# Multi-Robot Exploration Employing Harmonic Map Transformations
## 3 Methodology
### Robot Exploration
#### 3.3.2 Multi-Robot Setup  
The framework outlined for single robot exploration can be extended to accommodate multiple robots working cooperatively. This extension provides enhanced exploration efficiency and coverage, by leveraging collaborative efforts among multiple agents. The method detailed in Figure 5 is used for multi-robot exploration. First, utilizing SLAM, a local occupancy grid map is constructed. Then, if possible, maps from other robots are merged with the local map. Next, the boundaries are extracted and if frontiers are  
Figure 4: An example of inner frontier. (**a**) a workspace with a singular L-shaped inner obstacle. The explored region is colored in blue and the frontier in green, showing that the inner corner remains unexplored. (**b**) HMT of the explored region with the dipolar vector field overlayed on top. The inner obstacle is mapped to the green point and the white arrow shows the desired angle of approach. The dipolar vector field guides the robot through the desired angle. detected the robot navigates to the nearest frontier as explained previously. To reduce frontier overlap and increase efficiency, a frontier allocation process is used that will be discussed later. The approach is implemented in a distributed manner, since each robot has limited information about the others and can independently explore the workspace regardless of the state of the other agents. The information shared among robots involves their location, their map of the so far explored workspace, and their currently explored frontier. If the explored region of a robot intersects with another's, the new information is used to expand their explored region. Finally, overlapping between explored regions is minimized by the goal frontier selection, further optimizing the exploration process. By using a distributed approach, each robot can act autonomously, thus making the system robust to robot failure. It is important to consider the scalability of the system. Since the approach is distributed, when more agents are added the complexity of the system grows only in the amount of data shared and the number of maps to be merged. If all robots exchange information with each other, then \(n(n-1)\) information exchanges would be required, where \(n\) is the total number of robots, making the scaling of the system \(O(n^{2})\). The complexity is reduced by limiting the inter robot communication to a certain range, reducing the total number of information exchanges to \(O(nl)\), where \(l\leq n\) is the average number of agents  
Figure 5: The multi-robot exploration algorithm. within each robot's communication range. The amount of data in each exchange is mainly dependent on the size of the map being shared and its resolution. A big advantage of using multiple robots is the ability to combine many maps to produce a complete mapping of the workspace. This process is called map merging [26]. The merging process facilitates the creation of a unified and comprehensive map, thereby eliminating the need to fully map the workspace with each individual robot. In our approach, when a map is shared, it is aligned with the robot local map and added to the robot's merged map, as shown in Figure 6. Thus, each robot has an individual local and merged map. This approach ensures that the system does not rely on a central map merging node, although a common frame of reference is selected for all merged maps, so their resulting maps can be comparable. Moreover, the map merger checks if the added area is connected to the local explored region, in the sense that a clear path connecting the two regions exists in the merged explored region. If the two areas overlap, then the new region is added to the HMT. Notable is a fact that during the merging process when the local map overlaps with another, the data of the local map is selected as correct, so as to not interfere with the local explored region when a map is misaligned.