1. Introduction
Quadrotor unmanned aerial vehicles (UAVs) are well-suited for various applications, including vertical take-off 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 [3-7]. 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 [9-14]. 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-degree-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. While attempts have been made to address the shaking problem in existing methods by substituting the shaking term with saturation and tanh functions [
20], 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 [
21], refers to the utilization of electrical circuit elements in a patch of membrane within a biological neural system. Yang [
22] introduced this bioinspired neural dynamics to the field of robotics, while Zhu [
23] 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.
The control system for quadrotor unmanned aerial vehicles (UAVs) typically consists of two sub-systems: an outer-loop position control subsystem and an inner-loop attitude control subsystem. In their previous work [
11,
24], the authors introduced a PID controller for position control and an integral sliding mode controller for attitude control of the quadrotor UAV. Another study [
25] presented a terminal sliding mode position controller and a sliding mode attitude controller. Motivated by the characteristics of neural dynamics, this paper presents a robust nonlinear hybrid control strategy based on neural dynamics for quadrotor UAVs. The overall design of the control system includes based on biological neural dynamics backstepping position control subsystem and based on biological neural dynamics sliding mode attitude control subsystem. These subsystems are built upon the dual-loop control structure introduced in [
26,
27]. The proposed neural dynamics mitigate the speed jump issue commonly encountered in traditional backstepping control methods. Moreover, leveraging the filtering capability of neural dynamics, the control system effectively handles external noise and produces smooth velocity control signals. Furthermore, the integration of neural dynamics with the sliding mode control strategy addresses the control signal chattering problem inherent in traditional sliding mode control, resulting in continuous, smooth, and chattering-free torque control signals.
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 as well.
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.
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 [
26]. Subsequently, Grossberg [
30] further advanced this model to capture real-time adaptive behavior in individuals. Yang and Meng [
22] 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:
In this model,
denotes the neural activity of the i-th neuron.
represents the passive decay rate, while
and
represent the upper and lower bounds of the i-th neuron, respectively. The variables
and
indicate the excitatory and inhibitory inputs to the i-th neuron. By utilizing equation (7), the shunting model for the quadcopter can be defined as follows:
Where,
represents the inputs to the shunting model, which signifies the error between the desired pose and the quadrotor's state. The functions
and
are defined as
respectively.
denotes the output of the shunting model, and for any input
,
is constrained within the limits of
. Given the specific characteristics of the quadcopter drone model, we assume
. Moreover, it is important to highlight that the shunting model exhibits properties similar to those of a low-pass filter, imparting robustness against aerodynamic forces and external disturbances. Subsequent sections will delve into 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:
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
, combined with the velocity error obtained from equation (9) as
. Consequently, the position control inputs for the quadcopter, based on the backstepping control approach, are as follows:
and
, respectively. These parameters are defined as
and
. 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
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:
Where
and
represent the outputs of the shunting model for the quadcopter, replacing the position and velocity error inputs,
and
. The error terms in the virtual control commands must satisfy the following inequality:
and
represent positive constants, while
and
. Consequently, by carefully selecting appropriate desired velocity
and desired acceleration
, the virtual control command
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 generated by the bioinspired backstepping position control system, along with the desired yaw angle , are utilized as inputs to the attitude control system. Subsequently, the bioinspired sliding mode controller generates torque control signals to guide the quadcopter drone model towards the desired state.
The defined sliding surface is expressed as follows:
Where
and
are defined as positive constants. Taking the derivative of equation (13), we have:
From equation (14), we can see that when the system operates on the sliding surface
, we have:
By substituting the quadcopter dynamics model (5) into equation (15), we obtain:
Therefore, the design of the equivalent control law is given by:
Where
represents the compensation for aerodynamic moments by the sliding mode controller, satisfying:
and
, where
represents the upper bound of the aerodynamic moment and
represents the lower bound. Due to the computational challenges in calculating
as stated in equation (17), an approximation can be made by defining
, where
represents a feedback control gain. The conventional sliding mode control law can be expressed as follows:
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:
Where
and
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:
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:
Then, the derivative of
is denoted as:
Splitting the time derivative expression (23) of the candidate function into two components, we have:
and
. Based on the tracking errors and equation (10),
can be written as:
For the second part, ,
, of the Lyapunov function, considering the specific characteristics of the quadrotor unmanned aerial vehicle model, let's assume
and
. Based on the biological neural dynamics model equation (8), the second part can be written as:
Combining equations (24) and (25), we can infer that:
According to the equation (8), we can conclude that:
By using formula (27), we obtain:
From equations (22), (26), and (28), it can be inferred that as , and is negative definite. can only be zero when . 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:
Differentiating the Lyapunov equation, we obtain:
Combining the biological neural dynamics model (8) with equation (30), we have:
Combining formulas (5), formula (31) can be written as:
Where,
, rewriting equation (32), we have:
Based on formulas (27) and (28), we can also observe that
. From the biological neural dynamics model (8), as , , and also tends to 0. It follows that , is negative definite, and can only be zero when . 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:
The results obtained from equations (26) and (33) can prove that the entire control system is asymptotically stable.