  Recently there have been a lot of interests in introducing UAVs for a wide
range of applications, making ensuring safety of multi-vehicle systems a highly
crucial problem. Hamilton-Jacobi (HJ) reachability is a promising tool for
analyzing safety of vehicles for low-dimensional systems. However, reachability
suffers from the curse of dimensionality, making its direct application to more
than two vehicles intractable. Recent works have made it tractable to guarantee
safety for 3 and 4 vehicles with reachability. However, the number of vehicles
safety can be guaranteed for remains small. In this paper, we propose a novel
reachability-based approach that guarantees safety for any number of vehicles
while vehicles complete their objectives of visiting multiple targets
efficiently, given any K-vehicle collision avoidance algorithm where K can in
general be a small number. We achieve this by developing an approach to group
vehicles into clusters efficiently and a control strategy that guarantees
safety for any in-cluster and cross-cluster pair of vehicles for all time. Our
proposed method is scalable to large number of vehicles with little computation
overhead. We demonstrate our proposed approach with a simulation on 15
vehicles. In addition, we contribute a more general solution to the 3-vehicle
collision avoidance problem from a past recent work, show that the prior work
is a special case of our proposed generalization, and prove its validity.
