Swarm Intelligence-Based Multi-Robotics: A Comprehensive Review

2.1. Ant Colony Optimization (ACO)

2.2. Particle Swarm Optimization (PSO)

• A swarm of particles is initialized with random positions and velocities in the search space. Each particle i has a position vector ${\mathbf{x}}_i$ and a velocity vector ${\mathbf{v}}_i$.

  $$\begin{array}{cc}\hfill {\mathbf{x}}_i\left(0\right)& =\mathrm{rand}({\mathbf{x}}_{min},{\mathbf{x}}_{max})\hfill \end{array}$$

  (4)

  $$\begin{array}{cc}\hfill {\mathbf{v}}_i\left(0\right)& =\mathrm{rand}({\mathbf{v}}_{min},{\mathbf{v}}_{max})\hfill \end{array}$$

  (5)
• The velocity of each particle is updated based on its own best-known position ${\mathbf{p}}_i$ and the global best-known position $\mathbf{g}$.

  $$\begin{array}{c}\hfill {\mathbf{v}}_i(t+1)=w{\mathbf{v}}_i\left(t\right)+c_1{r}_1({\mathbf{p}}_i\left(t\right)-{\mathbf{x}}_i\left(t\right))+c_2{r}_2(\mathbf{g}\left(t\right)-{\mathbf{x}}_i\left(t\right))\end{array}$$

  (6)

  where:
  • w is the inertia weight.
  • $c_1$ and $c_2$ are acceleration coefficients.
  • $r_1$ and $r_2$ are random numbers uniformly distributed in $[0,1]$.
  • ${\mathbf{p}}_i\left(t\right)$ is the best-known position of particle i.
  • $\mathbf{g}\left(t\right)$ is the best-known position of the entire swarm.
• The position of each particle is updated based on its new velocity.

  $$\begin{array}{c}\hfill {\mathbf{x}}_i(t+1)={\mathbf{x}}_i\left(t\right)+{\mathbf{v}}_i(t+1)\end{array}$$

  (7)
• After updating the positions, the algorithm updates each particle’s and global best-known positions.

  $$\begin{array}{cc}\hfill \mathrm{if}f\left({\mathbf{x}}_i(t+1)\right)<f\left({\mathbf{p}}_i\left(t\right)\right)& \mathrm{then}{\mathbf{p}}_i(t+1)={\mathbf{x}}_i(t+1)\hfill \end{array}$$

  (8)

  $$\begin{array}{cc}\hfill \mathrm{if}f\left({\mathbf{x}}_i(t+1)\right)<f\left(\mathbf{g}\left(t\right)\right)& \mathrm{then}\mathbf{g}(t+1)={\mathbf{x}}_i(t+1)\hfill \end{array}$$

  (9)

2.3. Firefly Algorithm (FA)

• Its brightness determines a firefly’s attractiveness $\beta$ and decreases with distance. The attractiveness ${\beta }_{ij}$ of firefly i to firefly j is given by:

  $$\begin{array}{c}\hfill {\beta }_{ij}={\beta }_0{e}^{-\gamma r_{ij}^2}\end{array}$$

  (10)

  where:
  • ${\beta }_0$ is the initial attractiveness.
  • $\gamma$ is the light absorption coefficient.
  • $r_{ij}$ is the distance between firefly i and firefly j.
• The distance $r_{ij}$ between firefly i at position ${\mathbf{x}}_i$ and firefly j at position ${\mathbf{x}}_j$ is calculated using the Euclidean distance:

  $$\begin{array}{c}\hfill r_{ij}=\parallel {\mathbf{x}}_i-{\mathbf{x}}_j\parallel \end{array}$$

  (11)
• Firefly i moves towards a brighter firefly j according to:

  $$\begin{array}{c}\hfill {\mathbf{x}}_i={\mathbf{x}}_i+{\beta }_{ij}({\mathbf{x}}_j-{\mathbf{x}}_i)+\alpha (\mathrm{rand}-0.5)\end{array}$$

  (12)

  where:
  • $\alpha$ is a randomization parameter.
  • rand is a uniformly distributed random number in [0,1].
• The brightness I of a firefly is associated with the objective function value. For a minimization problem, the brightness $I_i$ of firefly i can be inversely related to the objective function $f\left({\mathbf{x}}_i\right)$:

  $$\begin{array}{c}\hfill I_i\propto {\displaystyle \frac{1}{f\left({\mathbf{x}}_i\right)}}\end{array}$$

  (13)

2.4. Cuckoo Search (CS)

• The position of a cuckoo is updated using a random walk called Levy flight, a step characterized by a heavy-tailed probability distribution. The new position ${\mathbf{x}}_i^{(t+1)}$ of cuckoo i is given by:

  $$\begin{array}{c}\hfill {\mathbf{x}}_i^{(t+1)}={\mathbf{x}}_i^{\left(t\right)}+\alpha ·\mathrm{Levy}\left(\lambda \right)\end{array}$$

  (14)

  where:
  • $\alpha$ is a step size scaling factor.
  • $\mathrm{Levy}\left(\lambda \right)$ is a Levy flight distribution.
• The Levy flight distribution is defined as:

  $$\begin{array}{c}\hfill \mathrm{Levy}\left(\lambda \right)=t^{-\lambda },(1<\lambda \le 3)\end{array}$$

  (15)
• The fitness $f\left({\mathbf{x}}_i\right)$ of each solution ${\mathbf{x}}_i$ is evaluated using the objective function of the optimization problem.
• A fraction $p_a$ of the worst nests (solutions) are replaced with new solutions:

  $$\begin{array}{c}\hfill {\mathbf{x}}_i^{(t+1)}={\mathbf{x}}_i^{\left(t\right)}+\alpha ·\mathrm{Levy}\left(\lambda \right)\mathrm{if}\mathrm{rand}<p_a\end{array}$$

  (16)

  where $p_a$ is the probability of discovering an alien egg.
• The best solutions are carried over to the next generation based on their fitness values.

2.5. Bat Algorithm (BA)

3.1. Search and Rescue

• Decentralized Coordination [52]: In SAR operations, multiple robots must explore large, often unknown, environments to find victims. Traditional centralized control approaches can be inefficient and prone to single points of failure. SI leverages decentralized coordination, where each robot operates based on local information and simple rules. This enhances robustness and scalability.
• Path Planning and Exploration [54]: SI algorithms excel in path planning and exploration, critical aspects of SAR missions. For instance, (i) ACO can simulate the behavior of ants foraging for food. Robots lay down virtual pheromone trails to mark paths that lead to successful discoveries (e.g., locating a victim). Other robots follow and reinforce these trails, optimizing the search paths over time; (ii) PSO can model the collective movement of birds or fish. Each robot adjusts its trajectory based on its own experience and the successes of its neighbors. This collaborative behavior helps cover more areas efficiently and quickly identify victims.

3.2. Environmental Monitoring

3.3. Agriculture

• Field Coverage and Monitoring [60,61]: In large agricultural fields, uniform coverage and monitoring are critical for soil analysis, crop health assessment, and pest detection. Multi-robot systems, guided by SI algorithms, can ensure comprehensive field coverage by distributing the robots evenly across the area.
• Planting and Seeding [39]: Precision planting and seeding are crucial for maximizing crop yield and minimizing waste. SI algorithms enable robots to coordinate their movements and actions to plant seeds at optimal distances and depths.
• Pest and Weed Control [9]: Managing pests and weeds is vital for healthy crop growth. Swarm robotics can precisely deploy robots to identify and target pests and weeds, reducing the need for blanket pesticide applications.
• Harvesting [62]: Harvesting crops is labor-intensive and time-sensitive. Swarm robots can collaborate to harvest crops efficiently, ensuring minimal damage and maximizing yield.

• Field Coverage: The agricultural field must be covered without significant overlaps or gaps.
• Obstacle Avoidance: Robots must navigate around irrigation systems, fences, and other obstacles.
• Battery Life: Each robot has limited battery life, requiring efficient path planning.
• Task Efficiency: Robots must perform tasks such as planting, spraying, and harvesting efficiently.
• Data Reliability: Data regarding crop health and environmental conditions must be accurate.

3.4. Space Exploration