A Novel Robust Hybrid Control Strategy for a Quadrotor Trajectory Tracking Aided with Bioinspired Neural Dynamics

Abstract

This paper introduces a novel hybrid control strategy for quadrotor UAVs inspired by neural dynamics. Our approach effectively addresses two common issues: the velocity jump problem in traditional backstepping control and the control signal chattering in conventional sliding mode control. The proposed system combines an outer-loop bioinspired backstepping controller with an inner-loop bioinspired sliding mode controller, ensuring smooth trajectory tracking even under external disturbances. We rigorously analyzed the system’s stability using Lyapunov stability theory. To validate our algorithm’s effectiveness, we conducted trajectory tracking experiments in both disturbance-free and step-disturbance conditions, comparing it with the traditional backstepping control, conventional sliding mode control, and saturated sliding mode control. The results demonstrate that our algorithm not only tracks trajectories more effectively but also significantly outperforms these methods in suppressing velocity jumps and signal chattering.

1. Introduction

Quadrotor unmanned aerial vehicles (UAVs) are well-suited to various applications, including vertical takeoff and landing [1] and flexible trajectory tracking [2]. Consequently, they have garnered significant attention from developers worldwide, stimulating extensive research in this field. Consequently, quadrotor UAVs have found extensive utilization across various civilian domains [3,4,5,6,7]. Nevertheless, the design of a UAV flight control system, be it for autonomous or non-autonomous flight, often presents challenges concerning sensor technology, hardware, and software design. The quadrotor UAV represents a highly complex nonlinear control system that has garnered considerable interest among researchers in the field of automatic control.

Trajectory tracking control has consistently been a focal point of research in the autonomous flight control of quadrotor UAVs. Among the commonly utilized control methods, linear control strategies [8] offer simplicity and ease of design and implementation. However, the inherent nonlinear and strongly coupled characteristics of quadrotor UAV dynamics impose numerous limitations on the effectiveness of these linear control strategies. Moreover, quadrotor UAVs are consistently exposed to external and internal disturbances during autonomous flight operations in complex scenarios, rendering the existing linear control strategies challenging to apply in practical settings.

To address the limitations of linear control strategies, previous studies have proposed various nonlinear control strategies [9,10,11,12,13,14]. Robust nonlinear control techniques, in general, enhance flight stability, ensuring satisfactory performance even in the face of unpredictable environmental changes. They enable flexible maneuverability and precise trajectory tracking. Initially, nonlinear control techniques were employed to govern the vertical takeoff and landing (VTOL) of three-degrees-of-freedom helicopters [15]. Controller design for quadrotor UAVs poses greater challenges due to the inherent strong coupling between the body frame dynamics and the yaw, pitch, and roll motion, in contrast to helicopters. Numerous studies have explored different nonlinear controllers for diverse autonomous flight control tasks of quadrotor UAVs, including backstepping controllers [10,11], sliding mode controllers [12,13], and other related adaptive nonlinear controllers [14].

The pioneering work of Reference [16] introduced a nonlinear controller for quadrotor aircraft. The authors developed a dynamic feedback controller for hovering flight, utilizing a technique known as exact feedback linearization. The simulation results presented in the paper demonstrated acceptable performance, even in the presence of external disturbances. However, the design of controllers using feedback linearization necessitates an exact model with stable zero dynamics to compensate for the nonlinear terms. However, in practice, the dynamic uncertainty and external disturbances that are inherent in quadrotor flight pose challenges in achieving accurate model predictions, rendering the feedback linearization control strategy infeasible for real-time trajectory tracking control. Consequently, to apply this feedback linearization control strategy in practical applications, it should be augmented with another control strategy that incorporates quadrotor model prediction, as suggested in Reference [17].

The backstepping control strategy [11] is widely employed in robot control as one of the most commonly used methods. Its fundamental principle involves recursively stabilizing a closed-loop backstepping system. Analysis and design of this strategy can be carried out using Lyapunov stability theory. When designing controllers for quadrotor unmanned aerial vehicles using the backstepping control strategy [18], the design of control laws is dependent on the tracking error. Consequently, a significant tracking error leads to a substantial speed jump in the control signal. The power system of the unmanned aerial vehicle cannot immediately realize such an unrealistic speed change. Employing a fuzzy logic control strategy [19] is a practical solution to address the speed jump issue within the aforementioned backstepping control strategy. Nevertheless, the design of fuzzy logic controllers relies on the designer’s prior knowledge, posing challenges in formulating satisfactory fuzzy rules tailored to specific environments. Neural network optimization strategy has been employed as a viable solution for addressing similar issues in unmanned aerial vehicles, enabling the computation of complex nonlinear relationships. However, this approach necessitates online learning training, costly computational resources, and a substantial volume of experimental data. Previous studies have explored the implementation of the backstepping control strategy to achieve position tracking control in quadrotor unmanned aerial vehicles; however, the majority of controller designs overlooked the presence of external disturbances.

Previous studies have focused on the development of robust controllers to mitigate the impact of external and internal disturbances on quadrotor unmanned aerial vehicles (UAVs) and ensure stable autonomous flight in challenging environments. One robust control method, the sliding mode control strategy [12], is known for its insensitivity to parameter variance, thereby enhancing the system’s resilience to disturbances. However, the control signals generated by the sliding mode control strategy exhibit significant shaking, which can lead to high-frequency variations detrimental to onboard hardware. Alattas [20] and Najafi [21], expanding on traditional sliding mode control, introduced non-singular terminal sliding mode control methods based on adaptive barrier functions and fast terminal sliding mode control techniques, respectively. These approaches tackle trajectory tracking challenges, particularly in scenarios involving bounded uncertain input constraints, external disturbances, parameter uncertainties, and actuator failures. Both methodologies have demonstrated remarkable effectiveness. While attempts have been made to address the shaking problem in existing methods by substituting the shaking term with saturation and tanh functions [22], these solutions compromise robustness to disturbances and necessitate meticulous parameter tuning to mitigate shaking. Furthermore, given the susceptibility of both backstepping control and sliding mode control to noise, control strategies inspired by biological systems can offer smoother control signals, thus effectively resolving the challenges associated with backstepping control and sliding mode control, which is particularly crucial in the context of limited actuators.

Biological neural dynamics, initially proposed by Hodgkin and Huxley [23], refers to the utilization of electrical circuit elements in a patch of membrane within a biological neural system. Yang [24] introduced these bioinspired neural dynamics to the field of robotics, while Zhu [25] subsequently applied it to the motion control of an underwater unmanned submarine. The overall control design was based on bioinspired neural dynamics and employed a hybrid control approach. A bioinspired backstepping control strategy was employed to address the velocity jump issue in traditional backstepping control, resulting in smooth position control. Additionally, a bioinspired sliding mode control strategy was utilized to tackle the control signal chattering problem in traditional sliding mode control, thereby minimizing the impact on onboard actuators.

Therefore, in this paper, a nonlinear hybrid control strategy for quadrotor UAVs aided with the bioinspired neural dynamics model is proposed based on the preliminary work in Refs. [26,27,28,29]. The system employs a dual closed-loop structure: the outer loop uses a biologically inspired neural dynamics-based backstepping method for position control, while the inner loop applies a biologically inspired neural dynamics-based sliding mode control for attitude control. The proposed method not only avoids the control signal jump problem in traditional backstepping control, but also effectively suppresses the control signal chattering problem in traditional sliding mode control.

The paper is organized as follows. Section 2 introduces the kinematic and dynamic models of the quadrotor unmanned aerial vehicle (UAV). In Section 3, a bioinspired neural dynamics model is presented as the foundation for the development of the bioinspired backstepping position controller and the bioinspired sliding mode attitude controller for the quadrotor UAV. The stability analysis of the proposed control system is conducted in Section 3. Section 4 provides the numerical simulation results, which are compared with other control systems to showcase their advantages and effectiveness. Finally, Section 5 summarizes the development philosophy and justifies the superiority of the proposed hybrid control strategy for quadrotor UAVs. The conclusion also includes suggestions for areas of improvement.

2. Dynamical Model

2.1. Quadrotor Model

To simplify control design while ensuring model accuracy, we make the following assumptions for the quadrotor UAV.

(1) The UAV is a rigid body with constant mass, moment of inertia, and center of gravity during flight. (2) The UAV structure is symmetrical. (3) Earth’s rotation and revolution effects are negligible. (4) The only external forces acting on the UAV are gravity, propeller-generated lift and torque, and external disturbances.

Using the Newton–Euler equations, the kinematic and dynamic models of the quadrotor unmanned aerial vehicle (UAV) are formulated in two coordinate reference frames: the inertial frame and the body frame [30]. In this study, the inertial frame is denoted as frame ${\widehat{F}}_E=\left\{\begin{array}{cc}\begin{array}{cc}{\widehat{f}}_{eo,}& {\widehat{f}}_{ex,}\end{array}& \begin{array}{cc}{\widehat{f}}_{ey,}& {\widehat{f}}_{ez}\end{array}\end{array}\right\}$, while the body frame is denoted as frame ${\widehat{F}}_B=\left\{\begin{array}{cc}\begin{array}{cc}{\widehat{f}}_{bo,}& {\widehat{f}}_{bx,}\end{array}& \begin{array}{cc}{\widehat{f}}_{by,}& {\widehat{f}}_{bz}\end{array}\end{array}\right\}$. Here, ${\widehat{f}}_{eo}$ represents the origin of the inertial coordinate system, and ${\widehat{f}}_{bo}$ represents the center of mass of the quadrotor UAV. These definitions are illustrated in Figure 1. The Euler angles of the quadrotor, denoted as ${\mathsf{\Theta }}^E={\left[\begin{array}{ccc}\varphi & \theta & \psi \end{array}\right]}^T\in {\mathbb{R}}^{3\times 1}$, represent its roll angle ($\varphi$), pitch angle ($\theta$), and yaw angle ($\psi$). These angles satisfy the conditions $\left|\varphi \right|<{\displaystyle \frac{\pi }{2}},\left|\theta \right|<{\displaystyle \frac{\pi }{2}},\left|\psi \right|<\pi$.The angular velocities of the drone, denoted as ${\mathsf{\Omega }}^B={\left[pqr\right]}^T\in {\mathbb{R}}^{3\times 1}$, correspond to its rotations along the X, Y, and Z axes of the body coordinate system, respectively. The position of the quadrotor in the world coordinate system ${\widehat{F}}_E$ is defined as $P^E={\left[\begin{array}{ccc}{p}_x^e& p_y^e& p_z^e\end{array}\right]}^T\in {\mathbb{R}}^{3\times 1}$, while the velocity of the quadrotor’s movement in the inertial coordinate system ${\widehat{F}}_E$ is defined as $V^E={\left[\begin{array}{ccc}{v}_x^e& v_y^e& v_z^e\end{array}\right]}^T\in {\mathbb{R}}^{3\times 1}$. Consequently, the kinematic and dynamic model of the quadrotor’s displacement and rotation can be expressed as follows:

$${\dot{P}}^E=V^E$$

$${\dot{V}}^E=m^{-1}\left[R_B^E{T}_B+F_d+dv\right]$$

$${\dot{\mathsf{\Theta }}}^E=R_{\mathsf{\Theta }}·{\mathsf{\Omega }}^B$$

$${\dot{\mathsf{\Omega }}}^B=I^{-1}\left[{\tau }_b+G_a-{\mathsf{\Omega }}^B\times \left(I·{\mathsf{\Omega }}^B\right)+M_d\right]$$

(1)

where $m$ represents the mass of the quadrotor, $I$ denotes the inertial matrix of the quadrotor in the inertial coordinate system, given by $I=diag(I_{xx},I_{yy},I_{zz})$, $R_B^E$ signifies the rotation matrix between the body-fixed coordinate system and the inertial coordinate system for the quadrotor’s translation, and $R_{\mathsf{\Theta }}$ represents the rotation matrix for the quadrotor’s rotation. These matrices satisfy the following conditions:

$$R_B^E=\left[\begin{array}{l}\begin{array}{ccc}cos\theta cos\psi & sin\varphi sin\theta -cos\varphi sin\psi & sin\varphi sin\psi +cos\varphi sin\theta cos\psi \end{array}\\ \begin{array}{ccc}cos\theta sin\psi & cos\varphi cos\psi +sin\varphi sin\theta sin\psi & cos\varphi sin\theta sin\psi -cos\psi sin\varphi \end{array}\\ \begin{array}{ccc}-sin\theta & sin\varphi cos\theta & cos\varphi cos\theta \end{array} \end{array}\right]$$

(2)

$$R_{\mathsf{\Theta }}=\left[\begin{array}{l}\begin{array}{ccc}1& sin\varphi tan\theta & cos\varphi tan\theta \end{array}\\ \begin{array}{ccc}0& cos\varphi & -sin\varphi \end{array}\\ \begin{array}{ccc}0& sin\varphi /cos\theta & cos\varphi /cos\theta \end{array}\end{array}\right]$$

(3)

$dv$ represents step-like increasing values of external disturbances. The variables $F_d$ and $M_d$ represent bounded variations of aerodynamic force and moment, respectively. $T_B$ satisfies $T_B=T_K{E}_3-mg{\left(R_B^E\right)}^T{E}_3$, where $g$ denotes the gravitational constant. $T_K$ represents the total thrust generated by the quadrotor at time $t_0$, and it can be expressed as $T_K=T_1+T_2+T_3+T_4$. Each $T_i(i=\mathrm{1,2},\mathrm{3,4})$ represents the thrust $T_i=k_{\omega }{\omega }_i^2$ generated by the i-th motor, where $k_{\omega }$ is a positive constant. The variable ${\tau }_b$ describes the external torques acting on the x, y, and z axes of the body frame, which satisfy:

$${\tau }_b=\left[\begin{array}{c}{l}_{b}b\left(T_1-T_2-T_3+T_4\right)\\ l_{b}b\left(T_1+T_2-T_3-T_4\right)\\ k_{\tau }\left(T_1-T_2+T_3-T_4\right)\end{array}\right]$$

(4)

where $l_b$ and $k_{\tau }$ represent the body radius and the conversion factor from force to torque, respectively.

2.2. Control Problem Description

In the above, $P_d^E={\left[\begin{array}{ccc}{p}_{dx}^e& p_{dy}^e& p_{dz}^e\end{array}\right]}^T$ and $V_d^E={\left[\begin{array}{ccc}{v}_{dx}^e& v_{dy}^e& v_{dz}^e\end{array}\right]}^T$ represent the desired position and velocity of the quadrotor in the inertial frame, respectively. ${\mathsf{\Theta }}_d^E={\left[\begin{array}{ccc}{\varphi }_d& {\theta }_d& {\psi }_d\end{array}\right]}^T$ represents the desired Euler angles of the quadrotor, and ${\mathsf{\Omega }}_d^B={\left[\begin{array}{ccc}{\mathrm{p}}_d& {\mathrm{q}}_d& {\mathrm{r}}_d\end{array}\right]}^T$ represents the desired angular velocity of the quadrotor in the body frame. Among these variables, $P_d^E$ and ${\psi }_d$ serve as inputs to the entire control system. The desired roll angle ${\varphi }_d$ and pitch angle ${\theta }_d$ are obtained by the outer-loop bioinspired backstepping position control system to achieve trajectory tracking. Specifically, the desired position $P_d^E$ and yaw angle ${\psi }_d$ of the quadrotor are determined based on the tracking trajectory at time $t_0$. Figure 2 depicts the relationship between tracking errors in the body and inertial coordinate systems during the tracking process. Here, the position error is represented by $e_P=P_d^E-P^E$, the velocity error by $e_V=V_d^E-V^E$, the Euler angle error by $e_{\mathsf{\Theta }}={\mathsf{\Theta }}_d^E-{\mathsf{\Theta }}^E$, and the body angle error by $e_{\mathsf{\Omega }}={\mathsf{\Omega }}_d^B-{\mathsf{\Omega }}^B$. The outputs of the relevant controllers are denoted as $u_P=\left[u_{pi}\right]\in {\mathbb{R}}^{3\times 1}$, $u_V=\left[u_{vi}\right]\in {\mathbb{R}}^{3\times 1}$, $u_{\mathsf{\Theta }}=\left[u_{\mathsf{\Theta }i}\right]\in {\mathbb{R}}^{3\times 1}$, and $u_{\mathsf{\Omega }}=\left[u_{\mathsf{\Omega }i}\right]\in {\mathbb{R}}^{3\times 1}$. Specifically, $u_P$, $u_{\mathsf{\Theta }}$, and $u_{vi}(i=\mathrm{1,2})$ correspond to the inputs of the velocity controller, angle controller, and angular velocity controller, respectively. They satisfy $V_d^E=u_P$, ${\varphi }_{cd}=u_{v1}$, ${\theta }_{cd}=u_{v2}$, and ${\mathsf{\Omega }}_d^B=u_{\mathsf{\Theta }}$. $u_{v3}$ and $u_{\mathsf{\Omega }}$ represent the control signal inputs for the quadrotor, which drive the four motors to generate total thrust and torque for following the desired trajectory. In this paper, the actual quadrotor dynamical system is treated as a nominal system with the addition of an equivalent disturbance. Therefore, the quadrotor dynamical model can be rewritten as follows:

$${\dot{P}}^E=u_P+{\mathsf{\Delta }}_p$$

$${\dot{V}}^E=G_V{u}_V-G+{\mathsf{\Delta }}_v$$

$${\dot{\mathsf{\Theta }}}^E=u_{\mathsf{\Theta }}+{\mathsf{\Delta }}_{\mathsf{\Theta }}$$

$${\dot{\mathsf{\Omega }}}^B=G_{\mathsf{\Omega }}{u}_{\mathsf{\Omega }}+I^{-1}\left[G_a-{\mathsf{\Omega }}^B\times \left(I·{\mathsf{\Omega }}^B\right)\right]+{\mathsf{\Delta }}_{\mathsf{\Omega }}$$

(5)

where $G={\left[\begin{array}{ccc}0& 0& g\end{array}\right]}^T$; $G_V=diag(g,-g,k_t/m)$; $G_{\mathsf{\Omega }}=I^{-1}diag(k_{\sigma p},k_{\sigma q},k_{\sigma r})$;$k_t$, $k_{\sigma p}$, $k_{\sigma q}$, and $k_{\sigma r}$ are gain constants of the controller satisfying $T_K=k_t{u}_{vz}$, ${\tau }_{bp}=k_{\sigma p}{u}_{\mathsf{\Omega }p}$,${\tau }_{bq}=k_{\sigma q}{u}_{\mathsf{\Omega }q}$, and${\tau }_{br}=k_{\sigma r}{u}_{\mathsf{\Omega }r}$; and ${\mathsf{\Delta }}_i(i=p,v,\mathsf{\Theta },\mathsf{\Omega })$ represents external disturbances and aerodynamics, satisfying:

$${\mathsf{\Delta }}_p=V^E-u_P$$

$${\mathsf{\Delta }}_v={\displaystyle \frac{1}{m}}\left(F_d+dv\right)$$

$${\mathsf{\Delta }}_{\Theta }=R_{\Theta }{\Omega }^B-u_{\Theta }$$

$${\mathsf{\Delta }}_{\Omega }=I^{-1}{M}_d$$

(6)

3. Quadrotor Control Design Aided with Bioinspired Neural Dynamics

In this section, a hybrid control strategy is developed for a quadcopter drone to address the issues of velocity jumps and control signal chattering in backstepping control and sliding mode control. Figure 3 illustrates the closed-loop control framework, comprising two cascaded control systems. The outer-loop position control system consists of a bioinspired backstepping position controller and velocity controller. The position controller generates smooth velocity commands to prevent abrupt velocity jumps caused by significant tracking errors and external disturbances, while the velocity controller provides desired angular commands. The inner-loop attitude control system utilizes the output angular commands from the outer-loop control system to generate torque control signals for trajectory tracking. To mitigate the control signal chattering problem encountered in traditional sliding mode control, the inner-loop attitude control system incorporates a sliding mode controller with bioinspired neural dynamics.

3.1. Bioinspired Neural Dynamic Model

Hodgkin and Huxley initially proposed a membrane model that utilized electrical elements to describe the behavior of a membrane [28]. Subsequently, Grossberg [31] further advanced this model to capture real-time adaptive behavior in individuals. Yang and Meng [24] were pioneers in applying this model to robots, enabling real-time collision-free motion planning in dynamic environments without the need for prior learning. Over time, this model has been extended to various other robot applications. Referred to as the shunting model, the bioinspired neural dynamics model is expressed as follows:

$${\dot{x}}_i=-A_i{x}_i+\left(B_i-x_i\right)S_i^{+}-\left(D_i+x_i\right)S_i^{-}$$

(7)

In this model, $x_i$ denotes the neural activity of the i-th neuron. $A_i$ represents the passive decay rate, while $B_i$ and $D_i$ represent the upper and lower bounds of the i-th neuron, respectively. The variables $S_i^{+}$ and $S_i^{-}$ indicate the excitatory and inhibitory inputs to the i-th neuron. Model (7) is a continuous differential equation. When we define $S_i^{+}$= max($x_i$,0) and $S_i^{-}$= max(−$x_i$,0) and examine the solution of this differential equation, we find that the model output can maintain the region [$B_i$, $D_i$] under any excitatory and inhibitory inputs. Furthermore, when the input experiences sudden changes, the model output remains smooth within this bounded interval, without any abrupt fluctuations. By utilizing Equation (7), the shunting model for the quadcopter can be defined as follows:

$${\dot{E}}_i=-A_i{E}_i+\left(B_i-E_i\right)f\left(e_i\right)-\left({E_i-D}_i\right)g\left(e_i\right)$$

(8)

where $e_i\left(i=P,V,\mathrm{S}\right)$ represents the inputs to the shunting model, which signifies the error between the desired pose and the quadrotor’s state. The functions $f\left(e_i\right)$ and $g\left(e_i\right)$ are defined as $f\left(e_i\right)=\max \left(e_i,0\right) \mathrm{a}\mathrm{n}\mathrm{d} g\left(e_i\right)=\mathrm{m}\mathrm{a}\mathrm{x}(-e_i,0)$, respectively. $E_i$ denotes the output of the shunting model, and for any input $e_i$, $E_i$ is constrained within the limits of ${(B}_i,D_i)$. Given the specific characteristics of the quadcopter drone model, we assume $\left|B_i\right|=\left|D_i\right|$. Moreover, it is important to highlight that the shunting model exhibits properties like those of a low-pass filter, imparting robustness against aerodynamic forces and external disturbances [31]. Subsequent sections will provide a detailed explanation of the design process for the entire control system.

3.2. Bioinspired Backstepping Position Control System

Designing a backstepping position controller for a quadcopter using Lyapunov functions is relatively straightforward. However, the drone’s state and external disturbances can cause unrealistic velocity changes that are not achievable for a drone with limited inputs. To address this issue, this section proposes a bioinspired backstepping position controller that combines the shunting model (Equation (8)) with the backstepping control strategy. The focus of this article is primarily on controlling the quadcopter drone to track a spiral ascent trajectory in three-dimensional space, as shown in Figure 2. Therefore, the relationship between the desired position state and the desired velocity state for the drone’s trajectory tracking is expressed as:

$$\left[\begin{array}{c}{v}_{dx}^e\\ v_{dy}^e\\ v_{dz}^e\end{array}\right]=\left[\begin{array}{c}{\dot{x}}_d\\ {\dot{y}}_d\\ {\dot{z}}_d\end{array}\right]$$

(9)

The objective of the outer-loop position control system is to produce control inputs for the thrust, roll angle, and pitch angle. The position controller in the system generates velocity control inputs to drive the quadcopter to a certain speed, thus minimizing the tracking error towards zero. Figure 2 shows the tracking position error in the X, Y, and Z axis directions of the inertial coordinate system as $e_P={\left[\begin{array}{ccc}{e}_x& e_y& e_z\end{array}\right]}^T$, combined with the velocity error obtained from Equation (9) as $e_V={\left[\begin{array}{ccc}{e}_{vx}& e_{vy}& e_{vz}\end{array}\right]}^T$. Consequently, the position control inputs for the quadcopter, based on the backstepping control approach, are as follows:

$$u_V=G_V^{-1}\left[{\ddot{P}}_d^E+G+k_V{e}_V\right]$$

$${ \dot{V}}^E={\ddot{P}}_d^E+k_V{e}_V$$

$${\dot{P}}^E=V^E={\dot{P}}_d^E+k_P{e}_P$$

(10)

$k_P$ and $k_{V }$refer to the position control gain and velocity control gain, respectively. These parameters are defined as $k_P=diag(k_{p1},k_{p2},k_{p3})$ and $k_V=diag(k_{v1},k_{v2},k_{v3})$. Equation (10) reveals that in scenarios where the drone is in its initial stage or affected by external disturbances, resulting in significant tracking errors in position and velocity, the terms $k_V{e}_V \mathrm{a}\mathrm{n}\mathrm{d} {k_{P}e}_P$may lead to abrupt changes in velocity during displacement motion. To mitigate this issue, a backstepping control strategy combined with the bioinspired neural dynamics model is proposed. The new virtual control commands are defined as follows:

$$u_V=G_V^{-1}\left[{\ddot{P}}_d^E+G+{k_{V}E}_V\right]$$

(11)

where $E_P$ and $E_V$ represent the outputs of the shunting model for the quadcopter, replacing the position and velocity error inputs, $e_P$ and $e_V$. The error terms in the virtual control commands must satisfy the following inequality:

$$k_P{D}_P<k_P{E}_P<k_P{B}_P , {k_{V}D}_V<k_V{E}_V<+k_V{B}_V$$

(12)

$B_P$ and $B_V$ represent positive constants, while $D_P={-B}_P$ and $D_V={-B}_V$. Consequently, by carefully selecting appropriate desired velocity ${\dot{P}}_d^E$ and desired acceleration ${\ddot{P}}_d^E$, the virtual control command $u_V$ remains within its dynamic constraints and is bounded. Furthermore, thanks to the filtering capability of the bioinspired neural dynamics model, the bioinspired backstepping controller can provide smooth control inputs, even in the presence of noise. The stability of the bioinspired backstepping controller is demonstrated in Section 3.4.

3.3. Bioinspired Sliding Attitude Control System

In this section, we present the development of a robust sliding mode control strategy for a quadcopter drone. The proposed attitude control system combines traditional sliding mode control with the bioinspired neural dynamics model (8) to effectively address the issue of signal chattering commonly encountered in traditional sliding mode controllers. The angle commands $u_{v1},u_{v2}$ generated by the bioinspired backstepping position control system, along with the desired yaw angle ${\psi }_d$, are utilized as inputs to the attitude control system. Subsequently, the bioinspired sliding mode controller generates torque control signals $u_{\mathsf{\Omega }}$ to guide the quadcopter drone model towards the desired state.

The defined sliding surface is expressed as follows:

$$S={\dot{e}}_{\Omega }+a_1{e}_{\Omega }+a_2\int e_{\Omega }$$

(13)

where $a_1$ and $a_2$ are defined as positive constants. Taking the derivative of Equation (13), we have:

$$\dot{S}={\ddot{e}}_{\Omega }+a_1{\dot{e}}_{\Omega }+a_2{e}_{\Omega }$$

(14)

From Equation (14), we can see that when the system operates on the sliding surface $\dot{S}=0$, we have:

$${\ddot{e}}_{\Omega }+a_1\left({\dot{\Omega }}_d^B-{\dot{\Omega }}^B\right)+a_2{e}_{\Omega }=0$$

(15)

By substituting the quadcopter dynamics model (5) into Equation (15), we obtain:

$${\ddot{e}}_{\Omega }+a_1\left({\dot{\Omega }}_d^B-(G_{\Omega }{u}_{\Omega }+I^{-1}\left[G_a-{\Omega }^B\times \left(I·{\Omega }^B\right)\right]+{∆}_{\Omega })\right)+a_2{e}_{\Omega }=0$$

(16)

Therefore, the design of the equivalent control law is given by:

$$u_q=G_{\Omega }^{-1}\left({\displaystyle \frac{1}{a_1}}\left({\ddot{e}}_{\Omega }+a_2{e}_{\Omega }\right)+{\dot{\Omega }}_d^B-I^{-1}\left[G_a-{\Omega }^B\times \left(I·{\Omega }^B\right)\right]-D_{\Omega }\right)$$

(17)

where $D_{\mathsf{\Omega }}$ represents the compensation for aerodynamic moments by the sliding mode controller, satisfying:

$$D_{\Omega }=d_{\Omega 2}-d_{\Omega 1}sgn\left(S\right)$$

(18)

$d_{\mathsf{\Omega }1}=({\delta }_U+{\delta }_L)/2$ and $d_{\mathsf{\Omega }2}=({\delta }_L-{\delta }_U)/2$, where ${\delta }_U$ represents the upper bound of the aerodynamic moment and ${\delta }_L$ represents the lower bound. Due to the computational challenges in calculating ${\ddot{e}}_{\mathsf{\Omega }}$ as stated in Equation (17), an approximation can be made by defining ${\ddot{e}}_{\mathsf{\Omega }}=k_e{\dot{e}}_{\mathsf{\Omega }}$, where $k_e$ represents a feedback control gain. The conventional sliding mode control law can be expressed as follows:

$$u_{\Omega }=u_q+k_{s}sgn\left(S\right)$$

(19)

Chattering in traditional sliding mode control signals is a result of the sign function sgn(·). To mitigate this issue, a commonly used approach is to substitute the sign function with a saturation function. Consequently, the sliding mode control law incorporating the saturation function can be expressed as follows:

$$u_{\Omega }=u_q+k_{s}sat\left(S\right)$$

$$sat\left(S\right)=\left\{\begin{array}{l}{B}_S, k_{s}S>B_S \\ k_{s}S, B_S\ge k_{s}S\ge \\ D_S, {D_S>k}_{s}S\end{array}\right.B_S$$

(20)

where $B_S$and${ D}_S$ denote the upper and lower bounds of the saturation function, respectively. Although the saturation function effectively reduces control signal chattering in the presence of small control signals, it may not be suitable for all applications in complex operating environments where larger control signals are necessary to ensure stable flight. To address this limitation, a combination of biologically inspired neural dynamics (Equation (8)) and sliding mode control is proposed. The resulting biologically inspired sliding mode control commands are defined as follows:

$$u_{\Omega }=u_q+E_S$$

(21)

The proposed biologically inspired sliding mode controller in this paper effectively mitigates the problem of control signal chattering that is commonly encountered in traditional sliding mode control methods. Moreover, it demonstrates excellent trajectory tracking performance even in complex scenarios.

3.4. Stability Analysis

This paper presents a novel approach to achieve progressive trajectory tracking control by integrating a bioinspired neural network control strategy with the aerodynamics and external disturbances of a quadrotor unmanned aerial vehicle. The asymptotic stability of both control systems is verified through rigorous Lyapunov stability analysis. Additionally, by constructing a global Lyapunov candidate function, the study establishes the global asymptotic stability of the proposed control system.

To demonstrate the stability of the bioinspired backstepping controller, the Lyapunov function is defined as follows:

$$V_P={\displaystyle \frac{1}{2}}{e}_{ P}^2+{\displaystyle \frac{1}{2}}{e}_{ V}^2+{\displaystyle \frac{k_P}{2B_P}}{E}_P^2+{\displaystyle \frac{k_V}{2B_V}}{E}_V^2$$

(22)

Then, the derivative of $V_P$ is denoted as:

$${\dot{V}}_P={e_P{\dot{e}}_P+e}_V{\dot{e}}_V+{\displaystyle \frac{k_P}{B_P}}{E}_P{\dot{E}}_P+{\displaystyle \frac{k_V}{B_V}}{E}_V{\dot{E}}_V$$

(23)

Splitting the time derivative expression (23) of the candidate function into two components, we have ${e_P{\dot{e}}_P+e}_V{\dot{e}}_V$ and ${\displaystyle \frac{k_P}{B_P}}{E}_P{\dot{E}}_P+{\displaystyle \frac{k_V}{B_V}}{E}_V{\dot{E}}_V$. Based on the tracking errors and Equation (10), ${e_P{\dot{e}}_P+e}_V{\dot{e}}_V$ can be written as:

$$\begin{gathered} {e_P{\dot{e}}_P+e}_V{\dot{e}}_V=e_P\left({\dot{P}}_d^E-{\dot{P}}^E\right)+e_V\left({\dot{V}}_d^E-{\dot{V}}^E\right) \\ ={-{k_{P}E}_{P}e}_P{-k_V{E}_{V}e}_V \end{gathered}$$

(24)

For the second part, ${\displaystyle \frac{k_P}{B_P}}{E}_P{\dot{E}}_P+{\displaystyle \frac{k_V}{B_V}}{E}_V{\dot{E}}_V$, of the Lyapunov function, considering the specific characteristics of the quadrotor unmanned aerial vehicle model, let us assume $B_P={-D}_P$ and $B_V={-D}_V$. Based on the biological neural dynamics model Equation (8), the second part can be written as:

$$\begin{gathered} {\displaystyle \frac{k_P}{B_P}}{E}_P{\dot{E}}_P+{\displaystyle \frac{k_V}{B_V}}{E}_V{\dot{E}}_V= {\displaystyle \frac{k_P}{B_P}}{E}_P\left(-A_P{E}_P+\left(B_P-E_P\right)f\left(e_P\right)-\left(E_P-D_P\right)g\left(e_P\right)\right) \\ +{\displaystyle \frac{k_V}{B_V}}{E}_V\left(-A_V{E}_V+\left(B_V-E_V\right)f\left(e_V\right)-\left(E_V-D_V\right)g\left(e_V\right)\right) \\ ={\displaystyle \frac{k_P}{B_P}}{E}_P^2\left(-A_P-f\left(e_P\right)-g\left(e_P\right)\right)+{k_{P}E}_P\left(f\left(e_P\right)-g\left(e_P\right)\right) \\ +{\displaystyle \frac{k_V}{B_V}}{E}_V^2\left(-A_V-f\left(e_V\right)-g\left(e_V\right)\right)+{k_{V}E}_V\left(f\left(e_V\right)-g\left(e_V\right)\right) \end{gathered}$$

(25)

Combining Equations (24) and (25), we can infer that:

$$\begin{gathered} {\dot{V}}_P= {\displaystyle \frac{k_P}{B_P}}{E}_P\left(-A_P{E}_P+\left(B_P-E_P\right)f\left(e_P\right)-\left(E_P-D_P\right)g\left(e_P\right)\right) \\ +{\displaystyle \frac{k_V}{B_V}}{E}_V\left(-A_V{E}_V+\left(B_V-E_V\right)f\left(e_V\right)-\left(E_V-D_V\right)g\left(e_V\right)\right) \\ ={\displaystyle \frac{k_P}{B_P}}{E}_P^2\left(-A_P-f\left(e_P\right)-g\left(e_P\right)\right)+{k_{P}E}_P\left(f\left(e_P\right)-g\left(e_P\right)-e_P\right) \\ +{\displaystyle \frac{k_V}{B_V}}{E}_V^2\left(-A_V-f\left(e_V\right)-g\left(e_V\right)\right)+{k_{V}E}_V\left(f\left(e_V\right)-g\left(e_V\right)-e_V\right) \end{gathered}$$

(26)

According to Equation (8), we can conclude that:

$$\left\{\begin{array}{c}f\left(e_i\right)=e_i g\left(e_i\right)=0 , e_i\ge 0 \\ f\left(e_i\right)=0 g\left(e_i\right)=-e_i , e_i<0\end{array}\right.$$

(27)

By using Formula (27), we obtain:

$$-A_i-f\left(e_i\right)-g\left(e_i\right)<0$$

$$f\left(e_i\right)-g\left(e_i\right)=e_i$$

(28)

From Equations (22), (26), and (28), it can be inferred that as $t\to \mathrm{\infty }$, $V_P\to 0$ and ${\dot{V}}_P$ is negative definite. $V_P$ can only be zero when $e_i=0^{3\times 1}$. Therefore, the bioinspired backstepping position control system designed in this paper is asymptotically stable.

To prove the stability of the bioinspired sliding mode controller, the designed Lyapunov function is defined as follows:

$$V_{\Theta }=-{\displaystyle \frac{1}{2}}{S}^{T}S+{\displaystyle \frac{1}{2 B_S}}{E}_S^T{E}_S$$

(29)

Differentiating the Lyapunov equation, we obtain:

$${\dot{V}}_{\Theta }=-S^T\dot{S}+{\displaystyle \frac{1}{B_S}}{\dot{E}}_S^T{E}_S$$

(30)

Combining the biological neural dynamics model (8) with Equation (30), we have:

$$\begin{gathered} {\dot{V}}_{\Theta }={-S}^T\left({\ddot{e}}_{\Omega }+a_1{\dot{e}}_{\Omega }+a_2{e}_{\Omega }\right) \\ +{\displaystyle \frac{1}{B_S}}{E}_S\left(-{A_S}^T{E_S}^T+\left({B_S}^T-{E_S}^T\right)f^T\left(S\right)-\left({E_S}^T-{D_S}^T\right)g^T\left(S\right)\right) \\ ={-S}^T\left({\ddot{e}}_{\Omega }+a_1{\dot{e}}_{\Omega }+a_2{e}_{\Omega }\right)+{\displaystyle \frac{1}{B_S}}{E}_S{E_S}^T\left(-{A_S}^T-f^T\left(S\right)-g^T\left(S\right)\right) \\ +{\displaystyle \frac{1}{B_S}}{E}_S\left({B_S}^T{f}^T\left(S\right)+{D_S}^T{g}^T\left(S\right)\right) \end{gathered}$$

(31)

Combining Formulas (5), Formula (31) can be written as:

$$\begin{gathered} {\dot{V}}_{\Theta }=-S^T{E}_S+{\displaystyle \frac{1}{B_S}}{E}_S{E_S}^T\left(-{A_S}^T-f^T\left(S\right)-g^T\left(S\right)\right) \\ +{\displaystyle \frac{1}{B_S}}{E}_S\left({B_S}^T{f}^T\left(S\right)+{D_S}^T{g}^T\left(S\right)\right) \end{gathered}$$

(32)

where ${B_{\mathrm{S}}}^T=-{D_{\mathrm{S}}}^T and {{B_{\mathrm{S}}}^T=B}_{\mathrm{S}}$, rewriting Equation (32), we have:

$${\dot{V}}_{\Theta }={\displaystyle \frac{1}{B_S}}{E}_S{E_S}^T\left(-{A_S}^T-f^T\left(S\right)-g^T\left(S\right)\right)+E_S\left(f^T\left(S\right)-g^T\left(S\right)-S^T\right)$$

(33)

Based on Formulas (27) and (28), we can also observe that

${\displaystyle \frac{1}{B_S}}{E}_{\mathrm{S}}{E_{\mathrm{S}}}^T\left(-{A_{\mathrm{S}}}^T-f^T\left(S\right)-g^T\left(S\right)\right)<0 and E_S\left(f^T\left(S\right)-g^T\left(S\right)-S^T\right)=0$. From the biological neural dynamics model (8), as $t\to \mathrm{\infty }$, $S\to 0$, and $E_{\mathrm{S}}$ also tends to 0. It follows that $V_{\mathsf{\Theta }}\to 0$, ${\dot{V}}_{\mathsf{\Theta }}$ is negative definite, and $V_{\mathsf{\Theta }}$ can only be zero when $S=0$. Therefore, the designed bioinspired sliding mode attitude control system is asymptotically stable.

To demonstrate the stability of the entire control system, the Lyapunov function of the system and its time derivative are designed as follows:

$$V=V_P+V_{\Theta }$$

$$\dot{V}={\dot{V}}_P+{\dot{V}}_{\Theta }$$

(34)

The results obtained from Equations (26) and (33) can prove that the entire control system is asymptotically stable.

4. Results

In this study, we validate the efficacy of the proposed hybrid control strategy through comprehensive numerical simulations conducted using MATLAB/Simulink R2022b. We employ the quadrotor unmanned aerial vehicle model described in Equation (1), and the specific model parameters are documented in

Table 1. Quadrotor model parameters.

Symbol	Value
m	0.18 kg
g	$9.81 \mathrm{m}/{\mathrm{s}}^2$
$l_b$	0.225 m
b	$2.9 \times {10}^{-6}$  $\mathrm{N}{\mathrm{s}}^2/{\mathrm{r}\mathrm{a}\mathrm{d}}^2$
$k_{\tau }$	$1.779 \times {10}^{-7}$  $\mathrm{N}{\mathrm{s}}^2/{\mathrm{r}\mathrm{a}\mathrm{d}}^2$
$I_{xx}$	$2.5 \times {10}^{-4}$  $\mathrm{m}/{\mathrm{s}}^2$
$I_{yy}$	$2.5 \times {10}^{-4}$  $\mathrm{m}/{\mathrm{s}}^2$
$I_{zz}$	$3.738 \times { 10}^{-4}$  $\mathrm{m}/{\mathrm{s}}^2$

. To evaluate the robustness and effectiveness of our control approach, we examine two scenarios: an ideal environment devoid of disturbances and a more intricate application setting featuring step-wise ascending disturbances. The simulations are conducted over a duration of 100 s.

4.1. Undisturbed Trajectory Tracking

To assess the trajectory tracking capabilities of our proposed hybrid control strategy, we assume a continuous and differentiable desired tracking trajectory. The chosen trajectory is defined as $x_d=8sin\left(0.1t\right),y_d=8sin\left(0.1t+{\displaystyle \frac{1}{2}}\pi \right), \mathrm{a}\mathrm{n}\mathrm{d} z_d=-8+0.2t$, with a constant desired yaw angle ${\psi }_d$ of zero. Initially, the body is positioned at $P^E={[0 0-8]}^T$, while the velocities $V^E={\mathsf{\Theta }}^E={\mathsf{\Omega }}^B=0^{3\times 1}$. We conduct numerical simulations in the absence of disturbances, utilizing the control parameters listed in

Table 2. Controller parameters.

Symbol	Value
$k_t$	0.6
$G_{\mathsf{\Omega }}$	$diag(1, 1, 1)$
$A_{\mathrm{P}}$	$diag(-0.4,-0.4,-0.4)$
$B_{\mathrm{P}}$	$diag(0.8 , 0.8 , 0.8)$
$D_{\mathrm{P}}$	$diag(-0.8 ,-0.8 ,-0.8)$
$A_{\mathrm{V}}$	$diag(-0.25,-0.25,-0.25)$
$B_{\mathrm{V}}$	$diag(1 , 1 , 1)$
$D_{\mathrm{V}}$	$diag(-1 ,-1 ,-1)$
$A_{\mathrm{S}}$	$diag(-0.825,-0.825,-0.825)$
$B_{\mathrm{S}}$	$diag(1.8 ,1.8 , 1.8)$
$D_{\mathrm{S}}$	$diag(-1.8 ,-1.8 ,-1.8)$
$K_{\mathrm{P}}$	$diag(1.48 , 1.48 , 1.2)$
$K_V$	$diag(6 , 6 , 4.8)$
$a_1$	$diag(0.1 , 0.1 , 0.1)$
$a_2$	$diag(0.01 , 0.01 , 0.01)$

. Figure 4 illustrates the performance of our proposed control method, which exhibits slight deviations compared to other control methods [32,33,34].

In comparison to other control methods, the proposed approach demonstrates superior trajectory tracking performance. While the traditional backstepping control strategy results in a shorter time for the unmanned aerial vehicle to reach the desired trajectory compared to the bioinspired backstepping control strategy, Figure 5 shows that the velocity commands generated by the former exhibit abrupt changes. On the other hand, the bioinspired hybrid control method produces smoother velocity commands without any jumps. The abrupt change in velocity commands is a crucial issue for practical applications in unmanned aerial vehicles because it demands the immediate attainment of unrealistic velocities and torques when there is a significant initial error, which is not achievable with existing actuators. Furthermore, this issue may pose a risk of crashes during flight. Therefore, the bioinspired backstepping control strategy effectively addresses this problem.

Furthermore, from Figure 6a, it can be observed that the traditional sliding mode control strategy suffers from control signal chattering, which imposes a significant burden on the onboard actuators. In contrast, the proposed bioinspired sliding mode controller effectively avoids this issue and provides smooth torque control inputs. The combination of saturation function and sliding mode control is a common approach that can effectively mitigate the chattering problem associated with some traditional sliding mode control methods. However, when the tracking error $e_{\mathsf{\Omega }}$ is outside the interval defined by $B_{\mathsf{\Omega }}$ and $D_{\mathsf{\Omega }}$, the control signal may still exhibit chattering. When the unmanned aerial vehicle operates in a complex environment, it is challenging to tune the control parameters of the saturation function to achieve satisfactory performance under varying flight conditions. As shown in Figure 6b, although the chattering problem is partially addressed, some chattering still persists. Hence, the saturation function alone cannot fully resolve the chattering issue in traditional sliding mode control, while the bioinspired sliding mode control strategy, as observed from Figure 6, demonstrates good performance in handling chattering.

4.2. Disturbed Trajectory Tracking

This section expands on the previous experiment by introducing step disturbances to the UAV’s flight along the x, y, and z axes, as illustrated in Figure 7. Figure 8 showcases the tracking performance, revealing that even under these step disturbances, the proposed controller maintains superior tracking capabilities.

Figure 9 shows the acceleration curves of the quadrotor UAV along the x, y, and z axes under three different algorithms.

Table 3. Comparison table of three-axis accelerations for three controllers during disturbances.

Disturbance Occurrence Time (s)	CB&SMC			CB&SAT			BB&BSMC
	x	y	z	x	y	z	x	y	z
0	85.06	0.15	40.56	82.52	0.01	41.96	16.17	−0.01	21.80
10	−6.95	−10.19	−3.60	−6.15	−3.91	5.57	−5.34	0.90	2.16
30	−6.82	−5.20	−1.26	−3.13	−2.98	1.89	4.43	0.18	2.25
40	−5.88	−3.90	−1.78	−3.38	−2.44	1.89	1.03	1.75	1.79
60	−4.69	−5.50	−1.72	−2.02	−3.35	1.88	−1.35	0.16	1.74

summarizes the acceleration values on the x, y, and z axes at the moment when disturbances appear in Figure 9. The results indicate that when step disturbances occur, our proposed algorithm produces the smallest acceleration values on each axis. This finding suggests that the proposed algorithm can effectively suppress the overly aggressive control actions that might be generated by the backstepping controller.

Figure 10a,b illustrate the comparison of thrust and three-axis torque generated by saturated SMC, traditional SMC, and our proposed algorithm. The figures clearly show that our algorithm’s total thrust chattering range is [2.08, 1.46], with nearly zero three-axis torque chattering. In contrast, traditional SMC exhibits chattering ranges of [2.24, 1.80] for total thrust and [0.019, −0.019], [0.02, −0.02], and [0.038, −0.031] for the three-axis torque, respectively. Saturated SMC displays thrust and three-axis torque chattering ranges of [2.21, 1.74], [0.017, −0.017], [0.014, −0.014], and [0.001, −0.001], respectively. These data convincingly demonstrate that our improved controller significantly mitigates the chattering problem inherent in SMC while achieving superior disturbance suppression compared to saturated SMC.

5. Conclusions

The control strategy based on the bioinspired neural dynamics model proposed in this paper demonstrates excellent trajectory tracking performance, as verified through comparative simulation experiments. This approach effectively addresses two key issues: the velocity jump problem inherent in backstepping control methods and the control signal chattering issue common in sliding mode control. As a result, it is better suited for operating motors with existing input–output constraints. Moreover, under disturbance conditions, this strategy can effectively eliminate tracking errors, making it ideal for precise tracking tasks in complex real-world environments. However, it is crucial to note that the assumptions made in modeling the multi-rotor UAV system and the effects of actuator delays in practical applications cannot be overlooked. When implementing this strategy, it is essential to evaluate the applicability of these assumptions based on specific situations and modify or extend them as necessary.