Finite-Time Altitude and Attitude Tracking of a Tri-Rotor UAV using

Modiﬁed Super-Twisting Second Order Sliding Mode

Yassine Kali

, Jorge Rodas

, Maarouf Saad

, Khalid Benjelloun

, Magno Ayala

and Raul Gregor

Ecole de Technologie Sup

erieure, Quebec University, Montreal, QC H3C 1K3, Canada

Laboratory of Power and Control Systems, Facultad de Ingenier

ıa, Universidad Nacional de Asunci

on, Paraguay

A2I Laboratory, Ecole Mohammadia d’Ing

enieurs, Mohammed V University, Rabat, Morocco

Keywords:

Altitude Tracking, Attitude Tracking, Finite-time Convergence, Lyapunov, Unmanned Aerial Vehicle, Second

Order Sliding Mode, Super-twisting Algorithm, Uncertainties.

Abstract:

This paper presents the problem of robust altitude and attitude trajectory tracking of a tri-rotor Unmanned Ae-

rial Vehicle (UAV) based on a ﬁnite-time second order sliding mode control algorithm. The chosen algorithm

is a modiﬁed super-twisting control with double closed-loop feedback regulation that provides fast ﬁnite-time

convergence even when the system trajectories are far from the sliding surface, robustness against a wide class

of uncertainties and disturbances. Moreover, this algorithm eliminates the major disadvantage of the classical

sliding mode, the well-known chattering phenomenon. The stability analysis of the closed-loop system and

the convergence time are given based on a strong Lyapunov function. To show the effectiveness of the used

method, simulation results of different scenarios are presented for the considered tri-rotor UAV.

1 INTRODUCTION

In recent years, control of aerial robots have become

a coveted ﬁeld of research. In fact, Unmanned Aerial

Vehicles (UAVs) are increasingly used in numerous

applications such as construction, visual inspection,

exploration, transportation and others (Nex and Re-

mondino, 2014; Sankaran et al., 2015; Segales et al.,

2016; Singh and Frazier, 2018). Apart from the fact

that UAVs are highly underactuated systems, as all

nonlinear systems, they are suffering from uncertain-

ties due to the variation of the inertia and mass (Yang

and Xian, 2017) (e.g.: the case of transportation) and

external disturbances due to environmental variations

as wind (Pﬂimlin et al., 2004; Ceccarelli et al., 2007)

(e.g.: the case of outdoor scenarios)

In literature, several nonlinear controllers have

been designed for the problem of tracking and/or

stabilization of UAVs such as feedback lineariza-

tion (Voos, 2009; Zhou et al., 2010), backstep-

ping (Ahmed et al., 2006; Lee et al., 2013), Sliding

Mode Control (SMC) (Runcharoon and Srichatrapi-

muk, 2013) and others. Among these controllers, the

best promising one is SMC which is famous for its

insensitivity to a wide class of uncertainties and dis-

turbances, its simplicity of design and its ﬁnite-time

convergence property (Utkin et al., 1999). SMC uses

discontinuous control inputs to force the system tra-

jectories to converge to the user-chosen sliding sur-

face. Nevertheless, to ensure all these good features,

the discontinuous signals must be high which cause

the chattering phenomenon (Fridman, 2001; Boiko

and Fridman, 2005). This phenomenon is conside-

red the major drawback of which SMC suffers from.

Consequently, the desired performances might be re-

duced and the system actuators cannot deal with the

chattering frequency and might be degraded.

To solve this problem, many works have been de-

veloped and published (Lee et al., 2009; Tseng and

Chen, 2010; Besnard et al., 2012; Kali et al., 2015).

The most popular one for second order systems is

the Second Order Sliding Mode (SOSM) introduced

in (Levant, 2003). The concept of SOSM is to make

the discontinuous signal acting on the derivative of the

control input signal, hence, the control input becomes

continuous (Kali et al., 2017b; Kali et al., 2017c).

SOSM has been extensively used for UAVs (Be-

nallegue et al., 2008; Zheng et al., 2014; Davila and

Salazar, 2017; Munoz et al., 2017a; Munoz et al.,

2017b). However, this method requires some infor-

mations (e.g.: ﬁrst time derivative of the sliding sur-

face) that are often not available for measurements.

Kali, Y., Rodas, J., Saad, M., Benjelloun, K., Ayala, M. and Gregor, R.

Finite-Time Altitude and Attitude Tracking of a Tri-Rotor UAV using Modiﬁed Super-Twisting Second Order Sliding Mode.

DOI: 10.5220/0006861904350442

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 435-442

ISBN: 978-989-758-321-6

435

As a solution to this limitation, the Super-Twisting

Algorithm (STA) has been proposed (Guzm

an and

Moreno, 2015; Gonz

alez-Hern

andez et al., 2017b;

Gonz

alez-Hern

andez et al., 2017a; Ibarra and Cas-

tillo, 2017; Kali et al., 2017a; Kali et al., 2018a; Kali

et al., 2018b). Indeed, apart from that STA ensures ro-

bustness, ﬁnite-time converge and chattering elimina-

tion, it does not require the measurements of the ﬁrst

time derivative of the sliding surface. However, when

the system trajectories are far from the selected swit-

ching surface, the convergence becomes slow (Mo-

reno, 2014) which is not desirable for fast robotic sy-

stems.

In order to improve the convergence speed, a mo-

diﬁed STA with double closed-loop feedback regula-

tion has been proposed (Yang and Xian, 2017). The

proposed algorithm was tested on a DC servo system.

The obtained results were satisfactory in comparison

with different proposed STA structures. To the aut-

hors’ best knowledge, this structure has never been

used for a tri-rotor UAV system. In this paper, this

algorithm will be derived for the altitude and attitude

tracking of an uncertain tri-rotor UAV system.

The rest of this paper is organized into four secti-

ons as follows. In the next section, the conside-

red tri-rotor UAV is described and its altitude and

attitude model are given. In Section 3, the modi-

ﬁed STA with double closed-loop feedback regula-

tion is designed for the problem of altitude and atti-

tude tracking in presence of uncertainties and its sta-

bility analysis is proved using the Lyapunov method.

In section 4, simulation results on the considered tri-

rotor UAV are given to demonstrate the effectiveness

of the used STA with double closed-loop feedback re-

gulation. The conclusion is given in the ﬁfth section.

2 TRI-ROTOR UAV MODEL

The considered tri-rotor UAV system is shown in

Fig. 1. The main advantage of this kind of UAVs is

that they require fewer motors than the other propo-

sed multi-rotors UAV systems such as four-rotors or

Figure 1: Tri-rotor UAV.

six-rotors. This advantage allows a reduction in vo-

lume, weight and energy consumption. The two ro-

tors placed in the forward part of the tri-rotor rotate in

opposite direction with respect to the third rotor pla-

ced in the backward part.

2.1 Altitude Model

The altitude model of the considered tri-rotor can be

expressed by the following equation:

X =

(sin(ψ)sin(θ)cos(φ) −cos(ψ)sin(φ))τ

(1)

Y =

(cos(ψ)sin(θ)cos(φ) + sin(ψ)sin(φ))τ

(2)

Z = −g +

cos(θ)cos(φ)τ

(3)

where m denotes the mass of the tri-rotor, g is the

constant of gravity, τ

is the collective or the vertical

force and φ, θ, ψ denote the Euler angles (roll φ, pitch

θ and yaw ψ).

2.2 Attitude Model

The attitude model of the considered tri-rotor can be

expressed by the following equation:

J W

Θ + J

Θ +



Θ ×J W



= τ (4)

where Θ = [φ, θ, ψ]

are the Euler angles, τ =

[τ

, τ

]

represents the roll, pitch and yaw torques,

J = diag(I

, I

) is the diagonal inertia matrix while

W is the Euler matrix. W and its ﬁrst-time derivative

W are deﬁned by:

W =





1 0 −sin(θ)

0 cos(φ) cos(θ) sin(φ)

0 −sin(φ) cos(θ) cos(φ)





(5)

Figure 2: Reference system for the tri-rotor UAV.

Moreover, the control torque inputs can be expres-

sed as follows:

= l

( f

− f

) (6)

= −l

( f

+ f

) + l

cos(α) (7)

= −l

sin(α) (8)

= f

+ f

cos(α) (9)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

436

where α represents the tilting angle of the third ro-

tor placed in the backward part, f

for i = 1, 2, 3 is

the thrust generated by the rotor i, l

for i = 1, 2, 3

are given in Fig. 2. The control torque inputs given

by (6), (7), (8) and (9) can be written in a matrix form

as follows:



















1 1 cos(α)

−l

cos(α)

0 0 −l

sin(α)







| {z }









3 MODIFIED SUPER-TWISTING

In this section, the proposed second order sliding

mode controller based on a modiﬁed STA for the

ﬁnite-time altitude and attitude tracking of the con-

sidered tri-rotor UAV will be designed. The control

objective is to ensure that the altitude and attitude po-

sitions deﬁned by Z, φ, θ, ψ track with high precision

even in presence of uncertainties and disturbances the

bounded desired trajectories Z

, φ

, θ

, ψ

Now, let us introduce x = [x

, x

]

as state va-

riables with x

= [Z, φ, θ, ψ]

and x

= [

φ,

θ,

ψ]

Then, the equations of motion are given by:



˙x

= x

˙x

= f (x) + g(x)u + h(t)

(10)

Comparing the above equation with (4) gives the fol-

lowing equivalences:

f (x) =



−g

−(JW )

−1



Θ +



Θ ×JW





g(x) =





cos(θ)cos(φ) 0

1×3

3×1

(JW )

−1





u =





and h(t) ∈ R

denotes the uncertain

vector caused by the wind disturbances, unmodelled

dynamics.

In the following, the control law that will force

the system trajectories x

to track with high accuracy

the known desired trajectories x

= [Z

, φ

, θ

, ψ

]

is designed based on the following assumptions:

• Assumption 1: The system trajectories x

and

their ﬁrst-time derivative x

are available for mea-

surements.

• Assumption 2: The desired trajectories x

and

their ﬁrst and second time derivatives x

, ˙x

are

known, bounded and limited to:

−π

< φ

−π

< θ

, −π < ψ

< π.

• Assumption 3: The Euler angles (roll, pitch and

yaw) are limited to:

−π

< φ <

−π

< θ <

, −π < ψ < π.

Based on all these assumptions, the uncertain

functions h

(t) for i = 1, 2, 3, 4 are globally Lip-

schitz (Derafa et al., 2012):



(t)



≤ δ

where δ

> 0 represents the Lipschitz constant.

Let e = x

−x

∈ R

be the trajectory tracking

error. Then, the ﬁrst step in the design procedure con-

sists on deﬁning the sliding surface. In this paper, the

integral sliding surface is selected:

S = ˙e + K

e + K

e dt (11)

where K

= diag(K

, K

) and K

diag(K

, K

) are diagonal positive deﬁnite

matrices. The ﬁrst time derivative of the above in-

tegral sliding surface is computed using the nominal

model as follows:

S = ¨e + K

˙e + K

= ˙x

− ¨x

+ K

˙e + K

= f (x) + g(x) u − ¨x

+ K

˙e + K

(12)

Hence, the modiﬁed super-twisting control algorithm

is obtained by resolving the following equation:



S = −K

Λ(S) sign(S) −K

S + ϖ

ϖ = −K

sign(S) −K

(13)

where Λ(S) = diag



0.5

, |S

0.5

, |S

0.5

, |S

0.5



= diag(K

, ··· , K

), K

= diag(K

, ··· , K

= diag(K

, ··· , K

) and K

= diag(K

, ··· , K

)

are diagonal positive matrices where the coefﬁ-

cients will be ﬁxed in the stability analysis and

sign(S) = [sign(S

), sign(S

), ··· , sign(S

)]

with:

sign(S

) =







1, if S

> 0

0, if S

= 0

−1, if S

< 0

(14)

Theorem 3.1. If the modiﬁed STA gains are chosen

for i = 1, 2, 3,4 as:

= γK

, K

= βK

, K

> 0

> max{C1,C2}

(15)

with:

γ >

|β −2|

√

2β

, γ 6= 2, β > 0, δ

> 0

C1 =

γβ

Finite-Time Altitude and Attitude Tracking of a Tri-Rotor UAV using Modiﬁed Super-Twisting Second Order Sliding Mode

437

C2 =

32 + γ

+ −8γ

β + 8γ

−4γ

4γ

β(2 −γ)

Then, the modiﬁed super-twisting control algorithm

with double closed-loop feedback regulation for the

considered uncertain tri-rotor UAV (10) is given by:

u = −g(x)

−1

[ f (x) + v] (16)

where v is deﬁned as:

v = −¨x

+ K

˙e + K

e + K

Λ(S) sign(S) + K

+ K

sign(S) dt −K

ϖdt

(17)

ensures ﬁnite-time altitude and attitude trajectory

tracking.

Proof. The stability of the closed loop error

will be analyzed using the same methodology used

in (Guzm

an and Moreno, 2015). First of all, sub-

stituting the modiﬁed super-twisting control algo-

rithm (16) in the equation of motion (10) leads to:

S = −K

Λ(S) sign(S) −K

S + ϖ

ϖ = −K

sign(S) −K

ϖ +

h(t)

(18)

The above closed-loop error dynamics can be decom-

posed into 4 sub-systems as:

(t) = −K

0.5

sign(S

) −K

+ ϖ

= −K

sign(S

) −K

(t).

(19)

Now, let us consider the following candidate positive

deﬁnite Lyapunov function:

V = ξ

Lξ (20)

where ξ = [ξ

]

with ξ

= |S

0.5

sign(S

) and

= ϖ

and L is a symmetric positive deﬁnite ma-

trix. The Lyapunov function in (20) is positive deﬁ-

nite, continuous and differentiable except when the

sliding surface is equal to zero S

= 0 and radially

bounded by choosing appropriate matrix L as:

L =







β +





−







(21)

where γ > 0 and β > 0. To calculate the ﬁrst-time deri-

vative of the Lyapunov function, we need to calculate

ﬁrst the ﬁrst-time derivative of the vector ξ. Remark

that |ξ

| = |S

0.5

. Then,

ξ = [

]

is as follows:

2|S

0.5

, and

. (22)

Hence:

ξ =

|ξ

Aξ +

|ξ

(t)|ξ

| (23)

where:

A =





−

+ K

|ξ

−K

|ξ





, B =

Moreover, the ﬁrst time derivative of the Lyapunov

function V is calculated as:

V =

(t)Lξ + ξ

ξ (24)

Substituting

ξ in

V leads to:

V =

|ξ



L + LA



ξ +

(t)

|ξ

Lξ

≤

|ξ



L + LA



ξ +

(t)|ξ

+ ξ

LBB

Lξ

≤

|ξ



L + LA + δ

C + LBB



≤ −ξ

Qξ

(25)

where C = [1 0]

. Considering that K

= γK

and

= βK

. Then, Q is calculated as follows:

Q = −

|ξ

L + LA + δ

C + LBB









(26)

where:



4β

+ β



|ξ



γβK

−δ

−



= Q

= −



2 + β



|ξ



−



= 2K

|ξ



−1



The above matrix Q can be splitted into two matrices

as follows:

Q = Q

|ξ

(27)

where









4β

+ β



−



2+β



−



2+β









(28)







γβK

−δ

−

−1







(29)

Since Q

and Q

are symmetrical. Then, they are po-

sitive deﬁnite if the four following conditions are met:



4β

+ β



> 0 (30)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

438

det(Q

) > 0 (31)

αβK

−δ

−

> 0 (32)

det(Q

) > 0 (33)

The condition in equation (30) is always veriﬁed. The

inequality in (31) is met if:

γ >

|β −2|

√

2β

(34)

While the inequalities in equations (32) and (33) are

veriﬁed if:

> max{C1,C2} (35)

with:

C1 =

γβ

C2 =

32 + γ

+ −8γ

β + 8γ

−4γ

4γ

β(2 −γ)

Therefore, verifying the conditions above,

V is nega-

tive deﬁnite. Hence, the stability of the closed-loop is

proven.

To prove the ﬁnite-time convergence, let us recall

the fact that the Lyapunov function is radially boun-

ded. Then:

min

{L}kξk

≤V ≤ λ

max

{L}kξk

(36)

with λ

min

{L} and λ

max

{L} denote respectively the

minimum and maximum eigenvalues of the matrix L

and kξk

is the Euclidean norm of ξ. Hence:

max

{L}

≤ kξk

≤

min

{L}

(37)

In the second part, equation (25) can be rewritten as:

V ≤ −ξ

Qξ

≤ −ξ

ξ −

|ξ

≤ −λ

min

}kξk

−

|ξ

min

}kξk

(38)

where λ

min

} is the minimum eigenvalue of Q

for

i = 1, 2. As |ξ

| ≤ kξk

. Then, the above equation

can be written as:

V ≤

min

}

max

{L}

V +

min

}λ

min

{L}

max

{L}

(39)

According to the equation above, the sliding surface

converges to zero in ﬁnite-time. This concludes the

proof.

4 SIMULATION RESULTS

In this section, simulation results are presented in

order to demonstrate the effectiveness of the propo-

sed controller based on a modiﬁed STA with double

closed-loop feedback regulation. The controller is si-

mulated on the altitude and attitude model (10) of the

considered tri-rotor UAV described in Section 2 using

Matlab/Simulink software. The physical parameters

of the used tri-rotor are given in Table 1.

Table 1: Physical parameters of the tri-rotor UAV.

Parameters Value

Mass, m 2.5 Kg

Nominal mass, ˆm 2.1 Kg

Moment of inertia, I

0.111132 Kg.m

Moment of inertia, I

0.13282 Kg.m

Moment of inertia, I

0.249039 Kg.m

Nominal moment of inertia,

0.1 Kg.m

Nominal moment of inertia,

0.1 Kg.m

Nominal moment of inertia,

0.2 Kg.m

Length, l

0.275 m

Length, l

0.42 m

Length, l

0.52 m

Gravity, g 9.81 m.s

−2

In this part, an altitude and attitude tracking simu-

lation has been performed. The initial altitude posi-

tions are chosen to be X(0) = 0 m, Y (0) = 0 m and

Z(0) = 0 m while initial Euler angles are chosen to be

φ(0) = 0 rad, θ(0) = 0 rad and ψ(0) = 0 rad. Moreo-

ver, the tracking is performed for the following desi-

red trajectories:

(t) = 10 m

(t) = 0.17 sin(πt) rad

(t) = −0.17 sin(πt) rad

(t) = 0.52 sin(πt) rad

For this scenario, the chosen controller gains are given

in Table 2

Table 2: Modiﬁed STA controller gains.

Gains Value

= diag(K

, ··· , K

) diag(5, 5, 5, 5)

= diag(K

, ··· , K

) diag(6.5, 6.5, 6.5, 6.5)

= γ K

diag(10, 10, 10, 10)

= β K

diag(2, 2, 2, 2)

= diag(K

, ··· , K

) diag(12, 12, 12, 12)

= diag(K

, ··· , K

) diag(1, 1, 1, 1)

The simulation results are given in Figs. 3-8. The

proposed modiﬁed super-twisting control algorithm

ensures the ﬁnite-time convergence of the altitude and

Finite-Time Altitude and Attitude Tracking of a Tri-Rotor UAV using Modiﬁed Super-Twisting Second Order Sliding Mode

439

Figure 6: Simulation results of control inputs.

Figure 3: Simulation results of 3D altitude and attitude

tracking.

attitude positions to their known desired position tra-

jectories with high precision due to the good rejection

of uncertainties and disturbances as shown in Figs. 4

and 7. This is conﬁrmed by the small values of the al-

titude and attitude tracking error as depicted in Figs. 4

and 8. Moreover, the range of variation of the tilt an-

gle α of the third rotor placed in the backward part in

Fig. 5 is ﬁtting the mechanical structure of the system.

Figure 6 shows that the control torque inputs effort is

very small and chattering free. The values of the con-

trol torque inputs are acceptable for our tri-rotor UAV

system.

5 CONCLUSIONS

In this paper, a robust modiﬁed second order sliding

mode control has been designed and successfully si-

mulated on a tri-rotor UAV system for the problem of

altitude and attitude tracking in presence of uncertain-

Figure 4: Simulation results of ﬁnite-time altitude tracking

and tracking error.

Figure 5: Simulation results of tilt angle variations.

ties and disturbances. The proposed controller ensure

robustness by rejecting the effects of the uncertainties

and allowing chattering elimination. The obtained si-

mulation results on the considered tri-rotor UAV sy-

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

440

Figure 7: Simulation results of ﬁnite-time attitude tracking.

stem show clearly the effectiveness of the proposed

modiﬁed super-twisting control algorithm in the alti-

tude and attitude tracking and disturbance rejection.

ACKNOWLEDGEMENTS

This work was supported by the Paraguayan Science

and Technology National Council - CONACYT

(PINV15-0136).

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