# Analysis of Cooperative Perception in Ant Traffic and Its Effects on Transportation System by Using a Congestion-Free Ant-Trail Model

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## Abstract

**:**

## 1. Introduction

## 2. Model and Simulation Scenario

- ${s}_{i}\left(t\right)$ is a binary variable, which either can be $0$ or 1 depending on whether the cell is empty ($0$) or occupied ($1$) by an ant at time step $t$.
- ${\sigma}_{i}$(t) is a numerical variable, which represents the pheromone concentration in the given cell. ${\sigma}_{i}\left(t\right)$ ranges from 0 to ${\sigma}_{sat}$, where ${\sigma}_{i}\left(t\right)=0$ means that there is no pheromone at time step $t$, whereas ${\sigma}_{i}\left(t\right)={\sigma}_{sat}$ means that the cell is saturated with pheromone at that time step. In a real-life AT study, pheromone concentration is measured in the number of molecules per cubic centimeter ($molecules/\mathsf{c}{\mathsf{m}}^{3}$). Whereas in ATM, pheromone concentration is measured in units of pheromone per cell ($punits/cell$).

- ${v}_{j}\left(t\right)$ is the instantaneous velocity of ant $j$ at time step $t,$ measured in cells per time step ($cells/timestep$). ${v}_{j}\left(t\right)$ is continuous and ranges from zero to one.
- ${p}_{j}\left(t\right)$ is the position of ant $j$ on the trail at time step $t$ and ranges from zero to $L$. Similar to ${v}_{j}\left(t\right),{p}_{j}\left(t\right)$ is also continuous.

#### 2.1. Stage I: Ant Motion

#### 2.2. Stage II: Pheromone Updating

- Evaporation:$${\sigma}_{i}^{\prime}\left(t+1\right)={\sigma}_{i}\left(t\right)-\left({\sigma}_{i}\left(t\right)\cdot {r}_{e}\right),\hspace{1em}if{\sigma}_{i}\left(t\right)0$$
- Accumulation:$${\sigma}_{i}\left(t+1\right)=\{\begin{array}{c}{\sigma}_{i}^{\prime}\left(t+1\right)+\tau ,\hspace{1em}if{s}_{i}\left(t\right)=1{\sigma}_{i}^{\prime}\left(t+1\right){\sigma}_{sat}{v}_{j}\left(t\right)0\\ {\sigma}_{sat},\hspace{1em}if{s}_{i}\left(t\right)=1{\sigma}_{i}^{\prime}\left(t+1\right)\ge {\sigma}_{sat}\end{array}$$

#### 2.3. Simulation Scenarios

## 3. Analysis of Pheromone Concentration and Its Implications for Cooperative Perception in the ATM

#### 3.1. Evaporation Rate and Fundamental Diagrams

#### 3.1.1. High-Medium Evaporation Rate ($0.5<{r}_{e}\le 1)$

#### 3.1.2. Meager Evaporation Rate ($0\le {r}_{e}\le 0.001$)

#### 3.1.3. Low Evaporation Rate ($0.005<{r}_{e}<0.1)$

#### 3.2. Pheromone Concentration and the Corresponding States of the CP&C in AT

#### 3.2.1. Minimal Pheromone State

#### 3.2.2. Inactive State

#### 3.2.3. Active State

#### 3.3. Pheromone Concentration and Cooperative Perception in AT

## 4. Analysis of Pheromone Dynamics in Cooperative Perceptions of AT

#### 4.1. Aggregation of Pheromone

#### 4.2. Depletion of Pheromone

- Minimal state: $1\ge {r}_{e}0.19$
- Active state: $0.19\ge {r}_{e}\ge 0.012$
- Inactive state: $0.012{r}_{e}\ge 0$

## 5. Analysis of Pheromone Emission Rate and Its Effect on CP&C System

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Validation of Analysis by Verifying the Non-Monotonic Behavior in ATM

${\mathit{r}}_{\mathit{e}}$ | ${\mathit{\sigma}}_{\mathit{X}}\left({\mathit{t}}_{\mathit{M}}\right)$ (Equation (9)) | $\mathit{D}$ (Equation (12)) | $\mathbf{Distance}\mathbf{Headway}=\mathit{D}\times 0.15$ |
---|---|---|---|

0.005 | 80.00 | 874.21 | 131.13 |

0.01 | 80.00 | 436.00 | 65.40 |

0.03 | 32.33 | 114.12 | 17.12 |

**Figure A1.**(

**a**) Headway ($cells$)-density and (

**b**) average velocity $\left(cells/timestep\right)$-density for different ${r}_{e}$ values are presented in order to predict and verify the rise-up phenomenon. In the headway-density plot, the averages of the distance headways measured of all of the platoons at low to medium density are plotted against the density at the time of measuring headways. The fitting curves based on the above mentioned low to medium density data are also presented to predict headways at higher densities. Using the predicted value of headway (Table A1) at which rise-up is hypothesized and fitting curves, we have calculated the density at which rise-up is expected (indicated by a dotted circle on the fitting curve). In (

**b**) average velocity from ATM simulations is presented against density to verify the rise-up phenomenon. (Simulation scenario: $L=1000cells,{\sigma}_{sat}=80punits/cell,{v}_{min}=0.15\mathrm{cells}/timestep,\tau =1punits/timestep)$.

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**Figure 2.**Fundamental diagrams for the ATM simulations with various ${r}_{e}$ values: (

**a**) average velocity ($cells/timestep$)$-$ density relationship and (

**b**) flow ($ants/timestep$)$-$ density relationship. Variables other than ${r}_{e}$ were kept constant: $L=1000cells,{\sigma}_{sat}=80punits/cell,{v}_{min}=0.15cells/timestep,and\tau =1punits$ [25].

**Figure 3.**Critical densities of fundamental diagrams from ATM simulations are plotted against evaporation rate (${r}_{e}$) of the corresponding simulations. For the simulations, variables other than ${r}_{e}$ were kept constant: $L=1000cells,{\sigma}_{sat}=80punits/cell,{v}_{min}=0.15cells/timestep,and\tau =1punits$.

**Figure 4.**Schematic representation of (

**a**) start of aggregation scenario ($time={t}_{0}$), and (

**b**) end of aggregation scenario ($time={t}_{M}$). In this scenario, we consider the $cel{l}_{X}$ (yellow cell), through which a considerable length of platoon will pass, leading to the aggregation of pheromone in that cell [29,30].

**Figure 5.**Approximate pheromone concentration at the end of the aggregation scenario (${t}_{M}$) based on Equation (9) against evaporation rate is presented (for the calculations $\tau =1punits/timestep,{\sigma}_{sat}=80punits/cell)$.

**Figure 6.**Approximate evaporation time required for the pheromone to decrease to the level below differentiable concentration (${\sigma}_{x}\le 1$) in depletion scenario is plotted against corresponding evaporation rate (${r}_{e}$) (for the calculation $\tau =1punits/timestep,{\sigma}_{sat}=80punit/cell$).

**Figure 7.**Schematic representation of the exclusion dynamics, which leads to creation of distance headway between adjoining ants from the same platoon in a single time step. In the diagram, the yellow cell represents the cell separating the rare end of the leading ant and the front end of the following ant.

**Figure 8.**A fundamental diagrams from the ATM simulations with various $\tau $ values: (

**a**) average velocity ($cells/timestep$)$-$ density relationship and (

**b**) flow ($ants/timestep$)$-$ density relationship. Variables other than $\tau $ were kept constant: $L=1000cells,{\sigma}_{sat}=80punits/cell,{v}_{min}=0.15cells/timestep,{r}_{e}=0.02$.

**Figure 9.**Approximate ${\sigma}_{sat}$ calculated for different $\tau $ is presented against evaporation rate (${r}_{e}$). The dotted line represents the ${r}_{e}$ that has been used for the analysis in the Section 5 (${r}_{e}=0.02,{\sigma}_{sat}=80punits/cell$).

Description | Symbol |
---|---|

Unique identity of a cell in the trail | $i$ |

Presence or absence of an ant in the trail $cel{l}_{i}$ at time t | ${s}_{i}\left(t\right)$ |

Pheromone concentration in the trail $cel{l}_{i}$ at time t | ${\sigma}_{i}\left(t\right)$ |

Pheromone concentration saturation level | ${\sigma}_{sat}$ |

Unique identity of an ant in the simulation | $j$ |

Velocity of the $an{t}_{j}$ at time $t$ | ${v}_{j}\left(t\right)$ |

Position of the $an{t}_{j}$ at time $t$ | ${p}_{j}\left(t\right)$ |

The minimum velocity of an ant towards the cell with no pheromone and no other ant | ${v}_{min}$ |

Number of ants in simulation | $N$ |

Stochastic parameter in velocity reduction scenario | $P$ |

Trail length | $L$ |

Evaporation rate | ${r}_{e}$ |

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**MDPI and ACS Style**

Kasture, P.; Nishimura, H.
Analysis of Cooperative Perception in Ant Traffic and Its Effects on Transportation System by Using a Congestion-Free Ant-Trail Model. *Sensors* **2021**, *21*, 2393.
https://doi.org/10.3390/s21072393

**AMA Style**

Kasture P, Nishimura H.
Analysis of Cooperative Perception in Ant Traffic and Its Effects on Transportation System by Using a Congestion-Free Ant-Trail Model. *Sensors*. 2021; 21(7):2393.
https://doi.org/10.3390/s21072393

**Chicago/Turabian Style**

Kasture, Prafull, and Hidekazu Nishimura.
2021. "Analysis of Cooperative Perception in Ant Traffic and Its Effects on Transportation System by Using a Congestion-Free Ant-Trail Model" *Sensors* 21, no. 7: 2393.
https://doi.org/10.3390/s21072393