Geometric and Control-Theoretic Limits on Drone Density in Bounded Airspace

This paper addresses the question of how many autonomous aerial vehicles (UAVs or drones) can safely operate within a bounded three-dimensional airspace. First, we derive the absolute mathematical limits on drone density using geometric arguments from sphere packing and covering theory. Then, we verify these limits empirically by simulating a swarm controlled via model predictive control. We incrementally increase the number of drones until motion becomes impossible. Each drone is modeled as a double-integrator system with a bounded speed and acceleration and is surrounded by a radius spherical safety zone r>0. The drones are controlled via model predictive control with hard separation constraints. We formalize complete blockage as the loss of any feasible non-trivial trajectory set, either due to geometric crowding or dynamic limitations. Using tools from discrete geometry, we establish absolute upper bounds on a safe population via sphere-packing results and sufficient conditions for total immobilization via sphere-covering arguments. We extend these static bounds by incorporating dynamics through stopping-distance analysis, leading to an inflated exclusion radius that captures the effect of finite control authority. In addition, we prove min-cut style flow-capacity bounds that limit feasible throughput across bottlenecks and derive horizon-dependent conflict-graph conditions that capture MPC infeasibility at high densities. These results provide a rigorous theoretical framework for determining the transition from feasible multi-drone operation to inevitable gridlock, offering explicit quantitative thresholds that can inform airspace design, drone density regulation, and the tuning of predictive controllers. We evaluate our theoretical findings with a simulation environment.