Nonlinear Dyn (2020) 102:2583–2596
https://doi.org/10.1007/s11071-020-06050-2

ORIGINAL PAPER

High-order sliding mode observer-based trajectory tracking
control for a quadrotor UAV with uncertain dynamics
Zhenhua Zhao · Dong Cao · Jun Yang ·
Huiming Wang

Received: 17 July 2020 / Accepted: 21 October 2020 / Published online: 3 November 2020
© Springer Nature B.V. 2020

Abstract This paper investigates the trajectory tracking problem of the quadrotor unmanned aerial vehicles (UAV) with consideration of both attitude and
position dynamics. First of all, the trajectory tracking
problem is divided into the commands tracking in
position and attitude loops by introducing the virtual
attitude angle commands. Secondly, the high-order
This work was supported by grants from the Natural Science
Foundation of China (Nos. 61903192, 61803059), Natural
Science Foundation of Jiangsu (No. BK20190402), Open
Project Program of Ministry of Education, Key Laboratory of
Measurement and CSE (No. MCCSE2019A01), Shenzhen
Science and Technology Innovation Committee
(JCYJ20190813152603594), Innovation Team Project of
Chongqing Education Committee (CXTDX201601019).
Z. Zhao (B)· D. Cao
College of Automation Engineering, Nanjing University of
Aeronautics and Astronautics, Nanjing 211106, People’s
Republic of China
e-mail: zzh@nuaa.edu.cn
D. Cao
e-mail: cdman@nuaa.edu.cn
J. Yang
School of Automation, Southeast University, Nanjing,
210096, Shenzhen Research Institute, Southeast University,
Shenzhen 518063, People’s Republic of China
e-mail: j.yang84@seu.edu.cn
H. Wang
Chongqing Key Laboratory of Complex Systems and
Bionic Control, Chongqing University of Posts and
Telecommunications, Chongqing 400065, People’s
Republic of China
e-mail: wanghm@cqupt.edu.cn

sliding mode observers (HSMOs) are introduced to
estimate the lumped disturbances in position loop and
the derivatives of the attitude angle tracking errors, the
lumped disturbances in the attitude loop. And then the
composite nonlinear dynamical inversion controller in
position loop and the composite nonsingular terminal
sliding mode controller in attitude loop are constructed
by introducing the estimation information of HSMOs
into controller design process. Finally, the simulations
based on the data of a practical UAV are carried out to
verify the effectiveness of the proposed method.
Keywords Quadrotor UAV · Sliding mode observer ·
Nonsingular terminal sliding mode · Disturbance
observer-based control (DOBC) · Trajectory tracking

1 Introduction
Unmanned aerial vehicles (UAVs) have the advantages
of low cost, high flexibility, no personal casualty, and
they are being utilized in more and more fields such
as aviation reconnaissance and rescue in disasters [1].
Due to its unique ability of hovering, vertical takeoff
and landing, quadrotor UAVs have been widely applied
in both the military and economic areas, such as tactics reconnaissance, pinpoint takeoff, and fixed point
monitoring. Since most missions of quadrotor UAVs
can be achieved by tracking a certain trajectory, highprecision trajectory tracking control is the core technique of quadrotor UAVs [2]. However, its characteris-

123

2584

tics of nature nonlinearity and strong couplings among
different channels bring much challenges to the controller design of quadrotor UAVs. Besides, with the
increasing complexity of the flight environment, multisource disturbances such as the external disturbances,
model uncertainties and unmodeled dynamics also seriously limit the tracking precision of quadrotor UAVs.
To deal with the challenges in the trajectory tracking control of quadrotor UAVs, many effective linear
control methods have been proposed. Based on the
linearized model, traditional control method such as
the proportional integral derivative (PID) and linear
quadratic regulation (LQR) methods was employed in
[3]. Although the PID and LQR methods realized the
hovering action of quadrotor UAVs, the tracking precision deteriorates dramatically when the reference is a
three-dimensional trajectory.
To attenuate the influence of nonlinearities and
multi-source disturbances, many nonlinear control
methods have been utilized in the quadrotor UAVs
based on the nominal nonlinear model [4–7]. The nonlinear robust control method is employed in [4] to realize the path tracking missions, and it guarantees null
tracking error and robustness in the presence of uncertainties. The adaptive output feedback control method
is proposed in [5] based on an optimized filter, and the
trajectory tracking experiment demonstrates its robustness to time delay and noises. The study in [6] proposed
a parameter-scheduled backstepping method by combining the command filtered backstepping method with
parameter scheduling technique, and the experiment
test verifies its effectiveness. In [7], the model predictive control and H∞ control are employed in position
and attitude loops of the quadrotor UAVs separately,
and the integral control scheme guarantees the null
steady-state tracking error for quadrotor UAVs in the
presence of disturbances. However, it should be noted
that all the above methods handle the multi-source disturbances in a robust way, which implies that the robustness to disturbances is achieved at the price of sacrificing the nominal control performance.
Recently, the disturbances observer-based control
(DOBC) method [8] has attracted wide attentions due
to its fast disturbance rejection ability [9]. Inspired by
the conception of DOBC, many composite trajectory
tracking methods have been developed for the trajectory tracking of quadrotor UAVs such as the composite PID control method based on linear disturbance
observer [10], the active disturbance rejection method

123

Z. Zhao et al.

based on extended state observer [11], the composite
dynamic surface control method based on sliding mode
observer [12], etc. These composite control methods
not only obtain strong disturbance rejection ability in
the presence disturbances but also guarantee the nominal performance of baseline controller.
Due to its simple design procedure and strong
robustness against uncertainty, the sliding mode (SMC)
methods have been widely employed in flight control systems such as the hypersonic vehicles [13,14],
Mars entry process [15,16], missile guidance systems [17,18], unmanned helicopter systems [19,20].
Recently, the SMC methods are also widely utilized
in the trajectory tracking missions of quadrotor UAVs
such as the traditional sliding mode controller [21],
integral sliding mode controller [22], terminal sliding
mode controller [23], fast terminal sliding mode controller [24], etc. Among the SMC community, the nonsingular terminal sliding mode (NTSM) control [25]
is a hot research topic since it guarantees not only the
finite-time reach of sliding manifold but also finite-time
convergence of system output. Inspired by the NTSM
method and the DOBC conception, the author in [26]
constructed a composite NTSM controller based on the
estimation of single-hidden-layer feedforward network
in the attitude loop and it guaranteed the finite-time
convergence of attitude tracking error. However, the
assumption that the attitude angles are small around
the balance position limits the method only applicable
to the simple trajectory cases.
In this paper, a dual composite control structure for
the complex trajectory tracking of a quadrotor UAV
is proposed. Firstly, the trajectory tracking problem is
divided into commands tracking in position and attitude
loops. Secondly, the high-order sliding mode observer
(HSMO) [27] is introduced to estimate the lumped
disturbances both in position and attitude loops. And
then, the composite nonlinear dynamical inversion controllers are constructed in position loop by introducing the disturbance estimations of HSMOs. Finally, the
composite NTSM controllers are constructed by introducing the estimation information of HSMOs into the
NTSM controller design process. Compared with the
existing studies, the proposed method has the following remarkable merits: (1) relaxing the restrictions on
reference trajectory; (2) asymptotical convergence of
position tracking error in the presence of time-varying
disturbances; (3) nominal performance recovery of the
baseline controller.

High-order sliding mode observer-based trajectory tracking control

The remainder of this paper is organized as follows.
The model of quadrotor UAVs is established, and the
control problem is formulated in Sect. 2. Sect. 3 gives
the controller design details. The comparative results
between the proposed method and traditional method
are shown in Sect. 4. Finally, a brief conclusion is summarized in Sect. 5.

2 Model description and problem formulation
2.1 Model of quadrotor UAVs
As shown in Fig. 1, a quadrotor UAV contains four
independent rotors and its dynamics are governed by
the angular velocities of the four rotors. Rotors 1, 3 and
Rotors 2, 4 spin in the counter-clockwise and clockwise
direction, respectively. The earth-based coordinate system {E} (Oe , X e , Ye , Z e ) and the body-fixed coordinate system {B} (Ob , X b , Yb , Z b ) are introduced
to describe the position [x, y, z] and attitude angles
[φ, θ, ψ] of the quadrotor UAV.
Define the angular velocities of the four rotors as
ω1 , ω2 , ω3 , ω4 , the total lift in the Z b direction as UL ,
and the torques around the body frame axis as τx , τ y ,
and τz , and then it can be obtained:
UL = kL (ω12 + ω22 + ω32 + ω42 ),
τx = lkL (ω22 − ω42 ),
τ y = lkL (−ω12 + ω32 ),
τz = b(−ω12 + ω22 − ω32 + ω42 ),

(1)

2585

where kL denotes the lift coefficient, b denotes the antitorque coefficient, and l is the length of the quadrotor arm. Considering the quadrotor UAV as a rigid
body and its structure is symmetrical, the model of the
quadrotor UAV can be obtained as:
1
ẍ = [UL (cos φ sin θ cos ψ + sin φ sin ψ)
m
−kt ẋ + Fdx ] ,
1
ÿ = [UL (cos φ sin θ sin ψ − sin φ cos ψ)
m

−kt ẏ + Fdy ,

1 
UL cos φ cos θ − kt ż − mg + Fdz ,
(2)
z̈ =
m
φ̇ = p + q sin φ tan θ + r cos φ tan θ,
θ̇ = q cos φ − r sin φ,
1
ψ̇ =
(3)
(q sin φ + r cos φ) ,
cos θ

1 
(Jz − Jy )qr + τx + τdx ,
ṗ =
Jx

1 
q̇ =
(Jx − Jz ) pr + τ y + τdy ,
Jy

1 
(Jy − Jx )qp + τz + τdz ,
(4)
ṙ =
Jz
where x, y, z represent the position in inertial frame,
m is the mass, g is the gravity coefficient, kt is the
drag coefficient; φ, θ, ψ represent the roll angle, pitch
angle and yaw angle, p, q, r denote the angular rates
around the axes of the body frame, Jx , Jy and Jz are
the moments of inertia; Fdx , Fdy , and Fdz denote the
external disturbance forces, τdx , τdy , and τdz denote the
external disturbance torques.
2.2 Problem formulation
Since position and heading angle are the most important
variables for a quadrotor UAV to realize certain tasks,
we set the control objectives of controller design as
to track a desired trajectory (xd , yd , z d ) with desired
heading angle ψd . To simplify the position dynamics
(2), we introduce the following virtual control action
in position loop:
UL
(cos φ sin θ cos ψ + sin φ sin ψ),
m
UL
a yu =
(cos φ sin θ sin ψ − sin φ cos ψ),
m
UL
azu =
cos φ cos θ − g.
m

axu =

Fig. 1 Rotors demonstration of a quadrotor UAV

(5)

123

2586

Z. Zhao et al.

Define the trajectory tracking error as:

Define the tracking error of attitude angles as:

e x = x − x d , e y = y − y d , ez = z − z d ,

eΘ = Θ − Θ d = [φ − φ d , θ − θ d , ψ − ψ d ] ,

where xd , yd , z d represent the desired position. With
the definition of ex , e y and ez in mind, combining
Eqs. (2) and (5) yields the position tracking error
dynamics in three channels as:

where Θ d = [φ d , θ d , ψ d ] are the desired attitude
angles, ψ d is the desired heading angle, and φ d and θ d
are calculated from the transfer in Eq. (7). The tracking
error dynamics of the attitude angles can be obtained
from Eq. (8) as follows:

Fdx
kt ẋ
+
− ẍ d ,
m
m
Fdy
kt ẏ
ë y = a yu −
+
− ÿ d ,
m
m
Fdz
kt ż
+
− z̈ d .
ëz = azu −
m
m

T

T

ëx = axu −

(6)

with

It can be observed from Eqs. (5) and (6) that the position control of the quadrotor UAV is realized by regulating the virtual acceleration axu , a yu and azu through
changing the attitude angles φ, θ and the lift force UL .
Therefore, the control objective in the attitude loop (3)–
(4) is set as to track the desired attitude angles required
in position loop. The desired value of φ, θ and UL can
be calculated based on Eq. (5) as:


axu sin ψ − a yu cos ψ
,
φ d = arcsin m
UL


axu cos ψ + a yu sin ψ
,
θ d = arctan
azu + g

UL = m axu 2 + a yu 2 + (azu + g)2 .

(7)

Thus the attitude dynamics (3)–(4) can be rewritten in
the following compact form:

123

D A = W τd − Θ̈ d ,
where Θ̇ d and Θ̈ d are the derivatives and second-order
derivatives of the desired attitude angles.
Therefore, the control objectives of trajectory tracking with a desired heading angle for quadrotor UAVs
are transferred to stabilize the tracking errors in position and attitude loops, i.e., stabilize ex , e y , ez in Eq. (6)
and eΘ in Eq. (9).
3 Controller design and stability analysis

To simplify the attitude dynamics (3)–(4), we introduce the following definitions:
⎡ ⎤
⎡ ⎤
⎡ ⎤
φ
p
τx
⎣
⎦
⎣
⎦
⎣
Θ = θ , Ω = q , τ = τy ⎦ ,
τz
ψ
r
⎤
⎡
⎤
⎡
Jx 0 0
τdx
τd = ⎣ τdy ⎦ , J = ⎣ 0 Jy 0 ⎦ ,
τdz
0 0 Jz
⎡
⎤
1 sin φ tan θ cot φ tan θ
W = ⎣0
cot φ
− sin φ ⎦ .
0 sin φ/cos θ cos φ/cos θ

Θ̇ = W Ω,
Ω̇ = −J −1 [Ω × (J Ω)] + J −1 τ + τd .

ėΘ = W Ω − Θ̇ d ,
ëΘ = W J −1 τ − W J −1 [Ω × (J Ω)] + Ẇ Ω + D A ,
(9)

3.1 High-order sliding mode observer design
Assumption 1 The external disturbance forces Fdx ,
Fdy , Fdz in the position loop (2) and the external disturbance torques τdx , τdy , τdz in attitude loop (4) are
differentiable, and their derivatives are bounded.
3.1.1 Observer design in position loop
It deduces from Assumption 1 that there exist positive
constants l po1 , l po2 , l po3 which satisfy:
|

Ḟdx
| ≤ l po1 ,
m

Ḟdy
| ≤ l po2 ,
m

|

Ḟdz
| ≤ l po3 .
m

(10)

To estimate the external disturbance force Fdx , Fdy and
Fdz , the HSMOs in the position loop are designed based
on Eqs. (2) and (5) as follows:
kt
ẋ + axu + vx1 , ż x2 = vx2 ,
m
ż x3 = −1.1l po1 sign(z x3 − vx2 ), F̂dx = mz x2 ,
ż x1 = −

(8)

|

High-order sliding mode observer-based trajectory tracking control
1/3

vx1 = −3l po1 |z x1 − ẋ|2/3 sign(z x1 − ẋ) + z x2 ,
1/2
vx2 = −1.5l po1 |z x2 − vx1 |1/2 sign(z x2 − vx1 ) + z x3 ,

(11)
kt
ż y1 = − ẏ + a yu + v y1 , ż y2 = v y2 ,
m
ż y3 = −1.1l po2 sign(z y3 − v y2 ), F̂dy = mz y2 ,
1/3

v y1 = −3l po2 |z y1 − ẏ|2/3 sign(z y1 − ẏ) + z y2 ,
1/2

v y2 = −1.5l po2 |z y2 − v y1 |1/2 sign(z y2 − v y1 ) + z y3 ,

(12)
kt
ż + azu − g + vz1 , ż z2 = vz2 ,
m
ż z3 = −1.1l po3 sign(z z3 − vz2 ), F̂dz = mz z2 ,
ż z1 = −

2587

where
⎡
⎤
⎡ ⎤
D A1
τ1
⎣ τ2 ⎦ = W J −1 τ, ⎣ D A2 ⎦ = D A ,
τ3
D A3
⎡
⎤
f A1
⎣ f A2 ⎦ = −W J −1 [Ω × (J Ω)] + Ẇ Ω.
f A3
It can be obtained from Assumption 1 that there exist
positive constants l Ao1 , l Ao2 , l Ao3 which satisfy:
| Ḋ A1 | ≤ l Ao1 , | Ḋ A2 | ≤ l Ao2 , | Ḋ A3 | ≤ l Ao3 .

1/3

vz1 = −3l po3 |z z1 − ż|2/3 sign(z z1 − ż) + z z2 ,
1/2
vz2 = −1.5l po3 |z z2 − vz1 |1/2 sign(z z2 − vz1 ) + z z3 .

(13)

(16)

To estimate ėφ , ėθ , ėψ and D A1 , D A2 , D A3 in Eq. (15),
the HSMOs in attitude loop are designed as:
1/3

v11 = −3l Ao1 |z 11 − eφ |2/3 sign(z 11 − eφ ) + z 12 ,
1/2

v12 = −1.5l Ao1 |z 12 − v11 |1/2 sign(z 12 − v11 ) + z 13 ,

Define the disturbance estimation errors as:

ż 11 = v11 , ż 12 = τ1 + f A1 + v12 ,
edx = F̂dx − Fdx , edy = F̂dy − Fdy , edz = F̂dz − Fdz .
(14)
Since l po1 , l po2 , l po3 satisfy Eq. (10), based on the theorem in [27], we obtain that edx , edy and edz converge
to zero in finite time, i.e., F̂dx , F̂dy and F̂dz approach
to Fdx , Fdy and Fdz , respectively, in finite time.

It should be noted that Θ̇ d in the attitude loop (9)
is unmeasured variable since φ d and θ d are obtained
based on Eq. (7) through calculations. As a result, ėΘ is
unmeasured, we need to estimate not only the lumped
disturbance D A but also ėΘ in the observer design of
the attitude loop.
To make the observer design more straightforward
in attitude loop, the observers in different channels are
designed separately and independently in the following
step. The attitude angles tracking error dynamics (9)
can be rewritten in a decoupling form as:
ëφ = τ1 + f A1 + D A1 ,
ëψ = τ3 + f A3 + D A3 ,

(17)

1/3
v21 = −3l Ao2 |z 21 − eθ |2/3 sign(z 21 − eθ ) + z 22 ,
1/2
v22 = −1.5l Ao2 |z 22 − v21 |1/2 sign(z 22 − v21 ) + z 23 ,

ż 21 = v21 , ż 22 = τ2 + f A2 + v22 ,
ż 23 = −1.1l Ao2 sign(z 23 − v22 ),
ėˆθ = z 22 , D̂ A2 = z 23 ,

(18)

1/3
v31 = −3l Ao3 |z 31 − eψ |2/3 sign(z 31 − eψ ) + z 32 ,
1/2
v32 = −1.5l Ao3 |z 32 − v31 |1/2 sign(z 32 − v31 ) + z 33 ,

3.1.2 Observer design in attitude loop

ëθ = τ2 + f A2 + D A2 ,

ż 13 = −1.1l Ao1 sign(z 13 − v12 ),
ėˆφ = z 12 , D̂ A1 = z 13 ,

(15)

ż 31 = v31 , ż 32 = τ3 + f A3 + v32 ,
ż 33 = −1.1l Ao3 sign(z 33 − v32 ),
ėˆψ = z 32 , D̂ A3 = z 33 .

(19)

Define the estimation errors of observers (17)–(19) as:

e A1 = ėˆφ − ėφ , e D1 = D̂ A1 − D A1 ,
e A2 = ėˆθ − ėθ ,

e D2 = D̂ A2 − D A2 ,

(20)

e A3 = ėˆψ − ėψ , e D3 = D̂ A3 − D A3 .
Considering l Ao1 , l Ao2 and l Ao3 satisfy Eq. (16), it
can be concluded based on the theorem in [27] that
e A1 , e A2 , e A3 and e D1 , e D2 , e D3 converge to zero in
finite time.

123

2588

Z. Zhao et al.

3.2 Composite controller design
Considering the fact that, the control action in the position loop of a quadrotor UAV is realized by changing
the attitude angles in the attitude loop, we divided the
controller design into two parts: position loop and attitude loop.
3.2.1 Controller design in position loop
Theorem 1 For the position loop subsystem (6), based
on the estimation information of HSMOs (11)–(13),
the following composite nonlinear dynamical inversion
(CNDI) controller:
F̂dx
kt
ẋ −
+ ẍ d − k x1 ex − k x2 ėx ,
m
m
F̂dy
kt
a yu = ẏ −
+ ÿ d − k 1y e y − k 2y ė y ,
m
m
F̂dz
kt
+ z̈ d − k z1 ez − k z2 ėz
azu = ż −
m
m

axu =

(21)

Theorem 2 For the attitude loop subsystem (15),
based on the estimation information of HSMOs (17)–
(19), the following composite nonsingular terminal
sliding mode (CNTSM) controller:
2m −n

m 1 ˆ m1 1 1
− k1 sign(σ1 ),
ė
τ1 = − f A1 − D̂ A1 −
c1 n 1 φ
2m −n
m 2 ˆ m2 2 2
− k2 sign(σ2 ),
τ2 = − f A2 − D̂ A2 −
ėθ
c2 n 2
2m −n
m 3 ˆ m3 3 3
τ3 = − f A3 − D̂ A3 −
− k3 sign(σ3 )
ėψ
c3 n 3
(23)
with k1 > 0, k2 > 0, k3 > 0, guarantees the attitude
angles tracking errors eφ , eθ and eψ converge to zero
in finite time.
It should be noted that the real control actions in the
attitude loop (8) are τx , τ y , τz and they can be calculated
by substituting τ1 , τ2 , τ3 into the following equation:
⎡

with k x1 > 0, k x2 > 0, k 1y > 0, k 2y > 0, k z1 > 0, k z2 >
0, guarantees the position tracking errors ex , e y and ez
converge to zero asymptotically.
It needs to point out that the virtual control action
axu , a yu and azu in the position loop (6) are regulated
by changing the attitude angles φ and θ . Therefore, the
reference of attitude angles can be obtained by substituting the virtual control action axu , a yu and azu into
the transfer (7).

⎤
⎡ ⎤
τx
τ1
⎣ τ y ⎦ = J W −1 ⎣ τ2 ⎦ .
τz
τ3

(24)

3.2.3 Control action reconstruction for full control
loop

3.2.2 Controller design in attitude loop

Since the control actuators of the quadrotor UAVs are
the four rotors, the real control actions for the full control loop are achieved by regulating the rotor speeds
ω1 , ω2 , ω3 and ω4 . It deduces from Eq. (1) that the
desired rotor speeds can be calculated based on the
value of UL , τx , τ y and τz as:

Define the dynamical sliding mode variables as:

⎡

σ1 = eφ + c1 ėˆφ
σ2 = eθ + c2 ėˆθ

n 1 /m 1

n 2 /m 2

σ3 = eψ + c3 ėˆψ

,

,

n 3 /m 3

(22)
,

where ėˆφ , ėˆθ , ėˆψ are obtained from HSMOs (17)–(19),
c1 , c2 , c3 are positive constants, m 1 , m 2 , m 3 , n 1 , n 2 , n 3
are positive odds and satisfy m 1 < n 1 < 2m 1 , m 2 <
n 2 < 2m 2 and m 3 < n 3 < 2m 3 .

123

⎤ ⎡
⎤ ⎡ UL ⎤
0.25 0 −0.5 −0.25
ω12
kL
τx ⎥
⎢ ω2 ⎥ ⎢ 0.25 0.5 0 0.25 ⎥ ⎢
⎢
⎢ 2⎥=⎢
⎥ ⎢ lkτ L ⎥
, (25)
⎣ ω2 ⎦ ⎣ 0.25 0 0.5 −0.25 ⎦ ⎣ y ⎥
3
lkL ⎦
τz
ω42
0.25 −0.5 0 0.25
b

where UL is calculated based on Eq. (7) and (21) and
τx , τ y and τz are calculated based on Eqs. (23) and (24).
It can be concluded that if Theorem 1 and Theorem 2
are established, the real control actions ω1 , ω2 , ω3 and
ω4 which are obtained based on Eq. (25) guarantee the
quadrotor UAV converges to its desired trajectory with
desired heading angle asymptotically. Therefore, it is

High-order sliding mode observer-based trajectory tracking control

only need to prove Theorem 1 and Theorem 2 in the
following stability analysis part.
3.3 Stability analysis
It can be found from Eqs. (6), (9), (21) and (23) that
both the dynamics and controllers in the three control
channels of position and attitude loops are symmetrical.
Therefore, it is only need to prove the stability in one
channel. Without loss of generality, we choose the proof
objectives as to prove following two statements:
(i) axu guarantees ex converges to zero asymptotically;
(ii) τ1 guarantees eφ converges to zero in finite time.
3.3.1 Proof in position loop
The proof is divided into following two steps: (i)
asymptotical convergence of system states after the
lumped disturbances Fdx being estimated accurately;
(ii) finite-time boundness of system states before the
lumped disturbances Fdx being estimated accurately.
(i) Asymptotical convergence of system states after
disturbance being estimated accurately;
With Eq. (14) in mind, substituting the control law (21)
into the position tracking error dynamics (6) yields:
ëx = −k x1 ex − k x2 ėx −

edx
.
m

(26)

Since the disturbance estimation error edx converges
to zero in finite time, there exists a bounded constant
tx which satisfies that when t ≥ tx , edx = 0 is kept.
When t ≥ tx , the close-loop dynamics of ex in Eq. (26)
is governed by the following dynamics:
ëx = −k x1 ex − k x2 ėx .

(27)

Define a Lyapunov function in terms of ex and ėx as:
Vx =

1 1 2
(k e + ė2x ).
2 x x

(28)

on the LaSalle theorem in [28], it concludes that the
dynamics (27) guarantee ex converges to zero asymptotically.
(ii) Finite-time boundness of system states;
Before the lumped disturbances Fdx being estimated
accurately, the tracking error dynamics of ex are governed by Eq. (26). Considering the fact that edx is the
estimation error of HSMO (11) and m is a constant, emdx
is bounded. Let us view the term emdx in Eq. (26) as the
system input, since emdx is bounded and the autonomous
system (27) is asymptotically stable, it can be obtained
from the bounded input and bounded output (BIBO)
theorem in [28] that system states ex and ėx in Eq. (26)
are bounded. This denotes the position tracking error
ex and its derivative ėx are bounded no matter Fdx is
accurately estimated or not.
It can be concluded from the above proof that under
the controller (21), system states of the tracking error
dynamic (6) are bounded, and after the disturbance
being estimated accurately, the tracking error ex converges to zero asymptotically. Therefore, the controller
(21) guarantees the tracking error ex converges to zero
asymptotically.
3.3.2 Proof in attitude loop
The proof is divided into following three steps: (i)
finite-time boundness of system states; (ii) finite-time
convergence of sliding variable; (iii) finite-time convergence of system states.
(i) Finite-time boundness of system states;
Substituting the control law (23) into the tracking error
dynamics of roll channel in Eq. (15) yields:
2m −n

m 1 ˆ m1 1 1
−k1 sign(σ1 ). (30)
ė
ëφ = (D A1 − D̂ A1 )−
c1 n 1 φ
Considering the definition of e A1 and e D1 in Eq. (20),
taking the derivative of σ1 along Eq. (22) yields:
1
c1 n 1 ˆ n1m−m
σ̇1 = −e A1 +
ėφ 1
m1


× −e D1 + eeëφ − k1 sign(σ1 )
(31)
with

Taking the derivative of Vx along Eq. (27) yields:
V̇x = −k x2 ė2x ≤ 0

2589

eeëφ = ė˙ˆφ − ëφ .
(29)

It can be deduced from Eqs. (27) and (29) that V̇x = 0
can be kept only at the point ex = 0, ėx = 0. Based

Define a finite time bounded function [29] in terms of
eφ , ėφ and σ1 as:
1
Veφ = (eφ2 + ėφ2 + σ12 ).
2

123

2590

Z. Zhao et al.

Taking the derivative of Veφ along Eqs. (30) and (31)
yields:
V̇eφ = eφ ėφ + ėφ ëφ + σ1 σ̇1
2m −n

m 1 ˆ m1 1 1
ė
ėφ − k1 sign(σ1 )ėφ
c1 n 1 φ
1 

c1 n 1 ˆ n1m−m
ėφ 1 −e D1 σ1 + eeëφ σ1 − k1 |σ1 | .
− e A1 σ1 +
m1

= eφ ėφ − e A1 ėφ −

Note that for arbitrary α ∈ (0 1), |x|α ≤ 1 + |x|
is always established and m 1 , n 1 are odds and satisfy
0 < m 1 < n 1 < 2m 1 , it can be deduced from above
equation that
m1
(1 + |e A1 + ėφ |)|ėφ |
c1 n 1


c1 n 1
+
(1 + |ėφ + e A1 |) |e D1 | + |eeëφ | + k1 |σ1 |
m1
+ k1 |ėφ | − e A1 σ1
1 2
1
< (eφ + 2ėφ2 + e2A1 ) +
(1 + 4ėφ2 + e2A1 )
2
2c1

V̇eφ ≤eφ ėφ − e A1 ėφ +

+ c1 (1 + ėφ2 + 3σ12 + e2A1 )(|e D1 | + |eeëφ | + k1 )
1
k1
(1 + ėφ2 ) + (e2A1 + σ12 )
2
2
2
k1
+ 3c1 k1 )(σ12 + eφ2 + ėφ2 )
< (1 +
+
c1
2
1
k1
1
1
+ c1 k1 + e2A1 ( +
+
+ c1 )
+
2c1
2
2
2c1
+

+ c1 (e2A1 + 1)(|e D1 | + |eeëφ |).

The above equation can be rewritten in a compact form:

Therefore, for any bounded time T, Veφ is bounded,
i.e., Veφ and eφ , ėφ will not escape to infinity when
t ≤ T.
(ii) Finite-time convergence of sliding variable;
Since the disturbance estimation errors e A1 and e D1 in
Eq. (17) will converge to zero in a finite time, system
(31) then reduces to
σ̇1 = −ρ2 (ėφ )k1 sign(σ1 )
with

n −m

c1 n 1 1m 1 1
ė
.
m1 φ
Since m 1 , n 1 are positive odds, ρ2 (ėφ ) > 0 for any ėφ
which satisfy ėφ = 0. For the case of ėφ = 0, it can
be followed from Eq. (34) that σ1 converges to zero in
finite time. For the case ėφ = 0, it is obtained from
(30) that ëφ = −k1 sign(σ1 ). Similar to the proof in
[25], it can be proved that ėφ = 0 is not an attractor.
Therefore, σ1 converges to zero in finite time.
(iii) Finite-time convergence of attitude angle tracking error;
Since σ1 converges to zero in finite time, there exists
a bounded constant tσ1 which satisfies that when t ≥
tσ1 , σ1 = 0 is kept. Since the disturbance estimation
error e A1 and e D1 converge to zero in finite time, there
exits a bounded constant t Ae which satisfies that when
t ≥ t Ae , e A1 = 0 and e D1 = 0 are kept. Define a
bounded constant tT as
ρ2 (ėφ ) =

tT = max{tσ1 , t Ae }.
V̇eφ ≤ K V Veφ + L V

(32)

with
2
k1
+ 3c1 k1 ),
+
c1
2
1
1
k1
1
+ c1 k1 ) + e2A1 ( +
LV = (
+
+ c1 )
2c1
2
2 2c1
+ c1 (e2A1 + 1)(|e D1 | + |eeëφ |).

n /m 1

Since c1 , k1 are positive bounded constants and e D1 ,
e A1 are the estimation errors of HSMO (17), eeëφ is
the derivative of e A1 , both K V and L V are positive and
bounded. It yields from Eq. (32) that


LV
LV
eKV t −
.
Veφ ≤ Veφ (0) +
KV
KV

123

(33)

(35)

Since tT is bounded, system states eφ (tT ) and ėφ (tT ) are
bounded. When t ≥ tT , the first equation in Eq. (22) is
reduced to
0 = eφ + ėφ1

K V = 2(1 +

(34)

.

(36)

In this case, the dynamics of eφ in system (9) are governed by Eq. (36). According to the conclusion in [30],
eφ will start from eφ (tT ) and then converge to zero in
finite time, i.e., the attitude angle tracking error eφ converges to zero in finite time, which completes the proof.
Similar to the above proof, it can be proved that
the proposed controller also guarantees the quadrotor
UAV to track its reference trajectory in y and z position channels. The whole control structure of the proposed sliding mode observer-based composite method
is demonstrated as Fig. 2.

High-order sliding mode observer-based trajectory tracking control

2591

Fig. 2 Block diagram of the proposed sliding mode observer-based composite controller

Remark 1 The observer gains l po1 , l po2 , l po3 and l Ao1 ,
l Ao2 , l Ao3 of HSMOs (11)–(13) and (17)–(19) only
need to be designed to satisfy Eqs. (10) and (16). If
there is no prior knowledge on Ḟdx , Ḟdy , Ḟdz and Ḋ A1 ,
Ḋ A2 , Ḋ A3 , we can choose relative big observer gains
l po1 , l po2 , l po3 and l Ao1 , l Ao2 , l Ao3 by trial and error.
Remark 2 The attitude angle references φ d , θ d

are
obtained based on the virtual control actions axu , a yu
and azu through Eq. (7). To guarantee the continuity of
φ d , θ d , the virtual control actions should be continuous. Two strategies are utilized in this paper to improve
the continuity degree of virtual control actions: (1)
the dynamical inversion controller rather than NTSM
controller is designed in position loop; (2) although
the second-order HSMOs can guarantee the finite-time
estimation of disturbance in position loop, the thirdorder HSMOs (11)–(11) are designed to improve the
continuity of disturbance estimation.

Table 1 Body parameters of the quadrotor UAV
Parameters

Value

Units

m

0.8

kg

Jx

5.445 × 10−3

kg · m2

Jy

5.445 × 10−3

kg · m2

Jz

1.089 × 10−2

kg · m2

l

0.165

m

b

2 × 10−6

1

kL

2.98 × 10−5

1

kt

9 × 10−2

1

UAV are set as:
x(0) = 0,

y(0) = 1 m,

z(0) = 0;

ẋ(0) = 0,

ẏ(0) = 0,

ż(0) = 0;

θ (0) = 0,

φ(0) = 0,

ψ(0) = 0;

p(0) = 0,

q(0) = 0,

r (0) = 0.

4 Simulation study
4.1 Simulation setting
The tracking performance of the proposed sliding
mode observer-based composite trajectory tracking
controller is tested in this part through numerical simulations. The traditional method without sliding mode
observer is also employed here as a comparison. The
body parameters of the quadrotor UAV in the simulation are set as Table 1. The initial states of the quadrotor

The reference trajectory is set as a complex cylindrical spiral curve which is defined as
x d = sin(0.5t) m,

y d = cos(0.5t) m,

z d = 2 + 0.1t m,

ψ d = 15◦ cos(0.5t).

The external disturbances in position loop dynamics
(2) and attitude loop dynamics (4) are set as:

123

2592

(a)
Tracking error in X axis (m)

(a)

Z. Zhao et al.

2
5
0
-5

1
0

-3

10

5

10

15

20

25

30

-1
-2

Composite controller
Traditional controller

-3
0

5

10

15

20

25

30

Time (s)

(b)
Tracking error in Y axis (m)

(b)

3

5

2

-5

-3

10

0
5

10

15

20

25

1
0
Composite controller
Traditional controller

-1
5

10

15

20

25

30

Time (s)

(c)
Tracking error in Z axis (m)

(c)

2
Composite controller
Traditional controller

1
0
5
0
-5
-10

-1
-2
0

5

10-4

5

10

10

15

15

20

25

20

25

30

30

Time (s)

(d)

Fig. 4 Trajectory tracking error response

In position loop, the HSMOs and composite controller are designed as Eqs. (11)–(13) and (21), and the
observer gains and controller parameters are chosen as:
l po1 = 20, l po2 = 20, l po3 = 40;
k x1 = 4, k x2 = 4, k 1y = 4, k 2y = 4, k z1 = 4, k z2 = 4.

Fig. 3 Trajectory tracking response

Fdx = −4[1 + sin(0.3π t)] N ,
Fdy = 2.4[1 + sin(0.3π t)] N ,
Fdz = −2.4[1 + sin(0.3π t)] N ,
τdx = −Jx [1 + sin(0.2π t)] N · m,
τdy = Jy [1 + sin(0.2π t)] N · m,
τdz = −2Jz [1 + sin(0.2π t)] N · m.

123

In attitude loop, the HSMOs and composite controller
are designed as Eqs. (17)–(19) and (22)–(23), and the
observer gains and controller parameters are chosen as:
l Ao1 = 200, l Ao2 = 500, l Ao3 = 50;
m 1 = m 2 = m 3 = 3; n 1 = n 1 = n 3 = 5;
c1 = c2 = c3 = 0.1; k1 = 2, k2 = 2, k3 = 0.5.
The traditional method has the similar structure and
control parameters with the proposed method except
for the disappear of the disturbance estimation terms
F̂dx , F̂dy , F̂dz in Eq. (21) and D̂ A1 , D̂ A2 , D̂ A3 in

High-order sliding mode observer-based trajectory tracking control

(a)

Heading angle response (deg)

(a)

20

20
0

15
-20
Reference
Composite controller
Traditional controller

-40

10

-60

5

Composite controller
Traditional controller

-80
-100
0

5

10

15
Time (s)

20

25

0

30

0

5

10

15

20

25

30

Time (s)

(b)
Heading angle tracking error (deg)

2593

(b)

0

0.2
0
-0.1
-0.2
-0.3

-20
-40

Composite controller
Traditional controller
5

10

15

20

25

30

0.1

-60

0

-80

Composite controller
Traditional controller

-100
0

5

10

15
Time (s)

20

25

30

Fig. 5 Tracking response of heading angle

-0.1
0

5

10

15

20

25

30

Time (s)

(c)
0.4
Composite controller
Traditional controller

0.2

Eq. (23). The traditional controller in position loop is
designed as:
kt
ẋ + ẍ d − k x1 ex − k x2 ėx ,
m
kt
a yu = ẏ + ÿ d − k 1y e y − k 2y ė y ,
m
kt
azu = ż + z̈ d − k z1 ez − k z2 ėz .
m
The traditional controller in attitude loop is designed
as:
axu =

-0.2
-0.4
0

5

10

15

20

25

30

Time (s)

(d)
Composite controller
Traditional controller

0.1

0

2m 1 −n 1
m1

m1 ˆ
− k1 sign(σ1 ),
ė
c1 n 1 φ
2m −n
m 2 ˆ m2 2 2
τ2 = − f A2 −
− k2 sign(σ2 ),
ėθ
c2 n 2
2m −n
m 3 ˆ m3 3 3
τ3 = − f A3 −
− k3 sign(σ3 ).
ėψ
c3 n 3

τ1 = − f A1 −

0

It should be noted that ėˆφ , ėˆθ and ėˆψ in the above traditional controller are obtained by self-contained differentiator of MATLAB.

4.2 Simulation results
The trajectory tracking responses under the proposed
composite controller and traditional controller are

-0.1
0

5

10

15

20

25

30

Time (s)

Fig. 6 Responses of virtual control action

shown in Fig. 3. As shown in Fig. 3a, the reference trajectory is a complex cylindrical spiral curve,
and the proposed controller guarantees the quadrotor
UAV track the reference with high precision, while the
quadrotor UAV under the traditional controller deviates its reference seriously. It also can be observed
from Fig. 3b–d that the composite controller guaran-

123

2594

Z. Zhao et al.

3000
2000
2600

Composite controller
Traditional controller

2400

1000

4

0
0

4.5

5

5

5.5

10

6

15

20

25

30

Time (s)

Speed of Rotor 2 (r/min)

-2
-4
-6
-8
Estimation
Real Value

-10
0

5

10

(b)

(b)
3500
3000
2500
2600

2000

Composite controller
Traditional controller

2400

1500

4

1000
0

4.5

5

5

5.5

10

6

15

20

25

30

Time (s)

Lumped disturbances in Z axis (N)

3000
2000

2600

Composite controller
Traditional controller

2400

1000

4

0
0

4.5

5

5

5.5

10

15

20

25

Time (s)

(d)

15
Time

20

25

30

8
Estimation
Real Value

6
4
2
0
0

5

10

(c)

(c)
Speed of Rotor 3 (r/min)

0

Lumped disturbances in Y axis (N)

Speed of Rotor 1 (r/min)

4000

Lumped disturbances in X axis (N)

(a)

(a)

15
Time

20

25

30

1
Estimation
Real

0
-1
-2
-3
-4
-5
0

5

10

15
Time

20

25

30

Speed of Rotor 4 (r/min)

4000

Fig. 8 Attitude angular rates and their estimations
3000
2800

2000

2600

Composite controller
Traditional controller

1000
2400
4

0
0

4.5

5

5

5.5

10

6

15

20

25

30

Time (s)

Fig. 7 Speed response of rotors

tees a better position tracking performance than the
traditional controller in all the three channels.
The heading angle tracking response is shown in
Fig. 5. It can be observed from the zoomed in figure of
Fig. 5b that under the proposed composite controller,
the setting time is less than 3 s and the steady-state
tracking error is smaller than 0.1◦ . While Fig. 5 also
shows that the traditional controller cannot guarantee
the heading angle converge to its reference.

123

Figure 4 illustrates the tracking error response of
the quadrotor UAV in three position channels, and
it clearly demonstrates that the composite controller
guarantees better tracking performance both in dynamical and steady-state response: (1) The maximum position tracking errors under the composite controller
in three channels are less than 0.6 m, 0.3 m and 2 m,
respectively, and the setting time is less than 5 s, while
under the traditional controller, the maximum position
tracking errors in three channels are 3 m, 3 m and 2 m,
respectively, and the setting time is greater than 10 s.
(2) The steady-state position tracking errors under the
proposed controller in the three channels are less than
0.005 m, 0.005 m and 0.001 m, respectively, while the
steady-state position tracking errors under the traditional controller are all greater than 1 m.
Figure 6 illustrates the virtual control action responses.
As shown in Fig. 6a, the response curve of lift force UL

High-order sliding mode observer-based trajectory tracking control

(a)
2
Estimation
Real Value
0
0.5

0

-2

0

-2

-0.5

-4
0

-4
0

0.5

5

1

2

1.5

10

15

3

4

20

25

3

4

5

30

Time (s)

(b)
10
5
0

5

-5
0

0.2

0.4

0.2
0
-0.2
-0.4

2

5

0
Estimation
Real Value
-5
0

5

10

15

20

25

30

2595

speed are less than 500 rad/s2 and this is also acceptable
in practical engineering. It concludes from the comparison between Figs. 6 and 7 that although the virtual
control actions τx , τ y and τz are nonsmooth, the real
control actions ω1 , ω2 , ω3 and ω4 are continuous and
they can be realized by practical quadrotor UAVs.
The estimation performance of the HSMOs is given
in Figs. 8 and 9. As shown in Fig. 8, the lumped disturbance in three channels of the position loop, i.e.,
Fdx , Fdy and Fdz , is estimated with high precision after
a short interval (less than 3 s), and the maximum estimation errors of the three channels are less than 5 N, 3 N
and 6 N, respectively. Figure 9 demonstrates the estimation performance of angular rates in attitude loop.
It can be observed from the zoomed in figure of Fig. 9
that the angular rates can be estimated with high precision. This verifies the effectiveness of the sliding mode
observer in position loop.

Time

(c)
2

Estimation
Real Value

5 Conclusion

0
0.5

0

-2

0

-2

-0.5

-4
0

-4
0

0.5

5

1

10

1.5

2

15

3

20

4

25

5

30

Time (s)

Fig. 9 Attitude angular rates and their estimations

under the proposed controller is smooth as the composite nonlinear dynamic inversion method is adopted
in position loop and there are no nonsmooth terms in
controller (21), while Fig. 6a–c demonstrates that the
response curves of the control actions in attitude loop,
i.e., τx , τ y and τz , are nonsmooth due to the existence of
the signal function in attitude controller (23). Figure 6
also demonstrates that the traditional method has similar control response curves with the proposed method,
and this denotes the comparison of the two methods is
reasonable.
Figure 7 illustrates the rotor speed response of the
four rotors. It can be observed from Fig. 7 that the maximum speed of the four rotors is less than 4000 r/min,
which is acceptable for the rotors. By zooming in the
response curves of the four rotors during the time period
4 ≤ t ≤ 6, it can be found that the response curves are
smooth and the maximum changing rates of the rotor

This paper has investigated the complex trajectory
tracking problem of a quadrotor UAV with consideration of both attitude and position loop dynamics. A new
composite control method based on HSMO techniques
has been proposed to handle the external disturbances
and serious couplings of the quadrotor UAV. The proposed method not only relaxes the restrictions on the
types of reference trajectory but also guarantees the
asymptotical convergence of trajectory tracking error
in the presence of multi-source disturbances. The simulation results based on the data of a practical quadrotor
UAV have validated the effectiveness of the proposed
method. Considering the wide applications of artificial
intelligence techniques in the driving systems [31,32],
we will try to introduce the artificial intelligence techniques to recognize the situations of quadrotor UAVs to
achieve the autonomous control of the quadrotor UAVs.

References
1. Cai, G.W., Chen, B.M., Lee, T.H.: Unmanned Rotorcraft
Systems, 1st edn. Springer, NY (2011)
2. Shen, Z., Li, F., Cao, X., et al.: Prescribed performance
dynamic surface control for trajectory tracking of quadrotor
UAV with uncertainties and input constraints. Int. J. Control
(2020). https://doi.org/10.1080/00207179.2020.1743366

123

2596
3. Bouabdallah, S., Noth, A., Siegwart, R.: PID vs LQ control
techniques applied to an indoor micro quadrotor. In: Proceedings of the 2004 IEEE/RSJ International Conference
on Intelligent Robots and Systems, Vol. 3, pp. 2451–2456
(2004)
4. Raffo, G.V., Ortega, M.G., Rubio, F.R.: Robust nonlinear
control for path tracking of a quad-rotor helicopter. Asian J.
Control 17(1), 142–156 (2015)
5. Jafarnejadsani, H., Sun, D., Lee, H., et al.: Optimized L1
adaptive controller for trajectory tracking of an indoor
quadrotor. J. Guid. Control Dyn. 40(6), 1415–1427 (2017)
6. Li, C., Zhang, Y., Li, P.: Full control of a quadrotor using
parameter-scheduled backstepping method: implementation
and experimental tests. Nonlinear Dyn. 89, 1259–1278
(2017)
7. Raffo, G.V., Ortega, M.G., Rubio, F.R.: An integral predictive/nonlinear H∞ control structure for a quadrotor helicopter. Automatica 46(1), 29–39 (2010)
8. Li, S., Yang, J., Chen, W.H., et al.: Disturbance ObserverBased Control: Methods and Applications. CRC Press, Boca
Raton (2014)
9. Guo, Z., Guo, J., Zhou, J., Chang, J.: Robust tracking
for hypersonic reentry vehicles via disturbance estimationtriggered control. IEEE Trans. Aerosp. Electron. Syst. 56(2),
1279–1289 (2020)
10. Wang, H., Chen, M.: Trajectory tracking control for an
indoor quadrotor UAV based on the disturbance observer.
Trans. Inst. Meas. Control 38(6), 675–692 (2016)
11. Zhang, Y., Chen, Z., Zhang, X., et al.: A novel control
scheme for quadrotor UAV based upon active disturbance
rejection control. Aerosp. Sci. Technol. 79, 601–609 (2018)
12. Lin, X., Yu, Y., Sun, C.: A decoupling control for quadrotor
UAV using dynamic surface control and sliding mode disturbance observer. Nonlinear Dyn. 97(1), 781–795 (2019)
13. Sun, H., Li, S., Sun, C.: Finite time integral sliding mode
control of hypersonic vehicles. Nonlinear Dyn. 73(1–2),
229–244 (2013)
14. Guo, Z., Guo, J., Chang, J., et al.: Coupling effect-triggered
control strategy for hypersonic flight vehicles with finitetime convergence. Nonlinear Dyn. 95(2), 1009–1025 (2019)
15. Zhao, Z., Yang, J., Li, S., et al.: Finite-time super-twisting
sliding mode control for Mars entry trajectory tracking. J.
Frankl. Inst. Eng. Appl. Math. 352(11), 5226–5248 (2015)
16. Zhao, Z., Yang, J., Li, S., et al.: Drag-based composite supertwisting sliding mode control law design for Mars entry
guidance. Adv. Space Res. 57(12), 2508–2518 (2016)
17. Zhang, L., Wei, C., Jing, L., et al.: Fixed-time sliding mode
attitude tracking control for a submarine-launched missile
with multiple disturbances. Nonlinear Dyn. 93(4), 2543–
2563 (2018)
18. Zhao, Z., Li, C., Yang, J., et al.: Output feedback continuous
terminal sliding mode guidance law for missile-target interception with autopilot dynamics. Aerosp. Sci. Technol. 86,
256–267 (2019)

123

Z. Zhao et al.
19. Fang, X., Liu, F.: High-order mismatched disturbance rejection control for small-scale unmanned helicopter via continuous nonsingular terminal sliding-mode approach. Int. J.
Robust Nonlinear Control 29(4), 935–2948 (2019)
20. Fang, X., Shang, Y.: Trajectory tracking control for smallscale unmanned helicopters with mismatched disturbances
based on a continuous sliding mode approach. Int. J. Aerosp.
Eng. 2019, Article ID 6235862. https://doi.org/10.1155/
2019/6235862
21. Zhang, J., Ren, Z., Deng, C., et al.: Adaptive fuzzy global
sliding mode control for trajectory tracking of quadrotor
UAVs. Nonlinear Dyn. 97(1), 609–627 (2019)
22. Mu, B.X., Zhang, K.W., Shi, Y.: Integral sliding mode flight
controller design for a quadrotor and application in a heterogeneous multi-agent system. IEEE Trans. Ind. Electron.
64(12), 9389–9398 (2017)
23. Wang, H., Ye, X., Tian, Y., et al.: Model-free-based terminal
smc of quadrotor attitude and position. IEEE Trans. Aerosp.
Electron. Syst. 52(5), 2519–2528 (2016)
24. Xiong, J., Zhang, G.: Global fast dynamic terminal sliding
mode control for a quadrotor UAV. ISA Trans. 66, 233–240
(2017)
25. Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding
mode control of rigid manipulators. Automatica 38(12),
2159–2167 (2002)
26. Xu, B.: Composite learning finite-time control with application to quadrotors. IEEE Trans. Syst. Man Cybern. Syst.
48(10), 1806–1815 (2018)
27. Levant, A.: Higher-order sliding modes, differentiation and
output-feedback control. Int. J. Control 76(9–10), 924–941
(2003)
28. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, NJ
(2002)
29. Li, S., Tian, Y.: Finite-time stability of cascaded timevarying systems. Int. J. Control 80(4), 646–657 (2007)
30. Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with
applications to finite-time stability. Math. Control Signal
Syst. 17(2), 101–127 (2005)
31. Yi, D., Su, J., Liu, C., Quddus, M., Chen, W.-H.: A machine
learning based personalized system for driving state recognition. Transp. Res. Pt. C Emerg. Technol. 105, 241–261
(2019)
32. Yi, D., Su, J., Hu, L., et al.: Implicit personalization in driving assistance: state-of-the-art and open issues. IEEE Trans.
Intell. Veh. 5(3), 397–413 (2020)
Publisher’s Note Springer Nature remains neutral with regard
to jurisdictional claims in published maps and institutional affiliations.

