Quaternion-Based Robust Sliding-Mode Controller for Quadrotor Operation Under Wind Disturbance

Abstract

This paper presents a quaternion-based robust sliding-mode controller for quadrotors operating under significant wind disturbances. The proposed control method improves the reliability and efficiency of quadrotor control by eliminating the singularity problem inherent in the Euler angle method. The quadrotor dynamics and wind environment are modeled, and dynamic analysis is performed via numerical simulation. A realistic wind model is used, similar to a combination of deterministic and statistical models. The Lyapunov stability theory is utilized to prove the convergence and stability of the proposed control system. The simulation results demonstrate that the quaternion-based controller enables the quadrotor to follow the desired path and remain stable, even under external wind disturbances. Specifically, both position and attitude converge to the desired values within 10 s, demonstrating stable performance despite the challenging wind disturbances in both scenarios. Scenario 1 features turbulence with an average wind speed of 12 m/s and changing wind directions, while Scenario 2 models an environment with wind speeds that change abruptly and discretely over time, coupled with temporal variations in wind direction. Additionally, a comparative analysis with the conventional PD controller highlights the superior performance of the proposed RSMC controller in terms of trajectory tracking, stability, and energy efficiency. The rotor speeds remain within a reasonable and hardware-feasible range, ensuring practical applicability.

1. Introduction

Recently, quadrotors have attracted attention in various fields due to their versatility and high maneuverability. Quadrotors are under-actuated systems that use four propeller inputs to stabilize their position and attitude in the air with six degrees of freedom (DOF). They have vertical takeoff and landing capabilities and can execute precise movements in various directions. These systems could be crucial in diverse applications, such as disaster relief, environmental monitoring, agriculture, and security monitoring [1,2,3,4]. However, quadrotors must be robust to the various external factors that may affect their operating environment, especially uncertain external forces such as wind.

Quadrotor control strategies span a wide spectrum—from classical linear methods (proportional-integral-derivative (PID) [5], linear quadratic regulator [6]) and nonlinear techniques (feedback linearization [7], backstepping [8], sliding-mode [9]) to advanced approaches like model predictive control [10], intelligent learning-based systems [11], and hybrid methods [12]—each offering unique trade-offs between simplicity, robustness, computational cost, and adaptability, particularly when dealing with challenging scenarios like wind disturbances [13,14,15].

Quaternions are a mathematical tool for representing rotation in three-dimensional space. Quaternion-based control enables more efficient handling of object rotation in three-dimensional space while avoiding singularity problems (such as gimbal lock) that can occur with angle representations [16,17]. For quadrotors, quaternions allow for more precise and stable flight control.

Robust sliding-mode control (RSMC) is a control technique that can respond robustly to system uncertainties and external disturbances. It allows the control system to maintain a predefined sliding-mode state, ensuring that the target state is reached, regardless of dynamic changes in the system or external disturbances [18,19]. When applied to quadrotors, RSMC ensures flight control that is robust to external disturbances such as wind.

Wind disturbance is one of the major challenges for flying vehicles such as quadrotors. Wind is unpredictable and volatile and can negatively affect the flight path and stability of a quadrotor [20]. Therefore, the development of a control system that can effectively respond to wind disturbances is essential for improving the practicability of quadrotors [15,21,22].

The main contributions of this research can be summarized as follows:

-

The quaternion representation eliminates singularity issues, enabling more reliable control of the quadrotor’s orientation.

-

The controller effectively compensates for the impact of wind disturbances, maintaining trajectory accuracy and stability.

-

The proposed control algorithm is numerically validated in a windy environment.

The remainder of the paper is organized as follows. Section 2 introduces the nonlinear mathematical model of the quadrotor and wind disturbance. Subsequently, the robust nonlinear control algorithm is described, and its stability is proven via the Lyapunov theorem in Section 3. Then, Section 4 presents and discusses the numerical simulation results for the RSMC algorithm for quadrotors, including wind disturbances. Finally, Section 5 presents the concluding remarks.

2. Theoretical Modeling

This paper proposes a quaternion-based RSMC that remains stable in the presence of wind disturbances. The controller is established for the dynamic system associated with quadrotor flight.

Figure 1 shows the system of the quadrotor model, which includes the Earth inertial frame, referred to herein as the E-frame. In Figure 1, a and b are, in order, the transverse and lateral distances from the center of gravity, $s_i^{\prime }$ is the local vector of the ith rotor position, ${\mathsf{\Omega }}_i$ is the angular velocity of the ith rotor, and $\psi$, $\theta$, and $\varphi$ are the angles in the yaw, pitch, and roll orientation, respectively. The local reference body frame, denoted as the B-frame, is fixed to the quadrotor body, and the origin of the B-frame is positioned at the quadrotor’s center of mass. The relative orientation between the B-frame and E-frame can be expressed as $R=R_{\psi }{R}_{\theta }{R}_{\varphi }$, which is the yaw–pitch–roll sequence Euler rotation matrix:

$$R=\left[\begin{array}{ccc}{R}_{11}& R_{12}& R_{13}\\ R_{21}& R_{22}& R_{23}\\ R_{31}& R_{32}& R_{33}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{c}\psi & -s\psi & 0\\ s\psi & \mathrm{c}\psi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\mathrm{c}\theta & 0& s\theta \\ 0& 1& 0\\ -s\theta & 0& \mathrm{c}\theta \end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{c}\varphi & -s\varphi \\ 0& \mathrm{s}\varphi & \mathrm{c}\varphi \end{array}\right],$$

(1)

where

$$\begin{array}{l}{R}_{11}=\cos \theta \cos \psi \\ R_{12}=-\cos \varphi \sin \psi +\sin \varphi \sin \theta \cos \psi \\ R_{13}=\sin \varphi \sin \psi +\cos \varphi \sin \theta \cos \psi \\ R_{21}=\cos \theta \sin \psi \\ R_{22}=\cos \varphi \cos \psi +\sin \varphi \sin \theta sin\psi \\ R_{23}=-\sin \varphi \cos \psi +\cos \varphi \sin \theta \sin \psi \\ R_{31}=-\sin \theta \\ R_{32}=\sin \varphi \cos \theta \\ R_{33}=\cos \varphi \cos \theta \end{array},$$

(2)

where cos is denoted by c, and sin is denoted by s for notational convenience.

With the rotation matrix defined, the vector from the quadrotor’s center of mass to any point within the quadrotor can be expressed as follows:

$$s=R{s}^{\prime },$$

(3)

where vectors marked with the prime (′) are defined with respect to the body frame ($G{x}^{″}{y}^{″}{z}^{″}$), and those without the prime are defined with respect to the inertial frame ($OXYZ$).

In Figure 1a, the rotors are numbered in counterclockwise order, starting from Rotor 1 on the top right in the B-frame. The local vectors of the position of each rotor in the B-frame are expressed sequentially as follows:

$$s_1^{\prime }={\left[\begin{array}{ccc}b& -a& 0\end{array}\right]}^T,$$

(4)

$$s_2^{\prime }={\left[\begin{array}{ccc}b& a& 0\end{array}\right]}^T,$$

(5)

$$s_3^{\prime }={\left[\begin{array}{ccc}-b& a& 0\end{array}\right]}^T,$$

(6)

$$s_4^{\prime }={\left[\begin{array}{ccc}-b& -a& 0\end{array}\right]}^T.$$

(7)

The moment of inertia matrix of the quadrotor can be written as

$$J^{\prime }=\left[\begin{array}{ccc}4m_r{a}^2& 0& 0\\ 0& 4m_r{b}^2& 0\\ 0& 0& 4m_r\left(a^2+b^2\right)\end{array}\right],$$

(8)

where $m_r$ is the mass of the rotor.

The gravitational force acting on the quadrotor can be represented by

$$f_g={\left[\begin{array}{ccc}0& 0& -mg\end{array}\right]}^T,$$

(9)

where $m=m_b+4m_r$, $m_b$ is the mass of the main body of the quadrotor, and $g$ is the gravitational acceleration.

The thrust force and moment of the quadrotor can be calculated as follows:

$$f_p^{\prime }={\displaystyle \sum _{i=1}^4{f}_i^{\prime }},$$

(10)

$$m_p^{\prime }={\displaystyle \sum _{i=1}^4{\tilde{s}}_i^{\prime }{f}_i^{\prime }},$$

(11)

where

$$f_i^{\prime }={\left[\begin{array}{ccc}0& 0& k_T{\mathsf{\Omega }}_i^2\end{array}\right]}^T,$$

(12)

where $k_T$ is the thrust coefficient, with the tilde (~) indicating a $3\times 3$ skew-symmetric matrix of the vector.

The torque generated around the rotor axis can be expressed as

$$m_r^{\prime }={\left[\begin{array}{ccc}0& 0& {\displaystyle \sum _{i=1}^4{(-1)}^i{k}_R{\mathsf{\Omega }}_i^2}\end{array}\right]}^T,$$

(13)

where $k_R$ is the torque coefficient.

The aerodynamic drag force and moment are

$$f_a^{\prime }={\displaystyle \frac{1}{2}}\rho C_D‖W^{\prime }‖W^{\prime },$$

(14)

$$m_a^{\prime }={\displaystyle \frac{1}{2}}\rho C_M‖W^{\prime }‖W^{\prime },$$

(15)

where $C_D$ and $C_M$ are the aerodynamic coefficients, $\rho$ is the air density, $W^{\prime }=W_{wind}^{\prime }-v^{\prime }$ is the relative wind vector, $W_{wind}$ is the wind vector, and $v$ is the velocity vector of the quadrotor’s mass center.

2.1. Dynamics

The equations of motion of the quadrotor can be expressed in 6-DOF, including quaternion vectors, as follows [17]:

$$\dot{r}=v,$$

(16)

$$m\dot{v}=F,$$

(17)

$$\dot{q}={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}0& -{{\omega }^{\prime }}^T\\ {\omega }^{\prime }& -{\tilde{\omega }}^{\prime }\end{array}\right]q,$$

(18)

$$J^{\prime }{\dot{\omega }}^{\prime }+{\tilde{\omega }}^{\prime }{J}^{\prime }{\omega }^{\prime }=M^{\prime },$$

(19)

where

$$F=f_g+R{f}_p^{\prime }+R{f}_a^{\prime },$$

(20)

$$M^{\prime }=m_p^{\prime }+m_r^{\prime }+m_a^{\prime },$$

(21)

where ${\omega }^{\prime }$ is the angular velocity of the quadrotor between the E-frame and the B-frame, and $q={\left[\begin{array}{cc}{e}_0& e^T\end{array}\right]}^T$ is a unit quaternion between the E-frame and the B-frame, with $e={\left[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right]}^T$.

The orientation angles calculated for $R$, defined for the yaw–pitch–roll sequence, are expressed as

$$\begin{array}{l}\varphi =\arctan \left(2\left(e_2{e}_3+e_0{e}_1\right)/1-2\left({e_1}^2+{e_2}^2\right)\right)\\ \theta =\arcsin \left(2\left(e_0{e}_3-e_1{e}_2\right)\right)\\ \psi =\arctan \left(2\left(e_1{e}_2+e_0{e}_3\right)/1-2\left({e_2}^2+{e_3}^2\right)\right)\end{array},$$

(22)

where

$$R=(2e_0^2-1)I+2(e{e}^T+e_0\tilde{e}),$$

(23)

where $I$ is the three-dimensional identity matrix.

The desired trajectory can be defined as a function of time in the following form: $r_d\left(t\right)$, ${\dot{r}}_d\left(t\right)$, and ${\ddot{r}}_d\left(t\right)$. Then, the equations of translational error become

$${\dot{r}}_e=\dot{r}-{\dot{r}}_d=v_e,$$

(24)

$${\dot{v}}_e={\displaystyle \frac{1}{m}}F-{\ddot{r}}_d={\displaystyle \frac{1}{m}}{u}_1+D-{\ddot{r}}_d,$$

(25)

where

$$u_1=F-D,$$

(26)

$$D={\displaystyle \frac{1}{m}}R{f}_a^{\prime },$$

(27)

where $r_e=r-r_d$ is the position error, and $D$ is the disturbance force calculated by the aerodynamic force $f_a$.

The desired rotation matrix can be defined as [16]

$$R_d=(2d_0^2-1)I+2(d{d}^T+d_0\tilde{d}),$$

(28)

where $q_d={\left[\begin{array}{cc}{d}_0& d^T\end{array}\right]}^T$ is the desired quaternion, in which $d={\left[\begin{array}{ccc}{d}_1& d_2& d_3\end{array}\right]}^T$. In this study, the desired rotation matrix was separated to match the heading angle of the flight attitude with the direction of the trajectory. As seen in Figure 2, the rotation matrix was separated into the firs yaw direction and second roll–pitch direction to match the heading angle of the flight attitude and the direction of the trajectory. The desired rotation matrix and quaternion are defined as follows:

$$R_d=R_1\left(i\right)R_2\left(w\right),$$

(29)

where

$$i={\left[\begin{array}{cccc}\cos ({\psi }_d/2)& 0& 0& \sin ({\psi }_d/2)\end{array}\right]}^T,$$

(30)

$$w={\left[\begin{array}{cccc}{w}_0& w_1& w_2& 0\end{array}\right]}^T,$$

(31)

where $R_1$ is the desired first rotation matrix, which is a function of $i$, the desired first quaternion; $R_2$ is the desired second rotation matrix, which is a function of $w$, the desired second quaternion; and ${\psi }_d$ is a predefined variable that depends on the desired trajectory.

The quaternion error is defined as the relative rotation between the desired rotation and the actual rotation. The relative rotation matrix and quaternion error can be expressed as [16]

$$R_e={R_d}^{T}R,$$

(32)

$$q_e=\left[\begin{array}{cc}{d}_0& d^T\\ -d& d_{0}I-\tilde{d}\end{array}\right]q,$$

(33)

where $q_e={\left[\begin{array}{cc}{p}_0& p^T\end{array}\right]}^T$ is the quaternion error, with $p={\left[\begin{array}{ccc}{p}_1& p_2& p_3\end{array}\right]}^T$.

Accordingly, the dynamic equations of attitude error are expressed as

$${\dot{q}}_e={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}0& -{{\omega }_e^{\prime }}^T\\ {\omega }_e^{\prime }& -{\tilde{\omega }}_e^{\prime }\end{array}\right]q_e,$$

(34)

$${\dot{\omega }}_e^{\prime }=-{\dot{\omega }}_d^{\prime }+{J^{\prime }}^{-1}{u}_2+T+{J^{\prime }}^{-1}\left(-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_d^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_d^{\prime }\right),$$

(35)

where

$$u_2=M^{\prime }-J^{\prime }T,$$

(36)

$$T={J^{\prime }}^{-1}{m}_a^{\prime },$$

(37)

where ${\omega }_e^{\prime }={\omega }^{\prime }-{\omega }_d^{\prime }$ is the angular velocity error; ${\omega }_d^{\prime }={R_e}^T{\omega }_d$ is the desired angular velocity, with ${\omega }_d\left(t\right)$ and ${\dot{\omega }}_d\left(t\right)$ being functions of time depending on the desired path; and $T$ is the disturbance moment calculated with the aerodynamic moment $m_a^{\prime }$.

2.2. Wind Modeling

Herein, the wind profile is defined as the sum of the deterministic and stochastic wind models. We ignore the deterministic wind blowing in the vertical direction.

$$W_{wind}^{\prime }=\left[\begin{array}{c}{U}_w\\ V_w\\ W_w\end{array}\right]=R^T{v}_{wind}+w_{wind}^{\prime },$$

(38)

where

$$v_{wind}=R_{\gamma }\left[\begin{array}{c}W\left(z\right)\\ 0\\ 0\end{array}\right],$$

(39)

$$R_{\gamma }=\left[\begin{array}{ccc}\mathrm{c}\gamma & -s\gamma & 0\\ s\gamma & \mathrm{c}\gamma & 0\\ 0& 0& 1\end{array}\right],$$

(40)

where $W\left(z\right)$ is the deterministic horizontal wind speed, with $z$ representing the height above the ground; $w_{wind}^{\prime }$ is the stochastic wind model, which follows a white Gaussian distribution; $w_{wind}^{\prime }\sim N\left(0,\mathsf{\Phi }\right)$, with $\mathsf{\Phi }$ denoting a power spectral density (PSD); and $\gamma$ is the wind direction angle.

The speed of the deterministic wind is calculated using the power law method [23]:

$$W\left(z\right)=W\left(z_0\right){\left({\displaystyle \frac{z}{z_0}}\right)}^{\alpha },$$

(41)

where $W\left(z_0\right)$ signifies the mean wind speed at a reference height of $z_0$, and $\alpha$ is the wind shear coefficient.

The spectral Dryden turbulence model is used as the stochastic wind model. The PSD functions are [24]

$${\mathsf{\Phi }}_u\left(\mathsf{\Omega }\right)={{\sigma }_u}^2{\displaystyle \frac{2L_u}{\pi }}{\displaystyle \frac{1}{1+{\left(L_u\mathsf{\Omega }\right)}^2}},$$

(42)

$${\mathsf{\Phi }}_v\left(\mathsf{\Omega }\right)={{\sigma }_v}^2{\displaystyle \frac{2L_v}{\pi }}{\displaystyle \frac{1+12{\left(L_v\mathsf{\Omega }\right)}^2}{{\left(1+4{\left(L_v\mathsf{\Omega }\right)}^2\right)}^2}},$$

(43)

$${\mathsf{\Phi }}_w\left(\mathsf{\Omega }\right)={{\sigma }_w}^2{\displaystyle \frac{2L_w}{\pi }}{\displaystyle \frac{1+12{\left(L_w\mathsf{\Omega }\right)}^2}{{\left(1+4{\left(L_w\mathsf{\Omega }\right)}^2\right)}^2}},$$

(44)

where $\sigma$ is the turbulence intensity, $L$ is the scale length, and $\mathsf{\Omega }$ is the spatial frequency. The turbulence intensities and scale lengths for heights less than 300 m can be determined using the ESDU data curves [25,26]. Specifically, the turbulence intensities are

$${\displaystyle \frac{{\sigma }_u}{W}}=3.51{\left({\displaystyle \frac{1}{z}}\right)}^{0.01}-3.23,$$

(45)

$${\displaystyle \frac{{\sigma }_v}{W}}=2.13{\left({\displaystyle \frac{1}{z}}\right)}^{0.01}-1.95,$$

(46)

$${\displaystyle \frac{{\sigma }_w}{W}}=1.17{\left({\displaystyle \frac{1}{z}}\right)}^{0.01}-1.06,$$

(47)

The scale lengths are

$$L_u=280{\left({\displaystyle \frac{z}{z_0}}\right)}^{0.35},$$

(48)

$$L_v=140{\left({\displaystyle \frac{z}{z_0}}\right)}^{0.48},$$

(49)

$$L_w=0.35z.$$

(50)

The system parameters of a quadrotor are substituted with appropriate values by referring to

Table 1. System parameters of quadrotor.

Property		Value	Unit
Geometrical parameters	$a$	0.35	m
	$b$	0.55	m
Total mass	$m$	135	kg
Rotor mass	$m_r$	15	kg
Thrust coefficient	$k_T$	0.0047
Torque coefficient	$k_R$	3.93 × 10−4
Air density	$\rho$	1.225	kg·m−3
Aerodynamic coefficients	$C_D$	1.35
	$C_M$	0.135

and the references [27,28].

3. Controller Design

The control input required to follow a desired path can be defined as [27]

$$\left\{\begin{array}{c}{U}_1\\ U_2\\ U_3\\ U_4\end{array}\right\}=\left[\begin{array}{cccc}{k}_T& k_T& k_T& k_T\\ -a{k}_T& a{k}_T& a{k}_T& -a{k}_T\\ -b{k}_T& -b{k}_T& b{k}_T& b{k}_T\\ -k_R& k_R& -k_R& k_R\end{array}\right]\left\{\begin{array}{c}{\mathsf{\Omega }}_1^2\\ {\mathsf{\Omega }}_2^2\\ {\mathsf{\Omega }}_3^2\\ {\mathsf{\Omega }}_4^2\end{array}\right\},$$

(51)

where

$$\left[\begin{array}{c}0\\ 0\\ U_1\end{array}\right]={\displaystyle \frac{1}{k_T}}{R_d}^{-1}\left(u_1-f_g\right),$$

(52)

$$\left[\begin{array}{c}{U}_2\\ U_3\\ U_4\end{array}\right]=u_2.$$

(53)

where $u_1$ is the required force and $u_2$ is the required torque. To facilitate the controller design, the following conditions are assumed:

-

The magnitude of the quadrotor velocity does not exceed the actuator speed constraint:

$$‖v‖\le 15.$$

(54)

-

The magnitude of the wind velocity does not exceed the actuator speed constraint:

$$‖W_{wind}‖\le 20.$$

(55)

Due to hardware limitations, quadrotors have limits to the speeds they can achieve. Their applicability is also limited to specific environments, and operation in harsh environments is challenging for the following reason [29,30]: a wind speed of $‖W_{wind}‖=20$ is considered to be a gale on the Beaufort scale and can have severely destructive effects [31]. Therefore, the assumptions in Equations (54) and (55) are reasonable.

In this paper, two types of wind scenarios are analyzed to compare with the traditional PD control method and demonstrate the robustness of the proposed controller. Scenario 1 simulates an environment with a mean wind speed of 12 m/s, representative of strong winds, where the wind direction varies over time. The wind profile for this scenario is described as follows:

$$W\left(z_0\right)=12,$$

(56)

$$z_0=1,$$

(57)

$$\alpha =0.143,$$

(58)

$$\gamma =\left\{\begin{array}{ll}0\hfill & t<30\hfill \\ \pi /180\left(t-30\right)\hfill & 30\le t<50\hfill \\ \pi /9\hfill & 50\le t<60\hfill \\ \pi /180\left(-t+80\right)\hfill & 60\le t<90\hfill \\ -\pi /18\hfill & t\ge 90\hfill \end{array}\right..$$

(59)

In contrast, Scenario 2 models an environment with wind speeds that change abruptly and discretely over time, coupled with temporal variations in wind direction. The corresponding wind profile is defined by Equations (57), (58), (60), and (61).

$$W\left(z_0,t\right)=\left\{\begin{array}{ll}3\hfill & t<30\hfill \\ 10\hfill & 30\le t<50\hfill \\ 5\hfill & 50\le t<60\hfill \\ 6\hfill & 60\le t\le 100\hfill \end{array}\right.,$$

(60)

$$\gamma =\left\{\begin{array}{ll}0\hfill & t<30\hfill \\ \pi /90\left(t-30\right)\hfill & 30\le t<50\hfill \\ 2\pi /9\hfill & 50\le t<60\hfill \\ \pi /180\left(-t+90\right)\hfill & 60\le t\le 100\hfill \end{array}\right.,$$

(61)

Figure 3 shows the profile of the wind model for Scenarios 1 and 2. In Figure 3a for Scenario 1, the wind profile considers a constant mean wind speed of the magnitude corresponding to strong winds, wind speed variation with altitude, wind direction variation over time, and wind turbulence. This setup shows conditions encountered by quadrotors operating in rapidly shifting horizontal winds under high-wind scenarios.

In contrast, Figure 3b for Scenario 2 features a wind profile characterized by abrupt, discrete changes in wind speed, wind speed variation with altitude, wind direction variation over time, and wind turbulence. This configuration similarly depicts challenging conditions for quadrotors navigating rapidly changing horizontal winds in high-wind environments.

3.1. PD Controller

To compare the performance with the proposed controller, a PD controller was introduced. The PID control is the most widely used control approach for quadrotors, because of its simplicity. However, in the presence of persistent or fluctuating disturbances, an integrator can continuously accumulate the error caused by the disturbance. In environments with large and variable disturbances, such as wind, this can lead to excessive error accumulation, negatively affecting the system’s performance. A PD controller avoids this issue, providing stable flight even under such conditions, which is why the integral component of the PID controller is not used.

The required force and torque calculated based on the conventional PD control method can be expressed as [32,33]

$$u_1=m\left(-K_{p,p}{r}_e-K_{d,p}{v}_e\right),$$

(62)

$$u_2=J^{\prime }\left(-K_{p,a}\mathrm{sgn}\left(p_0\right)p-K_{d,a}{\omega }_e^{\prime }\right),$$

(63)

where $K_{p,p}=\mathrm{diag}\left(\left[p_{1,p},p_{2,p},p_{3,p}\right]\right)$, $K_{d,p}=\mathrm{diag}\left(\left[d_{1,p},d_{2,p},d_{3,p}\right]\right)$, $K_{p,a}=\mathrm{diag}\left(\left[p_{1,a},p_{2,a},p_{3,a}\right]\right)$, and $K_{d,a}=\mathrm{diag}\left(\left[d_{1,a},d_{2,a},d_{3,a}\right]\right)$.

The PD controller design parameters are selected through manual tuning to ensure performance, without relying on external references, as shown in

Table 2. PD controller design parameter.

Property		Value
Position controller gain	$p_{1,p}=p_{2,p}=p_{3,p}$	2
	$d_{1,p}=d_{2,p}=d_{3,p}$	8
Attitude controller gain	$p_{1,a}=p_{2,a}=p_{3,a}$	10
	$d_{1,a}=d_{2,a}=d_{3,a}$	5

.

3.2. RSMC Position Controller

A sliding surface can be defined as

$$\mathbf{\xi }=v_e+{\mathsf{\Lambda }}_p{r}_e,$$

(64)

where ${\mathsf{\Lambda }}_p=diag\left(\left[{\lambda }_{1,p},{\lambda }_{2,p},{\lambda }_{3,p}\right]\right)$, and the time derivative of $\mathbf{\xi }$ is

$$\dot{\mathbf{\xi }}={\dot{v}}_e+{\mathsf{\Lambda }}_p{\dot{r}}_e={\displaystyle \frac{1}{m}}{u}_1+D-{\ddot{r}}_d+\mathsf{\Lambda }{\dot{r}}_e.$$

(65)

The RSMC can be designed based on these assumptions, and the derivative of $\mathbf{\xi }$ can be expressed as

$$\dot{\mathbf{\xi }}=-K_{\eta ,p}\mathrm{sgn}(\mathbf{\xi })-K_{\sigma ,p}\mathbf{\xi }-K_{\delta ,p}\mathrm{sgn}(\mathbf{\xi })+D,$$

(66)

where $K_{\eta ,p}=\mathrm{diag}\left(\left[{\eta }_{1,p},{\eta }_{2,p},{\eta }_{3,p}\right]\right)$, $K_{\sigma ,p}=\mathrm{diag}\left(\left[{\sigma }_{1,p},{\sigma }_{2,p},{\sigma }_{3,p}\right]\right)$, $K_{\delta ,p}=\mathrm{diag}\left(\left[{\delta }_{1,p},{\delta }_{2,p},{\delta }_{3,p}\right]\right)$, and all the disturbances are uniformly bounded. Specifically, $\left|D_1\right|\le {\delta }_{1,p}$, $\left|D_2\right|\le {\delta }_{2,p}$, and $\left|D_3\right|\le {\delta }_{3,p}$, with known positive constants ${\delta }_{1,p}$, ${\delta }_{2,p}$, and ${\delta }_{3,p}$, respectively. Then, the Lyapunov function is introduced in the positive-definite form:

$$V_p={\displaystyle \frac{1}{2}}{\mathbf{\xi }}^T\mathbf{\xi },$$

(67)

The negativity of the time derivative of $V_p$ is confirmed even in the presence of disturbances:

$$\begin{array}{l}{\dot{V}}_p={\mathbf{\xi }}^T\dot{\mathbf{\xi }}\\ ={\mathbf{\xi }}^T\left({\displaystyle \frac{1}{m}}{u}_1+D-{\ddot{r}}_d+{\mathsf{\Lambda }}_p{\dot{r}}_e\right)\\ ={\mathbf{\xi }}^T\left(-K_{\eta ,p}\mathrm{sgn}(\mathbf{\xi })-K_{\sigma ,p}\mathbf{\xi }-K_{\delta ,p}\mathrm{sgn}(\mathbf{\xi })+D\right)\\ ={\mathbf{\xi }}^T\left(-K_{\eta ,p}\mathrm{sgn}(\mathbf{\xi })-K_{\sigma ,p}\mathbf{\xi }\right)+{\mathbf{\xi }}^T\left(D-K_{\delta ,p}\mathrm{sgn}(\mathbf{\xi })\right)\\ \le {\mathbf{\xi }}^T\left(-K_{\eta ,p}\mathrm{sgn}(\mathbf{\xi })-K_{\sigma ,p}\mathbf{\xi }\right)\le 0\end{array},$$

(68)

where

$$\begin{array}{l}D={\displaystyle \frac{1}{m}}R{f}_a^{\prime }\\ \le {\displaystyle \frac{1}{m}}\left({\displaystyle \frac{1}{2}}\rho C_D{‖W^{\prime }‖}^{2}I\right)\\ \le {\displaystyle \frac{1}{m}}\left({\displaystyle \frac{1}{2}}\rho C_D\left({W_{air}}^T{W}_{air}+2‖W_{air}‖‖v‖+v^{T}v\right)I\right)\\ \le K_{\delta ,p}\end{array}.$$

(69)

Therefore, the control system is asymptotically stable, and the dynamic states reach the sliding surface, even with the disturbances. The signum function in Equation (66) can be replaced with a hyperbolic tangent function to reduce chattering, and the required control force can be calculated as

$$u_1=m\left({\ddot{r}}_d-{\mathsf{\Lambda }}_p{\dot{r}}_e-K_{\eta ,p}\tanh (\mathbf{\xi })-K_{\sigma ,p}\mathbf{\xi }-K_{\delta ,p}\tanh (\mathbf{\xi })\right),$$

(70)

where

$$\tanh (\mathbf{\xi })={\displaystyle \frac{e^{\mathbf{\xi }}-e^{-\mathbf{\xi }}}{e^{\mathbf{\xi }}+e^{-\mathbf{\xi }}}}.$$

(71)

3.3. RSMC Attitude Controller

As in the position controller design, a sliding surface can be defined as

$$\zeta ={\omega }_e^{\prime }+{\mathsf{\Lambda }}_a\mathrm{sgn}\left(p_0\right)p,$$

(72)

where ${\mathsf{\Lambda }}_a=diag\left(\left[{\lambda }_{1,a},{\lambda }_{2,a},{\lambda }_{3,a}\right]\right)$, and the time derivative of $\zeta$ is

$$\begin{array}{l}\dot{\zeta }={\dot{\omega }}_e^{\prime }+{\mathsf{\Lambda }}_a\mathrm{sgn}\left(p_0\right)\dot{p}\\ =-{\dot{\omega }}_d^{\prime }+{J^{\prime }}^{-1}{u}_2+T+{J^{\prime }}^{-1}\left(-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_d^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_d^{\prime }\right)+{\displaystyle \frac{{\mathsf{\Lambda }}_a}{2}}\mathrm{sgn}\left(p_0\right)\left(p_{0}I+\tilde{p}\right){\omega }_e^{\prime }\end{array}$$

(73)

Then, the derivative of $\zeta$ can be expressed as

$$\dot{\zeta }=-K_{\eta ,a}\mathrm{sgn}(\zeta )-K_{\sigma ,a}\zeta -K_{\delta ,a}\mathrm{sgn}(\zeta )+T,$$

(74)

where $K_{\eta ,a}=\mathrm{diag}\left(\left[{\eta }_{1,a},{\eta }_{2,a},{\eta }_{3,a}\right]\right)$, $K_{\sigma ,a}=\mathrm{diag}\left(\left[{\sigma }_{1,a},{\sigma }_{2,a},{\sigma }_{3,a}\right]\right)$, and $K_{\delta ,a}=\mathrm{diag}\left(\left[{\delta }_{1,a},{\delta }_{2,a},{\delta }_{3,a}\right]\right)$, and all the disturbances are uniformly bounded. Specifically, $\left|T_1\right|\le {\delta }_{1,a}$, $\left|T_2\right|\le {\delta }_{2,a}$, and $\left|T_3\right|\le {\delta }_{3,a}$, with known positive constants ${\delta }_{1,a}$, ${\delta }_{2,a}$, and ${\delta }_{3,a}$, respectively. Then, the Lyapunov function is introduced in the positive-definite form:

$$V_a={\displaystyle \frac{1}{2}}{\zeta }^T\zeta ,$$

(75)

$V_a$ satisfies the Lyapunov stability, including the moment disturbances, as follows:

$$\begin{array}{l}{\dot{V}}_a={\zeta }^T\dot{\zeta }\\ ={\zeta }^T\left(-{\dot{\omega }}_d^{\prime }+{J^{\prime }}^{-1}{u}_2+T+{J^{\prime }}^{-1}\left(-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_d^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_e^{\prime }-{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_d^{\prime }\right)+{\displaystyle \frac{{\mathsf{\Lambda }}_a}{2}}\mathrm{sgn}\left(p_0\right)\left(p_{0}I+\tilde{p}\right){\omega }_e^{\prime }\right)\\ ={\zeta }^T\left(-K_{\eta ,a}\mathrm{sgn}(\zeta )-K_{\sigma ,a}\zeta -K_{\delta ,a}\mathrm{sgn}(\zeta )+T\right)\\ ={\zeta }^T\left(-K_{\eta ,a}\mathrm{sgn}(\zeta )-K_{\sigma ,a}\zeta \right)+{\zeta }^T\left(T-K_{\delta ,a}\mathrm{sgn}(\zeta )\right)\\ \le {\zeta }^T\left(-K_{\eta ,a}\mathrm{sgn}(\zeta )-K_{\sigma ,a}\zeta \right)\le 0\end{array},$$

(76)

where

$$\begin{array}{l}T={J^{\prime }}^{-1}{m}_a^{\prime }\\ \le {\displaystyle \frac{1}{m}}\left({\displaystyle \frac{1}{2}}\rho C_M{‖W^{\prime }‖}^{2}I\right)\\ \le {\displaystyle \frac{1}{m}}\left({\displaystyle \frac{1}{2}}\rho C_M\left({W_{air}}^T{W}_{air}+2‖W_{air}‖‖v‖+v^{T}v\right)I\right)\\ \le K_{\delta ,a}\end{array}.$$

(77)

Therefore, the control system is asymptotically stable with the disturbances. The required control torque can thus be calculated as

$$u_2=J^{\prime }\left(-K_{\eta ,a}\tanh (\mathbf{\xi })-K_{\sigma ,a}\zeta -K_{\delta ,a}\tanh (\mathbf{\xi })-{\displaystyle \frac{{\mathsf{\Lambda }}_a}{2}}\mathrm{sgn}\left(p_0\right)\left(p_{0}I+\tilde{p}\right){\omega }_e^{\prime }+{\dot{\omega }}_d^{\prime }\right)+{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_e^{\prime }+{\tilde{\omega }}_e^{\prime }{J}^{\prime }{\omega }_d^{\prime }+{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_e^{\prime }+{\tilde{\omega }}_d^{\prime }{J}^{\prime }{\omega }_d^{\prime }.$$

(78)

The controller design parameters are manually tuned and chosen for good performance without reference, as shown in

Table 3. SMC controller design parameter.

Property		Value
Position controller gain	${\eta }_{1,p}={\eta }_{2,p}={\eta }_{3,p}$	1
	${\sigma }_{1,p}={\sigma }_{2,p}={\sigma }_{3,p}$	0.5
	${\delta }_{1,p}={\delta }_{2,p}={\delta }_{3,p}$	0.2
	${\lambda }_{1,p}={\lambda }_{2,p}={\lambda }_{3,p}$	0.4
Attitude controller gain	${\eta }_{1,a}={\eta }_{2,a}={\eta }_{3,a}$	2
	${\sigma }_{1,a}={\sigma }_{2,a}={\sigma }_{3,a}$	2
	${\delta }_{1,a}={\delta }_{2,a}={\delta }_{3,a}$	0.5
	${\lambda }_{1,a}={\lambda }_{2,a}={\lambda }_{3,a}$	6

.

4. Results

Considering the initial conditions $r_0={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$, $v_0={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$, $q_0={\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]}^T$, and ${\omega }_0^{\prime }={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$, numerical simulations were conducted using the fourth-order Runge–Kutta method. Figure 4 shows the desired simulation path for verification, which follows Equations (79) and (80):

$$r_d\left(t\right)={\left[\begin{array}{ccc}R\cos {\omega }_{0}t& R\sin {\omega }_{0}t& z_d\end{array}\right]}^T,$$

(79)

$${\psi }_d\left(t\right)={\omega }_{0}t+{\displaystyle \frac{\pi }{2}},$$

(80)

where $R=60$, $z_d=5$, and ${\omega }_0=\pi /18$.

The desired angles ${\theta }_d$ and ${\varphi }_d$ are calculated according to Equations (22) and (29)–(31), given the desired heading angle ${\psi }_d$ value and the posture required to reach the desired position.

4.1. Scenario 1

Figure 5 shows the position and attitude results obtained through the path-tracking simulation in Scenario 1. The desired trajectory and attitude are shown with red lines, while the actual responses are represented by black lines for the proposed RSMC controller and green lines for the PD controller.

Despite the presence of wind disturbances, the RSMC controller demonstrates stable convergence and accurate trajectory tracking. It took approximately 10 s for the system to converge from the initial maximum error value, begin flying, and follow the designated trajectory.

Notably, the roll and pitch angles exhibit rapid convergence within 10 s. However, due to the strong wind influence, the pitch angle shows a slightly slower response, which subsequently causes a minor delay in the yaw angle, as reflected in the behavior of $\psi$ in Figure 5b. This delayed pitch response induces a minor lag in the yaw angle, as observed in the oscillatory behavior of $\psi$. In fact, excessive attitude response can overwhelm the quadrotor’s required input capacity, hindering its ability to execute precise maneuvers and thus reducing its utility.

In comparison, the PD controller exhibits inferior performance across all axes. The roll and pitch angles controlled by the PD controller show more oscillations and slower convergence, reflecting reduced robustness to disturbances. For the yaw angle, the PD controller exhibits slightly slower convergence and more noticeable lag compared to the RSMC controller.

Overall, the RSMC controller outperforms the PD controller in terms of both stability and tracking accuracy. The RSMC controller effectively suppresses oscillations and ensures smooth convergence, whereas the PD controller’s larger tracking errors and oscillatory behavior could compromise the quadrotor’s stability in practical applications. These results emphasize the robustness and effectiveness of the quaternion-based RSMC controller in achieving precise and stable trajectory tracking under extreme wind scenarios.

Figure 6 shows the disturbance force and moment generated by the wind in Scenario 1. Figure 6a shows the disturbance forces in the x, y, and z directions, while Figure 6b shows the corresponding disturbance moments about the roll, pitch, and yaw axes. As mentioned earlier, the wind profile is defined relative to the quadrotor’s body frame.

The disturbance forces shown in Figure 6a are calculated with respect to the E-frame and fluctuate depending on the quadrotor’s position and attitude. These fluctuations reflect the interaction between the wind and the quadrotor’s motion. Notably, these forces remain bounded, confirming that the wind disturbances do not exceed the designed limits of the system.

On the other hand, the disturbance moments in Figure 6b are defined in the quadrotor’s body frame. As previously noted, the wind profile is defined relative to the quadrotor’s body frame; consequently, the disturbance moments depicted in Figure 6b tend to resemble those shown in Figure 3a. The magnitudes of the moment disturbances are also bounded to ensure the robustness of the controller.

To ensure robust performance against such wind disturbances, the controller parameters must be appropriately selected. The values of the controller parameters ${\delta }_{1,p}$, ${\delta }_{2,p}$, ${\delta }_{3,p}$, ${\delta }_{1,a}$, ${\delta }_{2,a}$, and ${\delta }_{3,a}$ must individually exceed the element values of $D$ and $T$ calculated from the disturbance. This ensures that the control system can effectively counteract disturbances while maintaining stability and precision in both trajectory tracking and attitude control.

Figure 7 shows the variations in the angular velocities of the rotors in response to the desired inputs. The responses indicate that after the initial position error converged, the system reached a stable equilibrium point despite the wind disturbances depicted in Figure 6. In Figure 7a, it is noted that the maximum angular velocity remained below 600 rad/s throughout the entire simulation duration, which is within the feasible specifications of the quadrotor [34].

However, Figure 7b shows a significantly higher peak rotor speed requirement for the PD controller, despite its inferior performance in trajectory tracking. This indicates that the PD controller demands greater input from the actuators to follow the desired trajectory, resulting in higher rotor speeds. In contrast, the RSMC controller achieves superior performance in trajectory tracking while requiring lower rotor speeds, highlighting its greater efficiency and reduced energy consumption.

4.2. Scenario 2

Figure 8 shows the position and attitude results obtained through the path-tracking simulation in Scenario 2. Similar to the results of Scenario 1, it takes approximately 10 s for the system to converge. In Figure 8b, the control of roll and pitch angles in Scenario 2 appears to respond more slowly than in Scenario 1 (Figure 5b). However, this behavior is a result of the controller handling abrupt disturbance moments caused by the discrete changes in wind speed under highly extreme conditions, as evidenced in Figure 9.

To address the slower convergence, increasing the controller gain could achieve faster results. However, this would lead to sudden increases in rotor thrust, potentially exceeding the hardware limits or reducing energy efficiency. The slower convergence of roll and pitch observed in Figure 8b is thus an intentional design choice, ensuring stable and efficient operation under significant wind disturbances. The controller settings were optimized to gradually reduce oscillation amplitude while maintaining system stability.

Additionally, Figure 8b shows a slightly longer delay in the yaw response compared to Scenario 1. This delay reflects the challenges of operating under rapidly changing wind conditions, where the controller prioritizes stability and robustness. Future work may explore adaptive gain tuning or advanced control strategies to enhance convergence speed for all attitude angles without compromising actuator performance.

Regarding the PD controller, similar to the results observed in Scenario 1, the proposed RSMC controller shows superior performance compared to the PD controller. As shown in Figure 8b, the PD controller exhibits a more sensitive response to disturbances, resulting in larger errors and more pronounced oscillations compared to the RSMC controller. Additionally, the lag in the yaw angle is also more pronounced with the PD controller.

Figure 9 shows the disturbance forces and moments generated by the wind in Scenario 2. Similar to Figure 6, Figure 9a presents the disturbance forces in the x, y, and z directions, while Figure 9b illustrates the corresponding disturbance moments about the roll, pitch, and yaw axes.

The wind profile in Scenario 2 is characterized by discrete changes in wind speed, which is clearly reflected in the results shown in Figure 9. With both the wind profile and disturbance moments defined in the B-frame, the disturbance moments in Figure 9b exhibit a pattern similar to those in Figure 3b.

As designed, the wind-induced disturbance forces and moments acting on the quadrotor stay within the controller’s robustness limits. This demonstrates the control system’s ability to mitigate disturbances effectively while ensuring stable and accurate performance in both trajectory tracking and attitude control.

Figure 10 illustrates the variations in the angular velocities of the rotors in response to the desired inputs in Scenario 2. Similar to the results of Scenario 1, the responses indicate that after the initial position error converged, the system achieved a stable equilibrium point despite the presence of wind disturbances, shown in Figure 10a.

Notably, the maximum angular velocity remained below 600 rad/s, and after convergence, it decreased to values below 300 rad/s. Consistent with the results of Scenario 1, the rotor velocities remained within the feasible operational specifications of the quadrotor throughout the entire simulation [34].

Similar to the results in Scenario 1, it is observed that the PD controller requires a peak rotor angular velocity approximately twice as high as that of the RSMC controller. This clearly demonstrates that the proposed RSMC controller outperforms the PD controller in both flight performance and energy efficiency.

5. Discussions

To verify the robustness and performance of the RSMC controller proposed in this paper, it is compared with the conventional PD control algorithm under two distinct wind scenarios. Scenario 1 simulates an environment with a constant wind speed of 12 m/s and changing wind directions, while Scenario 2 represents a situation where the wind speed fluctuates discretely, and the wind direction also varies. Both scenarios represent extreme wind conditions.

Simulations were carried out in the MATLAB R2023a environment, and the control and dynamic parameters used are provided. The quadrotor is commanded to stabilize its attitude and follow the desired path. The proposed RSMC control algorithm, which leverages a quaternion-based representation, offers several advantages over traditional Euler angle-based controllers. By avoiding the singularity issues inherent in Euler angles (gimbal lock), the quaternion-based controller ensures robust and precise attitude control even under significant disturbances. This allows the quadrotor to maintain stability and accurately track the desired trajectory despite the presence of wind disturbances.

However, due to the simulation being conducted under strong wind conditions, a relatively slow response time of the Euler angles is observed. While increasing the controller gain in the simulation could easily address this issue, excessively high gains may lead to hardware stress or exceed actuator limitations. This consideration is critical for real-world testing, so the controller was carefully tuned to ensure gradual oscillation reduction without overloading the actuators. Future work may explore adaptive gain tuning or nonlinear control strategies to achieve faster convergence while mitigating abrupt thrust changes.

In contrast, the PD control algorithm tracks the trajectory with larger errors and oscillations, which could affect the stability and performance of the quadrotor in practical applications. Overall, it is clear that the proposed algorithm outperforms the PD control method in terms of tracking accuracy, system performance, and stability

By incorporating a quaternion-based controller, the proposed RSMC algorithm not only addresses the stability and accuracy issues observed in the PD controller but also demonstrates enhanced robustness against extreme wind disturbances. This approach provides a solid foundation for future research and practical implementation in challenging flight environments.

6. Conclusions

This study presented a quaternion-based RSMC for quadrotors operating under significant wind disturbances. The primary goal was to enhance the stability and performance of quadrotors during flight in unpredictable windy environmental conditions. The proposed approach involved detailed modeling of the quadrotor dynamics and wind environment, along with controller design and dynamic analysis through simulation.

The use of quaternion representation eliminated the singularity issues inherent in Euler angle methods, thereby ensuring more reliable and efficient control of the quadrotor’s orientation. Based on the Lyapunov stability theory, we rigorously proved the convergence and stability of the proposed RSMC system. This verifies that the quadrotor would maintain its stability and follow the desired path accurately, even in the presence of external disturbances.

Through numerical simulations, we demonstrated that the proposed controller operates effectively under adverse wind conditions. In both Scenario 1, which involved turbulence with an average wind speed of 12 m/s and changing wind directions, and Scenario 2, featuring abrupt and discrete changes in wind speed along with temporal variations in wind direction, the quadrotor successfully maintained its attitude and tracked the desired trajectory. Additionally, a comparative analysis with the conventional PD controller highlighted the superior performance of the proposed RSMC. The RSMC exhibited faster convergence, more accurate trajectory tracking, and reduced rotor speed requirements, which directly translates into improved energy efficiency and operational reliability.

Future research should focus on experimental validation of the proposed control strategy in real-world flight tests. Furthermore, the robustness of the system can be enhanced by incorporating adaptive mechanisms that can dynamically adjust controller parameters based on varying environmental conditions. A robust controller should be able to handle not only wind disturbances, but also other unpleasant situations that may arise, and it is important to use the controller to deal with them safely. As further research, a robust controller should be considered for situations such as the quadrotor motor stalling in windy conditions.

In conclusion, the proposed quaternion-based RSMC offers a promising solution for improving quadrotor performance in wind-affected environments. Our findings have the potential to significantly elevate the operational reliability and efficiency of unmanned aerial vehicles in real-world applications in the future.