# Robust Integral Terminal Sliding Mode Control for Quadrotor UAV with External Disturbances

Authors: Moussa Labbadi; Mohamed Cherkaoui
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

The purpose of this paper is to solve the problem of controlling of the quadrotor exposed to external constant disturbances. The quadrotor system is partitioned into two parts: the attitude subsystem and the position subsystem. A new robust integral terminal sliding mode control law (RITSMC) is designed for stabilizing the inner loop and the quick tracking of the right desired values of the Euler angles. To estimate the disturbance displayed on thez-axis and to control the altitude position subsystem, an adaptive backstepping technique is proposed, while the horizontal position subsystem is controlled using the backstepping approach. The stability of the quadrotor subsystems is guaranteed by the Lyapunov theory. The effectiveness of the proposed methods is clearly comprehended through the obtained results of the various simulations effectuated on MATLAB/Simulink, and a comparison with another technique is presented.


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## Body

## 1. Introduction

### 1.1. Background and Motivation

Over the last decade, the robotic field has attracted great attention from researchers, in particular, the unmanned aerial vehicle (UAV). The quadrotor is a type of drone, which consists of four rotors. This last has a simple mechanical structure. The quadrotor can land vertically and take off. The quadrotor has been used in many applications such as military missions, journalism, disaster management, archaeology, geographical monitoring, taxi services, search/rescue missions, environmental protection, performing missions in oceans or other planets, mailing and delivery, and other miscellaneous applications [1, 2]. Therefore, the quadrotor is a strongly coupled and highly nonlinear system. The quadrotor is an underactuated system, because the six DOF (x, y, z, ϕ, θ, and ψ) should be regulated by the four controls.In order to control the quadrotor in a closed loop, many techniques have been designed such as the backstepping control, fuzzy logic based on intelligent control, sliding mode control, adaptive control approaches, neural network method, intelligent fuzzy logic control, model predictive control [3], hybrid finite-time control [4], proportional derivative sliding mode control [5], nonlinear PID controller [6], and adaptive fractional order sliding mode control [7].


### 1.2. Literature Review

In Reference [8], nonlinear control techniques are designed for the quadrotor’s position and attitude. The attitude loop uses the regular sliding mode controller. The backstepping technique is combined with a sliding mode control to design robust controllers for the outer loop and the yaw angle. In order to estimate the quadrotor UAV fault, an observer is developed. In Reference [9], a second-order sliding mode control technique is developed to control an underactuated quadrotor UAV. In order to select the best coefficient of this proposed controller, the Hurwitz analysis is used. The control approach allows converging the state variables to their reference values and ensuring the stability of the quadrotor system. In Reference [10], a global fast terminal sliding mode control technique is developed for a quadrotor UAV. The controller allows solving the chattering problem, stabilizing the vehicle, and converging all state variables to zero. In Reference [11], a nonsingular fast terminal SMC technique has been established for the stabilization and control of uncertain and nonlinear dynamical systems based on a disturbance observer. In Reference [12], an adaptive control method based on the sliding mode technique is developed for tracking control and for the stability of an uncertain quadrotor. The quadrotor unknown parameters are estimated at any moment. The proposed control laws guarantee the stability of the quadrotor system and recommend that the state variables converge to their origin values in finite time. A combination of sliding mode control and integral backstepping techniques is presented in Reference [13]. The control strategy allows tracking of the flight trajectory in the presence of the disturbances and stabilizing of the attitude of the UAV. In Reference [14], a nonsingular fast terminal sliding mode (NFTSM) method is designed to obtain the good performance of the quadrotor attitude. The tracking errors of the quadrotor are converged to zero through this controller. In the study by Reference [15], a fast terminal SMC approach is proposed for the tracking control problem of a nonlinear mass-spring system in the presence of noise, exterior disturbance, and parametric uncertainty. The proposed controller is confirmed via both experimental and simulation results. In Reference [16], a high-order sliding mode observer is combined with a nonsingular modified super-twisting algorithm to propose a solution for the trajectory-tracking problem of unmanned aerial systems (UAS). The proposed approach techniques offer an estimation for the translational velocities of the quadrotor and improve the robust performance ability of the UAV system against external disturbances. In Reference [17], an adaptive super-twisting based on the terminal sliding mode technique is applied on the fourth-order systems. The experimental results of the proposed control scheme are presented. In Reference [18], a terminal sliding mode controller for the yaw and altitude subsystems is designed. A sliding mode technique is developed to control the quadrotor underactuated subsystem. In Reference [19], two nonlinear control techniques (backstepping and sliding mode controllers) are suggested to solve the tracking trajectory problem in the presence of relatively high perturbations, problems of the quadrotor system affected by input delays, parametric uncertainties, unmolded uncertainties, and time-varying state and external disturbances; a robust nominal controller and a robust compensator are proposed in Reference [20] to solve the process.


### 1.3. Contribution

In this paper, a robust integral terminal sliding mode control method is proposed for a quadrotor attitude. Then, an integral terminal sliding mode surface is used for the attitude of the quadrotor to ensure tracking errors converge to zero in short finite time. The RITSMC scheme is designed based on the Lyapunov theory. However, an adaptive backstepping is designed to estimate the unknown external disturbance acting in thez-axis and to control the altitude subsystem. The backstepping technique is used to control the horizontal position of the quadrotor in the presence of external perturbations. The contribution of this paper is given by the following points:
(i)
A novel hybrid control structure of the quadrotor system subject to underactuated, kinematics-dynamics couplings, and external disturbances is proposed
(ii)
An RITSMC is proposed for the quadrotor attitude to eliminate singularities without adding any constraints compared to TSMC and to reduce the chattering effect in the conventional SMC
(iii)
An AB control approach commands the altitude of the quadrotor and estimates the disturbances at any moment simultaneously


### 1.4. Paper Organization

The rest of the paper is structured as follows: the formulation of the quadrotor system is given in Section2. The new RITSMC and the adaptive backstepping techniques are presented in Section 3. The numerical simulation results are prepared in Section 4. Finally, the conclusions are exposed in Section 5.


## 1.1. Background and Motivation

Over the last decade, the robotic field has attracted great attention from researchers, in particular, the unmanned aerial vehicle (UAV). The quadrotor is a type of drone, which consists of four rotors. This last has a simple mechanical structure. The quadrotor can land vertically and take off. The quadrotor has been used in many applications such as military missions, journalism, disaster management, archaeology, geographical monitoring, taxi services, search/rescue missions, environmental protection, performing missions in oceans or other planets, mailing and delivery, and other miscellaneous applications [1, 2]. Therefore, the quadrotor is a strongly coupled and highly nonlinear system. The quadrotor is an underactuated system, because the six DOF (x, y, z, ϕ, θ, and ψ) should be regulated by the four controls.In order to control the quadrotor in a closed loop, many techniques have been designed such as the backstepping control, fuzzy logic based on intelligent control, sliding mode control, adaptive control approaches, neural network method, intelligent fuzzy logic control, model predictive control [3], hybrid finite-time control [4], proportional derivative sliding mode control [5], nonlinear PID controller [6], and adaptive fractional order sliding mode control [7].


## 1.2. Literature Review

In Reference [8], nonlinear control techniques are designed for the quadrotor’s position and attitude. The attitude loop uses the regular sliding mode controller. The backstepping technique is combined with a sliding mode control to design robust controllers for the outer loop and the yaw angle. In order to estimate the quadrotor UAV fault, an observer is developed. In Reference [9], a second-order sliding mode control technique is developed to control an underactuated quadrotor UAV. In order to select the best coefficient of this proposed controller, the Hurwitz analysis is used. The control approach allows converging the state variables to their reference values and ensuring the stability of the quadrotor system. In Reference [10], a global fast terminal sliding mode control technique is developed for a quadrotor UAV. The controller allows solving the chattering problem, stabilizing the vehicle, and converging all state variables to zero. In Reference [11], a nonsingular fast terminal SMC technique has been established for the stabilization and control of uncertain and nonlinear dynamical systems based on a disturbance observer. In Reference [12], an adaptive control method based on the sliding mode technique is developed for tracking control and for the stability of an uncertain quadrotor. The quadrotor unknown parameters are estimated at any moment. The proposed control laws guarantee the stability of the quadrotor system and recommend that the state variables converge to their origin values in finite time. A combination of sliding mode control and integral backstepping techniques is presented in Reference [13]. The control strategy allows tracking of the flight trajectory in the presence of the disturbances and stabilizing of the attitude of the UAV. In Reference [14], a nonsingular fast terminal sliding mode (NFTSM) method is designed to obtain the good performance of the quadrotor attitude. The tracking errors of the quadrotor are converged to zero through this controller. In the study by Reference [15], a fast terminal SMC approach is proposed for the tracking control problem of a nonlinear mass-spring system in the presence of noise, exterior disturbance, and parametric uncertainty. The proposed controller is confirmed via both experimental and simulation results. In Reference [16], a high-order sliding mode observer is combined with a nonsingular modified super-twisting algorithm to propose a solution for the trajectory-tracking problem of unmanned aerial systems (UAS). The proposed approach techniques offer an estimation for the translational velocities of the quadrotor and improve the robust performance ability of the UAV system against external disturbances. In Reference [17], an adaptive super-twisting based on the terminal sliding mode technique is applied on the fourth-order systems. The experimental results of the proposed control scheme are presented. In Reference [18], a terminal sliding mode controller for the yaw and altitude subsystems is designed. A sliding mode technique is developed to control the quadrotor underactuated subsystem. In Reference [19], two nonlinear control techniques (backstepping and sliding mode controllers) are suggested to solve the tracking trajectory problem in the presence of relatively high perturbations, problems of the quadrotor system affected by input delays, parametric uncertainties, unmolded uncertainties, and time-varying state and external disturbances; a robust nominal controller and a robust compensator are proposed in Reference [20] to solve the process.


## 1.3. Contribution

In this paper, a robust integral terminal sliding mode control method is proposed for a quadrotor attitude. Then, an integral terminal sliding mode surface is used for the attitude of the quadrotor to ensure tracking errors converge to zero in short finite time. The RITSMC scheme is designed based on the Lyapunov theory. However, an adaptive backstepping is designed to estimate the unknown external disturbance acting in thez-axis and to control the altitude subsystem. The backstepping technique is used to control the horizontal position of the quadrotor in the presence of external perturbations. The contribution of this paper is given by the following points:
(i)
A novel hybrid control structure of the quadrotor system subject to underactuated, kinematics-dynamics couplings, and external disturbances is proposed
(ii)
An RITSMC is proposed for the quadrotor attitude to eliminate singularities without adding any constraints compared to TSMC and to reduce the chattering effect in the conventional SMC
(iii)
An AB control approach commands the altitude of the quadrotor and estimates the disturbances at any moment simultaneously


## 1.4. Paper Organization

The rest of the paper is structured as follows: the formulation of the quadrotor system is given in Section2. The new RITSMC and the adaptive backstepping techniques are presented in Section 3. The numerical simulation results are prepared in Section 4. Finally, the conclusions are exposed in Section 5.


## 2. The System Modeling

The quadrotor is equipped with four rotors as shown in Figure1. The body-fixed (Ob, Xb, Yb, and Zb) and earth-fixed (Oe, Xe, Ye, and Ze) frames are defined. The absolute position of the quadrotor is defined by the vector ξ=x,y,zT. The vector η=ϕ,θ,ψT represents the Euler angles. The angular and linear velocities of the quadrotor are defined by the vectors Ω=p,q,rT and V=u,v,wT, respectively.Figure 1
The quadrotor UAV with four rotors.In this context, the formalism of Newton-Euler is used to obtain the quadrotor model [19, 21, 22]:
(1)ξ̇=V,mV̇=RFe3−mge3−kii=1,2,3,Ṙ=RSΩ,JΩ̇=−Ω×JΩ+Mc−Ma+Mb.R is the rotation matrix
(2)R=CθCψSϕSθCψ−CθSψCϕSθCψ+SϕSψCθSψSϕCθSψ+CϕSψCϕSθSψ−SϕCψ−SθCϕCθCϕCθ.The coordinate transformation is given by the rotation matrix:(3)T=10−Sθ0CϕSϕCθ0−SϕCϕCθ.We useC. to cos. and S. to sin.. m represents the total mass of the quadrotor, I3×3 is a density matrix, and J=diagIx,Iy,Iz denotes the matrix moment of the inertia. F is the total thrust; this expression is given by F=F1+F2+F3+F4 where Fi=bΩi2, Ωi is the angular speed and b is a parameter that depends on the air density and blade geometry. S=p,q,rT denotes a skew-symmetric matrix. Ma, Mc, and Ma are the resultant of the aerodynamic friction torque, the resultant of the gyroscopic effect torque, and the torque developed by the four rotors of the quadrotor, respectively. These expressions can be given, respectively, as follows:
(4)Ma=diadk4ϕ̇2,k5θ̇2,k6ψ̇2,where k4, k5, and k6 represent the friction aerodynamics parameters.
(5)Mc=∑i=04Ω×Jr0,0,−1i+1ΩiT.Jr denotes the rotor inertia.
(6)Mb=lF3−F1lF4−F2d−Ω1+Ω2−Ω3+Ω4,d is the drag coefficient and l is the distance between a propeller and the center of the quadrotor. The complete quadrotor dynamics in the presence of external disturbances referred to in [22] is as
(7)ϕ¨=θ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2+1IxU2+dϕ,θ¨=ϕ̇ψ̇Iz−IxIy+ϕ̇JrIyΩr−k5Iyθ̇2+1IyU3+dθ,ψ¨=ϕ̇θ̇Ix−IyIz−k6Izψ̇2+1IzU4+dψ,x¨=1mCϕSθCψ+SϕSψU1+dx−k1mẋ,y¨=1mCϕSθSψ−SϕCψU1+dy−k2mẏ,z¨=1mCϕCθU1+dz−g−k3mż.The relationship between the quadrotor inputs and the propeller speeds of the quadrotor can be written as follows [22]:
(8)U1U2U3U4=bbbb−lb0lb00−lb0lb−d+d−ddΩ12Ω22Ω32Ω42.The dynamic model of the quadrotor position subsystem has three outputs (z, x, and y) and one control input U1. In order to solve the underactuated problem, three virtual control (v1, v2, and v3) inputs using adaptive backstepping and the backstepping technique are designed. The virtual control inputs are given as follows:
(9)v1v2v3=U1mCϕSθCψ+SϕSψU1mCϕSθSψ−SϕCψU1mCϕCθ−g.Therefore, the desired roll, pitch, and yaw angles (ϕd,θd) and the total thrust U1 can be obtained as follows:
(10)U1=mv12+v22+v3+g2,ϕd=arctanSθdv1Cψd−v2Sψdv3+g,θd=arctanv1Cψd+v2Sψdv3+g.


## 3. Controller Design and Stability Analysis

This section presents the robust nonlinear controller for the quadrotor UAV in the presence of external perturbations. The proposed control of the quadrotor is divided into the inner loop and the outer loop. For the outer loop, an adaptive backstepping (AB) controller for the altitude subsystem is designed. The AB method ensures the tracking of the desired altitudezd and an accurate estimation of the unknown perturbation dz acting on the altitude subsystem. The backstepping technique is used to obtain the virtual control (v1, v2). For the inner loop, a new robust integral terminal sliding mode control (RITSMC) is proposed. The attitude controller generates the rolling, pitching, and yawing torques to control the orientation of the quadrotor in the presence of external disturbances. Finally, the proposed control strategy solves the trajectory problem in short finite time with accuracy of the system performance. The general proposed control scheme is shown in Figure 2.Figure 2
The control general structure for the quadrotor UAV.


### 3.1. Adaptive Backstepping Control for Altitude Subsystem

Define tracking error of the altitude subsystem as(11)ez1=z−zd.Consider the Lyapunov candidate function:(12)Vz1=12ez12.Taking the time derivative of equation (12)
(13)V̇z1=ez1ėz1=ez1ż−żd.The virtual control law is(14)αz=−cz1ez1+żd,where cz1>0 is a positive constant.Define the tracking error of step 2 as(15)ez2=ż−αz.The virtual control and adaptive law of the altitude subsystem are given by equation (16) and equation (17), respectively:
(16)v3=−ez1+cz1ez2−cz1ez1+cz2ez2+k3mż−z¨d+d^z,(17)d^̇z=γzez2,where cz2 and γz are the positive parameters and d^z denotes the estimate of dz.Proof.
The Lyapunov function is considered as follows:(18)Vz2=12ez12+12ez22+12γzd~z2.
The time derivative ofVz2 is given as
(19)V̇z2=ez1ėz1+ez2ėz2+1γzd~̇zd~z=ez1ez2−cz1ez1+ez2z¨+cz1ėz1−z¨d−1γzd^̇zd~z=−cz1ez12+ez2−k3mz+v3+cz1ez2−cz1ez1−z¨d+ez2d^z+dzez2−d^̇zγz.
Considering the control law equation (16) and adaptive law equation (17), we get
(20)V̇z2=−cz1ez12−cz2ez22≤0.


### 3.2. Backstepping Control for Horizontal Position Subsystem

In this part, the backstepping technique is used to control the horizontal position of a quadrotor.Introduce the tracking errors of thex and y positions:
(21)ex1=x−xd,ey1=y−yd.Define the Lyapunov candidate function as(22)Vx1=12ex12,(23)Vy1=12ey12.The time derivatives of equations (23) and (24) are given by
(24)V̇x1=ex1ėx1=ex1ẋ−ẋd,(25)V̇y1=ey1ėy1=ey1ẏ−ẏd.From equation (25), the virtual control laws are
(26)αx=−cx1ex1+ẋd,αy=−cy1ey1+ẏd,where cx1,cy1>0 are the positive constants.Consider the tracking errors of step 2 are given as(27)ex2=ẋ−αx,ey2=ẏ−αy.The virtual control laws of the horizontal subsystems are given by(28)v1=−ex1+cx1ex2−cx1ex1+cx2ex2+k1mẋ−x¨d+dx,v2=−ey1+cy1ey2−cy1ey1+cy2ey2+k2mẏ−y¨d+dy.Proof.
Using the same, procedure presented in the subsection3.1 demonstrate the stability of horizontal subsystem.


### 3.3. Robust Integral Terminal Sliding Mode Control Laws for Attitude Subsystem

In order to solve the chattering phenomenon and to eliminate the singularity in a TSMC, a robust nonlinear controller based on the integral terminal sliding mode technique is designed for the quadrotor attitude.Introduce the tracking errors and its derivative, respectively, of the attitude subsystem:(29)eϕ=ϕ−ϕd,eθ=θ−θd,eψ=ψ−ψd,ėϕ=ϕ̇−ϕ̇d,ėθ=θ̇−θ̇d,ėψ=ψ̇−ψ̇d.Consider the ITSM surface function as of the quadrotor attitude as [23, 24]
(30)sϕ=ėϕ+∫αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕdt,sθ=ėθ+∫αθėθqθ/pθ+βθeθqθ/2pθ−qθdt,sψ=ėψ+∫αψėψqψ/pψ+βψeψqψ/2pψ−qψdt,where αii=ϕ,θ,ψ and βii=ϕ,θ,ψ are the positive constants and pii=ϕ,θ,ψ and qii=ϕ,θ,ψ the positive integers with pi>qi.The surface dynamics are given by(31)ṡϕ=e¨ϕ+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ,ṡθ=e¨θ+αθėθqθ/pθ+βθeθqθ/2pθ−qθ,ṡψ=e¨ψ+αψėψqψ/pψ+βψeψqψ/2pψ−qψ.Using the exponential reaching laws, we get(32)ṡϕ=−λϕsϕ−kϕsignsϕ,ṡθ=−λθsθ−kθsignsθ,ṡψ=−λψsψ−kψsignsψ.From equation (31) and equation (32), the control laws of the attitude subsystem are given as
(33)U2=Ix−θ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2−dϕ−λϕsϕ−kϕsignsϕ+ϕ¨d−αϕėϕqϕ/pϕ−βϕeϕqϕ/2pϕ−qϕ,U3=Iy−ϕ̇ψ̇Iz−IxIy+ϕ̇JrIyΩr−k5Iyθ̇2−dθ−λθsθ−kθsignsθ+θ¨d−αθėθqθ/pθ−βθeθqθ/2pθ−qθ,U4=Iz−ϕ̇θ̇Ix−IyIz−k6Izψ̇2−dψ−λψsψ−kψsignsψ+ψ¨d−αψėψqψ/pψ−βψeψqψ/2pψ−qψ.Proof.
In order to demonstrate the stability of the attitude subsystem, the Lyapunov candidate function of the roll subsystem is given as follows:(34)Vϕ=12sϕ2.The time derivative ofVϕ is
(35)V̇ϕ=sϕṡϕ=sϕϕ¨−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕθ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2+1IxU2+dϕ−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕθ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2+−θ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2−dϕ−λϕsϕ−kϕsignsϕ+ϕ¨d−αϕėϕqϕ/pϕ−βϕeϕqϕ/2pϕ−qϕ+dϕ−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕ−λϕsϕ−kϕsignsϕ=−λϕsϕ2−kϕsϕ≤0.


## 3.1. Adaptive Backstepping Control for Altitude Subsystem

Define tracking error of the altitude subsystem as(11)ez1=z−zd.Consider the Lyapunov candidate function:(12)Vz1=12ez12.Taking the time derivative of equation (12)
(13)V̇z1=ez1ėz1=ez1ż−żd.The virtual control law is(14)αz=−cz1ez1+żd,where cz1>0 is a positive constant.Define the tracking error of step 2 as(15)ez2=ż−αz.The virtual control and adaptive law of the altitude subsystem are given by equation (16) and equation (17), respectively:
(16)v3=−ez1+cz1ez2−cz1ez1+cz2ez2+k3mż−z¨d+d^z,(17)d^̇z=γzez2,where cz2 and γz are the positive parameters and d^z denotes the estimate of dz.Proof.
The Lyapunov function is considered as follows:(18)Vz2=12ez12+12ez22+12γzd~z2.
The time derivative ofVz2 is given as
(19)V̇z2=ez1ėz1+ez2ėz2+1γzd~̇zd~z=ez1ez2−cz1ez1+ez2z¨+cz1ėz1−z¨d−1γzd^̇zd~z=−cz1ez12+ez2−k3mz+v3+cz1ez2−cz1ez1−z¨d+ez2d^z+dzez2−d^̇zγz.
Considering the control law equation (16) and adaptive law equation (17), we get
(20)V̇z2=−cz1ez12−cz2ez22≤0.


## 3.2. Backstepping Control for Horizontal Position Subsystem

In this part, the backstepping technique is used to control the horizontal position of a quadrotor.Introduce the tracking errors of thex and y positions:
(21)ex1=x−xd,ey1=y−yd.Define the Lyapunov candidate function as(22)Vx1=12ex12,(23)Vy1=12ey12.The time derivatives of equations (23) and (24) are given by
(24)V̇x1=ex1ėx1=ex1ẋ−ẋd,(25)V̇y1=ey1ėy1=ey1ẏ−ẏd.From equation (25), the virtual control laws are
(26)αx=−cx1ex1+ẋd,αy=−cy1ey1+ẏd,where cx1,cy1>0 are the positive constants.Consider the tracking errors of step 2 are given as(27)ex2=ẋ−αx,ey2=ẏ−αy.The virtual control laws of the horizontal subsystems are given by(28)v1=−ex1+cx1ex2−cx1ex1+cx2ex2+k1mẋ−x¨d+dx,v2=−ey1+cy1ey2−cy1ey1+cy2ey2+k2mẏ−y¨d+dy.Proof.
Using the same, procedure presented in the subsection3.1 demonstrate the stability of horizontal subsystem.


## 3.3. Robust Integral Terminal Sliding Mode Control Laws for Attitude Subsystem

In order to solve the chattering phenomenon and to eliminate the singularity in a TSMC, a robust nonlinear controller based on the integral terminal sliding mode technique is designed for the quadrotor attitude.Introduce the tracking errors and its derivative, respectively, of the attitude subsystem:(29)eϕ=ϕ−ϕd,eθ=θ−θd,eψ=ψ−ψd,ėϕ=ϕ̇−ϕ̇d,ėθ=θ̇−θ̇d,ėψ=ψ̇−ψ̇d.Consider the ITSM surface function as of the quadrotor attitude as [23, 24]
(30)sϕ=ėϕ+∫αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕdt,sθ=ėθ+∫αθėθqθ/pθ+βθeθqθ/2pθ−qθdt,sψ=ėψ+∫αψėψqψ/pψ+βψeψqψ/2pψ−qψdt,where αii=ϕ,θ,ψ and βii=ϕ,θ,ψ are the positive constants and pii=ϕ,θ,ψ and qii=ϕ,θ,ψ the positive integers with pi>qi.The surface dynamics are given by(31)ṡϕ=e¨ϕ+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ,ṡθ=e¨θ+αθėθqθ/pθ+βθeθqθ/2pθ−qθ,ṡψ=e¨ψ+αψėψqψ/pψ+βψeψqψ/2pψ−qψ.Using the exponential reaching laws, we get(32)ṡϕ=−λϕsϕ−kϕsignsϕ,ṡθ=−λθsθ−kθsignsθ,ṡψ=−λψsψ−kψsignsψ.From equation (31) and equation (32), the control laws of the attitude subsystem are given as
(33)U2=Ix−θ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2−dϕ−λϕsϕ−kϕsignsϕ+ϕ¨d−αϕėϕqϕ/pϕ−βϕeϕqϕ/2pϕ−qϕ,U3=Iy−ϕ̇ψ̇Iz−IxIy+ϕ̇JrIyΩr−k5Iyθ̇2−dθ−λθsθ−kθsignsθ+θ¨d−αθėθqθ/pθ−βθeθqθ/2pθ−qθ,U4=Iz−ϕ̇θ̇Ix−IyIz−k6Izψ̇2−dψ−λψsψ−kψsignsψ+ψ¨d−αψėψqψ/pψ−βψeψqψ/2pψ−qψ.Proof.
In order to demonstrate the stability of the attitude subsystem, the Lyapunov candidate function of the roll subsystem is given as follows:(34)Vϕ=12sϕ2.The time derivative ofVϕ is
(35)V̇ϕ=sϕṡϕ=sϕϕ¨−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕθ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2+1IxU2+dϕ−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕθ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2+−θ̇ψ̇Iy−IzIx−θ̇JrIxΩr−k4Ixϕ̇2−dϕ−λϕsϕ−kϕsignsϕ+ϕ¨d−αϕėϕqϕ/pϕ−βϕeϕqϕ/2pϕ−qϕ+dϕ−ϕ¨d+αϕėϕqϕ/pϕ+βϕeϕqϕ/2pϕ−qϕ=sϕ−λϕsϕ−kϕsignsϕ=−λϕsϕ2−kϕsϕ≤0.


## 4. Simulation Results

To validate the performances of the proposed controllers, numerical simulations will be presented in this section. The parameters of the quadrotor used in the simulation are selected in Table1. The initial attitude and position of the quadrotor are chosen as [0, 0, 0] rad and [0, 0, 0] m. The desired trajectory of the yaw angle and the position is given in Table 2. The external disturbances used in the simulation are given as follows: dx=1N at t=5s; dϕ=1rad/s2 at t=10s; dy=1N at t=15s; dθ=1rad/s2 at t=20s; dz=1N at t=25s; and dψ=1rad/s2 at t=30s. Besides, the parameters of the proposed controller are listed in Table 3.Remark 1.
In order to achieve a smooth and quick tracking performance, the design parameters of the proposed AB-RITSMC, B-SMC, and SMC techniques have been tuned by using a toolbox optimization method in MATLAB/Simulink (see, e.g., [25]).
Furthermore, In order to highlight the superiority of the proposed control laws, comparisons with the backstepping sliding mode control and the first-order sliding mode control technique are done.Table 1
Quadrotor parameters.
Parameter
Value
Parameter
Value
gm/s2
9.81
k2N/m/s
5.567e‐4
mkg
0.486
k3N/m/s
6.354e‐5
Ixkg·m2
3.827e‐3
k4N/m/s
5.567e‐4
Iykg·m2
3.827e‐3
k5N/m/s
5.567e‐4
Izkg·m2
7.656e‐3
k6N/m/s
6.354e‐5
Jrkg·m2
2.838e‐5
bN·s2
2.984e‐3
k1N/m/s
5.567e‐4
dN·m/s2
3.232e‐2Table 2
The reference trajectories of the position and yaw angle.
Variables
Value
Time (s)
xd,yd,zd
0.6,0.6,0.6 m
0
0.3,0.6,0.6 m
10
0.3,0.3,0.6 m
20
0.6,0.6,0.6 m
30
0.6,0.6,0.6 m
40
0.6,0.6,0.6 m
50
ψd
0.5 rad
0
ψd
0.0 rad
50Table 3
Control system parameters.
Parameter
Value
Parameter
Value
cx1
1.68
cx2
2.34
cy1
1
cy2
2.25
cz1
6.40
cz2
20
αi
59.78
βi
974.57
pi
54
qi
52
λi
2
ki
10.66The simulation results are shown in Figures3–8. The desired and actual tracking positions x, y, and z are shown in Figure 3, where the proposed control strategy that drives the quadrotor to track the desired flight trajectory more rapidly and more accurately can be seen, then the classic sliding mode control method and the backstepping sliding mode controller. The constant disturbances are added in the position subsystem at t=5. It appears that the proposed control approach has managed to effectively hold the quadrotor’s position in finite time contrary to the SMC and B-SMC; the same behaviour can be observed at t=15 for the y position and at t=25 for the z position of the quadrotor system. Furthermore, trajectory attitudes ϕ, θ, and ψ are shown in Figure 4. It can be seen that the quadrotor attitude tracks the desired angles in short finite time. The attitude sliding variables (sϕ, sθ, and sψ) are shown in Figure 6; the convergence to zero in finite time of the sliding surfaces can be observed. The trajectory-tracking errors of the position are depicted in Figure 5. The estimate force acting in direction z is shown in Figure 8. In order to demonstrate the superiority of the proposed control strategy, the quadrotor trajectory path performance in 3-D space is shown in Figure 7. Clearly, it can be seen that the proposed control strategy can accurately track the square trajectory in the presence of external disturbances. The proposed control approach obtains better performance compared to the conventional SMC and B-SMC methods in terms of disturbance rejection and trajectory tracking.Figure 3
Response of quadrotor position.Figure 4
Response of quadrotor attitude.Figure 5
Position errors.Figure 6
Tracking errors (sϕ, sθ, and sψ), B-ITSMC.Figure 7
Flight trajectory tracking, B-RITSMC.Figure 8
The estimate of thedz parameter.


## 5. Conclusions

In this study, we have examined the problem of the flight trajectory-tracking phenomenon of the quadrotor with disturbances. The proposed control scheme is made up of three different parts: control of an altitude, a horizontal position, and an attitude subsystem. Firstly, the altitude subsystem is addressed based on adaptive backstepping. Also, the backstepping technique is designed to control the positionsx and y of a quadrotor. The control objectives of this loop are (i) obtaining the desired roll and pitch angles, (ii) tracking the desired flight trajectory in finite time, and (iii) generating the total thrust. Secondly, a new robust integral terminal sliding mode controller has been constructed to stabilize the attitude subsystem. The objectives of the RITSMC control scheme are
(i)
tracking the desired angles
(ii)
stabilizing the quadrotor attitude
(iii)
generating the rolling, pitching, and yawing torquesIn addition, the proposed controller achieves the fast and accurate tracking of the quadrotor trajectory. Finally, simulation results have demonstrated that the proposed controller is able to improve control performance of the quadrotor UAV system in the presence of external disturbances. The proposed control strategy has shown the effectiveness and superiority of the classical sliding mode control strategy and backstepping sliding mode control technique.The following points will be addressed in the future work:(i)
Taking into account the motor dynamics of the quadrotor
(ii)
Inclusion of a fault detection in the actuators and sensors
(iii)
Use of adaptive RITSMC approach in the improvement of the quadrotor control system



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*Source: 2016416-2019-08-19.xml*