*5.1. Simulation Results*

The analysis of the MOGWO exploration algorithm takes into consideration two aspects of the objective function: how it explores and how it improves the accuracy of the map. The experiment constraints influence the performance of the algorithm. Due to the GWO stochastic parameters, the decision-making process can be different in each simulation run. It leads us to test the algorithm performance several times with the same constraints. Based on the experiment parameters in Table 1, it can be calculated using Equation (8) that the iteration number should not be less than 60 and more than 120. In addition, for the waypoints, the range should be from 60 to 150 for three robots in a certain map size (Equations (9) and (10)). In this study, we selected the parameters as 60, 80, 100, and 120 iterations and 60, 80, 100, and 150 waypoints. Table 2 shows the results of map coverage in percentage, which is computed using the following equation:

$$\text{Map coverage } (\%) = \left( 100 - \frac{\text{sum of cell probability values after run time}}{\text{sum of cell probability values before run time}} \right) \times 100. \tag{11}$$


**Table 2.** Simulation results of the MOGWO exploration in several constraints.

In Table 2, it was emphasized that, for example, the maximum map coverage with a certain sequence of decisions and the constraints, 60 iterations by 60 waypoints, is 92.36%. Furthermore, the highest result among all the set of constraints used is 99.47% at the maximum allowable set of constraints.

Figure 6 shows one of the simulation-runs with the constraints: 120 iterations and 150 waypoints. In the *a* ≥ 1 stage, 87.57% of the environment was explored in half of the total number of iterations (61). For the map in Figure 6b, it can be concluded that the robots touched all the waypoints with the sensor rays. This means that the exploration ability of the algorithm is satisfied in this stage. Figure 6c demonstrates the completed result for the *a* ≤ 1 stage with a total of 99.06% map coverage. The trajectories of the robots can be observed through the blue, red, and green colored lines in Figure 6d.

**Figure 6.** The simulation of MOGWO exploration algorithm in (**a**) iteration 1 with a = 2; (**b**) iteration 61 with a = 1; (**c**) the last iteration, 120, with a = 0.016; and (**d**) the trajectories of three robots.

The decision-making process of each robot is presented in Figure 7. The values of alpha solutions in the simulation above (Figure 6) vary for the two stages: exploration and exploitation. It can be seen that when *a* ≥ 1, the trend goes up to maximum values, and when *a* ≤ 1, the simulation tries to achieve the minimum values.

The simulation of MOGWO exploration algorithm was implemented in MATLAB using the OccupancyGrid class of the Robotic System Toolbox [51,52]. The video of the simulation can be seen here [53].

**Figure 7.** Decision-making process of the alpha solutions: (**a**) robot 1, (**b**) robot 2, and (**c**) robot 3.

#### *5.2. The Pareto Optimality Analysis for MOGWO Exploration Algorithm*

Table 2 shows numerous exploration results, which are categorized by constraints. Considering only the percentage of map coverage, it is obvious that 99.47% is the best performance. However, the exploration with 120 iterations and 150 waypoints as constraints takes the longest time, which means it is not the optimal solution.

In this study, we take two factors that are important for the exploration: map coverage and time. In Figure 8, we searched for the trade-offs between the minimum number of iterations and the maximum number of map coverage. The plot was made based on the data in Table 2. The results for 120 and 60 iterations, which can be considered too long and too short for exploration, respectively, are extreme solutions. Thus, the solutions belong to the Pareto optimal front, which lies between two lines.

**Figure 8.** Pareto optimal set of the results of the MOGWO exploration algorithm.

It can be concluded here that the optimal set lies between 80 and 100 iterations with 150 waypoints. In the next subsection, the MOGWO exploration algorithm using 150 waypoints will be compared to two other algorithms using the same environment and map coverage computation using Equation (11).

#### *5.3. Comparison*

In this subsection, the proposed MOGWO exploration was compared with the original deterministic CME algorithm [22] and the hybrid stochastic exploration algorithm based on the GWO and CME [8]. The same map and experiment parameters (Table 1) were selected for the two algorithms with 60, 80, 100, and 120 iterations as it was implemented in the MOGWO exploration algorithm in Section 5.1. It should be noted that waypoints were not applied for the other two algorithms used in the comparison.

In the experiment, the CME algorithm was run only once for each iteration class (60, 80, 100, and 120) because it does not generate any random values even when tested multiple times. By its deterministic nature, CME implements differently for every modification in the environment. For instance, the simulation runs were aborted after the 98th iteration during our experiment when one of the robots got stuck next to the wall obstacle. For the purpose of completing the exploration, the initial position of robot 2 or r2 (from Table 1) was changed from (7,9) to (6,5). From these results, we can conclude that the exploration by the deterministic CME algorithm requires fine-tuning the map parameters for successful map coverage.

The hybrid stochastic algorithm is a stochastic approach, which uses the single-based GWO algorithm. During our experiments, the simulation-runs were aborted several times due to the robot's selection of inappropriate positions among the frontier cells. This situation occurs when the GWO parameters, *A* and *C,* oblige a robot to move into the wrong places, such as obstacles or another robot position. Fortunately, in this algorithm, the *A* and *C* parameters vary in each simulation-run, which allows us to obtain successful results.

Figure 9 shows the comparison of the results obtained using the original CME, the hybrid stochastic exploration, and the MOGWO exploration with 150 waypoints. It can be seen that the deterministic CME approach has the lowest values of map coverage among all the algorithms. The proposed MOGWO algorithm does not outperform the hybrid stochastic exploration algorithm in iteration classes, 60, 80, and 100. However, it surpasses the original CME in all iteration categories and the hybrid stochastic exploration algorithm in the category of 120 iterations. Additionally, aborted simulation-runs, which are drawbacks of the CME and hybrid stochastic exploration, did not occur in the MOGWO exploration. Thus, the MOGWO exploration proved more efficient and stable compared to the other algorithms studied in this subsection.

**Figure 9.** Comparison of three algorithms for the multi-robot exploration. The bars of the hybrid stochastic exploration algorithm based on GWO and coordinated multi-robot exploration (CME) are average values of 30 simulation-runs. The bars of the MOGWO exploration algorithm are average values taken from the row corresponding to 150 waypoints in Table 2.

#### **6. Conclusions**

This paper proposed a new method of solving the multi-robot exploration problem as a multi-objective problem. Two objective functions were formed: to search new terrain and to enhance the map accuracy. The use of the MOGWO algorithm enabled us to obtain high percent values of the map coverage without any aborted simulation-runs. The simulation results successfully demonstrated the capability of the MOGWO algorithm to build complete maps, which were completed within certain constraints: the number of waypoints and the number of iterations. Based on the results, the optimal solution was defined by the Pareto optimal set. Furthermore, the proposed MOGWO exploration algorithm was compared with the deterministic exploration and the hybrid stochastic exploration algorithms. The comparison showed that the proposed MOGWO exploration technique outperforms the deterministic exploration in all set of constraints and the hybrid stochastic exploration algorithm at 120 iterations and 150 waypoints.

**Author Contributions:** A.K. conceived and designed the algorithm. A.K., S.N. and D.Q. designed and performed the experiments. A.K., S.N. and D.Q. wrote the paper. A.K., D.Q. and S.G.L. formulated the mathematical model. S.G.L. supervised and finalized the manuscript for submission.

**Funding:** This work was supported by the Basic Science Research Program, through the National Research Foundation of Korea, Ministry of Science, under Grant 2017R1D1A3B04031864.

**Conflicts of Interest:** The authors declare no conflict of interest.
