## 3 Background Information
### Assumptions and Definitions
For realistic simulations of the RSSI for an underwater emitter, we use the model parameters in [28]. In Equation (5), \(P_{0}=100\) dB, \(E_{p}=2\), \(\gamma=0.05\), \(\sigma^{P}=1\) are used, according to [28]. In this case, Figure 4 shows the relationship between \(d_{I}\) and \(P(u_{i})\) using (5) and (6). As \(d_{I}\) increases, \(P(u_{i})\) decreases. It is assumed that if \(P(u_{i})>P_{thres}\), then SNR is sufficiently large and the intensity sensor can detect the LOS signal generated from the emitter. We say that a UUV \(u_{i}\) is a neighbor to UUV \(u_{j}\) if \(P(u_{i})>P_{thres}\) and \(P(u_{j})>P_{thres}\). Here, \(P_{thres}\) is a tuning parameter for determining a neighbor. The neighbor information can be set by mutual communication between \(u_{i}\) and \(u_{j}\). In the case where \(u_{i}\) and \(u_{j}\) are neighbors, \(\|\mathbf{u}_{i}-\mathbf{u}_{j}\|<r_{s}(P_{thres})\) and the LOS between \(u_{i}\) and \(u_{j}\) is not blocked by obstacles. Here, \(r_{s}(P_{thres})\) can be considered as the maximum sensing range for a UUV. Note that \(r_{s}(P_{thres})\) is determined by \(P_{thres}\). As \(P_{thres}\) decreases, \(r_{s}(P_{thres})\) increases using (5). For notation convenience, we use \(r_{s}\) instead of \(r_{s}(P_{thres})\). See Figure 4 for an illustration. If we set \(P_{thres}=52\) dB, then \(r_{s}=10\) m; if we set \(P_{thres}=20\) dB, then \(r_{s}=60\) m.  
We say that a UUV \(u_{i}\) encounters \(u_{j}\) if \(\|\mathbf{u}_{i}-\mathbf{u}_{j}\|<\epsilon\approx 0\). Let \(P^{e}\) denote the signal power measured when \(u_{i}\) encounters \(u_{j}\). Thus, using Equation (5), we obtain the following:  
\[P^{e}=P_{0}-10E_{p}log_{10}(\epsilon)+\gamma(\epsilon-1). \tag{7}\]  
If both \(P(u_{i})\) and \(P(u_{j})\) exceed \(P^{e}\), then we assume that \(u_{i}\) encountered \(u_{j}\). Let \(G=(V(G),E(G))\) indicate the graph presenting the networked system. Every node in \(V(G)\) indicates a UUV. An edge, e.g., \(\{u_{i},u_{j}\}\in E(G)\), indicates that \(u_{i}\) and \(u_{j}\) are neighboring UUVs. Recall that \(u_{i}\) is a neighbor to \(u_{j}\) if \(P(u_{i})>P_{thres}\) and \(P(u_{j})>P_{thres}\). Since we check both \(P(u_{i})\) and \(P(u_{j})\) for detecting neighbors, \(G\) is an undirected graph. Let \(G_{0}=(V_{0},E_{0})\) indicate the initial connectivity network (at sampling step \(t=0\)). Without loss of generality, it is assumed that \(G_{0}\) is connected. As far as we know, this initial connectivity assumption is required for any distributed rendezvous controls in the literature [34; 41; 51; 52]. This is due to the fact that distributed rendezvous controls are based on local interaction between neighboring agents. Figure 4: The relationship between \(d_{I}\) and \(P(u_{i})\) using Equations (5) and (6). As \(d_{I}\) increases, \(P(u_{i})\) decreases. Note that the initial connectivity assumption does not assure that the network connectivity is maintained during the maneuver of an agent. As agents move around, the communication link among them may be broken due to obstacles in the environments. Moreover, an agent cannot communicate with another agent if the relative distance between them is larger than \(r_{s}\). Thus, we need to develop distributed rendezvous controls, which assure that the network connectivity is maintained while all agents rendezvous at \(u_{N}\). In the inertial reference frame, let \(\mathbf{h}_{i}(k)\in\mathbb{R}^{3}\) present the heading direction of \(u_{i}\) at sampling step \(k\). Note that \(\mathbf{h}_{i}(k)\in\mathbb{R}^{3}\) is a unit vector. Let \(dt\) indicate the sampling interval in discrete-time systems. As a UUV's dynamic model in the inertial reference frame, we utilize the following equation:  
\[\mathbf{u}_{i}(k+1)=\mathbf{u}_{i}(k)+\mathbf{h}_{i}(k)*dt*s_{i}(k). \tag{8}\]  
This simple motion model is commonly utilized in the literature on multi-agent systems [51; 53; 54; 55; 56; 57; 58; 59; 60]. In Equation (8), \(s_{i}(k)\) denotes the linear speed of \(u_{i}\) at sampling step \(k\). We assume that \(s_{i}(k)\leq s^{max}\) for all \(k\).