# Robust Distributed Rendezvous Using Multiple Robots with Variable Range Radars
## 1 Introduction  
Networked robots can perform many tasks, such as formation controls [1; 2; 3] or workspace monitoring [4]. Such robot networks have become feasible due to advances in communication networking [5; 6; 7; 8]. Suppose that each robot does not know its position in the global coordinate system, i.e., a robot is not localized using global positioning systems (GPS). We require a _distributed_ controller such that every robot moves based only on relative position measurements. During the robots' maneuvering in formation controls, robots need to maintain 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 controls are the most basic coordination task for a network of mobile robots. Multi-robot rendezvous is an initial step for the creation of more complicated formation tasks and can be particularly useful for autonomous recharging or self-maintenance [9]. This article considers the distributed rendezvous problem of multiple robots, such that each robot can only measure nearby (neighbor) robots using a radar sensor, and each robot moves based on radar sensor measurements. Since GPS is not accessible, a robot cannot move towards a specified rendezvous point directly. For rendezvous controls, each robot uses a radar sensor, whose footprint is proportional to the signal intensity (power) of the transmitter [10]. Supply modulated transmitters for variable amplitude radar systems were addressed in [11]. Suppose that a robot, say \(A\), emits signals for detecting its neighbor robots. Once the signals generated from \(A\) are reflected from a nearby robot, say \(B\), then \(A\) can estimate the elevation angle, azimuthangle, and range to \(B\) using a 3D multiple signal classification (MUSIC) algorithm [12]. In this way, \(A\) can measure the relative position of its neighbor robot \(B\). Our article considers a practical scenario in which radar has a limited sensing range while containing measurement noise. Further, the environmental disturbance generates process noise in a robot's maneuvering. For instance, currents (for underwater robots) or wind gusts (for aerial robots) can generate process noise in robots' maneuvers. To the best of our knowledge, distributed rendezvous control in the literature assumed that all robots are initially deployed such that every robot is connected to any other robot utilizing multi-hop links. This assumption is called the _initial connectivity assumption_, which implies that the entire network is not split initially. Our 3D rendezvous control is developed to cope with a case where this assumption is not satisfied initially. The initial connectivity assumption may not hold in various scenarios. In practice, sensing measurement noise or process noise in a robot's motion can split the initial robot network. Moreover, complete failures of one or more robots can split the initial network. For example, Figure 1 plots a scenario where the failure of one robot splits the network. In this figure, a robot is marked with a dot, and the network connection between two robots is plotted with line segments. In the left subfigure, the robot with a red dot is completely broken. In this case, the networked system changes to the network in the right subfigure. See that the complete failure of one robot (red dot) splits the network. Considering 3D environments, the goal of our paper is to let all robots rendezvous in a distributed manner so that the network connectivity can be recovered even in the case where the network is split. In other words, we develop a 3D distributed rendezvous control so that all robots become connected to each other even after the network is split. In order to handle the case where the initial connectivity assumption does not hold, our paper assumes that a robot's radar sensor can enlarge its footprint by increasing the transmission power level (adjusting the amplifier in the transmitter).