Robust adaptive backstepping fast terminal sliding mode controller for uncertain quadrotor UAV

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Contents lists available at ScienceDirect

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Robust adaptive backstepping fast terminal sliding mode controller for uncertain quadrotor UAV

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Engineering for Smart and Sustainable Systems Research Center, Mohammadia School of Engineers (EMI), Mohammed V University, Rabat, Morocco

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Article history: Received 3 January 2019 Received in revised form 10 June 2019 Accepted 16 July 2019 Available online xxxx

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Moussa Labbadi ∗ , Mohamed Cherkaoui

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Keywords: Quadrotor unmanned aerial vehicle (UAV) Fast terminal sliding mode controller Robust control Newton-Euler Backstepping Adaptive laws Uncertainties and Disturbances

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The problem of controlling the quadrotor orientation and position is considered in the presence of parametric uncertainties and external disturbances. Previous works generally assume that the flight controller parameters are constants. In reality, these parameters depend on the desired trajectory. In this article, a complete mathematical model of a quadrotor UAV is presented based on the EulerNewton formulation. A robust nonlinear fast control structured for the quadrotor position and attitude trajectory tracking is designed. The position loop generates the actual thrust to control the altitude of the quadrotor and provides the desired pitch and roll angles to the attitude loop, which allow the control of the quadrotor center of gravity in the horizontal plane. The attitude loop generates the rolling, pitching and yawing torques that easily allow the insurance of the quadrotors stability. The outer loop (position loop) uses the robust adaptive backstepping (AB) control to get the desired Euler-angles and the control laws. The inner loop (attitude loop) employs a new controller based on a combination of backstepping technique and fast terminal sliding mode control (AB-ABFTSMC) to command the yaw angle and the tilting angles. In order to estimate the proposed controller parameters of the position and the upper bounds of the uncertainties and disturbances of the attitude, online adaptive rules are proposed. Furthermore, the Lyapunov analysis is used to warranty the stability of the quadrotor UAV system and to ensure the robustness of the controllers against variation. Finally, different simulations were performed in the MATLAB environment to show the efficiency of the suggested controller. The sovereignty of the proposed controller is highlighted by comparing its performance with various approaches such as classical sliding mode control, integral backstepping and second order sliding mode controls. © 2019 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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1.1. Background and motivations

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Over the last decade, there was an increase in flying robots development for both civilian and military applications [1,2]. Drone field became an interesting topic for researchers due to their effectiveness in the different missions. Unmanned aerial vehicles (UAVs) can accomplish repetitive tasks, dangerous and dirty missions. However, advancement in the remote control capabilities, power storage, material structures, electronics, real time processing and computing sciences has led to modern drones, the efficiency of these machines is due to their reduced weight and their small size, which saves money and increases mission capabilities [3]. The applications of drones cover a wide range in many ar-

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Corresponding author. E-mail address: [email protected] (M. Labbadi).

https://doi.org/10.1016/j.ast.2019.105306 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

eas such as landslide mapping and dimension estimation, video streaming in congested cellular macro-cells, analysis of environmental impacts, diagenetic dolomite geobody mapping, forest fire management, agricultural operations, search and rescue, monitoring, mailing and delivery, and planetary exploration [2,4–15]. The quadrotor is the platform for researchers in the field of drones, this vehicle is the most nascent drone and several projects have been realized on the quadrotor [16–18]. The quadrotor has a simple mechanical design and good maneuverability [19], as opposed to the conventional helicopter which retains variable pitch rotor and has a complex mechanical structure, the quadrotor can vertically take-off, land and do not need airflow over the blades to move forward. Quadrotors can be operated both indoors and outdoors, they have fast and agile platforms that perform demanding maneuvers, and they can also hover or move in small and cluttered areas [20]. The major problems in the quadrotor system are propeller rotation, blades flapping, the changing of propeller rotation speed and center of mass position [21]. The compensation of these

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sources needs to design robust nonlinear controllers. These controllers have three challenges must be overtaken in the design [22, 23]: (i) the quadrotor dynamics are multiple-input multiple-output (MIMO); (ii) the quadrotor dynamics involve external disturbances and parametric uncertainties; (iii) Delays input time and multiple time-varying state of the quadrotor. A series of nonlinear controllers have been designed to study these characteristics such as robust linear parameter varying (LPV) observer [24], adaptive sliding mode approach [25], model predictive control [26], nonlinear adaptive estimation techniques [27], nonlinear robust H-infinity PID controller [28], nonlinear robust adaptive hierarchical sliding mode control approach [29], LQR-control methods [30], nonlinear PID-type controller [31], and terminal sliding mode controller [32].

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1.2. Literature review

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In [33], the dynamical model of a tri-rotor is presented via the Newton-Euler approach. A nonlinear controller is proposed to control the position and attitude of the vehicle. This controller combines a fuzzy logic technique and a backstepping sliding mode control approach. An optimization method of gradient descent algorithm has been used to determine the controller coefficients. The reference [34] presents a multi-agent system (MAS) for the multiple quadrotors flight. The Newton-Euler method is used to describe the mathematical model of the quadrotors. These UAV are considered as agents and in order to control their dynamics, a robust super twisting algorithm is developed. In [23], a robust adaptive finite-time stabilizer and tracker are proposed for controlling the quadrotor dynamics. An online adaptation procedure is developed to know the upper bound of parametric uncertainties and to determine the coefficients of the proposed controller. In [35], a new adaptive fuzzy sliding mode control is proposed to control the unmanned quadrotors. This controller uses the analysis of the Lyapunov to guarantee the stability of the system. The intelligent fuzzy logic is used to determine the best coefficients of this controller. In [36], a new nonlinear control strategy is developed by combining the integral backstepping with the sliding mode control (integral B-SMC) to stabilize the quadrotor attitude, track the desired flight trajectory and avoid the chattering phenomenon of control inputs provocated by conventional sliding mode control (SMC). Similarly, in [37], a robust controller is designed for stabilizing a quadrotor attitude. This controller consists of a nominal controller and a robust signal-based compensator. The proposed control method parameters have been tuned to increase the performance of the system. In [38], a combination of fast terminal sliding mode control and backstepping control (BFTSMC) is used for controlling a ducted fan engine. The parameter uncertainties and external disturbances are eliminated by this controller. The adaptive law is considered to estimate the controller coefficients, The global stability of the closed-loop system is proved by Lyapunov analysis. In [39], the authors propose an adaptive neural network backstepping controller for the attitude system of a 12-rotor UAV. The work developed in [40] deals with the control problem of uncertain nonlinear systems, the backstepping sliding mode control method combined with neural-network-based adaptive gain scheduling is proposed in this paper. In [41], a new nonlinear internal model control (NLIMC) approach is designed and the stability of the quadrotor in the presence of actuator and sensor fault is guaranteed. The control performance is evaluated through an aerial vehicle in the presence of wind. In [32], a global fast terminal sliding mode control (TSMC) technique is developed to design a controller for a quadrotor UAV position and attitude in finitetime. The total stability in the vehicle is guaranteed and the all state variables converge to zero. The chattering problem caused by the switching control action is eliminated via the TSMC. In [42],

a novel control procedure for fourth order systems is developed based on both adaptive super-twisting and terminal sliding mode control approaches. These control techniques reduce the chattering problem, establish finite-time convergence of the system and offer a parameter-tuning law to eliminate the external perturbations. In [43], a new fast-specified finite-time nonsingular terminal sliding mode control scheme is proposed for trajectory tracking of robotic airships. In order to realize finite-time convergence and to guarantee the stability of the nonlinear third-order systems, a novel reaching law based on terminal sliding mode control has been designed [44]. In [45], a quadrotor dynamics are obtained by the Newton-Euler method. The advantage of the backstepping approach with finite-time convergence techniques are used to generate a control law for stabilizing a mini rotorcraft. The reference [46] focuses on the second order sliding mode control to command the quadrotor position and attitude. Hurwitz stability analysis is used to obtain the proposed controller coefficients. In [47], a continuous multivariable finite-time output feedback control algorithm is presented for a quadrotor. The controller and observer are designed in the attitude loop and the position loop. An observer to estimate the external disturbances and unknown states is developed. In [48], the quadrotor model is divided a fully actuated subsystem and an under-actuated subsystem. A controller of the altitude and the yaw subsystem is designed through a terminal sliding mode control (TSMC) algorithm and a controller for the under-actuated subsystem is designed through utilization the classic sliding mode control (SMC). The state variables of drone converge to their desired values in short time. In [49], an adaptation mechanism and the Nussbaum gain technique are used to control the rotorcraft attitude and position. These techniques attenuating immeasurable disturbances, compensating system parametric uncertainties, and accommodating actuation faults. In [50], the dynamics of former aerial/underwater are modeled by the NewtonEuler principle, taking into account the effects of the buoyancy and drag phenomenon. An hybrid controller is designed for trajectory tracking of the full system and the stability in closed-loop is provided by using hybrid Lyapunov analysis. In [51], a robust structured control system design for quadrotor attitude and position trajectory tracking is developed. The outer loop is providing the roll/pitch tilting commands in the inner loop. The position loop utilizes robust generalized dynamic inversion (RGDI). An adaptive non-singular terminal sliding mode (ANTSM) is used to control the tilting angles. In [52], a terminal sliding mode control is developed to control a second-order nonlinear systems in the presence of some perturbations. In order to cope the disturbances, nonlinearities, and uncertainties of nonlinear systems, a new adaptive global terminal sliding mode control approach is proposed in [53]. In [54], a nano quadrotor is employed for the autonomous flight control development in global positioning system denied environments. A nonlinear flight controller is designed to hold the quadrotor in a desired trajectory position and to guarantee the attitude stability. In [55], a mathematic model of a quadrotor is presented and a robust nonlinear controller, which combines the sliding mode control technique and the backstepping control technique. A sliding mode controller is designed for attitude subsystem and the backstepping technique is applied to the position loop. An adaptive observer is considered for taking off mode, this observer is based on a fault estimation. In [56], an adaptive controller is presented to offer increased robustness to parametric uncertainties and to be effective in mitigating the effects of a loss of thrust anomaly. In [57], an omni-directional multirotor vehicle has been designed, modelled, and have been controlled. This controller allows simultaneously tracking the desired trajectory vehicle position and attitude. The proposed control is based on multiple cascaded control loops, where the inner control loops can track the input commands arbitrarily and each control loop is designed through the feedback

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linearization. In [58], a control algorithm is proposed for the visual target tracking system which consists of fixed-wing UAV. Seven fuzzy controllers are used to stable UAV and to remove the external disturbances, the information obtained from the images makes generating the roll command possible. These proposed algorithms are able to accomplish a moving target. The work developed in [59] deals with a combination of backstepping technique and intelligent fuzzy to generate a controller for stabilizing the quadrotor attitude. The backstepping methodology is designed to guarantee the stability of the system, whereas the fuzzy logic is used as a compensator to attenuate the unknown uncertainties and disturbances. Based on backstepping sliding mode control approach, the author of in [60] propose an adaptive fuzzy control technique for stratospheric satellites under uncertainty and input constraints. In [61], the problem of trajectory tracking for robotic airships is addressed by using a neural network approximation-based nonsingular terminal sliding mode control approach. In [62], a robust controller is recommended for steering a quadrotor and for rejecting force disturbances. This controller is based on the adaptive backstepping and the Lyapunov analysis of which the goal is to guarantee the overall stability of the vehicle in the closed loop. The force disturbances are estimated through the utilization of a smooth projector operator. The experimental validation of this controller is developed. In [63], a sliding mode technique is proposed for controlling a quadrotor. To increase the performances of this controller and to eliminate the steady-state error an integral action in the sliding surface is included. At last, the simulations and experimental studies have demonstrated the robustness and effectiveness of the proposed controller in an outdoor environment. In [64], an extended Kalman filter is developed to control and estimate the state variables of the quadrotor, this estimation is based on noisy measurements. In [65], a nonlinear controller combines a nonsingular modified super-twisting controller with a high order sliding mode observer to enable a quadrotor tracking a desired trajectory in presence the unmodeled dynamics and external disturbances. The work presented in [19] treats a global comparative survey of nonlinear and adaptive intelligent control techniques for quadrotor UAV, in this study the adaptive terminal sliding mode control is the most suitable approach for quadrotor platform.

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1.3. Contribution

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The purpose of this research is to develop a robust fast controller to track a desired trajectory of a 6DOF quadrotor attitude and position. The attitude controller is based on adaptive backstepping fast terminal sliding mode control (ABFTSMC). The position controller is based on adaptive backstepping (AB). This involves the following points: a mathematical model of the UAV based on the Newton-Euler formalism, the development of a adaptive nonlinear control that satisfies the attitude stability, compensate the parametric uncertainties, external disturbances, and a fast convergence for state variables of quadrotor. Besides, an online adaptation procedure of the controller parameters was presented. The following points list the contributions of this paper:

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• Compared with the integral backstepping [66], the first order

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sliding mode controller, and the second order sliding mode control approach [46], the proposed AB-ABFTSM controller has a strong robustness against time-varying uncertainties, nonlinearities and external disturbances and has characteristics such as simplicity, and continuous control signals. • A new position and attitude controller is proposed for uncertain quadrotors with external disturbances. • Finite-time control, accurate tracking and fast convergence of a quadrotor can be achieved.

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Fig. 1. The drone configuration with four rotors.

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• An online estimation is done for the upper bounds of the uncertainties and disturbances of the attitude. 1.4. Paper organization The remaining part of the paper is structured as follows: The Section 2 presents the definition of coordinate system and a mathematical model of quadrotor in the presence of aerodynamics effects. An adaptive backstepping fast terminal sliding mode controller for attitude quadrotor is designed and an adaptive backstepping technique for position quadrotor is proposed in Section 3, these controllers generate the command for the quadrotor UAV. The results of the proposed controllers for the quadrotor position and attitude are presented in Section 4. Finally, the conclusion is given in Section 5.

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2. Mathematical modeling of flight

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The quadrotor is an under-actuated system because it has four inputs and six outputs. The translational motion in three directions is assured by three variables (x, y, z) and the variables (φ , θ , ψ ) include rotational motion around three axes. Fig. 1 is the schematic of the quadrotor with four rotors. Two pairs of rotors (1, 3) and (2, 4) turn opposite direction (See Fig. 1). The roll, pitch and yaw rotations can be obtained by changing the propeller speeds. The roll rotation is performed via changing the propeller speed of motors two and four. The pitch rotation is realized via modifying the propeller speed of motors one and three. Yaw rotation is done from counter torque difference between (1, 3) or (2, 4). The vertical motion is screamed by the increasing or decreasing the total speed of the rotors [23]. Using some subsequent assumptions to establish the model of the quadrotor [67]:

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Assumption 1. The quadrotor’s structure is symmetrical.

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Assumption 2. The quadrotor’s and propeller structures are rigid.

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Assumption 3. The thrust and drag are proportional to the square of the rotor speeds.

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2.1. The quadrotor dynamic

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The mathematical model of the quadrotor is obtained by using Assumptions (1), (2) and (3). As shown in Fig. 1 an inertial

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reference frame of coordinates E = { O E , e x , e y , e z } and a bodyfixed frame of coordinates B = { O B , b x , b y , b z } where b x is the longitudinal axis, b y is the lateral axis and b z is the vertical direction in hover conditions. The vector η = [φ θ ψ] T and the vector ξ = [x y z] T describing the rotorcraft position and orientation in frame O E , where φ denotes the angle of roll around the x-axis, θ denotes the pitch angle around the y-axis, and ψ denotes the yaw angle around the z-axis. These angles are bounded as follows: roll angle by (− π2 < φ < π2 ), pitch angle by (− π2 < θ < π2 ) and yaw angle by (−π < ψ < π ). The quadrotor’s orientation to a fixed inertial frame is defined by the rotation matrix.

⎡

⎡

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−r

1 S =⎣ r −q

0 p

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ωb = R r η˙

ωb = ( p q r )

T

and η˙ =

(3)

1 0 Rr = ⎣ 0 Cφ 0 −Sφ

−Sθ

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Sφ Cθ ⎦ Cφ Cθ

(4)

with C (.) and S (.) denote the cos(.) and sin(.) respectively. The system modeling is written based on the principle NewtonEuler as follows [23,68–70]:

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ξ˙ = v m v˙ = F f − F g − F d ⎩ ˙ J = − T × J + τb − τc − τa

(5)

where m is total mass of the system, J = diag ( I xx , I y y , I zz ) is a symmetric positive definite matrix with I xx , I y y , and I zz denote the rotary inertia respect to the O B b x , O B b y , and O B b z axes, respectively. F f is the total thrust produced via four propellers with respect to the body-fixed frame as:

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4 Cφ Sθ Cψ + Sφ Sψ F f = ⎣ Cφ Sθ Sψ − Sφ Cψ ⎦ Fi Cφ Cθ i =1

(6)

with F i = kp 2i , kp is a parameter depends on the blades geometry and air density and i is the angular rotor speed. F d denotes the resultant of the forces along (x, y , z) axis as:

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kdx Fd = ⎣ 0 0

0 kdy 0

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d( F 1 − F 3 ) d( F 2 − F 4 )

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⎦

(9)

cd ( 21 − 22 + 23 − 24 )

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d denotes the distance between the rotor axis and the center of mass of the quadrotor and cd is the drag coefficient. The terms τc and τa are the resultant torques due to the gyroscopic effects and the resultant of aerodynamic friction torque, the expressions are given by

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τa

kax =⎣ 0 0

(10)

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T J r ⎣

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0 0

(−1)i +1 i

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⎦

(11)

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⎤

0 0 ⎦ ξ˙ kdz

(7)

where J r is the rotor inertia and (kax , kay , kaz ) are aerodynamic friction coefficients. The quadrotor equations are given by

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X˙ = f ( X ) + g ( X )U

(12)

˙ θ, θ, ˙ ψ, ψ, ˙ x, x˙ , y , y˙ , z, z˙ ] ∈ 12 represents the where X = [φ, φ, state variables, f ( X ) and g ( X ) are nonlinear functions. The Eq. (12) is expended as follows: x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5 x˙ 6 x˙ 7 ⎪ ⎪ ⎪ ⎪ x˙ 8 ⎪ ⎪ ⎪ ⎪ x˙ 9 ⎪ ⎪ ⎪ ⎪ ˙ x ⎪ 10 ⎪ ⎪ ⎪ ˙ x ⎪ 11 ⎪ ⎩ x˙ 12

= = = = = = = = = = = =

x2 a1 x4 x6 + a2 x4 + a3 x22 + b1 τφ x4 a4 x2 x6 + a5 x2 + a6 x24 + b2 τθ x6 a7 x2 x4 + a8 x26 + b3 τψ x8 1 a9 x8 + m (C x1 S x3 C x5 + S x1 S x5 ) F f x10 1 a10 x10 + m (C x1 S x3 S x5 − S x1 C x5 ) F f x12 1 a11 x12 − g + m (C x1 C x3 ) F f

with: a1 = a6 =

−kay I yy

( I y y − I zz ) I xx

, a7 =

− kmdz , b1 =

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d , I xx

, a2 =

( I xx − I y y )

I zz b2 = I d yy

− r J r

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(13)

xx

, a8 =

−kaz

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1 I zz

I zz

yy

, a9 =

−kdx m

Ff

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kp ⎢ τφ ⎥ ⎢ k p ⎢ ⎥=⎢ ⎣ τθ ⎦ ⎣ 0 τψ cd

kp 0 kp −cd

kp −k p 0 cd

, a10 =

yy

−kdx m

, a11 =

and r = 1 − 2 + 3 − 4 .

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21 kp ⎢ 2⎥ 0 ⎥ ⎥ ⎢ 2 ⎥ −k p ⎦ ⎣ 23 ⎦ −cd 24

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J (I −I ) , a3 = −Ikax , a4 = zzI xx , a5 = I r r ,

I xx

, b3 =

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The control inputs of the quadrotor are defined as

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⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

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(14)

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3. Finite-time adaptive flight control of quadrotor

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0 0 ⎦  2 kaz

and

τc =

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0 kay 0

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(2)

The transformation matrix between ˙ T can be described as [55]. (φ˙ θ˙ ψ)

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τb = ⎣

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i =1

q −p ⎦ 0

where,

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⎤

Cφ Sθ Cψ + Sφ Sψ Cφ Sθ Sψ − Sφ Cψ ⎦ Cφ Cθ

R is an orthogonal matrix with determinant 1, nonsingular and satisfies that R˙ = R S (ωb ) where ωb = ( p q r ) T denotes the angular velocity with a body-fixed frame respect, S represents a skew symmetric matrix:

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Sφ Sθ Cψ − Cφ Sψ Sφ Sθ Sψ + Cφ Sψ Sφ Cθ

Cθ Cψ R = ⎣ Cθ Sψ −Sθ

quadrotor as

where kdx , kdy and kdz indicate the translation drag coefficient. F g is gravity force can be written as

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0 Fg = ⎣0⎦ g The term

(8)

τb denotes the moment developed by four rotors of the

This section is dedicated to the design of the flight controller for the quadrotor system. The goal of this control law is to design the quadrotor position (lateral and vertical) and attitude control. Under the flight controller, the stability in the finite-time is ensured in closed-loop and the trajectories track their references. Similar to [55,54], the considered quadrotor system in this work has an under-actuated dynamics, i.e., four control inputs are used

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for the control of six degrees of freedom (φ, θ, ψ, x, y , z). In what follows, a new controller by integrating fast terminal sliding mode controller and backstepping theories for attitude trajectory tracking control is proposed. The BFTSMC technique is designed for fast convergence of attitude state variables to stabilize the pitch/roll angles, and track the yaw angle trajectory in finite-time. The regular backstepping control technique is designed to ensure the trajectory tracking of the position flight path desired. The synoptic scheme presented in Fig. 2 depicts the different steps to control the quadrotor system.

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3.1. Robust adaptive backstepping control design for position trajectory tracking control

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The adaptive backstepping features is good robustness for the flight path tracking problems and allows to control of quadrotor UAV position in the presence of model uncertainties and external disturbances. AB is a recursive procedure for tracking the quadrotor position trajectory, handling uncertainties to a certain level and stabilizing the system based on Lyapunov approach. In this part, an adaptive backstepping for position trajectory tracking control is designed. Then the output vector for the quadrotor position is considered as [x, y , z]. The design procedure for the quadrotor position controller is as follows:

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Step 1: defining the position tracking errors as

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⎡

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⎤

e x1 x7 − x7d ⎣ e y1 ⎦ = ⎣ x9 − x9d ⎦ e z1 x11 − x11d

(15)

wherein x7d , x9d , and x11d are respectively the reference trajectories. The time derivative of the position tracking errors is

⎡

⎤

⎤

⎡

⎡

⎤

e˙ x1 x˙ 7 − x˙ 7d x8 − x˙ 7d ⎣ e˙ y1 ⎦ = ⎣ x˙ 9 − x˙ 9d ⎦ = ⎣ x10 − x˙ 9d ⎦ e˙ z1 x˙ 11 − x˙ 11d x12 − x˙ 11d

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⎤

⎡

⎤

⎡

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⎤

v˙ 7 e x1 e˙ x1 e x1 (x8 − x˙ 7d ) ⎣ v˙ 9 ⎦ = ⎣ e y1 e˙ y1 ⎦ = ⎣ e y1 (x10 − x˙ 9d ) ⎦ v˙ 11 e z1 e˙ z1 e z1 (x12 − x˙ 11d )

86

(18)

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Virtual control inputs x8d , x10d and x12d are selected [38] as follows:

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x8d = −c x1 e x1 + x˙ 7d ⎣ x10d = −c y1 e y1 + x˙ 9d ⎦ x12d = − z z1 e z1 + x˙ 11d

92

⎡

⎤

⎤

⎤

⎡

−c x1 e 2x1 v˙ 7 ⎣ v˙ 9 ⎦ = ⎣ −c y1 e 2y1 ⎦ ≤ 0 v˙ 11 −c z1 e 2

(20)

z1

As a result, the Lyapunov theorem will ensure the stability of the quadrotor position.

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⎡

108

⎤

⎡

⎤

e x2 x8 − x8d ⎣ e y2 ⎦ = ⎣ x10 − x10d ⎦ e z2 x12 − x12d

109

(21)

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In a similar way, the Lyapunov candidate functions of the step 2 are determined as:

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⎡

115

(16)

⎤

v8 ⎣ v 10 ⎦ = v 12

⎤ v 7 + 12 e 2x2 ⎣ v 9 + 1 e 2y2 ⎦ 2 v 11 + 12 e 2z2 ⎡

(22)

⎤

⎡

(17)

⎤

⎡

116 117 119 120

⎤

v˙ 8 v˙ 7 + e x2 e˙ x2 ⎣ v˙ 10 ⎦ = ⎣ v˙ 9 + e y2 e˙ y2 ⎦ v˙ 12 v˙ 11 + e z2 e˙ z2

⎡

114

118

The time derivative of the Lyapunov functions is:

⎡

⎤

The time derivative of Eq. (17) is Trajectories

98

Step 2: The second tracking errors are given by

121

(23)

122 123 124

⎡

2 z1

97

103

−c x1 e 2x1 + e x1 e x2 + e x2 (a9 x8 + v x − x¨ 7d ) v˙ 8 ⎣ v˙ 10 ⎦ = ⎣ −c y1 e 2y1 + e y1 e y2 + e y2 (a10 x10 + v y − x¨ 9d ) ⎦ v˙ 12 −c z1 e 2z1 + e z1 e z2 + e z2 (a11 x12 + v z − x¨ 11d )

⎡

96

100

From Eqs. (19), (21) and Eq. (23) one gets

⎤

94 95

where c x1 , c y1 and c z1 are a non-zero positive constants. Substituting (19) into (18) gives the derivatives of Lyapunov functions.

⎡

91 93

(19)

To ensure the position stability, the first Lyapunov candidate functions is given by 1 2 e v7 2 x1 ⎣ v 9 ⎦ = ⎣ 1 e 2y1 ⎦ 2 1 2 v 11 e

87

112

⎤

⎡

58 59

84

Fig. 2. The general block diagram for the quadrotor control.

125 126

⎤

127

(24)

The corresponding control laws for the quadrotor position are designed as follows:

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M. Labbadi, M. Cherkaoui / Aerospace Science and Technology ••• (••••) ••••••

6

1

v x =(−e x1 − cˆ x2 e x2 − c x1 (e x2 − c x1 e x1 ) − a9 x8 + x¨ 7d )

2

v y =(−e y1 − cˆ y2 e y2 − c y1 (e y2 − c y1 e y1 ) − a10 x10 + x¨ 9d )

3 4 5 6 7 8 9 10 11

(25)

v z =(−e z1 − cˆ z2 e z2 − c z1 (e z2 − c z1 e z1 ) − a11 x12 + x¨ 11d ) where cˆ x2 , cˆ y2 and cˆ z2 are the estimate of c x2 , c y2 and c z2 , respectively. Theorem 1. [38] If the control laws Eq. (25) with the following adaptation law Eq. (26) are applied to quadrotor position system, the asymptotic stability of the system is guaranteed.

12 13 14 15 16 17 18 19 20 21 22

Moreover, the following parameter adaptation laws are given as follows:

⎧ 2 ˙ ⎪ ⎨ cˆ x2 = γ7 e x2 ˙cˆ y2 = γ9 e 2 y2 ⎪ ⎩ ˙ˆ c z2 = γ11 e 2z2

(26)

25

where γ7 , γ9 and

γ11 are positive constants. Using the Barbalat  s Lemma to prove the Theorem 1. This lemma is needed.

26

Lemma 1. [38,71] if f (t ) is a uniformly continuous function and lim

t →+∞

t

67

⎧ ⎨ vx = v = ⎩ y vz =

72

1 (C x1 S x3 C x5 m 1 (C x1 S x3 S x5 m

+ S x1 S x5 ) F f − S x1 C x5 ) F f − g + m1 (C x1 C x3 ) F f

29 30 31 32 33 34 35 36 37 38 39

f (τ )dτ exists, then f (t ) converges to zero asymptotically.

0

v 78 = v 8 +

1 2γ 7

 c 2x2

(27)

where  c x2 represents the estimation error. The time derivative of Eq. (27) is

v˙ 78 =

− cˆ x2 e 2x2

−c x1 e 2x1

+

1

γ7

 c x2 c˙ x2

= −c x1 e 2x1 − (c x2 − c x2 )e 2x2 −

42 43 44 45 46 47 48 49

=

−c x1 e 2x1

− c x2 e 2x2

+ c x2 (e 2x2

1

γ7

 c x2 c˙ˆ x2

(28)

1 − c˙ˆ x2 )

γ7

In Eq. (28) the term  c x2 (e 2x2 − γ1 c˙ˆ x2 ) equals 0. By considering c x2 7 constant, the time derivative of  c x2 written as follows  c˙ x2 = 0 − c˙ˆ x2 . Hence, the Lyapunov function dynamic defining in Eq. (28) becomes as follows:

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

71

74 76 77 78 79 80

(33)

81 82 83

In this part, an adaptive backstepping law for position trajectory tracking control of a quadrotor UAV is designed. Adaptive laws are used to determine the second parameters of the proposed control. The stability of the position system is proved by Lyapunov analysis. On the basis of the preceding the backstepping and SMC laws, a robust adaptive backstepping fast terminal sliding mode controller for attitude trajectory tracking control is presented in the following part. 2

85 86 87 88 89 90 91 92 93

Proof. To prove the stability of the system and to determine cˆ x2 , cˆ y2 and cˆ z2 parameters, The Lyapunov approach is used. For cˆ x2 : introduce the Lyapunov candidate function of the position subsystem.

40 41

70

75

27 28

69

73

(32)

Based on (32), the actual total thrust F f and the desired attitude angles (φd θd ) can be written as follows:

⎧  2 2 2 ⎪ ⎪ ⎨ F f = m v x + v z S+ (vv−z C+ gv) ψd x ψd y φd = arctan(C θd ( )) v z +g ⎪ ⎪ ⎩ θ = arctan( C ψd v x + S ψd v y ) d v z+g

68

84

23 24

where c x1 , c y1 , c z1 , c x2 , c y2 , and c z2 are the position parameters. Thus, the stability of the position system is guaranteed by Eqs. (30) and (31), ensuring the flight trajectory tracking capability. By recalling the definition of the elements in model (13), these last are given by

v˙ 78 = −c x1 e 2x1 − c x2 e 2x2 ≤ 0

(29)

Therefore, the condition of the stability is satisfied thought Eq. (29). In order to ensure the position system stability. The Lyapunov candidate function for position system is chosen as:

1

1

v sp = (e 2x1 + e 2x2 +  c 2 + e 2y1 + e 2y2 + 2 γ7 x2 1 2  + e 2z2 + c z2 )

1

γ9

(30)

The time derivative of the Lyapunov position is

v˙ sp =

(−c x1 e 2x1

− c x2 e 2x2

− c y1 e 2y1

− c y2 e 2y2

− c z1 e 2z1

≤0

(31)

95

⎡

⎤

98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 115

⎤

⎡

97

114

Step 1: Introduce the attitude tracking errors as

116

eφ x1 − x1d ⎣ e θ ⎦ = ⎣ x3 − x3d ⎦ eψ x5 − x5d

(34)

117 118 119

wherein x1d , x3d and x5d are respectively the reference trajectories. The time derivative of the attitude tracking errors is

120

⎡

122

⎤

⎡

⎤

⎡

⎤

121 123

(35)

124 125 126

In order to ensure the attitude stability, let us introduce the first following Lyapunov candidate functions.

⎡ − c z2 e 2z2 )

94 96

The adaptive backstepping controller for the quadrotor position presented above is effective in term of flight trajectory tracking. It guarantees stability of the position subsystem and provides tracking capability of variables (x, y, z). However, the controller has some deficiencies in realizing attitude control. For that, a combination of fast terminal sliding mode control and backstepping control is proposed for controlling the quadrotor attitude. The TSMC has many advantages [19] removal of chattering phenomenon, finitetime convergence of the state variables, the nonlinear surfaces of TSMC provide faster convergence and compensate for parametric variations and external disturbances. This part presents an adaptive backstepping fast terminal sliding mode controller for attitude trajectory tracking control. The main objectives of the proposed controller are the insurance of the stability of the quadrotor and that the Euler angle trajectory (φ θ ψ) converges to the reference trajectories (φd θd ψd ). The ABFTSMC design of attitude trajectory tracking control is as follows:

e˙ φ x˙ 1 − x˙ 1d x2 − x˙ 1d ⎣ e˙ θ ⎦ = ⎣ x˙ 3 − x˙ 3d ⎦ = ⎣ x4 − x˙ 3d ⎦ e˙ ψ x˙ 5 − x˙ 5d x6 − x˙ 5d

 c 2y2 + e 2z1

γ11

3.2. Robust adaptive backstepping fast terminal sliding mode controller design for attitude trajectory tracking control

⎤

⎤

⎡

1 2 e v1 2 φ ⎢ ⎣ v 3 ⎦ = ⎣ 1 e2 ⎥ 2 θ ⎦ 1 2 v5 eψ 2

127 128 129 130

(36)

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M. Labbadi, M. Cherkaoui / Aerospace Science and Technology ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Deriving Eq. (36) as follows:

⎡

⎤

⎤

⎡

⎤

⎡

v˙ 1 e φ e˙ φ e φ (x2 − x˙ 1d ) ⎣ v˙ 3 ⎦ = ⎣ e θ e˙ θ ⎦ = ⎣ e θ (x4 − x˙ 3d ) ⎦ v˙ 5 e ψ e˙ ψ e ψ (x6 − x˙ 5d ) The system virtual input controls are selected [38] as follows:

⎡

⎤

x2d = sφ − c φ e φ + x˙ 1d ⎣ x4d = sθ − c θ e θ + x˙ 3d ⎦ x6d = sψ − c ψ e ψ + x˙ 5d

(38)

c φ , c θ and c ψ are positive constants. sφ , sθ and sψ are the attitude sliding surfaces. Step 2: Choose the attitude sliding surfaces: Using the fast terminal sliding surfaces [38,52] as follows:

16

⎡

17

⎢ ⎥ p /q ⎣ sθ = e˙ θ + αθ e θ + βθ e θ θ θ ⎦ p ψ /qψ sψ = e˙ ψ + αψ e ψ + βψ e ψ

18 19 20 21 22 23 24 25 26 27 28 29

p φ /qφ

sφ = e˙ φ + αφ e φ + βφ e φ

⎤ (39)

where (αφ , αθ , αψ , βφ , βθ , βψ ) are non-zero positive constants p p p and 0 < ( q φ , q θ , q ψ ) < 1. The time derivative of terminal sliding φ

θ

ψ

surfaces Eq. (39) is as follows:

⎡

s˙ φ = e¨ φ + αφ e˙ φ +

⎢ ⎢ s˙ θ = e¨ θ + αθ e˙ θ + ⎣ s˙ ψ = e¨ ψ + αψ e˙ ψ +

32 33 34 35 36 37 38

⎤

( p /q −1) pφ β e φ φ qφ φ φ ( p /q −1) ⎥ pθ ⎥ β e θ θ qθ θ θ ⎦ ( p ψ /qψ −1) pψ β e qψ ψ ψ

39

( p −q )/q

si = e˙ i + αi e i + βi (e i )

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

(e i ) =

61 62 63 64 65 66

pi qi

ei , ei ,

if if

where si = e˙ i + αi e i + stant, and i = (φ, θ, ψ).

(42)

⎤

⎡

v1 + v2 ⎣ v4 ⎦ = ⎢ ⎣ v3 + v6 v5 +

1 2 s 2 φ 1 2 s 2 θ 1 2 s 2 ψ

μi denotes a threshold small con-

⎤ ⎥ ⎦

(43)

⎤

⎡

⎤

v˙ 2 v˙ 1 + e φ s˙ φ ⎣ v˙ 4 ⎦ = ⎣ v˙ 3 + e θ s˙ θ ⎦ v˙ 6 v˙ 5 + e ψ s˙ ψ

(44)

Front Eq. (40), the time Lyapunov functions can be written as follows:

⎡

−c φ e φ + e φ sφ + sφ (¨e φ + αφ e˙ φ 2

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v˙ 2 ⎢ −c θ e θ + e θ sθ + sθ (¨e θ + αθ e˙ θ ⎥ ⎥ ⎣ v˙ 4 ⎦ = ⎢ ( p /q −1) ⎥ ⎢ + qpθθ βθ e θ θ θ ) ⎥ ⎢ v˙ 6 ⎢ −c e 2 + e s + s (¨e + α e˙ ⎥ ψ ψ ψ ψ ψ ψ⎦ ⎣ ψ ψ ( p /q −1) p + qψψ βψ eψ ψ ψ ) ⎡

⎤

69

ψ ψ

qψ

70 71 72

(46)

⎧ ⎨ τφ = τφ eq + τφ s τθ = τθ eq + τθ s ⎩ τψ = τψ eq + τψ s

74 76

ψ ψ

1 (−kφ sign(sφ )) b1 1 (−kθ sign(sθ )) b2 1 (−kψ sign(sψ )) b3

73 75 77

The quadrotor attitude switching control laws is choosed as follows:

⎧ ⎪ ⎨ τφ s = τθ s = ⎪ ⎩τ ψs =

68

78 79 80 81

(47)

82 83 84 85 86 87

(48)

88 89 90

So, the quadrotor attitude ABFTSMC controls are designed as follows:

⎧ τφ = b11 (−(a1 x4 x6 + a2 x4 + a3 x22 ) + x¨ 1d − αφ (sφ ⎪ ⎪ ⎪ ⎪ ( p /q −1) p ⎪ ⎪ −c φ e φ ) + qφφ βφ e φ φ φ − kˆ φ sign(sφ )) ⎪ ⎪ ⎪ ⎨ τθ = 1 (−(a4 x2 x6 + a5 x2 + a6 x2 ) + x¨ 3 − αθ (sθ 4 b2 ( p θ /qθ −1) pθ ⎪ − c e ) + β e − kˆ θ sign(sθ )) θ θ θ ⎪ θ qθ ⎪ ⎪ 1 2 ⎪ ¨ τ = (−( a x x + a x ) + x − αψ (sψ ⎪ ψ 7 2 4 8 6 5d ⎪ b3 ⎪ ⎪ ( p ψ /qψ −1) pψ ⎩ −c ψ e ψ ) + βψ e − kˆ ψ sign(sψ )) qψ

91 92 93 94 95 96

(49)

97 98 99 100 101

ψ

Remark 2. In order to solve the chattering problem, the discontinuous control component (sign(.) function) in Eq. (49) is replaced by the tanh(.) function.

103 104 105

In addition, the parameters of the adaptive laws are as follows:

⎧ ˙ ⎪ ⎪ ⎨ kˆ φ = γφ |sφ | ˙ kˆ θ = γθ |sθ | ⎪ ⎪ ⎩ k˙ˆ ψ = γψ | s ψ |

107 108 109 110

(50)

111 112 113

where (kˆ φ , kˆ θ , kˆ φ ) denote the online estimate of (kφ , kθ , kφ ) and

114

(γφ ,

115

γθ ,γφ ) are a non-zero positive constants.

116

The time derivative of Eq. (43) are

⎡

⎧ τφ eq = b11 (−(a1 x4 x6 + a2 x4 + a3 x22 ) + x¨ 1d − αφ (sφ ⎪ ⎪ ⎪ ⎪ ( p /q −1) p ⎪ ⎪ −c φ e φ ) + qφφ βφ e φ φ φ ) ⎪ ⎪ ⎪ ⎨ τθ eq = 1 (−(a4 x2 x6 + a5 x2 + a6 x2 ) + x¨ 3d − αθ (sθ 4 b2 ( p /q −1) ⎪ −c θ e θ ) + qpθθ βθ e θ θ θ ) ⎪ ⎪ ⎪ 1 2 ⎪ ⎪ ⎪ τψ eq = b3 (−(a7 x2 x4 + a8 x6 ) + x¨ 5d − αψ (sψ ⎪ ⎪ ( p /q −1) p ⎩ −c e ) + ψ β e ψ ψ )

106

In order to obtain stabilizing control laws of the quadrotor attitude, let us define the Lyapunov candidate functions.

⎡

67

102

si = 0 or si = 0, |e i | > μi si = 0, |e i |  μi p /q βi e i i i ,

59 60

(41)

The function (e i ) is defined as,

40 41

(40)

Remark 1. It is worth noting that the term e i i i i will approach to infinity if e i = 0, that is singularity. However, to avoid this problem, a sliding surface is modified as follows [72]:



According to Eq. (45), the quadrotor attitude equivalent control law is designed as follows:

Hence, the whole control inputs are

30 31

(37)

7

p + qφφ 2

( p /q −1) βφ e φ φ φ )

(45)

Proof. To prove the stability of the subsystem and to determine the adaption parameters. We’ll be based on a Lyapunov candidate function, for example the attitude roll and it will be given as follows:

v 12 = v 2 +

1

 k2 2γ φ φ

(51)

where  kφ is the corresponding estimate error. The time derivative of Eq. (51):

v˙ 12 = −c φ e 2φ + sφ e φ + sφ (a1 x4 x6 + a2 x4 + a3 x22 + b1 τφ pφ 1 ( p /q −1) + αφ (sφ − c φ e φ ) + βφ e φ φ φ ) +  kφ k˙ φ qφ γφ Substituting (49) into (52) gives

117 118 119 120 121 122 123 124 125 126 127 128

(52)

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6

72

7

73

8

74

9

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10

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11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85 86

20

Fig. 3. Block diagram of ABFTSMC for the quadrotor control.

21 22 23

v˙ 12 = −c φ e 2φ + sφ e φ + sφ (−kˆ φ sign(sφ )) +

24 25

= −c φ e 2φ − sφ (kφ −  kφ )sign(sφ ) −

26 27

≤0

32

37 38 39 40 41 42 43 44

lim si = lim si = [˙e i + αi e i + βi e p i /qi ] = 0

t →t f

qφ

αφ (qφ − p φ ) qθ

47

αθ (qθ − p θ )

48

αψ (qψ − p ψ )

50

53 54 55 56 57 58 59 60

ln

ln

qψ

49

αφ e(tr φ )(qφ − pφ )/qφ + βφ βφ

αθ e(tr θ )(qθ − pθ )/qθ + βθ

ln

βθ

65 66

Parameter

Value

91

g (m/s2 ) m (kg ) I xx (kg .m2 ) I y y (kg .m2 ) I zz (kg .m2 ) I r (kg .m2 ) k1 ( N /m/s)

9.81 0.486 3.827e-3 3.827e-3 7.6566e-3 2.8385e-5 5.5670e-4

k2 k3 k4 k5 k6 kp cd

5.5670e-4 5.5670e-4 5.5670e-4 5.5670e-4 5.5670e-4 2.9842e-3 3.2320e-2

92

( N /m/s) ( N /m/s) ( N /m/s) ( N /m/s) ( N /m/s) ( N .s2 ) ( N .m.s2 )

93 94 95 96 97 99 100

Table 2 The reference trajectories of the position and yaw angle. Variable

Value

αψ e(tr ψ

)(qψ − p ψ )/qψ

+ βψ

βψ

[xd , yd , zd ]

[0.6, 0.6, 0.6] [0.3, 0.6, 0.6] [0.3, 0.3, 0.6] [0.6, 0.3, 0.6] [0.6, 0.6, 0.6] [0.6, 0.6, 0.0] [0.5] rad [0.0] rad

t →t f

]

lim e˙ i (i = φ, θ, ψ) = 0

t →t f

m m m m m m

0 10 20 30 40 50 0 50

(56)

The ABFTSMC synoptic diagram for the quadrotor attitude is shown in Fig. 3. Finally, the control laws are obtained from Eqs. (25) and (49) via the adaptive backstepping control and adaptive fast terminal sliding mode controller techniques. However, these control laws ensure the quadrotor stability in the closed loop and ensure the obtaining of a better tracking of the flight path. 2 4. Results and discussion In this section, simulation results are presented in order to verify the effectiveness and the performances of the adaptive backstepping fast terminal sliding mode controller for the position

101 102 103 104 105 106 107 108 109 110 111 112 113

Table 3 Control ters.

By defining βi , αi , p i , qi  0, then it is obtained

lim e i (i = φ, θ, ψ) = 0,

Time(s)

(55)

,

63 64

Value

[ψd ] [ψd ]

,

61 62

90

Parameter

where t f = tr + t s , t s is the time interval that the initial error e (0) = [e φ (0), e θ (0), e ψ (0)] = 0, which is expressed as [74].

46

52

(54)

t →t f

t s =[

89

98

Define tr = [tr φ , tr θ , tr ψ ] as the reaching time, when si (i = φ, θ, ψ) reaches zero. The attitude states can reach the terminal sliding surfaces in finite-time tr . Using Eq. (39), we get

45

51

(53)

= −c φ e φ − kφ |sφ |

31

36

˙  kφ kˆ φ

2

30

35

Table 1 Quadrotor parameters.

γφ

29

34

γφ

γφ

88

 kφ k˙ φ

1 ˙ = −c φ e 2φ − kφ |sφ | +  kφ (|sφ | − kˆ φ )

28

33

1

1

87

114

system

parame-

115 116

Parameters

Value

117

cφ , cθ , cψ

0.0527 0.2146 9.3414 2.4429 4.3864 17.4513 2.0000 3.2270

118

βφ , βθ , βψ αφ , αθ , αψ pφ , pθ , pψ qφ , qθ , qψ γφ , γθ , γψ γ7 , γ9 , γ11 c x1 , c y1 , c z1

119 120 121 122 123 124 125 126

and attitude tracking of the quadrotor UAV. The initial state values of the quadrotor for the simulation tests are [0, 0, 0]rad and [0, 0, 0]m. Besides, the quadrotor’s physical parameters are listed in Table 1. Therefore, the proposed flight controllers parameters are given in Table 3.

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83

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19

85

20

86

21

87

22

88

23

89

24

90 91

25 26

Fig. 4. The quadrotor position tracking response in direction x, y and z.

92

27

93

28

94

29

95

30

96

31

97

32

98 99

33

Fig. 6. Linear velocities of the quadrotor.

34

In order, to demonstrate the robustness of the proposed flight controller, the nonlinear sliding mode controller, the integral backstepping method [66], and the second order sliding mode control technique [46] are considered for the sake of comparison. Also, the simulations are performed with parametric uncertainties and external disturbances caused by wind gusts and other factors.

36 37 38 39 40 41

4.1. Case 1

43 45 46 47 48 49 50 51 52 53

Fig. 5. The quadrotor attitude tracking response.

55 57 58 59 60 61

Remark 3. The gains of the proposed AB-ABFTSMC, integral backstepping, and 2-SMC controllers have been tuned to achieve a smooth and quick tracking performance by using a toolbox optimization in MATLAB/Simulink (Check Step Response Characteristics block) presented in [75].

62 63 64 65 66

103 104 105 106 107 109 110

44

56

102

108

42

54

100 101

35

The quadrotor is commanded to track the 3D square trajectory in the presence of aerodynamic forces and moments. More specifically, the reference trajectory for the position coordinates and the yaw angle are listed in Table 2.

In this case, the disturbances are neglected in the controller design. In order to highlight the superiority of the proposed ABABFTSMC method, comparisons with backstepping sliding mode control technique are conducted. The simulation results of the flight controller AB-ABFTSMC are shown in Figs. 4–11. These results demonstrate that the proposed controller has succeeded following the quadrotor’s position and attitude in finitetime as shown in Figs. 4 and 5. Even though the reference trajectories of the quadrotor’s horizontal position and altitude are modified, the proposed controller is able to maintain all state variables to the new trajectories (see Figs. 4, 5 and 9). The quadrotor’s position 3D trajectory in Fig. 9 shows the better tracking in a few seconds, similarly in Figs. 4 and 5, the position (variables x, y and z) trajectory show satisfactory performances. Fig. 5 shows the time evolution of the attitude variables (φ and θ ), which also coincide requires the quadrotor to operate and yaw angle desired. Figs. 6 and 7 represent the linear and angular velocities, it can be observed that these state variables converge to zero in finite-time. Fig. 8 depicts the actual thrust and the input torques for the quadrotor, which converge to their steady state values (4.768, 0, 0, 0), these results demonstrate the effectiveness and robustness of the proposed flight controller. The chattering

111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

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phenomenon caused by the switching control action is solved. Indeed, the inputs of system have better performance with respect to this problem. Furthermore, the pulses shown in Figs. 4–11 of the horizontal position, the roll/pitch angles and the rolling/pitching torques are caused by the highly coupled of the position and attitude state variables. The adaptive gains of the proposed controller are shown in Figs. 10 and 11, this estimation allows tracking the desired reference position and attitude with satisfactory accuracy. Finally, the robustness and supremacy of the proposed controller are demonstrated by comparing its performance with that of a conventional sliding mode controller. Furthermore, the ABABFTSMC control scheme has better performances (setting time, rise time, and overshoot) than A-SMC. The AB-ABFTSMC strategy has a good tracking of the 3D trajectory and the position of quadrotor (Fig. 4 and Fig. 9). Moreover, the amplitudes of the rolling, pitching, and yawing torques are greatly decreased compared to the results presented in the reference [73]. This last means that, the drone will be more stable.

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In this scenario, the external disturbances and uncertainties in the parameter values are considered. The terms di (i = x, y , z) = 0.01 cos(0.1t ) m/s2 and d j ( j = φ, θ, ψ) = 0.5 cos(0.7t ) rad/s2 are added in the position and attitude accelerations, respectively as disturbances. In addition, ±30% uncertainty in the values of mass m and inertias (I xx , I y y , I zz ) is taken into consideration to show the parameter robustness. Comparative simulations with the second order sliding mode control method proposed in [46] and the integral backstepping control approach in [66] are also given. The trajectory tracking (x, y , z, φ, θ, ψ), position tracking errors,

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and input signals of the quadrotor in the presence of parametric uncertainties and disturbances are illustrated in Figs. 12–15. The simulation results indicate that the both AB-ABFTSMC and 2-SMC methods can accomplish a robust tracking of the square trajectory. On the contrary, the robust trajectory tracking via the integral backstepping cannot be achieved in the presence of disturbances, however the proposed method can achieve a better tracking performance than the others. Figs. 12 and 13 show the position and attitude trajectories tracking respectively using three control approaches. It can be seen that the AB-ABFTSM controller is able to reject the disturbances and the uncertainties, moreover the integral backstepping tracks the desired position trajectories with large oscillations. These results have validated the effectiveness of the proposed control method when disturbances and other factors are considered. Fig. 14 shows the position tracking error. It can be observed that the proposed AB-ABFTSM control strategy achieves the best position tracking than 2-SMC and integral backstepping controls. Finally, the input signals, presented in Fig. 15 are easily to implement in real model and reach their steady values in short finite-time. As a result, the AB-ABFTSMC method provides a more accurate tracking and robustness against sustained time-varying disturbances and parametric uncertainties.

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We have successfully tested the application of an adaptive robust controller for flight path tracking and stabilization of a quadrotor UAV. In this paper, the Newton-Euler technique is used to find the dynamics of the quadrotor. Two new robust controllers have been designed to control the quadrotor UAV subject to parametric uncertainties, the first proposed controller is the adaptive backstepping (AB) to control the position loop, on the basis of this design and the SMC approach, we combine backstepping control and fast terminal sliding mode control to design a robust ABFTSMC to control attitude loop. However, fast terminal sliding surfaces are designed for finite-time convergence of the attitude quadrotor system. The quadrotor’s stability is guaranteed through the Lyapunov theory and analysis. The adaptive laws are devel-

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oped to estimate some parameters of the proposed controllers. Finally, the results verify the efficiency of the proposed controllers and show a great tracking of the 3D flight trajectory. Moreover, these results show that AB-ABFTSMC AB-ABFTSMC control strategy has high performances and good robustness against disturbance more than a classic SMC, 2-SMC, and integral backstepping techniques. In future work, the proposed AB-FTSMC approach will be validated by a real quadrotor UAV to achieve the trajectory-tracking

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missions. Also, the reaching phase and the control effort of the proposed AB-ABFTSMC method will be addressed.

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Declaration of Competing Interest

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None declared.

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