## 1 Introduction  
Networked robots have various applications, such as monitoring large environments, rescue missions, and target chasing [1; 2; 3; 4]. Multiple coordinated robots are also used to perform many specific tasks, such as rendezvous [5; 6; 7], spacecraft docking [8], environmental monitoring [9; 10; 11], underwater target chasing [12], and formation control [13; 14; 15]. This study addresses multi-robot rendezvous controls in cluttered underwater environments with many unknown obstacles. Unmanned Underwater Vehicles (UUVs) can be divided into two categories based on whether their bodies are streamlined. The UUV's shape is determined by the requirements of the application. For example, a streamlined shape reduces water resistance and is preferable if the UUV is required to move at high speeds. However, if underwater detection or operation tasks are the primary roles of a UUV, a non-streamlined shape is often preferred. Because of the good water pressure resistance of spherical objects, spherical UUVs can perform rotational motions with a 0 degree turn radius. In the literature, various spherical UUVs have been developed [16; 17; 18; 19; 20]. References [16; 18; 19; 20; 21] addressed a spherical UUV with hybrid propulsion devices including vectored water jet and propeller thrusters. The three Degree-of-Freedom (DoF) motions, including surging, heaving, and yawing, were performed in a swimming pool. References [16; 19; 21] further demonstrated that by adopting vectored water jets, a spherical UUV can be made to maneuver freely in any direction. Considering a spherical UUV, [22] used fuzzy proportional-integral-derivative (PID) controllers to independently control the robot's movement in all directions. Sincespherical UUVs are highly maneuverable, our study considers a spherical UUV [16; 17; 18; 19; 20] as our platform. We address practical application scenarios where multi-robot rendezvous controls are used. Recently, several studies [23; 24; 25] handled distributed formation control of multiple robots in environments with obstacles. For instance, a group of robots can be used for tracking a target while measuring the target's signal [26]. During the robots' maneuvering in formation controls, robots need to preserve the network connectivity. In order to make all robots gather close to each other, distributed rendezvous controls can be applied occasionally. In other words, distributed rendezvous algorithms can be used as a multi-robot control module to allow a team of robots to maintain network connectivity while they maneuver. For example, we can run rendezvous controls occasionally, so that all robots get closer to a root robot which leads the formation. Moreover, all robots need to get closer to the root robot, in the case where they move through a narrow tunnel. Furthermore, rendezvous algorithms can be utilized in multi-robot collection or charging scenarios [27]. Suppose that there is a charging station close to a robot, called the root. All the robots can be charged from the charging station, once they gather close to the root. The difficulty of the rendezvous problem decreases when a robot is equipped with Global Positioning Systems (GPS). However, GPS is unavailable in underwater environments. In difficult scenarios without GPS, robots cannot move toward a designated rendezvous point directly. Thus, we introduce a distributed rendezvous approach that does not require global localization. A rendezvous algorithm is considered distributed, when it depends on the local interaction between neighboring robots. A distributed rendezvous algorithm is practical, since a robot cannot communicate with another robot that is too far away. Furthermore, obstacles in cluttered 3D environments can easily block the communication links between robots. In this study, we propose distributed rendezvous algorithms for UUVs in 3D unknown environments with many obstacles. Here, unknown obstacles can lead to communication link failure or invisible robots.