# A Self-Organizing Area Coverage Method for Swarm Robots Based on Gradient and Grouping
## 2 Problem Description  
Assuming that the individual robot can complete the tasks within its coverage area. Through reasonable and uniform coverage, not only the maximum coverage of the task area is guaranteed, but also the number of robots is reduced as much as possible. Figure 1 shows a scenario in which a group of robots self-organize to cover an unknown task area. Swarm robots enter the task area through self-organization and form the pattern of the adaptive region as soon as possible in the task area. While avoiding collision, the distance between robots should not only maintain the communication connection but also cover the task area effectively. Table 1 shows the notation definitions for this article. Figure 1: The swarm robotic system covers the task area. **Definition 1** (_Swarm robotic system_). : _The swarm robotic system is defined as a set of \(n\) robots in a two-dimensional Euclidean space, i.e., \(A=\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{n}\}(n\geq 3)\). Each robot \(\alpha\) is a 6-tuple \(\langle\textbf{x},\textbf{v},\textbf{status},\textbf{g},\textbf{leder},\textbf{ grad}\rangle\), Where (1) **x** is the position of robot \(\alpha\); (2) **v** is the speed of Robot \(\alpha\); (3) status is the current status of robot \(\alpha\); (4) \(g\) is the group of robot \(\alpha\); (5) leder means whether robot \(\alpha\) is the leader or not; (6) grad is the gradient of robot \(\alpha\). The swarm robotic system has the following assumptions._  
* All robots are homogeneous and have the same speed. * Each robot is treated as a point of motion. * Robots communicate with each other locally and obtain the state of other robots within the communication range. **Definition 2** (_Covered area_). : _The covered area \(D\) is defined as a set of points on the two-dimensional environment \(\mathbb{R}\), \(D\subseteq\mathbb{R}^{2}\)._  
**Definition 3** (_Coverage rate_). : _Let the coverage area of swarm robotic system be \(S_{A}\).The coverage rate is defined as \(P(A,D)=S_{A}/S_{D}\times 100\%\). That is the percentage of the total target area covered by all robots, and the overlapped area of each robot should be eliminated._  
**Definition 4** (_Completion time of coverage_). : _When the coverage error is less than \(\varepsilon_{0}\), the coverage rate of the swarm robotic system is large enough to meet the requirements of task completion. The completion time of coverage is defined as \(t_{dpl}(A,D)\), i.e. :_  
\[\lim_{t\to t_{dpl}(A,D)}(1-P(A,D))\leq\varepsilon_{0} \tag{1}\]  
The swarm robots cover the unknown area, and the coverage of the individual robot is a circle with a center of the robot and a radius of \(r\). The adjacent robots can form a network through local communication. The shape of the covered area is usually irregular and unknown, which requires the swarm robots to cover the area as much as possible. At the same time, the urgency of each task point in the area requires the swarm robot to have a smaller coverage time. The more robots there are, the greater coverage for the static area, but it will lead to a waste of resources. Assuming that all the robots move at the same speed, the swarm robotic system needs to be able to cover the task area as quickly as possible by self-organizing. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Describe** \\ \hline \(\mathbb{R}^{2}\) & The two-dimensional environment \\ \(A\) & The swarm robotic system \\ \(\textbf{x}\) & The position of robot \\ \(\textbf{v}\) & The speed of Robot \\ _status_ & The current status of robot \\ \(g\) & The group of robot \\ _leader_ & The leader of the swarm robots \\ _grad_ & The gradient of robot \\ \(D\) & The covered area \\ \(S_{A}\) & The coverage area of swarm robotic system \\ \(S_{D}\) & Total area to be covered \\ \(P(A,D)\) & The coverage rate \\ \(t_{dpl}(A,D)\) & The completion time of coverage \\ \(d_{safe}\) & The safe distance \\ \(d_{cover}\) & The coverage distance \\ \(R_{c}\) & The communication distance \\ \hline \hline \end{tabular}
\end{table}