### floater structure

The floater used for the jellyfish movement characterization task is a 3-inch cylinder to which a hydrophone is attached to decipher changing depth commands. Pressure sensor to monitor floater depth. Salinity and temperature sensors to evaluate the speed of sound in water. Thrusters for the ascent of the floater. A weight to balance the floater in the water by -100 g. and acoustic tags. See illustration in Figure 7a. Controlling the thrusters is an Arduino card that turns the thrusters on and off based on the depth of the floater.In particular, to keep the floater near the target depth. *d*when the lower limit is reached, the thrusters operate, \(d+\delta ^{\text{l}}\)will be disabled once the limit is reached. \(d-\delta ^{\text{u}}\). In our tests, the floater was able to continue moving 0.5 m around the specified depth. Assisting in this action is a flexible fabric attached to the floater’s tube, which, like an umbrella, uses water drag to open as it descends and close as it rises. This fabric is used to cut thruster operating time by more than half and reduce power consumption. Floater recovery is performed by programming it to surface after a specified time. A safe weight drop mechanism based on a molten metal ring ensures levitation even in the event of a failure. Floaters are built to be approximately Lagrangian, meaning they float in a water stream with minimal drag.A series of field experiments proved this property^{27}. A photograph of the floater in operation is shown in Figure 7b.

The hydrophone mounted on the floater is sampled by a TLV320ADC6140 sound card connected to a Raspberry-Pi 3 controller. The controller continuously runs a decoder for messages that may be transmitted acoustically by the deployed vessel. These messages are interpreted by the floater as mission changes, such as changing depth or surfacing. In the context of jellyfish tracking, this underwater acoustic communication is used to match the depth of the floater with the depth of the jellyfish, thereby ensuring that both floater and jellyfish experience the same water flow. In particular, based on the jellyfish tag’s signal, the vessel operator can acoustically align the floater to the same depth.Underwater acoustic communication is based on the Janus standard^{38} For sending short messages.

### Acoustic localization of jellyfish and floating objects

#### Reception by at least 3 receivers

Acoustic localization of tags is performed based on ToA measurements detected by a subset of tags. \(R=6\) Receiving machine. These are merged into his TDoA measurements by the intersection of at least two isolines in the 2D plan.Regarding release at a certain point in time *t*indicates the 2D UTM position of the tag *I* as \(\textbf{s}^i_{{\text{tag}}}

(1)

where *c* is the speed of sound in water, \(e_{n,k}\) is the measurement noise \(\Tau _{n,k}\).

For all receiver pairs *n*, *k* Detect publication of the same tag *I*minimize the utility function.

$$\begin{aligned} \hat{\textbf{s}}^i_{{\text{tag}}}=\underset{\textbf{s}^i_{{\text{tag}}}}{{ \text{argmin}}} \sum _n\sum _k|\textbf{d}_{\text{n,k}}-\textbf{s}^{\text{i}}_{{\text{tag }}}|. \end{Align}$$

(2)

Finding the intersection of isovalue lines is a nonlinear optimization problem^{39}. Furthermore, the intersection of the contour lines may not converge to a single point due to receiver position uncertainty and measurement noise.^{40}. Therefore, if three or more receivers are detecting the same tag’s transmission, it is important to take advantage of the additional available information. In particular, we use an ordinary least squares (OLS) approach.

First, define one of the receivers as the reference receiver and set its 2D UTM position as follows: \(\textbf{s}^0_{{\text{rec}}}=[x^{0}_{{\text{rec}}}, y^{0}_{{\text{rec}}}]\). Considering the TDOA between \(n\)th receiver and reference receiver, \(\Tau _{\text{0,n}}\)we got

$$\begin{aligned} \tau _{\text{0,n}} = \frac{1}{c}\left( |\textbf{s}^{\text{i}}_{{\text {tag}}} – \textbf{s}_0| – |\textbf{s}^{\text{i}}_{{\text{tag}}} – \textbf{s}_{\text{n }}| \right) = \frac{1}{c}\left( \textbf{d}_{\text{0}} -\textbf{d}_{\text{n}}\right) . \end{Align}$$

(3)

difference in arrival distance *n*The th recipient is

$$\begin{aligned} \textbf{D}_{\text{0,n}} = \tau _{\text{0,n}} \cdot c = \textbf{d}_{\text{0 }} -\textbf{d}_{\text{n}},\end{aligned}$$

(Four)

like that

$$\begin{aligned} \textbf{d}_\text{0}^2 -\textbf{d}_\text{n}^2 = |\textbf{s} – \textbf{s}_0|^ 2 – |\textbf{s} – \textbf{s}_\text{n}|^2 = 2\textbf{d}_0 \textbf{D}_\text{0,n} – \textbf{D} _\text{0,n}^2. \end{Align}$$

(Five)

show

$$\begin{aligned} b_\text{n} = \frac{1}{2}\left( [x^0_{\text{rec}}]^2+[x^n_{\text{rec}}]^2+ [y^0_{\text{rec}}]^2+[y^n_{\text{rec}}]^2+\textbf{D}_\text{0,n}^2\right) , \end{aligned}$$

(6)

After sorting, it looks like this

$$\begin{aligned} b_\text{n}=(x^0_{\text{rec}}-x^n_{\text{rec}})x + (y^0_{\text{rec}} -y^0_{\text{rec}})y + \textbf{D}_\text{0,n}{} \textbf{d}_0. \end{Align}$$

(7)

This is a linear model for unknown tag position *X*, *y* and range \(\textbf{D}_0\). In matrix format, *N* When the receiver detects the transmission of the same tag,

$$\begin{aligned} \textbf{A}{} \textbf{X} = \textbf{b}, \end{aligned}$$

(8)

where

$$\begin{align} \textbf{A} = \left[ \begin{array}{ccc} x^0_{\text{rec}} – x^1_{\text{rec}} &{} y^0_{\text{rec}}-y^1_{\text{rec}} &{} \textbf{D}_{0,1} \\ x^0_{\text{rec}} – x^2_{\text{rec}} &{} y^0_{\text{rec}}-y^2_{\text{rec}} &{} \textbf{D}_{0,2} \\ &{} \vdots &{} \\ x^0_{\text{rec}} – x^N_{\text{rec}} &{} y^0_{\text{rec}}-x^N_{\text{rec}} &{} \textbf{D}_\text{0,N} \end{array} \right] ,\end{align}$$

(9)

\(\textbf{X}=\left[ x^i_{\text{tag}},y^i_{\text{rec}},\textbf{d}_0\right] ^\text{T}\)and \(\textbf{b}=\left[ b_0,\ldots , b_n\right] ^\text{T}\). The OLS solution of (8) is obtained as follows.

$$\begin{aligned} \textbf{X}_\text{ols} = \left( \textbf{A}^\text{T}{} \textbf{A}\right) ^{-1}{} \textbf{A}^\text{T}{} \textbf{b}. \end{Align}$$

(Ten)

#### If there are less than 3 receivers

If the tag transmission is of low rank and is decoded by fewer than three receivers, the above procedure introduces ambiguity. To resolve these, we rely on a constant TI between each tag’s emission and the assumption that both the jellyfish and the floater move at a fixed (unknown) velocity. \(v_x,v_y\).

As an example of time *t*the time of flight (ToF) is

$$\begin{aligned} \rho ^i_n

(11)

Indicates a fixed interval between transmissions from each tag *I* as \(\text{TI}^i\).For the same tag *I* and receiver *n*consider the ToA of consecutive measurements collected at one point in time. *t* and \(t+\text{TI}^i\), \(\text{ToA}^i_n

(12)