Research on Impact of Planned Path Length and Yaw Cost on Collaborative Search of Unmanned Aerial Vehicle Swarms

Abstract

To address the unclear impacts of a planned path length and yaw cost on search performance in large-scale Unmanned Aerial Vehicle (UAV) swarm collaborative search scenarios under complex and dynamic environments, a path grid determination algorithm is proposed, transforming the path-planning problem into an optimal waypoint selection problem, enabling UAVs to make rapid decisions using the Particle Swarm Optimization (PSO) algorithm. Simulation experiments were conducted for different planned path lengths with or without the inclusion of the yaw cost, analyzing indicators such as the coverage rate, target capture rate, average capture time, and communication and decision-making consumption. This research was conducted through simulation experiments, and the results demonstrate that increasing the planned path length significantly reduces communication and decision-making consumption while having no notable impact on the coverage rate or search performance. Incorporating the yaw cost slightly improves target search performance but also leads to a minor increase in communication and decision-making consumption.

1. Introduction

In recent years, unmanned system technologies have developed rapidly, with UAVs being widely applied in civil and military domains such as logistics transportation [1,2,3], emergency rescue [4,5,6], telemetry surveillance [7,8], and reconnaissance-strike missions [9,10]. UAV swarms, characterized by a low cost, small volume, and large quantity, have demonstrated significant potential advantages across various mission scenarios. Among diverse application scenarios, collaborative search stands as a representative research focus in UAV swarm studies, being commonly encountered in missions including search and rescue, disaster monitoring, and close-range reconnaissance.

The collaborative search mission features an unknown environment, multiple variables, and high dynamics, requiring UAV swarms to rapidly cover mission areas and maximize target detection within minimal timeframes. Additional constraints such as energy consumption and no-fly zones may apply depending on specific mission requirements, with varying priorities assigned to these indicators across different scenarios. To ensure that these mission indicators meet mission requirements, it is essential to establish a variable framework describing situation information within mission areas. In numerous mission scenarios, situational variables involved in UAV path-planning algorithms can generally be categorized into two classes: environmental model variables and aircraft platform performance parameters.

The description of mission environments predominantly employs variables such as target detection probability, environmental uncertainty, digital pheromones, or analogous indicators. Yang et al. [11] established a digital pheromone framework incorporating secretion, volatilization, and propagation mechanisms, alongside a target detection probability map updated via Bayesian principles. They further transformed the target detection probability map into an environmental uncertainty map using Shannon entropy calculations, serving as the foundation for swarm collaborative search path planning. Cheng et al. [12] utilized digital pheromones with only secretion and volatilization mechanisms, which cannot propagate to adjacent grids. Other studies [13,14] have selectively adopted these variables. A synthesis of the existing literature indicates no strictly unified standard for environmental modeling variables, although common emphases include target distribution patterns and environmental awareness as critical situational aspects.

The performance of UAV platforms themselves, including maneuverability and mission payload capabilities, significantly influences search effectiveness. Both Liu et al. [15] and Li et al. [16] considered the impact of onboard sensor mounting angles and the field of view (FOV) on detection ranges. Beyond sensor parameter quantification, Saadaoui et al. [17] and Yan et al. [18] incorporated remaining battery power into swarm cooperative decision-making algorithms, introducing energy cost indicators to investigate target search problems under energy-constrained conditions.

In the selection of collaborative search decision-making algorithms, numerous studies have adopted heuristic optimization algorithms represented by Ant Colony Optimization (ACO) [13,19], the Genetic Algorithm (GA) [13,20], and Particle Swarm Optimization (PSO) [21,22]. These algorithms are easy to implement and fast converging, yet prone to local optima. Neural networks demonstrate strong adaptability in modeling complex systems, hence their frequent application in swarm UAV decision making. Multiple studies employ deep learning combined with reinforcement learning methods [23,24] to enhance swarm adaptability in complex environments.

The aforementioned studies universally discretize mission areas into uniform square grid cells carrying situation information, where UAVs select appropriate grids from their immediate vicinity as waypoints. Upon reaching a waypoint, the situational data are immediately updated for subsequent decision making. While this approach minimizes environmental information utilization, it increases communication and decision-making consumption. Alternative research [25,26,27] proposes dividing areas based on global situational awareness, assigning specific sub-regions to individual UAVs for search missions. However, sub-region sizes typically exceed the coverage of single-path planning, and constrained by computational capacity and system robustness requirements, partitioning updates lag behind situation information updates, resulting in inadequate adaptability in complex environments.

To avoid the above problems, extending UAV path lengths while utilizing only situation information along planned trajectories presents a viable solution. The PSO algorithm has the characteristics of few parameters and fast computation. Although using this algorithm may lead to getting stuck in local optima, accurate decision making is unnecessary in the scenario of fast decision making in multi-UAV collaborative search tasks. When using PSO algorithm for route planning, when there are too many candidate waypoints, the computational load of the decision-making algorithm will become large. In this article, we will use path grid determination algorithm to replace the path with a small number of waypoints for planning, reducing the computational load. Furthermore, path planning must consider route curvature, which is typically reflected through yaw cost parameters in UAV path-planning algorithms. Current studies [28,29,30] primarily treat this parameter as a means to reduce energy consumption without investigating its impact on target search effectiveness. To address these two problems, this study will undertake the following work:

• Propose a waypoint update mechanism based on the path grid determination algorithm and multi-objective PSO algorithm: by using the path grid determination algorithm, the path-planning problem is transformed into a waypoint selection problem, allowing UAVs to make quick decisions using PSO;
• Analyze and study the impact of planned path length on collaborative search effectiveness, such as the coverage rate, target capture rate, capture time, and communication and decision-making consumption;
• Analyze and study the impact of the yaw cost on target search effectiveness and reveal its influence mechanism through simulation data analysis.

2. Problem Model

2.1. UAV Model

To ensure research feasibility while preserving generality, the UAV is abstracted as a two-dimensional mass point model:

$$\left\{\begin{array}{c}\dot{x}=V\cos y\\ \dot{y}=V\sin y\\ \omega =\dot{\psi }\end{array}\right.,$$

(1)

where $V$ represents the velocity magnitude, set at 50  $\mathrm{m}/\mathrm{s}$, and $\omega$ represents the heading angle rate relative to the geodetic coordinate system, set at 5  $\mathrm{deg}/\mathrm{s}$.

A flight state decision logic for UAVs is designed as shown in Figure 1. When the distance to the next waypoint is far, if the UAV is aligned with the next waypoint, it will fly in a straight line towards the waypoint. If not, it will adjust its heading by hovering. The UAV is considered to have reached the waypoint when the distance is less than $R_{ap}$, after which it exchanges situation information with other UAVs to replan paths and repeats the flight state decision process. This study incorporates two assumptions: first, UAVs avoid collisions by adjusting altitudes during encounters, ensuring their 2D trajectories remain unaffected by other UAVs; second, the UAV swarm maintains a fully connected communication topology. The detection range of the UAV is 1000 $\mathrm{m}$.

2.2. Environment Model

2.2.1. Mission Area Division

The mission area, as shown in Figure 2, is defined as a square with side length $L_x=L_y=$ 12 $\mathrm{km}$. To accurately characterize situation information for the area search and guide UAV, the mission area is uniformly divided into square grid cells with $\Delta L=$ 500 m side length. Each grid cell is uniquely identified by an ordered pair ${\mathit{G}}_{i,j}=\left(i,j\right)$ following a left-to-right and top-to-bottom numbering sequence, resulting in the grid cell set representation:

$$\mathit{G}=\left\{\left(i,j\right)\left|i\in \left[1,{\displaystyle \frac{L_x}{\Delta L}}\right]\right.\cap \mathbb{Z},j\in \left[1,{\displaystyle \frac{L_y}{\Delta L}}\right]\cap \mathbb{Z}\right\},$$

(2)

where $\mathbb{Z}$ is the set of integers.

The ordered pair $G_{i,j}=\left(i,j\right)$ is only used here for illustration and in the simulation model. For the convenience of subsequent description, the following discussion will only use one index to distinguish the grid, such as $g_i$.

2.2.2. Situation Information

The target occurrence is modeled as a binomial distribution event, where each grid cell in the mission area is assigned a target detection probability $p_i$. The collection of all $p_i$ values constitutes the target detection probability map (TDPM). When grid cell $g_i$ lies within the UAV detection range, its TDPM is updated according to the Bayesian criterion:

$$p_i\left(k+1\right)=\left\{\begin{array}{l}{\displaystyle \frac{p_d{p}_i\left(k\right)}{p_d{p}_i\left(k\right)+p_f\left[1-p_i\left(k\right)\right]}},s_i^d\left(k\right)=1\\ {\displaystyle \frac{\left(1-p_d\right)p_i\left(k\right)}{\left(1-p_d\right)p_i\left(k\right)+\left(1-p_f\right)\left[1-p_i\left(k\right)\right]}},s_i^d\left(k\right)=0\end{array}\right..$$

(3)

where $k$ is the index value of the time step. The target detection status $s_i^d\left(k\right)$ is assigned a value of 1 when a UAV detects a target in grid cell $g_i$ and 0 otherwise. The sensor’s detection probability $p_d$ and false alarm probability $p_f$ are set as 0.8 and 0.2, respectively. Under the mechanism of Equation (3), if the UAV detects the presence of a target, regardless of whether it is detected or a false alarm, the UAV will consider the probability of the target’s existence to be high and will then assign a greater probability of the target’s appearance to $g_i$. When this intelligence is broadcasted to other UAVs, they will have a greater chance of accessing $g_i$ for further reconnaissance, preventing missed detections or eliminating false alarms.

To reduce redundant search efforts, the digital pheromone map (DPM) method is employed. For any grid cell $g_i$ within the mission area, digital pheromone values are updated according to the following equation [31]:

$${\eta }_i\left(k+1\right)=\left(1-V_a\right)\left\{\left(1-P_a\right)\left[{\eta }_i\left(k\right)+W_i{o}_i\left(k+1\right)\right]+I_i\left(k+1\right)\right\},$$

(4)

where ${\eta }_i\left(k+1\right)$ represents the pheromone concentration at grid cell $g_i$ during the $k$-th step; $V_a$ and $P_a$ represent the propagation coefficient and volatilization coefficient, respectively; $W_i$ represents the secretion switch, which is typically set to 0 and changes to 1 when a UAV passes through the grid; $o_i\left(k+1\right)$ is the pheromone secreted by all UAVs in the grid, including itself, at $g_i$; and $I_i\left(k+1\right)$ is the pheromone entering $g_i$ from neighboring grids through propagation, calculated using Equation (5):

$$I_i\left(k+1\right)={\displaystyle \frac{P_a}{N\left({\mathit{G}}_s\right)}}{\displaystyle \sum _{j=1}^{N\left({\mathit{G}}_s\right)}\left[{\eta }_{s,j}\left(k\right)+o_{s,j}\left(k+1\right)\right]},$$

(5)

where ${\mathit{G}}_s$ represents the set of grid cells adjacent to $g_i$, and $N\left({\mathit{G}}_s\right)$ denotes the number of elements in ${\mathit{G}}_s$.

In Equation (4), for a map stored in a UAV, the pheromones possessed by grid $g_i$ between the $k$-th and $k+1$-th adjacent time points come from three parts: the pheromones ${\eta }_i\left(k\right)$ retained at the $k$-th time point, the pheromones $W_i{o}_i\left(k+1\right)$ secreted by $g_i$ by the UAV at the $k+1$-th time point, and the pheromones $I_i\left(k+1\right)$ propagated to $g_i$ by other grids around $g_i$. The first two terms have already evaporated a part of $P_a$ before the $k+1$-th time point, so $\left(1-P_a\right)\left[{\eta }_i\left(k\right)+W_i{o}_i\left(k+1\right)\right]+I_i\left(k+1\right)$ represents all the pheromones passing through $g_i$ between adjacent time points. Finally, these pheromones also propagate to the surrounding grids through a propagation mechanism with a proportion of $V_a$, so the pheromone of $g_i$ at the $k+1$-th time point is represented by Equation (4). From the perspective of individual UAVs, the only source of pheromones stored in their maps is the secretion of pheromones caused by UAV access to the grid, which has repulsive properties, and the UAV will prioritize areas with lower concentrations of pheromones. Under this mechanism, the digital pheromone used in this article can reduce the repetitive search of UAV swarms in a certain area and improve search efficiency.

For each grid cell $g_i$, an uncertainty measure ${\delta }_i$ is defined, which collectively forms the environmental cognition map (ECM) of the entire mission area. A smaller ${\delta }_i$ indicates a higher level of UAV awareness regarding $g_i$. The update rule for the ECM of a single UAV is as follows:

$${\delta }_i\left(k+1\right)=\left\{\begin{array}{l}{\left(1-\mu \right)}^{{\displaystyle \frac{t_s}{T_0}}}{\delta }_i\left(k\right),s_i^s=1\\ {\delta }_i\left(k\right),s_i^s=0\end{array}\right.,$$

(6)

where $s_i^s$ is set to 1 when the UAV detects grid cell $g_i$ and 0 otherwise; $\mu$ represents the uncertainty decay rate; $t_s$ denotes the total reconnaissance time of the swarm; and $T_0$ is the time constant.

Equations (3–6) are the methods by which a single UAV updates its intelligence based on the positions of other UAVs and the currently stored situational information before making a decision after reaching the waypoint. The UAV will use this information for the next decision.

2.2.3. Detection Interval Time

Due to limited detection range, UAVs estimate situation information in distant areas using Equations (3–6) or acquire intelligence from other UAVs, followed by information fusion. To address the challenge of intelligence fusion, this study defines the detection interval time (DIT), which represents the time elapsed since a grid cell was last detected. When a UAV detects a grid cell, the DIT for that grid is reset to zero in the UAV’s stored map; otherwise, the DIT accumulates over time. Based on this definition, a smaller DIT indicates that the UAV’s intelligence on the grid is more recent, and the likelihood of other UAVs having detected the grid during the undetected period is lower. Thus, the DIT serves as a measure of the credibility of the UAV’s intelligence on a specific grid. Information exchange among UAVs is prioritized and conducted in an orderly manner based on DIT values.

3. Decision Mechanism for Collaborative Search

3.1. Path Grid Determination Algorithm

To calculate the situation information cost along the UAV’s movement path, it is necessary to acquire the grids traversed by the path. As shown in Figure 3, assume that the center positions of the UAV’s current grid $g_L$ and the next candidate waypoint $W{P}_1$ are $\left(x_L,y_L\right)$ and $\left(x_{wp1},y_{wp1}\right)$, respectively. The line connecting these two points is denoted as $L_0$. By connecting the vertices of grid $g_L$ and grid $g_{wp1}$ (where $W{P}_1$ is located), two lines parallel to $L_0$, namely $L_1$ and $L_2$, are obtained:

$$\left\{\begin{array}{c}{L}_1:Ax+By=C_1\\ L_2:Ax+By=C_2\end{array}\right..$$

(7)

For the coordinates $\left(x_{g,i},y_{g,i}\right)$ of any grid $g_i$, the following holds:

$$C_{g,i}=A{x}_{g,i}+B{y}_{g,i}.$$

(8)

When the center position of grid $g_i$ satisfies Equation (9), it is considered to lie on the UAV’s future movement path:

$$\left\{\begin{array}{c}{C}_{g,i}\in \left[\min \left\{C_1,C_2\right\},\max \left\{C_1,C_2\right\}\right]\\ x_{g,i}\in \left[\min \left\{x_L,x_{wp1}\right\},\max \left\{x_L,x_{wp1}\right\}\right]\\ y_{g,i}\in \left[\min \left\{y_L,y_{wp1}\right\},\max \left\{y_L,y_{wp1}\right\}\right]\end{array}\right.,$$

(9)

by substituting $x_{wp1}$, $y_{wp1}$, $x_L$, and $y_L$ in Equation (9) with $x_{wp2}$, $y_{wp2}$, $x_{wp1}$, and $y_{wp1}$, the grids along the path from waypoint $w{p}_1$ to $w{p}_2$ can be determined.

3.2. Cost of Path Situation Information

If the set of path grids is represented as:

$${\mathit{G}}_{path}=\left\{g_{path,1},g_{path,2},\cdots ,g_{path,n}\right\},$$

(10)

then the cost of the probability of target detection probability is:

$$Cos{t}_{path,p}={\displaystyle \sum _{i=1}^{N\left({\mathit{G}}_{path}\right)}\frac{1-p_{path,i}}{N\left({\mathit{G}}_{path}\right)}},$$

(11)

the cost of digital pheromones is:

$$Cos{t}_{path,\eta }={\displaystyle \sum _{i=1}^{N\left({\mathit{G}}_{path}\right)}\frac{{\eta }_{path,i}}{N\left({\mathit{G}}_{path}\right){\eta }_L}},$$

(12)

and the cost of environmental cognition is:

$$Cos{t}_{path,\delta }={\displaystyle \sum _{i=1}^{N\left({\mathit{G}}_{path}\right)}\frac{1-{\delta }_{path,i}}{N\left({\mathit{G}}_{path}\right)}},$$

(13)

where $p_{path,i}$, ${\eta }_{path,i}$, and ${\delta }_{path,i}$ represent the target detection probability, pheromone concentration, and environmental uncertainty of the element $g_{path,i}$ in ${\mathit{G}}_{path}$, respectively, while $N\left({\mathit{G}}_{path}\right)$ denotes the number of elements in ${\mathit{G}}_{path}$.

The cost of path situation information is expressed as the weighted sum of the aforementioned three costs:

$$Cos{t}_{path}=Cos{t}_{path,p}+Cos{t}_{path,\eta }+Cos{t}_{path,\delta }.$$

(14)

3.3. Yaw Cost

The yaw cost is calculated using Equation (15):

$$Cos{t}_{turn}={\displaystyle \sum _{i=1}^{N\left(\mathit{W}\mathit{P}\right)}\frac{\Delta {\psi }_{i-1,i}}{N\left(\mathit{W}\mathit{P}\right)}},$$

(15)

where $\mathit{W}\mathit{P}=\left\{\begin{array}{cccc}w{p}_1& w{p}_2& \cdots & w{p}_n\end{array}\right\}$ represents the set of waypoints on the currently planned path. In this paper, the UAV plans two waypoints at a time; hence, $N\left(\mathit{W}\mathit{P}\right)=2$. $\Delta {\psi }_{i-1,i}$ denotes the angle between the line connecting waypoints $w{p}_{i-1}$ and $w{p}_i$ and the velocity direction of the UAV immediately after arriving at $w{p}_{i-1}$. Specifically, $\Delta {\psi }_{0,1}$ is the angle between the line connecting the UAV’s current position and the next waypoint and the current velocity direction of the UAV, as shown in Figure 4.

3.4. Path-Planning Algorithm

As shown in Figure 5, to balance the long-term benefits of UAV search, each decision made by the UAV determines two waypoints, which are located in two concentric annular regions centered on the UAV with different radii. The grids marked in red and blue represent the candidate grids for the first and second waypoints, respectively, denoted here by the sets ${\mathit{G}}_{wp1}$ and ${\mathit{G}}_{wp2}$. The elements of these sets satisfy the following conditions:

$$\left\{\begin{array}{c}{R}_{wp1}^{\min }\le D_{wp1}^i\le R_{wp1}^{\max }\\ R_{wp2}^{\min }\le D_{wp2}^i\le R_{wp2}^{\max }\end{array}\right.,$$

(16)

where $D_{wp1}^i$ and $D_{wp2}^i$ represent the distances from the centers of the elements in ${\mathit{G}}_{wp1}$ and ${\mathit{G}}_{wp2}$ to the current position of the UAV, respectively. For ease of subsequent analysis, $R_{wp1}^{\min }$ and $R_{wp1}^{\max }$ are expressed in units of grid width $\Delta L$, and the following convention is established: if $n\ge 2$ and $n\in \mathbb{Z}$, $\Delta n\in \left(0,1\right)$, $R_{wp1}^{\min }=\left(n-\Delta n\right)\cdot \Delta L$, $R_{wp1}^{\max }=\left(n+\Delta n\right)\cdot \Delta L$, and $D_{wpj}^i$ satisfies Equation (16), then the fuzzy definition $D_{wpj}^i=n\cdot \Delta L$ is considered valid. Under this definition, the candidate grids for the first waypoint are distributed as a single layer of grids around the UAV, and $n$ represents the multiple of the grid width corresponding to the distance from the UAV.

By using the index values $g_{1,i}^t$ and $g_{2,i}^t$ of ${\mathit{G}}_{wp1}$ and ${\mathit{G}}_{wp2}$ as the coordinates of particle $i$ at time $t$ in the PSO algorithm, denoted as ${\mathit{x}}_i^t=\left(g_{1,i}^t,g_{2,i}^t\right)$, the solution space can be represented as:

$$\mathit{S}=\left\{\left(g_{1,i}^t,g_{2,i}^t\right)\left|g_{1,i}^t\in \left[1,N\left({\mathit{G}}_1^{wp}\right)\right]\cap \mathbb{N},g_{2,i}^t\in \left[1,N\left({\mathit{G}}_2^{wp}\right)\right]\cap \mathbb{N}\right.\right\},$$

(17)

where $\mathbb{N}$ represents the set of natural numbers.

By using the index values of elements in set $\mathit{S}$ as the particle coordinates in the PSO algorithm and integrating the path grid determination algorithm, the optimal planned path can be obtained via the optimization of waypoint selection. The PSO algorithm is expressed as:

$$v_i^{t+1}=w\cdot v_i^t+c_1\cdot r_1\cdot \left(x_i^{best}-x_i^t\right)+c_2\cdot r_2\cdot \left(x_g^{best}-x_i^t\right),$$

(18)

$$x_i^{t+1}=x_i^t+v_i^{t+1},$$

(19)

where $v_i^t$ and $x_i^t$ represent the velocity and position of particle $i$ at time $t$, respectively. The parameter $w$ denotes the inertia weight, $c_1$ and $c_2$ are the learning factors for the individual historical best solution and the global historical best solution, respectively, and $r_1$ and $r_2$ are random numbers. Additionally, $x_i^{best}$ and $x_g^{best}$ correspond to the individual historical best solution and the global historical best solution, respectively.

The fitness function of all particles should be expressed as:

$$F_i=1-{\displaystyle \frac{1}{4}}\left(3Cos{t}_{path}+Cos{t}_{turn}\right).$$

(20)

Based on Equations (18–20), the UAV can complete a waypoint-planning decision. At this point, we have transformed the path-planning problem into a multi-objective optimization problem suitable for the PSO algorithm. In this paper, the number of particles is 25, the number of iterations is 15, and the learning factor and inertia weight are both set to 0.5. We stop iterating when the relative change in fitness function is less than 5%.

Each UAV initiates intelligence exchange and situation information updates with other UAVs only after reaching its first waypoint, followed by path replanning. Since the arrival times at waypoints vary across UAVs, at most one UAV acquires situation information and makes decisions at any given time. Under this mechanism, coordination among UAVs occurs asynchronously, thereby reducing communication and decision-making consumption for inter-UAV negotiation.

Based on the concept of a rolling optimization strategy, at each time step, the UAV has two waypoints, $W{P}_j=\left[\begin{array}{cc}w{p}_{j,1}& w{p}_{j,2}\end{array}\right]$, which it will sequentially pass through. Upon entering the approach threshold range $R_{ap}$ of the next waypoint, the UAV calculates a new set of waypoints $W{P}_{j+1}=\left[\begin{array}{cc}w{p}_{j+1,1}& w{p}_{j+1,2}\end{array}\right]$ based on the environmental information it has acquired. Here, $w{p}_{j,2}$ may not necessarily coincide with $w{p}_{j+1,1}$ in terms of location, but the influence of $w{p}_{j,2}$ is still considered when calculating the benefits of the waypoints.

4. Simulation Experiment and Result Analysis

During the simulation experiment, 60 moving targets were set up in the task area. They were uniformly generated around the task area at the beginning of the simulation and then passed through the task area at a constant speed along a straight line starting from the generated position. The moving speed was 20 $\mathrm{m}/\mathrm{s}$, and the motion trajectory was a straight line parallel to the grid line. Simulations were conducted under different conditions, such as different planned path lengths and whether yaw costs were added, and data analysis was conducted. In simulation experiments, the performance of sensors (detection distance, detection probability, false alarm probability, etc.) and UAVs (flight speed, yaw rate, etc.) come from typical values in practical scenarios.

4.1. The Impact of a Planned Path Length on Search Effectiveness

The straight-line distance flown by a UAV between two consecutive decision points, referred to as the planned path length, corresponds to the first waypoint interval $D_{wp,1}$ mentioned earlier. This section conducts a simulation analysis of the swarm’s collaborative search effectiveness under different $D_{wp,1}$ values without $Cos{t}_{turn}$, using the influence of UAV quantity $N_{UAV}$ on search effectiveness as a reference.

Figure 6 illustrates the variation trends of area coverage rate $r_{cover}^{i,j}$ under the conditions $N_{UAV}=i,i\in \left\{2,4,\dots ,8\right\}$ and $D_{wp,1}=j,j\in \left\{1000,1500,\dots ,4000\right\}$. To intuitively represent the influence of $N_{UAV}$ and $D_{wp,1}$ on $r_{cover}^{i,j}$, two variables, $r_{cover}^{\overline{D}}$ and $r_{cover}^{\overline{U}}$, are defined:

$$r_{cover}^{\overline{D}}={\displaystyle \sum _{j=1000,1500,\dots }^{4000}\frac{r_{cover}^{i,j}}{7}},$$

(21)

$$r_{cover}^{\overline{U}}={\displaystyle \sum _{i=2,4,\dots }^{12}\frac{r_{cover}^{i,j}}{6}}.$$

(22)

Figure 7 presents the variation trends of $r_{cover}^{\overline{D}}$ and $r_{cover}^{\overline{U}}$ with respect to $N_{UAV}$ and $D_{wp,1}$, respectively. As shown in the figure, when $N_{UAV}$ increases from 2 to 12, $r_{cover}^{\overline{D}}$ exhibits a variation range of 0.541, whereas $D_{wp,1}$ shows a significantly smaller variation range of only 0.056 as it increases from 1000 m to 4000 m.

Figure 8 illustrates the overall trends of target capture rate $r_{capture}^{i,j}$ with respect to $N_{UAV}$ and $D_{wp,1}$. Following the approach used for $r_{cover}^{\overline{D}}$ and $r_{cover}^{\overline{U}}$, two variables, $r_{capture}^{\overline{D}}$ and $r_{capture}^{\overline{U}}$, are defined for analyzing $r_{capture}^{i,j}$, and their corresponding variations are presented in Figure 9. Similar to the case of $r_{cover}^{i,j}$, $r_{capture}^{\overline{D}}$ exhibits a significant variation range of 0.457 when $N_{UAV}$ increases, whereas $r_{capture}^{\overline{U}}$ shows a much smaller variation range of only 0.044 when $D_{wp,1}$ increases.

When evaluating target search effectiveness, both the number of targets captured and the time spent on the search must be considered. Therefore, the variable of average capture time ${\overline{t}}_{capture}^{i,j}$ is introduced, defined as the average time elapsed from the start of the mission until target capture, with simulation results shown in Figure 10. Following the analytical approach used for $r_{cover}^{i,j}$ and $r_{capture}^{i,j}$, two variables, ${\overline{t}}_{capture}^{\overline{D}}$ and ${\overline{t}}_{capture}^{\overline{U}}$, are defined, and their variation trends are illustrated in Figure 11. When $N_{UAV}$ increases, ${\overline{t}}_{capture}^{\overline{D}}$ decreases significantly from 202.4 s to 79.8 s, a reduction of 122.6 s, whereas ${\overline{t}}_{capture}^{\overline{U}}$ shows no clear trend as $D_{wp,1}$ increases, with a much smaller variation range of only 28.2 s.

We define the average decision frequency per UAV as a metric to quantify communication and decision-making consumption:

$$N_{D,\overline{t}}={\displaystyle \frac{100N_D}{N_U\cdot t_s}},$$

(23)

where $N_D$ represents the total number of decisions made by all UAVs, $N_U$ denotes the number of UAVs, and $t_s$ is the total search duration.

As shown in Figure 12, $N_{UAV}$ has little impact on $N_{D,\overline{t}}$, whereas when $D_{wp,1}$ increases from 1000 m to 4000 m, $N_{D,\overline{t}}$ decreases from approximately 1.903 to 0.997, a reduction of about 47.6%. The decrease in $N_{D,\overline{t}}$ implies a slowdown in the Observation–Orientation–Decision–Action (OODA) cycle. Nevertheless, the increase in $D_{wp,1}$ also enables UAVs to consider situation information over a larger range during path planning, resulting in more globally optimal planning outcomes. Consequently, the overall search effectiveness remains almost unaffected by $D_{wp,1}$.

4.2. The Impact of the Yaw Cost on Search Effectiveness

The analysis of the yaw cost $Cos{t}_{turn}$ follows the same methodology used to examine the impact of $D_{wp,1}$ on search effectiveness, focusing on the effects of incorporating $Cos{t}_{turn}$ into the UAV decision-making algorithm. As shown in Figure 13 and Figure 14, similar to the case without $Cos{t}_{turn}$, the influence of $D_{wp,1}$ on the indicators is significantly smaller than that of $N_{UAV}$. However, Figure 14b and Figure 15b reveal that increasing $Cos{t}_{turn}$ slightly reduces the target capture rate while increasing the average capture time. As shown in Figure 15 and Figure 16, the trend of the influence of $N_{UAV}$ and $D_{wp,1}$ on the average capture time and average decision frequency with $Cos{t}_{turn}$ is consistent with that without $Cos{t}_{turn}$.

Table 1. Comparison of various indicators without and with $Cos{t}_{turn}$.

Conditions	Effectiveness Indicators	Result
Without $Cos{t}_{turn}$	${\overline{r}}_{cover}$	0.811
	${\overline{r}}_{capture}$	0.875
	${\overline{t}}_{capture}$	135.32
	${\overline{N}}_{D,\overline{t}}$	1.303
With $Cos{t}_{turn}$	${\overline{r}}_{cover}$	0.827
	${\overline{r}}_{capture}$	0.898
	${\overline{t}}_{capture}$	126.65
	${\overline{N}}_{D,\overline{t}}$	1.407

lists the average coverage rate ${\overline{r}}_{cover}$, average capture rate ${\overline{r}}_{capture}$, average capture time ${\overline{t}}_{capture}$, and average decision frequency ${\overline{N}}_{D,\overline{t}}$ for all combinations of $N_{UAV}$ and $D_{wp,1}$ values, both with and without $Cos{t}_{turn}$. Considering these four indicators collectively, the swarm’s collaborative search effectiveness is improved when UAV decision-making incorporates $Cos{t}_{turn}$, albeit at the cost of increased communication and decision-making consumption per UAV.

Figure 17a,b illustrate the ratio of average turning time to total flight duration $r_{turn}$ without and with $Cos{t}_{turn}$, respectively. The average values of $r_{turn}$ are 0.574 and 0.481, demonstrating that incorporating $Cos{t}_{turn}$ into the path-planning algorithm significantly reduces $r_{turn}$.

The difference in $r_{turn}$ explains the variation in search effectiveness with and without $Cos{t}_{turn}$. First, while the path grid determination algorithm extracts situation information from distant paths to calculate path costs, it does not limit the time UAVs spend circling near the previous waypoint. Incorporating $Cos{t}_{turn}$ into the cost function effectively mitigates this issue, reducing redundant searches caused by excessive circling. Second, as observed in Table 1, ${\overline{N}}_{D,\overline{t}}$ is higher with $Cos{t}_{turn}$ than without it. This is because minimizing the circling time near the previous waypoint allows the UAVs to reach the next waypoint more quickly via straight-line flight, facilitating faster information exchange and decision making, thereby accelerating the OODA cycle and improving search effectiveness.

4.3. Stability of Collaborative Search Algorithm

This paper explores the impact of GPS positioning errors and partial UAV loss on the collaborative search efficiency of UAV swarms through simulation experiments. During the experiment, eight UAVs were set up to simultaneously search for 60 moving targets within the task area, and the setting of moving targets was consistent with the previous text.

When considering GPS positioning error, the distance between the position obtained by UAV and the actual position follows a normal distribution:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}}$$

(24)

where $\mu$ is the actual position coordinate of UAV, and $\sigma$ is the overall standard deviation, which is taken as 20 m in this article. When considering the situation of UAV loss, we randomly select two UAVs to be lost at regular intervals.

Figure 18 shows the simulation results. As shown in the figure, when a GPS positioning error and UAV loss occur, there is no significant change in the coverage and target capture rates, but the average capture time increases by 45 s and 52 s, respectively, and the time spent capturing targets becomes longer. The proportion of decision consumption and average turning time still decreases with the changing trend. A GPS positioning error has no significant impact on decision consumption, while UAV loss will reduce decision consumption. This is because the two UAVs that are lost cannot share information with the original bee colony, resulting in a higher cost of path state information and yaw cost for all UAVs. The resulting path curvature is greater, increasing the flight time between waypoints and reducing the number of decisions.

In summary, positioning errors and UAV disconnection have no significant impact on coverage and target capture rates. Although these abnormal situations may increase the duration of target capture, UAV swarms can still complete the task.

5. Conclusions and Prospect

This study first establishes an environmental information model and an information fusion mechanism based on detection interval time, proposes a path grid determination algorithm to extract situation information along planned paths, and enables UAVs to use the PSO algorithm for rapid decision making with the goal of minimizing the path situation information cost and yaw cost $Cos{t}_{turn}$. Finally, the impacts of $D_{wp,1}$ and $Cos{t}_{turn}$ on swarm search effectiveness are investigated through multiple simulation comparisons. The main contributions and conclusions of this study are as follows:

• A path grid determination algorithm is proposed to acquire situation information from grid cells along planned paths and calculate path costs, transforming the path-planning problem into a waypoint selection problem, which facilitates UAV decision making using the PSO algorithm.
• The impact of $D_{wp,1}$ on search effectiveness is investigated through simulations. An analysis of the simulation results reveals that increasing $D_{wp,1}$ has no significant impact on search effectiveness but significantly reduces communication and decision-making consumption.
• The impact of $Cos{t}_{turn}$ on search effectiveness is comparatively analyzed. Incorporating $Cos{t}_{turn}$ into the path-planning algorithm reduces the time UAVs spend circling near the previous waypoint, avoiding redundant searches, while also decreasing the time to reach the next waypoint, accelerating the OODA cycle, and improving search effectiveness.
• Through simulation experiments, the impact of a GPS positioning error and UAV disconnection on collaborative search efficiency was analyzed. It was found that although the average capture time increased, the UAV swarm could still complete the task of searching for moving targets.

The work presented in this study focuses on real-time task allocation for UAVs, addressing the problem of determining optimal flight directions based on current situation awareness. However, the collaborative search algorithm discussed in this article is only applicable to situations where there is no no-fly zone and the shape of the task area is a regular quadrilateral, and it does not consider the two limiting factors of UAV energy and communication limitations. Therefore, further research will be conducted in the future to address these two deficiencies. After optimizing the collaborative search algorithm framework, the multi-UAV formation control law will be explored to achieve UAV cluster assembly, formation, turning, and obstacle avoidance in the task area and then collaborate to complete the search task.