# Information Theoretic Source Seeking Strategies for Multiagent Plume Tracking in Turbulent Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Methodology

Algorithm 1: Single robot search strategy. |

Input: Current estimate of the belief distribution over the possible position of the source. |

Output 1: Next waypoint on the search trajectory. |

Output 2: Updated estimate of the belief distribution over the possible position of the source. |

**Remark**

**1.**

**Remark**

**2.**

#### 3.1. Motion Model Uncertainty

- Randomly select one of the Gaussian models. Probability of selecting each model is to be proportional to the weight of Gaussian in the mixture model.
- Generate a sample from the selected Gaussian probability distribution.

#### 3.2. Gaussian Radial Basis Functions to Estimate the Probability Field

#### 3.3. Multi-Robot Collaborative Search Strategies

#### 3.3.1. Simple Collaborative Search

#### 3.3.2. Value of Information

- how probable that event is, considering the robot’s current estimate; and,
- how informative the event is as measured by the change it makes in the estimated probability field.

## 4. Simulations

#### 4.1. Multi-Robot Collaborative Search

#### 4.2. Experimental Results

^{−6}m

^{4}·s

^{−3}where g = 9.8 m·s

^{−2}is the gravity acceleration magnitude, ${w}_{i}$ = 4 cm·s

^{−1}is the inlet liquid velocity, ${A}_{i}=\pi {r}_{i}^{2}$ = 0.005 m

^{2}is the source cross-section area of radius ${r}_{i}$ and ${\alpha}_{b,i}=0.026$ is the inlet gas volume fraction. The initially unperturbed ambient fluid is thermally stratified with a constant slope $\zeta $ = 5.1 K·m

^{−1}and the system Coriolis parameter is set to f = 0.01 s

^{−1}. The cylindrical computational domain has a height H and diameter D of $H/{r}_{i}=D/{r}_{i}\approx 67$ with Dirichlet boundary condition at the bottom, no shear and no flux at the top for the momentum and scalars respectively and open lateral boundary conditions with numerical sponges to ensure numerical stability. The domain has been spatially discretized using spectral element methods into $K=7540$ conforming elements in which the solution is approximated with a 14th order polynomial expansion resulting in $\sim 21$ million nodes. The transport equations have been integrated using the nek5000 solver [27] that has demonstrated an excellent scalability on parallel machines [28]. The results used in this work correspond to the statistically stationary solution obtained after approximately 150, 000 core-hours on a Cray XE6 using 960 2.2 GHz AMD Magny-Cours cores.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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^{1}Consider the example where a robot is sitting in the middle of the plume and detects the presence of the contaminant material in all directions. In this scenario, it becomes impossible to determine a best search direction based on detection rate of the material plume alone.

**Figure 1.**The bar chart shows the mean number of timesteps per 10 trials to localize the plume source in a 200 × 200 unit workspace for a team of 3 robots. The robots start at predefined, randomly chosen, initial state. For each trial, robots start in the same initial state with a randomly generated initial belief distributions. The red line shows the percentage of the time the team successfully located the source for every 10 trials. None of the trials with 1.8 × 10

^{4}and 1.9 × 10

^{4}particles succeeded in localizing the source.

**Figure 2.**(

**a**–

**c**) The trajectory for a single robot searching for a source of chemical dispersion and the variation (

**d**) of the entropy of the estimated probability distribution. The red plot indicates the change in entropy the robot predicts upon the move, while the blue plot is the entropy of the estimation calculated after performing the observation. About 10, 000 particles are initially generated to estimate the probability distribution. The size of the particles does not represent the associated weight of the particle.

**Figure 3.**(

**a**–

**c**) Search trajectory for a single robot searching for a source of chemical dispersion. (

**d**) Variation of the entropy for the estimated probability distribution. The probability of detecting a chemical cue is proportional to the bearing of the robot with respect to the sources position. The chance of detecting a cue increases when the robot is moving toward the source. Red plot indicates the change in entropy the robot predicts upon the move, while the blue plot is the entropy of the estimation calculated after performing the observation. About 10, 000 particles are initially generated to estimate the probability distribution. Size of the particles does not represent the associated weight of the particle.

**Figure 4.**(

**a**–

**c**) Search trajectory for a single robot searching for a source of chemical dispersion. (

**d**) Variation of the entropy for the estimated probability distribution. The motion model of the robot injects uncertainty on the position of the robot, and consequently on the hypothesis of the particles. The sensing capability of the robot is assumed to be independent of the bearing of the robot, then the uncertainty in direction of the robot is not included in the estimation process. Red plot indicates the change in entropy the robot predicts upon the move, while the blue plot is the entropy of the estimation calculated after performing the observation. About 3000 particles are initially generated to estimate the probability distribution. Size of the particles does not represent the associated weight of the particle.

**Figure 5.**(

**a**–

**d**) The trajectory for a single robot searching for the source a of chemical dispersion. The probability field is spanned with 100 Gaussian functions each with a weight adjusted by each observation. (

**e**) Variation of the entropy for the estimated probability distribution. The sensing capability of the robot is assumed to be independent of the bearing of the robot. The red plot indicates the predicted change in entropy for the next move while the blue plot is the actual entropy of the estimation after integrating the observation.

**Figure 6.**(

**a**–

**c**) The trajectory of a member of a group of three robots searching for the source of chemical dispersion. Each observation is communicated to the group and each member of the group use the information it gets from the others to manipulate its estimation. The green stars indicate the position where the other member of the group make a positive observation. (

**d**) Variation of the entropy for the estimated probability distribution initially containing around 14, 000 particles. Note that the size of the particles does not represent the associated weight of the particle.

**Figure 7.**Search trajectory for a group of three robots searching for a source of chemical dispersion ((

**a**) Agent 1; (

**b**) Agent 2; (

**c**) Agent 3). The sensing capability of the robot is assumed to be independent of the bearing of the robot, then the uncertainty in direction of the robots is not included in the estimation process. The probability field is spanned by 300 Gaussian functions, which in this pictures are shown by a particles. Size of the particles does represent the associated weight of the particle. Each robot builds a probability distribution over the position of the source individually. At each step of time, every robot propagates the event it experiences. The other robots calculate the value of information encapsulate in that event and decide whether or not they contribute that sensor event to its estimation. The probability distribution for Robot 1 is pictured in 3D in Figure 8.

**Figure 8.**3D probability distribution estimated over the possible positions of the dispersion source by the Robot 1 ((

**a**) 10th Step; (

**b**) 20th Step). No particle is lost in the estimation process, and the incoming information stream is used to adjust the weights of the Gaussian functions.

**Figure 9.**A team of three robots searching for the source of chemical dispersion ((

**a**) 20th Step; (

**b**) 40th Step; (

**c**) 80th Step). The red dots indicates the positive sensor readings. The 2D dispersion model used in this simulations is developed to simulate the 2010 Deep Water Horizon Oil Spill. The dimensions of the simulation arena is 748 × 322 units, and the maximum reach of the robot at each time step is 5 units at each direction. The probability field is spanned by 100 Gaussian functions. At each step of time, every robot propagates the event it experiences. The other robots calculate the value of information encapsulate in that event and decide whether or not they contribute that sensor event to its estimation.

**Figure 10.**Variation of the entropy for the estimated probability distribution estimated by 100 weighted Gaussian functions. This entropy is related to the probability distribution estimated with one of the robots. The sensor detection with the robot is indicated with cyan diamonds, and the positive sensory events reported with other members of the team and contributed to the estimation is indicated with brown circles. As it can be seen these extra informations first disturbs the decreasing proceeding of the entropy of the estimation. However, eventually get in line with the general estimation of the robot.

**Figure 11.**The experimental setup (

**a**) to search for the source of oil spill. The robot is searching for the source of oil spill simulated to predict the behavior of the 2010 spill in Deep Horizon (

**b**). The position of the mobile robot is observed with motion capture cameras, and connected to the computer over the network. The Robot Operating System (ROS) environment is managing the data transfer and control of the robot. The algorithm is implemented by Matlab software.

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**MDPI and ACS Style**

Hajieghrary, H.; Mox, D.; Hsieh, M.A.
Information Theoretic Source Seeking Strategies for Multiagent Plume Tracking in Turbulent Fields. *J. Mar. Sci. Eng.* **2017**, *5*, 3.
https://doi.org/10.3390/jmse5010003

**AMA Style**

Hajieghrary H, Mox D, Hsieh MA.
Information Theoretic Source Seeking Strategies for Multiagent Plume Tracking in Turbulent Fields. *Journal of Marine Science and Engineering*. 2017; 5(1):3.
https://doi.org/10.3390/jmse5010003

**Chicago/Turabian Style**

Hajieghrary, Hadi, Daniel Mox, and M. Ani Hsieh.
2017. "Information Theoretic Source Seeking Strategies for Multiagent Plume Tracking in Turbulent Fields" *Journal of Marine Science and Engineering* 5, no. 1: 3.
https://doi.org/10.3390/jmse5010003