# A Bacterial Chemotaxis-Inspired Coordination Strategy for Coverage and Aggregation of Swarm Robots

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bacteria Inspiration

#### 2.1. Bacteria Behavior

#### 2.2. Applications

## 3. Problem Statements and Preliminary

#### 3.1. Problem Statements

- (1)
- Environment. The environment E ⊂ R
^{2}is unobstructed and bounded. The size denotes L × L. L is the number of units for the length of horizontal (ordinate) range. There are no chemical signals in the environment. - (2)
- Robots. The robot i is labeled as r
_{i}, i ∈ N, N = {1, 2, …, n}. Each robot is omnidirectional and can move in any direction. The physical structure of the robot is ignored. The robot can only sense the presence of other robots within a sense radius. The robots can collaborate with each other only by using the position information of their neighbors. - (3)
- Radius. As shown in Figure 3, we define Rs and Ri as the covered radius and sense radius, respectively, in which Ri = 2Rs. The robot can cover circular areas with a radius smaller than Rs. The robot can sense others within the Ri. Given robot i, the set of its neighbors is defined ${N}_{i}\triangleq \left\{s\in N|\left|\right|{x}_{s}-{x}_{i}\left|\right|\le Ri,i\ne s\right\},\mathrm{and}x\in {\mathbf{R}}^{2}$ is the position of the robot. N
_{i}follows the conditions: (i) ${N}_{i}\subset N$; (ii) $i\notin {N}_{i}$; (iii) $s\in {N}_{i}$.

_{i}(t) = (x

_{i}

_{,1}(t), x

_{i}

_{,2}(t), …, x

_{i}

_{,n}(t))

^{T}∈

**R**

^{n}is the state variable of robot i at time t, g(x

_{i}(t), [x

_{j}(t)]

_{j}

_{∈N}

_{i}

_{(t)}) is a coordinate strategy that robot i updates its state variable according to the state variables of itself and its neighbors at time t. The symbol ⊕ denotes the strategy g(x

_{i}(t), [x

_{j}(t)]

_{j}

_{∈N}

_{i}

_{(t)}) is performed on the state variable of the x

_{i}(t).

#### 3.2. Chaotic Model

_{i}, b

_{i}], then a new chaotic sequence z* can be obtained by the mapping and de-mapping according to Equations (3) and (4).

_{0}and ρ, which is used to find z

_{0}and ρ for chaotic sequences with good ergodicity and global searching ability.

_{0}= 0.9978 and ρ = 4, the Logistic chaotic mapping equation can form an averagely distributed chaotic sequence with good ergodicity, which can search space globally. Therefore, the initial value z

_{0}= 0.9978 and ρ = 4 are selected for chaotic model in the following text.

## 4. Proposed Method

- (1)
- Construct the fitness function, which is available for coverage and aggregation.
- (2)
- Compare the current fitness function value with previous ones.
- (3)
- The robot makes running or rotating depending on the comparison result in list (2).

#### 4.1. The BCCS for Coverage

^{2}is the range of horizontal and vertical range of the environment. Moreover, n elements with great differences are extracted from z* to form a new sequence ${x}^{\prime}$ as the initial positions of the robots. ${x}_{i}^{\prime}\in {x}^{\prime},{x}_{i}^{\prime}({x}_{i,1}^{\prime},{x}_{i,2}^{\prime})$ is the initial position of robot i. Chaotic preprocessing is optional without considering the interference of the robots.

_{k}(k ∈ Cell_Index) is the center of the cell k. If ||c

_{k}− x

_{i}(t)|| < Rs, then the cell k is coved by robot i at time t, as shown in the left part of Figure 6. Specially, the green cell is coved by robot i, but the yellow cell is not. When the robot i senses the robot j (d

_{ij}=||x

_{i}− x

_{j}|| < Ri), overlapping cells are generated, such as the gray cells in the right part of Figure 6. We defined the number of cells only covered by robot i at time t as F

_{i}(t), which also is the fitness function value of robot i, that is:

_{i}(t). O

_{i}(t) is the set of cells coved only by robot i at time t, and can be expressed by:

_{i}(t) is the neighbors of robot i at time t.

**r**is the integrated virtual repulsive force acting on the robot i from its neighbors at time t. A simple case is shown in Figure 7. It is evident that the robot moves far away from its neighbors by virtual forces.

_{i}, and the coverage will be achieved.

#### 4.2. The BCCS for Aggregation

_{1}and w

_{2}, $0\le {w}_{1}{,w}_{2}\le 1$ and w

_{1}+ w

_{2}= 1, are the weights of the neighbor impact and random impact, respectively. In this coordinate strategy, w

_{1}and w

_{2}can be dynamically adjusted according to the number of neighbors of the robot. The larger the value of w

_{1}(neighbor impact) gets, the smaller w

_{2}(random impact) is. For example, w

_{2}= 0, means that the robot can sense all the individuals in the swarm after iterations, which causes all the robots to be aggregate quickly.

_{i}and simultaneously monitor the number of neighbors in its sub-aggregation. When robots from multiple aggregations, w

_{2}can promote the exploration of individuals within each sub-aggregation. Fusion occurs when two sub-aggregations meet, and the aggregation will be achieved. In the whole process, there is no central controller or global network in the swarm robotic system and no direct communication between the robots.

## 5. Simulations and Discussion

#### 5.1. Simulations Setup

#### 5.2. Evaluation Criteria

_{i}for all the robots, which is shown as Equation (13). F corresponds to the performance measure from the perspective of the degree of the covered cells of the individual robot; (2) S is the number of cells covered by all the robots, which is shown as Equation (14). S corresponds to the performance measure from the perspective of the degree of the covered cells of the environment; (3) average and standard deviation of iterations; (4) the number of aggregations is another important performance measure for aggregation.

#### 5.3. Comparison for Coverage

_{i}of each robot over iterations. We can conclude that the F

_{i}of each robot increases as the number of iterations increases for two methods at the beginning of the coverage. In Figure 11a, the F

_{i}of some robots oscillates over a wide range for the existing bacteria-inspired controllers. As shown in Figure 11b, F

_{i}of each robot for A1 gradually converges after 80 iterations. It is because the robot evaluates the neighbors’ positions when performing rotating. In fact, the robot moves far away from its neighbors, and direction is calculated by Equation (8). The smaller the oscillation amplitude for all the robots is, the more stable the swarm robotic system is.

_{k}= 0.005. x′ (t + 1) = x (t + 1) + σ

_{k}is the actual moving position of robot at time t + 1. Figure 12 shows the evolution of F for all the robots with and without noise. For the noise free environment, although the robot based on the existing bacteria-inspired controller is more exploratory at the first 50 iterations, due to the randomness of the robot’s initial motion direction, the robot based A1 can quickly adjust their motion planning as the number of iterations increases. In Figure 12, although the noise reduces the stability of change of F for A1, the performance is better than that for the controller in Reference [27]. Meanwhile, after 60 iterations, the covered area of the environment for A1 is larger than that for the existing bacteria-inspired controller as the number of iterations increases, as shown in Figure 13.

#### 5.4. Comparison for Aggregation

_{2}helps a sub-aggregation explore space and search other sub-aggregations. The higher w

_{2}means the more obvious the exploration capability of the robot in a sub-aggregation. When the robot senses the others with Ri, w

_{2}is zero, and the robot loses the exploration capabilities and quickly gathers to its neighbors. Figure 15 shows the evolution of the performance measure F

_{i}of each robot over iterations for two methods. In Figure 15b, the F

_{i}of each robot for BCCS gradually converges after 60 iterations. In fact, each robot will not escape from its current aggregation because the w

_{2}of each robot is zero.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**The trajectory sequence distribution diagrams of five combinations of initial values and parameters.

**Figure 5.**Feedback system architecture for robot motion based bacterial chemotaxis-inspired coordination strategy (BCCS).

**Figure 10.**Simulation results for coverage in noise free environment: (

**a**–

**f**) The controller in Reference [27]; (

**g**–

**l**) A1.

**Figure 11.**The F

_{i}evolution of each robot for coverage in noise free environment: (

**a**) The controller in Reference [27]; (

**b**) A1.

**Figure 12.**The F evolution of two methods without and with noise for coverage. The red and purple lines are for the controller without and with noise in Reference [27], respectively. The blue and green lines are for A1 without and with noise, respectively.

**Figure 13.**The S evolution of two methods without and with noise for coverage. The red and purple lines are for the controller without and with noise in Reference [27], respectively. The blue and green lines are for A1 without and with noise, respectively.

**Figure 14.**Simulation results for aggregation in noise free environment: (

**a**–

**f**) The controller in Reference [27]; (

**g**–

**l**) BCCS.

**Figure 15.**The F

_{i}evolution of each robot for aggregation in noise free environment: (

**a**) The controller in Reference [27]; (

**b**) BCCS.

**Figure 16.**The F evolution of two methods without and with noise for aggregation. The red and purple lines are for the controller without and with noise in Reference [27], respectively. The blue and green lines are for BCCS without and with noise, respectively.

**Figure 17.**The S evolution of two methods without and with noise for aggregation. The red and purple lines are for the controller without and with noise in Reference [27], respectively. The blue and green lines are for BCCS without and with noise, respectively.

**Figure 18.**The comparison of the number of aggregations: (

**a**) n = 9; (

**b**) n = 16; (

**c**) n = 25. (

**a**–

**c**) are three simulation schemes under different number of robots. The vertical coordinate values of blue star points for BCCS maintain 1 for thirty simulations. The vertical coordinate values of the red circle for the controller in Reference [27] are from 1 to 5.

Related Works | Coverage | Aggregation | Multitask | |
---|---|---|---|---|

Related bacteria chemotaxis-inspired control strategies | [5] | ✓ | ||

[6] | ✓ | |||

[34] | ✓ | |||

[35] | ✓ | ✓ | ||

[27] | ✓ | ✓ | ✓ | |

This paper | ✓ | ✓ | ✓ |

Parameter | Value |
---|---|

Environments, E | 60 units × 60 units, 80 units × 80 units, 100 units × 100 units |

Cell size, k | 1 unit × 1 unit |

Number of robots, n | 9, 16, 25 |

Velocity of robot, v | 1 unit/iteration |

Maximum iterations, Imax | 2L |

Covered radius, Rs | 10 units |

Sense radius, Ri | 20 units |

Logistic equation iterations, I_{c} | 500 |

Rotation range, θ | 0–360° |

**Table 3.**Performance comparison results. Avg = average of iterations. Std = standard deviation of iterations. Succ = success rate.

The Controller in Reference [27] | A1 | BCCS | #Robots | #Environment | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Avg | Std | Succ (%) | Avg | Std | Succ (%) | Avg | Std | Succ (%) | (Unit × Unit) | |

84 | 10.88 | 93.33 | 73 | 9.35 | 100 | 55 | 8.81 | 100 | 9 | 60 × 60 |

117 | 17.19 | 76.67 | 98 | 13.17 | 93.33 | 71 | 8.47 | 100 | 16 | 80 × 80 |

162 | 15.88 | 60.00 | 126 | 19.55 | 90.00 | 108 | 22.88 | 100 | 25 | 100 × 100 |

Robot | x_{i}_{,1} | x_{i}_{,2} |
---|---|---|

1 | 10 | 10 |

2 | 30 | 10 |

3 | 50 | 10 |

4 | 10 | 30 |

5 | 30 | 30 |

6 | 50 | 30 |

7 | 10 | 50 |

8 | 30 | 50 |

9 | 50 | 50 |

The Controller in Reference [27] | BCCS | #Robots | #Environment | ||||
---|---|---|---|---|---|---|---|

Avg | Std | Succ (%) | Avg | Std | Succ (%) | (Unit × Unit) | |

104 | 8.53 | 46.67 | 56 | 9.85 | 100 | 9 | 60 × 60 |

137 | 8.26 | 30.00 | 100 | 14.67 | 100 | 16 | 80 × 80 |

178 | 10.36 | 23.33 | 125 | 7.76 | 100 | 25 | 100 × 100 |

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**MDPI and ACS Style**

Jiang, L.; Mo, H.; Tian, P.
A Bacterial Chemotaxis-Inspired Coordination Strategy for Coverage and Aggregation of Swarm Robots. *Appl. Sci.* **2021**, *11*, 1347.
https://doi.org/10.3390/app11031347

**AMA Style**

Jiang L, Mo H, Tian P.
A Bacterial Chemotaxis-Inspired Coordination Strategy for Coverage and Aggregation of Swarm Robots. *Applied Sciences*. 2021; 11(3):1347.
https://doi.org/10.3390/app11031347

**Chicago/Turabian Style**

Jiang, Laihao, Hongwei Mo, and Peng Tian.
2021. "A Bacterial Chemotaxis-Inspired Coordination Strategy for Coverage and Aggregation of Swarm Robots" *Applied Sciences* 11, no. 3: 1347.
https://doi.org/10.3390/app11031347