## 1 Introduction
The proposed rendezvous control is adaptive to communication link failure or invisible robots. Considering 3D underwater environments, this study addresses the problem of enabling a network of UUVs to rendezvous at a designated root robot. Any robot in the network can be set as the root robot. Thus, the rendezvous system in this study is referred to as the any-robot rendezvous system. The rendezvous controls in this study consider a UUV that can only measure the signal intensity emitted from its neighbor UUV. Thus, the proposed rendezvous control is suitable for a cheap UUV, which only has sensors for measuring the signal strength. Reference [28] addressed a received signal strength (RSS) sensor model for underwater sound propagation. We consider multiple UUVs such that each UUV has multiple signal intensity sensors surrounding it. Each intensity sensor can measure the RSS of sound generated from a neighbor UUV [28]. Multiple intensity sensors on a UUV can only measure the amplitude (intensity) of signals emitted from its neighboring UUVs. To enable a distributed rendezvous based on amplitude-only measurements, we utilize a method of making a UUV approach the source of the signal (neighbor UUV) by measuring the intensity of signal at multiple intensity sensors. We demonstrate that our distributed rendezvous algorithms are provably complete in achieving multi-robot rendezvous in cluttered 3D environments. The proposed 3D rendezvous controls assure the convergence to a designated UUV, while maintaining the connectivity of the time-varying and position-dependent communication network. Moreover, the proposed rendezvous controls can handle the case where some UUVs are broken. While controlling the networked system, any UUV, including the root UUV, can become faulty owing to various reasons, such as hardware malfunction. This study, thus, discusses a fault detection method, which detects faulty UUVs based on local sensingmeasurements. Once a UUV failure is sensed, then we update the network structure, so that healthy UUVs without faults can be controlled effectively. This study handles scenarios where multiple UUVs are deployed in cluttered 3D environments (e.g., [29]). In obstacle-rich environments, the communication (interaction) link between UUVs may be easily blocked owing to obstacles. Furthermore, invisible UUVs may occur, since line-of-sight can be blocked by unknown obstacles. In such scenarios, it is important to assure that the communication link is preserved, while every UUV maneuvers. The proposed distributed rendezvous control is unique in overcoming communication link failure or invisible UUVs. Note that communication link failure or invisible UUVs may happen due to unknown obstacles in cluttered 3D environments. We address a distributed rendezvous control for spherical UUVs with bounded speed, such that each UUV can only measure the strength of signals emitted from neighboring UUVs. The contributions of our study are summarized as follows. 1. As far as we know, this study is novel in developing 3D distributed rendezvous controls, considering a UUV that can only measure the strength of signals emitted from neighboring UUVs. 2. The proposed 3D rendezvous controls are provably complete, since we prove the rendezvous to the root UUV, while maintaining the connectivity of the time-varying and position-dependent communication network. 3. To the best of our knowledge, our study is unique in addressing a fault detection method that detects faulty UUVs based on local sensing measurements. In addition, the proposed rendezvous controls are adaptive to communication link failure or invisible UUVs. The study is organized as follows. Section 2 provides the literature review of this study. Section 3 provides the background information of this study. Section 4 discusses the 3D distributed rendezvous controls introduced in this study. Section 5 provides the simulation results of this study. Section 6 provides the conclusions of this study.