## 5 Simulations  
The MATLAB R2014a simulator is utilized to demonstrate the effectiveness of the proposed rendezvous controllers. The system environment includes Window 10, Intel CoreTM i5-7600K [EMAIL_ADDRESS] GHz. The controllers are implemented in a discrete-time system, and the sampling time interval to discretize the UUV's velocity control (8) is 0.3 s.  
We simulate a 3D underwater environment (200 \(\times\) 200 \(\times\) 200) with many obstacles. For realistic simulations of the RSSI for an underwater emitter, we use the model parameters used in [28]. In Equation (5), \(P_{0}=100\) dB, \(E_{p}=2\), \(\gamma=0.05\), and \(\sigma^{p}=1\) are used, according to [28]. Recall that we said that a UUV \(u_{i}\) is a neighbor to UUV \(u_{j}\) if \(P(u_{i})>P_{thres}\) and \(P(u_{j})>P_{thres}\). We set \(P_{thres}=52\) dB. Using Figure 4, the associated maximum sensing range \(r_{s}\) is 10 m. In this way, the relative distance between two neighbor UUVs is less than 10 m.  
The maximum speed of every UUV is set as \(s^{max}=2\) (m/s). Recall that if \(P(u_{i},p_{j})\) and \(P(u_{j},p_{i})\) exceeds \(P^{e}\), then we assume that \(u_{i}\) encountered \(p_{j}\). We set \(P^{e}=100\) (in dB), which is identical to \(P_{0}\). Using (7), \(P^{e}=100\) (in dB) is associated with \(\epsilon=1\) m.  
In this study, a UUV uses six intensity sensors. Figure 3 plots the local coordinates of every intensity sensors positioned on a UUV. \(dr\) is the relative distance between the first sensor and any other sensor. \(dr\) is a tuning parameter in our control. If \(dr\) is too small, then intensity measurements of six sensors on a UUV are not distinct from each other. In this case, the gradient direction at the UUV position cannot be estimated accurately. If \(dr\) is too large, then we cannot apply the Taylor expansion in Equation (10). In the simulations, six intensity sensors are installed such that \(dr=1\) m.  
To prove the robustness of the proposed methods, we run 100 Monte Carlo (MC) simulations, such that the initial network satisfies this initial connectivity assumption. As far as we know, this initial connectivity assumption is required for any distributed rendezvous controls in the literature [34; 41; 51; 52]. At the beginning of each MC simulation, 50 UUVs are randomly deployed until the following two conditions are met:  
1. No UUV is deployed inside an obstacle boundary. 2. The deployed UUVs satisfy the initial connectivity assumption. Once these two conditions are satisfied, then 50 UUVs begin to move under Algorithm 2. Otherwise, we keep deploying 50 UUVs until the above two conditions are met. Once these two conditions are satisfied, then 50 UUVs begin to maneuver under Algorithm 2. This maneuver indicates the beginning of a single MC simulation. In all MC simulations, rendezvous is achieved for every UUV. Considering one MC simulation, Figure 5 shows the initial position of every UUV. The initial position of every UUV is plotted with a green circle. This figure shows the obstacle boundaries with spheres. We also plot the tree graph \(T\) (blue line segments) generated initially. Figure 6 shows each UUV's maneuver until \(t=20\) s have passed. We also plot the tree graph \(T\) (blue line segments) generated at time 0. Every UUV's maneuver is illustrated as circles with a distinct color. After 148 s have passed, all UUVs encounter at the root UUV while avoiding collision with obstacles. Figure 7 shows each UUV's maneuver once the rendezvous is completed. Figure 8 shows the final position (green circle) of every UUV. Figure 5: The initial position of every UUV is plotted with a green circle. We also plot the tree graph \(T\) (blue line segments) generated at time 0 (one MC simulation). We can see that the initial connectivity assumption is satisfied. Obstacles are plotted with spheres. Figure 6: Every UUV’s maneuver until \(t=20\) s has passed (one MC simulation). Every UUV’s maneuver is illustrated as circles with a distinct color. We also plot the tree graph \(T\) (blue line segments) generated at time 0.