
order sliding mode controller for a practical quadrotor.
It begins with an introduction of refinements of
nonlinear dynamic equations for the drone. An
adaptive super-twisting sliding mode controller has
been implemented to ensure robustness with respect to
unknown bounded disturbances and uncertainties. The
control design is based on the selection of a sliding
surface and some parameters for adaptation of the
control gain taking account into chattering reduction.
Control performance is evaluated in simulation for the
cases of both external disturbances and system
uncertainties. The validity of the proposed control
scheme is also judged through comparison with the
accelerated twisting sliding mode, the conventional
sliding mode and PID controllers. Future plan will
focus on implementing the proposed controller to
demonstrate the feasibility of the proposed approaches.

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