2.1. Environmental Model

The rasterized method [21] is used to divide the search area into L_x × L_y cells; that is, the search area has L_x cells in each row and L_y cells in each column. The coordinates of the g^th cell are denoted by g = (x_g, y_g), where x_g ∈ {1, 2, ⋯L_x} and y_g ∈ {1, 2, ⋯L_y}. It is assumed that multiple UAVs take off from the airport with the same flight attitude into the search environment for target search, that each UAV flies at a uniform speed at each time step, and that at each time step, the UAVs can only move one cell to an adjacent cell.

2.2. Environment-Aware Map Model

2.3. UAV Model

In this study, the sensor detection radius of the UAV is D_r, the TPM of the grids within the detection range, and χ is updated locally according to the detection update rule in Section 2.2. The TPM of the undetected grids is updated using information diffusion, which is described in detail in Section 3.1. As shown in Figure 2, the detection area of the UAV at each moment is indicated by the surrounding green square.

2.4. Target Model

2.5. Obstacle Avoidance Model

3.1. Collaborative Search Framework in Poor Communication Environments

In the aforementioned problem, the search area is divided into multiple grids using the grid-based approach. Assuming the initial position of the target is known but its movement direction is unknown, a normal distribution is used to describe the initial distribution of the target’s position. This distribution affects the update of the TPMs. During the actual search process, the target is moving and obstacles may be encountered. This is reflected in the TPM by increasing the target existence probability in the grid where the target is located and reducing the target existence probability to zero in the grids where obstacles are detected. Therefore, the initial target distribution cannot be used continuously to update the TPM. This paper adopts an information diffusion approach to compensate for the impact of the aforementioned issues and dynamically adjust the target information acquired by the UAVs.

The effectiveness of UAV decisions can be determined by a reward function, where the reward function J_i(S_i(k), u_i(k)) represents the reward obtained by UAV_i when it is in the state S_i(k) and takes control input u_i(k). A higher value of J_i(S_i(k), u_i(k)) indicates better performance of u_i(k). However, in adverse communication environments, when UAVs package and transmit their locally sensed environmental information to other UAVs, some data may be lost, leading to significant errors in the information fusion results between UAVs. As a result, consensus cannot be reached among the UAVs, and the decisions made may lead to mutual repetition or conflicts, thereby reducing the overall efficiency of cooperative search. Therefore, it is necessary to address the issue of inconsistent information among UAVs in a degraded communication environment. In this paper, we employ RBFNN to recover the damaged information and utilize a weighted averaging method to perform a multisource fusion of information between UAVs, ultimately achieving consensus among all UAVs in the group.

After addressing the aforementioned issues, UAVs make decisions based on the acquired environmental information, specifically in the context of path planning. In this paper, we propose the use of the MPC-based IGWO algorithm for path planning of the UAV cluster. MPC considers future states during the search process, enabling UAVs to adapt better to the environment and tasks. Additionally, compared to other algorithms, IGWO incorporates strategies such as Levy flight and nonlinear convergence factors, which enhance its ability to escape local optima. Figure 4 illustrates the framework of the multi-UAV cooperative search under adverse communication conditions.

3.2. Update of Environment-Aware Map

As described above, the process of updating the TPM is influenced by the distribution of the target’s position and the presence of obstacles detected in the environment. In this section, an information diffusion model is used to simulate the effects of target movement, and the discovered obstacle grid cells are subjected to specific operations to reduce the probability of the target’s existence to zero. Additionally, a revisiting mechanism is incorporated into the update process to address the issue of the target potentially moving into areas that have already been searched due to the extended search duration.

3.2.1. Update of TPM

In the process of the UAV searching for a target, the probability distribution of the target UAV will change owing to the target movement, so we cannot always use the initial distribution of the target to update the local TPM of the UAV. Thus, we must make a reasonable prediction of the target movement to improve accuracy when the TPM is updated.

For the initial distribution of information on the target, we adopted the information diffusion mechanism [23] to predict the change in the probability distribution of the target. This approach is similar to the gas diffusion mechanism, which mainly describes the phenomenon in which information will diffuse from the current grid to the surrounding neighboring grids. The information of the neighboring grids will also diffuse into the current grid, the information will diffuse from the high-probability grids to the low-probability grids, and the probability distribution of the target will converge to a uniform distribution after a long period of time.

For example, the basic diffusion model of the TPM for grid g can be expressed as follows:

$$ ()

where λ_d is the diffusivity and $$ is the set of neighboring grids of grid g. In general, the number of neighboring grids is an integer; it was set to eight to ensure the local characterization of the information.

To ensure that the information diffusion process is stable, the diffusion rate must meet the following requirements:

$$ ()

Using the initial distribution information of the target and the above probability diffusion model, any grid target probability can be obtained at any time, and all UAVs have the same initial information and probability diffusion rules, which avoids the problem of inconsistent information among UAVs. In addition, using the predicted target probability distribution to update the target probability of the corresponding grids results in a higher level of confidence.

The above diffusion process corresponds to the case in which there are no obstacles in the search area. However, after detecting an obstacle through the sensors, the UAV needs to change the way in which the information diffuses in the obstacle area and its neighboring areas.

No possible target is assumed to exist in the obstacle region. Prior to identifying this zone, information is diffused in the same manner as in a normal grid. Once the obstacle is discovered, the normal (nonobstacle) grid around the obstacle does not diffuse the probability information to the obstacle grid, and the target probability that already exists in the obstacle grid needs to be diffused to the normal grid around it. This update method is used instead of directly dropping to zero. This approach ensures that the TPM of the overall region will not change abruptly and that the local characteristics of the region near the obstacle can be retained. The probability decrease using the diffusion mechanism is very fast; thus, the target probability of the obstacle region is reduced quickly and eventually approaches zero. The diffusion method can be expressed as follows:

$$ ()

where $$ is the normal grid adjacent to the obstacle.

The direction of the diffusion of grid probabilities is shown in Figure 5, where g₁ is the obstacle grid, g₂ is the normal grid adjacent to the obstacle, and g₃ is the normal grid. The red and green arrows show the direction of the probability of spreading outward and inward, respectively.

3.3. Group Consensus Among UAVs

The transmission of environmentally aware information in poor communication environments leads to partial data loss. As described in this section, we first compensate for the lost data and then fuse the UAV local information with the processed received information to make the environment-aware information consistent among all UAVs.

3.3.1. Information Recovery

Suppose that all information about grid g in the transmission message E_j(k) from UAV_j is lost. The following methods were adopted to recover lost information from other UAVs. Regarding the target probability at grid g, firstly, owing to the local correlation of the transmission information, the target probability of the grids around grid g can be collected using the k-nearest neighbors algorithm [24] (KNNA), and then the collected data are fitted to the lost probability of grid g using the RBFNN [25]. Regarding the uncertainty of the loss χ of grid g, χ_ig(k) can be set to one because the transmission information has been lost and the UAV knows nothing about the grid.

When using the KNNA to collect the neighbor grid information of grid g, the number of neighboring grids collected is limited to ensure that the collected information is relevant to the missing information. The distance threshold, which is the distance between grid g and its neighboring grids, is defined as follows:

$$ ()

If L ≤ L_d, then the position information $$ and target probability of g₋neighbor are recorded. In addition, because the target probability between the obstacle and nonobstacle regions has obvious differences, the acquisition information of the two must be distinguished to improve the accuracy of the subsequent operation, selected as follows:

$$ ()

Figure 6 illustrates the neighborhood grid information collection at L_d = 1, with the loss probability grid in red and the neighboring grids within the distance threshold in green.

After collecting sufficient valid information using the KNNA, this information must be used to fit the missing values. Assuming the target probability in grid g from UAV_j’s TPM is lost, because the information contained in the TPM consists of only the coordinates of the grid and the target probability, the fitting function can be designed as p_jg(k) = f_recover(x_g, y_g), where p_jg(k) represents the target presence probability of grid g recovered at time k and f_recover is a recovery function fitted using the coordinates and target presence probabilities of neighboring grids g₋neighbor in UAV_j. Because the RBFNN has the characteristics of a simple structure, fast learning speed, excellent approximation performance, and generalization ability, it was employed in this study to solve the above-fitting function.

The RBFNN is a commonly used three-layer feedforward network, the three layers are the input, implicit, and output layers. The implicit layer has multiple radial basis activation functions, which are used to map the input vectors from low-dimensional m to high-dimensional n; thus, an indistinguishable low-dimensional linear space is converted into a distinguishable high-dimensional linear space. The RBFNN is a local approximation network in which only a few connection weights in the local region of its input space affect the output. Compared with backpropagation and other global approximation networks, an RBFNN has a faster learning speed, which is conducive to meeting the real-time requirements of searching for targets in UAVs. The input of the RBFNN consists of the coordinates of neighboring grids and the target probability, whereas the output consists of the target probability of the damaged grid. Gaussian functions are used as the radial basis functions. The structure of the RBFNN is illustrated in Figure 7.

The parameter that must be determined in the RBFNN training process is the spread speed d, which mainly serves to adjust the sensitivity of the hidden layer to the data. The lower the spread speed d, the higher the sensitivity of the implicit layer to the input, and the more accurate the function approximation; however, the approximation process will be poorly smoothed, resulting in average network performance. The higher the d is, the smoother the function fitting will be, but the approximation error will be correspondingly larger.

3.4. Collaborative Target Search Algorithm

Section 3.3 presents UAV environment perception information with collective consensus. This section describes the combination of the MPC strategy with IGWO to find the optimal path for each UAV. Firstly, the UAVs exchange their respective environmental information such as position and target probability. Then, MPC is used to generate a set of feasible predicted paths for each UAV. Subsequently, IGWO evaluates the quality of each path using a path evaluation function and selects a path with a higher evaluation in each iteration. This newly generated path can be one of the original paths or a path generated by the algorithm. Through multiple iterations, the UAVs can achieve collaboration and obtain more efficient search paths.

3.4.1. Description of MPC

The basic idea of MPC is to predict the future dynamics of the UAV at each sampling moment, solve a finite-time open-loop optimization problem online, and consider the first element of the obtained control sequence as the decision of the UAV. This method of predicting the future state is helpful for enhancing the perception performance and adaptability to the environment of UAVs and for ensuring the optimality of the planned path to a certain extent; therefore, it is widely used in path planning problems.

The basic process of MPC can be described as follows: At moment k, UAV_i predicts that the control input sequence for the next m-steps as u[k : k + m − 1] = [u(k), ⋯, u(k + m − 1)], and J_i(S_i(k), u_i[k : k + m − 1]) can be used to evaluate the effect of the control input sequence. If J_i(S_i(k), u_i[k : k + m − 1]) is the highest, it means that the effect is the best among all the control input sequences, and the optimal control sequence can be expressed as follows:

$$ ()

Considering the accuracy of the prediction, only the first term, that is, $$, is executed. When the moment steps to k + 1, the above process is repeated, and the optimal control input, $$, is obtained at moment k + 1. Figure 8 shows the three-step MPC prediction process for m = 6, with the dashed line indicating the predicted trajectory, and the bold solid line indicating the actual action taken.

3.4.2. IGWO

3.4.2.1. Gray Wolf Optimizer (GWO) Algorithm

The GWO [26] is a classical and effective IOA that simulates the hunting behavior of gray wolves in nature, with the advantages of a simple structure and high solution accuracy. Gray wolves hunt in a hierarchy, as shown in Figure 9, where the lower ranked wolf ω is led by higher ranked wolves α, β, and δ to surround, approach, and capture prey.

Assuming that the number of gray wolves in the pack is Pop_size and the position of the gray wolf i at moment k is S_i(k) = (x_i(k), y_i(k)), the gray wolf with the highest fitness value is denoted as α (i.e., the one closest to the prey). The gray wolves with the second- and third-highest fitness values are denoted as β and δ, respectively, and the remaining gray wolves are denoted as ω. The mathematical description of the GWO is as follows:

$$ ()

$$ ()

In Equation (16), D denotes the bearing and unit distance moved by a gray wolf toward the target, k is the number of iterations, S_p(k) is the position of the prey at iteration to k generations, and C is a coefficient, expressed as:

$$ ()

where r₁ is a random value in the interval [0, 1]. Equation (17) indicates that the distance between the gray wolves and the target decreases and, eventually, the gray wolf pack forms an encircling posture on the target. The convergence factor A can be expressed as follows:

$$ ()

$$ ()

where r₂ in A is a random value in the interval [0, 1], a is a control parameter for the convergence factor A, and MAX is the maximum number of iterations.

3.4.2.2. Improvement Strategies

• 1.

  Levy flight strategy: Levy flight is stochastic wandering with a heavy-tailed probability distribution of the step length, which has a high probability of jumping out of the current search area during the wandering process. It then explores a larger search space and thus is helpful for solving the problem of falling into the local optimal solution, which can be mathematically described as follows:

$$ ()

where S(k) is the position of an individual gray wolf at the k^th iteration.

$$ ()

where p is the step size of the Levy flight; η is a random number belonging to [0,2], usually 1.5; S_b is the current optimal individual; and μ and v follow a normal distribution.

• 2.

  Nonlinear convergence factor strategy: The convergence factor A determines the search strategy of the gray wolf, and the control parameter a in the convergence factor A is optimally set to regulate the relationship between global and local searches more effectively throughout the iteration process.

$$ ()

As can be seen from Equation (23), we propose that the convergence factor decreases more gently in the preiteration period, which is conducive to the algorithm conducting a global search in a larger search space. The latter period decreases more sharply, which is conducive to the local search of the algorithm and can find the optimal solution in the candidate space more quickly.

• 3.

  Optimal individual information fusion strategy: To improve the ability of the community to find the optimal value in the global range further, the following strategy is designed such that the optimal individual in the k^th iteration is α_k, β_k, and δ_k; α, β, and δ are the individuals with the optimal fitness values up to this iteration, which is briefly referred to as the global optimal individual. Because the risk of falling into local optimality increases in the late iterations of the algorithm, it is considered that the optimal individuals α, β, and δ (e.g., in the k^th generation) and other stochastic individuals of this generation are integrated into the global optimal individuals in the form of differential evolution during the iteration process, which not only enhances the stochasticity and diversity of the community but also improves the solving efficiency of the algorithm. The mathematical description of this strategy is as follows

$$ ()

$$ ()

where S_α,β,δ denotes the positions of the global optimal individuals α, β, and δ; S_α(k), S_β(k), and S_δ(k) are the positions of the k^th iteration optimal individuals α_k, β_k, and δ_k, respectively; and S_rand i(k), i = 1, 2, 3 denotes the three randomly selected individuals in the k^th iteration. In Equation (25), K₁ is the positional update weight.

The pseudocode for IGWO is given in Algorithm 2.

Algorithm 2: Pseudocode for IGWO.

• 1: Input Population size Pop_size, Number of iterations MAX, Search area

• 2: Calculate the fitness of all gray wolves in the k^th iteration, and denote the top three individuals in fitness as α_k, β_k and δ_k

• 3: Update the global optimal individual position by using (24) and (25)

• 4: Update a and A by applying (23) and (19), update C according to Ref. [27]

• 5: Update all the individual positions by utilizing (16) and (17), apply Levy flight strategy to the individuals, and retain the optimal solution

4.1. Information Recovery and Fusion Performance Testing

The search area was a rectangular area of 9.5 km × 9.5 km, which was divided into 19 × 19 grids, each with a length and width of 0.5 km, in which five targets existed with initial positions of 3.25 km and 2.25 km, 4.75 km and 4.75 km, 6.75 km and 4.75 km, 3.75 km and 8.25 km, and 6.25 km and 6.75 km; in this area, the speed was 50 m/s for horizontal or vertical flight, the speed was √2 times the horizontal for oblique flight, the decision time interval was 10 s (i.e., the UAV was at the center of any grid each time it made a decision), and the maximum number of movement steps was S_MAX = 30; the information was lost during the transmission process, and the number of lost data grids was 10% of the total number of grids in the area.

Figure 12 shows a vertical view of TPM fusion for each UAV. Comparing Figures 12(a) and 12(b) demonstrates that numerous grids have significant missing data in the received information of UAV1, and the loss of these data hinders the subsequent decision-making process of UAV1. Figures 12(c) and 12(d) show the fused TPMs after recovering the target probabilities of the missing grids based on the strategies of RBFNN and [20] for UAV1, respectively. The RBFNN method of this study evidently has a better information recovery effect because [20] uses the least squares method for regression fitting, and the form of the fitting function used is generally adapted to the distribution of the target probability, whereas the RBFNN does not need to consider the form of the distribution of the fitting function. In addition, Li et al. [20] have a minimum restriction on the number of grids collected by the KNNA, but some of the grids that are at the boundaries of the obstacles are unable to collect enough information about neighboring grids. Figures 12(c) and 12(e) show the respective fused TPMs of the two UAVs, which are essentially the same as those of the fused TPM of the unlost data in Figure 12(a). The maximum value of the information error for the same grids in Figures 12(c) and 12(e) is 6.06 × 10⁻⁴, so that the information fusion strategy proposed in this study can enhance the consistency of information among UAVs and thus reach a better group consensus state.

To compare the two methods further, the recovery error of each lost data grid is shown in Figure 13, where the total number of lost data grids is 40. The maximum recovery error of the strategy in [20] is 0.00223, and the maximum recovery error of the proposed strategy is 0.00168. Moreover, among the 40 grids with lost data, the recovery results of the strategy used in this study are better than those of [20] in 28 out of 40 grids. These recovery results are better than those of [20], and in most of the grids with better recovery results, the error is less than 3 × 10⁻⁴; thus, the recovered fusion TPM can be assumed to be equal to the state of the unlost data. Therefore, the information recovery method adopted in this study can be assumed to reduce the information error caused by the data loss significantly.

4.2. Effect of UAV Parameters on Search Performance

For all three experiments, each set of experiments was run independently 20 times, and the results were averaged.

From Figure 14(a), when D_r = 0.5 or 1 km, all targets cannot be found at the end of the search, and the UAV cluster finds all targets when D_r > 1.5 km. Only one target is found in the whole search process when D_r = 0.5 km. In addition, the grid coverage increases owing to the increase in the detection radius of the UAV. When D_r is small, the cluster search efficiency is poor; therefore, the algorithm in this study is more sensitive to the detection radius of the sensors used, and sensors with smaller detection radii cannot be used.

From Figure 14(b), when N = 2, the UAV cluster finally searches for four targets; when N > 2, the UAV cluster can find all targets, and the grid coverage also improves with an increase in the number of UAVs.

From Figure 14(c), the UAV cluster, only the sensor with {p_d = 0.9, p_f = 0.1}, finds all the targets, and the search efficiency is higher. The sensors with smaller p_d and larger p_f all increase the grid coverage in the middle and late iterations to jump out of the local optimum to find more targets. However, they are not successful. The {p_d = 0.9, p_f = 0.1} sensors can find more targets by covering fewer grids, indicating that the sensor performance significantly affects the search results, and UAVs can achieve good search results only when the sensor performance is excellent.

4.3. Performance Comparison of Different Search Algorithms

To test the performance of the search algorithm proposed in this study, simulation experiments were conducted using six search algorithms, including the algorithm proposed in this study: GWO, distributed gray wolf optimizer (DGWO) [28], motion-encode particle swarm optimization (MPSO) [29], greedy [19], distributed MPC (DMPC) [30], and IGWO. GWO, DGWO, and MPSO are popular IOAs that apply their respective rules for reasoning and solving to predict the optimal target grids for the next moment. DGWO divides all individuals in the wolf pack into several islands, with each island containing multiple candidate solutions. The islands update their positions according to the updating rules of GWO, and they exchange their candidate solutions with each other at a certain frequency; MPSO encodes the search trajectory into motion paths of the UAV, which vary with the particle positions in the PSO algorithm, preserving group cognition and global consistency. The greedy method favors grids with the largest target probability in the next moment. DMPC uses MPC for the next m-steps of the UAV, seeking optimal, high-efficiency paths, which is computationally costly.

The simulation scenario is the multitarget scenario described in Section 4.1, with three UAVs with initial departure positions at 0.25 km and 1.25 km, 0.25 km and 4.25 km, and 0.25 km and 7.25 km, respectively. The search radius D_r was 1.5 km. The number of targets searched for, time used for searching, and coverage of detection grids were used to analyze the strengths and weaknesses of the method. In addition, the overlap of the UAV search path with the grid of high-probability regions can aid in analyzing the performance of the algorithm.

The comparison of the number of searched targets and the coverage of detected grids during the search process of the six methods is shown in Figure 15, which demonstrates that the IGWO algorithm outperforms the other algorithms in terms of the number of targets found, area coverage, and search efficiency. The performance of the IGWO is average in the presearch period, whereas the DMPC and DGWO have advantages in both the number of searched targets and coverage of detected grids because in the presearch period, UAVs have less information about the environment. In the DMPC method, the local information can be updated quickly by traversing all the predicted paths, whereas IOAs use their own updating rules for inference. Owing to the small difference in information between the grids, the degree of certainty of the grid information is low, which leads to a large error in the inference results. DGWO provides an additional exploration mechanism where the island model increases the probability of transforming low-quality candidate solutions on each island into better solutions. Compared to GWO, the computational complexity is significantly reduced, which is the reason for its efficient early-stage search. In the early stages of the search, IGWO directs UAVs toward high-probability areas. As the search progresses, each UAV begins to consider factors such as probability distribution and uncertainty, and based on their respective positions, they disperse their actions. The speed and quantity of target discoveries steadily increase, with all five moving targets found by 220 s. In contrast, other methods only found four targets by their maximum search time, demonstrating that the approach proposed in this paper has a stronger ability to escape local optima.

For the detection grid coverage, the proposed method maintains a low level in the first and middle periods. This occurs because the next moment of UAV travel to the grid predicted by this method has a higher reward value, which can accurately search for the target in a smaller detection area. This finding proves the feasibility of the strategy proposed in this study. At the late stage of the search, as the other methods get stuck in local optima, they fly around the local optimum for a long time, failing to find the last target, which slows the growth of coverage. However, the method proposed in this study promptly escapes local optima. After all the targets were found in 220 s, the UAV was unable to find the target each time it updated its position because the target disappeared; therefore, it continued to expand the search area to find the target, and the coverage rate increased rapidly in the process.

To demonstrate the superiority of the proposed method more intuitively, the comparison data of the six methods at different time periods are listed in Table 2. At t = 100 s, the number of targets searched by DGWO was two, while the other methods were all one, and the coverage of DMPC was higher than that of the remaining algorithms; at t = 150 s, the number of targets searched by DGWO was one, while the other methods were all one; at t = 200 s, IGWO and GWO can search four targets with lower coverage, whereas the greedy algorithm and DMPC only searched three targets, DGWO still only found four targets; at t = 250 s, only the IGWO searched all targets and the growth rate of its coverage was accelerated; and at t = 300 s, only the IGWO found all targets and achieved a coverage of 0.7875, which is higher than those of the other algorithms.

The overall search process is shown in Figure 10, with the final TPM of UAV1 in the bottom panel (Section 4.1 demonstrates that the TPMs of all UAVs are consistent), where the darker areas are the obstacles detected by the UAVs and the red and violet dots are the starting points of the UAV and targets, respectively. Notably, the trajectories of IGWO and GWO were the same in the early stage; however, the advantages of the strategy developed in this study became effective after the UAV gathered a certain amount of environmental information and avoided the local optima to locate all the targets. DGWO shows high efficiency in the early stage of search but tends to get trapped in local optima in the mid to late stages. The MPSO and greedy methods focused on high-probability regions, leading to local optima entrapment and hampering search efficiency. DMPC can escape local optimum; however, it has limitations with updating rules, impacting the overall information utilization and resulting in inferior search efficiency compared to that of IGWO.