# Enhanced Ant Colony Algorithm Based on Islands for Mobile Robot Path Planning
## 3 Brief Introduction of ACO
### Improved Distribution of the Initial Pheromones  
During the initial phase of ACO, the pheromone values are uniformly distributed throughout the environment. As a result, ants explore paths with a certain degree of randomness, leading to slow algorithm convergence. To address this issue, this article introduces an uneven initial pheromone distribution strategy, with its calculation principle illustrated in Figure 8.  
The rectangular region (with the adjacent island number \(L_{z}\) and \(L_{z+1}\) as vertices) is designated as the favorable region \(FG_{z}\). The equation of the straight line passing through points \(L_{z}\) and \(L_{z+1}\) is determined based on their coordinates:  
\[Ax+By+C=0, \tag{18}\]  
where \(A\), \(B\), and \(C\) refer to the relevant parameters of the linear equation.  
The formula for calculating the midpoint coordinate \(M(x_{M},y_{M})\) of the adjacent path nodes \(i(x_{i},y_{i})\) and \(j(x_{j},y_{j})\) within the rectangular area \(FG_{z}\) is expressed in Equation (19).  
Figure 8: Calculation principle of an uneven initial pheromone distribution.  
Figure 7: Distribution map of the angle coefficient for the next node.  
The local initialization pheromone concentration for this region is calculated by Equation (20). Subsequently, the local initialization pheromone concentration of other favorable regions in the environment is calculated in sequence. Finally, the non-uniform initial pheromone distribution for the entire map can be calculated using Equation (21):  
\[\begin{cases}x_{M}=\dfrac{x_{i}+x_{j}}{2},\\ y_{M}=\dfrac{y_{i}+y_{j}}{2}.\end{cases} \tag{19}\]  
\[\tau_{ij}^{{}^{\prime}}=\begin{cases}\dfrac{\sqrt{A^{2}+B^{2}}}{|Ax_{M}+By_{M} +C|},&\quad if\;i,j\in FG_{z},i\
eq j,\;Ax_{M}+By_{M}+C\
eq 0,\\ +\infty,&\quad otherwise\;if\;i,j\in FG_{z},i=j,\;Ax_{M}+By_{M}+C=0,\\ \dfrac{\sqrt{A^{2}+B^{2}}}{|Ax_{i}+By_{i}+C|},&\quad otherwise\;if\;i,j\in FG_ {z},i\
eq j,\;Ax_{M}+By_{M}+C=0.\end{cases} \tag{20}\]  
\[\tau_{ij}(0)=\begin{cases}\tau_{ij}^{{}^{\prime}},&\quad if\;i,j\in FG_{z}\; and\;\tau_{ij}^{{}^{\prime}}\
eq+\infty,\\ \lambda*\max\tau_{0},&\quad otherwise\;if\;i,j\in FG_{z}\;and\;\tau_{ij}^{{}^{ \prime}}=+\infty,\\ \min\tau_{0},&\quad otherwise\;if\;i,j\in FG_{z},\end{cases} \tag{21}\]  
where \(\tau_{ij}^{{}^{\prime}}\) indicates the local initialization pheromone concentration from node \(i\) to node \(j\) in the rectangular area \(FG_{z}\); \(\tau_{ij}(0)\) refers to the global pheromone concentration from node \(i\) to node \(j\) in any region of entire map; \(FG\) implies the grid set of all favorable region; \(\lambda\) represents the pheromone expansion coefficient; and \(\max\tau_{0}\) and \(\min\tau_{0}\) signify the maximum and minimum values of pheromone concentration in all favorable region, respectively. The latter can be expressed as follows:  
\[\left\{\begin{array}{ll}\max\tau_{0}=\arg\max(\tau_{ij}^{{}^{\prime}}),&\;if \;i,j\in FG\;and\;\tau_{ij}^{{}^{\prime}}\
eq+\infty,\\ \min\tau_{0}=\arg\min(\tau_{ij}^{{}^{\prime}}),&\;otherwise\;if\;i,j\in FG.\end{array}\right. \tag{22}\]  
A case of an uneven initial pheromone distribution is shown in Figure 9, where each red rectangle represents a favorable area between adjacent islands.  
Figure 9: Example of non-uniform initial pheromone distribution.