# Swarm Intelligence-Based Multi-Robotics: A Comprehensive Review
## 2 Swarm Intelligence
### Ant Colony Optimization (ACO)  
ACO [17,56] is a probabilistic technique inspired by the foraging behavior of ants, which use pheromone trails to communicate and find the shortest paths to food sources. In ACO, artificial ants build solutions by moving through a parameter space and depositing virtual pheromones, influencing the probability of future ants selecting specific paths. This approach is particularly effective for solving combinatorial optimization problems such as the Travelling Salesman Problem (TSP) and is increasingly being applied to multi-robot systems. The ACO algorithm is described as follows:  
1. The algorithm starts by initializing the pheromone trails and other parameters. \[\tau_{ij}(0)=\tau_{0}\] (1) where \(\tau_{ij}(0)\) is the initial pheromone level on edge \(ij\), and \(\tau_{0}\) is a small positive constant.
2. Each ant \(k\) constructs a solution by moving from one node to another. The probability \(p_{ij}^{k}(t)\) of moving from node \(i\) to node \(j\) at time \(t\) is given by: \[p_{ij}^{k}(t)=\frac{\left[\tau_{ij}(t)\right]^{\alpha}\left[\eta_{ij}\right]^{ \beta}}{\sum_{l\in\mathcal{N}_{l}^{k}}\left[\tau_{il}(t)\right]^{\alpha}\left[ \eta_{il}\right]^{\beta}}\] (2) where, \[\bullet\quad\tau_{ij}(t)\] is the pheromone level on edge \(ij\) at time \(t\). \[\bullet\quad\eta_{ij}=\frac{1}{d_{ij}}\] is the heuristic information (inverse of the distance \(d_{ij}\)). \[\bullet\quad\alpha\] and \[\beta\] are parameters that control the influence of pheromone and heuristic information, respectively. \[\bullet\quad\mathcal{N}_{l}^{k}\] is the set of feasible nodes for ant \[k\] when at node \[i\].
3. After all ants have constructed their solutions, the pheromone levels are updated. The pheromone on edge \(ij\) is updated as follows: \[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}\] (3) where, \[\bullet\quad\rho\] is the evaporation rate (0 < \[\rho\] < 1). \[\bullet\quad\Delta\tau_{ij}=\sum_{k=1}^{m}\Delta\tau_{ij}^{k}\] is the amount of pheromone deposited by ant \[k\] on edge \[ij\], with: \[\Delta\tau_{ij}^{k}=\begin{cases}\frac{Q}{L_{k}}&\text{if ant $k$ uses edge $ij$ in its solution}\\ 0&\text{otherwise}\end{cases}\] \[\bullet\quad Q\] is a constant. \[\bullet\quad L_{k}\] is the length of the tour constructed by ant \[k\].  
Consider a scenario where a team of robots must navigate a warehouse to collect and deliver items to specified locations. Each robot uses ACO to determine the shortest paths while avoiding collisions and dynamic obstacles. Firstly, robots initialize pheromone levels across the warehouse grid. Then, each robot selects paths based on pheromone levels and heuristic information (e.g., distance to target) following Equation (2). After completing their paths, robots update the pheromone levels based on the efficiency of their routes according to Equation (3). Robots refine their paths through iterations, improving the warehouse operation's efficiency. The decentralized nature of ACO ensures robustness and scalability as the system adapts to the number of robots and dynamic changes within the environment.