Adaptive Terminal Sliding Mode Control for a Quadrotor System with Barrier Function Switching Law

Abstract

This study presents a novel finite-time robust control framework for quadrotor systems subjected to model uncertainties and unknown external disturbances. A fast terminal sliding mode (FTSM) manifold is first constructed to achieve finite-time convergence of tracking errors. To address the challenges posed by uncertain system dynamics, a radial basis function neural network (RBFNN) is integrated for real-time approximation of unknown nonlinearities. In addition, an adaptive gain regulation mechanism based on a barrier Lyapunov function (BLF) is developed to ensure boundedness of system trajectories while enhancing robustness without requiring prior knowledge of disturbance bounds. The proposed control scheme guarantees finite-time stability, strong robustness, and precise trajectory tracking. Numerical simulations substantiate the efficacy and superiority of the proposed method in comparison with existing control approaches.

1. Introduction

In recent years, quadrotor unmanned aerial vehicles (UAVs) have garnered significant attention across various industries due to their compact structure and high maneuverability [1,2,3]. However, the inherent strong nonlinearity, high coupling, and time-varying dynamics of the quadrotor systems [4], combined with the uncertainties and unknown external disturbances commonly encountered in practical applications, pose significant challenges for achieving precise control [5,6]. Traditional control methods, such as proportional–integral–derivative (PID) control [7] and linear quadratic regulator (LQR) [8], often struggle to deliver satisfactory performance in the presence of model uncertainties and disturbances.

Sliding mode control (SMC) has emerged as an effective approach due to its strong robustness against uncertainties [9,10,11,12,13,14]. To tackle the challenge of quadrotor UAVs’ high sensitivity due to small moments of inertia and external disturbances, a synthesized SMC approach has been introduced [15]. This method integrates a novel adaptive law, improving convergence speed and robustness while efficiently mitigating external disturbances. In [16], to address the challenges of implementing adaptive SMC in quadrotors for real-world scenarios, a complete control architecture was proposed. In [17], to address the challenges of model uncertainties and unknown external disturbances in quadrotor UAVs, an event-triggered control technique based on adaptive barrier function higher-order global SMC was proposed. This approach aims to extend the sampling interval of control signal updates while ensuring system stability and improving overall efficiency. Building on fuzzy control’s well-documented capability for uncertainty handling [18], an innovative neuro-fuzzy sliding mode control scheme specifically designed for quadrotor UAV systems was introduced in [19]. However, conventional SMC methods may suffer from chattering and sensitivity to unknown dynamics, which can degrade overall control performance [20,21], this motivated us to conduct further research.

To enhance the robustness and adaptability of SMC, adaptive neural network-based control strategies have been extensively explored. RBFNNs have demonstrated excellent approximation capabilities, allowing for real-time compensation for system uncertainties and identification of unknown dynamics [22,23,24,25,26]. In [27], to address the attitude stabilization problem of UAVs, such as quadrotors, under uncertain inertia, external disturbances, and actuator faults within a predefined time, an adaptive RBFNN control approach was proposed. This approach ensures state convergence within a predefined time while effectively estimating system uncertainties to enhance control performance. In [28], to address the challenge of uncertain disturbances, all disturbances were treated as a lumped disturbance, and an RBFNN was proposed to compensate for the output of the controllers. In [29], to address the challenge of maintaining attitude control performance in quadrotor helicopters under varying operating conditions and time-varying disturbances, an RBFNN was proposed. The RBFNN, trained using gradient descent-based supervised learning, is implemented as a supervisor neural network controller to enhance robustness.

Furthermore, incorporating a BLF into the control design ensures that system states remain within predefined safe limits, thereby enhancing stability and robustness [30,31,32,33]. In [34], to address the challenge of obstacle avoidance in both the position and attitude control of quadrotor UAVs, a control scheme based on BLF was proposed. By incorporating BLFs into the backstepping process, the quadrotor UAV system was decoupled into position and attitude control subsystems, ensuring effective obstacle avoidance while maintaining trajectory tracking performance. In [35], to address the challenges of UAV tracking control in the presence of actuator faults and external disturbances, an advanced barrier function-based prescribed performance SMC method was proposed. The approach incorporates BLF to counteract the effects of actuator faults and disturbances, enhancing the UAV’s tracking robustness. The above results all indicate that the barrier function is beneficial for enhancing the robustness of the controller. However, how to fully leverage the advantages of both fast terminal sliding mode (TSMC) and linear sliding mode while addressing system uncertainties, and further incorporate the BLF to improve controller robustness, remains an open problem that warrants further investigation.

Motivated by these considerations, this paper proposes an adaptive neural barrier-based TSMC (BTSMC) strategy, with the following key contributions:

(i) A fast TSMC is developed to guarantee finite-time convergence of tracking errors, thereby significantly improving transient response and control precision.

(ii) An adaptive RBFNN compensator is designed to perform the online approximation of an unknown function, effectively addressing system uncertainties.

(iii) A BLF-based adaptive gain adjustment strategy is proposed to dynamically regulate control gains, improving the robustness and stability of the overall control system under varying conditions.

The remainder of this paper is structured as follows: Section 2 presents the mathematical modeling of the quadrotor system. Section 3 elaborates on the proposed control scheme, including sliding mode surface design, neural network compensation, and the BLF-based adaptive mechanism. Section 4 provides simulation results and performance analysis. Finally, Section 5 concludes the paper and discusses potential future research directions.

2. System Description

The Newton–Euler formulation, underpinned by assumptions of rigid-body mechanics, enables the derivation of quadrotor UAV dynamic equations expressed as follows [36,37]:

$$\left\{\begin{array}{c}m\ddot{\varkappa }={\mathit{P}}_{\mathit{f}}+{\mathit{P}}_{\mathit{d}}+{\mathit{P}}_{\mathit{g}}\hfill \\ \mathit{V}{{\mathit{R}}_{\mathit{v}}}^T=-{{\mathit{R}}_{\mathit{v}}}^T\mathit{V}{\mathit{R}}_{\mathit{v}}+{\mathit{Q}}_{\mathit{f}}+{\mathit{Q}}_{\mathit{a}}+{\mathit{Q}}_{\mathit{g}}\hfill \end{array}\right.$$

(1)

where $\varkappa$ represents the position vector from the center of inertia to the center of mass. The system mass is a positive real number $m\in {\mathbb{R}}^{+}$.

The inertia matrix $\mathit{V}\in {\mathbb{R}}^{3\times 3}$ is a constant symmetric positive-definite tensor, formulated as

$$\mathit{V}=\left[\begin{array}{ccc}{J}_x& 0& 0\\ 0& J_y& 0\\ 0& 0& J_z\end{array}\right]$$

(2)

where $J_i(i=x,y,z)$ denote the inertia values corresponding to the x, y, and z axes. Meanwhile, ${\mathit{R}}_{\mathit{v}}$ signifies the angular velocity of the quadrotor UAV. The expression is given as follows:

$${\mathit{R}}_{\mathit{v}}=\left[\begin{array}{ccc}1& 0& -sin\theta \\ 0& cos\varphi & sin\varphi cos\theta \\ 0& -sin\varphi & cos\varphi cos\theta \end{array}\right]\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]$$

(3)

where $\theta$, $\psi$, and $\Phi$ represent the pitch, yaw, and roll angles, respectively. ${\mathit{P}}_{\mathit{f}}$ indicates the total force defined as

$${\mathit{P}}_{\mathit{f}}=\left[\begin{array}{c}cos\varphi cos\psi sin\theta +sin\varphi sin\psi \\ cos\varphi sin\theta sin\psi -sin\varphi cos\psi \\ cos\theta cos\varphi \end{array}\right]{\displaystyle \sum _{i=1}^4}{P}_i$$

(4)

Meanwhile, $P_i=T_p{\omega }_i^2$, where $T_p$ is associated with lift and ${\omega }_i$ stands for the angular velocity of the four rotors. The composite force ${\mathit{P}}_{\mathit{d}}$ acting along the X, Y, and Z axes is given by

$${\mathit{P}}_{\mathit{d}}=\left[\begin{array}{ccc}-T_{fdx}& 0& 0\\ 0& -T_{fdy}& 0\\ 0& 0& -T_{fdz}\end{array}\right]\dot{{\mathit{R}}_{\mathit{v}}}$$

(5)

where $T_{fdx}>0$, $T_{fdy}>0$, and $T_{fdz}>0$ denote the translation drag coefficients. The gravity matrix ${\mathit{P}}_{\mathit{g}}$ is given by

$${\mathit{P}}_{\mathit{g}}=\left[\begin{array}{c}0\\ 0\\ -mg\end{array}\right]$$

(6)

where g stands for the gravitational acceleration, while ${\mathit{Q}}_{\mathit{f}}$ signifies the torque produced by the four rotors, which can be expressed as

$${\mathit{Q}}_{\mathit{f}}=\left[\begin{array}{c}n(P_3-P_1)\\ n(P_4-P_2)\\ H_M({\omega }_1^2-{\omega }_2^2+{\omega }_3^2-{\omega }_4^2)\end{array}\right]$$

(7)

where n is the distance from the propeller axis to the UAV center, $H_M$ is the drag coefficient, and ${\mathit{Q}}_{\mathit{a}}$ represents the aerodynamic friction torques, expressed as

$${\mathit{Q}}_{\mathit{a}}=\left[\begin{array}{ccc}{T}_{fax}& 0& 0\\ 0& T_{fay}& 0\\ 0& 0& T_{faz}\end{array}\right]{\parallel {\mathit{R}}_{\mathit{v}}\parallel }^2$$

(8)

In this context, $T_{fax}, T_{fay}$, and $T_{faz}$ represent the aerodynamic friction coefficients. Meanwhile, ${\mathit{Q}}_{\mathit{g}}$ signifies the resultant torques due to the gyroscopic effect, which is expressed as follows:

$${\mathit{Q}}_{\mathit{g}}=\sum _{i=1}^4{{\mathit{R}}_{\mathit{v}}}^T{V}_r\left[\begin{array}{c}0\\ 0\\ {(-1)}^{i+1}{\omega }_i\end{array}\right]$$

(9)

where $V_r$ represents the rotor inertia. The relationship between the control law and the propeller’s angular velocity is given by

$$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]=\left[\begin{array}{cccc}{T}_p& T_p& T_p& T_p\\ -T_p& 0& T_p& 0\\ 0& -T_p& 0& T_p\\ H_M& -H_M& H_M& -H_M\end{array}\right]\left[\begin{array}{c}{\omega }_1^2\\ {\omega }_2^2\\ {\omega }_3^2\\ {\omega }_4^2\end{array}\right]$$

(10)

In this context, $u_i {\omega }_i(i=1,2,3,4)$ and represents the control inputs and the angular velocity, respectively. They are defined as

$$\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ {\omega }_3\\ {\omega }_4\end{array}\right]={\left({\left[\begin{array}{cccc}{T}_p& T_p& T_p& T_p\\ -T_p& 0& T_p& 0\\ 0& -T_p& 0& T_p\\ H_M& -H_M& H_M& -H_M\end{array}\right]}^{-1}\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]\right)}^{{\displaystyle \frac{1}{2}}}$$

(11)

The dynamic model of a quadrotor UAV can be described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\ddot{x}={\displaystyle \frac{1}{m}}[-T_{fax}\dot{x}+(cos\varphi sin\theta cos\psi +sin\varphi sin\psi )u_1]\hfill \\ \ddot{y}={\displaystyle \frac{1}{m}}[-T_{fay}\dot{y}+(cos\varphi sin\theta sin\psi -sin\varphi cos\psi )u_1]\hfill \\ \ddot{z}={\displaystyle \frac{1}{m}}[-T_{faz}\dot{z}+(cos\varphi cos\theta )u_1]-g\hfill \\ \ddot{\varphi }={\displaystyle \frac{1}{I_x}}[(I_y-I_z)\dot{\psi }\dot{\theta }-T_{fax}{\dot{\varphi }}^2-V_r\overline{\Xi }\dot{\theta }+n{u}_2]\hfill \\ \ddot{\theta }={\displaystyle \frac{1}{I_y}}[(I_z-I_x)\dot{\psi }\dot{\varphi }-T_{fay}{\dot{\theta }}^2+V_r\overline{\Xi }\dot{\varphi }+n{u}_3]\hfill \\ \ddot{\psi }={\displaystyle \frac{1}{I_z}}[(I_x-I_y)\dot{\varphi }\dot{\theta }-T_{faz}{\dot{\psi }}^2+H_M{u}_4]\hfill \end{array}\right.\end{array}$$

(12)

where $\overline{\Xi }={\omega }_1-{\omega }_2+{\omega }_3-{\omega }_4.$

Define the system states ${\dot{x}}_i={\left[\dot{\varphi },\ddot{\varphi },\dot{\theta },\ddot{\theta },\dot{\psi },\ddot{\psi },\dot{z},\ddot{z}\right]}^T(i=1\dots 8)$. From Equation (12), we have

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=f_1{x}_4{x}_6+f_2{x}_2^2+f_3\overline{\Xi }{x}_4+b_1{u}_2+d_{\varphi }\left(t\right)\hfill \\ {\dot{x}}_3=x_4\hfill \\ {\dot{x}}_4=f_4{x}_2{x}_6+f_5{x}_4^2+f_6\overline{\Xi }{x}_2+b_2{u}_3+d_{\theta }\left(t\right)\hfill \\ {\dot{x}}_5=x_6\hfill \\ {\dot{x}}_6=f_7{x}_2{x}_4+f_8{x}_6^2+b_3{u}_4+d_{\psi }\left(t\right)\hfill \\ {\dot{x}}_7=x_8\hfill \\ {\dot{x}}_8=f_9{x}_8-g+{\displaystyle \frac{cos{x}_{1}cos{x}_3}{m}}{u}_1+d_z\left(t\right)\hfill \end{array}\right.$$

(13)

Here, $u_i(i=1,2,3,4)$ represent the control inputs, while $n_j(j=\varphi ,\theta ,\psi ,z)$ denote the external disturbances. The parameters are defined as

$$\begin{array}{cccccc}\hfill f_1& ={\displaystyle \frac{I_y-I_z}{I_x}},\hfill & \hfill f_2& ={\displaystyle \frac{-T_{fax}}{I_x}},\hfill & \hfill f_3& ={\displaystyle \frac{-V_r}{I_x}},\hfill \\ \hfill f_4& ={\displaystyle \frac{I_z-I_x}{I_y}},\hfill & \hfill f_5& ={\displaystyle \frac{-T_{fay}}{I_y}},\hfill & \hfill f_6& ={\displaystyle \frac{V_r}{I_y}},\hfill \\ \hfill f_7& ={\displaystyle \frac{I_x-I_y}{I_z}},\hfill & \hfill f_8& ={\displaystyle \frac{-T_{faz}}{I_z}},\hfill & \hfill f_9& ={\displaystyle \frac{-T_{fdz}}{m}},\hfill \\ \hfill b_1& ={\displaystyle \frac{n}{I_x}},\hfill & \hfill b_2& ={\displaystyle \frac{n}{I_y}},\hfill & \hfill b_3& ={\displaystyle \frac{H_M}{I_z}}.\hfill \end{array}$$

In order to facilitate the subsequent design of the controller, we further rewrite (13) as follows:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=f_{\varphi }+g_{\varphi }{u}_{\varphi }+d_{\varphi }\left(t\right)\hfill \\ {\dot{x}}_3=x_4\hfill \\ {\dot{x}}_4=f_{\theta }+g_{\theta }{u}_{\theta }+d_{\theta }\left(t\right)\hfill \\ {\dot{x}}_5=x_6\hfill \\ {\dot{x}}_6=f_{\psi }+g_{\psi }{u}_{\psi }+d_{\psi }\left(t\right)\hfill \\ {\dot{x}}_7=x_8\hfill \\ {\dot{x}}_8=f_z+g_z{u}_z+d_z\left(t\right)\hfill \end{array}\right.$$

(14)

where $f_{\varphi }=f_1{x}_4{x}_6+f_2{x}_2^2+f_3\overline{\Xi }{x}_4$, $g_{\varphi }=g_1$, $u_{\varphi }=u_2$,$f_{\theta }=f_4{x}_2{x}_6+f_5{x}_4^2+f_6\overline{\Xi }{x}_2$, $g_{\theta }=g_2$, $u_{\theta }=u_3$, $f_{\psi }=f_7{x}_2{x}_4+f_8{x}_6^2$, $g_{\psi }=g_3$, $u_{\psi }=u_4$, $f_z=f_9{x}_8-g$, $g_z={\displaystyle \frac{cos{x}_{1}cos{x}_3}{m}}$, and $u_z=u_1$.

Lemma 1

([38]). For any constants q and p, the following inequality holds:

$$qp\le {\displaystyle \frac{{\xi }^{\delta }}{\delta }}{|q|}^{\delta }+{\displaystyle \frac{1}{\iota {\xi }^{\iota }}}{|p|}^{\iota }$$

(15)

where $\xi >0$, $\delta >0$, $\iota >0$, and $(\delta -1)(\iota -1)=1$.

Lemma 2

([39]). For a Lyapunov function $V_L\left(t\right)>0$, if there exist $K_v<0$ and $\alpha >0$, such that the following is satisfied:

$${\dot{V}}_L\le -K_v{V}_L^{\alpha }$$

(16)

then the Lyapunov function $V_L\left(t\right)$ converges to zero neighbor in finite time.

3. Control Design

Define the tracking errors and their derivatives for the following attitude angles and altitudes:

$$\left\{\begin{array}{c}{e}_{\varphi }=x_1-x_{\varphi d}\hfill \\ {\dot{e}}_{\varphi }=x_2-{\dot{x}}_{\varphi d}\hfill \end{array}\right.\left\{\begin{array}{c}{e}_{\theta }=x_1-x_{\theta d}\hfill \\ {\dot{e}}_{\theta }=x_2-{\dot{x}}_{\theta d}\hfill \end{array}\right.\left\{\begin{array}{c}{e}_{\psi }=x_1-x_{\psi d}\hfill \\ {\dot{e}}_{\psi }=x_2-{\dot{x}}_{\psi d}\hfill \end{array}\right.\left\{\begin{array}{c}{e}_z=x_1-x_{zd}\hfill \\ {\dot{e}}_z=x_2-{\dot{x}}_{zd}\hfill \end{array}\right.$$

(17)

where $x_{\varphi d}$, $x_{\theta d}$, $x_{\psi d}$, and $x_{zd}$ correspond to the reference attitude angles $\varphi ,\theta ,\psi$ and altitude z, respectively, and these reference trajectories satisfy the Lipschitz condition.

Based on [14], the nonlinear terminal sliding surface is designed as follows:

$$\left\{\begin{array}{c}{s}_{\varphi }=a_{\varphi }{e}_{\varphi }+b_{\varphi }{exp}^{-{\lambda }_{\varphi }t}{e}_{\varphi }^{(1-2{\beta }_{\varphi })}+{\dot{e}}_{\varphi }\hfill \\ s_{\theta }=a_{\theta }{e}_{\theta }+b_{\theta }{exp}^{-{\lambda }_{\theta }t}{e}_{\theta }^{(1-2{\beta }_{\theta })}+{\dot{e}}_{\theta }\hfill \\ s_{\psi }=a_{\psi }{e}_{\varphi }+b_{\psi }{exp}^{-{\lambda }_{\psi }t}{e}_{\psi }^{(1-2{\beta }_{\psi })}+{\dot{e}}_{\psi }\hfill \\ s_z=a_z{e}_z+b_z{exp}^{-{\lambda }_{z}t}{e}_z^{(1-2{\beta }_z)}+{\dot{e}}_z\hfill \end{array}\right.$$

(18)

where $a_i,b_i,i=\varphi ,\theta ,\psi ,z$ are positive constants. ${\lambda }_i$ and $0<{\beta }_i<1$ are designed constants that are related to the convergence time of the sliding mode surface. When $2{\beta }_i{a}_i-{\lambda }_i>0,i=\varphi ,\theta ,\psi ,z$, the tracking errors can converge to zero in finite time, respectively, and the convergence time is given by

$$T_i={\displaystyle \frac{ln[1+\left({exp}^{2{\beta }_i{a}_{i}t}{V}_i^{\beta }\left(0\right)\right)/{\nu }_i]}{2{\beta }_i{a}_i-{\lambda }_i}},i=\varphi ,\theta ,\psi ,z$$

(19)

where ${\nu }_i=2^{(1-{\beta }_i)}{\beta }_i{b}_i/\left(2{\beta }_i{a}_1-{\lambda }_i\right)$.

Remark 1.

For the proposed sliding surface, the exponential term ${exp}^{-{\lambda }_{i}t}$ will converge to zero over time. As a result, the sliding surface will switch from a terminal sliding mode surface to a linear sliding mode surface. This design effectively combines the fast convergence of terminal sliding mode with the strong robustness of linear sliding mode.

Taking the derivative of (18) yields

$$\left\{\begin{array}{c}{\dot{s}}_{\varphi }=a_{\varphi }{\dot{e}}_{\varphi }+b_{\varphi }{\exp }^{-{\lambda }_{\varphi }t}{e_{\varphi }}^{-2{\beta }_{\varphi }}[(1-2{\beta }_{\varphi }){\dot{e}}_{\varphi }-{\lambda }_{\varphi }{e}_{\varphi }]+{\ddot{e}}_{\varphi }\hfill \\ {\dot{s}}_{\theta }=a_{\theta }{\dot{e}}_{\theta }+b_{\theta }{\exp }^{-{\lambda }_{\theta }t}{e_{\theta }}^{-2{\beta }_{\theta }}[(1-2{\beta }_{\theta }){\dot{e}}_{\theta }-{\lambda }_{\theta }{e}_{\theta }]+{\ddot{e}}_{\theta }\hfill \\ {\dot{s}}_{\psi }=a_{\psi }{\dot{e}}_{\psi }+b_{\psi }{\exp }^{-{\lambda }_{\psi }t}{e_{\psi }}^{-2{\beta }_{\psi }}[(1-2{\beta }_{\psi }){\dot{e}}_{\psi }-{\lambda }_{\psi }{e}_{\psi }]+{\ddot{e}}_{\psi }\hfill \\ {\dot{s}}_z=a_z{\dot{e}}_z+b_z{\exp }^{-{\lambda }_{z}t}{e_z}^{-2{\beta }_z}[(1-2{\beta }_{\varphi }){\dot{e}}_z-{\lambda }_z{e}_z]+{\ddot{e}}_z\hfill \end{array}\right.$$

(20)

Define ${\zeta }_i=b_i{\exp }^{-{\lambda }_{i}t}{e_i}^{-2{\beta }_i}[(1-2{\beta }_i){\dot{e}}_i-{\lambda }_i{e}_i],i=\varphi ,\theta ,\psi ,z$. Combining the UAV system (14), we further obtain

$$\left\{\begin{array}{c}{\dot{s}}_{\varphi }=a_{\varphi }{\dot{e}}_{\varphi }+{\zeta }_{\varphi }+f_{\varphi }+g_{\varphi }{u}_{\varphi }+d_{\varphi }-{\ddot{x}}_{\varphi d}\hfill \\ {\dot{s}}_{\theta }=a_{\theta }{\dot{e}}_{\theta }+{\zeta }_{\theta }+f_{\theta }+g_{\theta }{u}_{\theta }+d_{\theta }-{\ddot{x}}_{\theta d}\hfill \\ {\dot{s}}_{\psi }=a_{\psi }{\dot{e}}_{\psi }+{\zeta }_{\psi }+f_{\psi }+g_{\psi }{u}_{\psi }+d_{\psi }-{\ddot{x}}_{\psi d}\hfill \\ {\dot{s}}_z=a_z{\dot{e}}_z+{\zeta }_z+f_z+g_z{u}_z+d_z-{\ddot{x}}_{zd}\hfill \end{array}\right.$$

(21)

Considering that the nonlinear functions $f_i,i=\varphi ,\theta ,\psi ,z$ are unknown, we employ an RBFNN to approximate them, defined as $W_i^{*T}{S}_i\left({\Xi }_i\right)+{ϵ}_i=f_i$, where $W_i^{*T}$ is the optimal weight and $|{ϵ}_i|>0$ denotes bounded approximation errors. Then, the equivalent control laws are designed as

$$\left\{\begin{array}{c}{u}_{eq1}={\displaystyle \frac{1}{g_{\varphi }}}(-a_{\varphi }{\dot{e}}_{\varphi }-{\zeta }_{\varphi }+{\ddot{x}}_{\varphi d}-{\widehat{W}}_{\varphi }^T{S}_{\varphi }\left({\Xi }_{\varphi }\right)-k_{\varphi }sign\left(s_{\varphi }\right))\hfill \\ u_{eq2}={\displaystyle \frac{1}{g_{\theta }}}(-a_{\theta }{\dot{e}}_{\theta }-{\zeta }_{\theta }+{\ddot{x}}_{\theta d}-{\widehat{W}}_{\theta }^T{S}_{\theta }\left({\Xi }_{\theta }\right)-k_{\theta }sign\left(s_{\theta }\right))\hfill \\ u_{eq3}={\displaystyle \frac{1}{g_{\psi }}}(-a_{\psi }{\dot{e}}_{\psi }-{\zeta }_{\psi }+{\ddot{x}}_{\psi d}-{\widehat{W}}_{\varphi }^T{S}_{\psi }\left({\Xi }_{\psi }\right)-k_{\psi }sign\left(s_{\psi }\right))\hfill \\ u_{eq4}={\displaystyle \frac{1}{g_z}}(-a_z{\dot{e}}_z-{\zeta }_z+{\ddot{x}}_{zd}-{\widehat{W}}_z^T{S}_z\left({\Xi }_z\right)-k_{z}sign\left(s_z\right))\hfill \end{array}\right.$$

(22)

where ${\widehat{W}}_i,i=\varphi ,\theta ,\psi ,z$ denote the estimation weights.

Combining the concept of BLF, the switching control law is designed as follows:

$$\left\{\begin{array}{c}{u}_{sw1}=-{\varpi }_{\varphi }\sqrt{|s_{\varphi }|}sign\left(s_{\varphi }\right)-\int {\varpi }_{\varphi }^{2}sign\left(s_{\varphi }\right)dt\hfill \\ u_{sw2}=-{\varpi }_{\theta }\sqrt{|s_{\theta }|}sign\left(s_{\theta }\right)-\int {\varpi }_{\theta }^{2}sign\left(s_{\theta }\right)dt\hfill \\ u_{sw3}=-{\varpi }_{\psi }\sqrt{|s_{\psi }|}sign\left(s_{\psi }\right)-\int {\varpi }_{\psi }^{2}sign\left(s_{\psi }\right)dt\hfill \\ u_{sw4}=-{\varpi }_z\sqrt{|s_z|}sign\left(s_z\right)-\int {\varpi }_z^{2}sign\left(s_z\right)dt\hfill \end{array}\right.$$

(23)

where ${\varpi }_i,i=\varphi ,\theta ,\psi ,z$ are the time-varying parameters that are defined as

$${\varpi }_i(t,s)=\left\{\begin{array}{cc}{\kappa }_{i}t+{\varrho }_i,\hfill & 0\le t<t_a\hfill \\ {\varpi }_{bi}\left(s\right),\hfill & t\ge t_a\hfill \end{array}\right.,i=\varphi ,\theta ,\psi ,z$$

(24)

where ${\kappa }_i$ and ${\varrho }_i$ are positive constants. $k_a$ is the set time. $|{\varpi }_{bi}\left(s\right)|>0$ denotes the BLF, defined as

$${\varpi }_{bi}\left(s_i\right)={\displaystyle \frac{\sqrt{{\mu }_i}{\eta }_i}{({\mu }_i-|s_i{|)}^{\frac{1}{2}}}},s_i\in (-{\mu }_i,{\mu }_i)$$

(25)

where ${\eta }_i>0$ denotes a tuning parameter. ${\mu }_i$ stands for the boundary constraint such that $\underset{|s_i|\to {\mu }_i}{lim}{\varpi }_{bi}\left(s_i\right)=+\infty$.

Finally, the proposed controller is introduced as

$$\left\{\begin{array}{c}{u}_{\varphi }=u_{eq1}+u_{sw1}\hfill \\ u_{\theta }=u_{eq2}+u_{sw2}\hfill \\ u_{\psi }=u_{eq3}+u_{sw3}\hfill \\ u_z=u_{eq4}+u_{sw4}\hfill \end{array}\right.$$

(26)

In addition, the adaptive laws for the RBFNNs is given by

$$\left\{\begin{array}{c}{\dot{\widehat{W}}}_{\varphi }={\Gamma }_{\varphi }(s_{\varphi }{S}_{\varphi }\left({\Xi }_{\varphi }\right)+{\sigma }_{\varphi }{\widehat{W}}_{\varphi })\hfill \\ {\dot{\widehat{W}}}_{\theta }={\Gamma }_{\theta }(s_{\theta }{S}_{\theta }\left({\Xi }_{\theta }\right)+{\sigma }_{\theta }{\widehat{W}}_{\theta })\hfill \\ {\dot{\widehat{W}}}_{\psi }={\Gamma }_{\psi }(s_{\psi }{S}_{\psi }\left({\Xi }_{\psi }\right)+{\sigma }_{\varphi }{\widehat{W}}_{\psi })\hfill \\ {\dot{\widehat{W}}}_z={\Gamma }_z(s_z{S}_z\left({\Xi }_z\right)+{\sigma }_z{\widehat{W}}_z)\hfill \end{array}\right.$$

(27)

Remark 2.

Unlike conventional TSMC strategies, the proposed nonlinear sliding surface (18) design effectively combines the finite-time convergence property of TSMC with the steady-state robustness of linear SMC. Building upon this hybrid architecture, we further introduce an improved reaching law based on BLF methodology in (25), which enables the dynamic adaptation of the reaching law gain according to the instantaneous sliding variable magnitude. This novel integration strategy not only preserves the fast convergence characteristics during the reaching phase but also provides enhanced robustness against matched disturbances through its adaptive gain adjustment mechanism, ultimately achieving superior control performance across both transient and steady-state operating conditions. The flowchart of the proposed control scheme can be seen in Figure 1.

Theorem 1.

Given the closed-loop system with the UAV system (14), the controller (26), and the adaptive RBFNN laws (27), the sliding surface (18) can converge to zero neighborhood in finite time.

Proof. 

A candidate Lyapunov function for stability analysis can be applied as follows:

$$V={\displaystyle \frac{1}{2}}\sum _{i\in I}{s}_i^2+{\displaystyle \frac{1}{2}}\sum _{i\in I}{\tilde{W}}_i^T{\Gamma }_i^{-1}{\tilde{W}}_i$$

(28)

where $I=\{\varphi ,\theta ,\psi ,z\}$ is the defined set. ${\tilde{W}}_i=W_i^{*}-{\widehat{W}}_i$ is weight error.

Taking the time differential of V gives

$$\begin{array}{cc}\hfill \dot{V}=& \sum _{i\in I}{s}_i{\dot{s}}_i+\sum _{i\in I}{\tilde{W}}_i^T{\Gamma }_i^{-1}(-{\dot{\tilde{W}}}_i)\hfill \\ \hfill =& \sum _{i\in I}{s}_i[-k_{i}sign\left(s_i\right)-{\tilde{W}}_i^T{S}_i\left({\Xi }_i\right)+d_i]-\sum _{i\in I}{s}_i[{\varpi }_i\sqrt{|s_i|}sign\left(s_i\right)+\int {\varpi }_i^{2}sign\left(s_i\right)dt]\hfill \\ \hfill & -\sum _{i\in I}{\tilde{W}}_i^T[s_i{S}_i\left({\Xi }_i\right)+{\sigma }_i{\widehat{W}}_i]\hfill \\ \hfill =& -\sum _{i\in I}[k_i|s_i|+s_i{d}_i]-\sum _{i\in I}{\varpi }_i|s_i{|}^{{\displaystyle \frac{3}{2}}}-\int {\varpi }_i^2|s_i|dt-\sum _{i\in I}{\sigma }_i{\tilde{W}}_i^T{\widehat{W}}_i\hfill \\ \hfill \le & -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{\varpi }_i|s_i{|}^{{\displaystyle \frac{3}{2}}}-\int {\varpi }_i^2|s_i|dt-\sum _{i\in I}{\sigma }_i{\tilde{W}}_i^T{\widehat{W}}_i\hfill \end{array}$$

(29)

where $|d_i|\le {\overline{d}}_i$ is bounded, with ${\overline{d}}_i$ as a positive constant. ${\rho }_i=k_i-{\overline{d}}_i$ is positive by selecting a sufficiently large $k_i$.

Based on Lemma 1, we obtain

$$-\sum _{i\in I}{\sigma }_i{\tilde{W}}_i^T{\widehat{W}}_i\le -\sum _{i\in I}{\sigma }_i{\left({\tilde{W}}_i^T{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)$$

(30)

Then, from (29), we can further have

$$\dot{V}\le -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{\varpi }_i|s_i{|}^{{\displaystyle \frac{3}{2}}}-\int {\varpi }_i^2|s_i|dt-\sum _{i\in I}{\sigma }_i{\left({\tilde{W}}_i^T{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)$$

(31)

Since ${\varpi }_{bi}\left(s_i\right)$ is a piecewise function, the proof is divided into two parts for the separate analysis:

(i) When $0\le t<t_a$, $\dot{V}$ is given by

$$\begin{array}{cc}\hfill \dot{V}\le & -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{g}_i({\kappa }_{i}t+{\varrho }_i)|s_i{|}^{{\displaystyle \frac{3}{2}}}-\sum _{i\in I}{g}_i\int {({\kappa }_{i}t+{\varrho }_i)}^2|s_i|dt\hfill \\ \hfill & -\sum _{i\in I}{\sigma }_i{\left({\tilde{W}}_i^T{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \end{array}$$

(32)

Since $g_i>0$, ${\kappa }_i>0$, and ${\varrho }_i>0$, we further have

$$\begin{array}{cc}\hfill \dot{V}\le & -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{\sigma }_i{\left({\tilde{W}}_i^T{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \\ \hfill \le & -\sqrt{2}{\rho }_i\sum _{i\in I}{\left({\displaystyle \frac{1}{2}}{s}_i^2\right)}^{{\displaystyle \frac{1}{2}}}-\sqrt{{\Gamma }_i}{\sigma }_i\sum _{i\in I}{\left({\tilde{W}}_i^T{\Gamma }_i^{-1}{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \\ \hfill \le & -K_1{V}^{{\displaystyle \frac{1}{2}}}+C_1\hfill \end{array}$$

(33)

where $K_1=min\{\sqrt{2}{\rho }_i,\sqrt{{\Gamma }_i}{\sigma }_i\}$, and $C_1={\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)$.

(ii) When $t>t_a$, $\dot{V}$ is given by

$$\begin{array}{cc}\hfill \dot{V}\le & -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{g}_i{\displaystyle \frac{\sqrt{{\mu }_i}{\eta }_i}{({\mu }_i-|s_i{|)}^{\frac{1}{2}}}}|s_i{|}^{{\displaystyle \frac{3}{2}}}-\sum _{i\in I}{g}_i\int {({\displaystyle \frac{\sqrt{{\mu }_i}{\eta }_i}{({\mu }_i-|s_i{|)}^{\frac{1}{2}}}})}^2|s_i|dt\hfill \\ \hfill & -\sum _{i\in I}{\sigma }_i{({\tilde{W}}_i^T{\tilde{W}}_i)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \end{array}$$

(34)

Since $g_i>0$, ${\kappa }_i>0$, and ${\varrho }_i>0$, we further have

$$\begin{array}{cc}\hfill \dot{V}\le & -\sum _{i\in I}{\rho }_i|s_i|-\sum _{i\in I}{\sigma }_i{\left({\tilde{W}}_i^T{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \\ \hfill \le & -\sqrt{2}{\rho }_i\sum _{i\in I}{\left({\displaystyle \frac{1}{2}}{s}_i^2\right)}^{{\displaystyle \frac{1}{2}}}-\sqrt{{\Gamma }_i}{\sigma }_i\sum _{i\in I}{\left({\tilde{W}}_i^T{\Gamma }_i^{-1}{\tilde{W}}_i\right)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{{\sigma }_i}{2}}(W_i^{*T}{W}_i^{*}+1)\hfill \\ \hfill \le & -K_1{V}^{{\displaystyle \frac{1}{2}}}+C_1\hfill \end{array}$$

(35)

Finally, according to Lemma 2, the function V ultimately converges to a neighborhood of zero in finite time, meaning that the sliding surface $s_i$ also converges to a neighborhood of zero in finite time. Then, the trajectory tracking errors of the UAV’s attitude angles and altitude can both converge to zero. □

4. Simulations

In this section, in order to verify the effectiveness of the BTSMC control strategy proposed in this paper, four sets of comparative simulation experiments are designed, focusing on step response and sinusoidal tracking, and comparing the performance with traditional PID control and SMC control under the conditions of external interference and without external interference. These experiments evaluate the tracking performance of attitude angles $\varphi ,\theta ,\psi$ and altitude z under both step and sinusoidal reference inputs, with and without external disturbances.

In the step response experiment, the step values of $\varphi ,\theta ,\psi$, and z are set to 0.18 rad, 0.1 rad, 0.15 rad, and 0.02 rad, respectively. In the sinusoidal response experiment, the reference signals for $\varphi ,\theta ,\psi$, and z are defined as $sin\left(0.5t\right)$, $cos\left(0.5t\right)$, $sin\left(t\right)$, and $0.2t$, respectively. In the disturbance rejection experiment, step disturbances of varying magnitudes are applied to $\varphi ,\theta ,\psi$, and z at $t=10$, $t=12$, $t=14$, and $t=10$, respectively, during both the step response and sinusoidal response tests. In addition, the control parameters for the proposed BTSMC are as follows: ${\lambda }_i=4$, $a_i=50$, $b_i=85$, ${\beta }_i=0.5$, and $k_i=100$. The BLF function parameters are defined as ${\kappa }_i=2$, ${\mu }_i=2.5$, and ${\rho }_i=2$. The parameters for RBFNNs are selected as ${\Gamma }_i=2$ and ${\sigma }_i=0.001$. The system parameters are displayed in

Table 1. Parameter values.

Parameter	Value	Unit	Parameter	Value	Unit
m	1.4	kg	$T_{faz}$	6.6840 × 10−4	N/rad/s
d	0.225	m	$T_{fdx}$	5.3270 × 10−4	N/m/s
$J_x$	0.0211	N m/rad/s2	$T_{fdy}$	5.3270 × 10−4	N/m/s
$J_y$	0.0219	N m/rad/s2	$T_{fdz}$	6.6840 × 10−4	N/m/s
$J_z$	0.0366	N m/rad/s2	$K_P$	2.9742 × 10−3	N/m/rad/s
$T_{fax}$	5.3270 × 10−4	N/rad/s	$C_d$	3.4320 × 10−2	N m/rad/s
$T_{fay}$	5.3270 × 10−4	N/rad/s	$J_r$	2.8365 × 10−5	N m/rad/s2

.

Figure 2 illustrates the tracking performance of the attitude angles $\varphi ,\theta ,\psi$ and altitude z under a step reference input. Figure 3 presents the corresponding tracking errors, showcasing the deviations from the desired trajectories. Figure 4 presents the control input under step responses. Figure 5 depicts the tracking performance under a step response when an external disturbance is introduced at $t=10$ s, while Figure 6 provides the corresponding tracking errors in this disturbed scenario. Figure 7 demonstrates the tracking performance of $\varphi ,\theta ,\psi$, and z under a sinusoidal reference input, and Figure 8 presents the corresponding tracking errors. Figure 9 presents the control input under sinusoidal responses. Figure 10 shows the tracking performance under a sinusoidal reference input with an external disturbance applied at $t=10$ s, where as Figure 11 displays the corresponding tracking errors in the presence of the disturbance.

First, it can be observed that in the step response experiments in Figure 2 and Figure 3, the proposed BTSMC achieves faster convergence of the attitude angles $\varphi ,\theta ,\psi$ and altitude z to their desired values compared to PID and SMC [10], with significantly smaller overshoot, indicating better dynamic response characteristics. As shown in

Table 2. The RSME of tracking errors under step response.

RMSE	PID	SMC	BTSMC
$\varphi$	0.01335	0.01050	0.01028
$\theta$	0.00593	0.00592	0.00583
$\psi$	0.00981	0.00926	0.00732
z	0.00132	0.00117	0.00115

, when comparing the tracking performance of PID, SMC, and BTSMC control strategies, the proposed BTSMC demonstrates superior performance across all RMSE metrics. Specifically, BTSMC achieves the lowest errors in $\varphi$ RMSE (0.01028), $\theta$ RMSE (0.00583), $\psi$ RMSE (0.00732), and z RMSE (0.00115), indicating its enhanced precision and stability in dynamic tracking tasks. Furthermore, as shown in Figure 5 and Figure 6, under external disturbances, the PID controller exhibits a large steady-state error, and while SMC improves robustness, it still has a deviation from the predefined trajectory. In contrast, BTSMC, leveraging the adaptive gain adjustment strategy based on the BLF, effectively suppresses the influence of disturbances, resulting in better steady-state accuracy and robustness.

Secondly, for the sinusoidal tracking experiments in Figure 7 and Figure 8, the BTSMC controller also demonstrates superior tracking performance under complex dynamic tracking tasks. Compared to PID, which exhibits significant phase lag and steady-state errors, and SMC, which still suffers from latency issues, BTSMC ensures smoother trajectory tracking while maintaining system stability. The RMSE of tracking errors are displayed in

Table 3. The RSME of tracking errors under sinusoidal response.

RMSE	PID	SMC	BTSMC
$\varphi$	1.16690	0.12071	0.05263
$\theta$	1.44250	0.12393	0.05235
$\psi$	1.94510	0.11346	0.00685
z	0.22968	0.20343	0.19168

. Comparative analysis of the tracking error RMSE under three control strategies demonstrates the superior performance of BTSMC. For the attitude angles $\varphi$, $\theta$, and $\psi$, the BTSMC errors (0.05263, 0.05235, 0.00685) are 56.4–94.0% lower than those of SMC (0.12071, 0.12393, 0.11346) and over 95% lower than those of PID (1.16690, 1.44250, 1.94510). In altitude z control, BTSMC (0.19168) still achieves the lowest error, highlighting its stability advantage, particularly in dynamic tracking tasks. Moreover, as shown in Figure 10 and Figure 11, under external disturbances, BTSMC maintains strong disturbance rejection capability, enabling the system to closely follow the desired trajectory, whereas both PID and SMC exhibit larger deviations. In summary, the simulation results fully validate the advantages of the proposed BTSMC control strategy in improving tracking accuracy, enhancing robustness, and suppressing chattering, demonstrating its effectiveness and superiority in quadrotor attitude control.

Remark 3.

While the proposed BTSMC demonstrates superior tracking performance, its effectiveness relies heavily on the appropriate selection of control parameters. For example, an improperly selected constraint value of the barrier function may lead to the failure of the reaching law. Future work will explore dynamically adaptive parameters for the barrier function to further enhance the robustness of the controller.

5. Conclusions

In this paper, a novel BTSMC strategy was proposed for quadrotor attitude control under system uncertainties and external disturbances. A fast terminal sliding mode surface was designed to ensure finite-time convergence of tracking errors, and an RBF neural network was incorporated to compensate for system uncertainties. Additionally, an adaptive gain adjustment mechanism based on the BLF was introduced to enhance the robustness of the controller. Comprehensive simulation experiments were conducted to evaluate the proposed control strategy under both step and sinusoidal reference inputs, with and without external disturbances. The results demonstrated that, compared to conventional PID and SMC controllers, the proposed BTSMC achieved faster convergence, reduced overshoot, and improved tracking accuracy. Specifically, BTSMC outperformed PID and SMC controllers with average RMSE improvements of 15.74% and 6.57% in step responses, and 77.01% and 53.47% in sinusoidal responses, respectively. Furthermore, BTSMC effectively suppressed chattering and exhibited strong robustness against external disturbances, ensuring precise trajectory tracking even under uncertain conditions. Future work will also consider extending the method to multi-agent systems and investigating its scalability for more complex, high-dimensional control tasks.