Various missions and objectives employ different approaches to describe the multi-agent system management problem (Dorri et al., 2018). For instance, Markov decision process (Liu et al., 2020); MAPF problem formulation, which is a layered optimization framework (Stern et al., 2019); and sampling-based approaches such as Monte Carlo simulation (Dalmasso et al., 2021) are some of the approaches that can describe the uncertainties in the environment properly and result in more robust solutions.

Although problem formulation is a key element to reach the main objectives, different methods might be applicable to solve those problems that become impossible to solve analytically in most cases according to the complexity order. Regardless of the problem formulation, two prevalent attitudes toward finding solutions are reinforcement learning (Qie et al., 2019; Semnani et al., 2020) and evolutionary optimization (Biswas et al., 2017; Li et al., 2021; Bahaidarah et al., 2024). Although the structures of both approaches are quite similar, various studies demonstrate differences between those methods to establish the advantages and weaknesses of both in different applications (Di Mario et al., 2013; Das et al., 2016). One of the best descriptions for the multi-robot path-planning problem is that it exploits combinatorial optimization to convert the discretized states into an efficient framework for optimization (Yu and LaValle, 2013; Okubo and Takahashi, 2023). Although the combinatorial problems allow considering multiple states and effective features regarding the mission objectives, the optimization search space becomes highly complicated and nonlinear. Some studies demonstrate that reinforcement learning is capable of solving such problems (Bengio et al., 2021). However, evolutionary optimization and particularly the genetic algorithm are a dominant methods for solving such problems according to their reliable heuristic behavior and convergence capability (Blum et al., 2011; Juan et al., 2015). Therefore, if a well-defined combinatorial optimization is used to describe the multi-agent path planning, the genetic algorithm will be a considerable approach for finding the solution. Although some research studies developed promising frameworks employing customized forms of the genetic algorithm to solve the pathfinding problem in 2D workspaces (Kala, 2012; Nazarahari et al., 2019) or independent coverage for large areas (Sun et al., 2019), many robotic applications in real-world scenarios demand considering other aspects such as uncertainties, robustness, and cooperative performance to fulfill their objectives. Therefore, the development of new optimization frameworks to investigate other practical objectives facilitates the applicability of employing reliable multi-agent robotic systems in real-world scenarios.

Path planning in an uncertain environment, which increases the risk of collision with the static or dynamic obstacle, is another challenging aspect that has been investigated in many recent studies (Rafai et al., 2022). One of the promising solutions to confront the uncertainties, particularly with the environment, is the risk-aware framework to avoid collisions with the obstacles (Pereira et al., 2013; Cai et al., 2021). However, such risk-aware frameworks can consider various types of risks in dynamic environments to address not only the collision but also other risk factors that cause mission failure (Barbosa et al., 2021).

On the other hand, the basic framework for pathfinding has a significant influence on the system’s performance. Although many different approaches are studied for path planning in various applications (Galceran and Carreras, 2013), the probabilistic roadmap (PRM) algorithm (Kavraki et al., 1996) proposed a flexible foundation that is adaptable to a wide range of planning problem formulation and optimization techniques to make robust planning frameworks, particularly in uncertain environments (Agha-mohammadi et al., 2014; Ravankar et al., 2020).

This paper proposes a novel form of genetic optimization to find a solution for a multi-agent robotic system employing a risk-aware PRM scheme that was presented in Rekabi-Bana et al. (2024) as a robust framework for one robot to explore an area efficiently with minimum failure risk. The proposed planning algorithm utilizes the genetic algorithm in conjunction with the single-agent exploration method to find a conflict-free solution that enforces the agents exploring all subdomains together. Each subdomain is considered a district in the exploration area that includes at least one exploration point. Moreover, it considers the risk of collision with the obstacles according to a risk assessment function obtained from a priori knowledge of the environment and the robot’s dynamics. However, according to the problem definition, the risk assessment function can be defined to consider different sources of failure risk according to different scenarios. The proposed optimization framework allows incorporating the risk of failure according to the environmental conditions and, therefore, results in a reliable solution for exploration scenarios in workspaces with uncertainties. It allows the team to combine the data collected from each district to obtain as much information about the explored environment as possible. Therefore, the mission planning will include finding a proper formation shape for data collection, allocating the destinations to the agents, and collision-free pathfinding through the environment considering obstacle avoidance. The algorithm is capable of path planning in both 2D and 3D environments. Therefore, it will be applicable to use in the robotic system introduced in Rekabi-Bana et al. (2023) and small drones as well. Furthermore, the secondary objective is to develop a compatible path-planning algorithm with the distributed control framework designed for a multi-drone system (Rekabi-Bana et al., 2021) to make an entire autonomous robust framework for a group of flying robots to explore uncertain environments. The optimization scheme considers the traveling distance and the failure probability to determine the cost function. According to the problem definition, the exploration subdomain sequence, the agents’ destination in each district, and the priority order to find a collision-free set of trajectories are considered effective parameters that should be optimized to reach the mission objective. The combinatorial nature of the problem implies a particular structure to efficiently find the optimum point. Therefore, a customized structure is introduced to generate the chromosomes, reproduce the population, and analyze the convergence in each generation. The algorithm was designed as a centralized motion planning framework and has no dependencies on the agents’ communication. Therefore, this algorithm is applicable to cases where the robots are working close to the ground station and have a reliable network with it. The main contributions of this paper can be summarized as follows:

• Development of a multi-agent collision-free PRM-based path-planning algorithm for a group of robots working in an arbitrary 2D or 3D environment to explore the area cooperatively. The proposed algorithm allows the team to explore the area in a particular formation geometry for application in the multi-arm manipulator described in Rekabi-Bana et al., (2023).

According to the features defined for the final solution, the suggested chromosome structure is composed of the district identifiers, the agents’ identifiers allocated to the destination points, and the order of agents’ identifiers for collision-free pathfinding. Therefore, each chromosome can be described as a matrix that is structured as shown in Figure 2.

Algorithm 1

Collision-free pathfinding.

According to the proposed structure, the initial population is created by generating N pop random permutations for the districts’ sequence, agents’ identifiers for mapping to the destination points, and the sequence for pathfinding in the graph. The population size is determined according to Ahn and Ramakrishna (2002) to comply with the nature of the problem.

The first reproduction method is to transfer the N e l best chromosomes directly to the next generation. However, the proposed 2D chromosome structure and the combinatorial nature of the problem do not allow for using popular strategies for regeneration through crossover and mutation. Therefore, two new strategies are proposed and applied to create the next generation from the current parents. The first strategy is a one-parent crossover according to the following steps:

• Selection of N crs random chromosomes from the current population as the crossover parents.

There is no theoretical method to establish the final convergence for the genetic algorithm in an arbitrary problem. Therefore, it is necessary to stop the algorithm while it satisfies a set of predefined conditions. Different types of convergence analysis are applicable to the genetic algorithms in different problems (Safe et al., 2004). In this paper, the following criteria are considered to determine the algorithm’s convergence:

• Elite group sustainability: if the chromosomes that reached the top 5 % remain in their positions for at least 20 generations, it means the search is converged, and it is more likely for the algorithm to stay on the same point as the optimum solution in the next generations.

As is presented in Figure 1, the formation shape is considered an equilateral polygon regarding the number of agents in the exploration team. The size and orientation of the polygons are determined according to the exploration points’ distribution in each district and the data collection radius for the agents. Therefore, the more sporadic the exploration points, the larger polygon will be considered to place the agents in positions for a complete coverage.

The results obtained for the cost function’s best value at each generation are presented in Figure 4. The results demonstrated in Figure 4 show the first 75 iterations of the optimization for all the configurations. The result indicates that the algorithm structure is designed properly and the heuristic and optimization stability is not affected by the group size. However, the more agents in the group, the larger the optimization search domain, and accordingly, the more iteration is required to satisfy the convergence criteria. Furthermore, the simulation results obtained using the proposed planning algorithm for three different group sizes are presented in Figure 5, demonstrating the trajectories that each agent should pass to reach the allocated destination points in target formations.

Although Figure 5 demonstrates the proposed algorithm’s output for path planning, it is necessary to analyze the minimum agent-to-agent distance criterion to evaluate the collision-free feature of the proposed multi-agent path-planning algorithm. The results obtained for that criterion in four team configurations are demonstrated in Figure 6. The results depicted in Figure 6 show that in all the studied conditions, the minimum agent-to-agent distance is more than almost 0.5 m, and no collision has occurred between the agents. Furthermore, it is evident that the variation bound of the distance between the agents decreases with increase in the number of agents. Although the presented results establish the capability of the proposed optimization for multi-agent collision-free path planning in an arbitrary environment, it is necessary to evaluate the proposed algorithm’s performance in comparison with other methods. The comparison results with two benchmark scenarios based on the statistical data obtained from multiple simulations for different configurations are presented in Figure 7. The first benchmark is the ideal exploration, which means that the agents explore the area by moving directly between each pair of districts regardless of the existing obstacles and collision risk according to the best combination of the districts obtained according to a TSP problem. The second benchmark is the random exploration without any risk of collision with the obstacle and directly between districts, as considered in the first benchmark. According to the results demonstrated in Figure 7, the behavior of the proposed algorithm is similar to the ideal benchmarks considered for each configuration, although it is supposed to avoid collision with obstacles and find the trajectory through the PRM graph. Furthermore, the proposed algorithm outperforms the random exploration by preventing the agents from repeatedly visiting the districts, which is a problem with the random exploration method.

Although the proposed algorithm provides exploration plans similar to an ideal case, the constrained optimization framework requires a considerable computation resource. Algorithm speed was analyzed using its implementation in MATLAB 2022b on a computer with a 12th Gen Intel(R) Core(TM) i9-1200k processing unit and 64 Gb of RAM. The results obtained for different numbers of agents, exploration districts, and environment dimensionality are presented in Table 1. It is clear that compared to the 3D environments, path planning in two-dimensional environments achieves relatively short run times, even with a significantly larger graph. The results also indicate that increasing the number of agents or districts increases the run time approximately linearly.

The proposed algorithm is also applied to determine a proper set of trajectories for the multi-arm robot designed for the RoboRoyale project (Rekabi-Bana et al., 2023). In this study, a set of robotic agents, attached to arms on the manipulator, will interact with the honeybee queen (Stefanec et al., 2022). Each agent will be equipped with a miniature sensor to retrieve information about the queen and the comb. The robot configuration is presented in Figure 8.

When the queen is active, the agents follow her and monitor her activity, and they detect where she lays the eggs. To assess the healthiness of the brood, the agents have to visit the locations of the previous egg-laying events and investigate the progress in the brood development. This requires that the agents leave the queen during the resting times and gather the observations of the brood as quickly as possible to minimize the risk of missing important queen activities. During the exploration, the agents have to avoid disturbing the colony and prevent collisions with any elements of the hive.

Our experiment is based on real egg-laying data collected with respect to a honeybee observation hive. There, an image processing system was used to determine a set of egg-laying events (Žampachů et al., 2022), and the events gathered over the 24-h period were considered target points for workspace partitioning and target formation determination. A geometrical constraint was also added to the optimization framework to comply with the robot’s coupling and kinematic constraints. Accordingly, the obtained results for three sets of egg-laying spots are demonstrated in Figure 9.

The presented results in Figure 9 demonstrate the optimization results and workspace partitioning in three cases. In each case, 150 egg-laying samples are selected randomly to make a different distribution. Therefore, the workspace partitioning and each district value for exploration would be different according to a new sample distribution. Accordingly, the resulting trajectory becomes different in each case and complies with the new sample distribution. The robot’s endpoint footprint presented in Figure 9 demonstrates how the algorithm attempts to find the best placement for the robot arms’ endpoint according to the kinematic constraints. The robot arms’ mechanical constraints make solid restrictions such as maximum and minimum diameter for expansion and retraction and the overall rotation, which do not allow for full coverage of target points in each domain. However, the proposed algorithm tries to attain the maximum exploration score by reaching the position to cover the maximum number of the target points at each district. The performance of our method is compared with that of an efficient algorithm recently proposed for a joint area coverage scenario with a team of robots (Nawaz and Ornik, 2023). The performance of both algorithms was evaluated in different conditions with different numbers of exploration target points sampled randomly from the egg-laying events. The results obtained from the comparison are shown in Figure 10. The results in Figure 10 show that the two algorithms achieve similar performance. However, our algorithm finds optimal efficient solutions considering additional criteria such as kinematic coupling and the conflict-free path between the agents, which are not considered in the MA-MDP framework proposed in Nawaz and Ornik (2023). Therefore, the comparison result establishes the efficiency and optimality of the algorithm as it complies with the practical requirements of the robots for the exploration.

According to the results presented in previous sections, it is evident that the proposed algorithm has some limitations, particularly regarding the run-time performance. The results demonstrated in Table 1 show that the current implementation framework is time-consuming and limited to applications that do not demand fast planning and re-planning. For instance, the studied case of a multi-arm manipulator for egg-laying coverage in an observation hive is one of the applications not needing prompt planning, and the proposed algorithm is a good solution for that application. However, implementing the algorithm with a more efficient programming system and employing parallel computation will enhance the performance and reduce the run-time to cover more practical robotic applications. Moreover, the centralized framework for the algorithm limited its applicability to those cases that rely on a ground station with a reliable communication link between the agents and the main processing unit. Therefore, developing a distributed form of the algorithm will be one of the future solutions that allow utilizing the proposed method for applications that have less reliance on the ground station.