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 the -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.

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 (, , , , , and ) 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 the -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 Section 2. 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 Figure 1. The body-fixed (, , , and ) and earth-fixed (, , , and ) frames are defined. The absolute position of the quadrotor is defined by the vector . The vector represents the Euler angles. The angular and linear velocities of the quadrotor are defined by the vectors and , respectively.

In this context, the formalism of Newton-Euler is used to obtain the quadrotor model [19, 21, 22]:

is the rotation matrix

The coordinate transformation is given by the rotation matrix:

We use to and to . represents the total mass of the quadrotor, is a density matrix, and denotes the matrix moment of the inertia. is the total thrust; this expression is given by where , is the angular speed and is a parameter that depends on the air density and blade geometry. denotes a skew-symmetric matrix. , , and 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: where , , and represent the friction aerodynamics parameters.

denotes the rotor inertia. is the drag coefficient and 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

The relationship between the quadrotor inputs and the propeller speeds of the quadrotor can be written as follows [22]:

The dynamic model of the quadrotor position subsystem has three outputs (, , and ) and one control input . In order to solve the underactuated problem, three virtual control (, , and ) inputs using adaptive backstepping and the backstepping technique are designed. The virtual control inputs are given as follows:

Therefore, the desired roll, pitch, and yaw angles () and the total thrust can be obtained as follows:

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 altitude and an accurate estimation of the unknown perturbation acting on the altitude subsystem. The backstepping technique is used to obtain the virtual control (, ). 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.

3.1. Adaptive Backstepping Control for Altitude Subsystem

Define tracking error of the altitude subsystem as

Consider the Lyapunov candidate function:

Taking the time derivative of equation (12)

The virtual control law is where is a positive constant.

Define the tracking error of step 2 as

The virtual control and adaptive law of the altitude subsystem are given by equation (16) and equation (17), respectively: where and are the positive parameters and denotes the estimate of .

Proof. The Lyapunov function is considered as follows: The time derivative of is given as Considering the control law equation (16) and adaptive law equation (17), we get

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 the and positions:

Define the Lyapunov candidate function as

The time derivatives of equations (23) and (24) are given by

From equation (25), the virtual control laws are where are the positive constants.

Consider the tracking errors of step 2 are given as

The virtual control laws of the horizontal subsystems are given by

Proof. Using the same, procedure presented in the subsection 3.1 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:

Consider the ITSM surface function as of the quadrotor attitude as [23, 24] where and are the positive constants and and the positive integers with .

The surface dynamics are given by

Using the exponential reaching laws, we get

From equation (31) and equation (32), the control laws of the attitude subsystem are given as

Proof. In order to demonstrate the stability of the attitude subsystem, the Lyapunov candidate function of the roll subsystem is given as follows:

The time derivative of is

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 Table 1. 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: at ; at ; at ; at ; at ; and at . 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.

The simulation results are shown in Figures 38. The desired and actual tracking positions , , and 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 . 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 for the position and at for the position of the quadrotor system. Furthermore, trajectory attitudes , , and 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 (, , and ) 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 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.

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 positions and 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 torques

In 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

Data Availability

No new data were created during the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.