trajectory tracking control of a quadrotor uav based on

https://doi.org/10.15388/NA.2019.4.4Nonlinear Analysis: Modelling and Control, Vol. 24, No. 4, 545–560

eISSN: 2335-8963ISSN: 1392-5113

Trajectory tracking control of a quadrotor UAV basedon sliding mode active disturbance rejection control∗

Yong Zhanga, Zengqiang Chena,1, Mingwei Suna, Xinghui Zhangb

aCollege of Artificial Intelligence, Nankai University,Tianjin, 300350, [email protected] Sino-German University of Applied Sciences,Tianjin, 300350, [email protected]

Received: June 11, 2018 / Revised: December 25, 2018 / Published online: June 27, 2019

Abstract. This paper proposes a sliding mode active disturbance rejection control scheme to dealwith trajectory tracking control problems for the quadrotor unmanned aerial vehicle (UAV). Firstly,the differential signal of the reference trajectory can be obtained directly by using the trackingdifferentiator (TD), then the design processes of the controller can be simplified. Secondly, theestimated values of the UAV’s velocities, angular velocities, total disturbance can be acquired byusing extended state observer (ESO), and the total disturbance of the system can be compensated inthe controller in real time, then the robustness and anti-interference capability of the system can beimproved. Finally, the sliding mode controller based on TD and ESO is designed, the stability of theclosed-loop system is proved by Lyapunov method. Simulation results show that the control schemeproposed in this paper can make the quadrotor track the desired trajectory quickly and accurately.

Keywords: sliding mode control, active disturbance rejection control (ADRC), extended stateobserver (ESO), quadrotor UAV, trajectory tracking control.

1 Introduction

Quadrotor UAV is an underactuated system, which has some advantages such as verticaltakeoff and landing, autonomous hovering, high energy efficiency [13], autonomous con-trol, etc. Quadrotor is widely used in civil and military fields. In the civil field, quadrotorUAV can be used in power inspection, ground monitoring, aerial photography, pesticidespraying. In the military field, because of its high covertness and reliability, it is verysuitable for performing battlefield reconnaissance and assessment, target discovery andother military tasks [16]. However, the multivariable, nonlinear, strong coupling and sen-sitivity to disturbances of the quadrotor UAV system itself make it difficult to control [27].∗This research was supported by the National Natural Science Foundation of China under grant No.

61573197, 61573199.1Corresponding author.

c© 2019 Authors. Published by Vilnius University PressThis is an Open Access article distributed under the terms of the Creative Commons Attribution Licence,which permits unrestricted use, distribution, and reproduction in any medium, provided the original author andsource are credited.

mailto:[email protected]

mailto:[email protected]

https://www.vu.lt/leidyba/

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546 Y. Zhang et al.

Air resistance and other external disturbances are also encountered during the flight, sovarious external interference issues need to be considered when designing the controller.In addition, the dynamic model of the quadrotor UAV is complicated, and some of theseaerodynamic parameters are difficult to measure accurately, thus these unknown uncer-tainties further increase the difficulty of designing the controller for quadrotor UAV. Inconsequence, trajectory tracking control [11,29] of quadrotor UAV has attracted attentionsof the researchers widely.

At present, researchers have done a lot of algorithms for quadrotor UAV, these controlalgorithms can be divided into linear control algorithms, nonlinear control algorithmsand intelligent control algorithms. In [18, 28], PID controller is proposed for controllingthe quadrotor. In [23], LQR gain adjustment control is used to realize attitude andpath tracking control for quadrotor UAV. Backstepping and sliding mode control havebeen used in quadrotor UAV, which presented in [5, 10, 19, 22]. In [20], robust controlscheme is proposed to cope with the trajectory tracking problem for UAV. There aresome other control methods for quadrotor UAV such as adaptive control [3, 6], predictivecontrol [1], neural network control [8], fault tolerance control [4], etc. But these methodseither rely on precise mathematical models or require knowledge of the upper bounds ofuncertainties, which are easily influenced by internal uncertainties and unknown distur-bances. In [25], an adaptive neural output-feedback controller have been established fora class of uncertain nonlinear systems with unmodeled dynamics and the lack of full-state measurements. In [30], adaptive fuzzy control scheme have been used for uncertainnonsmooth nonlinear systems. In [26], Robust fuzzy adaptive control strategy is used fora class of nonaffine stochastic nonlinear switched systems. Despite the adaptive neuralcontrol or adaptive fuzzy control, the control design process is complicated, programmingis difficult, for multivariable and fast response system, is not very suitable. The controlmethod proposed in this paper has fast tracking speed and strong anti-disturbance ability,the control design process is more simplified, the controller has fewer control parameters,this method is more suitable for engineering applications.

Active disturbance rejection control technique [12, 14, 15] is a new control strategyemerged in the 80s of last century. The ESO is the core of the whole system, which canestimate the total disturbance including internal unknown and external disturbances, andthe total disturbance can be compensated by the state error feedback control law. ADRCis a control method that does not depend on accurate system model, it has fast trackingspeed, high control precision and strong anti-disturbance ability, it has been widely usedin many theoretical researches, experiments and engineering applications. Because thequadrotor UAV is continually affected by external disturbances and model uncertainties,this can destabilize the UAV easily. A better method to handle with these disturbances isADRC strategy.

In this paper, a sliding mode active disturbance rejection control strategy is proposedto deal with some difficult control problems in the quadrotor unmanned aerial vehiclesystem such as nonlinearity, strong coupling and sensitive to disturbance. This controlmethod neither depend on precise mathematical models nor require knowledge of theupper bounds of uncertainties, it not only can solve the difficulty of flight control, butalso can eliminate the complicated external interference and get good flight performance.

http://www.journals.vu.lt/nonlinear-analysis


Trajectory tracking control of a quadrotor UAV based on sliding mode ADRC 547

The main idea is to combine the advantages of sliding mode control method to track thedesired trajectory smoothly with the ability of ADRC to reject the model uncertaintiesand external disturbances. TD is used to obtain the differential signal of the referencetrajectory, which can simplify the design processes of the controller. ESO is used toobtain the estimated values of the vehicles velocities, angular velocities, total disturbance,and the total disturbance of the system can be compensated in the controller. Finally, thesliding mode controller based on ADRC is designed, and the stability of the closed-loopsystem is analyzed by Lyapunov method.

This paper is organized as follows: A typical quadrotor mathematical model is intro-duced in Section 2. Sliding mode control scheme based on ADRC for quadrotor systemis designed in Section 3. Stability of the quadrotor UAV system is proved in Section 4.Results of simulation experiment are illustrated in Section 5. Finally, some concludingremarks are given in Section 6.

2 System modelling

The scheme of the quadrotor UAV [7, 17, 21] is shown in Fig. 1. In this figure, Fi (i =1, 2, 3, 4) are lift forces of the four rotors, respectively. Ob = {Xb, Yb, Zb} is the bodyfixed frame, Oe = {Xe, Ye, Ze} is the fixed frame with respect to the earth. QuadrotorUAV is an underactuated system with six degrees of freedom and only four control inputs,and it is also a system with some characteristics such as multivariate, strong coupling,nonlinearity, sensitive to disturbance.

Assume that the quadrotor UAV is a rigid body and the body structure is symmetrical.The mathematical model of the quadrotor UAV system [2, 24] can be expressed by thefollowing equations:

x = 1m (Fx − kf x),

y = 1m (Fy − kf y),

z = 1m (Fz −mg − kf z),

θ =l(−F1+F2+F3−F4−kf θ)

Iy+ (Iz−Ix)

Iyφψ,

φ =l(−F1−F2+F3+F4−kf φ)

Ix+

(Iy−Iz)Ix

θψ,

ψ =kc(−F1+F2−F3+F4)−kf ψ

Iz+

(Ix−Iy)Iz

θφ,

(1)

where FxFyFz

= (F1 + F2 + F3 + F4)

cosψ sin θ cosφ+ sinψ sinφsinψ sin θ cosφ− sinφ cosψ

cos θ cosφ.

(2)

x, y, y are positions of the quadrotor, θ is the pitch angle, φ is the roll angle, and ψ is theyaw angle. F1, F2, F3, F4 are lift forces of the four rotors respectively, and other systemparameters are shown in Table 1.

In order to simplify the design procedures and the presentation of the quadrotor flightsystem, the virtual control variables of the whole system U1, U2, U3, U4 are introduced

Nonlinear Anal. Model. Control, 24(4):545–560

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548 Y. Zhang et al.

Figure 1. Free-body diagram of quadrotor.

Table 1. Quadrotor system parameters.

m Quality of the quadrotor 2 kgl Length between the centre of the aircraft and the rotor 0.35 mkc Torque coefficient 0.035 mkf Coefficient of air resistance 0.012 N s2/rad2

Ix Rotational inertia of the x-axis 1.25 Kg m2

Iy Rotational inertia of the y-axis 1.25 Kg m2

Iz Rotational inertia of the z-axis 2.5 Kg m2

g Gravitational acceleration 9.8 m/s2

to transform with the lift forces F1, F2, F3, F4. The variable transformation formula canbe expressed as follows:

U1

U2

U3

U4

=

1/m 1/m 1/m 1/m−l/Iy l/Iy l/Iy −l/Iy−l/Ix −l/Ix l/Ix l/Ix−kc/Iz kc/Iz −kc/Iz kc/Iz

F1

F2

F3

F4

. (3)

Combining (1), (2) and (3), the mathematical model of the quadrotor can be rewrittenas

x = U1(cosψ sin θ cosφ+ sinψ sinφ)− 1mkf x,

y = U1(sinψ sin θ cosφ− cosψ sinφ)− 1mkf y,

z = U1(cosφ cos θ)− g − 1mkf z,

θ = U2 − lkfIyθ + Iz−Ix

Iyφψ,

φ = U3 − lkfIxφ+

Iy−IzIx

θψ,

ψ = U4 − kfIzψ +

Ix−IyIz

θφ.

Since the position loop has only one control input U1, therefore, we can select twointermediate variables θ and φ from the attitude loop to control position loop indirectly.




Set Ux = U1(cosψ sin θ cosφ+ sinψ sinφ),

Uy = U1(sinψ sin θ cosφ− cosψ sinφ),

Uz = U1(cosφ cos θ).

(4)

Solving system (4), we get

U1 =Uz

cosφ cos θ, (5)

θ = arctan

(Ux cosψ + Uy sinψ

Uz

), (6)

φ = arctan

(cos θ

Ux sinψ − Uy cosψUz

). (7)

3 Control scheme

In consideration of the quadrotor dynamic characteristics, it is concluded that the quadro-tor system has two loops, and the two loops can be divided into six channels. The altitude zchannel and the yaw angle ψ channel are independent channels, the x − θ channels andthe y − φ channels are two cascade subsystems.

Firstly, the sliding mode controller based on ADRC is applied in the position loop fortranslational movements. Using the reference trajectory xd, yd, zd, the virtual inputs Ux,Uy , Uz can be obtained, then U1, θd, φd will be calculated through equations (5), (6),(7), respectively. Then, the same control strategy is used in attitude loop, θd, φd, ψd areused in the rotational subsystem as the Euler angle references. Finally, U2, U3, U4 can beobtained to achieve the quadrotor UAV stabilization. The block diagram of the quadrotorUAV control scheme is depicted in Fig. 2.

Figure 2. Control scheme.


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3.1 Tracking differentiator design

Since the quadrotor is an underactuated system, which has four control objectives: altitudez, position x, position y and yaw angel ψ, therefore the TD [25] should be designed forthese four channels. Taking the altitude channel as an example, the design algorithm isgiven by {

vz1 = vz2,

vz2 = fhan(vz1 − zd, vz2, r, h).

Here vz1 is the tracking signal for zd, and vz2 is the differential signal of vz1, r is thespeed factor, which can determine the convergence speed, and h is the filtering factor,fhan(·) denotes the optimal synthetic rapid control function, which is proposed as

d = rh2, a0 = hvz2, y = (vz1 − zd) + a0,

a1 =√d(d+ 8|y|), a2 = a0 + sign y a1−d2 ,

a = (a0 + y) fsg(y, d) + a2(1− fsg(y, d)

),

fsg(x, d) = 12

(sign(x+ d)− sign(x− d)

),

fhan = −r(ad

)fsg(a, d)− r sign a

(1− fsg(a, d)

).

3.2 Extended state observer design

In order to compensate the disturbances in the controller and to obtain the velocities andangular velocities of the UAV, let x1 = x, x2 = x, x3 = f1, y1 = y, y2 = y, y3 = f2,z1 = z, z2 = z, z3 = f3, θ1 = θ, θ2 = θ, θ3 = f4, φ1 = φ, φ2 = φ, φ3 = f5, ψ1 = ψ,ψ2 = ψ, ψ3 = f6, where fi (i = 1, 2, 3, 4, 5, 6) are total disturbances of the each channel,therefore the six channels’ ESO of the quadrotor UAV system can be designed as

˙x1 = x2 − β1(x1 − x1),˙x2 = x3 − β2(x1 − x1) + Ux,˙x3 = −β3(x1 − x1),

˙y1 = y2 − β1(y1 − y1),˙y2 = y3 − β2(y1 − y1) + Uy,˙y3 = −β3(y1 − y1),

˙z1 = z2 − β1(z1 − z1),˙z2 = z3 − β2(z1 − z1) + Uz,˙z3 = −β3(z1 − z1),

˙θ1 = θ2 − β1(θ1 − θ1),˙θ2 = θ3 − β2(θ1 − θ1) + U2,˙θ3 = −β3(θ1 − θ1),

˙φ1 = φ2 − β1(φ1 − φ1),˙φ2 = φ3 − β2(φ1 − φ1) + U3,˙φ3 = −β3(φ1 − φ1),

˙ψ1 = ψ2 − β1(ψ1 − ψ1),˙ψ2 = ψ3 − β2(ψ1 − ψ1) + U4,˙ψ3 = −β3(ψ1 − ψ1).

where β1, β2, β3 are observer gains, and β1 = 3ωo, β2 = 3ωo2, β3 = ωo

3, ωo is thebandwidth of the ESO.




3.3 Sliding mode controller design

TD can provide the differential signals of the target track. The velocities, angular veloci-ties and the total disturbance of the UAV are come from the ESO. Combining the advan-tages of the sliding mode control strategy to track the desired trajectory smoothly withthe ability of the ADRC to handle with model uncertainties and external disturbances,therefore the sliding mode controller based on TD and ESO can be designed as follows.

The sliding mode function of the quadrotor’s six channels can be written assx = kx2ex + ex,

sy = ky2ey + ey,

sz = kz2ez + ez,

sθ = kθ2eθ + eθ,

sφ = kφ2eφ + eφ,

sψ = kψ2eψ + eψ,

where kθ2 > 0, kφ2 > 0, kψ2 > 0, kx2 > 0, ky2 > 0, kz2 > 0, andex = x1 − xd,ey = y1 − yd,ez = z1 − zd,

eθ = θ1 − θd,eφ = φ1 − φd,eψ = ψ1 − ψd,

where xd, yd, zd, ψd are references of the system, θd, φd are obtained from equations (6)and (7), respectively. Then the sliding mode control law based on ADRC of the sixchannels can be designed as

Ux = −kx1sx − vx − x3,Uy = −ky1sy − vy − y3,Uz = −kz1sz − vz − z3,

U2 = −kθ1sθ − vθ − θ3,U3 = −kφ1sφ − vφ − φ3,U4 = −kψ1sψ − vψ − ψ3,

(8)

where kθ1 > 0, kφ1 > 0, kψ1 > 0, kx1 > 0, ky1 > 0, kz1 > 0, andsx = kx2ex + ˙ex,

sy = ky2ey + ˙ey,

sz = kz2ez + ˙ez,

sθ = kθ2eθ + ˙eθ,

sφ = kφ2eφ + ˙eφ,

sψ = kψ2eψ + ˙eψ,vx = kx2 ˙ex − xd,vy = ky2 ˙ey − yd,vz = kz2 ˙ez − zd,

vθ = kθ2 ˙eθ − θd,vφ = kφ2 ˙eφ − φd,vψ = kψ2 ˙eψ − ψd,

ex = x1 − xd,ey = y1 − ydez = z1 − zd,

eθ = θ1 − θd,eφ = φ1 − φd,eψ = ψ1 − ψd,

˙ex = x2 − xd,˙ey = y2 − yd,˙ez = z2 − zd,

˙eθ = θ2 − θd,˙eφ = φ2 − φd,˙eψ = ψ2 − ψd,

and through equation (5), the control variable U1 can be obtained.


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552 Y. Zhang et al.

4 System stability analysis

4.1 Convergence of the ESO

Because the quadrotor system can be divided into six channels, and the mathematicalmodel of the each channel can be written in a same state equation form, thereby, takingthe position x channel as an example, its state space equation can be expressed as

x1 = x2,

x2 = x3 + Ux,

x3 = f ,

(9)

where f is the total disturbance, Ux = U1(cosψ sin θ cosφ + sinψ sinφ), the linearextended state observer of equation (9) can be constructed as

˙x1 = x2 − β1(x1 − x1),˙x2 = x3 − β2(x1 − x1) + Ux,˙x3 = −β3(x1 − x1),

(10)

and L = [β1 β2 β3]T are observer gain parameters.

Let Ei(t) = xi(t) − xi(t), i = 1, 2, 3, combining equations (9) and (10), the ESOestimation error dynamic equation can be written as

E1 = −β1E1 + E2,

E2 = −β2E1 + E3,

E3 = −β3E1 + f .

(11)

Let ηi(t) = Ei(t)/ωoi−1, i = 1, 2, 3, then equation (11) can be rewritten as

η1(t) = − β1

ωo0 η1 + ωoη2,

η2(t) = − β2

ωo1 η1 + ωoη3,

η3(t) = − β3

ωo2 η1 +

fωo

2 .

(12)

Set β1 = 3ωo, β2 = 3ωo2, β3 = ωo

3, equation (12) can be expressed as

η(t) = ωoMη(t) +Nf

ωo2, (13)

where

M =

−3 1 0−3 0 1−3 0 0

, N =

001

.Theorem 1. Assuming that f is bounded, then there exist a positive constant εi > 0 anda finite time T > 0 such that |Ei(t)| 6 εi, i = 1, 2, 3, for all t > T > 0.




Proof. Solving (13), the following equation can be obtained:

η(t) = eωoMtη(0) +

t∫0

eωoM(t−τ)Nf

ωo2dτ.

Let

Q(t) =

t∫0

eωoM(t−τ)Nf

ωo2dτ.

Since f is bounded, |f | 6 D, D > 0, then∣∣Qi(t)∣∣ 6 D

ωo3[∣∣(M−1N)i

∣∣+ ∣∣(M−1eωoMtN)i

∣∣]. (14)

Since

M−1 =

0 0 −11 0 −30 1 −3

,then ∣∣(M−1N

)i

∣∣ = {1, i = 1,

3, i = 2, 3.(15)

Since M is a Hurwitz matrix, there exists a finite time T > 0 such that for all t > T ,i, j = 1, 2, 3, {∣∣(eωoMt

)ij

∣∣ 6 1ωo

3 ,∣∣(eωoMtN)i

∣∣ 6 1ωo

3 .

Since T depends on ωoM [31], then set

M−1 =

a11 a12 a13a21 a22 a23a31 a32 a33

, eωoMt =

b11 b12 b13b21 b22 b23b31 b32 b33

.For all t > T ,∣∣(M−1eωoMtN

)i

∣∣ = |ai1b13 + ai2b23 + ai3b33| 6

{1ωo

3 , i = 1,4ωo

3 , i = 2, 3.(16)

Combining (14), (15), (16), we get∣∣Qi(t)∣∣ 6 3D

ωo3+

4D

ωo6. (17)

Let ηs(0) = |η1(0)|+ |η2(0)|+ |η3(0)| and for all t > T , i = 1, 2, 3,∣∣[eωoMtη(0)]i

∣∣ 6 ηs(0)

ωo3, (18)

then ∣∣ηi(t)∣∣ 6 ηs(0)

ωo3+

3D

ωo3+

4D

ωo6. (19)


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554 Y. Zhang et al.

Let Es(0) = ‖E1(0)| + |E2(0)| + |E3(0)|, according to ηi(t) = Ei(t)/ωoi−1 and (17),

(18), (19), then ∣∣Ei(t)∣∣ 6 Es(0)

ωo3+

3D

ωo4−i+

4D

ωo7−i= εi. (20)

From equation (20), if the total disturbance is bounded, then the estimation errors ofthe ESO are convergent, and when increasing the bandwidth ωo of the ESO,Ei(t) is goingto zero.

4.2 System stability analysis

Assumption 1. The state variables of the quadrotor UAV are measurable and can be usedfor the sliding mode controller design [9].

Assumption 2. Reference trajectory xd, yd, zd, ψd are bounded.

Theorem 2. Assuming that fi (i = 1, 2, 3, 4, 5, 6) are bounded, then there exist constantskθi > 0 (i = 1, 2), kφi > 0 (i = 1, 2), kψi > 0 (i = 1, 2), kxi > 0 (i = 1, 2), kyi > 0(i = 1, 2), kzi > 0 (i = 1, 2) such that the closed-loop system is stable.

Proof. Define the Lyapunov function of the quadrotor system as

V =1

2sx

2 +1

2sy

2 +1

2sz

2 +1

2sθ

2 +1

2sφ

2 +1

2sψ

2,

then

V = sxsx + sy sy + sz sz + sθ sθ + sφsφ + sψ sψ

= sx(kx2ex + ex) + sy(ky2ey + ey) + sz(kz2ez + ez)

+ sθ(kθ2eθ + eθ) + sφ(kφ2eφ + eφ) + sψ(kψ2eψ + eψ)

= sx(kx2ex + Ux + f1 − xd) + sy(ky2ey + Uy + f2 − yd)+ sz(kz2ez + Uz + f3 − zd) + sθ(kθ2eθ + U2 + f4 − θd)+ sφ(kφ2eφ + U3 + f5 − φd)+sψ(kψ2eψ + U4 + f6 − ψd). (21)

Combining (8), (21), we have

V = sx(kx2ex − kx1sx − vx − x3 + x3 − xd)+ sy(ky2ey − ky1sy − vy − y3 + y3 − yd)+ sz(kz2ez − kz1sz − vz − z3 + z3 − zd)

+ sθ(kθ2eθ − kθ1sθ − vθ − θ3 + θ3 − θd)

+ sφ(kφ2eφ − kφ1sφ − vφ − φ3 + φ3 − φd)

+ sψ(kψ2eψ − kψ1sψ − vψ − ψ3 + ψ3 − ψd)= sx(kx2x2 + x3 − kx1sx) + sy(ky2y2 + y3 − ky1sy)+ sz(kz2z2 + z3 − kz1sz) + sθ(kθ2θ2 + θ3 − kθ1sθ)+ sφ(kφ2φ2 + φ3 − kφ1sφ) + sψ(kψ2ψ2 + ψ3 − kψ1sψ)




= −kx1sx2 + sx[kx1kx2x1 + (kx1 + kx2)x2 + x3

]− ky1sy2 + sy

[ky1ky2y1 + (ky1 + ky2)y2 + y3

]− kz1sz2 + sz

[kz1kz2z1 + (kz1 + kz2)z2 + z3

]− kθ1sθ2 + sθ

[kθ1kθ2θ1 + (kθ1 + kθ2)θ2 + θ3

]− kφ1sφ2 + sφ

[kφ1kφ2φ1 + (kφ1 + kφ2)φ2 + φ3

]− kψ1sψ2 + sψ

[kψ1kψ2ψ1 + (kψ1 + kψ2)ψ2 + ψ3

], (22)

where xi (i = 1, 2, 3), yi (i = 1, 2, 3), zi (i = 1, 2, 3), θi (i = 1, 2, 3), φi (i = 1, 2, 3), ψi(i = 1, 2, 3) are observed errors of the ESO. According to Theorem 1, since the ESO canmake the errors converge to zero, therefore equation (22) can be rewritten as

V ≈ −kx1sx2 − ky1sy2 − kz1sz2 − kθ1sθ2 − kφ1sφ2 − kψ1sψ2 6 0.

Therefore, if only kθi > 0 (i = 1, 2), kφi > 0 (i = 1, 2), kψi > 0 (i = 1, 2), kxi > 0

(i = 1, 2), kyi > 0 (i = 1, 2), kzi > 0 (i = 1, 2), then V 6 0, the closed-loop system ofthe quadrotor UAV is stable.

5 Simulation results

The trajectory tracking simulation research of quadrotor UAV system is carried on usingMATLAB. In order to illustrate the mobility and the validity of the proposed sliding modeADRC strategy, a cylindrical spiral as a reference trajectory is used in the simulation, theeffectiveness and advantages of the proposed control scheme are presented by compara-tive performances with the classical active disturbance rejection control.

The cylindrical spiral defined as: xd = 0.5 cos(0.5t) m, yd = 0.5 sin(0.5t) m, zd =2 + 0.1t m, ψd = 60 ◦. Set the initial position coordinates of the quadrotor UAV as[0, 0, 0], while the initial attitude is [0, 0, 0]. Denoting ω1(t), ω2(t) and ω3(t) as systemdisturbances and adding them to the attitude loop, we can write

ω1(t) = sign(sin(0.9t)

),

ω2(t) = sign(sin(0.9t)

)+ cos(0.3t),

ω3(t) = 0.5 sign(sin(0.5t)

)+ cos(0.3t) + 2 cos(0.9t).

The parameters of the controller in the simulation are chosen as r = 2, h = 0.01,ωo = 50, kx1 = 1, kx2 = 1, ky1 = 1, ky2 = 4, kz1 = 10, kz2 = 50, kθ1 = 0.1,kθ2 = 0.1, kφ1 = 0.1, kφ2 = 20, kψ1 = 10, kψ2 = 30. The simulation results are shownin Figs. 3–7.

The results of trajectory tracking are demonstrated in Fig. 3. The simulation resultsof position control and attitude control are shown in Figs. 4 and 5, respectively. Figure 6shows the error comparison of the two methods, and Fig. 7 shows the control inputs of theUAV system. In these figures, the black lines represent the reference trajectory, the blue


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556 Y. Zhang et al.

−1−0.5

00.5

1

−1

0

10

1

2

3

4

5

6

x/my/m

z/m

reference

ADRC

SM−ADRC

Figure 3. Trajectory tracking.

0 10 20 30 40−1

0

1

x/m

0 10 20 30 40−1

0

1

y/m

0 10 20 30 40−10

0

10

t/s

z/m

reference

ADRC

SM−ADRC

Figure 4. Position.

0 10 20 30 40−0.2

0

0.2

θ/r

ad

0 10 20 30 40−0.2

0

0.2

φ/r

ad

0 10 20 30 400

1

2

t/s

ψ/r

ad

reference

ADRC

SM−ADRC

Figure 5. Attitude angle.

lines and the red lines represent the responses of the quadrotor UAV in cases of applyingthe sliding mode ADRC and classical ADRC, respectively.

It can be seen from the Fig. 3 that although the initial value of the position is far awayfrom the reference trajectory, the control scheme that proposed in this paper can holdthe quadrotor UAV track the cylindrical spiral trajectory instantly and accurately, and theperformance of the sliding mode ADRC is better than the classical ADRC. The controlstrategy proposed in this paper has faster response, higher precision, less overshoot andless errors than classical ADRC. From Fig. 4, there is a higher overshoot in position xresponse controlled by classical ADRC than the response controlled by sliding modeADRC, the response speed of the proposed method in position z is faster than the classicalADRC obviously and the tracking errors of the three positions achieved by the proposedcontroller are lesser than the errors that achieved by classical ADRC.

The ways that the sliding mode ADRC and the classical ADRC make the UAV followthe rotational references are shown in Fig. 5. Although the yaw angle responses of thetwo methods can track its corresponding desired trajectory rapidly and accurately, but




0 20 40−0.5

0

0.5

t/s

err

or

x/m

0 20 40−0.1

−0.05

0

0.05

0.1

t/s

err

or

y/m

0 20 40−3

−2

−1

0

1

t/s

err

or

z/m

0 20 40−1.5

−1

−0.5

0

0.5

t/s

err

or

ψ/r

ad

SM−ADRC

ADRC

Figure 6. Errors.

0 10 20 30 400

5

10

15

20

t/s

U1

0 10 20 30 40−40

−20

0

20

40

t/s

U2

0 10 20 30 40−50

0

50

t/s

U3

0 10 20 30 40−8

−6

−4

−2

0

t/s

U4

SM−ADRC

ADRC

Figure 7. Control inputs.

the proposed scheme has better transient performances than classical ADRC, the pitchangle and the roll angle responses from the classical ADRC are more oscillatory at thebeginning of the trajectory than the sliding mode ADRC. It can be clearly observed thatthe variation ranges of pitch angel and roll angel of sliding mode ADRC are more narrowand the adjustment time are shorter than classical ADRC.

It is clearly that the errors of the proposed scheme are less than classical ADRC,especially the error x and error y of classical method are much bigger than sliding mode


https://doi.org/10.15388/NA.2019.4.4

558 Y. Zhang et al.

ADRC. From Fig. 6, it can be seen clearly that the performances are improved by slidingmode ADRC. Because there is no sign function in the controller, it can be seen fromthe Fig. 7 that the control inputs does not appear chattering in the sliding mode ADRCstrategy.

6 Concluding remarks

The sliding mode ADRC scheme is designed for the quadrotor UAV in this paper. Thedifferential signal of the reference trajectory can be obtained directly by using TD, theestimated values of the quadrotor’s velocities, angular velocities, total disturbance canbe acquired by ESO, and the total disturbance of the system can be compensated in thesliding mode controller.

Combining the sliding mode control and ADRC technique can maximize the advan-tages of the two control methods. Sliding mode control strategy can hold the quadrotorfollow the reference trajectory in a smooth way with the ability of ADRC to reject themodel uncertainties and external disturbances.

It can be seen from the simulation results that whether position tracking or angel track-ing can follow the reference trajectory quickly and accurately. As can be seen from thecomparison with classical ADRC, the performances of the quadrotor have been improveda lot. Therefore, it can be concluded that the control scheme proposed in this paper is aneffective control method with strong robustness and anti-disturbance ability.

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