Research on a Distributed Cooperative Guidance Law for Obstacle Avoidance and Synchronized Arrival in UAV Swarms

Abstract

In response to the issue where the original synchronization time becomes inapplicable for UAV swarms after temporal consistency convergence due to obstacle avoidance, a new distributed consultative temporal consistency guidance law that takes into account threat avoidance has been proposed. Firstly, a six-degree-of-freedom dynamic model and a guidance control model for unmanned aerial vehicles (UAVs) are established, and the guidance commands are decomposed into control signals for the pitch and yaw planes. Secondly, based on the theory of dynamic inversion control, a temporal consistency guidance law for a single UAV is constructed. On the other hand, an improved artificial potential field theory is used and integrated with a predictive correction network to generate guidance commands for threat avoidance. A threshold smoothing method is employed to integrate the two guidance systems, and a cluster consultation mechanism is introduced to design a two-layer temporal synchronization architecture, which negotiates to change the synchronization time of the swarm to achieve the convergence of consistency once again. Finally, in typical application scenarios, simulation verification demonstrates the effectiveness of the control method proposed in this paper. The proposed control method achieves the guidance of UAV formations to synchronize their arrival at the target location under complex threat conditions.

1. Introduction

In complex obstacle avoidance and threat area evasion scenarios, there is often a requirement for UAV (unmanned aerial vehicle) swarms to synchronize their arrival at targets from multiple directions and at various distances. Currently, interception techniques for low-cost, low-speed small UAVs have become relatively mature, which poses higher technical demands for UAV swarms to execute high-precision synchronized saturation strikes [1]. Moreover, when reconnaissance UAV swarms are performing dispersed surveillance missions, their dispersed positions and the potential for encountering sudden threats that need to be evaded during the approach to the target, all present greater challenges to the swarm’s collaborative combat capabilities. Therefore, exploring strategies for synchronized and precise targeting in a dynamic threat environment has become a cutting-edge issue and a key technical challenge in the field of collaborative combat research for UAV swarms. This paper focuses on the study of distributed collaborative arrival guidance methods for UAV swarms in scenarios with complex obstacles to be circumnavigated and threats to be evaded. Before this, we will provide a brief overview of the related works in the field of UAVs in three areas: the swarm’s constrained guidance laws, UAV route planning, and UAV swarm collision avoidance technology.

1.1. Related Works

Firstly, regarding the aspect of a swarm’s constrained guidance laws, guiding a swarm of drones to a target under certain constraints is actually a problem that traditional guidance laws need to solve. Driven by increasingly diverse combat requirements, many scholars have studied guidance laws that differ from the traditional proportional navigation guidance (PNG) and meet more complex constraints. Some of the more classic ones include the impact angle constraint guidance laws (IACGs) and the impact time control guidance laws (ITCGs). Currently, there is a lot of research on ITCG for constant speed UAVs, which can generally be divided into two categories. One category is to extend the guidance law that can achieve time consistency for a single UAV and a swarm UAV system like ref. [2]. In this case, in fact, each UAV does not exchange information, but adjusts its flight trajectory by calculating its own time to go, without achieving real-time online collaboration. The other category is that UAVs continuously exchange information during flight, adjusting their own flight time by obtaining the remaining time of the other UAVs. Ref. [3] uses a distributed strategy to design the guidance law, ensuring that each UAV in the swarm maintains a consistent time to go, which requires the UAV communication topology to be fully connected; Ref. [4] use a leader–follower architecture to keep the follower’s remaining time consistent with the leader’s, but there is still a problem with the inability to adjust the strike time. Due to the development and application of consensus-based collaborative control theory in recent years, the consensus convergence of the remaining flight time of multiple aircraft can be regarded as a typical application of consensus-based collaborative control, and there is a natural commonality between the two in terms of principles and ideas. For this reason, refs. [5,6,7] have constructed a dynamic model of the remaining flight time, transformed it into a low-order nonlinear model, and then used the theory of consensus tracking control to study the attack-time consensus collaborative guidance methods in leaderless/leader-following modes. Although the aforementioned research can be used to enable multiple drones to attack the same target simultaneously, it all requires the prior designation of a suitable attack time, without considering that drones can communicate with each other, as well as with the ground station, to modify this attack time. Moreover, during flight, each drone is inevitably subject to sudden threat situations or other disturbances, which may necessitate threat avoidance maneuvers or other actions, preventing the drone swarm from arriving at the target at the designated attack time and thus failing to achieve the expected effect. A better solution to this problem is a collaborative guidance method based on consensus algorithms, where the expected attack time depends on initial conditions or the leading UAV, possessing a certain degree of self-organization, and is relatively convenient to implement. However, it still cannot modify the synchronized arrival time online in a distributed manner.

Secondly, regarding the aspect of UAV route planning, in the research on trajectory planning for threat avoidance, most current studies are founded on a grid-based graph search (such as A*) like in refs. [8,9,10], discrete sampling search (such as RRT) like in refs. [11,12,13], or grid heuristic intelligent planning like in refs. [14,15,16,17,18,19,20]. The aforementioned studies are all founded on grid-based searches, which generally output a sequence of continuous waypoints that make up a trajectory. This requires the UAV’s onboard computer to process the waypoint information to output guidance control commands. The discrete waypoint control commands also bring about the issue of abrupt overload smoothing. Among these, Cai (ref. [21]), aiming to shorten the UAV’s target-tracking planning time and enhance planning stability, proposed an improved A*-based trajectory planning algorithm. Compared with trajectory planning using yaw control laws, this algorithm has a smaller oscillation amplitude and better results, to some extent, solving the control command oscillation problem brought about by grid-based searches. On the other hand, the aforementioned studies are based on offline trajectory search schemes. A*-type algorithms yield optimal trajectories but have longer search times; intelligent planning algorithms search are faster but have poor generalization; RRT-type algorithms balance search speed and generalization performance, but the resulting waypoints often have redundancy and more route smoothness issues. For real-time searches, artificial potential field theory quickly constructs repulsive and attractive force fields, allowing UAVs to avoid obstacles and plan safe and feasible flight trajectories, solving the problem of local path planning. The current improved artificial potential field algorithm [22] has the characteristics of a fast planning speed and good real-time performance online, but it also has the problem of only indicating the direction of threat avoidance, not providing specific acceleration control commands. Beyond the flight path planning algorithms themselves, from an engineering perspective, many scholars have studied the impact of different types of obstacles and environmental disturbances on the performance of the algorithms. For instance, as noted in the literature [23], complex threat areas like concave threat areas and wind disturbance can diminish the performance of path planning algorithms, and the literature [24] has considered the performance of path planning algorithms faced with moving obstacles.

Thirdly, regarding the aspect of collision avoidance technology, this can greatly enhance the safety and stability of the swarm, and has become an important research topic in the field of cooperative control technology for UAV swarms. Coordinated obstacle avoidance refers to a scenario within a swarm where some UAVs detect obstacles and, through communication within the swarm, those UAVs that have not sensed the obstacles can also avoid them, achieving the goal of overall obstacle avoidance by the swarm. Currently, commonly used coordinated obstacle avoidance technologies often adopt a form that integrates formation control methods with obstacle avoidance algorithms, such as consensus and artificial potential field methods, consensus and particle swarm optimization algorithms, pigeon-inspired optimization, and decentralized model predictive control methods. Ref. [25] improved the consensus method by adding maneuverability constraints for the UAVs and, based on the improved consensus method and particle swarm optimization algorithm, proposed the predictive control of the UAV motion model under different initial conditions, achieving an efficient obstacle avoidance capabilities for UAV swarms against both static and dynamic obstacles. Ref. [26] proposed a control algorithm that combines hybrid genetic simulated annealing with a consensus protocol, which can quickly determine the formation position of UAVs during formation reconstruction, reduce reconstruction time, and avoid collisions between UAVs. Ref. [27] designed a decentralized guidance control strategy aimed at maintaining a given connection topology and a specified distance between UAVs, and combined this with the model predictive control method, achieving obstacle avoidance and collision avoidance functions for the swarm in the presence of obstacles. Ref. [28], in response to the control and reconstruction problems of UAV swarms in a convex bounded polygonal area, proposed a decentralized algorithm based on a linear model predictive controller, which can automatically leave the formation and avoid collisions with other UAVs when some UAVs fail or are performing other tasks.

1.2. A Brief Summary of Our Work

In summary, in the context of complex obstacle avoidance and threat area circumnavigation, and aiming at resolving the problem of synchronized arrival for the swarm, there is still much work to be explored. This paper has designed a new consultative temporal consistency guidance law that takes into account threat avoidance. The main innovations of this paper are as follows: First, a control guidance law under the constraint of arrival time is designed using the method of dynamic inversion control. On the other hand, based on artificial potential field theory, an evasion potential field for sudden threat situations is established, and the direction of the potential field evasion is combined with predictive correction theory to generate acceleration guidance commands for threat avoidance, and a threat distance threshold is designed to effectively smooth the integration of the two guidance systems. Second, in response to the difficulty that the original synchronization time is not applicable after circumnavigation of sudden threat situations, a cluster consultation mechanism is introduced. After the time error of UAVs arriving at the target converges, if a sudden situation is encountered again, the simultaneous arrival time is delayed via communication, and ultimately, a distributed integrated collaborative guidance law for obstacle avoidance and synchronized arrival is realized.

2. Fixed-Wing UAV Dynamics and Guidance Control System Modeling

2.1. Six-DOF Fixed-Wing UAV Dynamic Model

According to ref. [29], a six-degree-of-freedom 12-state flight dynamics model equation system for unmanned aerial vehicles is established as follows:

(1)$\left(\begin{array}{c}{\dot{p}}_n\\ {\dot{p}}_e\\ {\dot{p}}_d\end{array}\right)=\left(\begin{array}{ccc}{c}_{\theta }{c}_{\psi }& s_{\varphi }{s}_{\theta }{c}_{\psi }-c_{\varphi }{s}_{\psi }& c_{\varphi }{s}_{\theta }{c}_{\psi }+s_{\varphi }{s}_{\psi }\\ c_{\theta }{s}_{\psi }& s_{\varphi }{s}_{\theta }{c}_{\psi }+c_{\varphi }{c}_{\psi }& c_{\varphi }{s}_{\theta }{c}_{\psi }-s_{\varphi }{s}_{\psi }\\ -s_{\theta }& s_{\varphi }{c}_{\theta }& c_{\varphi }{c}_{\theta }\end{array}\right)\left(\begin{array}{c}u\\ v\\ w\end{array}\right)$

(2)$\left(\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right)=\left(\begin{array}{c}rv-qw\\ pw-ru\\ qu-pv\end{array}\right)+{\displaystyle \frac{1}{m}}\left(\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right)$

(3)$\left(\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right)=\left(\begin{array}{ccc}1& \mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{t}\mathrm{a}\mathrm{n}\theta & \mathrm{c}\mathrm{o}\mathrm{s}\varphi \mathrm{t}\mathrm{a}\mathrm{n}\theta \\ 0& \mathrm{c}\mathrm{o}\mathrm{s}\varphi & -\mathrm{s}\mathrm{i}\mathrm{n}\varphi \\ 0& \mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{s}\mathrm{e}\mathrm{c}\theta & \mathrm{c}\mathrm{o}\mathrm{s}\varphi \mathrm{s}\mathrm{e}\mathrm{c}\theta \end{array}\right)\left(\begin{array}{c}p\\ q\\ r\end{array}\right)$

(4)$\left(\begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right)=\left(\begin{array}{c}{\Gamma }_{1}pq-{\Gamma }_{2}qr\\ {\Gamma }_{5}pr-{\Gamma }_6\left(p^2-r^2\right)\\ {\Gamma }_{7}pq-{\Gamma }_{1}qr\end{array}\right)+\left(\begin{array}{c}{\Gamma }_{3}l+{\Gamma }_{4}n\\ {\displaystyle \frac{1}{{\Gamma }_5}}m\\ {\Gamma }_{4}l+{\Gamma }_{8}n\end{array}\right)$

Here, $p_n,p_e,p_d$ represent the rectangular coordinates in the north-east-down coordinate system established with a certain point in space as the coordinate origin, $\varphi ,\theta ,\psi$, respectively, represent the roll angle, pitch angle, and yaw angle of the aircraft, $u,v,w$ represent the velocities in the directions of the aircraft’s three axes, $p,q,r$ are the angular velocities around the aircraft’s axes, $f_x,f_y,f_z$ are the forces acting on the aircraft’s axes, $f=f_a+f_b$, $l,m,n$ represent the moments around the aircraft’s axes, $m$ represents the mass of the aircraft, and ${\Gamma }_1$~${\Gamma }_9$ represent the elements of the inertia matrix.

The forces acting on the aircraft itself can be simplified and summarized as the components of gravity along the axes of the aircraft coordinate system, with thrust acting only on the x-axis of the aircraft coordinate system:

(5)$\left(\begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right)=\left(\begin{array}{c}{\Gamma }_{1}pq-{\Gamma }_{2}qr\\ {\Gamma }_{5}pr-{\Gamma }_6\left(p^2-r^2\right)\\ {\Gamma }_{7}pq-{\Gamma }_{1}qr\end{array}\right)+\left(\begin{array}{c}{\Gamma }_{3}l+{\Gamma }_{4}n\\ {\displaystyle \frac{1}{{\Gamma }_5}}m\\ {\Gamma }_{4}l+{\Gamma }_{8}n\end{array}\right)$

The aerodynamic forces and moments are described by the following model, where $\rho$ is described by the Jacchia–Robert atmospheric environment mathematical model:

(6)$\left(\begin{array}{c}{f}_{xa}\\ f_{ya}\\ f_{za}\end{array}\right)={\displaystyle \frac{1}{2}}\rho V_a^{2}S\left(\begin{array}{c}{C}_{\chi }\left(\alpha \right)+C_{{\chi }_q}\left(\alpha \right){\displaystyle \frac{c}{2V_a}}q+C_{{\chi }_{\delta e}}\left(\alpha \right){\delta }_e\\ C_{Y_0}+C_{Y_{\beta }}\beta +C_{Y_P}{\displaystyle \frac{b}{2V_a}}p+C_Y{\displaystyle \frac{b}{2V_a}}r+C_{Y_{{\delta }_a}}{\delta }_a+C_{Y_{{\delta }_r}}{\delta }_r\\ C_z\left(\alpha \right)+C_{Z_q}\left(\alpha \right){\displaystyle \frac{c}{2V_a}}q+C_{z_{{\delta }_e}}\left(\alpha \right){\delta }_e\end{array}\right)$

(7)$\left(\begin{array}{c}l\\ m\\ n\end{array}\right)={\displaystyle \frac{1}{2}}\rho V_a^{2}S\left(\begin{array}{c}b\left[C_{l_0}+C_{l_{\beta }}\beta +C_{l_p}{\displaystyle \frac{b}{2V_a}}p+C_{l_r}{\displaystyle \frac{b}{2V_a}}r+C_{l_{{\delta }_a}}{\delta }_a+C_{l_{{\delta }_r}}{\delta }_r\right]\\ c\left[C_{m_0}+C_{m_{\alpha }}\alpha +C_{m_q}{\displaystyle \frac{c}{2V_a}}q+C_{m{\delta }_e}{\delta }_e\right]\\ b\left[C_{n_0}+C_{n_{\beta }}\beta +C_{n_P}{\displaystyle \frac{b}{2V_a}}p+C_{n_r}{\displaystyle \frac{b}{2V_a}}r+C_{n_{{\delta }_a}}{\delta }_a+C_{n_{{\delta }_r}}{\delta }_r\right]\end{array}\right)$

Here, ${\delta }_a,{\delta }_e,{\delta }_r$ represent the deflection angles of the equivalent ailerons, elevators, and rudder of the UAV, respectively. $\alpha ,\beta$ are the angle of attack and sideslip angle of the UAV, $\rho$ is the current air density, $b,c,S$ are the aerodynamic characteristic coefficients of the UAV, and coefficients such as $C_{l_{\beta }}$ are aerodynamic derivatives. The main parameters of the UAV model and its detailed description can be seen in Figure 1 and the Abbreviations section.

2.2. Integrated Guidance and Control Model for UAV

The simultaneous arrival problem in a small airspace is decoupled into the pitch (altitude) plane and the yaw (horizontal) plane. In the pitch plane, a height controller is designed to ensure that the swarm completes altitude convergence before simultaneous arrival. Considering direct control of altitude, a PD altitude control law is designed, approximating the UAV dynamic system as a second-order system without steady-state error, designing the overload command below:

(8)$n_{kzc}=g\cdot \cos \theta +{\displaystyle \frac{-2\xi {\omega }_n\mathsf{\Delta }\dot{H}-{\omega }_n^2\mathsf{\Delta }H+\dot{V}\sin \theta }{\cos \theta }}$

Here, $n_{kzc}$ is the overload control command of the UAV along the z-axis in the trajectory coordinate system, $g\cdot \cos \theta$ is the gravity correction term, and $\xi {,\omega }_n$ approximates the damping and natural frequency of the control system and dynamic model as a whole, which are adjustable parameters.

The horizontal-plane control command $n_{kyc}$ is given by the simultaneous arrival guidance law in the yaw plane. Assuming the UAV is at a constant speed, $n_{kxc}$ is directly given by the velocity hold loop and the overall overload vector is represented as:

(9)$n_{kc}=\left[n_{kxc},n_{kyc},n_{kzc}\right]$

Using the bank to turn (BTT) control method, the main normal overload control command $n_c$ and the roll angle command ${\gamma }_c$ are calculated as follows:

(10)$\left\{\begin{array}{c}{n}_c=sign\left(n_{kzc}\right)\sqrt{n_{kyc}^2+n_{kzc}^2}\\ {\gamma }_c=\arctan {\displaystyle \frac{n_{kyc}}{{-n}_{kzc}}}\end{array}\right.$

The BTT control system structure is designed as shown in Figure 2, Figure 3 and Figure 4, where $n_c$ is the required main normal overload command for the pitch channel, ${\omega }_{y_1}$ is the pitch rate, ${\delta }_{yc}$ is the elevator deflection command, ${\delta }_y$ is the actual elevator deflection. $K_{PP},K_{PI},K_{PD}$ are adjustable parameters. The pitch overload control equation is described as follows:

(11)${\delta }_{jc}=K_{PP}\left(n_c-n\right)+K_{PI}\int \left(n_c-n\right)dt-K_{PD}{\omega }_{j_1}$

The roll attitude control system structure is designed as follows, where $\gamma$ is the roll angle of the UAV measured by the sensor, ${\gamma }_c$ is the roll angle command, $\dot{\gamma }$ is the roll rate, ${\delta }_{xc}$ is the aileron deflection command, ${\delta }_x$ is the actual aileron deflection, and $K_{RP},K_{RI},K_{RD}$ are adjustable parameters. The roll angle control equation is described as follows:

(12)${\delta }_{ic}=K_{RP}\left({\varphi }_c-\varphi \right)+K_{RI}\int \left({\varphi }_c-\varphi \right)dt-K_{RD}\dot{\varphi }$

The yaw channel adopts a stable control method that suppresses the sideslip angle; that is, the sideslip angle is controlled to be 0 throughout the process. The roll attitude control system’s structure is designed as follows, where $\beta$ is the sideslip angle of the UAV measured by the sensor, ${\omega }_{z1}$ is the yaw rate, ${\delta }_{zc}$ is the rudder deflection command, ${\delta }_z$ is the actual rudder deflection, and $K_{YP}$, $K_{YI}, K_{YD}$ are adjustable parameters. The sideslip inhibition control equation is described as follows:

(13)${\delta }_{ic}=-K_{YP}\mathsf{\beta }-K_{YI}\int \mathsf{\beta }dt-K_{YD}{\omega }_{k_1}$

3. Design of Distributed Guidance and Control Algorithms for Obstacle Avoidance and Impact Time Control

3.1. Design of an ITCG Law Based on Dynamic Inversion Control Method

This section introduces the simultaneous arrival guidance method to the yaw plane. The guidance law mainly focuses on situations where there is an error between the remaining time to reach the target position and the remaining time negotiated by the swarm. By using the guidance law overload command, the UAV is maneuvered appropriately in space to consume this error. Therefore, to simplify the research and based on the engineering background, the following reasonable assumptions are made:

• The speed of the UAV is constant;

• The field of view constraints of each UAV are not considered;

• The target or assembly position remains unchanged;

The relative position relationship between the single aircraft and the target is depicted in Figure 5, where $\theta$ and $q$ represent the angles of the drone’s velocity vector and the vector from the aircraft to the target with respect to the Ox-axis. $\phi$ is defined as the lead angles, with all being positive in a counterclockwise direction.

The remaining time for each UAV to reach the target in a straight line is calculated as follows:

(14)$t_{togo}={\displaystyle \frac{R}{V}}$

Here, $t_{togo}$ is an estimated value for the time taken to reach the target. There are many studies in this field that estimate time to go. For the convenience of subsequent guidance law design, we have adopted an estimation method based on ${\displaystyle \frac{R}{V}}$. From a physical perspective, this estimated value ignores the turning time during the UAV guidance process and estimates the remaining flight time after the UAV directly points towards the target. A guidance law designed based on this formula has been verified as feasible in subsequent simulation experiments. That is, it is the time required for the UAV to fly straight towards the target on the horizontal plane, and the remaining time error is expressed as:

(15)$t_{error}=t_{CA}-t-{\displaystyle \frac{R}{V}}$

Here, $t_{CA}$ represents the time agreed upon by the swarm for arrival at the target, and $t$ is the current time.

Taking the derivative of Equation (15), we obtain:

(16)${\dot{t}}_{error}=-1-{\displaystyle \frac{\dot{R}}{V}}=-1+\mathrm{c}\mathrm{o}\mathrm{s}\phi$

In plane kinematics, it is also easy to obtain:

(17)$\dot{\phi }={\displaystyle \frac{V\mathrm{s}\mathrm{i}\mathrm{n}\phi }{R}}-{\displaystyle \frac{a_n}{V}}$

The aim of arrival time control is for $t_{error}\to 0$, and the flight trajectory tends to be straight; that is, as $t_{error}\to 0$, $\phi \to 0$. Furthermore, according to Equation (16), $\phi \to 0$ is equivalent to ${\dot{t}}_{error}\to 0$. Equations (16) and (17) indicate that the control quantity, the normal acceleration $a_n$, directly controls $\phi$, and $\phi$ indirectly controls $t_{error}$. That is, Equation (16) represents the nonlinear slow subsystem, and Equation (17) represents the nonlinear fast subsystem.

The method of dynamic inversion control design with time scale separation is utilized, based on the aforementioned fast–slow subsystems, to design the normal acceleration control command $a_n$. The desired slow subsystem is:

(18)${\dot{t}}_{error}^{des}=-k_1{t}_{error}$

In the formula, $k_1$ represents the desired bandwidth of the slow subsystem. When the lead angle is equal to the desired lead angle command ${\phi }_c$, the slow subsystem satisfies Equation (13); that is:

(19)$\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_c=1-k_1{t}_{error}$

For the guidance system, it is desired that the UAV always approaches the target. By rearranging Equation (19) and taking its derivative, we obtain:

(20)${\dot{\phi }}_c=\left\{\begin{array}{c}{\displaystyle \frac{k_1\left(-1+\mathrm{c}\mathrm{o}\mathrm{s}\phi \right)}{\sqrt{1-{\left(1-k_1{t}_{error}\right)}^2}}}\phi \in \left[0,{\displaystyle \frac{\pi }{2}}\right]\\ -{\displaystyle \frac{k_1\left(1-\mathrm{c}\mathrm{o}\mathrm{s}\phi \right)}{\sqrt{1-{\left(1-k_1{t}_{error}\right)}^2}}}\phi \in \left(-{\displaystyle \frac{\pi }{2}},0\right)\end{array}\right.$

To achieve the desired dynamics (18) for the slow subsystem (16), it is necessary to make the lead angle converge quickly to the command value ${\phi }_c$. Therefore, the desired fast subsystem is designed as follows:

(21)${\dot{\phi }}_{des}={\dot{\phi }}^c-k_2\left(\phi -{\phi }^c\right)$

$k_2$ represents the bandwidth of the desired fast subsystem.

To make the real fast subsystem dynamics (17) approach the desired dynamics (21), let $\dot{\phi }={\dot{\phi }}_{des}$, and substitute into Equations (19) and (20), thus we obtain:

(22)$a_n=\left\{\begin{array}{c}{\displaystyle \frac{V^2\mathrm{s}\mathrm{i}\mathrm{n}\phi }{R}}+k_{2}V\left[\phi -\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(1-k_1{t}_{error}\right)-{\displaystyle \frac{k_{1}V\left(-1+\mathrm{c}\mathrm{o}\mathrm{s}\phi \right)}{\sqrt{1-{\left(1-k_1{t}_{error}\right)}^2}}}\right]\phi \in \left[0,{\displaystyle \frac{\pi }{2}}\right]\\ {\displaystyle \frac{V^2\mathrm{s}\mathrm{i}\mathrm{n}\phi }{R}}+k_{2}V\left[\phi +\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(1-k_1{t}_{error}\right)+{\displaystyle \frac{k_{1}V\left(-1+\mathrm{c}\mathrm{o}\mathrm{s}\phi \right)}{\sqrt{1-{\left(1-k_1{t}_{error}\right)}^2}}}\right]\phi \in \left(-{\displaystyle \frac{\pi }{2}},0\right)\end{array}\right.$

Equation (22) is the nonlinear dynamic inversion guidance law derived in this section for attack time control. The guidance law ensures that the fast–slow subsystems can achieve the desired dynamics, thereby making $t_{error}\to 0$. In this process, Equation (20) ensures that $\phi \to 0$. That is, as $t_{togo}\to {(t}_{CA}-t$), at the end of the flight, the UAV’s trajectory also gradually becomes straight.

3.2. Design of an Obstacle Avoidance Guidance Law Based on the Improved APF Method

When a UAV swarm approaches a target using the aforementioned guidance laws, the situational awareness of a single aircraft or the external situational awareness of the swarm detects an emergent threat that needs to be evaded, such as an air defense interception position or a high-rise building that was not surveyed in advance. In this case, it is necessary to calculate the overflight overload instruction on the horizontal plane to bypass the threat. Generally, threat areas can be abstracted and processed as circular or polygonal regions. To simplify the research, consider the danger area as a circular region with a specified radius, or a rectangular region with specified length and width. The repulsive function of the artificial potential field also has the characteristic of changing the magnitude and direction of its repulsive effect with distance, and by introducing the attractive effect of the assembly area on each UAV, an evasion guidance strategy based on artificial potential field theory can be formed.

The artificial potential field-based obstacle avoidance method is primarily applied in the field of obstacle path planning for vehicles or robots. The basic idea is to construct a repulsive field near obstacles and an attractive potential field near the target. The resultant force of attraction and repulsion then guides the motion of the controlled object, as shown in Figure 6, and Figure 7 shows the UAV trajectory guided by APF in the potential field.

The attractive potential field is constructed by the target and is mainly related to the distance between the aircraft and the target point. The greater the distance, the greater the attractive force exerted by the target on the aircraft; the smaller the distance, the smaller the attractive effect from the target. The attractive potential field function is generally described as:

(23)${\mathrm{U}}_{\mathrm{a}\mathrm{t}\mathrm{t}}\left(\mathrm{q}\right)={\displaystyle \frac{1}{2}}\mathsf{\eta }{\mathsf{\rho }}^2\left(\mathrm{q},{\mathrm{q}}_{\mathrm{g}}\right)$

In the formula, $\mathsf{\eta }$ is the positive proportional gain coefficient, $\mathsf{\rho }$ is a vector representing the Euclidean distance from the UAV’s position to the target’s position, directed from the UAV towards the target. The traction force experienced is the negative gradient of the UAV in this attractive field, representing the direction in which the attractive potential field function decreases most rapidly. The main goal is to guide the UAV to move quickly towards the target’s position.

(24)${\mathrm{U}}_{\mathrm{a}\mathrm{t}\mathrm{t}}\left(\mathrm{q}\right)=-\nabla {\mathrm{U}}_{\mathrm{a}\mathrm{t}\mathrm{t}}\left(\mathrm{q}\right)=-\mathsf{\eta }\mathsf{\rho }\left(\mathrm{q},{\mathrm{q}}_{\mathrm{g}}\right)$

The repulsive potential field is constructed based on the relationship between the threat area and the UAV. When the UAV is not within the influence range of the threat area, its potential energy value is 0. Once the UAV enters the influence range of the obstacle, the closer the distance between the two, the greater the potential energy value the UAV experiences, and vice versa, the greater the distance, the smaller the potential energy value. Trajectories of a swarm of UAVs guided by APF in a potential field is shown in Figure 8.

Traditional APF methods primarily suffer from issues of target inaccessibility and becoming trapped in local optima. To address these issues, an improved repulsive potential field function is designed to resolve the problems of local optima and target inaccessibility. According to the ref. [30], a regulatory factor is introduced into the obstacle repulsive field model of the traditional artificial potential field method, ensuring that the vehicle only experiences a simultaneous reduction in both attractive and repulsive forces to zero when it reaches the target point, thereby solving the issues of inaccessibility and local optima.

Improved repulsive field function is:

(25)${\mathrm{U}}_{\mathrm{r}\mathrm{e}\mathrm{q}}\left(\mathrm{q}\right)=\left\{\begin{array}{l}{\displaystyle \frac{1}{2}}\mathrm{k}{\left({\displaystyle \frac{1}{\rho (q,q_0)}}-{\displaystyle \frac{1}{{\rho }_0}}\right)}^2{\rho }_g^{\mathrm{n}},0\le \rho \left(\mathrm{q},{\mathrm{q}}_0\right)\le {\rho }_0\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\rho \left(\mathrm{q},{\mathrm{q}}_0\right)>{\rho }_0\end{array}\right.$

The term ${\rho }_g^{\mathrm{n}}$ represents the distance from the target, where $\mathrm{n}$ is a selectable positive constant.

Combining the attractive field function, we obtain:

(26)$\begin{array}{l}{\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}}={\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}1}+{\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}2}\\ {\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}1}=\mathrm{k}\left({\displaystyle \frac{1}{\rho (\mathrm{q},{\mathrm{q}}_0)}}-{\displaystyle \frac{1}{{\rho }_0}}\right){\displaystyle \frac{{\rho }_{\mathrm{g}}^{\mathrm{n}}}{{\rho }^2(\mathrm{q},{\mathrm{q}}_0)}}\\ {\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}2}={\displaystyle \frac{\mathrm{n}}{2}}\mathrm{k}{\left({\displaystyle \frac{1}{\rho (\mathrm{q},{\mathrm{q}}_0)}}-{\displaystyle \frac{1}{{\rho }_0}}\right)}^2{\rho }_{\mathrm{g}}^{\mathrm{n}-1}\end{array}$

The direction of ${\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}1}$ is from the obstacle pointing towards the UAV; the direction of ${\mathrm{F}}_{\mathrm{r}\mathrm{e}\mathrm{q}2}$ is from the UAV pointing towards the target point. The improved artificial potential field theory is shown in Figure 9.

It should be noted that the artificial potential field method and its improved versions directly apply forces in the form of vectors to the controlled unit, which is not suitable for the scenarios in this research. Therefore, it is necessary to consider integrating the threat evasion strategy calculated by the artificial potential field method into the guidance system to generate the normal overload required for the UAV to evade threats.

To quickly direct the actual movement direction of the UAV towards the threat evasion direction, the acceleration command is used as the output of the controller loop, with the actual movement direction and the desired direction ${\theta }_c$ as the loop input, to obtain the acceleration command required for threat evasion:

(27)$a_n=K_P\left({\theta }_c-\theta \right)+K_D{\displaystyle \frac{d\left({\theta }_c-\theta \right)}{dt}}$

The controller model’s output of the threat evasion overload command, together with the simultaneous arrival guidance law, forms a combined guidance law. A distance threshold is set, using the overload command for threat evasion provided by the artificial potential field when the UAV is close to a threat, and using the simultaneous arrival guidance law of the dynamic inversion control method when there are no threats around. At the same time, a command smoothing method is used to extend the system switch over a period of time to complete it, eliminating the kinematic impact response caused by different guidance systems, as follows:

(28)$a_n=\rho a_{nc,ITCG}+\left(1-\rho \right)\cdot a_{nc,APF}$

(29)$\rho =\left\{\begin{array}{cc}\begin{array}{l}0\\ {\displaystyle \frac{d-d_{c1}}{d_{c2}-d_{c1}}}\\ 1\end{array}& \begin{array}{l}d\le d_{c1}\\ d_{c1}<d\le d_{c2}\\ d>d_{c2}\end{array}\end{array}\right.$

This command can also be extended to normal acceleration commands in multiple dimensions, and the switching process is shown in Figure 10:

Integrating the design research from Section 3.1 and Section 3.2, the control process of the threat evasion and simultaneous assembly combined guidance law acting on the UAV is shown in Figure 11:

3.3. Architecture of a Two-Layer Time Synchronization Algorithm Based on Distributed Negotiation

Note that if $t_{error}<0$ in Equation (22), the overload command equation is no longer applicable, because the corresponding lead-angle command cannot be solved from Equation (19). From the perspective of planar kinematics, the physical meaning of $t_{error}<0$ is that even if the predicted trajectory of the aircraft directly points to the target without any other maneuvers, the aircraft at the current speed cannot reach the target position within the given time. Generally, the ITCG guidance laws are discussed and studied under the condition of $t_{error}>0$.

However, during flight, aircraft inevitably face external disturbances (this paper takes the avoidance of threats as an example), which may prevent one or more UAVs from reaching the target at the specified attack time according to the ITCG guidance law, thus failing to achieve the expected effect. As an alternative method to achieve the simultaneous arrival of a UAV swarm, the common arrival time can be dynamically adjusted based on the actual flight conditions of each UAV, to negotiate and complete the task of arriving at the target location simultaneously. To this end, it is necessary to appropriately delay the time of the entire swarm’s arrival at the target, ensuring that the predicted $t_{error}$ is always greater than 0.

On the other hand, the synchronization time provided by the ground control station often requires some UAVs to make more maneuvers to consume excess time in order to meet the time synchronization requirements. This is unacceptable from both the perspective of threat avoidance and energy conservation. Therefore, it is necessary for the swarm to negotiate the synchronization time, with the aim of allowing each UAV to arrive as required while keeping the range of maneuvers as small as possible.

Assuming that there are several UAVs participating in the synchronization task, the ITCG-APF designed in the second and third sections has a controllable quantity, which is the desired assembly time $t_{CA}$. Therefore, the coordination variable ξ is taken as the desired guidance time $t_{CA}$.

Obviously, if the desired guidance times for each UAV can reach a consensus; that is, they meet their current motion constraints, then under the guidance of ITCG-APF, each UAV will be able to arrive at the target location simultaneously while completing obstacle avoidance. The control energy $a_n^2$ for each UAV is taken as its own cost function. Since a centralized collaborative strategy is adopted, considering the cost functions of all UAVs comprehensively, the sum of the cost functions of each UAV is taken as the centralized coordination function; that is, the total cost is the sum of all control energies:

(30)$J_T=\stackrel{n}{\sum _{i=1}}{a}_{ni}^2$

The coordination variable $\xi$ is chosen as ${\xi }^{*}$, which minimizes the control energy. From a physical perspective, this means that the guidance time that minimizes the total control energy consumption of the system is the desired guidance time.

(31)${\xi }^{*}=arg\underset{\xi \in \cap t_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(x_i\right)}{\mathrm{m}\mathrm{i}\mathrm{n}}{J}_T=arg\underset{\xi \in \cap t_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(x_i\right)}{\mathrm{m}\mathrm{i}\mathrm{n}}\stackrel{n}{\sum _{i=1}}{a}_{ni}^2$

Inspired by the guidance equation, the analysis shows that the optimal value of the coordination variable is actually the generalized weighted average of the estimated remaining times for each UAV. Ignoring minor terms, a suboptimal value for the coordination variable can be obtained:

(32)${\xi }^{*}=\stackrel{n}{\sum _{i=1}}{\alpha }_i^2{T}_{togoi}/\stackrel{n}{\sum _{i=1}}{\alpha }_i^2$

In the formula, ${\alpha }_i={\displaystyle \frac{60V_i^5}{a_{pi}{R}_{togoi}^3}}$. It should be noted that it is not a strictly optimal solution. This paper uses a centralized coordination strategy to implement the time negotiation process, with the desired synchronization time calculated by the coordination unit.

Multi-UAV collaborative guidance is a control problem with high performance requirements. Collaborative control methods that have been successfully applied to other multi-robot systems often cannot meet the requirements of high real-time performance, low communication volume, and straight end trajectories. A two-layer control structure can be considered to implement collaborative guidance. This two-layer collaborative guidance structure includes: lower-level guidance control and upper-level coordination control. The lower-level guidance control is completed by the local guidance laws located on each UAV, while the upper-level coordination control is implemented by coordination strategies.

The ITCG-APF is used for local guidance control of each UAV, while the coordination algorithm in Equation (32) is used for time coordination. The combination of the two forms a collaborative guidance law. The collaborative framework is shown in the figure below. The ith UAV transmits the necessary information for collaboration to the centralized coordination unit. The centralized coordination unit calculates the coordination variable solution ${\xi }^{*}$ based on a centralized algorithm and then passes it on to the UAVs in the swarm. Each UAV then completes the collaborative task according to local guidance, simultaneously arriving at the target.

In summary, two-layer distributed time negotiation architecture is shown in Figure 12. This article considers simplifying Equation (32). The meaning of Equation (32) is that the coordination variable takes the current control energy of each UAV in the cluster as a weight, and the overall synchronization time will listen to the unit within the cluster that requires a larger control energy (normal overload). From a physical perspective, it can be seen as the cluster tolerating the UAV that is latest to arrive at the target. Based on this idea, directly regarding this coordination variable ${\xi }^{*}$ as the longest $t_{togo}$ among all UAVs in the cluster is considered.

So $t_{CA}$ can be updated by the following formula:

(33)$t_{CA}=max\{{\tilde{t}}_{arrive,1},{\tilde{t}}_{arrive,2},\dots ,{\tilde{t}}_{arrive,N},t_{CA}\}$

This two-layer collaborative guidance structure has a certain degree of universality and provides a feasible solution for the design of UAV collaborative guidance laws. By combining the corresponding guidance laws and coordination strategies for different collaborative missions, the appropriate collaborative guidance laws can be obtained. Integrating Section 3.1, Section 3.2 and Section 3.3, the pseudocode for the distributed guidance and control algorithm for obstacle avoidance and synchronized arrival (TNOA-ITCG) is shown in Algorithm 1:

Algorithm 1: TNOA-ITCG Algorithm

$\mathbf{Input}: \mathit{S}$, the state vector of UAVs in formation and Si represents the state vector of the i-th UAV, ${\mathit{S}}_i=\{x_i,y_i,V_i,{\psi }_i,\mathrm{t}$};    N, number of UAVs in formation;    ${\mathit{S}}_T$$, \mathrm{represents} \mathrm{the} \mathrm{state} \mathrm{vector} \mathrm{of} \mathrm{target}, {\mathit{S}}_T=\{x_t$$,y_t$};    $\mathit{O}\mathit{b}\mathit{s}$, obstacle vector, where the j-th obstacle is represented by $\{x_j,y_j,R_j$};     M, number of obstacles;$\mathbf{Output}: a_n$, represents the normal overload command on the horizontal plane in the current state of the UAV, $a_n=\{a_{n1},\dots ,a_{ni},\dots ,a_{nN}\}$1: for i←{1,…,N} do$2:\hspace{1em}\hspace{1em}\mathrm{Calculate} \mathrm{relationship} \mathrm{with} \mathrm{target} \mathrm{and} \mathrm{obstacle} {\mathit{G}}_i$$\leftarrow \{{\phi }_i$$,R_i$$,{\rho }_{ij}$}$3:\hspace{1em}\hspace{1em}\mathrm{Predict} \mathrm{minimum} \mathrm{arrive} \mathrm{time} {\tilde{t}}_{arrive,i}$$\mathrm{using} \mathrm{Equation} \left(12\right), \mathrm{suggests} t_{error}$ > 04:end$5:\mathrm{Update} t_{CA}$$\leftarrow \max \{{\tilde{t}}_{arrive,1}$$,{\tilde{t}}_{arrive,2}$$,\dots ,{\tilde{t}}_{arrive,N}$$,t_{CA}$}6: for i←{1,…,N} do$7:\hspace{1em}\hspace{1em}\mathrm{Calculate} a_{ni,ITCG}$ using Equation (19)$8:\hspace{1em}\hspace{1em}\mathrm{Calculate} a_{ni,OA}$ using Equation (24)$9:\hspace{1em}\hspace{1em}{a}_{ni}\leftarrow$$\mathrm{Mix}(a_{ni,ITCG}$$,a_{ni,OA}$) using Equation (26)10:end

4. Simulation Verification

4.1. Initial Condition Set

This simulation experiment is conducted on a computer equipped with an Intel Core i7-12700H processor, with a main frequency of 2.30 GHz and 32.00 GB of memory. Our aircraft/target model is modeled using Simulink and its code generation functionality. It is encapsulated into a dynamic library and simulated using a simulation platform built with the Python 3.8 interpreter.

Section 2, Section 3 and Section 4 of this paper, based on dynamic inversion control and artificial potential field theory, establish an avoidance and simultaneous arrival algorithm for a six-degree-of-freedom constant-speed fixed-wing UAV in a scenario of sudden threats emerging. This section carries out simulation for typical scenarios, in which six UAVs need to assemble simultaneously at a position located at (0,0). Each aircraft is scattered at a distance of 2.83 km to 4.24 km from the target point, the pitch overload limit of the drones is 15 G, and the yaw overload limit in the horizontal plane is 6 G, with speeds ranging from 60 m/s to 100 m/s. The specific initial parameters are shown in Table 1:

During the flight, the UAVs will encounter cylindrical threat areas that need to be circumnavigated. In the three sub-scenarios shown in Figure 13, Figure 14 and Figure 15, the threat areas range from simple to complex and are scattered within an area of 3 km by 3 km.

The initial simultaneous arrival time is artificially specified, set to be greater than the maximum of the current estimated times of each UAV, initially set at 48 s.

4.2. Simulation Results Analysis

As shown in Figure 13, Figure 14 and Figure 15, in the initial state, the UAV swarm maintains a stable flight state according to the given initial parameters, executing the dynamic inversion control simultaneous arrival–artificial potential field algorithm. When far from the threat area, each UAV receives control inputs according to the predetermined simultaneous arrival instruction, calculates the overload command, and the flight control system executes it. When encountering the threat area, the UAV performs overload control based on the artificial potential field, automatically avoiding the encountered threat area.

Figure 16 and Figure 17 illustrate the deviation from the scheduled arrival time and the line-of-sight angle information for each UAV in Scenario 3, the most complex one. During the flight, the third UAV in the figure has already completed time convergence and lead angle convergence under the simultaneous arrival guidance law, that is, it is flying straight towards the target. Under ideal conditions, it can arrive at the assembly point at the given time. However, after encountering a threat, the UAV had to circumnavigate and move away from the threat area. This maneuver causes it to fail to arrive at the given time. Therefore, it is necessary to use the dynamic negotiation of simultaneous arrival time discussed in Section 4. The swarm re-estimates the time at which each UAV will arrive at the target, postpones and updates $t_{CA}$, ensuring that the $t_{error}$ for each UAV is strictly greater than 0. As shown in Figure 17, in the scenario given in this paper, the swarm postponed the simultaneous arrival time by about 4 s, sacrificing time in exchange for the relative safety of each drone. In the end, the third UAV performed a circumnavigation maneuver to avoid the threat and employed the dynamic inversion control method again. Other UAVs, although not facing threats, also performed certain maneuvers to consume the remaining time due to the update of $t_{CA}$. The aforementioned process is repeated. Finally, the simultaneous arrival time was postponed by about 7 s, and at 55 s, the UAV swarm arrived at the target location simultaneously.

As shown in Figure 16, the scenario requires that the flight trajectory of each UAV in the final stage is relatively straight, without performing large maneuvers in the final stage (which is a strong constraint in many scenarios); that is, the lead angle in the final stage is required to maintain a small value. This was designed in the ITCG guidance law based on dynamic inversion control in the third section. At the initial state, due to the different flight velocity angles of the UAVs, the lead angles with the target are distributed at different values, but all converge to 0 under the control of the ITCG guidance law. Secondly, due to the execution of circumnavigation maneuvers and time-consuming maneuvers during the flight, the lead angle changes again. In the final stage of the flight, since there are no threat areas and the flight environment does not change dramatically, under the updated $t_{CA}$, each UAV once again executes the ITCG guidance law, achieving convergence of the lead angle and time.

In addition, the algorithm is designed at the fundamental level to support the tracking of moving targets and synchronized arrival. For this purpose, we have designed a simulation verification environment for a target moving at a constant speed in a straight line. A ground vehicle travels northeast at a constant speed of about 15 m/s (a typical off-road speed for special vehicles), as shown in Figure 18 and Figure 19. The simulation results demonstrate the algorithm’s tracking capability for moving targets, highlighting its potential value in target tracking missions. Compared to stationary targets, the algorithm requires more frequent time coordination and synchronization, as is shown in Figure 19. The final arrival time was extended to approximately 66 s.

This passage discusses the impact of UAV swarm size on the algorithm. The previously designed test cases were for six UAVs. On this basis, without changing the obstacles, we expanded the number of UAVs to 10 and 15 (also randomly distributed in the area, with random speeds and initial attitudes), and tested the performance of the algorithm. It can be seen that the algorithm still shows stable convergence on a larger scale and has good robustness; in terms of convergence speed, a larger UAV swarm means more frequent information communication and time synchronization delays. At the same time, the “emergencies” encountered by the UAV swarm during the search of the entire area also increase exponentially, as shown in Figure 20. When there are 15 UAVs, the algorithm completes the simulation of 78.7 s in real time within 8 s, with a super-real-time magnification of about 8 times. For 10 UAVs, this super-real-time magnification is 12 times, and for 6 UAVs, the super-real-time magnification is about 15 times. This result is in line with our expectations for the algorithm, and the real-time performance of the algorithm is still good for a swarm size of 15 UAVs.

This passage compares the differences between the algorithm proposed in this paper and other classic path planning algorithms. Firstly, in terms of algorithm speed and global optimality, as is shown in Table 2, graph search algorithms sacrifice a significant amount of search time to ensure their global optimality, losing real-time performance. Sampling-based search algorithms, to some extent, ensure the optimality of the trajectory (mathematically proven to be asymptotically optimal), but due to their inherent random sampling nature, they still cannot meet the requirements for real-time performance. The algorithm we propose inherits the characteristics of APF-like algorithms; the trajectory is not globally optimal but ensures obstacle avoidance and real-time performance. Secondly, in terms of trajectory smoothness, as mentioned in the first section, the trajectory generated by grid-based trajectory planning methods is not smooth, which is particularly evident in sampling search algorithms, as shown in Figure 21. For the TNOA-ITCG algorithm we propose, since the algorithm interacts directly with the aircraft’s low-level control system (output overload), the generated trajectory is smooth and continuous. Finally, in terms of cluster coordination and communication, the algorithm we propose integrates ideas from ITCG and distributed negotiation, requiring the aircraft to maintain certain communications to achieve time consistency upon arrival at the target.

5. Conclusions

This paper addresses the formation control problem of manned/unmanned aerial vehicle (UAV) formations and proposes a control method with a hierarchical control structure. Considering the command-and-control methods and combat processes of manned/unmanned systems, a control structure for the manned/unmanned formation system is designed. A dynamic surface control method is used to design a trajectory tracking controller for the unmanned aircraft, and a formation controller is designed for each UAV. Comparative simulations have verified the effectiveness of the proposed control method, which achieves the maintenance and transformation of manned/unmanned formations in three-dimensional space.

This paper addresses the problem of time synchronization difficulties during the simultaneous arrival process of UAV formations due to circumnavigating suddenly appearing threat areas, and proposes an integrated UAV control method with a hierarchical structure of time negotiation–time synchronization–threat avoidance. A synchronized arrival guidance law is designed using the dynamic inversion control method; a control command–avoidance strategy-integrated design loop is designed based on the improved artificial potential field method; finally, a double-layer time negotiation structure is used to adjust the $t_{error}<0$ problem caused by circumnavigating threat areas, achieving an integrated design structure of threat avoidance and synchronized arrival guidance law. Finally, the effectiveness of the control method proposed in this paper is verified by simulation in typical scenarios, and the proposed control method achieves the simultaneous arrival of UAV formations at the target location under complex threat conditions. We also tested the algorithm’s performance under scenarios with 6-, 10-, and 15-UAVs swarm sizes, confirming the algorithm’s quick response and robustness. We tested the algorithm’s tracking capabilities for dynamic targets, verifying the rationality of its foundational design. Lastly, we compared our algorithm with some fundamental trajectory planning algorithms. In contrast to traditional trajectory planning and obstacle avoidance algorithms, our algorithm also takes into account the issue of simultaneous arrival, offering high synchronization accuracy and a faster convergence rate.

In our forthcoming research, we intend to expand the current algorithm to accommodate more complex threat areas, such as polygonal and concave regions, while also taking into account dynamic factors like moving obstacles and wind disturbance. For instance, as noted in the literature [23], complex threats like concave threat areas and wind disturbance can diminish the performance of path planning algorithms, and the literature [24] has considered the performance of path planning algorithms under moving obstacles. Therefore, we will enhance the algorithm’s adaptability and responsiveness to these more complex environments in our subsequent work. Additionally, we will explore the algorithm’s adaptability to changes in wind speed and direction, which significantly impact the simultaneous arrival problem in dynamic environments. We will attempt to optimize the stability and path accuracy of drones by dynamically adjusting flight parameters, thereby improving the robustness of the algorithm. In the current work, the algorithm supports tracking of simple moving targets, and we look forward to incorporating the algorithm’s ability to track highly maneuverable targets in future work. Additionally, during the simulation verification process, we found that as the size of the UAV swarm increases, this leads to an increase in the frequency of coordination and time synchronization between UAVs, and there may also be difficulties in convergence. We will pursue control capabilities for larger (such as 30 or more) UAV swarms in subsequent work and to further improve the performance of the algorithm. Ultimately, we expect to verify the effectiveness and reliability of the new algorithm through technical integration and field testing, and explore its potential applications in various fields such as agricultural monitoring, environmental research, and disaster response. These studies will provide new solutions for the development and application of drone technology, significantly enhancing the autonomy and adaptability of drone swarms in complex environments.

Author Contributions

Conceptualization, X.L.; formal analysis, X.L.; funding acquisition, D.L.; investigation, X.L.; methodology, X.L.; project administration, D.L.; software, X.L.; supervision, D.L., Y.W., and X.Z.; validation, Y.W.; visualization, H.L.; writing—original draft, X.L.; writing—review and editing, Y.Z. and X.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The detailed definition of the main mathematical symbols used in this paper.

$m$	Mass of aircraft
$J$	Rotational inertia of aircraft
$V$	Speed of aircraft, can be decomposed into three directional velocities, u, v, and w
$p_n$	Rectangular coordinates in the north-east-down coordinate system (same: $p_e,p_d$)
$\omega$	Angular velocity of aircraft, can be decomposed into p, q, and r
$\varphi$	Tilt angle
$\vartheta$	Pitch angle
$\psi$	Yaw angle
$\theta$	Ballistic inclination
${\Gamma }_1$	The elements of the inertia matrix (same: ${\Gamma }_2~{\Gamma }_9)$
$C_{l_{\beta }}$	Aerodynamic derivatives
$M$	Torque acting on aircraft, can be decomposed into l, m, n
$F$	Aerodynamic forces acting on aircraft, can be decomposed into $f_x,f_y,f_z$
$\alpha$	Attack angle
$\beta$	Side-slip angle
$S_{REF}$	Aircraft reference area
$b, c$	Aircraft reference length
$\delta$	Aircraft rudder deflection angle, ${\delta }_a,{\delta }_e,$
$\rho$	Current air density, described by the Jacchia–Robert atmospheric environment mathematical model
$R$	The horizontal distance to the target point
$q_{it}$	Line of sight angle between the i-th UAV to the target
$\phi$	Lead angle

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Typical symbol representation for six-degree-of-freedom modeling of UAVs.

Figure 2. Structure diagram of the pitch channel overload control system.

Figure 3. The roll attitude control system structure diagram.

Figure 4. Structure diagram of the yaw stabilization control system.

Figure 5. The relative motion relationship between the UAV and the target.

Figure 6. Diagram indicating artificial potential field force.

Figure 7. UAV trajectory guided by APF in potential field.

Figure 8. Trajectories of a swarm of UAVs guided by APF in a potential field (2D).

Figure 9. Improved APF Threat Evasion Direction Illustration.

Figure 10. Schematic diagram of overload threshold smoothing algorithm.

Figure 11. Combined guidance law framework diagram.

Figure 12. Two-layer distributed time negotiation architecture.

Figure 13. The flight trajectory of the drone swarm in Scenario 1.

Figure 14. The flight trajectory of the drone swarm in Scenario 2.

Figure 15. The flight trajectory of the drone swarm in Scenario 3.

Figure 16. Line-of-sight angle (LOS) for each UAV.

Figure 17. Time error with [Forumla omitted. See PDF.] for each UAV.

Figure 18. The flight trajectory of the drone swarm in a scenario of tracking a moving target.

Figure 19. Time error with [Forumla omitted. See PDF.] for each UAV in scenario of tracking a moving target.

Figure 20. Performance of TNOA-ITCG algorithm in the cases of 6, 10, and 15 UAVs.

Figure 21. Path plan performance of TNOA-ITCG, RRT*, and APF algorithms.

Table 1

Formation initial parameters.

ID	Initial Position/(m)	Velocity/(m·s−1)	Velocity Angle/(°)
1	(−3000, −3000)	100	63
2	(2000, −3000)	85	91
3	(2000, 2000)	60	−132
4	(−3500, 2000)	90	5
5	(3500, 1000)	80	−9
6	(−1000, −3000)	80	83

Table 2

Comparison of performance for trajectory planning algorithms.

	Search-BasedPlanning	Sample-BasedPlanning	APF Method	Ours(TNOA-ITCG)
Real-time performance	slow	slow	fast	fast
Global optimality	√	Asym. Opt.	×	×
Trajectory smoothness	×	×	√	√
Account for impact time	×	×	×	√
Comm. network	no need	no need	no need	distributed

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