# 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

- Morrow, J.E. Schooling Behavior in Fishes. Q. Rev. Biol.
**1948**, 23, 27–38. [Google Scholar] [CrossRef] [PubMed] - Partridge, B.L.; Pitcher, T. The sensory basis of fish schools: Relative roles of lateral line and vision. J. Comp. Physiol.
**1980**, 135, 315–325. [Google Scholar] [CrossRef] - Triantafyllou, M.S.; Weymouth, G.D.; Miao, J. Biomimetic Survival Hydrodynamics and Flow Sensing. Annu. Rev. Fluid Mech.
**2016**, 48, 1–24. [Google Scholar] [CrossRef] [Green Version] - Ward, A.J.W.; Sumpter, D.J.T.; Couzin, I.D.; Hart, P.J.B.; Krause, J. Quorum decision-making facilitates information transfer in fish shoals. Proc. Natl. Acad. Sci. USA
**2008**, 105, 6948–6953. [Google Scholar] [CrossRef] [Green Version] - Puckett, J.G.; Pokhrel, A.R.; Giannini, J.A. Collective gradient sensing in fish schools. Sci. Rep.
**2018**, 8, 7587. [Google Scholar] [CrossRef] - Dykgraaf, S. Untersuchungen über die Funktion der Seitenorgane an Fischen. Zeitschrift für Vergleichende Physiologie
**1933**, 20, 162–214. [Google Scholar] [CrossRef] - Dykgraaf, S. The functioning and significance of the lateral-line organs. Biol. Rev. Camb. Philos. Soc.
**1963**, 38, 51–105. [Google Scholar] [CrossRef] - Bleckmann, H.; Przybilla, A.; Klein, A.; Schmitz, A.; Kunze, S.; Brücker, C. Station Holding of Trout: Behavior, Physiology and Hydrodynamics. In Nature-Inspired Fluid Mechanics: Results of the DFG Priority Programme 1207 ”Nature-Inspired Fluid Mechanics” 2006–2012; Springer: Berlin/Heidelberg, Germany, 2012; pp. 161–177. [Google Scholar] [CrossRef]
- Sutterlin, A.; Waddy, S. Possible Role of the Posterior Lateral Line in Obstacle Entrainment by Brook Trout (Salvelinus fontinalis). J. Fish. Res. Board Can.
**2011**, 32, 2441–2446. [Google Scholar] [CrossRef] - Akanyeti, O.; Venturelli, R.; Visentin, F.; Chambers, L.; Megill, W.M.; Fiorini, P. What information do Kármán streets offer to flow sensing? Bioinspir. Biomim.
**2011**, 6, 036001. [Google Scholar] [CrossRef] - Chambers, L.D.; Akanyeti, O.; Venturelli, R.; Ježov, J.; Brown, J.; Kruusmaa, M.; Fiorini, P.; Megill, W.M. A fish perspective: Detecting flow features while moving using an artificial lateral line in steady and unsteady flow. J. R. Soc. Interface
**2014**, 11. [Google Scholar] [CrossRef] [Green Version] - von Campenhausen, C.; Riess, I.; Weissert, R. Detection of stationary objects by the blind Cave FishAnoptichthys jordani (Characidae). J. Comp. Physiol.
**1981**, 143, 369–374. [Google Scholar] [CrossRef] - Hassan, E.S. Hydrodynamic Imaging of the Surroundings by the Lateral Line of the Blind Cave Fish Anoptichthys jordani. In The Mechanosensory Lateral Line; Coombs, S., Görner, P., Münz, H., Eds.; Springer: New York, NY, USA, 1989; pp. 217–227. [Google Scholar]
- Windsor, S.P.; Norris, S.E.; Cameron, S.M.; Mallinson, G.D.; Montgomery, J.C. The flow fields involved in hydrodynamic imaging by blind Mexican cave fish (Astyanax fasciatus). Part I: Open water and heading towards a wall. J. Exp. Biol.
**2010**, 213, 3819–3831. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Windsor, S.P.; Norris, S.E.; Cameron, S.M.; Mallinson, G.D.; Montgomery, J.C. The flow fields involved in hydrodynamic imaging by blind Mexican cave fish (Astyanax fasciatus). Part II: Gliding parallel to a wall. J. Exp. Biol.
**2010**, 213, 3832–3842. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hoekstra, D.; Janssen, J. Non-visual feeding behavior of the mottled sculpin, Cottus bairdi, in Lake Michigan. Environ. Biol. Fishes
**1985**, 12, 111–117. [Google Scholar] [CrossRef] - Pitcher, T.; Partridge, B.; Wardle, C. A blind fish can school. Science
**1976**, 194, 963–965. [Google Scholar] [CrossRef] - Satou, M.; Takeuchi, H.A.; Nishii, J.; Tanabe, M.; Kitamura, S.; Okumoto, N.; Iwata, M. Behavioral and electrophysiological evidences that the lateral line is involved in the inter-sexual vibrational communication of the himé salmon (landlocked red salmon, Oncorhynchus nerka). J. Comp. Physiol. A
**1994**, 174, 539–549. [Google Scholar] [CrossRef] - Huijbers, C.M.; Nagelkerken, I.; Lössbroek, P.A.C.; Schulten, I.E.; Siegenthaler, A.; Holderied, M.W.; Simpson, S.D. A test of the senses: Fish select novel habitats by responding to multiple cues. Ecology
**2012**, 93, 46–55. [Google Scholar] [CrossRef] [Green Version] - Montgomery, J.C.; Baker, C.F.; Carton, A.G. The lateral line can mediate rheotaxis in fish. Nature
**1997**, 389, 960–963. [Google Scholar] [CrossRef] - Coombs, S.; Janssen, J.; Webb, J.F. Diversity of lateral line systems: Evolutionary and functional considerations. In Sensory Biology of Aquatic Animals; Springer: Berlin/Heidelberg, Germany, 1988; pp. 553–593. [Google Scholar]
- Coombs, S.; Görner, P.; Münz, H. A Brief Overview of the Mechanosensory Lateral Line System and the Contributions to This Volume. In The Mechanosensory Lateral Line; Coombs, S., Görner, P., Münz, H., Eds.; Springer: New York, NY, USA, 1989; pp. 3–5. [Google Scholar]
- Denton, E.J.; Gray, J.A.B. Some Observations on the Forces Acting on Neuromasts in Fish Lateral Line Canals. In The Mechanosensory Lateral Line; Coombs, S., Görner, P., Münz, H., Eds.; Springer: New York, NY, USA, 1989; pp. 229–246. [Google Scholar]
- Coombs, S.; Braun, C.B. Information Processing by the Lateral Line System. In Sensory Processing in Aquatic Environments; Springer: New York, NY, USA, 2003; pp. 122–138. [Google Scholar] [CrossRef]
- Coombs, S.; Netten, S.V. The Hydrodynamics and Structural Mechanics of the Lateral Line System. In Fish Physiology; Elsevier: Amsterdam, The Netherlands, 2005; Volume 23, pp. 103–139. [Google Scholar] [CrossRef]
- Bleckmann, H. Peripheral and central processing of lateral line information. J. Comp. Physiol. A
**2008**, 194, 145–158. [Google Scholar] [CrossRef] - Jiang, Y.; Ma, Z.; Zhang, D. Flow field perception based on the fish lateral line system. Bioinspir. Biomim.
**2019**, 14, 041001. [Google Scholar] [CrossRef] - Engelmann, J.; Hanke, W.; Mogdans, J.; Bleckmann, H. Hydrodynamic stimuli and the fish lateral line. Nature
**2000**, 408, 1476–4687. [Google Scholar] [CrossRef] [PubMed] - Kottapalli, A.G.P.; Asadnia, M.; Miao, J.M.; Barbastathis, G.; Triantafyllou, M.S. A flexible liquid crystal polymer MEMS pressure sensor array for fish-like underwater sensing. Smart Mater. Struct.
**2012**, 21, 115030. [Google Scholar] [CrossRef] [Green Version] - Tao, J.; Yu, X. Hair flow sensors: From bio-inspiration to bio-mimicking—A review. Smart Mater. Struct.
**2012**, 21, 113001. [Google Scholar] [CrossRef] - Asadnia, M.; Kottapalli, A.G.P.; Miao, J.; Warkiani, M.E.; Triantafyllou, M.S. Artificial fish skin of self-powered micro-electromechanical systems hair cells for sensing hydrodynamic flow phenomena. J. R. Soc. Interface
**2015**, 12. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kottapalli, A.G.P.; Bora, M.; Sengupta, D.; Miao, J.; Triantafyllou, M.S. Hydrogel-CNT Biomimetic Cilia for Flow Sensing. In Proceedings of the 2018 IEEE SENSORS, New Delhi, India, 28–31 October 2018; pp. 1–4. [Google Scholar] [CrossRef]
- Wolf, B.J.; Morton, J.A.S.; MacPherson, W.N.; van Netten, S.M. Bio-inspired all-optical artificial neuromast for 2D flow sensing. Bioinspir. Biomim.
**2018**, 13, 026013. [Google Scholar] [CrossRef] [PubMed] - Yang, Y.; Chen, J.; Engel, J.; Pandya, S.; Chen, N.; Tucker, C.; Coombs, S.; Jones, D.L.; Liu, C. Distant touch hydrodynamic imaging with an artificial lateral line. Proc. Natl. Acad. Sci. USA
**2006**, 103, 18891–18895. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.; Nguyen, N.; Chen, N.; Lockwood, M.; Tucker, C.; Hu, H.; Bleckmann, H.; Liu, C.; Jones, D.L. Artificial lateral line with biomimetic neuromasts to emulate fish sensing. Bioinspir. Biomim.
**2010**, 5, 016001. [Google Scholar] [CrossRef] [Green Version] - Strokina, N.; Kämäräinen, J.; Tuhtan, J.A.; Fuentes-Pérez, J.F.; Kruusmaa, M. Joint Estimation of Bulk Flow Velocity and Angle Using a Lateral Line Probe. IEEE Trans. Instrum. Meas.
**2016**, 65, 601–613. [Google Scholar] [CrossRef] - Xu, Y.; Mohseni, K. A Pressure Sensory System Inspired by the Fish Lateral Line: Hydrodynamic Force Estimation and Wall Detection. IEEE J. Ocean. Eng.
**2017**, 42, 532–543. [Google Scholar] [CrossRef] - Sengupta, D.; Chen, S.H.; Kottapalli, A.G.P. Nature-Inspired Self-Powered Sensors and Energy Harvesters. In Self-Powered and Soft Polymer MEMS/NEMS Devices; Springer: Cham, Swizerland, 2019; pp. 61–81. [Google Scholar] [CrossRef]
- Zhang, X.; Shan, X.; Shen, Z.; Xie, T.; Miao, J. A New Self-Powered Sensor Using the Radial Field Piezoelectric Diaphragm in d33 Mode for Detecting Underwater Disturbances. Sensors
**2019**, 19, 962. [Google Scholar] [CrossRef] [Green Version] - Kruusmaa, M.; Fiorini, P.; Megill, W.; de Vittorio, M.; Akanyeti, O.; Visentin, F.; Chambers, L.; El Daou, H.; Fiazza, M.; Ježov, J.; et al. FILOSE for Svenning: A Flow Sensing Bioinspired Robot. IEEE Robot. Autom. Mag.
**2014**, 21, 51–62. [Google Scholar] [CrossRef] - DeVries, L.; Lagor, F.D.; Lei, H.; Tan, X.; Paley, D.A. Distributed flow estimation and closed-loop control of an underwater vehicle with a multi-modal artificial lateral line. Bioinspir. Biomim.
**2015**, 10, 025002. [Google Scholar] [CrossRef] [PubMed] - Ježov, J.; Akanyeti, O.; Chambers, L.D.; Kruusmaa, M. Sensing oscillations in unsteady flow for better robotic swimming efficiency. In Proceedings of the 2012 IEEE International Conference on Systems, Man, and Cybernetics (SMC), Seoul, Korea, 14–17 October 2012; pp. 91–96. [Google Scholar] [CrossRef]
- Yen, W.; Sierra, D.M.; Guo, J. Controlling a Robotic Fish to Swim Along a Wall Using Hydrodynamic Pressure Feedback. IEEE J. Ocean. Eng.
**2018**, 43, 369–380. [Google Scholar] [CrossRef] - Krieg, M.; Nelson, K.; Mohseni, K. Distributed sensing for fluid disturbance compensation and motion control of intelligent robots. Nat. Mach. Intell.
**2019**, 1, 216–224. [Google Scholar] [CrossRef] - Zheng, X.; Wang, M.; Zheng, J.; Tian, R.; Xiong, M.; Xie, G. Artificial lateral line based longitudinal separation sensing for two swimming robotic fish with leader-follower formation. In Proceedings of the 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Macau, China, 4–8 November 2019; pp. 2539–2544. [Google Scholar] [CrossRef]
- Ćurčić-Blake, B.; van Netten, S.M. Source location encoding in the fish lateral line canal. J. Exp. Biol.
**2006**, 209, 1548–1559. [Google Scholar] [CrossRef] [Green Version] - Ristroph, L.; Liao, J.C.; Zhang, J. Lateral Line Layout Correlates with the Differential Hydrodynamic Pressure on Swimming Fish. Phys. Rev. Lett.
**2015**, 114, 018102. [Google Scholar] [CrossRef] [Green Version] - Zhang, F.; Lagor, F.; Yeo, D.; Washington, P.; Paley, D. Distributed Flow Sensing Using Bayesian Estimation for a Flexible Fish Robot. In Proceedings of the ASME 2015 Dynamic Systems and Control Conference, Columbus, OH, USA, 28–30 October 2015. [Google Scholar] [CrossRef]
- Ahrari, A.; Lei, H.; Sharif, M.A.; Deb, K.; Tan, X. Design optimization of artificial lateral line system under uncertain conditions. In Proceedings of the 2015 IEEE Congress on Evolutionary Computation (CEC), Sendai, Japan, 25–28 May 2015; pp. 1807–1814. [Google Scholar] [CrossRef]
- Ahrari, A.; Lei, H.; Sharif, M.A.; Deb, K.; Tan, X. Reliable underwater dipole source characterization in 3D space by an optimally designed artificial lateral line system. Bioinspir. Biomim.
**2017**, 12, 036010. [Google Scholar] [CrossRef] - Boulogne, L.H.; Wolf, B.J.; Wiering, M.A.; van Netten, S.M. Performance of neural networks for localizing moving objects with an artificial lateral line. Bioinspir. Biomim.
**2017**, 12, 056009. [Google Scholar] [CrossRef] [Green Version] - Colvert, B.; Alsalman, M.; Kanso, E. Classifying vortex wakes using neural networks. Bioinspir. Biomim.
**2018**, 13, 025003. [Google Scholar] [CrossRef] [Green Version] - Wolf, B.J.; Pirih, P.; Kruusmaa, M.; Van Netten, S.M. Shape Classification Using Hydrodynamic Detection via a Sparse Large-Scale 2D-Sensitive Artificial Lateral Line. IEEE Access
**2020**, 8, 11393–11404. [Google Scholar] [CrossRef] - Wolf, B.; van de Wolfshaar, J.; van Netten, S. Three-dimensional multi-source localization of underwater objects using convolutional neural networks for artificial lateral lines. J. R. Soc. Interface
**2020**, 17, 20190616. [Google Scholar] [CrossRef] [PubMed] - Xu, D.; Lv, Z.; Zeng, H.; Bessaih, H.; Sun, B. Sensor placement optimization in the artificial lateral line using optimal weight analysis combining feature distance and variance evaluation. ISA Trans.
**2019**, 86, 110–121. [Google Scholar] [CrossRef] [PubMed] - Verma, S.; Papadimitriou, C.; Lüthen, N.; Arampatzis, G.; Koumoutsakos, P. Optimal sensor placement for artificial swimmers. J. Fluid Mech.
**2020**, 884, A24. [Google Scholar] [CrossRef] [Green Version] - Kern, S.; Koumoutsakos, P. Simulations of optimized anguilliform swimming. J. Exp. Biol.
**2006**, 209, 4841–4857. [Google Scholar] [CrossRef] [Green Version] - Gazzola, M.; Chatelain, P.; van Rees, W.M.; Koumoutsakos, P. Simulations of single and multiple swimmers with non-divergence free deforming geometries. J. Comput. Phys.
**2011**, 230, 7093–7114. [Google Scholar] [CrossRef] - Kern, S.; Chatelain, P.; Koumoutsakos, P. Modeling, Simulation and Optimization of Anguilliform Swimmers. In Bio-Mechanisms of Swimming and Flying: Fluid Dynamics, Biomimetic Robots, and Sports Science; Springer: Berlin/Heidelberg, Germany, 2008; p. 167. [Google Scholar]
- Carling, J.; Williams, T.L.; Bowtell, G. Self-propelled anguilliform swimming: Simultaneous solution of the two-dimensional Navier–Stokes equations and Newton’s laws of motion. J. Exp. Biol.
**1998**, 201, 3143–3166. [Google Scholar] - Angot, P.; Bruneau, C.H.; Fabrie, P. A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math.
**1999**, 81, 497–520. [Google Scholar] [CrossRef] - Coquerelle, M.; Cottet, G.H. A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys.
**2008**, 227, 9121–9137. [Google Scholar] [CrossRef] [Green Version] - Towers, J.D. Finite difference methods for approximating Heaviside functions. J. Comput. Phys.
**2009**, 228, 3478–3489. [Google Scholar] [CrossRef] - Chorin, A.J. Numerical Solution of the Navier–Stokes Equations. Math. Comput.
**1968**, 22, 745–762. [Google Scholar] [CrossRef] - Novati, G.; Verma, S.; Alexeev, D.; Rossinelli, D.; van Rees, W.M.; Koumoutsakos, P. Synchronised Swimming of Two Fish. arXiv
**2016**, arXiv:1610.04248. [Google Scholar] - Kroese, A.B.; Schellart, N.A. Velocity- and acceleration-sensitive units in the trunk lateral line of the trout. J. Neurophysiol.
**1992**, 68, 2212–2221. [Google Scholar] [CrossRef] [PubMed] - Bleckmann, H.; Zelick, R. Lateral line system of fish. Integr. Zool.
**2009**, 4, 13–25. [Google Scholar] [CrossRef] [PubMed] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberl, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dierckx, P. An algorithm for smoothing, differentiation and integration of experimental data using spline functions. J. Comput. Appl. Math.
**1975**, 1, 165–184. [Google Scholar] [CrossRef] [Green Version] - Huan, X.; Marzouk, Y.M. Simulation-based optimal Bayesian experimental design for nonlinear systems. J. Comput. Phys.
**2013**, 232, 288–317. [Google Scholar] [CrossRef] [Green Version] - Papadimitriou, C.; Lombaert, G. The effect of prediction error correlation on optimal sensor placement in structural dynamics. Mech. Syst. Signal Process.
**2012**, 28, 105–127. [Google Scholar] [CrossRef] - Simoen, E.; Papadimitriou, C.; Lombaert, G. On prediction error correlation in Bayesian model updating. J. Sound Vib.
**2013**, 332, 4136–4152. [Google Scholar] [CrossRef] - Ryan, K.J. Estimating Expected Information Gains for Experimental Designs With Application to the Random Fatigue-Limit Model. J. Comput. Graph. Stat.
**2003**, 12, 585–603. [Google Scholar] [CrossRef] - Papadimitriou, D.I.; Papadimitriou, C. Optimal sensor placement for the estimation of turbulence model parameters in CFD. Int. J. Uncertain. Quant.
**2015**, 5, 545–568. [Google Scholar] [CrossRef] [Green Version] - Papadimitriou, C. Optimal sensor placement methodology for parametric identification of structural systems. J. Sound Vib.
**2004**, 278, 923–947. [Google Scholar] [CrossRef]

**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