# Voronoi-GRU-Based Multi-Robot Collaborative Exploration in Unknown Environments
## 2 Background  
### Preliminaries  
Voronoi partitioning: Numerous studies have introduced this spatial partitioning method [26; 33; 34]. Consider \(n\) robots collaboratively exploring a region \(Q\). Let \(\mathbf{p}_{i}\) denote the position vector of robot \(i\), and let \(\mathcal{N}_{i}\) represent its neighbor set, where \(i=1,\cdots,n\). The Voronoi cell \(V(i)\) is defined in Equation (1):  
\[V(i)=\big{\{}\mathbf{q}\in Q\mid\|\mathbf{q}-\mathbf{p}_{i}\|\leq\|\mathbf{q}-\mathbf{p}_{j}\|, \forall j\in\mathcal{N}_{i}\big{\}}, \tag{1}\]  
ensuring that any point \(\mathbf{q}\) within \(V(i)\) is always closer to robot \(i\) than to any other neighboring robot \(j\). This geometrically intuitive decomposition enables efficient spatial task allocation by minimizing coverage overlaps between adjacent robots.  
Reinforcement learning: In reinforcement learning (RL), the reward is a scalar feedback signal that indicates how well an agent is performing in an environment. It is used to guide the learning process by providing immediate feedback after each action the agent takes. The objective of the reinforcement learning technique is to find an optimal shared policy \(\pi^{*}\)[35; 36] that maximizes the expected cumulative reward:  
\[\pi^{*}=\arg\max_{\pi}\mathbb{E}_{\pi}[R], \tag{2}\]  
where \(\pi\) represents the shared policy, and \(\mathbb{E}_{\pi}[R]\) denotes the expected cumulative reward. By learning this policy, the robots can effectively coordinate their actions to explore the region \(Q\) more efficiently through iterative environment interactions, balancing the exploration of unknown areas and the exploitation of gathered knowledge.  
Robot motion planning: Robot motion planning serves as the core technology for robotic autonomous navigation and task execution, aiming to generate safe and efficient motion trajectories for robots operating in complex environments. The key terminology employed in this paper is outlined as follows:* LiDAR: A laser-based sensor used to measure distances, used in this work for mapping and obstacle detection.
* Global Plan: The optimal path from the start to the goal computed using algorithms like A* [37], considering static obstacles and map constraints, as well as serving as a reference trajectory for navigation systems.
* A*: A heuristic search algorithm that efficiently finds minimum-cost paths by combining the actual traversal cost (\(g(n)\)) and the estimated remaining cost (\(h(n)\)) [38], being widely used in robotics for global path planning.
* Local Plan: Short-term path planning that handles immediate obstacles and dynamics, optimizing robot movement.
* TEB [39]: Timed Elastic Band, a local planner that optimizes robot trajectories as elastic bands, balancing path smoothness, obstacle clearance, and kinematic constraints, as well as supporting dynamic obstacle avoidance through receding-horizon optimization.
* Cartographer [40]: A graph-based SLAM system that uses submaps and loop closure detection for consistent 2D/3D mapping, supporting multi-sensor fusion (LiDAR and IMU) for robust localization.
* Multi-Robot Map Merge: The process of combining maps from multiple robots to create a unified map, essential for cooperative navigation.
* Optional Goal Set: A set of possible goals that robots can choose from, used in task allocation and planning.
* Relative Position: The position of one robot relative to another, based on coordinate systems, critical for coordination.
* Target Point: A specific point that a robot needs to reach, used in path planning and navigation.