# Robust Distributed Rendezvous Using Multiple Robots with Variable Range Radars
## 4 Distributed Rendezvous Controllers
### Updating the Sensing Range Adaptively  
The circumcenter algorithm was originally introduced in [13]. Consider the ideal case where there is no process noise or measurement noise. In addition, assume that the initial connectivity assumption is satisfied. In this ideal case, ref. [14] proved the convergence of the circumcenter algorithm for any switching sequence of adjacency graphs that contains a strongly connected graph every \(l\) instants of time, for some fixed \(l>0\).  
In practice, the initial connectivity assumption may not hold. Moreover, process noise, limited communication range, or measurement noise can disconnect the network frequently. In order to handle these practical problems, we need to increase network connectivity as much as possible.  
The proposed strategy is to increase the network connectivity by increasing the sensing range of each robot adaptively. By increasing the sensing range of each robot adaptively, we can make the switching sequence of adjacency graphs contain a strongly connected graph more frequently. Then, according to [14], rendezvous can be achieved more robustly. Once the rendezvous is achieved, the sensing range of every robot can shrink to reduce the power consumption of its radar.  
Recall that \(S_{i}(k)\) is the sensing range of \(u_{i}\) at time-step \(k\). In addition, recall that \(S^{u}\) is the upper bound for \(S_{i}(k)\) for all \(i\) and \(k\).  
Let \(N^{min}>0\) denote a lower threshold for the number of neighbors. Recall that \(N_{u}\) denotes the set of neighbors of \(u\). In the case where \(N_{u_{i}}<N^{min}\), we need to increase the number of neighbors for \(u_{i}\). In the case where \(N_{u_{i}}<N^{min}\), \(S_{i}(k)\) increases using  
\[S_{i}(k+1)=min(S_{i}(k)+\beta,S^{u}). \tag{9}\]  
Here, \(\beta>0\) is a constant. Using (9), the sensing range for \(u_{i}\) increases, until it reaches \(S^{u}\). Note that in the case where \(N_{u_{i}}\geq N^{min}\), (9) is not applied.  
In (9), \(\beta>0\) is a tuning parameter. As \(\beta\) increases, the sensing range of \(u_{i}\) can increase more rapidly. A large sensing range can increase the network connectivity, which improves the rendezvous performance. However, a large sensing range increases the power consumption of a robot.  
Moreover, \(N^{min}\) is a tuning parameter in our controls. As \(N^{min}\) increases, the network connectivity increases, which improves the rendezvous performance. However, increasing \(N^{min}\) increases the power consumption of each robot since the sensing range of a robot needs to increase using (9). The effect of changing \(N^{min}\) is discussed in the Simulation section.  
The original circumcenter algorithms in [13; 14; 15] require the initial connectivity assumption. For instance, consider the case where a robot is not connected to any other robot initially. This isolated robot cannot move based on local sensing measurements since it has no local interaction initially.  
In our paper, we can remove an isolated robot by increasing its radar sensing range. Thus, our paper does not require the initial connectivity assumption. Once the rendezvous is achieved, the sensing range of every robot can decrease in order to reduce the power consumption of its radar.