Sliding Mode Controller for Quadcopter UAVs: A Comprehensive Survey

Abstract

This paper provides a comprehensive investigation of nonlinear robust control methodologies, with a specific emphasis on the development of sliding mode controllers (SMCs) for quadcopter unmanned aerial vehicles (UAVs). Quadcopters are highly interconnected and underactuated and, thus, pose challenges in controlling them, especially in the presence of disturbances like wind. SMC is a widely employed approach that proves practical for managing the intricate nonlinear dynamics of UAVs with substantial coupling. The principal merit of SMC lies in its remarkable capability to reject external perturbations and uncertainties. This paper offers an extensive survey on robust control design techniques, specifically focusing on SMC design for quadcopter UAVs. This paper also delves into different SMC design approaches, such as classical SMC, super-twisting SMC (ST-SMC), terminal SMC(TSMC), adaptive SMC, backstepping SMC, event-triggered SMC, and neural network-based SMCs for quadcopters. This paper provides a detailed study of the different SMC designs to achieve various objectives for the UAV in the presence of uncertainties and disturbances. Simulations of the various SMCs are presented that demonstrate the comparative performance of the UAVs for different objectives. Finally, this article serves as an information foundation that covers various aspects of the SMC design for quadcopters.

1. Introduction

The adaptability of UAVs has sparked considerable interest in recent times. UAVs have a wide range of applications in both military and civilian sectors, including environmental monitoring, construction inspections, disaster relief, precision agriculture, aerial photography, security and surveillance, and carrying high-tech weapons in conflict zones [1]. Additionally, UAVs have been deployed on the front lines of the COVID-19 pandemic to perform tasks such as spraying antibacterial sanitizers and monitoring curfew conditions. Also, for tedious and dangerous tasks like border patrol or monitoring disaster areas—such as a damaged nuclear facility—UAVs are often preferred over conventional aircraft. Among UAVs, underactuated models like quadcopters, hexacopters, and octocopters have become more common in everyday life, standing out as eye-catching underactuated unmanned vehicles. Among the rotorcraft, quadcopters are widely used due to their ability for vertical takeoff and landing in confined spaces, straightforward design, small size, low weight, excellent dynamic agility, and ability to maneuver around obstacles. The increased number of rotors in hexacopters and octocopters leads to larger sizes and higher costs, making these rotorcraft suitable for specific purposes only [2]. Thus, underactuated quadcopter UAVs have attracted the interest of many researchers for applications in complex indoor and outdoor environments.

In recent times, various models of quadcopters have become available on the market, such as DJIF450, 3DR Solo drone, QBall2, DJIM100, Crazyfly, Iris, AR Parrot drone, etc. Typical quadcopter components include four rotors, an inertial measurement unit (IMU), electronic speed controllers (ESCs), and a microprocessor with a control algorithm. The vision-based system and the GPS are optional components. Researchers have explored various research domains of quadcopters, such as landing techniques for UAVs [3], modeling [4], strategies for optimized UAV surveillance [5], path planning [6], and landslide investigation and monitoring [7]. The application-based survey on quadcopters such as monitoring sugarcane crops, disaster management, chemical sensing, and agriculture monitoring survey articles can be found in Refs. [8,9,10,11]. The different important survey literature studies in various application domains of the quadcopter can be found in Refs. [12,13,14,15,16,17,18,19,20]. However, in the existing literature, there appears to be a gap in the exploration of how the control algorithm intricacies are addressed.

Also, like most highly non-linear systems, the dynamics of quadcopters present several challenges that make them difficult to control. These challenges include the non-linear and coupled dynamics of the quadcopter, as well as environmental effects such as wind, obstacles, sensor limitations, and system failures. In such scenarios, many linear and nonlinear motion control strategies for quadcopters have been proposed by researchers, including proportional–integral–derivative (PID) control [21], linear quadratic regulator (LQR) [22], backstepping [23], SMC [24], and model predictive control (MPC) [25]. Among those, SMC has become one of the most extensively used control design techniques for quadcopters. One of the pioneering and highly cited works on the design of SMC for the quadcopter can be found in Ref. [26]. This seminal paper by Bouabdallah et al. lays the foundational groundwork for applying SMC and backstepping techniques to quadcopter control, emphasizing robustness and performance in aerial maneuvers. SMC can also be adapted to fixed-wing aircraft, representing distinct types of systems. Various studies, such as [27,28,29], delved into the applications of SMC for fixed-wing aircraft. For a more comprehensive understanding, readers are referred to [30]. Since 2005, there have been significant advancements in the SMC designs for quadcopters. Researchers have developed classical SMC, ST-SMC, TSMC, adaptive SMC, event-triggered SMC, etc. Different surveys have been conducted to explore various controllers for quadcopters in the literature.

Pioneering survey papers discussing the design of controllers for UAVs can be found in Refs. [31,32]. In Ref. [31], the author described different linear and nonlinear learning-based controllers for quadcopter UAVs. Survey literature detailing path-following control strategies for UAVs is presented in Ref. [33], where the authors discuss various controllers. They provide a simulation comparison among backstepping and feedback linearization control-oriented algorithms, as well as NLGL and carrot-chasing geometric algorithms for quadcopters. The real-time implementation of controller design for quadcopters is a crucial aspect discussed in the literature, as highlighted in Ref. [34]. The author demonstrates the real-time implementation of various controllers, including PID, LQG, and SMC. Other recent important survey literature studies on controller design include Refs. [35,36,37,38,39,40]. Ref. [41] is just one of the recent notable contributions in the field of controller design for quadcopter UAVs; the author extensively examined the PID controller alongside various complementary techniques. The study delved into enhancing the stability and maneuverability of quadcopters through a comprehensive evaluation of control strategies. From

Table 1. A tabulation of survey literature on UAV quadcopters.

References	Contributions
[1]	Presents system configuration, collision avoidance algorithm, and fault tolerant control.
[3]	Provides a broad perspective on the status of the landing control problem and controller design.
[4]	Dynamic analysis and strategies.
[5]	Presents navigation algorithms and artificial intelligence technologies with UAV surveillance as well as the challenges of operating in complex environments.
[6]	Discusses the advantages, disadvantages, application challenges, and notable outcomes of each path-planning algorithm.
[7]	Addresses regulatory hurdles, hover time limitations, 3D reconstruction accuracy, and potential integration with technologies like UAV swarms.
[8]	Presents a method for sugarcane monitoring and management to improve yield and quality.
[9]	Deploys drones in mass disasters to empower and inspire possible future work.
[10]	Presents sensing platforms and algorithms as gas concentration mappings, source localization, and flux estimations.
[11]	UAV-based imagery, estimation of biomass, monitoring crop plant health and stress, detects pest or pathogen infestations.
[12]	Presents localization and navigation techniques.
[13]	Dynamic model, configuration, and design analysis,
[14]	Presents the key challenges, including charging challenges, collision avoidance, swarming challenges, networking, and security-related challenges.
[17]	Discusses drawbacks of classic physic-based dynamic modeling, control techniques, and challenges to augment or replace classic techniques with data-driven approaches.
[18]	Focuses on UAV autonomous features, network resource management, channel access, routing protocols, security, and privacy management.
[19]	Explores potential area communication, artificial intelligence, remote sensing, miniaturization, swarming, and cooperative control.
[20]	Explores UAV-based systems for traffic monitoring and management.

and

Table 2. A tabulation of survey literature on UAV quadcopter control strategies.

References	Contributions
[31]	Presents a brief overview of different control strategies,
[32]	Presents different linear and nonlinear control strategies,
[33]	Discusses control-oriented and geometric algorithms for the path following of UAVs.
[34]	Discusses real-time aspects of drone control as well as possible implementation of real-time flight control systems.
[35]	Discusses control strategies and details about simultaneous localization and mapping algorithms.
[36]	Comparative study of the control strategies (PID, LQR, MPC, SMC, FL) for attitude stabilization with simulations.
[37]	UAV architectures and different control strategies.
[38]	Presents different control strategies.
[39]	Presents a comprehensive analysis of control algorithms for quadrotor trajectory tracking.
[15]	Presents various robust attitude control strategies.
[16]	Brief discusses different linear and nonlinear control strategies.
[39]	Presents control strategies and challenges for rotorcraft.
[40]	Presents a detailed discussion on PID controllers for quadcopter UAVs.

, it can be observed that the survey literature on quadcopter UAVs has included many domains, especially different linear and nonlinear control strategies.

While existing literature surveys have not rigorously addressed the specialized domain of robust controller designs for quadcopters, this study has been specifically crafted to fill this gap, providing a comprehensive overview of SMC strategies for quadcopters. This paper represents the first survey, to the authors’ knowledge, that comprehensively examines various SMC designs specifically tailored for quadcopter UAVs. The principal aim of this research is to explore a wide array of SMC designs for quadcopter UAVs. This investigation entails a comprehensive examination of inherent disturbances within quadcopter systems, a critical analysis of published research outcomes, and a thorough evaluation of simulation results pertaining to the application of various SMC approaches in the context of quadcopter control. The contributions of the survey work that we present are as follows:

• A survey on the different SMC designs, their advantages, the effect of disturbances, and results analysis.
• The detailed analysis of the design of the different SMCs for the quadcopter.
• The advantages of the different SMCs, and the simulation results on the quadcopter for classical SMC, adaptive ST-SMC, ST-SMC, and TSMC.

The rest of this paper is organized as follows: A methodology of the detailed survey is presented in Section 2 and the descriptions of the quadcopter model, configuration, and dynamics are presented in Section 3.1. The details on the designs of different SMCs are proposed in Section 3.2. The different problems chosen for various quadcopter models and the design and details are discussed in Section 4. The simulation results of the various SMC types for the quadcopter model are discussed in Section 5. Section 6 presents the challenges and prospects for future endeavors. Finally, the work is concluded in Section 7.

2. Methodology

To conduct a comprehensive survey on robust quadcopter control techniques, a structured methodology was employed, focusing on the search and analysis of relevant scientific articles. The search was conducted using Web of Science, IEEE Xplore Digital Library, Sci-Hub, and Google Scholar, which were chosen for their extensive coverage and up-to-date scientific resources. This approach identified numerous papers published in reputable journals, all showcasing robust methodological frameworks. A detailed review of these articles enabled the identification of various effective control techniques, providing a thorough understanding of the current state of robust control strategies for drones. Given the large volume of search results, specific search and filtering criteria were established. Key search terms included “robust controller* for drones”, “sliding mode control for quadcopters”, “super-twisting SMC design”, “adaptive control for UAVs”, “hardware implementation of SMC for quadcopters”, “finite-time robust control policy”, “neural network-based SMC”, “backstepping control”, and “learning-based SMC”, focusing on titles, abstracts, or keywords. These terms were carefully selected to ensure they accurately reflected the core aspects of robust control strategies in the research topics. All terms were pre-filtered to guarantee relevance to robust control strategies. A block diagram representation of the key search criteria is provided in Figure 1.

After obtaining and filtering all relevant paper references based on the established criteria, a detailed analysis was conducted to classify various sliding mode controllers. This classification process identified specific categories, including classical SMC, ST-SMC, TSMC, NN-SMC, and advanced SMC. Figure 2 presents a graphical representation of this classification, highlighting the most widely used SMC types. This visualization provides a clear overview of how various SMC designs have been tailored to address different problems for quadcopters. Also, a graphical representation of the annual distribution of literature is shown in Figure 3 to understand the exploration of SMC designs in the last decade.

3. Background and Preliminaries

3.1. System Model and Dynamics

There are two different configurations of a quadcopter UAV, i.e., the ‘×’ configuration and the ‘+’ configuration. A quadrotor in the ‘×’ configuration is said to be more stable than one in the + configuration, which is more acrobatic. The schematic diagram of a quadcopter is shown in Figure 4, showcasing its four rotors, where rotor pairs (1,3) and (2,4) rotate in opposite directions. The roll, pitch, and yaw angles are represented by $\varphi$, $\theta$, and $\psi$, which describe the orientations of the quadcopter in the inertial frame and are bounded as $-{\displaystyle \frac{\pi }{2}}<\varphi <{\displaystyle \frac{\pi }{2}}$, $-{\displaystyle \frac{\pi }{2}}<\theta <{\displaystyle \frac{\pi }{2}}$, and $-\pi <\psi <\pi$. Precise control of the quadcopter’s position (in the x, y, and z directions) and attitude (pitch ($\theta$), roll ($\varphi$), and yaw($\psi$)) is achieved by manipulating the speed ($w_1,w_2,w_3,w_4$) of each rotor. The quadcopter is considered in the ‘×’ configuration and the control inputs can be represented in terms of the rotor’s angular speed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_z& =b(w_1^2+w_2^2+w_3^2+w_4^2),\hfill \\ \hfill u_{\varphi }& =bl(w_1^2-w_2^2-w_3^2+w_4^2),\hfill \\ \hfill u_{\theta }& =bl(-w_1^2-w_2^2+w_3^2+w_4^2),\hfill \\ \hfill u_{\psi }& =n(-w_1^2+w_2^2-w_3^2+w_4^2),\hfill \end{array}\end{array}$$

(1)

where $w_i$ represents the angular speed of the ith rotor. The thrust and drag factor and the distance between a rotor and the center of mass of the quadcopter are represented by b, n, and l, respectively. Here, vertical motion is generated by increasing or decreasing the speeds of all four propellers simultaneously. Conversely, adjusting the speed of propellers 2 and 4 induces roll rotation along with lateral motion while modifying the speed of propellers 1 and 3 produces pitch rotation and corresponding lateral motion. Yaw rotation results from the differential torques produced by each pair of propellers, a subtler effect. Despite having four actuators, the quadrotor remains an underactuated and dynamically unstable system, presenting a fascinating challenge for further exploration in this research.

To control the quadcopter, the control inputs for altitude, attitude, and position control need to be designed. The simplified dynamics [42] of the quadcopter model are obtained using small angle approximations as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \ddot{\varphi }& =\dot{\theta }\dot{\psi }\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)+{\displaystyle \frac{u_{\varphi }}{j_x}}+{\Lambda }_{\varphi }\hfill \\ \hfill \ddot{\theta }& =\dot{\varphi }\dot{\psi }\left({\displaystyle \frac{j_z-j_x}{j_y}}\right)+{\displaystyle \frac{u_{\theta }}{j_y}}+{\Lambda }_{\theta }\hfill \\ \hfill \ddot{\psi }& =\dot{\varphi }\dot{\theta }\left({\displaystyle \frac{j_x-j_y}{j_z}}\right)+{\displaystyle \frac{u_{\psi }}{j_z}}+{\Lambda }_{\psi }\hfill \\ \hfill \ddot{z}& ={\displaystyle \frac{u_z}{m}}(cos\varphi cos\theta cos\psi )-g+{\Lambda }_z\hfill \\ \hfill \ddot{x}& ={\displaystyle \frac{u_z}{m}}(cos\varphi sin\theta cos\psi +sin\varphi sin\psi )+{\Lambda }_x\hfill \\ \hfill \ddot{y}& ={\displaystyle \frac{u_z}{m}}(cos\varphi sin\theta cos\psi -sin\varphi cos\psi )+{\Lambda }_y.\hfill \end{array}\end{array}$$

(2)

When designing the controller, the dynamical model of the quadcopter (2) is segmented into six second-order dynamical subsystems, as outlined by [43]. The quadcopter model, described by Equation (2), exhibits 6 degrees of freedom and is equipped with four independent control inputs, classifying it as an underactuated system. These control inputs, denoted as $u_z$, $u_{\varphi }$, $u_{\theta }$, and $u_{\psi }$, correspond to the total thrust in the negative z direction and torques around the x, y, and z axes, respectively. Here, j represents the moment of inertia along the $x,y,z$ directions, ${\Lambda }_{\tau }$, $\tau \in (\varphi ,\theta ,\psi ,x,y,z)$ is the external disturbance and g is the acceleration due to gravity. The control inputs $u_z$ and $u_{\psi }$ are specifically employed for altitude and yaw control. To achieve positional control, a virtual control approach is implemented, involving the design of two virtual control inputs, $u_x$ and $u_y$, which directly influence the position. These virtual inputs serve as the foundation for generating reference commands, from which ${\varphi }_d$ and ${\theta }_d$ angle trajectories are derived. The tracking of these reference commands is then performed through the control inputs $u_{\varphi }$ and $u_{\theta }$. The underactuation mechanism is presented in Figure 5, which illustrates the inner and outer loop control mechanisms for the quadcopter system. In the forthcoming section, a detailed exploration of the quadcopter controller’s design variations will be provided, offering a deeper understanding of the control strategies employed for this complex system. The block diagram illustrating the sliding mode control system for the quadcopter is presented in Figure 6. From Figure 6, it is demonstrated that by using the two designed virtual sliding mode control inputs $u_x$ and $u_y$, the desired ${\varphi }_d$ and ${\theta }_d$ angle trajectories are derived. The tracking of reference commands ${\varphi }_d$ and ${\theta }_d$ is then performed through the control inputs $u_{\varphi }$ and $u_{\theta }$. The reference commands for $x_d,y_d,z_d,{\psi }_d$ are defined directly. The four control inputs $u_z,u_{\psi },u_{\varphi },u_{\theta }$ can be evaluated and generate the required rotor speed. For a comprehensive understanding of the problem formulation for the quadcopter system, let us examine the quadcopter model expressed in Equation (2) using the following configuration:

$$\begin{array}{c}\hfill \dot{\Theta }=f(\Theta )+g(\Theta )u+\Lambda ,\end{array}$$

(3)

where the terms $f(\Theta )$, $g(\Theta )$, and $\Lambda$ correspond to the system matrix, input matrix, and uncertainties, respectively. The state vector for the quadcopter model is written as $\Theta ={[\varphi ,\dot{\varphi },\theta ,\dot{\theta },\psi ,\dot{\psi },z,\dot{z},x,\dot{x},y,\dot{y}]}^{\top }$. Simultaneously, the control input vector is defined as $u={[u_z,u_{\varphi },u_{\theta },u_{\psi }]}^{\top }$. For brevity, the detailed examination of the design analysis will focus on the roll subsystem. It is understood that a similar procedure can be applied to the other subsystems. The equations governing the roll subsystem are as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\Theta }}_1& ={\Theta }_2,\hfill \\ \hfill {\dot{\Theta }}_2& ={\displaystyle \frac{j_y-j_z}{j_x}}{\nu }_4{\nu }_6+{\displaystyle \frac{u_2}{j_x}}+{\Lambda }_{\varphi }.\hfill \end{array}\end{array}$$

(4)

In the given scenario, ${\Lambda }_{\varphi }$ denotes an externally imposed perturbation that is constrained within certain bounds. Let the tracking error be defined as $e_{\varphi }={\varphi }_d-\varphi$, $e_{\theta }={\theta }_d-\theta$, $e_{\psi }={\psi }_d-\psi$, $e_z=z_d-z$, $e_x=x_d-x$, $e_y=y_d-y$, where the desired trajectories will be defined later.

3.2. Design of the Sliding Mode Control

SMC is a two-step design procedure; first, we need to design the appropriate sliding manifold, and second, the discontinuous switching control needs to be designed to guarantee the stability and performance of the system. The primary objective involves the design of a classical sliding mode controller equipped with a PID sliding surface for each subsystem. This design aims to enable the quadcopter to track a reference command while effectively dealing with external disturbances. By judiciously selecting the PID sliding surface parameters, it is possible to achieve convergence of the error trajectories, leading to reduced steady-state errors. This approach promises enhanced control performance and robustness for the quadcopter system, effectively addressing the complex dynamics and uncertainties inherent in its operation.

The PID sliding surface [44] is designed as follows:

$$\begin{array}{cc}\hfill s_{\tau }=& k_{p_{\tau }}{e}_{\tau }+k_{d_{\tau }}{\dot{e}}_{\tau }+k_{i_{\tau }}{\int }_0^t{e}_{\tau }\mathrm{d}\tau ,\hfill \end{array}$$

(5)

where $k_{d_{\tau }}$, $k_{d_{\tau }},k_{i_{\tau }}>0$ and $\tau \in (\varphi ,\theta ,\psi ,z,x,y)$. Next, to attain stability of the error system, the classical SMC can be represented as follows:

$$\begin{array}{cc}\hfill u_z& ={\displaystyle \frac{m}{k_{d_z}cos\varphi cos\theta }}(k_{1_z}{e}_z+k_{p_z}{\dot{e}}_z+k_{d_z}{\ddot{z}}_d\hfill \\ \hfill & +k_{d_z}g+{\lambda }_{1_z}\mathrm{sign}\left(s_z\right)),\hfill \\ \hfill u_{\varphi }& ={\displaystyle \frac{j_x}{k_{d_{\varphi }}}}({\lambda }_{1_{\varphi }}\mathrm{sign}\left(s_{\varphi }\right)+k_{i_{\varphi }}{e}_{\varphi }+k_{p_{\varphi }}{\dot{e}}_{\varphi }+k_{d_{\varphi }}{\ddot{\varphi }}_d\hfill \\ \hfill & -k_{d_{\varphi }}\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)\dot{\theta }\dot{\psi }),\hfill \\ \hfill u_{\theta }& ={\displaystyle \frac{j_y}{k_{d_{\theta }}}}({\lambda }_{1_{\theta }}\mathrm{sign}\left(s_{\theta }\right)+k_{i_{\theta }}{e}_{\theta }+k_{p_{\theta }}{\dot{e}}_{\theta }+k_{d_{\theta }}{\ddot{\theta }}_d\hfill \\ \hfill & -k_{d_{\theta }}\left({\displaystyle \frac{j_z-j_x}{j_y}}\right)\dot{\varphi }\dot{\psi }),\hfill \\ \hfill u_{\psi }& ={\displaystyle \frac{j_z}{k_{d_{\psi }}}}({\lambda }_{1_{\psi }}\mathrm{sign}\left(s_{\psi }\right)+k_{i_{\psi }}{e}_{\psi }+k_{p_{\psi }}{\dot{e}}_{\psi }+k_{d_{\psi }}{\ddot{\psi }}_d\hfill \\ \hfill & -k_{d_{\psi }}\left({\displaystyle \frac{j_x-j_y}{j_z}}\right)\dot{\theta }\dot{\varphi }),\hfill \end{array}$$

(6)

where ${\lambda }_{1\tau }$ represents the switching gain. In the context of quadcopter dynamics, the attainment of position control is achieved through indirect manipulation of the Euler angles $\varphi ,\theta$, and $\psi$. To facilitate this control strategy, two virtual control variables, denoted as $u_x$ and $u_y$, are introduced as follows:

$$\begin{array}{c}\hfill u_x=cos\varphi sin\theta cos\psi +sin\varphi sin\psi ,\end{array}$$

(7)

$$\begin{array}{c}\hfill u_y=cos\varphi sin\theta sin\psi -sin\varphi cos\psi .\end{array}$$

(8)

With virtual control variables, the SMCs for positions x and y are defined as follows:

$$\begin{array}{cc}\hfill {\tilde{u}}_x=& {\displaystyle \frac{m}{k_{d_x}{u}_z}}\left[{\lambda }_{1_x}\mathrm{sign}\left(s_x\right)+k_{i_x}{e}_x+k_{p_x}{\dot{e}}_x+k_{d_x}{\ddot{x}}_d\right],\hfill \\ \hfill {\tilde{u}}_y=& {\displaystyle \frac{m}{k_{d_y}{u}_z}}\left[{\lambda }_{1_y}\mathrm{sign}\left(s_y\right)+k_{i_y}{e}_y+k_{p_y}{\dot{e}}_y+k_{d_y}{\ddot{y}}_d\right].\hfill \end{array}$$

(9)

Keeping ${\psi }_d$ constant, the desired ${\varphi }_d$ and ${\theta }_d$ can be computed as follows:

$$\begin{array}{cc}\hfill {\varphi }_d=& arcsin\left({\tilde{u}}_{x}sin{\psi }_d-{\tilde{u}}_{y}cos{\psi }_d\right),\hfill \\ \hfill {\theta }_d=& arcsin\left({\displaystyle \frac{{\tilde{u}}_{x}cos{\psi }_d+{\tilde{u}}_{y}sin{\psi }_d}{cos{\psi }_d}}\right).\hfill \end{array}$$

(10)

The conventional SMC is a discontinuous control design technique where robustness cannot be guaranteed during the reaching phase, and there may be issues with chattering. In response, researchers in the sliding mode controller design community have devoted significant effort to developing nonlinear design techniques with better performance. This has led to the design of integral SMC (ISMC), second-order SMC, higher-order SMC, and advanced SMC.

3.2.1. Super-Twisting Sliding Mode Control (ST-SMC)

The continuous SMC class, which can significantly reduce the chattering, is called the ST-SMC [45]. The ST-SMC can achieve the second-order sliding mode [44], i.e., the sliding variable, s, and its time derivative are forced to zero ($s=\dot{s}=0$) in finite time [46,47] with the known information of the output. This second-order SMC class can ensure the finite time stability of the sliding trajectories and asymptotic convergence of the states.

Considering the same sliding surface (5) and system (2), the ST-SMCs designed for the attitude $\varphi$, $\theta$, $\psi$, and z are as follows:

$$\begin{array}{cc}\hfill u_z& ={\displaystyle \frac{m}{k_{d_z}cos\varphi cos\theta }}(k_{i_z}{e}_z+k_{p_z}{\dot{e}}_z+k_{d_z}{\ddot{z}}_d+k_{d_z}g\hfill \\ & +{\lambda }_{1_z}{|s_z|}^{1/2}\mathrm{sign}\left(s_z\right)+{\lambda }_{2_z}{\int }_0^t\mathrm{sign}\left(s_z\right)\mathrm{d}\tau ),\hfill \\ \hfill u_{\varphi }& ={\displaystyle \frac{j_x}{k_{d_{\varphi }}}}\left[{\lambda }_{1_{\varphi }}{|s_{\varphi }|}^{1/2}\mathrm{sign}\left(s_{\varphi }\right)+{\lambda }_{2_{\varphi }}{\int }_0^t\mathrm{sign}\left(s_{\varphi }\right)\mathrm{d}\tau +k_{i_{\varphi }}{e}_{\varphi }\right.\hfill \\ \hfill & \left.+k_{p_{\varphi }}{\dot{e}}_{\varphi }+k_{d_{\varphi }}{\ddot{\varphi }}_d-k_{d_{\varphi }}\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)\dot{\theta }\dot{\psi }\right],\hfill \\ \hfill u_{\theta }& ={\displaystyle \frac{j_y}{k_{d_{\theta }}}}\left[{\lambda }_{1_{\theta }}{|s_{\theta }|}^{1/2}\mathrm{sign}\left(s_{\theta }\right)+{\lambda }_{2_{\theta }}{\int }_0^t\mathrm{sign}\left(s_{\theta }\right)\mathrm{d}\tau \right.\hfill \\ & \left.+k_{i_{\theta }}{e}_{\theta }+k_{p_{\theta }}{\dot{e}}_{\theta }+k_{d_{\theta }}{\ddot{\theta }}_d-k_{d_{\theta }}\left({\displaystyle \frac{j_z-j_x}{j_y}}\right)\dot{\varphi }\dot{\psi }\right],\hfill \\ \hfill u_{\psi }& ={\displaystyle \frac{j_z}{k_{d_{\psi }}}}\left[{\lambda }_{1_{\psi }}{|s_{\psi }|}^{1/2}\mathrm{sign}\left(s_{\psi }\right)+{\lambda }_{2_{\psi }}{\int }_0^t\mathrm{sign}\left(s_{\psi }\right)\mathrm{d}\tau \right.\hfill \\ & \left.+k_{i_{\psi }}{e}_{\psi }+k_{p_{\psi }}{\dot{e}}_{\psi }+k_{d_{\psi }}{\ddot{\psi }}_d-k_{d_{\psi }}\left({\displaystyle \frac{j_x-j_y}{j_z}}\right)\dot{\theta }\dot{\varphi }\right],\hfill \end{array}$$

(11)

where ${\lambda }_{2\tau }$ is the switching gain. Using $u_x$ and $u_y$ from (7) and the sliding surface for position (5), the virtual ST-SMCs for positions are represented as follows:

$$\begin{array}{cc}\hfill {\tilde{u}}_{x_{\left(STSMC\right)}}=& {\displaystyle \frac{m}{k_{d_x}{u}_1}}[{\lambda }_{1_x}{|s_x|}^{1/2}\mathrm{sign}\left(s_x\right)+{\lambda }_{2_x}{\int }_0^t\mathrm{sign}\left(s_x\right)\mathrm{d}\tau \hfill \\ \hfill & +k_{i_x}{e}_x+k_{p_x}{\dot{e}}_x+k_{d_x}{\ddot{x}}_d],\hfill \\ \hfill {\tilde{u}}_{y_{\left(STSMC\right)}}=& {\displaystyle \frac{m}{k_{d_y}{u}_1}}[{\lambda }_{1_y}{|s_y|}^{1/2}\mathrm{sign}\left(s_y\right)+{\lambda }_{2_y}{\int }_0^t\mathrm{sign}\left(s_y\right)\mathrm{d}\tau \hfill \\ \hfill & +k_{i_y}{e}_y+k_{p_y}{\dot{e}}_y+k_{d_y}{\ddot{y}}_d].\hfill \end{array}$$

(12)

Similarly, the desired ${\varphi }_d$ and ${\theta }_d$ can be computed as in (10) considering virtual control (12).

To ensure finite-time convergence of the states, the researchers explore the development of another SMC class in the following subsection.

3.2.2. Terminal Sliding Mode Control (TSMC)

The finite-time stability of the state and the sliding surface can be achieved by a novel SMC named TSMC [48]. The terminal sliding mode (TSM) concept originates from the idea of terminal attractors [49], initially explored in the context of content-addressable memory in neural networks. The term “terminal” signifies the system’s ability to reach equilibrium within a finite time and the reaching time can be adjusted by appropriately tuning the surface parameters. Its first application in control design was introduced in Ref. [50], where a foundational TSM structure was developed for controlling second-order systems. The TSMC can be categorized as the singular terminal SMC (TSMC) [51] and nonsingular terminal SMC (NTSMC) [52]. The design of TSMC is especially useful for high-precision control as it can achieve faster convergence near the equilibrium point. To address the singularity issue in the controller design of TSMC systems, an integral TSM can be employed [53]. Also to ensure the fast finite time reaching, the sliding surface [42] can be designed as follows:

$$\begin{array}{cc}\hfill s_{\tau }=& {\alpha }_1{e}_{\tau }+{\dot{e}}_{\tau }+{\beta }_1{\left|e_{\tau }\right|}^{\delta }\mathrm{sign}\left(e_{\tau }\right),\hfill \end{array}$$

(13)

where $\alpha >0$, ${\beta }_1>0$ and $0<\delta <1$. Thus, considering the sliding surface (13), the terminal sliding mode control can be designed as follows:

$$\begin{array}{cc}\hfill u_z& ={\displaystyle \frac{m}{cos\varphi cos\theta }}({\alpha }_1{\dot{e}}_z+{\beta }_1\delta {\left|e_z\right|}^{\delta -1}{\dot{e}}_z\hfill \\ & +{\ddot{z}}_d+g+{\lambda }_{1_z}\mathrm{sign}\left(s_z\right)),\hfill \\ \hfill u_{\varphi }& =j_x({\lambda }_{1_{\varphi }}\mathrm{sign}\left(s_{\varphi }\right)+{\alpha }_1\dot{e_{\varphi }}+{\beta }_1\delta {\left|e_{\varphi }\right|}^{\delta -1}\dot{e_{\varphi }}\hfill \\ \hfill & +{\ddot{\varphi }}_d-\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)\dot{\theta }\dot{\psi })\hfill \\ \hfill u_{\theta }& =j_y({\lambda }_{1_{\theta }}\mathrm{sign}\left(s_{\theta }\right)+{\alpha }_1\dot{e_{\theta }}+{\beta }_1\delta {\left|e_{\theta }\right|}^{\delta -1}\dot{e_{\theta }}\hfill \\ \hfill & +{\ddot{\theta }}_d-\left({\displaystyle \frac{j_z-j_x}{j_y}}\right)\dot{\varphi }\dot{\psi }),\hfill \\ \hfill u_{\psi }& =j_z({\lambda }_{1_{\psi }}\mathrm{sign}\left(s_{\psi }\right)+{\alpha }_1\dot{e_{\psi }}+{\beta }_1\delta {\left|e_{\psi }\right|}^{\delta -1}\dot{e_{\psi }}\hfill \\ \hfill & +{\ddot{\psi }}_d-\left({\displaystyle \frac{j_x-j_y}{j_z}}\right)\dot{\theta }\dot{\varphi }),\hfill \end{array}$$

(14)

The virtual terminal sliding mode controller can be designed using the sliding surface (13) and system (2)

$$\begin{array}{cc}\hfill u_{x_{\left(TSMC\right)}}& ={\displaystyle \frac{m}{u_z}}\left[{\lambda }_{1_x}\mathrm{sign}\left(s_x\right)+{\alpha }_1\dot{e_x}+{\beta }_1\delta {\left|e_x\right|}^{\delta -1}\dot{e_x}+{\ddot{x}}_d\right],\hfill \\ \hfill u_{y_{TSMC}}& ={\displaystyle \frac{m}{u_z}}\left[{\lambda }_{1_y}\mathrm{sign}\left(s_y\right)+{\alpha }_1\dot{e_y}+{\beta }_1\delta {\left|e_y\right|}^{\delta -1}\dot{e_{\varphi }}+{\ddot{y}}_d\right].\hfill \end{array}$$

(15)

The switching gains ${\lambda }_{1_{\tau }}$ and sliding gains ${\beta }_1$, ${\alpha }_1$ in all schemes must be designed appropriately to ensure the finite time convergence of states. To ensure robustness from the initial state response, another type of SMC is explored, as discussed in the following subsection.

3.2.3. Integral Sliding Mode Control (ISMC)

In the context of SMC, there exists a technique known as integral sliding mode control (ISMC), which is employed to ensure the presence of sliding motion and robustness right from the system’s initial response, as documented in Ref. [54]. A key element of ISMC is the inclusion of an integral term within the sliding variable. The ISMC integrates a variable structure controller alongside a standard controller, ensuring that the closed-loop system maintains optimal performance despite external perturbations.

Let us consider the integral sliding surface as follows [55]:

$$\begin{array}{cc}\hfill s_{\tau }=& k_{p_{\tau }}{e}_{\tau }\left(t\right)+k_{d_{\tau }}{\dot{e}}_{\tau }\left(t\right)+k_{i_{\tau }}{\int }_{t_0}^t{e}_{\tau }\left(t\right)\mathrm{d}\tau -k_{p_{\tau }}{e}_{\tau }\left(t_0\right)-k_{d_{\tau }}{\dot{e}}_{\tau }\left(t_0\right),\hfill \end{array}$$

(16)

where the gains can be chosen as earlier. The integral controller can be designed considering the nominal proportional–derivative (PD) controller and a switching controller. Considering system (4), the ISMC can be designed as $u_{\varphi }=u_{{\varphi }_{nominal}}+u_{{\varphi }_{switching}}$. Thus, $u_{{\varphi }_{nominal}}={\displaystyle \frac{j_x}{k_{d_{\varphi }}}}\left(k_{i_{\varphi }}{e}_{\varphi }+k_{p_{\varphi }}{\dot{e}}_{\varphi }+k_{d_{\varphi }}{\ddot{\varphi }}_d\right)$ and $u_{{\varphi }_{switching}}={\displaystyle \frac{j_x}{k_{d_{\varphi }}}}\left({\lambda }_{1_{\varphi }}\mathrm{sign}\left(s_{\varphi }\right)-k_{d_{\varphi }}\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)\dot{\theta }\dot{\psi }\right)$. Similarly, the ISMC for $z,\psi ,\theta$ and the virtual control for $x,y$ can be designed as follows:

$$\begin{array}{cc}\hfill u_z& ={\displaystyle \frac{m}{k_{d_z}cos\varphi cos\theta }}(k_{1_z}{e}_z+k_{p_z}{\dot{e}}_z+k_{d_z}{\ddot{z}}_d\hfill \\ \hfill & +k_{d_z}g+{\lambda }_{1_z}\mathrm{sign}\left(s_z\right)),\hfill \\ \hfill u_{\varphi }& ={\displaystyle \frac{j_x}{k_{d_{\varphi }}}}({\lambda }_{1_{\varphi }}\mathrm{sign}\left(s_{\varphi }\right)+k_{i_{\varphi }}{e}_{\varphi }+k_{p_{\varphi }}{\dot{e}}_{\varphi }+k_{d_{\varphi }}{\ddot{\varphi }}_d\hfill \\ \hfill & -k_{d_{\varphi }}\left({\displaystyle \frac{j_y-j_z}{j_x}}\right)\dot{\theta }\dot{\psi }),\hfill \\ \hfill u_{\theta }& ={\displaystyle \frac{j_y}{k_{d_{\theta }}}}({\lambda }_{1_{\theta }}\mathrm{sign}\left(s_{\theta }\right)+k_{i_{\theta }}{e}_{\theta }+k_{p_{\theta }}{\dot{e}}_{\theta }+k_{d_{\theta }}{\ddot{\theta }}_d\hfill \\ \hfill & -k_{d_{\theta }}\left({\displaystyle \frac{j_z-j_x}{j_y}}\right)\dot{\varphi }\dot{\psi }),\hfill \\ \hfill u_{\psi }& ={\displaystyle \frac{j_z}{k_{d_{\psi }}}}({\lambda }_{1_{\psi }}\mathrm{sign}\left(s_{\psi }\right)+k_{i_{\psi }}{e}_{\psi }+k_{p_{\psi }}{\dot{e}}_{\psi }+k_{d_{\psi }}{\ddot{\psi }}_d\hfill \\ \hfill & -k_{d_{\psi }}\left({\displaystyle \frac{j_x-j_y}{j_z}}\right)\dot{\theta }\dot{\varphi }).\hfill \end{array}$$

(17)

and,

$$\begin{array}{cc}\hfill {\tilde{u}}_x=& {\displaystyle \frac{m}{k_{d_x}{u}_1}}\left[{\lambda }_{1_x}\mathrm{sign}\left(s_x\right)+k_{i_x}{e}_x+k_{p_x}{\dot{e}}_x+k_{d_x}{\ddot{x}}_d\right],\hfill \\ \hfill {\tilde{u}}_y=& {\displaystyle \frac{m}{k_{d_y}{u}_1}}\left[{\lambda }_{1_y}\mathrm{sign}\left(s_y\right)+k_{i_y}{e}_y+k_{p_y}{\dot{e}}_y+k_{d_y}{\ddot{y}}_d\right].\hfill \end{array}$$

(18)

While ISMC successfully guarantees system robustness from the system’s initial state, it is worth noting that it does not entirely eliminate the challenges associated with discontinuous control and chattering. As a result, there is an ongoing need for advancements within the SMC design community to address these issues comprehensively.

3.2.4. Other SMC Types

Other SMC types have been studied in the literature, such as the neural network [56], backstepping SMC [57], adaptive SMC [58], adaptive recursive SMC [59], event-triggered SMC [60], and prescribed SMC [61]. These advanced SMCs have many applications in UAVs. The above-mentioned SMC design techniques are designed for the different problems considered for the quadcopter. In the neural network-based SMC, the authors mainly used the neural network for the estimation of the external disturbances using the RBFNN network since SMC can tackle only matched bounded known disturbances. Other literature studies can be found based on the neural network-based SMC. Also, in the literature, a few other works can be found based on the backstepping SMC. In the backstepping-based SMC, the authors combined both the backstepping control and different SMC types for the tracking/stabilization of the quadcopter system. Furthermore, in the existing literature, researchers have extensively employed robust observers for estimating external disturbances affecting quadcopter systems. A notable approach involves utilizing a sliding mode observer, which incorporates the output estimation error through a nonlinear switching term. This method presents an appealing solution for handling uncertainties or disturbances.

Sliding Mode Observer (SMO): The sliding mode observer is designed to drive the output estimation error to zero within a finite timeframe, assuming a known bound on the disturbance amplitudes. Importantly, the observer states converge asymptotically to the system states even in the presence of uncertainties, demonstrating the efficacy of this technique in enhancing the quadcopter system’s stability and performance. Also, there are literature studies where authors have extensively used robust observers for estimating external disturbances acting on the quadcopter system. A sliding mode observer, which feeds back the output estimation error via a nonlinear switching term, provides an effective solution for handling uncertainties or disturbances. This observer can cause the output estimation error to converge to zero in a finite amount of time, provided the amplitude of the disturbances is known, while the observer states asymptotically converge to the system states. Also, studies on estimation, sensor fusion data processing, and filtering have been explored in the literature [62,63,64].

Event-triggered SMC: The event-triggered SMC has been extensively developed to ensure robustness and minimize resource utilization in systems. The core concept of the event-triggered technique is that the control signal is updated only when it is required, typically triggered by the system states violating a predefined threshold or condition known as the triggering condition. Various triggering policies have been developed, each offering distinct advantages such as longer inter-sampling times, reduced computational complexity, and minimal hardware requirements for implementation. This strategy reduces the communication load in the network and facilitates efficient usage of the resources by increasing the update intervals. Static event-triggering policy: The first form of triggering policy is the static-triggering mechanism, where the error $\mathsf{e}\left(t\right)$ is monitored until it reaches a constant threshold. The triggering policy is given by the following:

$$t_{i+1}=\inf \{t\in [t_i,+\infty ):\parallel \mathsf{e}\left(t\right)\parallel \ge \sigma \alpha \}$$

(19)

where $\sigma \in (0,1)$ and $\alpha >0$ are design constants. There are different types of triggering policies that can be observed, such as in Ref. [65].

Fractional order sliding mode control (FOSMC): This is an advanced control technique that extends the principles of traditional SMC by incorporating fractional calculus. Fractional calculus is a mathematical framework that generalizes integer-order differentiation and integration to non-integer (fractional) orders. Another SMC variant is based on the line of reducing the reaching phase in the sliding mode control domain. The sliding surface designed is the fractional order surface, such as in Ref. [66]. The prime design difference between FOSMC and the discussed SMC is that a fractional order surface needs to be designed, which can ensure the tunable reaching time with appropriate surface design parameters. Other important works have explored fractional order SMCs, such as Refs. [67,68,69].

3.2.5. Adaptive SMC

The primary challenges in implementing SMC revolve around two closely linked issues: chattering and excessive control activity. Chattering, which is the rapid switching of control values, is directly proportional to the magnitude of the abrupt control inputs. Addressing both of these concerns concurrently while still ensuring the existence of the sliding mode involves minimizing the magnitude of control inputs. The aim is to eliminate the need for predefined uncertainty limits and to dynamically adjust the control gain, aiming for the smallest value possible while still effectively mitigating uncertainties and disturbances.

Considering the sliding surface (5), control law (17), (18), and system (3), the adaptive SMC formulated for the states $\varphi$, $\theta$, $\psi$, x, y, and z can be outlined as follows. Here, the problem involves designing the adaptive sliding gain without the knowledge of the upper bound of the perturbations so that the finite time stability of the system trajectories can be ensured. The adaptive sliding gain dynamics [70] are chosen as follows:

$$\begin{array}{c}\hfill {\dot{\lambda }}_{1_{\tau }}=\left\{\begin{array}{cc}\varpi \sqrt{{\displaystyle \frac{\kappa }{2}}}{\lambda }_{1_{\tau }}\hfill & \mathrm{if}s\ne 0,\hfill \\ 0,\hfill & \mathrm{if}s=0.\hfill \end{array}\right.\end{array}$$

(20)

where $\varpi ,\kappa$ are arbitrary positive constants. There are many adaptive gain schemes in the literature, and one is shown.

A key consideration in the design of SMC for quadcopter systems involves mitigating the chattering phenomenon. Various techniques have been developed to reduce chattering, including the boundary layer method, and higher-order SMC approaches (ST-SMC, integral TSMC, multiphase SMC). These methods aim to minimize undesired oscillations or rapid switching near the sliding surface while preserving system robustness. For a detailed exploration of these chattering reduction strategies, readers can refer to [24,71,72].

Table 3. Sliding mode control scheme: advantages and disadvantages.

Control Scheme	Advantages	Disadvantages
Classical SMC	Simple, easy to implement	Chattering phenomena, discontinuous control
ST-SMC	Reduced chattering, continuous control	Complex design
TSMC	Finite-time stability	Singularity, nonlinear sliding surface design
ISMC	Sensitive to disturbance from initial state	Design of nominal control and switching control
Adaptive SMC	Relaxation of switching gain based on the bound of disturbance	Judicial choice of the adaptive parameter, extra computation
Event-triggered SMC	Minimal usage of resources	Extra hardware
NN SMC	Adaptability on the control gain, disturbance	Consumes more power-complex computations
Backstepping SMC	Improves tracking performance	The design of the SMC and backstepping control is more complex

shows the advantages and disadvantages of the different SMC designs.

A detailed analysis of the design of SMCs as well as the problems are explored in the following sections and subsections.

4. Control Problem Addressed Using SMC

In the development of the SMC design for a quadcopter, researchers have extensively explored various types of SMC designs and implementations. The block diagram for the quadcopter controller design system is presented in Figure 6. Different problems are chosen in this domain, such as tracking and stabilization, considering both the quadcopter’s dynamic and kinematic models. The tracking and stabilization problems chosen are as follows:

• Attitude (P1): The main objective is to design a controller for tracking the attitude—represented by the Euler angles [$\varphi$, $\theta$, $\psi$]—of the quadcopter, minimizing the tracking error regardless of the presence or absence of disturbances. Also, the objective is to ensure the stabilization of the attitude defined by Euler angles [$\varphi$, $\theta$, $\psi$] to the origin.
• Attitude and positions (P2): The control objective is to design a controller for the tracking/stabilization of the quadcopter for the inner loop attitude ($\varphi$, $\theta$, $\psi$) and the outer loop positions and altitude (x, y, z).
• Attitude and altitude (P3): This problem aims to design a controller to track the attitude and altitude of the quadcopter in the presence of external disturbance.

In the chosen problems, researchers have considered disturbances in attitude and position as constant, random, and stochastic disturbances. Many studies in the literature model disturbances as lumped disturbances, such as $D_1=sin\left(t\right)$ [73,74], $D_2=sin\left(nt\right)+cos\left(nt\right)$ [75] representing wind and moment disturbances. Also, estimated disturbances, aerodynamic forces, and air drag—consisting of attitude and altitude information—are often treated as external disturbances in the literature. In the literature, the results of the existing article’s demonstration include the tracking/stabilization performance on the real system (RS) as experimental validation, software-in-the-loop (SITL), and simulation (S) validations. In the next section, the chosen problems are discussed elaborately.

4.1. Attitude Control (P1)

In this section, the primary focus revolves around addressing key challenges related to quadcopter control, particularly pertaining to the stabilization of its attitude, including roll, pitch, and yaw, to achieve desired Euler angles [${\varphi }_d$, ${\theta }_d$, ${\psi }_d$]. Additionally, tracking the quadcopter’s roll, pitch, and yaw to match the desired values ${\varphi }_d$, ${\theta }_d$, ${\psi }_d$ is considered to ensure that the tracking error converges to zero. Numerous controllers have been developed in the existing literature to address the challenges associated with quadcopter attitude tracking and stabilization. These include PD [76], PID [21], Fuzzy PID [77], cascaded PD, and PI controllers [78]. However, a common challenge encountered across these controller designs is effectively handling disturbances that affect the quadcopter system. The presence of disturbances complicates the control task and necessitates the development of robust control methods capable of counteracting these disturbances. This is where the extensive utilization of SMC comes into play, i.e., in the context of quadcopter control. In the literature, numerous researchers have explored and designed different types of SMC strategies to address the challenges of attitude tracking and stabilization in quadcopters. Here, different sliding mode controllers are discussed in the context of problem P1. Also, this section is structured by first presenting the SMC design strategy validated on hardware, followed by a discussion of other SMC strategies supported by simulation results.

In recent research, several innovative approaches have been developed to enhance the performance of quadcopters in terms of attitude tracking and disturbance rejection. Notably, an adaptive recursive SMC method proposed by Chen et al. [79] has been introduced to improve quadcopter attitude tracking and bolster disturbance rejection capabilities. To assess its efficacy, a comparative analysis was conducted, evaluating convergence time and tracking errors in comparison to other control strategies. Real-world testing was performed using a DJI F450 quadrotor frame under windy conditions. Nguyen and colleagues delved into the domain of iterative learning SMC for the attitude tracking of quadcopters in their study [80]. To validate their approach’s advantages, they conducted field tests and employed a real-time experimental setup using a 3DR Solo drone. Another significant development in the field involved the creation of a dual-channel inner and outer disturbance rejection-based SMC, as detailed in Xiong’s work [81]. This innovative controller showcased its ability to enhance disturbance rejection in quadrotors, particularly in the presence of unknown roll, pitch, and yaw disturbances. The study incorporated simulations and experimental results to affirm the effectiveness of the newly designed control method. One important study, Ref. [82], explored a control algorithm that is robust to wind disturbances for quadrotor UAV attitude dynamics. The proposed approach employs a high-gain observer based on a discontinuous technique. The author also carried out a hardware implementation of the designed approach. Moreover, Hassani and colleagues presented a practical application-based design involving backstepping ST-SMC [83]. This flight control system was engineered to rapidly adjust the quadrotor’s attitude to reference values while minimizing the influence of external perturbations and model errors. Real-time experiments on the $F450$ quadrotor were conducted to demonstrate the viability and capabilities of the proposed control strategy for rejecting disturbances. In Ref. [84], the quadrotor attitude was controlled via an adaptive non-singular TSMC in the presence of parametric uncertainties and disturbances. Experiments on a self-made quadrotor testbed were performed to determine the control performances of the suggested controller, which strengthened the validity of the findings. An unknown system dynamic estimator (USDE)-based SMC was developed in Ref. [85] to improve the controller’s chattering problem. To accelerate the sluggish transient convergence, a continuous double hyperbolical reaching law (DHRL) was designed, which greatly alleviated control oscillations without sacrificing dynamic response. The authors presented simulations and experimental results to verify the salient features of the proposed method. Refs. [86,87] are other SMC literature studies that contributed to the design and experimental validation. These pioneering studies collectively demonstrate the continuous advancements in the field of quadcopter control, offering innovative solutions to enhance attitude tracking and disturbance rejection, validated through rigorous experiments and comparative analyses. In the following, simulation-based literature studies are explored.

Tao et al. [88] explored a novel control strategy for quadrotors operating in the presence of inertia uncertainties and external disturbances. This control approach is based on the implementation of an RBFNN for optimized performance. Similarly, in the study conducted by Yu et al. [89], a fuzzy extended state observer was devised to effectively handle unidentified lumped disturbances. Furthermore, a nonsingular fast TSMC algorithm was presented, which utilized estimation data from the fuzzy extended state observer. Comparative simulation results were offered to illustrate the enhancements achieved by the designed controller. In a pertinent investigation by Li et al. [90], a sliding mode tracking controller and observer were developed. A novel sliding mode observer was established to estimate external disturbances within a predefined settling time, and numerical simulation results were presented to corroborate its effectiveness. Ref. [91] presented another observer-based design. Cui et al. [92] introduced a fixed-time adaptive fast super-twisting disturbance observer for the estimation of unknown external disturbances and a fixed-time tracking controller. The superiority of this proposed controller is empirically validated through simulation results. Additionally, Wu et al. [93] addressed the challenges associated with mathematical modeling and attitude-tracking control of a quadrotor subject to time-varying mass and external disturbances. The proposed controller ensured robust tracking control performance during attitude maneuvering through simulation results.

An adaptive prescribed performance TSMC scheme was proposed in Ref. [75], where all state variables converged to their desired value in a short time while the prescribed performance function limited the convergence speed, maximum overshoot, and steady-state error. In another development, Coates et al. delved into the design of a generalized multivariable ST-SMC for attitude tracking in quadcopters, as discussed in Ref. [94]. The effectiveness of this design was rigorously demonstrated through a simulation study conducted under highly turbulent conditions. Furthermore, a novel event-triggered SMC approach was proposed in a recent article by Gao et al. [95]. Additionally, a fuzzy logic system was incorporated into an adaptive algorithm to counteract the adverse effects of aggregated disturbances. Simulation data were provided to showcase the performance of the suggested control strategy. Other literature studies on SMC designs can be found in Refs. [96,97,98,99]. These studies collectively contribute to the advancement of control strategies for quadcopters in the presence of disturbances and uncertainties, particularly in the context of finite-time convergence and robust performance. All the literature studies listed in problem P1 are tabulated in

Table 4. SMC-based controller for problem P1.

Approach	Ref.	Year	Method	Disturbances	Validation
SMC	[81] [93] [99] [82]	2021 2022 2012 2018	C C Fuzzy C	M-WG-E M-E WN W	RS S S RS
ST-SMC	[83] [94] [98]	2022 2021 2018	Backstepping multivariable ESO	wind-gust-E WG WG-AF	RS S RS
TSMC	[75] [79] [92] [88] [100] [90] [84] [86]	2020 2022 2022 2021 2018 2021 2019 2023	A-PP A-recursive Fast-DO ESO-NN-NS A-NS A-NS Adaptive NS TSMC	E E E MU-E WG E WG-E E	S RS S S RS S RS RS
Advance SMC	[80] [91] [85] [96] [87] [97]	2021 2021 2021 2024 2024 2023	Iterative learning FESO-NS USDE H B-NS-I A	MU E E E W-E W-E	RS S RS S RS RS

. The table highlights various SMC designs that have been explored for quadcopters with practical implementations. Notably, many studies involving hardware implementations focused on uniform wind gusts as the primary external disturbances. However, there were noticeable gaps in the current research, specifically in the areas of optimization-based, model-free, and application-specific controller designs for quadcopters.

In the next section, problems considering both attitude and position controls are discussed.

4.2. Attitudes, Positions, and Altitude Control (P2)

The development of control strategies for managing the positions, altitudes, and attitudes of quadcopters presents a particularly intricate challenge. Remarkably, this complexity arises from the need to control six states using only four control inputs, a significant hurdle in addressing this problem. Extensively explored in the literature is the pursuit of methods for tracking desired quadcopter trajectories and stabilizing the system while accommodating external disturbances and uncertainties that affect all states—roll, pitch, yaw, position, and altitude. Consequently, substantial research efforts have been directed toward the formulation of diverse controller designs specifically tailored to managing the attitude, position, and altitude control of quadcopters. Within the realm of the literature, the prevalent focus has gravitated toward the robust nonlinear control design of SMC when dealing with quadcopter systems. This has led to the extensive exploration of various SMC variants, each catering to the unique demands of quadcopter tracking and stabilization.

This section begins with descriptions of the classical SMC for problem P2 followed by ST-SMC and its variants. Then, we discuss TSMC and the combination of SMC with other controllers. Lastly, we present the neural network and learning-based SMC. In each type of SMC design, we first discuss the results that are validated in hardware and then other results. The observer-based SMC design is also discussed. The design of a classical SMC for this problem can be found in Refs. [101,102], where the authors demonstrated the hardware implementation of the proposed controller to validate its effectiveness. Also, an observer-based SMC design is shown in Ref. [103], where the authors designed an observer to estimate the disturbance. They portrayed the simulation results. Some other important literature studies are found in Refs. [104,105,106].

Within the realm of quadcopter control, the second-order ST-SMC has garnered substantial attention in the existing literature. An illuminating contribution to this field can be found in the work by Rios et al., as detailed in Ref. [107]. The authors present a finite-time sliding mode observer, a pivotal tool for estimating and identifying both the full system states and disturbances. The effectiveness of this continuous ST-SMC methodology is rigorously examined through experimental validation on the QBall2 quadcopter platform. In a parallel development, Tripathi et al. propose a fast terminal ST-SMC approach, as elaborated in Ref. [108], aimed at ensuring robust tracking of quadcopter trajectories. Experimental validation is conducted on the DJIM100 platform to affirm the effectiveness of this strategy.

Labbadi et al. [109] introduced a fractional order (FO)-enhanced ST-SMC for quadrotor systems. The proposed FO control approach showcases attributes such as fast convergence, high precision, and robustness against stochastic perturbations and uncertainties. Other important works on STSMC can be found in Refs. [110,111,112]. The study by Luque et al. offers a unique perspective [74]. The authors introduced a controller based on the block control approach and the super-twisting control algorithm. Furthermore, the study evaluated the system’s resilience against disturbances, specifically the wind parameter arising from aerodynamic forces, through extensive simulations. In Ref. [109], Moussa et al. introduced a super-twisting proportional–integral–derivative sliding-mode control (STPIDSMC) enhanced by FO techniques. Simulation results provided clear evidence of the controller’s robust performance and disturbance rejection capacity. Research on the design of ST-SMC can be found in Refs. [113,114,115].

The literature features an array of remarkable contributions in the realm of TSMC design, specifically tailored for tracking and stabilizing quadcopter systems. One such notable endeavor is presented in Ref. [42], where the authors proposed an adaptive finite-time fast TSMC approach, addressing attitude and position tracking of quadcopters. Notably, this research encompassed experimental validation conducted on the real-world DJIM100 system, confirming the effectiveness of the methodology in both tracking and regulation tasks. In a parallel investigation, detailed in Ref. [116], a novel adaptive robust backstepping fast TSMC strategy was introduced, designed to manage the attitude and position control of quadcopter, while also accommodating input saturation and external disturbances. Another significant contribution is Ref. [117], where an adaptive integral TSMC is proposed for quadrotor UAVs operating in the presence of disturbances. The authors introduced an adaptation law and a modified parameter-tuning integral TSMC scheme to effectively compensate for model uncertainties and external disturbances. The validation of this controller was accomplished through the presentation of simulation results. Furthermore, a pioneering approach was showcased in Ref. [118], where the authors introduced an adaptive nonsingular fast TSMC for steering quadrotor flight attitudes, complemented by a robust backstepping SMC for position management. This innovative combination ensured fast and accurate tracking, even in the presence of external disturbances. The exploration of adaptive nonsingular fast TSMC continued in Refs. [69,119,120,121,122,123,124,125].

Significant advancements in TSMC have been documented in the literature, addressing various aspects of quadcopter control. For instance, in Ref. [126], a non-singular TSMC approach was proposed for the complete actuation of quadcopter yaw and altitude systems, specifically targeting trajectory tracking in the presence of external disturbances. The authors not only presented simulation results to validate the efficacy of the approach but also conducted a comprehensive stability analysis. Similarly, Ref. [127] introduced a TSMC-based strategy to enhance trajectory tracking performance, robustness, and adaptability against wind disturbances. This research focused on constructing major control loops for position tracking and attitude stabilization, employing a finite-time adaptive integral backstepping fast TSMC approach. In a distinctive approach, Ref. [128] delved into the control of the attitude and position of a quadrotor equipped with a model that encompassed parameter fluctuations, uncertainties, and external disturbances. This study introduced a model-free-based MF-TSMC technique, which combined a sliding-mode control strategy with model-free control technology. The study provided numerical simulation results and compared them with findings from other control methods, such as PID, backstepping, and SMC, to underscore the performance and efficacy of the proposed MF-TSMC strategy.

In the context of quadrotor control, a disturbance observer specifically tailored for handling wind perturbations was introduced in Ref. [129]. This innovative observer was rooted in the principles of the nonsingular TSMC method. The study provided comparative findings with a previous research effort, showcasing the utility and robustness of the proposed disturbance observer. Other observer-based designs on TSMC can be found in Refs. [73,130]. In a separate study, Ref. [131] presented a robust flight control system comprising two proportional–derivative (PD) controllers and an adaptive fuzzy TSMC. The incorporation of adaptive Mamdani fuzzy systems enabled online identification of the quadrotor’s dynamics and external disturbances. The study underlined the effectiveness of this flight control system through a combination of simulation findings and real-time implementations. Furthermore, in Ref. [132], a TSMC design was proposed to address both position and attitude tracking for quadcopters. The authors asserted the promise of the presented simulation results in achieving precise control over the aircraft’s position and attitude tracking. In a related study, Ref. [133] introduced the global fast dynamic TSMC technique, aimed at designing a flight controller for achieving finite-time position and attitude tracking control of a small quadrotor. The study provided MATLAB simulation results to validate the effectiveness of the proposed controller. Addressing the intricate issues of strong coupling and underactuation in a quadrotor unmanned helicopter, Ref. [134] presented a novel robust TSMC approach. This research illustrated the composite application of TSMC and SMC for fully actuated and underactuated systems. The simulation results vividly demonstrated the utility of this composite control approach, particularly in the presence of external disturbances.

Next, we present a comprehensive exploration of research that delves deeply into the works that combine SMC controllers with other control algorithms. Notably, the integration of SMC controllers with different control strategies has been thoroughly examined. For instance, in the work by Xu et al. [135], a sophisticated control scheme combining backstepping SMC with active disturbance rejection was devised specifically for trajectory tracking. The study included rigorous experimental validation conducted on real systems. Labbadi et al. [136] introduced an FOSMC scheme tailored to the challenging task of quadrotor trajectory tracking under uncertain disturbances. Labbadi et al. [137] proposed an adaptive global SMC to tackle quadrotor tracking in the presence of external disturbances. The study substantiated its claims with comprehensive MATLAB simulation results, validating the efficacy of the controller design. Additionally, Shi et al. [138] introduced an adaptive FOSMC method engineered to handle quadcopter trajectory tracking in the face of both actuator faults and external disturbances. The proposed controller was rigorously validated through numerical simulations. Moreover, Ramirez et al. [139] developed a robust backstepping ISMC system. This system was engineered to combat external disturbances, including wind gusts and sideslip aerodynamics, demonstrating the robustness of the approach in adverse conditions. The effectiveness was proven by the simulation results. In Ref. [140], the authors explored the design of FO-B-SMC to track the position and attitude of the quadcopter.

The scholarly works discussed in this section significantly focused on advanced topics on SMC-based quadcopter control. In Ref. [141], Wu et al. investigated the application of adaptive finite-time SMC for robustly tracking quadcopter positions and attitude trajectories Their approach incorporated a disturbance observer to estimate external disturbances within the system. Additionally, Davoudi et al. presented another innovative application involving an adaptive finite-time SMC [142]; the focus was on a positionable rotor quadcopter system. The study revealed that this structural configuration enables the quadrotor to reduce power consumption during disturbance rejection while achieving superior performance against disturbances. Furthermore, an alternative approach to control was explored in the work by Zhang et al. [143]; the authors integrated a nonlinear disturbance observer methodology into a backstepping control method.

Comparative simulation results validate the effectiveness of this design. For researchers interested in ISMC for quadcopters, several valuable references are available. Eltayeb et al. [144] introduced a control law with a switching gain that allowed rapid adaptation and robustness against parameter uncertainty and external disturbances. The performance of the proposed ISMC controller was evaluated using the MATLAB/Simulink platform. In research by Eltayeb et al. [145], a control strategy consisting of an outer loop controller for position control and an inner loop controller for attitude control was presented. Simulation results demonstrated the reliability of this control strategy, even in the presence of uncertainties. Finally, in the work by Ramirez et al. [139], a novel robust backstepping controller based on ISMC was proposed. The results were rigorously analyzed through simulations to validate the approach’s efficacy. Important literature studies on the design based on the backstepping SMC can be found in Refs. [57,146,147,148,149]. In Ref. [150], the control system was structured as a combination of a fully actuated subsystem and an underactuated subsystem. The simulation results validated the effectiveness of the proposed controller design. Moving forward, the authors of Ref. [151] presented an innovative SMC strategy based on nonlinear sliding surfaces. The simulation results were provided to demonstrate the efficacy of the designed controller. In Ref. [152], an auto-tuning adaptive PID control system was proposed. This approach harnessed the adaptive mechanism of SMC to overcome the need for manual re-tuning of PID controllers. In Ref. [68], an implementation method was outlined for stabilizing a quadrotor amidst the presence of external random and time-varying disturbances. This was achieved through the utilization of an event-trigger-based FO-SMC strategy. Simulation results were employed to showcase the controller’s effectiveness, especially when compared to the adaptive SMC approach introduced in Ref. [153], which emphasized its practical applicability in tracking tasks. The focus was on adaptive SMC as an approach for stabilizing quadcopter UAV systems within a finite time frame, even in the presence of parametric uncertainties. In Ref. [154], the authors described the design of an intelligent nonlinear SMC method for the regulation of quadrotor motion. Real-time validation of the proposed approach was carried out using a DJI Matrice 100 in an outdoor setting. In Ref. [155], an advanced trajectory tracking controller was developed, seamlessly combining SMC technology with a model-free control technique. Simulation and experimental findings highlighted the satisfactory control performance of the suggested controller, emphasizing its superior resilience compared to PID and backstepping controllers. In Ref. [156], the authors proposed a non-cascade adaptive SMC technique for quadrotor trajectory tracking. Numerous indoor experiments were conducted to compare this innovative design with traditional cascade structure controllers.

There is limited research in the domain of SMC that incorporates neural networks and reinforcement learning algorithms. For several other works in this niche area, see Refs. [157,158,159]. In Ref. [158], a robust adaptive SMC tracking control approach was proposed, leveraging the power of RBFNN. The authors conducted a thorough examination of the system’s performance, substantiating the controller’s effectiveness through simulation and experimental results. Another innovative application of neural networks in SMC was discussed in Ref. [157], where a neural network-based adaptive ISMC system was introduced for the precise tracking of quadcopter positions and attitudes. This approach took into account the quadcopters’ unknown dynamics. Ref. [159] examined similar work. In Ref. [160], an innovative hierarchical controller was introduced, which combined the PD-SMC method with a robust integral of the signum of error approach. This approach amalgamated the advantages of PD control, SMC resilience, and the ability of RBFNN to approximate arbitrary functions. Numerical simulations were employed to assess the effectiveness of this control technique, and the results were compared against those obtained using conventional PD, PID, PD-SMC, and RBFNN-based controllers. See Refs. [161,162,163] for other significant literature studies based on neural networks.

Table 5. SMC-based controller for problem P2.

Approach	Ref.	Year	Method	Disturbances	Validation
SMC	[150] [151] [155] [142] [145] [101] [114] [102] [103]	2020 2020 2021 2022 2022 2024 2014 2018 2012	C C C C C C second C C-O	I-E E M-E E E W-E E W-E WG	S S RS S S RS S RS S
ST-SMC	[110] [74] [111] [109] [112] [115]	2023 2012 2022 2020 2022 2016	T - T FO-Order B ST	E AM-AF-M’ MU,E M-MI-E MU-E E	S S S S S S
TSMC	[126] [116] [127] [73] [154] [128] [117] [118] [69] [89] [125] [132] [133] [134] [164] [120] [129] [107] [42] [108] [119] [130] [95] [131] [165] [122] [124]	2018 2020 2020 2020 2022 2016 2021 2021 2021 2020 2021 2014 2017 2014 2019 2022 2022 2018 2021 2019 2020 2020 2020 2021 2021 2024 2024	NS B-Fast A-IB Fast DO-NS - Model Free A-I A-NS A-FO-NS SMO-NS A-ST-NS - Global Fast - I A-NS DO-NS SMO A-Fast Fast-ST A-DO- NN NS Fuzzy-NS A-Fuzzy A-NS A-NT NT	M-I-E E AM-GM-E E MU-E E E E M-MI-E E E E AM-AF-E E E E E MU-WG-E WG-E E MU-E E E E WGN E E	S S S S RS S S S S S S S S S S S S RS RS RS S S S RS S S S

and

Table 6. SMC-based controller for problem P2 Continued.

Approach	Ref.	Year	Method	Disturbances	Validation
Advance SMC	[146] [166] [56] [161] [57] [147] [148] [123] [113] [162] [106] [149] [140] [163] [105] [104] [157] [143] [141] [144] [121] [136] [137] [138] [152] [68] [153] [135] [158] [167] [139] [168] [159] [160]	2024 2024 2016 2019 2023 2023 2024 2024 2024 2023 2016 2018 2019 2012 2015 2015 2021 2018 2021 2020 2018 2020 2020 2020 2021 2022 2018 2020 2019 2018 2014 2020 2019 2019	A-B A- Recursive A-NN-I A-NN B B-FO A-B-ET-FO A-PP-R-NS NN-ST-NT B-NN D B FO-B FO C-II A A-Int-NN B-DO A-DO A Int FO A A-FO Adaptive-PID ET-FO A B A-NN A B-Int APF-NN Adaptive-fuzzy NN	E E AF-M-MU MU-E E E MU-E E E A W-E W-E E A E E MU E E M-E E E E WG MU-WG-E E MU WG E E WD-E AF-E E WD-E	RS S S S S SITL S RS S S S RS S S HIL RS S S S S S S S S S S S RS S S S S S S

provide a comprehensive overview of the discussed literature, categorized by various indices. It is evident from the tables that the majority of studies rely solely on simulation validations. Furthermore, many of these studies focus on disturbances affecting model parameters, while those incorporating practical implementations typically address uniform and non-uniform wind disturbances from external devices. Most practical experiments are conducted indoors. Over the past five years, there has been a growing interest in developing SMC for quadcopters. A critical gap identified is the lack of SMC designs tailored for resource-constrained scenarios and learning-based approaches, as well as practical implementations that account for varying wind conditions and outdoor environments. In the next section, problem 3 is discussed vividly.

4.3. Attitude and Altitude (P3)

This section, the problem of tracking and stabilization of the attitude (i.e., $\varphi$, $\theta$, $\psi$) and altitude z is addressed. Numerous articles in the literature have focused on the P3 problem. In Ref. [169], a robust adaptive global time-varying SMC is proposed for tracking the quadcopter altitude and attitude trajectories with disturbance. The authors presented the simulation results to validate the designed controller. An adaptive SMC and linear active disturbance rejection control (LADRC) is proposed in Ref. [170] for the attitude and altitude tracking of the quadcopter. The authors considered external load acting in the center of the quadcopter. See Ref. [171] to read up on the altitude and attitude stabilization of the quadcopter. In Ref. [172], gain-scheduled SMC was combined with a unique nonlinear disturbance observer to mitigate external disturbances and achieve reference nominal model performance. Also, numerical simulation results were displayed to show the effectiveness of the suggested regulation. A nonlinear adaptive SMC strategy was suggested, taking into account the quadrotor system’s underactuated and tightly coupled properties in Ref. [173]. The effectiveness and resilience of the suggested control scheme were then proved through simulations and tests, where the superiority of LQR and ADRC was amply shown.

Literature based on the ST-SMC addressing problem P3 can be found in Ref. [174]. In Ref. [174], an ST-SMC was designed for the altitude and attitude tracking of a quadcopter with a manipulator. Here, the authors demonstrated that the adaptive ST-SMC performed well with respect to the investigated system’s performance. Simulation results are provided to show the system’s quick convergence and strong robustness when using RBF and TSMC.

Also, there are very few articles that address the design of controllers for the desired tracking and stabilization of position, as well as position and yaw, in the presence of external disturbances. The main objective of a UAV controller in this context is to ensure the asymptotic convergence of the state vector (x, y, z, $\psi$), to the reference trajectories ($x_d$, $y_d$, $z_d$, ${\psi }_d$). In Ref. [175], a fixed time tracking controller and a multivariable fixed time disturbance observer were presented to track positions. The validation of the proposed design feature was ensured via the simulation results. Ref. [156] also addressed position tracking. The authors proposed a non-cascade adaptive SMC for the quadcopter’s trajectories. Numerical simulations and indoor flight experiments were performed to verify the effectiveness of the proposed adaptive SMC strategy.

The authors also addressed the tracking of positions and yaw, a topic that is rarely discussed in the literature. In Ref. [176], an event-triggered sliding mode position tracking controller for the quadcopter system was presented. The authors tested the designed controller on the actual Crazyflie 2.1 system to verify its performance without the influence of external disturbances. A real-world application-based article can be found in Ref. [177] where the author explored the ISMC technique for a quad-rotor aircraft vehicle to eliminate the steady-state error induced by the boundary layer and achieve asymptotic convergence to the desired altitude with a continuous control input. The simulations and experimental studies were supported by different tests to demonstrate the robustness and effectiveness of the designed approach. Ref. [178] was another real-world application-based literature study. Ref. [179] suggested a technique that combined RBF adaptive control and backstepping SMC. The goal of Ref. [180] was to design control of mini-quadrotors under wind perturbations. Taking into account a detailed unmanned aerial vehicle (UAV) model, the aim was to find a sliding mode control law, minimizing the impact of the wind field on UAV dynamics. For this purpose, an aerodynamic modelization of external disturbance was introduced. After that, the upper bounds of these disturbances were computed. Lastly, the sliding mode altitude and attitude controls were designed. The simulation results demonstrated that the method developed in this study achieves quicker adjustment times and reduces inaccuracies while accounting for both constant and variable external disturbances. Only a few other studies addressing problem $P3$ can be found in the literature, such as Ref. [181].

The existing literature is systematically presented in tabular format for enhanced clarity.

Table 7. Literature under problem P3.

Approach	Ref.	Year	Method	Disturbances	Validation
SMC	[170] [169] [172] [171] [180]	2022 2021 2020 2027 2017	C	M-E I-E Mo-E E E	S S S S S
ST-SMC	[182] [183] [174]	2019 2020 2019	DO Adaptive DO	E E E E	S S S S
TSMC	[184] [185] [186]	2020 2021 2022	DO NS NN	E MU-E E	S S RS
Adaptive SMC	[187] [181]	2019 2024	Fuzzy B-NS	Absent E	S S

provides a comprehensive overview of the research, offering a detailed analysis of the specific problems addressed, the types of SMC designs utilized, the disturbances under consideration, and the methods employed for result validation. To streamline the table’s content, various acronyms are employed, and their corresponding full forms are provided in the Abbreviations section, facilitating a more concise presentation of the information.

The next section is followed by the simulation results and analysis.

5. Simulation Results and Analysis

In this section, the simulation results are presented, considering different SMCs for the quadcopter model. The sets of simulation results are shown, considering problem P2 with classical SMC, super-twisting SMC, and adaptive ST-SMC. The simulation for problem P3 is presented with TSMC. The same quadcopter model and the appropriate parameters are chosen for each set of simulations. The simulation was conducted on the MATLAB simulator. The dynamic of the quadcopter is given by (2). The simulations utilized the 3DR Iris quadcopter model, with parameters shown in

Table 8. Parameter values.

Model parameters	$j_{xx}=2.9125\times {10}^{-2}$ Nm, $j_{yy}=2.9125\times {10}^{-2}$ Nm $j_{zz}=5.5225\times {10}^{-2}$ Nm, $m=1.5$ kg, $g=9.8m{sec}^{-2}$
SMC	$k_{p_{\varphi ,\theta ,\psi ,z}}=0.7$, $k_{d_{\varphi ,\theta ,\psi ,z}}=0.5$, $k_{i_{\varphi ,\theta ,\psi ,z}}=0.5$ $k_{p_y}=0.0001$, $k_{d_y}=0.05$, $k_{i_y}=0.0001$. ${\lambda }_{\varphi }=0.02$, ${\lambda }_{\theta }=0.02$, ${\lambda }_{\psi }=0.01$, ${\lambda }_z=0.15$, ${\lambda }_x=0.03$, ${\lambda }_y=0.03$
ST-SMC	$k_{p_{\varphi ,\theta ,\psi ,z}}=0.2$, $k_{d_{\varphi ,\theta ,\psi ,z}}=0.05$, $k_{i_{\varphi ,\theta ,\psi ,z}}=0.05$ $\lambda 1_{\varphi ,\theta ,\psi ,z}=0.2$$\lambda 2_{\varphi ,\theta ,\psi ,z}=0.1$, $\lambda 1_{x,y}=0.2$ and $\lambda 2_{x,y}=0.1$
AST-SMC	$\varpi =1,\kappa =1$, ${\lambda }_{2_m}=2ϵ{\lambda }_{1_m}$$ϵ=0.1$
TSMC	$\alpha =0.05$, ${\beta }_1=0.05$, $\delta =0.8$ $k_z=0.01$, $k_{\varphi }=0.01$, $k_{\theta }=0.02$ and $k_{\psi }=0.02$.

. The desired trajectories of positions for each set of simulations are defined as $x_d\left(t\right)=2cos\left(0.1t\right)$, $y_d\left(t\right)=2sin\left(0.2t\right)$, $z_d=5m$, and desired trajectories for $\varphi$ and $\theta$ are defined in (10) for SMC and ST-SMC, AST-SMC, TSMC, while keeping ${\psi }_d$ constant. Initial values for the trajectories were set at $\Theta ={\left[000000000000\right]}^{\top }$. The simulations encompassed constant and stochastic wind disturbances, generating forces along the quadcopter’s x and y subsystems, and are defined as $d\left(t\right)=0.2+0.3cos\left(0.1t\right)$ for each set of simulations. The units of the angles $\varphi$, $\psi$, and $\theta$ are in degrees, and units of the position $x,y$, and altitude z are in meters.

Classical SMC

The effectiveness of the classical SMC strategy for the quadcopter model has been verified through simulation results using MATLAB (R2024b). Employing the quadcopter dynamics (2), the sliding surface (5), and the control law (17), the chosen surface parameters and switching gains are presented in Table 8. The virtual sliding mode control law for positions x and y was developed in (18). The sliding surface trajectory responses were observed for the quadcopter’s attitude, altitude, and position, accounting for external disturbances $d\left(t\right)=0.2+0.3cos\left(0.1t\right)$ acting in positions x and y. Choosing suitable sliding parameters for the PID sliding surfaces and making a judicious selection of the switching gain can lead to a notable reduction in the steady-state tracking error.

Super-Twisting SMC

Considering the same quadcopter dynamic system (2) and the same designed sliding surfaces (5), the PID sliding surface gains were chosen, as presented in Table 8. The designed ST-SMC (11) strategies were implemented by choosing the switching gains, as presented in the table. Considering the same quadcopter system (2) and the same designed sliding surfaces (5), the same parameter was chosen for the adaptive ST-SMC, and the adaptive gain design parameters were chosen accordingly. All the control schemes successfully achieved tracking of the desired trajectories. Therefore, it can be inferred that the designed controllers meet the control objectives and perform well.

The evolution of state trajectories and the desired trajectories are presented in Figure 7. From Figure 7, it can be observed that the controller’s $SMC$, $ST$-$SMC$, $AST$-$SMC$ track the desired trajectories almost exactly. From Figure 7, one can see that the desired roll ${\varphi }_d$ and pitch ${\theta }_d$ angles are different for each controller since the desired roll and pitch commands are computed by each controller and their trackings are presented. Each controller is responsible for computing its own desired roll and pitch angles to ensure stability and the tracking of the trajectory on the x and y coordinates. It can be observed that the trajectories are tracked with minimum errors. The switching gains are kept the same as the classical SMC and super-twisting SMC. The adaptive gains are evaluated as per the adaptive law.

The simulation results demonstrate the effectiveness of various SMC designs. The tracking errors by each controller are shown in Figure 8. It can be observed that the tracking errors are significantly reduced and remain bounded after an initial transient phase. From Figure 8, it can be observed that the tracking error in the case of classical SMC is larger compared to those of ST-SMC and AST-SMC. Again, it depends on the judicious choice of the design parameters and the chosen adaptive law. It can be concluded that achieving minimal tracking error is contingent upon the appropriate choice of design parameters and adaptive laws for each controller.

The control effort plot for each controller is presented in Figure 9. It can be inferred that, among the three controllers (SMC, ST-SMC, and AST-SMC), reduced chattering and continuous control can be achieved by both ST-SMC and AST-SMC. In the classical SMC, the chattering problem can be a major challenge. Again, in the AST-SMC, the challenge may come in the design of appropriate adaptive law. Also, the implementation of classical SMC is relatively simpler and can achieve the desired objective with less computational burden, i.e., computational time. To achieve significantly reduced chattering, the adoption of ST-SMC and AST-SMC appears more viable. Again, the implementation of ST-SMC and AST-SMC becomes complex and requires more computational time. From Figure 7 and Figure 10, it can be seen that the total tracking error and total control effort in the case of classical SMC are 53.76 and 1473 for the simulation time 100 s. Similarly, the ST-SMC and AST-SMC can be calculated as (62.02, 72.89) and (1474, 1472). From, the numerical figure, it can be inferred that the total tracking error is less in classical SMC compared to ST-SMC and AST-SMC, and the total control efforts for the three approaches are almost the same. Again, with all these controllers, the tracking performances of the desired trajectories can be achieved. The choice of a particular SMC approach hinges on the specific requirements and outcomes of the chosen problem.

Terminal SMC

The sliding surface for the implementation of TSMC is designed as [42] in (13). The sliding surface gains and switching gains are given in Table 8. The evolution of state trajectories, the desired trajectories, and the tracking errors are demonstrated in Figure 11. The switching gains are kept the same as the classical SMC and super-twisting SMC.

It can be observed that the trajectories are tracked with minimal error. The control efforts are illustrated in Figure 10. From the plot, it can be inferred that while the control effort required for trajectory tracking is low, there is noticeable chattering behavior. To mitigate this chattering, higher-order TSMC (order > 3) could be investigated in future research. From the control plots, Figure 10 and Figure 11, it can be calculated from the simulations that the total tracking error and total control effort are 1495 and 151. It can also be concluded that effective tracking and hovering of the quadcopter system can be achieved with proper design parameters and the selected control approach.

6. Challenges and Prospects for Future Endeavors

Numerous solutions within the realm of SMC control have been crafted with a focus on theoretically formulated simulation-based, implemented on real quadcopters, and an analysis to uncover the limitations and advantages of the control strategy. We contend that there are still broader research opportunities within this domain. Subsequently, we will explore some issues that bring attention in future research. The future aspects can be seen as follows:

• Sliding mode predictive control (SMPC) [188] represents a novel fusion of SMC and MPC. Unlike many existing SMC approaches in the literature, SMPC addresses a notable gap by incorporating considerations for constraints and optimization, which are areas often overlooked in conventional SMC methods. While MPC has gained widespread acceptance in process control for its adeptness at achieving optimal control within constraint-laden environments, SMPC leverages the strengths of both SMC and MPC. The resultant approach, SMPC, combines robustness, straightforward implementation, and adaptability to both matched and unmatched disturbances and uncertainties—characteristics typical of SMC and MPC. Notably, SMPC also integrates optimization capabilities and the ability to handle constraints, making it well-suited for application in dynamic environments. Despite these advantages, the application of SMPC to the design and implementation of control systems for UAVs remains an open challenge.
• One of the important aspects within the robotics community revolves around the transport of loads via quadcopters, highlighted in the survey conducted by Villa et al. (2020) [189]. This interest has particularly heightened in today’s context, primarily driven by the demand for efficient package delivery in urban settings, the need for precise agricultural practices such as targeted pesticide application, and the facilitation of supply transport in conflict zones. This spans a wide spectrum of interests across commercial, military, and civilian domains. In the realm of load transportation, two primary approaches have been utilized: suspending the load using cables or directly attaching the cargo to the quadrotor’s body. Notably, affixing the load to the quadcopter introduces increased complexity to its stability, posing a more challenging task for stabilization. The design and implementation of SMC in such scenarios remain open problems, requiring further exploration and development.
• Regardless of the essentiality of the control signal, the time-driven controller consistently supplies it to the actuators. The event-driven controller diminishes the exertion of the actuators, thereby conserving computational time and power. Given the limited computing capacity onboard real systems, it becomes crucial to minimize processing costs, especially since only limited battery power is available, while still upholding control performance. Thus, the event-triggered SMC [190], which can ensure minimum resource utilization with robustness toward external perturbations, can be explored for the quadcopters. In particular, network-related challenges such as delays, packet dropouts, and jitter remain unexplored in the context of SMC design for quadcopter systems. Moreover, the design of various triggering policies for quadcopters is still an open research problem.
• By integrating artificial intelligence (AI) [191] techniques such as machine learning and neural networks into SMC frameworks, researchers will be able to develop controllers that not only improve robustness against disturbances but also adapt to complex and dynamic environments. AI-based SMC systems leverage predictive models and real-time data to fine-tune control strategies, making them more responsive to varying flight conditions and unforeseen disturbances. This approach will enable quadcopters to achieve higher precision in navigation and stability, even in challenging scenarios like fluctuating wind conditions or unexpected obstacles. The incorporation of AI into SMC for quadcopters will present a significant leap forward, offering more sophisticated and resilient control solutions that can handle a broader range of operational complexities. Also, chattering is a common issue in traditional SMC implementations. Machine learning techniques, like neural networks or reinforcement learning, can be applied to design smoother control laws, learning optimal switching strategies to minimize or eliminate chattering. Also, exploring model-free controller designs for quadcopters could provide a promising platform for further research investigation.

7. Conclusions

This article provides an in-depth survey on SMC design, specifically tailored for tracking and stabilizing quadcopter systems. The survey explores a broad spectrum of robust controllers utilizing various SMC strategies, with each category of SMC extensively examined regarding its applicability to quadcopter systems. The study also delves into the future prospects and practical implementation aspects of SMC design for quadcopter systems. A detailed presentation of existing research is presented in tabular format. Additionally, this article rigorously validates control strategies, including classical SMC, ST-SMC, adaptive ST-SMC, and TSMC through MATLAB simulations, ensuring the authenticity and effectiveness of each specific control design method for quadcopter systems.