# Optimal Flow Sensing for Schooling Swimmers

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow Simulations

#### 2.1.1. Schooling Formation

#### 2.1.2. Flow Sensors

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#### 2.2. Optimal Sensor Placement Based on Information Gain

#### 2.2.1. Bayesian Estimation of Swimmers

#### 2.2.2. Estimated Expected Utility for Continuous Random Variables: School Location

#### 2.2.3. Estimated Expected Utility for Discrete Random Variables: School Size

#### 2.2.4. Optimization of the Expected Utility Function

## 3. Results

#### 3.1. Utility Function for the First Sensor

#### 3.2. Sequential Sensor Placement

#### 3.3. Inference of the Environment

#### 3.4. Shear Stress Sensors

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Configurations

## Appendix B. The Posterior Is Not Symmetric

## References

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**Figure 1.**Parametrization of the swimmer surface as described in Equation (1).

**Figure 2.**Simulation setup used for determining the optimal sensor distribution on a fish-like body. The follower is initially located inside the rectangular area. The number of swimmers in the leading group is varied between one and eight. The sensor-placement algorithm attempts to find the arrangement of sensors $\mathit{s}$ that allows the follower to determine with lowest uncertainty the relative position $\mathbf{r}$ and the number of swimmers ${n}_{f}$ in the leading group of swimmers. For each sensor ${s}_{k}$ the swimmer collects measurements ${y}_{k}^{1}$ and ${y}_{k}^{2}$ at locations ${x}_{1}\left({s}_{k}\right)$ and ${x}_{2}\left({s}_{k}\right)$ on the skin, respectively.

**Figure 3.**Snapshots of the pressure field in the environment of the follower swimmer generated by one (

**a**), four (

**b**) and seven (

**c**) schooling swimmers. The snapshots are taken at the moment the measurement was performed for one particular location of the follower in the prior region. High pressure is shown in red and low pressure in blue.

**Figure 4.**Utility curves for the first sensor using pressure measurements. In (

**a**) the utility estimator for the “size of the leading school” experiment is presented. (

**b**) corresponds to the utility estimator for the “relative position” experiment. We show the resulting curves for one, three and seven swimmer in the leading group and the total expected utility. We observe that although the form does not drastically change, the total utility increases with increasing size of the leading group.

**Figure 5.**Optimal sensor placement for the pressure sensors and the “size of the leading school” experiment. In (

**a**) the utility estimator for the first five sensors and in (

**b**) the value of the utility estimator at the optimal sensor location for the first 20 sensors are presented. In (

**c**), the distribution of the sensors on the swimmer surface is presented. Here, the numbers associated to each sensor indicate that this location is the i-th sensor location chosen according to Equation (21).

**Figure 6.**Optimal sensor placement for the pressure gradient sensors for the “relative position” experiment. In (

**a**), the utility estimator for the first five sensors and in (

**b**) the value of the utility estimator at the optimal sensor location for the first 20 sensors are presented. In (

**c**), the distribution of the sensors on the swimmer surface is presented. Here, the numbers associated to each sensor indicate that this location is the i-th sensor location chosen according to Equation (21).

**Figure 7.**(

**a**) Estimated posterior probability for a single sensor optimally placed and a single configuration per group size. The posterior shows clear peaks at the correct number of swimmer for all cases, leading to perfect inference of the parameter of interest. The posterior probability for (

**b**) optimal and (

**c**) worst sensor location for multiple configurations per group size. Here, for the optimal sensor location and one, two, three and five swimmer we see a clear peak for the true size of the group. For the worst sensor location the posterior is almost uniform and does not allow to extract any information about the size of group.

**Figure 8.**Estimated posterior for the final location for the best (left column) and worst (right column) sensor-location for one (upper row) and three sensors (lower row). Light colors correspond to high probability density values. We marked the actual location with a black circle.

**Figure 9.**Optimal sensor locations for the shear stress measurements for the “size of the leading school” in (

**a**) experiment and “relative position” experiment in (

**b**).

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**MDPI and ACS Style**

Weber, P.; Arampatzis, G.; Novati, G.; Verma, S.; Papadimitriou, C.; Koumoutsakos, P.
Optimal Flow Sensing for Schooling Swimmers. *Biomimetics* **2020**, *5*, 10.
https://doi.org/10.3390/biomimetics5010010

**AMA Style**

Weber P, Arampatzis G, Novati G, Verma S, Papadimitriou C, Koumoutsakos P.
Optimal Flow Sensing for Schooling Swimmers. *Biomimetics*. 2020; 5(1):10.
https://doi.org/10.3390/biomimetics5010010

**Chicago/Turabian Style**

Weber, Pascal, Georgios Arampatzis, Guido Novati, Siddhartha Verma, Costas Papadimitriou, and Petros Koumoutsakos.
2020. "Optimal Flow Sensing for Schooling Swimmers" *Biomimetics* 5, no. 1: 10.
https://doi.org/10.3390/biomimetics5010010