1. Introduction

Unmanned air vehicle (UAV) which is further known as unmanned aircraft system (UAS) has reached a unique grades of evolution in military and civilian implementations (^{Ashry, 2019}; ^{Hou et al., 2017}). UAVs are capable of executing hazard missions with zero risk for human pilots (^{Madani & Benallegue, 2006}) and distinguished with their ease of deployment, high-motility, and low maintenance value (Shakhatreh et al., 2018). These characteristics open the door to researchers for new ranges in aerospace fields, where it allows using UAS in several assignments like forest meteorology (^{Gallay et al., 2016}), environmental observance (^{Alfeo et al., 2018}), temperature variation (^{Liu et al., 2018}), ice rivers dynamics (^{Ryan et al., 2017}), volcanic activities (^{Mori et al., 2016}; ^{Nakano et al., 2014}), atmospheric sampling (^{Greatwood et al., 2017}; ^{Techy et al., 2010}), border inspection patrol (^{Gupte et al., 2012}; ^{Ghazbi et al., 2016}), traffic regulation (^{Elloumi et al., 2018}), discrimination and surveillance vehicles (^{Finn & Wright, 2012}; ^{Li et al., 2008}), seeking-and-relief operations (^{Shakernia et al., 1999}).

UAVs are classified into different types according to their structure, one of the most demanded UAV structure nowadays is the rotor vertical take-off and landing (VTOL) craft referred to as quadrotor (^{Ghazbi et al., 2016}). It can be sorted as a rotating wing VTOL airliner due to its ability for vertical takeoff and landing, flight forward, and hover at a desired location (^{Martinez, 2007}). Compared with other types of UAVs, and by its singular features, quadrotor took the advantage to print its distinctive stamp in aircraft systems (Raymer, 2006; ^{Zhang et al., 2014}). Quadrotor is characterized by its ease of design, control simplicity, high maneuverability, small size, convenient payload capacity, silent operations, and high maneuver elasticity (^{Gupte et al., 2012}; ^{Kivrak, 2006}; ^{Li et al., 2015}). These attributes enable using quadrotors in vital applications such as seeking-and-relief tasks in difficult and dangerous areas beside rapid intervention assignments (^{Gharibi et al., 2016}; ^{Ghazbi et al., 2016}). The researchers were motivated to develop quadrotors control systems with the attraction of its simplicity. The simple shape of four mounted rotors symmetrically on equal perpendicular axis allow the designers to control quadrotor by adjusting the speed of each rotor (^{Ghazbi et al., 2016}; ^{Rahman et al., 2016}). Both attitude channels and linear velocity can be controlled through rotor speed (^{Bouabdallah & Siegwart, 2007}). This control approach is successfully applied in utilizing small quadrotors worked using electric motors. In this case, the mechanical complexity of the rotor is decreased (^{Özbek et al., 2015}), and different control techniques can be applied on the electric motors for semi-autonomous and full autonomous flight (^{Zulu & John, 2016}).

Due to modern tasks complexity and expansion nature of most targeted fields, the researchers start to use cooperative multiple quadrotors in performing most of the new missions.

This is known as swarm system (^{Jia et al., 2018}). The swarm system is inspired initially by observing biological phenomena. Observation results confirmed the fact that competence and performance can be enhanced, if biological creatures work together in a group (^{Zhu et al., 2016}). Multiple quadrotor swarm VTOL systems consist of at least couple of simple autonomous quadrotors working together to accomplish specific demanded mission (^{Ekawati et al., 2016}). This swarm of multiple cooperative quadrotors can work composing a certain geometric shape called formation (^{Wu et al., 2017}; ^{Yang, Ji et al., 2017}). The formation architectures are divide into a pre-designed formation and adaptively architectures (^{Oh et al., 2017}). According to the type of the formation, the cooperation between the team members is realized depending on the behavior of each quadrotor which takes into its consideration the behavior of the other group members in both free and in obstacle-loaded air domain (^{Farinelli et al., 2004}). This interaction, communication, and exchange information are essential to achieve an overall group operation (^{Zhang & Mehrjerdi, 2013}).

Swarming algorithms were inspired by the natural phenomena of animal cooperative aggregate motions (^{Reynolds, 1987}).

Swarming or grouping behaviors can be openly seen in droves of birds, colonies of ants, schools of fish, and flocks of animals (^{Okubo, 1986}; ^{Parrish et al., 2002}). Swarming becomes a field of interest for the researchers in the field of cooperative UAVs (^{Ashry et al., 2008}). Swarming algorithms are varied between classical algorithms and intelligence algorithms (^{Yang, Wang et al., 2017}). For instant, simple classical intelligence swarming algorithms were used to solve the swarming problem in affected by the behavior of ants where an optimized solution was introduced to solve the swarming problem (^{Colorni et al., 1991}; ^{Eberhart & Kennedy, 1995}).

In the last decade, several advanced intelligent swarming approaches were presented to tackle the swarming problem (^{Abbass, 2001}; ^{Eusuff & Lansey, 2003}; ^{Karaboga, 2005}; Pan, 2011; ^{Sun et al., 2014}; ^{Wu et al., 2013}; ^{Yang, 2010a}; ^2010b; ²⁰¹²; ^{Yang & Deb, 2009}). Several control techniques were used to solve the swarming problem for a cooperative UAVs team. One of the new valuable control approaches in the field of UAVs is the combination of backstepping and ANFIS controllers. This combination has been executed in the control of numerous sorts of non-linear systems. In (^{Khari et al., 2015}), a generalized backstepping approach combined with an ANFIS approach was applied to control a nonlinear Lorenz chaotic system. This introduced approach guarantees that the generalized backstepping approach achieves its optimal parameter using ANFIS. In ^{Asad et al. (2017)}, an MEMS gyroscope was controlled by a backstepping sliding approach based on type-2 fuzzy controller.

Moreover, the altitude stabilization of a hypersonic missile was controlled using a combination of backstepping and ANFIS controllers (^{Allahverdy & Fakharian, 2019}).

On one hand, the backstepping approach was used in the analysis of the missile altitude dynamics. On the other hand, the ANFIS approach was applied to calculate the uncertainty model parameters. In (^{Azar et al., 2020}), the stabilization of a dynamic framework of port-Hamiltonian systems is controlled by a combination of backstepping and ANFIS controllers. Also, ANFIS was utilized to estimate the ordinary backstepping sliding mode control law to enhance the performance of a non-linear system in (^{Tavoosi, 2020}).

Most backstepping control strategies are utilized as an upper hand controller (^{Fethalla et al., 2017}; ^{Rego et al., 2016}; ^{Vallejo-Alarcón et al., 2015}). Backstepping controller is an effective control strategy for extremely nonlinear frameworks (^{Basri et al., 2015}; ^{Lee et al., 2017}; ^{Liu et al., 2015}; ^{Mutawe et al., 2021}).

The rapid convergence rate, the stability, the capability to manage nonlinear frameworks using recursive operation of Lyapunov function, the robustness regardless presence of uncertainties, and the simplicity are the benefits of backstepping controller (^{Madani & Benallegue, 2006}).

Similarly, ANFIS controller is a moderate artificial intelligent control technique. It integrated the power of fuzzy logic (FL) approaches parallel processing and intelligent learning capacities of expert artificial neural networks (ANN). In its essence, ANFIS incorporates the benefits of the ANN and FL, and erases the drawbacks of the ANN and FL as well (^{Premkumar & Manikandan, 2018}).

In this paper, the benefits of backstepping and ANFIS are combined in the proposed approach which leads to improve the overall performance and robustness in the presence of disturbances and noise.

The main contribution of this paper lays in solving the problem of cooperative UAVs formation using a cascaded backstepping and ANFIS controller. This hybrid controller combines the backstepping-based PID controller with the ANFIS controller.

A proposed aggregate controller depending on the position controller is introduced to regulate the translational dynamics of cooperative UAV group to pursue the pre-planned formation path. Moreover, an attitude controller is designed to keep stability of the inner rotational dynamics.

This paper is organized as follows:

Section 2 introduces briefly the underlying quadrotor mathematical paradigm, and the hybrid controller design is demonstrated in Section 3. In Section 4, the simulation results for a group of cooperative quadrotors following a desired trajectory path using the designed hybrid controller are presented. The stability analysis of the proposed controller is discussed in Section 5, while in Section 6 a discussion about the underlying hybrid controller results is introduced. Finally, conclusion and future work are introduced in section 7.

2. Quadrotor schematic model

The underlying quadrotors are distinguished by a uniformed schematic. Each quadrotor consists of four rotors mounted at proportional distance from the center point and the rotors are set to provide necessary effort to all sort of quadrotor movements.

The quadrotor controller is accomplished by changing the speed of every rotor of the fourfold rotors (^{Mahfouz et al., 2018}). A voltage applied on every motor cause a net torque being utilized on the rotor shaft producing a thrust force Ti for ith quadrotor and jth rotor, where j = 1, 2, 3, 4 Once the rotor blade is spinning, there is a proportional speed difference among the rotor sharp edge and air during rotating clockwise and anti-clockwise causing a net moment around the roll hub. Further speed also creates a power drag on the rotor which is opposed to travel orientation Di, the thrust is realized through aerodynamic coefficients CTi as in (1) (^{Azzam & Wang, 2010}):

where ρi is the air density, Ai represents the area of the blade, Ωi performs the propeller's angular velocity, and Ri states the radius of the blade.

A group of fourfold control inputs parameters ηi are acquired like a function of normalized singular thrust and torques as in equations (2-4).

At hovering and semi hovering states when nonlinear components can be neglected as in (^{Mahfouz et al., 2019}), the thrust and forces are corresponding to the square of propellers' spinning speed. The thrust and drag forces can be calculated by ^{Azzam and Wang (2010)}, ^{Mahfouz et al. (2013)}:

where βT, and βD are constants. The thrust, rolling, pitching, and yawing moments are obtained in (4) (^{Mahfouz et al., 2013}):

In modeling strategy of the VTOL quadrotor, the rotational transmutations aren't performed within the identical formation to relocate from the earth to the body coordinate.

Naturally, the ultimate efficient technique performs the resultant rotation of the earth to the body transmutation over the thrust side. Assume a steady-earth coordinate E and a steady-body coordinate B. Utilizing Euler angles specifications; the air-coordinate direction in space is defined by a rotation matrix R. So, for the body to the earth transmutation, R is counted in (5) (^{Ashry, 2014}; ^{Mahfouz et al., 2013}):

where: ϕi,θi,ψi roll, pitch, and yaw angles for the ith quadrotor, respectively; s≡sin,c≡cos.

The evolution of a convenient attitude controller for the quadrotor targeted a precise active model to be progressed. Newton laws were chosen to estimate the quadrotor dynamics for the simplicity of the control goals.

The Newtonian approach is a definitive proper option for designing robust frameworks in 6-degree-of-freedom (^{Bang et al., 2004}). The Newtonian hypothesis exercised in order to produce a robust framework in 6DOF are well represented in (^{Hamel et al., 2002}; ^{Tayebi & McGilvray, 2006}).

Assuming the dynamics of a robust framework underneath forces Fib, and moments τib. These parameters are used at the middle point of mass and stated in the steady-body coordinate. It is represented in Newton-Euler approach as follows (^{Mahfouz et al., 2019}; ^{Murray et al., 1994}):

Where m represents the quadrotor overall body mass, vib performs the speed vector in the steady-body coordinate, ωib states the angular velocity at the steady-body coordinate, and Ii symbolizes the body moment of inertia matrix.

The quadrotor equations of motions are introduced in (7) (^{Bouabdallah, 2007}; ^{Bouabdallah & Siegwart, 2007}; Mahfouz, 2015; ^{Wierema, 2008}):

Where Ixxi,Iyyi,Izzi are body moment of inertia matrices, Qji is a drag moment, Jri is rotor moment of inertia, Hxji and Hyji are hub forces.

The proposed hybrid controller consists of position, attitude and altitude controllers.

The position and altitude controllers adjust the motion dynamics in order to track the pre-designed path for the UAV team.

The attitude controller keeps and executes the stability of the inner rotational dynamics.

The introduced control approach in this paper is composed of a two-loop controller layout technique. The outer control loop is a backstepping-based PID controller to trail proportional position orders and create body-axis rate orders for the internal loop.

The inner control loop is an ANFIS controller to generate desired angular rotations for every quadrotor separately, as seen in Figure 1.

One should notice that backstepping-based PID approach is utilized for position and altitude control, while ANFIS approach is used for attitude control.

The quadrotor control framework is organized in four unique controllers as outlined in Figure 2. VTOL controller yields the required elevation zd to altitude controller that is based on altitude sensor data to produce the required overall thrust Td.

Position controller gets quadrotor current location (x, y) and required thrust force. It yields required roll ϕd, and pitch θd, however, required yaw ψd gets through the client directly.

The desired motor speed is produced by the controller after the attitude control is adjusted.

3. Hybrid control design

4. Simulation results

A simulation atmosphere is settled under "SIMULINK-MATLAB” to approve the offered control strategy. The simulation is built on the entire non-linear model of a group of three quadrotors introduced by Equation (7). Each quadrotor in the underlying system contains two stages of controllers. The backstepping controller represents the higher controller where it controls the altitude and the position. The ANFIS controller represents the lower controller where it receives the altitude and the position from the backstepping to acquire the attitude controller to turn over the angels to execute the desired altitude and position. Also, it transmits the proper control signal for the propellers to accomplish the whole designed mission respecting the desired separating distance and desired velocity.

The designed ANFIS controller is composed of 16 membership functions and 16 fuzzy rules as shown in Figure 5.

The ANFIS paradigm is created using MATLAB's ANFIS GUI editor. The grid partition method generates optimized rules. The ANFIS paradigm architecture has the subsequent information as shown in Table 1.

The neural network uses the hybrid technique to minimize the error by optimizing the parameters and the membership functions.

This hybrid backstepping-ANFIS controller is studied earlier with a backstepping- backstepping controller for the selfsame framework in (^{Mahfouz et al., 2018}). For training approach, the input/output data sets of various points of operation are exercised.

The obtained data sets are divided equally; half of them is used for training, while the others are used for checking. Hybrid optimization technique generates the FIS structure based on obtained data sets. For the training approach, ANFIS finishes training at epoch two with zero tolerance.

For roll control channel, the error input is ranged from -0.3123 to +0.0156, the error derivative is ranged from -0.889 to +0.05057, and the output for roll moment is ranged from -6.844 to +0.9527. For pitch control channel, the error input ranges from -0.003404 to +0.4045, the error derivative is ranged from -0.6946 to +0.15, and the output for pitch moment is ranged from -24.11 to +3.344. Finally, for yaw control channel, the error input is ranged from -0.3 to +0.04915, the error derivative is ranged from -3.306 to +1.044, and the output for yaw moment is ranged from -67.58 to +199.6.

The ANFIS topology consists of four hidden layers having two inputs, one output, four premise parameters for each input, and sixteen membership functions that create sixteen rules in each ANFIS topology. Path trajectory is designed to test the ability of the underlying hybrid controller to pursue the path trajectory reserving the desired formation configuration of the group members of the cooperative quadrotors. The group is composed of three quadrotors. Every quadrotor began from various points of operation.

— Assumption 1

All the cooperative quadrotors under-study are rigids and have the same mathematical model.

— Assumption 2

The group of cooperative quadrotors under-study consists of three quadrotors in a triangular formation. Each follower quadrotor keeps a separating distance 5 m in the X and Y axis with respect to the leader, and a distance 10 m in the Y axis between each other.

— Assumption 3

All the altitude of the three cooperative quadrotors is constant.

The validation of the proposed controller is presented through the simulation results recorded in couple of scenarios in the presence and absence of stationary and moving obstacles.

— Scenario 1

In this scenario, a group of three cooperative quadrotors tracks a reference path along the X-axis in a triangular shape in a free obstacle environment. The initial position of the quadrotors are as follows: leader = (0,0), Follower 1 = (7,5) and Follower 2 = (-7,-7). The path of the group in the X — direction is demonstrated in Figure 6, where the followers’ quadrotors respect the separating distance with respect to the leader in order to pursue a pre-planned trajectory. Moreover, Figure 7 demonstrates the velocity component in the X — axis for each member of the group respecting the constrain velocity Vx≤2 (m/s) and the pitch control for all the team members is demonstrated in Figure 8.

The trajectory of the three cooperative quadrotors along the Y—axis is shown in Figure 9. The two followers succeeded in achieving the desired separating distance (10 m) between each other and the desired separating distance (5 m) with respect to the leader quadrotor. Moreover, Figure 10 demonstrates the velocity component in the Y—axis of each member of the group respecting the constrain Vy≤2 (m/s) and the roll control for all the group members is demonstrated in Figure 11.

The position of the quadrotors along the Z—axis is represented in Figure 12. Figure 13 represents the velocity components of the three quadrotors along the Z—axis. Finally, the yaw control for the three cooperative quadrotors is demonstrated in Figure 14.

—Scenario 2

In this scenario, the proposed controller is evaluated to solve the formation problem for a group of cooperative quadrotors in static/dynamic obstacles-loaded environment.

A couple of obstacles avoidance techniques are applied to guarantee the safety of all members of the group during tracking a reference desired trajectory in presence of stationary and moving obstacles in an obstacle-loaded dynamic environment.

Stationary obstacle avoidance approach was developed in (^{Mahfouz et al., 2018}; ^{Mahfouz et al., 2019}), while moving obstacle avoidance approach was introduced in (^{Mahfouz, Hafez, Ashry & Elnashar, 2020}; ^{Mahfouz, Hafez, Azzam, et al., 2020}).

Stationary obstacles

In order to demonstrate the underlying controller's ability to pursue the required route in an obstacle-filled environment, obstacles are designed as a rectangle shape. It is assumed that the quadrotors altitudes constant for simplification. Then it would reduce the difficulty of tracking the trajectory to a 2D problem. Two security zones are designed for complete flight safety reasons, by a 20m-radius safety zone and a 10m-detaching range identifying protected zone among every follower and the leader UAV (^{Mahjri et al., 2015}). These safety zones are activated when an obstacle is captured on the border of the safety zones. This causes an interruption for tracking the trajectory making a half circle maneuver to bypass the captured obstacle, and then resuming tracking the original trajectory. As sh the obstacle avoidance algorithm represented in this work guarantees the safety of all members of the fleet. In Figure 15, the leader of a team of cooperative quadrotors senses the presence of a stationary obstacle. The team reconfigures from a triangular formation into a line formation with a separation safety distance of 10 m between every successive quadrotor.

The designed obstacle avoidance technique in motion planning offered in this paper is a simple mathematical solution; this technique calculated the dimensions and the vertices of the captured obstacles to prevent collision.

Assume XObmin and XObmax symbolizes the minimum and maximum dimensions of the obstacle edges in x—axis. Similarly, ΥObmin and ΥObmax symbolizes the minimum and maximum dimensions of the obstacle edges in y—axis as shown in Figure 16.

Assume xc,yc the midpoint of the captured obstacle, this point can be acquired by equations (42, 43):

the radius Rc of the circumscribed circle can be calculated by equation (44):

the new trajectory can be generated by solving the drawn path mathematically through the mapping produced path from the series of (X,Υ) points to a new series of (Xn,Υn) points after gathering a safety distance δd as in Equations (45) and (46).

The new generated trajectory enables the group of the UAVs to avoid collision with the captured obstacle. The designed controller succeeded in controlling each unmanned quadrotor in the troop in a decentralized manner guaranteeing the reserving of the desired geometric formation in XYZ—plane is shown in Figure 17.

Figure 18 represents the separation distance between the leader and the Follower 1 and the separation distance between Follower 1 and Follower 2.

The proposed control approach was compared with a backstepping-backstepping controller. Figure 19 and Figure 20 demonstrate the trajectories of cooperative quadrotors in the presence of a single stationary obstacle in XY—axis and XYZ—plane, respectively. The two control approaches succeeded in guaranteeing the safety of all the members of the group.

Dynamic obstacles

In this subsection, the proposed hybrid control approach will be applied on three cooperative quadrotors in the presence of dynamic obstacles as shown in Figure 21.

The first obstacle is moving along the desired trajectory with a lower speed while the second obstacle is moving in an opposite direction to collide with the group.

The proposed control approach was compared with a backstepping - backstepping approach. Figure 22 and Figure 23 demonstrate the trajectories of cooperative quadrotors in the presence of a couple dynamic obstacles in XY—axis and XZ—axis, respectively.

The path of the group cooperative quadrotors in the presence of couple dynamic obstacles in XYZ—plane is introduced in Figure 24.

Figure 23 demonstrates the ANFIS controller versus backstepping controller in the presence of a couple of moving obstacles in XYZ—plane.

Finally, all the simulation results presented in Scenario 1 and Scenario 2 show that the proposed hybrid controller achieves a good track to pre-planned trajectories in the presence and absence of both stationary and moving obstacles.

5. Stability analysis of hybrid controller

For ease of control, the accurate perfect quadrotor model is reduced by decreasing the outer forces and torques to take on the real-time restrictions of the incorporated control cycle (^{Wierema, 2008}).

In this manner, the 12—states and 6—DOF framework i of the quadrotor can be reworded in state-space definition X˙i=f(Xi,ηi) taking into account the inputs parameters ηi and the state parameters Xi of the framework as (^{Bouabdallah & Siegwart, 2007}):

assuming the following:

by combining Equation (4) with Equations (47) and (49). The reduced quadrotor model is presented in (50):

were

from (50), the quadrotor rotation rate can be defined as:

Equations (4) and (52) empower configuring the quadrotor attitude control. Neglecting the gyroscopic parameters in (52):

supposing η2i*,η3i* and η4i* to obtain a linear framework where:

the nonlinear feedback for linearization can be acquired by:

utilizing (49) leads to transform (47) into linear and decoupled framework as:

considering U2i*=U3i*=U4i*=0 and the assumed points of operation x2i=x4i=x6i=0; the generated linearized closed-loop framework is constant with or without the gyroscopic parameters in (51).

Various methods discuss the stability of complex systems such as Routh-Hurwitz method, Bode diagram, Nyquist diagram, root-locus method, the Nichols chart, and Lyapunov stability criterions (^{Shinners, 1998}). The Lyapunov stability offers precise solutions close to the point of equilibrium of non-linear system. Exponential stability in Lyapunov guarantees a low decay rate which show an estimation of how rapidly the solutions converge (^{Fichera, 2013}; ^{Totoki et al., 2009}). Because of that Lyapunov stability criterions have been chosen to analyze and confirm the stability of the offered hybrid controller.

To prove the stability, the Lyapunov function Vix2i,x4i,x6i for the attitude controller Gci and positive designated about the point of operation (^{Wu, 2009}):

the initial derivative of the Lyapunov function V˙i is resolved by exercising (52), (54), and (55). In addition, if a1i=-a3i and a5i=0, a standard quadrotor cross configuration with Ixi=Iyi is gained. So, V˙i can be reached by:

it is obvious that V˙i is free from gyroscopic parameters as well. V˙i is negative if K2i,K3i,K4i<0. The points of operation concerning the linearized order of feedback are therefore asymptotically stable.

To check the stability of the proposed hybrid controller, an equal uniformed pulse disturbance is applied in different intervals of times to each quadrotor in the group as demonstrated in Figure 25.

Like Scenario 1 in Section IV, a group of three cooperative quadrotors moves along the X—axis with a desired separation distance in a free obstacle environment forming a triangular formation.

The sudden disturbances shown in Figure 25 are applied to the members of the group. Figure 26 demonstrates the path trajectory for the group of the cooperative quadrotors following a desired path along the X—axis. The result shows the ability of the hybrid controller to reject the applied disturbance.

The velocity components along the X—axis of the three quadrotors and their pitch control are demonstrated in Figure 27 and Figure 28 respectively. The results show that the hybrid controller succeeded in rejecting the effect of the sudden disturbances and guarantees the desired velocity.

Similarly, Figure 29, Figure 30, and Figure 31 demonstrate the trajectories of the three cooperative quadrotors along Y—axis, their velocity components and their roll control, respectively.

The results validate the success of the introduced hybrid control approach to reject the effect of the impulse and guarantee the desired separating distance and velocity.

However, the altitude of the three cooperative quadrotors is constant, sudden disturbance was applied to investigate their effects on the quadrotors. Figure 32, Figure 33, and Figure 34 demonstrate the trajectories of the three cooperative quadrotors along Z—axis, their velocity components and their yaw control, respectively. The results show the capacity of the hybrid controller to dismiss the applied disturbance.

In order to validate the proposed controller, the controller introduced in this paper is compared with a backstepping-backstepping controller developed earlier for the same underlying system (^{Mahfouz et al., 2018}). The trajectory of the leader quadrotor along the X—axis is shown in Figure 35 where both controllers under study succeeded to reject the effect of the applied sudden disturbances.

Figure 36, and Figure 37 demonstrate the velocity and the pitch control of the leader in the presence of disturbances, respectively.

The ANFIS controller can reject the effect of the disturbance more than the backstepping controller.

Similarly, Figure 38, Figure 39, and Figure 40 demonstrate the trajectory of the leader quadrotor along Y—axis, its velocity component and its roll control, respectively.

The results validate the success of the introduced hybrid control to reject the sudden disturbances over the backstepping controller.

Finally, Figure 41, Figure 42, and Figure 43 demonstrate the trajectory of the leader quadrotor along Z—axis, its velocity component and its yaw control, respectively. The results show the capacity of the ANFIS controller to dismiss the applied disturbances over the backstepping controller.

By investigating the results appear in the previously sketched figures, the proposed hybrid backstepping-ANFIS controller achieves good tracking of the reference in different scenarios. The comparison between the proposed hybrid controller and backstepping controller shows the superiority of the hybrid controller over the backstepping controller especially in stability analysis.