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Research Article

Second order sliding mode control for a quadrotor UAV En-Hui Zheng a, Jing-Jing Xiong a,n, Ji-Liang Luo b a b

College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou 310018, PR China College of Information Science and Engineering, Huaqiao University, Xiamen 361021, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 17 January 2014 Received in revised form 8 March 2014 Accepted 27 March 2014 This paper was recommended for publication by Dr. Jeff Pieper

A method based on second order sliding mode control (2-SMC) is proposed to design controllers for a small quadrotor UAV. For the switching sliding manifold design, the selection of the coefficients of the switching sliding manifold is in general a sophisticated issue because the coefficients are nonlinear. In this work, in order to perform the position and attitude tracking control of the quadrotor perfectly, the dynamical model of the quadrotor is divided into two subsystems, i.e., a fully actuated subsystem and an underactuated subsystem. For the former, a sliding manifold is defined by combining the position and velocity tracking errors of one state variable, i.e., the sliding manifold has two coefficients. For the latter, a sliding manifold is constructed via a linear combination of position and velocity tracking errors of two state variables, i.e., the sliding manifold has four coefficients. In order to further obtain the nonlinear coefficients of the sliding manifold, Hurwitz stability analysis is used to the solving process. In addition, the flight controllers are derived by using Lyapunov theory, which guarantees that all system state trajectories reach and stay on the sliding surfaces. Extensive simulation results are given to illustrate the effectiveness of the proposed control method. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Quadrotor UAV Second order sliding mode control Lyapunov theory Hurwitz stability

1. Introduction More recently, a growing interest in Unmanned Aerial Vehicles (UAVs) has been shown among the research community, including industry, government and academia [1–6]. This popularity may be owed to the ability to effectively carry out a wide range of applications such as, search and rescue missions, wild fire surveillance, law enforcement, mapping, aerial cinematography, power plants inspection, etc. [5]. The possibility of removing human pilots from danger as well as the size and cost of unmanned aircraft are indeed very attractive but have to be compared to the performances obtained by conventional manned aircraft in terms of mission capabilities, efficiency and flexibility [6]. The quadrotor UAV is a four-rotor Vertical Take-Off and Landing (VTOL) aircraft, which has all advantages of VTOL aircraft along with an increased payload capability, a stability in hover inherent to its design as well as an increased maneuverability. Furthermore, its most important advantage when compared to conventional aircraft, is its reduced mechanical complexity. Nowadays, the emerging and existing motions of the quadrotor UAV include: (a) the precession motion, however, neutralized by designing the front and rear rotors to spin in the opposite direction to the left

n

Corresponding author. E-mail addresses: [email protected] (E.-H. Zheng), [email protected] (J.-J. Xiong), [email protected] (J.-L. Luo).

and right propellers so eliminating any reactive torque around the vertical coordinate axis; (b) the hover motion, obtained by making the rotational velocity of each propellers same; (c) the roll and pitch motion, attained by applying a difference of rotational velocity between the opposing rotors forcing the vehicle to tilt towards the slowest propeller; (d) the yaw motion, generated by making the rotational velocity of neighboring rotors different from others thus forcing the vehicle to tilt towards the two slower propellers; (e) the vertical motion, acquired by increasing or decreasing the rotational velocity of all rotors by the same account; (f) the horizonal motion, gotten by making the vehicle roll or pitch firstly, so as to shift the direction of the thrust vector and then produce a forward component [6]. In this work, the position and attitude tracking control of a small quadrotor UAV is considered. In practical missions, the stability of the aircraft is easily affected by abruptly changed commands. The flight controller design capable of offering to the aircraft an accurate and robust control is crucial in the flight process. Many extended SMC methods have been proposed for the flight controller design for the quadrotor aircraft [7–12]. In [7], a robust second order sliding mode controller for the attitude stabilization of a four rotor helicopter (commonly known as quadrotor) to overcome the chattering phenomena in classical (first) sliding mode control, while preserving the invariance property of sliding mode, was proposed. A SMC approach was proposed to stabilize a class of underactuated systems which were in cascaded form [8],

http://dx.doi.org/10.1016/j.isatra.2014.03.010 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Zheng E-H, et al. Second order sliding mode control for a quadrotor UAV. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.03.010i

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2

where the dynamical model of a quadrotor helicopter was taken as an example for illustrating the proposed SMC. The use of those SMC strategies in the works is dictated by a necessity to compensate for the external disturbances. In addition, in order to illustrate the robustness of the control algorithm, the wind regarded as a specific disturbance has been taken into account the flight process of the quadrotor [6,13]. A second-order sliding mode control (2-SMC) was proposed for second-order uncertain plants using equivalent approach to improve the performance of control systems in [14]. An adaptive second order sliding mode (SOSM) controller with a nonlinear sliding surface was proposed in [15]. However, in the most of existing literatures and research efforts on the control of the quadrotor UAVs, the coefficients of the defined sliding manifolds are taken as special values and given directly in the simulations. In order to further explore the information about the characteristic of those coefficients, the condition of Hurwitz stability can be used to calculate the coefficients of sliding manifolds. Combining with the characteristics of 2-SMC, meanwhile, it is to obtain the good performance of tracking control of the quadrotor aircraft, the dynamics model is decomposed into the two subsystems. For the fully actuated subsystem in the 2-SMC, it is straightforward to conclude the convergence of the state variables from the linear switching surfaces when the state trajectories are in their linear sliding surfaces, but, for the underactuated subsystem, the linear sliding manifolds are not applicable because the system has fewer inputs than the independent variables to be controlled, as a result, a nonlinear sliding manifold or in general an internal dynamics must be stabilized by the proper selection of the switching sliding manifold coefficients [16]. The SMC along with a linear switching surface was proposed for similar underactuated system [16,17]. The linear sliding manifold is constructed by combining the position and velocity tracking errors of the two states of the underactuated subsystem in a linear form, which brings four coefficients associated with the four state variables. The 2-SMC law is derived by using Lyapunov theory, which guarantees the subsystem is stable. In the sliding mode, the sliding motion lies on the four coefficients, but, in a complex and highly nonlinear form. Therefore, it is difficult to directly choose the coefficients to obtain the desired sliding motion. To simplify the switching surface design, the nonlinear sliding manifold is linearized around the desired equilibrium points, the nonlinear coefficients are calculated by Hurwitz stability. The linearized switching manifold is equivalent to a normal linear system that is under a full state linear feedback control, through a mathematical transformation [17]. The rest of this paper is organized as follows. In Section 2 the dynamical model of the quadrotor is given. The problem is formulated in Section 3. The quadrotor flight controller design based on 2-SMC is shown in Section 4. The simulation results and conclusions are presented in Sections 5 and 6, respectively.

2. Quadrotor dynamical model The quadrotor aircraft is detailedly illustrated in Fig. 1. Its dynamical model is set up by the body-frame B(Oxyz) and the earth-frame e(Oxyz). Let a vector ½x; y; z0 denote the position of the center of the gravity of the quadrotor in the earth-frame while the vector ½u; v; w0 denotes its linear velocity in the earth-frame and the vector ½p; q; r0 represents its angular velocity in the bodyframe ms represents the total mass of the aircraft g denotes the acceleration of gravity l denotes the distance from the center of each rotor to the center of gravity. The orientation of the aircraft is given by the rotation matrix R: e-B, where R depends on the three Euler angles ½ϕ; θ; ψ 0 , which

F3 F4

B z ψ F1

O

y

x

f E

F2 θ

ms g

ey

O ez

ex

Fig. 1. Quadrotor aircraft.

represent the roll, the pitch and the yaw, respectively. Those angles are bounded as follows: roll angle, ϕ, by ð  π=2 o ϕ oπ=2Þ; pitch angle, θ, by ð  π=2 o θ o π=2Þ; and yaw angle, ψ, by ð π o ψ o πÞ. The rotation of the quadrotor's body must be compensated during position control. The compensation is attained by using the transpose of the rotation matrix R ¼ Rðϕ; θ; ψÞ ¼ Rðz; ψÞRðy; θÞRðx; ϕÞ 2 3 cos ψ  sin ψ 0 6 7 cos ψ 0 5; Rðz; ψÞ ¼ 4 sin ψ 0 0 1 2 3 cos θ 0 sin θ 6 7 0 1 0 5; Rðy; θÞ ¼ 4 2

 sin θ 1

6 Rðx; ϕÞ ¼ 4 0 0

cos θ

0

0 cos ϕ sin ϕ

0

3

 sin ϕ 7 5:

ð1Þ

cos ϕ

The kinematic equations of rotational and translation movements are obtained by means of the rotation matrix. The translational kinematic is written as ve ¼ R U v B

ð2Þ 0

0

where ve ¼ ½u0 ; v0 ; w0  and vB ¼ ½ub ; vb ; wb  are linear velocities of the mass center expressed in the earth-frame and body-frame, respectively. The rotational kinematics is obtained from the relationship between the rotation matrix and its derivative with a skewsymmetric matrix [6,18,19] as follows: _ ¼ Η  1Ω Φ 2_3 2 1 ϕ 6 θ_ 7 6 0 4 5¼4 ψ_

0

sin ϕ tan θ cos ϕ sin ϕ sec θ

32 3 p 6 7  sin ϕ 7 54 q 5 cos ϕ sec θ r

cos ϕ tan θ

ð3Þ

where Φ ¼ ½ϕ; θ; ψ 0 , and Ω ¼ ½p; q; r0 are the angular velocities in the body-frame. The translational movement is expressed by the following equation [19,20]: ms P€ þms Rj;3 ¼ f

ð4Þ

where P ¼ ½x; y; z0 , and f ¼ Rj;3 Uu1 þ a represents the translational force applied to the quadrotor due to the main control input u1 in _ K 3 U z_ 0 represents the air the z-axis direction, and a ¼ ½K 1 U x_ ; K 2 U y;

Please cite this article as: Zheng E-H, et al. Second order sliding mode control for a quadrotor UAV. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.03.010i

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drag vector, distributed in the ex, ey and ez axis, respectively. the term Rj;3 represents the third column of the rotation matrix. Substituting the state vector P into the Eq. (4), we have 8 _ > x€ ¼ m1s ð cos ϕ sin θ cos ψ þ sin ϕ sin ψÞu1  Km1sx > > < _ y K y€ ¼ m1s ð cos ϕ sin θ sin ψ  sin ϕ cos ψÞu1  m2s ð5Þ > > > : z€ ¼ 1 ð cos ϕ cos θÞu  g  K 3 z_ ms

1

ms

Considering that the quadrotor aircraft is a rigid body and symmetry, the rotational kinetic equation is expressed by d ðJΩÞ ¼ M dt

ð6Þ

where J ¼diag ½I x ; I y ; I z  denotes the inertia matrix of the quadrotor, I x ; I y and I z denote the inertias of the quadrotor, M represents the total torque. The torques of the quadrotor are mainly provided by the thrust generated by four rotors. The thrust generated by rotor i is given by F i ¼ bΩi 2

ð7Þ

The reactive torque caused by the rotor drag generated by rotor i, in free air, is M i ¼  kΩ2i

ð8Þ

where k 4 0 and b4 0 are two parameters depending on the density of air, the radius of the propeller, the number of blades and the geometry, lift and drag coefficients of the blade [6,21]. The rolling torque is given by M ϕ ¼ lð F 2 þ F 4 Þ

ð9Þ

The pitching torque is given by M θ ¼ lðF 1  F 3 Þ

3

state-space form is derived by 8 € ¼ qr Iy  Iz þ J r qΩr þ l u2  K 4 lp ϕ > > Ix Ix Ix Ix > < Jr K5l Ix l  pΩ þ u  θ€ ¼ pr Iz  r 3 Iy Iy Iy Iy q > > > Ix  Iy K6 C : ψ€ ¼ pq I z þ Iz u4  Iz r

ð15Þ

where Ki are the drag coefficients and positive constant, Ωr ¼  Ω1 þ Ω2  Ω3 þ Ω4 , Ωr is the overall residual rotor angular velocity, while Ωi correspond to the rotor's angular velocities.

3. Control problem formulation The control problem considered in this work is to preform asymptotic position and attitude tracking of the quadrotor by designing flight controllers depended on second order sliding mode technique. That's, under the controllers, P-P d and Φ-Φd . Based on the dynamical model in Eqs. (3), (5), and (15), the control system is divided into multiple subsystems (fully actuated € underactuated subsystem made subsystem composed of z€ and ψ, € and θ) € that is inspired by the sliding mode control up of x€ , y€ , ϕ approach [8,22]. For the fully actuated subsystem (or the underactuated subsystem), a switching sliding surface is constructed using a linear combination of position and velocity tracking errors of one (or two) state variable(s), the tacking errors are driven to zero in order to achieve the desired output tracking performance, performed by an independent controller.

4. Flight controller design

ð10Þ

This section mainly introduces the second order sliding mode control (2-SMC) method that is applied here to design the flight controller of the quadrotor as shown in Fig. 2.

ð11Þ

4.1. Controller design for fully actuated subsystem

The yawing torque generated by the four rotors is M ψ ¼ CðF 1  F 2 þ F 3 F 4 Þ

where C is the proportional coefficient. The gyroscopic torque additively caused by the motor rotor and the propeller is given by M g ¼ ΣΩ  H i

ð12Þ

where Hi is the rotational momentum moment, and it only appears in z-axis due to the angular velocity when the motor rotates. Thereafter, the rotational momentum moment meets H i ¼ ½0; 0; J r Ωi 0 , where Jr denotes the inertia of the z-axis. According to the above expressions, the total torque is 2 3 Mϕ 6 7 ð13Þ M ¼ M g þ 4 Mθ 5 Mψ The control inputs are calculated 2 3 2 T 3 2 u1 b b b 7 6 6 u2 7 6 M 6 7 ϕ lb 0  lb 6 7 6 7¼6 6 6 7¼6 4 4 u3 5 4 M θ 7 0  lb 0 5 Mψ u4 k k k

as: 32 2 3 Ω1 6 27 Ω 6 07 76 2 7 7 7 27 lb 56 4 Ω3 5 Ω24 k

A controller for the fully actuated subsystem of the quadrotor is designed by using 2-SMC. The objective is to ensure that the state variables ½z; ψ converge to the desired values ½zd ; ψ d . In addition, considering that the quadrotor is a rigid-body, according to the symmetry, I x ¼ I y is got. The sliding manifolds are defined as s1 ¼ cz ðzd  zÞ þ ðz_ d  z_ Þ .

z z

_ zd +

Altitude controller

.

ψ ψ _ ψd +

Yaw angle controller

.

b

ð14Þ

where u1 denotes the total thrust on the body in the z-axis; u2 and u3 represent the roll and pitch torques, respectively; and u4 represents a yawing torque. € θ; € ψ _ q_ ; r_ 0 , and the € 0 ¼ ½p; Invoking Eqs. (6), (13), and (14), let ½ϕ; air drag is also taken into account, therefore, the second order

ð16aÞ

θ θ _ xd θd +

Pitch angle controller

. _ x x y y. _ yd f d +

_

u1 Fully actuated subsystem

u4 Quadrotor aircraft

u3 Underactuated subsystem

Roll angle controller

f

f

u2

.

Fig. 2. The flight control architecture.

Please cite this article as: Zheng E-H, et al. Second order sliding mode control for a quadrotor UAV. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.03.010i

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4

_ s2 ¼ cψ ðψ d  ψÞ þ ðψ_ d  ψÞ

ð16bÞ

where the coefficients cz ; cψ 4 0. By making s_ i ¼  εi sgnðsi Þ  ηi si (i¼1, 2), the corresponding control laws are designed u 1 ¼ ms U

cz ðz_ d  z_ Þ þ z€ d þ g þ d1 þ ε1 sgnðs1 Þ þ η1 s1 cos ϕ cos θ

Iz _ þ ψ€ d þd2 þ ε2 sgnðs2 Þ þ η2 s2  u4 ¼ ½cψ ðψ_ d  ψÞ C

ð17aÞ ð17bÞ

where the coefficients of the exponential approach laws ε1 ; ε2 ; η1 ; η2 4 0 d1 ¼ K 3 z_ =ms and d2 ¼ K 6 r=I z are taken as disturbance terms. 4.2. Controller design for underactuated subsystem In this section, a controller for the underactuated subsystem of the quadrotor is design by using 2-SMC. The objective is to guarantee the state variables [x; θ] and [y; ϕ] converge to the desired values [xd ; θd ] and [yd ; ϕd ], respectively. The sliding manifolds are defined as [17] _ þ c4 ðθd  θÞ s3 ¼ c1 ðx_ d  x_ Þ þ c2 ðxd xÞ þ c3 ðθ_ d  θÞ

ð18aÞ

_ d  ϕÞ _ þc8 ðϕd  ϕÞ s4 ¼ c5 ðy_ d  y_ Þ þ c6 ðyd  yÞ þc7 ðϕ

ð18bÞ

where the coefficients ci (i¼1,…,8) will be obtained latter from the Hurwitz stability analysis. The time derivatives of the two sliding manifolds are obtained € þ c4 ðθ_ d  θÞ _ s_ 3 ¼ c1 ðx€ d  x€ Þ þ c2 ðx_ d  x_ Þ þ c3 ðθ€ d  θÞ

ð19aÞ

€ d  ϕÞ € þc8 ðϕ _ d  ϕÞ _ s_ 4 ¼ c5 ðy€ d  y€ Þ þ c6 ðy_ d  y_ Þ þc7 ðϕ

ð19bÞ

By making s_ i ¼  εi sgnðsi Þ  ηi si (i¼3, 4), the corresponding control laws are designed u3 ¼

  I y c1 c2 c4 _ þ d3 þ 1 ½ε3 sgnðs3 Þ þ η3 s3  ðx€ d  x€ Þ þ ðx_ d  x_ Þ þ θ€ d þ ðθ_ d  θÞ l c3 c3 c3 c3

ð20aÞ   I x c5 c6 c8 _ þ d4 þ 1 ½ε4 sgnðs4 Þ þ η4 s4  u2 ¼ ðy€ d  y€ Þ þ ðy_ d  y_ Þ þ ϕ€ d þ ðϕ_ d  ϕÞ l c7 c7 c7 c7

ð20bÞ where the coefficients of the exponential approach laws ε3 ; ε4 ; η3 ; η4 4 0. In addition, the disturbance terms are d3 ¼  prðI z  I x Þ=I y þ J r pΩr =I y þ K 5 lq=I y

and

d4 ¼  qrðI y  I z Þ=I x J r qΩr =I x þ K 4 lp=I x Theorem. Considering the dynamical model of the quad -rotor, the flight controller is designed as the Eqs. (17a), (17b), (20a) and (20b). Under the designed controllers, the nonlinear system is stable. Proof. Consider the Lyapunov function candidates: V i ¼ 12 s2i

ði ¼ 1; 2; 3; 4Þ

ð21Þ

Invoking the Eqs. (16a) and (17a), (16b) and (17b), (19a) and (20a), (19b) and (20b), the time derivatives of Vi are V_ i ¼ si U s_ i ¼  εi jsi j  ηi s2i r 0

4.3. Switching surface coefficients Considering that the coefficients of the sliding manifolds s3 and s4 are obtained using the same condition on Hurwitz stability, the following solving process of the coefficients ci (i ¼1, 2, 3, 4) is taken as an example for avoiding to repeat the same steps. Let s_ 3 ¼ 0. Replacing u3 with θ in Eq. (19a) yields c1 c2 c4 _ θ€ d  θ€ ¼  ðx€ d  x€ Þ  ðx_ d  x_ Þ  ðθ_ d  θÞ c3 c3 c3

ð23Þ

When s3 ¼ 0, c2 c3 _  c4 ðθd  θÞ; ðx  xÞ  ðθ_ d  θÞ c1 d c1 c1 2 €θd  θ€ ¼  c1 ðx€ d  x€ Þ þ c2 ðxd xÞ c3 c c   1 3 c2 c4 _ _ þ c2 c4 ðθd  θÞ  þ ðθd  θÞ c1 c3 c1 c3 x_ d  x_ ¼ 

ð24Þ

_ y3 ¼ xd  x. The cascaded form is Let y1 ¼ θd  θ, y2 ¼ θ_ d  θ, obtained y_ 1 ¼ y2 ; c2 c1 ðx€  x€ Þ þ 2 ðxd  xÞ c3 d c1 c3   c2 c4 _ _ þ c2 c4 ðθd  θÞ; ðθd  θÞ þ  c1 c3 c1 c3 c c _  c4 ðθd  θÞ: _y3 ¼  2 ðxd  xÞ  3 ðθ_ d  θÞ c1 c1 c1 y_ 2 ¼ 

ð25Þ

When the state variables are close to their equilibrium points, _ θ_ d , x-xd , x_ -x_ d , thus, y1 -0, y2 -0, y3 -0. After the i.e., θ-θd ; θlinearization around the equilibrium points, the new cascaded form is obtained y_ 1 ¼ y2 ; c1 u1 y_ 2 ¼  ½x€ d  ð  y1 cos ϕ cos ψ þ sin ϕ sin ψÞ þ d1  c3 ms   c2 2 c2 c4 _ _ þ c2 c4 ðθd  θÞ þ ðxd xÞ þ  ðθd  θÞ c1 c3 c1 c3 c1 c3 þξ1 y1 þ ξ2 y2 þ ξ3 y3 ; c2 c3 _  c4 ðθd  θÞ: y_ 3 ¼  ðxd  xÞ  ðθ_ d  θÞ c1 c1 c1 Let Y ¼ ½y1 y2 y3 0 , the matrix form is Y_ ¼ AY þ BY, 2 3 2 3 0 1 0 0 0 0 6 7 6 7 A ¼ 4 A21 A22 A23 5 and B ¼ 4 ξ1 ξ2 ξ3 5: a

b

c

0

0

0

The parameters ξi (i¼1, 2, 3) are small constant,λlef t ðAÞ denotes the real part of the leftmost eigenvalues of the matrix A in the negative half plane, when λlef t ðAÞ{0, i.e., the matrix A is Hurwitz, the system is asymptotically stable near the equilibrium points [8,17]. Thereafter, it is only necessary to consider the stability of Y_ ¼ AY. Assuming c1 a0, c3 a 0, the parameters are obtained c 1 u1 c2 c4 cos ϕ cos ψ þ ; c 3 ms c1 c3 c2 c4 c3 A23 ¼ 2 ; a ¼  ; b ¼  ; c1 c3 c1 c1

c2 c4  ; c1 c3 c2 c¼  c1

A21 ¼ 

Let jλ I  Aj ¼ 0 ;

ð22Þ

i:e:;

  λ     A21   a

A22 ¼

1 λ  A22 b

     A23  ¼ 0:  λc  0

The equation form is expressed 3

λ  ðA22 þ cÞλ2 þ ðcA22 A21  bA23 Þλ þ cA21  aA23 ¼ 0 Thus, under the control laws ui(i¼1, 2, 3, 4), all the system state trajectories can reach, and thereafter, stay on the corresponding sliding surfaces, respectively. □

ð26Þ

ð27Þ

Let the characteristic equation be ðλ þ 1Þðλ þ2Þðλ þ 3Þ ¼ 0

ð28Þ

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After the comparison above two equations, the coefficients ci (i¼1, 2, 3, 4) are obtained 8 c4 ¼6 > > < cc3 u 1 1 ð29Þ c3 ms cos ϕ cos ψ ¼ 11 : > > : c2 u1 cos ϕ cos ψ ¼ 6 c3 ms Let c3 ¼ 1 then c1 ¼ 11 ms =ðu1 cos ϕ cos ψÞ; c2 ¼ 6ms =ðu1 cos ϕ cos ψÞ; c4 ¼ 6: Note. The deviation terms ξi caused by the linearization around the state equilibrium points will bring uncertain deviations to the coefficient of u1 in the first equation of (5). However, it is overcome by the switching gain of the SMC laws (20a).

5

Table 2 Controller parameters. Variables

Values

Variables

Values

cz ε1 η1 c1 c2 c3 c4 ε1 η3

1 0.8 2 11ms =ðu1 cos ϕ cos ψÞ 6ms =ðu1 cos ϕ cos ψÞ 1 6 0.5 5

cψ ε2 η2 c5 c6 c7 c8 ε4 η4

1 0.8 2  11ms =ðu1 cos ψ Þ  6ms =ðu1 cos ψ Þ 1 6 0.5 5

Similarly, the coefficients ci (i ¼5, 6, 7, 8) are obtained through the same idea. Here it is simplified, and they are c5 ¼  11ms = ðu1 cos ψ Þ, c6 ¼  6ms =ðu1 cos ψÞ, c7 ¼ 1, c8 ¼ 6.

0.8

z (m)

0.6

5. Simulation results

0.4 0.2 0 0.8 0.6 0.4

y (m

)

Variables

Values

Units

ms l lx ¼ ly lz lr Ki (i¼ 1, 2, 3) Ki (i¼ 4, 5, 6) g b k C

1.1 0.21 1.22 2.2 0.2 0.1 0.12 9.81 5 2 1

kg m Ns2/rad Ns2/rad Ns2/rad Ns/m Ns/m m/s2 Ns2 N/ms2

0

0.2

0

0.8

)

x (m

x (m)

0.9 0.6 0.3 0

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40 T (s)

50

60

70

80

y (m)

0.9 0.6 0.3 0 0.9 0.6 0.3 0 -0.3

Table 1 Quadrotor model parameters.

0.2

0.6

0.4

Fig. 3. Quadrotor's path in set-point position and angle control.

z (m)

The proposed control method has been tested by the following simulations in order to verify the validity and efficiency of the control scheme utilized in this work, including the performance attained for the position and attitude tracking problem. Besides, simulations have been carried out with the more accurate model, i.e., the combination of Eqs. (3), (5), (14) and (15), which emulates a real quadrotor UAV. In addition, the aerodynamic forces and moments, and air drag are taken as external disturbances to demonstrate the robustness of the flight controller designed by using second order sliding mode technique. The simulations are performed on MATLAB7.1.0.246/Simulink, which is equipped with a computer consisting of a DUO E7200 2.53 GHz CPU with 2 GB of RAM and a 100 GB solid state disk drive. The initial position and angle values of the quadrotor for simulation tests are [0, 0, 0] m and [0, 0, 0] rad. In addition, the quadrotor's model variables are listed in Table 1 [23]. Besides, the controller parameters are listed in Table 2. The simulations results are shown in Figs. 3–9. The quaudrotor's path, shown in Fig. 3, is performed under the condition of set-point position and angle control. More specifically, in different moment, the different reference positions and angles are listed in Table 3. Even though the reference position and angle were changed in every moment, the proposed control scheme managed to effectively hold the quadrotor's position and attitude in finite-time, as shown in Figs. 4 and 5. It is shown that even though the quadrotor's position and attitude are affected by the abruptly changed reference positions and angles, the controller is able to

Fig. 4. The positions (x, y, z).

drive all these state variables back to the new reference position and angle within seconds. Moreover, the aerodynamic forces and moments, and air drag are taken into account the controller design. Those demonstrate the robustness of the designed controller and effectiveness of the proposed control scheme. The linear and angular velocities, shown in Figs. 6 and 7, respectively, exhibit the same behavior as the homologous positions and angles. Simultaneously, it is also shown that these state variables have coupling relationship, thus verifies the highlycoupled characteristic of the dynamical model of the quadrotor.

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0.4

0

0

-0.04

s1

φ (rad)

6

0

10

20

30

40

50

60

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θ (rad)

0 -0.04

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0

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0.3

0.5

s3

ψ (rad)

10

0

-0.3

80

1

0 -0.5

0

0.3

0.04

0 -0.3 0.3

0

10

20

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40

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s4

T (s)

0

Fig. 5. The angles (ϕ, θ, ψ,). -0.3

T (s)

Fig. 8. The sliding variables (s1, s2, s3, s4). 0

13 0

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w (m/s)

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8 0

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T (s)

Fig. 6. The linear velocities (u, v, w).

4 u4 (N)

0 -0.15

0 -8

0.15 p (rad/s)

0

0 -8

0 -0.5

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0.15 q (rad/s)

X: 66.94 Y: 10.79

8 -0.3

0

-4

T (s) 0 -0.15

Fig. 9. The controllers (u1, u2, u3, u4).

0

10

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0.4 r (rad/s)

X: 20.54 Y: 10.79

11 9

0

u2 (N)

v (m/s)

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u (m/s)

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0 -0.4

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T (s)

Fig. 7. The angular velocities (p, q, r).

As desired, these state variables are driven to their reference/ steady values. This, once again, demonstrates the effectiveness of the proposed control scheme.

The behavior of the sliding variables, shown in Fig. 8, follows the expectations as all these variables converge to their sliding surfaces. In addition, the convergence time of s1 and s2 is obviously faster than the convergence time of s3 and s4. Moreover, the position and velocity tracking errors of the system state variables are perfectly explained by the fluctuations of these sliding variables. Finally, the controllers, displayed in Fig. 9, are continuous as desired and easily applied to a real-life model. It is noted that although the controllers reach their steady states several times during the flight process, the stability of them or the quadrotor does not appear affected. In order to further verify that the real

Please cite this article as: Zheng E-H, et al. Second order sliding mode control for a quadrotor UAV. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.03.010i

E.-H. Zheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 3 The reference positions and angles. [2] Variables

Values

[xd , yd , zd ]

[0.6, [0.3, [0.3, [0.6, [0.6, [0.6,

[ϕd , θd , ψ d ]

0.6, 0.6, 0.3, 0.3, 0.6, 0.6,

Time (s) 0.6] m 0.6] m 0.6] m 0.6] m 0.6] m 0.0] m

[0.0, 0.0, 0.5] rad [0.0, 0.0, 0.0] rad

0 10 20 30 40 50 10 60

[3] [4]

[5] [6]

[7]

steady value and the theoretical value of u1 are consistent, the mark has been made in Fig. 9. 6. Conclusion In this work, a second order sliding mode control (2-SMC) method is proposed to solve the position and attitude tracking problem for a small quadrotor UAV. In order to further test the performance of the designed controller, the dynamical model of the quadrotor along with the controller is used to simulate on MATLAB/Simulink. The main conclusions are summarized as follows. (a) All the state variables converge to their reference values, respectively, even if their reference values are suddenly changed in different moment. (b) The various paths of the quadrotor are obtained by changing the reference positions, and the various attitudes are also obtained by changing the reference angles. (c) The position and velocity tracking errors of all the system state variables trend to zero, i.e., the sliding variables converge to their sliding surfaces. (d) The robustness of the designed controller are demonstrated, and the effectiveness of the proposed control scheme are also justified. All above, the presented simulation results are very promising. Acknowledgment

[8] [9]

[10]

[11] [12]

[13]

[14] [15] [16]

[17]

[18]

[19] [20]

This work was partially supported by the National Natural Science Foundation of China (60905034).

[21]

References

[22] [23]

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IASTED International Conference on Modelling, Simulation and OptimizationMSO 2003, Banff Canada: IASTED/ACTA Press; 2003. p. 320–5. Bouabdallah S., Murrieri P., Siegwart R. Design and control of an indoor micro quadrotor. In: Proceedings of IEEE international conference on robotics and automation, New Orlean, USA; 2004. Tayebi A, McGilvray S. Attitude stabilization of a VTOL quadrotor aircraft. IEEE Trans. Control Syst. Technol. 2006;14:562–71. Hérissé B, Hamel T, Mahony R, Russotto F-X. Landing a VTOL unmanned aerial vehicle on a moving platform using optical flow. IEEE Trans. Robot. 2012;28:77–89. Derafa L, Benallegue A, Fridman L. Super twisting control algorithm for the attitude tracking of a four rotors UAV. J. Frankl. Inst. 2012;349:685–99. Besnard L, Shtessel YB, Landrum B. Quadrotor vehicle control via sliding mode controller driven by sliding mode disturbance observer. J. Frankl. Inst. 2012;349:658–84. Bouchoucha M., Seghour S., Tadjine M. Classical and second order sliding mode control solution to an attitude stabilization of a four rotors helicopter: from theory to experiment. In: Proceedings of the 2011 IEEE international conference on mechatronics April 13–15, Istanbul, Turkey; 2011, p. 162–69. Xu R, Özgüner Ü. Sliding mode control of a class of underactuated systems. Automatica 2008;44:233–41. Xu R, Özgüner Ü. Sliding mode control of a quadrotor helicopter. In: Proceeding of the 45th IEEE conference on decision and control, San Diego, USA, December, 13–15; 2006 p. 4957–62. Mokhtari A, Benallegue A, Orlov Y. Exact linearization and sliding mode observer for a quadrotor unmanned aerial vehicle. Int. J. Robot. Autom. 2006;21:39–49. Benallegue A, Mokhtari A, Fridman L. High-order sliding-mode observer for a quadrotor UAV. Int. J. Robust Nonlinear Control 2008;18:427–40. Sharifi F., Mirzaei M., Gordon B.W., Zhang Y.M. Fault tolerant control of a quadrotor UAV using sliding mode control. In: Proceeding of the conference on control and fault tolerant systems, Nice, France; 2010 p. 239–44. Coza C, Nicol C, Macnab CJB, Ramirez-Serrano A. Adaptive fuzzy control for a quadrotor helicopter robust to wind buffeting. J. Intell. Fuzzy Syst. 2011;22: 267–83. Eker İ. Second-order sliding mode control with experimental application. ISA Trans. 2010;49:394–405. Mondal S, Mahanta C. A fast converging robust controller using adaptive second order sliding mode. ISA Trans. 2012;51:713–21. Guo ZQ, Xu JX, Lee TH. Design and implementation of a new sliding mode controller on an underactuated wheeled inverted pendulum. J. Frankl. Inst. 2014;351:2261–82. Ashrafiuon H, Erwin RS. Sliding mode control of underactuated multibody systems and its application to shape change control. Int. J. Control 2008;81: 1849–58. Olfati-Saber R. Nonli near control of underactuated mechanical systems with application to robotics and aerospace vehicles. Ph.D. Thesis. Mass. Inst. Technol. 2001. Raffo GV, Ortega MG, Rubio FR. An integral predictive/nonlinear H1 control structure for a quadrotor helicopter. Automatica 2010;46:29–39. Raffo G.V., Ortega M.G., Rubio F.R. Backstepping/nonlinear H1 control for path tracking of a quadrotor unmanned aerial vehicle. In: Proceeding of the 2008 American control conference-ACC2008, Seattle, USA; 2008, p. 3356–61. Prouty RW. Helicopter performance, stability and control. Krieger Publishing Company; 2001. Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Trans. 2014. http://dx.doi.org/10.1016/j.isatra.2014.01.004. Alexis K, Nikolakopoulos G, Tzes A. Model predictive quadrotor control: attitude, altitude and position experimental studies. IET Control Theory Appl. 2012;6(12):1812–27.

Please cite this article as: Zheng E-H, et al. Second order sliding mode control for a quadrotor UAV. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.03.010i

Second order sliding mode control for a quadrotor UAV.

A method based on second order sliding mode control (2-SMC) is proposed to design controllers for a small quadrotor UAV. For the switching sliding man...
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