## 3 Background Information
### Assumptions and Definitions  
This subsection discusses the assumptions and definitions used in this study. Suppose we have \(N\) UUVs in total. Let \(u_{i}\) indicate the \(i\)-th UUV (\(i\in\{1,2,\ldots,N\}\)). Let \(u_{N}\) indicate the root in the robot network. In our study, \(u_{N}\) is set as the root of a tree graph \(T\). In the inertial reference frame, \(\mathbf{u}_{i}\in\mathbb{R}^{3}\) is the 3D location of a UUV \(u_{i}\). A UUV has communication devices to enable information sharing with its neighboring UUVs. This local communication ability is fundamental for multi-agent controls as well as for handling UUV faults. Consider a UUV that has six signal intensity sensors surrounding it. Each intensity sensor can measure the received signal strength of sound generated from a neighbor UUV [28]. As a UUV, e.g., \(u_{i}\), detects a signal from its neighbor UUV \(u_{j}\), six intensity sensors on \(u_{i}\) can measure the strength of the signal emitted from \(u_{j}\). Figure 3 plots the local coordinates of every intensity sensors positioned on a UUV. The local coordinates are defined in the UUV's body\(-\)fixed frame. In Figure 3, the path of signal emitted from the emitter is illustrated with dotted arrow. The origin of the local coordinates frame is at the UUV's center. The numbering in front of local coordinates indicates the index of the associated intensity sensor. For instance, \(1:[dr,0,0]\) indicates that the first sensor has local coordinates \([dr,0,0]\). We can see that every intensity sensor is positioned at an equidistant point from the UUV's center. Here, \(dr\) is the relative distance between the UUV's center and any other sensor. In the body\(-\)fixed frame of \(u_{i}\), let \(\mathbf{c}_{i}^{r}\in\mathbb{R}^{3}\) indicate the local coordinates of the \(r\)-th intensity sensor (\(r\in\{1,2,\ldots,6\}\)) of \(u_{i}\). Let \(P(\mathbf{c}_{i}^{r})\in\mathbb{R}\) indicate the signal power received by the \(r\)-th intensity sensor (\(r\in\{1,2,\ldots,6\}\)) of \(u_{i}\). Recall that Figure 3 plots the local coordinates of every intensity sensor positioned on \(u_{i}\). In emitter localization, complexity increases owing to obstacles blocking the line-of-sight (LOS) path between an intensity sensor and an emitter [44; 45; 46; 47; 48; 49]. Considering an LOS emitter, the Received Signal Strength Indicator (RSSI) is modeled utilizing a log-normal shadowing model [28; 50]. Based on the log-normal shadowing model in [28], we use the following equation:  
\[P(\mathbf{c}_{i}^{r})=P_{0}-10E_{p}log_{10}(d_{I})+\gamma(d_{I}-1)+n^{p}. \tag{5}\]  
where \(d_{I}\) indicates the relative distance between the intensity sensor at the local coordinates \(\mathbf{c}_{i}^{r}\in\mathbb{R}^{3}\) and the emitter, \(P_{0}\) (dB) is the received signal power at 1 m, \(E_{p}\) is the propagation exponent, and \(n^{p}\) is a random variable with mean 0 and standard deviation \(\sigma^{p}\). In Equation (5), \(\gamma\) is the path loss exponent (PLE), which models the geometric spreading loss. The term \(\gamma\) (dB/m) is the frequency-dependent medium absorption, and \(\gamma\) can be determined using [PERSON]'s model, which is based on the experiments in [28]. Figure 3: The local coordinates of every intensity sensor positioned on a UUV. The origin of the local coordinates frame is at the UUV’s center. The numbering in front of local coordinates indicates the index of the associated intensity sensor. We can see that every intensity sensor is positioned at equidistant point from the UUV’s center. Here, \(dr\) is the relative distance between the UUV’s center and any other sensor. We consider the case where all sensors of a UUV are identical to those of another UUV. Let \(P(u_{i})\in\mathbb{R}\) denote the average intensity measurement associated with \(u_{i}\). We have  
\[P(u_{i})=\frac{1}{6}\sum_{r=1}^{6}P(\epsilon_{i}^{r}). \tag{6}\]  
It is assumed that \(P_{0}\), \(E_{p}\), and \(\sigma^{P}\) in Equation (5) are known a priori. This is feasible using experiments with UUVs [28].