# Robust Distributed Rendezvous Using Multiple Robots with Variable Range Radars
## 3 Assumptions and Definitions  
We introduce assumptions and definitions that are utilized in our distributed rendezvous control. \(G=(V(G),E(G))\) is a graph with vertex set \(V(G)\) and edge tuple \(E(G)\)[26]. Let \(min(a,b)\) denote a small value between \(a\) and \(b\). There are \(N\) robots in total. Let \(u_{i}\) denote the \(i\)-th robot, where \(i\leq N\). \(\mathbf{u}_{i}\subset\mathcal{R}^{3}\) is the 3D position of robot \(u_{i}\) where \(i\leq N\). We consider discrete-time systems. Let \(dt\) denote the sampling interval. \(\mathbf{u}_{i}(k)\) denotes \(\mathbf{u}_{i}\) at time-step \(k\). Considering the process noise, such as wind, the motion model of \(u_{i}\) is  
\[\mathbf{u}_{i}(k+1)=\mathbf{u}_{i}(k)+\mathbf{v}_{i}(k)*dt+\mathbf{n}, \tag{1}\]  
where \(\mathbf{n}=[randn*D,randn*D,randn*D]\) where \(D>0\) is a constant presenting the process noise strength, and \(randn\) generates a Gaussian noise with mean 0 and variance 1. In (1), \(\mathbf{v}_{i}(k)\) is the velocity input of \(u_{i}\) at time-step \(k\). This simple dynamic model (1) is commonly used in multi-robot systems [27; 28; 29; 30; 31; 32; 33]. This article further considers a practical scenario where a robot moves with a bounded speed. The maximum speed of \(u_{i}\) is \(M_{s}\). In other words, \(\|\mathbf{v}_{i}(k)\|\leq M_{s}\) for all \(k\) and \(i\). \(M_{s}\) is determined by the hardware specifications of a robot. Each robot has a radar with a limited sensing range in order to measure the relative position (elevation, azimuth, and range) of a nearby robot. We assume that a robot's radar sensor can increase its footprint by increasing the transmission power level (adjust the amplifier in the transmitter). In other words, we consider the case where a robot has a variable range radar, such that its sensor footprint is adjustable. Since each robot can change its radar sensing range, we consider heterogeneous robots with a distinct sensing range. Let us define \(S_{i}(k)\) as the radar sensing range of \(u_{i}\) at time-step \(k\). \(S^{u}\) is the upper bound for \(S_{i}(k)\) for all \(i\) and \(k\). Suppose that \(\|\mathbf{u}_{j}(k)-\mathbf{u}_{i}(k)\|\leq S_{i}(k)\). In this case, \(u_{i}\) can measure \(u_{j}\) using its radar. Note that increasing the footprint of the radar cannot cause a detection in the case of obstructed LOS. Let \(\mathbf{R}_{j,i}(k)\) denote the relative position (radar) measurement of \(u_{i}\) at time-step \(k\). \(\mathbf{R}_{j,i}(k)\) is given as the radar of \(u_{i}\) measures \(u_{j}\). Considering the radar measurement noise, we have  
\[\mathbf{R}_{j,i}(k)=\mathbf{u}_{j}(k)-\mathbf{u}_{i}(k)+\mathbf{n}_{R}, \tag{2}\]  
where \(\mathbf{n}_{R}=[randn*D_{R},randn*D_{R},randn*D_{R}]\) where \(D_{R}>0\) is a constant presenting the measurement noise strength, and \(randn\) generates a Gaussian noise with mean 0 and variance 1. For instance, \(\mathbf{n}_{R}\) can be generated due to the shape of a robot, since radar sensors measure the boundary of a nearby robot. We introduce the adjacency graph, \(A=(V(A),E(A))\), in which every vertex in \(V(A)\) represents a robot. Every directed edge, say \(\{u_{i},u_{j}\}\in E(A)\), indicates that \(u_{i}\) can detect \(u_{j}\) using its radar. \(\{u_{i},u_{j}\}\in E(A)\) implies that \(\|\mathbf{u}_{j}(k)-\mathbf{u}_{i}(k)\|\leq S_{i}(k)\). See that \(A\) is a directed graph with directed edges. \(\{u,v\}\) is distinct from \(\{v,u\}\), since an edge is directed. Even in the case where \(u\) can measure \(v\) using its radar, \(v\) may not measure \(u\) using its radar. This is due to the fact that the sensing range of \(u\) may be distinct from that of \(v\). We say that \(v\) is a _neighbor_ of \(u\) in the case where \(u\) can detect \(v\) using its radar. \(N_{u}\) denotes the set of neighbors of \(u\). We say that all robots achieved _rendezvous_ in the case where the relative distance between any two robots is less than a positive constant, say \(\delta>0\).