A Novel Design of a Sliding Mode Controller Based on Modified ERL for Enhanced Quadcopter Trajectory Tracking

Abstract

This paper introduces a new approach to obtain robust tracking performance, disturbance resistance, and input variation resistance, and eliminate chattering phenomena in the control signal and output responses of an unmanned aerial vehicle (UAV) quadcopter with parametric uncertainty. This method involves a modified exponential reaching law (ERL) of the sliding mode control (SMC) based on a Gaussian kernel function with a continuous nonlinear Smoother Signum Function (SSF). The smooth continuous signum function is proposed as a substitute for the signum function to prevent the chattering effect caused by the switching sliding surface. The closed-loop system’s stability is ensured according to Lyapunov’s stability theory. Optimal trajectory tracking is attained based on particle swarm optimization (PSO) to select the controller parameters. A comparative analysis with a classical hierarchical SMC based on different ERLs (sign function, saturation function, and SSF) is presented to further substantiate the superior performance of the proposed controller. The outcomes of the simulation prove that the suggested controller has much better effectiveness, unknown disturbance resistance, input variation resistance, and parametric uncertainty than the other controllers, which produce chattering and make the control signal range fall within unrealistic values. Furthermore, the suggested controller outperforms the classical SMC by reducing the tracking integral mean squared errors by 96.154% for roll, 98.535% for pitch, 44.81% for yaw, and 22.8% for altitude under normal flight conditions. It also reduces the tracking mean squared errors by 99.05% for roll, 99.26% for pitch, 40.18% for yaw, and 99.998% for altitude under trajectory tracking flight conditions in the presence of external disturbances. Therefore, the proposed controller can efficiently follow paths in the presence of parameter uncertainties, input variation, and external disturbances.

1. Introduction

Unmanned aerial vehicles (UAVs), especially small quadcopters, have advanced significantly in both military and civilian applications such as infrastructure management, the security sector, surveillance, and disaster assessment. With benefits including low cost, precise motions, and Vertical Take-Off and Landing (VTOL), these UAVs are being utilized increasingly for work in inaccessible areas. Unlike walking robots, UAVs are subjected to more substantial aerodynamic and gravitational forces, which significantly influence the dynamics of these airborne systems. Because of their complexity and six degrees of freedom (three for attitude and three for position), stability and control must be carefully considered. As a result, research on quadcopter UAVs has rapidly advanced, particularly in agriculture, logistics, and other domains that benefit from their mobility, simple mechanical design, and high maneuverability. However, achieving high-performance control remains difficult due to their dynamic complexities, model uncertainties, and external disturbances. Planar flying is necessary for quadcopter navigation in a number of application fields in order to track a moving object or follow a predetermined course under unpredictable circumstances. Various control algorithms have been proposed in existing research to enhance the tracking accuracy of quadcopter systems. One of the first academic works on the use of PID controllers in quadrotor systems was [1], which discussed gain tuning in PID control for quadrotors. A useful overview of quadcopter PID control is provided in [2]. Fractional-order PID controller design has been introduced in [3], which fully controls the quadrotor’s position and attitude using a modified Black–Nichols technique. In [4], the researchers examined and contrasted the tracking performance of type-1 and type-2 fuzzy neural networks using elliptical membership functions for the quadcopter VTOL aircraft trajectory tracking problem. As compared to a single PD controller, the experimental result demonstrates that the type-1 and type-2 fuzzy neural networks operating in tandem with a PD controller have noticeably lower steady-state errors. The suggested type-2 fuzzy neural network additionally exhibits superior noise ejection capabilities by utilizing elliptic membership functions. A multiple-layer hierarchical structure has been designed in [5] to control the velocity of the UAV using a hybrid control strategy that combines SMC and Linear Quadratic Regulator (LQR) techniques. To estimate the unmeasured states, a reduced-order observer is created and incorporated into the compensator. The controller achieved a short rising time and settling time, and slight overshoot. Other researchers implemented two adaptive controllers on both integer-order and fractional-order quadcopter UAV systems [6]. The results indicate that applying SMC and fractional-order SMC to the fractional-order quadcopter system yields superior control performance, robustness, and accuracy compared to the traditional integer-order quadcopter system. On the other hand, the data model is used to develop the model-free adaptive SMC for quadcopter attitude in [7], which guarantees rapid attitude angle convergence. To correct the detected disturbance and keep it from conflicting with the feedback control position, a disturbance observer has been added to the position controller. In [8], the control of the quadrotor UAV is achieved by using an enhanced integral backstepping SMC technique. This approach first adds the integral term to the virtual variable to improve the system’s performance; then the system is further treated to enhance its control performance in conjunction with SMC. A quadrotor UAV’s position and attitude tracking control in the presence of external disturbances and parametric uncertainties is proposed in [9] by combining sliding mode control with a neural network adaptive scheme using a novel approach. The suggested approach maintains the benefits of both approaches. Additionally, the path-following problem could be resolved in [10] using a hierarchical system that consists of a nonlinear H-infinity controller to stabilize the rotating movements and a model predictive controller to follow the trajectory reference. SMCs have many benefits, including robustness, adaptability, insensitivity to external disturbances, and fast responses; however, they also have drawbacks, such as chattering effects and high consumption of energy [11]. As a result, some authors have attempted to minimize chattering by replacing the typical SMC’s signum function with a continuous function, as seen in [12]. Next, an integral reinforcement learning approach is suggested, based on an optimal adaptive saturation function tuning approach. Also, the authors of [13,14] created the fast dynamic terminal SMC for attitude tracking control and tracking the position of a quadcopter. It can also achieve high-precision performance and minimize chattering produced by the switching control action. In [15], the scalar sign function technique is used in SMC and applied to an aluminum beam using a piezoceramic sensor and vibration control actuator. Furthermore, an excellent overview of various control strategies to address and fix a number of issues pertaining to autonomous quadcopters can be found in [16,17]. This work presents a novel Modified Sliding Mode Controller (MSMC) design that modifies the exponential reaching law of SMC using a kernel Gaussian formula and replaces the signum function with a continuous nonlinear function to eliminate the chattering effect and minimize the cost function. The proposed controller is applied to an underactuated, nonlinear quadcopter with Multiple-Input Multiple-Output (MIMO) dynamics for position and attitude tracking control considering different trajectories and external disturbances. Optimal trajectory tracking is achieved by online tuning of the controller parameters using the particle swarm optimization (PSO) algorithm. Four controllers are compared to evaluate the system’s performance. Simulation results show that both SMC based on the SSF and MSMC outperform the classical SMC in terms of robustness and performance, with MSMC achieving superior output performance compared to SMC based on the SSF. The novel contributions are hence summarized as follows:

1.

A new optimal adaptive sliding mode controller based on modification of ERL for an underactuated quadcopter system is developed that is employed under various simulation experiments like a trajectory tracking mission, automated take-off and landing, and altitude and attitude control in nominal mode, input variation mode, parametric uncertainty mode, and unknown disturbances mode.

2.

Elimination of the chattering effect caused by the switching sliding surface is conducted by employing a smooth continuous signum function as a substitute for the signum function. Then, the chattering effect is eliminated with the proposed SMC compared to that observed with classical SMC.

3.

All factors influencing the quadcopter’s dynamics, including gyroscopic effects, drag forces along the (x, y, z) axes, frictions from aerodynamic torque, and high-level non-holonomic limitations, were considered when designing the controller.

4.

Stability of the quadrotor’s trajectory tracking and attitude control system is demonstrated using Lyapunov’s theory.

5.

Our results demonstrate that the modified ERL improves the robustness of the SMC system when compared to previous research [11,13,14,18], and MATLAB R2021a simulations validate the theoretical framework and the suggested controller’s efficacy, robustness, and exact convergence.

This paper is structured as follows: The quadcopter UAV’s mathematical model is shown in Section 2. The problem statement is covered in Section 3. The controller architecture utilizing traditional SMC and an enhanced version of SMC is created in Section 4. In Section 5, the particle swarm optimization (PSO) technique is described, and a performance index is introduced. The simulation results and a comparative analysis are shown in Section 6. Finally, Section 7 provides the conclusions.

2. Nonlinear Quadcopter Dynamic Modeling

The underactuated quadrotor UAV, depicted in Figure 1, consists of four fixed-pitch rotors mounted symmetrically on a rigid cross-platform, with control electronics positioned at the center. The diametrically opposed rotors rotate in the same direction, and the vehicle’s movement is controlled by adjusting their rotational speeds. By strategically varying these speeds, the quadrotor can execute ascent and descent maneuvers, as well as rotational movements such as rolling (by an angle $\varphi$), pitching (by an angle $\theta$), and yawing (by an angle $\psi$). Each rotor generates a torque about its centre of rotation, and when all rotors spin at an equal angular velocity, the net aerodynamic torque acting on the quadrotor is zero. This occurs because the torques produced by the counter-rotating rotor pairs effectively cancel each other out. In order to produce yaw motion, one pair of rotors’ rotational speed is increased while the other pair’s speed is decreased. This causes an imbalance in the aerodynamic torque about the yaw axis. Each rotor generates an upward push perpendicular to the plane of blade rotation in addition to torque. The two pairs of rotors that revolve in the same direction use differential propulsion to control roll and pitch motions. The quadcopter rotates around the pitch or roll axis by increasing the speed of one of its two rotors while reducing the speed of the other [5,18].

As seen in Figure 1, let $I=(I_x,I_y,I_z)$ represents an inertial frame, and $B=(X_B,Y_B,Z_B)$ represents a frame that is firmly fixed to the quadrotor. This work makes the following assumptions in order to construct the quadcopter dynamic model and derive the subsequent control law:

1.

The quadcopter has a symmetrical architecture with a rigid body and rigid propellers.

2.

The body-fixed frame origin is precisely the quadcopter’s center of mass.

3.

Drag and thrust forces are proportionate to the rotor speed squared.

Given these assumptions, the dynamics of the fuselage can be described as those of a rigid body in space, with the additional aerodynamic forces resulting from the rotation of the rotors. Through use of the Newton–Euler formalism, the dynamic equations are expressed as follows [19]:

$$\begin{array}{cc}\hfill \dot{\xi }& =v\hfill \\ \hfill m\ddot{\xi }& =f_f+f_t+f_g\hfill \\ \hfill \dot{R}& =RS(\Omega )\hfill \\ \hfill J\dot{\Omega }& =-\Omega \wedge J\Omega +{\Gamma }_f-{\Gamma }_a-{\Gamma }_g\hfill \end{array}$$

(1)

where $\xi$ is the position of the quadcopter’s center of mass in the inertial frame, v is the linear velocity, m is the total mass, and $J\in {\mathbb{R}}^{3\times 3}$ is the symmetric positive definite inertia matrix of the quadcopter in the body frame with respect to B:

$$J=\left[\begin{array}{ccc}{I}_x& 0& 0\\ 0& I_y& 0\\ 0& 0& I_z\end{array}\right]$$

(2)

According to [20], R is the homogeneous rotation matrix and $\Omega$ is the angular velocity of the airframe. The rotation matrix R is defined as

$$R=\left[\begin{array}{ccc}{c}_{\theta }{c}_{\psi }& c_{\psi }{s}_{\varphi }{s}_{\theta }-s_{\psi }{c}_{\varphi }& c_{\psi }{c}_{\varphi }{s}_{\theta }+s_{\psi }{s}_{\varphi }\\ c_{\theta }{s}_{\psi }& s_{\psi }{s}_{\varphi }{s}_{\theta }+c_{\psi }{c}_{\varphi }& s_{\psi }{c}_{\varphi }{s}_{\theta }-c_{\psi }{s}_{\varphi }\\ -s_{\theta }& c_{\theta }{s}_{\varphi }& c_{\theta }{c}_{\varphi }\end{array}\right]$$

(3)

where c and s denote the cosine and sine functions, respectively. The angular velocity vector $\Omega$ is given by

$$\Omega =\left[\begin{array}{c}{\Omega }_1\\ {\Omega }_2\\ {\Omega }_3\end{array}\right]=\left[\begin{array}{ccc}1& 0& -s_{\theta }\\ 0& c_{\varphi }& s_{\varphi }{c}_{\theta }\\ 0& -s_{\varphi }& c_{\varphi }{c}_{\theta }\end{array}\right]\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]$$

(4)

The skew-symmetric matrix $S(\Omega )$ is defined as

$$S(\Omega )=\left[\begin{array}{ccc}0& -{\Omega }_3& {\Omega }_2\\ {\Omega }_3& 0& -{\Omega }_1\\ -{\Omega }_2& {\Omega }_1& 0\end{array}\right]$$

(5)

The total thrust force $f_f$ generated by the four rotors is given by

$$f_f=R·\left[\begin{array}{c}0\\ 0\\ \sum _{i=1}^4{F}_i\end{array}\right],\mathrm{where}{F}_i\approx b{\omega }_i^2$$

(6)

Here, b is the lift coefficient and ${\omega }_i$ is the angular speed of rotor i. The drag force $f_t$ along the body axes $(X_B,Y_B,Z_B)$ is given by

$$f_t=\left[\begin{array}{ccc}-K_{ftx}& 0& 0\\ 0& -K_{fty}& 0\\ 0& 0& -K_{ftz}\end{array}\right]\dot{\xi }$$

(7)

where $K_{ftx},K_{fty},$ and $K_{ftz}$ are the translational drag coefficients. The gravitational force $f_g$ is

$$f_g=\left[\begin{array}{c}0\\ 0\\ -mg\end{array}\right]$$

(8)

The moment generated by the quadcopter relative to the body-fixed frame, ${\Gamma }_f$, is given by

$${\Gamma }_f=\left[\begin{array}{c}l(F_3-F_1)\\ l(F_4-F_2)\\ d({\omega }_1^2-{\omega }_2^2+{\omega }_3^2-{\omega }_4^2)\end{array}\right]$$

(9)

where l is the distance between the propeller’s rotational axis and the quadcopter’s center of mass, and d is the drag coefficient. The aerodynamic friction torque ${\Gamma }_a$ is

$${\Gamma }_a=\left[\begin{array}{ccc}{K}_{fax}& 0& 0\\ 0& K_{fay}& 0\\ 0& 0& K_{faz}\end{array}\right]{\Omega }^2$$

(10)

where $K_{fax}$, $K_{fay}$, and $K_{faz}$ are aerodynamic friction coefficients. The gyroscopic torque ${\Gamma }_g$ is

$${\Gamma }_g=\sum _{i=1}^4\Omega \wedge J_r\left[\begin{array}{c}0\\ 0\\ {(-1)}^{i+1}{\omega }_i\end{array}\right]$$

(11)

where $J_r$ is the rotor inertia. The complete dynamic model of the quadcopter’s angular and translational motion is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \ddot{\varphi }& ={\displaystyle \frac{1}{I_x}}\left[l{U}_2+(I_y-I_z)(\dot{\psi }\cos \varphi \cos \theta -\dot{\theta }\sin \varphi )(\dot{\theta }\cos \varphi +\dot{\psi }\sin \varphi \cos \theta )\right.\hfill \\ \hfill & \left.-J_r{\Omega }_r(\dot{\psi }\sin \varphi \cos \theta +\dot{\theta }\cos \varphi )-K_{fax}({\dot{\varphi }}^2-2\dot{\varphi }\dot{\psi }{\sin }^2\theta )\right]\hfill \end{array}\end{array}$$

(12)

$$\begin{array}{cc}\hfill \ddot{\theta }& ={\displaystyle \frac{1}{I_y}}\left[l{U}_3+(I_z-I_x)(\dot{\psi }\cos \varphi \cos \theta -\dot{\theta }\sin \varphi )(\dot{\varphi }-\dot{\psi }\sin \theta )\right.\hfill \\ \hfill & \left.-J_r{\Omega }_r(\dot{\psi }\sin \theta -\dot{\varphi })-K_{fay}({\dot{\theta }}^2{\cos }^2\varphi +2\dot{\varphi }\dot{\psi }\sin \varphi \cos \varphi \cos \theta +{\dot{\psi }}^2{\sin }^2\varphi {\cos }^2\theta )\right]\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \ddot{\psi }& ={\displaystyle \frac{1}{I_z}}\left[U_4+(I_x-I_y)(\dot{\psi }\sin \varphi \cos \theta +\dot{\theta }\cos \varphi )(\dot{\varphi }-\dot{\psi }\sin \theta )\right.\hfill \\ \hfill & \left.-K_{faz}({\dot{\theta }}^2{\sin }^2\varphi -2\dot{\varphi }\dot{\psi }\sin \varphi \cos \varphi \cos \theta +{\dot{\psi }}^2{\cos }^2\varphi {\cos }^2\theta )\right]\hfill \end{array}$$

(14)

The translational accelerations are given by

$$\ddot{x}={\displaystyle \frac{1}{m}}\left[(\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )U_1-K_{ftx}\dot{x}\right]$$

(15)

$$\ddot{y}={\displaystyle \frac{1}{m}}\left[(\cos \varphi \sin \psi \sin \theta -\cos \psi \sin \varphi )U_1-K_{fty}\dot{y}\right]$$

(16)

$$\ddot{z}={\displaystyle \frac{1}{m}}\left[\cos \varphi \cos \theta U_1-mg-K_{ftz}\dot{z}\right]$$

(17)

To simplifying the dynamic equations of the quadcopter model (only used to simplify the formulation of the equations), small-angle approximations are used (i.e., $\cos x\approx 1$, $\sin x\approx 0$). Under these conditions, Equations (12)–(17) simplify as shown in [19]:

$$\ddot{\varphi }=a_1\dot{\psi }\dot{\theta }+b_1{U}_2+c_1{\Omega }_r\dot{\theta }+d_1{\dot{\varphi }}^2$$

(18)

$$\ddot{\theta }=a_2\dot{\varphi }\dot{\psi }+b_2{U}_3+c_2{\Omega }_r\dot{\varphi }+d_2{\dot{\theta }}^2$$

(19)

$$\ddot{\psi }=a_3\dot{\varphi }\dot{\theta }+b_3{U}_4+c_3{\dot{\psi }}^2$$

(20)

The simplified translational dynamics of the quadcopter are expressed as

$$\begin{array}{cc}\hfill \ddot{x}& =a_4\dot{x}+b_4(\cos \varphi \cos \psi \sin \theta +\sin \varphi \sin \psi )U_1\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \ddot{y}& =a_5\dot{y}+b_5(\cos \varphi \sin \psi \sin \theta -\sin \varphi \cos \psi )U_1\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \ddot{z}& =a_6\dot{z}+b_6\left(\cos \varphi \cos \theta \right)U_1-g\hfill \end{array}$$

(23)

The constants are defined as

$$\begin{array}{cccccccc}\hfill a_1& ={\displaystyle \frac{I_y-I_z}{I_x}},\hfill & \hfill b_1& ={\displaystyle \frac{l}{I_x}},\hfill & \hfill c_1& ={\displaystyle \frac{-J_r}{I_x}},\hfill & \hfill d_1& ={\displaystyle \frac{-K_{fax}}{I_x}}\hfill \\ \hfill a_2& ={\displaystyle \frac{I_z-I_x}{I_y}},\hfill & \hfill b_2& ={\displaystyle \frac{l}{I_y}},\hfill & \hfill c_2& ={\displaystyle \frac{-J_r}{I_y}},\hfill & \hfill d_2& ={\displaystyle \frac{-K_{fay}}{I_y}}\hfill \\ \hfill a_3& ={\displaystyle \frac{I_x-I_y}{I_z}},\hfill & \hfill b_3& ={\displaystyle \frac{1}{I_z}},\hfill & \hfill c_3& ={\displaystyle \frac{-K_{faz}}{I_z}}\hfill \\ \hfill a_4& ={\displaystyle \frac{-K_{ftx}}{m}},\hfill & \hfill b_4& ={\displaystyle \frac{1}{m}}\hfill \\ \hfill a_5& ={\displaystyle \frac{-K_{fty}}{m}},\hfill & \hfill b_5& ={\displaystyle \frac{1}{m}}\hfill \\ \hfill a_6& ={\displaystyle \frac{-K_{ftz}}{m}},\hfill & \hfill b_6& ={\displaystyle \frac{1}{m}}\hfill \\ \hfill {\Omega }_r& ={\omega }_1-{\omega }_2+{\omega }_3-{\omega }_4\hfill \end{array}$$

(24)

From Equations (18)–(23), it is evident that the quadcopter is a nonlinear underactuated system that has six outputs $(\varphi ,\theta ,\psi ,x,y,z)$ with four control inputs (U₁, U₂, U₃, U₄) described by

$$\left\{\begin{array}{c}{U}_1=b({\omega }_1^2+{\omega }_2^2+{\omega }_3^2+{\omega }_4^2)\hfill \\ U_2=b({\omega }_4^2-{\omega }_2^2)\hfill \\ U_3=b({\omega }_3^2-{\omega }_1^2)\hfill \\ U_4=d({\omega }_1^2-{\omega }_2^2+{\omega }_3^2-{\omega }_4^2)\hfill \end{array}\right.$$

(25)

The control signals $U_2$, $U_3$, and $U_4$ correspond to the roll $\left(\varphi \right)$, pitch $\left(\theta \right)$, and yaw $\left(\psi \right)$ angles, respectively. The position states x, y, and z are influenced by $\varphi$, $\theta$, and $U_1$. To independently control the output positions, three virtual control inputs $(U_x,U_y,U_z)$ are introduced:

$$\left\{\begin{array}{c}{U}_x=\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \hfill \\ U_y=\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \hfill \\ U_z=U_1\hfill \end{array}\right.$$

(26)

The desired roll and pitch angles are derived from the virtual controls:

$$\left\{\begin{array}{c}{\varphi }_d=arcsin\left({\displaystyle \frac{U_x\sin {\psi }_d-U_y\cos {\psi }_d}{\sqrt{U_x^2+U_y^2+U_z^2}}}\right)\hfill \\ {\theta }_d=arctan\left({\displaystyle \frac{U_x\cos {\psi }_d+U_y\sin {\psi }_d}{U_z}}\right)\hfill \end{array}\right.$$

(27)

3. Problem Statement

The state-space form of the quadcopter dynamic equations created in this paper can be rewritten as the following continuous-time system:

$$\ddot{X}\left(t\right)=f(X,t)+g\left(X\right)·U_i\left(t\right)+d_i\left(t\right)$$

(28)

where $d_i={[d_1,d_2,d_3,d_4,d_5,d_6]}^T$ represents the unknown external disturbances and some dynamics of the system (Equations (18)–(23)), $|d_i| \le {\delta }_i$ (upper bound), $i=1,\dots ,6$ represents the state-space number, $U_i={[U_2,U_3,U_4,U_x,U_y,U_1]}^T$ represents the control signals, and $X={[X_1,X_2,\dots ,X_6]}^T$ represents the state vector of the quadcopter system, such that

$$X_i={[\varphi ,\theta ,\psi ,x,y,z]}^T$$

(29)

The nonlinear functions $f\left(X\right)$ and $g\left(X\right)$, describing the system dynamics and control, respectively, are expressed as the following matrices:

$$f\left(X\right)=\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_4\\ f_5\\ f_6\end{array}\right]=\left[\begin{array}{c}{a}_1\dot{\psi }\dot{\theta }\\ a_2\dot{\varphi }\dot{\psi }\\ a_3\dot{\theta }\dot{\varphi }\\ 0\\ 0\\ -g\end{array}\right]$$

(30)

$$g\left(X\right)=\left[\begin{array}{cccccc}{b}_1& 0& 0& 0& 0& 0\\ 0& b_2& 0& 0& 0& 0\\ 0& 0& b_3& 0& 0& 0\\ 0& 0& 0& b_4& 0& 0\\ 0& 0& 0& 0& b_5& 0\\ 0& 0& 0& 0& 0& b_6\end{array}\right]$$

(31)

4. Control Design

The quadcopter-controlled system block diagram is displayed in Figure 2, where the first part refers to position and yaw angle control. The desired position $(x_d,y_d,z_d)$ and desired yaw angle ${\psi }_d$ are predefined by the user. Based on these inputs, the simulation program automatically calculates the required roll and pitch signals using sensor data to ensure the quadrotor remains stable during flight. Subsequently, the second part refers to attitude control. The main goal of the control system is to guide a trajectory state X in the direction of a desired reference trajectory: $X_{di}={\left[\begin{array}{cccccc}{X}_{d1}& X_{d2}& X_{d3}& X_{d4}& X_{d5}& X_{d6}\end{array}\right]}^T={\left[\begin{array}{cccccc}{\varphi }_d& {\theta }_d& {\psi }_d& x_d& y_d& z_d\end{array}\right]}^T$. The SMC design process consists of two parts. Designing a sliding surface is the initial stage in order to achieve the necessary system response for the plant that is confined to it. This indicates that a different set of equations, defining what is known as the switching surface, must be satisfied by the state variables of the plant dynamics. Building the switched feedback gains required to move the plant’s state trajectory toward the sliding surface is the second phase. These designs are based on the generalized Lyapunov stability theory.

4.1. Classical SMC Design Based on ERL

The initial step of constructing the SMC for the quadcopter UAV system involves defining the sliding surface as follows:

$$S_i\left(t\right)={\dot{e}}_i\left(t\right)+{\lambda }_i{e}_i\left(t\right)$$

(32)

where $e_i\left(t\right)=X_i\left(t\right)-X_{di}\left(t\right)$ represents the tracking error, ${\dot{e}}_i\left(t\right)={\dot{X}}_i\left(t\right)-{\dot{X}}_{di}\left(t\right)$ represents the derivative tracking error, and ${\lambda }_i$ is the sliding surface coefficient that must be positive and selected carefully to satisfy the Hurwitz condition. The time derivative of Equation (32) can be determined as follows:

$${\dot{S}}_i\left(t\right)={\ddot{e}}_i\left(t\right)+{\lambda }_i{\dot{e}}_i\left(t\right)$$

(33)

By substituting Equation (28) into Equation (33), the time derivative of the sliding surface can be rewritten as

$${\dot{S}}_i\left(t\right)=\left(f_i(X,t)+b_i{U}_i\left(t\right)+d_i\left(t\right)\right)-{\ddot{X}}_{di}\left(t\right)+{\lambda }_i{\dot{e}}_i\left(t\right)$$

(34)

When the state reaches the sliding manifold and approaches equilibrium, the expected motion constraints are shown by the sliding mode manifold $S_i$. The controller will ensure that the trajectories of $X_i$ do not escape from the sliding surface in the ideal scenario, i.e., that the state values remain on the manifold $S_i=0$ and are driven to the sliding surface in finite time. This can be achieved by using various types of reaching control law techniques, such as conventional ERL [21], which can be expressed as follows:

$${\dot{S}}_i\left(t\right)=-k_i\mathrm{sign}\left(S_i\left(t\right)\right)-{\lambda }_i{S}_i\left(t\right)+d_i\left(t\right),k_i>0,{\lambda }_i>0$$

(35)

The approach rate is represented by the coefficient $k_i$, and the signum function (S_i) is defined as

$$\mathrm{sign}\left(S_i\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}{S}_i<0\hfill \\ 0\hfill & \mathrm{if}{S}_i=0\hfill \\ 1\hfill & \mathrm{if}{S}_i>0\hfill \end{array}\right.$$

(36)

The equivalent sliding control law can be obtained by solving Equations (34) and (35) as follows:

$$U_i\left(t\right)={\displaystyle \frac{1}{b_i}}\left({\ddot{X}}_{di}\left(t\right)-f_i\left(t\right)-k_i\mathrm{sign}\left(S_i\left(t\right)\right)-2{\lambda }_i{\dot{e}}_i\left(t\right)-{\lambda }_i^2{e}_i\left(t\right)\right)$$

(37)

The Lyapunov stability theory was applied to ensure the system’s stability; hence, the Lyapunov candidate was chosen as

$$V_i\left(S_i\right)={\displaystyle \frac{1}{2}}{S}_i^2$$

(38)

By applying Equation (34) to the derivative of the Lyapunov candidate, we obtain

$${\dot{V}}_i\left(S_i\right)=S_i{\dot{S}}_i=S_i\left(f_i\left(t\right)+b_i{U}_i\left(t\right)+d_i\left(t\right)-{\ddot{X}}_{di}\left(t\right)+{\lambda }_i{\dot{e}}_i\left(t\right)\right)$$

(39)

Also, Equation (37) can be substituted into Equation (39) to obtain

$${\dot{V}}_i\left(S_i\right)=-{\lambda }_i{S}_i^2-k_i{S}_i\mathrm{sign}\left(S_i\right)+S_i{d}_i\left(t\right)=-{\lambda }_i{S}_i^2-k_i\left|S_i\right|+S_i{d}_i\left(t\right)$$

(40)

Remark 1.

The quadcopter UAV’s trajectory tracking is guaranteed to be stable under SMC based on the signum function if the controller design parameters are selected such that $k_i\ge {\delta }_i$$\forall S_i\ne 0$. This condition ensures that the Lyapunov candidate is positive definite and its derivative ${\dot{V}}_i\le 0$ is negative definite or semi-negative definite, leading the sliding surfaces to converge to zero as t →∞.

4.2. SMC Design Based on ERL Using Saturation Function

Some studies have attempted to use the saturation function for the SMC’s switching surfaces, such as [22], for the precision positioning of a nano-positioning stage. Other researchers have used the hyperbolic tangent function [23] for the switched reluctance motor. To improve sliding mode controller performance, the discontinuous sign function is replaced by a saturation function to reduce chattering. The saturation function utilized in the control law is defined as

$$\mathrm{sat}\left(S_i\right)=\left\{\begin{array}{cc}\mathrm{sign}\left(S_i\right)\hfill & \mathrm{if}|S_i|>\mu \hfill \\ {\displaystyle \frac{S_i}{\mu }}\hfill & \mathrm{if}|S_i|\le \mu \hfill \end{array}\right.$$

(41)

Switch control is applied outside the boundary layer ($\mu$), while linear feedback control is used within it. In this case, the control signal of the SMC can be stated as follows:

$$U_i\left(t\right)={\displaystyle \frac{1}{b_i}}\left({\ddot{X}}_{di}\left(t\right)-f_i\left(t\right)-k_i\mathrm{sat}\left(S_i\left(t\right)\right)-2{\lambda }_i{\dot{e}}_i\left(t\right)-{\lambda }_i^2{e}_i\left(t\right)\right)$$

(42)

Through use of the same Lyapunov candidate and the same procedure utilized previously, quadcopter stability can be achieved using the following equation:

$${\dot{V}}_i\left(S_i\right)=-{\lambda }_i{S}_i^2-k_i{S}_i\mathrm{sat}\left(S_i\right)+S_i{d}_i\left(t\right)=-{\lambda }_i{S}_i^2-k_i|S_i\left|\mathrm{sat}\right(|S_i\left|\right)+S_i{d}_i\left(t\right)$$

(43)

Remark 2.

The quadcopter stability is guaranteed under SMC based on the saturation function, if the controller design parameters are selected such that $\mathrm{sat}\left(\right|S_i\left|\right)\ge {\delta }_i/k_i$, $\forall S_i$, and this ensures that ${\dot{V}}_i\le 0$ always. The ultimate bound will be $|S_i|\le (\mu /k_i){\delta }_i$. The saturation function used was able to stabilize the system as t → ∞, and the upper and lower bounds were selected based on heuristic experience. However, the system is still struggling to decrease sliding chattering and increase approaching speed. In the next subsection, we will propose a solution to overcome these difficulties.

4.3. Adaptive SMC Design Based on Modified ERL

The classical ERL offers good convergence, but the system experiences significant chattering when approaching the sliding mode. To prevent the chattering effect caused by the switching sliding surface, a smooth continuous nonlinear function is proposed and can be used to approximate the discontinuous switching function. Therefore, the signum function $\mathrm{sign}\left(S_i\right)$ in the traditional ERL is replaced by a nonlinear switching surface function (SSF) in the new reaching law, which can be expressed by the following equation:

$$\mathrm{SSF}\left(S_i\right)={\displaystyle \frac{S_i}{\sqrt{S_i^2+e^{-1/\epsilon }}}}$$

(44)

where $\epsilon$ is a positive tuning value between 0 and 1 for smoothing the discontinuity. It is manually calibrated to reduce chattering noise. The varied SSFs based on different values of $\epsilon$ are presented in Figure 3. On the other hand, the approaching speed to the sliding surface and system performance can be improved by modifying the traditional ERL based on the Gaussian kernel function (GK). Consequently, the modified ERL will be created as follows:

$${\dot{S}}_i\left(t\right)=-k_i\mathrm{GK}\left(S_i\right)·{\displaystyle \frac{S_i}{\sqrt{S_i^2+e^{-1/\epsilon }}}}-{\lambda }_i{S}_i\left(t\right)$$

(45)

Presume that $\mathrm{GK}\left(S_i\right)=A·e^{-\frac{1}{2}{S}_i^2}$, where $A={\left(2\pi \right)}^{-D/2}$ and D is the space dimension. To stabilize and converge to the sliding surface within a specified time, the condition ${\dot{V}}_i\le 0$ must be met. In this case, substituting Equation (45) into the derivative of the Lyapunov candidate (Equation (38)), we get

$${\dot{V}}_i\left(S_i\right)=-{\lambda }_i{S}_i^2-k_i\mathrm{GK}\left(S_i\right)·{\displaystyle \frac{S_i^2}{\sqrt{S_i^2+e^{-1/\epsilon }}}}+S_i{d}_i\left(t\right)$$

(46)

Remark 3.

The quadcopter stability is guaranteed under adaptive SMC based on the modified ERL if the controller design parameters are selected such that $|S_i|\ge {\delta }_i/k_i$, $\forall S_i$ and the $\mathrm{GK}\left(S_i\right)$ remains strictly positive for all values of $S_i$. Therefore, these conditions ensure that ${\dot{V}}_i\le 0$ and the sliding surface converges to zero as t → ∞. The overall control law of the adaptive SMC based on the modified ERL can be expressed as follows:

$$U_i\left(t\right)={\displaystyle \frac{1}{b_i}}\left({\ddot{X}}_{di}\left(t\right)-f_i\left(t\right)-k_i\mathrm{GK}\left(S_i\right)·{\displaystyle \frac{S_i^2}{\sqrt{S_i^2+e^{-1/\epsilon }}}}-2{\lambda }_i{\dot{e}}_i\left(t\right)-{\lambda }_i^2{e}_i\left(t\right)\right)$$

(47)

5. Optimal SMC and Performance Indices

5.1. Basics of Particle Swarm Optimization

In 1995, Eberhart and Kennedy first proposed particle swarm optimization (PSO), a stochastic optimization method [24,25]. The PSO algorithm mimics the social behavior of various animals, such as fish, birds, insects, and herds. These swarms work together to find food, and each member continuously modifies the search strategy based on its own and other members’ learning experiences. The PSO can be expressed mathematically by applying a swarm size of N and the position vector of each particle in a space of dimension d as ${\mathbf{x}}_i=(x_{i1},x_{i2},\dots ,x_{id})$. While the velocity vector is represented by ${\mathbf{V}}_i=(V_{i1},V_{i2},\dots ,V_{id})$, the optimal position of an individual is represented by ${\mathbf{P}}_i=(P_{i1},P_{i2},\dots ,P_{id})$, and the optimal swarm position is represented by ${\mathbf{P}}_g=(P_{g1},P_{g2},\dots ,P_{gd})$, which corresponds to the position with the lowest fitness value. Each particle represents a candidate solution within the search space and updates its position over each iteration guided by changes in its velocity. Initially, particle locations and velocities are initialized randomly. Then, each particle attempts to enhance its fitness by maintaining momentum in the direction of its current velocity and is attracted toward its own best-known position $P_{id}(t+1)$. The update formula for the individual’s optimal position in the original PSO method is as follows [26]:

$$P_{id}(t+1)=\left\{\begin{array}{cc}{x}_{id}(t+1)\hfill & \mathrm{if}f\left(x_i(t+1)\right)<f\left(P_i\left(t\right)\right)\hfill \\ P_{id}\left(t\right)\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(48)

Finally, the particle’s position is updated for the next iteration based on its newly computed velocity. Therefore, the optimum position of the swarm is equal to the optimal positions of each individual. The following equations represent the updated velocity and position formulas, respectively:

$$V_{id}(t+1)=V_{id}\left(t\right)+c_1·\mathrm{rand}·(P_{id}\left(t\right)-x_{id}\left(t\right))+c_2·\mathrm{rand}·(P_{gd}\left(t\right)-x_{id}\left(t\right))$$

(49)

$$x_{id}(t+1)=x_{id}\left(t\right)+V_{id}(t+1)$$

(50)

Here, $c_1$ and $c_2$ represent the acceleration coefficients, and the uniformly distributed random number $\mathrm{rand}\in [0,1]$ adds stochastic behavior. Shortly after the original PSO technique was proposed, a modified version of the algorithm [27] was developed since the original version was not very effective in solving optimization problems. After adding an inertia weight w to the velocity update calculation, the resulting formula becomes

$$V_{id}(t+1)=w·V_{id}\left(t\right)+c_1·\mathrm{rand}·(P_{id}\left(t\right)-x_{id}\left(t\right))+c_2·\mathrm{rand}·(P_{gd}\left(t\right)-x_{id}\left(t\right))$$

(51)

5.2. Optimal SMC Based on PSO

The optimal SMC parameters for the outputs of the quadcopter system $(\varphi ,\theta ,\psi ,x,y,z)$ can be obtained online using the PSO algorithm technique. The optimal estimations of ${\lambda }_i$ and $k_i$ guarantee the optimal tracking control performance under both normal flight conditions and flight conditions affected by external disturbances. The optimal SMC parameters for the six controllers in the first and second scenarios, where Equations (37) and (42) were utilized as control signals, are given by (${\lambda }_1^{*}$, ${\lambda }_2^{*}$, ${\lambda }_3^{*}$, ${\lambda }_4^{*}$, ${\lambda }_5^{*}$, ${\lambda }_6^{*}$, $k_1^{*}$, $k_2^{*}$, $k_3^{*}$, $k_4^{*}$, $k_5^{*}$, $k_6^{*}$). In the third scenario, where Equation (47) was used as a control signal, the optimal adaptive SMC parameters are (${\lambda }_1^{*}$, ${\lambda }_2^{*}$, ${\lambda }_3^{*}$, ${\lambda }_4^{*}$, ${\lambda }_5^{*}$, ${\lambda }_6^{*}$, $k_1^{*}$, $k_2^{*}$, $k_3^{*}$, $k_4^{*}$, $k_5^{*}$, $k_6^{*}$, $R_1^{*}$, $R_2^{*}$, $R_3^{*}$, $R_4^{*}$, $R_5^{*}$, $R_6^{*}$), where $k_i^{*}=A·{\tilde{k}}_i$ and $R_i^{*}$ is an optimal positive Gaussian kernel constant. Figure 4 shows the block diagram of the optimal proposed SMC based on the modified ERL. The optimal control signal can be rewritten as

$$U_i\left(t\right)={\displaystyle \frac{1}{b_i}}\left({\ddot{X}}_{di}\left(t\right)-f_i\left(t\right)-k_i^{*}·S_i^2·e^{-R_i^{*}{S}_i^2}·{\displaystyle \frac{1}{\sqrt{S_i^2+e^{-1/\epsilon }}}}-2{\lambda }_i^{*}{\dot{e}}_i\left(t\right)-{\left({\lambda }_i^{*}\right)}^2{e}_i\left(t\right)\right)$$

(52)

The PSO algorithm uses 200 particles within a search space from 0 to 500. It operates in 12 dimensions for the first and second scenarios, and 18 dimensions for the third. A weight factor of w = 0.8 is applied to balance local exploitation and global exploration. The acceleration coefficients are set to C1 = 0.8 and C2 = 1.2 to prevent premature convergence. The algorithm stops after a maximum of 50 iterations, which acts as the stopping criterion. Additionally, the particle with the highest fitness value is kept throughout the process.

5.3. Performance Indices

The performance of the quadcopter’s responses with the proposed SMC controllers is evaluated using cost functions (CF₁), which include mean squared error (MSE) or integral mean squared error (IMSE), settling time ($t_s$), and maximum peak overshoot ($M_p$). The following cost function is applied during hovering and trajectory tracking without external disturbances:

$$\begin{array}{cc}\hfill C{F}_1& =10·IMS{E}_1+10·IMS{E}_2+IMS{E}_3\hfill \\ \hfill & +\left(IMS{E}_4+0.5M_{px}+0.3t_{sx}\right)\hfill \\ \hfill & +\left(IMS{E}_5+0.5M_{py}+0.3t_{sy}\right)\hfill \\ \hfill & +\left(IMS{E}_6+0.5M_{pz}+0.3t_{sz}\right)\hfill \end{array}$$

(53)

where ${\mathrm{ISE}}_i={\int }_0^t{e}_i{\left(t\right)}^{2}dt=[{\mathrm{ISE}}_1,{\mathrm{ISE}}_2,{\mathrm{ISE}}_3,{\mathrm{ISE}}_4,{\mathrm{ISE}}_5,{\mathrm{ISE}}_6]$, corresponding to roll, pitch, yaw, longitude, latitude, and altitude, respectively. Also, the cost function that is used in the presence of external disturbances is expressed as follows:

$${\mathbf{CF}}_2=\sum _{i=1}^6{\mathrm{MSE}}_i$$

(54)

where ${\mathrm{MSE}}_i=\mathrm{mean}\left(e_i^2\right)=[{\mathrm{MSE}}_1,{\mathrm{MSE}}_2,{\mathrm{MSE}}_3,{\mathrm{MSE}}_4,{\mathrm{MSE}}_5,{\mathrm{MSE}}_6]$, corresponding to roll, pitch, yaw, longitude, latitude, and altitude, respectively.

6. Simulation and Results

The simulation of the controlled quadcopter system is executed for 20 s with a sampling time equal to 0.001 to calculate the fitness function and to prove the capability of our controller design to stabilize the quadcopter in less time and eliminate the chattering in both control signal and output responses under quadrotor hovering at a fixed spot, and under hovering and landing during tracking missions. The symbols and values of the parameters utilized in the quadcopter UAV’s dynamic model are summarized in

Table 1. Quadcopter dynamic model parameters [18,19,20].

Parameter	Definition	Value
b	Thrust coefficient	$2.9842\times {10}^{-5}$ N·m/rad/s
d	Drag coefficient	$3.232\times {10}^{-7}$ N·m/rad/s
g	Gravity acceleration	$9.806 {\mathrm{m}/\mathrm{s}}^2$
$I_x$	Inertia around x-axis	$0.0075 {\mathrm{kg·m}}^2$
$I_y$	Inertia around y-axis	$0.0075 {\mathrm{kg·m}}^2$
$I_z$	Inertia around z-axis	$0.013 {\mathrm{kg·m}}^2$
$J_r$	Rotor inertia	$2.8385\times {10}^{-5}$${ \mathrm{N·m}/\mathrm{rad}/\mathrm{s}}^2$
$K_{fax}$	Aerodynamic friction coefficient in x	$5.567\times {10}^{-4}$ N/rad/s
$K_{fay}$	Aerodynamic friction coefficient in y	$5.567\times {10}^{-4}$ N/rad/s
$K_{faz}$	Aerodynamic friction coefficient in z	$6.354\times {10}^{-4}$ N/rad/s
$K_{ftx}$	Translational drag coefficient in x	$5.567\times {10}^{-4}$ N/m/s
$K_{fty}$	Translational drag coefficient in y	$5.567\times {10}^{-4}$ N/m/s
$K_{ftz}$	Translational drag coefficient in z	$6.354\times {10}^{-4}$ N/m/s
l	Quadcopter arm length	$0.23$ m
m	Quadcopter mass	$0.65$ kg

. The nonlinear quadcopter dynamic system is implemented in MATLAB/Simulink with SMC based on classical ERL, SMC based on the saturation function in ERL (S-SMC), SMC based on the SSF in ERL (SSF-SMC), and SMC based on modified ERL (MSMC) controllers.

Table 2. Optimal controller parameters for the quadcopter UAV system based on different functions in ERL.

Quadcopter Output	Optimal Parameter	Sign Function	Saturation Function	SSF	Modified ERL
$\varphi$	$k_1^{*}$	3.75243	0.00001	7.99816	7.94719
	${\lambda }_1^{*}$	67.03035	212.80940	124.86942	153.14592
	$R_1^{*}$	–	–	–	1.48729
$\theta$	$k_2^{*}$	0.00001	2.89651	1.27273	0.08776
	${\lambda }_2^{*}$	82.44825	175.80519	173.86590	132.69362
	$R_2^{*}$	–	–	–	4.17532
$\psi$	$k_3^{*}$	3.74008	0.81539	1.41473	6.96150
	${\lambda }_3^{*}$	49.26941	73.23008	185.83627	171.37256
	$R_3^{*}$	–	–	–	0.27411
x	$k_4^{*}$	0.00001	6.00547	2.89196	18.15754
	${\lambda }_4^{*}$	2.13331	3.69706	3.40799	3.04928
	$R_4^{*}$	–	–	–	1.57180
y	$k_5^{*}$	0.00001	2.48297	3.97825	48.46309
	${\lambda }_5^{*}$	3.91519	4.26076	4.22160	2.79202
	$R_5^{*}$	–	–	–	0.00001
z	$k_6^{*}$	0.53839	1.26397	2.36012	35.56046
	${\lambda }_6^{*}$	4.99033	6.17652	5.69886	6.81947
	$R_6^{*}$	–	–	–	1.40125

presents the optimal parameter values for the quadcopter across the various controllers discussed above.

6.1. Hovering Flight Under Attitude Stabilization

The main goal of the proposed control system is to guide a trajectory state (X) in the direction of a desired reference trajectory. The initial trajectory state of the quadcopter system is assumed to be zero, while the desired reference trajectory state vector is considered to be

$$X_d={[{\varphi }_d,{\theta }_d,{\psi }_d,x_d,y_d,z_d]}^T={[{\varphi }_d,{\theta }_d,5^{\circ },1\mathrm{m},2\mathrm{m},2\mathrm{m}]}^T$$

and $\epsilon$ in the SSF has been specified as 0.1 to obtain better smoothness. It is worth noting that the desired roll and pitch angles are not fixed but are dynamically derived from the control efforts of the x-axis and y-axis controllers, denoted by $U_x$ and $U_y$, respectively. As a result, the values of ${\varphi }_d$ and ${\theta }_d$ are directly influenced by the outputs of these controllers.

For this scenario, the optimal parameter values for quadcopter controllers are presented in Table 2. The system output responses for the six state variables are shown in Figure 5, while the corresponding control inputs are illustrated in Figure 6. The output responses demonstrate that the proposed controller, incorporating the SSF approach, achieves superior performance. Specifically, it exhibits faster convergence to the desired states with minimal steady-state error and reduced overshoot and chattering compared to alternative methods.

The Performance Improvement ($PI$) between the controllers can be quantified as follows:

$$PI=\left(1-{\displaystyle \frac{C{F}_{\mathrm{Controller}\mathrm{A}}}{C{F}_{\mathrm{Controller}\mathrm{B}}}}\right)\times 100\%$$

(55)

The improvement in the proposed controller’s performance relative to the classical SMC can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{2.9630}{5.2544}}\right)\times 100\%=43.6092\%.$$

Comparison of the proposed controller’s performance with the S-SMC can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{2.9630}{4.1762}}\right)\times 100\%=29.015\%.$$

Meanwhile, the improvement in the proposed controller’s performance relative to the proposed SSF-SMC can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{2.9630}{3.4748}}\right)\times 100\%=14.729\%.$$

Furthermore, the suggested controller outperforms classical SMC by decreasing tracking errors under hovering flight conditions to 96.154% for roll, 98.535% for pitch, 44.81% for yaw, and 22.8% for altitude.

6.2. Trajectory Tracking Flight in the Presence and Absence of Disturbances

The objective of this scenario is to guide the quadrotor in tracking a desired circular trajectory while maintaining a stable hover, ensuring attitude stabilization, and executing a landing maneuver accompanied by a 5° yaw rotation. In this scenario, the value of $\epsilon$ is considered to be 0.2 to test the role of this parameter in improving the smoothness of the SSF. Therefore, the quadcopter was required to track the desired trajectory defined for $t\ge 0$:

$$[x_d,y_d]=\left[1+\sin \left({\displaystyle \frac{2\pi t}{10}}\right),2+2\sin \left({\displaystyle \frac{2\pi t}{10}}-{\displaystyle \frac{\pi }{2}}\right)\right]$$

(56)

$$z_d=\left\{\begin{array}{cc}0.625t\hfill & \mathrm{for}0\le t<4\hfill \\ 2.5\hfill & \mathrm{for}4\le t\le 16\hfill \\ -0.625t\hfill & \mathrm{for}16<t<20\hfill \end{array}\right.$$

(57)

The external disturbances are applied to all six measured outputs, represented by disturbance terms $d_i=\sin \left(0.5t\right)={[d_1,d_2,d_3,d_4,d_5,d_6]}^T$. The simulation results for the quadcopter’s output responses in the absence and presence of external disturbances are presented in Figure 7 and Figure 8, respectively. The results demonstrate that the quadrotor is capable of accurately tracking the desired trajectory while effectively compensating for external disturbances. Additionally, Figure 7 and Figure 8 offer a comparative analysis of the position and attitude tracking performance achieved by the proposed controller against alternative control strategies. The proposed control–observer framework exhibits superior performance in both subsystems. Figure 9 shows the trajectory tracking positions in three-dimensional planes for the optimal controllers under disturbance conditions. The proposed controllers demonstrate a strong ability to reject disturbances, respond quickly, greatly reduce chattering issues, and deliver excellent tracking accuracy.

The improvement in the proposed controller’s performance relative to classical SMC in the presence of external disturbances can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{0.0835}{7.7022}}\right)\times 100\%=98.92\%.$$

Meanwhile, the comparison of the proposed controller’s performance with the S-SMC in the presence of external disturbances can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{0.0835}{2.1689}}\right)\times 100\%=96.16\%.$$

Also, the improvement in the proposed controller’s performance relative to the proposed SSF-SMC in the presence of external disturbances can be evaluated using the following:

$$PI=\left(1-{\displaystyle \frac{0.0835}{0.2128}}\right)\times 100\%=60.8\%.$$

Furthermore,

Table 3. Performance results for SMC, S-SMC, SSF-SMC, and MSMC controllers based on the PSO method.

Quadcopter Output	Performance Index	SMC	S-SMC	SSF-SMC	MSMC
$\varphi$	ISE1	0.0208	0.0026	0.0083	0.0008
	MSE1	2.3744	1.2538	0.0540	0.0226
$\theta$	ISE2	0.0273	0.0098	0.0076	0.0004
	MSE2	4.5995	0.7766	0.0521	0.0343
$\psi$	ISE3	0.6287	0.4217	0.1634	0.0347
	MSE3	0.0437	0.0457	0.0370	0.0262
x	ISE4	0.3388	0.3157	0.2782	0.2492
	MSE4	0.0041	1.2738 × 10−4	2.0740 × 10−4	1.3440 × 10−4
y	ISE5	1.7165	1.5138	1.2161	1.0743
	MSE5	0.4863	0.0911	3.5796 × 10−4	2.1168 × 10−4
z	ISE6	1.0957	0.8760	0.9091	0.8459
	MSE6	0.1941	0.0016	1.9995 × 10−4	3.9379 × 10−6
Total CF1		5.2544	4.1762	3.4748	2.9630
Total CF2		7.7022	2.1689	0.2128	0.0835

provides a summary comparison of the performance achieved by the optimal SMC controllers based on classical ERL, saturation ERL, SSF-ERL, and MSMC methods. The suggested controller outperforms classical SMC by reducing tracking errors during trajectory tracking flight conditions, even with disturbances, to 99.05% for roll, 99.26% for pitch, 40.18% for yaw, and 99.998% for altitude.

6.3. Trajectory Tracking Flight Under Input Variation in the Presence of Disturbances

The objective of this scenario is to test the quadrotor’s ability to follow a specified circular trajectory under the influence of varying input magnitudes and frequencies for the X, Y, and Z axes, and execute a landing maneuver accompanied by a 10° yaw rotation. The quadcopter was required to track the desired trajectory defined for $t\ge 0$:

$$[x_d,y_d]=\left[1+2\sin \left({\displaystyle \frac{2\pi t}{6.28}}\right),2+4\sin \left({\displaystyle \frac{2\pi t}{6.28}}-{\displaystyle \frac{\pi }{2}}\right)\right]$$

(58)

$$z_d=\left\{\begin{array}{cc}1.25t\hfill & \mathrm{for}0\le t<4\hfill \\ 5\hfill & \mathrm{for}4\le t\le 16\hfill \\ -1.25t\hfill & \mathrm{for}16<t<20\hfill \end{array}\right.$$

(59)

The same disturbances are applied uniformly to all six measured outputs. Therefore, the simulation results for the quadcopter’s output responses under input variation and the presence of disturbances are presented in Figure 10.

A more challenging test is conducted by reducing the landing angle to 2.5° yaw rotation and halving the nominal values of the desired X, Y, and Z inputs, as follows:

$$[x_d,y_d]=\left[1+0.5\sin \left({\displaystyle \frac{2\pi t}{6.28}}\right),2+\sin \left({\displaystyle \frac{2\pi t}{6.28}}-{\displaystyle \frac{\pi }{2}}\right)\right]$$

(60)

$$z_d=\left\{\begin{array}{cc}0.375t\hfill & \mathrm{for}0\le t<4\hfill \\ 1.5\hfill & \mathrm{for}4\le t\le 16\hfill \\ -0.375t\hfill & \mathrm{for}16<t<20\hfill \end{array}\right.$$

(61)

The simulation results for the quadcopter’s output responses under another input variation test are presented in Figure 11. Figure 10 and Figure 11 prove that the proposed controller accurately tracks the desired trajectory and outperforms alternative control strategies in both position and attitude under input variation.

6.4. Trajectory Tracking Flight Under Random Noise and Parameter Variation in the Presence of Disturbances

To evaluate the robustness of the proposed controller against random noises, white band-limited noise with a power of 0.01, a sampling time of 0.3 s, and a seed value of 23341 was introduced to the quadcopter outputs. Figure 12 illustrates the resulting output responses for both position and attitude under the influence of this noise. It can be observed that the other control methods fail to compensate for the applied noise, resulting in significant deviations from the desired trajectory. In contrast, the proposed controller remains close to the reference value and achieves accurate tracking performance.

Furthermore, to perform a more extensive and complex test, all quadcopter parameters are varied by $\pm 200\%$ of their nominal values, while applying the same input variations used previously and the external disturbance. The output responses are shown in Figure 13 for the $\pm 200\%$ parameter variation scenario. These results highlight the robustness and adaptability of the control unit under complex conditions and varying system parameters. Such variations arise from the need to handle changing values in multiple scenarios. The findings confirm the effectiveness, reliability, and resilience of the control system.

7. Conclusions

This study concludes that the modified ERL based on SMC (MSMC) is an effective approach for controlling quadcopter UAV systems and ensuring accurate trajectory tracking. Compared to conventional SMC, MSMC offers enhanced precision and robustness, guarantees system convergence quickly, and eliminates the chattering effect. Simulation results confirmed that the proposed controller performs effectively under various flight conditions, including the presence of disturbances, noise, input variations, and parametric uncertainties. These findings demonstrate that the MSMC provides strong robustness, eliminates chattering, and achieves high-precision trajectory tracking with reduced settling and rise times. Although the MSMC shows excellent performance in simulations, the current study is limited to a simulated environment and specific unknown bounded disturbance scenarios. Real-world implementation may involve additional uncertainties such as sensor noise, wind gusts, and model inaccuracies. Future work should include experimental validation on a physical quadcopter platform, testing under more diverse environmental conditions, and further optimization of the control parameters to enhance adaptability and energy efficiency.